text
stringlengths 14
1.76M
|
|---|
# Progressive Co-Attention Network for Fine-Grained Visual Classification
Tian Zhang1, Dongliang Chang1, Zhanyu Ma1,2,∗, and Jun Guo1 ∗ Corresponding
author 1 Pattern Recognition and Intelligent System Lab.,
Beijing University of Posts and Telecommunications, Beijing, China 2 Beijing
Academy of Artificial Intelligence, Beijing, China
###### Abstract
Fine-grained visual classification aims to recognize images belonging to
multiple sub-categories within a same category. It is a challenging task due
to the inherently subtle variations among highly-confused categories. Most
existing methods only take an individual image as input, which may limit the
ability of models to recognize contrastive clues from different images. In
this paper, we propose an effective method called progressive co-attention
network (PCA-Net) to tackle this problem. Specifically, we calculate the
channel-wise similarity by encouraging interaction between the feature
channels within same-category image pairs to capture the common discriminative
features. Considering that complementary information is also crucial for
recognition, we erase the prominent areas enhanced by the channel interaction
to force the network to focus on other discriminative regions. The proposed
model has achieved competitive results on three fine-grained visual
classification benchmark datasets: CUB-$200$-$2011$, Stanford Cars, and FGVC
Aircraft.
###### Index Terms:
Fine-grained visual classification, channel interaction, attention mechanism
## I Introduction
In the past few years, convolutional neural networks (CNNs) [1, 2, 3, 4] have
achieved remarkable success in general image classification tasks. However,
recognizing fine-grained object categories (e.g., bird species [5], car [6]
and aircraft [7] models) is still a challenging task due to high intra-class
variances and low inter-class variances, which attracts extensive research
attentions.
The research of fine-grained visual classification has changed from multi-
stage framework with hand-crafted features [8, 9, 10, 11] to multi-stage
framework based on CNN features and then to end-to-end methods. Some works
design a localization sub-network to locate key parts, and then a
classification sub-network follows for identification. Such as STN [12], RA-
CNN [13], NTS-Net [14]. They first detected the local areas, and then cropped
the detected areas on the original image to learn the key parts. However,
little or no effort has been made to guarantee maximum discriminability in
these areas [15] and the training procedures of these methods are
sophisticated due to the complex architecture designs [16]. Others directly
learn a more discriminative feature representation by developing powerful deep
models [17, 18, 19, 20, 21]. Among them, the most representative method is
bilinear CNNs [17], which used two deep convolutional networks to fuse output
features, so that it could encode high-order statistics of convolutional
activation. More recent advances reduce high feature dimensions [18, 19] or
use kernel methods [20, 21] to extract higher order information. However,
these works do not have an effective design for fine-grained visual
classification in terms of the relationship between categories and the number
of key parts. Meanwhile, most of the methods mentioned above only take an
individual image as input, which will limit their ability to identify
contrastive clues from different images for fine-grained visual
classification. Generally, humans often recognize fine-grained objects by
comparing image pairs, instead of checking single image alone [22]. As shown
in Figure 1, by comparing a pair of same-category images, we can easily
capture their common and prominent features, such as head and wings. In this
way, we can learn the discriminative features precisely.
Figure 1: The motivation of the proposed method. Most previous methods only
took an individual image as input, and the relationship between images was not
explored. In our work, we input a pair of same-category images and model the
channel interaction between them to capture their common features.
Inspired by this observation, we propose a co-attention module (CA-Module) to
model the channel interaction between a pair of same-category images. By
capturing the contrastive features between channels, the model can better
learn the commonality of same-category images, thereby force the network to
focus on the common discriminative features. However, only focusing on common
features of same-category images will cause the network to ignore
complementary features that are essential for highly-confused categories. In
order to tackle this problem, we propose an attention erase module (AE-Module)
to learn complementary features by erasing the most prominent area found in
CA-Module. With the combination of these two modules, the proposed method can
capture more relevant areas to improve the model performance.
Figure 2: The framework of the progressive co-attention network (PCA-Net). The
CA-Module can model the channel-wise interaction within a pair of same-
category images to focus on the prominent areas with common characteristics.
The AE-Module can erase the images to distract attention to other areas to
capture complementary features.
## II METHODOLOGY
In this section, we present the progressive co-attention network (PCA-Net) for
fine-grained visual classification, as illustrated in Figure 2. It is made up
of the CA-Module and the AE-Module to dig distinguishing features that are
both prominent and complementary.
### II-A Co-Attention Module
Each channel of feature maps can be considered as the representation of a
certain feature, and its value is the response to the current feature’s
strength. Convolutional layer is to model the interaction between channels
within an image to generate more abundant features. Gao et al. [23] proposed a
self-channel interaction (SCI) module to model the interplay between different
channels within an image more effectively. Since the intra-class variances of
fine-grained images are very high, it is important to compare a pair of same-
category images to obtain common features. Hence, unlike [23] modeling self-
channel interaction, we propose a co-attention module (CA-Module) to model the
interaction between channels within two same-category images, forcing the
network to focus on the common areas.
Specifically, given an image pair, the two images are first processed by a
convolutional network to generate a pair of feature maps
${F_{1},F_{2}}\in\mathbb{R}^{c\times h\times w}$. The $c$, $h$ and $w$
indicate channel numbers, height, and width respectively. We reshape the
feature maps $F_{1},F_{2}$ to
$F^{{}^{\prime}}_{1},F^{{}^{\prime}}_{2}\in\mathbb{R}^{c\times l},l=h\times
w$. Then the channel-wise similarity is calculated by performing a bilinear
operation on $F^{{}^{\prime}}_{1}$ and $F_{2}^{\mathrm{{}^{\prime}T}}$ to
obtain a bilinear matrix $F^{{}^{\prime}}_{1}F_{2}^{\mathrm{{}^{\prime}T}}$ as
$M$. Then we take the negative value of it and get the weight matrix through a
softmax function:
$\displaystyle W_{ij}=\frac{exp({-M}_{ij})}{\sum_{k=1}^{c}exp({-M}_{ik})}.$
(1)
Where $i$ represents the $i^{th}$ channel of the first image, and $j$
represents the $j^{th}$ channel of the second image. We weight the weight
matrix $W$ to the original feature maps. Then the interacted feature maps can
be obtained as:
$\displaystyle F_{W1}={W\times F_{1}}\in\mathbb{R}^{c\times h\times w},$
$\displaystyle F_{W2}={W\times F_{2}}\in\mathbb{R}^{c\times h\times w}.$ (2)
Moreover, we use the cross-entropy loss for classification based on the
predictions that are generated by the features $F_{W}$.
After weighting the feature maps, the channels corresponding to similar
features are enhanced to find the commonality of this pair of images. The
response values of similar parts increase, and the response values of other
parts decrease.
### II-B Attention Erase Module
The CA-Module aims to focus on the common features of same-category images,
but it ignores other slight clues containing complementary information.
Because there are subtle differences between highly-confused categories in
fine-grained images, it is necessary to pay attention to subtle features.
Hence, we propose an attention erase module (AE-Module) to capture the subtle
complementary features by erasing the prominent areas in the image.
We perform global average pooling on the feature maps weighted by CA-Module.
Then we select the channel of the feature maps corresponding to the maximum
value as attention map, and up-sample it to the original image size:
$\displaystyle A(x,y)=Upsample(\underset{m}{max}\ [F_{W}(m,x,y)]).$ (3)
Where $m$ denotes the channel index, $x$ and $y$ denote the spatial indexs.
We obtain a drop mask $M$ by setting the elements $A(x,y)$ larger than
threshold $\theta$ to $0$, and setting other elements to $1$:
$\displaystyle M(x,y)=\begin{cases}0,&\mbox{if }A(x,y)>\theta\\\ 1,&\mbox{else
}\end{cases}.$ (4)
Overlay the drop mask on the original image to obtain a new image with partial
areas erased:
$\displaystyle I_{e}=I(x,y)\otimes M(x,y),$ (5)
where $\otimes$ denotes element-wise multiplication, $I$ denotes the original
image, $I_{e}$ denotes the erased image.
TABLE I: Statistics of benchmark datasets Datesets | Class | Training | Testing
---|---|---|---
CUB-$200$-$2011$ | $200$ | $5997$ | $5794$
Stanford Cars | $196$ | $8144$ | $8041$
FGVC Aircraft | $100$ | $6667$ | $3333$
As the prominent areas of the image are erased, the attention is distracted
and the network is forced to learn discriminative information from other
areas. It can also reduce the dependence on training samples to improve the
robustness of the model.
TABLE II: Comparative experiment between recent methods with our method on CUB-$200$-$2011$, Stanford Cars, and FGVC Aircraft. The best result is colored in red, and the second best result is colored in blue. Method | Backbone | Input size | CUB-$200$-$2011$ (%) | Stanford Cars (%) | FGVC-Aircraft (%)
---|---|---|---|---|---
FT VGG-$19$ [24] | VGG-$19$ | $448\times 448$ | $77.8$ | $84.9$ | $84.8$
RA-CNN [13] | VGG-$19$ | $448\times 448$ | $85.3$ | $92.5$ | $-$
MA-CNN [25] | VGG-$19$ | $448\times 448$ | $86.5$ | $92.8$ | $89.9$
FT ResNet-$50$ [24] | ResNet-$50$ | $448\times 448$ | $84.1$ | $91.7$ | $88.5$
RAM [26] | ResNet-$50$ | $448\times 448$ | $86.0$ | $93.1$ | -
DFL-CNN [24] | ResNet-$50$ | $448\times 448$ | $87.4$ | $93.1$ | $91.7$
NTS-Net [14] | ResNet-$50$ | $448\times 448$ | $87.5$ | $93.9$ | $91.4$
MC-Loss [27] | ResNet-$50$ | $448\times 448$ | $87.3$ | $93.7$ | $92.6$
TASN [28] | ResNet-$50$ | $448\times 448$ | $87.9$ | $93.8$ | -
Cross-X [29] | ResNet-$50$ | $448\times 448$ | $87.7$ | $94.6$ | $92.6$
ACNet [30] | ResNet-$50$ | $448\times 448$ | $88.1$ | $94.6$ | $92.4$
MAMC [16] | ResNet-$50$ | $448\times 448$ | $86.2$ | $92.8$ | -
MAMC [16] | ResNet-$101$ | $448\times 448$ | $86.5$ | $93.0$ | -
CIN [23] | ResNet-$50$ | $448\times 448$ | $87.5$ | $94.1$ | $92.6$
CIN [23] | ResNet-$101$ | $448\times 448$ | $88.1$ | $94.5$ | $92.8$
API-Net [22] | ResNet-$50$ | $448\times 448$ | $87.7$ | 94.8 | 93.0
API-Net [22] | ResNet-$101$ | $448\times 448$ | 88.6 | 94.9 | 93.4
Ours | ResNet-$50$ | $448\times 448$ | $88.3$ | $94.3$ | $92.4$
Ours | ResNet-$101$ | $448\times 448$ | 88.9 | $94.6$ | $92.8$
### II-C Fine-Grained Feature Learning
In order to make the learned features more aggregated in embedding space, we
introduce the center loss [31]. For each category, a feature vector is
calculated as the center of the corresponding category, and it will be
continuously updated during training phase. By penalizing the deviation
between the bilinear feature vector of each sample and the center of the
corresponding category, the samples belonging to the same category are grouped
together as many as possible, which enhances the discrimination of the learned
features.
Benefiting from global and local informative features of an image, we define
the feature representations for each image: $F_{J}=\\{F_{O},F_{W},F_{E}\\}$,
where $F_{O}$ denotes the feature maps extracted from the original image,
$F_{W}$ denotes the weighted feature maps of the original image, and $F_{E}$
denotes the feature maps extracted from the erased image. These features are
then fed to a fully-connection layer with a softmax function for the final
classification.
During training phase, the whole model is optimized by losses defined as
$\displaystyle L=\sum_{i\in I}L_{cls}(Y^{i},Y^{*})+\lambda
L_{cen}(Y^{i},Y^{c}),$ (6)
where $L_{cls}$ denotes the cross-entropy loss, $L_{cen}$ denotes the center
loss and $\lambda$ denotes the weight of $L_{cen}$. $Y^{i}$ is the predicted
bilinear label vector based on features $F_{O}$, $F_{W}$ and $F_{E}$. $Y^{*}$
is the ground-truth label vector and $Y^{c}$ is the learned center vector of
category.
## III EXPERIMENTS AND DISCUSSIONS
We evaluate the proposed approach on three publicly competitive fine-grained
visual classification datasets, including CUB-$200$-$2011$ [5], Stanford Cars
[6], and FGVC Aircraft [7]. The detailed statistics with category numbers and
the standard training/testing splits are summarized in Table I. We employ
top-1 accuracy as evaluation metric.
Figure 3: The first line is the original images, the second line is the
features learned by the base model, and the third line is the features learned
by the proposed method. It can be observed that the proposed method has
learned more feature information than the base model.
### III-A Implementation details
In all experiments, we use ResNet-$50$ [4] and ResNet-$101$ [4] as our base
networks. We remove the last pooling layer and fully-connected layer, and add
a billinear pooling [17] to generate a more powerful feature representation.
Then send the feature representation to a new fully-connected layer. The
convolutional layers are pretrained on ImageNet [32] and the fully connected
layer is randomly initialized.
The input image size of all experiments is $448\times 448$, which is the same
as the most state-of-the-art fine-grained classification approaches [24, 14,
28, 16, 23]. In order to improve the generalization ability of the model, we
implement data augmentation including random cropping and random horizontal
flipping during training phase. Only center cropping is involved in inference
phase. All experimental models are trained for $180$ epochs, using SGD as
optimizer. The initial learning rate is set to $0.01$, which annealed by $0.9$
every $2$ epoches. We use a batch size of $32$, and the last $16$ samples
belong to the same category as the first $16$ samples. The weight decay is set
to $10^{-5}$ and the momentum is set to $0.9$.
### III-B Experimental results
The proposed method is compared with several representative methods without
extra annotations on three fine-grained benchmark datasets of
CUB-$200$-$2011$, Stanford Cars, and FGVC Aircraft, as shown in Table II. The
proposed method performs competitive results compared with the state of-the-
art methods.
Specifically, the top-1 accuracy of the proposed method on CUB-$200$-$2011$
dataset is $88.9\%$, which is better than $88.6\%$ of API-Net [22]. API-Net
performs well because it also takes a pair of images as input, and learns a
mutual feature vector by modeling the interaction between them to capture the
semantic difference in the input pair. Our method is to capture the
semantically similarity of a pair of same-category images, and disperse the
network’s attention to each discriminative area of the image.
On the other two datasets, our method also performs well, which achieves the
result with $94.6\%$ top-1 accuracy for Stanford Cars dataset and $92.8\%$
top-1 accuracy for FGVC-Aircraft dataset. However, our method does not obtain
the best results on these two datasets. Since cars and aircrafts are rigid
objects, the intra-class variances of them are not assignificant as birds. And
our method works better for objects that greatly vary within the category.
TABLE III: Ablation studies CA-Module | AE-Module | center loss | accuracy(%)
---|---|---|---
$\checkmark$ | | | $86.8$
| $\checkmark$ | | $86.5$
$\checkmark$ | | $\checkmark$ | $87.9$
| $\checkmark$ | $\checkmark$ | $87.5$
$\checkmark$ | $\checkmark$ | $\checkmark$ | $88.3$
### III-C Ablation studies
In order to verify the effectiveness of each component in the model, we
conduct ablation studies on CUB-$200$-$2011$ dataset using ResNet-$50$ with
some of components removed for better understanding the behavior of the model,
as shown in Table III.
It can be observed that CA-Module generates weighted feature maps through
modeling the channel-wise interaction of same-category images, so that the
network can learn their common features. AE-Module erases the most prominent
area based on the weighted feature maps, making the informative areas learned
by the network diversified. The center loss makes the extracted features more
discriminative by constraining the distance between the feature vector and the
center vector of the corresponding category. Each component of the model
contributes to performance.
### III-D Visualizations
In order to further evaluate the effectiveness of our method, we apply Grad-
CAM [33] to visualize the images of the CUB-$200$-$2011$ dataset. Grad-CAM is
formed by weighted summation of feature maps, which can show the importance of
each area to its classification. We compare the visualization results of our
method with the base model (ResNet-$50$), as shown in Figure 3. It can be
observed that the base model only learns the most prominent area of the image,
such as the bird’s beak. Our method can learn more abundant and discriminative
features, including wings and claws. This is because that our method can
distribute attention to each area to make the prediction more comprehensive,
which can not only focus on the salient features, but also capture the subtle
and fine-grained features.
## IV CONCLUSIONS
We propose an effective fine-grained visual classification method, namely
progressive co-attention network. Among them, the co-attention module learns
discriminative features by comparing same-category images, and the attention
erase module learns subtle complementary features of images by erasing the
most prominent area. We have conducted experiments on CUB-$200$-$2011$,
Stanford Cars, and FGVC Aircraft datasets, which are superior to most existing
methods.
Acknowledgments: This work was supported in part by the National Key R & D
Program of China under Grant $2019$YFF$0303300$ and under Subject II
No.$2019$YFF$0303302$, in part by National Natural Science Foundation of China
(NSFC) No.$61922015$,$61773071$, U$19$B$2036$, in part by Beijing Natural
ScienceFoundation Project No. Z$200002$.
## References
* [1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems, 2012.
* [2] Karen Simonyan and Andrew Zisserman, “Very deep convolutional networks for large-scale image recognition,” arXiv preprint arXiv:1409.1556, 2014.
* [3] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich, “Going deeper with convolutions,” in CVPR, 2015.
* [4] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, “Deep residual learning for image recognition,” in CVPR, 2016.
* [5] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie, “The caltech-ucsd birds-200-2011 dataset,” Tech. Rep., 2011.
* [6] Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei, “3d object representations for fine-grained categorization,” in ICCV workshops, 2013.
* [7] Subhransu Maji, Esa Rahtu, Juho Kannala, Matthew Blaschko, and Andrea Vedaldi, “Fine-grained visual classification of aircraft,” Tech. Rep., 2013.
* [8] Thomas Berg and Peter Belhumeur, “Poof: Part-based one-vs.-one features for fine-grained categorization, face verification, and attribute estimation,” in CVPR, 2013.
* [9] Bangpeng Yao, Gary Bradski, and Li Fei-Fei, “A codebook-free and annotation-free approach for fine-grained image categorization,” in CVPR, 2012.
* [10] Yuning Chai, Victor Lempitsky, and Andrew Zisserman, “Symbiotic segmentation and part localization for fine-grained categorization,” in ICCV, 2013.
* [11] Philippe-Henri Gosselin, Naila Murray, Hervé Jégou, and Florent Perronnin, “Revisiting the fisher vector for fine-grained classification,” Pattern Recognition Letters, vol. 49, pp. 92–98, 2014.
* [12] Max Jaderberg, Karen Simonyan, Andrew Zisserman, et al., “Spatial transformer networks,” in NeurIPS, 2015.
* [13] Jianlong Fu, Heliang Zheng, and Tao Mei, “Look closer to see better: Recurrent attention convolutional neural network for fine-grained image recognition,” in CVPR, 2017.
* [14] Ze Yang, Tiange Luo, Dong Wang, Zhiqiang Hu, Jun Gao, and Liwei Wang, “Learning to navigate for fine-grained classification,” in ECCV, 2018.
* [15] Ruoyi Du, Dongliang Chang, Ayan Kumar Bhunia, Jiyang Xie, Yi-Zhe Song, Zhanyu Ma, and Jun Guo, “Fine-grained visual classification via progressive multi-granularity training of jigsaw patches,” in ECCV, 2020.
* [16] Ming Sun, Yuchen Yuan, Feng Zhou, and Errui Ding, “Multi-attention multi-class constraint for fine-grained image recognition,” in ECCV, 2018.
* [17] Tsung-Yu Lin, Aruni RoyChowdhury, and Subhransu Maji, “Bilinear cnn models for fine-grained visual recognition,” in ICCV, 2015.
* [18] Yang Gao, Oscar Beijbom, Ning Zhang, and Trevor Darrell, “Compact bilinear pooling,” in CVPR, 2016, pp. 317–326.
* [19] Shu Kong and Charless Fowlkes, “Low-rank bilinear pooling for fine-grained classification,” in CVPR, 2017.
* [20] Yin Cui, Feng Zhou, Jiang Wang, Xiao Liu, Yuanqing Lin, and Serge Belongie, “Kernel pooling for convolutional neural networks,” in CVPR, 2017.
* [21] Sijia Cai, Wangmeng Zuo, and Lei Zhang, “Higher-order integration of hierarchical convolutional activations for fine-grained visual categorization,” in ICCV, 2017.
* [22] Peiqin Zhuang, Yali Wang, and Yu Qiao, “Learning attentive pairwise interaction for fine-grained classification.,” in AAAI, 2020.
* [23] Yu Gao, Xintong Han, Xun Wang, Weilin Huang, and Matthew R. Scott, “Channel Interaction Networks for Fine-Grained Image Categorization,” arXiv e-prints, 2020.
* [24] Yaming Wang, Vlad I Morariu, and Larry S Davis, “Learning a discriminative filter bank within a cnn for fine-grained recognition,” in CVPR, 2018.
* [25] Heliang Zheng, Jianlong Fu, Tao Mei, and Jiebo Luo, “Learning multi-attention convolutional neural network for fine-grained image recognition,” in ICCV, 2017.
* [26] Zhichao Li, Yi Yang, Xiao Liu, Feng Zhou, Shilei Wen, and Wei Xu, “Dynamic computational time for visual attention,” in ICCV, 2017.
* [27] Dongliang Chang, Yifeng Ding, Jiyang Xie, Ayan Kumar Bhunia, Xiaoxu Li, Zhanyu Ma, Ming Wu, Jun Guo, and Yi-Zhe Song, “The devil is in the channels: Mutual-channel loss for fine-grained image classification,” IEEE Transactions on Image Processing, 2020.
* [28] Heliang Zheng, Jianlong Fu, Zheng-Jun Zha, and Jiebo Luo, “Looking for the devil in the details: Learning trilinear attention sampling network for fine-grained image recognition,” in CVPR, 2019.
* [29] Wei Luo, Xitong Yang, Xianjie Mo, Yuheng Lu, Larry S Davis, Jun Li, Jian Yang, and Ser-Nam Lim, “Cross-x learning for fine-grained visual categorization,” in ICCV, 2019.
* [30] Ruyi Ji, Longyin Wen, Libo Zhang, Dawei Du, Yanjun Wu, Chen Zhao, Xianglong Liu, and Feiyue Huang, “Attention convolutional binary neural tree for fine-grained visual categorization,” arXiv preprint arXiv:1909.11378, 2019.
* [31] Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao, “A discriminative feature learning approach for deep face recognition,” in ECCV, 2016.
* [32] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in CVPR, 2009.
* [33] Ramprasaath R. Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra, “Grad-cam: Visual explanations from deep networks via gradient-based localization,” in ICCV, 2017.
|
# Electric dipole polarizability in neutron-rich Sn isotopes as a probe of
nuclear isovector properties
Z. Z. Li Y. F. Niu<EMAIL_ADDRESS>W. H. Long School of Nuclear Science
and Technology, Lanzhou University, Lanzhou 730000, China
###### Abstract
The determination of nuclear symmetry energy, and in particular, its density
dependence, is a long-standing problem for nuclear physics community. Previous
studies have found that the product of electric dipole polarizability
$\alpha_{D}$ and symmetry energy at saturation density $J$ has a strong linear
correlation with $L$, the slope parameter of symmetry energy. However, current
uncertainty of $J$ hinders the precise constraint on $L$. We investigate the
correlations between electric dipole polarizability $\alpha_{D}$ (or times
symmetry energy at saturation density $J$) in Sn isotopes and the slope
parameter of symmetry energy $L$ using the quasiparticle random-phase
approximation based on Skyrme Hartree-Fock-Bogoliubov. A strong and model-
independent linear correlation between $\alpha_{D}$ and $L$ is found in
neutron-rich Sn isotopes where pygmy dipole resonance (PDR) gives a
considerable contribution to $\alpha_{D}$, attributed to the pairing
correlations playing important roles through PDR. This newly discovered linear
correlation would help one to constrain $L$ and neutron-skin thickness $\Delta
R_{\textnormal{np}}$ stiffly if $\alpha_{D}$ is measured with high resolution
in neutron-rich nuclei. Besides, a linear correlation between $\alpha_{D}J$ in
a nucleus around $\beta$-stability line and $\alpha_{D}$ in a neutron-rich
nucleus can be used to assess $\alpha_{D}$ in neutron-rich nuclei.
###### keywords:
Electric dipole polarizability , Slope parameter of symmetry energy , Neutron-
skin thickness
## 1 Introduction
The determination of nuclear equation of state (EoS) at high density is a
challenge for both experimental and theoretical nuclear physics [1, 2], which
is crucial for constraining current theoretical models [3, 4] and
understanding many phenomena in astrophysics [5, 6]. The biggest uncertainty
of EoS comes from its isovector parts, which are governed by the nuclear
symmetry energy $\mathcal{S}(\rho)$. The symmetry energy can be expanded as a
function of $\varepsilon=(\rho-\rho_{0})/3\rho_{0}$ by
$\mathcal{S}(\rho)=J+L\varepsilon+\dfrac{1}{2}K_{\textnormal{sym}}\varepsilon^{2}+...$
(1)
where $J=\mathcal{S}(\rho_{0})$ is the symmetry energy at saturation density
$\rho_{0}$, while
$L=3\rho_{0}\Big{(}\dfrac{\partial\mathcal{S}}{\partial\rho}\Big{)}\Big{|}_{\rho=\rho_{0}}$
and
$K_{\textnormal{sym}}=9\rho_{0}^{2}\Big{(}\dfrac{\partial^{2}\mathcal{S}}{\partial\rho^{2}}\Big{)}\Big{|}_{\rho=\rho_{0}}$
correspond to the slope and curvature parameters at saturation density,
respectively.
The slope parameter of symmetry energy $L$ determines the behavior of symmetry
energy at high density, however, it varies a lot in different nuclear models.
Constraints on $L$ can be obtained from heavy-ion collisions [1, 7],
properties of neutron stars [5, 8], and nuclear properties of ground state and
excited states of finite nuclei [9]. For example, it is revealed that $L$ is
proportional to the neutron-skin thickness $\Delta R_{np}$ by droplet model
[10, 11], which is further conformed by many microscopic models [12, 13].
However, the obstacle in the measurements of neutron radius hinders the access
to high-resolution neutron skin data. As an alternative, charge radii
difference $\Delta R_{c}$ between mirror nuclei is proposed as another
possible way to constrain $L$ [14, 15, 16], which also faces difficulties in
the measurements of charge radius in proton-rich nucleus.
The electric dipole $(E1)$ excitation in nucleus is mainly composed of the
giant dipole resonance (GDR), which is formed by the relative dipole
oscillation between neutrons and protons, thus reflecting asymmetry
information in nuclear EoS. The electric dipole polarizability $\alpha_{D}$,
being proportional to the inverse energy-weighted sum rule of $E1$ excitation,
can be served as a possible probe for nuclear isovector properties.
Theoretically, (quasiparticle) random-phase approximation [(Q)RPA] approach is
widely used to describe small oscillations of nucleus, such as $E1$
excitations. The self-consistent (Q)RPA models have been developed based on
Skyrme density functionals [17, 18, 19], Gogny density functionals [20, 21],
and relativistic density functionals [22, 23, 24, 25]. Global properties of
GDR, such as centroid energies and electric dipole polarizabilities, can be
well described within this approximation.
Based on these self-consistent (Q)RPA models, the correlations between
electric dipole polarizability $\alpha_{D}$ and other nuclear isovector
properties have been investigated in recent years. Calculations performed by
RPA model based on Skyrme density functionals SV-min series [26] and
relativistic density functionals RMF-$\delta$-t series in 208Pb suggested a
strong linear correlation between $\alpha_{D}$ and neutron-skin thickness
$\Delta R_{\textnormal{np}}$ [27]. However, when one combines the results from
a host of different nuclear density functionals, this linear correlation is
not universal anymore [28]. Starting from droplet model, and further supported
by RPA calculations based on many different Skyrme and relativistic density
functionals in 208Pb, the product of dipole polarizability and symmetry energy
at saturation density $\alpha_{D}J$ was suggested to be much better correlated
with neutron-skin thickness and symmetry energy slope parameter $L$ than
$\alpha_{D}$ alone is [29]. Based on this correlation,
$L=43\pm(6)_{\textnormal{expt}}\pm(8)_{\textnormal{theor}}\pm(12)_{\textnormal{est}}$
MeV was given by using the experimental $\alpha_{D}$ value in 208Pb [29], and
the intervals $J=30-35$ MeV and $L=20-66$ MeV were further obtained by
combining the measured polarizabilities in 68Ni, 120Sn and 208Pb [30]. Below
saturation density, $\alpha_{D}$ in 208Pb was also found to be sensitive to
both the symmetry energy $\mathcal{S}(\rho_{c})$ and slope parameter
$L(\rho_{c})$ at the subsaturation cross density $\rho_{c}=0.11$fm-3 [31].
Since $\mathcal{S}(\rho_{c})$ is well constrained, $L(\rho_{c})$ can be
strongly constrained from experimental $\alpha_{D}$ in 208Pb [31]. At
$\rho_{r}=\rho_{0}/3$, another linear correlation was built between
$\alpha_{D}^{-1}$ and $\mathcal{S}(\rho_{r})$ [32]. Besides, $\alpha_{D}$
between two different nuclei [33], as well as $\alpha_{D}J$ between two
different nuclei [30], were also shown to have good linear correlations.
In recent years, the electric dipole polarizabilities $\alpha_{D}$ in 208Pb
[34], 48Ca [35], and stable Sn isotopes [33, 36, 37] were measured with high
resolution via polarized proton inelastic scattering at extreme forward angles
[38]. For unstable nucleus 68Ni, $\alpha_{D}$ was also extracted by Coulomb
excitation in inverse kinematics [39]. However, there are problems when one
uses these high-resolution dipole polarizability data to constrain isovector
properties: the constraints on $L$ or $\Delta R_{\textnormal{np}}$ is either
with big uncertainties due to the uncertainty of $J$ or in model-dependent
ways. One way to solve the problem and constrain $L$ stiffly is to find a
direct and model-independent correlation between $\alpha_{D}$ and $L$.
Although the previous studies have shown that the model-independent linear
correlation only exists between $\alpha_{D}J$ and $L$, it was only limited to
stable nuclei or nuclei near $\beta$-stability line. It is well known that
exotic phenomena will present when approaching to nuclei far from
$\beta$-stability line, such as novel shell structures [40, 41, 42, 43, 44],
new types of excitations [45, 23, 46], and so on. For $E1$ excitations, the
pygmy dipole resonance (PDR) appears in neutron-rich nuclei [45, 23, 46],
which would cause different characteristics of $E1$ excitations compared to
the ones around $\beta$-stability line, and further affect $\alpha_{D}$. So an
interesting question is if the linear correlation between $\alpha_{D}J$ and
$L$ observed in stable nuclei still holds and new correlations would appear in
neutron-rich nuclei.
Therefore, in our study we will explore the correlations between $\alpha_{D}$
and nuclear isovector properties such as slope parameter $L$ and neutron-skin
thickness $\Delta R_{np}$ in even-even Sn isotopes from neutron-deficient
100Sn to neutron-rich 164Sn. The calculations are performed by QRPA based on
Skyrme Hartree-Fock-Bogoliubov (HFB) model, in which the spherical symmetries
are imposed. The linear correlations are evaluated by a least-square
regression analysis. Based on the newly discovered correlations, constraints
on $L$ and neutron-skin thickness will be discussed.
## 2 Theoretical Framework
We carry out a self-consistent HFB$+$QRPA calculation of $E1$ strength using
19 Skyrme functionals: SIII, SIV, SV, SVI [47], SLy230a, SLy230b, SLy4, SLy5,
SLy8 [52, 53], SAMi [54], SAMi-J30, SAMi-J31, SAMi-J32, SAMi-J33 [55], SGI,
SGII [48], SkM [49], SkM* [50], Ska [51]. The detailed formulas of QRPA on top
of HFB can be found in Ref. [18]. The density-dependent zero-range surface
pairing force is implemented in the particle-particle channel,
$V_{pp}(\boldsymbol{r}_{1},\boldsymbol{r}_{2})=V_{0}\Big{[}1-\dfrac{\rho(\boldsymbol{r})}{\rho_{0}}\Big{]}\delta(\boldsymbol{r}_{1}-\boldsymbol{r}_{2})$
(2)
where $\boldsymbol{r}=(\boldsymbol{r}_{1}+\boldsymbol{r}_{2})/2$, and
$\rho_{0}=0.16$fm-3 is the nuclear saturation density, while $V_{0}$ is
adjusted by fitting neutron pairing gaps of 116∼130Sn according to the five-
point formula [56]. The electric dipole polarizability $\alpha_{D}$ is given
by
$\alpha_{D}=\dfrac{8\pi e^{2}}{9}m_{-1},\quad
m_{-1}=\sum\limits_{\nu}\dfrac{\big{|}\langle\psi_{\nu}|F_{1\mu}^{\textnormal{(IV)}}|\psi_{0}\rangle\big{|}^{2}}{E_{\nu}}$
(3)
where $\psi_{\nu}$ and $E_{\nu}$ are the eigenstates and eigenvalues of QRPA
equations, and $\psi_{0}$ is the ground state. $m_{-1}$ is the inverse energy-
weighted sum rule (EWSR), which is calculated using the isovector dipole
operator
$F_{1\mu}^{\textnormal{(IV)}}=\dfrac{N}{A}\sum\limits_{p=1}^{Z}r_{p}Y_{1\mu}-\dfrac{Z}{A}\sum\limits_{n=1}^{N}r_{n}Y_{1\mu}$
(4)
where $A$, $N$, $Z$ denote mass number, neutron number, proton number, and
$Y_{1\mu}$ are the spherical harmonics. In our calculations, the quasiparticle
energy cutoff $E_{\textnormal{cut}}$ is set as 90 MeV and the total angular
momentum cutoff of quasiparticle $j_{\textnormal{max}}$ is set as $21/2$ to
ensure the convergence of numerical results.
## 3 Results and Discussions
### 3.1 Correlations between $\alpha_{D}$ and nuclear isovector properties
Table 1: Pearson’s coefficient $r$ between the product of dipole polarizability and saturated symmetry energy $\alpha_{D}J$ and the slope parameter of symmetry energy $L$ in Sn isotopes, as well as the corresponding slope $k$ of the regression line ($\alpha_{D}J$ as a function of $L$), calculated by QRPA based on HFB with 19 Skyrme density functionals. Nucleus | 100Sn | 110Sn | 120Sn | 130Sn | 140Sn | 150Sn | 160Sn
---|---|---|---|---|---|---|---
$r$ | 0.965 | 0.966 | 0.974 | 0.961 | 0.966 | 0.940 | 0.937
$k$ (fm3) | 0.844 | 1.066 | 1.383 | 1.543 | 2.272 | 2.880 | 3.541
Figure 1: (Color online) Plots for dipole polarizability $\alpha_{D}$ against
slope parameter of symmetry energy $L$ in Sn isotopes calculated by QRPA based
on HFB with 19 Skyrme density functionals: SIII, SIV, SV, SVI (blue up
triangles); SLy230a, SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi, SAMi-J30,
SAMi-J31, SAMi-J32, SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*, Ska
(black squares). A regression line (red solid line) is obtained by a least-
square linear fit of the calculated $\alpha_{D}$ as a function of $L$. $r$ is
Pearson’s coefficient and $k$ (fm${}^{3}/$MeV) is the slope of the regression
line. Figure 2: (Color online) The same as Fig. 1 but for 120,140,150,160Sn
without the pairing correlations.
First of all, we study if the previously discovered linear correlation between
$\alpha_{D}J$ and $L$ holds in the whole tin isotopes from neutron-deficient
ones to neutron-rich ones. So in Tab. 1, Pearson correlation coefficients (or
Pearson’s coefficients) $r$ between $\alpha_{D}J$ and $L$ in even-even Sn
isotopes from 100Sn to 160Sn , as well as the corresponding slopes $k$ of the
regression lines are shown based on the HFB+QRPA calculations using 19 Skyrme
density functionals. Pearson’s coefficient $r$ is a statistic that measures
linear correlation between two variables, which is defined by the covariance
of two variables divided by the product of their standard deviations. A value
of $|r|=1$ means that the two observables are fully linearly correlated while
$r=0$ are totally uncorrelated. From Tab. 1, one can see the Pearson’s
coefficients $r$ in the whole Sn isotopes are all above $0.9$, showing strong
linear correlations between $\alpha_{D}J$ and $L$. So it further proofs this
linear correlation is a universal one which exists not only in stable nuclei
as revealed in previous studies [29] but also in neutron-deficient and
neutron-rich nuclei. The corresponding slope $k$ of the regression line shows
a clear increase trend with the increase of neutron number. The larger $k$
value means a more rapid increase of $\alpha_{D}J$ as a function of $L$, which
gives a smaller range of $L$ under the same uncertainty of $\alpha_{D}J$. So
the slope $k$ of the regression line is an important quantity to select good
candidate nuclei as probes of nuclear isovector properties, which will be
discussed in details in Sec. 3.2.
Although the above correlation is universal, it cannot provide a stiff
constraint on the slope parameter of symmetry energy $L$ due to the
uncertainty in the symmetry energy $J$. For example, by adopting $J=31\pm 2$
MeV, Roca-Maza et al. obtained
$L=43\pm(6)_{\textnormal{expt}}\pm(8)_{\textnormal{theor}}\pm(12)_{\textnormal{est}}$
MeV, where the uncertainty $\pm 12$ MeV comes from the uncertainty of $J$
[29]. So it would be better to find a direct correlation between $\alpha_{D}$
and $L$. Previous studies have shown that $L$ and $\alpha_{D}$ have a good
linear correlation within some specific parameter family [27], however, by
including different parameter families, this correlation becomes bad, for
example, in 208Pb the Pearson’s coefficient r was given as $r=0.62$ [29] and
$r=0.77$ [28]. Here we recheck the correlation between the dipole
polarizability $\alpha_{D}$ and the slope parameter $L$ of symmetry energy for
the whole tin isotopes from neutron-deficient ones to neutron-rich ones, as
shown in Fig. 1, to see if the previous conclusions still hold. In stable
nucleus 120Sn, for some specific Skyrme parameter family, such as SAMi (green
diamonds) or SIII-SVI (up blue triangles), one can observe a good linear
correlation, in agreement with Ref. [27]. However, when one includes more
different Skyrme parameter sets, the linear correlation becomes poor, and the
Pearson coefficient $r$ is around 0.8, again in agreement with the case in
208Pb [29, 28]. Similar situations still exist in nuclei not far from the
stability line such as 100,110,130Sn.
However, the cases become totally different in the neutron-rich nuclei. The
coefficients are above $0.9$ for the isotopes with mass number $A\geq 140$,
which present strong correlations between $\alpha_{D}$ and $L$ in the neutron-
rich Sn isotopes. After $A\geq 146$, the correlation between $\alpha_{D}$ and
$L$ is even better than the one between $\alpha_{D}J$ and $L$. We stress here
the assessments are carried out by different Skyrme functional families. For
the neutron-rich nuclei of $A\geq 140$ with a clear linear correlation, we
further give the slopes $k$ of the regression lines. It is seen that $k$
becomes larger with the increase of neutron number, which implies that the
more neutron rich the nucleus is, the better probe it can be served as for
nuclear isovector properties, seeing detailed discussions in Sec. 3.2.
Figure 3: (Color online) The dipole polarizabilities as functions of mass
number $A$ in even-even Sn isotopes calculated by QRPA (square line) and RPA
(circle line) using Skyrme functional SLy4. The total dipole polarizabilities
(red) and the contributions from PDR (blue) are shown respectively. Figure 4:
(Color online) Plots for dipole polarizability contributed by PDR against
slope parameter of symmetry energy in 134,140,150,160Sn isotopes calculated by
QRPA based on HFB with 19 Skyrme density functionals: SIII, SIV, SV, SVI (blue
up triangles); SLy230a, SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi,
SAMi-J30, SAMi-J31, SAMi-J32, SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*,
Ska (black squares). A linear fit is done for each nucleus (red solid line)
with a corresponding Pearson’s coefficient $r$. Figure 5: (Color online) Plots
for neutron-skin thickness $\Delta R_{\textnormal{np}}$ against dipole
polarizability $\alpha_{D}$ in 150,160Sn calculated by QRPA based on HFB with
19 Skyrme density functionals: SIII, SIV, SV, SVI (blue up triangles);
SLy230a, SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi, SAMi-J30, SAMi-J31,
SAMi-J32, SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*, Ska (black
squares). A regression line (red solid line) is obtained by a least-square
linear fit of the calculated $\Delta R_{\textnormal{np}}$ as a function of
$\alpha_{D}$. $r$ is Pearson’s coefficient and $k$ (fm-2) is the slope of
regression line.
To understand the above strong linear correlations in neutron-rich Sn
isotopes, we first investigate the role of pairing correlations. So in Fig. 2
the correlations between $\alpha_{D}$ and $L$ in 120,140,150,160Sn are studied
without considering pairing effects. For stable nucleus 120Sn, the correlation
between $\alpha_{D}$ and $L$ is similar as the case with pairing correlations,
where the Pearson’s coefficient is only slightly reduced without the inclusion
of pairing correlations. However, for these three neutron-rich nuclei
140,150,160Sn, the linear correlations become much worse, where the Pearson’s
coefficients are largely reduced to the values $0.817$, $0.873$ and $0.867$,
respectively, being all below $0.9$. It shows that the pairing correlations
play important roles in keeping strong linear correlations between
$\alpha_{D}$ and $L$ in neutron-rich Sn isotopes.
On the other hand, for neutron-rich nuclei, the PDR appears in the low-energy
part of $E1$ transition strength distribution, which would give big
contributions to the dipole polarizability. Since PDR represents an
oscillation between neutron skin and nearly isospin-saturated core, the
correlations between its strengths and symmetry energy were also explored [57,
27, 58, 59], although it is still an open question. Inspired by this, we
extract the contributions of PDR to $\alpha_{D}$ in Sn isotopes in Fig. 3,
where the total dipole polarizabilities and contributions from PDR as
functions of mass number $A$ in even-even Sn isotopes calculated by QRPA and
RPA using Skyrme functional SLy4 are plotted. According to the dipole strength
distributions and the transition densities, different energies are selected as
the upper boundaries of PDR for different Skyrme functionals, which are $9.0$
MeV for SVI, $10.0$ MeV for SIII, SLy family, SkM, SkM*, SGII, $10.5$ MeV for
Ska, SAMi family, $11.0$ MeV for SGI, $12.0$ MeV for SIV, and $13.0$ MeV for
SV.
Starting from 132Sn, the PDR appears and starts to contribute to the dipole
polariziability $\alpha_{D}$. With the neutron number increases, the
contribution from PDR becomes larger and larger, which dominates the evolution
trend with mass number of the total $\alpha_{D}$. With the pairing
correlations being turned off, the contribution from PDR to $\alpha_{D}$ is
greatly reduced, which almost keeps a small constant with the increase of
neutron number. As a result, the total $\alpha_{D}$ is also reduced a lot, and
its increase trend with mass number becomes as slow as that before 132Sn.
Before 132Sn, the pairing correlations only have very small influences on
$\alpha_{D}$. Therefore, it can be seen that the pairing correlations play
their important roles on dipole polariziabilities and further the linear
correlations between $\alpha_{D}$ and $L$ through PDR.
In Fig. 4 we further study the correlation between dipole polarizabilities
$\alpha_{D}$ contributed by PDR and the slope parameter $L$ of symmetry energy
in 134Sn, 140Sn, 150Sn, 160Sn isotopes. It shows that polarizability
$\alpha_{D}$ of PDR has a good correlation with the slope parameter $L$ in
general, which enhances the linear correlations between the total $\alpha_{D}$
and symmetry energy slope parameter $L$.
Apart from the correlation between $\alpha_{D}$ and $L$, the correlation
between $\alpha_{D}$ and another important isovector property, i.e., neutron-
skin thickness, is also investigated, and the plots for neutron-skin thickness
against dipole polarizability in 150,160Sn are shown in Fig. 5. Not
surprisingly, the linear correlations between $\Delta R_{np}$ and $\alpha_{D}$
in 150Sn and 160Sn are strong with $r=0.939$ and $r=0.947$ respectively, since
the neutron-skin thickness $\Delta R_{np}$ and $L$ are reported to have a good
linear correlation when $|N-Z|$ is large [15]. The slopes $k$ of regression
lines, fitted by $\Delta R_{\textnormal{np}}$ as a function of $\alpha_{D}$,
are generally small in these neutron-rich nuclei, suggesting that $\alpha_{D}$
in neutron-rich nuclei can provide an effective constraints on neutron-skin
thickness of the corresponding nuclei.
### 3.2 $\alpha_{D}$ as a probe of nuclear isovector properties
Table 2: Constraints on the slope parameter of symmetry energy $L$ from experimental dipole polarizability values $\alpha_{D}^{\textnormal{exp.}}$ [34, 39, 35, 36, 30] using linear correlation between $\alpha_{D}J$ and $L$ obtained by skyrme QRPA calculations using 19 Skyrme functionals. The Pearson’s coefficient $r$ and the slope $k$ of the regression line fitted by $\alpha_{D}J$ as a function of $L$ are also given. $J=31.7\pm 3.2$MeV is adopted [2]. $\Delta L_{\textnormal{min}}$ denotes the uncertainty coming from the uncertainty of $J$. Nucleus | $\alpha_{D}^{\textnormal{exp.}}$ (fm3) | $r$ | $k$ (fm3) | $L$ (MeV) | $\Delta L_{\textnormal{min}}$ (MeV)
---|---|---|---|---|---
208Pb | $19.6\pm 0.60$ | 0.97 | 2.68 | $39.45\pm 34.15$ | $\pm 23.44$
68Ni | $3.88\pm 0.31$ | 0.96 | 0.56 | $33.25\pm 40.75$ | $\pm 22.12$
48Ca | $2.07\pm 0.22$ | 0.96 | 0.33 | $14.75\pm 44.25$ | $\pm 19.97$
112Sn | $7.19\pm 0.50$ | 0.97 | 1.10 | $12.80\pm 34.80$ | $\pm 20.87$
114Sn | $7.29\pm 0.58$ | 0.97 | 1.15 | $10.50\pm 36.00$ | $\pm 20.22$
116Sn | $7.52\pm 0.51$ | 0.97 | 1.23 | $12.25\pm 32.75$ | $\pm 19.52$
118Sn | $7.91\pm 0.87$ | 0.97 | 1.32 | $18.75\pm 40.75$ | $\pm 19.24$
120Sn | $8.08\pm 0.60$ | 0.97 | 1.38 | $17.90\pm 33.10$ | $\pm 18.70$
124Sn | $7.99\pm 0.56$ | 0.98 | 1.47 | $8.50\pm 31.50$ | $\pm 17.42$
Figure 6: (Color online) The dipole polarizability $\alpha_{D}$ (a) in 208Pb
and (b) in 160Sn as a function of the dipole polarizability in 150Sn. The
dipole polarizability $\alpha_{D}$ (c) in 208Pb and (d) in 124Sn times the
symmetry energy at saturation density $J$ as a function of the dipole
polarizability in 150Sn. Calculations are done by QRPA based on HFB with 19
Skyrme density functionals: SIII, SIV, SV, SVI (blue up triangles); SLy230a,
SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi, SAMi-J30, SAMi-J31, SAMi-J32,
SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*, Ska (black squares). $r$ is
the Pearson’s coefficient. Utilizing the experimental values of $\alpha_{D}$
in 208Pb [34, 30] and in 124Sn [36], and assuming $J=31.7\pm 3.2$ MeV [2], the
dipole polarizability of 150Sn is predicted to be between 14.13 and 16.25 fm3.
In Sec. 3.1, the correlations between $\alpha_{D}$ (or $\alpha_{D}J$) and
nuclear isovector properties, e.g., $L$, $\Delta R_{\textnormal{np}}$, are
investigated for the whole tin isotopes, so in the following, we will analyse
what information we can obtain from these correlations, and which nucleus
could be treated as a proper probe of nuclear isovector properties in terms of
dipole polarizabilities.
Experimentally, the dipole polarizabilities of 208Pb [34], 68Ni [39], 48Ca
[35], and stable Sn isotopes [36] were measured with high resolution. The
correlations between $\alpha_{D}J$ and $L$ are always strong for both stable
nuclei and nuclei far from stability line from previous studies and our
results in Sec. 3.1. So in Tab. 2, we show the constraints on the slope
parameter of symmetry energy $L$ from experimental dipole polarizability
values $\alpha_{D}^{\textnormal{exp.}}$ using correlation between
$\alpha_{D}J$ and $L$ in these experimentally measured nuclei. The
correlations between $\alpha_{D}J$ and $L$ are obtained by QRPA calculations
using 19 Skyrme density functionals as done in Sec.3.1. The corresponding
Pearson’s coefficients $r$ and slopes $k$ of regression lines fitted by
$\alpha_{D}J$ as a function of $L$ are also given in the table. It can be seen
that the linear correlations are well kept for all these nuclei with $r>0.95$.
$J=31.7\pm 3.2$ MeV from the statistic analysis of various available
constraints [2] is adopted for deducing the $L$ value. The uncertainty of $L$
is determined by $\Delta L=\big{(}J\Delta\alpha_{D}+\alpha_{D}\Delta
J\big{)}/k$, where $\Delta\alpha_{D}$ and $\Delta J$ are the uncertainties of
$\alpha_{D}$ and $J$, respectively. From Tab. 2, it can be seen that $L$ have
a remarkable uncertainties which are all larger than $\pm 30$ MeV. In the
limiting case $\Delta\alpha_{D}=0$, the uncertainty of slope parameter $\Delta
L_{\textnormal{min}}$ comes only from the the uncertainty of $J$, which is
also given in Tab. 2. It shows the uncertainty of $J$ contributes more than
half of the total uncertainties of $L$, which hinders the effective
constraints on $L$ from the correlation between $\alpha_{D}J$ and $L$.
However, with the increase of neutron number in Sn isotopes, $\Delta
L_{\textnormal{min}}$ has the tendency to become smaller. This is because the
slope $k$ of regression line increases faster than the dipole polarizabity
$\alpha_{D}$ with the increase of neutron number, and hence $\alpha_{D}/k$
becomes smaller. So the uncertainty caused by $\Delta J$ would become small if
one finds a nucleus with a small $\alpha_{D}/k$ value.
Based on the analysis of neutron-rich Sn isotopes in Sec. 3.1, a strong
correlation between $\alpha_{D}$ and $L$ appears in neutron-rich nuclei
(seeing Fig. 1) where the PDR gives a considerable contribution to the inverse
energy-weighted sum rule $m_{-1}$. So it provides a more effective way to
constrain $L$ directly from dipole polarizability. Moreover, the slope $k$ of
regression line (in Fig. 1) becomes larger with the increase of neutron
number, which makes the constraints on $L$ from this correlation in neutron-
rich nuclei more stiff. For example, an uncertainty of $\pm 0.5$ fm3 in
$\alpha_{D}$ of 140Sn, which is about the present accuracy for experimental
measurement in dipole polarizability, could constrain $L$ within $\pm 10$ MeV,
while with the same uncertainty of $\alpha_{D}$ in 160Sn, $L$ can be
constrained within $\pm 6$ MeV. However, for these neutron-rich nuclei, the
experimental data for dipole polarizabilities is still unavailable, so we
first need to make predictions on $\alpha_{D}$ in neutron-rich nuclei.
Table 3: Predictions of the dipole polarizabilities in neutron-rich Sn isotopes from experimental dipole polarizabilities of 208Pb [34, 30] and 124Sn [36, 37] using the correlations shown in Fig. 6 (c) and (d). The constrained values of slope parameter of symmetry energy $L$ and neutron-skin thickness of neutron-rich Sn isotopes are also given from the correlations shown in Figs. 1 and Figs. 5. The Pearson’s coefficients $r$ and slopes of regression line $k$ fitted by dipole polarizability $\alpha_{D}$ as a function of $L$, as well as by neutron-skin thickness $\Delta R_{\textnormal{np}}$ as a function of $\alpha_{D}$, are also shown respectively. Nuclei | $\alpha_{D}^{P}$ (fm3) | $\alpha_{D}$ as a function of $L$ | $\Delta R_{\textnormal{np}}$ as a function of $\alpha_{D}$
---|---|---|---
$r$ | k (fm3/MeV) | $L$ (MeV) | r | k (fm-2) | $\Delta R_{\textnormal{np}}$(fm)
140Sn | $11.97\pm 0.91$ | 0.91 | 0.050 | $18.5\pm 18.1$ | 0.89 | 0.032 | $0.295\pm 0.029$
142Sn | $12.60\pm 0.96$ | 0.93 | 0.054 | $19.4\pm 17.7$ | 0.90 | 0.033 | $0.316\pm 0.031$
144Sn | $13.25\pm 0.99$ | 0.94 | 0.057 | $20.3\pm 17.3$ | 0.91 | 0.033 | $0.338\pm 0.033$
146Sn | $13.91\pm 1.02$ | 0.96 | 0.060 | $21.1\pm 16.9$ | 0.92 | 0.034 | $0.358\pm 0.034$
148Sn | $14.56\pm 1.04$ | 0.97 | 0.063 | $21.7\pm 16.5$ | 0.93 | 0.034 | $0.377\pm 0.035$
150Sn | $15.19\pm 1.06$ | 0.98 | 0.065 | $22.3\pm 16.2$ | 0.94 | 0.034 | $0.396\pm 0.036$
152Sn | $15.79\pm 1.09$ | 0.98 | 0.068 | $22.7\pm 16.0$ | 0.94 | 0.034 | $0.414\pm 0.037$
154Sn | $16.35\pm 1.12$ | 0.99 | 0.071 | $23.1\pm 15.7$ | 0.95 | 0.033 | $0.431\pm 0.038$
156Sn | $16.84\pm 1.16$ | 0.99 | 0.075 | $23.5\pm 15.5$ | 0.94 | 0.032 | $0.447\pm 0.038$
158Sn | $17.37\pm 1.21$ | 0.99 | 0.078 | $23.5\pm 15.5$ | 0.95 | 0.032 | $0.461\pm 0.039$
160Sn | $17.81\pm 1.27$ | 0.99 | 0.082 | $23.5\pm 15.5$ | 0.95 | 0.031 | $0.474\pm 0.040$
In Fig. 6(a), we study the correlations of $\alpha_{D}$ between 208Pb and
150Sn. Although it was found that $\alpha_{D}$ between two stable nuclei,
e.g., between 208Pb and 120Sn, have a good linear correlation [33, 30], this
correlation is no longer well kept when it is extended to $\alpha_{D}$ between
one stable nucleus and one neutron-rich nucleus, e.g., between 208Pb and
150Sn, as seen in Fig. 6(a). The correlation between two neutron-rich nuclei,
e.g., between 160Sn and 150Sn, is further checked in Fig. 6(b), and it becomes
strong again. So one fails to predict $\alpha_{D}$ of neutron-rich nuclei from
$\alpha_{D}$ of stable nuclei directly. Since both $\alpha_{D}J$ in stable
nuclei and $\alpha_{D}$ in neutron-rich nuclei linearly correlate with $L$,
$\alpha_{D}J$ in stable nuclei should also linearly correlate with
$\alpha_{D}$ in neutron-rich nuclei. This is checked by our calculations in
Fig. 6, where $\alpha_{D}J$ in 208Pb (c) and in 124Sn (d) as a function of
$\alpha_{D}$ in 150Sn are plotted. Good linear correlations with $r=0.950$ and
$0.964$ are found respectively, which can be used for the predictions of
$\alpha_{D}$ in 150Sn as well as other neutron-rich nuclei. Utilizing the
experimental $\alpha_{D}$ values of 208Pb and 124Sn, shown in Tab. 2, and
adopting $J=31.7\pm 3.2$ [2], $\alpha\in[12.26,16.25]$ fm3 and
$\alpha_{D}\in[14.13,18.29]$ fm3 are obtained for 150Sn. The overlap
$\alpha_{D}\in[14.13,16.25]$ fm3 is finally taken as the predicted value for
150Sn.
The same process can be done for other neutron-rich nuclei. The predicted
$\alpha_{D}$ from 140Sn to 160Sn are given in Tab. 3, with which the
corresponding constraints on $L$ and neutron-skin thickness $\Delta
R_{\textnormal{np}}$ are deduced and presented in Tab. 3 from the correlations
between $\alpha_{D}$ and $L$, as well as between $\Delta R_{\textnormal{np}}$
and $\alpha_{D}$. The corresponding Pearson’s coefficients $r$ of both
correlations are shown in the table, and it can be seen that the linear
correlations are very well kept for all these neutron-rich nuclei. Here since
the $L$ values are constrained from the linear correlation between
$\alpha_{D}$ and $L$ directly, the uncertainties become much smaller compared
to those shown in Tab. 2. With the increase of neutron number, the slope of
regression line fitted by $\alpha_{D}$ as a function of $L$ becomes larger,
and as a result, the uncertainty of $L$ also becomes smaller until 156Sn even
with an increasing uncertainty in the predicted $\alpha_{D}^{P}$. For the
neutron-skin thickness, the slope of regression line fitted by $\Delta
R_{\textnormal{np}}$ as a function of $\alpha_{D}$ keeps almost a constant
with increasing neutron numbers, yet the uncertainties of constrained neutron-
skin thickness are becoming larger caused by the increasing uncertainties in
$\alpha_{D}^{P}$. Due to the lack of experimental data of $\alpha_{D}$ in
neutron-rich nuclei, the present constraints on $L$ shown in Tab. 3 in fact
don’t give new information compared to the $L$ values obtained from the
correlation between $L$ and $\alpha_{D}J$ in 208Pb and in 124Sn. However, the
direct correlation between $\alpha_{D}$ and $L$ would show its special
importance and effectiveness in constraining nuclear isovector properties when
the experimental data of $\alpha_{D}$ in neutron-rich tin isotopes are
available, so the measurements of dipole polarizability towards neutron-rich
nuclei are strongly called for.
## 4 Summary
The correlations between electric dipole polarizability $\alpha_{D}$ (or times
symmetry energy at saturation density $J$) and slope parameter of symmetry
energy $L$ are studied in Sn isotopes preformed by QRPA based on Skyrme HFB
theory. The previously found correlation between $\alpha_{D}J$ and $L$ is
confirmed in the whole Sn isotopes from neutron-deficient ones to neutron-rich
ones. The linear correlation between $\alpha_{D}$ and $L$ is not strong in
stable tin isotopes and their surroundings, however, it becomes better for
mass number $A>132$, and strong correlations are found when $A\geq 140$ with
the correlation coefficients $r>0.9$, where PDR gives a considerable
contribution to $\alpha_{D}$. The enhancement of this correlation between
$\alpha_{D}$ and $L$ is attributed to the pairing correlations, which play
important roles through PDR.
With the available high-resolution data of $\alpha_{D}$, the constraints on
$L$ are obtained from the correlation between $\alpha_{D}J$ and $L$. Large
uncertainties of $L$ are found, where more than half are contributed by the
uncertainty from symmetry energy $\Delta J=\pm 3.2$ MeV. A proper candidate
nucleus for constraining $L$ is the one with a small $\alpha_{D}/k$ value,
where $k$ is the slope of regression line fitted by $\alpha_{D}J$ as a
function of $L$. In stable Sn isotopes, the $\alpha_{D}/k$ becomes smaller
towards neutron-rich side.
With the strong correlation between $\alpha_{D}$ and $L$ in neutron-rich Sn
isotopes, $L$ can be constrained directly and more stiffly if experimental
data of $\alpha_{D}$ with high resolution in these nuclei are known. At the
moment, $\alpha_{D}$ in neutron-rich nuclei are predicted using the linear
correlation between $\alpha_{D}J$ in a stable nucleus with experimental data
and $\alpha_{D}$ in a neutron-rich nucleus. The measurements of electric
dipole polarizability towards neutron-rich nuclei are called for.
## Acknowledgement
This work is partly supported by National Natural Science Foundation of China
under Grant No. 12075104, 11675065 and 11875152, Fundamental Research Funds
for the Central Universities under Grant No.lzujbky-2019-11, and Strategic
Priority Research Program of Chinese Academy of Sciences, Grant No.
XDB34000000.
## References
* Li et al. [2008] B.-A. Li, L.-W. Chen, and C. M. Ko, Phys. Rep. 464, 113 (2008).
* Oertel et al. [2017] M. Oertel, M. Hempel, T. Klahn, and S. Typel, Rev. Mod. Phys. 89, 015007 (2017).
* Dutra et al. [2012] M. Dutra, O. Lourenco, J. S. Martins, A. Delfino, J. stone, and P. Stevenson, Phys. Rev. C 85, 035201 (2012).
* M.Dutra et al. [2014] M.Dutra, O.Lourenco, S.S.Avancini, A.Delfino, D.P.Menezes, C.Providencia, S.Typel, and J.R.Stone, Phys. Rev. C 90, 055203 (2014).
* Lattimer and Prakash [2001] J. M. Lattimer and M. Prakash, Astrophys. J 550, 426 (2001).
* Liu et al. [2018] Z. W. Liu, Z. Qian, R. Y. Xing, J. R. Niu, and B. Y. Sun, Phys. Rev. C 97, 025801 (2018).
* Tsang et al. [2009] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. Steiner, Phys. Rev. Lett. 102, 122701 (2009).
* Lattimer and Prakash [2016] J. M. Lattimer and M. Prakash, Phys. Rep. 621, 127 (2016).
* Roca-Maza and Paar [2018] X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. 101, 96 (2018).
* W.D.Myers and W.J.Swiatecki [1980] W.D.Myers and W.J.Swiatecki, Nucl. Phys. A 336, 267 (1980).
* M.Warda et al. [2009] M.Warda, X.Vinas, X.Roca-Maza, and M. Centelles, Phys. Rev. C 80, 024316 (2009).
* Brown [2000] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).
* Chen et al. [2005] L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev. C 72, 064309 (2005).
* Wang and Li [2013] N. Wang and T. Li, Phys. Rev. C 88, 011301(R) (2013).
* Brown [2017] B. A. Brown, Phys. Rev. Lett. 119, 122502 (2017).
* Yang and Piekarewicz [2018] J. Yang and J. Piekarewicz, Phys. Rev. C 97, 014314 (2018).
* Colo et al. [2013] G. Colo, L. Cao, N. V. Giai, and L. Capelli, Comp. Phys. Comm. 184, 142 (2013).
* Terasaki et al. [2005] J. Terasaki, J. Engel, M. Bender, and J. Dobaczewski, Phys. Rev. C 71, 034310 (2005).
* E.Khan and Giai [2000] E.Khan and N. V. Giai, Phys. Lett. B 472, 253 (2000).
* G.Giambrone et al. [2003] G.Giambrone, S. Scheit, F. Barranco, P. Bortignon, G.Colo, D.Sarchi, and E.Vigezzi, Nucl. Phys. A 726, 3 (2003).
* Martini et al. [2011] M. Martini, S. Peru, and M. Dupuis, Phys. Rev. C 83, 034309 (2011).
* N.Paar et al. [2003] N.Paar, P.Ring, T.Niksic, and D.Vretenar, Phys. Rev. C 67, 034312 (2003).
* Paar et al. [2007] N. Paar, D. Vretenar, E. Khan, and G. Colo, Rep. Prog. Phys. 70, 691 (2007).
* Ring et al. [2001] P. Ring, Z. yu Ma, N. V. Giai, D. Vretenar, A. Wandelt, and L. gang Cao, Nucl. Phys. A 694, 249 (2001).
* T.Niksic et al. [2002] T.Niksic, D.Vretenar, and P.Ring, Phys. Rev. C 66, 064302 (2002).
* Klupfel et al. [2009] P. Klupfel, P.-G. Reinhard, T. Burvenich, and J. Maruhn, Phys. Rev. C 79, 034310 (2009).
* Reinhard and Nazarewicz [2010] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 81, 051303(R) (2010).
* Piekarewicz et al. [2012] J. Piekarewicz, B. K. Agrawal, G. Colo, W. Nazarewicz, N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vretenar, Phys. Rev. C 85, 041302(R) (2012).
* Roca-Maza et al. [2013a] X. Roca-Maza, M. Brenna, G. Colo, M. Centelles, X. Vinas, B. Agrawal, N. Paar, D. Vretenar, and J. Piekarewicz, Phys. Rev. C 88, 024316 (2013a).
* Roca-Maza et al. [2015] X. Roca-Maza, X. Vinas, M. Centelles, B. Agrawal, G. Colo, N. Paar, J. Piekarewicz, and D. Vretenar, Phys. Rev. C 92, 064304 (2015).
* Zhang and Chen [2014] Z. Zhang and L.-W. Chen, Phys. Rev. C 90, 064317 (2014).
* Zhang and Chen [2015] Z. Zhang and L.-W. Chen, Phys. Rev. C 92, 031301 (2015).
* Hashimoto et al. [2015] T. Hashimoto, A. M. Krumbholz, P.-G. Reinhard, A. Tamii, P. von Neumann-Cosel, T. Adachi, N. Aoi, C. A. Bertulani, H. Fujita, Y. Fujita, et al., Phys. Rev. C 92, 031305 (2015).
* Tamii et al. [2011] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertulani, J. Carter, M. Dozono, H. Fujita, K. Fujita, et al., Phys. Rev. Lett. 107, 062502 (2011).
* Birkhan et al. [2017] J. Birkhan, M. Miorelli, S. Bacca, S. Bassauer, C. A. Bertulani, G. Hagen, H. Matsubara, P. von Neumann-Cosel, T. Papenbrock, N. Pietralla, et al., Phys. Rev. Lett. 118, 252501 (2017).
* Bassauer et al. [2020a] S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi, C. A. Bertulani, P. Y. Chan, A. D. Alessio, H. Fujioka, H. Fujita, et al., Phys. Rev. C 102, 034327 (2020a).
* Bassauer et al. [2020b] S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi, C. A. Bertulani, P. Y. Chan, A. D. Alessio, H. Fujioka, H. Fujita, et al., Phys. Lett. B 810, 135804 (2020b).
* von Neumann-Cosel and Tamii [2019] P. von Neumann-Cosel and A. Tamii, Eur. Phys. J. A 55, 110 (2019).
* Rossi et al. [2013] D. M. Rossi, P. Adrich, F. Aksouh, H. Alvarez-Pol, T. Aumann, J. Benlliure, M. Bohmer, K. Boretzky, E. Casarejos, M. Chartier, et al., Phys. Rev. Lett. 111, 242503 (2013).
* Wienholtz et al. [2013] F. Wienholtz, D. Beck, K. Blaum, C. Borgmann, M. Breitenfeldt, R. B. Cakirli, S.George, F. Herfurth, J. D.Holt, M. Kowalska, et al., Nature 498, 346 (2013).
* Liu et al. [2020] J. Liu, Y. F. Niu, and W. H. Long, Phys. Lett. B 806, 135524 (2020).
* Li et al. [2019a] J. J. Li, W. H. Long, J. Margueron, and N. V. Giai, Phys. Lett. B 788, 192 (2019a).
* Li et al. [2019b] Z. Z. Li, S. Y. Chang, Q. Zhao, W. H. Long, and Y. F. Niu, Chinese Physics C 43, 074107 (2019b).
* Grasso [2014] M. Grasso, Phys. Rev. C 89, 034316 (2014).
* Savran et al. [2013] D. Savran, T.Aumann, and A.Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013).
* Aumann [2019] T. Aumann, Eur. Phys. J. A 55, 234 (2019).
* Beiner et al. [1976] M. Beiner, H. Flocard, N. V. Giai, and P. Quentin, Nucl. Phys. A 238, 29 (1976).
* Giai and Sagawa [1981] N. V. Giai and H. Sagawa, Phys. Lett. B 106, 379 (1981).
* Krivine et al. [1980] H. Krivine, J. Treiner, and O. Bohigas, Nucl. Phys. A 336, 155 (1980).
* Bartel et al. [1982] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B. Hakansson, Nucl. Phys. A 386, 79 (1982).
* Kohler [1976] H. Kohler, Nucl. Phys. A 258, 301 (1976).
* Chabanat et al. [1997] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A 627, 710 (1997).
* Chabanat et al. [1998] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A 635, 231 (1998).
* Roca-Maza et al. [2012] X. Roca-Maza, G. Colo, and H. Sagawa, Phys. Rev. C 86, 031306 (2012).
* Roca-Maza et al. [2013b] X. Roca-Maza, M. Brenna, B. K. Agrawal, P. F. Bortignon, G. Colo, L.-G. Cao, N. Paar, and D. Vretenar, Phys. Rev. C 87, 034301 (2013b).
* Bender et al. [2000] M. Bender, K. Rutz, P.-G. Reinhard, and J. Maruhn, Eur. Phys. J. A 8, 59 (2000).
* J.Piekarewicz [2006] J.Piekarewicz, Phys. Rev. C 73, 044325 (2006).
* Carbone et al. [2010] A. Carbone, G. Colo, A. Bracco, L.-G. Cao, P. F. Bortignon, F. Camera, and O. Wieland, Phys. Rev. C 81, 041301(R) (2010).
* D.Vretenar et al. [2012] D.Vretenar, Y.F.Niu, N.Paar, and J.Meng, Phys. Rev. C 85, 044317 (2012).
|
11institutetext: Riccardo Gatto 22institutetext: University of Bern,
Department of Mathematics an Statistics, Institute of Mathematical Statistics
and Actuarial Science, Alpeneggstrasse 22, 3012 Bern, Switzerland
Email<EMAIL_ADDRESS>
2010 Mathematics Subject Classification
62M20 Inference from stochastic processes: prediction; filtering
62H11 Multivariate analysis: directional data; spatial statistics
# Information theoretic results for stationary time series and the Gaussian-
generalized von Mises time series
Riccardo Gatto
January 21 2021
###### Abstract
This chapter presents some novel information theoretic results for the
analysis of stationary time series in the frequency domain. In particular, the
spectral distribution that corresponds to the most uncertain or unpredictable
time series with some values of the autocovariance function fixed, is the
generalized von Mises spectral distribution. It is thus a maximum entropy
spectral distribution and the corresponding stationary time series is called
the generalized von Mises time series. The generalized von Mises distribution
is used in directional statistics for modelling planar directions that follow
a multimodal distribution. Furthermore, the Gaussian-generalized von Mises
times series is presented as the stationary time series that maximizes
entropies in frequency and time domains, respectively referred to as spectral
and temporal entropies. Parameter estimation and some computational aspects
with this time series are briefly analyzed.
## 1 Introduction
Nonstationary data typically have mean, variance, and covariances that change
significantly over the time. It is consequently difficult to make reliable
predictions or forecasts directly from these data. For this reason,
nonstationary data are transformed to stationary data, viz. data that possess
constant mean, constant variance and constant covariance between any two
observations that are separated by any fixed time lag. Stationary data are
often analyzed in the frequency domain, where the spectral distribution plays
the central role: it characterizes the correlations between the values of the
time series and it allows for linear predictions. The analysis in the
frequency domain is particularly interesting for the identification of
periodicities of the data. The first developments of the theory of stationary
processes appeared at the end of the 19-th century with the analysis of data
in the frequency domain, which is called the spectral analysis. The
alternative analysis in the time domain, viz. based on the covariance
function, appeared only later. The first statistical theory for periodic
phenomena was developed by Fisher (1929). Other early leading contributions to
the theory of stationary processes are Cramér (1942), Rice (1944 and 1945) as
well as the volumes Cramér and Leadbetter (1967) and Yaglom (1962). An up to
date volume on stationary processes is Lindgren (2012) and an historical
review can be found in Brillinger (1993). This chapter provides various
information theoretic results for spectral distributions of stationary
processes with discrete time, i.e. stationary time series. It recasts the
generalized von Mises (GvM) distribution, which was introduced in directional
statistics as a model for planar directions, in the context of the spectral
analysis of time series. It shows that the spectral distribution that
corresponds to the most uncertain or unpredictable time series and whose
autocovariance function agrees with some few first predetermined values, for
example estimated from a sample, is the GvM spectral distribution. It is thus
a maximum entropy spectral distribution and the corresponding stationary time
series can be called the generalized von Mises time series. The Gaussian
stationary time series with GvM spectral distribution is called Gaussian-GvM.
This time series follows the maximal entropy principle w.r.t. time and
frequency. Although some estimation and other computational aspects are
briefly analyzed, this chapter is only a first study of the GvM time series.
Let $\\{X_{j}\\}_{j\in\mathbb{Z}}$ be a complex-valued time series whose
elements belong to a common Hilbert space ${\cal L}_{2}$ of square integrable
random variables, thus ${\sf E}\left[|X_{j}|^{2}\right]<\infty$, $\forall
j\in\mathbb{Z}$. Its autocovariance function (a.c.v.f.) is given by
$\displaystyle\psi(j+r,j)$ $\displaystyle={\rm
cov}\left(X_{j+r},X_{j}\right)={\sf
E}\left[X_{j+r}\overline{X_{j}}\right]-{\sf E}[X_{j+r}]{\sf
E}\left[\overline{X_{j}}\right],\;\forall j,r\in\mathbb{Z}.$
We assume that the time series is weakly stationary, which will be shortened
to stationary, precisely that ${\sf E}[X_{j}]$ and $\psi(j+r,j)$ do not depend
on $j$, $\forall j,r\in\mathbb{Z}$. In this case we denote $\mu={\sf
E}[X_{j}]$,
$\displaystyle\psi(r)$ $\displaystyle=\psi(r,0)=\psi(j+r,j),\;\forall
j,r\in\mathbb{Z},$
and $\sigma^{2}=\psi(0)$, for some $\sigma\in(0,\infty)$. A stronger type of
stationarity is the strict stationarity, which requires that the double finite
dimensional distributions (f.d.d.) of the time series are invariant after a
fixed time shift, i.e. $\forall j_{1}<\ldots<j_{n}\in\mathbb{Z}$,
$r\in\mathbb{Z}$ and $n\geq 1$,
$\displaystyle\left(U_{j_{1}},\ldots,U_{j_{n}},V_{j_{1}},\ldots,V_{j_{n}}\right)\sim\left(U_{j_{1}+r},\ldots,U_{j_{n}+r},V_{j_{1}+r},\ldots,V_{j_{n}+r}\right),$
(1)
where $U_{j}=\mbox{\rm Re}\hskip 2.27621ptX_{j}$ and $V_{j}=\mbox{\rm
Im}\hskip 2.27621ptX_{j}$, $\forall j\in\mathbb{Z}$. As usual, $E_{1}\sim
E_{2}$ means that the random elements $E_{1}$ and $E_{2}$ follow the same
distribution. Stationary time series can be analyzed in the frequency domain,
precisely through the spectral distribution. A spectral distribution function
(d.f.) is any nondecreasing function $F_{\sigma}:[-\pi,\pi]\to[0,\infty)$ that
satisfies $F_{\sigma}(-\pi)=0$ and $F_{\sigma}(\pi)=\sigma^{2}$. Thus, the
total mass of $F_{\sigma}$ is $\sigma^{2}$. This d.f. relates to the a.c.v.f.
through the equation
$\displaystyle\psi(r)$ $\displaystyle=\int_{(-\pi,\pi]}{\rm e}^{{\rm i}rv}{\rm
d}F_{\sigma}(v),\;\forall r\in\mathbb{Z}.$
The simplest nontrivial stationary time series $\\{X_{j}\\}_{j\in\mathbb{Z}}$
is called white noise if it has mean zero and a.c.v.f.
$\psi(r)=\begin{cases}\sigma^{2},&\;\mathrm{if}\,\,r=0,\\\
0,&\;\mathrm{if}\,\,r=\pm 1,\pm 2,\ldots,\end{cases}$
for some $\sigma>0$. All frequencies of $\\{X_{j}\\}_{j\in\mathbb{Z}}$ are
equally represented, because its spectral density is the uniform one with
total mass $\sigma^{2}$, namely $f_{\sigma}(\theta)=\sigma^{2}/(2\pi)$,
$\forall\theta\in(-\pi,\pi]$. The term white noise originates from the fact
that white color reflects all visible wave frequencies of light. Real-valued
time series are used in many applied sciences; refer e.g. to Brockwell and
Davis (1991) or Chatfield (2013). However, complex-valued time series are
often preferred representations of bivariate signals, mainly because their
compact formulation. They have been applied in various technical domains, such
as magnetic resonance imaging (cf. e.g. Rowe, 2005) or oceanography (cf. e.g.
Gonella, 1972).
Spectral distributions of complex-valued time series can be viewed as rescaled
circular distributions. For real-valued time series, the spectral distribution
is a rescaled axially symmetric circular distribution. We recall that a
circular distribution is a probability distribution over the circle that is
used for modelling planar directions as well as periodic phenomena. Two major
references are Mardia and Jupp (2000) and Jammalamadaka and SenGupta (2001).
For a short introduction cf. e.g. Gatto and Jammalamadaka (2015). During the
last few decades, there has been a considerable amount of theoretical and
applied research on circular distributions.
Let $k\in\\{1,2,\ldots\\}$. A class of circular distributions that possess
important theoretical properties has densities given by
$\displaystyle
f_{1}^{(k)}\left(\theta\mid\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k}\right)=$
(2) $\displaystyle\frac{1}{2\pi
G_{0}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k}\right)}\exp\left\\{\sum_{j=1}^{k}\kappa_{j}\cos
j(\theta-\mu_{j})\right\\},$
$\forall\theta\in(-\pi,\pi]$ (or any other interval of length $2\pi$), where
$\mu_{j}\in(-\pi/j,\pi/j]$, $\kappa_{j}\geq 0$, for $j=1,\ldots,k$,
$\displaystyle
G_{0}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\exp\\{\kappa_{1}\cos\theta+\kappa_{2}\cos
2(\theta+\delta_{1})+\ldots+\kappa_{k}\cos k(\theta+\delta_{k-1})\\}{\rm
d}\theta,$
and where $\delta_{j}=(\mu_{1}-\mu_{j+1})\mbox{\rm mod}(2\pi/(j+1))$, for
$j=1,\ldots,k-1$, whenever $k\geq 2$. The circular density (2) for $k\geq 2$
was thoroughly analyzed by Gatto and Jammalamadaka (2007) and Gatto (2009),
who called it “generalized von Mises density of order $k$” (${\rm GvM}_{k}$).
Let us denote a circular random variable $\theta$ with that density as
$\theta\sim{\rm GvM}_{k}(\mu_{1},$ $\ldots,$ $\mu_{k},\kappa_{1},$ $\ldots,$
$\kappa_{k})$. The ${\rm GvM}_{1}$ density is the well-known circular normal
or von Mises (vM) density, which represents within circular statistics what
the normal distribution represents in linear statistics. It is given by
$f_{1}^{(1)}(\theta\mid\mu_{1},\kappa_{1})=\\{2\pi
I_{0}(\kappa_{1})\\}^{-1}\exp\\{\kappa_{1}\cos(\theta-\mu_{1})\\},$
$\forall\theta\in(-\pi,\pi]$, where $\mu_{1}\in(-\pi,\pi]$, $\kappa_{1}\geq 0$
and where $I_{n}(z)=(2\pi)^{-1}\int_{0}^{2\pi}\cos n\theta$
$\exp\\{z\cos\theta\\}{\rm d}\theta$, $\forall z\in\mathbb{C}$, is the
modified Bessel function of the first kind and integer order $n$ (see e.g.
Abramowitz and Stegun, 1972, p. 376). Compared to the vM, which is axially
symmetric and unimodal whenever $\kappa_{1}>0$, the ${\rm GvM}_{2}$
distribution allows for substantially higher adjustability, in particular in
terms of asymmetry and bimodality. This makes it a practical circular
distributions that has found various applications. Some recent ones are: Zhang
et al. (2018), in meteorology, Lin and Dong (2019), in oceanography, Astfalck
et al. (2018), in offshore engineering, and in Christmas (2014), in signal
processing. The ${\rm GvM}_{k}$ spectral density is given by
$f_{\sigma}^{(k)}=\sigma^{2}f_{1}^{(k)}$, for some $\sigma\in(0,\infty)$: it
is the ${\rm GvM}_{k}$ circular density $f_{1}^{(k)}$ given in (2) that is
rescaled to have any desired total mass $\sigma^{2}$. When the ${\rm GvM}_{k}$
spectral density is axially symmetric around the null axis, then the
corresponding time series $\\{X_{j}\\}_{j\in\mathbb{Z}}$ is real-valued. As
shown in Salvador and Gatto (2020a), the ${\rm GvM}_{2}$ density with
$\kappa_{1},\kappa_{2}>0$ is axially symmetric iff $\delta_{1}=0$ or
$\delta_{1}=\pi/2$. In both cases, the axis of symmetry has angle $\mu_{1}$
with respect to (w.r.t.) the null direction. The ${\rm GvM}_{2}$ spectral
density has a practical role time series because of its uni- and bimodal
shape. A complete analysis of the modality of the ${\rm GvM}_{2}$ distribution
is given in Salvador and Gatto (2020b). Note that in some situations a three-
parameter version of the ${\rm GvM}_{2}$ distribution introduced by Kim and
SenGupta (2013) appears sufficient to model both asymmetric and bimodal data.
The densities of this subclass are obtained by setting $\delta_{1}=\pi/4$ and
$k=2$ in the ${\rm GvM}_{k}$ density (2). However, this subclass does not
possess the optimality properties of the ${\rm GvM}_{2}$ distribution that are
presented in Section 2. It is worth mentioning that the GvM spectral
distribution has many similarities with the exponential model of Bloomfield
(1973), which is a truncated Fourier series of the logarithm of some spectral
distribution. Bloomfield motivates the low truncation of the Fourier series by
the fact that “the logarithm of an estimated spectral density function is
often found to be a fairly well-behaved function”. A closely related reference
is Healy and Tukey (1963). We should however note that Bloomfield’s model if
given for real-valued time series only.
The estimation of the spectral distribution is one of the important problems
in the analysis of stationary time series and of stationary stochastic
processes with continuous time. Information theoretic quantities like
Kullback-Leibler’s information (cf. Kullback and Leibler, 1951) or Shannon’s
entropy (cf. Shannon, 1948) are very useful in this context. These quantities
are usually defined for probability distributions but they can be considered
for distributions with finite mass. These are spectral distributions and we
assume them absolutely continuous (w.r.t. the Lebesgue measure). Thus, let
$f_{\sigma}$ and $g_{\sigma}$ be two spectral densities whose integrals over
$(-\pi,\pi]$ are both equal to $\sigma^{2}$. The spectral Kullback-Leibler
information of $f_{\sigma}$ w.r.t. $g_{\sigma}$ is given by
$I(f_{\sigma}|g_{\sigma})=\int_{-\pi}^{\pi}\log\frac{f_{\sigma}(\theta)}{g_{\sigma}(\theta)}f_{\sigma}(\theta){\rm
d}\theta=\sigma^{2}I(f_{1}|g_{1}),$ (3)
where $0\log 0=0$ is assumed and where the support of $f_{\sigma}$ is included
in the support of $g_{\sigma}$, otherwise $I(f_{\sigma}|g_{\sigma})=\infty$.
It follows from Gibbs inequality that $I(f_{\sigma}|g_{\sigma})$ is
nonnegative, precisely
$I(f_{\sigma}|g_{\sigma})\geq 0,$
for all possible spectral densities $f_{\sigma}$ and $g_{\sigma}$, with
equality iff $f_{1}=g_{1}$ a.e. The Kullback-Leibler information is also
called relative entropy, Kullback-Leibler divergence or distance, eventhough
it is not a metric. Thus (3) is a measure of divergence for distributions with
same total mass $\sigma^{2}$. Shannon’s entropy can be defined for the
spectral density $f_{\sigma}$ by
$S(f_{\sigma})=-\int_{-\pi}^{\pi}\log\frac{f_{\sigma}(\theta)}{(2\pi)^{-1}\sigma^{2}}f_{\sigma}(\theta){\rm
d}\theta=-I(f_{\sigma}|u_{\sigma})=-\sigma^{2}I(f_{1}|u_{1}),$ (4)
where $u_{\sigma}$ is the uniform density with total mass $\sigma^{2}$ over
$(-\pi,\pi]$, viz. $u_{\sigma}=\sigma^{2}/(2\pi){\sf I}_{(-\pi,\pi]}$, ${\sf
I}_{A}$ denoting the indicator of set $A$. Shannon’s entropy of the circular
density $f_{1}$ over $(-\pi,\pi]$ is originally defined as
$-\int_{-\pi}^{\pi}\log f_{1}(\theta)f_{1}(\theta){\rm
d}\theta=-(2\pi)^{-1}-I(f_{1}|u_{1})$. It measures the uncertainty inherent in
the probability distribution with density $f_{1}$. Equivalently, $S(f_{1})$
measures the expected amount of information gained on obtaining an observation
from $f_{1}$, based on the principle that the rarer an event, the more
informative its occurrence. The spectral entropy defined in (4) differs
slightly differs the original formula of Shannon’s entropy for probability
distributions: inside the logarithm, $f_{\sigma}$ is divided by the uniform
density with total mass $\sigma^{2}$. With this modification the spectral
entropy becomes scale invariant w.r.t. $\sigma^{2}$, just like the spectral
Kullback-Leibler information (3). The spectral entropy satisfies
$S(f_{\sigma})\leq 0$, with equality iff $f_{\sigma}=u_{\sigma}$ a.e. This
follows from Gibbs inequality.
The topics of the next sections of this chapter are the following. Section 2
provides information theoretic results for spectral distributions and
introduces the related GvM and the Gaussian-GvM time series. Section 2.1 gives
general definitions and concepts. Section 2.2 provides the optimal spectral
distributions under constraints on the a.c.v.f. The GvM spectral distribution
appears as the one that maximizes Shannon’s entropy under constraints on the
first few values of the a.c.v.f. Section 2.3 motivates the Gaussian-GvM time
series from the fact that it follows the maximal entropy principle in both
time and frequency domains. Section 3 provides some computational aspects.
Section 3.1 gives some series expansions for integral functions appearing in
the context of the ${\rm GvM}_{2}$ time series, which is one the most relevant
case. The estimation problem of the GvM spectral distribution is presented in
Section 3.2. After presenting some general consideration and some related
results on maximum entropy spectral estimation, an estimator to the parameters
of the GvM spectral distribution is provided. Section 3.3 provides an
expansion for the GvM spectral d.f. Some short concluding remarks are given in
Section 4.
## 2 The GvM and the Gaussian-GvM time series
Section 2.1 gives some general facts on time series and defines the GvM time
series. Section 2.2 provides information theoretic results for spectral
distributions. An important result is that the GvM spectral distribution
maximizes the entropy under constraints on the a.c.v.f. Section 2.3 proposes
the Gaussian-GvM time series based on the fact that it follows the maximal
entropy principle in both time and frequency domains, under the same
constraints.
### 2.1 General considerations
Two central theorems of spectral analysis of time series are the following.
The first one is Herglotz theorem:
* $\psi:\mathbb{Z}\rightarrow\mathbb{C}$ is nonnegative definite (n.n.d.)111 The function $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is n.n.d. if $\sum_{i=1}^{n}\sum_{j=1}^{n}$ $c_{i}\overline{c}_{j}f(x_{i}-x_{j})\geq 0$, $\forall x_{1},\ldots,x_{n}\in\mathbb{R}$, $c_{1},\ldots,c_{n}\in\mathbb{C}$ and $n\geq 1$.
Any n.n.d. function $f$ is Hermitian, i.e. $f(-x)=\overline{f(x)}$, $\forall
x\in\mathbb{R}$.$\Leftrightarrow$ $\psi(r)=\int_{(-\pi,\pi]}{\rm e}^{{\rm
i}rv}{\rm d}F_{\sigma}(v)$, $\forall r\in\mathbb{Z}$, for some d.f.
$F_{\sigma}$ over $[-\pi,\pi]$, with $F_{\sigma}(-\pi)=0$ and
$\sigma^{2}=F_{\sigma}(\pi)\in(0,\infty)$.
The second theorem is a characterization of the a.c.v.f.:
* $\psi:\mathbb{Z}\rightarrow\mathbb{R}$ is the a.c.v.f. of a (strictly) stationary complex-valued time series $\Leftrightarrow$ $\psi$ is n.n.d.
These two theorems tell that if we consider the spectral d.f.
$F^{(k)}_{\sigma}=\sigma^{2}F_{1}^{(k)}$, where $F_{1}^{(k)}$ is the ${\rm
GvM}_{k}$ d.f. with density $f_{1}^{(k)}$ given by (2), then there exists a
stationary time series $\\{X_{j}\\}_{j\in\mathbb{Z}}$ with spectral d.f.
$F_{\sigma}^{(k)}$ and density $f_{\sigma}^{(k)}=\sigma f_{1}^{(k)}$ that we
call GvM or, more precisely, ${\rm GvM}_{k}$ time series. Thus the ${\rm
GvM}_{k}$ time series is stationary by definition, it has variance
$F_{\sigma}^{(k)}(\pi)=\sigma^{2}$ and it is generally complex-valued, unless
the ${\rm GvM}_{k}$ spectral distribution is axially symmetric around the
origin.
The complex-valued ${\rm GvM}_{k}$ stationary time series
$\\{X_{j}\\}_{j\in\mathbb{Z}}$ can be chosen with mean zero, variance
$\sigma^{2}$ and Gaussian, meaning that the double f.d.d. given in (1) are
Gaussian. In this case, the distribution of $\\{X_{j}\\}_{j\in\mathbb{Z}}$ is
however not entirely determined by its a.c.v.f. $\psi^{(k)}$ or,
alternatively, by its spectral d.f. $F_{\sigma}^{(k)}$. (The formula for the
a.c.v.f. is given later in Corollary 1.4.) In order to entirely determine this
distribution, one also needs the so-called pseudo-covariance ${\sf
E}[X_{j+r}X_{j}]$, $\forall j,r\in\mathbb{Z}$. So an arbitrary Gaussian, with
mean zero and (weakly) stationary time series $\\{X_{j}\\}_{j\in\mathbb{Z}}$
is not necessarily strictly stationary: $\\{X_{j}\\}_{j\in\mathbb{Z}}$ is
strictly stationary iff the covariance ${\sf
E}\left[X_{j+r}\overline{X_{j}}\right]$ and the pseudo-covariance ${\sf
E}[X_{j+r}X_{j}]$ do not depend on $j\in\mathbb{Z}$, $\forall r\in\mathbb{Z}$.
This is indeed equivalent to the independence on $j\in\mathbb{Z}$ of
$\displaystyle\psi_{UU}(r)$ $\displaystyle={\sf
E}[U_{j+r}U_{j}],\;\psi_{VV}(r)={\sf E}[V_{j+r}V_{j}],$
$\displaystyle\psi_{UV}(r)$ $\displaystyle={\sf E}[U_{j+r}V_{j}]\;\text{ and
}\psi_{VU}(r)={\sf E}[V_{j+r}U_{j}],\;\forall r\in\mathbb{Z},$ (5)
where $U_{j}=\mbox{\rm Re}\hskip 2.27621ptX_{j}$ and $V_{j}=\mbox{\rm
Im}\hskip 2.27621ptX_{j}$, $\forall j\in\mathbb{Z}$. Under this independence
on $j\in\mathbb{Z}$, we have $\psi_{VU}(r)=\psi_{UV}(-r)$, $\forall
r\in\mathbb{Z}$. However, according Herglotz theorem, if the a.c.v.f.
$\psi^{(k)}$ is obtained by Fourier inversion of the ${\rm GvM}_{k}$ spectral
density, then it is n.n.d. By the above characterization of the a.c.v.f., a
strictly stationary ${\rm GvM}_{k}$ time series always exists. The existence
of a particular (precisely radially symmetric) strictly stationary
Gaussian-${\rm GvM}_{k}$ time series that satisfies some constraints on the
a.c.v.f. is shown in Section 2.3.
Next, for any given Gaussian-${\rm GvM}_{k}$ time series with spectral d.f.
$F_{\sigma}^{(k)}$, there exists a spectral process
$\\{Z_{\theta}\\}_{\theta\in[-\pi,\pi]}$ that is complex-valued and Gaussian.
We remind that the process of the frequencies
$\\{Z_{\theta}\\}_{\theta\in[-\pi,\pi]}$ is defined through the mean square
stochastic integral
$\displaystyle X_{j}$ $\displaystyle=\int_{(-\pi,\pi]}{\rm e}^{{\rm i}\theta
j}{\rm d}Z_{\theta},\;\text{ a.s.,}\;\forall j\in\mathbb{Z},$ (6)
and by the following conditions: ${\sf E}[Z_{\theta}]=0$,
$\forall\theta\in[-\pi,\pi]$, ${\sf
E}\left[\left(Z_{\theta_{2}}-Z_{\theta_{1}}\right)\overline{\left(Z_{\theta_{4}}-Z_{\theta_{3}}\right)}\right]$
$=0$, $\forall-\pi\leq\theta_{1}<\theta_{2}<\theta_{3}<\theta_{4}\leq\pi$,
viz. it has orthogonal increments, and
$\displaystyle{\sf
E}\left[\left|Z_{\theta_{2}}-Z_{\theta_{1}}\right|^{2}\right]$
$\displaystyle=F_{\sigma}^{(k)}(\theta_{2})-F_{\sigma}^{(k)}(\theta_{1}),\;\forall-\pi\leq\theta_{1}<\theta_{2}\leq\pi.$
(7)
There are several reasons for considering the Gaussian-GvM time series. A
practical one is that their simulation can be done with the algorithms
presented in Chapter XI of Asmussen and Glynn (2007). One of these algorithms
the decomposition (6). A theoretical reason for considering normality is that
it leads to a second maximal entropy principle, this one no longer in the
frequency domain but in the time domain. We pursue this explanation on the
temporal entropy in Section 2.3.
### 2.2 Spectral Kullback-Leibler information and entropy
Let $g_{\sigma}$ be the spectral density of some stationary time series with
variance $\sigma^{2}$, for some $\sigma\in(0,\infty)$. For a chosen
$k\in\\{1,2,\ldots\\}$, consider the $r$-th a.c.v.f. condition or constraint
$\displaystyle{\cal C}_{r}:$ $\displaystyle\int_{-\pi}^{\pi}{\rm e}^{{\rm
i}r\theta}g_{\sigma}(\theta){\rm d}\theta=\psi_{r},$ (8)
for some $\psi_{r}\in\mathbb{C}$ satisfying $|\psi_{r}|\leq\sigma^{2}$, for
$r=1,\ldots,k$, and such that the $(k+1)\times(k+1)$ matrix
$\displaystyle\left(\begin{matrix}\sigma^{2}&\psi_{1}&\ldots&\psi_{k}\\\
\overline{\psi_{1}}&\sigma^{2}&\ldots&\psi_{k-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
\overline{\psi_{k}}&\overline{\psi_{k-1}}&\ldots&\sigma^{2}\end{matrix}\right)$
(9)
is n.n.d., for $k=1,2,\ldots$. One can re-express these conditions as
$\displaystyle{\cal C}_{r}:$ $\displaystyle\int_{-\pi}^{\pi}\cos
r\theta\,g_{\sigma}(\theta){\rm d}\theta=\nu_{r}\;\text{ and
}\;\int_{-\pi}^{\pi}\sin r\theta\,g_{\sigma}(\theta){\rm d}\theta=\xi_{r},$
(10)
where $\nu_{r}=\mbox{\rm Re}\hskip 2.27621pt\psi_{r}$ and $\xi_{r}=\mbox{\rm
Im}\hskip 2.27621pt\psi_{r}$, giving thus
$\nu_{r}^{2}+\xi_{r}^{2}\leq\sigma^{2}$, for $r=1,2,\ldots$, and with n.n.d.
matrix (9), for a chosen $k\in\\{1,2,\ldots\\}$. One can encounter the two
following practical problems.
In an applied field where a specific spectral density $h_{\sigma}$ is
traditionally used (refer to comments in Section 4), one may search for the
the spectral density $g_{\sigma}$ that satisfies ${\cal C}_{r}$, given in (8),
for $r=1,\ldots,k$, and that is the closest to the traditional density
$h_{\sigma}$.
Alternatively, the spectral density $g_{\sigma}$ is unknown but the values of
$\psi_{1},\ldots,\psi_{k}$ are available, either because they constitute a
priori knowledge about the time series or because they are obtained from a
sample of the stationary time series. In this second case, the values of
$\psi_{1},\ldots,\psi_{k}$ can be obtained by taking them equal to the
corresponding values of the empirical or sample a.c.v.f. For the sample
$X_{1},\ldots,X_{n}$ of the time series, the sample a.c.v.f. is given by the
Hermitian function
$\displaystyle\hat{\psi}_{n}(r)=\frac{1}{n}\sum_{j=1}^{n-r}(X_{j+r}-M_{n})\overline{(X_{j}-M_{n})}\;\;\mathrm{and}\;\;\hat{\psi}_{n}(-r)=\overline{\hat{\psi}_{n}(r)},\;\;\text{
for }\,r=0,\ldots,n-1,$ (11)
where $M_{n}=n^{-1}\sum_{j=1}^{n}X_{j}$. Thus we set
$\psi_{r}=\hat{\psi}_{n}(r)$, for $r=1,\ldots,k$ and for $k\leq n-1$. Note
that the matrix (9) is n.n.d. in this case.
Theorem 2.1 below addresses the first of these two problems and it is the
central part of this article. The second problem is addressed by Corollary 1.
The following definitions are required. For $k=1,2,\ldots$ and for an
arbitrary circular density $g_{1}$, define the following integral functions:
$G_{r}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)=\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\int_{0}^{2\pi}\cos r\theta\,\exp\left\\{\kappa_{1}\cos\theta+\kappa_{2}\cos
2(\theta+\delta_{1})+\ldots+\kappa_{k}\cos
k(\theta+\delta_{k-1})\right\\}g_{1}(\theta){\rm d}\theta,$
$H_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1})=\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\int_{0}^{2\pi}\sin r\theta\,\exp\\{\kappa_{1}\cos\theta+\kappa_{2}\cos
2(\theta+\delta_{1})+\ldots+\kappa_{k}\cos
k(\theta+\delta_{k-1})\\}g_{1}(\theta){\rm d}\theta,$ $\displaystyle
A_{r}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)$
$\displaystyle=\frac{G_{r}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)}{G_{0}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)}$
and
$\displaystyle
B_{r}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)$
$\displaystyle=\frac{H_{r}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)}{G_{0}^{(k)}\left(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};g_{1}\right)},$
for $r=1,\ldots,k$, where $\delta_{j}=(\mu_{1}-\mu_{j+1})\mbox{\rm
mod}(2\pi/(j+1))$, for $j=1,\ldots,k-1$ and $\kappa_{1},\ldots,\kappa_{k}$
$\geq 0$. For these constants we make the conventions that the arguments
$\delta_{1},\ldots,\delta_{k-1}$ vanish when $k=1$ and that the argument
$g_{1}$ is omitted when equal to the circular uniform density $u_{1}$. For
example, $G_{0}^{(1)}(\kappa_{1})=(2\pi)^{-1}\int_{0}^{2\pi}{\rm
e}^{\kappa_{1}\cos\theta}{\rm d}\theta=I_{0}(\kappa_{1})$. Define the matrix
of counter-clockwise rotation of angle $\alpha$ as
$\mathbf{R}(\alpha)=\left(\begin{array}[]{cc}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{array}\right).$ (12)
###### Theorem 2.1 (Kullback-Leibler closest spectral distribution)
Let $\sigma\in(0,\infty)$ and let $g_{\sigma}$ and $h_{\sigma}$ be two
spectral densities with total mass $\sigma^{2}$.
1. 1.
The spectral density $g_{\sigma}$ that satisfies ${\cal C}_{r}$, given in (8),
for $r=1,\ldots,k$, and that is the closest to another spectral density
$h_{\sigma}$, in the sense of minimizing the Kullback-Leibler information
$I(g_{\sigma}|h_{\sigma})$, is the exponential tilt of $h_{\sigma}$ that takes
the form
$\displaystyle g_{\sigma}(\theta)=$ (13)
$\displaystyle\frac{1}{G^{(k)}_{0}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})}\exp\left\\{\sum_{j=1}^{k}\kappa_{j}\cos
j(\theta-\mu_{j})\right\\}h_{\sigma}(\theta),$
$\forall\theta\in(-\pi,\pi]$, where $\delta_{j}=(\mu_{1}-\mu_{j+1})\mbox{\rm
mod}(2\pi/(j+1))$, for $j=1,\ldots,k-1$, $\mu_{j}\in(-\pi/j,\pi/j]$ and
$\kappa_{j}\geq 0$, for $j=1,\ldots,k$. The values of these parameters are the
solutions of
$\left(\begin{array}[]{c}\nu_{r}\\\
\xi_{r}\end{array}\right)=\sigma^{2}\mathbf{R}(r\mu_{1})\left(\begin{array}[]{c}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})\\\
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})\end{array}\right),$
(14)
where $\mathbf{R}(r\mu_{1})$ denotes the rotation matrix (12) at
$\alpha=r\mu_{1}$ and where $\nu_{r}$ and $\xi_{r}$ are given by (10), for
$r=1,\ldots,k$.
2. 2.
For any spectral density $g_{\sigma}$ that satisfies ${\cal C}_{r}$, for
$r=1,\ldots,k$, the minimal Kullback-Leibler information of $g_{\sigma}$
w.r.t. $h_{\sigma}$ is given by
$\displaystyle-\sigma^{2}\log
G^{(k)}_{0}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})+\sum_{r=1}^{k}\kappa_{r}(\nu_{r}$
$\displaystyle\cos r\mu_{r}+\xi_{r}\sin r\mu_{r})$ $\displaystyle\leq
I(g_{\sigma}|h_{\sigma}),$ (15)
with equality iff $g_{\sigma}$ is a.e. given by (13), where the values of the
parameters $\mu_{j}\in(-\pi/j,\pi/j]$ and $\kappa_{j}\geq 0$, for
$j=1,\ldots,k$, are solutions of (14).
Theorem 2.1 is a rather direct consequence or generalization of Theorem 2.1 of
Gatto (2007), in which the trigonometric moments are replaced by the a.c.v.f.
and the circular distribution is replaced by the spectral distribution.
Indeed, along with the generalization of the circular distribution to the
spectral distribution, the a.c.v.f. of a stationary time series generalizes
the trigonometric moment. Precisely, the $r$-th trigonometric moment of the
circular random variable $\theta$ with density $g_{1}$ is given by
$\displaystyle\varphi_{r}$ $\displaystyle=\gamma_{r}+{\rm i}\sigma_{r}={\sf
E}\left[{\rm e}^{{\rm i}r\theta}\right]=\int_{-\pi}^{\pi}{\rm e}^{{\rm
i}r\theta}g_{1}(\theta){\rm d}\theta,$ (16)
for some $\gamma_{r},\sigma_{r}\in\mathbb{R}$ and $\forall r\in\mathbb{Z}$,
whereas the a.c.v.f. of the stationary time series with the spectral density
$g_{\sigma}=\sigma^{2}g_{1}$ is given by
$\displaystyle\psi(r)$
$\displaystyle=\sigma^{2}\varphi_{r}=\sigma^{2}(\gamma_{r}+{\rm
i}\sigma_{r}),\;\forall r\in\mathbb{Z}.$ (17)
Clearly, $\psi(0)=\sigma^{2}$ and $|\psi(r)|\leq\psi(0)$, $\forall
r\in\mathbb{Z}$. The claim that (17) is indeed the a.c.v.f. of a stationary
time series is rigorously justified by the above mentioned Herglotz theorem
and characterization of the a.c.v.f.
In the context of the justification of Theorem 2.1.1, we can note that an
equivalent expression for (14) is given by
$\displaystyle\psi_{r}$ $\displaystyle=\sigma^{2}{\rm e}^{{\rm
i}r\mu_{1}}\left\\{A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})+{\rm
i}B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k};h_{1})\right\\},$
(18)
which can be seen equivalent to ${\cal C}_{r}$, for $r=1,\ldots,k$.
A major consequence of Theorem 2.1 is that the ${\rm GvM}_{k}$ spectral
distribution is a maximum entropy distribution. This fact and related results
are given in Corollary 1.
###### Corollary 1 (Maximal Shannon’s spectral entropy distribution)
Let $\sigma\in(0,\infty)$ and $g_{\sigma}$ a spectral density with total mass
$\sigma^{2}$.
1. 1.
The spectral density $g_{\sigma}$ that maximizes Shannon’s entropy
$S(g_{\sigma})$ under ${\cal C}_{r}$, given in (8), for $r=1,\ldots,k$, is the
${\rm GvM}_{k}(\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$ density
multiplied by $\sigma^{2}$, viz. $f_{\sigma}^{(k)}$
$(\cdot|\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$, where
$\mu_{j}\in(-\pi/j,\pi/j]$ and $\kappa_{j}\geq 0$, for $j=1,\ldots,k$. The
values of these parameters are determined by (14).
2. 2.
If $g_{\sigma}$ is a spectral density satisfying ${\cal C}_{r}$, for
$r=1,\ldots,k$, then its entropy is bounded from above as follows,
$S(g_{\sigma})\leq\sigma^{2}\log
G_{0}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})-\sum_{r=1}^{k}\kappa_{r}(\nu_{r}\cos
r\mu_{r}+\xi_{r}\sin r\mu_{r}),$
with equality iff
$g_{\sigma}=f_{\sigma}^{(k)}(\cdot|\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$
a.e. The values of the parameters are determined by (14) with $h_{1}=u_{1}$,
i.e. the circular uniform density, where $\nu_{r}$ and $\xi_{r}$ are given by
(10), for $r=1,\ldots,k$.
3. 3.
The entropy of the ${\rm
GvM}_{k}(\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$ spectral
density with total mass $\sigma^{2}$ is given by the formula
$\displaystyle S\left(f_{\sigma}^{(k)}\right)=$
$\displaystyle\sigma^{2}\Bigg{\\{}\log
G_{0}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})-\kappa_{1}A_{1}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})$
$\displaystyle-\sum_{r=2}^{k}\kappa_{r}\big{[}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\cos
r\delta_{r-1}$ $\displaystyle-
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\sin
r\delta_{r-1}\big{]}\Bigg{\\}},$
where $\sum_{r=2}^{k}$ is defined as null whenever $k<2$.
4. 4.
The a.c.v.f. $\psi^{(k)}$ of the ${\rm
GvM}_{k}(\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$ spectral
distribution can be obtained by
$\left(\begin{array}[]{c}\mbox{\rm Re}\hskip 2.27621pt\psi^{(k)}(r)\\\
\mbox{\rm Im}\hskip
2.27621pt\psi^{(k)}(r)\end{array}\right)=\sigma^{2}\mathbf{R}(r\mu_{1})\left(\begin{array}[]{c}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\\\
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\end{array}\right)$
and $\psi^{(k)}(-r)=\overline{\psi^{(k)}(r)}$, for $r=1,2,\ldots$.
Corollary 1 can be obtained from Theorem 2.1 as follows. Theorem 2.1.1 and the
relation between Kullback-Leibler information and entropy (4) tell that the
${\rm GvM}_{k}$ spectral distribution maximizes the entropy, under the given
constraints on the a.c.v.f. The upper bound for the entropy of a circular
distribution satisfying the given constraints is provided by Theorem 2.1.2.
Thus, by considering $h_{1}=u_{1}$ , we obtain the parts 1 and 2. The part 3
is a consequence of the part 2. It is obtained by replacing $\nu_{r}$ and
$\xi_{r}$, for $r=1,\ldots,k$, that appear in the upper bound of the entropy,
by expressions depending on the parameters of the ${\rm GvM}_{k}$
distribution, through the identity (14).
Thus, when partial prior information in the form of ${\cal C}_{r}$, for
$r=1,\ldots,k$, is available and it is desired to determine the most
noninformative spectral distribution that satisfies the known prior
information, then the ${\rm GvM}_{k}$ spectral distribution is the optimal
one. It is in fact the most credible distribution, or the one that nature
would have generated, when the prior information and only that information
would be available. Maximal entropy distributions are important in many
contexts. In statistical mechanics, the choice of a maximum entropy
distribution subject to constraints is a classical approach referred to as the
maximum entropy principle. One can find various studies on spectral
distributions with maximal entropy. It is explained in Section 3.2 that the
autoregressive model of order $k$ (AR($k$)) maximizes an alternative entropy
among all time series satisfying ${\cal C}_{r}$, for $r=1,\ldots,k$. Franke
(1985) showed that the autoregressive and moving average time series (ARMA)
maximizes that entropy among all time series satisfying these same constraints
on the a.c.v.f. and additional constraints on the impulse responses. Further
properties on these optimal ARMA time series can be found in Huang (1990).
There are many other references on spectral distributions with maximal
entropy: Burg (1978), Kay and Marple (1981), Laeri (1990), etc.
The simplest situation is the following.
###### Example 1 (vM spectrum)
Corollary 1.3 with $k=1$ yields the entropy of the vM spectral distribution,
$\displaystyle S\left(f_{\sigma}^{(1)}\right)$
$\displaystyle=\sigma^{2}\left\\{\log
G_{0}^{(1)}(\kappa_{1})-\kappa_{1}A_{1}^{(1)}(\kappa_{1})\right\\}=\sigma^{2}\left\\{\log
I_{0}(\kappa_{1})-\kappa_{1}\frac{I_{1}(\kappa_{1})}{I_{0}(\kappa_{1})}\right\\},$
for $\kappa_{1}\geq 0$. By noting that $B_{r}^{(1)}(\kappa_{1})=0$, for
$r=1,2,\ldots$, Corollary 1.4 with $k=1$ gives the a.c.v.f. of the vM spectral
distribution as
$\displaystyle\left(\begin{array}[]{c}\mbox{\rm Re}\hskip
2.27621pt\psi^{(1)}(r)\\\ \mbox{\rm Im}\hskip
2.27621pt\psi^{(1)}(r)\end{array}\right)=\sigma^{2}A_{r}^{(1)}(\kappa_{1})\left(\begin{array}[]{c}\cos
r\mu_{1}\\\ \sin
r\mu_{1}\end{array}\right)=\sigma^{2}\frac{I_{r}(\kappa_{1})}{I_{0}(\kappa_{1})}\left(\begin{array}[]{c}\cos
r\mu_{1}\\\ \sin r\mu_{1}\end{array}\right),$ (25)
and $\psi^{(1)}(-r)=\overline{\psi^{(1)}(r)}$, for $r=1,2,\ldots$. When
$\kappa_{1}>0$, the vM spectral distribution is axially symmetric about the
origin iff $\mu_{1}=0$. In other terms and according to (25), the ${\rm
GvM}_{1}$ or vM time series is real-valued iff $\mu_{1}=0$.
### 2.3 Temporal entropy
This section provides a strictly stationary Gaussian-GvM time series that
follows the maximal entropy principle in the time domain, in addition to the
maximal entropy principle in the frequency domain, under the previous
constraints on the a.c.v.f. Consider the complex-valued Gaussian time series
$\\{X_{j}\\}_{j\in\mathbb{Z}}$ in ${\cal L}_{2}$ that is strictly stationary
with mean zero. This time series is introduced at the end of Section 2.1.
Define $U_{j}=\mbox{\rm Re}\hskip 2.27621ptX_{j}$ and $V_{j}=\mbox{\rm
Im}\hskip 2.27621ptX_{j}$, $\forall j\in\mathbb{Z}$. Let $n\geq 1$ and
$j_{1}<\ldots<j_{n}\in\mathbb{Z}$. Consider the random vector
$(U_{j_{1}},\ldots,U_{j_{n}},V_{j_{1}},\ldots,V_{j_{n}})$ and denote by
$p_{j_{1},\ldots,j_{n}}$ its joint density. Thus $p_{j_{1},\ldots,j_{n}}$ is
the $2n$-dimensional normal density with mean zero and $2n\times 2n$
covariance matrix
$\displaystyle\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$ $\displaystyle={\rm
var}\left(\left(U_{j_{1}},\ldots,U_{j_{n}},V_{j_{1}},\ldots,V_{j_{n}}\right)\right)={\sf
E}\left[\left(\begin{array}[]{cc}\mathbf{U}{\mathbf{U}}^{\top}&{\mathbf{U}}{\mathbf{V}}^{\top}\\\
\mathbf{V}{\mathbf{U}}^{\top}&\mathbf{V}{\mathbf{V}}^{\top}\end{array}\right)\right],$
(28)
where $\mathbf{U}=\left(U_{j_{1}},\ldots,U_{j_{n}}\right)^{\top}$ and
$\mathbf{V}=\left(V_{j_{1}},\ldots,V_{j_{n}}\right)^{\top}$. According to
(2.1), the elements of $\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$ are given by
$\displaystyle{\sf E}[U_{j_{l}}U_{j_{m}}]$
$\displaystyle=\psi_{UU}(j_{l}-j_{m}),\;{\sf
E}[V_{j_{l}}V_{j_{m}}]=\psi_{VV}(j_{l}-j_{m}),$ $\displaystyle{\sf
E}[U_{j_{l}}V_{j_{m}}]$ $\displaystyle=\psi_{UV}(j_{l}-j_{m})\;\text{ and
}\;{\sf E}[V_{j_{l}}U_{j_{m}}]=\psi_{VU}(j_{l}-j_{m}),$
with $\psi_{VU}(j_{l}-j_{m})=\psi_{UV}(j_{m}-j_{l})$, for $l,m=1,\ldots,n$.
Because $\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$ depends on $j_{1},\ldots,j_{n}$
only through $l_{1}=j_{2}-j_{1},\ldots,l_{n-1}=j_{n}-j_{n-1}$, we consider the
alternative notation
$\mathbf{\Sigma}^{l_{1},\ldots,l_{n-1}}=\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$.
An important subclass of complex-valued normal random vectors is made by the
radially symmetric ones, which is obtained by setting the mean and the pseudo-
covariance matrix equal to zero. That is, the Gaussian vector
$\mathbf{X}=(X_{j_{1}},\ldots,X_{j_{n}})^{\top}$, where
$X_{j_{l}}=U_{j_{l}}+{\rm i}V_{j_{l}}$, for $l=1,\ldots,n$, is radially
symmetric iff ${\sf E}[\mathbf{X}]=\mathbf{0}$ and ${\sf
E}\left[\mathbf{X}\mathbf{X}^{\top}\right]=\mathbf{0}$. A radially symmetric
complex normal random vector $\mathbf{X}$ is characterized by the fact that,
$\forall\theta\in(-\pi,\pi]$, ${\rm e}^{{\rm
i}\theta}\mathbf{X}\sim\mathbf{X}$. Because these vectors and the related
processes are often used in signal processing, we consider them in this
section.
More generally, by assuming neither stationarity nor normality, we define the
temporal entropy of the complex-valued time series
$\\{X_{j}\\}_{j\in\mathbb{Z}}$ at times $j_{1}<\ldots<j_{n}\in\mathbb{Z}$ in
terms of Shannon’s entropy of $\left(U_{j_{1}},\ldots,U_{j_{n}},\right.$
$\left.V_{j_{1}},\ldots,V_{j_{n}}\right)$, precisely as
$\displaystyle T_{j_{1},\ldots,j_{n}}=-\int_{-\infty}^{\infty}$
$\displaystyle\ldots\int_{-\infty}^{\infty}\log
p_{j_{1},\ldots,j_{n}}(u_{1},\ldots,u_{n},v_{1},\ldots,v_{n})$ $\displaystyle
p_{j_{1},\ldots,j_{n}}(u_{1},\ldots,u_{n},v_{1},\ldots,v_{n}){\rm
d}u_{1}\ldots{\rm d}u_{n}{\rm d}v_{1}\ldots{\rm d}v_{n},$ (29)
whenever the density $p_{j_{1},\ldots,j_{n}}$ exists. Under strict
stationarity, the temporal entropy (2.3) becomes invariant under time shift
and we can thus define the alternative notation
$T^{l_{1},\ldots,l_{n-1}}=T_{j_{1},\ldots,j_{n}}$.
Let us now mention two known and important information theoretic results for
the Gaussian distribution. The first one is the formula of the Gaussian
entropy:
* if $p_{j_{1},\ldots,j_{n}}$ is the $2n$-dimensional Gaussian density with arbitrary mean and covariance matrix $\Sigma_{j_{1},\ldots,j_{n}}$, then the temporal entropy (2.3) is given by
$\displaystyle T_{j_{1},\ldots,j_{n}}$
$\displaystyle=\left\\{1+\log(2\pi)\right\\}n+\frac{1}{2}\log\det\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}.$
(30)
The second result is the maximum entropy property of the Gaussian
distribution:
* among random vectors $\left(U_{j_{1}},\ldots,U_{j_{n}},V_{j_{1}},\ldots,V_{j_{n}}\right)$ having arbitrary density with fixed covariance matrix $\Sigma_{j_{1},\ldots,j_{n}}$, the one that is normally distributed maximizes Shannon’s entropy (2.3). The maximum of the entropy is given by (30).
We now consider the previous constraints on the a.c.v.f. (8) and search for
the (strictly) stationary time series, with mean and pseudo-covariances null,
that maximizes the temporal entropy.
###### Theorem 2.2 (Maximal Shannon’s temporal entropy distribution)
Consider the class of complex-valued and stationary time series
$\\{X_{j}\\}_{j\in\mathbb{Z}}$ with mean null, variance $\sigma^{2}$, for some
$\sigma\in(0,\infty)$, and pseudo-covariances null. Denote by $\psi$ the
a.c.v.f. of $\\{X_{j}\\}_{j\in\mathbb{Z}}$, $\nu=\mbox{\rm Re}\hskip
2.27621pt\psi$ and $\xi=\mbox{\rm Im}\hskip 2.27621pt\psi$.
1. 1.
If the a.c.v.f. $\psi$ satisfies ${\cal C}_{r}$ given in (8) or in (10), for
$r=1,\ldots,k$, thus $\psi(1)=\psi_{1}=\nu_{1}+{\rm
i}\,\xi_{1},\ldots,\psi(k)=\psi_{k}=\nu_{k}+{\rm i}\,\xi_{k}$, then the time
series $\\{X_{j}\\}_{j\in\mathbb{Z}}$ in the above class that maximizes
Shannon’s temporal entropy (2.3) with $n=k+1$ and $j_{1}=1,\ldots,j_{k+1}=k+1$
is the one for which the corresponding double f.d.d. (1) with
$j_{1}=1,\ldots,j_{k+1}=k+1$ is Gaussian, with mean zero and with
$2(k+1)\times 2(k+1)$ covariance matrix
$\mathbf{\Sigma}(k)=\mathbf{\Sigma}^{1,\ldots,1}$ given by (28) with
$\displaystyle\psi_{UU}(r)=\psi_{VV}(r)=\frac{\nu_{r}}{2}\;\text{ and
}\;\psi_{UV}(r)=\psi_{VU}(-r)=-\frac{\xi_{r}}{2},$
for $r=1,\ldots,k$.
2. 2.
The corresponding value of the temporal entropy is given by
$\displaystyle T(k)$
$\displaystyle=\left\\{1+\log(2\pi)\right\\}(1+k)+\frac{1}{2}\log\det\mathbf{\Sigma}(k).$
###### Proof
1.a. This initial part of the proof shows that for any a.c.v.f. $\psi$, there
exists a complex-valued Gaussian time series that is strictly stationary,
centered and radially symmetric. Let $n\geq 1$, $u_{j},v_{j}\in\mathbb{R}$,
$c_{j}=u_{j}-{\rm i}v_{j}$, for $j=1,\ldots,n$, let
$j_{1}<\ldots<j_{n}\in\mathbb{Z}$, $\mathbf{u}=(u_{1},\ldots,u_{n})^{\top}$,
$\mathbf{v}=(v_{1},\ldots,v_{n})^{\top}\in\mathbb{R}^{n}$ and define
$\displaystyle q(\mathbf{u},\mathbf{v})$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{n}\sum_{m=1}^{n}c_{l}\overline{c_{m}}\psi(j_{l}-j_{m}).$
Then $q(\mathbf{u},\mathbf{v})\geq 0$ implies
$\displaystyle q(\mathbf{u},\mathbf{v})$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{n}\sum_{m=1}^{n}(u_{l}-{\rm
i}v_{l})(u_{m}+{\rm i}v_{m})\\{\nu(j_{l}-j_{m})+{\rm i}\xi(j_{l}-j_{m})\\}$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{n}\sum_{m=1}^{n}(u_{l}u_{m}+v_{l}v_{m})\nu(j_{l}-j_{m})-(u_{l}v_{m}-v_{l}u_{m})\xi(j_{l}-j_{m}).$
(31)
Define $\mathbf{U}=(U_{j_{1}},\ldots,U_{j_{n}})^{\top}$ and
$\mathbf{V}=(V_{j_{1}},\ldots,V_{j_{n}})^{\top}$. Assume
$\left(\mathbf{U}^{\top},\mathbf{V}^{\top}\right)$ normally distributed with
mean zero and covariance matrix $\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$, as in
(28).
A particular choice of $\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}$ can be obtained
by setting
$\displaystyle\varphi(\mathbf{u},\mathbf{v})$ $\displaystyle={\sf
E}\left[\exp\left\\{{\rm
i}\left(\mathbf{u}^{\top},\mathbf{v}^{\top}\right){\mathbf{U}\choose\mathbf{V}}\right\\}\right]=\exp\left\\{-\frac{1}{2}q(\mathbf{u},\mathbf{v})\right\\},$
leading to
$\displaystyle\left(\mathbf{u}^{\top},\mathbf{v}^{\top}\right)\mathbf{\Sigma}_{j_{1},\ldots,j_{n}}{\mathbf{u}\choose\mathbf{v}}$
$\displaystyle=q(\mathbf{u},\mathbf{v}).$
This, (2.3) and (28) yield
$\displaystyle{\sf E}[U_{j_{l}}U_{j_{m}}]$
$\displaystyle=\frac{1}{2}\nu(j_{l}-j_{m}),\;{\sf
E}[V_{j_{l}}V_{j_{m}}]=\frac{1}{2}\nu(j_{l}-j_{m}),$ $\displaystyle{\sf
E}[U_{j_{l}}V_{j_{m}}]$ $\displaystyle=-\frac{1}{2}\xi(j_{l}-j_{m}),\;{\sf
E}[V_{j_{l}}U_{j_{m}}]=\frac{1}{2}\xi(j_{l}-j_{m})$ (32)
and therefore $\xi(j_{l}-j_{m})=-\xi(j_{m}-j_{l})$, for $l,m=1,\ldots,n$.
Define $\mathbf{X}=(X_{j_{1}},\ldots,X_{j_{n}})^{\top}$, where
$X_{j_{l}}=U_{j_{l}}+{\rm i}V_{j_{l}}$, for $l=1,\ldots,n$. We obtain the
covariance matrix
$\displaystyle{\rm var}(\mathbf{X})$ $\displaystyle={\sf
E}\left[\mathbf{X}\overline{\mathbf{X}}^{\top}\right]={\sf
E}\left[(\mathbf{U}+{\rm i}\mathbf{V})(\mathbf{U}-{\rm
i}\mathbf{V})^{\top}\right]={\sf
E}\left[\mathbf{U}\mathbf{U}^{\top}+\mathbf{V}\mathbf{V}^{\top}+{\rm
i}\left(\mathbf{V}\mathbf{U}^{\top}-\mathbf{U}\mathbf{V}^{\top}\right)\right]$
$\displaystyle=\frac{1}{2}\left(\nu(j_{l}-j_{m})+\nu(j_{l}-j_{m})+{\rm
i}\left[\xi(j_{l}-j_{m})-\\{-\xi(j_{l}-j_{m})\\}\right]\right)_{l,m=1,\ldots,n}$
$\displaystyle=\left(\nu(j_{l}-j_{m})+{\rm
i}\xi(j_{l}-j_{m})\right)_{l,m=1,\ldots,n}=\left(\psi(j_{l}-j_{m})\right)_{l,m=1,\ldots,n}$
and the pseudo-covariance matrix
$\displaystyle{\sf E}\left[\mathbf{X}\mathbf{X}^{\top}\right]$
$\displaystyle={\sf E}\left[(\mathbf{U}+{\rm i}\mathbf{V})(\mathbf{U}+{\rm
i}\mathbf{V})^{\top}\right]={\sf
E}\left[\mathbf{U}\mathbf{U}^{\top}-\mathbf{V}\mathbf{V}^{\top}+{\rm
i}\left(\mathbf{V}\mathbf{U}^{\top}+\mathbf{U}\mathbf{V}^{\top}\right)\right]$
$\displaystyle=\frac{1}{2}\left(\nu(j_{l}-j_{m})-\nu(j_{l}-j_{m})+{\rm
i}\left[\xi(j_{l}-j_{m})+\\{-\xi(j_{l}-j_{m})\\}\right]\right)_{l,m=1,\ldots,n}$
$\displaystyle=\left(0+{\rm i}0\right)_{l,m=1,\ldots,n}=\mathbf{0},$
as desired. We have thus established the existence of a complex-valued
Gaussian time series $\\{X_{t}\\}_{t\in\mathbb{R}}$ that is strictly
stationary, radially symmetric and centered.
1.b. Consider $n=k+1$ and $j_{1}=1,\ldots,j_{k+1}=k+1$. Under ${\cal C}_{r}$,
for $r=1,\ldots,k$, ${\rm var}(\mathbf{X})$ is entirely determined: it is the
$(k+1)\times(k+1)$, n.n.d. and Toeplitz matrix (9). The pseudo-covariance
matrix and the mean vector are null and thus also determined. We know from
(2.3) that, for $r=1,\ldots,k+1$,
$\displaystyle\psi_{UU}(r)=\psi_{UV}(r)=\frac{1}{2}\nu(r)=\frac{\nu_{r}}{2}\;\text{
and }\;\psi_{UV}(r)=\psi_{VU}(-r)=-\frac{1}{2}\xi(r)=-\frac{\xi_{r}}{2}.$
So the covariance matrix of $(\mathbf{U}^{\top},\mathbf{V}^{\top})$ is
entirely determined by ${\cal C}_{r}$, for $r=1,\ldots,k$, and it is the
$2(k+1)\times 2(k+1)$ matrix $\Sigma_{1,\ldots,k+1}=\Sigma^{1,\ldots,1}$.
Clearly, ${\sf E}\left[(\mathbf{U}^{\top},\mathbf{V}^{\top})\right]=0$. The
second information theoretic result for the Gaussian distribution, just above,
concludes the proof Theorem 2.2.1.
2\. The second information theoretic result for the Gaussian distribution,
viz. (30), leads directly to the entropy formula in Theorem 2.2.2.
So when $\\{X_{j}\\}_{j\in\mathbb{Z}}$ is the strictly stationary Gaussian-GvM
time series, both spectral and temporal Shannon’s entropies are maximized
under the constraints ${\cal C}_{r}$, for $r=1,\ldots,k$.
## 3 Some computational aspects
The following computational aspects are studied: the computation of the
integral functions of the ${\rm GvM}_{2}$ time series in Section 3.1, the
estimation of the ${\rm GvM}_{k}$ spectral distribution in Section 3.2 and the
computation of the ${\rm GvM}_{k}$ spectral d.f. in Section 3.3.
### 3.1 Integral functions of the $\mathbf{GvM_{2}}$ time series
This section provides some series expansions for some integral functions
appearing with the ${\rm GvM}_{k}$ spectral distribution. Indeed, the
information theoretic results of Section 2 require the constants or integral
functions $G_{r}^{(k)}$, for $r=0,\ldots,k$, and $H_{r}^{(k)}$, for
$r=1,\ldots,k$. They are integrals over a bounded domain of smooth integrands
and therefore numerical integration should perform well. Alternatively, one
can evaluate these integral functions by Fourier series expansions. Gatto
(2007) provides expansions for some of these constants and in particular for
those with $k=2$, that are given here. Define
${\rm ep}_{r}=\left\\{\begin{array}[]{ll}1,&{\rm if}\;r\;\text{ is even and
positive},\\\ 0,&{\rm otherwise}.\end{array}\right.$
Let $\delta\in[0,\pi)$ and $\kappa_{1},\kappa_{2}\geq 0$. Then the following
expansions hold for $r=0,1,\ldots$,
$\displaystyle G_{r}^{(2)}(\delta,\kappa_{1},\kappa_{2})=$
$\displaystyle\;I_{0}(\kappa_{1})I_{\frac{r}{2}}(\kappa_{2})\cos r\delta\;{\rm
ep}_{r}+I_{0}(\kappa_{2})I_{r}(\kappa_{1})$
$\displaystyle+\sum_{j=1}^{\infty}\cos
2j\delta\;I_{j}(\kappa_{2})\left\\{I_{2j+r}(\kappa_{1})+I_{\mid
2j-r\mid}(\kappa_{1})\right\\},$ (33)
and
$\displaystyle H_{r}^{(2)}(\delta,\kappa_{1},\kappa_{2})=$ $\displaystyle-
I_{0}(\kappa_{1})I_{\frac{r}{2}}(\kappa_{2})\sin r\delta\;{\rm ep}_{r}$
$\displaystyle+\sum_{j=1}^{\infty}\sin
2j\delta\;I_{j}(\kappa_{2})\left\\{I_{2j+r}(\kappa_{1})-I_{\mid
2j-r\mid}(\kappa_{1})\right\\}.$ (34)
We can deduce from the two above expansions that $G_{r}$ and $H_{r}$ inherit
the asymptotic behavior of the Bessel function $I_{r}$, for large $r$. It
follows from Abramowitz and Stegun (1972, 9.6.10, p. 375) that
$I_{r}(z)=(z/2)^{r}\\{r\Gamma(r)\\}^{-1}\left\\{1+{\rm
O}\left(r^{-1}\right)\right\\}$, as $r\to\infty$. This and the Stirling
approximation yield $I_{r}(z)=(2\pi r)^{-1/2}$ $\\{{\rm
e}z/(2r)\\}^{r}\left\\{1+{\rm O}\left(r^{-1}\right)\right\\}$, as
$r\to\infty$. Hence $I_{r}$ decreases rapidly to zero as $r$ increases. We see
that the same holds for $G_{r}$ and $H_{r}$.
### 3.2 Estimation of the GvM spectral distribution
This section concerns the estimation problem. After reviewing some classical
results of spectral estimation, it presents an estimator to the parameters of
the GvM spectral distribution.
A classical estimator of the spectral density is the periodogram. We remember
that it is based on the discrete Fourier transform of the sample a.c.v.f.,
i.e. on
$\Lambda_{n}(j)=\sum_{r=-(n-1)}^{n-1}\hat{\psi}_{n}(r){\rm e}^{-{\rm
i}\frac{2\pi jr}{n}},$
for $j=\lfloor(n-1)/2\rfloor,\ldots,-1,1,\ldots,\lfloor n/2\rfloor$,
$\hat{\psi}_{n}$ being the sample a.c.v.f. (11), $n$ the sample size and
$\lfloor\cdot\rfloor$ the floor function. Because of its nonparametric nature,
the periodogram is well-suited for detecting particular features such as a
periodicity, which may not be identified by a parametric estimator. However,
its irregular nature may not be desirable in some contexts and it does not
result from an important optimality criterion.
One of the earliest studies on maximum entropy spectral distributions is Burg
(1967), who considered $B(f_{\sigma})=\int_{-\pi}^{\pi}\log
f_{\sigma}(\theta){\rm d}\theta$ as measure of entropy of the spectral density
$f_{\sigma}$. This entropy is different than our adaptation of Shannon’s
entropy, viz. $S(f_{\sigma})$ given in (4). We can easily relate Burg’s
entropy of the spectral density $f_{\sigma}$ to the Kullback-Leibler
information as follows,
$\displaystyle B(f_{\sigma})$
$\displaystyle=2\pi\left\\{\log\frac{\sigma^{2}}{2\pi}-\frac{1}{\sigma^{2}}I(u_{\sigma}|f_{\sigma})\right\\}=2\pi\left\\{\log\frac{\sigma^{2}}{2\pi}-I(u_{1}|f_{1})\right\\}.$
(35)
This shows that maximizing Burg’s entropy amounts to minimize the re-directed
Kullback-Leibler information, instead of the usual Shannon’s entropy. For
real-valued time series, it turns out that the spectral density estimator that
maximizes the entropy (35) subject to the constraints ${\cal C}_{r}$, for
$r=1,\ldots,k$, with $k\leq n-1$, is equal to the autoregressive estimator of
order $k$. This autoregressive estimator is given by the formula of the
spectral density the AR($k$) model which has been fitted to the sample of $n$
consecutive values of the time series. For more details refer e.g. to p.
365-366 of Brockwell and Davis (1991).
Estimators of the parameters of the ${\rm GvM}_{k}$ spectral distribution can
be obtained from a direct generalization of the trigonometric method of
moments estimator for the ${\rm GvM}_{k}$ circular distribution, which is
introduced by Gatto (2008). This estimator is the circular version of the
method of moments estimators. Consider the ${\rm GvM}_{k}$ spectral
distribution with unknown parameters $\mu_{1},\ldots,\mu_{k}$ and
$\kappa_{1},\ldots,\kappa_{k}$, for some $k\in\\{1,\ldots,n-1\\}$. Consider
the $r$-th a.c.v.f. condition ${\cal C}_{r}$ in which the spectral density
$g_{r}$ is taken equal to the ${\rm GvM}_{k}$ spectral density with variance
$\sigma^{2}$, viz. $\sigma^{2}$ times the circular density (2), and in which
the quantity $\psi_{r}\in\mathbb{C}$ is replaced by the sample a.c.v.f. at
$r$, namely by $\hat{\psi}_{n}(r)$, $r=1,\ldots,k$, cf. (11). The resulting
$r$-th equation can be re-expressed in a similar way to (18), which in turn
leads to
$\displaystyle\left(\begin{array}[]{c}\mbox{\rm Re}\hskip
2.27621pt\hat{\psi}_{n}(r)\\\ \mbox{\rm Im}\hskip
2.27621pt\hat{\psi}_{n}(r)\end{array}\right)=\sigma^{2}\mathbf{R}(r\mu_{1})\left(\begin{array}[]{c}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\\\
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\end{array}\right),$
(40)
for $r=1,\ldots,k$, with $k\leq n-1$. This gives a system of $2k$ real
equations and $2k$ unknown real parameter. The values of $\mu_{1}$,
$\delta_{1},\ldots,\delta_{k-1}$, $\kappa_{1},\ldots,\kappa_{k}$ that solve
this system of equations are the resulting estimators and they can be denoted
$\hat{\mu}_{1}$, $\hat{\delta}_{1},\ldots,\hat{\delta}_{k-1}$,
$\hat{\kappa}_{1},\ldots,\hat{\kappa}_{k}$. We now give two examples.
###### Example 2 (vM spectrum)
When $k=1$ we have the basic vM unimodal spectral distribution. Because
$B_{1}^{(1)}(\kappa_{1})=0$, we have the estimating equation
$\displaystyle\left(\begin{array}[]{c}\mbox{\rm Re}\hskip
2.27621pt\hat{\psi}_{n}(1)\\\ \mbox{\rm Im}\hskip
2.27621pt\hat{\psi}_{n}(1)\end{array}\right)=\sigma^{2}A_{1}^{(1)}(\kappa_{1})\left(\begin{array}[]{c}\cos\mu_{1}\\\
\sin\mu_{1}\end{array}\right)=\sigma^{2}\frac{I_{1}(\kappa_{1})}{I_{0}(\kappa_{1})}\left(\begin{array}[]{c}\cos\mu_{1}\\\
\sin\mu_{1}\end{array}\right),$
giving two equations and two unknown values, namely $\mu_{1}$ and
$\kappa_{1}$. The solutions are the estimators $\hat{\mu}_{1}$ and
$\hat{\kappa}_{1}$. For $\kappa_{1}>0$, if $\mu_{1}=0$ is given, then we have
axial symmetry about the origin and so the corresponding time series is real-
valued. The two estimating equations reduce to the single equation
$\displaystyle\frac{\hat{\psi}_{n}(1)}{\sigma^{2}}$
$\displaystyle=A_{1}^{(1)}(\kappa_{1}),$ (41)
whose solution is the estimator $\hat{\kappa}_{1}$. Note that Amos (1974)
showed that $A_{1}^{(1)}$ has positive derivative over $(0,\infty)$. It
follows essentially from this fact that $A_{1}^{(1)}$ is a strictly increasing
and differentiable probability d.f. over $[0,\infty)$. So its inverse function
is easily computed.
###### Example 3 (${\rm GvM}_{2}$ spectrum)
When $k=2$ we retrieve the practical ${\rm GvM}_{2}$ unimodal or bimodal
spectral distribution. In the estimating equations (40) with $k=2$, we can use
the series expansions of the constants given by (3.1) and (3.1). As previously
mentioned, with $\kappa_{1},\kappa_{2}>0$, the ${\rm GvM}_{2}$ distribution is
axially symmetric around the axis $\mu_{1}$ iff $\delta_{1}=\delta^{(1)}=0$ or
$\delta_{1}=\delta^{(2)}=\pi/2$. With these values of $\delta_{1}$ and with
$\mu_{1}=0$, the axial symmetry is about the origin and so the corresponding
time series is real-valued. We note that
$B_{r}^{(2)}\left(\delta^{(j)},\kappa_{1},\kappa_{2}\right)=0$, for $r=1,2$
and for the cases $j=1,2$. Because of these equalities and because $\mbox{\rm
Im}\hskip 2.27621pt\hat{\psi}_{n}(r)=0$, for $r=1,2$, the estimating equations
(40) simplify to
$\displaystyle\frac{\hat{\psi}_{n}(r)}{\sigma^{2}}$
$\displaystyle=A_{r}^{(2)}\left(\delta^{(j)},\kappa_{1},\kappa_{2}\right),$
for $r=1,2$ and for the two cases $j=1,2$. These estimating equations appear
as the natural generalization of the estimation equation (41) of the real-
valued vM time series. For each one of these two cases, we have two equations
and two unknown values, namely $\kappa_{1}$ and $\kappa_{2}$. The solutions
are the estimators $\hat{\kappa}_{1}$ and $\hat{\kappa}_{2}$.
### 3.3 GvM spectral distribution function
A formula for the ${\rm GvM}_{k}$ spectral d.f. can be obtained in terms of a
series as follows. Let $\psi^{(k)}$ denote the a.c.v.f. and let
$f_{\sigma}^{(k)}$ denote the spectral density of the ${\rm GvM}_{k}$ time
series with variance $\sigma^{2}$. It follows from $\mbox{\rm Re}\hskip
2.27621pt\psi^{(k)}(-r)=\mbox{\rm Re}\hskip 2.27621pt\psi^{(k)}(r)$ and
$\mbox{\rm Im}\hskip 2.27621pt\psi^{(k)}(-r)=-\mbox{\rm Im}\hskip
2.27621pt\psi^{(k)}(r)$, for $r=1,2,\ldots$, and from (14) that
$\displaystyle
f_{\sigma}^{(k)}(\theta|\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{r=-\infty}^{\infty}\psi^{(k)}(r)\exp\\{-{\rm
i}r\theta\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left(1+2\sum_{r=1}^{\infty}\mbox{\rm Re}\hskip
2.27621pt\psi^{(k)}(r)\cos r\theta+\mbox{\rm Im}\hskip
2.27621pt\psi^{(k)}(r)\sin r\theta\right)$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2\pi}\left(1+2\sum_{r=1}^{\infty}(\cos
r\theta,\sin
r\theta)\mathbf{R}(r\mu_{1})\left(\begin{array}[]{c}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\\\
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\end{array}\right)\right),$
$\forall\theta\in(-\pi,\pi]$. Pointwise convergence is due to Dirichlet’s
theorem (cf. e.g. Pinkus and Zafrany, 1997, p. 47). The term by term
integration of the Fourier series of a piecewise continuous function converges
uniformly towards the integral of the original function (cf. e.g. Pinkus and
Zafrany, 1997, p. 77). So the ${\rm GvM}_{k}$ spectral d.f. admits the series
representation
$\displaystyle
F_{\sigma}^{(k)}(\theta|\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k})$
$\displaystyle=$
$\displaystyle\int_{-\pi}^{\alpha}f_{\sigma}(\alpha|\mu_{1},\ldots,\mu_{k},\kappa_{1},\ldots,\kappa_{k}){\rm
d}\alpha$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2\pi}\Bigg{(}\theta+2\sum_{r=1}^{\infty}\frac{1}{r}\big{[}A_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\\{\sin
r(\theta-\mu_{1})+\sin r\mu_{1}\\}$ $\displaystyle-
B_{r}^{(k)}(\delta_{1},\ldots,\delta_{k-1},\kappa_{1},\ldots,\kappa_{k})\\{\cos
r(\theta-\mu_{1})-\cos
r\mu_{1}\\}\big{]}\Bigg{)},\;\forall\theta\in(-\pi,\pi],$
where the convergence is uniform. The order of the $r$-th summand is $r$ times
smaller w.r.t. the original Fourier series, so we can expect rapid
convergence. When $k=2$, we can use the series expansions (3.1) and (3.1) for
the computation this d.f. This expansion of the spectral d.f. can be used in
conjunction with (7) for the computation of the ${\cal L}_{2}$-norm of the
increments of the spectral process of the ${\rm GvM}_{k}$ time series.
## 4 Concluding remarks
This chapter presents an innovative application of the GvM distribution of
directional statistics to the analysis of stationary time series. As already
mentioned, these developments are merely a first analysis to the GvM and
Gaussian-GvM time series. The scope is limited to the presentation of some
first few results on the GvM spectrum and further developments are required.
For example, simulation algorithms for the Gaussian-GvM time series could be
developed. The GvM spectrum is motivated by rather theoretical considerations
and one is aware that ad hoc spectra are often used in applied domains. For
example, the Pierson-Moskowitz spectrum is widely used for the stationary
modelling of ocean waves, in the context of naval construction. We refer e.g.
to p. 315-316 of Lindgren (2012) for a list of commonly used spectra (that
includes the Pierson-Moskowitz spectrum). These spectra correspond to
continuous time stationary models, but wrapping them around the circle of
circumference $2\pi$ leads to the spectra of these processes sampled at
integer times, viz. of time series. Lastly, we note that the connection
between spectral and circular distributions is well-known: the wrapped Cauchy
circular distribution is the normalized spectral distribution of the the AR(1)
time series and the cardioid distribution is the normalized spectral
distribution of the first order moving average (MA(1)) time series. This
connection is exploited by Tanigichi et al. (2020) in order to construct new
circular distributions. It is therefore in the opposite direction that this
chapter exploits this connection.
## 5 References
Abramowitz, M., Stegun, I. E. (1972), Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, Dover Publications (9-th printing
conform to the 10-th original printing).
Amos, D. E. (1974), “Computation of modified Bessel functions and their
ratios”, Mathematics of Computation, 28, 239-251.
Asmussen, S., Glynn P. W. (2007), Stochastic Simulation. Algorithms and
Analysis, Springer.
Astfalck, L., Cripps, E., Gosling, J., Hodkiewicz, M., Milne, I. (2018),
“Expert elicitation of directional metocean parameters”, Ocean Engineering,
161, 268-276.
Bloomfield, P. (1973), “An exponential model for the spectrum of a scalar time
series”, Biometrika, 60, 217-226.
Bogert, B. P., Healy, M. J. R., Tukey, J. W., Rosenblatt, M. (1963), “The
quefrency analysis of time series for echoes: cepstrum, pseudoauto-covariance,
cross-cepstrum and saphe cracking”, Proceedings of the Symposium on Time
Series Analysis, editor Rosenblatt, Wiley and Sons, 209-243.
Brillinger, D. R. (1993), “The digital rainbow: some history and applications
of numerical spectrum analysis”, Canadian Journal of Statistics, 21, 1-19.
Brockwell, P. J., Davis, R. A. (1991), Time Series: Theory and Methods, second
edition, Springer.
Burg, J. P. (1967), “Maximum entropy spectral analysis”, unpublished
presentation, 37-th meeting of the Society of Exploration Geophysicists,
Oklahoma City, Oklahoma.
Burg, J. P. (1978), “Maximum entropy spectral analysis”, Modern Spectrum
Analysis, editor Childers. Wiley and Sons, 34-41.
Chatfield, C. (2013), The Analysis of Time Series. Theory and Practice,
Springer.
Christmas, J. (2014), “Bayesian Spectral Analysis With Student-$t$ Noise”,
IEEE Transactions on Signal Processing, 62, 2871-2878.
Cramér, H. (1942), “On harmonic analysis in certain functional spaces”, Arkiv
för Matematik, Astronomi och Fysik, 28B, 12.
Cramér, H., Leadbetter, M. R. (1967), Stationary and Related Stochastic
Processes, Wiley, New York.
Fisher, R. A. (1929), “Tests of significance in harmonic analysis”,
Proceedings of the Royal Society of London, Series A, 125, 796, 54-59.
Gatto, R. (2008), “Some computational aspects of the generalized von Mises
distribution”, Statistics and Computing, 18, 321-331.
Gatto, R. (2009), “Information theoretic results for circular distributions”,
Statistics, 43, 409-421.
Gatto, R., Jammalamadaka, S. R. (2007), “The generalized von Mises
distribution”, Statistical Methodology, 4, 341-353.
Gatto, R., Jammalamadaka, S. R. (2015), “Directional statistics:
introduction”, StatsRef: Statistics Reference Online, editors Balakrishnan et
al., Wiley and Sons, 1-8.
Gonella, J. (1972), “A rotary-component method for analysing meteorological
and oceanographic vector time series”, Deep-Sea Research, 19, 833-846.
Huang, D. (1990), “On the maximal entropy property for ARMA processes and ARMA
approximation”, Advances in Applied Probability, 612-626.
Jammalamadaka, S. R., SenGupta, A. (2001), Topics in Circular Statistics,
World Scientific Press.
Kay, S. M., Marple, S. L. (1981), “Spectrum analysis: a modern perspective”,
Proceedings of the IEEE, 69, 1380-1419.
Kim, S., SenGupta, A. (2013), “A three-parameter generalized von Mises
distribution”, Statistical Papers, 54, 685-693.
Kullback, S. (1954), “Certain inequalities in information theory and the
Cramér-Rao inequality”, Annals of Mathematical Statistics, 25, 745-751.
Kullback, S., Leibler, R. A. (1951), “On information and sufficiency”, Annals
of Mathematical Statistics, 22, 79-86.
Laeri, F. (1990), “Some comments on maximum entropy spectral analysis of time
series”, Computers in Physics, 4, 627-636.
Lin, Y., Dong, S. (2019), “Wave energy assessment based on trivariate
distribution of significant wave height, mean period and direction”, Applied
Ocean Research, 87, 47-63.
Lindgren, G. (2012), Stationary Stochastic Processes: Theory and Applications,
Chapman and Hall/CRC.
Mardia, K. V., Jupp, P. E. (2000), Directional Statistics, Wiley.
Pinkus, A., Zafrany, S. (1997), Fourier Series and Integral Transforms,
Cambridge University Press.
Rice, S. O. (1944), “Mathematical analysis of random noise”, Bell Systems
technical Journal, 23, 282-332.
Also in Rice (1954), Selected Papers on Noise and Stochastic Processes, editor
Wax, Dover Publications, p. 133-295.
Rice, S. O. (1945), “Mathematical analysis of random noise”, Bell Systems
technical Journal, 24, 46-156.
Also in Rice (1954), “Mathematical analysis of random noise”, Selected Papers
on Noise and Stochastic Processes, editor Wax, Dover Publications, p. 133-295.
Rowe, D. B. (2005), “Modeling both the magnitude and phase of complex-valued
fMRI data”, NeuroImage, 25, 1310–1324.
Salvador, S., Gatto, R. (2020a), “Bayesian tests of symmetry for the
generalized von Mises distribution”, preprint, Institute of mathematical
statistics and actuarial science, University of Bern.
Salvador, S., Gatto, R. (2020b), “A Bayesian test of bimodality for the
generalized von Mises distribution”, preprint, Institute of mathematical
statistics and actuarial science, University of Bern.
Shannon, C. E. (1948), “A mathematical theory of communication”, Bell System
Technical Journal, 27, 379-423, 623-656.
Taniguchi, M., Kato, S., Ogata, H., Pewsey, A. (2020), “Models for circular
data from time series spectra”, Journal of Time Series Analysis, 41, 808-829.
Yaglom, A. M. (1962), An Introduction to the Theory of Stationary Random
Functions, Prentice-Hall.
Zhang, L., Li, Q., Guo, Y., Yang, Z., Zhang, L. (2018), “An investigation of
wind direction and speed in a featured wind farm using joint probability
distribution methods”, Sustainability, 10.
|
# Exploiting the wide dynamic range of Silicon photomultipliers for Quantum
Optics applications
S. Cassina Department of Science and High Technology, University of Insubria,
Via Valleggio 11, I-22100 Como (Italy), A. Allevi
<EMAIL_ADDRESS>Department of Science and High Technology,
University of Insubria, and Institute for Photonics and Nanotechnologies, IFN-
CNR, Via Valleggio 11, I-22100 Como (Italy), V. Mascagna Department of
Science and High Technology, University of Insubria, Via Valleggio 11, I-22100
Como (Italy), and INFN Section of Milano Bicocca, Piazza della Scienza 3,
I-20126 Milano (Italy), M. Prest Department of Science and High Technology,
University of Insubria, Via Valleggio 11, I-22100 Como (Italy), and INFN
Section of Milano Bicocca, Piazza della Scienza 3, I-20126 Milano (Italy), E.
Vallazza INFN Section of Milano Bicocca, Piazza della Scienza 3, I-20126
Milano (Italy), M. Bondani Institute for Photonics and Nanotechnologies,
IFN-CNR, Via Valleggio 11, I-22100 Como (Italy).
###### Abstract
Silicon photomultipliers are photon-number-resolving detectors endowed with
hundreds of cells enabling them to reveal high-populated quantum optical
states. In this paper, we address such a goal by showing the possible
acquisition strategies that can be adopted and discussing their advantages and
limitations. In particular, we determine the best acquisition solution in
order to properly reveal the nature, either classical or nonclassical, of
mesoscopic quantum optical states.
###### pacs:
42.50.-p Quantum Optics, 42.50.Ar Photon statistics and coherence theory,
42.65.Lm Parametric down conversion and production of entangled photons,
85.60.Gz Photodetectors
## I Introduction
Silicon photomultipliers (SiPMs) are photon-number resolving detectors
characterized by hundreds of pixels (or cells) operated in the
Geiger-M$\ddot{\rm u}$ller regime and read in parallel, in order to yield a
single output akindinov ; bondarenko ; saveliev ; piemonte ; renker . By
assuming that each cell is fired by at most one photon, the number of fired
cells should correspond to the number of impinging photons. However, the non-
ideal quantum efficiency and the presence of some drawbacks, such as dark
count, optical cross-talk effect and afterpulses, prevent this correspondence.
While the efficiency of the detector can be only slightly modified acting on
the bias voltage, we have recently demonstrated that it is possible to make
the drawbacks negligible by properly acquiring the output of the detector,
taking advantage of their different occurence in time scirep19 . Moreover, in
Ref. OL19 we have investigated in which way the drawbacks affect the
observation of nonclassical correlations between the two parties of a multi-
mode twin-beam state.
In this paper, we focus on the temporal development of the output signal in
order to reduce the spurious contributions to the final signal, and to select
only the information on the light. To do this, we follow two approaches: 1) we
use the minimum integration gate; 2) we select the peak values and check the
quality of the extracted information by calculating a relevant parameter (the
noise reduction factor) for both classically- and nonclassically-correlated
light states. To this aim, we consider and compare different kinds of
amplifiers and digitizers used to sample the detector output and show the
advantages and limitations of all the employed devices. Through this analysis,
we also study the limits imposed by nonlinearities and saturation effects of
some parts of the acquisition chain, having in mind the exploitation of the
high dynamic range of SiPMs to detect well-populated states of light. Indeed,
reliably detecting the number of photons in every pulse of highly-populated
states is the key resource to implement homodyne-like schemes with a
mesoscopic local oscillator (up to 50 mean photon numbers), as required to
achieve an optimal quantum state reconstruction NJP19 ; PLA20 . Increasing the
dynamic range is also required if the considered states of light are
characterized by large fluctuations, such as in the case of superthermal
states of light OL15 ; qmetro17 . Moreover, since the ultimate aim of our
research activity is the exploitation of SiPMs for the reconstruction of
nonclassical states of light, in our work we do not limit ourselves to the
reconstruction of statistical properties, but we also analyze the shot-by-shot
measurements, which are the main ingredient of the calculation of photon-
number correlations and nonclassicality criteria.
## II Methods
In Fig. 1 we plot a number of single-shot detector outputs of the S13360 SiPM
series of Hamamatsu. This SiPM series is characterized by low values of cross
talk and dark counts and negligible afterpulse probability.
Figure 1: Main: Typical single-shot detector outputs of MPPC S13360-1350CS
SiPMs. Inset: Zoom of the fast component of some of the waveforms.
The rising edge of the signal, corresponding to the charge process, is very
fast, lasting less than 1 ns, while the falling edge, corresponding to the
discharge process, is much longer (hundreds of ns depending on the specific
model). We assume that the output charge, given by the sum of the signals from
all the fired cells, is proportional to the number of detected photons,
modified by the presence of dark counts and cross talk. The value of the
output charge is given by the integral of the output signal, that is the area
under the curves in Fig. 1. Since the presence of cross-talk effects manifests
at delayed times with respect to the main detection peak, limiting the
integrated area reduces such spurious contributions. The main question
starting our analysis is thus if even a portion of the area or the peak height
alone contain the same information on light as the entire area. Indeed, in
some previous works of ours scirep19 ; OL19 , we have demonstrated that
integrating the signal over a gate shorter than the entire curve reduces the
incidence of spurious effects and makes it possible the observation of the
nonclassical character of quantum states of light. However, to understand if
such a procedure has general value, one should model the signal output. In
Refs. corsi ; seifert the rising edge of the output is described as a single
exponential or, more properly, by two exponentials with the same time
constant, while the falling edge is modelled as the sum of two exponentials
with distinct decay times. This means that, formally, there is a
proportionality between the area under the rising edge and the height of the
peak. On the contrary, no perfect proportionality between the total area and a
portion of the area under the falling edge is expected. Thus, to successfully
exploit SiPMs for applications, such as for the realization of a homodyne-like
detection scheme PLA20 , it is important that the signal output is properly
acquired and analyzed. All these observations led us to test different
detection chains based on different devices and to compare their performance.
In the following, we describe all the investigated acquisition chains by
emphasizing their advantages and limitations with respect to the above-
mentioned goal, that is the detection of well-populated states of light.
### II.1 The sensors
The SiPMs we used are the MPPC S13360-1350CS produced by Hamamatsu Photonics
hama . They consist of 667 pixels in a 1.3 $\times$ 1.3 mm2 photosensitive
area, with a pixel pitch equal to 50$\leavevmode\nobreak\ \mu$m and a maximum
quantum efficiency of 40$\%$ at 460 nm piemonte ; frach . In our experiment we
operated at 523 nm, where the quantum efficiency is still good ($\sim 38\%$).
As anticipated, the main drawbacks that affect such detectors are given by the
dark counts, the optical cross talk and the afterpulses. Dark counts consist
of spurious avalanches triggered by thermally-generated charge carriers dinu ;
akiba , whereas cross-talk effect is due to the spontaneous emission of
secondary infrared photons inside the silicon substrate after an avalanche
process is generated in a cell by a detection event. The secondary photons can
be detected by another neighboring cell, generating the same kind of signal as
the primary photons lacaita ; buzhan ; du ; gola . This effect can be
simultaneous to the light signal (prompt cross talk) or retarded (delayed
cross talk) nagy . While the contribution of the delayed cross talk to the
output signal can be removed by reducing the integration gate, the prompt one
is indistinguishable from the light signal. Finally, afterpulses are
avalanches produced by the photoelectrons captured by the geometric
imperfections of the device structure and released at a later time with
respect to the light signal cova . The model of SiPMs we considered is endowed
with a moderate dark-count rate (the typical value reported in the datasheet
is $\sim$ 90 kHz), a low cross-talk probability ($\sim$3 $\%$) and a
negligible afterpulse probability (less than 1 $\%$).
### II.2 The amplifiers
In general, the SiPM output is externally amplified. We consider amplifiers of
two different kinds. The first device is a fast inverting amplifier embedded
in the computer-based Caen SP5600 Power Supply and Amplification Unit (PSAU)
caen_ampli . Once amplified, the output temporal shape is essentially the same
as that of the SiPM. Such an amplifier is quite versatile since its gain can
be changed from 1 dB up to 40 dB in unit steps.
The second amplifier is a home-made circuit including a slow non-inverting
amplifier with two amplifying stages opamp , each one having a gain of 5.5 for
a total gain of 29.6 dB AD828 . As detailed in the following, this second
choice allowed us to better select the peak of the detector output since it
stretches the rising edge from less than 1 ns to 50-ns. For a fair comparison,
in Fig. 2 we show two typical outputs of the two amplifiers, both digitized
with the DRS4 digitizer described below. Panel (a) displays the output of the
Caen amplifier, while panel (b) that of the slow amplifier.
Figure 2: (a): typical output of the Caen PSAU amplifier. (b): typical output
of the slow amplifier. Both signal outputs have been sampled at 5 GS/s by a
DRS4 digitizer (see text).
By observing the two different temporal behaviors, we can assess that the fast
amplifier is more indicated to collect the entire output, while the slow one
is more suitable for catching the peak, as it will appear clearer in the next
Section.
### II.3 The digitizers
To acquire information from the signal trace, two possible strategies can be
implemented: an analogical integration or a signal digitalization followed by
an offline integration. In Ref. OL19 , the best results in terms of
nonclassicality detection were achieved by amplifying the signals with a PSAU
unit and integrating them with boxcar-gated integrators over a very short gate
(10-ns long). However, as we remarked in that work, this procedure can be
fragile when the integration gate is short, as it gives no direct control on
the unwanted presence of electronic signal jitter.
To avoid the problems of analogical integration, in this work we decided to
proceed with an offline analysis of the digitized amplified traces. We used
two different models of digitizers: the first one is the computer-based Caen
DT5720 desktop waveform digitizer, a two-channel device endowed with 12-bit
resolution, a full scale range of 2 V peak-to-peak, and a sampling rate
ranging from 31.25 to 250 MS/s caen_digi . Since its minimum sampling time is
4 ns, this device is suitable for the acquisition of rather long signals, such
as the entire amplified output. The second digitizer we used is the model DRS4
produced by the Paul Scherrer Institute Ritt ; drs4 . Also this device has
12-bit resolution, whereas its peak-to-peak voltage range is limited to 1 V,
but its sampling frequency can be changed from 1 up to 5 GS/s. Two typical
signals, both amplified by the PSAU unit and digitized by the devices
described above, are shown in the two panels of Fig. 3.
Figure 3: (a): typical output of a Caen-amplified signal digitized at 250MS/s
by the Caen digitizer (PSAU+DT5720). (b): typical output of a Caen-amplified
signal digitized at 5 GS/s by the DRS4 digitizer (PSAU+DRS4). The signal
traces are negative because of the Caen amplifier.
By inspecting the two panels, it clearly appears that the signal acquired with
the Caen digitizer lacks in most details characterizing the signal acquired
with DRS4 digitizer. Moreover, the resulting shape of the peak is completely
different, indicating an undersampling of the trace that does not allow
following the fast part of the SiPM signal.
### II.4 The light sources
In order to test the different acquisition chains described in the previous
Sections, we generated different classical and quantum optical states. The
light source is a mode-locked Nd:YLF laser regeneratively amplified at 500 Hz,
emitting the fundamental beam at 1047 nm, the second harmonic at 523 nm and
the third-harmonic at 349 nm. In the following Section, we show the results
obtained by considering three kinds of optical states: coherent states,
pseudo-thermal states and multi-mode twin-beam states.
Figure 4: Sketch of the experimental setup. (a): Scheme used to produce and
detect single-mode pseudo-thermal states. (b): Scheme used to produce and
detect multi-mode twin-beam states. See the text for details.
To produce the classical states (coherent and pseudo thermal), we exploited
the second-harmonic of the laser to match the sensitivity region of SiPMs. The
coherent state was obtained by taking a portion of the laser light, while a
rotating ground glass disk combined with a pin-hole to select a single speckle
was used to generate pseudo-thermal light (see Fig. 4(a)). The light was then
split into two beams by means of an adjustable beam splitter (BS) made of a
polarizing beam splitter preceded by a half-wave plate allowing a careful
balancing of the mean values in the two beams. At the two outputs of the BS,
two achromatic doublets focused the two beams into multi-mode fibers (1-mm
core diameter) to deliver them to SiPMs. A set of neutral density filters was
used in front of the BS to change the mean value of the light.
The quantum states were twin-beam states generated by pumping a
$\beta$-barium-borate crystal (BBO2 in Fig. 4(b)) with the fourth harmonic of
the laser (at 262 nm, 3.5-ps pulse duration). We selected two portions of the
generated twin beam at frequency degeneracy (523 nm) both in space (by means
of irises 7-mm wide) and in spectrum (by means of bandpass filters 10-nm wide)
and sent each of them to a SiPM by focusing it into a multi-mode fiber (1-mm
core diameter) with an achromatic doublet. A half-wave plate and a polarizing
beam splitter were used to change the pump energy and, consequently, the mean
number of photons of the twin beam.
## III Results
### III.1 Pulse-height spectra
First of all, we investigate the reconstruction of the statistical properties
of some optical states. SiPMs are photon-number-resolving detectors since
their pulse-height spectrum is characterized by a multi-peak structure peaks .
Upon a proper normalization procedure, each peak of the spectrum of PNR
detectors can be interpreted as the probability that a given number of photons
is detected JMO ; PNR14 . Here, we want to compare the pulse-height spectra
obtained by measuring a given state and integrating the detector output over
either different amplifier gains or different gate widths. In Fig. 5, we
compare the pulse-height spectra of a pseudo-thermal state plotted as a
function of $x_{\rm out}/\bar{\gamma}$, $x_{\rm out}$ being the detection
output and $\bar{\gamma}$ the mean peak-to-peak distance, representing the
gain of the detection chain. As extensively discussed in previous papers of
ours (see $e.g.$ Ref. PNR14 ), $\bar{\gamma}$ includes all the amplification
stages of the detection apparatus, and is assumed to be sharp enough PRA09 .
Note that $\bar{\gamma}$ can be determined in different ways: by the self-
consistent method explained in Ref. JMO , by a multi-Gaussian fitting
procedure, as described in Ref. JOSABramilli , or as the mean distance between
consecutive peaks of the pulse-height spectrum PNR14 .
The mean value of the state in Fig. 5 is $x_{\rm out}/\bar{\gamma}\sim 14$,
measured combining the amplifier of the PSAU Unit and the digitizer DRS4
operated at 5 GS/s. The gain of the amplifier was varied, while the offline
integration gate was kept fixed at $\tau=70$ ns.
Figure 5: Pulse-height spectra for a pseudo-thermal state obtained with the
system PSAU+DRS4 and integrated over $\tau=70$ ns for the different amplifier
gains $g$ indicated in the figure panels. The mean value of the pseudo-thermal
state is $x_{\rm out}/\bar{\gamma}\sim 14$. The corresponding visibility
values are (a): $v=0.39\pm 0.04$, (b): $v=0.43\pm 0.05$, (c): $v=0.49\pm 0.04$
and (d): $v=0.70\pm 0.06$.
We note that, as expected, the quality of the spectra seems to improve at
increasing gain values, and the noise between neighboring peaks visibly
decreases. In order to quantify the quality of the spectra, we define the
visibility as
$v=\sum_{i=1}^{N}\frac{(M_{i}-m_{i})}{N(M_{i}+m_{i})},$ (1)
in which $i$ is the i-th peak of the pulse-height spectrum, $N$ is the total
number of visible peaks, whereas $M_{i}$ and $m_{i}$ are the values of the
i-th peak height and its consecutive valley, respectively. For the plots in
the figure we obtained $v=0.39\pm 0.04$ for 8-dB gain, $v=0.43\pm 0.05$ for
10-dB gain, $v=0.49\pm 0.04$ for 12-dB gain, and $v=0.70\pm 0.06$ for 24-dB
gain. All these values prove the qualitative impression about the resolution.
Indeed, for the low gain values the results are rather similar, while for the
highest gain value $v$ is definitely larger, even if in this last case the
peaks corresponding to large values of $x_{\rm out}/\bar{\gamma}$ appear less
resolved probably because of some saturation effects of the amplifier. To
check the presence of saturation, in Fig. 6 we plot the peak-to-peak distance,
$\gamma$, which represents the overall gain of the detection chain, as a
function of the peak number. For a perfectly linear system, $\gamma$ should be
constant. In Fig. 6 we observe that $\gamma$ is constant for the lower gain
values (panels (a)-(c)), while for the highest amplifier gain $\gamma$
dramatically decreases after about 10 peaks (panel (d)) due to amplifier
saturation. This implies that detected-photon values exceeding 10 cannot be
reliably recovered.
Figure 6: Plot of the peak-to-peak distance in the pulse-height spectra as a
function of the peak number for the data in Fig. 5 (blue lines). The dashed
magenta line represents $\bar{\gamma}$, the mean peak-to-peak distance used to
calibrate the abscissa in all pulse-height spectra figures.
To avoid saturation issues, in the following we consider pulse-height spectra
obtained with gain values limited to 12 dB, which represent a good compromise
between peak resolution and amplifier linearity.
In Fig.s 7 and 8 we show the results achieved when the SiPM outputs are
amplified by the PSAU Unit ($g=12$ dB) and then digitized by the DRS4
digitizer operated at 5 GS/s. The mean value of the reconstructed state is
$x_{\rm out}/\bar{\gamma}\sim 1$ in Fig. 7 and $x_{\rm out}/\bar{\gamma}\sim
13$ in Fig. 8. In both cases, the offline integration gate widths are
$\tau=2.4$, 48, 70, and 100 ns. By comparing the two multipanel figures, it is
clear that the best reconstructed pulse-height spectrum is differently
achieved in the two cases. Indeed, for $x_{\rm out}/\bar{\gamma}\sim 1$, the
best performing gate width seems to be $\tau=48$ ns. This impression is
quantitatively proved by the values of $v$, which are $0.74\pm 0.07$, $0.97\pm
0.01$, $0.95\pm 0.01$, $0.88\pm 0.01$, respectively. On the contrary, for what
concerns Fig. 8, the shorter the gate the less resolved the tail of the
spectrum: the obtained values of $v$ are equal to $0.32\pm 0.05$, $0.60\pm
0.06$, $0.71\pm 0.04$, $0.63\pm 0.04$, respectively.
The comparison performed so far demonstrates that reducing the integration
gate is not the correct strategy at increasing mean number of photons. This
observation is a clear evidence of what we have already noticed in Section II:
when the light state is quite populated, the integral of just a part of the
falling edge of the output signal is not proportional to the integral of the
whole output.
In general, the plots in Fig.s 7 and 8 prove that, when the mean value of the
light is quite large, integrating the signal output of the detection chain can
be critical. In particular, it seems that the best resolution is achieved by
considering the entire output trace, even if this choice entails the presence
of delayed spurious effects.
Figure 7: Pulse-height spectra for a coherent state obtained with the system
PSAU+DRS4 at the gain g $=12$ dB for the different integration gate widths
indicated in the figure panels. The mean value of the detected coherent state
is $\langle x_{\rm out}\rangle/\bar{\gamma}\sim 1$. The corresponding
visibility values are $v=0.74\pm 0.07$ for $\tau=2.4$ ns, $v=0.97\pm 0.01$ for
$\tau=48$ ns, $v=0.95\pm 0.01$ for $\tau=70$ ns, and $v=0.88\pm 0.01$ for
$\tau=100$ ns.
We thus consider a different strategy and substitute the fast PSAU amplifier
with a slow amplifier and digitize the amplified signal with the DRS4 at 5
GS/s. We then integrate a portion of the rising edge or select the peak value.
The data are presented in Fig. 9.
Figure 8: Same as in Fig. 7 for a coherent state having mean value $\langle
x_{\rm out}\rangle/\bar{\gamma}\sim 13$. The corresponding visibility values
are $v=0.32\pm 0.05$ for $\tau=2.4$ ns, $v=0.60\pm 0.06$ for $\tau=48$ ns,
$v=0.71\pm 0.04$ for $\tau=70$ ns, and $v=0.63\pm 0.04$ for $\tau=100$ ns.
The best resolved spectrum is that obtained from the peak values for which
$v=0.82\pm 0.06$. Such a value coincides with that from the integration over
$\tau=2$ ns. On the contrary, for larger integration widths the spectra are
noisier ($v=0.75\pm 0.08$ for $\tau=10$ ns and $v=0.57\pm 0.11$ for $\tau=18$
ns), even if definitely better than those in Fig.s 5 \- 7.
The analysis performed so far proves that the proper reconstruction of the
light statistics depends on the integration gate width and that either the
entire trace signal or the peak values should be considered. Obviously, the
latter choice requires a proper shaping of the detector output, but offers the
advantage of avoiding the main drawbacks of SiPMs.
### III.2 Classical correlations
For Quantum Optics applications, the reconstruction of the statistical
properties is not enough to guarantee that the information about the light
state under examination is properly acquired. In particular, in many
situations, such as in state-discrimination protocols OLcorr ; OE17 , it is
crucial that each single pulse is correctly detected. Hereafter we compare the
different acquisition strategies discussed above for the calculation of shot-
by-shot photon-number correlations and nonclassicality criteria.
Figure 9: Pulse-height spectra for a coherent state obtained with the system
slow amplifier+DRS4 obtained either by selecting the peak value or integrating
the rising edge of the signal over different integration widths as displayed
in the figure panels. The mean value of the detected coherent state is
$\langle x_{\rm out}\rangle/\bar{\gamma}\sim 3.6$. The corresponding
visibility values are $v=0.82\pm 0.06$ for the peak value, $v=0.82\pm 0.06$
for $\tau=2$ ns, $v=0.75\pm 0.08$ for $\tau=10$ ns, and $v=0.57\pm 0.11$ for
$\tau=18$ ns.
As the experimental estimator we consider the noise reduction factor, a
quantity usually employed to quantify the nonclassical character of quantum-
correlated bipartite states of light, such as twin-beam states. The noise
reduction factor is defined as
$R=\frac{\sigma^{2}(n_{1}-n_{2})}{\langle n_{1}\rangle+\langle n_{2}\rangle}$
(2)
$n_{j}$ being the number of photons in the two components of the bipartite
state. According to the definition, $R$ is the ratio between the variance of
the photon-number difference at the two outputs and the shot-noise level. It
is well-known that $R$ must be less than 1 in case of quantum-correlated
optical states, while it is identically equal to 1 in case of classical states
of light jointdiff . Thus, we expect that for pseudo-thermal states, both
single-mode and multi-mode, the noise reduction factor is unitary. Actually,
the value of $R$ that can be experimentally measured is affected by non-unit
quantum efficiency, dark counts, cross-talk and imbalance between the
components of the bipartite state. For this reason, the experimental value of
$R$ is
$R=\frac{\sigma^{2}(k_{1}-k_{2})}{\langle k_{1}\rangle+\langle k_{2}\rangle}$
(3)
$k_{j}$ being the number of detected photons including all the experimental
effects.
To derive an analytical expression for $R$ that takes all the real effects
into account, we exploit the calculation of the correlation function jointdiff
and implement the detector model introduced in Ref. JOSABramilli . The
resulting general expression is OL19 111Note that Eq. (2) in Ref. OL19
contains a misprint: the factor $\langle k_{1}\rangle+\langle k_{2}\rangle$
should be added as the denominator of the last term.
$\displaystyle R$ $\displaystyle=$ $\displaystyle
1+\frac{1}{\mu}\frac{(\langle k_{1}\rangle-\langle k_{2}\rangle)^{2}}{\langle
k_{1}\rangle+\langle k_{2}\rangle}$ (4) $\displaystyle+$
$\displaystyle\frac{2\epsilon_{1}}{1+\epsilon_{1}}\frac{\langle
k_{1}\rangle}{\langle k_{1}\rangle+\langle
k_{2}\rangle}+\frac{2\epsilon_{2}}{1+\epsilon_{2}}\frac{\langle
k_{2}\rangle}{\langle k_{1}\rangle+\langle k_{2}\rangle}$ $\displaystyle-$
$\displaystyle\frac{2}{\mu}\left[(1+\epsilon_{1})\langle m_{\rm
1dc}\rangle-(1+\epsilon_{2})\langle m_{\rm 2dc}\rangle\right]\frac{\langle
k_{1}\rangle-\langle k_{2}\rangle}{\langle k_{1}\rangle+\langle k_{2}\rangle}$
$\displaystyle+$
$\displaystyle\frac{1}{\mu}\frac{\left[(1+\epsilon_{1})\langle m_{\rm
1dc}\rangle-(1+\epsilon_{2})\langle m_{\rm 2dc}\rangle\right]^{2}}{\langle
k_{1}\rangle+\langle k_{2}\rangle}$ $\displaystyle-$ $\displaystyle
2\sqrt{\eta_{1}\eta_{2}}\sqrt{\frac{(1+\epsilon_{1})\left[\langle
k_{1}\rangle-(1+\epsilon_{1})\langle m_{\rm 1dc}\rangle\right]}{\langle
k_{1}\rangle+\langle k_{2}\rangle}}$ $\displaystyle\ \ \ \ \ \ \times\
\sqrt{\frac{(1+\epsilon_{2})\left[\langle k_{2}\rangle-(1+\epsilon_{2})\langle
m_{\rm 2dc}\rangle\right]}{\langle k_{1}\rangle+\langle k_{2}\rangle}},$
where $\mu$ is the number of multi-thermal modes, $\eta_{j}$ is the detection
efficiency of the detection chains in the two arms, $\langle
k_{j}\rangle=(\eta_{j}\langle n_{j}\rangle+\langle m_{\rm
jdc}\rangle)(1+\epsilon_{j})$ is the mean value of the detector output,
$\langle m_{\rm jdc}\rangle$ the mean value of dark counts and $\epsilon_{j}$
the cross-talk probability. According to the model in Ref. JOSABramilli , the
dark-count probability distribution is assumed to be Poissonian, so that Eq.
(4) also describes the case in which some stray light is detected
simultaneously with the signal. Note that the last term vanishes for
classically correlated light highorder .
Since, in spite of all the preliminary alignment procedure, it is not possible
to exclude the presence of an imbalance between the detected photons in the
two arms of the bipartite state, we introduce the imbalance coefficient
$t\in[0,1]$ defined so that $\langle k_{1}\rangle\equiv\langle k\rangle$ and
$\langle k_{2}\rangle=t\langle k\rangle$ JOSABramilli . Under this assumption,
Eq. (4) simplifies to:
$\displaystyle R$ $\displaystyle=$ $\displaystyle
1+\frac{1}{\mu}\frac{(1-t)^{2}}{1+t}\langle
k\rangle+\frac{2}{1+t}\left(\frac{\epsilon_{1}}{1+\epsilon_{1}}+\frac{\epsilon_{2}}{1+\epsilon_{2}}t\right)$
(5) $\displaystyle-$
$\displaystyle\frac{2}{\mu}\frac{1-t}{1+t}\left[(1+\epsilon_{1})\langle m_{\rm
1dc}\rangle-(1+\epsilon_{2})\langle m_{\rm 2dc}\rangle\right]$
$\displaystyle+$ $\displaystyle\frac{1}{\mu}\frac{1}{(1+t)\langle
k\rangle}\left[(1+\epsilon_{1})\langle m_{\rm
1dc}\rangle-(1+\epsilon_{2})\langle m_{\rm 2dc}\rangle\right]^{2}$
$\displaystyle-$ $\displaystyle
2\frac{\sqrt{\eta_{1}\eta_{2}}}{1+t}\sqrt{(1+\epsilon_{1})\left[1-(1+\epsilon_{1})\frac{\langle
m_{\rm 1dc}\rangle}{\langle k\rangle}\right]}$ $\displaystyle\ \ \ \ \ \
\times\ \ \sqrt{(1+\epsilon_{2})\left[t-(1+\epsilon_{2})\frac{\langle m_{\rm
2dc}\rangle}{\langle k\rangle}\right]}.$
To test the model we start considering classically-correlated light states,
namely a single-mode pseudo-thermal state divided at a beam splitter, and
evaluate the noise reduction factor from data acquired with three different
detection chains. In more detail, we consider SiPMs followed by the three
different amplification and acquisition chains introduced above: (i)
PSAU+DT5720 ($g=12$ dB), (ii) PSAU+DRS4 ($g=10$ dB) and (iii) slow
amplifier+DRS4. For each case, we also investigate the role played by the
integration gate width by comparing the results achieved with two different
values of $\tau$. The experimental data are shown in Fig.s 10, 11 and 12,
respectively. In each figure, the results corresponding to the shortest $\tau$
are shown as black dots + error bars, while those corresponding to the longest
one are shown as red dots + error bars. In the same figures, the theoretical
expectations are presented as colored curves with the same color choice.
We notice that the experimental values of $R$ are always larger than 1 and
that the largest values are obtained in Fig. 10, that is the case of
acquisition with the Caen system PSAU+DT5720. In this case, the two datasets
refer to $\tau=48$ ns (black dots) and $\tau=80$ ns (red dots). The
theoretical expectations were obtained according to Eq. (5), in which we set
the number of modes $\mu=1$, and left the cross-talk probability, the dark-
counts and the imbalance as free parameters. In particular, we assumed
$\epsilon_{1}=\epsilon_{2}$, and $\langle m_{\rm 1dc}\rangle\neq\langle m_{\rm
2dc}\rangle$. By looking at the fitting parameters, we can notice that the
value of the cross-talk probability is compatible with that indicated in the
datasheets and that, as expected, the fitting value slightly increases at
increasing the gate width. Moreover, in both cases the two BS outputs exhibit
a quite different dark-count contribution, whereas the imbalance coefficient
is very close to 1.
A similar behavior is achieved for the case (ii) with the second acquisition
chain, that is based on the amplifiers of Caen Unit ($g=10$ dB) and the DRS4
digitizer. The experimental values of $R$ are shown in Fig. 11 as colored dots
+ error bars, whereas the theoretical models according to Eq. (5) are plotted
as colored curves with the same color choice. In particular, the two datasets
refer to $\tau=70$ ns (black dots) and $\tau=110$ ns (red dots). The fitting
procedure yields values of cross-talk comparable with the data in Fig. 10. As
to the dark-count contribution, we obtain different values in the two arms, as
expected from the decreasing trend of the data. We notice that, according to
the values of the reduced $\chi^{(2)}$, the best fit is obtained for
$\tau=110$ ns, that is by integrating the entire signal.
Figure 10: Noise reduction factor for the single-mode thermal state as a
function of the mean number of photons detected at a BS output. Colored dots
and error bars: experimental data obtained with the CAEN system PSAU ($g=12$
dB) and DT5720 digitizer integrating the traces over $\tau=48$ ns (black dots)
and $\tau=80$ ns (red dots); colored lines: theoretical expectations according
to Eq. (5). The fitting parameters for $\tau=48$ ns are: $\epsilon=0.029$,
$\langle m_{\rm 1dc}\rangle=0.231$, $\langle m_{\rm 2dc}\rangle=0.569$,
$t=0.999$ and $\chi^{(2)}=4.8$, while those for $\tau=80$ ns are:
$\epsilon=0.031$, $\langle m_{\rm 1dc}\rangle=0.205$, $\langle m_{\rm
2dc}\rangle=0.595$, $t=0.999$ and $\chi^{(2)}=3.6$.
For the case (iii), based on the shaping amplifiers and the DRS4 digitizer,
the situation is definitely different. The data are shown as colored dots and
error bars in Fig. 12 together with the theoretical expectations according to
Eq. (5) with the same color choice. The black dots are obtained by selecting
the peak values, whereas the red dots correspond to the integral of the signal
over $\tau=2$ ns before the peak value.
Figure 11: Noise reduction factor for the single-mode thermal state as a
function of the mean number of photons detected at a BS output. Colored dots
and error bars: experimental data obtained with the system PSAU ($g=10$ dB)
and DRS4 digitizer integrating the traces over $\tau=70$ ns (black dots) and
$\tau=110$ ns (red dots); colored lines: theoretical expectations according to
Eq. (5). The fitting parameters for $\tau=70$ ns are: $\epsilon=0.026$,
$\langle m_{\rm 1dc}\rangle=1.406$, $\langle m_{\rm 2dc}\rangle=0.594$,
$t=0.958$ and $\chi^{(2)}=1.5$, while those for $\tau=110$ ns are:
$\epsilon=0.038$, $\langle m_{\rm 1dc}\rangle=1.515$, $\langle m_{\rm
2dc}\rangle=0.485$, $t=0.930$ and $\chi^{(2)}=1.2$.
In this case, the behavior of $R$ as a function of the mean values exhibits a
weak increasing trend. For what concerns the cross-talk probability, we notice
that the smallest value is obtained for the integration over $\tau=2$ ns. This
is not surprising since integrating the signal amplified by the slow amplifier
over 2 ns (10 points) is equivalent to a smoothing operation that reduces
possible irregularities given by the simple peak selection. The obtained
cross-talk probability is in this case very small, thus proving that the
acquisition of the smoothed peak is the best solution among those considered
so far.
Figure 12: Noise reduction factor for the single-mode thermal state as a
function of the mean number of photons detected at a BS output. Colored dots
and error bars: experimental data obtained with the slow amplifier and DRS4
digitizer by selecting the peak values (black dots) or integrating over
$\tau=2$ ns before the peak (red dots); colored lines: theoretical
expectations according to Eq. (5). The fitting parameters for the peak are:
$\epsilon=0.026$, $\langle m_{\rm 1dc}\rangle=0.278$, $\langle m_{\rm
2dc}\rangle=0.422$, $t=0.924$ and $\chi^{(2)}=3.0$, while those for $\tau=2$
ns are: $\epsilon=0.018$, $\langle m_{\rm 1dc}\rangle=0.275$, $\langle m_{\rm
2dc}\rangle=0.425$, $t=0.907$ and $\chi^{(2)}=2.6$.
As a general comment on the estimated values of dark counts, we can safely
assess that they are larger than the expected values from the sensor
datasheet. In fact, at room temperature, the maximum dark-count contribution
can be evaluated as $\langle m_{\rm dc}\rangle=270$ kHz$\times 110$ ns
$=0.029$. To account for the experimental results we can assume the presence
of some residual Poissonian infrared light, that is the fundamental of the
laser, we could not eliminate completely.
To further check the validity of solution (iii), we consider the more
interesting case of quantum-correlated optical states, namely the multi-mode
twin-beam states.
### III.3 Quantum correlations
As anticipated in Section II.4, we generated multi-mode twin-beam states by
sending the fourth harmonic of a Nd:YLF laser to BBO2 of Fig. 4(b) to produce
parametric downconversion.
For a fair comparison with the classical case of light discussed in the
previous Section, we consider the noise reduction factor in Eq. (5). We expect
$R<1$ for nonclassical light. The experimental values of $R$, calculated
according to Eq. (3), are shown as colored dots and error bars as a function
of the mean number of photons detected in the two arms. The black dots are
obtained by selecting the peak values, whereas the red dots correspond to the
integral of the signal over $\tau=2$ ns before the peak value. The fitting
procedure yields a low value of cross-talk probability and a non-negligible
dark-count contribution that is the same in the two arms. The estimated values
of cross-talk probability are very close to those for classical light case
(iii) 2 ns gate (see captions of Fig.s 12 and 13). Note that, at variance with
the case of single-mode pseudo-thermal light, for the multi-mode twin-beam
states the difference between selecting the peak and smoothing over 2 ns is
less evident, probably due to reduced fluctuations of the multi-mode twin-beam
statistics. As in the case of classical light, the estimated dark count values
are larger than expected due to the presence of spurious infrared light.
Figure 13: Noise reduction factor for a multi-mode twin-beam state as a
function of the mean number of photons on one of the arms. Colored dots and
error bars: experimental data obtained with the slow amplifier and DRS4
digitizer by selecting the peak values (black dots) or integrating over
$\tau=2$ ns before the peak (red dots); colored lines: theoretical
expectations according to Eq. (5). The fitting parameters for the peak are:
$\eta=0.182$, $\mu=9256$, $\epsilon=0.019$, $\langle m_{\rm dc}\rangle=0.349$,
$t=0.913$ and $\chi^{(2)}=3.2$, while those for $\tau=2$ ns are: $\eta=0.181$,
$\mu=9990$, $\epsilon=0.020$, $\langle m_{\rm dc}\rangle=0.348$, $t=0.924$ and
$\chi^{(2)}=6.2$.
Finally, we emphasize that the data presented in Fig. 13 are of better quality
than those already reported in Ref. OL19 : comparing the same kind of
acquisition chains, we see that the mean number of photons is definitely
larger (up to 11 instead of 3.5) and the absolute values of $R$ are smaller
(0.86 instead of 0.9).
## IV Discussion and conclusions
Let us summarize the results presented in the previous Sections and draw some
conclusions.
With the aim of detecting and properly characterizing mesoscopic optical
states, namely light states endowed with sizeable numbers of photons, by means
of SiPMs, we explored and compared different devices, obtained by combining
two different kinds of amplifiers and two different digitizers. In order to
choose the best detection chain, we proceeded in two steps. First of all, we
analyzed the quality of the pulse-height spectra, introducing the visibility
$v$ as a figure of merit for the quality of the spectrum. The analysis of the
spectra taken for different integration gate widths indicates that, in the
case of well-populated light states, to extract the proper amount of
information from the measurements requires the integration of the entire
output trace. However, this choice does not exclude all the drawbacks
affecting SiPMs since also dark counts, cross talk and afterpulses are
acquired together with the light signal.
To reduce the incidence of drawbacks, a different strategy can be implemented,
consisting in acquiring only a portion of the rising edge of the signal
output. Moreover, we have demonstrated that integrating only a small portion
of the area under the rising edge or selecting the peak height are equivalent,
even if the latter solution is definitely simpler and allows a complete
rejection of spurious effects. This is why we devised a proper detection chain
in order to correctly catch the peak. The solution addressed in this paper is
based on a shaping amplifier directly connected to the SiPM output and a fast
digitizer.
In the second part of the paper, we analyzed the problem of light acquisition
from the different perspective of getting the correct number of photons shot-
by-shot in order to calculate the noise reduction factor $R$. We measured
pseudo-thermal light using the three different acquisition chains: the best
results are given by the acquisition of the peak values, for which the values
of $R$ are closer to unity. The same acquisition strategy results optimal also
in the case of the twin-beam states, which are quantum correlated.
In conclusion, the performed investigation leads us to conclude that, provided
a proper shaping of the amplified signal and its fast enough digitalization,
the strategy of selecting the peak value is the most reliable and even the
simplest. On the one side, it avoids the effect of drawbacks because only
spurious effects simultaneous to the light signal are acquired, on the other
side it holds for any mean photon number unless saturation effects of the
detection chain occur. Increasing the dynamic range of the detectors is
important to avoid saturation.
Finally, we emphasize that the chosen chain is rather compact and can be made
portable for possible applications in open air, such as for Quantum
Communication.
## V Acknowledgements
We acknowledge Giovanni Chesi for useful discussions.
## VI Funding
No funding.
## VII Abbreviations
SiPM = Silicon photomultiplier, PSAU = Power Supply and Amplification Unit, BS
= beam splitter, BBO = $\beta$-barium-borate crystal.
## VIII Availability of data and materials
The datasets used and analysed during the current study are available from the
corresponding author on reasonable request.
## IX Competing interests
The authors declare that they have no competing interests.
## X Authors contributions
SC, AA and MB conceptualized the work, VM, MP and EV designed the detection
chain to properly detect the peak, SC and AA performed the measurements, SC
and VM wrote a new code to prepare data for analysis, AA and MB analysed and
interpreted the data, AA and MB drafted the work, SC, VM, MP and EV
substantively revised it. All the authors have read and approved the final
manuscript.
## References
* (1) A. V. Akindinov, A. N. Martemianov, P. A. Polozov, V. M. Golovin, and E. A. Grigoriev, Nucl. Instrum. Methods Phys. Res. A 387, (1997) 231–234.
* (2) G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, Nucl. Physics B (proc. Suppl.) 61B, (1998) 347-352.
* (3) V. Saveliev and V. Golutvin, Nucl. Instrum. Methods Phys. Res. A 442, (2000) 223.
* (4) C. Piemonte, Nucl. Instrum. Methods Phys. Res. A 568, (2006) 224.
* (5) D. Renker and E. Lorenz, JINST 4, (2009) P04004.
* (6) G. Chesi, L. Malinverno, A. Allevi, R. Santoro, M. Caccia, A. Martemiyanov, and M. Bondani, Sci. Rep. 9, (2019) 7433.
* (7) G. Chesi, L. Malinverno, A. Allevi, R. Santoro, M. Caccia, and M. Bondani, Opt. Lett. 44, (2019) 1371-1374.
* (8) S. Olivares, A.Allevi, G. Caiazzo, M. G. A. Paris, and M. Bondani, New J. Phys. 21, (2019) 103045.
* (9) S. Olivares, A. Allevi, and M. Bondani, Phys. Lett. A. 384, (2020) 126354.
* (10) A. Allevi and M. Bondani, Opt. Lett. 40, (2015) 3089-3092.
* (11) A. Allevi, S. Cassina, and M. Bondani, Quantum Meas. Quantum Metrol. 4, (2017) 26-34.
* (12) F. Corsi, A. Dragone, C. Marzocca, A. Del Guerra, P. Delizia, N. Dinuc, C. Piemonte, M. Boscardin, and G. F. Dalla Betta, Nucl. Instrum. Methods Phys. Res. A 572, (2007) 416–418.
* (13) S. Seifert, H. T. van Dam, J. Huizenga, R. Vinke, P. Dendooven, H. Loehner, and D. R. Schaart, IEEE TNS 56, (2009) 3726-3733.
* (14) http://www.hamamatsu.com/us/en/S13360-1350CS.html.
* (15) T. Frach, JINST 7, (2012) C01112.
* (16) N. Dinu, Z. Amara, C. Bazin, V. Chaumat, C. Cheikali, G. Guilhem, V. Puill, C. Sylvia, and J. F. Vagnucci, Nucl. Instr. and Meth. in Phys. Res. A 610, (2009) 423-426.
* (17) M. Akiba, K. Tsujino, K. Sato, and M. Sasaki, Opt. Express 17, (2009) 16885-16897.
* (18) A. L. Lacaita, F. Zappa, S. Bigliardi, and M. Manfredi, IEEE TED 40, (1993) 577-582.
* (19) P. Buzhan, B. Dolgoshein, A. Ilyin, V. Kantserov, V. Kaplin, A. Karakash, A. Pleshko, E. Popova, S. Smirnov, Yu. Volkov, L. Filatov, S. Klemin, and F. Kayumov, ICFA Instrum. Bull. 23, (2001) 28-41.
* (20) Y. Du and F. Reti$\grave{\rm e}$re, Nucl. Instrum. Methods Phys. Res. A 596, (2008) 396-401.
* (21) A. Gola, A. Ferri, A. Tarolli, N. Zorzi, and C. Piemonte, Phys. Med. Biol. 59, (2014) 3615–3635.
* (22) F. Nagy, M. Mazzillo, L. Renna, G. Valvo, D. Sanfilippo, B. Carbone, A. Piana, G. Fallica, and J. Molnar, Nucl. Instrum. Methods Phys. Res. A 759, (2014) 44–49.
* (23) S. Cova, A. Lacaita, and G. Ripamonti, IEEE EDL 12, (1991) 685-687.
* (24) https://www.caen.it/products/sp5600e/.
* (25) http://www.ti.com/lit/an/sboa092b/sboa092b.pdf.
* (26) https://www.analog.com/en/products/ad828.html.
* (27) https://www.caen.it/products/dt5720/.
* (28) S. Ritt, R. Dinapoli, and U. Hartmann, Nucl. Instrum. Meth. A 623 (2010), 486-488.
* (29) https://www.psi.ch/en/drs/evaluation-board.
* (30) R. Klanner, Nucl. Instrum. Methods Phys. Res. A 926, (2019) 36-56.
* (31) M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, (2009) 226-231.
* (32) A. Allevi and M. Bondani, J. Opt. Soc. Am. B 31, (2014) B14-B19.
* (33) A. Andreoni and M. Bondani, Phys. Rev. A 80, (2009) 013819.
* (34) M. Ramilli, A. Allevi, V. Chmill, M. Bondani, M. Caccia, and A. Andreoni, J. Opt. Soc. Am. B 27, (2010) 852-862.
* (35) A. Allevi, M. Bondani, and A. Andreoni, Opt. Lett. 35, (2010) 1707-1709.
* (36) M. Bina, A. Allevi, M. Bondani, and S. Olivares, Opt. Express 25, (2017) 10685-10692.
* (37) A. Agliati, M. Bondani, A. Andreoni, G. D. Cillis, and M. G. A. Paris, J. Opt. B Quantum Semiclassical Opt. 7, (2005) S652.
* (38) A. Allevi, S. Olivares, and M. Bondani, Phys. Rev. A 85, (2012) 063835.
|
# Measurements of pulse jitter and single-pulse variability in millisecond
pulsars using MeerKAT
A. Parthasarathy1,2,3, M. Bailes1,3, R.M. Shannon1,3, W. van Straten4, S.
Osłowski1,5, S. Johnston6, R. Spiewak1,3,7, D. J. Reardon1,3, M. Kramer2, V.
Venkatraman Krishnan2, T. T. Pennucci8,9, F. Abbate2, S. Buchner10, F.
Camilo10, D. J. Champion2, M. Geyer10,B. Hugo10,11, A. Jameson1,3 A.
Karastergiou12, M. J. Keith7, M. Serylak10
1Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia
2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,
Germany
3 OzGrav: Australian Research Council Centre of Excellence for Gravitational
Wave Discovery.
4 Institute for Radio Astronomy & Space Research, Auckland University of
Technology, Private Bag 92006, Auckland 1142, New Zealand
5 Gravitational Wave Data Centre, Swinburne University of Technology, P.O. Box
218, Hawthorn, VIC 3122, Australia
6 CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO
Box 76, Epping NSW 1710, Australia
7 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, UK
8 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA,
9 Institute of Physics, Eötvös Loránd University, Pázmány P. s. 1/A, 1117
Budapest, Hungary
10 South African Radio Astronomy Observatory, Cape Town, 7925, South Africa
11 Department of Physics and Electronics, Rhodes University, Artillery Road,
Grahamstown, South Africa,
12 Oxford Astrophysics, Denys Wilkinson Building, Keble Road, OX1 3RH, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Using the state-of-the-art SKA precursor, the MeerKAT radio telescope, we
explore the limits to precision pulsar timing of millisecond pulsars
achievable due to pulse stochasticity (jitter). We report new jitter
measurements in 15 of the 29 pulsars in our sample and find that the levels of
jitter can vary dramatically between them. For some, like the 2.2 ms pulsar
PSR J2241–5236, we measure an implied jitter of just $\sim$ 4 ns/hr, while
others like the 3.9 ms PSR J0636–3044 are limited to $\sim$ 100 ns/hr. While
it is well known that jitter plays a central role to limiting the precision
measurements of arrival times for high signal-to-noise ratio observations, its
role in the measurement of dispersion measure (DM) has not been reported,
particularly in broad-band observations. Using the exceptional sensitivity of
MeerKAT, we explored this on the bright millisecond pulsar PSR J0437–4715 by
exploring the DM of literally every pulse. We found that the derived single
pulse DMs vary by typically 0.0085 cm-3 pc from the mean, and that the best DM
estimate is limited by the differential pulse jitter across the band. We
postulate that all millisecond pulsars will have their own limit on DM
precision which can only be overcome with longer integrations. Using high-time
resolution filterbank data of 9 $\mu$s, we also present a statistical analysis
of single pulse phenomenology. Finally, we discuss optimization strategies for
the MeerKAT pulsar timing program and its role in the context of the
International Pulsar Timing Array (IPTA).
###### keywords:
stars: neutron; pulsars: general; methods: data analysis;
††pubyear: 2020††pagerange: Measurements of pulse jitter and single-pulse
variability in millisecond pulsars using MeerKAT–Measurements of pulse jitter
and single-pulse variability in millisecond pulsars using MeerKAT
## 1 Introduction
The precise monitoring of periodic radio pulses emitted from millisecond
pulsars (MSPs) has enabled some of the most stringent tests of fundamental
physics. It has been used to test the theory of general relativity (Taylor &
Weisberg 1982; Kramer et al. 2006b), alternative theories of gravity (Zhu et
al. 2015; Voisin et al. 2020), constrain neutron star equations of state
(Demorest et al. 2010; Antoniadis et al. 2013; Cromartie et al. 2020), measure
irregularities in terrestrial time standards (Petit & Tavella 1996; Hobbs et
al. 2020), detect planetary-mass companions (Wolszczan & Frail 1992) and can
potentially be used to detect and characterise nHz-frequency gravitational
radiation (Hellings & Downs 1983; Foster & Backer 1990). Precision long-term
timing of an ensemble of the most stable MSPs to $<$ 100 ns root-mean-square
(rms) residuals has led to placing upper limits on the stochastic
gravitational wave background (Shannon et al. 2015; Lentati et al. 2016;
Aggarwal et al. 2019; Perera et al. 2019), allowing constraints on the
formation and evolutionary scenarios of supermassive black holes and their
host galaxies (Taylor et al. 2017).
Pulsar timing residuals, which are the differences between the observed pulse
times-of-arrival (ToAs) and those predicted by a timing model, are a
fundamental diagnostic tool in assessing the quality of the timing model.
Numerous studies since the 1970s have shown that the scatter in timing
residuals are larger than that expected from the formal uncertainties (i.e.,
the uncertainties reported from a match-filtered based arrival time
determination algorithm) alone (Groth 1975; Cordes & Downs 1985; Osłowski et
al. 2011; Shannon et al. 2014; Lam et al. 2019). This excess noise can be
categorised into a time-correlated, red-noise component and an uncorrelated
white-noise component. One of the main contributing sources to the red noise
is caused by rotational irregularities in the pulsar’s spin period also known
as spin noise or timing noise (Boynton et al. 1972; Cordes 1980). Spin noise
manifests as a low-frequency process in pulsar timing residuals over
timescales of months to years. Many studies have attempted to characterise the
strength and non-stationarity of spin noise across the pulsar population
(Shannon & Cordes 2010; Parthasarathy et al. 2019) finding that although it is
widespread in pulsars, it is weaker in millisecond pulsars (Lam et al. 2017).
Additional contributions to the red-noise component can arise from quasi-
periodic processes, due to magnetospheric torque variations (Kramer et al.
2006a; Lyne et al. 2010), unmodelled planetary companions (Shannon et al.
2013; Kerr et al. 2015), unmodelled dispersion measure (DM) variations (Keith
et al. 2013), turbulence in the interstellar medium (Cordes & Shannon 2010;
Lam et al. 2017) and uncertainties in the solar system ephemeris (Champion et
al. 2010; Caballero et al. 2018).
On timescales of minutes to hours, the excess noise in the timing residuals is
typically dominated by the uncorrelated white-noise component. This excess
white noise, in addition to radiometer noise, can arise from several sources,
the most significant of which is from differences between the integrated pulse
profile (the averaged phase-resolved light curve of the pulsar) and a template
profile (average of a finite number of pulses). This difference contributes
directly to the excess white noise which results in observed ToA uncertainties
being higher than the predicted formal values. The formal uncertainty in the
arrival time is derived from the template-matching algorithm, which models the
integrated pulse profile ($P$) as a scaled (by a factor $A$) and offset (by a
constant $B$) version of the template ($O$) rotated by a phase shift $\phi$
with additional white noise $N(t)$, expressed as (Taylor 1992),
$P(t)=AO(t-\phi)+B+N(t).$ (1)
Observations of single pulses from pulsars with high flux densities have
exhibited variations in their pulse morphology along with correlated
variations in their phase and amplitudes (Drake & Craft 1968; Helfand et al.
1975; Jenet et al. 1998). Aside from the fact that pulse profile variability
adversely affects the attainable precision in pulsar timing experiments with
MSPs, it is important to acknowledge that the general phenomenology has been
extensively studied across the pulsar population on both short (seconds to
hours) and long (months to years) timescales. It has been known that emission
changes can occur in pulsars on short timescales (Backer 1970b, c); where
pulse profiles were observed to switch between two or more distinct emission
states (known as mode changing) or where the profile was either in a weak
emission state or completely turned ‘off’ (known as nulling). Pulsars
categorized as intermittent have been observed to cycle between quasi-periodic
phases in which the radio emission is either clearly present or invisible
(Kramer et al. 2006a; Camilo et al. 2012; Lyne et al. 2017) and in all cases
the observed emission changes have been correlated with the pulsar’s
rotational behaviour. Changes in pulse morphology have also been attributed to
precession of the pulsar’s spin axis, which causes different regions of the
emission beam to orient along our line-of-sight (Weisberg et al. 1989; Kramer
1998). In an important study, Cordes & Downs (1985) analysed 24 pulsars and
concluded that pulse shape variations or jitter were significant in a large
fraction of their sample and proposed that it is likely to occur in all
pulsars with varying degrees of importance.
Profile changes in MSPs have been reported in very few cases. Hotan et al.
(2004) & van Straten (2013) showed that previous reports of profile
instabilities in PSR J1022+1001 (Kramer et al. 1999) are possibly due to
instrumental polarimetric calibration errors while a recent study suggests
that calibration alone cannot account for the observed profile variations
(Padmanabh et al. 2021). Shannon et al. (2016) reported a broad-band profile
change in PSR J1643–1224 (also observed by Brook et al. 2016) which may have
been due to a disturbance or a state change in the pulsar magnetosphere. Very
recently, the brightest and closest MSP, PSR J0437–4715 exhibited a
significant change in its integrated pulse profile (Kerr et al. 2020),
indicating that such abrupt profile changes may be common among MSPs as well.
In contrast to observations of young pulsars, the profile changes in both MSPs
were not accompanied with measurable changes in their spin down rate.
A few studies have recently focused on studying jitter noise in MSPs and its
effect on limiting the attainable precision through pulsar timing (Osłowski et
al. 2011; Liu et al. 2012; Shannon et al. 2014; Lam et al. 2019). Jitter noise
is a stochastic process common to all pulsars, arising from intrinsic self-
noise in the pulsar emission mechanism and is thought to be a wide-band
phenomenon (Taylor et al. 1975; Rickett 1975). Since single pulse morphology
changes stochastically from pulse to pulse, and can be measurable in high
signal-to-noise (S/N) observations, the averaged pulse profile ($P$) will have
a shape that is different from the template, thus causing an excess scatter in
the measured ToA uncertainty. This excess scatter can be measured from its
contribution to the rms of the ToAs, expressed as $\sigma_{\rm J}(N_{\rm p})$,
where $N_{\rm p}$ is the number of averaged pulses or can also characterised
with a dimensionless jitter parameter ($f_{\rm j}$), as the ratio of
$\sigma_{\rm J}(N_{\rm p})$ to the pulse period ($P$). Unlike the formal ToA
uncertainty ($\sigma_{\mathrm{S}/\mathrm{N}}$), $\sigma_{\rm J}$ is
independent of the S/N. Furthermore, additional scatter in the ToAs can also
result from narrow-band diffractive interstellar scintillation (DISS) along
with the frequency dependence of the pulse profile, especially if the profiles
are averaged over large bandwidths Demorest et al. (2013); Shannon & Cordes
(2017). A second effect related to diffractive scintillation, is caused by
stochasticity in the pulse broadening and has been termed the finite-scintle
effect Cordes et al. (1990); Cordes & Shannon (2010). This effect can be is
greater for pulsars which show larger degrees of scatter broadening, so is
especially important for high dispersion-measure pulse observed at lower
frequencies.
Osłowski et al. (2011) analysed 25 hours of high-precision timing data of PSR
J0437–4715 (using the Murriyang/64-m Parkes radio telescope, at an observing
frequency of $\sim$ 1400 MHz) reporting that in one hour ($N_{\rm p}$ $\sim$
$10^{6}$ pulses), pulse jitter limits the attainable timing precision to
$\sim$ 30 ns. Shannon & Cordes (2012) similarly reported the jitter in PSR
J1713+0747 to be $\sim$ 20 ns in an hour. Jitter measurements for a sample of
22 MSPs as part of the Parkes Pulsar Timing Array (PPTA; Manchester et al.
2013) project were reported by Shannon et al. (2014) with PSR J1909–3744
showing the lowest levels of jitter noise of $\sim$ 10 ns in an hour. They
modelled the contribution of jitter as a function of observing time ($T$) as
$\approx$ $0.5W_{\rm eff}\sqrt{P/T}$. More recently, Lam et al. (2019)
detected jitter in 43 MSPs as part of timing program of the North American
Nanohertz Observatory for Gravitational Waves (NANOGrav, Arzoumanian et al.,
2018) and found significant frequency dependence of jitter in 30 of them.
These studies clearly show that pulse jitter is a generic property of MSPs,
that it dominates the white noise budget when observing with increased S/N and
that robust characterisation of jitter noise is vital in the search for
stochastic nHz gravitational wave background using pulsar timing arrays.
The MeerKAT radio telescope is several times more sensitive than the Parkes
radio telescope and offers the opportunity to detect and constrain jitter in
many more MSPs at declinations $<$ $+$40∘ . In Section 2 we describe the
MeerTime MSP observing program, the relevant data processing by the observing
backend and the various processing steps implemented in the data reduction
pipeline. We discuss the methodology used to estimate jitter, report jitter
measurements for 29 MSPs and discuss its frequency dependence in Section 3. In
Section 4 we discuss fundamental limits imposed on DM measurements in PSR
J0437–4715. Statistical studies of single pulses allow us to link timing
variations and shape variations thus providing further insights into
characterising jitter noise. In Section 5, we describe the statistical
properties of MSP single pulses and discuss these results in the context of
timing variations. Finally, in Section 6, we discuss the implications of our
results and the role of the MeerTime Pulsar Timing Array (MPTA) programme in
aiding high precision pulsar timing and PTA experiments.
## 2 Observations and data reduction
The MeerKAT radio telescope is the South African precursor for the Square
Kilometre Array (SKA) mid radio telescope located in the Great Karoo region,
and is capable of observing the large population of known pulsars in the
Southern and Northern hemispheres. The MeerTime collaboration (Bailes et al.
2020) is one of the MeerKAT Large Survey Projects, which has been
provisionally awarded many months of observing time. The MPTA is one of the
four major science themes as part of MeerTime, which focuses on the precision
timing of MSPs and is poised to contribute to an internationally coordinated
effort to detect the stochastic gravitational wave background using pulsar
timing arrays through the International Pulsar Timing Array (IPTA, Hobbs et
al. 2010; Verbiest et al. 2016; Perera et al. 2019). The current observing
strategy for the MPTA is to attain sub-microsecond formal timing precision on
as many pulsars as possible with integration times less than 2048 seconds. For
each pulsar, the integration time is set to match the median expected time
either based on previous observations from the MeerKAT MSP census program or
previous monitoring campaigns. A maximum integration time of 2048 seconds is
imposed per epoch and a minimum integration time of 256 seconds is used if
sub-microsecond timing precision is achievable in less than that time. The
MPTA, since commencing observations from February 2019, attains timing
precision of $<$ 1$\mu s$ on $\sim$ 70 MSPs in a total integration time of
approximately 11 hours. The Parkes Pulsar Timing Array, in comparison,
achieves sub-microsecond precision in 22 MSPs in 24 hours and NANOGrav
achieves the same precision on 47 MSPs (Alam et al. 2020a, b).
For the jitter analysis presented here, we selected MSPs observed with MeerKAT
that exceeded a S/N per pulse of unity following Shannon & Cordes (2012).
These included $\sim$ 350 observations of 29 pulsars over a span of about a
year (starting from February 2019). Approximately 80% of these observations
had an integration time of $<$ 900 seconds. We used the L-band receiver with a
system temperature of $\sim$ 18 K, operating at a frequency range between 856
MHz and 1712 MHz. The dual-polarization and channelized signal from the
beamformer were processed in the Pulsar Timing User Supplied Equipment (PTUSE)
using the dspsr111http://dspsr.sourceforge.net/ software library (van Straten
& Bailes 2011). dspsr provides two signal processing pipelines and produces
fold-mode and search-mode data products. Pulsar timing applications use the
fold-mode pipeline to produce frequency- and phase-resolved averages of the
polarised flux for the target pulsar. Search-mode applications (for single
pulses) on the other hand, use digifits to produce high time resolution
spectra (filterbanks) data products. A detailed description of the MeerKAT
pulsar timing infrastructure including polarisation calibration is provided in
Bailes et al. 2020. For the analysis presented here however, we only used the
total intensity profiles.
The coherently dedispersed folded archives and search-mode data products
produced by PTUSE are automatically ingested by the MeerKAT kat-archive and
transferred to the OzStar HPC facility at Swinburne University of Technology
through an authenticated download for post-processing. PTUSE produces folded
archives of 8 second integrations with 4 Stokes parameters, 1024 frequency
channels across the observing bandwidth with 1024 phase bins. Although the
receiver has a bandwidth of 856 MHz, a portion of the bandwidth is ignored due
to bandpass filter roll-off which reduces the usable bandwidth to 775.75 MHz
and thus the number of frequency channels to 928. These archives are further
processed by the fold-mode processing pipeline
meerpipe222https://bitbucket.org/meertime/meerpipe/src/master/, which
generates RFI-excised full-frequency and time resolution psrfits based
archives (van Straten et al. 2010) along with decimated products with user-
specified frequency and time resolution and associated ToAs.
The fragmented 8 second folded archives are integrated using psrchive tool
psradd and aligned using an up-to-date pulsar ephemeris. The ephemerides are
automatically checked using a series of pre-defined standards which is then
followed by a manual vetting process. These are automatically version
controlled and used by PTUSE , ensuring accountability. The noise-free
templates used for timing are generated either manually (using the psrchive
tool paas) or automatically wavelet-smoothed using the psrchive tool
psrsmooth. Frequency-dependent template profiles (or portraits) are generated
for pulsars which show significant profile evolution across frequency using
PulsePortraiture333https://github.com/pennucci/PulsePortraiture (Pennucci
2019).
To study single pulses, the filterbanks are recorded at a time-resolution of
9$\mu s$ across 768 frequency channels and are stored in psrfits format. For
the analysis presented here, the post-processing of single pulses is done
using dspsr.
RFI excision is implemented using a modified version of coastguard (Lazarus et
al. 2016) which uses a template profile to identify the phase-bins containing
the pulsar signal and computes profile residuals by subtracting the observed
profile from the template. The folded data cube with full-frequency and time
resolution after integrating the four Stokes parameters is used to excise RFI.
Using metrics as described in Lazarus et al. (2016), RFI mitigation is
performed on the Fourier transform of the profile residuals. For the analysis
presented here, only frequency integrated template profiles are used for RFI
excision. Furthermore, manual inspection of the output archives is performed
to ensure that no residual RFI is present in the data.
The reference signal produced by the incoherent sum of the noise diode signals
in the MeerKAT array cannot be used to calibrate the absolute gain,
differential gain and differential phase of the system as it deviates from
100% linear polarization (Bailes et al. 2020). However, the differential gain
is calibrated to within 1% during the procedure that is used to phase up the
tied array, leaving only the absolute gain and differential phase to be
calibrated. The absolute gain can be calibrated using a separate set of flux
calibration observations of a radio source known to have constant flux
density, such as PKS B1934$-$638 and in a subset of MeerTime observations, the
differential phase also happens to be very close to zero. Therefore, for these
observations, it is sufficient to perform only the feed hand (basis) and
parallactic angle (projection) corrections which are implemented in the
processing pipeline using pac.
The cleaned and vetted archives are then decimated into various data products
containing different number of frequency channels and sub-integrations. The
ToAs are then calculated by cross correlating these sub-banded observations
with a frequency integrated template profile in the Fourier domain. The formal
uncertainties produced by this method assumes that the only source of noise in
the measurement is white radiometer noise and thus, underestimates the true
ToA uncertainty. The meta-data associated with each ToA follows the IPTA
convention (Verbiest et al. 2016). Wideband ToAs that account for frequency-
dependent profile evolution are also produced (Pennucci et al. 2014). For the
results presented here, we used profiles that are frequency-averaged to 32
channels with 8 second subintegrations.
## 3 Jitter measurements
In this section we first describe the methodology used to estimate jitter
using frequency averaged ToAs followed by the use of wideband templates to
examine the frequency dependence of jitter.
### 3.1 Methodology
The sub-banded ToAs produced by meerpipe are fitted to curated pulsar
ephemerides to obtain sub-banded timing residuals. Since the observations
typically have a 5 minute duration, only the spin-frequency ($\nu$) and DM are
fitted for and only ToAs derived from 8 s profiles with S/N $>$ 10 are
retained in the analysis444We have cross-checked these measurements with a
wideband template as well and did not find any discrepancies.. To compute the
rms uncertainty of jitter in $T_{\rm{sub}}$($=8$s), we frequency average the
sub-banded timing residuals and compute the quadrature difference of the rms
of the frequency-averaged residuals and the rms expected from ideal simulated
data sets as expressed in equation,
$\sigma_{\rm
J}^{2}\left(T_{\rm{sub}}\right)=\sigma_{\mathrm{obs}}^{2}\left(T_{\rm{sub}}\right)-\sigma_{\mathrm{sim}}^{2}\left(T_{\rm{sub}}\right).$
(2)
The uncertainties from the observed, frequency-averaged ToAs are induced in an
idealised ToA data set to generate the simulated data. Using the tempo2 fake
tool (Hobbs et al. 2006), we simulate $\sim$ 1000 realisations of the data set
for each pulsar (for every epoch) to obtain the variance of
$\sigma_{\mathrm{sim}}^{2}\left(T_{\rm{sub}}\right)$. We assume that all
excess noise observed in the arrival time measurements are caused due to
jitter since other effects that lead to such short-timescale perturbations
vary more strongly with observing frequency and cause perturbations in the
ToAs on longer ($\sim$ hours) timescales than what is reported here (Shannon &
Cordes 2012). Distortions in the pulse profile caused due to imperfect
polarization calibration typically tend to vary with the parallactic angle of
the receiver and are caused on much longer timescales than a few minutes. The
methodology presented here is similar to that discussed in Shannon et al.
(2014), except that we use simulated ToAs rather than simulated profiles.
Since jitter is expected to scale proportionally to $1/\sqrt{N_{\rm p}}$,
where $N_{\rm p}$ is the number of pulses, we can estimate the implied jitter
in one hour to be,
$\sigma_{\rm J}\left(\rm{1hour}\right)=\sigma_{\rm
J}\left(T_{\rm{sub}}\right)/\sqrt{3600/\left(T_{\rm{sub}}\right)}.$ (3)
We also use the Bayesian pulsar timing package, temponest (Lentati et al.
2014) as a consistency check for our jitter measurements. To estimate jitter
using temponest, we determine the standard white noise parameter, EQUAD, used
commonly in pulsar timing analyses555Since we use frequency-averaged ToAs,
EQUAD and ECORR result in similar estimates of jitter noise. The ECORR
parameter models short-timescale noise processes that result in correlated
sub-banded TOAs within an epoch/observation, but which are otherwise
uncorrelated between epochs/observations. NANOGrav Collaboration et al.
(2015). . EQUAD represents a source of time-independent noise which could
arise from stochastic shape variations in the integrated pulse profile.
We claim to have detected jitter in a pulsar if
$\sigma_{\mathrm{obs}}^{2}\left(T_{\rm{sub}}\right)$ is greater than 95% of
the simulated $\sigma_{\mathrm{sim}}^{2}\left(T_{\rm{sub}}\right)$ values,
which suggests that the rms of the frequency-averaged residuals is higher due
to excess scatter in the observed ToAs than that caused due to radiometer
noise.
### 3.2 Jitter measurements from frequency-averaged ToAs
Table 1 reports the jitter measurements for 29 MSPs in our sample using
frequency averaged ToAs. We constrain jitter in 13 pulsars and report upper
limits for the remaining. Pulsars with either new jitter measurements or new
upper limits are highlighted in the table. PSR J2241–5236 has the lowest level
of jitter hitherto reported. Our measurements are consistent within
uncertainties to previously published values which are also reported in the
table.
In Figure 1, we show a representative sample of frequency averaged timing
residuals of eight pulsars with different levels of jitter. To highlight the
markedly different strengths of jitter noise across the population, the timing
residuals are all plotted on the same y-scale.
Table 1: Jitter measurements and upper limits for 29 MSPs in our sample. For each pulsar, the parameters reported here are corresponding to the brightest observation, i.e, with the highest average S/N per pulse. For reference, the median S/N per pulse computed from all selected observations per pulsar is also reported. Columns two and three report the period and the DM while columns six to eight report the integration time, the mean ToA error and the weighted RMS values in 8 s sub-integrations. The last two columns report the implied jitter in one hour and a reference to a previously published jitter measurement where available. S+2014 refers to Shannon et al. (2014) and L+2019 refers to Lam et al. (2019) (values are scaled from single-pulse rms values). Pulsar names in bold represent either a new detection or a new upper limit. PSR | P | DM | S/Npulse,max | S/Npulse,med | $\rm{T}_{\rm obs}$ | $\sigma_{\rm ToA}$ | WRMS | Implied $\sigma_{\rm J}(\rm{hr})$ | Previous $\sigma_{\rm J}(\rm{hr})$
---|---|---|---|---|---|---|---|---|---
| (ms) | (pc cm-3) | | | (s) | ($\mu$s) | ($\mu$s) | (ns) | (ns)
J0030+0451 | 4.9 | 4.4 | 1.9 | 1.4 | 370 | 0.298 | 1.765 | $<$60 | 60$\pm$5 (L+2019)
J0125–2327 | 3.7 | 9.6 | 3.6 | 3.6 | 256 | 0.183 | 1.108 | 48$\pm$13 | -
J0437–4715 | 5.8 | 2.6 | 139 | 112 | 180 | 0.019 | 0.868 | 50$\pm$10 | 48.0$\pm$0.6 (S+2014)
J0636–3044 | 3.9 | 16.0 | 1.6 | 1.1 | 1170 | 1.285 | 3.555 | 100$\pm$30 | -
J0711–6830 | 5.5 | 18.4 | 2.1 | 1.6 | 256 | 0.619 | 1.284 | 60$\pm$20 | $<90$ (S+2014)
J0900–3144 | 11.1 | 75.7 | 2.9 | 2.5 | 256 | 0.554 | 3.687 | $<$130 | -
J1017–7156 | 2.3 | 94.2 | 2.4 | 2.2 | 256 | 0.149 | 0.259 | $<$10 | $<100$ (S+2014)
J1022+1001 | 16.5 | 10.3 | 23.3 | 5.6 | 256 | 0.167 | 2.054 | 120$\pm$20 | 280$\pm$140 (S+2014), 265$\pm$20 (L+2019)
J1024–0719 | 5.2 | 6.5 | 2.1 | 1.8 | 256 | 0.204 | 0.886 | $<$30 | 18$\pm$10 (L+2019)
J1045–4509 | 7.5 | 58.1 | 2.9 | 2.0 | 256 | 0.555 | 3.368 | 130$\pm$75 | $<900$ (S+2014)
J1157–5112 | 43.6 | 39.7 | 1.7 | 1.5 | 3064 | 3.675 | 32.848 | $<$690 | -
J1600–3053 | 3.6 | 52.3 | 1.9 | 1.4 | 256 | 0.118 | 0.711 | $<$30 | $<200$ (S+2014)
J1603–7202 | 14.8 | 38.0 | 8.1 | 3.8 | 512 | 0.297 | 3.947 | 180$\pm$40 | 300$\pm$56 (S+2014)
J1622–6617 | 23.6 | 87.9 | 1.6 | 1.5 | 256 | 2.158 | 26.291 | $<$300 | -
J1629–6902 | 6.0 | 29.5 | 1.5 | 1.4 | 256 | 0.331 | 1.917 | $<$60 | -
J1643–1224 | 4.6 | 62.4 | 2.4 | 2.0 | 256 | 0.303 | 1.647 | $<$60 | $<500$ (S+2014), 31$\pm$12 (L+2019)
J1730–2304 | 8.1 | 9.6 | 3.0 | 1.7 | 256 | 0.451 | 1.973 | 80$\pm$45 | $<400$ (S+2014)
J1744–1134 | 4.1 | 3.1 | 6.6 | 2.6 | 512 | 0.026 | 0.686 | 30$\pm$6 | 37.8$\pm$0.8 (+2014), 44$\pm$1 (L+2019)
J1756–2251 | 28.5 | 121.2 | 2.1 | 1.7 | 1345 | 0.926 | 21.176 | $<$500 | -
J1757–5322 | 8.9 | 30.8 | 1.9 | 1.4 | 1242 | 0.704 | 4.196 | 130$\pm$45 | -
J1802–2124 | 12.6 | 149.6 | 1.3 | 1.4 | 256 | 0.545 | 8.485 | $<$80 | -
J1909–3744 | 2.9 | 10.4 | 9.6 | 3.2 | 256 | 0.021 | 0.199 | 9$\pm$3 | 8.6$\pm$0.8 (S+2014), 14$\pm$0.5 (L+2019)
J1918–0642 | 7.6 | 26.5 | 2.1 | 2.1 | 512 | 0.681 | 1.645 | $<$55 | -
J1946–5403 | 2.7 | 23.7 | 1.2 | 1.2 | 256 | 0.090 | 0.359 | $<$9 | -
J2010–1323 | 5.2 | 22.2 | 1.5 | 1.5 | 256 | 0.224 | 1.048 | $<$80 | 59$\pm$3 (L+2019)
J2039–3616 | 3.3 | 24.0 | 1.1 | 1.1 | 512 | 0.166 | 0.937 | $<$25 | -
J2129–5721 | 3.7 | 31.9 | 3.3 | 3.3 | 360 | 0.285 | 0.493 | $<$11 | $<400$ (S+2014)
J2145–0750 | 16.1 | 9.0 | 11.1 | 3.6 | 256 | 0.331 | 3.883 | 200$\pm$20 | 192$\pm$6 (S+2014), 173$\pm$4 (L+2019)
J2241–5236 | 2.2 | 11.4 | 10.3 | 2.3 | 512 | 0.011 | 0.086 | 3.8$\pm$0.8 | $<50$ (S+2014)
Figure 1: Frequency averaged timing residuals for eight pulsars from our
sample showing varying levels of jitter. All y-axes are plotted on the same
scale for ease of comparison.
#### 3.2.1 Contribution of scattering noise to the jitter measurements
The other short term noise source that can contribute to excess white noise is
related to propagation in the interstellar medium. Stochasticity in the pulse-
broadening function, referred to as the finite-scintle effect Cordes et al.
(1990); Cordes & Shannon (2010); Lam et al. (2016) will cause arrival time
variations that can be significant for highly scattered pulsars. The strength
of this effect depends on the pulse broadening time $\tau$, and the number of
scintles in the observation $N_{s}$, $\sigma_{\rm FS}=\tau/\sqrt{N_{s}}$. The
numbers of scintles is $N_{s}=(1+\eta\Delta\nu/\nu_{d})(1+\eta\Delta
T/t_{d})$, where $\Delta T$ and $\Delta\nu$ are the observing time and
bandwidth while $t_{d}$ and $\nu_{d}$ are the diffractive scintillation time
and bandwidth, and $\eta\approx 0.3$ is the scintillation filling factor
Cordes & Shannon (2010). Of the pulsars for which we have detected excess
white noise, only one, PSR J1045$-$4509 has a dispersion measure above $50$ pc
cm-3. Based upon its diffractive scintillation bandwidth and time scale we
expect that the contribution of the finite-scintle effect to be $40$ ns, which
is lower than the measured level of jitter noise. The high dispersion measure
pulsar PSR J1017$-$7156 has a tight constraint on excess noise $\sigma_{\rm
J}<10$ ns. This pulsar however has an under-turbulent line of sight with a
scintillation bandwidth of $\Delta\nu\approx 2$ MHz and time scale of $10$ min
at 1.4 GHz (Coles et al. 2015). As a result, the strength of the effect is
estimated to be $\sim$ 5 ns in a 1 h observations, consistent with our jitter
limit. We defer further analysis of this effect to future work, noting that
studies of the effect would be particularly amenable with the MeerKAT UHF
system.
### 3.3 Frequency dependence of jitter
By utilizing the large fractional bandwidth of MeerKAT, we are able to study
the frequency dependence of jitter in our sample of MSPs. Based on the S/N of
the observation we generate ToAs per sub-band to compute $\sigma_{\rm J}$ as a
function of frequency channel. We do not detect significant frequency
dependence of jitter in any other pulsars except in PSR J0437–4715. Higher S/N
observations of these pulsars, especially during scintillation maxima might
enable such studies.
For PSR J0437–4715, we find that jitter decreases with increasing observing
frequency. In the lower part of the band at $\sim$ 910 MHz, we measure
$\sigma_{\rm J}$ to be 63$\pm$25 ns, at $\sim$ 1300 MHz, we measure
$\sigma_{\rm J}$ to be 50$\pm$15 ns, while at $\sim$ 1660 MHz, we measure
$\sigma_{\rm J}$ to be only 24$\pm$20 ns. This is consistent with Shannon et
al. (2014), who reported jitter to be modestly greater at lower frequencies.
However, it is important to note that PSR J0437–4715 shows significant
frequency-dependent profile evolution which could likely bias the
uncertainties on the jitter measurements at lower and higher bands due to
template fitting errors. To account for these limitations, we generate a
wideband template that models the frequency evolution of the profile.
Using PulsePortraiture and the methodology described in Pennucci (2019), we
create a frequency-dependent smoothed template for PSR J0437–4715. Using this,
we compute sub-banded ToAs with uncertainties that account for the frequency-
dependent profile evolution. Figure 2 shows the frequency dependence of jitter
using frequency-averaged and wideband templates. Using these ToAs, at $\sim$
910 MHz, we measure $\sigma_{\rm J}$ to be 64$\pm$20 ns, at $\sim$ 1300 MHz,
we measure $\sigma_{\rm J}$ to be 50$\pm$13 ns, while at $\sim$ 1660 MHz, we
measure $\sigma_{\rm J}$ to be 42$\pm$12 ns. The deviations in the
measurements of $\sigma_{\rm J}$ using frequency-averaged and sub-banded ToAs
are consistent within uncertainties. It must be noted that as the template
deviates from an accurate description of the profile, we get an artificial
lower value of jitter. The varying levels of jitter in higher and lower
frequency bands in PSR J0437–4715 can most likely be attributed to the
narrowing of the pulse profile at higher frequencies.
Figure 2: Jitter as a function of observing frequency for PSR J0437–4715,
using ToAs generated from frequency-averaged (blue) and wideband templates
(purple). The frequency-averaged points are offset by 5 MHz for clarity. The
observing bandwidth is averaged to 32 frequency channels. There is a moderate
dependence of jitter on observing frequency.
Owing to our ability to measure jitter in individual sub-bands and also due to
the high flux density of the pulsar, we can estimate the degree of correlation
of jitter between the bands. We see a high degree of correlation ($\sim$0.9)
between adjacent frequency bands, while the degree of correlation
significantly reduces ($\sim$ 0.4) between the lowest and highest frequency
bands. Assuming a reference frequency of $\sim 910$ MHz, we compute the
correlation strength ($r_{\rm j}$) as a function of increasing channel
separation as shown in Figure 3. The total observing bandwidth is divided into
32 channels and for each frequency channel we compute a mean Spearman
correlation coefficient by bootstrapping the corresponding 8 s sub-
integrations. It is clear that with increasing channel separation, the ToAs
begin to decorrelate, implying that the bandwidth of the process that causes
jitter is comparable to the bandwidth of our observations.
Figure 3: ToA correlation as a function of observing frequency (in MHz) for
PSR J0437–4715 showing the increasing decorrelation of jitter with increasing
channel separation. The lowest frequency channel at a frequency of $\sim$ 900
MHz is considered as the reference frequency.
## 4 Limits on DM measurements in PSR J0437–4715
The emergence of decorrelation of jitter with observing frequency in PSR
J0437–4715, implies that at the lower and higher frequency bands, the emission
statistics are increasingly independent of each other. In the top panel of
Figure 4, we show the post-fit timing residuals (generated using a wideband
template) from each frequency channel plotted serially in time across a 256
second observation. Each cluster of ToAs is 8 s long with the color
representative of the frequency band of observation. A striking feature is the
varying frequency dependence of the arrival times on a timescale of $\sim$ 8
seconds which can naively be interpreted as a change in the DM. It must be
noted that if a wideband template is not used to compute the ToAs, the
dominant frequency-dependent term in the timing residuals is caused by profile
evolution. An implication of this time-varying frequency dependence is that
the measured values of DM appear to vary on such short timescales. Measuring
the DM independently from each 8 s cluster of ToAs, we obtain a median value
of 2.6419 cm-3 pc with a standard deviation of 2.7$\times$10-4 cm-3 pc as
shown in the left panel of Figure 4. We also investigate this effect by
analysing the single pulses from this pulsar. For each pulse with full
frequency resolution, we compute the arrival times using a wideband template
and estimate the DM. The right panel of Figure 4 shows the distribution of
estimated DMs with a median value of 2.643 cm-3 pc and a standard deviation of
8.5$\times$10-3 cm-3 pc, which is consistent with what is expected from
extrapolating the 8-s subintegrations.
Figure 4: Top panel: Post-fit wideband timing residuals of PSR J0437–4715
estimated using 8 s integrated profiles containing 32 frequency channels over
a 256 s observation. Each set of ToAs are colored based on the frequency
channel as indicated in the plot. The timing residuals are plotted serially
(as ToA numbers) to showcase the varying frequency dependence for each ToA
set. Left panel: Estimated values of DM for every 8 s subintegration from fits
to the timing residuals shown in the top panel. The horizontal dashed line
represents the median estimated DM value of 2.6419 cm-3 pc. Right panel:
Distributions of measured DM values from $\sim$ 47000 single pulses for PSR
J0437–4715.
The left panel of Figure 5 shows a single 8 s integration of wideband timing
residuals selected from Figure 4 but after fitting for DM, and the right panel
shows the corresponding profile residuals. The clear presence of structures in
the post-fit timing residuals suggests that there may be other (possibly
intrinsic) processes that produce a spectral dependence on the ToAs. It must
be noted that the fit for DM also absorbs the $1/\nu^{2}$ contributions from
such processes, thereby introducing a large scatter in the DM estimates.
Analysing consecutive single pulses, we find that the characteristics of the
spectral structures changes from pulse to pulse, potentially causing the
varying frequency dependence as shown in Figure 4. This is likely to be
“spectral jitter", a phenomena where the amount of jitter varies
stochastically across the observing band.
To understand why such a spectral structure arises, we analysed phase-resolved
modulation index of the pulsar using its single pulses. We find that the
modulation index of the main component (C1; see Figure 6) is much higher than
the other (wings) parts of the profile. The shape of the 8-second integrations
is thus dependent on the instantaneous modulation of each profile components,
which causes the profile shape to significantly deviate from the average
profile.
An alternative, more speculative perspective is that the observed apparent DM
variations could arise due to changes in the plasma density inside or nearby
the pulsar magnetosphere. We note that the magnitude of the observed DM
variations, 8.5$\times$10-3 cm-3 pc, would in principle allow for such a
possibility. Previous work, for instance, by Wu & Chian (1995) or Luo (1998),
have proposed that DM variations arising from non-linearities in the pulsar
magnetosphere may lead to a strong dependence between radio luminosity and DM
fluctuations. Since we do not find any correlation between single-pulse DM
estimates and S/N and that the observed arrival time variations do not show
$\nu^{-2}$ scaling, the argument for a dispersive origin for the frequency
dependent variations is not strongly supported.
In summary, it is evident that in PSR J0437–4715, spectral jitter places a
fundamental limit on the precision of DM estimates on short timescales and it
is likely that this phenomena can be observed in other bright nearby pulsars.
A similar effect has been observed in PSR J1713$+$0747\. In the highest S/N
observation obtained with the Arecibo telescope in a band spanning 1.1-1.7 GHz
(Lam et al. 2016), sub-banded arrival times showed temporally uncorrelated,
frequency-dependent linear variations in arrival times in both $10$-s and
$80$-s sub-integrations. In $80$ sub-integrations, the arrival time variations
had a typical value of $\sim 0.3~{}\mu$s MHz-1, which extrapolates to
$0.9\,\mu$s MHz-1 for $8$ s subintegrations. In J0437$-$4715 $8$ s
observations we observe a typical drift666The drifts in the MeerKAT
observations of PSR J0437$-$4715 often depart markedly from a linear trends.
of $\approx~{}1\,\mu$s GHz-1. The similarity in the values can be explained by
the pulsars having comparable levels of jitter noise and pulse profile
evolution.
Figure 5: Left panel: Sub-banded timing residuals of a single 8 s integrated
profile of PSR J0437-4715 plotted serially in time, after fitting for DM.
Right panel: Pulse profile residuals as a function of observing frequency
computed by subtracting an average pulse frequency evolution model from the
same 8 s integrated profile.
## 5 Single-pulse phenomenology
The phenomenon of jitter can be better understood by examining shape
variations of single pulses. Single pulses from MSPs have been studied in
relatively few cases compared to more slower, normal pulsars, owing to their
low S/N per pulse. Out of the 29 pulsars listed in Table 1, we present a
statistical analysis of single pulses for eight of them with detected jitter
measurements. Some pulsars with inferred jitter measurements (from Section 3)
did not have associated search mode observations to enable single pulse
analysis and others with upper limits on jitter had very low S/N single
pulses.
For each pulsar, a set of statistical properties, derived from frequency-
averaged pulse profiles were determined to study and compare their emission
properties. These are described below:
Characterising variations in single pulse morphology: Examining the brightest
pulses and their phase-resolved modulation can provide insights into the state
of plasma emission. We examine a number of statistical properties to
characterise single pulse amplitude and shape variability.
We compute the phase-resolved modulation index as,
$m_{I}(\phi)=\frac{\sqrt{\sigma_{I}^{2}(\phi)-\sigma_{\mathrm{off}}^{2}}}{I(\phi)},$
(4)
where $\sigma_{I}(\phi)$ and $I(\phi)$ are the rms and the mean intensity
computed at phase $\phi$ and $\sigma_{\mathrm{off}}$ is the rms intensity
computed from an off-pulse window.
We also investigate temporal correlations in intensities of single pulse
emission, in particular to check whether emission shows the pulse-pulse
correlation observed in many young pulsars (e.g., Backer, 1970a). We
implemented this through analysis of the time series of maximum intensity of
the individual pulsars. In particular we calculated the auto-correlation
function (ACF) of the time series.
Pulse energy distributions: Analysis of single pulse energy distributions
provide a measurement of energy contained in a pulse or sub-component(s) of a
pulse and the type of distribution allows us to probe the pulse emission
mechanism. We measure the integrated S/N by defining windows around the main
component and various sub-components of the main pulse and interpulse
depending on the profile morphology. In cases where multiple sub-components
were present, the size of the windows were chosen to be similar (wherever
reasonable) to enable a more direct statistical comparison. The instantaneous
S/N over a selected window is computed as,
$S/N=\frac{\sum_{i=1}^{N_{\text{window
}}}\left(A_{i}-B\right)}{\sqrt{N_{\text{window }}}\sigma_{\text{off }}},$ (5)
where $A_{i}$ is the pulse flux density at the $i$-th bin, $B$ is the mean
off-pulse flux density, $N_{\rm window}$ is the number of phase bins in the
selected component window and $\sigma_{\rm off}$ is the off-pulse rms flux
density. RFI excision on single pulses is implemented using standard psrchive
tools and coastguard, whenever necessary. Residual RFI was manually inspected
and removed from the analysis. A least-squares minimisation of the data,
fitted by a model for the pulse energy distribution was performed using log-
normal and Gaussian distributions. The log-normal model was defined as,
$P(S)=\frac{1}{S\sigma_{\ell}\sqrt{2\pi}}\exp\left[\frac{-\left(\log_{10}(S)-\mu_{\ell}\right)^{2}}{2\sigma_{\ell}^{2}}\right],$
(6)
where $S$ is the S/N , and $\mu_{\ell}$ and $\sigma_{\ell}$ parameterise the
distribution. For the Gaussian model, we fit for the mean ($\mu_{g}$) and
standard deviation ($\sigma_{g}$) of the pulse energy distribution. A
$\chi^{2}$ statistic was used for quantifying the goodness of fit of the
preferred distribution.
Timing properties of single pulses: By computing frequency-averaged single
pulse ToAs and investigating correlations between single pulse properties and
corresponding ToAs, we study the timing properties of single pulses. We also
measure jitter as a function of the number of pulses integrated ($N_{\rm p}$)
and show that the arrival time uncertainties obtained from averaged profiles
due to template fitting are consistent with the measured pulse-to-pulse
variations and that jitter typically scales as $1/\sqrt{N_{\rm p}}$.
Figure 6: Single pulse S/N histograms for six MSPs. The various histograms
shown for each pulsar correspond to selected windows across the pulse profile.
The windows used for each pulsar are shown in the respective sub-plot
containing the integrated profile (solid line) and the mean profile which is
derived from averaging the brightest 100 or 1000 single pulses (dashed line).
The bottom axis for these sub-plots represent the number of phase bins used
while the top axis shows the corresponding phase in turns. The S/N across a
selected window is computed using Equation 5. Figure 7: Phase-resolved
modulation index for PSRs J1022+1001 and J2145–0750 shown across the on-pulse
region in the upper panel. The middle panel shows the integrated profile from
15000 single pulses and the lower panel shows the mean profile from averaging
the brightest 100 pulses. The vertical grey dashed line represents the phase
of maximum S/N for the profile shown in the lower panel.
### 5.1 PSR J0437–4715
The single pulses of PSR J0437–4715 have been studied extensively and owing to
its high flux density, pulse shape variations cause excess timing uncertainty,
at least four times greater than that predicted from radiometer noise alone
(Osłowski et al. 2014). We analysed $\sim$ 47000 pulses and detected every
single pulse (the lowest S/N was $\sim$ 20), indicating that the pulse
emission in this pulsar likely does not exhibit nulling. Its profile has
multiple components that span the majority of the pulse phase. The S/N
distribution was analysed by selecting different windows spanning the pulse
profile as shown in Figure 6. The profile formed by averaging the brightest
100 pulses is shown in the subplot along with the integrated profile. The S/N
histograms for the C1, C2 and the full profile windows show log-normal
distributions. It can also be seen that the brightest pulses coincide in phase
with the leading edge of the integrated profile and that the brightest pulses
are narrower than the rest and do not exhibit strong emission in the wings of
the profile. The phase-resolved modulation index is highest towards the
leading edge of the main component, consistent with previous results (Osłowski
et al. 2014)777The off-pulse variance was not subtracted to unbias the
measured modulation index in Osłowski et al. (2014). We computed the ACF from
the peak flux intensities of single pulses and find no evidence of temporal
correlations amongst the pulses. The level of jitter noise estimated by
integrating increasing number of pulses from 10 to $\sim$10000 shows that
jitter scales proportionally with 1$/\sqrt{\rm N_{\rm p}}$ and is consistent
within uncertainties over multiple epochs. Our statistical analysis of single
pulses from PSR J0437–4715 show that they are consistent with previously
published analyses (Jenet et al. 2001, Osłowski et al. 2014, and Shannon et
al. 2014).
### 5.2 PSR J0125–2327
PSR J0125–2327 is a binary pulsar with a spin-period of $\sim$ 3.6 ms,
discovered in the Green Bank North Celestial Cap (GBNCC) survey (McEwen et al.
2020). With MeerKAT, the observed median S/N per pulse was $\sim$4 with an
estimated $\sigma_{\rm J}$ to be 48$\pm$13 ns in an hour. Statistical
properties of single pulses from PSR J0125–2327 have not been previously
reported. A total of $\sim$ 15000 pulses were analysed and the maximum
observed S/N was $\sim$ 15 per pulse. In the observed band, the pulse profile
has multiple components with a main strong component towards the leading edge
of the profile and multiple weaker components towards the trailing edge. The
distribution of S/N computed from single pulses follows a log-normal
distribution as shown in Figure 6 for two selected windows over the full on-
pulse region and the peak component (C1). Unlike PSR J0437$-$4715, bright
pulses from this pulsar span the entire on-pulse phase region.
### 5.3 PSR J1022+1001
PSR J1022+1001 is a relatively bright MSP with a rotation period of $\sim$ 16
ms. Owing to its rotational stability and low mean arrival time errors, it is
a part of the Parkes Pulsar Timing Array program (PPTA) (Manchester et al.
2013). Previous measurements of pulse shape variations have shown that there
is excess scatter in ToAs, larger than that expected from only radiometer
noise (Liu et al. 2015). The pulse profile at 20 cm wavelengths has a double
peaked structure and each component has a different spectral index, resulting
in strong evolution of the pulse profile across the observing band. We
analysed $\sim$ 20000 pulses and found the maximum S/N to be $\sim$ 30\. In
the left panel of Figure 7, we show the phase-resolved modulation index
computed across the on-pulse region. The modulation index is a measure of
intensity variations from pulse to pulse. In this pulsar, the modulation index
grows increasingly stronger across the pulse profile and becomes strongest
towards the trailing edge of the second peak component suggesting high levels
of amplitude modulation in the pulses originating from these phase ranges. The
modulation index of the first component is $\sim$ two times smaller than the
second component. The brightest pulses are dominated by emission from the
second peak component and coincide in phase with the trailing edge of the
second component as shown by the vertical dotted lines. From the ACF of pulse
peak flux intensities, we find no evidence of temporal correlations amongst
pulses. The single pulse S/N distributions of the pulse profile and its peak
components are shown in Figure 6 and follow a log-normal distribution. While
the integrated pulse profile formed from 100 brightest pulses shows evidence
of both the components, the second component (C2) is much more prominent than
the first (C1) which is also supported by the S/N distributions which show
that the brightest pulses originate from C2. We estimate $\sigma_{\rm J}$ to
be 130$\pm$20 ns in an hour for PSR J1022+1001 and find that our measurements
of jitter scale as $1/\sqrt{N_{\rm p}}$ which are consistent over multiple
epochs.
### 5.4 PSR J1603–7202
PSR J1603–7202 has a spin period of $\sim$ 15 ms and at 20 cm wavelengths has
two sub-components connected by a more dominant bridge of emission. We
analysed $\sim$ 20000 pulses and found the maximum S/N to be $\sim$ 17\.
Unlike in PSR J1022+1001, the modulation index of this pulsar is strongest
towards the leading edge of first component and is much weaker towards the
second component. The S/N distribution of both the components follow a log-
normal distribution. We estimate $\sigma_{J}$ to be 180$\pm$40 ns in an hour
and find that it scales as $1/\sqrt{N_{\rm p}}$ similar to other pulsars.
### 5.5 PSR J1744–1134
PSR J1744–1134 has a spin period of $\sim$ 4 ms and has a relatively narrow
main pulse profile with an interpulse at 20 cm wavelengths. We analysed $\sim$
35000 single pulses and found the maximum S/N to be $\sim$ 12\. Unlike pulsars
discussed so far, the S/N distributions for both the peak pulse component and
the interpulse tends towards a Gaussian distribution consistent with low
amplitude modulation as shown in Figure 6. In contrast to PSR J1022+1001,
where the modulation index peaks towards the trailing edge of the second
component, in PSR J1744–1134, it gets weaker towards the peak component
implying low levels of amplitude modulation. We do not detect single pulse
emission from the interpulse (C1). We estimate $\sigma_{\rm J}$ to be 30$\pm$6
ns in an hour and similar to other pulsars it scales as $1/\sqrt{N_{\rm p}}$.
### 5.6 PSR J1909–3744: pulse nulling?
Figure 8: Left panel: Consecutive single pulses of PSR J1909–3744 shown
across the on-pulse region indicating potential pulse nulling behaviour as
depicted by the two horizontal dashed lines. Right top panel: Single pulse S/N
histograms for PSR J1909$-$3744 similar to Figure 6. We note that the window
sizes for the interpulse and the off-pulse are different which leads to a
broader S/N distribution for the interpulse relative to that of the off-pulse.
Right bottom panel: Profile formed from integrating the amplitudes of eight
consecutive pulses over the region highlighted in the left panel is shown in
purple relative to an integrated profile formed from adding eight pulses in
the emission region (black dashed line) for reference.
PSR J1909–3744 is one of the most precisely timed pulsars owing to its
extremely stable rotational behaviour, narrow pulse profile and high flux
density. At 20 cm wavelengths, the pulse profile consists of a narrow main
component with an interpulse. We analysed $\sim$ 35000 pulses and found the
maximum S/N to be $\sim$ 20 but do not detect single pulses from the much
fainter interpulse. In the left panel of Figure 8, we show $\sim$ 200
consecutive pulses in which the pulse emission appears to null occasionally as
highlighted by two horizontal dashed lines at $\sim$ 0.28 seconds. This is the
first reported detection of a nulling phenomenon in an MSP. Integrating the
flux densities over the nulling region reveals no detection of emission as
shown in the bottom right panel. However, it must be noted that analysing
longer observations using high time resolution search-mode data might help
identify any weak emission modes. The S/N distribution of single pulses of the
main component appears to follow a Gaussian distribution similar to PSR
J1744$-$1134 although with a broad spread about the mean value (i.e. a
platykurtic distribution) as shown in the top right panel. It is also
interesting to note that the mean profile formed from integrating the
brightest 100 pulses has relatively more flux density towards its leading edge
as compared to the integrated profile resulting in the brightest pulses (S/N
$>$ 18) having later times of arrival than pulses with average S/N (5 $<$ S/N
$<$ 18) as shown in Figure 9. We estimate the value of $\sigma_{\rm J}$ to be
9$\pm$3 ns in an hour and find that when integrating from 10 to 10000 pulses,
jitter scales as $1/\sqrt{N_{\rm p}}$.
Figure 9: ToAs computed from a randomly chosen subset of $\sim$ 10000 single
pulses plotted against their corresponding S/N for PSRs J1909–3744 (left) and
J2241-5236 (right). The blue dashed horizontal line represents the S/N
threshold for selecting the brightest pulses which are indicated in the top
and middle panels. The vertical dashed line in the middle panel represents the
median of the histogram, while the vertical dotted line across all three
panels is at zero $\mu s$.
### 5.7 PSR J2145–0750
PSR J2145–0750 has a spin period of $\sim$ 15 ms and is a relatively bright
pulsar at 20 cm wavelengths. Its pulse profile is dominated by two main
components connected by a bridge of emission and a precursor. We detect single
pulses from both the main components and none from the precursor. We analysed
$\sim$ 16000 pulses and found the maximum S/N to be $\sim$ 50\. The modulation
index shows complex structures, with high levels of modulation towards the
leading edge of the first component and the trailing edge of the second
component with the bridge also exhibiting a high modulation index as shown in
the right panel of Figure 7. The S/N histograms of the two components (C1 and
C2) and the full on-pulse window show log-normal distributions as shown in
Figure 6. The wide variability in phase and amplitude exhibited by the single
pulses results in large levels of jitter noise. We estimate a $\sigma_{\rm J}$
value of 200$\pm$20 ns in an hour and find that jitter scales as
$1/\sqrt{N_{\rm p}}$.
### 5.8 PSR J2241–5236
PSR J2241–5236 has a spin period of $\sim$ 2.2 ms and is in a 3.5 hour orbit
with a low mass companion (Keith et al. 2011). The higher flux density, low
dispersion measure and rotational stability makes it an excellent candidate
for high precision pulsar timing experiments. Consecutive single pulses from
PSR J2241–5236 exhibit very high levels of stability in phase. Its pulse
profile at the observed band has a narrow main component with an interpulse,
similar to PSR J1909–3744. We analysed $\sim$ 24000 pulses and found the
maximum S/N to be $\sim$ 15\. We did not detect any single pulses from the
interpulse. The S/N histograms of the single pulses follow an approximately
Gaussian distribution as shown in Figure 6. Its modulation index is weakest
towards the peak of the pulse profile, similar to PSR J1909–3744 indicative of
low levels of amplitude modulation. Accordingly, this pulsar exhibits the
lowest levels of jitter hitherto reported of just 3.8$\pm$0.8 ns in an hour.
The brightest pulses (S/N $>$ 7) largely resemble the integrated profile
implying that their ToAs are expected to have similar arrival times as pulses
with average S/N (5 $<$ S/N $<$ 7) as shown in Figure 9.
The S/N of single pulses were not sufficient to estimate the scaling relation
of jitter. The observation that was used to measure jitter from the folded
archives (in Section 3) had an estimated median single pulse S/N of $\sim$
10\. However, there was no associated single pulse data for that particular
observation. In comparison, the median single pulse S/N that are presented
here are only $\sim$ 4\. Analysis of single pulses during scintillation maxima
would prove very interesting for precision timing analysis.
## 6 Discussions and Conclusions
### 6.1 High-precision pulsar timing with MeerKAT
We have presented the first short-term high-precision pulsar timing results
using the MeerKAT radio telescope including a study of single pulse
phenomenology for a selection of MSPs. We found that in the highest S/N
observations, stochastic pulse shape and amplitude variations cause excess
scatter in the ToAs, which are higher than that expected from radiometer noise
alone. Out of the 29 MSPs in our sample, we reported new jitter measurements
for 15 pulsars, out of which six pulsars had constraint measurements while
upper limits were reported for the remaining. In the remaining 14 pulsars, we
found that our measurements are either consistent with previously reported
values or have tighter upper limits. PSR J2241–5236 has the lowest levels of
jitter reported in any pulsar of just $\sim$ 4 ns in an hour followed by PSR
J1909–3744 with a jitter level of $\sim$ 9 ns in the same time. The upper
limit on PSRs J1017–7156, J1946–5403 and J2129–5721 is $<$ 11 ns in an hour,
which promise to be excellent pulsars for high-precision timing experiments.
New detections of jitter in MSPs clearly confirm that jitter is a generic
property in all pulsars and our ability to detect jitter in many more pulsars
will grow with increasing sensitivity of radio telescopes. It also shows that
the levels of jitter vary markedly across the population, even between pulsars
with similar spin periods and pulse profiles.
We find evidence for the frequency dependence of jitter in PSR J0437–4715 and
showed that jitter decreases only moderately with increasing observing
frequency. We also found that jitter decorrelates over the observing bandwidth
of MeerKAT implying that the pulse emission statistics will become
increasingly independent to each other with wider bandwidths. This is
consistent with analysis of the pulsar obtained with the 64-m Parkes radio
telescope, which showed no correlation in jitter in observations obtained at
bands centred at $730$ MHz and $3100$ MHz Shannon et al. (2014). The
decorrelated nature of emission also results in individual pulses having
spectral structures that vary from pulse-to-pulse and owing to the very high
S/N of each pulse across the observing band, we detect this effect in the
pulsar timing residuals as time-varying frequency dependence. Measurements of
DMs on short-timescales will thus have a large scatter, placing a fundamental
limit on the precision with which pulsar DMs can be measured. In addition to
the limitations posed to the measurement of the gravitational wave background
signal, this effect can also lead to limiting the precision of orbital
parameters in highly relativistic systems, especially due to covariances
between the DM and timing model parameters when combining observation from
multiple orbits.
We speculate that with more sensitive radio telescopes with larger bandwidths,
pulse jitter will dominate the error budget.
### 6.2 Pulse shape variability in MSPs
We reported the first single pulse study of eight MSPs using MeerKAT. Single
pulse statistics for PSRs J0125-2327 and J2241–5236 have not been previously
studied. We also reported the first observation of pulse nulling in an MSP,
seen in PSR J1909–3744. Most pulsars in our sample with detected single pulses
showed log-normal distributions consistent with previous results with MSPs
(Shannon et al. 2014) and slow spinning pulsars (Burke-Spolaor et al. 2012).
However, pulsars with the lowest levels of jitter noise, PSRs J2241–5236,
J1909–3744 and J1744–1134 showed approximately Gaussian distributions.
Studying energy distributions enables us to distinguish between different
models of plasma behaviour based upon theoretical predictions. For example,
self-organised criticality (Bak 1996; Bak et al. 1988) describes self-
consistent systems that interact without any preferred distance or timescales.
It predicts a power-law distribution of pulse intensities. However, models
based on stochastic growth theory (Robinson et al. 1992; Robinson 1995)
describe interactions in an independent homogeneous medium with preferred
distances and timescales. This model predicts a log-normal distribution of
pulse intensities. Cairns et al. (2004) found Gaussian energy distributions at
the edges of the pulse profile in slow spinning pulsars. They attribute this
to either irregularities in the density of the plasma or a superposition of
multiple weak emission components. Following from this, it can perhaps be
speculated that for the three pulsars in our sample showing Gaussian energy
distributions, our line of sight traverses the edge of the emission region. It
is also interesting to note that these three pulsars have single component
Gaussian profiles with an interpulse.
In pulsars that show high levels of jitter noise, we found that the phase-
resolved modulation index increased towards the main emission component, such
as in PSRs J0437–4715, J1022+1001, J1603–7202 which have wide profiles with
multiple components; whereas in pulsars with the lowest levels of jitter
noise, the modulation index showed the opposite trend and approached a minima
towards the main emission component, such as in PSRs J1744–1134, J1909–3744
and J2241–5236 which have narrow profiles with single components and a weak
interpulse.
The phase-resolved modulation index can be used to distinguish between pulsar
emission models and previous theoretical works have suggested that the
modulation index depends upon some function of the pulsar period and its
period derivative. It is however, somewhat arbitrary to define a modulation
index from the phase-resolved modulation index profile because it is highly
dependent on the pulse phase. In this case, we chose a value that is
representative of the modulation index near the main emission component of the
integrated pulse profile. We used a Spearman correlation coefficient to
determine the correlations between the modulation index and other pulsar
parameters and found moderate correlations of 0.62$\pm$0.08 and 0.54$\pm$0.03
between the period, period derivative and the modulation index of the pulsar
respectively. These measurements are consistent with previous reports (Jenet &
Gil 2003, 2004; Edwards & Stappers 2003) which may imply that such a
relationship is a consequence of the ‘sparking gap’ model first theorized by
Ruderman & Sutherland (1975).
Future observations of more pulsars will enable us to identify correlations
with the various ‘complexity parameters’ (Gil & Sendyk 2000; Lou 2001) that
aid in distinguishing between different emission models which will prove
powerful in probing deeper into understanding pulsar emission models. We also
measure the correlation of $\sigma_{\rm J}$ (measured in 1 hr) with various
estimates of the pulse widths at 50% ($W_{\rm{50}}$), 10% ($W_{\rm{10}}$) and
the effective pulse width ($W_{\rm{eff}}$). We estimate the effective pulse
width following the definition in Downs & Reichley (1983) and Cordes & Shannon
(2010),
$W_{\mathrm{eff}}=\frac{\Delta\phi}{\sum_{i}\left[P\left(\phi_{i+1}\right)-P\left(\phi_{i}\right)\right]^{2}},$
(7)
where $\Delta\phi$ is the phase resolution of the pulse profile in units of
time and $P$ is the pulse profile which is normalised to a peak intensity of
unity. We find a moderate correlation of 0.64$\pm$0.09 with $W_{\rm 50}$ and
no significant relationship with $W_{\rm eff}$. These results provide
excellent confirmation to those reported in Lam et al. (2019) and suggest that
the level of pulse jitter does not significantly depend on the ‘sharpness’ of
the pulse profile.
### 6.3 Contributing to the PTA network
Although the MeerKAT MSP data set is not yet sensitive to gravitational wave
radiation due to its short timing baselines, there are a number of ways in
which it can contribute to high precision pulsar timing and PTA experiments.
Limitations in precision timing experiments are typically due to an incomplete
understanding of systematics in both the instrument and analysis methodologies
or due to lack of sensitivity. Precise estimates of DM variations are crucial
for PTA experiments, which can be improved by conducting near simultaneous
global observing campaigns of a few MSPs with telescopes like MeerKAT, GBT,
Parkes, Effelsberg and the Five Hundred Meter Aperture Spherical Radio
Telescope. Such campaigns, as was conducted previously on PSR J1713$+$0747 in
2013 Dolch et al. (2014) not only place a strong constraint on DM variations
but also help characterize the instrumental response and post-processing
pipelines. With new upcoming receivers such as the Ultra High Frequency (UHF)
receiver (580 MHz to 1015 MHz) and the S-band receiver (1750 MHz to 3500 MHz)
(Kramer et al. 2016) at MeerKAT, high sensitivity can be achieved owing to
steep spectral indices of pulsars and by dynamically observing pulsars at
their scintillation maxima.
The results presented here are very relevant to current and future radio
telescopes in the context of precision pulsar timing experiments and in
optimizing observing strategies. For pulsars that are limited by jitter noise,
it would be better to sub-array MeerKAT to observe multiple pulsars
simultaneously over longer durations than use the full array to observe
pulsars one by one. The pulsar timing backend at MeerKAT has the potential to
form up to four tied-array beams on the sky leading to a significant
improvement in the efficiency of the timing program. Sub-arrays consisting of
a small number of antennas ($\sim$ 4 to 8) could be deployed to scan the skies
to look for pulsars in a scintillation maxima, which could enhance the
sensitivity of pulsar timing observations and aiding high precision pulsar
timing experiments. In the context of the IPTA, observations could potentially
be optimised to have sensitive telescopes avoid observing pulsars that are
jitter limited and instead observe relatively faint pulsars to improve the
overall network sensitivity Lee et al. (2012). High precision pulsar timing
will greatly benefit from utilizing MeerKAT in conjunction with established
radio telescopes and observing programs from around the globe.
## 7 Acknowledgements
The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory, which is a facility of the National Research Foundation, an
agency of the Department of Science and Innovation. This work made use of the
gSTAR and OzSTAR national HPC facilities. gSTAR is funded by Swinburne and the
Australian Government Education Investment Fund. OzSTAR is funded by Swinburne
and the National Collaborative Research Infrastructure Strategy (NCRIS). This
work is supported through Australian Research Council (ARC) Centre of
Excellence CE170100004. A.P. acknowledges support from CSIRO Astronomy and
Space Science. R.M.S. acknowledges support through ARC grant CE170100004. M.B,
S.O, and R.M.S. acknowledge support through ARC grant FL150100148. R.M.S. also
acknowledges funding support through Australian Research Council Future
Fellowship FT190100155. FA gratefully acknowledge support from ERC Synergy
Grant “BlackHoleCam” Grant Agreement Number 610058. T.T.P. is supported
through a NANOGrav Physics Frontiers Center Postdoctoral Fellowship from the
National Science Foundation Physics Frontiers Center Award Number 1430284.
This work also made use of standard Python packages (Oliphant 2006, Jones et
al. 2001, McKinney 2010, Hunter 2007), Chainconsumer (Hinton 2016) and Bokeh
(Bokeh Development Team 2018).
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
## References
* Aggarwal et al. (2019) Aggarwal K., et al., 2019, ApJ, 880, 116
* Alam et al. (2020a) Alam M. F., et al., 2020a, arXiv e-prints, p. arXiv:2005.06490
* Alam et al. (2020b) Alam M. F., et al., 2020b, arXiv e-prints, p. arXiv:2005.06495
* Antoniadis et al. (2013) Antoniadis J., et al., 2013, Science, 340, 448
* Arzoumanian et al. (2018) Arzoumanian Z., et al., 2018, ApJS, 235, 37
* Backer (1970a) Backer D. C., 1970a, Nature, 227, 692
* Backer (1970b) Backer D. C., 1970b, Nature, 228, 1297
* Backer (1970c) Backer D. C., 1970c, Nature, 228, 42
* Bailes et al. (2020) Bailes M., et al., 2020, arXiv e-prints, p. arXiv:2005.14366
* Bak (1996) Bak P., 1996, How nature works : the science of self-organized criticality
* Bak et al. (1988) Bak P., Tang C., Wiesenfeld K., 1988, Phys. Rev. A, 38, 364
* Bokeh Development Team (2018) Bokeh Development Team 2018, Bokeh: Python library for interactive visualization. https://bokeh.pydata.org/en/latest/
* Boynton et al. (1972) Boynton P. E., Groth E. J., Hutchinson D. P., Nanos G. P. J., Partridge R. B., Wilkinson D. T., 1972, ApJ, 175, 217
* Brook et al. (2016) Brook P. R., Karastergiou A., Johnston S., Kerr M., Shannon R. M., Roberts S. J., 2016, MNRAS, 456, 1374
* Burke-Spolaor et al. (2012) Burke-Spolaor S., et al., 2012, MNRAS, 423, 1351
* Caballero et al. (2018) Caballero R. N., et al., 2018, MNRAS, 481, 5501
* Cairns et al. (2004) Cairns I. H., Johnston S., Das P., 2004, MNRAS, 353, 270
* Camilo et al. (2012) Camilo F., Ransom S. M., Chatterjee S., Johnston S., Demorest P., 2012, ApJ, 746, 63
* Champion et al. (2010) Champion D. J., et al., 2010, ApJ, 720, L201
* Coles et al. (2015) Coles W. A., et al., 2015, ApJ, 808, 113
* Cordes (1980) Cordes J. M., 1980, ApJ, 237, 216
* Cordes & Downs (1985) Cordes J. M., Downs G. S., 1985, ApJS, 59, 343
* Cordes & Shannon (2010) Cordes J. M., Shannon R. M., 2010, arXiv e-prints, p. arXiv:1010.3785
* Cordes et al. (1990) Cordes J. M., Wolszczan A., Dewey R. J., Blaskiewicz M., Stinebring D. R., 1990, ApJ, 349, 245
* Cromartie et al. (2020) Cromartie H. T., et al., 2020, Nature Astronomy, 4, 72
* Demorest et al. (2010) Demorest P. B., Pennucci T., Ransom S. M., Roberts M. S. E., Hessels J. W. T., 2010, Nature, 467, 1081
* Demorest et al. (2013) Demorest P. B., et al., 2013, ApJ, 762, 94
* Dolch et al. (2014) Dolch T., et al., 2014, ApJ, 794, 21
* Downs & Reichley (1983) Downs G. S., Reichley P. E., 1983, ApJS, 53, 169
* Drake & Craft (1968) Drake F. D., Craft H. D., 1968, Nature, 220, 231
* Edwards & Stappers (2003) Edwards R. T., Stappers B. W., 2003, A&A, 407, 273
* Foster & Backer (1990) Foster R. S., Backer D. C., 1990, ApJ, 361, 300
* Gil & Sendyk (2000) Gil J. A., Sendyk M., 2000, ApJ, 541, 351
* Groth (1975) Groth E. J., 1975, ApJS, 29, 453
* Helfand et al. (1975) Helfand D. J., Manchester R. N., Taylor J. H., 1975, ApJ, 198, 661
* Hellings & Downs (1983) Hellings R. W., Downs G. S., 1983, ApJ, 265, L39
* Hinton (2016) Hinton S. R., 2016, The Journal of Open Source Software, 1, 00045
* Hobbs et al. (2006) Hobbs G. B., Edwards R. T., Manchester R. N., 2006, MNRAS, 369, 655
* Hobbs et al. (2010) Hobbs G., et al., 2010, Classical and Quantum Gravity, 27, 084013
* Hobbs et al. (2020) Hobbs G., et al., 2020, MNRAS, 491, 5951
* Hotan et al. (2004) Hotan A. W., Bailes M., Ord S. M., 2004, MNRAS, 355, 941
* Hunter (2007) Hunter J. D., 2007, Computing In Science & Engineering, 9, 90
* Jenet & Gil (2003) Jenet F. A., Gil J., 2003, ApJ, 596, L215
* Jenet & Gil (2004) Jenet F. A., Gil J., 2004, ApJ, 602, L89
* Jenet et al. (1998) Jenet F. A., Anderson S. B., Kaspi V. M., Prince T. A., Unwin S. C., 1998, ApJ, 498, 365
* Jenet et al. (2001) Jenet F. A., Anderson S. B., Prince T. A., 2001, ApJ, 546, 394
* Jones et al. (2001) Jones E., et al., 2001, SciPy: Open source scientific tools for Python
* Keith et al. (2011) Keith M. J., et al., 2011, MNRAS, 414, 1292
* Keith et al. (2013) Keith M. J., et al., 2013, MNRAS, 429, 2161
* Kerr et al. (2015) Kerr M., Johnston S., Hobbs G., Shannon R. M., 2015, ApJ, 809, L11
* Kerr et al. (2020) Kerr M., et al., 2020, arXiv e-prints, p. arXiv:2003.09780
* Kramer (1998) Kramer M., 1998, ApJ, 509, 856
* Kramer et al. (1999) Kramer M., Lange C., Lorimer D. R., Backer D. C., Xilouris K. M., Jessner A., Wielebinski R., 1999, ApJ, 526, 957
* Kramer et al. (2006a) Kramer M., Lyne A. G., O’Brien J. T., Jordan C. A., Lorimer D. R., 2006a, Science, 312, 549
* Kramer et al. (2006b) Kramer M., et al., 2006b, Science, 314, 97
* Kramer et al. (2016) Kramer M., et al., 2016, in Proceedings of MeerKAT Science: On the Pathway to the SKA. 25-27 May. p. 3
* Lam et al. (2016) Lam M. T., et al., 2016, ApJ, 819, 155
* Lam et al. (2017) Lam M. T., et al., 2017, ApJ, 834, 35
* Lam et al. (2019) Lam M. T., et al., 2019, ApJ, 872, 193
* Lazarus et al. (2016) Lazarus P., Karuppusamy R., Graikou E., Caballero R. N., Champion D. J., Lee K. J., Verbiest J. P. W., Kramer M., 2016, MNRAS, 458, 868
* Lee et al. (2012) Lee K. J., Bassa C. G., Janssen G. H., Karuppusamy R., Kramer M., Smits R., Stappers B. W., 2012, MNRAS, 423, 2642
* Lentati et al. (2014) Lentati L., Alexander P., Hobson M. P., Feroz F., van Haasteren R., Lee K. J., Shannon R. M., 2014, MNRAS, 437, 3004
* Lentati et al. (2016) Lentati L., et al., 2016, MNRAS, 458, 2161
* Liu et al. (2012) Liu K., Keane E. F., Lee K. J., Kramer M., Cordes J. M., Purver M. B., 2012, MNRAS, 420, 361
* Liu et al. (2015) Liu K., et al., 2015, MNRAS, 449, 1158
* Lou (2001) Lou Y.-Q., 2001, ApJ, 563, L147
* Luo (1998) Luo Q., 1998, Brazilian Journal of Physics, 28, 191
* Lyne et al. (2010) Lyne A., Hobbs G., Kramer M., Stairs I., Stappers B., 2010, Science, 329, 408
* Lyne et al. (2017) Lyne A. G., et al., 2017, ApJ, 834, 72
* Manchester et al. (2013) Manchester R. N., et al., 2013, Publ. Astron. Soc. Australia, 30, e017
* McEwen et al. (2020) McEwen A. E., et al., 2020, ApJ, 892, 76
* McKinney (2010) McKinney W., 2010, Proceedings of the 9th Python in Science Conference, 51-56
* NANOGrav Collaboration et al. (2015) NANOGrav Collaboration et al., 2015, ApJ, 813, 65
* Oliphant (2006) Oliphant T., 2006, NumPy: A guide to NumPy, USA: Trelgol Publishing
* Osłowski et al. (2011) Osłowski S., van Straten W., Hobbs G. B., Bailes M., Demorest P., 2011, MNRAS, 418, 1258
* Osłowski et al. (2014) Osłowski S., van Straten W., Bailes M., Jameson A., Hobbs G., 2014, MNRAS, 441, 3148
* Padmanabh et al. (2021) Padmanabh P. V., Barr E. D., Champion D. J., Karuppusamy R., Kramer M., Jessner A., Lazarus P., 2021, MNRAS, 500, 1178
* Parthasarathy et al. (2019) Parthasarathy A., et al., 2019, MNRAS, 489, 3810
* Pennucci (2019) Pennucci T. T., 2019, ApJ, 871, 34
* Pennucci et al. (2014) Pennucci T. T., Demorest P. B., Ransom S. M., 2014, ApJ, 790, 93
* Perera et al. (2019) Perera B. B. P., et al., 2019, MNRAS, p. 2468
* Petit & Tavella (1996) Petit G., Tavella P., 1996, A&A, 308, 290
* Rickett (1975) Rickett B. J., 1975, ApJ, 197, 185
* Robinson (1995) Robinson P. A., 1995, Physics of Plasmas, 2, 1466
* Robinson et al. (1992) Robinson P. A., Cairns I. H., Gurnett D. A., 1992, ApJ, 387, L101
* Ruderman & Sutherland (1975) Ruderman M. A., Sutherland P. G., 1975, ApJ, 196, 51
* Shannon & Cordes (2010) Shannon R. M., Cordes J. M., 2010, ApJ, 725, 1607
* Shannon & Cordes (2012) Shannon R. M., Cordes J. M., 2012, ApJ, 761, 64
* Shannon & Cordes (2017) Shannon R. M., Cordes J. M., 2017, MNRAS, 464, 2075
* Shannon et al. (2013) Shannon R. M., et al., 2013, ApJ, 766, 5
* Shannon et al. (2014) Shannon R. M., et al., 2014, MNRAS, 443, 1463
* Shannon et al. (2015) Shannon R. M., et al., 2015, Science, 349, 1522
* Shannon et al. (2016) Shannon R. M., et al., 2016, ApJ, 828, L1
* Taylor (1992) Taylor J. H., 1992, Philosophical Transactions of the Royal Society of London, 341, 117
* Taylor & Weisberg (1982) Taylor J. H., Weisberg J. M., 1982, ApJ, 253, 908
* Taylor et al. (1975) Taylor J. H., Manchester R. N., Huguenin G. R., 1975, ApJ, 195, 513
* Taylor et al. (2017) Taylor S. R., Simon J., Sampson L., 2017, Phys. Rev. Lett., 118, 181102
* Verbiest et al. (2016) Verbiest J. P. W., et al., 2016, MNRAS, 458, 1267
* Voisin et al. (2020) Voisin G., Cognard I., Freire P. C. C., Wex N., Guillemot L., Desvignes G., Kramer M., Theureau G., 2020, A&A, 638, A24
* Weisberg et al. (1989) Weisberg J. M., Romani R. W., Taylor J. H., 1989, ApJ, 347, 1030
* Wolszczan & Frail (1992) Wolszczan A., Frail D. A., 1992, Nature, 355, 145
* Wu & Chian (1995) Wu X., Chian A. C. L., 1995, ApJ, 443, 261
* Zhu et al. (2015) Zhu W. W., et al., 2015, ApJ, 809, 41
* van Straten (2013) van Straten W., 2013, ApJS, 204, 13
* van Straten & Bailes (2011) van Straten W., Bailes M., 2011, Publ. Astron. Soc. Australia, 28, 1
* van Straten et al. (2010) van Straten W., Manchester R. N., Johnston S., Reynolds J. E., 2010, Publ. Astron. Soc. Australia, 27, 104
|
# Eliminate Deviation with Deviation for Data Augmentation and a General
Multi-modal Data Learning Method
Yunpeng Gong Liqing Huang Lifei Chen
College of Computer and Cyber Security, Fujian Normal University, P. R. China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
One of the challenges of computer vision is that it needs to adapt to color
deviations in changeable environments. Therefore, minimizing the adverse
effects of color deviation on the prediction is one of the main goals of
vision task. Current solutions focus on using generative models to augment
training data to enhance the invariance of input variation. However, such
methods often introduce new noise, which limits the gain from generated data.
To this end, this paper proposes a strategy eliminate deviation with
deviation, which is named Random Color Dropout (RCD). Our hypothesis is that
if there are color deviation between the query image and the gallery image,
the retrieval results of some examples will be better after ignoring the color
information. Specifically, this strategy balances the weights between color
features and color-independent features in the neural network by dropouting
partial color information in the training data, so as to overcome the effect
of color devitaion. The proposed RCD can be combined with various existing
ReID models without changing the learning strategy, and can be applied to
other computer vision fields, such as object detection. Experiments on several
ReID baselines and three common large-scale datasets such as Market1501,
DukeMTMC, and MSMT17 have verified the effectiveness of this method.
Experiments on Cross-domain tests have shown that this strategy is significant
eliminating the domain gap. Furthermore, in order to understand the working
mechanism of RCD, we analyzed the effectiveness of this strategy from the
perspective of classification, which reveals that it may be better to utilize
many instead of all of color information in visual tasks with strong domain
variations.
## 1 Introduction
Figure 1: The retrieval results of the model trained with visible (RGB) image
and the model trained with grayscale image on Market1501[1] are displayed. It
shows that the color deviation between the query image and gallery image will
affect the retrieval results, and the retrieval results of some samples will
be better after ignoring the color information. The numbers on the images
indicate the rank of similarity in the retrieval results, the red and green
numbers denote the wrong and correct results, respectively.
Person re-identification (ReID) is to match the same person across diferent
cameras and scenes[2, 1, 3, 4]. This technology have been widely applied to
video surveillance[5, 6, 7], image retrieval[8, 9], criminal investigation[8],
target tracking[10] and others. The challenge of this task is that images
captured by different cameras often contain significant intra-class variation
caused by variations in viewpoint, human pose changes, occlusions, and color
deviation under variable camera conditions, etc. As a result, the appearance
of the same pedestrian image with great changes, making intra-class (the same
pedestrian) metric distance larger than inter-class (different pedestrians).
ReID usually combines representation learning[11, 12, 13, 14, 15, 16] with
metric learning[17, 18, 19, 20, 21, 22, 23], and combines classification
loss[16, 13, 11] with triplet loss[23, 22] in the training stage to optimize
the neural network. In the inference stage, it is only necessary to use Cosine
distance or Euclidean distance to measure the similarity between the query
image and the gallery image, and then rank the gallery images according to the
similarity, and finally use the re-ranking technique[24] to further refine the
search results.
The complexity of the inherent challenge of ReID means that its demand for
data has the same complexity, and the complex data demand is difficult to meet
and balance by the training set, which is also accompanied by the potential
problem that the model overfits the only training data and lacks robustness.
The dataset is hard to cover different camera environments and all their
variations at different times, so the trained models tend to overfit the given
training set and lack robustness to additional scenarios. It is no doubt that
color features are important discriminative features, but color features
instead limit the model to make correct predictions in some cases. For
example, because white and gray, black and dark blue, and brown and yellow are
similar under some lighting conditions, it is difficult for the model to make
correct predictions for negative samples that are similar to target after
overfitting the color deviation variations. As shown in (a) to (c) in Figure
1, the prediction results made by the model trained with grayscale images in
this case are better after discarding the color bias.
Realistically, color deviations cause domain gaps exist both between datasets
and within datasets[25, 26]. These color biases are practically inexhaustible.
Instead of generating a variety of data to let the model ”see” these
variations (especially intra-class variations)[25] during training to enhance
the robustness to input variations, it is better to balance the weight between
color features and other important discriminant features implicitly. Aiming at
the inherent color deviation problem that the images obtained under different
shooting conditions, this paper proposes a strategy to eliminate deviation
with deviation based on the assumption that the retrieval results of some
samples will be better when discarding color information, which is named
random color dropout (RCD). RCD balances the weight between color features and
other important discriminant features in neural network by discarding part of
the color information in the training data, so as to overcome the influence of
color deviation. This strategy exists in various forms. For example, color
deviation can be overcome with biased grayscale information or sketch
information (or contour information). Taking grayscale as an example, it can
be randomly selected a rectangular area in the RGB image and replace its
pixels with the same rectangular area in the corresponding grayscale image ,
thus it generates a training image with different areas of biases during ReID
model training. Compared with existing methods based on generative adversarial
networks (GANs)[27, 25, 26, 28, 29, 30, 31], the proposed method is more
lightweight and effective because it not only does not introduce new noise but
also saves a large amount of computational resources. At the same time, this
strategy enables the model to naturally have cross-modal retrieval[8, 9, 32,
33, 34, 35] capabilities. For example, when taking the contour information as
the intermediary to overcome the color deviation, the cross-modal retrieval
between sketch and RGB visible image can be realized.
In addition, this paper analyzes the relationship between RCD and the
generalization ability of neural networks from the perspective of
classification, and reveals the intrinsic reasons that networks trained with
RCD may outperform ordinary networks. Experiments show that the proposed
method not only increases the robustness of the model to color deviation but
also bridges the domain gap between different datasets, which has significant
advantages over the existing state-of-the-art method. Taking the grayscale as
an example, the RCD strategy proposed in this paper includes global grayscale
transformation, local grayscale transformation, and a combination of these
two. The method has the following advantages:
It is a lightweight approach which does not require any additional parameter
learning or memory consumption. It can be combined with various CNN models
without changing the learning strategy. It is a complementary approach to
existing data augmentation.
The main contributions of this paper are summarized as follows:
$\bullet$ This paper proposes a learning strategy which against color
deviation with information deviation, which decreases the overfitting and
increases generalization ability of the model.
$\bullet$ A simple and effective cross-modal retrieval method is proposed,
which does not need complex network design.
$\bullet$ This paper proves that the network trained with RCD may be better
than the ordinary network from the perspective of classification.
$\bullet$ The strategy proposed in this paper is proved to be effective in
improving ReID performance through extensive experiments and analysis. The
effectiveness of the proposed method is verified on several baselines and
representative datasets.
This work was previously published as a preprint on Arxiv and extended on the
basis of it, including related demonstration and cross-modal retrieval.
Figure 2: Framework diagram of our Random Color Dropout (RCD): The application
of global grayscale transformation and local grayscale transformation in the
framework.
## 2 Related Work
The complexity of the inherent challenge of ReID means that its demand for
data has the same complexity, and the failure to fully meet the complex demand
of the data is the source of the problem of overfitting and insufficient
generalization of the model to the training data. Improving generalization
ability is the focus of research in convolutional neural networks (CNNs).
Therefore, data augmentation is effective in improving the generalization
ability of the model.
### 2.1 Classic Data Augmentation
Many data augmentation[36] methods have been proposed, such as random
cropping[36], flipping[37], which are well known to play an important role in
classification, detection and ReID. CutMix[38] replaces one patch of an image
with a patch from another image. Random erasing or cutout[39, 40] adds noise
block to the image to regularize the network, while it helps to solve the
occlusion problem in the ReID. The above methods are regarded as indispensable
methods, and they are applied to various baselines[41, 42, 43, 44]. These
techniques have been proved to be effective in improving the prediction
accuracy, and they are complementary to each other[41].
In solving the problem of color deviation, the early work[45] used the filter
and the maximum grouping layer to learn the illumination transformation,
divided the pedestrian image into more small pieces to calculate the
similarity, and uniformly handled the problems of misalignment, occlusion and
illumination variation under the deep neural network; [46] performed pre-
processing before feature extraction and used multiscale Retinex algorithm to
enhance the color information of light shaded regions to improve the color
changes caused by lighting condition changes.
With the increasing maturity of GANs, GANs-based approaches for data
augmentation have become an active research field.
### 2.2 Data Augmentation Based on GANs
The goal of these methods is to mitigate the effect of color deviation or
human-pose variation, and to improve the robustness of the model by learning
the invariant features from the variation of the input. The appearance details
and the emphases generated by different GANs-based methods are also different,
but their goal is all to compensate for the difference between the source and
target domains. For example, CamStyle [52] generates new data for transferring
different camera styles to learn invariant features between different cameras
to increase the robustness of the model to camera style changes; CycleGAN[47]
was applied in[29, 48] to transfer pedestrian image styles from one dataset to
another; StarGAN[49] was used by[50] to generate pedestrian images with
different camera styles. Wei et al.[26] proposed PTGAN to achieve pedestrian
image transfer across different ReID datasets. It uses semantic segmentation
to extract foreground masks to assist style transfer, and converts the
background into the desired style of the dataset while keeping the foreground
unchanged. different from global style transfer, DGNet[25] utilizes GANs to
transfer clothing among different pedestrians by manipulating appearance and
structural details to generate more diverse data to reduce the impact of color
changes on the model, which effectively improves the generalization ability of
the model. In addition,[51] uses 3D engine and environment rendering
technology to build a virtual pedestrian data set with multiple lighting
conditions, which is combined with other large real data sets to jointly pre-
train a model.
The method proposed in this paper has been partially validated in other works.
[52] proved that the lack of robustness to color deviation is one of the main
reasons why the model is vulnerable to adversarial metric attacks[53, 54, 55],
and enhanced the model’s adversarial defense using the method proposed in this
paper; It is adopted in the new baseline proposed in[56, 57] to help increase
the generalization of the model. In addition, [58] showed that the method
proposed in this paper is also suitable for object detection.
## 3 Proposed Methods
The RCD strategy proposed in this paper includes global transformation, local
transformation, and a combination of the two. Taking grayscale as an example,
it includes global grayscale transformation, local grayscale transformation,
and combinations of the two. At the end of this subsection, we give the
corresponding analysis of the proposed method. The framework of this method is
showed in Figure $\color[rgb]{1,0,0}2$.
### 3.1 Global Grayscale Transformation
In the data loading, it randomly samples K identities and M images of per
person to constitute a training batch which size equals to $B=K\times M$. The
set is denoted as $x^{v}=\\{x_{i}^{v}|i=1,2,...,M\times K\\}$, where
$x_{i}^{v}=\\{x_{i}^{v}|y_{i}\\}$ represents the i-th sample image of the
training batch, and $y_{i}$ represents the class label of the pedestrian.
Taking the grayscale as an example, this method randomly performs global
grayscale transformation on the training batch with a probability, and then
inputs into the model for training. This process can be defined as:
$I_{g}=t(R,G,B)$ (1)
where $t(\bullet)$ is the grayscale image conversion function, which is
implemented by performing pixel-by-pixel accumulation calculations on the R,
G, and B channels of the original visible RGB image; y is the label of the
sample, the converted grayscale image label of $x^{g}$ are the same as the
original ones:
$(x^{g}|y)=(x^{v}|y)$ (2)
the procedure of LGT is shown in Algorithm.$\color[rgb]{1,0,0}1$.
Input : Input image $I$;
Graycale transformation probability $p$;
.
Output : Grayscale images $I^{\ast}$.
Initialization: $p_{1}\leftarrow$ Rand (0, 1).
if _$p_{1}\geq p$_ then
$I^{\ast}\leftarrow I$;
return _$I^{\ast}$_.
else
$I^{\ast}\leftarrow$ t($I$);
return _$I^{\ast}$_.
end if
Algorithm 1 Global Graycale Transformation
### 3.2 Local Grayscale Transformation
In addition to transforming the data globally, we also consider transforming
the data locally so that the model adapt better to the significantly varying
bias due to color dropout from the local variation.
The local grayscale transformation (LGT) for each visible image $x^{v}$ can be
achieved by the following equations:
$x^{g}=t(x^{v}),$ (3)
$rect=RandPosition(x^{v}),$ (4)
$x^{lg}=LGT(x^{v},x^{g},rect)$ (5)
and
$(x^{lg}|y)=(x^{v}|y)$ (6)
where $x^{g}$ is the grayscale images, and $t(\bullet)$ is the grayscale
transformation funtion; $RandPosition(\bullet)$ is used to generate a random
rectangle in the image, and the function of $LGT(\bullet)$ is to give the
pixels in the rectangle corresponding to the $x^{g}$ image to the $x^{v}$
image; $x^{lg}$ is the sample after local grayscale transformation, and $y$ is
the label of the transformed image.
Input : Input image $I$;
Image size $W$ and $H$;
Area of image $S$;
Transformation probability $p$;
Area ratio range $s_{l}$ and $s_{h}$;
Aspect ratio range $r_{1}$ and $r_{2}$.
Output : Transformed image $I^{\ast}$.
Initialization: $p_{1}\leftarrow$ Rand (0, 1).
if _$p_{1}\geq p$_ then
$I^{\ast}\leftarrow I$;
return _$I^{\ast}$_.
else
while _True_ do
$S_{t}\leftarrow$ Rand $(s_{l},s_{h})$$\times S$;
$r_{t}\leftarrow$ Rand $(r_{1},r_{2})$;
$H_{t}\leftarrow\sqrt{S_{t}\times r_{t}}$,
$W_{t}\leftarrow\sqrt{\frac{S_{t}}{r_{t}}}$;
$x_{t}\leftarrow$ Rand $(0,W)$, $y_{t}\leftarrow$ Rand $(0,H)$;
if _$x_{t}+W_{t}\leq W$ and $y_{t}+H_{t}\leq H$_ then
$Position\leftarrow(x_{t},y_{t},x_{t}+W_{t},y_{t}+H_{t})$;
$I(Position)\leftarrow$ $t(Position)$;
$I^{\ast}\leftarrow I$;
return _$I^{\ast}$_.
end if
end while
end if
Algorithm 2 Local Graycale Transformation
In the process of model training, we conduct LGT randomly transformation on
the training batch with a probability. For an image $I$ in a batch, denote the
probability of it undergoing LGT be $p_{r}$, and the probability of it being
kept unchanged be $1-p_{r}$. In this process, it randomly selects a
rectangular region in the image and replaces it with the pixels of the same
rectangular region in the corresponding grayscale image. Thus, training images
which include regions with different levels of grayscale are generated. Among
them, $s_{l}$ and $s_{h}$ are the minimum and maximum values of the ratio of
the image to the randomly generated rectangle area, and the $S_{t}$ of the
rectangle area limited between the minimum and maximum ratio is obtained by
$S_{t}$ ← $Rand(s_{l},s_{h})\times S$, $r_{t}$ is a coefficient used to
determine the shape of the rectangle. It is limited to the interval ($r_{1}$,
$r_{2}$ ). $x_{t}$ and $y_{t}$ are randomly generated by coordinates of the
upper left corner of the rectangle. If the coordinates of the rectangle exceed
the scope of the image, the area and position coordinates of the rectangle are
re-determined. When a rectangle that meets the above requirements is found,
the pixel values of the selected region are replaced by the corresponding
rectangular region on the grayscale image converted from RGB image. As a
result, training images which include regions with different levels of
grayscale are generated, and the object structure is not damaged. The
procedure of LGT is shown in Algorithm.$\color[rgb]{1,0,0}2$.
### 3.3 Loss function
ReD usually combines classification loss and triplet loss to train the
model[59, 60]. Here, we use $x_{i}^{v}$ to denote the $i$-th RGB image in a
training batch, and $x_{i}^{g}$ to denote the image obtained after GGT or LGT
conversion. Thenthe features of $x_{i}^{v}$ and $x_{i}^{g}$ can be expressed
as:
$\left\\{\begin{array}[]{ll}f_{i}^{v}=f(x_{i}^{v})\\\
f_{i}^{g}=f(f_{i}^{g})\end{array}\right.$ (7)
The Euclidean distance between two samples $x_{i}^{g}$ and $x_{j}^{g}$ is
denoted as $D(x_{i}^{g},x_{j}^{g})$, where The subscript $\\{i,j\\}$ denotes
the image index in the training batch. Formally, let $x_{i}^{g}$ be the anchor
sample, the triple $\\{x_{i}^{g},x_{j}^{g},x_{k}^{g}\\}$ is selected in the
following way:
$P_{i,j}^{g}=\max_{\forall y_{i}=y_{j}}D(x_{i}^{g},x_{j}^{g})$ (8)
$N_{i,k}^{g}=\min_{\forall y_{i}\neq y_{j}}D(x_{i}^{g},x_{k}^{g})$ (9)
For each anchor point $x_{i}^{g}$, the above strategy selects the positive
sample pair with the same pedestrian class label and the farthest the most
distant positive sample pair with the same pedestrian category label and the
nearest negative sample pairs, forming a triplet
$\\{x_{i}^{g},x_{j}^{g^{+}},x_{k}^{g^{-}}\\}$ for mining grayscale
information. In general, the use of The boundary parameter $\varepsilon$ is
used to control the spacing of the positive and negative sample pairs. In
summary, we can define the following triadic loss for training:
$L_{g}=\frac{1}{n}\sum_{i=1}^{n}\max[\varepsilon+D(x_{i}^{g},x_{j}^{g^{+}})+D(x_{i}^{g},x_{k}^{g^{-}})]$
(10)
In addition, $x^{v}$ and $x^{g}$ are trained using a shared identity
classifier $\phi$. The predicted probability of identity labels $y_{i}$ is
define as $p(y_{i}|x_{i}^{g};\phi)$. The ID loss is represented as follows:
$L_{ide}=-\frac{1}{n}\sum_{i=1}^{n}log(p(y_{i}|x_{i}^{g};\phi))$ (11)
Therefore, the overall loss during random grayscale transformation is:
$L_{total}=L_{g}+L_{ide}$ (12)
### 3.4 Analysis of Random Color Dropping Policy
Here suppose there are $m$ instances, the expected output, i.e.
$D=[d_{1},d_{2},…,d_{m}]^{T}$ where $d_{j}$ denotes the expected output on the
$j$-th instance, and the actual output of the $i$-th component neural network,
i.e. $F_{i}=[f_{i1},f_{i2},…,f_{im}]^{T}$ where $f_{ij}$ denotes the actual
output of the $i$-th component network on the $j$-th instance. $D$ and $F_{i}$
satisfy that $d_{j}$ $\in\\{-1,+1\\}(j=1,2,…,m)$ and
$f_{ij}$$\in\\{-1,+1\\}(i=1,2,…,N;j=1,2,…,m)$ respectively. It is obvious that
if the actual output of the $i$-th component network on the $j$-th instance is
correct according to the expected output then $f_{ij}d_{j}=+1$, otherwise
$f_{ij}d_{j}=-1$. Thus the generalization error of the $i$-th component neural
network on those $m$ instances is:
$E_{i}=\frac{1}{m}\sum_{j=1}^{m}{Error(f_{ij}d_{j})}$ (13)
where $Error(x)$ is a function defined as:
$Error(x)=\left\\{\begin{array}[]{ll}1,\quad\quad if\quad x=-1\\\ 0.5,\quad
if\quad x=0\\\ 0,\quad\quad if\quad x=1\end{array}\right.$ (14)
Here we introduce a vector $Sum=[Sum_{1},Sum_{2},…,Sum_{m}]^{T}$ where
$Sum_{j}$ denotes the sum of the actual output of all the component neural
networks on the $j$-th instance, i.e.
$Sum_{j}=\sum_{i=1}^{N}f_{ij}$ (15)
Then the output of the neural network ensemble on the j-th instance is:
$\hat{f_{j}}=Sgn(Sum_{j})$ (16)
where $Sgn(x)$ is a function defined as:
$Sgn(x)=\left\\{\begin{array}[]{ll}1,\quad\quad if\quad x>0\\\ 0,\quad\quad
if\quad x=0\\\ -1,\quad if\quad x<0\end{array}\right.$ (17)
It is obvious that $\hat{f}_{j}\in\\{-1,0,+1\\}(j=1,2,…,m)$ . If the actual
output of the ensemble on the $j$-th instance is correct according to the
expected output then $\hat{f_{j}}d_{j}=+1$; if it is wrong then
$\hat{f}_{j}d_{j}=-1$; otherwise $\hat{f}_{j}d_{j}=0$, which means that there
is a tie on the j-th instance, e.g. three component networks vote for +1 while
other three networks vote for -1. Thus the generalization error of the
ensemble is:
$\hat{E}=\frac{1}{m}\sum_{j=1}^{m}{Error(\hat{f_{j}}d_{j})}$ (18)
Here suppose that the k-th component neural network is trained using grayscale
images. Then the output of the new set on the $j$-th instance is:
$\hat{f_{j}^{{}^{\prime}}}=Sgn(Sum_{j(j\neq k)}-f_{kj})$ (19)
and the generalization error of the new ensemble is:
$\hat{E^{{}^{\prime}}}=\frac{1}{m}\sum_{j=1}^{m}{Error(\hat{f_{j}^{{}^{\prime}}}d_{j})}$
(20)
It is assumed that a certain number of networks with deviations will not
affect the performance of the overall neural network, and the retrieval
results of some examples will be better after ignoring the color information.
$Error(\hat{f_{j}^{{}^{\prime}}}d_{j})\leqslant Error(\hat{f_{j}}d_{j})$ (21)
From Eq.$\color[rgb]{1,0,0}(18)$ and Eq.$\color[rgb]{1,0,0}(20)$ we can derive
that if Eq.$\color[rgb]{1,0,0}(21)$ is satisfied then $\hat{E}$ is not smaller
than $\hat{E^{{}^{\prime}}}$, which means that the ensemble including k-th
component neural network which is trained using grayscale images is better
than the one no including:
$\begin{split}\sum_{j=1}^{m}\\{Error(Sgn(Sum_{j})d_{j})-\\\
Error(Sgn(Sum_{j(j\neq k)+f{kj}})d_{j})\\}\geq 0\end{split}$ (22)
## 4 Comparison and Analysis
Figure 3: Performance of GGT under different hyperparameters on Market1501.
### 4.1 Datasets and Evaluation criteria
Datasets. Market-1501 [1] includes 1,501 pedestrians captured by six cameras
(five HD cameras and one low-definition camera). DukeMTMC [61] is a large-
scale multi-target, multi-camera tracking dataset, a HD video dataset recorded
by 8 synchronous cameras, with more than 2,700 individual pedestrians. The
above two datasets are widely used in ReID studies. MSMT17[26], created in
winter, was presented in 2018 as a new, larger dataset closer to real-life
scenes, containing a total of 4,101 individuals and covering multiple scenes
and time periods.
These three datasets are currently the largest datasets of ReID, and they are
also the most representative because they collectively contain multi-season,
multi-time, HD, and low-definition cameras with rich scenes and backgrounds as
well as complex lighting variations.
Sketch ReID dataset[8] contains 200 persons, each of which has one sketch and
two photos. Photos of each person were captured during daytime by two cross-
view cameras. It cropped the raw images (or video frames) manually to make
sure that every photo contains the one specific person. It have a total of 5
artists to draw all persons’sketches and every artist has his own painting
style.
Evaluation criteria. Following existing works [1], Rank-k precision and mean
Average Precision (mAP) are adapted as evaluation metrics. Rank-1 denotes the
average accuracy of the first return result corresponding to each query image.
mAP denotes the mean of average accuracy, the query results are sorted
according to the similarity, the closer the correct result is to the top of
the list, the higher the score.
Figure 4: Performance of LGT under different hyperparameters on Market1501.
### 4.2 Hyper-Parameter Setting
During CNN training, two hyper-parameters need to be evaluated. One of them is
GGT probability $p$. Firstly, we take the hyper-parameter $p$ as 0.01, 0.03,
0.05, 0.07, 0.1, 0.2, 0.3,…, 1 for the GGT experiments. Then we take the value
of each parameter for three independent repetitions of the experiments.
Finally, we calculate the average of the final result. The results of
different p are shown in Fig.$\color[rgb]{1,0,0}3$. We can see that when
$p=0.05$, the performance of the model reaches the maximum value in Rank-1 and
mAP. If we do not specify, the hyper- parameter is set $p=0.05$ in the next
experiments.
Another hyper-parameter is LGT probability $p_{r}$. We take the hyper-
parameter $p_{r}$ as the same as above for the LGT experiments, whose
selection process is similar to the above $p$. The results of different
$p_{r}$ are shown in Fig.$\color[rgb]{1,0,0}4$.
Obviously, when $p_{r}=0.4$ or $p_{r}=0.7$, the model achieves better
performance. And the best performance is achieved when $p_{r}=0.4$. If we do
not specify, the hyper- parameter is set $p_{r}=0.4$ in the subsequent
experiments.
Figure 5: Performance of combining GGT with LGT under different
hyperparameters on Market1501.
Evaluation of GGT and LGT. Compared with the best results of GGT on
baseline[44], the accuracy of LGT is improved by 0.5% and 1.4% on Rank-1 and
mAP, respectively. Under the same conditions using re-Ranking[24], the
accuracy of Rank-1 and mAP is improved by 0% and 0.4%, respectively.
Therefore, the advantages of LGT are more obvious when re-Ranking is not used.
However, Fig.$\color[rgb]{1,0,0}4$ also shows that the performance improvement
brought by LGT is not stable enough because of the obvious fluctuation in LGT,
while the performance improvement brought by GGT is very stable. Therefore, we
improve the stability of the method by combining GGT with LGT.
Evaluation by Combining GGT with LGT. First, we fix the hyper-parameter value
of GGT to $p=0.05$, then keep the control variable unchanged to further
determine the hyper-parameter of LGT. Finally, we take the hyper-parameter pr
of LGT to be 0.1, 0.2, ···, 0.7 to conduct combination experiments of GGT and
LGT, and conduct 3 independent repeated experiments for each parameter $p_{r}$
to get the average value. The result is shown in $\color[rgb]{1,0,0}5$. It can
be seen that the performance improvement brought by the combination of GGT and
LGT is more stable and with less fluctuation, and the comprehensive
performance of the model is the best when the hyper-parameter value of LGT is
$p_{r}=0.4$.
Figure 6: Diagram of Global Sketch Transformation (GST) and Local Sketch
Transformation (LST).
### 4.3 Comparison Experiments
Performance comparison and analysis. We first evaluate baseline [44] on the
Market-1501 dataset[1]. To be consistent with recent works, we follow the new
training/testing protocol to conduct our experiments by k-reciprocal re-
ranking (RK) [24]. As can be seen from Fig.$\color[rgb]{1,0,0}3$ and
Fig.$\color[rgb]{1,0,0}4$, our method improves by 1.2% on Rank-1 and 3.3% on
mAP on the baseline, and 1.5% on Rank-1 and 2.1% on mAP above baseline in the
same conditions using the re-Ranking[24].
Secondly, we further test the method in this paper on the baselines[59, 60]
with better performance. As we can see from Table.$\color[rgb]{1,0,0}1$ to
Table.$\color[rgb]{1,0,0}3$, the best results of our method improve by 0.6%
and 1.3% on the Rank-1 and mAP on the strong baseline[59], respectively, and
0.8% and 0.5% Rank-1 and mAP above baseline under the same conditions using
the re-Ranking[24], respectively. On fastReID[60], our method is 0.2% higher
and 0.9% than baseline in Rank-1 and mAP, respectively, and higher 0.1% and
0.3% than baseline under using re-Ranking.
The default configuration on the Strong Baseline[59] and FastReID[60] uses
data augmentation such as random flipping[37], cropping[36], and erasing[39].
The method proposed in this paper further improves the model accuracy on the
basis of using them, which shows that our method can be combined with other
data augmentation methods.
Table 1: Performance comparison on Market1501 dataset. Methods | Market1501
---|---
Rank-1(%) | mAP(%)
IANet[62] | 94.4 | 83.1
DGNet[25] | 94.8 | 86.0
SCAL[63] | 95.8 | 89.3
Circle Loss[64] | 96.1 | 87.4
SB[59] | 94.5 | 85.9
SB[59] \+ RK[24] | 95.4 | 94.2
SB + GGT(ours) | 94.6 | 85.7
SB + GGT+ RK(ours) | 96.2 | 94.7
SB + LGT(ours) | 95.1 | 87.2
SB + LGT + RK(ours) | 95.9 | 94.4
FastReID[60] | 96\. 3 | 90.3
FastReID + RK | 96.8 | 95.3
FastReID + GGT(ours) | 96.5 | 91.2
FastReID + GGT + RK(ours) | 96.9 | 95.6
Table 2: Performance comparison on DukeMTMC dataset. Methods | DukeMTMC
---|---
Rank-1(%) | mAP(%)
IANet[62] | 87.1 | 73.4
DGNet[25] | 86.6 | 74.8
SCAL[63] | 89.0 | 79.6
SB[59] | 86.4 | 76.4
SB + RK[24] | 90.3 | 89.1
SB + GGT(ours) | 87.8 | 77.3
SB + GGT+ RK(ours) | 90.9 | 89.2
SB + LGT(ours) | 87.3 | 77.3
SB + LGT + RK(ours) | 91 | 89.4
FastReID[60] | 92.4 | 83.2
FastReID + RK | 94.4 | 92.2
FastReID + LGT(ours) | 92.8 | 84.2
FastReID + LGT + RK(ours) | 94.3 | 92.7
Table 3: Performance comparison on MSMT17 dataset. Methods | MSMT17
---|---
Rank-1(%) | mAP(%)
IANet[62] | 75.5 | 46.8
DGNet[25] | 77.2 | 52.3
RGA-SC[65] | 80.3 | 57.5
SCSN[66] | 83.8 | 58.5
AdaptiveReID[67] | 81.7 | 62.2
FastReID[60] | 85.1 | 63.3
FastReID + GGT(ours) | 86.2 | 65.3
FastReID + GGT&LGT(ours) | 86.2 | 65.9
Cross-domain tests. Cross-domain person re-identification aims at adapting the
model trained on a labeled source domain dataset to another target domain
dataset without any annotation. It is pointed out by[59] that the higher
accuracy of the model does not mean that it has better generalization
capacity. In response to the above potential problems, we use cross-domain
tests to verify the robustness of the model. Experiments show that the
proposed method effectively enhances the generalization capacity of the model,
and the Table $\color[rgb]{1,0,0}2$ shows the cross-domain experiments of the
proposed method between two datasets, Market-1501[1] and DukeMTMC[61]. In
order to further explore the effectiveness of the proposed method in cross-
domain experiments, we use GGT to conduct the following cross-domain
experiments on strong baseline[59]. The experiments are shown in
Table.$\color[rgb]{1,0,0}4$.
In Table 4, +REA means that the trick of Random Erasing is used in model
training, -REA means turning it off. Experimental results show that random
erasing[39] can also significantly improve the performance of the ReID model,
but it will cause a significant drop in cross-domain performance. The proposed
method can not only significantly improve the cross-domain performance of the
ReID model, but also be more robust because of learning more discriminative
features.
Table 4: The performance of different models is evaluated on cross-domain dataset. M→D means that we train the model on Market1501[1] and evaluate it on DukeMTMC[61]. Methods | Cross-Domain
---|---
M→D | D→M
Rank-1/mAP(%) | Rank-1/mAP(%)
SB[59]+REA[39]+RK | 33.6/24.3 | 51.6/32.3
SB+REA+GGT+RK(ours) | 37.8/27.8 | 55.4/35.7
SB-REA+RK | 45.5/37.0 | 58.2/37.8
SB-REA+GGT+RK(ours) | 48.2/37.9 | 65.0/43.7
Table 5: Performance comparison between our LGT conversion and DGNet data augmentation on Market1501. Methods | Market1501
---|---
Rank-1 | mAP(%)
Baseline[44] | 88.8 | 71.6
Baseline + DGNet[25] | 88.9 | 72.1
Baseline+ LGT(ours) | 90.0 | 74.9
Table 6: Cross-domain performance comparison between our LGT and DGNet on Market1501. Methods | Market1501→DukeMTMC
---|---
Rank-1 | mAP
Baseline[44] | 37.8 | 27.0
Baseline + DGNet[25] | 36.7 | 25.6
Baseline+ LGT(ours) | 39.7 | 27.9
Table 7: Performance comparison on Market1501 dataset. Methods | Market1501
---|---
Rank-1(%) | mAP(%)
Baseline[44] | 88.8 | 71.6
Baseline + RK[24] | 90.5 | 85.2
Baseline+ GST+LST(ours) | 88.9 | 72.6
Baseline+ GST+LST+RK(ours) | 91.2 | 86.8
Table 8: Performance comparison between our RCD and Adversarial Feature Learning on Sketch ReID dataset. Sketch ReID dataset | Rnak-1(%) | Rnak-5(%) | Rank-10(%)
---|---|---|---
AFL[8] | 34.0 | 56.3 | 72.5
GST+LST(ours) | 42.5 | 70.0 | 87.5
Figure 7: t-SNE [68] visualization of six randomly selected images with
different identities on Market1501[1]. Each image corresponds to the randomly
generated images with color deviation. The same color means that they are
obtained by transformation of the same image. Dots means original example.
Comparison of state-of-the-arts. A comparison of the performance of our method
with the state-of-the-art methods DGNet[25] on Market1501[1] is shown in
Table.$\color[rgb]{1,0,0}5$ and Table.$\color[rgb]{1,0,0}6$. As can be seen
from Table.$\color[rgb]{1,0,0}5$, our method delivers a performance
improvement that far exceeds that of DGNet, the state-of-the-art GAN-based
method, by more than 2.7 percentage points on mAP, which suggests that the
proposed method is superior to existing data augmentation.
As can be seen from Table.$\color[rgb]{1,0,0}6$, the generalization ability of
the proposed method in cross-domain tests is improved by 1.9 percentage points
in the Rank-1 compared with the baseline[44], which further shows that the
proposed method is better than the existing data augmentation based on
generative models. It is worth noting that when the data generated by DGNet is
used for model training, the cross-domain performance of the model is poorly,
which confirms the point of this paper that color deviation is is difficult to
exhaust and that instead of enhancing robustness to input changes by
generating a variety of data for the model to ”see” during training, it is
better to implicitly reduce the weight of the model in the discriminant
feature of color information.
Cross-modal retrival. Another form of strategy proposed in this paper is to
take sketch image as the intermediary of balancing weight. By applying the
proposed global homogeneity transformation and local homogeneity
transformation, the sketch image is transformed as a homogeneous image, as
shown in Figure.$\color[rgb]{1,0,0}6$. It can not only improve the robustness
of the model, but also realize the sketch-based ReID. We can see this clearly
from Table.$\color[rgb]{1,0,0}.7$ and Table.$\color[rgb]{1,0,0}.8$.
In terms of cross-modal retrieval[69, 8, 70], in order to match images of
different modalities, existing approaches usually achieve cross-modal
retrieval with the help of attention mechanisms[69], multi-stream
networks[70], and generative adversarial networks[8]. Lu et al. proposed a
cross-domain adversarial feature learning (AFL) method for sketch re-
identification and contributed the sketch character re-identification
dataset[8].
In order to make a fair comparison, as same as AFL[8], the method proposed in
this paper is firstly trained on the Market-1501 dataset, and then fine tuned
on sketch ReID dataset. In parameter setting, this paper set 5% Global Sketch
Transformation and 70% Local Sketch Transformation. The experiment result
shows that the performance improvement in the Sketch Re-identification more
than 8%. This experiment also shows the generality of the proposed method.
Figure 8: Comparison of Grad-CAM [71] activation map between normally trained
model and our proactive defense model.
Visualization analysis. As the show in Figure $\color[rgb]{1,0,0}7$, the model
trained by DCR is robust to color variations. Therefore, we can observe that
the features of examples with color deviation exhibit clustering effects
better.
Grad-CAM [71] uses the gradient information flowing into the last
convolutional layer of the CNN to visualize the importance of each neuron in
the output layer for the final prediction, by which it is possible to
visualize which regions of the image have a significant impact on the
prediction of a model. As shown in Figure $\color[rgb]{1,0,0}8$, we can see
that the the normally trained model activates irrelevant parts in the case of
severe color deviation, while the model trianed with RCD is still effectively
activating some important parts.
## 5 Conclusion
In this paper, a simple, effective and general strategy that can be applied in
computer vision to overcome color deviation. Neither does the method require
large scale training like GAN, nor introduces any noise. The method uses a
random homogeneous transformation to realize the modeling of different modal
relationships. The model balances the weights between color features and
discriminative non-color features by fitting differentiated homogeneous
information in a mixed domain with information bias during the training
process, thus reducing the negative impact of color deviation on ReID. In
addition, this paper reveals the intrinsic reasons why networks trained with
RCD outperform ordinary networks from a classification perspective. At the
same time, experiments on several datasets and baselines show that the
proposed method is effective and outperforms the state of the arts, and extra
experiments show that the proposed strategy has natural cross modal
properties.
## References
* [1] Liang Zheng, Liyue Shen, Lu Tian, Shengjin Wang, Jingdong Wang, and Qi Tian. Scalable person re-identification: A benchmark. In Proceedings of the IEEE international conference on computer vision, pages 1116–1124, 2015.
* [2] Mang Ye, Jianbing Shen, Gaojie Lin, Tao Xiang, Ling Shao, and Steven C.H. Hoi. Deep learning for person re-identification: A survey and outlook. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021\.
* [3] Zhedong Zheng, Liang Zheng, and Yi Yang. A discriminatively learned cnn embedding for person reidentification. IEEE Transactions on Information Forensics and Security, 16:728–739, 2017.
* [4] Yulin Li, Jianfeng He, Tianzhu Zhang, Xiang Liu, Yongdong Zhang, and Feng Wu. Diverse part discovery: Occluded person re-identification with part-aware transformer. In CVPR, pages 2898–2907, 2021.
* [5] Ruibing Hou, Hong Chang, Bingpeng Ma, Rui Huang, and Shiguang Shan. Bicnet-tks: Learning efficient spatial-temporal representation for video person re-identification. In CVPR, pages 2014–2023, 2021.
* [6] Xudong Tian, Zhizhong Zhang, Shaohui Lin, Yanyun Qu, Yuan Xie, and Lizhuang Ma. Farewell to mutual information: Variational distillation for cross-modal person re-identification. In CVPR, pages 1522–1531, 2021.
* [7] Xuehu Liu, Pingping Zhang, Chenyang Yu, Huchuan Lu, and Xiaoyun Yang. Watching you: Global-guided reciprocal learning for video-based person re-identification. In CVPR, pages 13334–13343, 2021.
* [8] Lu Pang, Yaowei Wang, YiZhe Song, Tiejun Huang, and Yonghong Tian. Cross-domain adversarial feature learning for sketch re-identification.
* [9] Yi Li, Timothy M. Hospedales, Yizhe Song, and Shaogang Gong. Fine-grained sketch-based image retrieval by matching deformable part models. BMVC, pages 1–12, 2014.
* [10] Lucas Beyer, Stefan Breuers, Vitaly Kurin, and Bastian Leibe. Towards a principled integration of multi-camera re-identification and tracking through optimal bayes filters. In CVPRW, 2017.
* [11] Zhedong Zheng, Liang Zheng, and Yi Yang. A discriminatively learned cnn embedding for person reidentification. ACM transactions on multimedia computing, communications, and applications (TOMM), 14(1):1–20, 2017.
* [12] Tetsu Matsukawa and Einoshin Suzuki. Person re-identification using cnn features learned from combination of attributes. In 2016 23rd international conference on pattern recognition (ICPR), pages 2428–2433. IEEE, 2016.
* [13] Xing Fan, Wei Jiang, Hao Luo, and Mengjuan Fei. Spherereid: Deep hypersphere manifold embedding for person re-identification. Journal of Visual Communication and Image Representation, 60:51–58, 2019.
* [14] Yutian Lin, Liang Zheng, Zhedong Zheng, Yu Wu, Zhilan Hu, Chenggang Yan, and Yi Yang. Improving person re-identification by attribute and identity learning. Pattern Recognition, 95:151–161, 2019.
* [15] Liang Zheng, Yi Yang, and Alexander G Hauptmann. Person re-identification: Past, present and future. arXiv preprint arXiv:1610.02984, 2016.
* [16] Hantao Yao, Shiliang Zhang, Richang Hong, Yongdong Zhang, Changsheng Xu, and Qi Tian. Deep representation learning with part loss for person re-identification. IEEE Transactions on Image Processing, 28(6):2860–2871, 2019.
* [17] Xinshao Wang, Yang Hua, Elyor Kodirov, Guosheng Hu, Romain Garnier, and Neil M Robertson. Ranked list loss for deep metric learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5207–5216, 2019.
* [18] Rahul Rama Varior, Mrinal Haloi, and Gang Wang. Gated siamese convolutional neural network architecture for human re-identification. In European conference on computer vision, pages 791–808. Springer, 2016.
* [19] Rahul Rama Varior, Bing Shuai, Jiwen Lu, Dong Xu, and Gang Wang. A siamese long short-term memory architecture for human re-identification. In European conference on computer vision, pages 135–153. Springer, 2016.
* [20] Florian Schroff, Dmitry Kalenichenko, and James Philbin. Facenet: A unified embedding for face recognition and clustering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 815–823, 2015.
* [21] Hao Liu, Jiashi Feng, Meibin Qi, Jianguo Jiang, and Shuicheng Yan. End-to-end comparative attention networks for person re-identification. IEEE Transactions on Image Processing, 26(7):3492–3506, 2017.
* [22] De Cheng, Yihong Gong, Sanping Zhou, Jinjun Wang, and Nanning Zheng. Person re-identification by multi-channel parts-based cnn with improved triplet loss function. In Proceedings of the iEEE conference on computer vision and pattern recognition, pages 1335–1344, 2016.
* [23] Alexander Hermans, Lucas Beyer, and Bastian Leibe. In defense of the triplet loss for person re-identification. arXiv preprint arXiv:1703.07737, 2017.
* [24] Zhun Zhong, Liang Zheng, Donglin Cao, and Shaozi Li. Re-ranking person re-identification with k-reciprocal encoding. In CVPR, 2017.
* [25] Zhedong Zheng, Xiaodong Yang, Zhiding Yu, Liang Zheng, Yi Yang, and Jan Kautz. Joint discriminative and generative learning for person re-identification. In proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2138–2147, 2019.
* [26] Longhui Wei, Shiliang Zhang, Wen Gao, and Qi Tian. Person transfer gan to bridge domain gap for person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 79–88, 2018.
* [27] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in neural information processing systems, 27, 2014.
* [28] Zhun Zhong, Liang Zheng, Zhedong Zheng, Shaozi Li, and Yi Yang. Camera style adaptation for person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5157–5166, 2018.
* [29] Weijian Deng, Liang Zheng, Qixiang Ye, Guoliang Kang, Yi Yang, and Jianbin Jiao. Image-image domain adaptation with preserved self-similarity and domain-dissimilarity for person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 994–1003, 2018.
* [30] Jinxian Liu, Bingbing Ni, Yichao Yan, Peng Zhou, Shuo Cheng, and Jianguo Hu. Pose transferrable person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4099–4108, 2018.
* [31] Xuelin Qian, Yanwei Fu, Tao Xiang, Wenxuan Wang, Jie Qiu, Yang Wu, Yu-Gang Jiang, and Xiangyang Xue. Pose-normalized image generation for person re-identification. In Proceedings of the European conference on computer vision (ECCV), pages 650–667, 2018.
* [32] Yuanxin Zhu, Zhao Yang, Li Wang, Sai Zhao, Xiao Hu, and Dapeng Tao. Hetero-center loss for cross-modality person re-identification. Neurocomputing, 386:97–109, 2020.
* [33] Zhipeng Huang, Jiawei Liu, Liang Li, Kecheng Zheng, and Zheng-Jun Zha. Modality-adaptive mixup and invariant decomposition for rgb-infrared person re-identification. arXiv preprint arXiv:2203.01735, 2022.
* [34] Mang Ye, Zheng Wang, Xiangyuan Lan, and Pong C Yuen. Visible thermal person re-identification via dual-constrained top-ranking. In IJCAI, volume 1, page 2, 2018.
* [35] Diangang Li, Xing Wei, Xiaopeng Hong, and Yihong Gong. Infrared-visible cross-modal person re-identification with an x modality. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 4610–4617, 2020.
* [36] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems, 25, 2012.
* [37] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
* [38] Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE/CVF international conference on computer vision, pages 6023–6032, 2019.
* [39] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data augmentation. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 13001–13008, 2020.
* [40] Terrance DeVries and Graham W Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.
* [41] Hao Luo, Youzhi Gu, Xingyu Liao, Shenqi Lai, and Wei Jiang. Bag of tricks and a strong baseline for deep person re-identification. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition workshops, pages 0–0, 2019.
* [42] Lingxiao He, Xingyu Liao, Wu Liu, Xinchen Liu, Peng Cheng, and Tao Mei. Fastreid: A pytorch toolbox for general instance re-identification. arXiv preprint arXiv:2006.02631, 2020.
* [43] Zhedong Zheng, Tao Ruan, Yunchao Wei, Yi Yang, and Tao Mei. Vehiclenet: Learning robust visual representation for vehicle re-identification. IEEE Transaction on Multimedia (TMM), 2020.
* [44] Zhedong Zheng, Liang Zheng, and Yi Yang. A discriminatively learned cnn embedding for person reidentification. ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM), 14(1):13, 2018.
* [45] Wei Li, Rui Zhao, Tong Xiao, and Xiaogang Wang. Deepreid: Deep filter pairing neural network for person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 152–159, 2014.
* [46] Shengcai Liao, Yang Hu, Xiangyu Zhu, and Stan Z Li. Person re-identification by local maximal occurrence representation and metric learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2197–2206, 2015.
* [47] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223–2232, 2017.
* [48] Zhun Zhong, Liang Zheng, Zhiming Luo, Shaozi Li, and Yi Yang. Invariance matters: Exemplar memory for domain adaptive person re-identification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 598–607, 2019.
* [49] Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8789–8797, 2018.
* [50] Zhun Zhong, Liang Zheng, Shaozi Li, and Yi Yang. Generalizing a person retrieval model hetero-and homogeneously. In Proceedings of the European conference on computer vision (ECCV), pages 172–188, 2018.
* [51] Slawomir Bak, Peter Carr, and Jean-Francois Lalonde. Domain adaptation through synthesis for unsupervised person re-identification. In Proceedings of the European conference on computer vision (ECCV), pages 189–205, 2018.
* [52] Yunpeng Gong, Liqing Huang, and Lifei Chen. Person re-identification method based on color attack and joint defence. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, pages 4313–4322, June 2022.
* [53] Quentin Bouniot, Romaric Audigier, and Angelique Loesch. Vulnerability of person re-identification models to metric adversarial attacks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 794–795, 2020.
* [54] Song Bai, Yingwei Li, Yuyin Zhou, Qizhu Li, and Philip HS Torr. Metric attack and defense for person re-identification. 2019\.
* [55] Hongjun Wang, Guangrun Wang, Ya Li, Dongyu Zhang, and Liang Lin. Transferable, controllable, and inconspicuous adversarial attacks on person re-identification with deep mis-ranking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 342–351, 2020.
* [56] Xingyang Ni and Esa Rahtu. Flipreid: Closing the gap between training and inference in person re-identification. In 2021 9th European Workshop on Visual Information Processing (EUVIP), pages 1–6. IEEE, 2021.
* [57] Minghui Chen, Zhiqiang Wang, and Feng Zheng. Benchmarks for corruption invariant person re-identification. arXiv preprint arXiv:2111.00880, 2021.
* [58] Seong-Eun Ryu and Kyung-Yong Chung. Detection model of occluded object based on yolo using hard-example mining and augmentation policy optimization. Applied Sciences, 11(15):7093, 2021.
* [59] Hao Luo, Youzhi Gu, Xingyu Liao, Shenqi Lai, and Wei Jiang. Bag of tricks and a strong baseline for deep person re-identification. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition workshops, pages 0–0, 2019.
* [60] Lingxiao He, Xingyu Liao, Wu Liu, Xinchen Liu, Peng Cheng, and Tao Mei. Fastreid: a pytorch toolbox for general instance re-identification. arXiv:2006.02631, 2020.
* [61] Zhedong Zheng, Liang Zheng, and Yi Yang. Unlabeled samples generated by gan improve the person re-identification baseline in vitro. In Proceedings of the IEEE international conference on computer vision, pages 3754–3762, 2017.
* [62] Feng Zheng, Cheng Deng, Xing Sun, Xinyang Jiang, Xiaowei Guo, Zongqiao Yu, Feiyue Huang, and Rongrong Ji. Pyramidal person re-identification via multi-loss dynamic training. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8514–8522, 2019.
* [63] Binghui Chen, Weihong Deng, and Jiani Hu. Mixed high-order attention network for person re-identification. In Proceedings of the IEEE/CVF international conference on computer vision, pages 371–381, 2019.
* [64] Yifan Sun, Changmao Cheng, Yuhan Zhang, Chi Zhang, Liang Zheng, Zhongdao Wang, and Yichen Wei. Circle loss: A unified perspective of pair similarity optimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6398–6407, 2020.
* [65] Zhizheng Zhang, Cuiling Lan, Wenjun Zeng, Xin Jin, and Zhibo Chen. Relation-aware global attention for person re-identification. In Proceedings of the ieee/cvf conference on computer vision and pattern recognition, pages 3186–3195, 2020.
* [66] Xuesong Chen, Canmiao Fu, Yong Zhao, Feng Zheng, Jingkuan Song, Rongrong Ji, and Yi Yang. Salience-guided cascaded suppression network for person re-identification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3300–3310, 2020.
* [67] Xingyang Ni, Liang Fang, and Heikki Huttunen. Adaptive l2 regularization in person re-identification. In 2020 25th International Conference on Pattern Recognition (ICPR), pages 9601–9607. IEEE, 2021.
* [68] L Maaten and G Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, pages 2579–2605, 2008.
* [69] Jifei Song, Qian Yu, Yi-Zhe Song, Tao Xiang, and Timothy M Hospedales. Deep spatial-semantic attention for fine-grained sketch-based image retrieval. In Proceedings of the IEEE international conference on computer vision, pages 5551–5560, 2017.
* [70] Emrah Basaran, Muhittin Gökmen, and Mustafa E Kamasak. An efficient framework for visible–infrared cross modality person re-identification. Signal Processing: Image Communication, 87:115933, 2020.
* [71] Ramprasaath R. Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, pages 618–626, 2017.
|
# High harmonic spectroscopy of disorder-induced Anderson localization
Adhip Pattanayak Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, India Á. Jiménez-Galán Max-Born Straße 2A,
D-12489 Berlin, Germany Misha Ivanov Max-Born Straße 2A, D-12489 Berlin,
Germany Department of Physics, Humboldt University, Newtonstraße 15, D-12489
Berlin, Germany. Blackett Laboratory, Imperial College London, South
Kensington Campus, SW7 2AZ London, United Kingdom. Gopal Dixit
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, India
###### Abstract
Exponential localization of wavefunctions in lattices, whether in real or
synthetic dimensions, is a fundamental wave interference phenomenon.
Localization of Bloch-type functions in space-periodic lattice, triggered by
spatial disorder, is known as Anderson localization and arrests diffusion of
classical particles in disordered potentials. In time-periodic Floquet
lattices, exponential localization in a periodically driven quantum system
similarly arrests diffusion of its classically chaotic counterpart in the
action-angle space. Here we demonstrate that nonlinear optical response allows
for clear detection of the disorder-induced phase transition between
delocalized and localized states. The optical signature of the transition is
the emergence of symmetry-forbidden even-order harmonics: these harmonics are
enabled by Anderson-type localization and arise for sufficiently strong
disorder even when the overall charge distribution in the field-free system
spatially symmetric. The ratio of even to odd harmonic intensities as a
function of disorder maps out the phase transition even when the associated
changes in the band structure are negligibly small.
Disorder is an ubiquitous effect in crystals Evers and Mirlin (2008). The
seminal work by Anderson Anderson (1958) predicted that above a critical
disorder value, the electronic wavefunction will change from being delocalized
across the lattice to exponentially localized (insulating state) due to the
interference of multiple quantum paths originating from the scattering with
random impurities and defects. Anderson localization is a fundamental wave
phenomenon and thus permeates many branches of physics; it has been observed
in matter waves Billy _et al._ (2008), light waves Wiersma _et al._ (1997),
and microwaves Dalichaouch _et al._ (1991). Anderson localization also finds
direct analogues in periodically driven systems, with time-periodic dynamics
taking the role of space-periodic structure. While periodically driven
classical systems can develop chaotic behaviour for sufficiently strong
driving fields, leading to delocalization of the original ensemble across the
whole phase space, their quantum counterpart shows exponential localization of
the light-dressed states Casati _et al._ (1987).
Dramatic changes in a wavefunction during transition from a delocalized to a
localized state may lead to changes in the nonlinear optical response of the
system. In this context, symmetry–forbidden harmonics of the driving field are
an appealing bellwether candidate. Indeed, while even-order harmonics are
known to be forbidden in systems with inversion symmetry Boyd (2008), they are
also known to arise in such systems if and when charges localize Silva _et
al._ (2016); Bandrauk and Lu (2005). The required symmetry breaking can then
be triggered by even a small asymmetry in the oscillating electric field of
the driving laser pulse. Such asymmetry is natural in short laser pulses and
is controlled by the phase of the electric field oscillations under the pulse
envelope, i.e., the carrier-envelope phase (CEP) Krausz and Ivanov (2009).
Exponentially localized states in symmetric multiple well potentials appear to
be particularly sensitive to even small field asymmetries, leading to even
harmonics in the nonlinear response even for pulses encompassing tens of
cycles Silva _et al._ (2016).
High harmonic generation (HHG) is a powerful tool for ultrafast spectroscopy
Smirnova and Ivanov (2013); Lein (2007). Extremely large coherent bandwidth of
harmonic spectra enables sub-femtosecond resolution. HHG is a sensitive probe
of Cooper minima Bertrand _et al._ (2012), Auger decay Leeuwenburgh _et al._
(2013), attosecond dynamics of optical tunnelling Shafir _et al._ (2012);
Pedatzur _et al._ (2015) and the dynamics of electron exchange Sukiasyan _et
al._ (2009) in atoms, ultrafast hole dynamics Eckart _et al._ (2018); Gaal
_et al._ (2007); Bruner _et al._ (2016); Smirnova and Ivanov (2013), nuclear
motion Baykusheva _et al._ (2014); Baker _et al._ (2006); Lan _et al._
(2017) in small molecules, and enantio-sensitive electronic response in more
complex chiral molecules Cireasa _et al._ (2015); Ayuso _et al._ (2018);
Neufeld _et al._ (2019); Baykusheva _et al._ (2019). In solids, high
harmonic spectroscopy has allowed observation of dynamical Bloch oscillations,
band structure tomography Vampa _et al._ (2015), probing of defects in solids
Pattanayak _et al._ (2020); Mrudul _et al._ (2020a), sub-fs monitoring of
core excitons Luu _et al._ (2015), optical measurement of the valley
pseudospin Langer _et al._ (2018); Jimenez-Galan _et al._ (2020); Mrudul
_et al._ (2020b), tracking of van Hove singularities Uzan _et al._ (2020),
picometer resolution of valence band electrons Lakhotia _et al._ (2020),
imaging internal structures of a unit cell Mrudul _et al._ (2019), monitoring
of light-driven insulator-to-metal transitions Silva _et al._ (2018), and
probing of topological effects Bauer and Hansen (2018); Silva _et al._
(2019).
In this work, we employe HHG to track phase transition between delocalized and
localized states in the Aubry-André (AA) system Aubry and Andre (1980)
(similar to that proposed in Ref. Harper (1955)), where localization occurs
only above a critical value of disorder, already in one dimension. This model
captures the metal-to-insulator transition, is a workhorse to study non-
trivial topology, and has been realized in optical lattices and photonic
quasi-crystals Sanchez-Palencia and Lewenstein (2010); Roati _et al._ (2008).
The model system is described by the following tight-binding Hamiltonian,
$\hat{H}=-t_{0}\sum_{j=1}^{L-1}\left(c_{j}^{\dagger}c_{j+1}+\textrm{h.c.}\right)+V\sum_{j=1}^{L}\cos(2\pi\sigma
j)c_{j}^{\dagger}c_{j},$ (1)
where $t_{0}$ is the nearest neighbour hopping term, $V$ is the strength of
the potential, $c_{j}^{\dagger}$ and $c_{j}$ are, respectively, the fermionic
creation and annihilation operators at site $j$, h.c. stands for the hermitian
conjugate, $L$ is the total number of lattice sites and $\sigma$ determines
the periodicity of the potential. A rational value of $\sigma$ corresponds to
a periodic potential and consequently to delocalized electronic wavefunctions.
If $\sigma$ is irrational, the potential becomes quasi-periodically disordered
(for finite systems, this may also happen for rational $\sigma$). For a
disordered potential, the system undergoes the localization phase transition
at $V/t_{0}=2$. For $V/t_{0}>2$ all states are exponentially localized on one
site, while for $V/t_{0}<2$ the states are delocalized.
Figures 1(a) and (b) show the eigenspectrum of our system, for
$\sigma=\frac{\sqrt{5}+1}{2}$, with 100 lattice sites, in the delocalized
($V/t_{0}=1.9$) and localized ($V/t_{0}=2.1$) phases, respectively. The
hopping term $t_{0}=0.26$ eV and the lattice constant $a_{0}=7.56$ atomic unit
of length ($\sim$0.4 nm) are used throughout. The Fermi energy is $E_{F}=-0.2$
eV, so that red-colored states correspond to fully occupied valence band
states, while the blue-colored states are unoccupied. Differences in the
eigenspectrum for both phases are indiscernible. In contrast, the individual
eigenstates of the system present a clear localization phase transition: Figs.
1(c) and (d) show the occupation number of the eigenstate with index=10 (other
eigenstates show similar behaviour). The eigenstate is delocalized for
$V/t_{0}=1.9$ but fully localized for $V/t_{0}=2.1$. These differences between
the charge distribution in individual eigenstates disappear completely when we
consider the fully-filled valence band, see Figs. 1(e) and (f): the
differences between both phases are barely visible.
Figure 1: Electronic structure of the model system, for
$\sigma=\frac{\sqrt{5}+1}{2}$. Left panels (a,c,e) show the delocalized phase
($V/t_{0}=1.9$), right panels (b,d,f) show the localized phase
($V/t_{0}=2.1$). (a,b) – eigenspectrum; (c,d) – electron density per site for
one eigenstate $(m=10)$, (e,f) lattice site occupation numbers in a fully-
filled valence band. Red and blue curves in (a) and (b) represent valence and
conduction states, respectively.
The question is: will the non-linear optical response be sensitive to the
phase transition? To address this question, we consider a non-resonant, low-
frequency field polarized along the 1D chain,
$F(t)=F_{0}f(t)\cos(\omega t+\phi),$ (2)
with $\phi$ the carrier-envelope phase and $f(t)$ the sine-squared envelope
with a full duration of 10 optical cycles. We include the laser-matter
interaction via the time-dependent Peierls phase: $t_{0}\to
t_{0}\,e^{ia_{0}eA(t)}$, where $A(t)$ is the field vector potential,
$F(t)=-{dA(t)}/{dt}$, and $e$ is the electron charge. The carrier
$\omega=0.136$ eV is set well below the bandgap $\Delta\simeq 0.4$ eV.
Two characteristic regimes describe laser-induced electron dynamics in such
low-frequency fields.
In the localized phase, efficient resonant tunnelling between localized states
at different sites, including transitions from the valence to the conduction
band, becomes possible when the peak voltage between the adjacent sites
$F_{0}a_{0}$ approaches and/or exceeds the characteristic energy gap $\Delta$,
$F_{0}a_{0}\simeq\Delta$. This regime enables rapid energy gain by the system
within a few laser cycles, allowing it to climb to the top of the energy scale
Ivanov _et al._ (1996). In our case, its signature would be the emission of
harmonics up to the maximum transition frequency of the system (harmonic 10),
for all fields enabling resonant tunnelling (RT). The onset of this regime
corresponds to $F_{0,\text{RT}}\sim\Delta/a_{0}\simeq 0.4$ MV/cm in our
system. In the localized phase, exponential sensitivity of resonant tunnelling
to positions of individual states and to the field strength create ideal
conditions for symmetry breaking, leading to generation of even harmonics; the
direction in which the symmetry is broken is controlled by the pulse CEP.
In the delocalized case, the delocalized states adiabatically follow
oscillations of the low-frequency driving field, preventing symmetry breaking.
The latter requires light-induced electron localization, which occurs when
$F_{0}a_{0}\omega\geq\Delta^{2}$ Dietrich _et al._ (1996), i.e., when
$F_{0}\geq 1$ MV/cm in our system. Even harmonic generation should therefore
only emerge around $F_{0}\simeq 1$ MV/cm.
Our numerical simulations below fully confirm all of these expectations. The
time-dependent Schrödinger equation is solved independently for the $m$
normalized eigenstates $|\psi_{m}(t=t_{i})\rangle$ of the field-free
Hamiltonian Eq. (1) that lie below $E_{F}$,
$|\psi_{m}(t)\rangle=e^{-i\int_{t_{i}}^{t}\hat{H}(t^{\prime})dt^{\prime}}|\psi_{m}(t=t_{i})\rangle$,
where $\hat{H}(t)$ is the Hamiltonian in Eq. (1) with the time-dependent
Peierls substitution. The current from a single eigenstate $m$ is calculated
as
$j_{m}(t)=\left\langle\psi_{m}(t)|\hat{J}(t)|\psi_{m}(t)\right\rangle$ (3)
where the current operator is defined as
$\hat{J}(t)=-iea_{0}t_{0}\sum_{j=1}^{L}\left(e^{-ia_{0}eA(t)}c_{j}^{\dagger}c_{j+1}-e^{ia_{0}eA(t)}c_{j+1}^{\dagger}c_{j}\right).$
(4)
The total current from all the valence band eigenstates $m$, i.e., the fully-
filled valence band, is,
$j(t)=\sum_{m}j_{m}(t).$ (5)
The harmonic spectra are then calculated from the Fourier transform of the
time derivative of the total current. Prior to the Fourier transform, we apply
an envelope to the current that coincides with the laser pulse envelope Wu
_et al._ (2015), to filter out emission after the end of the laser pulse. We
consider 100 lattice sites and 0.01 atomic unit of time-step ($\sim$0.25 as)
to get the converged spectra.
Figure 2: (a,b) High harmonic spectra for the system in the (a, c, e, g, i)
delocalized and (b, d, f, h, j) localized phase: (a, b) HHG from the 10th
eigenstate only for a field strength $F_{0}=0.6$ MV/cm. HHG from the fully-
filled valence band for a field strength (c, d) $F_{0}=0.6$ MV/cm, (e, f)
$F_{0}=2.2$ MV/cm, (e, f) $F_{0}=0.4$ MV/cm, and (i, j) $F_{0}=1.0$ MV/cm.
Figure 2 shows the HHG spectrum for different initial states and field
strengths in the localized and delocalized phases. First, in Figs. 2(a) and
(b), we consider a single valence band eigenstate as our initial state
($m=10$, shown in Fig. 1(c,d)). The charge distribution of the initial state
is strongly asymmetric in both phases, which breaks the left-right symmetry of
the chain in the HHG process and leads to the appearance of even harmonics,
both in the delocalized [Fig.2(a)] and localized [Fig.2(b)] phases.
Figure 3: High harmonic spectra in the delocalized (top panels, red curve) and
the localized (lower panels, green curve) phase calculated for different
carrier-envelope phases (CEP) of the field: (a,d) CEP=$+\pi/2$, (b,e)
CEP=$-\pi/2$ and (d,f) the coherent superposition of CEP=$+\pi/2$ and
CEP=$-\pi/2$.
However, the equilibrium initial state corresponds to the fully-filled valence
band, with the charge distributed relatively evenly between all sites; the
distribution is virtually identical between the two phases [Figs. 1(e) and
(f)]. With this initial condition, the HHG spectra of the two phases, shown in
Figs. 2(c) and (d) for a field strength of $F_{0}=0.6$ MV/cm, are now
strikingly different.
Destructive interference from different initial states completely suppresses
even harmonics in the delocalized phase [Fig. 2(c)], but they remain prominent
in the localized phase [Fig. 2(d)]. This result follows from our previous
discussion. The field strength $F_{0}=0.6$ MV/cm is above the threshold for
resonant tunneling between localized states in our system
($F_{0,\text{RT}}=0.4$ MV/cm), where the instantaneous field brings the energy
levels into resonance, generating coherence between all sites. Resonant
tunneling between the sites depends sensitively on the instantaneous field
strength, leading to CEP-dependent symmetry breaking and the appearance of
even harmonics in the localized phase. We find that even harmonics emerge
already at $F_{0}\simeq 0.4$ MV/cm. As resonant tunnellng induces coherences
between the localized sites, it enables population of the highest band and
generation of higher harmonics in the localized phase than in the delocalized
phase [Figs. 2(c) and (d)]. In the latter, the system follows adiabatically
the field oscillations and transitions to the highest states are suppressed.
The field strength used in Figs. 2(c) and (d) is below the threshold field for
laser-induced localization in the delocalized phase, which is $F_{0}\sim 1$
MV/cm for our system. Therefore, the harmonic spectrum in the delocalized
regime shows no sign of even harmonics and a smaller cut-off energy [Fig.
2(c)]. Even harmonics emerge as soon as the field amplitude crosses this
threshold. At $F_{0}=2.2$ MV/cm the harmonic spectra in both phases become
very similar [Figs. 2(e) and (f)]. For this field strength, the cut-off is the
same in both phases, and corresponds to the (Stark-shifted) maximum transition
frequency of the system.
To confirm the origin of symmetry breaking and even harmonics emission, Fig. 3
shows the harmonic spectra for two values of the pulse carrier-envelope phase
(CEP) shifted by $\pi$, and their coherent superposition in Figs. 3(c) and
(f). In the delocalized case, even harmonics are absent regardless of the CEP
[Figs. 3(a) and (b)]. In the localized case, they are identical in both cases,
Figs. 3(d) and (e), but with opposite phase: upon coherent addition even
harmonics are completely washed out [Fig. 3(f)]. The reason is that the laser-
induced asymmetry in the electron charge distribution at CEP=$+\pi/2$ is
exactly opposite to that at CEP=$-\pi/2$, as graphically illustrated by the
cancellation of the emission upon interference.
Figure 4: Even-to-odd harmonic peak ratio (R) for different $V/t_{0}$ ranging
from 1.7 to 2.3. The ratio R is calculated as
$\mathrm{R}=\mathrm{log_{10}}(\bar{H}_{even})/\mathrm{log_{10}}(\bar{H}_{odd})$,
where $\bar{H}=H^{peak}/H^{min}$. Blue tiangles (orange circles) represent the
R value correspond to 4th (6th) harmonic and 5th (7th) harmonic. The increment
of R value at $V/t_{0}>2$ indicates the phase transition phenomenon.
Figure 4 shows that the relative intensity of even harmonics does indeed track
the metal to insulator (delocalized – localized) phase transition in the
system. The Figure plots the ratio of even to odd harmonics for different
values of $V/t_{0}$, at a field strength of $F_{0}=0.6$ MV/cm, i.e., that
which generates resonant tunneling in the localized phase but not laser-
induced localization in the delocalized phase. The even–odd ratio increases
dramatically at the phase transition $V/t_{0}=2$, showing that HHG, driven by
a phase-stable CEP pulse, is able to map disorder-induced electron
localization in a solid.
In conclusion, we have used high harmonic spectroscopy to track delocalized to
localized phase transition in the Aubry-André system. For aperiodic potentials
($\sigma=\frac{\sqrt{5}+1}{2}$), the localized and delocalized phases show
almost identical eigenspectra and site occupation numbers for a state with
fully-filled valence band. Yet, high harmonic spectra between the two phases
show striking differences, especially in the appearance of forbidden (even)
harmonics. This effect is a consequence of dynamical symmetry breaking induced
by the field but enabled by initial electron localization, which promotes
resonant tunneling and leads to CEP-dependent symmetry breaking even in the
low-frequency regime. Our work shows that the localisation-delocalisation
phase transition, which can be driven by a very small modification of the on-
site energy, can be effectively traced by HHG spectroscopy.
A. P. acknowledges sandwich doctoral fellowship from Deutscher Akademischer
Austauschdienst (DAAD, reference no. 57440919). M.I. and A. J-G acknowledge
support of the FET-Open Optologic grant. M.I. acknowledges support from the
Deutsche Forschungsgemeinschaft (DFG) Quantum Dynamics in Tailored Intense
Fields (QUTIF) grant. G. D. acknowledges financial support from Science and
Engineering Research Board (SERB) India (Project No. ECR/2017/001460).
## References
* Evers and Mirlin (2008) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
* Anderson (1958) P. W. Anderson, Phys. Rev. 109, 1492 (1958).
* Billy _et al._ (2008) J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
* Wiersma _et al._ (1997) D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
* Dalichaouch _et al._ (1991) R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature 354, 53 (1991).
* Casati _et al._ (1987) G. Casati, B. V. Chirikov, D. L. Shepelyansky, and I. Guarneri, Physics Reports 154, 77 (1987).
* Boyd (2008) R. W. Boyd, “Nonlinear optics, third edition,” (2008).
* Silva _et al._ (2016) R. E. F. Silva, P. Riviére, F. Morales, O. Smirnova, M. Ivanov, and F. Martín, Scientific Reports 6, 32653 (2016).
* Bandrauk and Lu (2005) A. D. Bandrauk and H. Lu, Journal of Physics B: Atomic, Molecular and Optical Physics 38, 2529 (2005).
* Krausz and Ivanov (2009) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
* Smirnova and Ivanov (2013) O. Smirnova and M. Ivanov, “Multielectron high harmonic generation: simple man on a complex plane,” (2013), arXiv:1304.2413 [physics.atom-ph] .
* Lein (2007) M. Lein, Journal of Physics B: Atomic, Molecular and Optical Physics 40, R135 (2007).
* Bertrand _et al._ (2012) J. B. Bertrand, H. J. Wörner, P. Hockett, D. M. Villeneuve, and P. B. Corkum, Phys. Rev. Lett. 109, 143001 (2012).
* Leeuwenburgh _et al._ (2013) J. Leeuwenburgh, B. Cooper, V. Averbukh, J. P. Marangos, and M. Ivanov, Phys. Rev. Lett. 111, 123002 (2013).
* Shafir _et al._ (2012) D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, Nature 485, 343 (2012).
* Pedatzur _et al._ (2015) O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. D. Bruner, A. J. Uzan, D. S. Brambila, A. Â. G. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, Nature Physics 11, 815 (2015).
* Sukiasyan _et al._ (2009) S. Sukiasyan, C. McDonald, C. Destefani, M. Y. Ivanov, and T. Brabec, Phys. Rev. Lett. 102, 223002 (2009).
* Eckart _et al._ (2018) S. Eckart, M. Kunitski, M. Richter, A. Hartung, J. Rist, F. Trinter, K. Fehre, N. Schlott, K. Henrichs, L. P. H. Schmidt, T. Jahnke, M. Schöffler, K. Liu, I. Barth, J. Kaushal, F. Morales, M. Ivanov, O. Smirnova, and R. Dörner, Nature Physics 14, 701 (2018).
* Gaal _et al._ (2007) P. Gaal, W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, Nature 450, 1210 (2007).
* Bruner _et al._ (2016) B. D. Bruner, Z. Mašín, M. Negro, F. Morales, D. Brambila, M. Devetta, D. Faccialà, A. G. Harvey, M. Ivanov, Y. Mairesse, S. Patchkovskii, V. Serbinenko, H. Soifer, S. Stagira, C. Vozzi, N. Dudovich, and O. Smirnova, Faraday Discuss. 194, 369 (2016).
* Baykusheva _et al._ (2014) D. Baykusheva, P. M. Kraus, S. B. Zhang, N. Rohringer, and H. J. Wörner, Faraday Discuss. 171, 113 (2014).
* Baker _et al._ (2006) S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilă, M. Lein, J. W. G. Tisch, and J. P. Marangos, Science 312, 424 (2006).
* Lan _et al._ (2017) P. Lan, M. Ruhmann, L. He, C. Zhai, F. Wang, X. Zhu, Q. Zhang, Y. Zhou, M. Li, M. Lein, and P. Lu, Phys. Rev. Lett. 119, 033201 (2017).
* Cireasa _et al._ (2015) R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf, N. Thiré, A. Ferré, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi, F. Légaré, V. Blanchet, B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj, Nature Physics 11, 654 (2015).
* Ayuso _et al._ (2018) D. Ayuso, P. Decleva, S. Patchkovskii, and O. Smirnova, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 124002 (2018).
* Neufeld _et al._ (2019) O. Neufeld, D. Ayuso, P. Decleva, M. Y. Ivanov, O. Smirnova, and O. Cohen, Phys. Rev. X 9, 031002 (2019).
* Baykusheva _et al._ (2019) D. Baykusheva, D. Zindel, V. Svoboda, E. Bommeli, M. Ochsner, A. Tehlar, and H. J. Wörner, Proceedings of the National Academy of Sciences 116, 23923 (2019).
* Vampa _et al._ (2015) G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, Phys. Rev. Lett. 115, 193603 (2015).
* Pattanayak _et al._ (2020) A. Pattanayak, M. M. S., and G. Dixit, Phys. Rev. A 101, 013404 (2020).
* Mrudul _et al._ (2020a) M. S. Mrudul, N. Tancogne-Dejean, A. Rubio, and G. Dixit, npj Computational Materials 6, 1 (2020a).
* Luu _et al._ (2015) T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
* Langer _et al._ (2018) F. Langer, C. P. Schmid, S. Schlauderer, M. Gmitra, J. Fabian, P. Nagler, C. Schüller, T. Korn, P. G. Hawkins, J. T. Steiner, U. Huttner, S. W. Koch, M. Kira, and R. Huber, Nature 557, 76 (2018).
* Jimenez-Galan _et al._ (2020) A. Jimenez-Galan, R. Silva, O. Smirnova, and M. Ivanov, “Sub-cycle valleytronics: control of valley polarization using few-cycle linearly polarized pulses,” (2020), arXiv:2005.10196 [physics.optics] .
* Mrudul _et al._ (2020b) M. S. Mrudul, Á. Jiménez-Galán, M. Ivanov, and G. Dixit, arXiv preprint arXiv:2011.04973 (2020b).
* Uzan _et al._ (2020) A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. F. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, Nature Photonics 14, 183 (2020).
* Lakhotia _et al._ (2020) H. Lakhotia, H. Y. Kim, M. Zhan, S. Hu, S. Meng, and E. Goulielmakis, Nature 583, 55 (2020).
* Mrudul _et al._ (2019) M. S. Mrudul, A. Pattanayak, M. Ivanov, and G. Dixit, Physical Review A 100, 043420 (2019).
* Silva _et al._ (2018) R. E. F. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, Nature Photonics 12, 266 (2018).
* Bauer and Hansen (2018) D. Bauer and K. K. Hansen, Phys. Rev. Lett. 120, 177401 (2018).
* Silva _et al._ (2019) R. E. F. Silva, Á. Jiménez-Galán, B. Amorim, O. Smirnova, and M. Ivanov, Nature Photonics 13, 849 (2019).
* Aubry and Andre (1980) S. Aubry and G. Andre, Ann. Israel Phys. Soc 3, 18 (1980).
* Harper (1955) P. G. Harper, Proceedings of the Physical Society. Section A 68, 874 (1955).
* Sanchez-Palencia and Lewenstein (2010) L. Sanchez-Palencia and M. Lewenstein, Nature Physics 6, 87 (2010).
* Roati _et al._ (2008) G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
* Ivanov _et al._ (1996) M. Ivanov, T. Seideman, P. Corkum, F. Ilkov, and P. Dietrich, Phys. Rev. A 54, 1541 (1996).
* Dietrich _et al._ (1996) P. Dietrich, M. Y. Ivanov, F. A. Ilkov, and P. B. Corkum, Phys. Rev. Lett. 77, 4150 (1996).
* Wu _et al._ (2015) M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, Phys. Rev. A 91, 043839 (2015).
|
# Activity Graph Transformer for Temporal Action Localization
Megha Nawhal1, Greg Mori1,2
1 Simon Fraser University, Burnaby, Canada
2 Borealis AI, Vancouver, Canada
###### Abstract
We introduce Activity Graph Transformer, an end-to-end learnable model for
temporal action localization, that receives a video as input and directly
predicts a set of action instances that appear in the video. Detecting and
localizing action instances in untrimmed videos requires reasoning over
multiple action instances in a video. The dominant paradigms in the literature
process videos temporally to either propose action regions or directly produce
frame-level detections. However, sequential processing of videos is
problematic when the action instances have non-sequential dependencies and/or
non-linear temporal ordering, such as overlapping action instances or re-
occurrence of action instances over the course of the video. In this work, we
capture this non-linear temporal structure by reasoning over the videos as
non-sequential entities in the form of graphs. We evaluate our model on
challenging datasets: THUMOS14, Charades, and EPIC-Kitchens-100. Our results
show that our proposed model outperforms the state-of-the-art by a
considerable margin.
## 1 Introduction
Figure 1: Main Idea. Given an untrimmed human activity video, we directly
predict the set of action instances (label, start time, end time) that appear
in the video. We observe that human activity videos contain non-sequential
dependencies (illustrated by the ground truth instances as colored bars). In
this work, we propose Activity Graph Transformer that captures this non-
sequential structure by reasoning over such videos as graphs. Overall, the
network receives a video and directly infers a set of action instances. The
network achieves this by transforming a set of graph-structured abstract
queries into contextual embeddings which are then used to provide predictions
of action instances. It is trained end-to-end using classification and
regression losses.
Visual understanding of human activities in untrimmed videos involves
reasoning over multiple action instances with varying temporal extents. This
problem has been formally studied in the setup of temporal action
localization, _i.e_., given a human activity video, the goal is to predict a
set of action labels and their corresponding start and end timestamps
indicating their occurrence in the video. Reasoning over untrimmed human
activity videos for action localization is particularly challenging due to the
idiosyncrasies of the videos such as: (1) overlap - the action instances may
have overlaps in their temporal extents indicating non-sequential temporal
ordering of the instances; (2) non-sequential dependencies - some action
instances may have temporal dependencies but are separated by other unrelated
action instances and/or durations of no action; and (3) re-occurrence -
instances belonging to same category may appear more than once over the course
of the video. In this work, we propose a novel end-to-end learnable model for
temporal action localization that receives a video as an input and directly
predicts the set of action instances that appear in the video.
Existing approaches for the task of temporal action localization predominantly
fall into two paradigms. First is the local-then-global paradigm where the
video-level predictions are obtained by postprocessing of local (_i.e_. frame-
level or snippet-level) predictions using sequence modeling techniques such as
recurrent neural networks, temporal convolutions and temporal pooling [41, 65,
11, 47, 32, 63, 28, 39, 25, 39]. Second is the proposal-then-classification
paradigm which involves generation of a sparse set of class agnostic segment
proposals from the overall video followed by classification of the action
categories for each proposal using either two-stage learning [4, 3, 18, 12,
44, 42, 69, 68] or end-to-end learning [8, 14, 7, 61, 66].
The local-then-global paradigm does not utilize the overall temporal context
provided by the activity in the video as the local predictions are solely
based on visual information confined to the frame or the snippet. For
instance, consider the example in Figure 1, these approaches would miss out on
important relevant information provided by ‘mix pasta’ in predicting ‘put
spoon’ or may produce imprecise predictions when the temporal extents of
instances ‘take colander’ or ‘open cupboard’ overlap.
Alternatively, the proposal-then-classification paradigm generates a subset of
proposals by processing the video as a sequence. As a result, these approaches
suffer from limited receptive field for incorporating temporal information,
and do not capture non-sequential temporal dependencies effectively. This
problem is further aggravated in the case of overlapping action instances. For
instance, in the example in Figure 1, ‘open cupboard’ and ‘close cupboard’
share information but are separated by other, potentially overlapping, action
instances such as ‘take colander’. Due to such ordering, when generating
proposals corresponding to ‘close cupboard’, these approaches are unlikely to
capture the dependency with the visual information pertaining to ‘open
cupboard’. Furthermore, these approaches use heuristics to perform non-maximal
suppression of proposals that might result in imprecise localization outcomes
when the action instances vary widely in their temporal extents.
As such, both these types of approaches process videos sequentially to either
generate direct local predictions or action proposals and are problematic when
action instances reoccur, overlap, or have non-sequential dependencies. These
observations suggest that although a video has a linear ordering of frames,
the reasoning over the video need not be sequential. We argue that modeling
the non-linear temporal structure is a key requirement for effective reasoning
over untrimmed human activity videos. In this work, we seek a temporal action
localization model that: (1) captures the temporal structure in complex human
activity videos, (2) does not rely on heuristics or postprocessing of the
predictions, and (3) is trained end-to-end.
Towards this goal, we formulate temporal action localization as a direct set
prediction task. We propose a novel temporal action localization model,
Activity Graph Transformer (AGT), an end-to-end learnable model that receives
a video as input and predicts the set of action instances that appear in the
video. In order to capture the non-linear temporal structure in videos, we
reason over videos as non-sequential entities, specifically, learnable graph
structures. Particularly, we map the input video to graph-structured
embeddings using an encoder-decoder transformer architecture that operates
using graph attention. A final feed forward network then uses these embeddings
to directly predict the action instances. Thus, we propose a streamlined end-
to-end training process that does not require any heuristics.
To summarize, our contributions are as follows: (1) we propose an encoder-
decoder transformer based model Activity Graph Transformer that reasons over
videos as graphs and can be trained end-to-end, and (2) we achieve state-of-
the-art performance on the task of temporal action localization on challenging
human activity datasets, namely, THUMOS14 [23], Charades [45], and EPIC-
Kitchens100 [10].
## 2 Related Work
In this section, we discuss the prior work relevant to temporal action
localization and graph based modeling in videos.
Temporal Action Localization. Early methods for temporal action localization
use temporal sliding windows and design hand-crafted features to classify
action within each window [64, 13, 22, 50, 35]. However, these approaches are
computationally inefficient as they apply classifiers on windows of all
possible sizes and locations in the entire video.
With the advances in convolutional neural networks, recent approaches fall
into two dominant paradigms: (1) local-then-global, (2) proposal-then-
classification. The methods following the local-then-global paradigm rely on
obtaining temporal boundaries of actions based on local (_i.e_. frame-level or
snippet-level) predictions and perform video-level reasoning using temporal
modeling techniques such as explicit modeling of action durations or
transitions [41, 65], recurrent neural networks [11, 47, 32, 63], temporal
pooling [25], temporal convolutions [28, 39], and temporal attention [39].
However, these approaches does not utilize the overall temporal context of the
videos as local predictions are computed using only the frame/snippet
information.
The methods based on proposal-then-classification paradigm formulate temporal
action localization as the mirror problem of object detection in the temporal
domain. Inspired by the progress in object detection [16] techniques, some
methods employ a two-stage training framework [4, 3, 18, 12, 44, 42, 69, 68] –
they generate a set of class-agnostic segment proposals in the first stage and
predict an action label for each proposal in the second stage. Most recent
methods in this direction focus on improving the proposal generation stage [4,
3, 18, 12, 29, 30, 68, 1], while a few propose a more accurate classification
stage [42, 69].
Recently, some end-to-end trainable architectures have also been proposed [8,
14, 7, 61, 66, 62]. However, these methods also process the video as a
sequence and, thus, have limited receptive field for capturing temporal
information. They do not capture non-sequential temporal dependencies in
action instances. Moreover, these approaches use heuristics during training
(_e.g_. intersection-over-union thresholds) to perform non-maximal suppression
in the set of proposals. This might lead to poor localization performance when
the action instances vary widely in their temporal extents as it might skip
some highly overlapping proposals. To address these problems in object
detection, [5] propose a transformer based end-to-end learnable architecture
that implicitly learns the non-max suppression and perform object detection
using proposals as abstract encodings.
In contrast to the above approaches, we formulate temporal action localization
as a direct set prediction task. We propose to reason over untrimmed videos as
non-sequential entities (_i.e_. graphs) as opposed to existing methods that
perform sequential reasoning. Our approach is inspired by [5] in that we
propose an end-to-end learnable transformer based model for direct set
prediction. But unlike [5], the transformer model in our approach operates
graphs.
Additionally, there are other realms of work on temporal action localization
in weakly supervised setting [43, 58, 21] and spatio-temporal action
localization [48, 24, 17, 15]. These are beyond the scope of this paper.
Action Recognition. Action recognition methods operate on short video clips
that are trimmed such that a single action instance spans the video duration
and, hence, are not suitable for untrimmed videos containing multiple actions.
Nonetheless, models pretrained for the task of action recognition provide
effective feature representations for tasks related to untrimmed videos. A
wide variety of action recognition approaches have been proposed ranging from
earlier methods based on hand-crafted features [27, 9, 55] to convolutional
models such as I3D [46], 3D-CNN [51] through to advanced temporal modeling
[57, 59, 67] and graph modeling [60, 20] techniques. In this paper, we use I3D
[46] pretrained on the Kinetics dataset [6] for feature extraction.
Graph-based Modeling for Videos. The advances in graph convolutional networks
(GCNs) [26] have inspired several recent approaches for video based tasks [36,
60, 34]. Most of the graph based approaches for videos represent either the
input space (_i.e_. videos or derived visual information) as graphs [36, 60,
34, 19] or the output space (_i.e_. labels) as graphs [52]. In contrast, we
design our model based on the insight that both the input space (_i.e_.
features derived from videos) and the output space (_i.e_. labels and
timestamps for the action) are graph-structured for the task of temporal
action localization. Specifically, we propose an encoder-decoder transformer
architecture to learn the mapping between the input and output space.
Furthermore, GCNs require the information pertaining to the nodes and edges a
priori. In contrast, we learn the graph structure (_i.e_. both nodes and
edges) from the data itself using self-attention.
## 3 Proposed Approach
Figure 2: Model Overview. Activity Graph Transformer (AGT) receives a video as
input and directly predicts a set of action instances that appear in the
video. The input video is fed into a backbone network to obtain a compact
representation (Section 3.1.1). Then, the encoder network (Section 3.1.2)
receives the compact video-level representation from the backbone network and
encodes it to a latent graph representation context graph. The decoder network
(Section 3.1.3) receives the context graph along with graph-structured
abstract query encodings action query graph. The decoder transforms the action
query graph to a graph-structured set of embeddings. Each node embedding of
the decoder output is fed into a prediction head (Section 3.1.4). The network
is trained end-to-end (Section 3.1.5) using classification and regression
losses for the action labels and timestamps of the action instances
respectively.
In this section, we present the problem formulation and provide a detailed
description of our proposed approach.
Problem Formulation. The task of temporal action localization involves
prediction of the category labels as well as start and end timestamps of the
actions that occur in a given video. In this work, we formulate this task as a
direct set prediction problem wherein each element in the predicted set
denotes an action instance in a video. Specifically, given a video $V$, the
goal is to predict a set $\mathcal{A}$ where the $i$-th element
$a^{(i)}=(c^{(i)},t^{(i)}_{s},t^{(i)}_{e})$ denotes an action instance in the
video depicting action category $c^{(i)}$ that starts at time $0\leq
t^{(i)}_{s}\leq T$, ends at time $0\leq t^{(i)}_{e}\leq T$, for
$i\in\\{1,2,\ldots,|\mathcal{A}|\\}$. Here, $|\mathcal{A}|$ is the number of
action instances present in the video and $T$ is the duration of the video.
Thus, $|\mathcal{A}|$ and $T$ vary based on the input video.
Towards this goal, we propose Activity Graph Transformer (AGT), an end-to-end
learnable model that receives a video as input and directly infers the set of
action instances (label, start time, end time) in the video. Our approach
consists of: (1) a network that predicts a set of action instances in a single
forward pass; and (2) a loss function to train the network by obtaining a
unique alignment between the predicted and ground truth action instances. We
contend that effective reasoning over untrimmed human activity videos requires
modeling the non-linear temporal structure in the videos. In our approach, we
seek to capture this structure by employing graphs. Specifically, we propose a
novel encoder-decoder transformer network that leverages graph based self-
attention to reason over the videos. We describe the details of our approach
below.
### 3.1 Activity Graph Transformer
As shown in Figure 2, Activity Graph Transformer (AGT) consists of three
components: (1) backbone network to obtain a compact representation of the
input video; (2) transformer network consisting of an encoder network and a
decoder network that operates over graphs; and (3) prediction heads for the
final prediction of action instances of the form (label, start time, end
time). The encoder network receives the compact video-level representation
from the backbone network and encodes it to a latent graph representation,
referred to as context graph. The decoder network receives graph-structured
abstract query encodings (referred to as action query graph) as input along
with the context graph. The decoder uses the context graph to transform the
action query graph to a graph-structured set of embeddings. Each node
embedding of this decoder output is fed into a feed forward network to obtain
predictions of action instances. The whole AGT network is trained end-to-end
using a combination of classification and regression losses for the action
labels and timestamps respectively. Refer to Algorithm 1 for an overview of
one training iteration of AGT. We provide detailed description of the
components below.
#### 3.1.1 Backbone
To obtain a compact representation for the input video $V$ containing $T$
frames, any 3D convolutional network can be used to extract the features. In
our implementation, we chunk the videos into short overlapping segments of 8
frames and use an I3D model pretrained on the Kinetics[6] dataset to extract
features of dimension $C$ ($=2048$) from the segments, resulting in video-
level feature
$\mathbf{v}=[\mathbf{v}^{(1)},\mathbf{v}^{(2)}\ldots\mathbf{v}^{(N_{v})}]$
where $N_{v}$ is the number of chunks used in the feature extraction.
#### 3.1.2 Transformer Encoder
The backbone simply provides a sequence of local features and does not
incorporate the overall context of the video or the temporal structure in the
video. Therefore, we use an encoder network that receives the video-level
feature as input and encodes this video representation to a graph (referred to
as the context graph). Intuitively, the encoder is designed to model the
interactions among the local features using self-attention modules.
The context graph is initialized with video-level feature $\mathbf{v}^{(i)}$
(of dimension $C=2048$) as the $i$-th node for $i\in\\{1,2,\ldots,N_{v}\\}$.
Usually, transformer networks use fixed positional encoding [37] to provide
position information of each element in the input sequence. In contrast, in
our setting, we contend that the video features have a non-linear temporal
structure. Thus, we provide the positional information using learnable
positional encodings $\mathbf{p}_{v}$ as additional information to the video
feature $\mathbf{v}$. The positional encoding $\mathbf{p}_{v}^{(i)}$
corresponds to the $i$-th node in the graph and is of the same dimension as
the node. Next, the graph nodes are from the same video and hence, they are
related to each other. However, their connection information (edges) is not
known a priori. Thus, we model the interactions among these nodes as learnable
edge weights. This is enabled by the graph self-attention module (described
below).
We design the transformer encoder network $\mathbf{E}$ as a sequence of
$L_{e}$ blocks, wherein, an encoder block $\mathbf{E}_{\ell}$ for
$\ell\in\\{1,2,\ldots,L_{e}\\}$ consists of a graph self-attention module
followed by a feed forward network. The output of the encoder network is the
context graph
$\mathbf{h}_{L_{e}}=[\mathbf{h}^{(1)}_{L_{e}},\mathbf{h}^{(2)}_{L_{e}}\ldots\mathbf{h}^{(N_{v})}_{L_{e}}]$
where $\mathbf{h}^{(i)}_{L_{e}}$ is the $i$-th node and is of dimension $d$
(same for each block). The output of the $\ell$-th encoder block
$\mathbf{h}_{\ell}$ and the final output of the encoder $\mathbf{h}_{L_{e}}$
are defined as:
$\begin{split}\mathbf{h}_{0}&=\mathbf{v}\\\
\mathbf{h}_{\ell}&=\mathbf{E}_{\ell}(\mathbf{h}_{\ell-1},\mathbf{p}_{v})\\\
\mathbf{h}_{L_{e}}&=\mathbf{E}_{L_{e}}\circ\cdots\circ\mathbf{E}_{1}(\mathbf{v},\mathbf{p}_{v}).\end{split}$
(1)
Graph Self-Attention. This module aims to model interactions among graph
structured variables along with learnable edge weights. Here, we describe the
graph self-attention module in the context of the $(\ell+1)$-th encoder block
$\mathbf{E}_{\ell+1}$. For simplicity of notation, let $\mathbf{x}$ be the
output of the $\ell$-th block of the encoder, _i.e_.,
$\mathbf{x}=\mathbf{E}_{\ell}(\mathbf{h}_{\ell-1},\mathbf{p}_{v})$.
$\mathbf{x}$ is a graph contains $N_{v}$ nodes
$\mathbf{x}^{(1)},\mathbf{x}^{(2)},\ldots,\mathbf{x}^{(N_{v})}$ which are
connected using learnable edge weights. The graph self-attention module first
performs graph message passing (as described in [54]) to produce the output
$\mathbf{x}^{\prime}$, with the $i$-th node of the output defined as
$\mathbf{x^{\prime}}^{(i)}=\mathbf{x}^{(i)}+\Big{|}\Big{|}_{k=1}^{K}\sigma\Big{(}\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}^{k}\mathbf{W}_{g}^{k}\mathbf{x}^{(j)}\Big{)},$
(2)
where $\Big{|}\Big{|}$ represents concatenation operator, $K$ is the number of
parallel heads in the self-attention module, $\sigma$ is a non-linear function
(leaky ReLU in our case), $\mathcal{N}_{i}$ represents the set of neighbours
of the $i$-th node, $\mathbf{W}_{g}^{k}$ is the learnable transformation
weight matrix. $\alpha_{ij}^{k}$ are the self attention coefficients computed
by the $k$-th attention head described as:
$\alpha_{ij}^{k}=\frac{\exp(f(\mathbf{w}_{a,k}^{T}[\mathbf{W}_{g}^{k}\mathbf{x}^{(i)}||\mathbf{W}_{g}^{k}\mathbf{x}^{(j)}]))}{\sum_{m\in\mathcal{N}_{i}}\exp(f(\mathbf{w}_{a,k}^{T}[\mathbf{W}_{g}^{k}\mathbf{x}^{(i)}||\mathbf{W}_{g}^{k}\mathbf{x}^{(m)}]))},$
(3)
where $\cdot^{T}$ represents a transpose operator, $f$ is a non-linear
activation (leaky ReLU in our case) and $\mathbf{w}_{a,k}$ is the attention
coefficients. $\alpha_{ij}^{k}$ is the attention weight and denotes the
strength of the interaction between $i$-th and $j$-th node of the input graph
of the module. Subsequent to the message passing step, we apply batch
normalization and a linear layer. This is then followed by a standard multi-
head self-attention layer (same as in [53]). Overall, the graph self-attention
module models interactions between the nodes, _i.e_., local features derived
from the video.
Algorithm 1 A training iteration of AGT model
1:video $V$ containing $T$ frames; number of action query encodings $N_{o}$;
ground truth action instances
$\mathcal{A}=\\{a^{(i)}\\}_{i=1}^{|\mathcal{A}|}$
2:Backbone initialized to I3D model [6] pretrained on Kinetics dataset;
initialize action query graph $\mathbf{q}$ with $N_{o}$ random encodings
3:compute features using backbone
4:compute context graph using encoder (see Eq. 1)
5:compute output embeddings using decoder (see Eq. 4)
6:for $i\in 1,2,\ldots,N_{o}$ do
7: predict action instance $\tilde{a}^{(i)}$ using prediction head
8:end for
9:compute optimal matching $\hat{\phi}$ between
$\\{a^{(i)}\\}_{i=1}^{|\mathcal{A}|}$ and
$\\{\tilde{a}^{(i)}\\}_{i=1}^{N_{o}}$ using matcher
10:compute final loss $\mathcal{L}_{H}$ between
$\\{a^{(i)}\\}_{i=1}^{|\mathcal{A}|}$ and
$\\{\tilde{a}^{(\hat{\phi}(i))}\\}_{i=1}^{|\mathcal{A}|}$ using Eq. 9
11:backpropagate $\mathcal{L}_{H}$
#### 3.1.3 Transformer Decoder
Based on the observation that the action instances in the video have a non-
linear temporal structure, we design the decoder to learn a graph-structured
set of embeddings which would subsequently be used for predicting the action
instances. Intuitively, the output graph provided by the decoder serves as the
latent representation for the set of action instances depicted in the video.
The inputs of the transformer decoder are: (1) a graph-structured abstract
query encodings, referred to as action query graph $\mathbf{q}$, containing
$N_{o}$ nodes wherein each node is a learnable positional encoding of
dimension $d$ (same as the dimension used in the encoder); and (2) the context
graph $\mathbf{h}_{L_{e}}$ containing $N_{v}$ nodes (obtained from the
encoder). We assume that the number of nodes in the action query graph $N_{o}$
is fixed and is sufficiently larger than the maximum number of action
instances per video in the dataset. This idea of using representations of
prediction entities as positional query encodings is inspired from [5].
However, unlike the independent queries in [5], we use graph-structured
encodings for the decoder. To learn the interactions among the graph-
structured query embeddings, we use graph self-attention modules (same module
as used in transformer encoder). Additionally, we use graph-to-graph attention
module (described below) to learn interactions between the context graph,
_i.e_., the latent representation of the input video, and the graph-structured
query embeddings, _i.e_., the latent representations of the action queries.
The overall decoder network $\mathbf{D}$ consists of $L_{d}$ blocks, wherein,
a decoder block $\mathbf{D}_{\ell^{\prime}}$ for
$\ell^{\prime}\in\\{1,2,\ldots,L_{d}\\}$ consists of a graph self-attention
module followed by a graph-to-graph attention module, and then a feed forward
network. The decoder block $\mathbf{D}_{\ell^{\prime}}$ has an output
$\mathbf{y}_{\ell^{\prime}}$ and the final output of the decoder
$\mathbf{y}_{L_{d}}=[\mathbf{y}^{(1)}_{L_{d}},\mathbf{y}^{(2)}_{L_{d}},\ldots,\mathbf{y}^{(N_{o})}_{L_{d}}]$.
They are defined as:
$\begin{split}\mathbf{y}_{0}&=\mathbf{q}\\\
\mathbf{y}_{\ell^{\prime}}&=\mathbf{D}_{\ell^{\prime}}(\mathbf{y}_{\ell^{\prime}-1},\mathbf{h}_{L_{e}})\\\
\mathbf{y}_{L_{d}}&=\mathbf{D}_{L_{d}}\circ\cdots\circ\mathbf{D}_{1}(\mathbf{q},\mathbf{h}_{L_{e}})\end{split}$
(4)
Graph-to-Graph Attention. The graph-to-graph attention module aims to learn
the interactions between two different graphs referred to as a source graph
and a target graph. Here, we describe this module in the context of the
decoder block $\mathbf{D}_{\ell^{\prime}+1}$. The input to this block is the
output $\mathbf{y}_{\ell^{\prime}}$ of the previous decoder block
$\mathbf{D}_{\ell^{\prime}}$. This is fed to the graph self-attention module
in the block $\mathbf{D}_{\ell^{\prime}+1}$, and the output is used as the
target graph for the graph-to-graph attention module. The source graph for
this module (in any decoder block) is the context graph $\mathbf{h}_{L_{e}}$.
For simplicity of notation, let $\mathbf{x}_{s}$ denote the source graph
(_i.e_. $\mathbf{h}_{L_{e}}$) and $\mathbf{x}_{t}$ denote the target graph.
Here, the source and target graphs may contain different number of nodes. In
our case, $\mathbf{x}_{s}$ contains $N_{v}$ nodes and $\mathbf{x}_{t}$
contains $N_{o}$ nodes. The graph-to-graph attention module first performs
message passing from source graph to target graph to provide an output
$\mathbf{x^{\prime}}_{t}$, with the $i$-th node defined as
$\mathbf{x^{\prime}}_{t}^{(i)}=\mathbf{x}_{t}^{(i)}+\Big{|}\Big{|}_{k=1}^{K}\sigma\Big{(}\sum_{j\in\mathcal{N}_{i}}\beta_{ij}^{k}\mathbf{W}_{s}^{k}\mathbf{x}_{s}^{(j)}\Big{)},$
(5)
where $\mathbf{W}_{s}^{k}$ is the learnable transformation weight matrix for
the source graph. Other symbols denote the same entities as in graph self-
attention (Section 3.1.2). $\beta_{ij}^{k}$ is the attention coefficient for
$k$-th attention head between $i$-th node of the source graph and $j$-th node
of the target graph computed as:
$\beta_{ij}^{k}=\frac{\exp(f(\mathbf{w}^{st}{{}_{a,k}^{T}}[\mathbf{W}_{s}^{k}\mathbf{x}_{s}^{(i)}||\mathbf{W}_{t}^{k}\mathbf{x}_{t}^{(j)}]))}{\sum_{m\in\mathcal{N}_{i}}\exp(f(\mathbf{w}^{st}{{}_{a,k}^{T}}[\mathbf{W}_{s}^{k}\mathbf{x}_{s}^{(i)}||\mathbf{W}_{t}^{k}\mathbf{x}_{t}^{(m)}]))}$
(6)
where $\mathbf{w}^{st}_{a,k}$ is the graph-to-graph attention coefficients,
and $\mathbf{W}_{s}^{k}$ and $\mathbf{W}_{t}^{k}$ are the learnable
transformation weight matrices for source and target graphs respectively.
Other symbols denote the same entities as in graph self-attention. Similar to
the transformer encoder, subsequent to the message passing step, we apply
batch normalization and a linear layer. This is then followed by a standard
multi-head self-attention layer (same as in [53]). Overall, the graph-to-graph
attention module models the interactions between the latent representations of
the input video and the action queries.
#### 3.1.4 Prediction Heads
The decoder network provides a set of embeddings where the embeddings serve as
the latent representations for the action instances in the video. This output
graph $\mathbf{y}_{L_{d}}$ contains $N_{o}$ nodes. We use these $N_{o}$ node
embeddings to obtain predictions for $N_{o}$ action instances using prediction
heads. The prediction heads consist of a feed forward network (FFN) with ReLU
activation which provides the start time and end time of the action instance
normalized with respect to the overall video duration. Additionally, we use a
linear layer with a softmax function to predict the categorical label
corresponding to the action instance.
Therefore, when provided with the $i$-th node embedding
$\mathbf{y}_{L_{d}}^{(i)}$, the prediction head provides prediction
$\tilde{a}^{(i)}=(\tilde{c}^{(i)},\tilde{t}^{(i)}_{s},\tilde{t}^{(i)}_{e})$
where $\tilde{c}^{(i)}$, $\tilde{t}^{(i)}_{s}$ and $\tilde{t}^{(i)}_{e}$ are
the category label, start time and end time for the $i$-th action instance for
$i\in\\{1,2,\ldots,N_{o}\\}$. Note that the ground truth set would contain a
variable number of action instances, whereas $N_{o}$ is larger than the
maximum number of action instances per video in the dataset. This calls for a
need to suppress irrelevant predictions. We do this by introducing an
additional class label $\varnothing$ indicating no action (similar to [5]). As
such, this non-maximal suppression (typically performed using heuristics in
existing methods [7]) is learnable in our model.
#### 3.1.5 Loss functions
To train the overall network, we align the predictions with the ground truth
action instances using a matcher module which optimizes a pair-wise cost
function. This provides a unique matching between the predicted and ground
truth action instances. Subsequently, our model computes losses corresponding
to these matched pairs of predicted and ground truth action instances to train
the overall network end-to-end.
Matcher. The matcher module finds an optimal matching between the predicted
set of action instances (that contains fixed number of elements for every
video) and the ground truth set of action instances (that contains a variable
number of elements depending on the video). To obtain this matching, we design
a matching cost function and employ the Hungarian algorithm to obtain an
optimal matching between the two sets as described in prior work [49].
Formally, let $\mathcal{A}$ be the ground truth set of action instances
$\mathcal{A}=\\{a^{(i)}\\}^{|\mathcal{A}|}_{i=1}$, where
$a^{(i)}=(c^{(i)},t^{(i)}_{s},t^{(i)}_{e})$ and $\mathcal{\tilde{A}}$ be the
predicted set of action instances
$\mathcal{\tilde{A}}=\\{\tilde{a}^{(i)}\\}^{N_{o}}_{i=1}$ where
$\tilde{a}^{(i)}=(\tilde{c}^{(i)},\tilde{t}^{(i)}_{s},\tilde{t}^{(i)}_{e})$.
In our model, we assume that $N_{o}$ is larger than the number of actions in
any video in the dataset. Therefore, we assume that ground truth set
$\mathcal{A}$ also is a set of size $N_{o}$ by padding the remaining
$(N_{o}-|\mathcal{A}|)$ elements with $\varnothing$ element indicating no
action. The optimal bipartite matching between the two sets reduces to
choosing the permutation of $N_{o}$ elements $\hat{\phi}$ from the set of all
possible permutations $\Phi_{N_{o}}$ that results in lowest value of the
matching cost function $\mathcal{L}_{m}$. Thus,
$\hat{\phi}=\operatorname*{argmin}_{\phi\in\Phi_{N_{o}}}\mathcal{L}_{m}(a^{(i)},\tilde{a}^{(\phi(i))})$,
where $\mathcal{L}_{m}(a^{(i)},\tilde{a}^{(\phi(i))})$ is the matching cost
function between ground truth $a^{(i)}$ and prediction with index $\phi(i)$.
The matching cost function incorporates the class probabilities of the action
instances and the proximity between predicted and ground truth timestamps.
Specifically, we define the cost function as:
$\begin{split}\mathcal{L}_{m}(a^{(i)},\tilde{a}^{(\phi(i))})&=-\mathbbm{1}_{\\{c^{(i)}\neq\varnothing\\}}\tilde{p}_{\phi(i)}(c^{(i)})\\\
&+\mathbbm{1}_{\\{c^{(i)}\neq\varnothing\\}}\mathcal{L}_{s}(s^{(i)},\tilde{s}^{(\phi(i))}),\end{split}$
(7)
where $s^{(i)}=[t^{(i)}_{s},t^{(i)}_{e}]$ and
$\tilde{s}^{(\phi(i))}=[\tilde{t}^{(\phi(i))}_{s},\tilde{t}^{(\phi(i))}_{e}]$,
and $\tilde{p}_{\phi(i)}(c^{(i)})$ is the probability of class $c^{(i)}$ for
prediction $\phi(i)$ and $\mathcal{L}_{s}$ represents segment loss that
measures proximity in the timestamps of the instances. The segment loss is
defined as a weighted combination of an $L_{1}$ loss (sensitive to the
durations of the instances) and an IoU loss (invariant to the durations of the
instances) between the predicted and ground-truth start and end timestamps. It
is expressed as:
$\mathcal{L}_{s}=\lambda_{iou}\mathcal{L}_{iou}(s^{(i)},\tilde{s}^{(\phi(i))})+\lambda_{L1}||s^{(i)}-\tilde{s}^{(\phi(i))}||_{1},$
(8)
where $\lambda_{iou},\lambda_{L1}\in\mathbbm{R}$ are hyperparameters.
Subsequent to obtaining the optimal permutation $\hat{\phi}$, we compute the
Hungarian loss $\mathcal{L}_{H}$ over all the matched pairs as follows:
$\mathcal{L}_{H}=\sum_{i=1}^{N_{o}}\Big{[}-\log\tilde{p}_{\hat{\phi}}(c^{(i)})+\mathbbm{1}_{\\{c^{(i)}\neq\varnothing\\}}\mathcal{L}_{s}(s^{(i)},\tilde{s}^{(\hat{\phi}(i))})\Big{]}.$
(9)
This loss is used to train our AGT model end-to-end. We provide further
implementation details of the model in the supplementary material.
In summary, our proposed Activity Graph Transformer performs temporal action
localization using an encoder-decoder based architecture leveraging graph
based attention modules. We jointly optimize all parameters of our model to
minimize the regression loss for the start and end timestamps of the action
instances and the cross entropy losses for the corresponding action labels.
## 4 Experiments
We conducted several experiments to demonstrate the effectiveness of our
proposed approach. In this section, we report the results of our evaluation.
Datasets. We use three benchmark datasets for evaluation. They vary in their
extent of overlap in action instances, the number of action instances per
video, and the number of action categories in the dataset. Thus, these
datasets together serve as a challenging testbed for our model.
THUMOS14 [23] contains 200 videos in training set and 213 videos in testing
set for the the task of action localization. This dataset has 20 action
categories. The videos contain an average of 15 action instances per video
with an average of 8% overlapping with other instances.
Charades [45] is large scale dataset containing 9848 videos of daily indoor
activities. This dataset has 157 action categories. Videos in the dataset
contain an average of 6 action instances per video with an average of 79% of
overlapping instances in a video. This dataset is challenging because of the
high degree of overlap in the action instances.
EPIC-Kitchens100 [10] contains 700 egocentric videos of daily kitchen
activities. This dataset contains 289 noun and 97 verb classes. Videos in the
dataset contain an average of 128 action instances per video with an average
of 28% overlapping instances in a video.
Table 1: Comparison with state-of-the-art (THUMOS14). We report the mean
average precision at different intersection over union thresholds (mAP@tIoU)
for tIoU$\in\\{0.1,0.2,0.3,0.4,0.5\\}$. $\uparrow$ indicates higher is better.
Method | mAP@tIoU $\uparrow$
---|---
0.1 | 0.2 | 0.3 | 0.4 | 0.5
Oneata _et al_. [35] | 36.6 | 33.6 | 27.0 | 20.8 | 14.4
Wang _et al_. [56] | 18.2 | 17.0 | 14.0 | 11.7 | 08.3
Caba _et al_. [4] | - | - | - | - | 13.5
Richard _et al_. [41] | 39.7 | 35.7 | 30.0 | 23.2 | 15.2
Shou _et al_. [44] | 47.7 | 43.5 | 36.3 | 28.7 | 19.0
Yeung _et al_. [63] | 48.9 | 44.0 | 36.0 | 26.4 | 17.1
Yuan _et al_. [64] | 51.4 | 42.6 | 33.6 | 26.1 | 18.8
Buch _et al_. [3] | - | - | 37.8 | - | 23.0
Shou _et al_. [42] | - | - | 40.1 | 29.4 | 23.3
Yuan _et al_. [65] | 51.0 | 45.2 | 36.5 | 27.8 | 17.8
Buch _et al_. [2] | - | - | 45.7 | - | 29.2
Gao _et al_. [14] | 60.1 | 56.7 | 50.1 | 41.3 | 31.0
Dai _et al_. [8] | - | - | - | 33.3 | 25.6
Xu _et al_. [61] | 54.5 | 51.5 | 44.8 | 35.6 | 28.9
Zhao _et al_. [69] | 66.0 | 59.4 | 51.9 | 41.0 | 29.8
Lin _et al_. [30] | - | - | 53.5 | 45.0 | 36.9
Chao _et al_. [7] | 59.8 | 57.1 | 53.2 | 48.5 | 42.8
Zeng _et al_. [66] | 69.5 | 67.8 | 63.6 | 57.8 | 49.1
Xu _et al_. [62] | 66.1 | 64.2 | 54.5 | 47.6 | 40.2
AGT (Ours) | 72.1 | 69.8 | 65.0 | 58.1 | 50.2
Comparison with state-of-the-art. We compare the performance of our proposed
AGT with the state-of-the-art methods. We use mean average precision as the
metric to evaluate the model. To ensure fair comparison, we use the same
evaluation protocol as used by state-of-the-art methods for each of the
datasets. Table 1 shows that the our AGT achieves upto 3.5% improvement over
state-of-the-art for THUMOS14 dataset and consistently shows performance
improvement across all IoU thresholds. Table 2 shows the comparisons with
state-of-the-art methods on Charades dataset. Our model achieves 13%
improvement in the Charades dataset. We also perform comparison on recenty
released EPIC-Kitchens100 dataset for classification of verb, noun, and action
(_i.e_. both verb and noun) classes. Table 3 indicates that our model performs
consistently for all three tasks for EPIC-Kitchens100 datasets across all IoU
thresholds. Overall, these results clearly show that our proposed method AGT
outperforms the state-of-the-art methods by a considerable margin.
Table 2: Comparison with state-of-the-art (Charades). We report mean average
precision (mAP) computed using Charades_v1_localize setting in [45].
$\uparrow$: higher is better.
Method | mAP $\uparrow$
---|---
Predictive-corrective (Dave _et al_. [11]) | 08.9
Two-stream (Siggurdson _et al_. [45]) | 08.9
Two-stream + LSTM (Siggurdson _et al_. [45]) | 09.6
R-C3D (Xu _et al_. [61]) | 12.7
SSN (Zhao _et al_. [69]) | 16.4
I3D baseline [38] | 17.2
Super-events (Piergiovanni _et al_. [39]) | 19.4
TGM (Piergiovanni _et al_. [39]) | 22.3
Mavroudi _et al_. [33] | 23.7
3D ResNet-50 + super-events (Piergiovanni _et al_. [40]) | 25.2
AGT (Ours) | 28.6
Table 3: Comparison with state-of-the-art (EPIC-Kitchens100). We report mean
average precision at different intersection over union thresholds (mAP@tIoU)
for tIoU$\in\\{0.1,0.2,0.3,0.4,0.5\\}$. We use the validation split in the
original dataset for testing. $\uparrow$ indicates higher is better.
Method | Task | mAP@tIoU $\uparrow$
---|---|---
0.1 | 0.2 | 0.3 | 0.4 | 0.5
| Verb | 10.51 | 09.24 | 07.67 | 06.40 | 05.12
Damen | Noun | 10.71 | 08.73 | 06.75 | 05.05 | 03.35
_et al_. [10] | Action | 06.78 | 06.03 | 04.94 | 04.04 | 03.35
| Verb | 12.01 | 10.25 | 08.15 | 07.12 | 06.14
AGT | Noun | 11.63 | 09.33 | 07.05 | 06.57 | 03.89
(Ours) | Action | 07.78 | 06.92 | 05.53 | 04.22 | 03.86
Impact of graph based reasoning. To demonstrate the importance of reasoning
over videos as graphs, we conducted ablation studies by removing the graph
based reasoning components from either the encoder or the decoder or both
(_i.e_. overall transformer network) in our model. Specifically, this is
implemented by removing the graph message passing layers from the attention
modules (_i.e_., graph self-attention module and graph-to-graph attention
module) in the encoder and/or decoder blocks in the network. Intuitively, when
the graph message passing module is removed from the whole transformer
network, the transformer encoder treats the input as a sequence and the
transformer decoder treats the action queries as independent. Table 4 shows
the performance of these ablated versions of our model. The results clearly
show that eliminating the graph-based reasoning module hurts the localization
performance. The results also suggest that graph-based modeling is more useful
in the encoder than in the decoder. We believe this is because the graph
reasoning performed by the encoder is more useful in capturing the non-
sequential dependencies as it operates directly on the video features. For
better readability, here, we provide the mAP values averaged over the various
intersection-over-union thresholds (tIoU) for THUMOS14 and EPIC-Kitchens100.
For mAP values at specific thresholds, refer to the supplementary.
Table 4: Ablation Study (Impact of graph based reasoning). We report
performance of ablated versions of our AGT model. We report mAP for evaluation
performance (higher is better). We remove graph reasoning in the encoder
($\mathbf{E}$) and/or decoder ($\mathbf{D}$) of the transformer. ✓ and ✗
indicates whether a component (encoder or decoder) contains graph message
passing module or not respectively. EPIC(A), EPIC (V), EPIC (N) indicates task
‘Action’, ‘Verb’, ‘Noun’ classification on EPIC-Kitchens100.
Dataset | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✗ | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✓ | $\mathbf{E}$: ✓/ $\mathbf{D}$: ✗ | $\mathbf{E}$: ✓/ $\mathbf{D}$: ✓
---|---|---|---|---
THUMOS14 | 55.6 | 56.3 | 58.3 | 63.0
Charades | 18.2 | 19.2 | 22.5 | 28.6
EPIC (A) | 03.0 | 03.3 | 05.1 | 05.9
EPIC (V) | 05.7 | 06.1 | 07.4 | 08.7
EPIC (N) | 04.9 | 05.3 | 06.3 | 07.7
Table 5: Ablation Study (Impact of temporal resolution). Performance of our
AGT for different temporal resolutions of input video. Here, SR indicates
sampling rate of frames for feature extraction, _i.e_., SR=1/$k$ means frames
sampled at 1/$k$ -th factor of the original frame rate. EPIC(A), EPIC (V),
EPIC (N) indicate tasks ‘Action’, ‘Verb’, ‘Noun’ on dataset EPIC-Kitchens100.
We report mAP for evaluation (higher is better).
Dataset | SR = 1/8 | SR=1/4
---|---|---
THUMOS14 | 60.2 | 63.0
Charades | 27.3 | 28.6
EPIC (A) | 4.1 | 5.9
EPIC (V) | 7.2 | 8.7
EPIC (N) | 6.4 | 7.7
Impact of temporal resolution. To evaluate the impact of temporal resolution,
we experimented with different frame rates for the input video. Table 5 shows
the results suggesting higher resolution leads to better performance as the
higher temporal resolution provides more information in the input. However,
our results also show that lower resolution does not lead to any major drop in
performance. For better readability, here, we provide the mAP values averaged
over the various intersection-over-union thresholds (tIoU) for THUMOS14 and
EPIC-Kitchens100. mAP values at specific thresholds are available in the
supplementary.
Qualitative Results. We visualize the predictions of the model on two
different samples in Figure 3. The visualizations indicate that our model is
able to predict the correct number of action instances as well as correct
action categories with minimal errors in start and end timestamps. We believe
this is because video content around the start and end timestamps in some
instances do not contain enough information pertaining to the action. We
provide additional visualizations of predictions in the supplementary.
Additionally, refer to the supplementary for experiments on performance of our
model with varied number of layers and heads in the transformer and ablations
of loss functions.
---
Figure 3: Qualitative Results. Visualization of ground truth and predicted
action instances.
## 5 Conclusion
In this paper, we proposed a novel end-to-end learnable encoder-decoder
transformer model for the task of temporal action localization in untrimmed
human activity videos. Our approach aims to model the non-linear temporal
structure in such videos by reasoning over the videos as graphs using graph
self-attention mechanisms. The experimental evaluation showed that our model
achieves state-of-the-art performance on the task of temporal action
localization on challenging human activity datasets. Overall, this work
highlights the importance of reasoning over videos as non-sequential entities
and shows that graph-based transformers are an effective means to model
complex activity videos.
## References
* [1] Yueran Bai, Yingying Wang, Yunhai Tong, Yang Yang, Qiyue Liu, and Junhui Liu. Boundary content graph neural network for temporal action proposal generation. 2020\.
* [2] Shyamal Buch, Victor Escorcia, Bernard Ghanem, Li Fei-Fei, and Juan Carlos Niebles. End-to-end, single-stream temporal action detection in untrimmed videos. In Proceedings of the British Machine Vision Conference (BMVC), 2017\.
* [3] Shyamal Buch, Victor Escorcia, Chuanqi Shen, Bernard Ghanem, and Juan Carlos Niebles. Sst: Single-stream temporal action proposals. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2017.
* [4] Fabian Caba Heilbron, Juan Carlos Niebles, and Bernard Ghanem. Fast temporal activity proposals for efficient detection of human actions in untrimmed videos. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2016.
* [5] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In Proceedings of the European Conference on Computer Vision (ECCV), 2020.
* [6] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [7] Yu-Wei Chao, Sudheendra Vijayanarasimhan, Bryan Seybold, David A Ross, Jia Deng, and Rahul Sukthankar. Rethinking the faster r-cnn architecture for temporal action localization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
* [8] Xiyang Dai, Bharat Singh, Guyue Zhang, Larry S Davis, and Yan Qiu Chen. Temporal context network for activity localization in videos. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017.
* [9] Navneet Dalal, Bill Triggs, and Cordelia Schmid. Human detection using oriented histograms of flow and appearance. In Proceedings of the European Conference on Computer Vision (ECCV), 2006.
* [10] Dima Damen, Hazel Doughty, Giovanni Maria Farinella, , Antonino Furnari, Jian Ma, Evangelos Kazakos, Davide Moltisanti, Jonathan Munro, Toby Perrett, Will Price, and Michael Wray. Rescaling egocentric vision. CoRR, abs/2006.13256, 2020.
* [11] Achal Dave, Olga Russakovsky, and Deva Ramanan. Predictive-corrective networks for action detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [12] Victor Escorcia, Fabian Caba Heilbron, Juan Carlos Niebles, and Bernard Ghanem. Daps: Deep action proposals for action understanding. In Proceedings of the European Conference on Computer Vision (ECCV), 2016.
* [13] Adrien Gaidon, Zaid Harchaoui, and Cordelia Schmid. Temporal localization of actions with actoms. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 2013.
* [14] Jiyang Gao, Zhenheng Yang, Kan Chen, Chen Sun, and Ram Nevatia. Turn tap: Temporal unit regression network for temporal action proposals. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017.
* [15] Rohit Girdhar, Joao Carreira, Carl Doersch, and Andrew Zisserman. Video action transformer network. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
* [16] Ross Girshick. Fast r-cnn. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015.
* [17] Georgia Gkioxari and Jitendra Malik. Finding action tubes. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR), 2015.
* [18] Fabian Caba Heilbron, Wayner Barrios, Victor Escorcia, and Bernard Ghanem. Scc: Semantic context cascade for efficient action detection. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2017.
* [19] Noureldien Hussein, Efstratios Gavves, and Arnold WM Smeulders. Videograph: Recognizing minutes-long human activities in videos. Proceedings of the ICCV Workshop on Scene Graph Representation and Learning (ICCV-W), 2019.
* [20] Ashesh Jain, Amir R Zamir, Silvio Savarese, and Ashutosh Saxena. Structural-rnn: Deep learning on spatio-temporal graphs. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [21] Mihir Jain, Amir Ghodrati, and Cees GM Snoek. Actionbytes: Learning from trimmed videos to localize actions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
* [22] Mihir Jain, Jan Van Gemert, Hervé Jégou, Patrick Bouthemy, and Cees GM Snoek. Action localization with tubelets from motion. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2014.
* [23] Y.-G. Jiang, J. Liu, A. Roshan Zamir, G. Toderici, I. Laptev, M. Shah, and R. Sukthankar. THUMOS challenge: Action recognition with a large number of classes. http://crcv.ucf.edu/THUMOS14/, 2014.
* [24] Vicky Kalogeiton, Philippe Weinzaepfel, Vittorio Ferrari, and Cordelia Schmid. Action tubelet detector for spatio-temporal action localization. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017.
* [25] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video classification with convolutional neural networks. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR), 2014.
* [26] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations (ICLR), 2017\.
* [27] Ivan Laptev. On space-time interest points. International Journal of Computer Vision(IJCV), 2005.
* [28] Colin Lea, Michael D Flynn, Rene Vidal, Austin Reiter, and Gregory D Hager. Temporal convolutional networks for action segmentation and detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [29] Tianwei Lin, Xiao Liu, Xin Li, Errui Ding, and Shilei Wen. Bmn: Boundary-matching network for temporal action proposal generation. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2019.
* [30] Tianwei Lin, Xu Zhao, Haisheng Su, Chongjing Wang, and Ming Yang. Bsn: Boundary sensitive network for temporal action proposal generation. In Proceedings of the European Conference on Computer Vision (ECCV), 2018.
* [31] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. 2017\.
* [32] Shugao Ma, Leonid Sigal, and Stan Sclaroff. Learning activity progression in lstms for activity detection and early detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [33] Effrosyni Mavroudi, Benjamín Béjar Haro, and René Vidal. Representation learning on visual-symbolic graphs for video understanding. In Proceedings of the European Conference on Computer Vision (ECCV), 2020.
* [34] Tushar Nagarajan, Yanghao Li, Christoph Feichtenhofer, and Kristen Grauman. Ego-topo: Environment affordances from egocentric video. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
* [35] Dan Oneata, Jakob Verbeek, and Cordelia Schmid. Action and event recognition with fisher vectors on a compact feature set. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2013.
* [36] Boxiao Pan, Haoye Cai, De-An Huang, Kuan-Hui Lee, Adrien Gaidon, Ehsan Adeli, and Juan Carlos Niebles. Spatio-temporal graph for video captioning with knowledge distillation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
* [37] Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Łukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Image transformer. Proceedings of the International Conference on Machine Learning (ICML), 2018.
* [38] AJ Piergiovanni and Michael Ryoo. Temporal gaussian mixture layer for videos. In Proceedings of the International Conference on Machine Learning (ICML), 2019.
* [39] AJ Piergiovanni and Michael S Ryoo. Learning latent super-events to detect multiple activities in videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
* [40] AJ Piergiovanni and Michael S Ryoo. Avid dataset: Anonymized videos from diverse countries. In Advances in Neural Information Processing Systems (NeurIPS), 2020\.
* [41] Alexander Richard and Juergen Gall. Temporal action detection using a statistical language model. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [42] Zheng Shou, Jonathan Chan, Alireza Zareian, Kazuyuki Miyazawa, and Shih-Fu Chang. Cdc: Convolutional-de-convolutional networks for precise temporal action localization in untrimmed videos. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2017.
* [43] Zheng Shou, Hang Gao, Lei Zhang, Kazuyuki Miyazawa, and Shih-Fu Chang. Autoloc: Weakly-supervised temporal action localization in untrimmed videos. In Proceedings of the European Conference on Computer Vision (ECCV), 2018.
* [44] Zheng Shou, Dongang Wang, and Shih-Fu Chang. Temporal action localization in untrimmed videos via multi-stage cnns. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2016.
* [45] Gunnar A Sigurdsson, Gül Varol, Xiaolong Wang, Ali Farhadi, Ivan Laptev, and Abhinav Gupta. Hollywood in homes: Crowdsourcing data collection for activity understanding. In Proceedings of the European Conference on Computer Vision (ECCV), 2016.
* [46] Karen Simonyan and Andrew Zisserman. Two-stream convolutional networks for action recognition in videos. In Advances in Neural Information Processing Systems (NIPS), 2014\.
* [47] Bharat Singh, Tim K Marks, Michael Jones, Oncel Tuzel, and Ming Shao. A multi-stream bi-directional recurrent neural network for fine-grained action detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [48] Gurkirt Singh, Suman Saha, Michael Sapienza, Philip HS Torr, and Fabio Cuzzolin. Online real-time multiple spatiotemporal action localisation and prediction. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017.
* [49] Russell Stewart, Mykhaylo Andriluka, and Andrew Y Ng. End-to-end people detection in crowded scenes. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [50] Kevin Tang, Bangpeng Yao, Li Fei-Fei, and Daphne Koller. Combining the right features for complex event recognition. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2013.
* [51] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015.
* [52] Yao-Hung Hubert Tsai, Santosh Divvala, Louis-Philippe Morency, Ruslan Salakhutdinov, and Ali Farhadi. Video relationship reasoning using gated spatio-temporal energy graph. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
* [53] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NIPS), 2017\.
* [54] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.
* [55] Heng Wang and Cordelia Schmid. Action recognition with improved trajectories. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2013.
* [56] Limin Wang, Yu Qiao, and Xiaoou Tang. Action recognition and detection by combining motion and appearance features. THUMOS14 Action Recognition Challenge, 2014.
* [57] Limin Wang, Yu Qiao, and Xiaoou Tang. Action recognition with trajectory-pooled deep-convolutional descriptors. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015.
* [58] Limin Wang, Yuanjun Xiong, Dahua Lin, and Luc Van Gool. Untrimmednets for weakly supervised action recognition and detection. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [59] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua Lin, Xiaoou Tang, and Luc Van Gool. Temporal segment networks: Towards good practices for deep action recognition. In Proceedings of the European Conference on Computer Vision (ECCV), 2016.
* [60] Xiaolong Wang and Abhinav Gupta. Videos as space-time region graphs. In Proceedings of the European Conference on Computer Vision (ECCV), 2018.
* [61] Huijuan Xu, Abir Das, and Kate Saenko. R-c3d: Region convolutional 3d network for temporal activity detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [62] Mengmeng Xu, Chen Zhao, David S Rojas, Ali Thabet, and Bernard Ghanem. G-tad: Sub-graph localization for temporal action detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2020.
* [63] Serena Yeung, Olga Russakovsky, Greg Mori, and Li Fei-Fei. End-to-end learning of action detection from frame glimpses in videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2678–2687, 2016.
* [64] Jun Yuan, Bingbing Ni, Xiaokang Yang, and Ashraf A Kassim. Temporal action localization with pyramid of score distribution features. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [65] Zehuan Yuan, Jonathan C Stroud, Tong Lu, and Jia Deng. Temporal action localization by structured maximal sums. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
* [66] Runhao Zeng, Wenbing Huang, Mingkui Tan, Yu Rong, Peilin Zhao, Junzhou Huang, and Chuang Gan. Graph convolutional networks for temporal action localization. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2019.
* [67] Bowen Zhang, Limin Wang, Zhe Wang, Yu Qiao, and Hanli Wang. Real-time action recognition with enhanced motion vector cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [68] Peisen Zhao, Lingxi Xie, Chen Ju, Ya Zhang, Yanfeng Wang, and Qi Tian. Bottom-up temporal action localization with mutual regularization. In Proceedings of the European Conference on Computer Vision (ECCV), 2020.
* [69] Yue Zhao, Yuanjun Xiong, Limin Wang, Zhirong Wu, Xiaoou Tang, and Dahua Lin. Temporal action detection with structured segment networks. In Proceedings of the IEEE International Conference on Computer Vision (CVPR), 2017.
## A Appendix
We report additional quantitative results and qualitative analysis and provide
implementation details of our model. Specifically, this document contains the
following.
* •
Code provided in the folder agt_code.zip
* •
Additional quantitative evaluation
* –
Section A.1.1: Supplemental tables for Table 4 and Table 5 from the main paper
to report mAP values at specific IoU thresholds
* –
Section A.1.2: Ablation study of loss function (Eq. 9 in the main paper).
* –
Section A.1.3: Impact of different number of layers in transformer encoder and
decoder.
* –
Section A.1.4: Impact of different number of heads in attention modules of the
transformer.
* –
Section A.1.5: Impact of different number of nodes in the action query graph.
* •
Additional qualitative analysis
* –
Section A.2.1: Visualization of predictions
* –
Section A.2.2: Visualization of graphs learned by the model
* –
Section A.2.3: Analysis of AGT predictions based on the duration of action
instances
* •
Technical details
* –
Section A.3.1: Details of the architecture of AGT
* –
Section A.3.2: Details of initialization, data augmentation, and
hyperparameters.
### A.1 Additional Quantitative Evaluation
In this section, we report the quantitative evaluation of our proposed AGT
model to supplement the quantitative evaluation in the main paper.
#### A.1.1 Supplemental Tables
In the main paper, we only reported the mAP averaged over different IoU
thresholds for THUMOS14 and EPIC-Kitchens100 dataset (Table 4 and Table 5 in
main paper). For completeness, we report mAP at specific IoU thresholds in
Table T1 and Table T2.
Table T1: Supplemental Tables: Impact of graph based reasoning. We report
performance of ablated versions of our AGT model. We remove graph reasoning in
the encoder ($\mathbf{E}$) and/or decoder ($\mathbf{D}$) of the transformer. ✓
and ✗ indicates whether a component (encoder or decoder) contains graph
message passing module or not respectively. EPIC (A), EPIC (V), EPIC (N)
indicates task ‘Action’, ‘Verb’, ‘Noun’ classification on EPIC-Kitchens100. We
report the mean average precision at different intersection over union
thresholds (mAP@tIoU) for tIoU$\in\\{0.1,0.2,0.3,0.4,0.5\\}$. $\uparrow$
indicates higher is better.
Dataset | Model | mAP@tIoU $\uparrow$
---|---|---
0.1 | 0.2 | 0.3 | 0.4 | 0.5
| $\mathbf{E}$: ✗/ $\mathbf{D}$: ✗ | 64.6 | 60.8 | 59.1 | 51.2 | 40.3
THUMOS14 | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✓ | 65.1 | 62.4 | 60.3 | 52.4 | 41.3
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✗ | 67.1 | 64.4 | 62.5 | 53.6 | 44.9
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✓ | 72.1 | 69.8 | 65.0 | 58.1 | 50.2
| $\mathbf{E}$: ✗/ $\mathbf{D}$: ✗ | 9.4 | 6.9 | 5.2 | 4.5 | 2.5
EPIC (V) | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✓ | 9.9 | 7.5 | 5.5 | 4.9 | 2.7
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✗ | 11.4 | 9.0 | 6.9 | 6.3 | 3.4
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✓ | 12.0 | 10.3 | 8.2 | 7.1 | 6.1
| $\mathbf{E}$: ✗/ $\mathbf{D}$: ✗ | 8.9 | 5.4 | 4.9 | 3.6 | 1.7
EPIC (N) | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✓ | 9.2 | 6.0 | 5.1 | 4.2 | 2.0
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✗ | 10.1 | 8.0 | 6.8 | 5.2 | 2.3
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✓ | 11.6 | 9.3 | 7.1 | 6.6 | 3.9
| $\mathbf{E}$: ✗/ $\mathbf{D}$: ✗ | 4.8 | 4.1 | 2.9 | 2.1 | 1.5
EPIC (A) | $\mathbf{E}$: ✗/ $\mathbf{D}$: ✓ | 5.1 | 4.3 | 3.2 | 2.3 | 1.8
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✗ | 7.3 | 6.1 | 5.0 | 3.9 | 3.7
| $\mathbf{E}$: ✓/ $\mathbf{D}$: ✓ | 7.8 | 6.9 | 5.5 | 4.2 | 3.9
Table T2: Supplemental Tables: Impact of temporal resolution. Performance of
our AGT for different temporal resolutions of input video. Here, SR indicates
sampling rate of frames for feature extraction, _i.e_., SR=1/$k$ means frames
sampled at 1/$k$ -th factor of the original frame rate. EPIC (A), EPIC (V),
EPIC (N) indicate tasks ‘Action’, ‘Verb’, ‘Noun’ classification on dataset
EPIC-Kitchens100. We report the mean average precision at different
intersection over union thresholds (mAP@tIoU) for
tIoU$\in\\{0.1,0.2,0.3,0.4,0.5\\}$. $\uparrow$ indicates higher is better.
Dataset | Model | mAP@tIoU $\uparrow$
---|---|---
0.1 | 0.2 | 0.3 | 0.4 | 0.5
THUMOS14 | SR=1/8 | 70.4 | 65.8 | 62.3 | 54.1 | 48.6
| SR=1/4 | 72.1 | 69.8 | 65.0 | 58.1 | 50.2
EPIC (V) | SR=1/8 | 10.9 | 8.5 | 6.3 | 5.5 | 4.9
| SR=1/4 | 12.0 | 10.3 | 8.2 | 7.1 | 6.1
EPIC (N) | SR=1/8 | 10.2 | 8.1 | 5.9 | 5.2 | 2.8
| SR=1/4 | 11.6 | 9.3 | 7.1 | 6.6 | 3.9
EPIC (A) | SR=1/8 | 7.0 | 5.0 | 4.1 | 2.5 | 1.9
| SR=1/4 | 7.8 | 6.9 | 5.5 | 4.2 | 3.9
#### A.1.2 Ablation Study: Loss function
Note that for any version of the loss function, the model requires cross
entropy loss to be able to classify the action label pertaining to an
instance. The model also requires some form of regression loss to produce
predictions pertaining to the start and end timestamps of an action instance.
Recall, our overall loss (see Eq. (9) in the main paper) is a combination of
cross-entropy loss and regression loss, _i.e_., segment loss
$\mathcal{L}_{s}$. The segment loss contains two components: $L_{1}$ loss and
IoU loss $\mathcal{L}_{iou}$. Table T3 shows the results of the performance of
our model when trained with ablated versions of the segment loss. The results
indicate that the models trained with only $L_{1}$ loss perform better than
the ones trained with only IoU loss $\mathcal{L}_{iou}$. Additionally, models
trained with both losses are better than the ones trained with only one of the
losses. Nonetheless, all versions of our AGT model perform better than state-
of-the-art methods. We only provide the mAP values averaged over the various
intersection-over-union thresholds (tIoU) for THUMOS14 and EPIC-Kitchens100.
Table T3: Ablation Study: Loss function. We report performance of our AGT
model when trained with ablated versions of the loss function. We report mAP
for evaluation performance (higher is better). We train the model with a
combination of cross-entropy loss and segment loss containing $L_{1}$ loss
and/or IoU loss $\mathcal{L}_{iou}$. ✓ and ✗ indicate whether the specific
component of the segment loss is used or not respectively. EPIC(A), EPIC (V),
EPIC (N) indicate tasks ‘Action’, ‘Verb’, ‘Noun’ classification on EPIC-
Kitchens100.
Dataset | $L_{1}$: ✓ | $L_{1}$: ✗ | $L_{1}$: ✓
---|---|---|---
| $\mathcal{L}_{iou}$: ✗ | $\mathcal{L}_{iou}$: ✓ | $\mathcal{L}_{iou}$: ✓
THUMOS14 | 61.3 | 59.6 | 63.0
Charades | 26.0 | 25.3 | 28.6
EPIC (A) | 04.8 | 03.7 | 05.9
EPIC (V) | 07.1 | 06.4 | 08.7
EPIC (N) | 06.3 | 05.0 | 07.7
#### A.1.3 Impact of number of layers
Table T4 shows the results of the performance of our model with different
number of layers in encoder and decoder component of the transformer. While
increase in number of layers increases the training time, we did not observe
much difference in the performance of the model with increased depth of the
transformer components. We only provide the mAP values averaged over the
various intersection-over-union thresholds (tIoU) for THUMOS14 and EPIC-
Kitchens100.
Table T4: Impact of number of layers. We report performance of our AGT model
with different number of layers in encoder and decoder. We report mAP for
evaluation performance (higher is better). EPIC (A), EPIC (V), EPIC (N)
indicates task ‘Action’, ‘Verb’, ‘Noun’ classification on EPIC-Kitchens100. #E
indicates number of layers in encoder and #D indicates number of layers in
decoder.
Dataset | #E | #D | mAP
---|---|---|---
| 4 | 2 | 62.5
THUMOS14 | 4 | 4 | 63.0
| 2 | 4 | 62.7
| 4 | 2 | 28.0
Charades | 4 | 4 | 28.6
| 2 | 4 | 28.2
| 4 | 2 | 5.5
EPIC (A) | 4 | 4 | 5.9
| 2 | 4 | 5.6
| 4 | 2 | 8.5
EPIC (V) | 4 | 4 | 8.7
| 2 | 4 | 8.6
| 4 | 2 | 7.2
EPIC (N) | 4 | 4 | 7.7
| 2 | 4 | 7.3
#### A.1.4 Impact of number of heads
Table T5 shows the results of the performance of our AGT model with different
number of heads in the attention modules of the transformer. The results
suggest a slight improvement with more number of heads in the transformer
network. We only provide the mAP values averaged over the various
intersection-over-union thresholds (tIoU) for THUMOS14 and EPIC-Kitchens100.
Table T5: Impact of number of heads. We report performance of our AGT model
with different number of heads in the attention modules of the transformer
network. We report mAP for evaluation performance (higher is better). EPIC
(A), EPIC (V), EPIC (N) indicates task ‘Action’, ‘Verb’, ‘Noun’ classification
on EPIC-Kitchens100. #heads indicates number of heads in attention modules of
the transformer.
Dataset | #heads | mAP
---|---|---
THUMOS14 | 8 | 63.0
| 4 | 61.4
Charades | 8 | 28.6
| 4 | 26.4
EPIC (A) | 8 | 5.9
| 4 | 5.2
EPIC (V) | 8 | 8.7
| 4 | 8.4
EPIC (N) | 8 | 7.7
| 4 | 7.1
#### A.1.5 Impact of action query graph size
Table T6 shows the results of the performance of our AGT model with different
number of node encodings in the action query graph. Intuitively, a very large
size of action query graph implies the model will require more time to learn
the non-maximal suppression of the irrelevant predictions. On the other hand,
a very small size of action query graph might limit the ability of model to
learn complex structure in the action instances. Note that, any value used for
our experiments is higher than the maximum number of action instances per
video in the dataset. The results suggest minor improvement with more number
of nodes in the action query graph, however, the models with more number of
nodes require longer training times. Our experiments also suggest that when
the size of the action query graph is reduced, the localization performance of
our model degrades. We only provide the mAP values averaged over the various
intersection-over-union thresholds (tIoU) for THUMOS14 and EPIC-Kitchens100.
Table T6: Impact of action query graph size. We report performance of our AGT
model with different number of nodes in the action query graph. We report mAP
for evaluation performance (higher is better). EPIC (A), EPIC (V), EPIC (N)
indicates task ‘Action’, ‘Verb’, ‘Noun’ classification on EPIC-Kitchens100.
#queries indicates number of nodes in the action query graph.
Dataset | #queries | mAP
---|---|---
| 150 | 59.1
THUMOS14 | 300 | 63.0
| 900 | 63.2
| 30 | 24.0
Charades | 50 | 28.6
| 100 | 28.6
| 900 | 4.3
EPIC (A) | 1500 | 5.9
| 2000 | 7.0
| 900 | 7.0
EPIC (V) | 1500 | 8.7
| 2000 | 8.8
| 900 | 6.1
EPIC (N) | 1500 | 7.7
| 2000 | 7.9
### A.2 Additional Qualitative Analysis
In this section, we visualize the results of our proposed model AGT to
supplement the qualitative analysis in the main paper.
#### A.2.1 Visualization: Predictions
---
Figure F1: Visualization: Predictions. Visualization of predictions and
groundtruth action instances
We provide additional visualizations of the predictions of our AGT on several
diverse samples in Figure F1. The visualizations indicate that our model is
able to predict the correct number of action instances as well as most of the
correct action categories with minimal errors in start and end timestamps for
videos containing overlapping instances with varying temporal extents.
#### A.2.2 Visualization: Learned Graphs
|
---|---
Figure F2: Visualization: Learned Graphs. Visualizations of embeddings
corresponding to the last layer of the decoder and ground truth instances. The
thickness of edges show the strength of interaction between the nodes. For
ease of visibility, the nodes have been numbered based on the order of their
predictions sorted with respect to the start time (_i.e_., node 0 represents
the instance that starts first). These visualizations demonstrate that the
model indeed learns non-linear dependencies between the action instances in a
video. The legend below each figure shows the action labels corresponding to
the color coded elements. For details on the visualization process, please
refer to Section A.2.2
.
We visualize the learned action query graph in Figure F2. by observing the
graph embeddings obtained from the last layer of decoder. For better
visibility, we do not plot the nodes (or their edges) that are classified as
no action (_i.e_. class label $\varnothing$) by the prediction head. Note that
the edge matrix is also learnable in our model. For the purpose of this
visualization, we obtained the edge weights from the attention coefficients in
the self-attention based graph message passing module . We show samples with
reoccurring and/or overlapping action instances. The visualizations
demonstrate that the model indeed learns non-linear dependencies among the
action instances that appear in the video.
#### A.2.3 Analysis: Effect of Action Instance Durations
We conduct further analysis to study the performance of our model in terms of
the durations of the action instances. Figure F3 shows the trend of
segmentation error, _i.e_., $L_{1}$ norm computed between the ground truth and
predicted timestamps of actions instances plotted against the duration of the
ground truth instances (normalized with respect to the video duration). The
error is computed over normalized values of the timestamps. This analysis
indicates that action instances with larger durations (with respect to the
whole video duration) have lower segmentation errors in their predictions as
compared to the instances with smaller durations.
Figure F3: Analysis (THUMOS14). Analysis of segmentation error (L1 loss) with
respect to the duration of corresponding ground truth instances. All the
values are normalized with respect to the overall video duration. We observe
that the action instances of longer durations have lower segmentation errors
in their predictions. Figure F4: Detailed Architecture Architecture of
Activity Graph Transformer. Please see Section A.3.1 for details. ‘Q’,‘K’,‘V’
are query, key and value to the self-attention layer as described in [53].
### A.3 Technical Details
In this section, we provide additional implementation details to supplement
the model section in the main paper.
#### A.3.1 Additional details
Detailed Architecture. Figure F4 presents the architecture of our AGT in
detail. Activity Graph Transformer (AGT) consists of three components: (1)
backbone network to obtain features corresponding to the input video; (2)
transformer network consisting of an encoder network and a decoder network
that operates over graphs; and (3) prediction heads for the final prediction
of action instances of the form (label, start time, end time). The encoder
network receives the compact video-level representation from the backbone
network and encodes it to a latent graph representation, referred to as
context graph. The decoder network receives graph-structured abstract query
encodings (referred to as action query graph) as input along with the context
graph. The decoder uses the context graph to transform the action query graph
to a graph-structured set of embeddings. Each node embedding of this decoder
output is fed into a prediction head to obtain predictions of action
instances. The whole AGT network is trained end-to-end using a combination of
classification and regression losses for the action labels and timestamps
respectively.
Positional Encoding. Positional encoding layer consists of a layer that
retrieves encodings based on an integer index provided to it. In our case,
given a video feature
$\mathbf{v}=[\mathbf{v}^{(1)},\mathbf{v}^{(2)}\ldots\mathbf{v}^{(N_{v})}]$,
the positional encoding layer receives input $i$ and provides an embedding
$\mathbf{p}_{v}^{(i)}$ corresponding to the $i$-th element of the video
feature $\mathbf{v}^{(i)}$ where $i\in{1,2,\ldots,N_{v}}$. In our
implementation, the embedding size is same as that of the video feature so as
to allow addition of the positional encodings and input video features. Since
the weights of the layer are learnable during training, the positional
encoding layer is learnable. We use torch.nn.Embedding in Pytorch to implement
it. This layer initialization requires maximum possible value of $N_{v}$ in
the features corresponding to the video.
Action Query Graph. Similar to positional encoding layer, the $N_{o}$
encodings in the action query graph $\mathbf{q}$ is obtained using an
embedding layer. Specifically, the layer receives $i$ as input to provide
$i$-th node $\mathbf{q}^{(i)}$ of the query graph where
$i\in{1,2,\ldots,N_{o}}$. In our implementation, we use torch.nn.Embedding in
Pytorch to implement this. The weights of this layer are learnable during
training.
Losses. For completeness, we describe the IoU loss ($\mathcal{L}_{iou}$) which
is used as a component of segment loss $\mathcal{L}_{s}$ to train our model.
The segment loss is described as:
$\mathcal{L}_{s}=\lambda_{iou}\mathcal{L}_{iou}(s^{(i)},\tilde{s}^{(\phi(i))})+\lambda_{L1}||s^{(i)}-\tilde{s}^{(\phi(i))}||_{1},$
(10)
where $\lambda_{iou},\lambda_{L1}\in\mathbbm{R}$ are hyperparameters.
$\mathcal{L}_{iou}(s^{(i)},\tilde{s}^{(\phi(i))})=1-\frac{|s^{(i)}\cap\tilde{s}^{(\phi(i))}|}{|s^{(i)}\cup\tilde{s}^{(\phi(i))}|}$
(11)
where $|.|$ is the duration of the instance, _i.e_., difference between end
and start timestamp.
#### A.3.2 Training Details
Feature Extraction & Data augmentation. To obtain I3D features corresponding
to an input video $V$ containing $T$ frames sampled at a specific sample rate,
we first divide the video into short overlapping segments of 8 frames with an
overlap of 4 frames resulting in $T^{\prime}$ chunks. We use an I3D model
pretrained on the Kinetics [6] dataset to extract features of dimension $C$
($=2048$). In our implementation, we obtain two-stream features (both RGB and
flow streams). We obtain features for these $T^{\prime}$ chunks to obtain a
tensor of size $T^{\prime}\times 2048$. Here, the length of the video $T$
depends on the duration of the video, and, hence the size of the temporal
channel (_i.e_. $T^{\prime}$) of the feature tensor varies based on the input.
To prevent severe overfitting, we perform data augmentation to train our model
on the features directly obtained from I3D model (described above). We use a
hyperparameter $N_{v}^{max}$ as the maximum size of temporal channel used for
training. This helps in stabilizing the training as the video datasets contain
high variance in their duration. If the size of temporal channel of the video
tensor $T^{\prime}$ is less than $N_{v}^{max}$, we repeat each element in the
temporal channel $\gamma$ times ($\gamma=4$) in our implementation to obtain a
modified tensor of size $\gamma T^{\prime}\times 2048$ and then randomly
sample $T^{\prime}$ elements from the modified tensor. If the size of temporal
channel of the video tensor $T^{\prime}$ is more than $N_{v}^{max}$, we just
randomly sample $T^{\prime}$ elements from the modified tensor. Note that,
positional encoding is applied on this feature of size
$N_{v}=\min(T^{\prime},N_{v}^{max})$.
We find such data augmentation during training to be crucial to prevent
overfitting and obtain good performance of our model, especially for smaller
datasets such as THUMOS14. During testing, if the size of temporal channel of
the video tensor $T^{\prime}$ is less than $N_{v}^{max}$, we don’t perform any
augmentation. If the size of temporal channel of the video tensor $T^{\prime}$
is more than $N_{v}^{max}$, we uniformly sample $T^{\prime}$ elements from the
feature in order to match the maximum index of the positional encoding layer.
Furthermore, to perform training in minibatches, we apply $0$-padding to
ensure all elements have the same size as the largest element of the batch.
For training efficiency and minimizing the amount of $0$-padding, we sort all
the dataset based on the duration of the video. We observe that this type of
batch formation leads to improvement in training speed without affecting the
model performance.
Hyperparameters. We provide the hyperparameters used to train our model below.
We train all our models using AdamW optimizer [31] with a learning rate of
1e-5 and a weight decay of 1e-5 for 3000k steps. We reduce the learning rate
by factor of 10 after 2000k steps. The hyperparameters in the loss functions
$\lambda_{L1}$ and $\lambda_{iou}$ are set to 5 and 3 respectively for all our
experiments. All the learnable weights are initialized using Xavier
initialization.
For our experiments, we sample frames at 1/4 of the original frame rate and
obtain the I3D features as decribed earlier. We do not finetune the I3D model.
We mention the dataset specific hyperparameters below.
THUMOS14. We do not use dropout for this dataset. We use maximum number of
nodes in the context graph $N_{v}^{max}$ equal to 256. The size of the action
query graph is 300 for our experiments (except when conducting ablation on the
size of action query graph). We use base model dimension in the transformer as
512 and set the number of encoder and decoder layers as 4 (except when
conducting ablation on the number of layers).
Charades. We use dropout with default probability $0.1$. We use maximum
number of nodes in the context graph $N_{v}^{max}$ equal to 64. The size of
the action query graph is 100 for our experiments (except when conducting
ablation on the size of action query graph). We use base model dimension in
the transformer as 512 and set the number of encoder and decoder layers as 4
(except when conducting ablation on the number of layers).
Epic-Kitchens100. We do not use dropout for this dataset. We use maximum
number of nodes in the context graph $N_{v}^{max}$ equal to 1024. The size of
the action query graph is 1200 for our experiments (except when conducting
ablation on the size of action query graph). We use base model dimension in
the transformer as 512 and set the number of encoder and decoder layers as 4
(except when conducting ablation on the number of layers).
|
# Approximation of Discontinuous Signals by Exponential Sampling Series
###### Abstract.
We analyse the behaviour of the exponential sampling series $S_{w}^{\chi}f$ at
jump discontinuity of the bounded signal $f.$ We obtain a representation lemma
that is used for analysing the series $S_{w}^{\chi}f$ and we establish
approximation of jump discontinuity functions by the series $S_{w}^{\chi}f.$
The rate of approximation of the exponential sampling series $S_{w}^{\chi}f$
is obtained in terms of logarithmic modulus of continuity of functions and the
round-off and time-jitter errors are also studied. Finally we give some
graphical representation of approximation of discontinuous functions by
$S_{w}^{\chi}f$ using suitable kernels.
Keywords: Exponential Sampling Series, Discontinuous Functions, Logarithmic
Modulus of Smoothness, Rate of Approximation, Round-off and Time Jitter Errors
Mathematics Subject Classification(2010): 41A25, 26A15, 41A35.
SATHISH KUMAR ANGAMUTHU ${}^{1},$ PRASHANT KUMAR2 and DEVARAJ PONNAIAN3
1,2Department of Mathematics, Visvesvaraya National Institute of Technology
Nagpur, Nagpur, Maharashtra-440010, India
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
3School of Mathematics, Indian Institute of Science Education and Research,
Thiruvananthapuram, India.
<EMAIL_ADDRESS>
## 1\. Introduction and Preliminaries
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers and let
$\chi$ be a real valued function defined on $\mathbb{R}^{+}.$ For
$\nu\in\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\},$ the algebraic moments of order
$\nu$ is defined by
$m_{\nu}(\chi,u):=\sum_{k=-\infty}^{+\infty}\chi(e^{-k}u)(k-\log
u)^{\nu},\hskip 14.22636pt\forall\ u\in\mathbb{R}^{+}.$
In a similar way, we can define the absolute moment of order $\nu$ as
$M_{\nu}(\chi,u):=\sum_{k=-\infty}^{+\infty}|\chi(e^{-k}u)||k-\log
u|^{\nu},\hskip 14.22636pt\forall\ u\in\mathbb{R}^{+}.$
We define $\displaystyle
M_{\nu}(\chi):=\sup_{u\in\mathbb{R}^{+}}M_{\nu}(\chi,u).$ We say that $\chi$
is a kernel if it satisfies the following conditions:
* (i)
for every $u\in\mathbb{R}{{}^{+}},$
$\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}u)=1,$
* (ii)
for some $\nu>0,$ $\displaystyle
M_{\nu}(\chi,u)=\sup_{u\in\mathbb{R}^{+}}\sum_{k=-\infty}^{+\infty}|\chi(e^{-k}u)||k-\log
u|^{\nu}<+\infty.$
Let $\Phi$ denote the set of all functions satisfying conditions (i) and (ii).
For $t\in\mathbb{R}^{+},$ $\chi\in\Phi$ and $w>0,$ the exponential sampling
series for a function $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ is defined by
([6])
$(S_{w}^{\chi}f)(t)=\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})f(e^{\frac{k}{w}}).$
(1.1)
It is easy to see that the series $S_{w}^{\chi}f$ is well defined for $f\in
L^{\infty}(\mathbb{R}^{+}).$ Using the above sampling series $S_{w}^{\chi}f$
one can reconstruct the functions which are not Mellin-band limited. Recently,
Bardaro et.al. [4] pointed out that the study of Mellin-band limited functions
are different from that of Fourier-band limited functions. Mamedov was the
first person who studied the Mellin theory in [17] and then Butzer et.al.
further developed the Mellin’s theory and studied its approximation properties
in [10, 11, 9, 13]. The reconstruction using exponential sampling formula was
first studied by Butzer and Jansche in [9]. The pointwise and uniform
convergence of the series $S_{w}^{\chi}f$ for continuous functions was
analysed in [6] and the convergence of $S_{w}^{\chi}f$ was studied in Mellin-
Lebesgue spaces. Recently Bardaro et.al. studied various approximation results
using Mellin transform which can be seen in [5, 2, 3, 4, 6]. To improve the
rate of convergence, a linear combination of $S_{w}^{\chi}f$ was taken in [1].
The exponential sampling series with the sample points which are exponentially
spaced on $\mathbb{R}{{}^{+}}$ has been obtained as solution of some
mathematical model related to light scattering, Fraunhofer diffraction and
radio astronomy (see [8, 15, 16, 18]).
The approximation of discontinuous functions by classical sampling operators
was first initiated by Butzer et.al.[12]. Further, the Kantorovich sampling
series for discontinuous signals was analysed in [14]. Inspired by these works
we analyse the behaviour of exponential sampling series (1.1) as
$w\rightarrow\infty$ for discontinuous functions at the jump discontinuities,
i.e., at a point $t$ where the limits
$f(t+0):=\lim_{p\rightarrow 0^{+}}f(t+p),$
and
$f(t-0):=\lim_{p\rightarrow 0^{+}}f(t-p)$
exists and are different. For a kernel $\chi,$ we define the functions
$\psi_{\chi}^{+}(u):=\sum_{k<\log u}\chi(ue^{-k}),$
and
$\psi_{\chi}^{-}(u):=\sum_{k>\log u}\chi(ue^{-k}).$
We observe that $\psi_{\chi}^{+}(u)$ and $\psi_{\chi}^{-}(u)$ are recurrent
functions with fundamental interval $[1,e]$. Now we shall recall the
definition of Mellin transform. Let $L^{p}(\mathbb{R}^{+}),$ $p\in[1,\infty)$
be the set of all Lebesgue measurable and $p$-integrable functions defined on
$\mathbb{R}^{+}.$ For $c\in\mathbb{R}$, we define the space
$X_{c}=\\{f:\mathbb{R}^{+}\rightarrow\mathbb{C}:f(\cdot)(\cdot)^{c-1}\in
L^{1}(\mathbb{R}^{+})\\}$
equipped with the norm
$\|f\|_{X_{c}}=\|f(\cdot)(\cdot)^{c-1}\|_{1}=\int_{0}^{+\infty}|f(y)|y^{c-1}dy.$
For $f\in X_{c},$ it’s Mellin transform is defined by
$\widehat{[f]_{M}}(s):=\int_{0}^{+\infty}y^{s-1}f(y)\ dy\ ,\,\
(s=c+it,t\in\mathbb{R}).$
We say that a function $f\in X_{c}\cap C(\mathbb{R}^{+}),c\in\mathbb{R}$ is
Mellin band-limited in the interval $[-\kappa,\kappa],$ if
$\widehat{[f]_{M}}(c+it)=0$ for all $|t|>\kappa,\ \kappa\in\mathbb{R}^{+}.$
The paper is organized as follows. In section 2, we prove the representation
lemma for the exponential sampling series (1.1) and using this lemma we
analyse the approximation of discontinuous functions by $S_{w}^{\chi}f$ in
Theorem 2, Theorem 3 and Theorem 5. Further we analyse the degree of
approximation for the sampling series (1.1) in-terms of logarithmic modulus of
smoothness in section 3. In section 4, we study the round-off and time jitter
errors for these sampling series. In example section, we have given a
construction of a family of Mellin-band limited kernels such that $\chi(1)=0$
for which $S_{w}^{\chi}f$ converge at any jump discontinuities. Further, the
convergence at discontinuity points of the sampling series $S_{w}^{\chi}{f}$
has been tested numerically and numerical results are provided in Tables 1, 2
and 3.
## 2\. Approximation of Discontinuous Signals
For any given bounded function $f:\mathbb{R}^{+}\rightarrow\mathbb{R},$ we
first prove the following representation lemma for the sampling series
$S_{w}^{\chi}f.$ Throughout this section we assume that the right and left
limits of $f$ at $t\in\mathbb{R}^{+}$ exist and are finite.
###### Lemma 1.
For a given bounded function $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ and a
fixed $t\in\mathbb{R}^{+},$ let $h_{t}:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be
defined by
$\displaystyle h_{t}(x)=\left\\{\begin{array}[]{ll}{f(x)-f(t-0),}&\mbox{if}\
x<t\\\ {f(x)-f(t+0),}&\mbox{if}\ x>t\\\ {0,}&\mbox{if}\
x=t.\end{array}\right.$
Then the following holds:
$\displaystyle(S_{w}^{\chi}f)(t)=(S_{w}^{\chi}h_{t})(t)+f(t-0)+\psi_{\chi}^{-}(t^{w})[f(t+0)-f(t-0)]+\chi(1)[f(t)-f(t-0)],$
if $w\log(t)\in\mathbb{Z}$ and
$\displaystyle(S_{w}^{\chi}f)(t)=(S_{w}^{\chi}h_{t})(t)+f(t-0)+\psi_{\chi}^{-}(t^{w})[f(t+0)-f(t-0)],$
if $w\log(t)\notin\mathbb{Z}.$
###### Proof.
Let $w\log(t)\in\mathbb{Z},$ and $w>0.$ Then, we can write
$\displaystyle(S_{w}^{\chi}h_{t})(t)$ $\displaystyle=$
$\displaystyle\sum_{k<w\log(t)}\chi(e^{-k}t^{w})(f(e^{\frac{k}{w}})-f(t-0))+\sum_{k>w\log(t)}\chi(e^{-k}t^{w})(f(e^{\frac{k}{w}})-f(t+0))$
$\displaystyle=$
$\displaystyle\sum_{k<w\log(t)}\chi(e^{-k}t^{w})f(e^{\frac{k}{w}})+\sum_{k\geq
w\log(t)}\chi(e^{-k}t^{w})f(e^{\frac{k}{w}})-f(t-0)\sum_{k<w\log(t)}\chi(e^{-k}t^{w})$
$\displaystyle-f(t+0)\sum_{k>w\log(t)}\chi(e^{-k}t^{w})-\chi(1)f(t)$
$\displaystyle=$
$\displaystyle(S_{w}^{\chi}f)(t)-f(t-0)\sum_{k<w\log(t)}\chi(e^{-k}t^{w})-f(t+0)\sum_{k>w\log(t)}\chi(e^{-k}t^{w})-\chi(1)f(t).$
Adding and subtracting $\displaystyle f(t-0)\sum_{k\geq
w\log(t)}\chi(e^{-k}t^{w})$ in the above equation and rearranging all terms,
we obtain
$\displaystyle(S_{w}^{\chi}f)(t)$ $\displaystyle=$
$\displaystyle(S_{w}^{\chi}h_{t})(t)+f(t-0)\bigg{(}\sum_{k<w\log(t)}\chi(e^{-k}t^{w})+\sum_{k\geq
w\log(t)}\chi(e^{-k}t^{w})\bigg{)}$
$\displaystyle+[f(t+0)-f(t-0)]\sum_{k>w\log(t)}\chi(e^{-k}t^{w})+\chi(1)f(t)-f(t-0)\chi(1)$
$\displaystyle=$
$\displaystyle(S_{w}^{\chi}h_{t})(t)+f(t-0)\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})+[f(t+0)-f(t-0)]\sum_{k>w\log(t)}\chi(e^{-k}t^{w})$
$\displaystyle\chi(1)[f(t)-f(t-0)].$
Hence using the condition that
$\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}u)=1,$ we can easily obtain
$\displaystyle(S_{w}^{\chi}f)(t)$ $\displaystyle=$
$\displaystyle(S_{w}^{\chi}h_{t})(t)+f(t-0)+\psi_{\chi}^{-}(t^{w})[f(t+0)-f(t-0)]+\chi(1)[f(t)-f(t-0)].$
Now let $w\log(t)\notin\mathbb{Z}$ and $w>0.$ Then repeating the same
computations, we easily obtain
$\displaystyle(S_{w}^{\chi}f)(t)$ $\displaystyle=$
$\displaystyle(S_{w}^{\chi}h_{t})(t)+f(t-0)+[f(t+0)-f(t-0)]\sum_{k\geq
w\log(t)}\chi(e^{-k}t^{w})$ $\displaystyle=$
$\displaystyle(S_{w}^{\chi}h_{t})(t)+f(t-0)+[f(t+0)-f(t-0)]\psi_{\chi}^{-}(t^{w}).$
∎
Before proving the approximation of discontinuous functions by
$S_{w}^{\chi}{f},$ we recall the following theorem proved in ([6]) for
continuous functions on $\mathbb{R}^{+}.$
###### Theorem 1.
Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded function and
$\chi\in\Phi.$ Then $(S_{w}^{\chi}{f})(t)$ converges to $f(t)$ at any point
$t$ of continuity. Moreover, if $f\in C(\mathbb{R}^{+}),$ then we have
$\lim_{w\rightarrow\infty}\|f-S_{w}^{\chi}{f}\|_{\infty}=0.$
Now we analyse the behaviour of the exponential sampling series at jump
discontinuity at $t\in\mathbb{R}^{+}$ when $w\log(t)\in\mathbb{Z}.$
###### Theorem 2.
Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded signal and let
$t\in\mathbb{R}^{+}$ be a point of non-removable jump discontinuity of $f.$
For a given $\alpha\in\mathbb{R},$ the following statements are equivalent:
1. (i)
$\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\alpha
f(t+0)+[1-\alpha-\chi(1)]f(t-0)+\chi(1)f(t),\hskip 7.11317pt$
2. (ii)
$\psi_{\chi}^{-}(1)=\alpha,$
3. (iii)
$\psi_{\chi}^{+}(1)=1-\alpha-\chi(1).$
###### Proof.
First, we prove that $(i)\Longleftrightarrow(ii)$. In view of the
representation Lemma 1, we have
$\displaystyle(S_{w}^{\chi}f)(t)=(S_{w}^{\chi}h_{t})(t)+f(t-0)+\psi_{\chi}^{-}(t^{w})[f(t+0)-f(t-0)]+\chi(1)[f(t)-f(t-0)],$
for any $w>0$ such that $w\log(t)\in\mathbb{Z}.$ Since $h_{t}$ is bounded and
continuous at zero and using Theorem 1, we obtain
$\displaystyle\lim_{w\rightarrow\infty}(S_{w}^{\chi}h_{t})(t)=0.$
Thus, we have
$\displaystyle\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=f(t-0)+\left(\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})\right)[f(t+0)-f(t-0)]+\chi(1)[f(t)-f(t-0)].$
Now, we have
$\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})=\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}\left(\sum_{k>w\log(t)}\chi(e^{-k}t^{w})\right).$
As $\psi_{\chi}^{-}$ is recurrent with fundamental domain $[1,e],$ we get
$\psi_{\chi}^{-}(t^{w})=\psi_{\chi}^{-}(1),\,\ \forall w,t\,\ \mbox{such
that}\,\ w\log(t)\in\mathbb{Z}.$
Therefore, we have
$\displaystyle\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\psi_{\chi}^{-}(1)f(t+0)+[1-\psi_{\chi}^{-}(1)-\chi(1)]f(t-0)+\chi(1)f(t).$
$\mbox{Now}\,\ (i)\Longleftrightarrow\alpha
f(t+0)+[1-\alpha-\chi(1)]f(t-0)+\chi(1)f(t)$
$\displaystyle=$
$\displaystyle\psi_{\chi}^{-}(1)f(t+0)+[1-\psi_{\chi}^{-}(1)-\chi(1)]f(t-0)+\chi(1)f(t)$
$\displaystyle\Longleftrightarrow$
$\displaystyle\psi_{\chi}^{-}(1)(f(t+0)-f(t-0))=\alpha(f(t+0)-f(t-0))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\psi_{\chi}^{-}(1)=\alpha$
$\displaystyle\Longleftrightarrow$ $\displaystyle(ii)\,\ \mbox{holds}.$
Since $\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})=1,$ we have
$\psi_{\chi}^{+}(1)=1-{\chi}(1)-\psi_{\chi}^{-}(1).$
This implies that $(ii)\Longleftrightarrow(iii).$ Hence, the proof is
completed. ∎
Next we analyse the behaviour of the exponential sampling series at jump
discontinuity at $t\in\mathbb{R}^{+}$ when $w\log(t)\notin\mathbb{Z}.$
###### Theorem 3.
Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded signal and let
$t\in\mathbb{R}^{+}$ be a point of non-removable jump discontinuity of $f.$
Let $\alpha\in\mathbb{R}.$ Then the following statements are equivalent:
1. (i)
$\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\alpha
f(t+0)+(1-\alpha)f(t-0),$
2. (ii)
$\psi_{\chi}^{-}(u)=\alpha,\hskip 14.22636ptu\in(1,e)$
3. (iii)
$\psi_{\chi}^{+}(u)=1-\alpha,\hskip 14.22636ptu\in(1,e).$
###### Proof.
Using representation Lemma 1, we obtain
$\displaystyle\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=f(t-0)+\left(\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})\right)[f(t+0)-f(t-0)]$
$\displaystyle(i)$ $\displaystyle\Longleftrightarrow$ $\displaystyle\alpha
f(t+0)+(1-\alpha)f(t-0)=f(t-0)+\left(\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})\right)[f(t+0)-f(t-0)]$
$\displaystyle\Longleftrightarrow$
$\displaystyle\alpha[f(t+0)-f(t-0)]=\left(\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})\right)[f(t+0)-f(t-0)]$
$\displaystyle\Longleftrightarrow$
$\displaystyle\alpha=\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}\psi_{\chi}^{-}(t^{w})$
$\displaystyle\Longleftrightarrow$
$\displaystyle\alpha=\psi_{\chi}^{-}(u),\hskip 14.22636pt\forall u\in(1,e)$
$\displaystyle\Longleftrightarrow$ $\displaystyle(ii)\,\ \mbox{holds}.$
Let $w\log(t)\notin\mathbb{Z}.$ Then, we have
$\psi_{\chi}^{+}(t^{w})+\psi_{\chi}^{-}(t^{w})=1.$
Thus, we obtain
$(ii)\Longleftrightarrow\psi_{\chi}^{+}(u)=1-\alpha,\hskip
14.22636ptu\in(1,e).$
∎
The results in the above Theorem 3 was proved by assuming that
$\psi_{\chi}^{-}(u)$ is constant on $(1,e).$ In what follows, we show that if
$\psi_{\chi}^{-}(u)$ is not a constant on $(1,e),$ then the exponential
sampling series can not converge at jump discontinuities.
###### Theorem 4.
Let $\chi$ be a kernel such that $\psi_{\chi}^{-}(u)$ is not constant on
$(1,e).$ Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded signal with
a non-removable jump discontinuity at $t\in\mathbb{R}^{+}.$ Then
$(S_{w}^{\chi}f)(t)$ does not converge point-wise at $t$.
###### Proof.
Suppose not. Then $\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\ell,$ for
some $\ell\in\mathbb{R}^{+}.$ By the uniqueness of the limit and Lemma 1, we
obtain
$\displaystyle\ell=f(t-0)+\displaystyle\lim_{w\rightarrow\infty}\psi_{\chi}^{-}(t^{w})[f(t+0)-f(t-0)].$
Since $f(t+0)-f(t-0)\neq 0,$ we obtain
$\displaystyle\frac{\ell-f(t-0)}{f(t+0)-f(t-0)}=\displaystyle\lim_{w\rightarrow\infty}\psi_{\chi}^{-}(t^{w}).$
The above expression gives a contradiction. Indeed, if
$\displaystyle\displaystyle\lim_{w\rightarrow\infty}\psi_{\chi}^{-}(t^{w})=C,$
where $C$ is a constant, then it fails to satisfy that $\psi_{\chi}^{-}$ is
recurrent and not a constant, hence the theorem proved. ∎
Finally, the more general theorem of the exponential sampling series at jump
discontinuity at $t\in\mathbb{R}^{+}$ for any bounded signal can be proved.
###### Theorem 5.
Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded signal and let
$t\in\mathbb{R}^{+}$ be a point of non-removable jump discontinuity of $f.$
Let $\alpha\in\mathbb{R}.$ Suppose that the kernel $\chi$ satisfies the
additional condition that $\chi(1)=0.$ Then, the following statements are
equivalent:
1. (i)
$\displaystyle\lim_{w\rightarrow\infty}(S_{w}^{\chi}f)(t)=\alpha
f(t+0)+(1-\alpha)f(t-0),$
2. (ii)
$\psi_{\chi}^{-}(u)=\alpha,\hskip 14.22636ptu\in[1,e)$
3. (iii)
$\psi_{\chi}^{+}(u)=1-\alpha,\hskip 14.22636ptu\in[1,e).$
Moreover, if in addition we assume that $\chi$ is continuous on
$\mathbb{R}^{+},$ then the above statements are equivalent to the following
statements:
1. (iv)
$\displaystyle\int_{0}^{1}\chi(u)u^{2k\pi i}\frac{du}{u}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\ \ \ k\neq 0\\\
\alpha,&\ \ \ \mbox{if}\ \ \ k=0\end{array}\right.$
2. (v)
$\displaystyle\int_{1}^{\infty}\chi(u)u^{2k\pi i}\frac{du}{u}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\
\ \ k\neq 0\\\ 1-\alpha,&\ \ \ \mbox{if}\ \ \ k=0.\end{array}\right.$
###### Proof.
Proceeding along the lines proof of Theorem 2 and Theorem 3, we see that
$(i),(ii)$ and $(iii)$ are equivalent. Let $\chi$ be continuous on
$\mathbb{R}^{+}$ and let
$\displaystyle\chi_{0}(u)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\chi(u),&\ \ \ \mbox{for}\ \ \ u<1\\\
0,&\ \ \ \mbox{for}\ \ \ u\geq 1.\end{array}\right.$
Then, we have
$\displaystyle\psi_{\chi}^{-}(u)=\sum_{k>\log
u}\chi(ue^{-k})=\sum_{k\in\mathbb{Z}}\chi_{0}(ue^{-k}).$
Therefore, $\psi_{\chi}^{-}$ is recurrent continuous function with the
fundamental interval $[1,e].$ Using Mellin-Poisson summation formula, we
obtain
$\displaystyle\psi_{\chi}^{-}(u)=\sum_{k=-\infty}^{+\infty}\widehat{[\chi_{0}]_{M}}(2k\pi
i)\ u^{-2k\pi i}=\sum_{k=-\infty}^{+\infty}\left(\int_{0}^{1}\chi(u)u^{2k\pi
i}\frac{du}{u}\right)u^{-2k\pi i}.$
Therefore, we obtain
$\displaystyle\psi_{\chi}^{-}(u)$ $\displaystyle=$
$\displaystyle\alpha,\forall u\in[1,e)$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{[\chi]_{M}}(2k\pi i)=\left\\{\begin{array}[]{ll}0,&\ \
\ \mbox{if}\ \ \ k\neq 0\\\ \alpha,&\ \ \ \mbox{if}\ \ \
k=0\end{array}\right.$ $\displaystyle\Longleftrightarrow$
$\displaystyle\int_{0}^{1}\chi(u)u^{2k\pi
i}\frac{du}{u}=\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\ \ \ k\neq 0\\\
\alpha,&\ \ \ \mbox{if}\ \ \ k=0.\end{array}\right.$
This implies that $(ii)\Longleftrightarrow(iv).$ Finally using the condition
$\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}u)=1\Longleftrightarrow\widehat{[\chi]_{M}}(2k\pi
i)=\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\ \ \ k\neq 0\\\ 1,&\ \ \
\mbox{if}\ \ \ k=0\end{array}\right.$
the equivalence between $(iv)$ and $(v)$ can be established easily. Thus the
proof is completed. ∎
###### Remark 1.
Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a bounded signal with a
removable discontinuity $t\in\mathbb{R}^{+}$, i.e. $f(t+0)=f(t-0)=\ell.$ Then
we have
1. (i)
$\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\in\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\ell+\chi(1)[f(t)-\ell],$
2. (ii)
$\displaystyle\lim_{\stackrel{{\scriptstyle
w\rightarrow\infty}}{{w\log(t)\notin\mathbb{Z}}}}(S_{w}^{\chi}f)(t)=\ell,$
3. (iii)
If $\chi(1)=0,$ then
$\displaystyle\lim_{w\rightarrow\infty}(S_{w}^{\chi}f)(t)=\ell.$
## 3\. Degree of Approximation
In this section, we estimate the order of convergence of the exponential
sampling series by using the logarithmic modulus of continuity. Let
$C(\mathbb{R}^{+}$) denote the space of all real valued bounded continuous
functions on $\mathbb{R}^{+}$ equipped with the supremum norm
$\|f\|_{\infty}:=\sup_{x\in\mathbb{R}^{+}}|f(x)|.$ We say that a function
$f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ is log-uniformly continuous if the
following hold: for a given $\epsilon>0,$ there exists $\delta>0$ such that
$|f(p)-f(q)|<\epsilon$ whenever $|\log p-\log q|<\delta,$ for any
$p,q\in\mathbb{R}^{+}.$ The subspace consisting of all bounded log-uniformly
continuous functions on $\mathbb{R}^{+}$ is denoted by
$\mathcal{C}(\mathbb{R}^{+}).$ Let $f\in C(\mathbb{R}^{+}).$ Then the
logarithmic modulus of continuity is defined by
$\omega(f,\delta):=\sup\\{|f(p)-f(q)|:\ \mbox{whenever}\
|\log(p)-\log(q)|\leq\delta,\ \ \delta\in\mathbb{R}^{+}\\}.$
The logarithmic modulus of continuity satisfies the following properties:
* (a)
$\omega(f,\delta)\rightarrow 0,$ as $\delta\rightarrow 0.$
* (b)
$\omega(f,c\delta)\leq(c+1)\omega(f,\delta),$ for every $\delta,c>0.$
* (c)
$|f(p)-f(q)|\leq\omega(f,\delta)\left(1+\dfrac{|\log p-\log
q|}{\delta}\right).$
Further properties of logarithmic modulus of continuity can be seen in [2]. In
the following theorem, we obtain the order of convergence for the exponential
sampling series when $M_{\nu}(\chi)<\infty$ for $0<\nu<1.$
###### Theorem 6.
Let $\chi\in\Phi$ be a kernel such that $M_{\nu}(\chi)<\infty$ for $0<\nu<1$
and $f\in\mathcal{C}(\mathbb{R}^{+})$. Then for sufficiently large $w>0,$ the
following hold:
$|(S_{w}^{\chi}f)(t)-f(t)|\leq\omega(f,w^{-\nu})[M_{\nu}(\chi)+2M_{0}(\chi)]+2^{\nu+1}\|f\|_{\infty}M_{\nu}(\chi)w^{-\nu},$
for every $t\in\mathbb{R}^{+}.$
###### Proof.
Let $t\in\mathbb{R}^{+}$ be fixed. Then using the condition
$\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})=1,$ we obtain
$\displaystyle|(S_{w}^{\chi}f)(t)-f(t)|$ $\displaystyle=$
$\displaystyle\bigg{|}\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})(f(e^{\frac{k}{w}})-f(t))\bigg{|}$
$\displaystyle\leq$ $\displaystyle\left(\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}+\sum_{\big{|}k-w\log
t\big{|}\geq\frac{w}{2}}\right)\big{|}\chi(e^{-k}t^{w})\big{|}|f(e^{\frac{k}{w}})-f(t)|$
$\displaystyle:=$ $\displaystyle I_{1}+I_{2}.$
Let $0<\nu<1.$ Then we have
$\omega\left(f,\Big{|}\frac{k}{w}-\log
t\Big{|}\right)\leq\omega\left(f,\Big{|}\frac{k}{w}-\log
t\Big{|}^{\nu}\right).$
Therefore, using the above inequality and the property $(c),$ we obtain
$\displaystyle I_{1}$ $\displaystyle\leq$ $\displaystyle\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\omega\left(f,\Big{|}\frac{k}{w}-\log
t\Big{|}^{\nu}\right)$ $\displaystyle\leq$ $\displaystyle\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\left(1+w^{\nu}\Big{|}\frac{k}{w}-\log
t\Big{|}^{\nu}\right)\omega(f,w^{-\nu})$ $\displaystyle\leq$
$\displaystyle\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\omega(f,w^{-\nu})$
$\displaystyle+\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}w^{\nu}\Big{|}\frac{k}{w}-\log
t\Big{|}^{\nu}\omega(f,w^{-\nu})$ $\displaystyle\leq$
$\displaystyle\omega(f,w^{-\nu})\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}+\omega(f,w^{-\nu})\sum_{\big{|}k-w\log
t\big{|}<\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\Big{|}k-w\log
t\Big{|}^{\nu}.$
In view of the conditions $M_{0}(\chi)$ and $M_{\nu}(\chi),$ we easily obtain
$I_{1}\leq\omega(f,w^{-\nu})[M_{0}(\chi)+M_{\nu}(\chi)].$
Now we estimate $I_{2}.$ Since $\big{|}k-w\log t\big{|}\geq\frac{w}{2},$ we
have
$\frac{1}{\big{|}k-w\log t\big{|}^{\nu}}\leq 2^{\nu}w^{-\nu},\,\,\ 0<\nu<1.$
Hence, we obtain
$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
2\|f\|_{\infty}\sum_{\big{|}k-w\log
t\big{|}\geq\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\leq
2\|f\|_{\infty}\sum_{\big{|}k-w\log
t\big{|}\geq\frac{w}{2}}\frac{\big{|}k-w\log t\big{|}^{\nu}}{\big{|}k-w\log
t\big{|}^{\nu}}\big{|}\chi(e^{-k}t^{w})\big{|}$ $\displaystyle\leq$
$\displaystyle 2^{\nu+1}\|f\|_{\infty}w^{-\nu}\sum_{\big{|}k-w\log
t\big{|}\geq\frac{w}{2}}\big{|}\chi(e^{-k}t^{w})\big{|}\big{|}k-w\log
t\big{|}^{\nu}\leq 2^{\nu+1}\|f\|_{\infty}w^{-\nu}M_{\nu}(\chi)<\infty.$
On combining the estimates $I_{1}$ and $I_{2},$ we get the desired estimate. ∎
## 4\. Round-Off and Time Jitter Errors
This section is devoted to analyse round off and time jitter errors connected
with exponential sampling series (1.1). The round-off error arises when the
exact sample values $f(e^{\frac{k}{w}})$ are replaced by approximate close
ones $\bar{f}(e^{\frac{k}{w}})$ in the sampling series (1.1). Let
$\xi_{k}=f(e^{\frac{k}{w}})-\bar{f}(e^{\frac{k}{w}})$ be uniformly bounded by
$\xi,$ i.e., $\mid\xi_{k}\mid\leq\xi,$ for some $\xi>0.$ We are interested in
analysing the error when $f(t)$ is approximated by the following exponential
sampling series:
$\displaystyle(S_{w}^{\chi}\bar{f})(t)=\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})\bar{f}(e^{\frac{k}{w}}).$
The total round-off or quantization error is defined by
$(Q_{\xi}f)(t):=\mid(S_{w}^{\chi}{f})(t)-(S_{w}^{\chi}\bar{f})(t)\mid.$
###### Theorem 7.
For $f\in C(\mathbb{R}^{+}),$ the following hold:
1. (i)
$\parallel(Q_{\xi}f)\parallel_{C(\mathbb{R}^{+})}\leq\xi M_{0}(\chi)$
2. (ii)
$\parallel f-S_{w}^{\chi}\bar{f}\parallel_{C(\mathbb{R}^{+})}\leq
C\omega\left(f,\dfrac{1}{w}\right)+\xi M_{0}(\chi),$ where
$C=M_{0}(\chi)+{M_{1}(\chi)}.$
###### Proof.
The error in the approximation can be splitted as
$\displaystyle\mid f(t)-(S_{w}^{\chi}\bar{f})(t)\mid$ $\displaystyle\leq$
$\displaystyle\mid
f(t)-(S_{w}^{\chi}f)(t)\mid+(Q_{\xi}f)(t):=I_{1}+(Q_{\xi}f)(t).$
The term $I_{1}$ is the error arising if the actual sample value is used and
the total round-off or quantization error can be evaluated by
$\displaystyle\parallel(Q_{\xi}f)\parallel_{C(\mathbb{R}^{+})}$
$\displaystyle=$
$\displaystyle\displaystyle\sup_{t\in\mathbb{R}^{+}}{\left|\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}})-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w})\bar{f}(e^{\frac{k}{w}})\right|}$
$\displaystyle=$
$\displaystyle\displaystyle\sup_{t\in\mathbb{R}^{+}}{\sum_{k=-\infty}^{+\infty}\bigg{|}\xi_{k}\chi(e^{-k}t^{w})\bigg{|}}\leq\xi
M_{0}(\chi).$
In view of Theorem 4 ([6], page no.7), we have
$I_{1}\leq
M_{0}(\chi)\omega(f,\delta)+\frac{\omega(f,\delta)}{w\delta}M_{1}(\chi).$
On combining the estimates $I_{1}$ and $I_{2},$ we get
$\displaystyle\parallel f-S_{w}^{\chi}\bar{f}\parallel_{C(\mathbb{R}^{+})}$
$\displaystyle\leq$
$\displaystyle\bigg{(}M_{0}(\chi)+\frac{M_{1}(\chi)}{w\delta}\bigg{)}\omega(f,\delta)+\xi
M_{0}(\chi).$
Choosing $\delta=\displaystyle\frac{1}{w},$ we obtain
$\displaystyle\parallel f-S_{w}^{\chi}\bar{f}\parallel_{C(\mathbb{R}^{+})}$
$\displaystyle\leq$ $\displaystyle C\omega(f,\frac{1}{w})+\xi M_{0}(\chi),$
where $C=M_{0}(\chi)+{M_{1}(\chi)}.$ Hence, the proof is completed. ∎
The time-jitter error occurs when the function $f(t)$ being approximated from
samples which are taken at perturbed nodes, i.e., the exact sample values
$f(e^{\frac{k}{w}})$ are replaced by $f(e^{\frac{k}{w}}+\varrho_{k})$ in the
sampling series (1.1). So we are interested in analysing time jitter error and
the approximation behaviour when $f(t)$ is approximated by the sampling series
$\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}}+\varrho_{k}).$
We assume that the values $\varrho_{k}$ are bounded by a small number
$\varrho,$ i.e., $\mid\varrho_{k}\mid\leq\varrho,$ for all $k\in\mathbb{Z}$
and for some $\varrho>0.$ The total time jitter error is defined by
$J_{\varrho}f(t):=\left|\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}})-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}}+\varrho_{k})\right|.$
###### Theorem 8.
For $f\in C^{(1)}(\mathbb{R}^{+}),$ the following hold:
1. (i)
$\|J_{\varrho}f\|_{C(\mathbb{R}^{+})}\leq\varrho\,\parallel
f^{{}^{\prime}}\parallel_{C(\mathbb{R}^{+})}M_{0}(\chi)$
2. (ii)
$\bigg{\|}f(.)-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}(.)^{w}){f}(e^{\frac{k}{w}}+\varrho_{k})\bigg{\|}_{C(\mathbb{R}^{+})}\leq\,\
C\omega\left(f,\dfrac{1}{w}\right)+\varrho\parallel
f^{{}^{\prime}}\parallel_{C(\mathbb{R}^{+})}M_{0}(\chi),$ where
$C=M_{0}(\chi)+{M_{1}(\chi)}.$
###### Proof.
Applying the mean value theorem, error can be estimated by
$\displaystyle\|J_{\varrho}f\|_{C(\mathbb{R}^{+})}$ $\displaystyle\leq$
$\displaystyle\displaystyle\sup_{k\in\mathbb{Z}}\\{\sup_{t\in\mathbb{R}^{+}}|{f}(e^{\frac{k}{w}})-{f}(e^{\frac{k}{w}}+\varrho_{k})|\\}\sup_{t\in\mathbb{R}^{+}}{\sum_{k=-\infty}^{+\infty}|\chi(e^{-k}t^{w})|}$
$\displaystyle\leq$ $\displaystyle\mid\varrho_{k}\parallel
f^{{}^{\prime}}\parallel_{C(\mathbb{R}^{+})}\mid\sup_{t\in\mathbb{R}^{+}}{\sum_{k=-\infty}^{+\infty}|\chi(e^{-k}t^{w})|}\leq\varrho\,\parallel
f^{{}^{\prime}}\parallel_{C(\mathbb{R}^{+})}M_{0}(\chi).$
From the above estimates it is clear that the jitter error essentially depends
on the smoothness of function $f.$ For $f\in C^{(1)}(\mathbb{R}^{+}),$ the
associated approximation error is estimated by
$\displaystyle\bigg{|}f(t)-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}}+\varrho_{k})\bigg{|}$
$\displaystyle\leq$
$\displaystyle\bigg{|}f(t)-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}})\bigg{|}$
$\displaystyle+\bigg{|}\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}})-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}t^{w}){f}(e^{\frac{k}{w}}+\varrho_{k})\bigg{|}$
$\displaystyle\leq$ $\displaystyle|f(t)-S_{w}^{\chi}{f}(t)|+J_{\varrho}f(t).$
Again using Theorem 4 ([6], page no.7), we have
$|f(t)-S_{w}^{\chi}{f}(t)|\leq
M_{0}(\chi)\omega(f,\delta)+\frac{\omega(f,\delta)}{w\delta}M_{1}(\chi).$
Using the above estimate and $J_{\varrho}f,$ we obtain
$\displaystyle\bigg{\|}f(.)-\sum_{k=-\infty}^{+\infty}\chi(e^{-k}(.)^{w}){f}(e^{\frac{k}{w}}+\varrho_{k})\bigg{\|}_{C(\mathbb{R}^{+})}\leq
M_{0}(\chi)\omega(f,\delta)+\frac{\omega(f,\delta)}{w\delta}M_{1}(\chi)+\varrho\parallel
f^{{}^{\prime}}\parallel_{C(\mathbb{R}^{+})}M_{0}(\chi).$
Hence, the proof is completed. ∎
## 5\. Examples of the Kernels
In this section, we provide certain examples of the kernel functions which
will satisfy our assumptions. First we give the family of Mellin-B spline
kernels. The Mellin B-spline of order $n$ is given by
$\bar{B}_{n}(x):=\frac{1}{(n-1)!}\sum_{j=0}^{n}(-1)^{j}{n\choose
j}\bigg{(}\frac{n}{2}+\log x-j\bigg{)}_{+}^{n+1},\,\ x\in\mathbb{R}^{+}$
It can be easily seen that $\bar{B}_{n}(x)$ is compactly supported for every
$n\in\mathbb{N}.$ The Mellin transform of $\bar{B}_{n}$ (see [6]) is
$\displaystyle\widehat{[\bar{B}_{n}]_{M}}(c+it)=\bigg{(}\frac{\sin(\frac{t}{2})}{(\frac{t}{2})}\Bigg{)}^{n},\
\ \hskip 14.22636ptt\neq 0.$
The Mellin’s-Poisson summation formula [11]) is defined by
$(i)^{j}\sum_{k=-\infty}^{+\infty}\chi(e^{k}x)(k-\log
u)^{j}=\sum_{k=-\infty}^{+\infty}\frac{d^{j}}{dt^{j}}\widehat{[\chi]_{M}}(2k\pi
i)\ x^{-2k\pi i},\ \ \ \ \ \ \mbox{for}\ k\in\mathbb{Z}.$
We need the following lemma (see [6]).
###### Lemma 2.
The condition $\displaystyle\sum_{k=-\infty}^{+\infty}\chi(e^{-k}x^{w})=1$ is
equivalent to
$\displaystyle\widehat{[\chi]_{M}}(2k\pi
i)=\left\\{\begin{array}[]{ll}1,&\mbox{if}\ k=0\\\
0,&\mbox{otherwise.}\end{array}\right.$
Moreover $m_{j}(\chi,u)=0$ for $j=1,2,\cdots,n$ is equivalent to
$\dfrac{d^{j}}{dt^{j}}\widehat{[\chi]_{M}}(2k\pi i)=0$ for $j=1,2,\cdots,n$
and $\forall\ k\in\mathbb{Z}.$
Using the above Lemma, we obtain
$\displaystyle\widehat{[\bar{B}_{n}]_{M}}(2k\pi
i)=\left\\{\begin{array}[]{ll}1,&\mbox{if}\ k=0\\\
0,&\mbox{otherwise.}\end{array}\right.$
Using Mellin’s-Poisson summation formula, it is easy to see that
$\bar{B}_{n}(x)$ satisfies the condition (i). As $\bar{B}_{n}(x)$ is compactly
supported, the condition (ii) is also satisfied. Next we consider the Mellin
Jackson kernels. For $x\in\mathbb{R}^{+},\beta\in\mathbb{N},\gamma\geq 1,$ the
Mellin Jackson kernels are defined by
$J_{\gamma,\beta}^{-1}(x):=d_{\gamma,\beta}x^{-c}sinc^{2\beta}\left(\frac{\log
x}{2\gamma\beta\pi}\right),$
where
$d_{\gamma,\beta}^{-1}:=\int_{0}^{\infty}sinc^{2\beta}\left(\frac{\log
x}{2\gamma\beta\pi}\right)\frac{du}{u}.$
One can easily verify that the Mellin Jackson kernels also satisfies
conditions (i) and (ii) (see [6]). We can analyse the convergence of the
exponential sampling series with jump discontinuity associated with these
kernels only for the case given in Theorem 2 and we observe that $\chi(1)\neq
0$. So Theorem 5 can not be applied for these kernels. In order to obtain the
convergence of the exponential sampling series at jump discontinuity
$t\in\mathbb{R}^{+}$ of the given bounded signal
$f:\mathbb{R}^{+}\rightarrow\mathbb{R}$, we need to construct suitable
kernels. One such construction is given in the following theorem.
###### Theorem 9.
Let $\chi_{a},\chi_{b}$ be two continuous kernels supported respectively in
the intervals ${[e^{-a},e^{a}]}$ and ${[e^{-b},e^{b}]}.$ Let
$\alpha\in\mathbb{R}$ be fixed. We define
$\chi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ by
$\chi{(u)}:=(1-\alpha)\chi_{a}(2ue^{-a-1})+\alpha\chi_{b}(2ue^{b}),\,\,\
u\in\mathbb{R}^{+}.$
Then $\chi$ is a kernel satisfying conditions (i), (ii) and $\chi(1)=0.$
Moreover, the corresponding exponential sampling series $S_{w}^{\chi}{f},w>0$
based upon $\chi$ satisfy (i) of Theorem 5 with parameter $\alpha$ for a given
bounded signal $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ at any non-removable
discontinuity $t\in\mathbb{R}^{+}$ of $f.$
###### Proof.
The Mellin transform of $\chi(u)$ is
$\displaystyle\widehat{[\chi]_{M}}(s)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}(1-\alpha)\chi_{a}(2te^{-a-1})t^{s-1}dt+\int_{0}^{\infty}\alpha\chi_{b}(2te^{b})t^{s-1}dt$
$\displaystyle=$
$\displaystyle(1-\alpha)\widehat{[\chi_{a}]_{M}}(s)\left(\frac{e^{(1+a)}}{2}\right)^{s}+\alpha\widehat{[\chi_{b}]_{M}}(s)\left(\frac{e^{-b}}{2}\right)^{s}.$
It is simple to check that $\chi$ satisfies condition (ii). Now we show that
kernel satisfies the condition (i). We obtain
$\displaystyle\widehat{[\chi]_{M}}(2k\pi i)$ $\displaystyle=$
$\displaystyle(1-\alpha)\widehat{[\chi_{a}]_{M}}(2k\pi
i)\left(\frac{e^{(1+a)}}{2}\right)^{2k\pi
i}+\alpha\widehat{[\chi_{b}]_{M}}(2k\pi i)\left(\frac{e^{-b}}{2}\right)^{2k\pi
i}.$
As $\chi_{a}$ and $\chi_{b}$ satisfies condition (i), we have
$\displaystyle\widehat{[\chi_{a}]_{M}}(2k\pi i)=\widehat{[\chi_{b}]_{M}}(2k\pi
i)=\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\ \ \ k\neq 0\\\ 1,&\ \ \
\mbox{if}\ \ \ k=0\end{array}\right.$
For suitable choices of $a$ and $b$, we obtain
$\displaystyle\widehat{[\chi]_{M}}(2k\pi i)=\left\\{\begin{array}[]{ll}0,&\ \
\ \mbox{ if}\ \ \ k\neq 0\\\ 1,&\ \ \ \mbox{if}\ \ \ k=0.\end{array}\right.$
Therefore, $\chi$ satisfies condition (i) and we can easily see that
$\chi(1)=0.$ Now, we obtain
$\displaystyle\int_{0}^{1}\chi(u)u^{2k\pi i-1}du$ $\displaystyle=$
$\displaystyle\alpha\int_{0}^{1}\chi_{b}(2e^{b}u)u^{2k\pi i-1}du$
$\displaystyle=$ $\displaystyle\alpha\widehat{[\chi_{b}]_{M}}(2k\pi
i)\left(\frac{e^{-b2k\pi i}}{2^{2k\pi i}}\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0,&\ \ \ \mbox{if}\ \ \ k\neq 0\\\
\alpha,&\ \ \ \mbox{if}\ \ \ k=0.\end{array}\right.$
Therefore, the condition (iv) of Theorem 5 is satisfied, hence the proof is
completed. ∎
Now we test numerically the approximation of discontinuous function
$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\dfrac{11}{2t^{2}+1},&\ \ \ \ \
t<\dfrac{3}{2}\\\ \vspace{0.25 cm}3,&\ \ \ \ \ \ \dfrac{3}{2}\leq
t<\dfrac{7}{2}\\\ \vspace{0.25 cm}\par 2,&\ \ \ \ \ \ \dfrac{7}{2}\leq
t<\dfrac{11}{2}\\\ \vspace{0.25 cm}\par\dfrac{12}{1+2t},&\ \ \ \ \ \
t\geq\dfrac{11}{2}\end{array}\right.$
by exponential sampling series at its jump discontinuities $t=\dfrac{3}{2},$
$t=\dfrac{7}{2}$ and $t=\dfrac{11}{2}.$ We consider a linear combination of
Mellin B-spline kernels defined by (see Fig. 1)
$\displaystyle\chi(t)=\dfrac{1}{4}\bar{B}_{2}(2te^{-2})+\dfrac{3}{4}\bar{B}_{2}(2te),$
where $\bar{B}_{2}$ is given by
$\displaystyle\bar{B}_{2}(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1-\log t,&\ \ \ \ \ 1<t<e\\\
\vspace{0.25 cm}1+\log t,&\ \ \ \ \ \ \dfrac{1}{e}<t<1\\\ \vspace{0.25 cm}0,&\
\ \ \ \ \ \mbox{otherwise.}\end{array}\right.$
Figure 1. Plot of the kernel
$\chi(t)=\dfrac{1}{4}\bar{B}_{2}(2te^{-2})+\dfrac{3}{4}\bar{B}_{2}(2te).$
Clearly the exponential sampling series $S_{w}^{\chi}{f}$ based on $\chi(t)$
satisfies the conditions (i), (ii) and $\chi(1)=0.$ We also observe that the
condition (i) of Theorem 5 is satisfied with $\alpha=\dfrac{3}{4}.$ From
Theorem 5 and Theorem 9, we have that at the discontinuity points of $f,$ the
sampling series $S_{w}^{\chi}{f}$ converges to
$\dfrac{3}{4}f(t+0)+\dfrac{1}{4}f(t-0).$ The convergence of the sampling
series $S_{w}^{\chi}{f}$ at discontinuity points $t=\dfrac{3}{2},$
$t=\dfrac{7}{2}$ and $t=\dfrac{11}{2}$ of the function $f$ has been tested and
numerical results are presented in Tables 1, 2 and 3.
Figure 2. Approximation of $f(t)$ by $S_{w}^{\chi}{f}$ based on
$\chi(t)=\dfrac{1}{4}\bar{B}_{2}(2te^{-2})+\dfrac{3}{4}\bar{B}_{2}(2te)$ for
$w=5.$ Figure 3. Approximation of $f(t)$ by $S_{w}^{\chi}{f}$ based on
$\chi(t)=\dfrac{1}{4}\bar{B}_{2}(2te^{-2})+\dfrac{3}{4}\bar{B}_{2}(2te)$ for
$w=10.$
###### Table 1.
Approximation of $f$ at the jump discontinuity point $t=\dfrac{3}{2}$ by the
exponential sampling series $S_{w}^{\chi}{f}$ based on $\chi(t)$ for different
values of $w>0.$ The theoretical limit of
$(S_{w}^{\chi}{f})\left(\dfrac{3}{2}\right)$ as $w\rightarrow\infty$ is
$\dfrac{3}{4}f\left(\dfrac{3}{2}+0\right)+\dfrac{1}{4}f\left(\dfrac{3}{2}-0\right)=2.75.$
$w$ | $5$ | $10$ | $20$ | $50$ | $100$ | $200$
---|---|---|---|---|---|---
$S_{w}^{\chi}{f}$ | $3.0036$ | $2.8669$ | $2.8059$ | $2.7717$ | $2.7608$ | $2.7554$
###### Table 2.
Approximation of $f$ at the jump discontinuity point $t=\dfrac{7}{2}$ by the
exponential sampling series $S_{w}^{\chi}{f}$ based on $\chi(t)$ for different
values of $w>0.$ The theoretical limit of
$(S_{w}^{\chi}{f})\left(\dfrac{7}{2}\right)$ as $w\rightarrow\infty$ is
$\dfrac{3}{4}f\left(\dfrac{7}{2}+0\right)+\dfrac{1}{4}f\left(\dfrac{7}{2}-0\right)=2.25.$
$w$ | $5$ | $10$ | $20$ | $50$ | $100$ | $200$
---|---|---|---|---|---|---
$S_{w}^{\chi}{f}$ | $2.25$ | $2.25$ | $2.25$ | $2.25$ | $2.25$ | $2.25$
###### Table 3.
Approximation of $f$ at the jump discontinuity point $t=\dfrac{11}{2}$ by the
exponential sampling series $S_{w}^{\chi}{f}$ based on $\chi(t)$ for different
values of $w>0.$ The theoretical limit of
$(S_{w}^{\chi}{f})\left(\dfrac{11}{2}\right)$ as $w\rightarrow\infty$ is
$\dfrac{3}{4}f\left(\dfrac{11}{2}+0\right)+\dfrac{1}{4}f\left(\dfrac{11}{2}-0\right)=1.25.$
$w$ | $5$ | $10$ | $20$ | $50$ | $100$ | $200$
---|---|---|---|---|---|---
$S_{w}^{\chi}{f}$ | $1.0492$ | $1.1420$ | $1.1939$ | $1.2271$ | $1.2384$ | $1.2442$
Acknowledgments. The first two authors are supported by DST-SERB, India
Research Grant EEQ/2017/000201. The third author P. Devaraj has been supported
by DST-SERB Research Grant MTR/2018/000559.
## References
* [1] S. Balsamo, I. Mantellini, On linear combinations of general exponential sampling series, Results Math. 74, 180, 1-19 (2019).
* [2] C. Bardaro, I. Mantellini, On Mellin convolution operators: a direct approach to the asymptotic formulae, Integral Transforms Spec. Funct. 25 (3) 182-195 (2014).
* [3] C. Bardaro, P.L. Butzer, I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics, Integral Transforms Spec. Funct. 27 (1) 17-29 (2016).
* [4] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley-Wiener theorem in the Mellin transform setting, J. Approx. Theory 207, 60-75 (2016).
* [5] C. Bardaro, P.L. Butzer and I. Mantellini, The exponential sampling theorem of signal analysis and the reproducing kernel formula in the Mellin transform setting, Sampl. Theory Signal Image Process. 13 (1), 35-66 (2014).
* [6] C. Bardaro, L. Faina, I. Mantellini, A generalization of the exponential sampling series and its approximation properties, Math. Slovaca. 67(6), 1481-1496 (2017).
* [7] C. Bardaro, I. Mantellini, G. Schmeisser, Exponential sampling series: convergence in Mellin-Lebesgue spaces, Results Math. 74 , no. 3, Art. 119, 20 pp, (2019).
* [8] M. Bertero, E. R. Pike, Exponential-sampling method for Laplace and other dilationally invariant transforms. II. Examples in photon correlation spec- troscopy and Fraunhofer diffraction, Inverse Probl. 7(1), 21–41 (1991).
* [9] P.L. Butzer, S. Jansche, The exponential sampling theorem of signal analysis, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996), Atti. Sem. Mat. Fis. Univ. Modena, Suppl. 46, 99-122 (1998).
* [10] P.L. Butzer, S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl. 3, 325-376 (1997).
* [11] P.L. Butzer, S. Jansche, The finite Mellin transform, Mellin-Fourier series, and the Mellin-Poisson summation formula. Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. I (Acquafredda di Maratea, 1996). Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. I (1998), 55-81.
* [12] P.L. Butzer, S. Ries and R.L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50, 25-39 (1987).
* [13] P.L. Butzer, R.L. Stens, A self contained approach to Mellin transform analysis for square integrable functions;applications, Integral Transform. Spec. Funct. 8 (3-4), 175-198 (1999).
* [14] D. Costarelli, A. M. Minotti, G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl. 450 (2017) 1083-1103.
* [15] D. Casasent, Optical Data Processing, pp. 241–282. Springer, Berlin (1978).
* [16] F. Gori, Sampling in optics. In: Marks, R.J. (ed.) Advanced topics in Shan- non sampling and interpolation theory, Springer Texts Electrical Engineering, Springer, New York, NY (1993).
* [17] R.G. Mamedov, The Mellin transform and approximation theory (in Russian) ”Elm”, Baku, 1991, 273 pp, ISBN:5-8066-0137-4.
* [18] N. Ostrowsky, D. Sornette, P. Parke, E. R. Pike, Exponential sampling method for light scattering polydispersity analysis, Opt. Acta. 28, 1059-1070 (1994).
|
# Real-space representation of winding number for one-dimensional chiral-
symmetric topological insulator
Ling Lin Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing
$\&$ School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus),
Zhuhai 519082, China State Key Laboratory of Optoelectronic Materials and
Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275,
China Yongguan Ke<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory
of Quantum Metrology and Sensing $\&$ School of Physics and Astronomy, Sun
Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China Chaohong Lee
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum
Metrology and Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen
University (Zhuhai Campus), Zhuhai 519082, China State Key Laboratory of
Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou
Campus), Guangzhou 510275, China
###### Abstract
The winding number has been widely used as an invariant for diagnosing
topological phases in one-dimensional chiral-symmetric systems. We put forward
a real-space representation for the winding number. Remarkably, our method
reproduces an exactly quantized winding number even in the presence of
disorders that break translation symmetry but preserve chiral symmetry. We
prove that our real-space representation of the winding number, the winding
number defined through the twisted boundary condition, and the real-space
winding number derived previously in [Phys. Rev. Lett. 113, 046802 (2014)],
are equivalent in the thermodynamic limit at half filling. Our method also
works for the case of filling less than one half, where the winding number is
not necessarily quantized. Around the disorder-induced topological phase
transition, the real-space winding number has large fluctuations for different
disordered samples, however, its average over an ensemble of disorder samples
may well identify the topological phase transition. Besides, we show that our
real-space winding number can be expressed as a Bott index, which has been
used to represent the Chern number for two-dimensional systems.
## I Introduction
Topological states have attracted a tremendous amount of studies in various
systems involving electrons, cold atoms, and photons, etc. Most of the non-
interacting topological states can be successfully explained by topological
band theory [1] based on perfect translation symmetry [2, 3, 4, 5]. The first
well-known example is the integer quantum Hall effect [6], in which the
quantized Hall conductivity is related to a topological TKNN invariant [7]
defined with Bloch wavefunctions of filling bands. However, in reality, there
always exists disorder that breaks translation symmetry and hence the
consequent Bloch wavefunctions. It becomes problematic for the calculation of
topological invariants with Bloch wavefunctions. Numerous topological states
immune to disorder [8, 9, 10, 11, 12, 13, 14, 15] indicate that the lack of
Bloch wavefunctions should not be a hindrance for defining topological
invariants.
To circumvent the absence of translation symmetry, one may consider a real-
space representation of the topological invariants. One of the well-known
examples is the real-space representation of the Chern number. The
construction of real-space representation of the Chern number is through
transforming the momentum-space formula to the real-space one [16, 17, 18], or
considering a Bott index [19, 20]. The real-space representation of the Chern
number has been widely used in studying disorder effects in two-dimensional
topological systems [21, 22, 23, 24]. Another example is that the polarization
in one dimension can be calculated in the real space as well via the projected
position operator approach [25, 26, 27, 28]. Recently, a real-space
representation of the winding number is proposed for one-dimensional (1D)
topological insulators with chiral symmetry [29], in which the momentum-space
formula of winding number is transformed to a real-space formula. Provided
that disorder does not break the chiral symmetry, such a real-space
representation of the winding number has been proved to be valid and widely
used in exploring the topological Anderson insulator [30, 31, 32, 33], and
particularly, for detection of the winding number in experiment [34, 35]. On
the other hand, we note the momentum-space winding number for 1D systems can
be written as the ‘skew’ polarization [29, 36], that is, the difference of
polarizations (Berry phases) between two sublattices. As the usual
polarization for 1D lattices can be obtained via the projected position
operator in real space, _can we derive the real-space winding number in views
of the skew polarization_?
In this paper, we propose a real-space representation of the winding number
for 1D chiral-symmetric topological insulators. We use the singular value
decomposition (SVD) method to derive a formula for calculating the difference
of polarization between two sublattices. Our formula can be written in form of
the Bott index [37, 38], which produces a strictly quantized winding number.
We prove that our formula is exactly equivalent to the momentum-space winding
number in the presence of translation symmetry. We also prove that our real-
space representation of the winding number, the winding number defined through
the twisted boundary condition (TBC), and the real-space winding number
derived previously in Ref. [29], are equivalent in the thermodynamic limit at
half filling. Provided the chiral symmetry is preserved, our formula is self-
averaging and satisfies the bulk-edge correspondence in the presence of
disorder. We have verified numerically that our results are in agreement with
the previous works [29, 30] on the disordered model after averaging over many
random realizations. However, our method gives exactly quantization of winding
number in each realization of disorder, in stark contrast to previous methods
[29, 30]. Away from topological transition points, our method has advantages
over the previous method [29, 30] for higher accuracy and less fluctuation.
Furthermore, we show that our formula can work for the case of filling less
than one half. Also, we find our real-space representation of the winding
number in one dimension can be written as the Bott index [19, 23], which was
used to define the real-space representation of the Chern number in two
dimensions.
The rest of this paper is organized as follows: In Sec. II, we review the
projected position operator approach, and show that it is related to the
Wilson loop. In Sec. III, we employ SVD for chiral-symmetric Hamiltonian to
obtain the flattened Hamiltonian, and then construct a real-space
representation of the winding number. We will prove the equivalence of our
real-space representation of winding number and the twisted-boundary winding
number. In Sec. IV, we apply our arguments to a 1D toy model belonging to BDI
classe. Finally, in Sec. V, we make a summary and discussion.
## II Bulk polarization and projected position operator
In this section, we give a brief review on calculating the bulk polarization
in the 1D system via the projected position operator.
Firstly, we consider a finite 1D lattice with $L$ cells under periodic
boundary conditions. When translational symmetry is present, quasi-momentum
$k$ becomes a good quantum number, and the Hamiltonian has a block-diagonal
structure. The eigenstates are Bloch waves labeled by quasi-momentum $k$. The
basis of momentum space is the Fourier transform of real space basis. In the
following context, we use $|l,\alpha\rangle$ to refer to the state that a
particle is located at the $\alpha$-th sublattice (orbital) of the $l$th cell.
Thus, the basis of momentum space reads
$|k,\alpha\rangle=\frac{1}{\sqrt{L}}\sum\limits_{l=1}^{L}{{e^{ikl}}|l,\alpha\rangle},$
(1)
and the Bloch waves can be written as the linear combinations of momentum
basis
$|{\psi_{k,n}}\rangle=\sum\limits_{\alpha}{u_{k,\alpha}^{n}|k,\alpha\rangle},$
(2)
where $n$ is the index of the band.
The bulk polarization can be calculated through the following formula [28]
$p=\frac{1}{{2\pi i}}\log\det{{\cal W}_{k+2\pi\leftarrow k}},$ (3)
where ${{\cal W}_{k+2\pi\leftarrow k}}$ is the so-called Wilson loop
${{\cal W}_{k+2\pi\leftarrow k}}={F_{k+2\pi-\delta k}}{F_{k+2\pi-2\delta
k}}...{F_{k}}.$ (4)
The matrix element of $F_{k}$ is ${\left({{F_{k}}}\right)_{m,n}}=\langle
u_{k+\delta k}^{m}|u_{k}^{n}\rangle$, in which $m,n$ are the indices of
occupied bands. We will also use this notation in the following context.
Next, we show the Wilson loop can be derived from the projected position
operator ${P_{{\rm{occ}}}}\mathcal{X}{P_{{\rm{occ}}}}$, where
${P_{{\rm{occ}}}}=\sum_{n=1}^{{n_{{\rm{occ}}}}}{\sum_{k}{|{\psi_{n,k}}\rangle\langle{\psi_{n,k}}|}}$
is the projector onto occupied bands, and $|{\psi_{n,k}}\rangle$ is the
eigenstate of system at the $n$th band with quasi-momentum $k$. A quantum-
mechanical position operator [39] $\hat{\mathcal{X}}=\text{exp}(i\delta
k\hat{X})$, in which $\hat{X}=\sum\nolimits_{x}{x{{{\hat{n}}}_{x}}}$ is the
general position operator, is introduced for a lattice with periodic boundary
condition. Here, $\delta k=2\pi/L$ is the increment of discrete quasi-momentum
$k$. Note that operator $\mathcal{X}$ is actually the translation operator for
quasi-momentum $k$: $\mathcal{X}|k,\alpha\rangle=|k+\delta k,\alpha\rangle$,
By expanding the expression of projected position operator, we have
$\displaystyle{P_{{\rm{occ}}}}\mathcal{X}{P_{{\rm{occ}}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{n,n^{\prime}=1}^{{n_{{\rm{occ}}}}}{\sum\limits_{k,k^{\prime}}{|{\psi_{n^{\prime},k^{\prime}}}\rangle\langle{\psi_{n^{\prime},k^{\prime}}}|\mathcal{X}|{\psi_{n,k}}\rangle\langle{\psi_{n,k}}|}}$
(5) $\displaystyle=$
$\displaystyle\sum\limits_{n,n^{\prime}=1}^{{n_{{\rm{occ}}}}}\sum\limits_{k}{\langle
u_{k+\delta k}^{n^{\prime}}|u_{k}^{n}\rangle|{\psi_{n^{\prime},k+\delta
k}}\rangle\langle{\psi_{n,k}}|}$
where we have used the relation [40, 41]
$\displaystyle\langle{\psi_{n^{\prime},k^{\prime}}}|\mathcal{X}|{\psi_{n,k}}\rangle$
$\displaystyle=$
$\displaystyle\sum\limits_{\alpha,\alpha^{\prime}}{{{\left({u_{k^{\prime},\alpha^{\prime}}^{n^{\prime}}}\right)}^{*}}u_{k,\alpha}^{n}\langle
k^{\prime},\alpha^{\prime}|\mathcal{X}|k,\alpha\rangle}$ (6) $\displaystyle=$
$\displaystyle\sum\limits_{\alpha,\alpha^{\prime}}{{{\left({u_{k^{\prime},\alpha^{\prime}}^{n^{\prime}}}\right)}^{*}}u_{k,\alpha}^{n}\langle
k^{\prime},\alpha^{\prime}|k+\delta k,\alpha\rangle}$ $\displaystyle=$
$\displaystyle\sum\limits_{\alpha,\alpha^{\prime}}{{\delta_{k^{\prime},k+\delta
k}}{\delta_{\alpha,\alpha^{\prime}}}{{\left({u_{k^{\prime},\alpha^{\prime}}^{n^{\prime}}}\right)}^{*}}u_{k,\alpha}^{n}}$
$\displaystyle=$ $\displaystyle{\delta_{k^{\prime},k+\delta k}}\langle
u_{k^{\prime}}^{n^{\prime}}|u_{k}^{n}\rangle.$
Then, we seek the eigenvalues of the projected position operator (5). Assuming
its eigenstates are the superposition of occupied Bloch states
$|\Psi\rangle=\sum\nolimits_{k,n}{{\Psi_{n,k}}|{\psi_{n,k}}\rangle}$, the
eigenvalue problem reads
$\displaystyle{P_{{\rm{occ}}}}\mathcal{X}{P_{{\rm{occ}}}}|\Psi\rangle=\lambda|\Psi\rangle.$
(7)
Combining Eq. (5) and Eq. (7), we obtain the following iterative relation
$\sum\limits_{n=1}^{{n_{{\rm{occ}}}}}{\langle u_{k+\delta
k}^{n^{\prime}}|u_{k}^{n}\rangle{\Psi_{n,k}}}=\lambda{\Psi_{n^{\prime},k+\delta
k}},$ (8)
which can be further written in a more compact form
${F_{k}}{\Psi_{k}}=\lambda{\Psi_{k+\delta k}}$ (9)
where $\Psi_{k}=(\Psi_{1,k},\Psi_{2,k},...,\Psi_{n_{\rm{occ}},k})^{\rm{T}}$,
$[F_{k}]_{n^{\prime},n}=\langle u_{k+\delta k}^{n^{\prime}}|u_{k}^{n}\rangle$.
Repeating the iterative relation for $L$ times, we have
${{\cal W}_{k+2\pi\leftarrow k}}{\Psi_{k}}{\rm{=}}{\lambda^{L}}{\Psi_{k}},$
(10)
which reveals that the Wilson loop is related to the eigenvalues of the
projected position operator. We may obtain the bulk polarization directly
through the projected position operator. In fact, the eigenstates of projected
position operator are Wannier states, while its eigenvalues are center of mass
of Wannier states [25, 26, 27].
To summarize, one may obtain the polarization from projected position
operators directly in real space. This is beneficial for investigating the
disorder system since the translation symmetry is broken and the Wilson loop
method Eq. (3) is not applicable.
## III Winding number of 1D chiral-symmetric topological insulator
### III.1 Chiral-symmetric system and sigular-value decomposition
In this section, we shall firstly review some properties of the chiral-
symmetric system. Due to the special form of the chiral-symmetric Hamiltonian,
we will introduce the singular value decomposition (SVD) for the Hamiltonian
and construct a real-space representation of the winding number.
A lattice with chiral symmetry can be classified into two kinds of
sublattices, namely $A$ and $B$. Thus, the chiral symmetry is also called the
sublattice symmetry. The Hilbert space of the system can be written as the
direct sum of the two subspaces ${\cal H}={{\cal H}_{A}}\oplus{{\cal H}_{B}}$.
The chiral symmetry manifests that $\Gamma H{\Gamma}=-H$, where
$\Gamma=\sum\limits_{l,\alpha\in A}{|l,\alpha\rangle\langle
l,\alpha|}-\sum\limits_{l,\beta\in B}{|l,\beta\rangle\langle l,\beta|}.$ (11)
In the canonical representation, where the chiral operator $\Gamma$ is
diagonal, the Hamiltonian has the following structure
$H=\left({\begin{array}[]{*{20}{c}}0&h\\\
{{h^{\dagger}}}&0\end{array}}\right)$ (12)
where $h$ is a $L_{A}\times L_{B}$ matrix. Here, $L_{A}$ and $L_{B}$ are
respectively the total numbers of $A$ and $B$ sublattices. By decomposing the
eigenstates into two sectors
$|\psi_{n}\rangle=(\psi^{A}_{n},\psi^{B}_{n})^{T}$, the eigenvalue equation
$H|\psi_{n}\rangle=E_{n}|\psi_{n}\rangle$ leads to the coupled equations
$\begin{array}[]{l}h{\psi^{B}_{n}}=E{\psi^{A}_{n}}\\\
{h^{\dagger}}{\psi^{A}_{n}}=E{\psi^{B}_{n}},\end{array}$ (13)
which can be further written as
$\begin{array}[]{l}\left({h{h^{\dagger}}}\right)\psi_{n}^{A}={E^{2}}\psi_{n}^{A}\\\
\left({{h^{\dagger}}h}\right)\psi_{n}^{B}={E^{2}}\psi_{n}^{B}\end{array}$ (14)
Now we may obtain $\psi_{n}^{A}$ and $\psi_{n}^{B}$ by calculating the
eigenvectors of ${h{h^{\dagger}}}$ and ${{h^{\dagger}}h}$ respectively. Note
that both $(\psi^{A}_{n},\pm\psi^{B}_{n})^{T}$ are the eigenvectors of
Hamiltonian (12), and they have opposite eigenenergies. This is a consequence
of the chiral symmetry.
The expression of Eq. (14) reminds us of the singular value decomposition
(SVD). We can make the SVD for the off-diagonal block
$h=U_{A}\Sigma{U_{B}^{-1}}$, in which $U_{A}$ and $U_{B}$ are both unitary
matrices, and $\Sigma$ is a diagonal matrix. The diagonal elements of $\Sigma$
are called singular values. We note that
$\displaystyle U_{A}^{-1}h{h^{\dagger}}{U_{A}}$ $\displaystyle=$
$\displaystyle{\Sigma^{2}},$ $\displaystyle U_{B}^{-1}{h^{\dagger}}h{U_{B}}$
$\displaystyle=$ $\displaystyle{\Sigma^{2}},$ (15)
which reveals that $U_{A}$ and $U_{B}$ respectively diagonalize
$h{h^{\dagger}}$ and ${h^{\dagger}}h$. Therefore, the singular values are
identified as the non-negative eigenenergies of the system. The column of
unitary matrices $U_{A}$ and $U_{B}$ are respectively the eigenvectors
$\psi_{n}^{A}$ and $\psi_{n}^{B}$ (up to a normalization factor $1/\sqrt{2}$).
We have not assumed the numbers of the two kinds of sublattices $L_{A},L_{B}$
in the above derivation. Actually, $L_{A}$ and $L_{B}$ can be different, and
the matrix $h$ is not necessarily squared. If $L_{A}\neq L_{B}$, there must be
at least $|L_{A}-L_{B}|$ zero singular values. However, throughout this paper,
we only consider the numbers of $A$ and $B$ sublattices are equal. More
precisely, we are interested in the situation where $h$ is not singular. This
further requires the system is gapped at $E=0$.
When all singular values are non-zero, one may deform the singular values to
arbitrary positive values and still maintain the same eigenstates. Hence, it
is convenient to set $\Sigma$ to be identity. We denote the new Hamiltonian as
$Q=\left({\begin{array}[]{*{20}{c}}0&q\\\ {{q^{-1}}}&0\end{array}}\right),$
(16)
where $q=U_{A}U_{B}^{-1}$ is a unitary matrix. Now the energy spectrum of the
Hamiltonian becomes completely flat and only takes values of $+1$ and $-1$.
$Q$ is called the flattened Hamiltonian [36]. Later we will see the SVD is
convenient for calculating and understanding the winding number.
### III.2 Winding number in momentum space
When translation symmetry is present, we can work in momentum space by
introducing the Fourier transformation
$|k,\alpha\rangle=\int{\frac{{dk}}{{2\pi}}\exp\left({ikl}\right)|l,\alpha\rangle}.$
Then we can block diagonalize the flattened Hamiltonian (16) as
${Q(k)}=\left({\begin{array}[]{*{20}{c}}0&{q\left(k\right)}\\\
{q{{\left(k\right)}^{-1}}}&0\end{array}}\right).$ (17)
The eigenstates can be written as
$|{\psi_{n,k}}\rangle=\sum_{\alpha}{u_{n,k}^{\alpha}|k,\alpha\rangle}$. There
is a map from Brillouin Zone to the unitary matrices $q(k)$. The map is
classified by the first homotopy group ${\pi_{1}}[U(n)]\cong\mathbb{Z}$ and
characterized by the winding number [36, 42]. The winding number in 1D can be
calculated via [36, 42]
$\nu=\frac{i}{{2\pi}}\int_{-\pi}^{\pi}{dk{{\rm{Tr}}\left[{{q}\left(k\right)^{-1}{\partial_{k}}q\left(k\right)}\right]}}.$
(18)
By inserting $q(k)=U_{A}(k)U_{B}^{-1}(k)$ introduced in previous subsection
into Eq. (18), we find
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{i}{{2\pi}}\int_{-\pi}^{\pi}{dk{{\rm{Tr}}\left[{{U_{A}}\left(k\right)^{-1}{\partial_{k}}U_{A}\left(k\right)}\right]}}$
(19)
$\displaystyle\;\;-\frac{i}{{2\pi}}\int_{-\pi}^{\pi}{dk{{\rm{Tr}}\left[{U_{B}\left(k\right)^{-1}{\partial_{k}}{U_{B}}\left(k\right)}\right]}},$
where we have used a fact that
$\mathrm{Tr}\left({U_{q}}^{-1}{\partial_{q}}{U_{q}}\right)=-\mathrm{Tr}\left({U_{q}}{\partial_{q}}U_{q}^{-1}\right)$
when $U_{q}$ is a unitary matrix. As mentioned above, the column of unitary
matrix $U_{\sigma}(k)$ is the eigenvector $u_{n,k}^{\sigma}\;(\sigma=A,B)$ up
to a normalization factor. One can find that Eq. (19) is exactly the “skew”
polarization. The above expression can be considered as the difference of the
polarization between two sublattices. In other words, the winding number
(divided by 2) measures the difference of polarization between $A$ and $B$
sublattices. In addition, one can find that the summation of the winding of
$U_{A}(k)$ and $U_{B}(k)$ divided by 2 mode 1 leads to the polarization. This
implies a relation between polarization and winding number in chiral-symmetric
topological insulators: $p=\nu/2\;\mathrm{mod}\;1$.
### III.3 Winding number in real space
Recently, the winding number is generalized from momentum-space formula [Eq.
(18)] to real-space formula [29, 34, 30]. The idea introduced in Ref. [29] is
to replace the integral and the derivative versus quasi-momentum $k$ by its
real-space representation. For a 1D system, the real-space winding number
reads [29]
$\nu={\cal T}\left\\{{{Q_{BA}}\left[{X,{Q_{AB}}}\right]}\right\\},$ (20)
where $\cal T$ refers to trace per volume,
${Q_{BA}}={\Gamma_{B}}Q{\Gamma_{A}},{Q_{AB}}={\Gamma_{A}}Q{\Gamma_{B}}$, and
${{\Gamma}_{\sigma}}=\sum\nolimits_{l,\alpha\in\sigma}{|l,\alpha\rangle\langle
l,\alpha|}$ is the projector onto the $\sigma=A,B$ subspaces [34]. The real-
space formula is still valid even in the presence of disorder given that the
chiral symmetry is preserved [29, 30].
Here, we present a quite different formula to calculate the winding number in
real space. Later, we will prove that these two formulas are equivalent in the
thermodynamic limit at half filling. As stated in the previous section, the
winding number is related to the relative polarization of $A$ and $B$
sublattice. We shall follow the projected position operator approach described
in Sec. II to derive the winding number in real space.
To illustrate our idea, we consider a finite system with $L$ cells and
discretize the integral in Eq. (19) as
$\nu=\frac{1}{{2\pi
i}}\sum\limits_{k}{{\rm{Tr}}\left[{{\rm{log}}\left({{F}_{k}^{A}{{{{F}_{k}^{B}}}^{\dagger}}}\right)}\right]},$
(21)
where
$F_{k}^{\sigma}=U_{\sigma}^{\dagger}\left({k}\right){U_{\sigma}}\left(k-\delta
k\right),\;\sigma=A,B$ and $k=2n\pi/L,n\in\mathbb{Z}$ (see Appendix A for
detailed derivation). Since the $n$-th column of $U_{\sigma}(k)$ is the vector
$|u^{\sigma}_{n,k}\rangle$, it can be found that the matrix elements of
$F_{k}^{\sigma}$ are ${\left[{F_{k}^{\sigma}}\right]_{m,n}}=\langle
u_{m,k}^{\sigma}|u_{n,k-\delta k}^{\sigma}\rangle$.
Next, we propose the following equivalent formula to calculate the winding
number
$\nu=\frac{1}{{2\pi
i}}{\rm{Tr}}\left[{\log\left({{{\mathcal{X}}_{A}}\mathcal{X}_{B}^{-1}}\right)}\right],$
(22)
where
${{\mathcal{X}}_{\sigma}}=U_{\sigma}^{-1}\Gamma_{\sigma}\mathcal{X}\Gamma_{\sigma}{U_{\sigma}}$
($\sigma=A,B$) are unitary matrices. ${{\mathcal{X}}_{\sigma}}$ can be
considered as the position operator projected onto the $\sigma$ sector of the
eigenstate in the occupied band. The quantum-mechanical position operator can
be chosen as
$\mathcal{X}=\sum\limits_{l,\alpha\in A,\beta\in
B}{{{e}^{i\frac{2\pi}{L}l}}\left(|l,\alpha\rangle\langle
l,\alpha|+|l,\beta\rangle\langle l,\beta|\right)},$ (23)
Then this operator have the same form when it is projected to the $A$ and $B$
sectors. For convenience, we denote the projected operators as
$\mathcal{\tilde{X}}\equiv\Gamma_{A}\mathcal{X}{{\Gamma}_{A}}={{\Gamma}_{B}}\mathcal{X}{{\Gamma}_{B}}$
in the following context.
To prove the equivalence between Eqs. (22) and (21), we note that
$\displaystyle\left[{{\mathcal{X}}_{\sigma}}\right]_{(m,k^{\prime}),(n,k)}$
$\displaystyle=$
$\displaystyle\langle\psi_{m,k^{\prime}}^{\sigma}|\mathcal{X}|\psi_{n,k}^{\sigma}\rangle$
(24) $\displaystyle=$ $\displaystyle{\delta_{k^{\prime},k+\delta k}}\langle
u_{m,k^{\prime}}^{\sigma}|u_{n,k}^{\sigma}\rangle,$
where
$|\psi_{n,k}^{\sigma}\rangle={\sum_{\alpha\in\sigma}}u_{n,k}^{\sigma}|k,\alpha\rangle$
is the $\sigma$ sector of eigenstate (up to $1/\sqrt{2}$ normalization
factor). Then, we have
$\displaystyle{\left[{{{\mathcal{X}}_{A}}\mathcal{X}_{B}^{-1}}\right]_{(m,k^{\prime}),(l,k^{\prime\prime})}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n,k}{{{\left[{{{\mathcal{X}}_{A}}}\right]}_{(m,k^{\prime}),(n,k)}}{{\left[{\mathcal{X}_{B}^{-1}}\right]}_{(n,k),(l,k^{\prime\prime})}}}$
(25) $\displaystyle=$
$\displaystyle\sum\limits_{n,k}{{\delta_{k^{\prime},k+\delta
k}}{\delta_{k,k^{\prime\prime}-\delta k}}\langle
u_{m,k^{\prime}}^{A}|u_{n,k}^{A}\rangle\langle
u_{n,k}^{B}|u_{l,k^{\prime\prime}}^{B}\rangle}$ $\displaystyle=$
$\displaystyle{\delta_{k^{\prime},k^{\prime\prime}}}\sum\limits_{n}{\langle
u_{m,k^{\prime}}^{A}|u_{n,k^{\prime}-\delta k}^{A}\rangle\langle
u_{n,k^{\prime\prime}-\delta k}^{B}|u_{l,k^{\prime\prime}}^{B}\rangle}$
$\displaystyle=$
$\displaystyle{\delta_{k^{\prime},k^{\prime\prime}}}\sum\limits_{n}{{\left[{F_{k^{\prime}}^{A}}\right]_{m,n}}{\left[{\left({F_{k^{\prime}}^{B}}\right)^{\dagger}}\right]_{n,l}}}.$
Now we can see that matrix ${{{\mathcal{X}}_{A}}\mathcal{X}_{B}^{-1}}$ has a
block-diagonal structure. Each block is associated with certain quasi-momentum
$k$, and is exactly equal to the matrix
${{F}_{k}^{A}{{{{F}_{k}^{B}}}^{\dagger}}}$. Therefore, one can immediately
recognize that Eqs. (22) and (21) are equivalent.
Note that Eq. (21) should reproduce a strictly quantized winding number
$\nu\in\mathbb{N}$. This can be proved by noting that
$\det\left({{\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}}\right)=1$, since
$U_{\sigma}$ and $\mathcal{\tilde{X}}$ are unitary matrices. Then, tracing the
logarithm in Eq. (22) gives an integer multiple of $2\pi i$, and therefore the
resulting winding number is an integer number.
We have to emphasize that this quantization occurs only when the system is
half-filled, i.e. the Fermi energy lies in the bandgap. Generally, when the
Fermi surface lies in the bands (i.e. the filling is less than one half), the
momentum-space winding number in Eq. (18) is ill-defined. However, the real-
space formula (22) enables us to calculate the ‘winding number’ at arbitrary
fractional filling less than one half. As mentioned above, each column of
$U_{\sigma}$ is the $\sigma$ sector of the eigenstates. Thus, one may select
certain columns of $U_{\sigma}$ to construct a reduced matrix
$\tilde{U}_{\sigma}$, and then use Eq. (22) to calculate the ‘winding number’.
Physically, it can be understood as projecting the position operator onto a
certain subspace. For example, we may choose the eigenstates whose energies
are below the Fermi energy $E_{n}<E_{\mathrm{F}}$ and calculate the
corresponding fractional winding number via Eq. (22). As $\tilde{U}_{\sigma}$
is no more a unitary matrix after the reduction, the result may give a
fractional value. We will verify this method numerically in Sec. IV.2.
In addition, we shall show that our real-space representation of the winding
number can be written in a form of the Bott index. By transforming
${{{\mathcal{X}}_{A}}\mathcal{X}_{B}^{-1}}$ via unitary matrix $U_{B}$ and
using $q=U_{A}U_{B}^{-1}$, we have
$\displaystyle{U_{B}}\left({{\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}}\right)U_{B}^{-1}$
$\displaystyle=$
$\displaystyle{U_{B}}\left({U_{A}^{-1}\mathcal{\tilde{X}}{U_{A}}U_{B}^{-1}{\mathcal{X}^{-1}}{U_{B}}}\right)U_{B}^{-1}$
(26) $\displaystyle=$
$\displaystyle\left({U_{B}}U_{A}^{-1}\right)\mathcal{\tilde{X}}\left({U_{A}}{U_{B}^{-1}}\right){\mathcal{\tilde{X}}^{-1}}$
$\displaystyle=$
$\displaystyle{q^{-1}}\mathcal{\tilde{X}}q{\mathcal{\tilde{X}}^{-1}}.$
Hence, Eq. (22) can be written as
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi
i}}{\rm{Tr}}\log\left({{q^{-1}}\mathcal{\tilde{X}}q{\mathcal{\tilde{X}}^{-1}}}\right)$
(27) $\displaystyle=$
$\displaystyle\mathrm{Bott}(q^{-1},\mathcal{\tilde{X}}),$
which is exactly the form of Bott index introduced in [37, 38]. In previous
works, the Bott index is related to the real-space Chern number of the 2D
topological insulator. To the best of our knowledge, the Bott index has not
been applied to the real-space winding number of 1D chiral-symmetric
topological insulators. Although the Bott indices for the Chern number and
winding number have similar forms, they are fundamentally distinct.
### III.4 Winding number defined through twisted boundary condition
In the previous subsection, we obtain a real-space representation of the
winding number for the 1D chiral-symmetric topological insulator. However,
some of the properties, such as the self-averaging nature and the bulk-edge
correspondence, are still vague. In this subsection, we introduce the winding
number defined through the twisted boundary condition (TBC). Then, we prove
the bulk-edge correspondence for the TBC winding number, which indicates that
the TBC winding number is self-averaging in the presence of disorder in the
thermodynamic limit. Next, we will prove that our real-space representation of
winding number Eq. (22) is equivalent to the TBC winding number.
The TBC manifests that the two ends of the 1D lattice are glued together, but
the particle will gain a phase $\Phi$ when they move through the boundary.
Thus, the TBC is also called the generalized periodic boundary condition [43,
44]. The TBC can be equivalently expressed as a result of the magnetic flux
$\Phi$ piercing through the periodic chain. General 1D Hamiltonian under TBC
can be written as
$\displaystyle H\left(\Phi\right)=-\sum\limits_{\alpha,\beta,n\leq
m}{t_{m,n}^{\alpha,\beta}{{e}^{i{{\Phi}_{m,n}}}}c_{\alpha,m}^{\dagger}{{c}_{\beta,n}}+h.c.}$
(28) $\displaystyle{{\Phi}_{m,n}}=\left\\{\begin{matrix}\Phi,&\langle
m,n\rangle\ \text{cross the boundary};\\\ 0,&\text{otherwise}.\\\
\end{matrix}\right.,$
in which $c^{\dagger}_{\alpha,m}\;(c_{\alpha,m})$ is the creation
(annihilation) operator of the $m$-th cell, $\alpha$ is the index of
sublattice, and $t_{m,n}^{\alpha,\beta}$ is the corresponding tunneling
strength.
#### III.4.1 Winding number and bulk-edge correspondence
The bulk-edge correspondence is a well-known principle in topological band
theory. Non-trivial bulk topological invariant will lead to gapless
excitations in the ground state at the edges. There have been some rigorous
mathematical proofs of bulk-edge correspondence in 1D chiral-symmetric
topological insulator [45, 46, 47]. Here, we use the TBC and follow the idea
in Ref. [14] to derive this principle.
When the system possesses the chiral symmetry, as mentioned before, the
flattened Hamiltonian is parametrized by the flux $\Phi$
$Q\left(\Phi\right)=\left(\begin{matrix}0&q\left(\Phi\right)\\\
{{q}^{-1}}\left(\Phi\right)&0\\\ \end{matrix}\right).$ (29)
In Ref. [48], it has been proved that the excitation gap will not be affected
by the twist angle $\Phi$ in the thermodynamic limit. This means
$q\left(\Phi\right)$ is non-singular for arbitrary $\Phi\in[0,2\pi]$ as long
as the chiral-symmetric system is gapped at $E=0$. Then, the TBC winding
number can be safely defined as
$\tilde{\nu}=\frac{1}{2\pi i}\int\limits_{0}^{2\pi}{d\Phi\
\text{Tr}\left[{{q}^{-1}}\left(\Phi\right){{\partial}_{\Phi}}q\left(\Phi\right)\right]}.$
(30)
Next, we show that non-trivial winding number $\tilde{\nu}\neq 0$ under PBC
results in the zero-energy modes, and the number of zero-energy modes is twice
the winding number. Inspired by Ref. [14], we modify the boundary condition by
adding a parameter $\eta\in[0,1]$ onto the tunneling which crosses the
boundary
$t_{m,n}^{\alpha,\beta}\left(\eta\right)=\left\\{\begin{matrix}\eta
t_{m,n}^{\alpha,\beta},&\langle m,n\rangle\ \text{cross the boundary}\\\
t_{m,n}^{\alpha,\beta},&\text{otherwise}\\\ \end{matrix}\right..$ (31)
The system has open boundary when $\eta=0$, and restores the usual TBC when
$\eta=1$. Now the Hamiltonian are parameterized by $(\Phi,\eta)$. Then, we
introduce the $U(1)$ phase field
$z\left(\Phi,\eta\right)=\det q\left(\Phi,\eta\right)\in U(1).$ (32)
Provided $z\left(\Phi,\eta\right)\neq 0$, the TBC winding number Eq. (30) can
be written as
$\tilde{\nu}=\frac{1}{2\pi i}\oint\limits_{\eta=1}{{{z}^{-1}}dz}.$ (33)
For non-trivial winding number $\tilde{\nu}\neq 0$, Eq. (33) implies some
poles of $z$ reside in the circle of $\eta=1$, and the number of the poles
should be equal to the absolute value of winding number. Recall that the twist
angle will not affect the excitation gap in the thermodynamic limit, the
system should be either gapped or gapless at $E=0$ for arbitrary twist angle
$\Phi\in[0,2\pi]$. This argument also holds for $\eta<1$. Hence, there should
be an infinite number of poles inside the circle if
$z\left(\Phi,\eta\right)=0$ for $0<\eta<1$, which is impossible when $\nu$ is
well-defined and the system is away from the phase transition. Consequently,
the poles of $z\left(\Phi,\eta\right)$ can only occur at $\eta=0$, which
corresponds to the open boundary condition. This proof is similar to the
discussion about the bulk-edge correspondence of the quantum Hall effect in
Ref. [14].
Then, we prove that the appearance of the zero-energy modes is associated to
the non-trivial winding number. Note that $z\left(\Phi,\eta\right)=\det
q\left(\Phi,\eta\right)$ is proportional to the following products
$z\left(\Phi,\eta\right)=\det
q\left(\Phi,\eta\right)\propto\prod\limits_{n}{{{\xi}_{n}\left(\Phi,\eta\right)}}$
(34)
in which $\\{\xi_{n}\left(\Phi,\eta\right)\\}$ are the singular values of
$q\left(\Phi,\eta\right)$. We can learn from the above discussions that the
number of poles is exactly equal to the number of zero singular values.
Meanwhile, as mentioned in Sec. III.1, the singular values of the off-diagonal
block $h$ are half of the eigenvalues of the corresponding chiral-symmetric
Hamiltonian. Therefore, we can conclude that the number of zero-energy modes
is twice the winding number under open boundary condition.
Since we have assumed the system is gapped under PBC, the zero-energy modes
are the in-gap modes and should be localized at the edge, which is known as
the bulk-edge correspondence. The above considerations are still valid if we
replace the flattened Hamiltonian $Q$ by the original Hamiltonian $H$.
Importantly, this bulk-edge correspondence holds for the disordered case given
that chiral symmetry persists. The bulk-edge correspondence also implies that
the TBC winding number is self-averaging in the thermodynamic limit and away
from the phase transition point.
#### III.4.2 Equivalence of the TBC winding number and the real-space
representation of the winding number
Now, we would like to prove that our real-space representation of winding
number Eq. (22) is equivalent to the TBC winding number in the thermodynamic
limit at half filling. The TBC Hamiltonian (28) can be transformed to [48, 49]
$\displaystyle\tilde{H}\left(\Phi\right)$ $\displaystyle=$
$\displaystyle{{\mathcal{U}}_{\Phi}}H\left(\Phi\right){{\mathcal{U}}_{\Phi}}^{-1}$
(35) $\displaystyle=$ $\displaystyle-\sum\limits_{\alpha,\beta,n\leq
m}{t_{m,n}^{\alpha,\beta}{{e}^{i\frac{\Phi}{L}\left(m-n\right)}}c_{\alpha,m}^{\dagger}{{c}_{\beta,n}}+h.c.}$
where
${{\mathcal{U}}_{\Phi}}={{e}^{i\frac{\Phi}{L}\hat{X}}},\ \
\hat{X}=\sum\limits_{\alpha,m}{mc_{\alpha,m}^{\dagger}{{c}_{\alpha,m}}}.$ (36)
We shall use the tilde notation to distinguish these two unitarily equivalent
Hamiltonians in the following discussions. For a sufficiently large system
$L\to\infty$, and assuming the range of tunneling is finite, we can expand Eq.
(35) up to the leading order of $\Phi/L$
$\displaystyle\tilde{H}\left(\Phi\right)=H\left(0\right)+\frac{\Phi}{L}\mathcal{J}+O\left(\frac{1}{{{L}^{2}}}\right)$
$\displaystyle\mathcal{J}=i\sum\limits_{\alpha,\beta,n\leq
m}{\left[\left(m-n\right)t_{m,n}^{\alpha,\beta}c_{\alpha,m}^{\dagger}{{c}_{\beta,n}}-h.c.\right]}.$
(37)
One may notice that $\mathcal{J}$ is the current operator.
Similarly, we can expand $\tilde{q}(\Phi)$ up to the leading order of
$\epsilon=\Phi/L$
$\tilde{q}\left(\epsilon\right)=q\left(0\right)+\epsilon{{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}+O\left({{\epsilon}^{2}}\right)$
(38)
where we have expressed $\tilde{q}(\Phi)$ as $\tilde{q}(\epsilon)$ for
clarity. With this approximation, the winding number Eq. (30) can be written
as
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int\limits_{0}^{2\pi/L}{d\epsilon\
\text{Tr}\left[{{\tilde{q}}^{-1}}\left(\epsilon\right){{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}$
(39) $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int\limits_{0}^{2\pi/L}{d\epsilon\
\text{Tr}\left[{{q}^{-1}}\left(0\right){{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}\right]}+O\left({{\epsilon}^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{iL}\text{Tr}\left[{{q}^{-1}}\left(0\right){{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}\right]$
where we have used the fact that
$\left[{{q}^{-1}}\left(0\right){{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}\right]$
is independent on $\epsilon$.
On the other hand, noting that $\mathcal{U}_{2\pi}=\mathcal{X}$, there is
$H\left(0\right)=H\left(2\pi\right)=\mathcal{X}^{-1}\tilde{H}\left(2\pi\right){{\mathcal{X}}}$
according to Eq. (35). This relation can be also generalized to the flattened
Hamiltonian
$Q\left(0\right)=\mathcal{X}^{-1}\tilde{Q}\left(2\pi\right){{\mathcal{X}}}$.
Since the matrix $q(\Phi)$ can be obtained from projecting the Q matrix via
$\tilde{q}\left(\Phi\right)={{\Gamma}_{A}}\tilde{Q}\left(\Phi\right){{\Gamma}_{B}}$.
We can derive a similar relation for the $q(0)$ and $\tilde{q}(2\pi)$
$\displaystyle q\left(0\right)$ $\displaystyle=$
$\displaystyle{{\Gamma}_{A}}Q\left(0\right){{\Gamma}_{B}}={{\Gamma}_{A}}\mathcal{X}^{-1}\tilde{Q}\left(2\pi\right){{\mathcal{X}}}{{\Gamma}_{B}}$
(40) $\displaystyle=$
$\displaystyle\left({{\Gamma}_{A}}{{\mathcal{X}}^{-1}}{{\Gamma}_{A}}\right)\left[{{\Gamma}_{A}}\tilde{Q}\left(2\pi\right){{\Gamma}_{B}}\right]\left({{\Gamma}_{B}}\mathcal{X}{{\Gamma}_{B}}\right)$
$\displaystyle=$
$\displaystyle{{{\tilde{\mathcal{X}}}}^{-1}}\tilde{q}\left(2\pi\right)\tilde{\mathcal{X}}$
where we have used the fact that $\mathcal{X}$ is diagonal in the position
space and
${{\Gamma}_{\sigma}}\mathcal{X}={{\Gamma}_{\sigma}}\mathcal{X}{{\Gamma}_{\sigma}},\
\ {{\Gamma}^{2}_{\sigma}}={{\Gamma}_{\sigma}}$, ($\sigma=A,B$).
Let $\epsilon=2\pi/L$ in Eq. (38), we can combine it with Eq. (40) and obtain
the following relation
$\displaystyle{\mathcal{\tilde{X}}}q\left(0\right)\mathcal{\tilde{X}}^{-1}$
$\displaystyle=$ $\displaystyle\tilde{q}\left(\epsilon=\frac{2\pi}{L}\right)$
(41) $\displaystyle=$ $\displaystyle
q\left(0\right)+\frac{2\pi}{L}{{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}+O(\frac{1}{L^{2}}).$
Therefore, we can rewrite Eq. (39) as
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{1}{iL}\text{Tr}\left\\{{{q}^{-1}}\left(0\right){{\left[{{\partial}_{\epsilon}}\tilde{q}\left(\epsilon\right)\right]}_{\epsilon=0}}\right\\}$
(42) $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\text{Tr}\left\\{{{q}^{-1}}\left(0\right)\left[{{\mathcal{\tilde{X}}}}q\left(0\right)\mathcal{\tilde{X}}^{-1}-q\left(0\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\text{Tr}\left({{q}^{-1}}\mathcal{\tilde{X}}q{{\mathcal{\tilde{X}}}^{-1}}-I\right).$
where $I$ is the identity matrix and we have dropped the dependence on the
twist angle in the last row. From Eq. (41), one can find that
${{q}^{-1}}\mathcal{\tilde{X}}q{{\mathcal{\tilde{X}}}^{-1}}$ is close to the
identity for sufficiently large $L$. In the thermodynamic limit $L\to\infty$,
Eq. (42) can be written as the matrix logarithm [50]
$\nu=\frac{1}{2\pi
i}\text{Tr}\log\left({{q}^{-1}}\mathcal{\tilde{X}}q{{\mathcal{\tilde{X}}}^{-1}}\right)$
(43)
which is in agreement with Eq. (27). Thus, we have proved that the real-space
representation of the winding number is equivalent to the winding number
defined through the TBC in the thermodynamic limit at half filling.
Generally speaking, to obtain the winding number defined through the TBC, one
needs to change the twist angle $\Phi$ from $0$ to $2\pi$, which requires
great computational resources. Here, we have shown that the winding angle of
$\tilde{q}(\Phi)$ changes linearly with the twist angle:
$\arg\left[\det\tilde{q}\left(\Phi\right)\right]\sim\Phi$. It is such a kind
of linear dependence that leads to our efficient real-space representation of
the winding number. Moreover, as discussed above, the TBC winding number
should be a self-averaging quantity, and we can therefore conclude that our
real-space representation of the winding number is also self-averaging.
In addition, we can further prove that the TBC winding number [Eq. (30)], as
well as the real-space representation of the winding number [Eq. (22)] in our
work, are equivalent to the formula [Eq. (20)] obtained in Ref. [29]. To show
this, we note that in the thermodynamic limit, there is
$\displaystyle\left[X,H\left(0\right)\right]$ $\displaystyle=$
$\displaystyle\sum\limits_{\alpha,\beta,n\leq
m}{\left[\left(m-n\right)t_{m,n}^{\alpha,\beta}c_{\alpha,m}^{\dagger}{{c}_{\beta,n}}-h.c.\right]}$
(44) $\displaystyle=$ $\displaystyle-i\mathcal{J},$
where $X=\sum\nolimits_{\alpha,m}{mc_{\alpha,m}^{\dagger}{{c}_{\alpha,m}}}$.
On the other hand, the off-diagonal block $h(\Phi)$ matrix and its inverse
read as
$\tilde{h}\left(\Phi\right)={{\Gamma}_{A}}\tilde{H}\left(\Phi\right){{\Gamma}_{B}},\;\;\tilde{h}\left(\Phi\right)^{-1}={{\Gamma}_{B}}\tilde{H}\left(\Phi\right)^{-1}{{\Gamma}_{A}}.$
(45)
Hence, using Eq. (III.4.2), the derivative with respect to $\Phi$ reads as
$\displaystyle{{\left[{{\partial}_{\epsilon}}\tilde{h}\left(\epsilon\right)\right]}_{\epsilon=0}}$
$\displaystyle=$ $\displaystyle
L{{\left[{{\partial}_{\Phi}}\tilde{h}\left(\Phi\right)\right]}_{\Phi=0}}$ (46)
$\displaystyle=$ $\displaystyle{{\Gamma}_{A}}\mathcal{J}{{\Gamma}_{B}}$
$\displaystyle=$ $\displaystyle i{{\Gamma}_{A}}[X,H(0)]{{\Gamma}_{B}}.$
The winding number [Eq. (39)] can be equivalently written as
$\tilde{\nu}=\frac{1}{iL}h{{\left(0\right)}^{-1}}{{\left[{{\partial}_{\epsilon}}\tilde{h}\left(\epsilon\right)\right]}_{\epsilon=0}}=\frac{1}{L}{{\Gamma}_{B}}H^{-1}{{\Gamma}_{A}}\left[X,H\right]{{\Gamma}_{B}},$
(47)
where we have dropped the dependence on the twist angle $\Phi$ in the last
term for simplicity. Then, it is tempting to replace the Hamiltonian by the
flattened Hamiltonian, and we have
$\displaystyle\tilde{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{L}{{\Gamma}_{B}}Q{{\Gamma}_{A}}\left[X,Q\right]{{\Gamma}_{B}}$
(48) $\displaystyle=$
$\displaystyle\frac{1}{L}{{Q}_{BA}}\left[X,{{Q}_{AB}}\right],$
in which we have used the relation $\Gamma_{\sigma}^{2}=\Gamma_{\sigma}$,
$\Gamma_{\sigma}X=\Gamma_{\sigma}X\Gamma_{\sigma}$ ($\sigma=A,B$) and
$Q^{-1}=Q$. We find that Eq. (48) is identical to Eq. (20). Therefore, we can
conclude that the TBC winding number [Eq. (30)], the real-space representation
of the winding number [Eq. (22)] in our work, and the real-space formula [Eq.
(20)] obtained in Ref. [29], are equivalent in the thermodynamic limit. Note
that Eq. (48) can be also derived from Eq. (42) by using the Baker-Campbell-
Hausdorff formula to expand $\mathcal{\tilde{X}}q\mathcal{\tilde{X}}^{-1}$ up
to the first term, but the convergence of the expansion seems to be unclear in
that way.
### III.5 Winding number defined through path connection
Previously, we obtain the real-space winding number by considering the
sublattice polarization. We find that our formula can be expressed as the Bott
index. For the Bott index, there is an alternative method to express the
winding number by a path connection [37], which clearly shows the meaning of
winding in real space. We shall give a simple illustration here.
Since unitary matrix is always diagonalizable, we can make an eigenvalue
decomposition
${\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}=T\mathcal{X}_{\mathrm{diag}}T^{-1}$
where $T$ is an unitary matrix and
$\mathcal{X}_{\mathrm{diag}}=\mathrm{diag}\left\\{{{e^{i{\theta_{1}}}},{e^{i{\theta_{2}}}},\ldots,{e^{i{\theta_{n}}}}}\right\\}$
is a diagonal matrix with all elements lying on an unit circle in complex
plane $\mathbb{C}$. Then, Eq. (22) can be written as
$\nu=\frac{1}{{2\pi
i}}{\rm{Tr}}\left[{\log\left({{\mathcal{X}_{{\rm{diag}}}}}\right)}\right]=\frac{1}{{2\pi}}\sum\limits_{k}{{\theta_{k}}}.$
(49)
In other words, the information of the winding number is encoded in these
phases.
Recall that the matrix logarithm is a multi-valued matrix function. To
‘unwind’ the logarithm in Eq. (22), we consider a continuous path (homotopy)
$\phi:[0,1]\to{\mathrm{GL}}\left({n,\mathbb{C}}\right)$ such that
$\phi(0)={\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}$ and $\phi(1)=I$. The function
$\phi(r)$ can be written as
$\displaystyle\phi\left(r\right)$ $\displaystyle=$ $\displaystyle
T{\mathcal{X}_{{\rm{diag}}}}\left(r\right){T^{-1}}$ (50) $\displaystyle=$
$\displaystyle
T{\rm{diag}}\left\\{{{e^{i{\tilde{\theta}_{1}}\left(r\right)}},{e^{i{\tilde{\theta}_{2}}\left(r\right)}},\ldots,{e^{i{\tilde{\theta}_{n}}\left(r\right)}}}\right\\}{T^{-1}}$
where $\tilde{\theta}_{k}(r)\in[0,2\pi),r\in[0,1]$ is a continuous real
function with $\tilde{\theta}_{k}(0)=\theta_{k}$ and
$\tilde{\theta}_{k}(1)=0$. The winding number can be defined as
$\nu=-\frac{1}{2\pi}\int_{0}^{1}{dr}\frac{\partial}{{\partial
r}}\arg\left[{\det\phi\left(r\right)}\right],$ (51)
which is indeed related to the real-space winding number since
$\displaystyle-$
$\displaystyle\frac{1}{2\pi}\int_{0}^{1}{dr}\frac{\partial}{{\partial
r}}\arg\left[{\det\phi\left(r\right)}\right]$ (52)
$\displaystyle\;\;\;\;=-\frac{1}{2\pi}\sum_{k}{\int_{0}^{1}{dr\frac{\partial}{{\partial
r}}{{\tilde{\theta}}_{k}}\left(r\right)}}=\frac{1}{2\pi}\sum\limits_{k}{{\theta_{k}}},$
where we have used $\int_{0}^{1}{dr\frac{\partial}{{\partial
r}}{\tilde{\theta}_{k}}\left(r\right)}=-\theta_{k}$.
When the winding number is non-trivial $\nu>0$, the continuous path $\phi$
connects different branches of complex logarithm. The winding number can only
be changed discontinuously when the system undergoes a phase transition. This
is accompanied by the sudden changes of some phase $\theta_{k}$, and the
function $\mathrm{arg}[\det\phi(r)]$ becomes discontinuous and
indifferentiable. Note that $\det\phi(r)$ may be still continuous and well-
defined. The singularity of $\mathrm{arg}[\det\phi(r)]$ is due to the multi-
valued nature of the argument. For example, when a phase grows continuously
from $\theta_{0}$ to $\theta_{0}+2\pi$, the complex function $e^{i\theta_{0}}$
is continuous, while its principal argument $\mathrm{arg}(e^{i\theta_{0}})$ is
discontinuous. We will show this later with numerical calculation.
In the presence of translation symmetry, we have proved that our formula (22)
is exactly equal to the momentum-space formula (18). Thus, it is natural that
the winding number [Eq. (51)] is also equivalent to the momentum-space formula
[Eq. (18)]. Also, it is always possible to elaborate a path that the behavior
of the winding number defined in Eq. (51) is identical to the momentum-space
winding number [Eq. (18)] when translation symmetry is present. In other
words, the loop $\det\phi(r)$ is homotopic to $\det h(k)$ and $\det q(k)$ in
the presence of translation symmetry.
## IV Application to 1D BDI class model
In the previous section, we have introduced a representation of the real-space
winding number. Our formula ensures that the winding number is an integer at
half-filling case. For arbitrary fractional filling less than one-half, our
formula is still applicable but does not necessarily give an integer number.
On the other hand, we define a winding number through a continuous path from
the special unitary matrix to the identity matrix. In this section, we will
apply these arguments to a toy model in the 1D BDI class. We shall also
consider a disorder on tunneling to see the robustness and validity of the
real-space winding number. These calculations can be easily extended to the
AIII class.
### IV.1 Extended Su-Schrieffer-Heeger model and disordered tunneling
We introduce a 1D lattice model described by the following second quantized
Hamiltonian
$H=-\sum\limits_{l}{\left({{t_{1,l}}\hat{a}_{l}^{\dagger}{\hat{b}_{l}}+{t_{2,l}}\hat{a}_{l+1}^{\dagger}{\hat{b}_{l}}+{t_{3,l}}\hat{a}_{l+2}^{\dagger}{\hat{b}_{l}}}\right)}+H.c.,$
(53)
where $\hat{a_{l}}^{{\dagger}}$ ($\hat{b_{l}}^{{\dagger}}$) creates a particle
at the A (B) sublattice of the $l$th cell.
$t_{n,l}=t_{n}+W_{n}\varepsilon_{n,l},n=1,2,3$ is the tunneling strength,
$\varepsilon_{n,l}\in(-1/2,1/2)$ is a random strength distributed uniformly,
and $W_{n}$ is the strength of disorder. This is an extended Su-Schrieffer-
Heeger (SSH) model up to a next-next-nearest-neighbor (NNNN) term. A schematic
illustration of this toy model is shown in Fig. 1. With this special NNNN
tunneling, the system still preserves chiral symmetry. According to the ten-
fold classification, this model belongs to BDI class. There exist topological
phases characterized by winding number $\nu=0,1,2$ in this model.
Figure 1: Schematic illustration of the extended SSH model. Besides the
general nearest-neighbor tunneling in the SSH model, we include special long-
range tunneling indicated by the dashed line.
As a benchmark, we adapt a similar configuration in Ref. [30], and set
$W=W_{1}=2W_{2}$, $W_{3}=0$ and $t_{1}=0,t_{2}=1,t_{3}=-2$. We use the two
approaches Eq. (20) and Eq. (22) to numerically calculate the winding number
as a function of disorder strength $W$. Here the position operator reads as
$\mathcal{\hat{X}}=\exp[i\delta
k\sum_{i}{l(\hat{a}^{\dagger}_{l}\hat{a}_{l}+\hat{b}^{\dagger}_{l}\hat{b}_{l})}]$.
The results are shown in Fig. 2. It can be seen that the winding numbers
obtained from Eq. (22) stay quantized in each random realization (grey dots).
We have examined that the averaged results (blue circles) agree well with the
averaged results obtained from Eq. (20). Away from the phase transition point,
there is almost no fluctuation of winding number because each realization is
perfectly the same quantization. Around the phase transition point, we find
that the fluctuation of winding number becomes large, which can be served as a
signature of the topological phase transition in disorder systems. The phase
transition point can be determined with higher accuracy as the lattice length
increases. The scaling with different lattice lengths is presented in
Appendix. B. However, we note that the real-space winding number obtained from
Eq. (20) is not strictly quantized for the finite system. As shown in the
inset of Fig. 2, the real-space winding number calculated via Eq. (20)
slightly deviates from the integer.
Figure 2: Winding number $\nu$ as a function of disorder strength $W$ in
disordered extended SSH model in Eq. (53). Grey dots are the results from 100
random realizations based on Eq. (22). Blue circles are averaged over the grey
dots. _Inset_ : Comparison between Eq. (20) and Eq. (22). Light-purple dots
are the results from 100 random realization based on Eq. (20). Red squares are
the averaged results of the purple dots. The above calculations are
implemented with $L=1001$ cells. The parameters are
$t_{1}=0,t_{2}=1,t_{3}=-2$, and the configuration of disorder is
$W=W_{1}=2W_{2}$, $W_{3}=0$.
Next, based on Eq. (51), we use Eq. (51) to graphically show the winding
number in the presence of disorder. Practically, we can simply assume
$\phi\left(r\right)={\left({1-r}\right){\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}+rI},\quad
r\in[0,1].$ (54)
The determinant is
$\displaystyle\det\phi\left(r\right)$ $\displaystyle=$
$\displaystyle\det\left[{\left({1-r}\right){\mathcal{X}_{A}}\mathcal{X}_{B}^{-1}+rI}\right]$
(55) $\displaystyle=$
$\displaystyle\det\left\\{{T\left[{\left({1-r}\right){\mathcal{X}_{{\rm{diag}}}}+rI}\right]{T^{-1}}}\right\\}$
$\displaystyle=$
$\displaystyle\det\left[{\left({1-r}\right){\mathcal{X}_{{\rm{diag}}}}+rI}\right]$
$\displaystyle=$
$\displaystyle\prod\limits_{k}{\left[{\left({1-r}\right){e^{i{\theta_{k}}}}+r}\right]}.$
The graphical illustrations of real-space winding numbers are shown in Figs. 3
(a-c). It can be seen that the winding around the singularity point
($\det\phi(r)=0$) in the complex plane coincides well with the real-space
winding number.
Besides, it can be found that $\det\phi\left(r\right)$ only takes zero value
at $r=1/2$ and $\theta_{k}=\pi$. As stated in the previous section, we may
identify that $\theta_{k}=\pi$ is related to the discontinuous change of
argument, and thus corresponds to phase transition. To show this, we
numerically calculate the arguments $\left\\{{{\theta_{k}}}\right\\}$ from the
diagonal entries of $X_{\mathrm{diag}}$ as a function of disorder strength $W$
in a _single_ random realization $\left\\{{{\varepsilon_{n,l}}}\right\\}$. As
presented in Figs. 3 (d-e), given a random realization
$\left\\{{{\varepsilon_{n,l}}}\right\\}$, all the phases
$\left\\{{{\theta_{k}}}\right\\}$ change continuously with disorder strength
except for the phase transition point. This is in agreement with our
discussion.
Figure 3: (a-c) show the trajectory of $\det\phi(r)$ with $W=1,10,25$ in a
single random realization. The red star represents the zero point. (d) shows
the real-space winding number $\nu$ as a function of disorder strength $W$
given a certain set of random factor $\left\\{{{\varepsilon_{n,l}}}\right\\}$
in a single random realization. (e) shows the argument
$\left\\{{{\theta_{k}}}\right\\}$ of the diagonal entries of
$\mathcal{X}_{\mathrm{diag}}$ as a function of $W$ in the same random
realization as (d). Other parameters are identical to the parameters in Fig.
2.
### IV.2 Winding number under different filling factors
Now, we calculate the winding number for various fillings (less than one-half)
in the extended SSH model; see Fig. 4. It can be seen that the winding number
is zero when the filling is empty. As the Fermi energy increases, the winding
number changes continuously, and finally reaches an integer value when the
system is half-filled (the Fermi energy $E_{\mathrm{F}}$ lies in the spectral
gap). This result may be understood by considering the winding number as the
difference of polarization of $A$ and $B$ sublattices. The difference in
sublattice polarization will change with the filling numbers.
Notice that there appears discontinuity in the first derivative of the Fermi
energy. This is because there are some local minima in the band structure, as
shown in the inset of Fig. 4. This method can be also applied to the
disordered case.
Figure 4: Winding number as a function of Fermi energy $E_{F}$ for three
different parameters $t_{1}=0.8,2,4$ in the clean limit $W_{1}=W_{2}=W_{3}=0$.
The inset shows the corresponding band structure. Other parameters are chosen
as $t_{2}=1,t_{3}=-2$.
## V Summary and Discussion
In summary, we propose a real-space formula for calculating the winding number
of 1D chiral-symmetric systems. Our real-space representation of the winding
number is inspired by the projected position operator approach since the
winding number can be written as the difference of polarization between two
sublattices. We show that our approach is equivalent to the momentum-space
representation of the winding number in the clean limit. Even in the presence
of disorder, our formula produces a quantized value. We have also shown that
our method works for the case of fillings less than one-half.
With the help of TBC, we have proved the bulk-edge correspondence principle
for the TBC winding number. We further prove that our real-space
representation of winding number is equivalent to the TBC winding number and
the real-space formula proposed in Ref. [29] in the thermodynamic limit at
half filling. Therefore our real-space representation of winding number also
satisfies the bulk-edge correspondence and the self-averaging property.
Meanwhile, compared with the general TBC method, where includes the integral
of twist angle $\Phi$, our formula can be obtained from a single Hamiltonian,
which is more efficient.
Interestingly, we find that our real-space winding number can be expressed as
a Bott index. However, the Bott index is usually employed for the real-space
Chern number [19, 51, 52], which is quite different from the winding number in
1D chiral-symmetric topological insulator. We also show that the Bott index
has deep connection to the twisted boundary condition. Our work may provide
another concrete example for investigating the Bott index. Meanwhile, one may
find the position operator plays a crucial role in the construction of real-
space representation of topological invariants, such as the Chern number [16,
17, 18, 19, 20], the Zak-Berry phase [25, 26, 27, 28] and the winding number
[29, 30, 32, 31, 33, 34, 35]. Thus, it is intriguing to generalize the
application of position operators to other topological systems in other
topological classes or higher dimension in the future.
We also note that recently the SVD has been applied to non-Hermitian systems
[53], where the singular values of a non-Hermitian Hamiltonian obey the bulk-
edge correspondence. The study on the interplay between non-Hermitian systems
and disorder has received many interests recently [54, 55]. With the formula
of winding number for the non-Hermitian system [56], it is intriguing to
generalize our formula to the non-Hermitian case. The non-Hermitian
Hamiltonian can be considered as the off-diagonal block of some chiral-
symmetric Hermitian Hamiltonians. Then our real-space representation of the
winding number may be extended to non-Hermitian systems.
###### Acknowledgements.
This work has been supported by the National Natural Science Foundation of
China (12025509,11874434), the Key-Area Research and Development Program of
Guangdong Province (2019B030330001), and the Science and Technology Program of
Guangzhou (201904020024). Y.K. is partially supported by the National Natural
Science Foundation of China (Grant No. 11904419).
Figure 5: Winding number as a function of disorder strength $W$ with
different length $L$ of system. The results are averaged over 100 random
realizations. Other parameters are the same as Fig. 2.
## Appendix A Derivation of discretized formula of winding number
Here we give a detailed derivation of Eq. (21). Approximately, we have
$U_{\sigma}^{-1}\left(k\right){\partial_{k}}{U_{\sigma}}\left(k\right)\simeq
U_{\sigma}^{-1}\left(k\right)\frac{{{U_{\sigma}}\left({k+\delta
k}\right)-{U_{\sigma}}\left({k}\right)}}{{\delta k}},$ (56)
which leads to
$1+U_{\sigma}^{-1}\left(k\right){\partial_{k}}{U_{\sigma}}\left(k\right)\delta
k\simeq U_{\sigma}^{-1}\left(k\right){U_{\sigma}}\left({k+\delta k}\right).$
(57)
Take the logarithm on both sides
$\displaystyle\log\left({1+U_{\sigma}^{-1}\left(k\right){\partial_{k}}{U_{\sigma}}\left(k\right)\delta
k}\right)$
$\displaystyle\simeq\log\left({U_{\sigma}^{-1}\left(k\right){U_{\sigma}}\left({k+\delta
k}\right)}\right),$ (58)
and use the approximation $\mathrm{log}(1+x)\approx x$ when $x\rightarrow 0$,
we obtain
$U_{\sigma}^{-1}\left(k\right){\partial_{k}}{U_{\sigma}}\left(k\right)\delta
k\simeq\log\left({U_{\sigma}^{-1}\left(k\right){U_{\sigma}}\left({k+\delta
k}\right)}\right).$ (59)
In the thermodynamic limit, the quantities in two sides of Eq. (59) are
equivalent. Thus, the discretized form of winding number can be written as
$\nu=\frac{1}{{2\pi i}}\sum\limits_{k}{{\rm{Tr}}\left({\log F_{k}^{A}-\log
F_{k}^{B}}\right)},$ (60)
where
$F_{k}^{\sigma}=U_{\sigma}^{\dagger}\left({k}\right){U_{\sigma}}\left(k-\delta
k\right),\;\sigma=A,B$. Alternatively, we can use the fact that
${\rm{Tr}}\log\left(A\right)=\log\det\left(A\right)$ when
$\left\|A\right\|<\pi$ [57], and then obtain the expression in Eq. (21).
## Appendix B Scaling of the phase transition
To examine the finite-size effect on the disorder-induced topological phase
transition, we present the winding number as a function of disorder strength
$W$ in Fig. 5. The results show that the boundary of phase transition tends to
be clear when $L\to\infty$.
## References
* Kane [2013] C. L. Kane, Topological band theory and the $\mathbb{Z}_{2}$ invariant, _Contemporary Concepts of Condensed Matter Science_ 10.1016/B978-0-444-63314-9.00001-9 (2013).
* Haldane [1988] F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
* Kane and Mele [2005] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005).
* Bernevig _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
* Bernevig and Zhang [2006] B. A. Bernevig and S.-C. Zhang, Quantum Spin Hall Effect, Phys. Rev. Lett. 96, 106802 (2006).
* Klitzing _et al._ [1980] K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980).
* Thouless _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
* Bernevig and Hughes [2013] B. A. Bernevig and T. L. Hughes, _Topological insulators and topological superconductors_ (Princeton university press, 2013).
* Fu _et al._ [2007] L. Fu, C. L. Kane, and E. J. Mele, Topological Insulators in Three Dimensions, Phys. Rev. Lett. 98, 106803 (2007).
* Nomura and Nagaosa [2011] K. Nomura and N. Nagaosa, Surface-Quantized Anomalous Hall Current and the Magnetoelectric Effect in Magnetically Disordered Topological Insulators, Phys. Rev. Lett. 106, 166802 (2011).
* Sheng _et al._ [2005] L. Sheng, D. N. Sheng, C. S. Ting, and F. D. M. Haldane, Nondissipative Spin Hall Effect via Quantized Edge Transport, Phys. Rev. Lett. 95, 136602 (2005).
* Xu and Moore [2006] C. Xu and J. E. Moore, Stability of the quantum spin hall effect: Effects of interactions, disorder, and ${\mathbb{z}}_{2}$ topology, Phys. Rev. B 73, 045322 (2006).
* Wu _et al._ [2006] C. Wu, B. Bernevig, and S.-C. Zhang, Helical Liquid and the Edge of Quantum Spin Hall Systems, Phys. Rev. Lett. 96, 106401 (2006).
* Qi _et al._ [2006] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74, 045125 (2006).
* Kobayashi _et al._ [2013] K. Kobayashi, T. Ohtsuki, and K.-I. Imura, Disordered Weak and Strong Topological Insulators, Phys. Rev. Lett. 110, 236803 (2013).
* Prodan _et al._ [2010] E. Prodan, T. L. Hughes, and B. A. Bernevig, Entanglement Spectrum of a Disordered Topological Chern Insulator, Phys. Rev. Lett. 105, 115501 (2010).
* Prodan [2011] E. Prodan, Disordered topological insulators: a non-commutative geometry perspective, Journal of Physics A: Mathematical and Theoretical 44, 113001 (2011).
* Bianco and Resta [2011] R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84, 241106 (2011).
* Loring and Hastings [2010] T. A. Loring and M. B. Hastings, Disordered topological insulators via $C^{\ast}$-algebras, EPL (Europhysics Letters) 92, 67004 (2010).
* Yi-Fu _et al._ [2013] Z. Yi-Fu, Y. Yun-You, J. Yan, S. Li, S. Rui, S. Dong-Ning, and X. Ding-Yu, Coupling-matrix approach to the Chern number calculation in disordered systems, Chinese Physics B 22, 117312 (2013).
* Bandres _et al._ [2016] M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport, Phys. Rev. X 6, 011016 (2016).
* Mross _et al._ [2018] D. F. Mross, Y. Oreg, A. Stern, G. Margalit, and M. Heiblum, Theory of Disorder-Induced Half-Integer Thermal Hall Conductance, Phys. Rev. Lett. 121, 026801 (2018).
* Loring [2015] T. A. Loring, K-theory and pseudospectra for topological insulators, Annals of Physics 356, 383 (2015).
* Xue and Prodan [2012] Y. Xue and E. Prodan, Noncommutative Kubo formula: Applications to transport in disordered topological insulators with and without magnetic fields, Phys. Rev. B 86, 155445 (2012).
* Kivelson [1982] S. Kivelson, Wannier functions in one-dimensional disordered systems: Application to fractionally charged solitons, Phys. Rev. B 26, 4269 (1982).
* Niu [1991] Q. Niu, Theory of the quantized adiabatic particle transport, Modern Physics Letters B 5, 923 (1991).
* Marzari _et al._ [2012] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
* Benalcazar _et al._ [2017] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017).
* Mondragon-Shem _et al._ [2014] I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Prodan, Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder, Phys. Rev. Lett. 113, 046802 (2014).
* Song and Prodan [2014] J. Song and E. Prodan, AIII and BDI topological systems at strong disorder, Phys. Rev. B 89, 224203 (2014).
* Hua _et al._ [2019] C.-B. Hua, R. Chen, D.-H. Xu, and B. Zhou, Disorder-induced Majorana zero modes in a dimerized Kitaev superconductor chain, Phys. Rev. B 100, 205302 (2019).
* Claes and Hughes [2020a] J. Claes and T. L. Hughes, Disorder driven phase transitions in weak AIII topological insulators, Phys. Rev. B 101, 224201 (2020a).
* Lin _et al._ [2020] L. Lin, S. Kruk, Y. Ke, C. Lee, and Y. Kivshar, Topological states in disordered arrays of dielectric nanoparticles, Phys. Rev. Research 2, 043233 (2020).
* Meier _et al._ [2018] E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, and B. Gadway, Observation of the topological Anderson insulator in disordered atomic wires, Science 362, 929 (2018).
* Maffei _et al._ [2018] M. Maffei, A. Dauphin, F. Cardano, M. Lewenstein, and P. Massignan, Topological characterization of chiral models through their long time dynamics, New Journal of Physics 20, 013023 (2018).
* Chiu _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
* Exel and Loring [1991] R. Exel and T. A. Loring, Invariants of almost commuting unitaries, Journal of Functional Analysis 95, 364 (1991).
* Hastings and Loring [2010] M. B. Hastings and T. A. Loring, Almost commuting matrices, localized Wannier functions, and the quantum Hall effect, Journal of mathematical physics 51, 015214 (2010).
* Resta [1998] R. Resta, Quantum-Mechanical Position Operator in Extended Systems, Phys. Rev. Lett. 80, 1800 (1998).
* Marzari and Vanderbilt [1997] N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
* Alexandradinata _et al._ [2016] A. Alexandradinata, Z. Wang, and B. A. Bernevig, Topological Insulators from Group Cohomology, Phys. Rev. X 6, 021008 (2016).
* Ryu _et al._ [2010] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New Journal of Physics 12, 065010 (2010).
* Niu and Thouless [1984] Q. Niu and D. J. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction, Journal of Physics A: Mathematical and General 17, 2453 (1984).
* Niu _et al._ [1985] Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
* Prodan and Schulz-Baldes [2016] E. Prodan and H. Schulz-Baldes, Bulk and boundary invariants for complex topological insulators, K (2016).
* Graf and Shapiro [2018] G. M. Graf and J. Shapiro, The bulk-edge correspondence for disordered chiral chains, Communications in Mathematical Physics 363, 829 (2018).
* Shapiro [2020] J. Shapiro, The bulk-edge correspondence in three simple cases, Reviews in Mathematical Physics 32, 2030003 (2020).
* Watanabe [2018] H. Watanabe, Insensitivity of bulk properties to the twisted boundary condition, Phys. Rev. B 98, 155137 (2018).
* Watanabe and Oshikawa [2018] H. Watanabe and M. Oshikawa, Inequivalent berry phases for the bulk polarization, Phys. Rev. X 8, 021065 (2018).
* Higham [2008] N. J. Higham, _Functions of matrices: theory and computation_ (SIAM, 2008).
* Loring [2014] T. A. Loring, Quantitative K-theory related to spin Chern numbers, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 10, 077 (2014).
* Toniolo [2017] D. Toniolo, On the equivalence of the Bott index and the Chern number on a torus, and the quantization of the Hall conductivity with a real space Kubo formula (2017), arXiv:1708.05912 [cond-mat.mes-hall] .
* Herviou _et al._ [2019] L. Herviou, J. H. Bardarson, and N. Regnault, Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition, Phys. Rev. A 99, 052118 (2019).
* Velury _et al._ [2021] S. Velury, B. Bradlyn, and T. L. Hughes, Topological crystalline phases in a disordered inversion-symmetric chain, Phys. Rev. B 103, 024205 (2021).
* Claes and Hughes [2020b] J. Claes and T. L. Hughes, Skin effect and winding number in disordered non-Hermitian systems, arXiv preprint arXiv:2007.03738 (2020b).
* Gong _et al._ [2018] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological Phases of Non-Hermitian Systems, Phys. Rev. X 8, 031079 (2018).
* Aprahamian and Higham [2014] M. Aprahamian and N. J. Higham, The matrix unwinding function, with an application to computing the matrix exponential, SIAM Journal on Matrix Analysis and Applications 35, 88 (2014).
|
# Optimal convergence rates for the invariant density estimation of jump-
diffusion processes
Chiara Amorino and Eulalia Nualart Université du Luxembourg, L-4364 Esch-Sur-
Alzette, Luxembourg. CA gratefully acknowledges financial support of ERC
Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional
Diffusions”.Universitat Pompeu Fabra and Barcelona School of Economics,
Department of Economics and Business, Ramón Trias Fargas 25-27, 08005
Barcelona, Spain. EN acknowledges support from the Spanish MINECO grant
PGC2018-101643-B-I00 and Ayudas Fundacion BBVA a Equipos de Investigación
Científica 2017.
###### Abstract
We aim at estimating the invariant density associated to a stochastic
differential equation with jumps in low dimension, which is for $d=1$ and
$d=2$. We consider a class of fully non-linear jump diffusion processes whose
invariant density belongs to some Hölder space. Firstly, in dimension one, we
show that the kernel density estimator achieves the convergence rate
$\frac{1}{T}$, which is the optimal rate in the absence of jumps. This
improves the convergence rate obtained in [2], which depends on the
Blumenthal-Getoor index for $d=1$ and is equal to $\frac{\log T}{T}$ for
$d=2$. Secondly, when the jump and diffusion coefficients are constant and the
jumps are finite, we show that is not possible to find an estimator with
faster rates of estimation. Indeed, we get some lower bounds with the same
rates $\\{\frac{1}{T},\frac{\log T}{T}\\}$ in the mono and bi-dimensional
cases, respectively. Finally, we obtain the asymptotic normality of the
estimator in the one-dimensional case for the fully non-linear process.
Keywords: Minimax risk, convergence rate, non-parametric statistics, ergodic
diffusion with jumps, Lévy driven SDE, invariant density estimation
## 1 Introduction
Solutions to Lévy-driven stochastic differential equations have recently
attracted a lot of attention in the literature due to its many applications in
various areas such as finance, physics, and neuroscience. Indeed, it includes
some important examples from finance such as the well-known Kou model in [32],
the Barndorff-Nielsen-Shephard model ([8]), and the Merton model ([37]) to
name just a few. An important example of application of jump-processes in
neuroscience is the stochastic Morris-Lecar neuron model presented in [25]. As
a consequence, statistical inference for jump processes has recently become an
active domain of research.
We consider the process $X=(X_{t})_{t\geq 0}$ solution to the following
stochastic differential equation with jumps:
$X_{t}=X_{0}+\int_{0}^{t}b(X_{s})ds+\int_{0}^{t}a(X_{s})dB_{s}+\int_{0}^{t}\int_{\mathbb{R}^{d}_{0}}\gamma(X_{s^{-}})z(\nu(ds,dz)-F(z)dzds),$
(1)
where $(B_{t})_{t\geq 0}$ is a $d$-dimensional Brownian motion and $\nu$ is a
Poisson random measure on $\mathbb{R}_{+}\times\mathbb{R}^{d}$ associated to a
Lévy process $(L_{t})_{t\geq 0}$ with Lévy density function $F$. We focus on
the estimation of the invariant density $\mu$ associated to the jump-process
solution to (1) in low dimension, which is for $d=1$ and $d=2$. In particular,
assuming that a continuous record of $(X_{t})_{t\in[0,T]}$ is available, our
goal is to propose a non-parametric kernel estimator for the estimation of the
stationary measure and to discuss its convergence rate for large $T$.
The same framework has been considered in some recent papers such as [2], [23]
(Section 5.2), and [3]. In the first paper, it is shown that the kernel
estimator achieves the following convergence rates for the pointwise
estimation of the invariant density: $\frac{\log T}{T}$ for $d=2$ and
$\frac{(\log T)^{(2-\frac{(1+\alpha)}{2})\lor 1}}{T}$ for $d=1$ (where
$\alpha$ is the Blumenthal-Getoor index). We recall that, in the absence of
jumps, the optimal convergence rate in the one-dimensional case is
$\frac{1}{T}$, while the one found in [2] depends on the jumps and belongs to
the interval $(\frac{\log T}{T},\frac{(\log T)^{\frac{3}{2}}}{T})$.
In this paper, we wonder if such a deterioration on the rate is because of the
presence of jumps or the used approach. Indeed, our purpose is to look for a
new approach to recover a better convergence rate in the one-dimensional case
(hopefully the same as in the continuous case) and to discuss the optimality
of such a rate. This new approach will also lead to the asymptotic normality
of the proposed estimator. After that, we will discuss the optimality of the
convergence rate in the bi-dimensional case. This will close the circle of the
analysis of the convergence rates for the estimation of the invariant density
of jump-diffusions, as the convergence rates and their optimality in the case
$d\geq 3$ have already been treated in detail in [3].
Beyond these works, to our best knowledge, the literature concerning non-
parametric estimation of diffusion processes with jumps is not wide. One of
the few examples is given by Funke and Schmisser: in [28] they investigate the
non parametric adaptive estimation of the drift of an integrated jump
diffusion process, while in [40], Schmisser deals with the non-parametric
adaptive estimation of the coefficients of a jumps diffusion process. To name
other examples, in [24] the authors estimate in a non-parametric way the drift
of a diffusion with jumps driven by a Hawkes process, while in [4] the
volatility and the jump coefficients are considered.
On the other hand, the problem of invariant density estimation has been
considered by many authors (see e.g. [38], [20], [10], [45], and [5]) in
several different frameworks: it is at the same time a long-standing problem
and a highly active current topic of research. One of the reasons why the
estimation of the invariant density has attracted the attention of many
statisticians is the huge amount of numerical methods to which it is
connected, the MCMC method above all. An approximation algorithm for the
computation of the invariant density can be found for example in [33] and
[39]. Moreover, invariant distributions are essential for the analysis of the
stability of stochastic differential systems (see e.g. [29] and [5]).
In [5], [6], and [11] some kernel estimators are used to estimate the marginal
density of a continuous time process. When $\mu$ belongs to some Hölder class
whose smoothness is $\beta$, they prove under some mixing conditions that
their pointwise $L^{2}$ risk achieves the standard rate of convergence
$T^{\frac{2\beta}{2\beta+1}}$ and the rates are minimax in their framework.
Castellana and Leadbetter proved in [15] that, under condition CL below, the
density can be estimated with the parametric rate $\frac{1}{T}$ by some non-
parametric estimators (the kernel ones among them).
In order to introduce condition CL it is necessary to request that the process
$X$ belongs to a class of real processes with common marginal density $\mu$
with respect to the Lebesgue measure on $\mathbb{R}$ and such that the joint
density of $(X_{s},X_{t})$ exists for all $s\neq t$, it is measurable and
satisfies $\mu_{(X_{s},X_{t})}=\mu_{(X_{t},X_{s})}=\mu_{(X_{0},X_{t-s})}$ and
it is denoted by $\mu_{|t-s|}$ for all $s,t\in\mathbb{R}$. We also denote by
$g_{u}$ the function $g_{u}(x,y)=\mu_{u}(x,y)-\mu(x)\mu(y)$. Then, condition
CL writes as follows:
* CL:
$u\mapsto\left\|g_{u}\right\|_{\infty}$ is integrable on $(0,\infty)$ and
$g_{u}(\cdot,\cdot)$ is continuous for each $u>0$.
In our context, $g_{u}(x,y)=\mu(x)p_{u}(x,y)-\mu(x)\mu(y)$, where $p_{u}(x,y)$
is the transition density. More precisely, they shed light to the fact that
local irregularities of the sample paths provide some additional information.
Indeed, if the joint distribution of $(X_{0},X_{t})$ is not too close to a
singular distribution for $|t|$ small, then it is possible to achieve the
superoptimal rate $\frac{1}{T}$ for the pointwise quadratic risk of the kernel
estimator. Condition CL can be verified for ergodic continuous diffusion
processes (see [44] for sufficient conditions). The paper of Castellana and
Leadbetter led to a lot of works regarding the estimation of the common
marginal distribution of a continuous time process. In [9], [10], [14], [21],
and [7] several related results and examples can be found.
An alternative to the kernel density estimator is given by the local time
density estimator, which was proposed by Kutoyants in [22] in the case of
diffusion processes and was extended by Bosq and Davydov in [12] to a more
general context. The latest have proved that, under a condition which is
mildly weaker than CL, the mean squared error of the local time estimator
reaches the full rate $\frac{1}{T}$. Leblanc built in [34] a wavelet estimator
of a density belonging to some general Besov space and proved that, if the
process is geometrically strong mixing and a condition like CL is satisfied,
then its $L^{p}$-integrated risk converges at rate $\frac{1}{T}$ as well. In
[18] the authors built a projection estimator and showed that its
$L^{2}$-integrated risk achieves the parametric rate $\frac{1}{T}$ under a
condition named WCL, which is blandly different compared to CL.
* WCL:
There exists a positive integrable function $k$ (defined on $\mathbb{R}$) such
that
$\sup_{y\in\mathbb{R}}\int_{0}^{\infty}|g_{u}(x,y)|du\leq k(x),\qquad\text{
for all }x\in\mathbb{R}.$
In this paper, we will show that our mono-dimensional jump-process satisfies a
local irregularity condition WCL1 and an asymptotic independence condition
WCL2 (see Proposition 1), two conditions in which the original condition WCL
can be decomposed. In this way, it will be possible to show that the $L^{2}$
risk for the pointwise estimation of the invariant measure achieves the
superoptimal rate $\frac{1}{T}$, using our kernel density estimator. Moreover,
the same conditions will result in the asymptotic normality of the proposed
estimator. Indeed, as we will see in the proof of Theorem 2, the main
challenge in this part is to justify the use of dominated convergence theorem,
which will ensured by conditions WCL1 and WCL2. We will find in particular
that, for any collection $(x_{i})_{1\leq i\leq m}$ of real numbers, we have
$\sqrt{T}(\hat{\mu}_{h,T}(x_{i})-\mu(x_{i}),\,1\leq i\leq
m)\xrightarrow{\mathcal{D}}N^{(m)}(0,\Sigma^{(m)})\text{ as
}T\rightarrow\infty,$
where $\hat{\mu}_{h,T}$ is the kernel density estimator and
$\Sigma^{(m)}:=(\sigma(x_{i},x_{j}))_{1\leq i,j\leq
m},\qquad\sigma(x_{i},x_{j}):=2\int_{0}^{\infty}g_{u}(x_{i},x_{j})du.$
We remark that the precise form of the equation above allows us to construct
tests and confidence sets for the density.
We have found the convergence rate $\left\\{\frac{1}{T},\frac{\log
T}{T}\right\\}$ for the risk associated to our kernel density estimator for
the estimation of the invariant density for $d=1$ and $d=2$. Then, some
questions naturally arise: are the convergence rates the best possible or is
it possible to improve them by using other estimators? In order to answer, we
consider a simpler model where both the volatility and the jump coefficient
are constant and the intensity of the jumps is finite. Then, we look for a
lower bound for the risk at a point $x\in\mathbb{R}^{d}$ defined as in
equation (9) below. The first idea is to use the two hypothesis method (see
Section 2.3 in [43]). To do that, the knowledge of the link between the drift
$b$ and the invariant density $\mu_{b}$ is essential. In absence of jumps such
link is explicit, but in our context it is more challenging. As shown in [19]
and [3], it is possible to find the link knowing that the invariant measure
has to satisfy $A^{*}\mu_{b}=0$, where $A^{*}$ is the adjoint of the generator
of the considered diffusion. This method allows us to show that the
superoptimal rate $\frac{1}{T}$ is the best possible for the estimation of the
invariant density in $d=1$, but it fails in the bi-dimensional case (see
Remark 1 below for details). Finally, we use a finite number of hypotheses to
prove a lower bound in the bi-dimensional case. This requires a detailed
analysis of the Kullback divergence between the probability laws associated to
the different hypotheses. Thanks to that, it is possible to recover the
optimal rate $\frac{\log T}{T}$ in the two-dimensional case.
The paper is organised as follows. In Section 2 we give the assumptions on our
model and we provide our main results. Section 3 is devoted to state and prove
some preliminary results needed for the proofs of the main results. To
conclude, in Section 4 we give the proof of Theorems 1, 2, 3, and 4, where our
main results are gathered.
Throughout all the paper $c$ and $\lambda$ are constants that may change from
line to line. Their dependence on $T$ or other fixed constants will be implied
from the statements.
## 2 Model assumption and main results
We consider the following stochastic differential equation with jumps
$X_{t}=X_{0}+\int_{0}^{t}b(X_{s})ds+\int_{0}^{t}a(X_{s})dB_{s}+\int_{0}^{t}\int_{\mathbb{R}^{d}_{0}}\gamma(X_{s^{-}})z(\nu(ds,dz)-F(z)dzds),$
(2)
where $t\geq 0$, $d\in\\{1,2\\}$,
$\mathbb{R}^{d}_{0}=\mathbb{R}^{d}\backslash\left\\{0\right\\}$, the initial
condition $X_{0}$ is a $\mathbb{R}^{d}$-valued random variable, the
coefficients $b:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$,
$a:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and
$\gamma:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ are
measurable functions, $(B_{t})_{t\geq 0}$ is a $d$-dimensional Brownian
motion, and $\nu$ is a Poisson random measure on
$\mathbb{R}_{+}\times\mathbb{R}^{d}$ associated to a Lévy process
$(L_{t})_{t\geq 0}$ with Lévy density function $F$. All sources of randomness
are mutually independent.
We consider the following assumptions on the coefficients and on the Lévy
density $F$:
1. A1
The functions $b$, $\gamma$ and $aa^{T}$ are globally Lipschitz and bounded.
Moreover, $\inf_{x\in{\color[rgb]{0,0,0}\mathbb{R}^{d}}}aa^{T}(x)\geq c$Id,
for some constant $c>0$, where Id denotes the $d\times d$ identity matrix and
$\inf_{x\in{\color[rgb]{0,0,0}\mathbb{R}^{d}}}\text{det}(\gamma(x))>0$.
2. A2
$\langle x,b(x)\rangle\leq-c_{1}|x|+c_{2}$, for all $|x|\geq\rho$, for some
$\rho,c_{1},c_{2}>0$.
3. A3
Supp$(F)=\mathbb{R}^{d}_{0}$ and for all $z\in\mathbb{R}^{d}_{0}$,
$F(z)\leq\frac{c_{3}}{|z|^{d+\alpha}}$, for some $\alpha\in(0,2),c_{3}>0$.
4. A4
There exist $\epsilon_{0}>0$ and $c_{4}>0$ such that
$\int_{\mathbb{R}^{d}_{0}}|z|^{2}e^{\epsilon_{0}|z|}F(z)dz\leq c_{4}$.
5. A5
If $\alpha=1$, $\int_{r<|z|<R}zF(z)dz=0$, for any $0<r<R<\infty$.
Assumption A1 ensures that equation (2) admits a unique càdlàg adapted
solution $X=(X_{t})_{t\geq 0}$ satisfying the strong Markov property, see e.g.
[1]. Moreover, it is shown in [2, Lemma 2] that if we further assume
Assumptions A2-A4, then the process $X$ is exponentially ergodic and
exponentially $\beta$-mixing. Therefore the process is stationary and, in
particular, it has a unique invariant distribution $\pi$, which we assume it
has a density $\mu$ with respect to the Lebesgue measure. Finally, Assumption
A5 ensures the existence of the transition density of $X$ denoted by
$p_{t}(x,y)$ which satisfies the following upper bound (see [2, Lemma 1]): for
all $T\geq 0$, there exist $c>0$ and $\lambda>0$ such that for any $t\in[0,T]$
and $x,y\in\mathbb{R}^{d}$,
$p_{t}(x,y)\leq
c\left(t^{-d/2}e^{-\lambda\frac{|y-x|^{2}}{t}}+\frac{t}{(t^{1/2}+|y-x|)^{d+\alpha}}\right).$
(3)
We assume that the process is observed continuously $X=(X_{t})_{t\in[0,T]}$ in
a time interval $[0,T]$ such that $T$ tends to $\infty$. In the paper [2]
cited above, the nonparametric estimation of $\mu$ is studied via the kernel
estimator which is defined as follows. We assume that $\mu$ belongs to the
Hölder space $\mathcal{H}_{d}(\beta,\mathcal{L})$ where
$\beta=(\beta_{1},\ldots,\beta_{d})$, $\beta_{i}\geq 1$ and
$\mathcal{L}=(\mathcal{L}_{1},\ldots,\mathcal{L}_{d})$, $\mathcal{L}_{i}>0$,
which means that for all $i\in\\{1,\ldots,d\\}$,
$k=0,1,\ldots,\lfloor\beta_{i}\rfloor$ and $t\in\mathbb{R}$,
$\left\|D^{(k)}_{i}\mu\right\|_{\infty}\leq\mathcal{L}\quad\text{ and
}\quad\left\|D_{i}^{(\lfloor\beta_{i}\rfloor)}\mu(.+te_{i})-D_{i}^{(\lfloor\beta_{i}\rfloor)}\mu(.)\right\|_{\infty}\leq\mathcal{L}_{i}|t|^{\beta_{i}-\lfloor\beta_{i}\rfloor},$
where $D^{(k)}_{i}$ denotes the $k$th order partial derivative of $\mu$ w.r.t
the $i$th component, $\lfloor\beta_{i}\rfloor$ is the integer part of
$\beta_{i}$, and $e_{1},\ldots,e_{d}$ is the canonical basis of
$\mathbb{R}^{d}$. That is, all the partial derivatives of $\mu$ up to order
$\lfloor\beta\rfloor$ are bounded and the $\lfloor\beta\rfloor$th partial
derivative is Hölder continuous of order $\beta-\lfloor\beta\rfloor$ in any
direction. We recall that it is natural in our context to assume that the
invariant density belongs to a Hölder class as above. In fact, the proof of
the bias bound (6) stated below gives a direct application of this assumption,
see the proof of Proposition 2 in [2]. Other examples of nonparametric
estimation over Hölder classes can be found in [30], [31], [35], and [42].
We set
$\hat{\mu}_{h,T}(x)=\frac{1}{T\prod_{i=1}^{d}h_{i}}\int_{0}^{T}\prod_{i=1}^{d}K\left(\frac{x_{i}-X^{i}_{t}}{h_{i}}\right)dt=:\frac{1}{T}\int_{0}^{T}\mathbb{K}_{h}(x-X_{t})dt,$
where $x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$, $h=(h_{1},\ldots,h_{d})$ is a
bandwidth and $K:\mathbb{R}\rightarrow\mathbb{R}$ is a kernel function
satisfying
$\int_{\mathbb{R}}K(x)dx=1,\quad\left\|K\right\|_{\infty}<\infty,\quad\mbox{supp}(K)\subset[-1,1],\quad\int_{\mathbb{R}}K(x)x^{i}dx=0,$
for all $i\in\left\\{0,\ldots,M\right\\}$ with $M\geq\max_{i}\beta_{i}$.
We first consider equation (2) with $d=1$ and show that the kernel estimator
reaches the optimal rate $T^{-1}$, as it is for the stochastic differential
equation (2) without jumps. For this, we need the following additional
assumption on $a$.
1. A6
If $d=1$, $a^{2}\in C^{2}_{b}(\mathbb{R})$, that is, $a^{2}$ is twice
continuously differentiable with bounded first and second derivatives.
Assumption A6 is needed in order to show the results gathered in Theorems 1
and 2, while for the other results only assumptions A1 \- A5 will be required.
###### Theorem 1.
Let $X$ be the solution to (2) on $[0,T]$ with $d=1$. Suppose that Assumptions
A1-A6 hold and $\mu\in\mathcal{H}_{1}(\beta,\mathcal{L})$, with $\beta\geq 1$.
Then there exists $c>0$ such that for all $T>0$, $h\leq 1$, and
$x\in\mathbb{R}$,
$\mathbb{E}[|\hat{\mu}_{h,T}(x)-\mu(x)|^{2}]\leq
ce^{\epsilon|x|}(h^{2\beta}+\frac{1}{T}),$ (4)
where
$0<\epsilon\leq\min(\frac{\epsilon_{0}}{\left\|\gamma\right\|_{\infty}},\epsilon_{0})$,
with $\epsilon_{0}>0$ as in Assumption A4 In particular, choosing
$h(T)=\frac{1}{\sqrt{T}}$, we conclude that for $T\geq 1$,
$\mathbb{E}[|\hat{\mu}_{h,T}(x)-\mu(x)|^{2}]\leq\frac{ce^{\epsilon|x|}}{T}.$
(5)
We observe that both the bandwidth and the upper bound do not depend on the
unknown smoothness of the invariant density $\beta$, so there is no need to
propose a data driven bandwidth adaptive selection procedure as in the case
$d>2$ (see [2]).
Theorem 1 improves the upper bound obtained in [2] which was of the form
$\frac{(\log T)^{(2-\frac{1+\alpha}{2})\vee 1}}{T}$. The price to pay is that
the constant in the upper bound depends on $x$ (see Remark 1 below). However,
we are able to find a convergence rate which is optimal, as we will see in
Theorem 3. As in [2], we will use the bias-variance decomposition (see [17,
Proposition 1])
$\begin{split}\mathbb{E}[|\hat{\mu}_{h,T}(x)-\mu(x)|^{2}]&\leq{\color[rgb]{0,0,0}|\mathbb{E}[\hat{\mu}_{h,T}(x)]-\mu(x)|^{2}+\mathbb{E}[|\hat{\mu}_{h,T}(x)-\mathbb{E}[\hat{\mu}_{h,T}(x)]|^{2}]}\\\
&\leq
c\left(h^{2\beta}+T^{-2}\text{Var}\left(\int_{0}^{T}\mathbb{K}(x-X_{t})dt\right)\right),\end{split}$
(6)
for some constant $c>0$. For the proof of the bias bound $ch^{2\beta}$ in the
same setting of this paper see the proof of Proposition 2 in [2].
Then in [2] bounds on the transition semigroup and on the transition density
(see (3) above) give an upper bound for the variance depending on the
bandwidth. Here, we use a similar approach as in [15] and [18] to obtain a
bandwidth-free rate for the variance of smoothing density estimators (which
include the kernel estimator). For Markov diffusions, the sufficient
conditions can be decomposed into a local irregularity condition WCL1 plus an
asymptotic independence condition WCL2. There exist two positive integrable
functions $k_{1}$ and $k_{2}$ (defined on $\mathbb{R}$) and $u_{0}>0$ such
that
$\displaystyle\mbox{\bf WCL1:
}\sup_{y\in\mathbb{R}}\int_{0}^{u_{0}}|g_{u}(x,y)|\,du<k_{1}(x),\qquad\text{for
all }x\in\mathbb{R},$ $\displaystyle\mbox{\bf WCL2:
}\sup_{y\in\mathbb{R}}\int_{u_{0}}^{\infty}|g_{u}(x,y)|\,du<k_{2}(x),\qquad\text{for
all }x\in\mathbb{R}$
where $g_{u}(x,y):=\mu(x)p_{u}(x,y)-\mu(x)\mu(y)$. In order to show these
conditions, some further bounds on the transition density $p_{t}(x,y)$
involving partial derivatives are needed (see Lemma 1 below), for which the
additional condition A6 is required.
###### Remark 1.
The term $e^{\epsilon|x|}$ that appears in the bounds (4) and (5) comes from
the fact that we are able to show condition WCL2 with
$k_{2}(x)=\mu(x)(1+f^{\ast}(x))$, where $f^{\ast}$ is the Lyapunov function
constructed in [2], defined as a $C^{\infty}$ approximation of
$e^{\epsilon|x|}$ (see the proof of Proposition 1). We know that
$\int_{\mathbb{R}}\mu(x)f^{\ast}(x)dx<\infty$, as shown in [36], but this is
not sufficient as it was in [18] in order to bound the variance term in (6)
since here we are dealing with the kernel estimator. In order to remove the
term $e^{\epsilon|x|}$ an additional assumption would be needed that ensures
that $\sup_{x\in\mathbb{R}}\mu(x)f^{\ast}(x)<\infty$.
As shown in [13], conditions WLC1 and WLC2 are also useful to show the
asymptotic normality of the kernel density estimator, as proved in the next
theorem.
###### Theorem 2.
Let $X$ be the solution to (2) on $[0,T]$ with $d=1$. Suppose that Assumptions
A1-A6 hold and $\mu\in\mathcal{H}_{1}(\beta,\mathcal{L})$, with $\beta\geq 1$.
Consider the bandwidth $h(T)=(\frac{1}{T})^{\frac{1}{2}-\epsilon}$, where
$\epsilon\in(0,\frac{1}{2})$. Then, for any collection $(x_{i})_{1\leq i\leq
m}$ of distinct real numbers
$\sqrt{T}(\hat{\mu}_{h,T}(x_{i})-\mathbb{E}[\hat{\mu}_{h,T}(x_{i})],\,1\leq
i\leq m)\xrightarrow{\mathcal{D}}N^{(m)}(0,\Sigma^{(m)})\text{ as
}T\rightarrow\infty,$ (7)
where
$\Sigma^{(m)}:=(\sigma(x_{i},x_{j}))_{1\leq i,j\leq
m},\qquad\sigma(x_{i},x_{j}):=2\int_{0}^{\infty}g_{u}(x_{i},x_{j})du.$
Observe that using the choice of $h(T)=(\frac{1}{T})^{\frac{1}{2}-\epsilon}$,
with $\epsilon>0$ in the bias bound (6), we get that for any $x\in\mathbb{R}$
and $T\geq 1$,
$\sqrt{T}|\mathbb{E}[\hat{\mu}_{h,T}(x)]-\mu(x)|\leq
cT^{-\frac{1}{2}(\beta-1-2\beta\epsilon)}.$
Therefore, choosing $\beta>1$ and $\epsilon<\frac{\beta-1}{2\beta}$ and
applying Theorem 2, we conclude that as $T\rightarrow\infty$
$\sqrt{T}(\hat{\mu}_{h,T}(x_{i})-\mu(x_{i}),\,1\leq i\leq
m)\xrightarrow{\mathcal{D}}N^{(m)}(0,\Sigma^{(m)}).$
We are also interested in obtaining lower bounds in dimension $d\in\\{1,2\\}$.
For the computations of the lower bounds we consider the particular case of
equation (2) given by
$X_{t}=X_{0}+\int_{0}^{t}b(X_{s})ds+aB_{t}+\int_{0}^{t}\int_{\mathbb{R}^{d}_{0}}\gamma
z(\nu(ds,dz)-F(z)dzds),$ (8)
where $a$ and $\gamma$ are $d\times d$ constant matrices and the rest of terms
are as in equation (2).
We next introduce the following set of drift functions of equation (8). We say
that a bounded and Lipschitz function
$b:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ belongs to
$\Sigma(\beta,\mathcal{L})$ if the unique invariant density $\mu_{b}$ of the
solution $X=(X_{t})_{t\geq 0}$ to (8) belongs to
$\mathcal{H}_{d}(\beta,{\color[rgb]{0,0,0}2}\mathcal{L})$ for some
$\beta,\mathcal{L}\in\mathbb{R}^{d}$, $\beta_{i}\geq 1$, $\mathcal{L}_{i}>0$.
A detailed description of the set $\Sigma(\beta,\mathcal{L})$ will be given in
Section 4.3, where two explicit examples of drift coefficients $b_{0}$ and
$b_{1}$ belonging to $\Sigma(\beta,\mathcal{L})$ will be introduced.
We denote by $\mathbb{P}_{b}^{(T)}$ and $\mathbb{E}_{b}^{(T)}$ the law and
expectation of the solution $(X_{t})_{t\in[0,T]}$. We define the minimax risk
at a point $x\in\mathbb{R}^{d}$ by
$\mathcal{R}^{x}_{T}(\beta,\mathcal{L}):=\inf_{\tilde{\mu}_{T}}\mathcal{R}(\tilde{\mu}_{T}(x)):=\inf_{\tilde{\mu}_{T}}\sup_{b\in\Sigma(\beta,\mathcal{L})}\mathbb{E}_{b}^{(T)}[(\tilde{\mu}_{T}(x)-\mu_{b}(x))^{2}],$
(9)
where the infimum is taken on all possible estimators of the invariant
density.
The following lower bounds hold true.
###### Theorem 3.
Let $X$ be the solution to (8) on $[0,T]$ with $d=1$. Suppose that Assumptions
A1-A5 hold, that $\int_{\mathbb{R}}F(z)dz<\infty$ and that
$\mu_{b}\in\mathcal{H}_{1}(\beta,\mathcal{L})$, with $\beta\geq 1$. Then,
there exists $T_{0}>0$ and $c>0$ such that, for all $T\geq T_{0}$,
$\inf_{x\in\mathbb{R}}\mathcal{R}^{x}_{T}(\beta,\mathcal{L})\geq\frac{c}{T}.$
###### Theorem 4.
Let $X$ be the solution to (8) on $[0,T]$ with $d=2$. Suppose that Assumptions
A1-A5 hold, that $\int_{\mathbb{R}^{2}}F(z)dz<\infty$ and that
$\mu_{b}\in\mathcal{H}_{2}(\beta,\mathcal{L})$, with $\beta_{i}\geq 1$ for
$i=1,2$. Assume that for all $i\in\\{1,2\\}$ and $j\neq i$,
$|(aa^{T})_{ij}(aa^{T})^{-1}_{jj}|\leq\frac{1}{2}.$ (10)
Then, there exists $T_{0}>0$ and $c>0$ such that, for $T\geq T_{0}$,
$\inf_{\tilde{\mu}_{T}}\sup_{b\in\Sigma(\beta,\mathcal{L})}\mathbb{E}^{(T)}_{b}\left[\sup_{x\in\mathbb{R}^{2}}(\tilde{\mu}_{T}(x)-\mu_{b}(x))^{2}\right]\geq
c\frac{\log T}{T}.$
Recall for these two theorems, $a$ and $\gamma$ are $d\times d$ constant
matrices. In this case, when $d=1$, Assumption A1 is equivalent to say that
$a\neq 0$ and $\gamma>0$, while when $d=2$, it is equivalent to say that
$\det(a)\neq 0$ and $\det(\gamma)>0$. Moreover, hypotheses A3-A5 imply that
the unique solution to equation (8) admits a unique invariant measure
$\pi_{b}$, which we assume has a density $\mu_{b}$ with respect to the
Lebesgue measure, as before.
Comparing these lower bounds with the upper bound of Theorem 1 for the case
$d=1$ and Proposition 4 in [2] for the two-dimensional case, we conclude that
the convergence rate $\\{\frac{1}{T},\frac{\log T}{T}\\}$ are the best
possible for the estimation of the invariant density in dimension
$d\in\\{1,2\\}$.
The proof of Theorem 3 follows along the same lines as that of Theorem 2 in
[3], where a lower bound for the estimation of the invariant density for the
solution to (8) for $d\geq 3$ is obtained. The proof is based on the two
hypotheses method, explained for example in Section 2.3 of [43]. However, this
method does not work for the two-dimensional case as explained in Remark 2
below. Instead, we use the Kullback’s version of the finite number of
hypotheses method as stated in Lemma C.1 of [41], see Lemma 2 below. Observe
that this method gives a slightly weaker lower bound as we get a $\sup_{x}$
inside the expectation, while the method in [3] provides an $\inf_{x}$ outside
the expectation.
## 3 Preliminary results
The proof of Theorems 1 and 2 will use the following bounds on the transition
density.
###### Lemma 1.
Let $X$ be the solution to (2) on $[0,T]$ with $d=1$. Suppose that Assumptions
A1-A6 hold. Then, there exist jointly continuous processes $Z$, $A$ and $B$ on
$\mathbb{R}_{+}\times\mathbb{R}^{2}$ such that for all $t\geq 0$ and
$x,y\in\mathbb{R}$,
$p_{t}(x,y)=Z_{t}(x,y)+A_{t}(x,y)+B_{t}(x,y)$ (11)
satisfying that for all $T>0$, there exist $c>0$ and $\lambda>0$ such that for
any $x,y\in\mathbb{R}$ and $t\in[0,T]$
$\bigg{|}\frac{\partial^{2}}{\partial y^{2}}Z_{t}(x,y)\bigg{|}\leq
c\,t^{-3/2}e^{-\lambda\frac{|y-x|^{2}}{t}},$ (12) $|A_{t}(x,y)|\leq
c\,(t^{3/2}(|y-x|+\sqrt{t})^{-1-\alpha}+e^{-\lambda\frac{|y-z|^{2}}{t}}),$
(13)
and
$|B_{t}(x,y)|\leq c\,(1+t^{2-\alpha/2})(|y-x|+\sqrt{t})^{-1-\alpha}.$ (14)
###### Proof.
By Duhamel’s formula (1.12) of [16], the transition density of the solution to
(2) satisfies that for all $t\geq 0$ and $x,y\in\mathbb{R}$,
$p_{t}(x,y)=Z_{t}(x,y)+A_{t}(x,y)+B_{t}(x,y)$
where $Z_{t}(x,y)$ is the transition density of the solution to (2) with
$b=\gamma=0$, and $A_{t}$ and $B_{t}$ are defined as follows
$A_{t}(x,y):=\int_{0}^{t}\int_{\mathbb{R}}p_{r}(x,z)\,b(z)\frac{\partial}{\partial
z}Z_{t-r}(z,y)\,dz\,dr,$
and
$\begin{split}B_{t}(x,y)&:=\int_{0}^{t}\int_{\mathbb{R}}p_{r}(x,z)\int_{\mathbb{R}}\big{(}Z_{t-r}(z+\xi,y)-Z_{t-r}(z,y)\\\
&\qquad\qquad\qquad-{\bf 1}_{|\xi|\leq 1}\,\xi\,\frac{\partial}{\partial
z}Z_{t-r}(z,y)\big{)}\frac{k(z,\xi)}{|\xi|^{1+\alpha}}\,d\xi\,dz\,dr,\end{split}$
where $k(z,\xi)=\frac{1}{\gamma(z)}|\xi|^{1+\alpha}F(\frac{\xi}{\gamma(z)})$.
This shows the decomposition formula (11).
By (6.1) in Theorem 7 of [27], using the fact that $a^{2}$ is bounded together
with A6, we have that for all $T>0$, there exist $c,\lambda>0$ such that for
all $x,y\in\mathbb{R}$ and $t\in[0,T]$
$\bigg{|}\frac{\partial^{2}}{\partial y^{2}}Z_{t}(x,y)\bigg{|}\leq
c\,t^{-3/2}e^{-\lambda\frac{|y-x|^{2}}{t}},$
(which proves (12)) and
$\bigg{|}\frac{\partial}{\partial x}Z_{t}(x,y)\bigg{|}\leq
c\,t^{-1}e^{-\lambda\frac{|y-x|^{2}}{t}}.$ (15)
In particular, using (15) and the fact that $b$ is bounded, we get that
$|A_{t}(x,y)|\leq
c\int_{0}^{t}\int_{\mathbb{R}}p_{r}(x,z)(t-r)^{-1}e^{-\lambda\frac{|y-z|^{2}}{t-r}}\,dz\,dr.$
Moreover, using (3) together with (2.6) and (2.8) of [16] with $\gamma_{1}=-1$
and $\gamma_{2}=2$, and $\gamma_{1}=0$ and $\gamma_{2}=-1$, respectively, we
conclude that (13) holds true.
On the other hand, appealing to Corollary 2.4(i) of [16], from hypotheses A1,
A3 and A5, we get that for all $T>0$, there exists $c>0$ such that for all
$x,y\in\mathbb{R}$ and $t\in[0,T]$,
$|B_{t}(x,y)|\leq
c\int_{0}^{t}\int_{\mathbb{R}}p_{r}(x,z)(|y-z|+\sqrt{t-r})^{-1-\alpha}\,dz\,dr.$
Finally, using again (3) together with (2.5) and (2.6) of [16] with
$\gamma_{1}=0$ and $\gamma_{2}=2$, and $\gamma_{1}=0$ and $\gamma_{2}=0$,
respectively, we obtain (14).
The proof of the Lemma is completed. ∎
The key point of the proof of Theorem 1 consists in showing that conditions
WCL1 and WCL2 hold true, which is proved in the next proposition.
###### Proposition 1.
Let $X$ be the solution to (2) on $[0,T]$ with $d=1$. Suppose that Assumptions
A1-A6 hold. Then, conditions WCL1 and WCL2 are satisfied.
###### Proof.
We start considering WCL1. The density estimate (3) yields
$p_{t}(x,y)\leq
ct^{-\frac{1}{2}}+\tilde{c}t^{\frac{1-\alpha}{2}}\leq\bar{c}t^{-\frac{1}{2}}\qquad
0<t\leq 2,$ (16)
which combined with $\sup_{y\in\mathbb{R}}\mu(y)<\infty$ gives WCL1 with
$k_{1}(x)=\mu(x)$ and $u_{0}=2$. In order to show WCL2, we set
$\varphi(\xi):=\mathbb{E}[\exp(i\xi X_{t})]$ and
$\varphi_{x}(\xi,t):=\mathbb{E}[\exp(i\xi X_{t})|X_{0}=x]$ and we claim that
there exists $\hat{c}>0$ such that for all $\xi\in\mathbb{R}$,
$|\varphi(\xi)|\leq{\color[rgb]{0,0,0}\hat{c}}(1+|\xi|)^{-2}.$ (17)
Moreover, there exists $\tilde{c}>0$, such that for all $t\geq 2$,
$x\in\mathbb{R}$, and $\xi\in\mathbb{R}$,
$|\varphi_{x}(\xi,t)|\leq{\color[rgb]{0,0,0}\tilde{c}}(1+|\xi|)^{-2}.$ (18)
Recall from Lemma 2 in [2] and its proof that the process $X$ is exponentially
$\beta$-mixing and there exists $\rho>0$ such that for all $x\in\mathbb{R}$
and $t>0$,
$\left\|P_{t}(x,\cdot)-\mu(\cdot)\right\|_{TV}\leq(1+f^{*}(x))e^{-\rho t},$
(19)
where $(P_{t})_{t\in\mathbb{R}}$ is the transition semigroup of our process
$X$, $\left\|\cdot\right\|_{TV}$ is the total variation norm and $f^{*}(x)$ is
a Lyapounov function. Specifically, $f^{*}(x)$ is defined as $e^{\epsilon|x|}$
for $|x|\geq 1$, with
$\epsilon\leq\min(\frac{\epsilon_{0}}{\left\|\gamma\right\|_{\infty}},\epsilon_{0})$
($\epsilon_{0}>0$ as in Assumption A4). In order to avoid any regularity
problem in $0$, $f^{*}$ is introduced as piecewise function. For $|x|<1$ it is
defined as a $C^{\infty}$ approximation of $e^{\epsilon|x|}$, such that
$f^{*}$ is $C^{\infty}$ on $\mathbb{R}$.
We now prove that inequalities (17), (18) and (19) imply WCL2. Using the
inverse Fourier transform, we have
$2\pi(p_{t}(x,y)-\mu(y))=\int_{\mathbb{R}}\exp(-i\xi
y)(\varphi_{x}(\xi,t)-\varphi(t))d\xi.$
Then, using (17) and (18) we get, for $t\geq 2$,
$2\pi|p_{t}(x,y)-\mu(y)|\leq
2(\tilde{c}+\hat{c})^{\frac{p-1}{p}}(\sup_{\xi\in\mathbb{R}}|\varphi_{x}(\xi,t)-\varphi(\xi)|)^{\frac{1}{p}}\int_{\mathbb{R}^{+}}(1+\xi)^{-2\frac{p-1}{p}}d\xi,$
where we have used that $1=\frac{1}{p}+\frac{p-1}{p}$. We can choose $p>2$, so
that $2\frac{p-1}{p}>1$. We get that there exists a finite constant $c$ such
that, for all $t\geq 2$ and $x,y\in\mathbb{R}$,
$|g_{t}(x,y)|=\mu(x)|p_{t}(x,y)-\mu(y)|\leq
c\mu(x)(\sup_{\xi\in\mathbb{R}}|\varphi_{x}(\xi,t)-\varphi(\xi)|)^{\frac{1}{p}},$
where we observe that the right hand side is independent of $y$. By using the
fact that
$\sup_{\lambda\in\mathbb{R}}|\varphi_{x}(\lambda,t)-\varphi(\lambda)|\leq\left\|P_{t}(x,\cdot)-\mu(\cdot)\right\|_{TV}$
together with (19) we obtain that there exist $c>0$ and $\rho>0$ such that for
all $x,y\in\mathbb{R}$ and $t\geq 2$,
$\displaystyle|g_{t}(x,y)|$ $\displaystyle\leq c\mu(x)(1+f^{*}(x))e^{-\rho
t},$
as $f^{\ast}$ is positive, and so
$\sup_{y\in\mathbb{R}}\int_{2}^{\infty}|g_{t}(x,y)|\,dt\leq
c\mu(x)(1+f^{*}(x))\int_{2}^{\infty}e^{-\rho t}\,dt,$
which implies WCL2 with $k_{2}(x)=c\mu(x)(1+f^{*}(x))$.
We are left to show (17) and (18). We start showing (18). Using (11) and
integrating by parts yields
$\begin{split}|\varphi_{x}(\xi,t)|&=\bigg{|}\int_{\mathbb{R}}\exp(i\xi
y)p_{t}(x,y)dy\bigg{|}\\\
&=\bigg{|}\int_{\mathbb{R}}\int_{\mathbb{R}}\exp(i\xi
y)p_{t-1}(x,z)p_{1}(z,y)dy\,dz\bigg{|}\\\
&=\bigg{|}\int_{\mathbb{R}}\int_{\mathbb{R}}\exp(i\xi
y)p_{t-1}(x,z)\left(Z_{1}(z,y)+A_{1}(z,y)+B_{1}(z,y)\right)dy\,dz\bigg{|}\\\
&\leq|\xi|^{-2}\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)\bigg{|}\frac{\partial^{2}}{\partial
y^{2}}Z_{1}(z,y)\bigg{|}dy\,dz\\\
&\qquad+\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)|A_{1}(z,y)|dy\,dz+\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)|B_{1}(z,y)|dy\,dz\\\
&=:|\xi|^{-2}(I_{1}+I_{2}+I_{3}).\end{split}$
Appealing to (12), we obtain that
$\begin{split}I_{1}\leq
c\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)e^{-\lambda|y-z|^{2}}dy\,dz=c,\end{split}$
where $c$ is independent of $t$ and $x$ as
$\int_{\mathbb{R}}p_{t-1}(x,z)dz=1$. Using (13), we get that
$\begin{split}I_{2}&\leq
c\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)(|y-z|+1)^{-1-\alpha}+e^{-\lambda|y-z|^{2}})dy\,dz=c,\end{split}$
as the $dy$ integral is finite since $\alpha\in(0,2)$. Similarly, by (14),
$\begin{split}&I_{3}\leq
c\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-1}(x,z)(|y-x|+1)^{-1-\alpha}dy\,dz\leq
c.\end{split}$
Thus, we have proved that $|\varphi_{x}(\xi,t)|\leq c|\xi|^{-2}$. Since
$|\varphi_{x}(\xi,t)|\leq 1$, this implies (18). Similarly,
$\begin{split}|\varphi(\xi)|&=\bigg{|}\int_{\mathbb{R}}\exp(i\xi
y)\mu(y)dy\bigg{|}\\\ &\leq\bigg{|}\int_{\mathbb{R}}\int_{\mathbb{R}}\exp(i\xi
y)\mu(z)Z_{1}(z,y)dy\,dz\bigg{|}\\\
&\qquad+\int_{\mathbb{R}}\int_{\mathbb{R}}\mu(z)|A_{1}(z,y)|dy\,dz+\int_{\mathbb{R}}\int_{\mathbb{R}}\mu(z)|B_{1}(z,y)|dy\,dz\\\
&\leq|\xi|^{-2}\int_{\mathbb{R}}\int_{\mathbb{R}}\mu(z)\bigg{|}\frac{\partial^{2}}{\partial
y^{2}}Z_{1}(z,y)\bigg{|}dy\,dz\\\
&\qquad+\int_{\mathbb{R}}\int_{\mathbb{R}}\mu(z)|A_{1}(z,y)|dy\,dz+\int_{\mathbb{R}}\int_{\mathbb{R}}\mu(z)|B_{1}(z,y)|dy\,dz\\\
&\leq c|\xi|^{-2},\end{split}$
which implies (18) since $|\varphi(\xi)|\leq 1$. The proof of the proposition
is now completed. ∎
Theorem 2 is an application of the following central limit theorem for
discrete stationary sequences. Let $Y_{n}=(Y_{n,i},\,i\in\mathbb{Z})$, $n\geq
1$ be a sequence of strictly stationary discrete time $\mathbb{R}^{m}$ valued
random process. We define the $\alpha$-mixing coefficient of $Y_{n}$ by
$\alpha_{n,k}:=\sup_{A\in\sigma(Y_{n,i},\,i\leq 0),\quad
B\in\sigma(Y_{n,i},\,i\geq k)}{\color[rgb]{0,0,0}\big{(}\mathbb{P}(A\cap
B)-\mathbb{P}(A)\mathbb{P}(B)\big{)}}$
and we set $\alpha_{k}:=\sup_{n\geq 1}\alpha_{n,k}$ (see also Section 1 in
[26]). We denote by $Y^{(r)}$ the r-th component of an $m$ dimensional random
vector $Y$.
###### Theorem 5 (Theorem 1.1 [13]).
Assume that
1. (i)
$\mathbb{E}[Y_{n,i}^{(r)}]=0$ and $|Y_{n,i}^{(r)}|\leq M_{n}$ for every $n\geq
1$, $i\geq 1$ and $1\leq r\leq m$, where $M_{n}$ is a constant depending only
on $n$.
2. (ii)
$\sup_{i\geq 1,1\leq r\leq m}\mathbb{E}[(Y_{n,i}^{(r)})^{2}]<\infty.$
3. (iii)
For every $1\leq r,s\leq m$ and for every sequence $b_{n}\rightarrow\infty$
such that $b_{n}\leq n$ for every $n\geq 1$, we have
$\lim_{n\rightarrow\infty}\frac{1}{b_{n}}\mathbb{E}\left[\sum_{i=1}^{b_{n}}Y_{n,i}^{(r)}\sum_{j=1}^{b_{n}}Y_{n,j}^{(s)}\right]=\sigma_{r,s}.$
4. (iv)
There exists ${\color[rgb]{0,0,0}\gamma_{0}}\in(1,\infty)$ such that
$\sum_{k\geq
1}k\alpha_{k}^{\frac{{\color[rgb]{0,0,0}\gamma_{0}}-1}{{\color[rgb]{0,0,0}\gamma_{0}}}}<\infty$.
5. (v)
For some constant $c>0$ and for every $n\geq 1$, $M_{n}\leq
cn^{\frac{{\color[rgb]{0,0,0}\gamma_{0}}^{2}}{(3{\color[rgb]{0,0,0}\gamma_{0}}-1)(2{\color[rgb]{0,0,0}\gamma_{0}}-1)}}$.
Then,
$\frac{\sum_{i=1}^{n}Y_{n,i}}{\sqrt{n}}\xrightarrow{\mathcal{D}}N(0,\Sigma)\quad\mbox{as
}n\rightarrow\infty,$
where $\Sigma=(\sigma_{r,s})_{1\leq r,s\leq m}$.
The proof of Theorem 4 is based on the following Kullback version of the main
theorem on lower bounds in [43], see Lemma C.1 of [41]:
###### Lemma 2.
Fix $\beta,\mathcal{L}\in(0,\infty)^{2}$ and assume that there exists
$f_{0}\in\mathcal{H}_{2}(\beta,\mathcal{L})$ and a finite set $J_{T}$ such
that one can find $\left\\{f_{j},\,j\in
J_{T}\right\\}\subset\mathcal{H}_{2}(\beta,\mathcal{L})$ satisfying
$\left\|f_{j}-f_{k}\right\|_{\infty}\geq 2\psi>0\qquad\forall j\neq k\in
J_{T}.$ (20)
Moreover, denoting $\mathbb{P}_{j}^{(T)}$ the probability measure associated
with $f_{j}$, $\forall j\in J_{T}$,
$\mathbb{P}^{(T)}_{j}\ll\mathbb{P}^{(T)}_{0}$ and
$\frac{1}{|J_{T}|}\sum_{j\in
J_{T}}KL(\mathbb{P}^{(T)}_{j},\mathbb{P}^{(T)}_{0})=\frac{1}{|J_{T}|}\sum_{j\in
J_{T}}\mathbb{E}^{(T)}_{j}\left[\log\left(\frac{d\mathbb{P}^{(T)}_{j}}{d\mathbb{P}^{(T)}_{0}}(X^{T})\right)\right]\leq{\color[rgb]{0,0,0}\delta}\log(|J_{T}|)$
(21)
for some ${\color[rgb]{0,0,0}\delta}\in(0,\frac{1}{8})$. Then, for $q>0$, we
have
$\inf_{\tilde{\mu}_{T}}\sup_{\mu_{b}\in\mathcal{H}_{2}(\beta,\mathcal{L})}(\mathbb{E}^{(T)}_{b}[\psi^{-q}\left\|\tilde{\mu}_{T}-\mu_{b}\right\|_{\infty}^{q}])^{1/q}\geq
c({\color[rgb]{0,0,0}\delta})>0,$
where the infimum is taken over all the possible estimators $\tilde{\mu}_{T}$
of $\mu_{b}$.
## 4 Proof of the main results
### 4.1 Proof of Theorem 1
By the symmetry of the covariance operator and the stationarity of the
process,
$\begin{split}&T\,\text{Var}(\hat{\mu}_{h,T}(x))=\frac{1}{T}\int_{0}^{T}\int_{0}^{T}\text{Cov}(\mathbb{K}_{h}(x-X_{t}),\mathbb{K}_{h}(x-X_{s}))ds\,dt\\\
&\qquad=\frac{2}{T}\int_{0}^{T}(T-u)\text{Cov}(\mathbb{K}_{h}(x-X_{u}),\mathbb{K}_{h}(x-X_{0}))du\\\
&\qquad=2\int_{0}^{T}(1-\frac{u}{T})\int_{\mathbb{R}}\int_{\mathbb{R}}\mathbb{K}_{h}(x-y)\mathbb{K}_{h}(x-z)g_{u}(y,z)dy\,dz\,du\\\
&\qquad\leq\int_{\mathbb{R}}|\mathbb{K}_{h}(x-y)|\sup_{z\in\mathbb{R}}\int_{0}^{\infty}|g_{u}(y,z)|du\,dy\int_{\mathbb{R}}|\mathbb{K}_{h}(x-z)|dz.\end{split}$
In the proof of Proposition 1 we have shown that
$\sup_{z\in\mathbb{R}}\int_{0}^{\infty}|g_{u}(y,z)|du\leq
c(1+\mu(y)(1+f^{*}(y))).$
It follows that
$T\,\text{Var}(\hat{\mu}_{h,T}(x))\leq
c\int_{\mathbb{R}}|\mathbb{K}_{h}(x-y)|(1+\mu(y)(1+f^{*}(y)))dy,$
since, by the definition of the kernel function,
$\int_{\mathbb{R}}|\mathbb{K}_{h}(x-z)|dz=\int_{x-h}^{x+h}|\mathbb{K}_{h}(x-z)|dz\leq\left\|\mathbb{K}_{h}\right\|_{\infty}h\leq\frac{\left\|K\right\|_{\infty}}{h}h=\left\|K\right\|_{\infty}.$
Then, by the definition of $\mathbb{K}_{h}$, we get that
$\begin{split}&\int_{\mathbb{R}}|\mathbb{K}_{h}(x-y)|(1+\mu(y)(1+f^{*}(y)))dy\\\
&\qquad\qquad=\frac{1}{h}\int_{x-h}^{x+h}|K(\frac{x-y}{h})|(1+\mu(y)(1+f^{*}(y)))dy\\\
&\qquad\qquad\leq\left\|K\right\|_{\infty}\int_{-1}^{1}(1+\mu(x-h\tilde{y})(1+f^{*}(x-h\tilde{y})))d\tilde{y},\end{split}$
where we have applied the change of variable $\tilde{y}:=\frac{x-y}{h}$. Now
we observe that, if $|x-h\tilde{y}|\leq 1$, then $f^{*}(x-h\tilde{y})$ is
bounded by construction. Otherwise, for $|x-h\tilde{y}|>1$, we have
$\displaystyle f^{*}(x-h\tilde{y})$
$\displaystyle=e^{\epsilon|x-h\tilde{y}|}\leq e^{\epsilon|x|}e^{\epsilon
h|\tilde{y}|}\leq e^{\epsilon|x|}e^{\epsilon},$
where in the last inequality we have used the fact that both $h$ and
$|\tilde{y}|$ are smaller than $1$. Therefore, we have shown that
$T\,\text{Var}(\hat{\mu}_{h,T}(x))\leq ce^{\epsilon|x|},$
where $c$ is independent of $T$, $h$ and $x$. Finally, from the bias-variance
decomposition (6) we obtain (4), which concludes the desired proof.
### 4.2 Proof of Theorem 2
We aim to apply Theorem 5. For this, we split the interval $[0,T]$ into $n$
intervals $[t_{i-1},t_{i}]$, where $t_{i}=i\Delta$ for any
$i\in\left\\{0,\ldots,n\right\\}$, $n\Delta=T$, and $n=\lfloor T\rfloor$ with
$T\geq 1$, which implies that $1\leq\Delta<2$.
For each $n\geq 1$ and $1\leq r\leq m$, we consider the sequence
$(Y_{n,i}^{(r)})_{i\geq 1}$ defined as
$Y_{n,i}^{(r)}:=\frac{1}{\sqrt{\Delta}}\left(\int_{t_{i-1}}^{t_{i}}\mathbb{K}_{h}(x_{r}-X_{u})du-\mathbb{E}\left[\int_{t_{i-1}}^{t_{i}}\mathbb{K}_{h}(x_{r}-X_{u})du\right]\right),$
for $x_{r}\in\mathbb{R}$. We denote by $Y_{n,i}$ the $\mathbb{R}^{m}$ valued
random vector defined by $Y_{n,i}=(Y_{n,i}^{(1)},\ldots,Y_{n,i}^{(m)})$. By
construction,
$\frac{\sum_{i=1}^{n}Y_{n,i}}{\sqrt{n}}=\sqrt{T}(\hat{\mu}_{h,T}(x)-\mathbb{E}[\hat{\mu}_{h,T}(x)]),$
where $\hat{\mu}_{h,T}(x)-\mathbb{E}[\hat{\mu}_{h,T}(x)]$ is the vector
$(\hat{\mu}_{h,T}(x_{1})-\mathbb{E}[\hat{\mu}_{h,T}(x_{1})],\ldots,\hat{\mu}_{h,T}(x_{m})-\mathbb{E}[\hat{\mu}_{h,T}(x_{m})]).$
It is clear that $\mathbb{E}[Y_{n,i}]=0$ for all $n\geq 1$ and $i\geq 1$.
Moreover, for all $i\geq 1$, $1\leq r\leq m$ and $n\geq 1$ we have
$|Y_{n,i}^{(r)}|\leq\frac{1}{\sqrt{\Delta}}\left\|\mathbb{K}_{h}\right\|_{\infty}\Delta\leq{\color[rgb]{0,0,0}\frac{\left\|K\right\|_{\infty}}{h(T)}\sqrt{2}}.$
We choose
$h(T):=(\frac{1}{T})^{\frac{1}{2}-\epsilon}=(\frac{1}{n\Delta})^{\frac{1}{2}-\epsilon}\geq
c(\frac{1}{n})^{(\frac{1}{2}-\epsilon)}$, for some
$\epsilon\in(0,\frac{1}{2})$. Hence, assumption (i) holds true with
$M_{n}:=cn^{\frac{1}{2}-\epsilon}$. Concerning assumption (ii) we remark that,
for any $i\geq 1$ and any $1\leq r\leq m$,
$\begin{split}\mathbb{E}[(Y_{n,i}^{(r)})^{2}]&=\text{Var}\left(\frac{1}{\sqrt{\Delta}}\int_{0}^{\Delta}\mathbb{K}_{h}(x_{r}-X_{u})du\right)=\text{Var}(\sqrt{\Delta}\hat{\mu}_{h,\Delta}(x_{r}))\\\
&=\Delta\text{Var}(\hat{\mu}_{h,\Delta}(x_{r}))\leq\Delta\frac{c}{\Delta}=c,\end{split}$
where in the last inequality we have used (4.1). We next check condition
(iii). Let $b_{n}$ be a sequence of integers such that
$b_{n}\rightarrow\infty$ and $b_{n}\leq n$ for every $n$. For every $1\leq
r\leq m$ and $1\leq s\leq m$, we have
$\begin{split}&\frac{1}{b_{n}}\mathbb{E}\left[\sum_{i=1}^{b_{n}}Y_{n,i}^{(r)}\sum_{j=1}^{b_{n}}Y_{n,j}^{(s)}\right]=\frac{1}{{\color[rgb]{0,0,0}\Delta}b_{n}}\int_{0}^{{\color[rgb]{0,0,0}\Delta}b_{n}}\int_{0}^{{\color[rgb]{0,0,0}\Delta}b_{n}}\text{Cov}(\mathbb{K}_{h}(x_{r}-X_{u}),\mathbb{K}_{h}(x_{s}-X_{v}))du\,dv\\\
&=2\int_{0}^{{\color[rgb]{0,0,0}\Delta}b_{n}}(1-\frac{u}{{\color[rgb]{0,0,0}\Delta}b_{n}})\int_{\mathbb{R}}\int_{\mathbb{R}}\mathbb{K}_{h}(x_{r}-z_{1})\mathbb{K}_{h}(x_{s}-z_{2})g_{u}(z_{1},z_{2})dz_{1}\,dz_{2}\,du\\\
&=2\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{{\color[rgb]{0,0,0}\Delta}b_{n}}(1-\frac{u}{{\color[rgb]{0,0,0}\Delta}b_{n}})K(w_{1})K(w_{2})g_{u}(x_{r}-h(T)w_{1},x_{s}-h(T)w_{2})du\,dw_{1}\,dw_{2},\end{split}$
where we have used Fubini’s theorem and the change of variables
$w_{1}:=\frac{x_{r}-z_{1}}{h(T)}$, $w_{2}:=\frac{x_{s}-z_{2}}{h(T)}$. Using
dominated convergence and the fact that $h(T)\rightarrow 0$ for
$T\rightarrow\infty$ and $\Delta b_{n}\rightarrow\infty$ for
$n\rightarrow\infty$ as $\Delta\geq 1$, we obtain
$\begin{split}\lim_{n\rightarrow\infty}\frac{1}{b_{n}}\mathbb{E}\left[\sum_{i=1}^{b_{n}}Y_{n,i}^{(r)}\sum_{j=1}^{b_{n}}Y_{n,j}^{(s)}\right]&=2\int_{\mathbb{R}}K(w_{1})\int_{\mathbb{R}}K(w_{2})\int_{0}^{\infty}g_{u}(x_{r},x_{s})du\,dw_{2}\,dw_{1}\\\
&=2\int_{0}^{\infty}g_{u}(x_{r},x_{s})du=:\sigma(x_{r},x_{s}),\end{split}$
which proves (iii). Remark that it is possible to use dominated convergence
theorem since we have shown in the proof of Proposition 1 that
$\sup_{y\in\mathbb{R}}|g_{u}(x,y)|\leq c\left(u^{-1/2}{\bf 1}_{\\{u\leq
2\\}}+\mu(x)(1+f^{\ast}(x))e^{-\rho u}{\bf 1}_{\\{u>2\\}}\right),$
for some positive constants $c$ and $\rho$. In particular, we have
$\displaystyle|(1-\frac{u}{\Delta
b_{n}})K(w_{1})K(w_{2})g_{u}(x_{r}-h(T)w_{1},x_{s}-h(T)w_{2})1_{[0,b_{n}]}(u)1_{\mathbb{R}^{2}}(w_{1},w_{2})|$
$\displaystyle\leq c\left(u^{-1/2}{\bf 1}_{\\{u\leq
2\\}}+e^{\epsilon(|x_{r}|+|w_{1}|)}e^{-\rho u}{\bf
1}_{\\{u>2\\}}\right)|K(w_{1})K(w_{2})|\in
L^{1}(\mathbb{R}^{+}\times\mathbb{R}^{2}),$
as $K$ has support on $[-1,1]$.
We now check (iv). We remark that if a process is $\beta$-mixing, then it is
also $\alpha$-mixing and the following estimation holds (see Theorem 3 in
Section 1.2.2 of [26])
$\alpha_{k}\leq\beta_{Y_{n,i}}(k)=\beta_{X}(k)\leq ce^{-\gamma_{1}k}.$
Therefore, it suffices to show that there exists
${\color[rgb]{0,0,0}\gamma_{0}}\in(1,\infty)$ such that
$\sum_{k\geq
1}ke^{-k\gamma_{1}\frac{({\color[rgb]{0,0,0}\gamma_{0}}-1)}{{\color[rgb]{0,0,0}\gamma_{0}}}}<\infty,$
which is true for any ${\color[rgb]{0,0,0}\gamma_{0}}>1$, so (iv) is
satisfied.
We are left to show (v). Set
$f({\color[rgb]{0,0,0}\gamma_{0}}):=\frac{{\color[rgb]{0,0,0}\gamma_{0}}^{2}}{(3{\color[rgb]{0,0,0}\gamma_{0}}-1)(2{\color[rgb]{0,0,0}\gamma_{0}}-1)}$
and observe that $f(1)=\frac{1}{2}$ and for $\gamma_{0}>1$, $f$ is continuous,
strictly decreasing, and $\frac{1}{6}<f(\gamma_{0})<\frac{1}{2}$. Therefore,
given $\epsilon\in(0,\frac{1}{2})$, there always exists $\gamma_{0}>1$ such
that for all $n\geq 1$,
$n^{\frac{1}{2}-\epsilon}\leq n^{f(\gamma_{0})}.$
Thus, condition (v) is satisfied. We can then apply Theorem 5 which directly
leads us to (7) and concludes the desired proof.
### 4.3 Proof of Theorem 3
The proof of of Theorem 3 follows as the proof of the lower bound for $d\geq
3$ obtained in Theorem 3 of [3]. Therefore, we will only explain the main
steps and the principal differences.
Step 1 The first step consists in showing that given a density function $f$,
we can always find a drift function $b_{f}$ such that $f$ is the unique
invariant density function of equation (8) with drift coefficient $b=b_{f}$.
We give the statement and proof in dimension $d=1$, as in Propositions 2 and 3
of [3] it is only done for $d\geq 2$.
###### Proposition 2.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a $\mathcal{C}^{2}$ positive
probability density satisfying the following conditions
1. 1.
$\lim_{y\rightarrow\pm\infty}f(y)=0$ and
$\lim_{y\rightarrow\pm\infty}f^{\prime}(y)=0$.
2. 2.
There exist $\hat{c}_{1}>0$ and $0<\epsilon<\frac{\epsilon_{0}}{|\gamma|}$,
where $\epsilon_{0}$ is as in Assumption A4 such that, for any
$y,z\in\mathbb{R}$,
$f(y\pm z)\leq\hat{c}_{1}e^{\epsilon|z|}f(y).$
3. 3.
For $\epsilon>0$ as in 2. there exists $\hat{c}_{2}(\epsilon)>0$ such that
$\sup_{y<0}\frac{1}{f(y)}\int_{-\infty}^{y}f(w)dw<\hat{c}_{2}\;\text{ and
}\;\sup_{y>0}\frac{1}{f(y)}\int_{y}^{\infty}f(w)dw<\hat{c}_{2}.$
4. 4.
There exists
$0<\tilde{\epsilon}<\frac{a^{2}}{2\gamma^{2}c_{4}\hat{c}_{2}\hat{c}_{4}\hat{c}_{1}}$
and $R>0$ such that for any $|y|>R$,
$\frac{f^{\prime}(y)}{f(y)}\leq-\tilde{\epsilon}\operatorname{sgn}(y)$, where
$c_{4}$ is as in Assumption A4. Moreover, there exists $\hat{c}_{3}$ such that
for any $y\in\mathbb{R}$, $|f^{\prime}(y)|\leq\hat{c}_{3}f(y)$.
5. 5.
For any $y\in\mathbb{R}$ and $\tilde{\epsilon}$ as in 4.
$|f^{\prime\prime}(y)|\leq\hat{c}_{4}\tilde{\epsilon}^{2}f(y).$
Then there exists a bounded Lipschitz function $b_{f}$ which satisfies A2 such
that $f$ is the unique invariant density to equation (8) with drift
coefficient $b=b_{f}$.
###### Proof.
Let $A_{d}$ be the discrete part of the generator of the diffusion process $X$
solution of (8) and let $A^{*}_{d}$ its adjoint. We define $b_{f}$ as
$b_{f}(x)=\begin{cases}\frac{1}{f(x)}\int_{-\infty}^{x}(\frac{1}{2}a^{2}f^{\prime\prime}(w)+A^{*}_{d}\,f(w))dw,\quad&\mbox{if
}x<0;\\\
-\frac{1}{f(x)}\int_{x}^{\infty}\frac{1}{2}a^{2}f^{\prime\prime}(x)(w)+A^{*}_{d}\,f(w)dw,\quad&\mbox{if
}x>0,\end{cases}$
where
$A^{*}_{d}\,f(x)=\int_{\mathbb{R}}[f(x-\gamma z)-f(x)+\gamma
zf^{\prime}(x)]F(z)dz.$
Then, following Proposition 3 in [3], one can check that $b_{f}$ is bounded,
Lipschitz, and satisfies A2. Moreover, if we replace $b$ by $b_{f}$ in
equation (8), then $f$ is the unique invariant density. ∎
Step 2 The second step consists in defining two probability density functions
$f_{0}$ and $f_{1}$ in $\mathcal{H}_{1}(\beta,\mathcal{L})$.
We first define $f_{0}(y)=c_{\eta}f(\eta|y|)$, where $\eta\in(0,\frac{1}{2})$,
$c_{\eta}$ is such that $\int f_{0}=1$, where $f$ is defined as follows. We
first consider the piecewise function
$g(x)=\begin{cases}e^{-|x|},&\text{ if }|x|\geq 1\\\
e^{-4(|x|-\frac{1}{2})^{2}},&\text{ if }\frac{1}{2}<|x|<1\\\ 1,&\text{ if
}|x|\leq\frac{1}{2}.\end{cases}$
Observe that $g$ is continuous, satisfies $\frac{1}{2}e^{-|x|}\leq g(x)\leq
2e^{-|x|}$ for all $x\in\mathbb{R}$, and each piece belongs to $C^{\infty}$
and has bounded derivatives. We define $f$ as a $\mathcal{C}^{\infty}$
approximation of $g$, with bounded derivatives of all orders and satisfying
$\frac{1}{2}e^{-|x|}\leq f(x)\leq 2e^{-|x|},\quad|f^{\prime}(|x|)|\leq
5e^{-|x|},\quad\text{and}\quad|f^{\prime\prime}(|x|)|\leq 14e^{-|x|}.$ (22)
Observe that the two latter inequalities are satisfied by $g$ piecewise.
It is easy to see that $\eta$ can be chosen small enough so that
$f_{0}\in\mathcal{H}_{1}(\beta,\mathcal{L})$. Indeed, first, it is clear that
all the derivatives of $f_{0}$ can be bounded by the constant $\mathcal{L}$
for $\eta$ small enough. Furthermore, the following bounds hold true for any
$x$ and $t$ in $\mathbb{R}$
$\displaystyle|D^{\lfloor\beta\rfloor}f_{0}(x+t)-D^{\lfloor\beta\rfloor}f_{0}(x)|$
$\displaystyle\leq|D^{\lfloor\beta\rfloor}f_{0}(x+t)-D^{\lfloor\beta\rfloor}f_{0}(x)|^{\beta-\lfloor\beta\rfloor}(2\left\|D^{\lfloor\beta\rfloor}f_{0}\right\|_{\infty})^{1-(\beta-\lfloor\beta\rfloor)}$
$\displaystyle\leq\left\|D^{\lfloor\beta\rfloor+1}f_{0}\right\|_{\infty}^{\beta-\lfloor\beta\rfloor}(2\left\|D^{\lfloor\beta\rfloor}f_{0}\right\|_{\infty})^{1-(\beta-\lfloor\beta\rfloor)}\,|t|^{\beta-\lfloor\beta\rfloor}.$
Again, it suffices to choose $\eta$ small enough to ensure that
$\left\|D^{\lfloor\beta\rfloor+1}f_{0}\right\|_{\infty}^{\beta-\lfloor\beta\rfloor}(2\left\|D^{\lfloor\beta\rfloor}f_{0}\right\|_{\infty})^{1-(\beta-\lfloor\beta\rfloor)}\leq\mathcal{L},$
which shows that
$f_{0}\in\mathcal{H}_{1}(\beta,\mathcal{L})\subset\mathcal{H}_{1}(\beta,2\mathcal{L})$.
We also ask that the constant $c_{4}$ in Assumption A4 is such that
$c_{4}<\frac{a^{2}}{2\gamma^{2}4^{2}28}.$ (23)
This means that the jumps have to integrate an exponential function. The bound
depends on the coefficients $a$ and $\gamma$ and so it depends only on the
model.
Under the conditions above it is easy to see that $f_{0}$ satisfies the
assumptions of Proposition 2 with $\hat{c}_{1}=4$, $\epsilon=\eta$,
$\hat{c}_{2}=\frac{4}{\eta}$, $R=\frac{1}{\eta}$, $\tilde{\epsilon}=\eta$
$\hat{c}_{3}=28\eta$, and $\hat{c}_{4}=28$. Indeed, point 1 of Proposition 2
clearly holds true from the definition of $f_{0}$. To show the second point we
observe that, thanks to (22), we have
$f_{0}(y\pm z)=c_{n}f(\eta|y\pm z|)\leq 2c_{n}e^{-\eta|y|}e^{\eta|z|}\leq
4f_{0}(y)e^{\eta|z|},$
which implies point 2 with $\hat{c}_{1}=4$ and $\epsilon=\eta$, since we can
choose $\eta$ small enough to make the condition on $\epsilon$ satisfied. In
order to prove point 3 we use again (22). It follows that, for any $y<0$,
$\displaystyle\frac{1}{f_{0}(y)}\int_{-\infty}^{y}f_{0}(w)dw$
$\displaystyle=\frac{1}{c_{n}f(\eta|y|)}\int_{-\infty}^{y}c_{n}f(\eta|w|)dw$
$\displaystyle\leq 2e^{\eta|y|}\int_{-\infty}^{y}2e^{-\eta
w}dw=4e^{\eta|y|}\frac{e^{-\eta|y|}}{\eta}=\frac{4}{\eta}.$
For $y>0$ an analogous reasoning applies. Thus, $f_{0}$ satisfies the third
point with $\hat{c}_{2}(\epsilon)=\hat{c}_{2}(\eta)=\frac{4}{\eta}$. For the
fourth point, we observe that, for $|y|>\frac{1}{\eta}$,
$f_{0}(y)=-\eta\,\text{sgn}(y)f_{0}(y).$
That is, the first part of point 4 holds true for $|y|>R$, taking
$R=\frac{1}{\eta}$ and $\tilde{\epsilon}=\eta$. Moreover, we observe that
using (22) we have, for $k=1,2$,
$|f_{0}^{(k)}(y)|=|c_{n}f^{(k)}(\eta|y|)|\leq 14c_{n}\eta^{k}e^{-\eta|y|}\leq
28\eta^{k}f_{0}(y).$
This shows that both the fourth and the fifth points hold true, with
$\hat{c}_{3}(\eta)=28\eta$ and $\hat{c}_{4}=28$. Finally, we need to check
that the condition on $\tilde{\epsilon}$ given in the fourth point which
writes as
$\tilde{\epsilon}=\eta<\frac{a^{2}}{2\gamma^{2}c_{4}\hat{c}_{2}\hat{c}_{4}\hat{c}_{1}}=\frac{a^{2}\,\eta}{2\gamma^{2}c_{4}\,4\,28\,4},$
which is equivalent to (23). Hence, $f_{0}$ satisfies all the assumptions in
Proposition 2.
Therefore, $b_{0}:=b_{f_{0}}$ belongs to $\Sigma(\beta,\mathcal{L})$. Recall
that $b_{0}$ belongs to $\Sigma(\beta,\mathcal{L})$ if and only if $f_{0}$
belongs to $\mathcal{H}_{1}(\beta,{\color[rgb]{0,0,0}2}\mathcal{L})$ and
$b_{0}$ is bounded, Lipschitz and satisfies the drift condition A2.
We next define
$f_{1}(x)=f_{0}(x)+\frac{1}{M_{T}}\hat{K}\left(\frac{x-x_{0}}{{\color[rgb]{0,0,0}H}}\right),$
(24)
where $x_{0}\in\mathbb{R}$ is fixed and
$\hat{K}:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^{\infty}$ function with
support on $[-1,1]$ such that
$\hat{K}(0)=1,\quad\int_{-1}^{1}\hat{K}(z)dz=0.$
Here $H$ is a constant and $M_{T}$ will be calibrated later and satisfies that
$M_{T}\rightarrow\infty$ as $T\rightarrow\infty$. Observe that in the proof of
the lower bound for the case $d\geq 3$ presented in [3], $H$ is a function of
$T$ converging to 0 as $T\rightarrow\infty$. For the case $d=1$, it suffices
to chose it constant and we will see below that the same computations done in
[3] will work in this case and it suffices to calibrate $M_{T}$.
Then it can be shown as in [3, Lemma 3] that if for all $\epsilon>0$ and $T$
sufficiently large,
$\frac{1}{M_{T}}\leq\epsilon
H^{\beta}\qquad\mbox{and}\qquad\frac{1}{H}=o(M_{T})$ (25)
as $T\rightarrow\infty$, then if $\epsilon>0$ is small enough we have that
$b_{1}:=b_{f_{1}}$ belongs to $\Sigma(\beta,\mathcal{L})$ for $T$ sufficiently
large. Indeed, on one hand, (25) is clearly true when $H$ is a constant. On
the other hand, the same argument used in [3, Lemma 3] applies to show that
$f_{1}$ belongs to $\mathcal{H}_{1}(\beta,2\mathcal{L})$ when $H$ is a
constant, up to choose $\epsilon$ in (25) smaller than a constant depending on
$\mathcal{L}$ and $H$.
Step 3 As $b_{0},b_{1}\in\Sigma(\beta,\mathcal{L})$, we can write
$R(\tilde{\mu}_{T}(x_{0}))\geq\frac{1}{2}\mathbb{E}_{1}^{(T)}[(\tilde{\mu}_{T}(x_{0})-f_{1}(x_{0}))^{2}]+\frac{1}{2}\mathbb{E}_{0}^{(T)}[(\tilde{\mu}_{T}(x_{0})-f_{0}(x_{0}))^{2}],$
where $\mathbb{E}_{i}^{(T)}$ denotes the expectation with respect to $b_{i}$.
Then, following as in [3], using Girsanov’s formula, we can show that if
$\sup_{T\geq 0}T\frac{1}{M_{T}^{2}{\color[rgb]{0,0,0}H}}<\infty,$ (26)
then for sufficiently large $T$,
$R(\tilde{\mu}_{T}(x_{0}))\geq\frac{C}{8\lambda}\frac{1}{M_{T}^{2}},$ (27)
where the constants $C$ and $\lambda$ are as in Lemma 4 of [3] and they do not
depend on the point $x_{0}$. We finally look for the larger choice of
$\frac{1}{M^{2}_{T}}$ for which both (25) and (26) hold true. It suffices to
choose $M_{T}=\sqrt{T}$ to conclude the proof of Theorem 3.
###### Remark 2.
The two hypothesis method used above does not work to prove the 2-dimensional
lower bound of Theorem 4. Indeed, following as above, we can define
$f_{1}(x)=f_{0}(x)+\frac{1}{M_{T}}\hat{K}\left(\frac{x-x_{0}}{H_{1}(T)}\right)\hat{K}\left(\frac{x-x_{0}}{H_{2}(T)}\right).$
Then, it is possible to show that (27) still holds and, therefore, we should
take $M_{T}$ such that $\frac{1}{M^{2}_{T}}=\frac{\log T}{T}$. On the other
hand, condition (26) now becomes
$\sup_{T\geq
0}T\frac{1}{M^{2}_{T}}\left(\frac{H_{2}(T)}{H_{1}(T)}+\frac{H_{1}(T)}{H_{2}(T)}\right)<\infty.$
The optimal choice of the bandwidth is achieved for $H_{2}(T)=H_{1}(T)$ which
yields to $\sup_{T\geq 0}T\frac{1}{M^{2}_{T}}<\infty,$ which is clearly not
satisfied when $\frac{1}{M^{2}_{T}}=\frac{\log T}{T}$.
### 4.4 Proof of Theorem 4
We will apply Lemma 2 with $\psi:=v\sqrt{\frac{\log T}{T}}$, where $v>0$ is
fixed. As above we divide the proof into three steps.
Step 1 As in the one-dimensional case, the first step consists in showing that
given a density function $f$, we can always find a drift function $b_{f}$ such
that $f$ is the unique invariant density function of equation (8) with drift
coefficient $b=b_{f}$, which is proved in Propositions 2 and 3 of [3]. We
remark that condition (10) is needed in Proposition 3 to ensure that the terms
on the diagonal of the volatility coefficient $a$ dominate on the others,
which is crucial to get that $b_{f}$ satisfies the drift condition A2. Step 2
We next define the probability density
$f_{0}\in\mathcal{H}_{2}(\beta,\mathcal{L})$, the finite set $J_{T}$, and the
set of probability densities $\left\\{f_{j},\,j\in
J_{T}\right\\}\subset\mathcal{H}_{2}(\beta,\mathcal{L})$ needed in order to
apply Lemma 2.
We first define $f_{0}$ as $\pi_{0}$ in Section 7.2 of [3], which is the two-
dimensional version of $f_{0}$ defined in the proof of Theorem 3, that is,
$f_{0}(x)=c_{\eta}f(\eta(aa^{T})^{-1}_{11}|x_{1}|)f(\eta(aa^{T})^{-1}_{22}|x_{2}|),\quad
x=(x_{1},x_{2})\in\mathbb{R}^{2},$ (28)
where $f$ is as in Step 2 of the proof of Proposition 2. The density $f_{0}$
belongs to $\mathcal{H}_{2}(\beta,\mathcal{L})$ by construction.
We then set
$J_{T}:=\left\\{1,\ldots,\lfloor\frac{1}{\sqrt{H_{1}}}\rfloor\right\\}\times\left\\{1,\ldots,\lfloor\frac{1}{\sqrt{H_{2}}}\rfloor\right\\},$
(29)
where in order to lighten the notation we will write $H_{1}$ and $H_{2}$ for
$H_{1}(T)$ and $H_{2}(T)$, respectively, which are two quantities that
converge to $0$ as $T\rightarrow\infty$ and need to be calibrated.
Finally, for $j:=(j_{1},j_{2})\in J_{T}$, we define
$x_{j}:=(x_{j,1},x_{j,2})=(2j_{1}H_{1},2j_{2}H_{2})$ and we set
$f_{j}(x):=f_{0}(x)+{\color[rgb]{0,0,0}2}v\sqrt{\frac{\log
T}{T}}\hat{K}\left(\frac{x_{1}-x_{j,1}}{H_{1}}\right)\hat{K}\left(\frac{x_{2}-x_{j,2}}{H_{2}}\right),$
where recall that $v>0$ is fixed and $\hat{K}$ is as in (24).
Acting as in Lemma 3 of [3], recalling that the rate $\frac{1}{M_{T}}$ therein
is now replaced by $\sqrt{\frac{\log T}{T}}$ (see also points 1. and 3. in the
proof of Proposition 3 below), it is easy to see that if there exists
$\epsilon>0$ sufficiently small such that for large $T$,
$\sqrt{\frac{\log T}{T}}\leq\epsilon H_{1}^{\beta_{1}},\qquad\sqrt{\frac{\log
T}{T}}\leq\epsilon H_{2}^{\beta_{2}},$ (30)
then, for any $j\in J_{T}$ and large $T$, $b_{j}\in\Sigma(\beta,\mathcal{L}).$
In particular, $f_{j}\in\mathcal{H}_{2}(\beta,\mathcal{L}).$ Therefore,
$\left\\{f_{j},\,j\in
J_{T}\right\\}\subset\mathcal{H}_{2}(\beta,\mathcal{L})$.
In order to evaluate the difference between $f_{j}$ and $f_{k}$ we remark
first of all that, as $\hat{K}$ has support on $[-1,1]$,
$\prod_{l=1}^{2}\hat{K}(\frac{x_{l}-x_{j,l}}{H_{l}})$ is different from $0$
only if $|\frac{x_{l}-x_{j,l}}{H_{l}}|\leq 1$ for any $l\in\\{1,2\\}$. Then,
$\displaystyle\left\|f_{j}-f_{k}\right\|_{\infty}$
$\displaystyle\geq|f_{j}(x_{j})-f_{k}(x_{j})|$
$\displaystyle=2v\sqrt{\frac{\log
T}{T}}[\prod_{l=1}^{2}\hat{K}(\frac{x_{j,l}-x_{j,l}}{H_{l}})-\prod_{l=1}^{2}\hat{K}(\frac{x_{j,l}-x_{k,l}}{H_{l}})]$
$\displaystyle=2v\sqrt{\frac{\log
T}{T}}\prod_{l=1}^{2}\hat{K}(0)=2v\sqrt{\frac{\log T}{T}}=2\psi,$
where we have used that, as $j\neq k$, there is a $l_{0}\in\\{1,2\\}$ such
that $l_{0}\neq k_{0}$ and so in particular, by construction,
$|j_{l_{0}}-k_{l_{0}}|\geq 1$. It follows that
$|\frac{x_{j,l_{0}}-x_{k,l_{0}}}{H_{l_{0}}}|=|\frac{2j_{l_{0}}H_{l_{0}}-2k_{l_{0}}H_{l_{0}}}{h_{l_{0}}}|\geq
2$
and so the kernel evaluated in this point is null. This proves the first
condition of Lemma 2.
Step 3 We are left to show the remaining conditions of Lemma 2. The absolute
continuity $\mathbb{P}^{(T)}_{j}\ll\mathbb{P}^{(T)}_{0}$ and the expression
for $\frac{d\mathbb{P}^{(T)}_{j}}{d\mathbb{P}^{(T)}_{0}}(X^{T})$ are both
obtained by Girsanov formula, as in Lemma 4 of [3]. We have,
$KL(\mathbb{P}^{(T)}_{j},\mathbb{P}^{(T)}_{0})=\mathbb{E}^{(T)}_{j}\left[\log\left(\frac{f_{j}}{f_{0}}(X^{T})\right)\right]+\frac{1}{2}\mathbb{E}^{(T)}_{j}\left[\int_{0}^{T}|a^{-1}(b_{0}(X_{u})-b_{j}(X_{u}))|^{2}du\right],$
where the law of $X^{T}=(X_{t})_{t\in[0,T]}$ under $\mathbb{P}^{(T)}_{j}$ is
the one of the solution to equation (8) with $b=b_{0}$.
By the definition of the $f_{j}$’s it is easy to see that the first term is
$o(1)$ as $T\rightarrow\infty$. In fact, as $\hat{K}$ is supported in
$[-1,1]$,
$\begin{split}\mathbb{E}^{(T)}_{j}\left[\log\left(\frac{f_{j}}{f_{0}}(X^{T})\right)\right]&=\int_{\mathbb{R}^{2}}\log\bigg{(}1+\frac{{\color[rgb]{0,0,0}2}v\sqrt{\frac{\log
T}{T}}\hat{K}\left(\frac{x_{1}-x_{j,1}}{H_{1}}\right)\hat{K}\left(\frac{x_{2}-x_{j,2}}{H_{2}}\right)}{f_{0}(x)}\bigg{)}f_{0}(x)dx\\\
&\leq\bigg{|}\log\bigg{(}1+c_{\ast}v\sqrt{\frac{\log
T}{T}}\|\hat{K}\|^{2}_{\infty}\bigg{)}\bigg{|},\end{split}$
which tends to zero as $T\rightarrow\infty$, where
$c_{\ast}:=\frac{{\color[rgb]{0,0,0}8}}{c_{\eta}}e^{4\eta\,k}$, $c_{\eta}$ is
the constant of normalization introduced in the definition of $f_{0}$, and
$k:=\max_{i=1,2}(aa^{T})^{-1}_{ii}$. In fact, this follows from the definition
of $f_{0}$ in (28). Since $f(x)\geq\frac{1}{2}e^{-|x|}$, we obtain
$\frac{1}{f_{0}(x)}\leq\frac{1}{c_{\eta}}\frac{2}{e^{-\eta(aa^{T})^{-1}_{11}|x_{1}|}}\frac{2}{e^{-\eta(aa^{T})^{-1}_{22}|x_{2}|}}\leq\frac{4}{c_{\eta}}e^{\eta
k(|H_{1}|+|x_{j,1}|+|H_{2}|+|x_{j,2}|)},$
where we have also used the fact that, as $\hat{K}$ is supported in $[-1,1]$,
we have
$x\in[x_{j,1}-H_{1},x_{j,1}+H_{1}]\times[x_{j,2}-H_{2},x_{j,2}+H_{2}]$.
Finally, by the definition of $x_{j}$ and the fact that $H_{i}\rightarrow 0$
as $T\rightarrow\infty$ for $i=1,2$ (and so for $T$ large enough they are
smaller than 1), we get
$\frac{1}{f_{0}(x)}\leq\frac{4}{c_{\eta}}e^{4\eta k}\quad\mbox{for any
}x\in[x_{j,1}-H_{1},x_{j,1}+H_{1}]\times[x_{j,2}-H_{2},x_{j,2}+H_{2}].$ (31)
Regarding the second term, using the stationarity of the process $X^{T}$, we
have
$\mathbb{E}^{(T)}_{j}\left[\int_{0}^{T}|a^{-1}(b_{0}(X_{u})-b_{j}(X_{u}))|^{2}du\right]=T\int_{\mathbb{R}^{2}}|a^{-1}(b_{0}(x)-b_{j}(x))|^{2}f_{0}(x)dx.$
Then, the following asymptotic bound will be proved at the end of this
Section.
###### Proposition 3.
For $T$ large enough,
$\displaystyle\int_{\mathbb{R}^{2}}|a^{-1}(b_{0}(x)-b_{j}(x))|^{2}f_{0}(x)dx\leq
64\frac{e^{8\eta
k}}{c_{\eta}^{2}}k^{2}v^{2}H_{1}H_{2}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right)^{2}\frac{\log
T}{T}.$
Taking the optimal choice for the bandwidth in Proposition 3, which is
$H_{1}=H_{2}$, we get that
$\int_{\mathbb{R}^{2}}|a^{-1}(b_{0}(x)-b_{j}(x))|^{2}f_{0}(x)dx\leq
64\frac{e^{8\eta k}}{c_{\eta}^{2}}k^{2}v^{2}4\frac{\log T}{T}.$
In particular, after having ordered $\beta_{1}\leq\beta_{2}$, we choose
$H_{1}=H_{2}=(\frac{\log T}{T})^{{\color[rgb]{0,0,0}\alpha}}$ with
${\color[rgb]{0,0,0}\alpha}\leq\frac{1}{2\beta_{2}}=(\frac{1}{2\beta_{1}}\land\frac{1}{2\beta_{2}})$
so that condition (30) is satisfied. We therefore get
$\begin{split}KL(\mathbb{P}^{(T)}_{j},\mathbb{P}^{(T)}_{0})\leq
128\frac{e^{8\eta k}}{c_{\eta}^{2}}k^{2}\,v^{2}\log T\leq 128\frac{e^{8\eta
k}}{c_{\eta}^{2}{\color[rgb]{0,0,0}\alpha}}k^{2}\,v^{2}\log(|J_{T}|),\end{split}$
being the last estimation a consequence of the fact that, by construction,
$\log(|J_{T}|)\geq{\color[rgb]{0,0,0}\alpha}\log\left(\frac{T}{\log
T}\right)={\color[rgb]{0,0,0}\alpha}\log(T)(1+o(1)).$
It is therefore enough to choose $v$ such that $128\frac{e^{8\eta
k}}{c_{\eta}^{2}{\color[rgb]{0,0,0}\alpha}}k^{2}\,v^{2}<\frac{1}{8}$ (ie
$v^{2}<\frac{c_{\eta}^{2}{\color[rgb]{0,0,0}\alpha}}{1024\,k^{2}e^{8\eta k}}$)
and apply Lemma 2 to conclude the proof of Theorem 4.
### 4.5 Proof of Proposition 3
The proof of Proposition 3 follows similarly as Proposition 4 of [3]. Indeed,
we first define the set
$K_{T}^{j}:=[x_{j,1}-H_{1},x_{j,1}+H_{1}]\times[x_{j,2}-H_{2},x_{j,2}+H_{2}],$
where we recall that we write $H_{1}$ and $H_{2}$ for $H_{1}(T)$ and
$H_{2}(T)$, respectively, in order to simplify the notation. Then we show the
following points for $T$ large enough:
1. 1.
There exists a constant $c>0$ such that, for any $x$ in the complementary set
of $K_{T}$, that we denote as $K_{T}^{j\,c}$, and for any
$i\in\left\\{1,2\right\\}$,
$|b^{i}_{j}(x)-b^{i}_{0}(x)|\leq c\,v\,\sqrt{\frac{\log T}{T}}.$
2. 2.
There exists a constant $c>0$ such that, for any $i\in\left\\{1,2\right\\}$,
$\int_{K_{T}^{j\,c}}|b^{i}_{j}(x)-b^{i}_{0}(x)|f_{0}(x)dx\leq
c\,v\,\sqrt{\frac{\log T}{T}}H_{1}H_{2}.$
3. 3.
For any $x\in K_{T}^{j}$ and $i\in\left\\{1,2\right\\},$
$|b^{i}_{j}(x)-b^{i}_{0}(x)|\leq\frac{8}{c_{\eta}}e^{4\eta
k}kv\sqrt{\frac{\log T}{T}}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right).$
The proof of the first two points follows exactly the one in Proposition 4 of
[3], remarking that
$d_{T}(x):=\pi_{1}(x)-\pi_{0}(x)=\frac{1}{M_{T}}\prod_{l=1}^{d}K\left(\frac{x_{l}-x_{0}^{l}}{h_{l}(T)}\right)$
in [3] is now replaced by
$d_{T}^{j}(x):=f_{j}(x)-f_{0}(x)={\color[rgb]{0,0,0}2}v\,\sqrt{\frac{\log
T}{T}}\hat{K}\left(\frac{x_{1}-x_{j,1}}{H_{1}}\right)\hat{K}\left(\frac{x_{2}-x_{j,2}}{H_{2}}\right),$
and the set
$K_{T}:=[x_{0}^{1}-h_{1}(T),x_{0}^{1}+h_{1}(T)]\times\cdots\times[x_{0}^{d}-h_{d}(T),x_{0}^{d}+h_{d}(T)]$
introduced in [3] is now replaced by $K_{T}^{j}$. We recall that $K$ and
$\hat{K}$ are exactly the same kernel function. The proof of Proposition 4 of
[3] is based on the fact that $d_{T}(x)$ and its derivatives are null for
$x\in K_{T}^{c}$. In the same way, $d_{T}^{j}(x)$ and its derivatives are null
for $x\in K_{T}^{j\,c}$. Then, acting as in [3], it is easy to see that the
first two points above hold true.
Comparing the third point above with the third point of Proposition 4 of [3],
it is clear that our goal is to show that the constant $c$ that appears in the
third point of Proposition 4 of [3] is explicit and equal to
$\frac{8}{c_{\eta}}e^{4\eta k}k$ when $d=2$. Keeping the notation in [3], we
first introduce the following quantities:
$\tilde{I}^{i}_{1}[f_{0}](x):=\frac{1}{2}\sum_{j=1}^{2}(aa^{T})_{ij}\frac{\partial
f_{0}}{\partial
x_{j}}(x),\qquad\tilde{I}^{i}_{2}[f_{0}](x)=\int_{-\infty}^{x_{i}}A^{*}_{d,i}f_{0}(w_{i})dw.$
We moreover introduce the notation
$\tilde{I}^{i}[f_{0}](x)=\tilde{I}^{i}_{1}[f_{0}](x)+\tilde{I}^{i}_{2}[f_{0}](x).$
According with the definition of $b$, we have
$b^{i}_{0}(x)=\frac{1}{f_{0}(x)}\tilde{I}^{i}[f_{0}](x),\qquad
b^{i}_{j}(x)=\frac{1}{f_{j}(x)}\tilde{I}^{i}[f_{j}](x).$
Since the operator $f\rightarrow\tilde{I}^{i}[f]$ is linear, we deduce that
$b^{i}_{j}(x)=\frac{1}{f_{j}(x)}\tilde{I}^{i}[f_{j}](x)=\frac{1}{f_{j}(x)}\tilde{I}^{i}[f_{0}](x)+\frac{1}{f_{j}(x)}\tilde{I}^{i}[d_{T}^{j}](x).$
(32)
Therefore,
$b^{i}_{j}-b^{i}_{0}=(\frac{1}{f_{j}}-\frac{1}{f_{0}})\tilde{I}^{i}[f_{0}]+\frac{1}{f_{j}}\tilde{I}^{i}[d_{T}^{j}]=\frac{f_{0}-f_{j}}{f_{j}}\frac{1}{f_{0}}\tilde{I}^{i}[f_{0}]+\frac{1}{f_{j}}\tilde{I}^{i}[d_{T}^{j}]=\frac{d_{T}^{j}}{f_{j}}b^{i}_{0}+\frac{1}{f_{j}}\tilde{I}^{i}[d_{T}^{j}].$
We need to evaluate such a difference on the compact set $K_{T}^{j}$. For
this, we will use that fact that $f_{j}=f_{0}+d_{T}^{j}$, and obtain a lower
bound away from $0$. Specifically, from the definition of $d_{T}^{j}$, we get
$\left\|d_{T}^{j}\right\|_{\infty}\leq{\color[rgb]{0,0,0}2}v\sqrt{\frac{\log
T}{T}}\|\hat{K}\|_{\infty}^{2}={\color[rgb]{0,0,0}2}v\sqrt{\frac{\log T}{T}}.$
(33)
In particular,
$f_{j}\geq f_{0}-|d_{T}^{j}|\geq f_{0}-{\color[rgb]{0,0,0}2}v\sqrt{\frac{\log
T}{T}}\geq\frac{f_{0}}{2},$
since $\sqrt{\frac{\log T}{T}}\rightarrow 0$ as $T\rightarrow\infty$, so for
$T$ large enough we have ${\color[rgb]{0,0,0}2}v\sqrt{\frac{\log
T}{T}}\leq\frac{f_{0}}{2}$. Then, for any $x\in K_{T}^{j}$, using (31) we have
$\frac{1}{f_{j}(x)}\leq\frac{2}{f_{0}}\leq\frac{8}{c_{\eta}}e^{4\eta k}.$
Moreover, as $b_{0}$ is bounded, we deduce that for all $x\in K^{j}_{T}$,
$|b^{i}_{j}(x)-b^{i}_{0}(x)|\leq\frac{{\color[rgb]{0,0,0}16}v}{c_{\eta}}e^{4\eta
k}\left\|b_{0}^{i}\right\|_{\infty}\sqrt{\frac{\log T}{T}}+\frac{8e^{4\eta
k}}{c_{\eta}}\tilde{I}^{i}[d_{T}^{j}](x).$ (34)
We therefore need to evaluate
$\tilde{I}^{i}[d_{T}^{j}](x)=\tilde{I}_{1}^{i}[d_{T}^{j}](x)+\tilde{I}_{2}^{i}[d_{T}^{j}](x)$
on $K_{T}^{j}$. As
$\bigg{\|}\frac{\partial d_{T}^{j}}{\partial
x_{j}}\bigg{\|}_{\infty}\leq\frac{{\color[rgb]{0,0,0}2}v}{H_{j}}\sqrt{\frac{\log
T}{T}},$ (35)
it clearly follows that
$\tilde{I}_{1}^{i}[d_{T}]^{j}(x)\leq{\color[rgb]{0,0,0}2}kv\sqrt{\frac{\log
T}{T}}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right).$ (36)
Regarding $\tilde{I}_{2}^{i}[d_{T}^{j}](x)$, we can act exactly as in the
third point of Proposition 4 of [3]. As $x\in K_{T}^{j}$,
$x_{i}\in[x_{j,i}-H_{i},x_{j,i}+H_{i}]$ for $i=1,2$. Therefore, using also the
definition of $d_{T}^{j}$, the first integral is between $x_{j,i}-H_{i}$ and
$x_{i}$. We enlarge the domain of integration to
$[x_{j,i}-H_{i},x_{j,i}+H_{i}]$ and then, appealing to (33) and (35) and the
fact that the intensity of the jumps is finite, we get
$\begin{split}|\tilde{I}_{2}^{i}[d_{T}^{j}](x)|&\leq\int_{x_{j,i}-H_{i}}^{x_{j,i}+H_{i}}\int_{\mathbb{R}^{2}}|d_{T}^{j}(\tilde{w}_{i})-d_{T}^{j}(\tilde{w}_{i-1})+(\gamma\cdot
z)_{i}\frac{\partial}{\partial x_{i}}d_{T}^{j}(w_{i})|F(z)dzdw\\\ &\leq
2\,(\int_{\mathbb{R}^{2}}F(z)dz)\int_{x_{j,i}-H_{i}}^{x_{j,i}+H_{i}}\left\|d_{T}^{j}\right\|_{\infty}dw\\\
&\qquad+\int_{x_{j,i}-H_{i}}^{x_{j,i}+H_{i}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}|(\gamma\cdot
z)_{i}|\bigg{\|}\frac{\partial d_{T}^{j}}{\partial
x_{i}}\bigg{\|}_{\infty}F(z)dzdw\\\ &\leq cH_{i}\sqrt{\frac{\log
T}{T}}+\frac{cH_{i}}{H_{i}}\sqrt{\frac{\log T}{T}},\end{split}$
for some $c>0$. Using this together with (34) and (36) it follows that, for
any $x\in K^{j}_{T}$,
$\displaystyle|b_{j}(x)-b_{0}(x)|$ $\displaystyle\leq c\sqrt{\frac{\log
T}{T}}+\frac{8e^{4\eta k}}{c_{\eta}}kv\sqrt{\frac{\log
T}{T}}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right)$
$\displaystyle\qquad+cH_{i}\sqrt{\frac{\log T}{T}}+c\sqrt{\frac{\log T}{T}}$
$\displaystyle\leq\frac{8e^{4\eta k}}{c_{\eta}}kv\sqrt{\frac{\log
T}{T}}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right),$
where the last inequality is a consequence of the fact that, $\forall
i\in\left\\{1,2\right\\}$, $H_{i}\rightarrow 0$ as $T\rightarrow\infty$ and
so, for $T$ large enough, all the terms are negligible when compared to the
second one. Hence, the three points listed at the beginning of the proof hold
true. We deduce that
$\displaystyle\int_{\mathbb{R}^{2}}|b_{0}(x)-b_{j}(x)|^{2}f_{0}(x)dx$
$\displaystyle\qquad=\int_{K_{T}^{j}}|b_{0}(x)-b_{j}(x)|^{2}f_{0}(x)dx+\int_{K_{T}^{j\,c}}|b_{0}(x)-b_{j}(x)|^{2}f_{0}(x)dx$
$\displaystyle\qquad\leq c\,v^{2}\frac{\log T}{T}H_{1}H_{2}+\frac{64e^{8\eta
k}}{c_{\eta}^{2}}k^{2}v^{2}\frac{\log
T}{T}\left(\frac{1}{H_{1}}+\frac{1}{H_{2}}\right)^{2}|K_{T}^{j}|.$
We recall that $|K_{T}^{j}|=H_{1}H_{2}$ and that, as $T\rightarrow\infty$,
$H_{i}\rightarrow 0$. Thus, the first term is negligible compared to the
second one. The desired result follows.
## Acknowledgements
The authors would like to thank the anonymous referees for their helpful
remarks that helped to improve the first version of the paper.
## References
* [1] Applebaum, D. (2009). Lévy processes and stochastic calculus. Cambridge university press.
* [2] Amorino, C., Gloter, A. (2021). Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes. Journal of Statistical Planning and Inference 213, 106-129.
* [3] Amorino, C. (2021). Rate of estimation for the stationary distribution of jump-processes over anisotropic Hölder classes. Electronic Journal of Statistics, to appear.
* [4] Amorino, C., Dion, C., Gloter, A., Lemler, S. (2020). On the nonparametric inference of coefficients of self-exciting jump-diffusion. arXiv preprint arXiv:2011.12387.
* [5] Banon, G. (1978). Nonparametric identification for diffusion processes, SIAM J. Control Optim. 16, 380–395.
* [6] Banon, G., Nguyen, H.T. (1981). Recursive estimation in diffusion model, SIAM J. Control Optim. 19, 676–685.
* [7] Blanke, D. (1996) Estimation de la densité pour des trajectoires non directement observables, Publ. Inst. Statist. Univ. Paris 40, 21–36.
* [8] Barndorff-Nielsen, O. E., Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc., Ser. B, Stat. Methodol., 63, 167-241.
* [9] Bosq, D. (1997). Parametric rates of nonparametric estimators and predictors for continuous time processes, Ann. Statist. 25, 982–1000.
* [10] Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes. (Second edition), Lecture Notes Statist., 110, New York: Springer-Verlag.
* [11] Bosq, D. (1998) Minimax rates of density estimators for continuous time processes, Sankhya. Ser. A 60, 18–28.
* [12] Bosq, D., Davydov, Yu. (1999). Local time and density estimation in continuous time, Math. Methods Statist. 8, 22–45.
* [13] Bosq, D., Merlevède, F., Peligrad, M. (1999). Asymptotic normality for density kernel estimators in discrete and continuous time. J. Multivar. Anal. 68, 78-95.
* [14] Cheze-Payaud, N. (1994). Nonparametric regression and prediction for continuous-time processes, Publ. Inst. Statist. Univ. Paris 38, 37–58.
* [15] Castellana, J. V., Leadbetter, M. R. (1986). On smoothed probability density estimation for stationary processes. Stoch. Process Their Appl. 21(2), 179-193.
* [16] Chen, Z. Q., Hu, E., Xie, L., Zhang, X. (2017). Heat kernels for non-symmetric diffusion operators with jumps. J Differ Equ., 263(10), 6576-6634.
* [17] Comte, F., Lacour, C. (2013). Anisotropic adaptive kernel deconvolution. In Annales de l’IHP Probabilités et statistiques (Vol. 49, No. 2, pp. 569-609).
* [18] Comte, F., Merlevède, F. (2005). Super optimal rates for nonparametric density estimation via projection estimators. Stoch. Process Their Appl., 115, 797-826
* [19] Delattre, S., Gloter, A., Yoshida, N. (2020). Rate of Estimation for the Stationary Distribution of Stochastic Damping Hamiltonian Systems with Continuous Observations. arXiv preprint arXiv:2001.10423.
* [20] Delecroix, M. (1980). Sur l’estimation des densités d’un processus stationnaire á temps continu. Publications de l’ISUP, XXV, 1-2, 17-39.
* [21] Kutoyants, Y.A. (1997). Some problems of nonparametric estimation by observations of ergodic diffusion process, Statist. and Probab. Lett. 32, 311–320.
* [22] Kutoyants, Y.A. (1998). Efficient density estimation for ergodic diffusion processes, Stat. Inference Stoch. Process. 1, 131–155.
* [23] Dexheimer, N., Strauch, C., Trottner, L. (2020). Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm. arXiv preprint arXiv:2011.00308.
* [24] Dion, C., Lemler, S. (2020). Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process. Statistical Inference for Stochastic Processes 23, 489-515.
* [25] Ditlevsen, S., Greenwood, P. (2013). The Morris–Lecar neuron model embeds a leaky integrate-and-fire model. Journal of Mathematical Biology 67 239-259.
* [26] Doukhan, P. (2012). Mixing: properties and examples (Vol. 85). Springer Science and Business Media.
* [27] Friedman, A. (1964). Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J.
* [28] Funke, B., Schmisser, E. (2018). Adaptive nonparametric drift estimation of an integrated jump diffusion process. ESAIM: Probability and Statistics 22, 236-260.
* [29] Has’minskii, R. Z. (1980). Stability of differential equations. Germantown, MD: Sijthoff and Noordhoff.
* [30] Höpfner, R., Hoffmann, M., & Löcherbach, E. (2002). Non‐parametric Estimation of the Death Rate in Branching Diffusions. Scandinavian journal of statistics, 29(4), 665-692.
* [31] Juditsky, A., & Nemirovski, A. (2002). On nonparametric tests of positivity/monotonicity/convexity. Annals of statistics, 498-527.
* [32] Kou, S.G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48, 1086-1101.
* [33] Lamberton, D., Pages, G., (2002). Recursive computation of the invariant distribution of a diffusion. Bernoulli 8(3), pp.367-405.
* [34] Leblanc, F. (1997). Density estimation for a class of continuous time processes, Math. Methods Statist. 6, 171–199.
* [35] Lepski, O. V., & Spokoiny, V. G. (1997). Optimal pointwise adaptive methods in nonparametric estimation. The Annals of Statistics, 2512-2546.
* [36] Masuda, H. (2007). Ergodicity and exponential beta - mixing bounds for multidimensional diffusions with jumps. Stochastic processes and their applications, 117(1), 35-56.
* [37] Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, 125-144.
* [38] Nguyen, H. T. (1979). Density estimation in a continuous-time Markov processes. Ann. Statist. 7, 341-348.
* [39] Panloup, F. (2008). Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process. The Annals of Applied Probability 18(2), 379-426.
* [40] Schmisser, E. (2019). Non parametric estimation of the diffusion coefficients of a diffusion with jumps. Stochastic Processes and their Applications 129(12), 5364-5405.
* [41] Strauch, C. (2018). Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. The Annals of Statistics 46(6B), 3451-3480.
* [42] Tsybakov, A. B. (1997). On nonparametric estimation of density level sets. The Annals of Statistics, 25(3), 948-969.
* [43] Tsybakov, A. B. (2008). Introduction to nonparametric estimation. Springer Science and Business Media.
* [44] Veretennikov, A. Yu. (1999). On Castellana-Leadbetter’s condition for diffusion density estimation. Stat. Inference Stoch. Process. 2, 1, 1-9.
* [45] Van Zanten, H. (2001). Rates of convergence and asymptotic normality of kernel estimators for ergodic diffusion processes. Nonparametric Statist. 13 (6), 833-850.
|
Applied Mathematics, Mechanics
A. J. Hutchinson
# Prandtl’s extended mixing length model applied to the two-dimensional
turbulent classical far wake
A. J. Hutchinson1 N. Hale2 K. Born3 and D. P. Mason4 1,3,4School of Computer
Science and Applied Mathematics, University of the Witwatersrand,
Johannesburg, Private Bag 3, Wits 2050, South Africa. 1,4DSI-NRF Centre of
Excellence in Mathematical and Statistical Sciences, South Africa. 2Department
of Mathematical Sciences, Stellenbosch University, Stellenbosch, 7602, South
Africa<EMAIL_ADDRESS>
###### Abstract
Despite its limitations, Prandtl’s mixing length model is widely applied in
modelling turbulent free shear flows. Prandtl’s extended model addresses many
of the shortfalls of the original model, but is not so widely used, in part
due to additional mathematical complexities that arise in its derivation and
implementation. Furthermore, in both models Prandtl neglects the kinematic
viscosity on the basis that it is much smaller in magnitude than the turbulent
viscosity. Recent work has shown that including the kinematic viscosity in the
original model has both mathematical and physical advantages. In the present
work, a novel derivation of the extended model is provided, and it is
demonstrated that similar advantages are again obtained when the kinematic
viscosity is included. Additionally, through the use of scaling techniques,
similarity mean velocity profiles of the extended model are derived, resulting
in a single nonlinear ordinary differential equation that is solved
numerically with a Hermite spectral method. The computed profiles for the
normalised similarity mean velocity and shear stress are compared to
experimental observations and shown to be in excellent agreement.
###### keywords:
Prandtl’s mixing length, turbulent classical wake, eddy viscosity, mean
velocity deficit
## 1 Introduction
Turbulent flows are ubiquitous in both nature and industry [1, 2]. In many
applications, reliable models are needed to predict the behaviour of complex
free shear flows. For example, understanding the development of turbulent
wakes behind wind turbines allows informed decision-making regarding turbine
placement and wake steering control operations [3]. Although the use of
numerical techniques to simulate turbulence, such as large eddy simulations
(LES) and direct numerical simulations (DNS) [4, 5, 6, 7, 8, 9], is
increasing, simple analytic models, such as those investigated in the present
work, continue to be widely used in the context of free shear flows [10, 11,
12].
A common approach to modelling turbulence is to derive equations for the mean
flow variables [13, 14]. The Reynolds decomposition, in which a turbulent flow
is represented as the combination of a mean flow and a fluctuation, is
substituted into the Navier–Stokes equations and the time average is taken.
This procedure gives rise to the Reynolds-averaged Navier–Stokes (RANS)
equations, which contain unknown turbulent stress terms known as the Reynolds
(or apparent) stresses [13]. To calculate the mean flow variables from the
RANS equations, a closure model is needed. Boussinesq proposed the eddy
viscosity approach, which relates the turbulent stresses to the mean rates of
deformation [15, pp. 23-46].
In this work, particular attention is paid to simple analytic models using the
eddy viscosity approach applied to the two-dimensional turbulent classical far
wake. Classical far wake studies for laminar flows date back to the 1930’s and
are largely accredited to Goldstein [16]. A classical wake develops in the
region downstream of a stationary solid body placed in a laminar mainstream
flow. For sufficiently large Reynolds numbers, these laminar flows become
turbulent [14]. In algebraic closure models, the effective viscosity is
defined as the sum of the kinematic viscosity, which is an intrinsic property
of the fluid, and a turbulent or eddy viscosity, which is a characteristic of
the flow. Various closure models may then be used to describe the eddy
viscosity. The simplest closure model, where the eddy viscosity is taken to be
constant, is often used as a baseline to compare against the other models [2,
11, 1]. However, upon comparison with experimental results, the constant eddy
viscosity (CEV) model fails to capture the correct behaviour near the
boundaries of the wake [17].
To improve upon the CEV model, Prandtl introduced the concept of a mixing
length [18], and Prandtl’s mixing length (PML) closure model has since been
used extensively to describe the eddy viscosity. In this model, the eddy
viscosity is written in terms of a mixing length and the gradient of the mean
velocity deficit perpendicular to the axis of the wake. Although an
improvement on the CEV model, the PML model still fails to capture some of the
important physics observed in experimental data [19] and has the nonphysical
property of the eddy viscosity vanishing on the centre line of the wake [18].
Prandtl realised the limitations of his closure model and proposed a
modification, which we refer to as the extended Prandtl mixing length (EPML)
model, to address some of them. Here, two mixing lengths are introduced and
the eddy viscosity is expressed as a function of these mixing lengths and both
the gradient and curvature of the mean velocity deficit perpendicular to the
centre line of the wake [18]. As a result, the eddy viscosity no longer
vanishes on the axis of the wake and one of the limitations of the PML model
is resolved. However, as with the PML model, it is not possible to obtain the
form of the mixing lengths without imposing an additional hypothesis; for
example, that the mixing length is proportional to the width of the wake [20].
Furthermore, although simpler versions of the EPML model have received some
attention [21], and an initial study pertaining to wake flows has been
undertaken [22], the model is seldom used in full due to the additional
mathematical complexity that results from its implementation [12].
Many of the mathematical and physical limitations of the PML and EPML models
arise from one important assumption: the kinematic viscosity can be neglected
in comparison to the turbulent viscosity. Mathematically, this assumption does
not hold on the axis of the wake for the PML model nor at the wake boundaries
for both the PML and EPML models, and as a result, the width of the wake is
underestimated [18].
Prandtl’s original model was recently modified to include the kinematic
viscosity [23]. It was shown that the mixing length can be derived without
imposing any additional hypotheses and that the wake boundary predicted from
this model lies outside of the underestimated boundary obtained from the
original PML model where the kinematic viscosity is neglected. In the present
work, we study the effect of including the kinematic viscosity in the EPML
model. In particular, we provide a detailed derivation of the EPML model and
show that if (and only if) the kinematic viscosity is included then a form for
both of the mixing lengths can be obtained without imposing any additional
assumptions. Furthermore, the derivation is unified in the sense that the PML
model appears as a special case, allowing for a convenient comparison.
Similarity solutions are used to reduce the governing partial differential
equation (PDE) to a second order nonlinear ordinary differential equation
(ODE), which can be solved analytically in one case (PML with no kinematic
viscosity) and numerically in others (PML with kinematic viscosity, and EPML).
The resulting self-similar mean velocity and shear stress profiles are
compared with experimental data from the literature, and the EPML model is
shown to give excellent correspondence.
The practical use of Lie groups in turbulence modelling has been previously
demonstrated, such as the similarity transforms derived by Cantwell [24] for
the two-dimensional unsteady, stream function equation, and the many notable
studies showing significant progress in symmetry methods applied to turbulent
flows [25, 26, 27, 28, 29]. The similarity methods employed in the current
work present yet another example and establish the potential for the general
application of Lie Group Theory in evaluating turbulence models.
The outline of this paper is as follows. In Section 2 we present a
mathematical model for a two-dimensional turbulent classical far wake. In
Section 3 we present a new derivation of the EPML model with kinematic
viscosity included, and in Section 4 identify a scaling solution that reduces
the model to a one-dimensional ODE. In Section 5 we compare the numerically
computed solution of the EPML model to that of various other closure models
and with experimental results from the literature. A short summary is
presented in Section 6.
## 2 Mathematical model for a two-dimensional turbulent classical far wake
Consider a turbulent wake downstream of a slender stationary object which is
referred to as a classical wake. We focus here on symmetric wakes which
develop when the object is aligned with the laminar mainstream flow. We define
the Cartesian coordinate system, $(x,y)$, so that the velocity of the
mainstream flow is $(U,0)$, where $U$ is the constant mainstream speed. The
origin of the coordinate system is placed at the trailing edge of the slender
object. Because the object is slender, any length variation in the
$y$-direction may be neglected, and we approximate its location as a finite
line along a section of the negative $x$-axis. The fluid has constant density
and dynamic (molecular) viscosity, denoted by $\rho$ and $\mu$, respectively.
The velocity components $(u,v)$ are decomposed into mean velocity components,
$(\bar{u},\bar{v})$, and turbulent fluctuations,
$({{u}^{\prime}},{{v}^{\prime}})$, so that $u=\bar{u}+{{u}^{\prime}}$ and
$v=\bar{v}+{{v}^{\prime}}$. The pressure $p$ is also similarly decomposed into
$\bar{p}+p^{\prime}$. We assume that these mean quantities are independent of
time. Flows of this kind are called steady turbulent flows [30, p. 502]. We
consider the far downstream wake which behaves self-similarly.111The
downstream distance at which self-similar behaviour is observed depends on the
type of wake generator [17]. In this region, a mean velocity deficit,
$\bar{w}$, defined by $\bar{u}=U-\bar{w}$, is used to describe the flow and
the inertia terms in the RANS equations can be linearised. A turbulent wake is
illustrated in Figure 1.
Figure 1: A two-dimensional turbulent classical wake behind a slender object
$O$ aligned with the mainstream flow. The laminar mainstream flow has constant
velocity $(U,0)$ and the mean velocity in the wake in the stream-wise
direction is denoted by $\bar{u}$. In the far wake region, we define a mean
velocity deficit $\bar{w}$.
### 2.1 Governing equations
To derive the governing equations for the mean velocity components in the
turbulent far wake, the RANS equations are used as a starting point. These are
obtained by substituting the flow variables
$(\bar{u}+{{u}^{\prime}},\bar{v}+{{v}^{\prime}},\bar{p}+p^{\prime})$ into the
Navier–Stokes equations, and then taking the time average. Although the time
averages of the fluctuations are zero, averages of products of these
fluctuations are nonzero [31, 2].
The $i$-$th$ component of the time-averaged momentum equation for steady
turbulent flows is
$\rho\bar{u}_{j}\dfrac{\partial\bar{u}_{i}}{\partial
x_{j}}=\dfrac{\partial}{\partial
x_{j}}\left[-\bar{p}\delta_{ij}+\mu\left(\dfrac{\partial\bar{u}_{i}}{\partial
x_{j}}+\dfrac{\partial\bar{u}_{j}}{\partial
x_{i}}\right)-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right],\ \ i=1,2,$
(1)
where $u_{1}=u$ and $u_{2}=v$. Three types of stresses arise: the isotropic
stresses, $-\bar{p}\delta_{ij}$, the viscous stresses
$\bar{\tau}_{ij}=\mu\left(\dfrac{\partial\bar{u}_{i}}{\partial
x_{j}}+\dfrac{\partial\bar{u}_{j}}{\partial x_{i}}\right),$ (2)
and the Reynolds or apparent stresses [2]
$\bar{\tau}^{T}_{ij}=-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}.$ (3)
The Reynolds stresses are unknown resulting in more unknowns than equations,
leading to the closure problem. To overcome the closure problem, these
stresses must be specified. In the eddy viscosity model [15], these Reynolds
stresses are incorporated into the viscous stresses by defining a kinematic
eddy viscosity $\nu_{T}=\mu_{T}/\rho$, where $\mu_{T}$ is the dynamic eddy
viscosity, i.e.
$\bar{\tau}^{T}_{ij}=\mu_{T}\left(\dfrac{\partial\bar{u}_{i}}{\partial
x_{j}}+\dfrac{\partial\bar{u}_{j}}{\partial
x_{i}}\right)-\dfrac{1}{3}\rho\overline{u_{k}^{\prime}u_{k}^{\prime}}\delta_{ij}.$
(4)
It is convenient to define an effective kinematic viscosity, $E$, which in
algebraic closure models is taken as the sum of the kinematic viscosity and
the turbulent or eddy kinematic viscosity, i.e. $E=\nu+\nu_{T}$. The form of
$\nu_{T}$ (and hence $E$) depends on the flow geometry and also on the closure
model used. Various closure models — which in general may depend on $x$, $y$,
$\bar{u}$, $\partial\bar{u}/\partial y$, and higher partial derivatives of
$\bar{u}$ — are discussed in Section 3. A detailed derivation of the governing
equations is given in [32]. For completeness, an outline is provided below.
We begin by defining some characteristic quantities. Let $L$ be the length in
the $x$-direction beyond which the reduction in velocity is small enough to be
neglected, and $E_{C}$ the characteristic effective kinematic viscosity. The
turbulent or modified Reynolds number is then
$Re_{T}=\frac{UL}{E_{C}},$ (5)
and is related to the Reynolds number, $Re=UL/\nu$, by
$Re_{T}=Re\dfrac{\nu}{E_{C}}.$ (6)
A turbulent region downstream of the object can develop for large $Re$, but in
order for a turbulent boundary layer to exist, terms of order $1/Re_{T}$ must
be neglected [32]. This is similar to the condition for a laminar boundary
layer to exist, which requires terms of order $1/Re$ to be neglected [31].
Therefore, the turbulent Reynolds number must be large in order for the
boundary layer approximation to be applied. In this work, we restrict our
attention to large $Re_{T}$ and focus on fully developed turbulent boundary
layers.
Introducing the dimensionless variables
$x^{*}=\dfrac{x}{L},\quad
y^{*}=\dfrac{\sqrt{Re_{T}}}{L}y,\quad{{\bar{u}}}^{*}=\frac{{\bar{u}}}{U},\quad{{\bar{v}}}^{*}={\bar{v}}\frac{\sqrt{Re_{T}}}{U},\quad{\bar{p}}^{*}=\frac{p}{\rho
U^{2}},$ (7)
the boundary layer equations for the two-dimensional turbulent classical wake
in terms of the dimensionless mean velocity components are [23]
$\dfrac{\partial\bar{u}^{*}}{\partial{x^{*}}}+\dfrac{\partial\bar{v}^{*}}{\partial{y^{*}}}=0,$
(8)
$\bar{u}^{*}\dfrac{\partial\bar{u}^{*}}{\partial{x^{*}}}+\bar{v}^{*}\dfrac{\partial\bar{u}^{*}}{\partial{y^{*}}}=\dfrac{\partial}{\partial{y^{*}}}\left(E^{*}\dfrac{\partial\bar{u}^{*}}{\partial{y^{*}}}\right),$
(9)
where $E^{*}=E/E_{C}$ is the nondimensionalised effective kinematic viscosity.
The only surviving stress terms are the shear stresses
$\bar{\tau}^{*}_{x^{*}y^{*}}+\bar{\tau}^{*T}_{x^{*}y^{*}}=\dfrac{1}{\sqrt{Re_{T}}}E^{*}\dfrac{\partial\bar{u}^{*}}{\partial{y^{*}}},$
(10)
where $\rho U^{2}$ is used to scale the shear stresses. Note that it is the
$y^{*}$-derivative of this expression that appears in (9), and that this term
is of $\mathcal{O}(1)$.
As a consequence of choosing $U$ to nondimensionalise $\bar{u}$, the
mainstream velocity is now unity in the $x^{*}$-direction. That is, far
downstream from the body,
$\bar{u}^{*}(x^{*},y^{*})=1-\bar{w}^{*}(x^{*},y^{*}),$ (11)
where $\bar{w}^{*}(x^{*},y^{*})$ is the dimensionless mean velocity deficit in
the $x^{*}$-direction. In the far wake region,
$|\bar{w}^{*}|\ll{{\color[rgb]{0,0,0}{1}}}$ and $|\bar{v}^{*}|\ll 1$, so
products and powers of small terms can be neglected, and substituting (11)
into (8) and (9) gives
$-\dfrac{\partial\bar{w}}{\partial x}+\dfrac{\partial\bar{v}}{\partial y}=0,$
(12) $\dfrac{\partial\bar{w}}{\partial x}=\dfrac{\partial}{\partial
y}\left(E\dfrac{\partial\bar{w}}{\partial y}\right),$ (13)
where the stars ∗ have been suppressed for convenience. Similarly, the shear
stress terms (10) become
$\bar{\tau}_{xy}+\bar{\tau}^{T}_{xy}=-\dfrac{1}{\sqrt{Re_{T}}}E\dfrac{\partial\bar{w}}{\partial{y}}.$
(14)
### 2.2 Boundary conditions and the conserved quantity
The momentum and conservation of mass equations, (13) and (12), must be solved
subject to appropriate boundary conditions. Consider first the mean velocity
deficit $\bar{w}$ in the $x$-direction. When using the boundary layer theory
approximation for the turbulent wake region, the effective viscosity is
neglected everywhere except in the shear layer [31]. Boundary conditions are
obtained by ensuring a smooth transition from the wake region to the inviscid
mainstream flow at the boundary of the wake, which we denote by $\pm
y_{b}(x)$. Although a finite wake boundary is nonphysical, we shall see later
that $y_{b}(x)$ may be finite, or infinite, depending on the closure model
used and in particular whether the kinematic viscosity is neglected or not. As
a consequence of studying symmetric wakes, we restrict our attention to the
upper half of the wake only, i.e. $y\geq 0$.
Mainstream matching provides two conditions. First, as $y$ tends to $y_{b}(x)$
the mean velocity $\bar{u}$ tends to unity, and therefore the mean velocity
deficit $\bar{w}$ will tend to zero, i.e.
$\bar{w}(x,y_{b}(x))=0.$ (15)
The second matching condition is given by Hutchinson [33]. At the boundary of
the wake, $y=y_{b}(x)$, the mean vorticity, $\bar{\omega}$, must vanish to
match that of the inviscid mainstream flow. In the boundary layer
approximation we have
$\mathbf{\bar{\omega}}=\dfrac{\partial\bar{u}}{\partial{y}}=-\dfrac{\partial\bar{w}}{\partial{y}},$
(16)
and hence,
$\dfrac{\partial\bar{w}}{\partial y}(x,y_{b}(x))=0.$ (17)
A further condition on $\bar{w}$ is imposed by the fact that the mean velocity
deficit $\bar{w}(x,y)$ is a maximum with respect to $y$ at each point on the
positive $x$-axis, which must hold for wakes symmetric about the $x$ axis.
Therefore,
$\dfrac{\partial\bar{w}}{\partial y}(x,0)=0,\quad x>0.$ (18)
As the governing equations and boundary conditions are homogeneous (and
$y_{b}(x)$ is unknown), an extra condition is required to complete the
solution. This condition comes from a conserved quantity, which for the
classical wake is the drag force [16]. The conserved quantity imposes the
constraint [23]
$\int^{y_{b}(x)}_{0}\bar{w}(x,y)dy=\dfrac{D}{2},$ (19)
where the dimensionless drag force per unit breadth $D$ is independent of $x$.
Consider now $\bar{v}$, the mean velocity component in the $y$-direction. By
the symmetry condition, $\bar{v}$ is zero along the positive $x$-axis, i.e.
$\bar{v}(x,0)=0,\quad x>0.$ (20)
To derive an expression for $\bar{v}$, we substitute (13) into (12) and
integrate with respect to $y$ to obtain
$\bar{v}(x,y)=E\dfrac{\partial\bar{w}}{\partial y}+A(x),$ (21)
where $A(x)$ is an arbitrary function of $x$. If $E$ remains finite at $y=0$
then it follows from (18) and (20) that $A(x)\equiv 0$, and therefore,
$\bar{v}(x,y)=E\dfrac{\partial\bar{w}}{\partial y}.$ (22)
Furthermore, it follows from (17) that if $E$ remains finite as $y\rightarrow
y_{b}(x)$ then $\bar{v}(x,y_{b}(x))=0$. Hence, if the effective viscosity is
finite both on the centre line and the wake boundary, then this theoretical
model predicts that there is no fluid entrainment. Large scale turbulent
motions, which are not a feature of this model, are responsible for
entrainment [24]. The importance of $\bar{v}$ can be seen from Equation (14)
and Equation (22), in that we can write
$-\dfrac{1}{\sqrt{Re_{T}}}\bar{v}(x,y)=-\dfrac{1}{\sqrt{Re_{T}}}E\dfrac{\partial\bar{w}}{\partial
y}=\bar{\tau}_{xy}+\bar{\tau}^{T}_{xy},$ (23)
and find the shear stress can be determined by solving for $\bar{v}$.
## 3 Derivation of the eddy viscosity for the EPML model
Application of the RANS equations to turbulent wake flows requires an
appropriate closure model to complete the system (12)–(13) [15]. In algebraic
closure models, the effective kinematic viscosity is expressed as the sum of
the kinematic viscosity $\nu$ and the kinematic eddy viscosity $\nu_{T}$ [2]:
$E=\frac{\mu+\mu_{T}}{\rho}=\nu+\nu_{T}.$ (24)
Introducing nondimensional variables,
$E^{*}=\frac{E}{E_{C}},\quad\nu_{T}^{*}=\dfrac{\nu_{T}}{\nu_{TC}},$ (25)
where $E_{C}=\nu+\nu_{TC}$ and $\nu_{TC}$ is the characteristic kinematic eddy
viscosity, the dimensionless effective viscosity, $E^{*}$, is given by
$E^{*}=\dfrac{\nu}{\nu+\nu_{TC}}+\dfrac{\nu_{TC}}{\nu+\nu_{TC}}\nu^{*}_{T}.$
(26)
Previous work investigates the class of models that can be described by a
kinematic eddy viscosity of the form
$\nu_{T}=\nu_{T}\left(x,y,\partial\bar{u}/\partial y\right)$ [32, 34, 23, 35].
In this paper, we extend the work conducted in [23] by considering effective
kinematic viscosities of the form
$E=E\left(x,\dfrac{\partial\bar{u}}{\partial{y}},\dfrac{\partial^{2}\bar{u}}{\partial{y}^{2}}\right).$
(27)
This form is convenient not only in that it extends the range of closure
models that can be applied, but it also incorporates the CEV model, the PML
model, the EPML model, and any variations thereof, as special cases, allowing
for a unified derivation and a direct comparison.
Note that the boundary conditions (15), (17), and (18) are all independent of
the closure model used to define the effective viscosity. The only condition
that must be imposed directly on the closure models is that $\nu_{T}$ is
finite at $y=0$ and at $y=y_{b}(x)$, which shows that no fluid entrainment
occurs. Because a finite-valued effective viscosity on the entire domain is
required to describe a turbulent flow — large effective viscosities would
decrease the turbulent Reynolds number $Re_{T}$, and the flow would no longer
satisfy the condition for the existence of a turbulent boundary layer — this
seems reasonable.
For free shear flows, a viscous superlayer separates the turbulent flow region
from the mainstream flow [2]. However, this layer is thin compared to the
boundary layer thickness $\delta$ and so the wake boundary is essentially a
well-defined interface. Since the wake boundary is where the turbulent wake
merges with the laminar mainstream flow, the eddy viscosity $\nu_{T}$ should
vanish there. Consider then the momentum equation (13) with an effective
viscosity of the form (26). If $\nu_{T}\rightarrow 0$ as $y\rightarrow\pm
y_{b}(x)$ then for $y$ sufficiently close to $\pm y_{b}(x)$, $\nu\gg\nu_{T}$.
Hence, as $y\rightarrow\pm y_{b}(x)$, Equation (13) becomes
$\dfrac{\partial\bar{w}}{\partial
x}={\dfrac{\nu}{E_{C}}}\dfrac{\partial^{2}\bar{w}}{\partial y^{2}},$ (28)
and the exponential solution for a laminar wake will apply [16]. Therefore,
$\bar{w}$ can not reach zero for any finite value of $y_{b}$ and we must have
that $y_{b}(x)=\infty$. A finite wake can therefore only exist if $\nu=0$
everywhere in the approximation. We shall see later that the condition
$\nu_{T}\rightarrow 0$ as $y\rightarrow\pm y_{b}(x)$ is satisfied for the PML
and EPML models, but not for the CEV model.
### 3.1 Prandtl’s extended mixing length model
In an attempt to improve upon the accuracy of the results obtained when the
constant eddy viscosity model is applied, Prandtl introduced a mixing length
model (PML).222A detailed description of the origins of the ideas pertaining
to the PML model can be found in [36]. Prandtl then extended this model to
address some of its shortcomings [18]. However, to the best of the authors’
knowledge, the extended model (EPML) was not accompanied by a detailed
explanation or derivation. We now present a derivation of the EPML model,
which relies on an adapted derivation of the PML model for parallel mean flow
as provided by Schlichting and Schlichting & Gersten [31, 30].
Consider a turbulent flow with parallel mean flow as shown in Equation (2). In
turbulent flows, fluid particles coalesce forming lumps of fluid, which then
travel as a whole in both the $x$ and $y$ directions. These lumps of fluid
remain intact and retain their momentum, travelling some distance before
mixing in with the surrounding fluid once again. In parallel laminar flows,
the $y$-component of the velocity is zero and so fluid elements cannot travel
in the transverse direction. However, in parallel turbulent flows, turbulent
fluctuations displace fluid lumps in the transverse direction. These
displacements are random. The standard deviation, (with a multiplicative
constant that can be absorbed), of these displacements is known as the mixing
length, from which this model derives its name.
In the turbulent flow shown in Equation (2), the parallel mean velocity is
denoted by $(\bar{u}(y),0)$ and the random velocity fluctuations by
$(u^{\prime},v^{\prime})$. The fluid velocity is then
$(u(x,y,t),v(x,y,t))=(\bar{u}(y)+u^{\prime}(x,y,t),v^{\prime}(x,y,t)).$ (29)
Consider a mean flow where $\mathrm{d}\bar{u}/\mathrm{d}y>0$. Fluid lumps are
displaced by the turbulent fluctuations from position $y$ to position
$y+\ell^{\prime}$, whilst retaining their original momentum in the
$x$-direction.333In [31], turbulent fluctuations are caused by fluid lumps
arriving at a layer from layers above and below, and in [30], fluctuations
cause fluid lumps to leave a layer and move into a neighbouring layer. In
general, the random variable $\ell^{\prime}$ is a function of $x$ and $y$ and
time $t$, and can take on both positive and negative values. For the purposes
of clear illustration we will consider positive $\ell^{\prime}$ values, and
then later relax this condition. The size of $\ell^{\prime}$ gives an
indication as to the strength of the fluctuations that resulted in a fluid
lump moving from $y$ to $y+\ell^{\prime}$. Because the fluid lumps retain
their momentum and hence velocity in the $x$-direction while being displaced,
they arrive at the new layer $y+\ell^{\prime}$ with a lower velocity of
$\bar{u}(y)$ than that of the surrounding fluid, $\bar{u}(y+\ell^{\prime})$.
Figure 2: Explanation of the mixing length concept (diagram adapted from
Schlichting [30, p 539]). Fluid lumps at a position $y$ are displaced by
random turbulent fluctuations to position $y+{{\ell}^{\prime}}$. These lumps
retain their momentum, and hence velocity. The mean velocity of the fluid is
denoted by $(\bar{u}(y),0)$ and the turbulent velocity fluctuations by
$({{u}^{\prime}},{{v}^{\prime}})$. The difference in mean velocity of the
fluid lumps and that of the surrounding fluid is used to estimate the strength
of the turbulent fluctuations. Note that, as a result of conservation of mass,
${{u}^{\prime}}{{v}^{\prime}}<0$.
To estimate the strength of these fluctuations, we use the difference between
the velocity of the surrounding fluid and the newly arrived fluid lump:
$\triangle u=\bar{u}(y+\ell^{\prime})-\bar{u}(y).$ (30)
Assuming that ${{\ell}^{\prime}}$ is small, expanding
$\bar{u}(y+\ell^{\prime})$ as a Taylor series gives
$\triangle
u={{\ell}^{\prime}}\dfrac{d\bar{u}}{dy}+\dfrac{{{\ell}^{\prime}}^{2}}{2}\dfrac{d^{2}\bar{u}}{dy^{2}}+\dfrac{{{\ell}^{\prime}}^{3}}{6}\dfrac{d^{3}\bar{u}}{dy^{3}}+\dfrac{{{\ell}^{\prime}}^{4}}{24}\dfrac{d^{4}\bar{u}}{dy^{4}}+\mathcal{O}({{{\ell}^{\prime}}^{5}}).$
(31)
In the PML model, only the first term in Equation (31) is retained. However,
for points where $d\bar{u}/dy=0$, the first term in (31) vanishes, and
$\triangle u$ is zero if higher order terms are neglected. In the EPML model,
points at which $d\bar{u}/dy=0$ are taken into account, and additional terms
must be included. In particular, terms up to and including
${\cal{O}}({{{\ell}^{\prime}}^{4}})$ are retained in the EPML model.
Before we can express the fluctuations ${{u}^{\prime}}$ and ${{v}^{\prime}}$
in terms of $\triangle u$, we must first determine their signs and order of
magnitude. In order for a fluid lump to be displaced in the positive
$y$-direction, the velocity fluctuation, ${{v}^{\prime}}$, must be positive.
Furthermore, because a fluid lump initially at $y$ which is displaced to
$y+\ell^{\prime}$ has a lower velocity than that of its surroundings,
${{u}^{\prime}}$ must be negative (see Figure 2). Therefore, ${{u}^{\prime}}$
and ${{v}^{\prime}}$ must have opposite signs. For small-scale turbulent
motions (the type which are considered here), it is reasonable to assume that
characteristic length scales in the $x$ and $y$ directions are
comparable.444This assumption may not hold for large-scale motions. By
comparing the magnitude of the terms in the conservation of mass equation,
$\frac{\partial{{u}^{\prime}}}{\partial
x}+\frac{\partial{{v}^{\prime}}}{\partial y}=0,$ (32)
we find that the fluctuations in the $x$ and $y$ directions must also be of
the same magnitude, i.e. $|{{v}^{\prime}}|\sim|{{u}^{\prime}}|$ [31]. We may
therefore write
${{v}^{\prime}}=-c{{u}^{\prime}}>0,$ (33)
where $c$ is a positive constant of order $1$.
Recalling that, for the moment, we are considering $\ell^{\prime}>0$ and
$d\bar{u}/dy>0$, the leading order term in (31) is positive and hence, for
sufficiently small $\ell^{\prime}$ we have $\triangle u>0$. Now, since
$\triangle u$ is an estimate of the difference in mean velocity between a
fluid lump and its surroundings (see (30)), we assume
${{u}^{\prime}}=-\triangle u,$ (34)
where the minus comes from comparing the signs of $u^{\prime}$ and $\triangle
u$ as described above.
Using the Reynolds decomposition, we may write the $x$-component of the
velocity, $u$, at position $(x,y)$ and time $t$ as
$u(x,y,t)=\bar{u}(y)+{{u}^{\prime}}=\bar{u}(y)-\ell^{\prime}\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}-\dfrac{{{\ell}^{\prime}}^{2}}{2}\dfrac{d^{2}\bar{u}}{dy^{2}}-\dfrac{{{\ell}^{\prime}}^{3}}{6}\dfrac{d^{3}\bar{u}}{dy^{3}}-\dfrac{{{\ell}^{\prime}}^{4}}{24}\dfrac{d^{4}\bar{u}}{dy^{4}},$
(35)
and the $y$-component as
$v(x,y,t)=v^{\prime}=-c{{u}^{\prime}}=+c\left({{\ell}^{\prime}}\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}+\dfrac{{{\ell}^{\prime}}^{2}}{2}\dfrac{d^{2}\bar{u}}{dy^{2}}+\dfrac{{{\ell}^{\prime}}^{3}}{6}\dfrac{d^{3}\bar{u}}{dy^{3}}+\dfrac{{{{\ell}^{\prime}}^{4}}}{24}\dfrac{d^{4}\bar{u}}{dy^{4}}\right).$
(36)
Allowing ${{\ell}^{\prime}}$ to now take on both positive and negative values
and averaging over time (see (3)), the Reynolds shear stress,
$\bar{\tau}_{xy}^{T}$, is given by
$\bar{\tau}^{T}_{xy}=-\rho\overline{{{u}^{\prime}}{{v}^{\prime}}}=\rho
c\left[\overline{{{{\ell}^{\prime}}^{2}}}\left(\dfrac{d\bar{u}}{dy}\right)^{2}+\overline{{{{\ell}^{\prime}}^{3}}}\dfrac{d\bar{u}}{dy}\dfrac{d^{2}\bar{u}}{dy^{2}}+\dfrac{\overline{{{{\ell}^{\prime}}^{4}}}}{4}\left(\dfrac{d^{2}\bar{u}}{dy^{2}}\right)^{2}+\dfrac{\overline{{{{\ell}^{\prime}}^{4}}}}{3}\dfrac{d\bar{u}}{dy}\dfrac{d^{3}\bar{u}}{dy^{3}}\right]+\mathcal{O}(\overline{{{\ell}^{\prime}}^{5}}).$
(37)
The EPML model specifically aims to improve on the approximation used by the
PML model by taking into account points at which $d\bar{u}/dy$ vanishes.
Therefore, we retain only the lowest order term and those higher order terms
that do not depend on $d\bar{u}/dy$, giving
$\bar{\tau}^{T}_{xy}=-\rho\overline{{{u}^{\prime}}{{v}^{\prime}}}=\rho
c\left[\overline{{{{\ell}^{\prime}}^{2}}}\left(\dfrac{d\bar{u}}{dy}\right)^{2}+\dfrac{\overline{{{\ell}^{\prime}}^{4}}}{4}\left(\dfrac{d^{2}\bar{u}}{dy^{2}}\right)^{2}\right]+\mathcal{O}(\overline{{{\ell}^{\prime}}^{5}}).$
(38)
Setting $\ell_{1}^{2}=c\overline{{{{\ell}^{\prime}}^{2}}}$,
$\ell_{2}^{2}=\overline{{{\ell}^{\prime}}^{4}}/\overline{{{\ell}^{\prime}}^{2}}$,
and neglecting terms of $\mathcal{O}(\overline{{{\ell}^{\prime}}^{5}})$ leads
to
$\bar{\tau}^{T}_{xy}=\rho\ell_{1}^{2}\left[\left(\dfrac{d\bar{u}}{dy}\right)^{2}+\dfrac{\ell_{2}^{2}}{4}\left(\dfrac{d^{2}\bar{u}}{dy^{2}}\right)^{2}\right],$
(39)
where the first mixing length, $\ell_{1}$, is simply a constant multiple of
the standard deviation, and the second mixing length, $\ell_{2}$, is the
kurtosis.
We will consider flows where $d\bar{u}/dy>0$ almost everywhere, except at a
finite number of points where $d\bar{u}/dy=0$. Therefore, over the majority of
the flow domain,
$\left(\dfrac{d\bar{u}}{dy}\right)^{2}\gg\ell_{2}^{2}\left(\dfrac{d^{2}\bar{u}}{dy^{2}}\right)^{2},$
(40)
which is a reasonable assumption when $\ell_{2}$ is small compared to the
boundary layer thickness, $\ell_{2}/\delta\ll 1$. A reformulated version of
the EPML model using this concept has been applied to turbulent pipe flows
[12] and flows in circular tubes [21]. However, to obtain Prandtl’s version of
the extended mixing length model, which can be written in the form [18],
$\bar{\tau}^{T}_{xy}=\rho\ell_{1}^{2}\sqrt{\left(\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}\right)^{2}+\dfrac{\ell_{2}^{2}}{2}\left(\dfrac{\mathrm{d}^{2}\bar{u}}{\mathrm{d}y^{2}}\right)^{2}}\dfrac{d\bar{u}}{dy},$
(41)
we simply note that when (40) holds, substituting the binomial approximation
$\sqrt{\left(\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}\right)^{2}+\dfrac{\ell_{2}^{2}}{2}\left(\dfrac{\mathrm{d}^{2}\bar{u}}{\mathrm{d}y^{2}}\right)^{2}}\approx\dfrac{d\bar{u}}{dy}+\dfrac{\ell_{2}^{2}}{4\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}}\left(\dfrac{\mathrm{d}^{2}\bar{u}}{\mathrm{d}y^{2}}\right)^{2},$
(42)
into (41), results in (39).
Looking now at Equation (4), the normal stresses, $\bar{\tau}^{T}_{xx}$ and
$\bar{\tau}^{T}_{yy}$, vanish (a consequence of the mean parallel flow), and
so the only remaining term is the shear stress
$\bar{\tau}^{T}_{xy}=\mu_{T}\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}.$ (43)
Comparing this expression to (41) leads to
$\mu_{T}=\rho\ell_{1}^{2}\sqrt{\left(\dfrac{\mathrm{d}\bar{u}}{\mathrm{d}y}\right)^{2}+\ell_{2}^{2}\left(\dfrac{\mathrm{d}^{2}\bar{u}}{\mathrm{d}y^{2}}\right)^{2}},$
(44)
where the constant factor of $1/2$ has been absorbed into $\ell_{2}$ for
convenience. This result also holds for the case where $d\bar{u}/dy<0$.
In boundary layer flows,
$\left|\dfrac{\partial\bar{v}}{\partial
x}\right|\ll\left|\dfrac{\partial\bar{u}}{\partial y}\right|,$ (45)
and the Reynolds stress, $\bar{\tau}^{T}_{xy}$, depends only on
$\partial\bar{u}/\partial y$ as shown in Equation (10). Therefore, the result
in Equation (44) can be extended to boundary layers. For wakes, we replace the
ordinary derivative $\mathrm{d}\bar{u}/\mathrm{d}y$ with
$-\partial\bar{w}/\partial y$:
$\nu_{T}=\dfrac{\mu_{T}}{\rho}=\ell_{1}^{2}\sqrt{\left(\dfrac{\partial\bar{w}}{\partial{y}}\right)^{2}+\ell_{2}^{2}\left(\dfrac{\partial^{2}\bar{w}}{\partial{y}^{2}}\right)^{2}}.$
(46)
In Prandtl’s mixing length model for free shear flows, the mixing length is a
function of $x$ alone. Define dimensionless lengths
$\ell_{1}^{*}(x^{*})=\dfrac{\ell_{1}(x)}{\delta},\ \ \
\ell_{2}^{*}(x^{*})=\dfrac{\ell_{2}(x)}{\delta},$ (47)
where $\delta=L/\sqrt{Re_{T}}$ is the boundary layer thickness. In
dimensionless form, Equation (46) is
$\nu_{T}^{*}={\ell_{1}^{*}}^{2}(x^{*})\sqrt{\left(\dfrac{\partial\bar{w}^{*}}{\partial{y^{*}}}\right)^{2}+{\ell_{2}^{*}}^{2}(x^{*})\left(\dfrac{\partial^{2}\bar{w}^{*}}{\partial{y^{*}}^{2}}\right)^{2}},$
(48)
and the effective viscosity, Equation (26), is
$E^{*}=\dfrac{\nu}{\nu+\nu_{TC}}+\dfrac{\nu_{TC}}{\nu+\nu_{TC}}{\ell_{1}^{*}}^{2}(x^{*})\sqrt{\left(\dfrac{\partial\bar{w}^{*}}{\partial{y^{*}}}\right)^{2}+{\ell_{2}^{*}}^{2}(x^{*})\left(\dfrac{\partial^{2}\bar{w}^{*}}{\partial{y^{*}}^{2}}\right)^{2}},$
(49)
where $\nu_{TC}=U\delta$.
To simplify the notation, let
$\beta=\dfrac{\nu}{\nu+\nu_{TC}},\quad\alpha=\dfrac{\nu_{TC}}{\nu+\nu_{TC}}.$
(50)
In terms of (50), Equation (49) is
$E\left(x,\dfrac{\partial
w}{\partial{y}},\dfrac{\partial^{2}w}{\partial{y}^{2}}\right)=\beta+\alpha\ell_{1}^{2}(x)\left[\left(\dfrac{\partial
w}{\partial{y}}\right)^{2}+\ell_{2}^{2}(x)\left(\dfrac{\partial^{2}w}{\partial{y}^{2}}\right)^{2}\right]^{1/2},$
(51)
where the bars and stars have been suppressed for further convenience.
Equation (13) with the effective viscosity defined by Equation (51) is then
$\dfrac{\partial
w}{\partial{x}}=\dfrac{\partial}{\partial{y}}\left[\beta\dfrac{\partial
w}{\partial{y}}+\alpha\ell_{1}^{2}(x)\left[\left(\dfrac{\partial
w}{\partial{y}}\right)^{2}+\ell_{2}^{2}(x)\left(\dfrac{\partial^{2}w}{\partial{y}^{2}}\right)^{2}\right]^{1/2}\dfrac{\partial
w}{\partial{y}}\right].$ (52)
The PML model is a special case of the extended model that is obtained when
$\beta$ is set to zero and the term containing the second mixing length
$\ell_{2}$ is neglected. Prandtl excluded the kinematic viscosity on the basis
that it is much smaller than the turbulent viscosity. However, from (17) and
(18), the eddy viscosity vanishes at the wake boundary and on the wake axis,
and this is no longer true. It was shown in [32] that including the kinematic
viscosity leads to an infinite wake boundary, and the form of the mixing
length can be obtained without imposing additional restrictions. Whilst
idealised boundary conditions on unbounded domains may seem a long way from
realistic experiments, we shall see in Section 5 that in practice the computed
velocity profiles decay rapidly – exponentially in some cases – indicating
that the assumption of an unbounded domain has little effect on the behaviour
model.
In the next section, scaling solutions will be investigated for the case where
the kinematic viscosity is neglected ($\beta=0$), and for when it is included
($\beta>0$) for the EPML model. In Section 5 we show that the inclusion of the
second mixing length in the EPML model can significantly improve upon the PML
model when comparing the resulting mean velocity profiles to experimental
data.
## 4 Scaling solutions
We now investigate when the PDE (52) is invariant under the scaling
transformation
$x=\lambda^{a}\bar{x},\quad y=\lambda^{b}\bar{y},\quad
w=\lambda^{c}\bar{w},\quad\ell_{1}=\lambda^{d}\bar{\ell}_{1},\quad\ell_{2}=\lambda^{e}\bar{\ell}_{2},$
(53)
and thereby reduce Equation (52) to an ODE. In the scaling transformation
(53), only the ratios of $a,b,c,d,$ and $e$ need to be determined. If $a\neq
0$, we therefore need to determine only the four ratios $b/a,c/a,d/a,$ and
$e/a$ and only four conditions need to be found. Without loss of generality,
we set $a=1$. The boundary conditions are invariant under the transformation
(53) because they are homogeneous. Therefore, the conditions for invariance
are obtained from the equation itself, (52), and the conserved quantity.
We first examine the case in which the kinematic viscosity $\nu$ is neglected
as considered by Prandtl. We will see that only three conditions are obtained
and to determine the scaling transformation completely, one additional
condition needs to be imposed. We will impose Prandtl’s hypothesis [20]. We
then extend this work and find the scaling transformation when $\nu$ is
included in Equation (52). We find that four conditions are obtained and the
scaling transformation is completely determined. An additional condition is
not required. Analytical and numerical solutions of the ODE obtained in the
reduction, subject to the boundary conditions and the conserved quantity, are
derived.
### 4.1 Extended Prandtl model without kinematic viscosity
In Prandtl’s original version of the extended mixing length model, the
kinematic viscosity is neglected. The eddy viscosity for this model is a
special case of Equation (51) with $\beta=0$. Equation (52) becomes
$\dfrac{\partial
w}{\partial{x}}=\alpha\ell_{1}^{2}(x)\dfrac{\partial}{\partial{y}}\left[\left[\left(\dfrac{\partial
w}{\partial{y}}\right)^{2}+\ell_{2}^{2}(x)\left(\dfrac{\partial^{2}w}{\partial{y}^{2}}\right)^{2}\right]^{1/2}\dfrac{\partial
w}{\partial{y}}\right].$ (54)
In terms of the scalings defined in Equation (53) with $a=1$, Equation (54)
becomes
$\dfrac{\partial\bar{w}}{\partial{\bar{x}}}=\lambda^{1-3b+c+2d}\alpha\bar{\ell}_{1}^{2}\dfrac{\partial}{\partial{\bar{y}}}\left[\left[\left(\dfrac{\partial\bar{w}}{\partial{\bar{y}}}\right)^{2}+\lambda^{-2b+2e}\bar{\ell}_{2}^{2}\left(\dfrac{\partial^{2}\bar{w}}{\partial{\bar{y}}^{2}}\right)^{2}\right]^{1/2}\dfrac{\partial\bar{w}}{\partial{\bar{y}}}\right],$
(55)
and the conserved quantity (19) is given by
$\dfrac{D}{2}=\lambda^{c+b}\int_{0}^{\bar{y}_{b}}\bar{w}(\bar{x},\bar{y})d\bar{y},$
(56)
where $\bar{y}_{b}(\bar{x})=y_{b}(x)/\lambda^{b}$. Now, Equations (55) and
(56) are invariant under the transformation (53) provided
$1-3b+c+2d=0,\ \ b-e=0,\ \ b+c=0,$ (57)
that is, provided
$c=-b,\quad d=\dfrac{1}{2}\left(4b-1\right),\quad e=b.$ (58)
Only three conditions have been obtained relating the four unknowns. One
further condition is required. A condition that we will consider is Prandtl’s
hypothesis which states that the mixing length is proportional to the width of
the boundary layer [20]. In addition to Prandtl’s hypothesis, we consider two
other cases.
Case $1$: $\ell_{1}(x)\propto y_{b}(x)$. Imposing Prandtl’s hypothesis on the
first mixing length $\ell_{1}(x)$ gives
$\ell_{1}(x)=k_{1}y_{b}(x),$ (59)
where $k_{1}$ is a constant of proportionality. In terms of the scaled
variables,
$\bar{\ell}_{1}=\lambda^{b-d}k_{1}\bar{y}_{b},$ (60)
and we see that $d=b$ for invariance. Therefore, this condition, combined with
the ones in Equation (58) results in
$b=\dfrac{1}{2},\quad c=-\dfrac{1}{2},\quad d=\dfrac{1}{2},\quad
e=\dfrac{1}{2},$ (61)
and the scaling transformation (53), which is now completely determined, takes
the form
$x=\lambda\bar{x},\quad y=\lambda^{1/2}\bar{y},\quad
w=\lambda^{-1/2}\bar{w},\quad\ell_{1}=\lambda^{1/2}\bar{\ell}_{1},\quad\ell_{2}=\lambda^{1/2}\bar{\ell}_{2}.$
(62)
Case $2$: $\ell_{2}(x)\propto y_{b}(x)$. Now suppose that we apply Prandtl’s
hypothesis on the second mixing length, $\ell_{2}(x)$. Then
$\ell_{2}(x)=k_{2}y_{b}(x),$ (63)
where $k_{2}$ is a constant of proportionality. In terms of the scaled
variables we have
$\bar{\ell}_{2}=\lambda^{b-e}k_{2}\bar{y}_{b}.$ (64)
However, from Equation (64) we find $b=e$ for invariance which is a repeat
condition and we are still short of one condition. We see that Prandtl’s
hypothesis must be applied to the first mixing length.
Case $3$: $\ell_{1}(x)\propto\ell_{2}(x)$. In this case, we have
$\ell_{1}(x)=k_{3}\ell_{2}(x),$ (65)
where $k_{3}$ is a constant of proportionality. In terms of the scaled
variables, Equation (65) becomes
$\bar{\ell}_{1}=\lambda^{e-d}k_{3}\bar{\ell}_{2}.$ (66)
From Equation (66) we see that $d=e$ for invariance and so using (58) we have
$d=b$ which results in the same scaling as in Equation (62).
Prandtl’s hypothesis is satisfied when the mixing lengths are assumed to be
proportional. Thus, Prandtl’s hypothesis is verified for a special case of a
two-dimensional turbulent classical wake described by Prandtl’s extended model
for the eddy viscosity in which the two mixing lengths are proportional.
### 4.2 Extended Prandtl model with kinematic viscosity
In this section we consider an improved version of the EPML model where the
kinematic viscosity is included. Here, the two mixing lengths are taken to be
distinct and we will show that for an invariant solution to exist the two
mixing lengths must be proportional.
Consider again a scaling transformation given by Equation (53). Equation (52)
transforms to
$\dfrac{\partial\bar{w}}{\partial{\bar{x}}}=\dfrac{\partial}{\partial{\bar{y}}}\left[\lambda^{1-2b}\beta\dfrac{\partial\bar{w}}{\partial{\bar{y}}}+\lambda^{1-3b+c+2d}\alpha\bar{\ell}_{1}^{2}\left[\left(\dfrac{\partial\bar{w}}{\partial{\bar{y}}}\right)^{2}+\lambda^{-2b+2e}\bar{\ell}_{2}^{2}\left(\dfrac{\partial^{2}\bar{w}}{\partial{\bar{y}}^{2}}\right)^{2}\right]^{1/2}\dfrac{\partial\bar{w}}{\partial{\bar{y}}}\right].$
(67)
Thus Equation (67) and the conserved quantity (56) are invariant provided
$1-2b=0,\ \ 1-3b+c+2d=0,\ \ b-e=0,\ \ b+c=0.$ (68)
The additional condition, $1-2b=0$ is obtained. Hence, we again obtain the
result in Equation (61). Including the kinematic viscosity leads to the same
result, (61), without the requirement that Prandtl’s hypothesis be imposed as
was done in Case 1 or the assumption that $\ell_{1}(x)\propto\ell_{2}(x)$ as
was done in Case 3.
The additional condition that needed to be imposed for $\nu=0$ to obtain the
complete scaling solution can be formulated alternatively as follows: The
solution for $\nu\neq 0$ must match with the solution for $\nu=0$ in the limit
as $\nu\rightarrow 0$. This again gives $b=1/2$. We can use this to replace
Prandtl’s hypothesis as these conditions are equivalent. We now see that
because the original extended model with $\beta=0$ is a special case of the
extended model with $\beta\neq 0$, we need only to consider the scaling
solution in Equation (62) and reduce the PDE to an ODE. By using the method
described in the text [37], it can be shown that the invariant solution under
the scaling transformation (53) with $a=1$ is of the form
$w(x,y)=x^{c}F(\xi),\ \ \ \ \xi=\dfrac{y}{x^{b}},$ (69)
$\ell_{1}(x)=K_{1}x^{d},\ \ \ell_{2}(x)=K_{2}x^{e},$ (70)
where $F(\xi)$ is an arbitrary function to be determined, and $K_{1}$ and
$K_{2}$ are constants. Hence, from (61), the invariant solution is
$w(x,y)=\dfrac{F(\xi)}{\sqrt{x}},\ \ \ \ \xi=\dfrac{y}{\sqrt{x}},$ (71)
$\ell_{1}(x)=K_{1}\sqrt{x},\ \ \ell_{2}(x)=K_{2}\sqrt{x}.$ (72)
This applies for Case 1 and Case 3 of the extended model with $\beta=0$ and
for the extended model with $\beta\neq 0$. It is readily seen that
$\ell_{1}\propto\ell_{2}$ and therefore by including the kinematic viscosity,
the two mixing lengths are found to be proportional without any additional
hypothesis. From Equation (71), we see that the invariant solution only
applies for finite $x$. As $x\rightarrow\infty$, the velocity deficit tends to
zero and the flow reverts to the undisturbed mainstream flow.
Expressing Equation (52) in terms of the similarity variables (71)–(72) gives
$\dfrac{\mathrm{d}}{\mathrm{d}\xi}\left[\beta\dfrac{\mathrm{d}F}{\mathrm{d}\xi}+\alpha
K_{1}^{2}\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\left[\left(\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\right)^{2}+K_{2}^{2}\left(\dfrac{\mathrm{d}^{2}F}{\mathrm{d}\xi^{2}}\right)^{2}\right]^{1/2}\right]+\dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}\xi}\left[\xi
F\right]=0.$ (73)
Note that the $y$-component of the velocity, Equation (22), in terms of the
similarity variables (71) - (72) is
$v(x,y)=\dfrac{1}{x}\left(\beta\dfrac{\mathrm{d}F}{\mathrm{d}\xi}+\alpha
K_{1}^{2}\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\left[\left(\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\right)^{2}+K_{2}^{2}\left(\dfrac{\mathrm{d}^{2}F}{\mathrm{d}\xi^{2}}\right)^{2}\right]^{1/2}\right),$
(74)
which can be written in the form
$v(x,y)=\dfrac{G(\xi)}{x},$ (75)
where
$G(\xi)=\beta\dfrac{\mathrm{d}F}{\mathrm{d}\xi}+\alpha
K_{1}^{2}\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\left[\left(\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\right)^{2}+K_{2}^{2}\left(\dfrac{\mathrm{d}^{2}F}{\mathrm{d}\xi^{2}}\right)^{2}\right]^{1/2}.$
(76)
The conserved quantity in Equation (19) transforms to
$\int_{0}^{\dfrac{y_{b}(x)}{\sqrt{x}}}Fd\xi=\dfrac{D}{2}.$ (77)
Now, $D$ is a constant, independent of $x$, provided
$\dfrac{y_{b}(x)}{\sqrt{x}}=\text{constant}=\xi_{b}.$ (78)
Hence, the boundary of the wake is given by
$y_{b}(x)=\xi_{b}\sqrt{x},\quad\xi_{b}=\text{constant}.$ (79)
If the boundary of the wake extends to infinity at some finite distance
downstream, then $\xi_{b}=\infty$. For $x>0$, the boundary conditions in (18)
(15), and (17) are now given in terms of the similarity variables by
$F(\xi_{b})=0,\ \ \dfrac{\mathrm{d}F}{\mathrm{d}\xi}(\xi_{b})=0,\ \
\dfrac{\mathrm{d}F}{\mathrm{d}\xi}(0)=0,$ (80)
respectively.
Equation (73) is an example of the double reduction theorem of Sjöberg [38]
that if a PDE is reduced to an ODE by a symmetry associated with a conserved
vector of the PDE, then the ODE can be integrated at least once. Integrating
Equation (73) once yields
$\xi F+2\beta\dfrac{\mathrm{d}F}{\mathrm{d}\xi}+2\alpha
K_{1}^{2}\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\left[\left(\dfrac{\mathrm{d}F}{\mathrm{d}\xi}\right)^{2}+K_{2}^{2}\left(\dfrac{\mathrm{d}^{2}F}{\mathrm{d}\xi^{2}}\right)^{2}\right]^{1/2}=a_{1},$
(81)
where $a_{1}$ is a constant of integration. Using the third boundary condition
in (80), we find $a_{1}=0$.
From Equation (81) with $a_{1}=0$, Equation (76) can be written as
$G(\xi)=-\dfrac{1}{2}\xi F(\xi),$ (82)
and we see from (75) that $G$ describes the similarity profile for $v(x,y)$.
From Equation (23),
$-\dfrac{1}{\sqrt{Re_{T}}}\dfrac{G(\xi)}{x}=\bar{\tau}_{xy}+\bar{\tau}^{T}_{xy},$
(83)
and so $-G$ also gives the similarity profile for the shear stresses scaled by
$1/\sqrt{Re_{T}}$.
To recover characteristic lengths, we introduce further scalings
$F=A\bar{F},\quad\xi=B\bar{\xi},\quad G=AB\bar{G},$ (84)
so that (81) and (77) become
$\bar{\xi}\;\bar{F}+\frac{2\beta}{B^{2}}\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}+\frac{2\alpha
AK_{1}^{2}}{B^{3}}\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}\left[\left(\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}\right)^{2}+\frac{K_{2}^{2}}{B^{2}}\left(\dfrac{\mathrm{d}^{2}\bar{F}}{\mathrm{d}\bar{\xi}^{2}}\right)^{2}\right]^{1/2}=0,$
(85)
and
$AB\int_{0}^{{\bar{\xi_{b}}}}\bar{F}d\bar{\xi}=\dfrac{D}{2},$ (86)
respectively, where $\bar{\xi_{b}}=\xi_{b}/B$. Choosing
$A=\frac{D^{3/4}}{2\alpha^{1/4}K_{1}^{1/2}},\qquad
B=D^{1/4}\alpha^{1/4}K_{1}^{1/2},$ (87)
and defining
$\tilde{\beta}=\frac{2\beta}{D^{1/2}\alpha^{1/2}K_{1}},\quad\tilde{K}_{2}^{2}=\frac{K^{2}_{2}}{D^{1/2}\alpha^{1/2}K_{1}},\quad$
(88)
Equations (85) and (86) simplify to
$\bar{\xi}\;\bar{F}+\tilde{\beta}\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}+\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}\left[\left(\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}\right)^{2}+\tilde{K}_{2}^{2}\left(\dfrac{\mathrm{d}^{2}\bar{F}}{\mathrm{d}\bar{\xi}^{2}}\right)^{2}\right]^{1/2}=0,$
(89)
and
$\int_{0}^{\bar{\xi_{b}}}\bar{F}d\bar{\xi}=1,$ (90)
respectively. The two remaining boundary conditions in (80) become
$\bar{F}(\bar{\xi_{b}})=0,\ \
\dfrac{\mathrm{d}\bar{F}}{\mathrm{d}\bar{\xi}}(\bar{\xi_{b}})=0.$ (91)
Note that Prandtl’s original version of the extended model (i.e., where
kinematic viscosity neglected) is recovered when $\tilde{\beta}=0$. Whether
$\tilde{\beta}$ is zero or not, we see from the definitions in (88) that the
wake profile depends only on the product $DK_{1}^{2}$, and not on the
conserved quantity $D$ and the proportionality constant $K_{1}$ independently.
Other choices of $A$ and $B$ do not seem to reveal any other interesting
length scales.
Finally, with the scalings (87), Equation (82) becomes
$\bar{G}(\bar{\xi})=-\dfrac{1}{2}\bar{\xi}\bar{F}(\bar{\xi}),$ (92)
and from (83), the scaled similarity profile for the shear stress, which we
denote by $\bar{g}(\bar{\xi})$, is
$\bar{g}(\bar{\xi})=\dfrac{1}{2\sqrt{Re_{T}}}\bar{\xi}\bar{F}(\bar{\xi}).$
(93)
## 5 Results
### 5.1 Similarity solutions
From data generated from wind tunnel experiments, Wygnanski et al. [17]
demonstrated that regardless of the wake generator — examples of which
included cylinders, symmetric airfoils, and flat plates — the shape of the
normalised mean velocity profile far downstream is the same. In particular,
they show that the curve
$F_{N}(\xi_{N})=\exp{\left[-0.637\xi_{N}^{2}-0.056\xi_{N}^{4}\right]},$ (94)
provides a good fit to the mean velocity profile of the far wake when the
similarity variables are normalised so that $F_{N}(0)=1$ and $F_{N}(1)=1/2$.
This universality is not true of the shear stresses, (93), whose profiles
depend on the wake generator. From [17], the normalised Reynolds shear stress
$g_{N}$ is given by
$g_{N}(\xi_{N})=S\xi_{N}F_{N}(\xi_{N}),$ (95)
where the constant $S$ depends on the wake generator (compare with Equation
(93)). The data and the curves (94) and (95) are shown in Figure 3.
AirfoilSolid strip
Figure 3: Left: Experimental data and fitted curve (94) for the normalised
mean velocity profile. Different icons correspond to different wake
generators, which collapse to a single curve after appropriate normalisation.
Equation (94) provides a reasonable fit to the data, but is a heuristic
approximation, rather than the result of a mathematical model. The raw data
used to produce the figure is no longer available and the coloured dots
represent digitally extracted values used in our analysis below. Values were
only extracted when the corresponding data point was clearly identifiable.
Right: Shear stress measurements for two different generators. The
universality in the mean velocity profile is not present here, however the
shear stress can be related to the normalised mean velocity by (95). (Both
figures are reproduced with permission of the corresponding author of [17].)
We now compare the various closure models presented previously with this
experimental data. For each model, we solve for the normalised mean velocity
profile, $F_{N}(\xi_{N})$, and use Equation (95) to determine the shear
stresses. The airfoil and solid strip in Figure 3 correspond to $S=0.103$ and
$S=0.072$, respectively. Unfortunately, the raw data used to produce the
original version of Figure 3 is no longer available, so we manually extract
values from the figures in [17] to allow a quantitative comparison. The
extracted values used have been made available online [39].
### 5.2 Solving the models
The CEV model, which results in the boundary value problem [2]
$\dfrac{\mathrm{d}^{2}F}{\mathrm{d}\xi^{2}}+\dfrac{\mathrm{d}}{\mathrm{d}\xi}\left[\xi
F(\xi)\right]=0,\quad F(\infty)=0,\ \ \ F^{\prime}(\infty)=0,$ (96)
has a closed-form solution, and the resulting normalised mean velocity profile
is given by [17]
$F_{N}(\xi_{N})=\exp\left[-\ln(2)\xi_{N}^{2}\right].$ (97)
An analytic solution of (89)–(91) can be obtained for the case
$\tilde{\beta}=\tilde{K}_{2}=0$ (i.e. PML with $\tilde{\beta}=0$), giving the
normalised profile [11]
$F_{N}(\xi_{N})=\left[\left(\dfrac{\xi_{N}}{\xi_{b}}\right)^{3/2}-1\right]^{2},\
\ \xi_{b}=\left(2\left(3+2\sqrt{2}\right)\right)^{1/3}.$ (98)
Notice here that the condition $F_{N}(\xi_{b})=0$ results in a finite wake
boundary, and hence the wake is bounded in the $y$-direction. This is unlike
in the CEV model where the exponential solution does not vanish for finite
$\xi_{b}$.
Unfortunately, other than this special case, a closed form solution of
(89)–(91) seems beyond reach and the ODE must be solved numerically. For both
the PML model with $\tilde{\beta}>0$ and for the EPML model, we discretise the
equations using a Hermite pseudospectral method [40].555Although a Laguerre
spectral method would be more typical for a semi-infinite domain, we find the
Hermite to be more accurate and reliable in solving (89)–(91). We suspect this
is due to the super-exponential decay of the solution (more closely matching
the Hermite weight) and because the Laguerre method requires enforcing an
additional boundary condition at the origin. MATLAB code to solve (89)–(91)
and reproduce Figure 4 is available in an online repository [39]. The
resulting nonlinear system is then solved using a Newton iteration with
initial guess (94). The results are shown in Figure 4, along with the data and
fit from [17], and the CEV solution (97). For both PML and EPML we find that
increasing $\tilde{\beta}$ increases the effective width of the wake, but not
significantly, and so fix $\tilde{\beta}=0.01$.
Figure 4: Normalised mean velocity profiles of various closure models compared
to experimental data from [17]. The CEV and PML ($\tilde{\beta}=0.01$) models
provide a poor fit to the experimental data, particularly near the tail of the
wake. As $\tilde{K}_{2}$ is increased from zero, the EPML
($\tilde{\beta}=0.01$) solutions increase for $\xi_{N}<1$ and decrease for
$\xi_{N}>1$, more closely matching the heuristic fit (94) and the data. It is
interesting to note that the EPML solution with $\tilde{K}_{2}=0.5$ matches
almost exactly the solution from the CEV model for $0\leq\xi_{N}\leq 1$. The
best fit to the data is obtained when $\tilde{K}_{2}$ is around 0.375 (see
Table 1).
We see, as expected, that the CEV model provides a good fit near the centre
line of the wake, but overestimates near the wake boundary. The PML model also
overestimates near the wake boundary, and underestimates near the centre line
of the wake. As $\tilde{K}_{2}$ is increased from 0 the EPML solution
increases near the centre line, and decreases near the wake boundary, hence
providing a significantly improved match to both the curve (94) and the
experimental results from [17]. Increasing $\tilde{K}_{2}$ beyond 0.5 might
give an even closer match to (94), but the numerical solution begins to break
down for $\tilde{K}_{2}>0.5$, we suspect due to a lack of regularity. Since
(94) is only a heuristic approximation, and values of $0.25<\tilde{K}_{2}<0.4$
give a better fit to the data itself (see Table 1), we do not pursue this
further. However, it is interesting to note that for $0\leq\xi_{N}\leq 1$ the
EPML model solution with $\tilde{K}_{2}=0.5$ corresponds almost exactly with
the CEV model solution.
Model: | (94) | CEV | PML | $\tilde{K}_{2}=0.1$ | $\tilde{K}_{2}=0.2$ | $\tilde{K}_{2}=0.3$ | $\tilde{K}_{2}$ = 0.4 | $\tilde{K}_{2}=0.5$
---|---|---|---|---|---|---|---|---
$\ell_{2}$ error: | 0.181 | 0.194 | 0.206 | 0.186 | 0.160 | 0.144 | 0.141 | 0.148
Table 1: The 2-norm errors of the various closure models (CEV, PML, and EPML
with $\tilde{K}_{2}=0.1,0.2,\ldots,0.5$) applied to the experimental data from
[17]. For values of $\tilde{K}_{2}$ in the range $[0.25,5]$ the EPML model
provides a significantly better fit (up to 27.3%) than both the CEV and PML
models, and is even an improvement (22.1%) on the heuristic fit (94). For
completeness, we report that the smallest error was obtained with
$\tilde{K}_{2}$ around 0.375.
The success of the EPML model here may be attributed to two factors: First,
from the asymptotic solution as $y\rightarrow y_{b}(x)$ derived in Section 3,
the second derivative, $F_{N}^{\prime\prime}$, tends to zero as
$\xi\rightarrow\infty$, which is supported by the numerical results. Hence the
eddy viscosity vanishes at the boundary of the wake, capturing the physical
behaviour correctly. Second, there are two free parameters that can be chosen
to fit the data. Although the PML model has the free parameter
$\tilde{\beta}$, we have seen this has a minimal affect on the shape of the
normalised mean velocity profile. The new parameter, $\tilde{K}_{2}$,
corresponding to the second scaled mixing length in the EPML model, has a far
more significant impact on the shape of the normalised profile. Because the
EPML model was derived using physical considerations and provides a
significantly improved fit to the experimental data, we conclude that it
outperforms the other models considered in this work when predicting the mean
velocity profile.
In Figure 5 we perform a similar comparison, but now for the shear stresses.
In particular, for each of the closure models (and the fit (94)) we substitute
the obtained normalised mean velocity profiles into (95) and compare to the
experimental data from Figure 3. Whilst it is clear that the CEV model
significantly overestimates the stresses at the wake boundaries, other
conclusions are harder to draw. For the solid strip generator, all but the CEV
give a reasonable fit to the data, with the fit (94) being slightly better
than the rest; particularly towards the tails of the wake. For the airfoil
generator, all of the curves significantly overestimate the maximum magnitude
of the stress. The fit (94) is still the best towards the wake boundaries, but
overestimates more than the other models near the centre line. Of the curves
corresponding to mathematical models, the EPML (here with $K_{2}=0.375)$
improves on that of the PML (and CEV) models, but the improvement is not so
pronounced as for the velocity profiles in Figure 4. The fact that the models
gave a good fit for the normalized velocity profiles but not the Reynolds
shear stresses suggests that further investigation into the relationship (95)
is required. Although we do not pursue this here, it raises an important
question as to whether eddy viscosity models will ever be capable of
accurately predicting the shear stresses.
AirfoilSolid strip Figure 5: Normalised shear stress profiles for the
different closure models compared to experimental data from [17]. For each
closure model, the shear stresses are computed from the normalised mean
velocity profiles using (95), with $S$ = 0.103 for the airfoil and $S$ = 0.072
for the strip. Similarly to the mean velocity profile, the CEV model
significantly overestimates the magnitude of the stress towards the wake
boundaries. Equation (94) gives a good fit for the solid strip generator, but
overestimates the maximum magnitude of the stress for the airfoil. The PML and
EPML models also overestimate the stress for the airfoil.
## 6 Summary
The eddy viscosity closure model was used to complete the system of equations
for the mean flow variables describing the far downstream two-dimensional
turbulent classical wake, and the governing equations were expressed in terms
of the mean velocity deficit in the $x$-direction. The boundary conditions on
the mean velocity deficit were obtained by imposing matching conditions
between the turbulent wake region with the inviscid laminar mainstream flow,
and the condition that the velocity deficit is maximised on the axis of the
wake. Since the boundary conditions were independent of the choice of closure
model, the conserved quantity was also independent. The $y$-component of the
mean velocity was then derived and it was shown that there was no entrainment
at the far wake boundary for finite-valued eddy viscosities. This depends
critically on the linear approximation made for the inertia in Equation (13).
An outline of the derivation of Prandtl’s mixing length model was provided,
which was then modified to introduce a new derivation of the extended Prandtl
mixing length model. Scaling solutions admitted by the governing equations of
the extended model were obtained, and it was shown when kinematic viscosity is
neglected that Prandtl’s hypothesis is necessary to obtain a similarity
solution. Conversely, when kinematic viscosity is included, no additional
constraints are required. The similarity variables were used to reduce the PDE
to an ODE, and exact solutions were found for special cases. Numerical results
were calculated for the remaining cases, and the normalised similarity mean
velocity and shear stress profiles were compared for various closure models.
For the mean velocity profiles, it was shown that the extended Prandtl mixing
length model gives a significantly improved fit to experimental data when
compared with the other considered models. The shear stress profiles, which
are computed from the mean velocity profiles, did not provide such a good fit
to the experimental data, suggesting that more work is needed in relating the
two.
The work presented here also sets the stage for the development of new closure
models and application of similarity methods in turbulence modelling. Unlike
the CEV model, the PML and EPML models both satisfy $\nu_{T}\rightarrow 0$ as
$y\rightarrow\pm y_{b}(x)$, which adheres to the condition of mainstream
matching between the laminar mainstream flow region and the turbulent wake. We
saw that for models satisfying this property, the solution tends to the
exponential solution for the laminar wake. The exponential solution satisfies
$\partial^{n}\bar{w}/\partial y^{n}\rightarrow 0$ as $y\rightarrow\pm\infty$
for $n\in\mathbb{Z}^{+}$, meaning that an entire class of closure models with
nonzero kinematic viscosity and depending on partial derivatives higher than
$\partial^{2}\bar{w}/\partial y^{2}$ that satisfy the mainstream matching
condition might be produced. This is an interesting consideration for future
work.
AJH lead the development of the work and was responsible for many of the ideas
presented in Sections 2, 3. NH produced the numerical solutions and analysis,
and with AJH, lead the structuring and writing of the paper. KB produced all
diagrams in Sections 2 and 3, obtained the scaling solutions in Section 4, and
provided valuable input into the other sections. DPM, as the primary subject
matter expert, made many corrections to all sections, significantly improving
upon the quality of the work.
DPM thanks the National Research Foundation, Pretoria, South Africa, for
financial support. Grant number: 96270
The authors are grateful to I. Wygnanski (University of Arizona) for
permitting the reproduction of Figure 3 and to the anonymous referees for
their useful feedback. We also thank P. Broadbridge (La Trobe University) for
his valuable comments.
## References
* [1] Tennekes H, Lumley JL. 1971 A First Course in Turbulence. Cambridge, Massachusetts, and London: MIT Press.
* [2] Pope SB. 2000 Turbulent Flows. Cambridge University Press.
* [3] Howland MF, Lele SK, Dabiri JO. 2019 Wind farm power optimization through wake steering. Proceedings of the National Academy of Sciences 116, 14495–14500.
* [4] Breton SP, Sumner J, Sørensen JN, Hansen KS, Sarmast S, Ivanell S. 2017 A survey of modelling methods for high-fidelity wind farm simulations using large eddy simulation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160097.
* [5] Mehta D, van Zuijlen AH, Koren B, Holierhoek JG, Bijl H. 2014 Large Eddy Simulation of wind farm aerodynamics: A review. Journal of Wind Engineering and Industrial Aerodynamics 133, 1–17.
* [6] Jiménez Á, Crespo A, Migoya E. 2010 Application of a LES technique to characterize the wake deflection of a wind turbine in yaw. Wind Energy 13, 559–572.
* [7] Abkar M, Dabiri JO. 2017 Self-similarity and flow characteristics of vertical-axis wind turbine wakes: an LES study. Journal of Turbulence 18, 373–389.
* [8] Wu YT, Porté-Agel F. 2015 Modeling turbine wakes and power losses within a wind farm using LES: An application to the Horns Rev offshore wind farm. Renewable Energy 75, 945 – 955.
* [9] Göçmen T, van der Laan P, Réthoré PE, Diaz AP, Larsen GC, Ott S. 2016 Wind turbine wake models developed at the technical university of Denmark: A review. Renewable and Sustainable Energy Reviews 60, 752–769.
* [10] Hutter K, Wang Y. 2016 Turbulent Mixing Length Models and Their Applications to Elementary Flow Configurations. Springer International Publishing.
* [11] Cafiero G, Obligado M, Vassilicos JC. 2020 Length scales in turbulent free shear flows. Journal of Turbulence 21, 243–257.
* [12] Doshi MR, Gill WN. 1970 A note on the mixing length theory of turbulent flow. AIChE Journal 16, 885–888.
* [13] Reynolds O. 1895 IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philosophical Transactions of the Royal Society of London. (A.) 186, 123–164.
* [14] Liepmann HW. 1952 Aspects of the turbulence problem. Zeitschrift für angewandte Mathematik und Physik 3, 407–426.
* [15] Boussinesq J. 1877 Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l’Académie des Sciences XXIII.
* [16] Goldstein S. 1933 On the two-dimensional steady flow of a viscous fluid behind a solid body. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 142, 545–562.
* [17] Wygnanski I, Champagne F, Marasli B. 1986 On the large-scale structures in two-dimensional, small-deficit, turbulent wakes. Journal of Fluid Mechanics 168, 31–71.
* [18] Prandtl L. 1925 Bericht über Untersuchenden zur ausgebildeten Turbulenz. Zeitschrift für angewandte Mathematik und Mechanik 5, 136–139. English: NACA-TM-1231.
* [19] Swain LM. 1929 On the turbulent wake behind a body of revolution. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 125, 647–659.
* [20] Prandtl L. 1927 Über die ausgebildete Turbulenz. In Verhandlungen des II. Internationalen Kongresses für Technische Mechanik 1926 pp. 62–75. Zürich:Füßli-Verlag.
* [21] Luo X, Liu Pl, Luo Ha. 2008 Improvement of Prandtl mixing length theory and application in modeling of turbulent flow in circular tubes. Journal of Central South University of Technology 15, 774–778.
* [22] Hutchinson AJ. 2019-20 The extended Prandtl closure model applied to the two-dimensional turbulent classical far wake. MATRIX Annals, MATRIX Book Series, Springer 4. To appear.
* [23] Hutchinson AJ, Mason DP. 2015 Revised Prandtl mixing length model applied to the two-dimensional turbulent classical wake. International Journal of Non-Linear Mechanics 77, 162–171.
* [24] Cantwell BJ. 1978 Similarity transformations for the two-dimensional, unsteady, stream-function equation. Journal of Fluid Mechanics 85, 257–271.
* [25] Oberlack M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. Journal of Fluid Mechanics 427, 299–328.
* [26] Razafindralandy D, Hamdouni A, Oberlack M. 2007 New turbulence models preserving symmetries. In 5th International Symposium on Turbulence and Shear Flow Phenomena.
* [27] Oberlack M, Wacławczyk M, Rosteck A, Avsarkisov V. 2015 Symmetries and their importance for statistical turbulence theory. Mechanical Engineering Reviews 2, 15–157.
* [28] Ibragimov NH, Ünal G. 1994 Lie groups in turbulence: I. Kolmogorov’s invariant and the algebra $l_{\tau}$. Lie Groups and their Applications 1, 98–103.
* [29] Ünal G. 1994 Application of equivalence transformations to inertial subrange of turbulence. Lie Groups and their Applications 1, 232–240.
* [30] Schlichting H, Gersten K. 2017 Boundary-Layer Theory. Springer Berlin Heidelberg ninth edition.
* [31] Schlichting H. 1979 Boundary-Layer Theory. Springer Berlin Heidelberg seventh edition.
* [32] Hutchinson AJ, Mason DP, Mahomed FM. 2015 Solutions for the turbulent classical wake using Lie symmetry methods. Communications in Nonlinear Science and Numerical Simulation 23, 51–70.
* [33] Hutchinson AJ. 2018 Application of a modified Prandtl mixing length model to the turbulent far wake with a variable mainstream flow. Physics of Fluids 30, 095102.
* [34] Hutchinson AJ, Mason DP. 2016 Lie symmetry methods applied to the turbulent wake of a symmetric self-propelled body. Applied Mathematical Modelling 40, 3062–3080.
* [35] Hutchinson AJ. 2016 A unified theory for turbulent wake flows described by eddy viscosity. International Journal of Non-Linear Mechanics 81, 40 – 54.
* [36] Bradshaw P. 1974 Possible origin of Prandt’s mixing-length theory. Nature 249, 135–136.
* [37] Dresner L. 1983 Similarity solutions of nonlinear partial differential equations. Pitman, Boston.
* [38] Sjöberg A. 2007 Double reduction of PDEs from the association of symmetries with conservation laws with applications. Applied Mathematics and Computation 184, 608–616.
* [39] Hale N. 2020 MATLAB code for the EPML model. https://github.com/nickhale/EPML.
* [40] Weideman J, Reddy S. 2000 A MATLAB Differentiation Matrix Suite. ACM Trans. Math. Softw. 26, 465–519.
|
# A Generic Slater-Koster Description of the Electronic Structure of
Centrosymmetric Halide Perovskites
Ravi Kashikar Mayank Gupta B. R. K. Nanda<EMAIL_ADDRESS>Condensed
Matter Theory and Computational Lab, Department of Physics, Indian Institute
of Technology Madras, Chennai - 36, India
###### Abstract
The halide perovskites have truly emerged as efficient optoelectronic
materials and show the promise of exhibiting nontrivial topological phases.
Since the bandgap is the deterministic factor for these quantum phases, here
we present a comprehensive electronic structure study using first-principle
methods by considering nine inorganic halide perovskites CsBX3 (B = Ge, Sn,
Pb; X = Cl, Br, I) in their three structural polymorphs (cubic, tetragonal and
orthorhombic). A series of exchange-correlations (XC) functionals are examined
towards accurate estimation of the bandgap. Furthermore, while thirteen
orbitals are active in constructing the valence and conduction band spectrum,
here we establish that a four orbital based minimal basis set is sufficient to
build the Slater-Koster tight-binding model (SK-TB), which is capable of
reproducing the bulk and surface electronic structure in the vicinity of the
Fermi level. Therefore, like the Wannier based TB model, the presented SK-TB
model can also be considered as an efficient tool to examine the bulk and
surface electronic structure of halide family of compounds. As estimated by
comparing the model study and DFT band structure, the dominant electron
coupling strengths are found to be nearly independent of XC functionals, which
further establishes the utility of the SK-TB model.
††preprint: APS
## I Introduction
Halide perovskites of the form ABX3 (where A is an organic or inorganic
monovalent entity, B is a divalent cation such as Pb, Sn, Ge and X is a
halogen (Cl, Br, I and F)), have brought a paradigm shift in the photovoltaic
applications because of parity allowed direct transitions between the band
extrema Trots and Myagkota (2008); Fujii _et al._ (1974a); Hirotsu _et al._
(1974a); Yang _et al._ (2017a). In recent times along with the optical
properties, these perovskites have shown the ferroelectrically driven spin-
texture, and topological quantum phase transition under external forces in
both centrosymmetric and noncentrosymmetric phases Yang _et al._ (2012); Liu
_et al._ (2016); Kashikar, Khamari, and Nanda (2018a); Jin, Im, and Freeman
(2012); Shi _et al._ (2015); Song _et al._ (2017); Kepenekian _et al._
(2015); Yao, Xiao, and Niu (2008). While the entity A predominantly decides
the structural stability and centrosymmetricity, B and X governs the
electronic properties of these compounds Borriello, Cantele, and Ninno (2008);
Kashikar, Khamari, and Nanda (2018a). The cubic phase of halide perovskites
exhibit structural phase transition with temperature and pressure, and the
resulted lower symmetry crystal structures are characterized by in-plane as
well as out-of-plane octahedral rotations Yang _et al._ (2017b); Yu _et al._
(2013); Yang _et al._ (2017c). The schematic representation of the orbital
overlap in high symmetric and lower symmetric phases is illustrated in the
Fig. 1.
Over the last decade, both experimental and density functional theory (DFT)
methods have been employed to unravel the orbital and crystal interplay to
study the optoelectronic and other intriguing properties of these perovskite
materials. The previous electronic structure studies on these compounds
suggest that the band spectrum consists of anti-bonding and bonding states,
along with the X-p dominated non-bonding states arise out of strong covalent
hybridization between B-{s, p} and X-p statesHuang and Lambrecht (2016);
Borriello, Cantele, and Ninno (2008); Kashikar, Khamari, and Nanda (2018a).
Nature of the band structure remains similar for a particular phase with
varying bandwidth for a family of halide perovskites. Beyond this basic
observations, there are several issues which need to be addressed to construct
a comprehensive picture of the electronic structure of this important class of
compounds. For example choice of exchange correlation functional is one of the
debatable issue Jishi, Ta, and Sharif (2014); Traoré _et al._ (2019).
As the bandgap varies widely in this class of compounds and the materials
applicability in optoelectronic devices as well as in inducing non-trivial
topological phases primarily depend on the bandgap, and we know that the
bandgap highly sensitive to type of exchange-correlation functionals employed
within the DFT formalism. Therefore, it is pertinent that a relation between
the bandgap and the type of exchange correlation functional be established,
which could capture the experimental observations. Furthermore, this class of
compounds exhibit three temperature dependent structural phases governed by
anionic displacements Yu _et al._ (2013); Yang _et al._ (2017c). The bandgap
of these phases differ significantly from each other. Therefore, it is crucial
to identify the chemical interactions that govern the bandgap in this family.
Also, it has been observed that electronic structure is sensitive to both B
and X. For example if B is Sn for any X, we observe a lower bandgap as
compared Pb and Ge and the reason has not been able to explain through
parameter-free density functional calculations. As a whole comprehensive first
principles calculations and formulation of model Hamiltonians are required to
provide a generic description of the electronic structure of the halide
perovskites.
Figure 1: Schematic representation of chemical bonding of B-{s, p}-X-p
orbitals in various polymorphs of halide perovskites.
In the literature, from the model Hamiltonian perspective, only a handful of
studies have examined the band structure of halide perovskites. Boyer-Richard
et al., envisaged the fourteen orbital basis based tight-binding (TB)
Hamiltonian without incorporating the second neighbour interaction and Jin et
al. have studied the continuum model based on the four orbital basis TB
Hamiltonian Boyer-Richard _et al._ (2016); Jin, Im, and Freeman (2012). In
our recent work, the four orbital basis model is employed to lower symmetry
polymorphs to examine the band topology of these perovskites systems Kashikar,
Khamari, and Nanda (2018a). However, all these studies were carried out on
individual members, and so far, there has been no systematic study of TB
Hamiltonian for all the members of the halide perovskites family to understand
sheer variety of properties with varying B and X elements. This is the first
attempt to bring the generic picture of halide perovskites, and for that, we
have considered the nine members and three structural phases to analyze the
role of each entity of inorganic ABX3 compounds.
## II Computational details
In the present work, we have employed both pseudopotential method with the
plane-wave basis set as implemented Vienna Abinitio Simulation Package (VASP),
and full-potential linearized augmented plane wave (FP-LAPW) method as
implemented in WIEN2k simulation tool for DFT calculations Kresse and
Furthmüller (1996); Hamann (1979); Blaha _et al._ (2001). In both the
methods, the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional
within the generalized gradient approximation (GGA) is considered. However,
the PBE underestimates the bandgap severely as compared to the experimental
bandgap. Therefore, the PBE-GGA approximation along with modified Becke-
Johnson (mBJ) potential as well as hybrid functionals are used to take into
account the exchange-correlation effect and for comparison purpose Perdew,
Burke, and Ernzerhof (1996); Tran and Blaha (2009). All the band structures in
the present work are presented from PBE+Tran-Blaha mBJ potential. The self-
consistent field (SCF) calculations comprise of augmented plane waves of the
interstitial region and localized orbitals: B-{ns, np} (B = Ge, Sn, Pb) and
X-p (X = Cl, Br, I). Here, n and m vary with B and X. The R${}_{MT}^{X}$ set
to 7.0 for all the compounds. The Brillouin zone integration is carried out
with a Monkhorst-Pack grid. We used a $k$-mesh of 10$\times$10$\times$10
(yielding 35 irreducible points), 6$\times$6$\times$4 (yielding 35 irreducible
points), and 8$\times$8$\times$6 (yielding 100 irreducible points) for the
$\alpha$, $\beta$ and $\gamma$-phases respectively. To build a Slater-Koster
tight binding (SK-TB) Hamiltonian, we consider the appropriate basis set out
of the orbitals dominating the bands at the Fermi level and this information
is obtained from the first-principle based density functional calculations.
The details of the Hamiltonian and orbital basis is discussed in details in
section IIIB.
Table 1: Structural parameters of various CsBX3 perovskites as obtained from DFT calculations and in comparison with the experimental results. The compounds exist in different crystal polymorphs at different temperature ranges. The $\theta_{ab}$ and $\theta_{c}$ is 180∘ for $\alpha$ phase. Phase | B | X | Lattice Parameter (Å) | Octahedral angle | Temperature (K) | Ref.
---|---|---|---|---|---|---
| | | DFT | Expt. | $\theta_{ab}$, $\theta_{c}$ | |
| | Cl | 5.34 | 5.47 | | 443,449 | Thiele, Rotter, and Schmidt (1987); Yamada _et al._ (1994)
| Ge | Br | 5.6 | 5.69 | | 543 | Thiele, Rotter, and Schmidt (1987)
| | I | 6.0 | 6.05 | | 573 | Thiele, Rotter, and Schmidt (1987)
| | Cl | 5.62 | 5.55 | | 293 | Yang _et al._ (2017d)
$\alpha$ | Sn | Br | 5.88 | 5.8 | | 292,300 | Fabini _et al._ (2016)
| | I | 6.27 | 6.21 | | 426 | Yamada _et al._ (1991)
| | Cl | 5.71 | 5.6 | | 320 | Fujii _et al._ (1974b)
| Pb | Br | 5.98 | 5.87 | | 403 | Hirotsu _et al._ (1974b)
| | I | 6.38 | 6.29 | | 554 | Marronnier _et al._ (2018)
| Sn | Br | 8.27 8.27 5.92 | 8.18, 8.18, 5.82 | 162.8 ∘, 180 ∘ | 270-300 | Fabini _et al._ (2016)
$\beta$ | | I | 8.81 8.81 6.31 | 8.77, 8.77, 6.26 | 162.6 ∘, 180 | 351-426 | Yamada _et al._ (1991)
| Pb | Br | 8.35, 8.35, 6.04 | | 156.3∘, 180∘ | 361-403 | Hirotsu _et al._ (1974b)
| | I | 8.89 8.89 6.44 | 8.82, 8.82, 6.3 | 155.9 ∘, 180∘ | 457-554 | Marronnier _et al._ (2018)
| Sn | Br | 8.15 8.37 11.77 | 8.19, 11.58, 8.02 | 157.4 ∘, 163.5∘ | 270 | Fabini _et al._ (2016)
$\gamma$ | | I | 8.66 8.92 12.53 | 8.68, 8.64, 12.37 | 154.6 ∘, 161.4∘ | 426 | Yamada _et al._ (1991)
| Pb | Br | 8.24 8.47 11.93 | 8.21, 8.25, 11.76 | 153.4 ∘, 160.6∘ | 361 | Hirotsu _et al._ (1974b)
| | I | 8.76 9.04 12.7 | 8.62, 8.85, 12.5 | 152.3 ∘, 159.2∘ | 457 | Marronnier _et al._ (2018)
## III Results and Discussion
### III.1 DFT study
Figure 2: (a) Molecular orbital picture of halide perovskites envisaged from
the B-{s, p}-X-p atomic orbitals, produces the bonding and antibonding
orbitals along with the nonbonding orbitals. (b-d) Band structure of cubic,
tetragonal and orthorhombic band structure of representative compound CsPbI3.
To start with, we will first discuss the electronic structure of CsPbI3 and
explain it through a generic molecular orbital picture as shown in Fig. 2.
Here, we outline some of the standard features of band spectrum of halide
perovskites: (I) The band spectrum in general consist of four antibonding and
bonding bands along with the five nonbonding bands arising due to the B-{s,
p}-X-p covalent hybridization in BX6 octahedron. Thus, the eigenfunctions of
antibonding and bonding states are the linear combination of B-s, B-p and X-p
orbitals. Their strength varies with the type of atom at B, halogens as well
as crystal symmetry. The nonbonding states are from X-p orbitals. The
molecular orbital picture presented in Fig. 2a is for Oh point group symmetry
which exists at R and $\Gamma$ points of cubic Brillouin zones. The X-p
orbitals undergo symmetry adapted linear combinations and form the molecular
orbitals with B-{s, p} orbitals. (II) The Fermi level in the electronic
spectrum is solely determined by the valence electron count (VEC) which is the
total number of valence electrons available per formula unit. In single halide
perovskites, the atom at $A$ contributes one electron, while atoms $B$ and $X$
contributes four and five electrons respectively. Thus, the VEC for halide
perovskites turns out to be 20. This makes the states up to $\sigma^{*}_{s-p}$
occupied and hence the Fermi level lies in the gapped region. (III) All of the
inorganic halide perovskites have direct bandgap in nature. The bandgap value
varies with the crystal symmetry and chemical composition of ABX3 and spin-
orbit coupling (SOC) strength of $B$ site atom.(IV) The parities of the
valence band and conduction band edges for three polymorphs in terms of Koster
notations is denoted in Fig. 2. The analysis indicates that all the halide
perovskites exhibit parity allowed transitions due to the opposite of parity
of band edges.
The instability of the high-temperature cubic phase introduces the octahedral
rotations in the unitcell, and there occurs the crystal phase transition from
the cubic to tetragonal to orthorhombic phases with temperature as listed in
Table 1. The tetragonal phase is characterized by only inplane octahedral
rotation, whereas the orthorhombic phase is characterized both inplane and out
of plane octahedral rotations. These rotations increases the unitcell volume
as well as the number of atoms. The Ge based halide perovskites exhibit single
phase transition and crystallize in rhombohedral unitcell in room temperature
phase.
In Fig. 3 and Fig. 4 we have compared the bandgap of cubic and lower symmetry
halide perovskites for various exchange-correlation functionals. It is known
that GGA-PBE underestimates the bandgap significantly, the correction is added
through several other approximations. The modified Becke-Johnson (mBJ)
proposed by Tran-Blaha improves the bandgap significantly in most of the
halide perovskites. The hybrid functionals with mixing parameter $\alpha=0.25$
(HSE06) and $\alpha=0.3$ (HSE) offer better estimation of bandgap in few
compounds and in others the values are similar to those obtained using TB-mBJ
functional. The recently proposed mBJ potential by Jishi et al. provides the
bandgap close to experimental one in the case of Pb based compounds Jishi, Ta,
and Sharif (2014); Traoré _et al._ (2019). In few cases; CsGeI3 and CsSnI3
hybrid functional minimizes the error as compared to Jishi-mBJ. The bandgap
for lower symmetry polymorphs for PBE. mBJ and Jishi-mBJ are shown in Fig. 4,
in comparison with available experimental values. It is observed that the gap
decreases from Cl to Br to I, irrespective of atoms at A and B sites, and Sn-
based halide perovskites have lower bandgap values as compared to the Pb and
Ge based perovskites. A broad explanation to this end can be given by
comparing the free atomic energy eigenvalues. The difference between the
onsite energies of Sn valence orbitals ($E_{p1/2}-E_{s1/2}$ = 6.35 eV) is
lowest as compared to that of Pb (7.04 eV) and Ge (7.53 eV). Whereas the
onsite energy halogen valence $p$ orbital decreases from I (-7.84 eV) to Br
(-8.96 eV) to Cl (-9.96 eV) SS The bandgap also increases as we move from
cubic to tetragonal to orthorhombic phases due to octahedral rotations which
decrease the orbitals overlap as shown in Fig. 1. The valence band maximum
(VBM) and conduction band minimum (CBM) are predominantly of B-s and B-p
character. The VBM and CBM occurs at R (0.5, 0.5, 0.5), Z (0, 0, 0.5) and
$\Gamma$ (0, 0, 0) $k$-points of cubic, tetragonal and orthorhombic Brillouin
zone respectively.
Figure 3: Bar chart representation of bandgap of cubic halide perovskites
under various exchange correlation functionals. Here, HSE06 and HSE are
calculated with pseudopotential methods with mixing parameter 0.25 and 0.3,
respectively. Figure 4: Bar chart representation of bandgap of tetragonal (a)
and orthorhombic (b) halide perovskites under various exchange correlation
functionals. The experimental bandgap values of tetragonal and orthorhombic
systems are obtained from Huang and Lambrecht (2013); Tao _et al._ (2019);
Straus, Guo, and Cava (2019).
The DFT calculations offers a limited quantitative understanding to establish
a generic description of the electronic structure. Specifically, the type and
strength of the interactions that govern the valence and conduction bands in
the vicinity of the Fermi level needs to be determined. Therefore, in the next
section, we build the Slater-Koster based tight-binding Hamiltonian using a
thirteen orbitals (one B-s, three B-p and nine X-p) basis set. Subsequently,
the basis set will be reduced to four.
### III.2 Model Hamiltonian for Halide Perovskites
The appropriate SK-TB Hamiltonian for the family of CsBX3, in the second
quantization notation is expressed as
$H=\sum_{i,m}\epsilon_{im}c_{im}^{\dagger}c_{im}+\sum_{\langle\langle
ij\rangle\rangle;m,n}t_{imjn}(c_{im}^{\dagger}c_{jn}+h.c)+\lambda\textbf{L}\cdot\textbf{S}.$
(1)
Here, i (j) and $\alpha$ ($\beta$ ) are site and the orbitals indices
respectively. The parameters $\epsilon_{i\alpha}$ and $t_{i\alpha j\beta}$
respectively, represent the on-site energy and hopping integrals. The spin-
orbit coupling (SOC) is included in the third term of the Hamiltonian with
$\lambda$ denoting the SOC strength. The inclusion of SOC doubles the Hilbert
space. Adopting a two centre integral approach J. C. Slater and G. F. Koster,
expressed the $t_{i\alpha j\beta}$ with direction cosines (DCS) ($l$, $m$,
$n$) of the vector joining site ($i$) to other site ($j$) Slater and Koster
(1954). For $s$ and $p$ orbitals, which forms the basis for halide
perovskites, the generic expressions are provided below
$\displaystyle E_{s,s}$ $\displaystyle=t_{ss\sigma}$ (2) $\displaystyle
E_{s,p_{x}}$ $\displaystyle=lt_{sp\sigma}$ $\displaystyle E_{s,p_{y}}$
$\displaystyle=mt_{sp\sigma}$ $\displaystyle E_{s,p_{z}}$
$\displaystyle=nt_{sp\sigma}$ $\displaystyle E_{p_{x},p_{x}}$
$\displaystyle=l^{2}t_{pp\sigma}+(1-l^{2})t_{pp\pi}$ $\displaystyle
E_{p_{x},p_{m}}$ $\displaystyle=lmt_{pp\sigma}-lmt_{pp\pi}$
Using Eq. 1 and 2, we now develop TB Hamiltonian for the halide perovskites.
Thus, the spin independent TB Hamiltonian matrix, with the basis set in the
order $\\{|s^{B}\rangle$, $|p^{B}_{x}\rangle$, $|p^{B}_{y}\rangle$,
$|p^{B}_{z}\rangle$, $|p^{X1}_{x}\rangle$, $|p^{X1}_{y}\rangle$,
$|p^{X1}_{z}\rangle$, $|p^{X2}_{x}\rangle$, $|p^{X2}_{y}\rangle$,
$|p^{X2}_{z}\rangle$, $|p^{X3}_{x}\rangle$, $|p^{X3}_{y}\rangle$,
$|p^{X3}_{z}\rangle$$\\}$, can be written as
$H^{FB}_{TB}=\left(\begin{array}[]{cc}M_{4\times 4}^{B-B}&M_{4\times
9}^{B-X}\\\ \\\ (M_{4\times 9}^{B-X})^{{\dagger}}&M_{9\times
9}^{X-X}\end{array}\right).$ (3)
Here, FB indicates the full basis. The individual blocks of this matrix are as
follows, The $M_{4\times 4}^{B-B}$ is given by
$\left(\begin{array}[]{cccc}\epsilon_{s}+h_{1}(\vec{k})&2it_{sp\sigma}^{B-B}S_{x}&2it_{sp\sigma}^{B-B}S_{y}&2it_{sp\sigma}^{B-B}S_{z}\\\
-2it_{sp\sigma}^{B-B}S_{x}&\epsilon_{p1}+h_{2}(\vec{k})&0&0\\\
-2it_{sp\sigma}^{B-B}S_{y}&0&\epsilon_{p1}+h_{3}(\vec{k})&0\\\
-2it_{sp\sigma}^{B-B}S_{z}&0&0&\epsilon_{p1}+h_{4}(\vec{k})\\\
\end{array}\right)$ (4)
$M_{4\times 9}^{B-X}=\left(\begin{array}[]{ccc}M^{4\times 3}M^{4\times
3}M^{4\times 3}\end{array}\right)$ (5)
$M_{4\times 3}^{B-X}=\left(\begin{array}[]{ccc}t_{sp\sigma}^{B-X}S_{x}&0&0\\\
t_{pp\sigma}^{B-X}C_{x}&0&0\\\ 0&t_{pp\pi}^{B-X}C_{x}&0\\\
0&0&t_{pp\pi}^{B-X}C_{x}\\\ \end{array}\right)$ (6)
From the Fig. 2 it is observed that X-p dominated bands are very narrow
$(<1.0$ eV), suggesting negligible X-{p}-X-{p} second interactions. Hence the
block $M_{9\times 9}^{X-X}$ can be approximated as
$M_{9\times 9}^{X-X}=\epsilon_{p2}I_{9\times 9}$ (7)
Here, $\epsilon_{s}$, $\epsilon_{p1}$ and $\epsilon_{p2}$ are on-site energies
of B-s, B-p and X-p orbitals respectively. The terms $C_{x}$ and $S_{x}$ are
short notations for $2\cos(k_{x}a/2)$ and $2i\sin(k_{x}a/2)$ respectively. The
dispersion term $g_{i}$ $(i=1,2,3,4)$, arising from B-B second neighbour
interactions are given by
$\displaystyle h_{1}(\vec{k})$ $\displaystyle=$ $\displaystyle
2t_{ss}^{B-B}(cos(k_{x}a)+cos(k_{y}a)+cos(k_{z}a))$ $\displaystyle
h_{2}(\vec{k})$ $\displaystyle=$ $\displaystyle
2t_{pp\sigma}^{B-B}cos(k_{x}a)+2t_{pp\pi}^{B-B}[cos(k_{y}a)+cos(k_{z}a)]$
$\displaystyle h_{3}(\vec{k})$ $\displaystyle=$ $\displaystyle
2t_{pp\sigma}^{B-B}cos(k_{y}a)+2t_{pp\pi}^{B-B}[cos(k_{x}a)+cos(k_{z}a)]$ (8)
$\displaystyle h_{4}(\vec{k})$ $\displaystyle=$ $\displaystyle
2t_{pp\sigma}^{B-B}cos(k_{z}a)+2t_{pp\pi}^{B-B}[cos(k_{x}a)+cos(k_{y}a)]$
The analytical expression for eigenvalues at time reversal invariant momentum
(TRIM) $R$ $(\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{a})$, which is of
particular interest as the valence band maximum (VBM) and conduction band
minimum (CBM) are observed at this point, and are given as
$\displaystyle E_{1}[1]$
$\displaystyle=\frac{(E_{p}^{X}+E_{s}^{B})}{2}-3t_{ss}^{B-B}-\eta$
$\displaystyle E_{2}[8]$ $\displaystyle=E_{p}^{X}$ $\displaystyle E_{3}[3]$
$\displaystyle=E_{p}^{B}-2t_{pp\sigma}^{B-B}-4t_{pp\pi}^{B-B}$ $\displaystyle
E_{4}[1]$ $\displaystyle=\frac{(E_{p}^{X}+E_{s}^{B})}{2}-3t_{ss}^{B-B}+\eta.$
(9)
Where
$\eta=\frac{\sqrt{(E_{p}^{X}-E_{s}^{B}+6t_{ss}^{B-B})^{2}+48(t_{sp}^{B-X})^{2}}}{2}$
The number in square bracket indicates the degeneracy of the eigenvalues. In
presence of SOC, operating $H_{soc}=\lambda\textbf{L.S}$ on B-$p$ orbital
basis with the order ($p_{x\uparrow}^{B}$, $p_{y\uparrow}^{B}$,
$p_{z\downarrow}^{B}$,$p_{x\downarrow}^{B}$, $p_{y\downarrow}^{B}$,
$p_{z\uparrow}^{B}$), we get the following matrix
$H_{SOC}=\lambda\left(\begin{array}[]{cccccc}0&-i&1&&&\\\ i&0&-i&&0&\\\
1&-i&0&&&\\\ &&&0&i&1\\\ &0&&-i&0&i\\\ &&&1&i&0\end{array}\right).$ (10)
As a case study, the thirteen band model is applied to nine cubic CsBX3 (B =
Ge, Sn, Pb; X = Cl, Br, I) perovskites and corresponding bands are fitted with
DFT bands to obtain onsite and hopping interactions, which are listed in Table
2. The DFT and TB bands are shown in Fig. 5, suggesting the excellent
agreement between each other. The table infers that (I) tin-based halide
perovskites exhibit higher onsite energies (lowest Es-Ep energy), agrees with
the atomic energies mentioned in section-II. A similar trend is observed for
halogen orbital energies. The hopping parameter $t_{sp}^{\sigma}$ varies from
Cl to Br to I indicating a decrease in the bandwidth of uppermost valence band
for all the compounds as we move from Cl to Br to I. The onsite energies
provided in Table 2 agrees qualitatively well with the recent work of Hoffmann
et al. Goesten and Hoffmann (2018). The other hopping parameters
$t_{sp}^{B-X}$ and $t_{pp\sigma}^{B-B}$ governs the bandwidth of uppermost
valence band and conduction band, and these parameters increase with Cl-Br-I.
The other parameters remain more or less constant for all the perovskites.
Table 2: On-site energy, hopping parameters and GGA-mBJ bandgap of CsBX3 perovskites in units of eV B | X | $E_{B-s}$ | $E_{B-p}$ | $E_{X-p}$ | $t_{sp}^{B-X}$ | $t_{pp\sigma}^{B-X}$ | $t_{pp\pi}^{B-X}$ | $t_{ss}^{B-B}$ | $t_{sp\sigma}^{B-B}$ | $t_{pp\sigma}^{B-B}$ | $t_{pp\pi}^{B-B}$ | $\lambda$
---|---|---|---|---|---|---|---|---|---|---|---|---
| Cl | -3.17 | 5.35 | -0.18 | -1.20 | 1.94 | -0.58 | 0.02 | -0.16 | 0.31 | 0.03 | 0.07
Ge | Br | -3.55 | 4.76 | 0.40 | -1.16 | 1.94 | -0.56 | 0.02 | -0.15 | 0.32 | 0.02 | 0.07
| I | -3.97 | 4.29 | 0.92 | 1.00 | 1.92 | -0.52 | 0.02 | -0.15 | 0.37 | 0.02 | 0.07
| Cl | -0.71 | 6.42 | -0.06 | -1.29 | 1.94 | -0.52 | -0.08 | -0.2 | 0.24 | 0.06 | 0.16
Sn | Br | -1.43 | 5.71 | 0.36 | -1.27 | 1.96 | -0.53 | -0.07 | -0.19 | 0.34 | 0.05 | 0.16
| I | -2.34 | 4.79 | 0.92 | -1.12 | 1.9 | -0.53 | -0.02 | -0.17 | 0.38 | 0.01 | 0.14
| Cl | -1.61 | 7.63 | 0.25 | -1.18 | 1.88 | -0.57 | -0.03 | -0.13 | 0.31 | 0.06 | 0.53
Pb | Br | -3.15 | 5.84 | 0.43 | -1.13 | 1.86 | -0.53 | -0.02 | -0.12 | 0.27 | 0.04 | 0.53
| I | -4.11 | 4.75 | 0.96 | -0.94 | 1.82 | -0.45 | 0.01 | 0.12 | 0.25 | 0.02 | 0.5
Figure 5: Thirteen orbital basis based TB band structure of CsBX3 in comparison with DFT band structure obtained from GGA+mBJ exchange-correlation functional. Figure 6: Schematic representation hopping interaction in thirteen orbital basis based TB model and four orbital basis based TB model. Here, Fermi level is set to zero. Table 3: Orbital weight in-terms of % at conduction band minimum and valence band maximum of different halide perovskites. | | CBM | VBM |
---|---|---|---|---
B | X | B-p | B-s | X-p
| Cl | 100 | 65 | 35
Ge | Br | 100 | 63 | 37
| I | 100 | 65 | 35
| Cl | 100 | 69 | 31
Sn | Br | 100 | 67 | 33
| I | 100 | 67 | 33
| Cl | 100 | 66 | 34
Pb | Br | 100 | 61 | 39
| I | 100 | 60 | 40
Having understood about the full band spectrum of cubic halide perovskites,
now we aim to apply for lower symmetry polymorphs. However, looking at the
size and number atoms in the lower symmetry crystal structures, it is
difficult to trace and analyze a large number of interactions in these
systems. Therefore, it is necessary to develop minimal basis Hamiltonian
having less number of interaction without losing essential physics. The bands
forming the VBM and CBM at high symmetry points are primarily created by the
four B-{s, p} orbitals. From our calculations, we found that the contribution
B-p orbital at conduction band minimum is 100%. Whereas the valence band
maximum is made up of a linear combination of X-p and B-s characters as listed
in Table 3. As we can see, the contribution of X-p orbitals is approximately
half of B-s. Thus, there is room to minimize the number of interactions.
Therefore, we choose the basis set from B-{s, p}, where the interactions
between B-X-B will be included in B-B interactions as shown in Fig. 6. Thus,
the SOC incorporated four-band TB Hamiltonian in the matrix form can be
written as
$H_{TB}^{MB}=\left(\begin{array}[]{cc}H_{\uparrow\uparrow}&H_{\uparrow\downarrow}\\\
H_{\downarrow\uparrow}^{\dagger}&H_{\downarrow\downarrow}\end{array}\right),$
(11)
$\footnotesize
H_{{\uparrow\uparrow}}=\left(\begin{array}[]{cccc}\epsilon_{s}+h_{1}(\vec{k})&2i(t_{sp}^{x})^{AA}S_{x}&2i(t_{sp}^{x})^{AA}S_{y}&2i(t_{sp}^{z})^{AA}S_{z}\\\
-2i(t_{sp}^{x})^{AA}S_{x}&\epsilon_{p}^{x}+h_{2}(\vec{k})&-i\lambda&0\\\
-2i(t_{sp}^{x})^{AA}S_{y}&i\lambda&\epsilon_{p}^{x}+h_{3}(\vec{k})&0\\\
-2i(t_{sp}^{z})^{AA}S_{z}&0&0&\epsilon_{p}^{z}+h_{4}(\vec{k})\end{array}\right)$
(12)
Here, MB refers to minimal basis. The $\epsilon^{A}$s are band centres of the
anti-bonding bands, and $t^{A}$s are the second neighbour hopping integrals.
The dispersion functions $f_{i}$ ($i=1,2,3,4)$ are expressed in Eq. 8. The TB
bands obtained from this four-band model are shown in the Fig. 7 for nine
cubic halide perovskites and fitting parameters are listed in Table 4, and
they completely agree with DFT bands along the broad M-R-X $k$-path of the
cubic Brillouin zone. Thus, it validates the minimal basis set based model
which captures the essential features of halide perovskites around the Fermi
level with the less parametric quantities. In the next section, we further
validate this minimal basis set based TB model to lower symmetry polymorphs.
Figure 7: Band structure of various halide perovskites obtained from four orbital based TB Hamiltonian and compared with the DFT band structure. Here, Fermi level is set to zero. Table 4: Interaction parameters ($\epsilon^{A}$s and $t^{A}$s) and SOC strength $\lambda$ in units of eV. B | X | $\epsilon_{s}$ | $\epsilon_{p}$ | $t_{ss}$ | $t_{sp}$ | $t_{pp\sigma}$ | $t_{pp\pi}$ | $\lambda$
---|---|---|---|---|---|---|---|---
| Cl | 1.17 | 6.47 | -0.26 | 0.47 | 0.75 | 0.09 | 0.07
Ge | Br | 1.46 | 6.10 | -0.23 | 0.48 | 0.84 | 0.09 | 0.06
| I | 1.70 | 5.55 | -0.16 | 0.48 | 0.86 | 0.09 | 0.06
| Cl | 2.08 | 8.68 | -0.25 | 0.45 | 0.74 | 0.10 | 0.52
Sn | Br | 1.73 | 7.11 | -0.21 | 0.50 | 0.77 | 0.11 | 0.52
| I | 1.68 | 6.23 | -0.15 | 0.48 | 0.83 | 0.10 | 0.49
| Cl | 2.46 | 7.57 | -0.31 | 0.49 | 0.72 | 0.10 | 0.16
Pb | Br | 2.36 | 6.88 | -0.30 | 0.52 | 0.79 | 0.11 | 0.16
| I | 2.18 | 6.05 | -0.22 | 0.48 | 0.85 | 0.10 | 0.14
Figure 8: DFT fitted TB band structure of tetragonal and orthorhombic crystal
phases of various halide perovskites. Here, Fermi level is set to zero.
Figure 9: Comparison of various TB parameters of cubic halide perovskites
obtained from fitting with the DFT band structure under various exchange-
correlation functionals.
In the present section, we extend our minimal basis set TB model to tetragonal
and orthorhombic systems. In the tetragonal phase, the in-plane rotation of
the octahedra produces two in-equivalent B atoms. We denote them as BA and BB.
Now the basis set include eight eigenstates, viz, $|s^{B_{A}}\rangle$,
$|p^{B_{A}}_{x}\rangle$, $|p^{B_{A}}_{y}\rangle$,
$|p^{B_{A}}_{z}\rangle$,$|s^{B_{B}}\rangle$, $|p^{B_{B}}_{x}\rangle$,
$|p^{B_{B}}_{y}\rangle$, $|p^{B_{B}}_{z}\rangle$. The corresponding SOC
included 16$\times$16\. Thus, the spin-orbit coupled TB Hamiltonian is
envisaged on 16 orbital basis set. For the complete Hamiltonian ref, Kashikar,
Khamari, and Nanda (2018b). As the tetragonal phase having a pseudo-cubic
structure and creates anisotropic interactions in in-plane and out of plane
directions. Thus, there are two sets of TB parameters exist corresponding to
in-plane and out of plane directions. The DFT fitted TB band structures of
tetragonal phase for various halide perovskites are listed in Table 5. As can
be seen, the in-plane interactions’ strength is weak compared to out of plane
interactions. Similarly, in the case of the orthorhombic system, the rotation
along $c$ direction creates inequivalence of B atoms, and thus the Hamiltonian
matrix is designed for 32 orbitals basis set. In Fig. 8, we have presented the
DFT fitted TB band structure and fitting parameters are listed in Table 5.The
table infers that the interaction strength has decreased from that of the
cubic phase, due to octahedral rotations.
Table 5: Interaction parameters ($\epsilon$ and $t$) for unstained equilibrium configurations in the units of eV. The SOC strength ($\lambda$) is estimated to be 0.14 eV. Phase | Interaction | $\epsilon_{s}$ | $\epsilon_{p}$ | $t_{ss}$ | $t_{sp}$ | $t_{pp\sigma}$ | $t_{pp\pi}$
---|---|---|---|---|---|---|---
| Path | | | | | |
$\beta-$CsSnBr3 | $\hat{x}$,$\hat{y}$ | -1.52 | 2.99 | -0.27 | 0.47 | 0.8 | 0.09
| $\hat{z}$ | | 3.08 | -0.25 | 0.49 | 0.72 | 0.09
$\beta-$CsSnI3 | $\hat{x}$,$\hat{y}$ | -1.24 | 2.59 | -0.21 | 0.45 | 0.8 | 0.085
| $\hat{z}$ | | 2.64 | -0.21 | 0.43 | 0.74 | 0.08
$\beta-$CsPbBr3 | $\hat{x}$,$\hat{y}$ | -1 | 4.2 | -0.18 | 0.38 | 0.76 | 0.08
| $\hat{z}$ | | 4.24 | -0.16 | 0.38 | 0.66 | 0.07
$\beta-$CsPbI3 | $\hat{x}$,$\hat{y}$ | 0.76 | 3.54 | -0.13 | 0.38 | 0.74 | 0.08
| $\hat{z}$ | | 3.6 | -0.13 | 0.32 | 0.64 | 0.07
$\gamma-$CsSnBr3 | $\hat{x}$,$\hat{y}$ | -1.63 | 3.07 | -0.26 | 0.4 | 0.68 | 0.07
| $\hat{z}$ | | | -0.24 | 0 | 0.665 | 0.06
$\gamma-$CsSnI3 | $\hat{x}$,$\hat{y}$ | -1.23 | 2.66 | -0.18 | 0.38 | 0.64 | 0.08
| $\hat{z}$ | | | -0.24 | 0 | 0.665 | 0.06
$\gamma-$CsPbBr3 | $\hat{x}$,$\hat{y}$ | -0.9 | 4.23 | -0.14 | 0.4 | 0.68 | 0.05
| $\hat{z}$ | | | -0.16 | 0.04 | 0.7 | 0.07
$\gamma-$CsPbI3 | $\hat{x}$,$\hat{y}$ | -0.75 | 3.59 | -0.12 | -0.32 | 0.64 | 0.04
| $\hat{z}$ | | | -0.14 | 0.02 | 0.66 | 0.07
### III.3 Effect of Exchange-Correlation functionals on TB parameters.
Having validated about the minimal basis set TB model, now we examine how the
TB parameters are sensitive towards the XC functionals employed for DFT
calculations. From the Eq. 9, the bandgap can be defined as E3-E4, which is
thus the function of TB parameters. In the minimal basis set TB model, it
turns out to be a Eg =
$E_{p}^{B}-E_{s}^{B}-2t_{pp\sigma}^{B}-4t_{pp\pi}^{B}+6t_{ss}^{B}$. It can be
seen from the Fig. 3 and 4 that the bandgap of halide perovskites is highly
sensitive to the type of exchange-correlation functionals used in performing
DFT calculations. Thus, we would like to examine the the effect of XC
functionals on these effective parameters. In Fig. 9, we have shown the
effective TB parameters for three different XC viz., GGA, mBJ and Jishi-mBJ,
functional for cubic halide perovskites to know their dependency on XC
functionals. The figure infers that difference between the onsite energies
increases with the bandgap and exhibit maximum values for Jishi-mBJ XC
functional. Among all the hopping interactions, tss influence is profound on
the band spectrum and small change in the values affect the band topology
significantly.
### III.4 Role of $A$ Site Atom on the Electronic Structure.
To know the role of $A$ site atoms, we have investigated the band structure of
cubic RbPbX3 by placing the Rb atom at Cs site without relaxing the structure
and same is shown in the Fig. 10. Our results show a tiny increase in the
bandgap value of 0.02 eV for RbPbI3 as compared with the CsPbI3 and no
significant changes have been observed in band spaghetti. Structural
relaxation carried out on the RbPbX3 shows the reduced lattice parameters of
the cubic lattice as compared to CsPbX3. In contrast, the structural
relaxation study with the organic molecule at A site shows the larger volume
as compared to the CsPbX3. However, the organic molecule introduces structural
distortion in the octahedral cage, and compounds show noncentrosymmetric
nature and this, in turn, affects the band structure profoundly.
Figure 10: Band structure of CsPbI3 and RbPbI3 perovskites for the similar
lattice parameter.
### III.5 Surface Band Structure
Figure 11: Surface band structure of CsPbI3 polymorphs obtained from the SK-
TB formalism (upper panel) and compared with band structure of Wannier
function based TB model (lower panel).
The bulk electronic structure study has enabled us to analyze the bulk
properties of the halide perovskites. However, in many cases, the study
requires the surface electronic structure to understand the surface and
interface phenomena of these systems. The DFT based calculations on slab
structure require a huge amount of computational time and memory to analyze
these properties. Thus to overcome such difficulty, we build Slab TB model and
obtain the surface electronic structure. The slab TB model is envisaged by
taking four orbital basis of the proposed bulk TB model and according the
surface bands are estimated from the bulk Hamiltonian by using the bulk TB
parameters.
Here, the slab consists of alternative stacking of CsI and BI2 layers.
However, as discussed in section-II, there are no contribution X-p orbitals at
the Fermi level, the model is restricted to B-{s, p} orbital basis. The
appropriate Hamiltonian for the slab consisting of $n$ unit cells along (001)
direction. We employ bulk TB parameters and construct the Hamiltonian in the
desired direction. The matrix form the TB Hamiltonian is given by
$H_{TB}^{Slab}=\left(\begin{array}[]{cc}H_{\uparrow\uparrow}&H_{\uparrow\downarrow}\\\
H_{\downarrow\uparrow}&H_{\downarrow\downarrow}\end{array}\right).$ (13)
Where,
$H_{\uparrow\uparrow}=\left(\begin{array}[]{cccccc}H_{11}&H_{12}&0&0&0&\dots\\\
H_{21}&H_{22}&H_{23}&0&0&\dots\\\ 0&H_{32}&H_{33}&H_{34}&0&\dots\\\
\vdots&&\ddots&\ddots&\ddots\\\ \dots&0&0&H_{n-1n-2}&H_{n-1n-1}&H_{n-1n}\\\
\dots&0&0&0&H_{nn-1}&H_{nn}\end{array}\right)$ (14)
Here, $H_{jj}$ is the Hamiltonian for $j^{th}$ layer of the slab, and it is
given by,
$H_{jj}=\left(\begin{array}[]{cccc}\epsilon_{s}+f_{0}&2it_{sp}^{x}S_{x}&2it_{sp}^{x}S_{y}&0\\\
-2it_{sp}^{x}S_{x}&\epsilon_{p}^{x}+f_{1}&-i\lambda&0\\\
-2it_{sp}^{x}S_{y}&i\lambda&\epsilon_{p}^{x}+f_{2}&0\\\
0&0&0&\epsilon_{p}^{z}+f_{3}\end{array}\right)$ (15)
Here,
$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle
2t_{ss}^{x}cos(k_{x}a)+2t_{ss}^{x}cos(k_{y}a),$ $\displaystyle f_{1}$
$\displaystyle=$ $\displaystyle
2t_{pp\sigma}^{x}cos(k_{x}a)+2t_{pp\pi}^{x}cos(k_{y}a),$ $\displaystyle f_{2}$
$\displaystyle=$ $\displaystyle
2t_{pp\sigma}^{x}cos(k_{y}a)+2t_{pp\pi}^{x}cos(k_{x}a),$ $\displaystyle f_{3}$
$\displaystyle=$ $\displaystyle
2t_{pp\pi}^{x}cos(k_{x}a)+2t_{pp\pi}^{x}cos(k_{y}a).$ (16)
The block Hjj-1 describe the interaction between layer j and j-1.
$H_{j-1j}=(H_{jj-1})^{T}=\left(\begin{array}[]{cccc}t_{ss}^{z}&0&0&t_{sp}^{z}\\\
0&t_{pp\pi}^{z}&0&0\\\ 0&0&t_{pp\pi}^{z}&0\\\
-t_{sp}^{z}&0&0&t_{pp\sigma}^{z}\end{array}\right)$ (17)
The off diagonal block emerging from the SOC is of the form
$H_{\uparrow\downarrow}=\left(\begin{array}[]{cccccc}G_{11}&0&0&0&0&\dots\\\
0&G_{22}&0&0&0&\dots\\\ 0&0&G_{33}&0&0&\dots\\\
\vdots&&\ddots&\ddots&\ddots\\\ \dots&0&0&0&G_{n-1n-1}&0\\\
\dots&0&0&0&0&G_{nn}\end{array}\right)$ (18)
$G_{jj}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&\lambda\\\
0&0&0&-i\lambda\\\ 0&\lambda&-i\lambda&0\end{array}\right).$ (19)
The band spectrum obtained from the diagonalization surface TB matrix for
cubic, tetragonal and orthorhombic phases are shown in Fig. 11 and is compared
with Wannier functions based TB band structure.
## IV Conclusions
In conclusion, the present work, we have analyzed the family of halide
perovskites using both parameter free Density functional calculations and
parametric tight-binding (TB) model through the Slater-Koster description. The
analysis has unravelled the dominant orbital overlapping interactions that
govern the bandgap and play an important role in exploring the optoelectronic
applications and topological phases. Various exchange-correlation (XC)
functionals (PBE, PBE+mBJ, HSE06, HSE and PBE+J-mBJ) were employed on nine
halide perovskites in three structural polymorphs. While HSE and PBE+mBJ
underestimate the bandgap by a similar magnitude, PBE+J-mBJ either gives
bandgap close to the experimental value or overestimate it Jishi, Ta, and
Sharif (2014); Traoré _et al._ (2019). The corrections mBJ and J-mBJ adopts
the same formalism, but with varying parameters. Furthermore, the study
reveals that, though thirteen orbitals are involved in the chemical bonding, a
four orbital based tight-binding model is good enough to capture energy
dispersion in the momentum space in the vicinity of the Fermi level. The
successful extension of the present minimal basis TB model to other lower
symmetry crystal polymorphs and slab structure for various experimentally
synthesized compounds provides the effectiveness of this model. Interestingly,
the strength of the dominant electron hopping integrals are found to be nearly
independent of the XC functional adopted and therefore, the proposed TB model
has become more universal. Also, since it excellently reproduces the surface
band structures, it can be further improved to study the transport phenomenon
as the Wannier formalism does.
Acknowledgement: The work is funded by the Department of Science and
Technology, India, through Grant No. CRG/2020/004330. We acknowledge the use
of the computing resources at HPCE, IIT Madras.
Data availability: The data that support the findings of this study are
available from the corresponding author upon reasonable request.
## References
* Trots and Myagkota (2008) D. Trots and S. Myagkota, “High-temperature structural evolution of caesium and rubidium triiodoplumbates,” Journal of Physics and Chemistry of Solids 69, 2520 – 2526 (2008).
* Fujii _et al._ (1974a) Y. Fujii, S. Hoshino, Y. Yamada, and G. Shirane, “Neutron-scattering study on phase transitions of cspb${\mathrm{cl}}_{3}$,” Phys. Rev. B 9, 4549–4559 (1974a).
* Hirotsu _et al._ (1974a) S. Hirotsu, J. Harada, M. Iizumi, and K. Gesi, “Structural phase transitions in cspbbr3,” Journal of the Physical Society of Japan 37, 1393–1398 (1974a).
* Yang _et al._ (2017a) R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, and A. Walsh, “Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites cssnx3 and cspbx3 (x = f, cl, br, i),” The Journal of Physical Chemistry Letters 8, 4720–4726 (2017a).
* Yang _et al._ (2012) K. Yang, W. Setyawan, S. Wang, M. Buongiorno Nardelli, and S. Curtarolo, “A search model for topological insulators with high-throughput robustness descriptors,” Nature Materials 11, 614–619 (2012).
* Liu _et al._ (2016) S. Liu, Y. Kim, L. Z. Tan, and A. M. Rappe, “Strain-induced ferroelectric topological insulator,” Nano Letters 16, 1663–1668 (2016).
* Kashikar, Khamari, and Nanda (2018a) R. Kashikar, B. Khamari, and B. R. K. Nanda, “Second-neighbor electron hopping and pressure induced topological quantum phase transition in insulating cubic perovskites,” Phys. Rev. Materials 2, 124204 (2018a).
* Jin, Im, and Freeman (2012) H. Jin, J. Im, and A. J. Freeman, “Topological insulator phase in halide perovskite structures,” Phys. Rev. B 86, 121102 (2012).
* Shi _et al._ (2015) W.-J. Shi, J. Liu, Y. Xu, S.-J. Xiong, J. Wu, and W. Duan, “Converting normal insulators into topological insulators via tuning orbital levels,” Phys. Rev. B 92, 205118 (2015).
* Song _et al._ (2017) G. Song, B. Gao, G. Li, and J. Zhang, “First-principles study on the electric structure and ferroelectricity in epitaxial cssni3 films,” RSC Adv. 7, 41077–41083 (2017).
* Kepenekian _et al._ (2015) M. Kepenekian, R. Robles, C. Katan, D. Sapori, L. Pedesseau, and J. Even, “Rashba and dresselhaus effects in hybrid organic–inorganic perovskites: From basics to devices,” ACS Nano 9, 11557–11567 (2015).
* Yao, Xiao, and Niu (2008) W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77, 235406 (2008).
* Borriello, Cantele, and Ninno (2008) I. Borriello, G. Cantele, and D. Ninno, “Ab initio investigation of hybrid organic-inorganic perovskites based on tin halides,” Phys. Rev. B 77, 235214 (2008).
* Yang _et al._ (2017b) R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, and A. Walsh, “Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites cssnx3 and cspbx3 (x = f, cl, br, i),” The Journal of Physical Chemistry Letters 8, 4720–4726 (2017b).
* Yu _et al._ (2013) C. Yu, Y. Ren, Z. Chen, and K. Shum, “First-principles study of structural phase transitions in cssni3,” Journal of Applied Physics 114, 163505 (2013).
* Yang _et al._ (2017c) R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, and A. Walsh, “Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites cssnx3 and cspbx3 (x = f, cl, br, i),” The Journal of Physical Chemistry Letters 8, 4720–4726 (2017c).
* Huang and Lambrecht (2016) L.-y. Huang and W. R. L. Lambrecht, “Electronic band structure trends of perovskite halides: Beyond pb and sn to ge and si,” Phys. Rev. B 93, 195211 (2016).
* Jishi, Ta, and Sharif (2014) R. A. Jishi, O. B. Ta, and A. A. Sharif, “Modeling of lead halide perovskites for photovoltaic applications,” The Journal of Physical Chemistry C 118, 28344–28349 (2014), https://doi.org/10.1021/jp5050145 .
* Traoré _et al._ (2019) B. Traoré, G. Bouder, W. Lafargue-Dit-Hauret, X. Rocquefelte, C. Katan, F. Tran, and M. Kepenekian, “Efficient and accurate calculation of band gaps of halide perovskites with the tran-blaha modified becke-johnson potential,” Phys. Rev. B 99, 035139 (2019).
* Boyer-Richard _et al._ (2016) S. Boyer-Richard, C. Katan, B. Traoré, R. Scholz, J.-M. Jancu, and J. Even, “Symmetry-based tight binding modeling of halide perovskite semiconductors,” The Journal of Physical Chemistry Letters 7, 3833–3840 (2016).
* Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
* Hamann (1979) D. R. Hamann, “Semiconductor charge densities with hard-core and soft-core pseudopotentials,” Phys. Rev. Lett. 42, 662–665 (1979).
* Blaha _et al._ (2001) P. Blaha, K. Schwartz, G. Madsen, D. Kvasnicka, and J. Luitz, _WIEN2k An Augmanted Plane Wave+Local Orbitals Program for Calculating Crystal Properties_ (Karlheinz Schwartz, Tech. Universitt Wien, Austria, 2001).
* Perdew, Burke, and Ernzerhof (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996).
* Tran and Blaha (2009) F. Tran and P. Blaha, “Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential,” Phys. Rev. Lett. 102, 226401 (2009).
* Thiele, Rotter, and Schmidt (1987) G. Thiele, H. W. Rotter, and K. D. Schmidt, “Kristallstrukturen und Phasentransformationen von Caesiumtrihalogenogermanaten(II) CsGeX3 (X = Cl, Br, I),” ZAAC ‐ Journal of Inorganic and General Chemistry 545, 148–156 (1987).
* Yamada _et al._ (1994) K. Yamada, K. Isobe, T. Okuda, and Y. Furukawa, “Successive Phase Transitions and High Ionic Conductivity of Trichlorogermanate(II) Salts as Studied by 35Cl NQR and Powder X-Ray Diffraction,” Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 49, 258–266 (1994).
* Yang _et al._ (2017d) R. X. Yang, J. M. Skelton, E. L. Da Silva, J. M. Frost, and A. Walsh, “Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites CsSnX3 and CsPbX3 (X = F, Cl, Br, I),” Journal of Physical Chemistry Letters 8, 4720–4726 (2017d).
* Fabini _et al._ (2016) D. H. Fabini, G. Laurita, J. S. Bechtel, C. C. Stoumpos, H. A. Evans, A. G. Kontos, Y. S. Raptis, P. Falaras, A. der Ven, M. G. Kanatzidis, and R. Seshadri, “Dynamic Stereochemical Activity of the Sn2+ Lone Pair in Perovskite CsSnBr3,” Journal of the American Chemical Society 138, 11820–11832 (2016).
* Yamada _et al._ (1991) K. Yamada, S. Funabiki, H. Horimoto, T. Matsui, T. Okuda, and S. Ichiba, “Structural phase transitions of the polymorphs of cssni3 by means of rietveld analysis of the x-ray diffraction,” Chemistry Letters 20, 801–804 (1991), https://doi.org/10.1246/cl.1991.801 .
* Fujii _et al._ (1974b) Y. Fujii, S. Hoshino, Y. Yamada, and G. Shirane, “Neutron-scattering study on phase transitions of CsPbCl3,” Physical Review B 9, 4549–4559 (1974b).
* Hirotsu _et al._ (1974b) S. Hirotsu, J. Harada, M. Iizumi, and K. Gesi, “Structural phase transitions in cspbbr3,” Journal of the Physical Society of Japan 37, 1393–1398 (1974b), https://doi.org/10.1143/JPSJ.37.1393 .
* Marronnier _et al._ (2018) A. Marronnier, G. Roma, S. Boyer-Richard, L. Pedesseau, J. M. Jancu, Y. Bonnassieux, C. Katan, C. C. Stoumpos, M. G. Kanatzidis, and J. Even, “Anharmonicity and Disorder in the Black Phases of Cesium Lead Iodide Used for Stable Inorganic Perovskite Solar Cells,” ACS Nano 12, 3477–3486 (2018).
* (34) http://savrasov.physics.ucdavis.edu/mindlab/MaterialResearch/Databases/indexatoms.htm .
* Huang and Lambrecht (2013) L. Y. Huang and W. R. Lambrecht, “Electronic band structure, phonons, and exciton binding energies of halide perovskites CsSnCl3, CsSnBr3, and CsSnI3,” Physical Review B - Condensed Matter and Materials Physics 88, 1–12 (2013), arXiv:1510.01834 .
* Tao _et al._ (2019) S. Tao, I. Schmidt, G. Brocks, J. Jiang, I. Tranca, K. Meerholz, and S. Olthof, “Absolute energy level positions in tin- and lead-based halide perovskites,” Nature Communications 10, 1–10 (2019), arXiv:1902.06646 .
* Straus, Guo, and Cava (2019) D. B. Straus, S. Guo, and R. J. Cava, “Kinetically Stable Single Crystals of Perovskite-Phase CsPbI3,” Journal of the American Chemical Society 141, 11435–11439 (2019).
* Slater and Koster (1954) J. C. Slater and G. F. Koster, “Simplified lcao method for the periodic potential problem,” Phys. Rev. 94, 1498–1524 (1954).
* Goesten and Hoffmann (2018) M. G. Goesten and R. Hoffmann, “Mirrors of bonding in metal halide perovskites,” Journal of the American Chemical Society 140, 12996–13010 (2018), pMID: 30207152, https://doi.org/10.1021/jacs.8b08038 .
* Kashikar, Khamari, and Nanda (2018b) R. Kashikar, B. Khamari, and B. R. K. Nanda, “Second-neighbor electron hopping and pressure induced topological quantum phase transition in insulating cubic perovskites,” Phys. Rev. Materials 2, 124204 (2018b).
|
# Rapidity window dependence of ridge correlations in the glasma
Donghai Zhang Key Laboratory of Quark and Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
Yeyin Zhao Key Laboratory of Quark and Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China
Mingmei Xu<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics
(MOE) and Institute of Particle Physics, Central China Normal University,
Wuhan 430079, China Xue Pan School of Electronic Engineering, Chengdu
Technological University, Chengdu 611730, China Yuanfang Wu
<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China
###### Abstract
We study ridge correlations of the glasma in pp collisions at
$\sqrt{s_{\mathrm{NN}}}=7$ TeV by using the color glass condensate (CGC)
formalism. The azimuthal collimation at long range rapidity is intrinsic to
glasma dynamics and is reproduced here. When rapidity window enlarges, ridge
correlations in two dimensional $\Delta y$-$\Delta\phi$ distribution and one
dimensional $\Delta\phi$ distribution at long range rapidity gap are enhanced.
The enhancements are demonstrated to be the contributions of source gluons.
The quantum evolution of the gluons presents unique correlation patterns in
differential correlation function. These characters of two gluon correlations
open a way of testing the production mechanism from experimental measurements.
## I Introduction
The long range rapidity ridge correlation in small systems is one of the most
unexpected discoveries CMS-pp-JHEP-2010 ; CMS-pPb-PLB-2013 ; CMS-pPb-PbPb-
PLB-2013 ; ATLAS-pPb-PRL-2013 ; ALICE-pPb-PLB-2013 ; ALICE-pPb-PLB-2016 ;
ridge-RHIC-small-1 ; ridge-RHIC-small-2 . The structure of ridge in two
dimensional $\Delta\eta$-$\Delta\phi$ correlations is twofold: a long ridge in
$\Delta\eta$ direction and two peaks in $\Delta\phi$ direction. The two peaks
at $\Delta\phi=0$ and $\pi$ are also termed the collimation production,
revealing the collectivity at long range rapidity in small systems. The
physics underlying the collectivity at long range rapidity is open yet and
under extensive studies.
Final state interactions, especially the hydrodynamic description, combined
with a variety of initial conditions, attribute the long range ridge to the
fluid nature pp-hydro1 ; pp-hydro2 ; pp-hydro3 ; pp-hydro4 ; pp-hydro5 ; pp-
hydro6 . The transverse expansion of the fluid driven by the pressure gradient
can explain the azimuthal anisotropy of final state particles. The
difficulties of hydrodynamic description are that it can not reproduce the
multi-particle cumulant $c_{2}\\{4\\}$ hydro-cumu-1 ; hydro-cumu-2 and the
elliptic flow of heavy flavor particles hydro-flow-hf-1 ; hydro-flow-hf-2 . In
these respects, CGC calculations reproduce the data well cgc-c2 ; cgc-heavy .
Only initial correlations from CGC can explain the long range ridge yield very
well, too cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III . The trouble of CGC is that it
can not generate the right ordering of Fourier harmonics $v_{n}$ in the three
system sizes of pAu, dAu and HeAu cgc-vn-ordering . In a word, the two
mechanisms, i.e. hydrodynamics and CGC, can only describe some of the data in
small systems, respectively. Therefore, they both need further studies.
CGC describes the initial state of a colliding hadron or nucleus. The gluon
density grows rapidly and gets saturated as the collision energy increases or
as Feynman variable $x$ decreases. At saturation the small-$x$ gluons have
large occupation numbers (of order $1/\alpha_{s}$), so their mutual
interactions can be treated as the classical field with a strong field
strength (of order $1/\sqrt{\alpha_{s}}$). Thus two colliding hadrons or
nuclei at very high energies are two sheets of colored glass approaching one
another. After the collision, strong longitudinal color electric and magnetic
fields are formed in the region between the nuclei, which is called glasma.
The glasma fields are localized in the transverse scale (of order $1/Q_{s}$)
that are smaller than the nucleon size, which gives the picture of glasma flux
tube. The strong longitudinal color fields in glasma flux tubes are
approximately boost invariant and multi-particle productions in the small $x$
region naturally generate long range longitudinal correlations cgc-NPA-2008 ;
cgc-NPA-2010 .
The correlations between gluons in high energy scatterings are caused by the
quantum evolution, which is illustrated in Fig. 1(a). The short lines
represent valence quarks, which radiate gluons in the form of gluon cascades.
Besides the gluon radiation, when the gluon density gets large, recombinations
of radiated gluons (non-linear effects) are also important and shown in the
evolution. They result in the saturation phenomenon of the gluon distribution
at $k_{\perp}\ll Q_{s}$, while the gluon distribution at $k_{\perp}\gg Q_{s}$
follows a behavior of linear approximation. Therefore, $Q_{s}$, called
saturation momentum, as a scale separating linear and non-linear behavior, is
the only parameter which can determine the physics in a given collision. This
indicates an equivalence between nuclei and protons. The universality of
hadronic interactions at high energies in the CGC effective field theory
succeeds in explaining long range azimuthal correlations in pp, pPb
systematically cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III and may account for the
similarity between PbPb and pPb with the same multiplicity Dusling-PRL-2018 .
The CGC framework succeeds in explaining not only the multiplicity and
transverse momentum dependence, but also the $\Delta\eta$ acceptance
dependence of the long range azimuthal correlations in the LHC pPb data at
5.02 TeV cgc-PRD-III . With a common set of parameters, ref cgc-PRD-III gets
good fits to the CMS, ALICE and ATLAS data of different acceptances
simultaneously. The overall agreement with data is a spectacular achievement
of CGC. Because they considers the combined effect of rapidity acceptance and
normalizations of different LHC experiments, an explicit dependence on the
rapidity acceptance alone is not presented yet.
In high energy limit, both rapidity and transverse momentum of a gluon are
related to its Feynman $x$. For a right moving projectile, the relation reads
$x=\frac{\mathrm{p_{\perp}}}{\sqrt{s}}e^{y},$ (1)
with $y$ representing rapidity, $\mathrm{p_{\perp}}$ transverse momentum and
$\sqrt{s}$ center-of-mass energy. The dependence on rapidity is much more
sensitive due to the exponential function. At $\sqrt{s}=7$ TeV and
intermediate $\mathrm{p_{\perp}}$, e.g. 2 GeV$/$c, the rapidity region
$y\leqslant 1.2$ corresponds to $x\lesssim 10^{-3}$, while $y\leqslant 3.5$
corresponds to $x\lesssim 10^{-2}$. It means that gluons at central rapidity
region ($|y|\leqslant 1.2$) reflect properties of the small-$x$ ($x\lesssim
10^{-3}$), as the green band shows in Fig. 1(b). In that rapidity region
quantum evolutions are essential. In contrast, gluons at middle rapidity
regions ($1.2<|y|\leqslant 3.5$), as the blue bands show in Fig. 1(b), present
features of moderate-$x$ ($10^{-3}<x<10^{-2}$) degree of freedom. Within this
region, radiated gluons still dominate, but contributions of color sources
begin to show up. The large-$x$ ($x\gtrsim 10^{-2}$) degrees of freedom, i.e.
gluons at large rapidity ($|y|>3.5$) , act as sources and are referred to as
source gluons, which are denoted by red bands in Fig. 1(b).
It has been demonstrated that, the strong correlation between radiated gluons
and source gluons can explain long range rapidity correlations zhao-1 ; zhao-2
. However, ridge correlations in experiments are usually measured within
certain rapidity or pseudorapidity windows. For example, the pseudorapidity
windows of ALICE and CMS are [-0.9, 0.9] and [-2.4, 2.4], respectively. A
quantitative comparison of results from different experiments requires a study
of the rapidity window dependence of ridge correlations.
On the theoretical side, different rapidity windows are expected to reveal
gluon dynamics at different $x$ values. Fig. 2 demonstrates that different
rapidity windows ($Y_{\rm W}$ for short) pick pairs of different $x$ when a
given rapidity gap like $\Delta y=2$ is being concerned. A narrow rapidity
window only include contributions of radiated gluons, while a wider rapidity
window can further include correlations between radiated gluons and source
gluons. So, rapidity window dependence of long range ridge correlations is
sensitive to the quantum evolution of gluon saturation dynamics.
Considering the above reasons, it would be valuable to study the rapidity
window dependence of the ridge correlations systematically. This paper is
organized as follows. In section II, the definition of correlation function
and some related formulae of single- and double-gluon inclusive production in
CGC framework are given. The formulae in this manuscript follow those in Ref.
cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III and are identical with those at gluon
level without fragmentation functions. The new aspects of analysis method here
lie in the exploration of contributions of different $x$ degrees of freedom.
Results of correlations are shown and discussed in section III, where the
sensitivity to rapidity windows are carefully compared. Section IV gives the
summary and discussion.
Figure 1: (a) The quantum evolution of gluons in the right moving projectile.
The black axis represents the longitudinal momentum fraction $x$ of partons in
the projectile, and the blue axis roughly indicates corresponding rapidity
$y$. (b) The quantitative correspondence between $x$ regions and rapidity
regions for the case of $\mathrm{p_{\perp}}=$ 2 GeV$/$c in the collision of
$\sqrt{s}=7$ TeV. Figure 2: A schematic plot for pairs with rapidity
separation $\Delta y=2$. Rapidity window ($Y_{\rm W}$) of [-1, 1] only
includes pairs with $y_{\rm p}=-1$ and $y_{\rm q}=1$, as the green curve
shows. Rapidity window of [-2, 2] counts more pairs including pairs with
$y_{\rm p}=0$ and $y_{\rm q}=\pm 2$, as blue curves show. Rapidity window of
[-4, 4] further includes pairs with $y_{\rm p}=\pm 2$ and $y_{\rm q}=\pm 4$,
as red curves show.
## II Two-gluon $\Delta y$-$\Delta\phi$ correlations from high energy QCD
evolution
In a high energy collision, both the projectile and the target are regarded as
high parton density sources. When they pass through each other, strong
longitudinal color electric and magnetic fields are formed. The framework that
describes the physics of high parton densities and strong color fields is the
CGC effective field theory (CGC EFT) Gribov-1983 ; Iancu-2003 ; Weigert-2005 ;
Gelis-2010 . The effective degrees of freedom in this framework are color
sources $\rho$ at large $x$ and classical gauge fields $\mathcal{A}_{\mu}$ at
small $x$, as Fig.1(a) shows. The classical gauge field $\mathcal{A}_{\mu}$
can be obtained by numerically solving Yang-Mills equations with a given
configuration of color source. For a given initial configuration of color
source, fields in the nuclear wave functions evolve with Feynman $x$, which is
described by the Jalilian-Marian-Iancu-McLerran-Weigert-Kovner (JIMWLK)
renormalization group equations Jalilian-Marian-1997 ; Jalilian-Marian-PRD ;
Iancu-2001 . In mean field approximation and large-$N_{c}$ limit, the JIMWLK
equation is reduced to the Balitsky-Kovchegov (BK) equation Balitsky1996 ;
Balitsky1999 ; Kovchegov1999 , which describes the quantum evolution in Fig.
1(a).
Supposing two gluons are produced with transverse momentum
$\mathbf{p}_{\perp}$, $\mathbf{q}_{\perp}$ and rapidity $y_{\rm p}$, $y_{\rm
q}$, the correlation function is defined as,
$\displaystyle C(\mathbf{p}_{\bot},y_{\rm p};\mathbf{q}_{\bot},y_{\rm q})$
$\displaystyle=$
$\displaystyle\frac{\frac{dN_{2}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}}{\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}\frac{dN_{1}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}-1=\frac{\frac{dN_{\mathrm{2}}^{\rm corr}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}}{\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}\frac{dN_{1}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}},$ (2)
where $\frac{dN_{2}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}}$,
$\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}$ are the double- and single-
gluon inclusive productions and $\frac{dN_{2}^{\rm
corr}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}}$
is the correlated double-gluon production. Subscripts “p” and “q” are used to
mark the two gluons.
In the framework of CGC EFT, for a given collision, the observable under the
leading-log approximation is factorized as cgc-NPA-2008 ,
$\displaystyle\left\langle\mathcal{O}\right\rangle_{\mathrm{LLog}}$
$\displaystyle=$
$\displaystyle\int[D\rho_{1}][D\rho_{2}]W[\rho_{1}]W[\rho_{2}]\mathcal{O}[\rho_{1},\rho_{2}]_{\mathrm{LO}},$
(3)
where $\mathcal{O}[\rho_{1},\rho_{2}]_{\mathrm{LO}}$ is leading-order single-
or double- gluon inclusive distribution for a fixed distribution of color
sources, and the integration denotes an average over different distributions
of color sources with the weight functional $W[\rho_{1,2}]$. In general,
$W[\rho_{1,2}]$ encodes all possible color charge configurations of the
projectile and target, and obeys the JIMWLK renormalization group equations
Jalilian-Marian-1997 ; Jalilian-Marian-PRD ; Iancu-2001 . All the quantum
evolution of the projectile/target is absorbed into the distribution
$W[\rho_{1,2}]$.
The averaging over color sources can be done under the McLerran-Venugopalan
(MV) model with a Gaussian weight functional. According to Ref. cgc-PRD-I ;
cgc-PRD-II ; cgc-PRD-III , the correlated two-gluon production can be
expressed by unintegrated gluon distributions (uGD) as,
$\displaystyle\frac{dN_{\mathrm{2}}^{\mathrm{corr}}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm
q}}}=\frac{C_{2}}{\mathbf{p}^{2}_{\bot}\mathbf{q}^{2}_{\bot}}\left[\int\frac{d^{2}\mathbf{k}_{\bot}}{(2\pi)^{2}}(D_{1}+D_{2})+\sum_{j=\pm}\left[D_{3}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})+\frac{1}{2}D_{4}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})\right]\right],$
(4)
where
$C_{2}=\frac{\alpha_{s}(\rm{p}_{\bot})\alpha_{s}(\rm{q}_{\bot})N_{c}^{2}S_{\bot}}{\pi^{8}(N_{c}^{2}-1)^{3}}$,
and
$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle\Phi_{A_{1}}^{2}(y_{\rm
p},\mathbf{k}_{\bot})\Phi_{A_{2}}(y_{\rm
p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot})\left[\Phi_{A_{2}}(y_{\rm
q},\mathbf{q}_{\bot}+\mathbf{k}_{\bot})+\Phi_{A_{2}}(y_{\rm
q},\mathbf{q}_{\bot}-\mathbf{k}_{\bot})\right],$ $\displaystyle D_{2}$
$\displaystyle=$ $\displaystyle\Phi_{A_{2}}^{2}(y_{\rm
q},\mathbf{k}_{\bot})\Phi_{A_{1}}(y_{\rm
p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot})[\Phi_{A_{1}}(y_{\rm
q},\mathbf{q}_{\bot}+\mathbf{k}_{\bot})+\Phi_{A_{1}}(y_{\rm
q},\mathbf{q}_{\bot}-\mathbf{k}_{\bot})].$ (5)
Here, $\Phi_{A_{1(2)}}(y,\mathbf{k}_{\bot})$ denotes the uGD of projectile
$A_{1}$ or target $A_{2}$. In Eq.(4),
$D_{3}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})=\delta^{2}(\mathbf{p}_{\bot}+j\mathbf{q}_{\bot})\left[\mathcal{I}_{1}^{2}+\mathcal{I}_{2}^{2}+2\mathcal{I}_{3}^{2}\right],$
(6)
with
$\displaystyle\mathcal{I}_{1}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2})^{2}}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(7) $\displaystyle\mathcal{I}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{|\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot}|^{2}}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(8) $\displaystyle\mathcal{I}_{3}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2})|\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot}|}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
(9)
and
$\displaystyle D_{4}(\mathbf{p}_{\bot},j\mathbf{q}_{\bot})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}_{1\bot}}{(2\pi)^{2}}\Phi_{A1}(y_{\rm
p},\mathbf{k}_{1\bot})\Phi_{A1}(y_{\rm p},\mathbf{k}_{2\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})\Phi_{A2}(y_{\rm
q},\mathbf{p}_{\bot}-\mathbf{k}_{2\bot})$ (10) $\displaystyle\times$
$\displaystyle\frac{(\mathbf{k}_{1\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}^{2}){(\mathbf{k}_{2\bot}\cdot\mathbf{p}_{\bot}-\mathbf{k}_{2\bot}^{2})}+(\mathbf{k}_{1\bot}\times\mathbf{p}_{\bot})\cdot(\mathbf{k}_{2\bot}\times\mathbf{p}_{\bot})}{\mathbf{k}_{1\bot}^{2}(\mathbf{p}_{\bot}-\mathbf{k}_{1\bot})^{2}}$
$\displaystyle\times$ $\displaystyle\frac{(\mathbf{k}_{1\bot}\cdot
j\mathbf{q}_{\bot}-\mathbf{k}_{1\bot}^{2}){(\mathbf{k}_{2\bot}\cdot
j\mathbf{q}_{\bot}-\mathbf{k}_{2\bot}^{2})}+(\mathbf{k}_{1\bot}\times\mathbf{q}_{\bot})\cdot(\mathbf{k}_{2\bot}\times\mathbf{q}_{\bot})}{\mathbf{k}_{2\bot}^{2}(j\mathbf{q}_{\bot}-\mathbf{k}_{1\bot})^{2}},$
where
$\mathbf{k}_{2\bot}\equiv\mathbf{p}_{\bot}-\mathbf{k}_{1\bot}+j\mathbf{q}_{\bot}$.
The single-gluon inclusive production reads
$\displaystyle\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm p}}$
$\displaystyle=$ $\displaystyle\frac{\alpha_{s}({\rm
p}_{\bot})N_{c}S_{\bot}}{\pi^{4}(N_{c}^{2}-1)}\frac{1}{\mathbf{p}_{\bot}^{2}}\int\frac{d\mathbf{k}_{\bot}^{2}}{(2\pi)^{2}}\Phi_{A}(y_{\rm
p},\mathbf{k}_{\bot})\Phi_{A}(y_{\rm p},\mathbf{p}_{\bot}-\mathbf{k}_{\bot}).$
Based on correlation function $C(\mathbf{p}_{\bot},y_{\rm
p};\mathbf{q}_{\bot},y_{\rm q})$, the associated yield per trigger is defined
as
$\displaystyle Y(\Delta\phi,\Delta
y)=\frac{1}{N_{\rm{Trig}}}\frac{d^{2}N_{\rm{Assoc}}}{d\Delta\phi d\Delta y},$
(12)
where
$\displaystyle\frac{d^{2}N_{\rm{Assoc}}}{d\Delta\phi d\Delta y}$
$\displaystyle=$ $\displaystyle\int_{y^{\rm min}}^{y^{\rm max}}dy_{\rm
p}\int_{y^{\rm min}}^{y^{\rm max}}dy_{\rm q}\delta(y_{\rm q}-y_{\rm p}-\Delta
y)\int_{0}^{2\pi}d\phi_{\rm p}\int_{0}^{2\pi}d\phi_{\rm q}\delta(\phi_{\rm
q}-\phi_{\rm p}-\Delta\phi)$ (13) $\displaystyle\times\int^{{\rm
p}^{\rm{max}}_{\bot}}_{{\rm p}^{\rm{min}}_{\bot}}\frac{d{\rm
p}^{2}_{\bot}}{2}\int^{{\rm q}^{\rm{max}}_{\bot}}_{{\rm
q}^{\rm{min}}_{\bot}}\frac{d{\rm
q}^{2}_{\bot}}{2}\frac{dN^{\rm{corr}}_{\mathrm{2}}}{{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}{d^{2}\mathbf{q}_{\bot}dy_{\rm q}}},$
and
$\displaystyle N_{\rm{Trig}}$ $\displaystyle=$
$\displaystyle\iiint_{\mathrm{Acceptance}}dyd^{2}\mathbf{p}_{\bot}\frac{dN_{1}}{d^{2}\mathbf{p}_{\bot}dy_{\rm
p}}.$ (14)
The associated yield per trigger $Y(\Delta\phi,\Delta y)$ describes
correlations of two gluons with rapidity separation $\Delta y$ and azimuthal
separation $\Delta\phi$ in given transverse momentum intervals
($\mathrm{p^{min}_{\bot}}$, $\mathrm{p^{max}_{\bot}}$),
($\mathrm{q^{min}_{\bot}}$, $\mathrm{q^{max}_{\bot}}$) and rapidity window
$(y^{\rm min},y^{\rm max})$. In the following we do not distinguish between
trigger gluons and associated gluons, but we still use the name “associated
yield per trigger” for simplicity.
The framework is valid to leading-logarithmic accuracy in $x$ and momentum
${\rm p}_{\bot}$, ${\rm q}_{\bot}\mathrm{\gtrsim}Q_{s}$. The important
ingredient in the above expressions is uGD ($\Phi$), which can be obtained by
solving the BK equation with running coupling corrections with a given initial
condition. To avoid repetition, details can be found in Ref. cgc-NPA-2010 and
our previous paper zhao-1 ; zhao-2 . For pp collision at $7\ \mathrm{TeV}$,
$Q^{2}_{s0}$ (with $Q_{s0}$ the initial value of $Q_{s}$ at $x_{0}$) is chosen
to be $0.168\ \mathrm{GeV^{2}}$ cgc-PRD-I .
The solution of the BK equation is reliable when $x\leqslant 0.01$. For larger
values of $x$, the BK description breaks down. A phenomenological
extrapolation for the unintegrated gluon distribution has the form
$\Phi(x,\mathbf{k}_{\bot})=(\frac{1-x}{1-x_{0}})^{\beta}\Phi(x_{0},\mathbf{k}_{\bot}),\mbox{
for }x>x_{0},$ (15)
where $x_{0}=0.01$ and the parameter $\beta=4$cgc-NPA-2010 .
Both the experimental data CMS-pp-JHEP-2010 ; ALICE-pPb-PLB-2016 and the CGC
cgc-PRD-I ; cgc-PRD-II ; cgc-PRD-III show that the correlation at near side
gets strongest only in an intermediate $\mathrm{p_{\bot}}$ interval,
approximately $1<\mathrm{p_{\bot}}<3\ \mathrm{GeV/c}$. A calculation from the
CGC points out that the correlation function gets a maximum when the
transverse momenta are close to $\mathrm{2Q_{sp}}\sim 2\ \mathrm{GeV/c}$ for
minimum-bias pp collisions, where $\mathrm{Q_{sp}}$ denotes the saturation
momentum of proton zhao-1 . To obtain the strongest correlation, the
correlation function is integrated within
$1\leq\mathrm{p_{\bot}}(\mathrm{q_{\bot}})\leq 3\ \mathrm{GeV/c}$ here.
## III Rapidity window dependence of the ridge correlations
| |
---|---|---
Figure 3: The associated yield per trigger on $\Delta y$-$\Delta\phi$ plane
for 7 TeV pp collisions with transverse momentum integrated within $1\leq
p_{\perp}(q_{\perp})\leq 3$GeV$/$c and rapidity integrated in (a) $-0.9\leq
y_{\rm p}(y_{\rm q})\leq 0.9$, (b) $-2.4\leq y_{\rm p}(y_{\rm q})\leq 2.4$ and
(c) $-4.2\leq y_{\rm p}(y_{\rm q})\leq 4.2$, respectively.
Ridge correlations are usually illustrated by two dimensional $\Delta
y$-$\Delta\phi$ distribution and one dimensional $\Delta\phi$ distribution. In
this paper we start with the calculation of the $\Delta y$-$\Delta\phi$
distribution, i.e. the associated yield per trigger in Eq. (12), which is
widely measured in CMS, ALICE and ATLAS experiments.
In order to study the rapidity window dependence of the long range ridge
correlations, three rapidity windows, i.e. $[-0.9,0.9],[-2.4,2.4]$ and
$[-4.2,4.2]$, are used. The correlations in pp collisions at $\sqrt{s}$ = 7
TeV with transverse momentum integrated within
$1\leq\mathrm{p_{\bot}}(\mathrm{q_{\bot}})\leq 3\ \mathrm{GeV/c}$ are plotted
on $\Delta y$-$\Delta\phi$ plane in Fig. 3.
The rapidity window $[-0.9,0.9]$ is chosen to be the same with the ALICE
acceptance. As Fig. 3(a) shows, the correlations have two moderate peaks of
equal height at $\Delta\phi=0$ and $\pi$ for a fixed $\Delta y$. For a fixed
$\Delta\phi$, the correlations show a downward trend as $|\Delta y|$
increases.
As rapidity window extends to $[-2.4,2.4]$, i.e. the CMS acceptance, the peaks
at $\Delta\phi=0$ and $\pi$ get pronounced. The double ridges, i.e. two raised
lines along $\Delta y$ direction at $\Delta\phi=0$ and $\pi$ on the surface
plot, are clearly seen in Fig. 3(b). The range of $\Delta y$ is only plotted
to $\pm 2$ so that a direct comparison can be made with Fig. 3(a). For longer
range ($|\Delta y|>2$) the ridges extend continuously which are not shown in
the plot.
When rapidity window extends to $[-4.2,4.2]$, the surface in Fig. 3(c) is
overall uplifted compared to Fig. 3(a) and 3(b). The correlations at long
range rapidity are uplifted more, as the color at large $|\Delta y|$ changes
from blue in Fig. 3(a) to light blue in Fig. 3(b) to green in Fig. 3(c). When
it comes to the $\Delta\phi$ direction, the amplitude of $\Delta\phi$
distribution, both at short range and long range rapidity gap, increases
significantly with increasing rapidity window.
Fig. 3 demonstrates an enhancement of correlations at long range rapidity gap
for larger $Y_{\rm W}$. As $Y_{\rm W}$ increases, rapidity correlations at
fixed $\Delta\phi$ change from a steep to a relatively flat shape, indicating
that long range rapidity correlations increase with rapidity window. On the
other hand, the amplitude of $\Delta\phi$ distribution at fixed $\Delta y$
also increases with rapidity window.
The downward trend in $\Delta y$ direction is gentle in the ALICE measurement
of the associated yield per trigger in pPb collisions at a center-of-mass
energy of 5.02 TeV ALICE-pPb-PLB-2013 . Compared to the ALICE results, the
downward trend here is much more steep. When rapidity window extends to
$[-2.4,2.4]$, ridges that can extend to $|\Delta y|=4$ are not as flat as
those obtained in CMS measurements, either CMS-pp-JHEP-2010 . It shows that
glasma graphs have significant short range rapidity correlations zhao-2 .
Of particular interest in studies of ridge is the long range (large $|\Delta
y|$) structure of two-particle correlation functions. In order to get a better
view of the ridge correlation at long range rapidity and its enhancement, we
plot one dimensional $\Delta\phi$ distribution at fixed long range rapidity
gap in Fig. 4(a). For $Y_{\rm W}$ of $[-0.9,0.9]$, the $\Delta\phi$
distribution at $\Delta y=1.8$ (1.8 units are the maximum rapidity gap for
that rapidity window) is straight, as the black curve shows. In fact, the two
peaks are still there. They are not visible in the plot just because the
amplitude is small. For $Y_{\rm W}$ of $[-2.4,2.4]$, the $\Delta\phi$
distribution at $\Delta y=2$ has a larger amplitude and the peaks at 0 and
$\pi$ is clearly seen, as the blue curve shows. As rapidity window gets larger
to $[-4.2,4.2]$, the peaks get more pronounced and the amplitude of the
distribution grows larger, as the red curve shows.
|
---|---
Figure 4: (a) The $\Delta\phi$ distribution at a fixed long range rapidity
gap. (b) The $\Delta\phi$ distribution with a constant background subtracted
by the ZYAM method.
According to the zero-yield-at-minimum (ZYAM) method note , the integrated
associated yield is defined as the area under the peak of the $\Delta\phi$
distribution above a constant background. The $\Delta\phi$ distribution with a
constant background subtracted by the ZYAM method is shown in Fig. 4(b). ZYAM
subtraction scheme makes the signals cleaner. The biggest amplitude is the red
curve, corresponding to the widest rapidity window. The area under the peaks
for the three curves qualitatively shows that the integrated associated yield
grows drastically with rapidity window.
The two peak structure of $\Delta\phi$ distribution at long range rapidity for
the three rapidity windows confirms an intrinsic collimation production in the
CGC formalism. Quantitative calculations about the near side ridge yield in pp
and pPb collisions are consistent with CMS, ATLAS and ALICE data cgc-PRD-I ;
cgc-PRD-II ; cgc-PRD-III . It means that small $x$ dynamics can explain the
ridge yield. The enhancement at wider rapidity window, demonstrated in Fig.
4(b) of this paper, indicates that large $x$ gluons, i.e. source gluons, make
significant contributions to the ridge yield.
The ridge structure is twofold. The azimuthal collimation in CGC comes from
Cauchy-Schwartz inequality Dumitru-PLB-2011 , which states that the azimuthal
correlations get maximum when the two transverse momenta are parallel or
antiparallel. The ridge at long range rapidity is due to the contribution of
color sources. Beyond that, we also demonstrate that source gluons enhance the
azimuthal collimation, indicating an interplay between longitudinal and
transverse directions.
The rapidity window of track reconstruction from the ALICE detector is very
different from the CMS and ATLAS. Whether the data from the three experiments
at the LHC shows the increasing trend with rapidity window as shown in Fig.
4(b) is hard to conclude at present. As mentioned before, long range angular
correlations are sensitive to centrality class and $\mathrm{p_{\bot}}$ window.
In this paper we show that long range angular correlations are also sensitive
to rapidity window. In order to see the trend with rapidity window, the
correlations should be compared at the same centrality and the same
$\mathrm{p_{\bot}}$ window. The ALICE collaboration reports long range angular
correlations in a centrality bin that is different from CMS and ATLAS CMS-pPb-
PLB-2013 ; ALICE-pPb-PLB-2013 ; ATLAS-pPb-PRL-2013 , which makes a direct
comparison between the three experiments impossible at the moment.
Enlarging the rapidity window is one way of seeing the contributions of source
gluons. Obviously, the integration over a given rapidity window includes
different combinations of gluons. In order to further clarify the
contributions of source gluons, a more direct observable is the differential
correlation function $C(\mathbf{p}_{\bot},y_{\rm p};\mathbf{q}_{\bot},y_{\rm
q})$, i.e. Eq.(2) with specific $y_{\rm p}$ and $y_{\rm q}$, which is shown in
Fig.5.
Figure 5: The differential correlation function at
$\mathrm{p_{\bot}}=\mathrm{q_{\bot}}=2.0$ GeV/c with $\phi_{\rm p}=\phi_{\rm
q}=0$. Black, blue and red curves denote $y_{\rm p}=0$, 2 and 3.5,
respectively.
A similar figure has already been shown in ref zhao-1 . The motivation of
presenting it here again is to understand the integrated correlations by the
differential correlations. The enhanced correlations originating from source
gluons is reflected more directly by the rapidity location dependence of the
differential correlation function.
The differential correlation function in Fig. 5 shows some interesting
correlation patterns. In the case of $y_{\rm p}=0$, i.e. the trigger particle
is at small $x$, the correlation function is almost flat and shows a rising
trend when $\Delta y$ exceeds 3 as the black curve shows. The solid, dash and
dash-point lines denote $x_{\rm q}<10^{-3}$, $10^{-3}<x_{\rm q}<10^{-2}$ and
$x_{\rm q}>10^{-2}$, labeling the results of the small, moderate and large-$x$
degrees of freedom of the associated particle, respectively. The point
connecting the dash line and dash-point line represents $x_{0}=0.01$. Since
quantitative results of the extrapolation Eq. (15) is unreliable, we only take
the dash-point lines that are near the connection points as large-$x_{\rm q}$.
A rapidity difference of only 2 units is sensitive to the small $x$ evolution,
denoted by the solid lines in Fig. 5. A wider region is sensitive to the
moderate and large $x$ degrees of freedom. The black curve indicates that the
correlation strength between radiated gluons (r for short), moderate-$x$
gluons (m for short) and source gluons (s for short) has
$C_{\rm rr}\approx C_{\rm rm}<C_{\rm rs},$ (16)
where each subscript represents a kind of gluon. It shows that correlations
between radiated gluons are not as strong as those with source gluons.
Similarly, the rising trend of the blue curve in Fig. 5 indicates
$C_{\rm mr}<C_{\rm mm}<C_{\rm ms},$ (17)
and the sharp rise of the red curve indicates
$C_{\rm sr}\ll C_{\rm sm}\ll C_{\rm ss}.$ (18)
$C_{\rm rr}$, $C_{\rm rm}$ and $C_{\rm ms}$ represents the green, blue and red
curves in Fig. 2, respectively. The strong correlations between the
moderate-$x$ gluon and the source gluon, i.e. $C_{\rm ms}$, make the ridge
yield enhanced significantly, as the red curve in Fig. 4(b) shows.
The ordering in Eqs. (16)-(18) demonstrates that correlations with source
gluons are the strongest for each type of trigger gluon. For trigger gluons at
non-central rapidity, like the case of $y_{\rm p}=2$ and 3.5, correlations
with small-x, moderate-x and large-x gluons show a continuously increasing
trend. The strongest correlations exist between source gluons. These ordering
patterns, especially the continuously rising trend of correlations between
trigger particle at non-central rapidity and associated particle at other
rapidities demonstrate characteristic features of CGC.
## IV Summary and discussion
According to the relation of Feynman $x$ to rapidity $y$, different rapidity
regions reveal gluon dynamics at different $x$ and thus the rapidity window
dependence of long range ridge correlations is sensitive to the quantum
evolution of gluon saturation dynamics.
The ridge structure is twofold, i.e. the longitudinal and transverse
structures. Two dimensional $\Delta y$-$\Delta\phi$ distributions illustrate
that ridge structures in both directions have rapidity window dependence. As
the rapidity window enlarges, the longitudinal structure has a rising and
flattening trend at long range rapidity gap. It indicates an enhancement of
long range rapidity correlations. On the other hand, the angular distribution
at long range rapidity gap shows larger amplitudes as rapidity window
enlarges. It indicates the enhancement of azimuthal correlations and the
resulting ridge yields. In summary, ridge correlations in both longitudinal
and transverse directions get stronger as the rapidity window enlarges.
Within the CGC framework, the azimuthal collimation comes from Cauchy-Schwartz
inequality which requires parallel or antiparallel of two gluon transverse
momenta to get the maximum correlations. The long range rapidity correlations
is due to the contribution of color sources. As the rapidity window enlarges,
source gluons are included and begin to play a role. The enhancement in both
longitudinal and transverse correlations demonstrates that source gluons
enhance the long range rapidity correlations and azimuthal collimation.
While the origin of the ridge in small systems is inconclusive, it is already
widely known that the color flux tube picture naturally explains the long
range rapidity correlation and angular collimation within the CGC framework.
This paper notices that the rapidity of gluons is related to the quantum
evolution and the inclusion of large rapidity, or equivalently the source
gluons, will enhance the long range rapidity ridge correlations significantly.
So far the correlation patterns calculated are for gluons. It is shown in Ref.
Dusling-PRL-2012 that fragmentation functions only have a major impact on
transverse momentum dependence. The correlation patterns as a function of
rapidities would remain in the final state.
Since the quantum evolution of the CGC gives characteristic rapidity location
dependence of the differential correlation, differential measurements are more
direct for probing the CGC and are highly needed in the future measurement.
The correlation patterns of one particle at non-central rapidities with the
other particle at other rapidities can provide sensitive tests of this
picture. Similar suggestions have been made in Ref. Dumitru-PLB-2011 .
The work of Ref.Schenke-PRL-2016 where IP-glasma model is combined with Lund
string fragmentation demonstrates similar ridge structures. Flatter ridges
along rapidity direction shown there are a result of an exactly boost
invariant distribution implemented. The similarity in ridge structures
indicates that the long range rapidity ridge correlation in high energy
hadronic collisions is intrinsic to glasma dynamics.
Ultra-long range rapidity ridge, up to nearly 8 units of pseudorapidity
separation, has been observed in a preliminary result of ALICE experiment
ultra-long . The picture of classical fields has a restricted range of
validity. In a larger range of rapidity, quantum fluctuations become
important, and the description in terms of classical fields breaks down. For
large rapidity, one has less theoretical control within this framework. The
contribution of quarks should be considered.
## Acknowledgement
This work is supported in part by the NSFC of China under Grant No. U1732271.
## References
* (1) CMS Collaboration, JHEP 09, 091 (2010).
* (2) CMS Collaboration, Phys. Lett. B 718, 795 (2013).
* (3) CMS Collaboration, Phys. Lett. B 724, 213 (2013).
* (4) ATLAS Collaboration, Phys. Rev. Lett. 110, 182302 (2013).
* (5) ALICE Collaboration, Phys. Lett. B 719, 29 (2013).
* (6) ALICE Collaboration, Phys. Lett. B 753, 126 (2016).
* (7) A. Adare et al.(PHENIX Collaboration), Phys. Rev. Lett. 114, 192301 (2015).
* (8) A. Adare et al.(PHENIX Collaboration), Phys. Rev. Lett. 115, 142301 (2015).
* (9) K. Werner, I. Karpenko, and T. Pierog, Phys. Rev. Lett. 106, 122004 (2011).
* (10) P. Bozek and W. Broniowski, Phys. Lett. B 718, 1557(2013).
* (11) G.Y. Qin and B. Muller, Phys. Rev. C 89, 044902 (2014).
* (12) A. Bzdak, B. Schenke, P. Tribedy, and R. Venugopalan, Phys. Rev. C 87, 064906 (2013).
* (13) K. Werner, M. Bleicher, B. Guiot, I. Karpenko, and T. Pierog, Phys. Rev. Lett. 112, 232301 (2014).
* (14) K. Werner, B. Guiot, I. Karpenko, and T. Pierog, Phys. Rev. C 89, 064903 (2014).
* (15) W. Zhao, Y. Zhou, H. Xu, W. Deng, H. Song, Phys. Lett. B 780, 495 (2018).
* (16) B. Schenke, C. Shen, and P. Tribedy, arXiv:1908.06212.
* (17) CMS Collaboration, Phys. Lett. B 791, 172 (2019).
* (18) X. Du and R. Rapp, JHEP 03, 015 (2019).
* (19) K. Dusling, M. Mace, and R. Venugopalan, Phys. Rev. Lett. 120, 042002 (2018).
* (20) C. Zhang, C. Marquet, G.Y. Qin, S.Y. Wei, and B.W. Xiao, Phys. Rev. Lett. 122, 172302 (2019).
* (21) K. Dusling and R. Venugopalan, Phys. Rev. D 87, 051502(R) (2013).
* (22) K. Dusling and R. Venugopalan, Phys. Rev. D 87, 054014 (2013).
* (23) K. Dusling and R. Venugopalan, Phys. Rev. D 87, 094034 (2013).
* (24) M. Mace, V.V. Skokov, P. Tribedy, and R. Venugopalan, Phys. Rev. Lett. 123, 039901(E) (2019).
* (25) A. Dumitru, F. Gelis, L. McLerran, and R. Venugopalan, Nucl. Phys. A 810, 91 (2008).
* (26) K. Dusling, F. Gelis, T. Lappi and R. Venugopalan, Nucl. Phys. A 836, 159 (2010).
* (27) K. Dusling, M. Mace, and R. Venugopalan, Phys. Rev. Lett. 120, 042002 (2018).
* (28) Ye-Yin Zhao, Ming-Mei Xu, Heng-Ying Zhang and Yuan-Fang Wu, Nucl. Phys. A 955, 88 (2016).
* (29) Ye-Yin Zhao, Ming-Mei Xu, Heng-Ying Zhang and Yuan-Fang Wu, arXiv:1709.08678.
* (30) L.V. Gribov, E. M. Levin, and M. G. Ryskin, Phys. Rep. 100, 1 (1983).
* (31) E. Iancu and R. Venugopalan, arXiv:hep-ph/0303204.
* (32) H. Weigert, Prog. Part. Nucl. Phys. 55, 461 (2005).
* (33) F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, Annu. Rev. Nucl. Part. Sci. 60, 463 (2010).
* (34) J. Jalilian-Marian, A. Kovner, L.D. McLerran, and H. Weigert, Phys. Rev. D 55, 5414 (1997).
* (35) J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Phys. Rev. D 59, 014014 (1998).
* (36) E. Iancu, A. Leonidov, and L.D. McLerran, Phys. Lett. B 510, 133 (2001).
* (37) I. Balitsky, Nucl. Phys. B 463, 99 (1996).
* (38) I. Balitsky, Phys. Rev. D 60, 014020 (1999).
* (39) Y.V. Kovchegov, Phys. Rev. D 60, 034008 (1999).
* (40) An implementation of the zero-yield-at-minimum (ZYAM) method firstly calculates the integral of $Y(\Delta y,\Delta\phi)$ over $2.0<|\Delta y|<4.8$ to give $R(\Delta\phi)$. Then a second-order polynomial is fitted to $R(\Delta\phi)$ in the region $0.5<|\Delta\phi|<1.5$. The location of the minimum of the polynomial in this region is denoted $\Delta\phi_{\rm ZYAM}$. Using the position of the minimum, the integrated associated yield is then calculated as the integral of $R(\Delta\phi)$ minus its value at $\Delta\phi_{\rm ZYAM}$ over the region $0<|\Delta\phi|<\Delta\phi_{\rm ZYAM}$.
* (41) Kevin Dusling and Raju Venugopalan, Phys. Rev. Lett. 108, 262001 (2012).
* (42) A. Dumitru, K. Dusling, F. Gelis, J. Jalilian-Marian, T. Lappi, R. Venugopalan, Phys. Lett. B 697, 21 (2011).
* (43) B. Schenke, S. Schlichting, P. Tribedy, R. Venugopalan, Phys. Rev. Lett. 117, 162301 (2016).
* (44) M. Weber for the ALICE Collaboration, Highlights from ALICE, talk given in Quark Matter 2019.
|
# Correlated Power Time Series of Individual Wind Turbines: a Data Driven
Model Approach
Tobias Braun
Faculty of Physics
University of Duisburg-Essen
Lotharstrasse 1, 47057 Duisburg, Germany
& Matthias Waechter
ForWind $\&$ Institute of Physics
Carl-von-Ossietzky University Oldenburg
Küpkersweg 70, 26129 Oldenburg, Germany & Joachim Peinke
ForWind $\&$ Institute of Physics
Carl-von-Ossietzky University Oldenburg
Küpkersweg 70, 26129 Oldenburg, Germany & Thomas Guhr
Faculty of Physics
University of Duisburg-Essen
Lotharstrasse 1, 47057 Duisburg, Germany Complexity Science (RD4), Potsdam
Institute for Climate Impact Research, 14473 Potsdam, Germany, Tel.:
+49-331-28820744<EMAIL_ADDRESS>
###### Abstract
Wind farms can be regarded as complex systems that are on the one hand coupled
to the nonlinear, stochastic characteristics of weather and on the other hand
strongly influenced by supervisory control mechanisms. One crucial problem in
this context today is the predictability of wind energy as an intermittent
renewable resource with additional non–stationary nature. In this context, we
analyse power time series measured in an offshore wind farm for a total period
of one year with a time resolution of ten minutes. Applying Detrended
Fluctuation Analysis, we characterize the autocorrelation of power time series
and find a Hurst exponent in the persistent regime with crossover behaviour.
To enrich the modelling perspective of complex large wind energy systems, we
develop a stochastic reduced–form model of power time series. Observed
transitions between two dominating power generation phases are reflected by a
bistable deterministic component while correlated stochastic fluctuations
account for the identified persistence. The model succeeds to qualitatively
reproduce several empirical characteristics such as the autocorrelation
function and the bimodal probability density function.
_Keywords_ wind energy $\cdot$ power generation $\cdot$ stochastic modelling
$\cdot$ correlation $\cdot$ detrended fluctuation analysis
## 1 Introduction
In the context of anthropogenic climate change, the challenge of reducing
carbon emissions is of central importance. Renewable sources of energy are
considered to be one of the most promising solutions in the electricity sector
to cover an increasing energy demand without exacerbating the high carbon
emissions coupled to it. Wind energy in particular appears to be one of the
most strongly increasing sources of renewable energy [1, 2] but demands an
extraordinary adaption of grids and related power systems due to its
intermittent nature [3, 4]. It consequently raises the need for a profound
understanding of this intermittency and the opportunity to perform extensive
studies on the reliability of power systems by suitable models. Such models
can only be calibrated with respect to empirical data while different
approaches are required to reflect features on multiple spatial and temporal
scales [5, 6]. Moreover, they can be used to study the impact of certain well
known statistical characteristics on the resulting dynamics. This provides
wind farm controllers with a rich set of tools to analyse variable scenarios
taking influences into account that are known to be of importance.
In this work, we focus on the dynamics of single wind turbine (WT) power
generation in an offshore wind farm. While especially power generation of
aggregated wind farms or even complexes of several wind farms have received
considerable attention, the challenge of intermittent and stochastic
characteristics is particularly high on the scale of single turbines [7].
Still, it does not vanish for larger units like wind farms or national wide
wind power generation [8]. How generated power fluctuations from
geographically separated wind farms are dampened when aggregated [9, 10] is of
crucial importance for large scale grid stability [11, 12].
Our perspective on this problem is to study the autocorrelation of power time
series as an indicator of their intermittent and nonstationary nature in a
first step. In most modelling approaches, information on the complex temporal
evolution of variations is included and hence autocorrelation is of great
general importance. Nonstationarity is another feature that is frequently
encountered dealing with complex systems [13]. We address this within the
analysis of power output correlations using the popular method of Detrended
Fluctuation Analysis (DFA) [14]. It can deal with polynomial nonstationarities
in a generalized fashion. The fluctuations around these polynomial trends are
subject to the correlation analysis which yields a Hurst exponent as an
indicator for the strength and nature of the autocorrelation.
The idea that power time series should exhibit scaling behaviour is based on
the finding that the underlying wind speed dynamics are governed by
atmospherical turbulence [15]. The traditional tool to capture this aspect is
the power spectrum obtained by Fourier analysis [16]. Several studies have
examined the power spectral density both for wind speed [17, 18] and power
time series [19, 20], also uncovering how aggregation on different spatial
scales impacts power fluctuations [21, 10]. Other classical methods range from
structure functions to estimators for the Hurst exponent based on scale
dependent variance calculation on the respective time series [22]. Still,
these methods do not take the above mentioned nonstationarity into account. To
this extent, several works study the autocorrelation of wind speed time series
$u(t)$ applying DFA and Multifractal DFA (MFDFA) and conclude that it behaves
persistent with a multifractal dependence [23, 24, 25]. While most research in
this context focuses on local wind speed measurements of single sites, some
studies reveal multifractality of wind speed on a larger spatial scale. For
instance, [26] incorporates wind speed data spatially distributed all over
Switzerland and finds multifractal dependence of persistence for the
cooperative behaviour in the context of monitoring systems. Similar research
for power time series appears more limited. The authors of [27, 28] reveal a
high degree of multifractality for both wind speed and power time series of an
aggregated wind farm and join both in a description via generalized
correlation functions. Furthermore, aggregated power output of wind farms is
known to show complex correlations in terms of persistence and
multifractality. The authors in [29] uncover multifractality for power time
series of an aggregated wind farm in South Australia with data on a similar
time scale. They classify power time series in the persistent regime,
especially on short time scales of several minutes. Yet to our best knowledge,
no research has employed DFA or multifractal methods focusing on single WTs’
power output.
After we have obtained a better understanding of autocorrelations, we put
forward a model based on these theoretical insights and an important feature
of the empirical probability density function (PDF) of power output. A broad
range of approaches is considered for wind power generation models in the
respective literature. A general distinction of models can be based on whether
power is directly or indirectly modelled, e.g. by mapping certain variables
like measured wind speed on power via a distinct transformation. Our work
contributes to direct power modelling of time series for single WTs. Most
models aim at finding a precise point forecast of time series for a certain
time scale and horizon [30]. In contrast to that, other effort is put into
modelling different properties related to power output such as the power curve
[31, 32], wind power ramps [33] or power density estimates [34]. Models also
vary in terms of the time scale [35] that ranges from ultra–short–term
$(\mathrm{ms}$-$\mathrm{s})$[36] to long–term forecasts (months) [37].
The model proposed in this work aims at modelling short–term power time series
without the objective to give precise point forecasts. Instead, we aim at
deepening the systemic understanding of a WT due to its stochastic nature on
the one hand and the impact of control mechanisms (e.g. curtailment) on the
other hand. Following this motivation, the model is based on a single
nonlinear stochastic differential equation that is split into two components
in a Langevin fashion. Since WTs are directly coupled to the complex
atmospherical dynamics of wind, it is paramount to incorporate a stochastic
component that allows for a certain degree of complex diffusive behaviour. A
sufficient approach to include such complex diffusive dynamics without a loss
of simple applicability is given by Fractional Gaussian Noise (FGN) [38]. Time
series generated as FGN entail a certain degree of correlation and yield
Fractional Brownian Motion when cumulated. Thus we are able to reflect on the
results from the correlation analysis from a model perspective. The second
component of our model takes the impact of control mechanisms on power output
into account. We address this feature in two steps: a deterministic component
in the differential equation accounts for the characteristical shape of the
PDF as a first mechanism to focus the power values around the respective fixed
points. The second step incorporates control mechanisms such as curtailing in
a simplified numerical fashion. Finally, we include a simple first approach to
account for a time dependent seasonal drive of power output aswell. By
constructing our model in such a way, its easily distinguishable components
and parameters give way to a better understanding of how certain theoretical
features affect power time series qualitatively, such as the degree of
autocorrelation. It further yields the opportunity to test parametrized
scenarios and may be used in large power network simulations based on
simplified models of single wind turbine dynamics.
The paper is organised as follows: We present the data set and perform a
cleansing procedure on it in sec. 2 such that we can get a first impression of
its fundamental characteristics afterwards. In sec. 3, we identify the
autocorrelation of power time series via the traditional autocorrelation
function and the method of DFA. The stochastic model we introduce in sec. 4 is
based on these empirical findings and will enlarge upon our understanding of
how varying autocorrelations have an impact on fundamental statistical
features by comparison with the data. When we present the results, we will
find a sufficient agreement between the empirical and the modelled features.
We summarize our results in sec. 5.
## 2 Data Treatment and Characteristic Features
We briefly introduce the data set in 2.1. The data cleansing procedure
described in 2.2 is important to focus only on a reasonable subset of the
empirical data [39]. Finally, we get a first glance of the most substantial
characteristic features of the wind farm data in 2.3.
### 2.1 Data Set
The data set we analyse comprises time series of 30 WTs located at the German
offshore windfarm RIFFGAT. Several observables are measured via a SCADA
(Supervisory Control and Data Acquisition) system, however we will focus only
on the active power output of the WTs. The respective time series cover a
total period of one year between 01/03/2014 and 28/02/2015. All analysis is
carried out on ten minute average values calculated from data points measured
with 1 Hz frequency. Thus the timestamp precision is limited to ten minutes
and we obtain 52560 values in total.
### 2.2 Data Cleansing
In the following we briefly present the applied data cleansing procedure. We
detect outliers and ensure that we only consider a reasonable subset of the
initially measured data. Every data value we consider to be erroreneous will
be set to NA (not available) and is not included in any upcoming analysis. In
a first step we ensure that there are no redundant timestamps in any time
series and look for consecutive identical values. Since one ten minute average
value is based on 600 measured values, it can be rated as a highly unlikely
case that two consecutive 10–minute averages are identical with a five digit
precision in the data. This could only be conceivable if e.g. constant rated
power is generated for ten minutes without any variation. To rule out this
case, we also took the maximum and minimum values for the respective ten
minute intervals into account.
Apart from the ten minute average values, we inspect the respective standard
deviations. Any data value with a vanishing ten minute standard deviation is
set to NA. In a last step we analyse if there are unreasonable changes of
consecutive power values which we will call power increments. We consider each
power increment to be unreasonably high (regardless of its direction) if it
meets both of the following criteria: the power increment
$\Xi_{+}=\left.\left(P(t+1)-P(t)\right)\middle/P_{+}\right.$ relative to the
so called rated power output $P_{+}$ of the WT is higher than a certain
threshold $\Xi_{0}$ and the respective minimum power $P_{\mathrm{min}}$ in a
ten minute time interval is higher than the maximum power $P_{\mathrm{max}}$
in the previous time interval by a certain factor $q$:
$P_{\mathrm{min}}\,>\,qP_{\mathrm{max}}$ which yields unphysical data. We
choose the threshold value $\Xi_{0}$ and the factor $q$ in a way that limits
extreme power increments to a typical value found in the respective literature
[40]. This yields $\Xi_{0}=0.67$ and $q=0.99$. The higher these values are
chosen, the more unphysical ramps are still kept in the data which results in
a higher number of strong jumps between low and high power generation. If we
apply a too strict choice, some of the true strong ramps that resemble
intermittent fluctuations are spuriously eliminated, also biasing results for
temporal correlations. After applying the stated cleansing steps, we obtain
$9.58\%$ of NA values in the data. We will only consider pairs of values
containing no NA value in every calculation of correlations between time
series.
### 2.3 Bimodality and Power Increments
To achieve a sufficient understanding of wind power data, some basic facts
about the conversion of wind speed $u$ into active power output $P$ and the
control of WTs have to be outlined [41, 42]. Although an increase in wind
speed obviously leads to a higher gain of generated power in general, WTs only
run within a certain operating range. This limitation is due to the finite
performance of power generators. In fact, the operation of WTs is limited by a
lower cut-in value $u_{-}$ and an upper cut-off value $u_{+}$ of wind speed
$u$. Below $u_{-}$, there is simply not enough wind energy for an economic use
of the turbine so that $P=P_{-}(=0\,\mathrm{kW}\,)$. When $u$ exceeds $u_{+}$,
power is controlled to remain constant at the rated power output
$P=P_{+}(=3600\,\mathrm{kW}\,)$. For even higher wind speeds, it becomes
essential to avoid mechanical damage and the WTs are turned down by a
continuous adjustment of the rotor blades. Within $u_{-}\leq u\leq u_{+}\,$,
the generated power of an ideal WT increases proportionally to $u^{3}$. For
this work the most important conclusion to draw from these control mechanisms
is that power time series have to be at least in parts artificially flattened.
We expect power values to be constant at $P=P_{-}$ or $P=P_{+}$ for certain
time periods.
This manifests itself in fig.1(a) as a striking bimodal pattern in the five
displayed time series and a striking bimodality of the empirical PDF in
fig.1(b), visualized as a histogram. Most power values are concentrated around
zero power generation $P_{-}$ and active power output $P_{+}$. In fact, the
intervals $50\,\mathrm{kW}<P<200\,\mathrm{kW}$ and
$3550\,\mathrm{kW}<P<3800\,\mathrm{kW}$ around the two peaks sum up to
$41.62\%$ of all values. This feature also governs the dynamics observed from
the time series and is observed in several different works [43, 34, 44, 45].
Nevertheless, other analyses [46, 47] also show unimodal distributions around
$P_{-}$ or flatter, less concentrated distributions. While the first
observation is related to the different efficiencies of WTs, the latter is
mostly found for aggregated power of several wind farms where WTs with
different rated power outputs are combined. Despite of this bimodal shape of
the PDF, also values exceeding the rated power can be observed. Finally,
strong downward ramps to zero power output can be observed for some of the
WTs.
(a) Power time series.
(b) Empirical PDF as histogram of all power values.
Figure 1: Exemplary power time series and average power output (black)
covering two weeks. The respective PDF plotted as a histogram is shown for all
WTs covering the entire time period.
We will characterize such power ramps by the increments
$\Xi_{k}(t)=P_{k}(t+\Delta t)-P_{k}(t)$ which are a fundamental property for
the understanding and management of wind farms. We define them as the
standardized, dimensionless differences
$\displaystyle\tilde{\Xi}_{k}(t)\,=\,\frac{\Xi_{k}(t)\ -\
\frac{1}{T}\sum_{t=1}^{T}\Xi_{k}(t)}{\sqrt{\frac{1}{T}\sum_{t=1}^{T}\left(\Xi_{k}(t)-\mu_{k}\right)^{2}}}$
(1)
For our data, we only analyse $\Delta t=10\,\mathrm{min}$. A visual inspection
of fig.2(a) shows an example of this property for one WT in a time period of
two weeks. Apparently, increments of similar size cluster in time, indicating
some degree of correlation. This generally indicates that simple random walk
models do not yield a sufficient description that captures the complex
temporal correlations of time series. Power seems to fluctuate symmetrically
but clearly in a non–Gaussian fashion as it can be seen in the PDF of all
$\Xi_{k}(t)$ for the total time period in fig.2(b). Here we compare the
empirical distribution to a Gaussian (black) with the same mean value and
standard deviation. For the aggregated wind farm, the strong fluctuations
appear slightly dampened. These findings are in accordance with results in the
literature for increments on even shorter time scales[8].
(a) Power increment time series.
(b) Single–logarithmic histogram of all power increments.
Figure 2: Exemplary power increment time series covering two weeks and
respective single–logarithmic histogram for all WTs covering the entire time
period, compared to a Gaussian distribution. The dashed line includes power
increments of the aggregated wind farm.
If the PDF deviates from Gaussain statistics we speak of intermittent behavior
after Kolmogorov 1962, expressed by a heavy tailed distribution and
multifractal statistics. Note this turbulent intermittency term is not the
same as the alternative denotation of intermittency, i.e. nonstationary time
series switching between different flow states like between laminar and
turbulent flows.
## 3 Autocorrelation
Both the bimodality of power, persisting at values around zero and rated power
output, and the complex dynamics of power increments suggest that an analysis
of correlations can be fruitful. As a first step, we display linear
autocorrelations for power time series $P_{k}(t)$.
To this extent, we use the Pearson Correlation with a time delay $\tau$
defined by
$\displaystyle\Theta(\tau)\,=\,\left.\left[\langle
X(t)X(t+\tau)\rangle_{t}\,-\,\left(\langle
X(t)\rangle_{t}\right)^{2}\right]\middle/\langle(X(t))^{2}\rangle\right.$ (2)
Quantifying autocorrelations of power time series yields valuable information
on how power can be generally modelled (sec. 4). In this paper, we only
consider autocorrelation of power time series. With $\Theta(\tau)$ as a
correlation measure we only account for linear dependence. Moreover, it is
sensible to outliers and only gives sufficient information for time series
with finite variance. Figure 3(a) shows $\Theta(\tau)$ for power time series
of all 30 WTs. It is shown up to a lag of seven days which is approximately
the point where they drop below a significant level (dashed horizontal lines).
As such we use a simple surrogate approach and shuffle the time series,
eliminating temporal information but preserving the PDF. We identify a
(constant) confidence band by calculating its width as $2\sigma$ of the
autocorrelation after the shuffeling process, averaged over all time series.
The autocorrelation of all $P_{k}(t)$ slowly decreases over three orders of
magnitude and thus indicates long–range dependence. Nevertheless,
$\Theta(\tau)$ does not show a typical power law decay but runs through
several local maxima, reflecting the inherent nonstationarity. In [48] an
explanation for this observation is provided that corroborates our results:
the autocorrelation of wind speed decreases with slightly visible maxima due
to weather related seasonalities. Since power is closely coupled to wind speed
and persists at an almost constant level for values around $u_{-}$ and
$u_{+}$, the local maxima are not only sustained but amplified. The displayed
autocorrelations do finally not show significant differences between single
WTs even though it is known that relative positions of WTs play a role for
power generation, e.g. through wind shear effect [49]. Such differences could
potentially become visible in lagged cross–correlations between WTs of varying
relative position which is not within the scope of this work though.
(a) Linear autocorrelations of all $P_{k}(t)\,$.
(b) Fourier power spectrum of exemplary $P_{k}(t)\,$.
Figure 3: Autocorrelation and Fourier spectrum of power time series. (a):
$\Theta(\tau)$ of all 30 WTs power time series for $\tau\leq
1\,\mathrm{week}\,$. The average autocorrelation is drawn as a black dashed
line. The dashed horizontal lines represent a level of significance. (b):
Fourier power spectrum of an exemplary power time series with log-log-scale.
The time series was split into ten equally sized segments for which the power
spectra were computed (example: gray) and averaged subsequently (black). The
orange line shows the linear regression performed within the fitted region
(dashed lines), the red line displays a scaling with $\beta=5/3$. The three
additional spectra are equivalently computed for the optimized model time
series (see. fig.8) with the respective Hurst exponent and shifted for better
visibility.
The scaling behaviour of power time series is expected to be related to that
of wind time series. For the latter, it is well known that the Fourier power
spectrum obtained yields a power law decay $E(f)\propto f^{-\beta}$ with
$\beta=\left.5\middle/3\right.$ estimated through linear regression in the
log-log plot. This finding is in accordance with results one obtains from
Kolmogorov’s turbulence model. A simple way to confirm that similar scaling
can be found for the power time series, we estimate $\beta$ from a respective
Fourier power spectrum in fig.3(b). To this extent, we compute a smoothed
spectrum (black) by splitting the full time series $P(t)$ of an exemplary WT
into ten chunks and average over spectral power values obtained for each of
the single segments (example in gray). The linear MLE regression (orange)
yields $\beta=1.58\approx\left.5\middle/3\right.$ within a linear region of
sufficient width and thus matches both the expected universal scaling (red)
and results from similar data well [50].
The presence of nonstationarity leads to biased or spurious detection of
autocorrelations [51]. To uncover autocorrelations in presence of
nonstationary we apply DFA [52] to the power time series $P_{k}(t)$. The main
idea of DFA is to eliminate nonstationarity in a generalized fashion by
subtracting polynomial trends and consequently focusing on correlations that
are present in the remaining noise. The method aims at calculating the Hurst
exponent $H$, introduced by Hurst in 1951 [53]. We distinguish between
persistent ($0.5<H<1$) and antipersistent ($0<H<0.5$) behaviour of a time
series. The special case $H=0.5$ yields diffusive behaviour i.e. uncorrelated
white noise with $\langle X^{H}(t)X^{H}(t^{\prime})\rangle=0$ for the time
series $X(t)$ at arbitrary times $t$ and $t^{\prime}$. Cumulated white noise
entails Brownian Motion. The persistent regime $H>0.5$ results in
super–diffusive dynamics and long–range dependence of the increments $X(t)$.
Antipersistence yields the opposite case, i.e. a time series changes its
direction more frequently than a diffusive time series. The respective
extension of a white noise process is called Fractional Gaussian Noise (FGN)
and will be adressed later.
We now briefly sketch the method of DFA and refer to [52] for a more detailed
description. As a first step we calculate the integrated, mean adjusted power
time series. A time series of equal length is obtained that is splitted into
$\,N_{s}=\lfloor T/s\rfloor\,$ disjunct subsets of equal length $s$. The
brackets round the value of $T/s$ down to an integer value. We repeat this
step with the reversed time series to incorporate all subsets. Subsequently, a
polynomial quadratic detrending via MLE–regression (Maximum Likelihood
Estimation) is applied for all subsets respectively. We then calculate the
standard deviation $F_{\nu}(s)$ of all detrended subsets and from that derive
the average standard deviation. Repeating this calculation for different $s$,
we obtain the fluctuation function
$\displaystyle F(s)\ =\
\sqrt{\frac{1}{2N_{s}}\sum\limits_{\nu=1}^{2N_{s}}F_{\nu}^{2}(s)}$ (3)
that represents the scale dependent fluctuation strength. We estimate the
scaling exponent $\alpha$ from $F(s)\propto s^{\alpha}$ empirically via linear
regression in a double–logarithmic plot since we expect $F(s)$ to increase as
a power law. For stationary time series (e.g. FGN [22]), we can identify
$\alpha$ with the Hurst exponent $H$. In the more general case frequently
encountered for real data, even for the detrended time series some
nonstationarity remains present. In this case, the Hurst exponent can only be
estimated as $H\approx\alpha-1$ which only strictly holds for a Fractional
Brownian Motion (FBM) process. This distinction is sometimes not pointed out
distinctively in the literature [54]. In general, $\alpha$ is also linked to
the Fourier scaling exponent $\beta$ via $\beta=2\alpha-1=1+2H$, enabling us
to assess the consistency of our results.
For applications, $s$ must not be chosen too small for a significant outcome.
The DFA method is known to cause misleading finite size effects with respect
to short time scales $s$ and the applied detrending generally needs to be
regarded critically [55]. Another typical issue are values at the limits or
outside of the range $0<\alpha<1$. For $\alpha>1$, we have to consider the
time series as an integrated process with more complex nonstationarities
($H=1.5$ equals Brownian Motion) [56, 57]. Apart from that, the resulting
increase of $F(s)$ does not have to be completely linear but can contain
crossovers with different slopes [58]. In many cases, these crossovers are
meaningful and uncover different scaling regions that entail different
correlations [14]. As we see in fig.4, the displayed fluctuation functions
follow power laws with similar scaling exponents $\alpha$. We checked that the
displayed results are robust to different orders of polynomial detrending.
Three shifted curves are shown for single WTs and one for the aggregated
windfarm. A linear increase with one clearly visible crossover
$s_{\mathrm{c}}$ at a time scale of approximately three days can be identified
for all of the curves. The resulting scaling exponents are $\alpha=1.33$ for
$s\leq s_{\mathrm{c}}$ and $\alpha=0.80$ for $s>s_{\mathrm{c}}\,$, averaged
over all obtained $\alpha$ for the different turbines. The values are almost
identical for the aggregated windfarm. The dashed line emphasizes that all
found results in fact give meaningful information about the correlations and
not only the distribution. The line refers to a stationary random surrogate
time series we obtained with the same procedure explained above and applying
DFA afterwards. A value of $H_{\mathrm{surr}}=0.49$ reflects diffusive
behaviour. Consequently, for short time scales $s\leq s_{\mathrm{c}}$ power
time series have to be regarded as a highly nonstationary stochastic process
with trends that can not be sufficiently eliminated by DFA. If we suppose the
validity of $H\approx\alpha-1$ for $s\leq s_{\mathrm{c}}$, we obtain $H\approx
0.33$ matching $H\approx 0.5(\beta-1)=0.29$ derived from the Fourier spectrum
in fig.3(b). For $s>s_{\mathrm{c}}\,$, the scaling exponent $\alpha=0.80$
yields persistence and a Hurst exponent $H\approx-0.20$. The latter finding is
consistent with the displayed autocorrelation functions and values of $\alpha$
found in the literature [24, 29, 50].
The same type of crossover behaviour is also found for hourly wind speed data
at geographically different sites [54]. The authors conjecture that this
scaling behaviour might arise from the different scales of weather patterns
which would manifest itself in a multifractal spectrum of different scaling
exponents. Yet, the crossover is not critically reflected on, even though
misleading crossovers in DFA are known to appear with several known causes. In
[14] it is suggested to test whether different orders of polynomial detrending
change the crossover position which we ensured not to occur. Furthermore, the
number of NA–values does not have a significant impact on our results as
supposed in [59]. Although the scaling law of $F(s)$ is still valid for
$\alpha>1$ [60], this special case hints that unidentified trends remain after
the detrending procedure. Such trends are also known as a source of erroneous
crossovers [61, 62] which we can not fully rule out. If such trends were the
cause, similar trends would also be present in the wind speed data in [54]
though. The consequences of such a crossover still seem to be of potential
importance for possible modelling approaches which will be adressed in sec. 4.
.
Figure 4: Fluctuation function $F(s)$ for three exemplary WTs and aggregated
WF (crosses). $s$ is given in units of 10 min. The curves are shifted in
$y$–direction for better visibility. The red line displays the MLE–regression
with shaded error region. The gray lines visualize the crossover. The dashed
line shows $F(s)$ for a surrogate of randomly shuffled data.
## 4 Stochastic Model
The identified features of single WT power time series motivate a model
approach that could be calibrated with empirical data. We now put forward such
a time series modelling approach that has the potential to be used in a larger
simulation, aiming at a multi-scale model of wind power generation. In sec.
4.1, we introduce our model approach. We compare our model results to the
empirical data in sec. 4.2 to gain first qualitative insights into the scope
of the model.
### 4.1 Model Approach
Numerous approaches of power time series $P(t)$ simulations are employed in
the literature. A major fraction tends to simulate wind farm power time series
only and does not concentrate on single WTs as we intend to do [63, 64, 65,
35]. Another central distinction is the aim of the model. Our reduced-form
approach does not intend to reproduce temporally ordered forecasts but only
statistical features such as distributions and long–term averages. Several
models in the literature manage to give precise forecasts of different time
horizons of $P(t)$ using black-box models with a high number of parameters or
simple regression parameters that often lack a comprehensive contextual
meaning. Typical model approaches in this context are Markov processes [66,
48, 67] and ARIMA (Autoregressive integrated moving average) processes [37]
which both focus on the autocorrelation of $P(t)$. Further approaches are
based on nonlinear models [68], stochastic models [43] and stochastic
drift–diffusion–models [20]. In contrast to the stochastic process approach in
[20] which aims to model the stochastic wind speed / wind power relation in
seconds by the estimators of Kramers–Moyal coefficients, we aim here to
achieve a stochastic modelling of the power output based on 10 minute values.
We will try to introduce our model parameters with a comprehensible meaning.
Furthermore, our model equation itself is not a regression formula but is
based on our qualitative understanding of power time series. The simulations
are carried out on a 10–minute time scale. The essence of our model comes down
to one fundamental stochastic differential equation (SDE) based on central
statistical features we identified in sec. 2 and sec. 3. We do not model NA-
values separately and aim at simulating the raw uncleansed data that is
directly obtained from the SCADA system [69]. For the sake of simplicity all
model-related equations are formulated in a notation for continuous systems,
yet baring in mind that we are dealing with discrete data. We now briefly
introduce the stochastic model equation and summarize the related mathematical
concepts. In our modelling framework, we will regard power generation as an
autocorrelated stochastic process with a deterministic and a stochastic
component. These drift–diffusion–types of models are often expressed by a
Langevin equation of the general form
$\displaystyle\dfrac{\mathrm{d}P}{\mathrm{d}t}(t)\ =\
-k\,\dfrac{\mathrm{d}V(P)}{\mathrm{d}P}\,+\,D\,\xi_{H}(t)\qquad\quad.$ (4)
Here, $P(t)$ denotes the power time series. The first term equals the
deterministic drift component with a drift parameter $k$ and a potential
function $V(P)$. The second component incorporates stochastic fluctuations
$\xi_{H}(t)$ with a constant diffusion strength $D$. The index $H$ denounces
the Hurst exponent which hints at the fact that we can include arbitrary power
law-like correlations into our model. Thus, $\xi_{H}(t)$ itself is a solution
to a simple FGN–stochastic process [70] with the defining property
$\displaystyle\langle\xi_{H}(t)\xi_{H}(t^{\prime})\rangle\ =\
\frac{1}{2}\left(|t|^{2H}\,+\,|t^{\prime}|^{2H}\,-\,|t-t^{\prime}|^{2H}\right)\,.$
(5)
In this way, we can transfer our empirical knowledge about the autocorrelation
of $P(t)$ from sec. 3 to the model approach. Furthermore, we choose the
deterministic potential function $V(P)$ in a way that focuses on the time
series dynamics around the fundamental fixed points. Since we have observed
that power generation mostly concentrates around zero power output $P_{-}$ and
rated power $P_{+}$, we choose a bimodal approach with the double-well
potential function
$\displaystyle V(P)\ =\
\frac{1}{4a^{4}}(P-P_{0})^{4}\,-\,\frac{1}{2a^{2}}(P-P_{0})^{2}\,.$ (6)
While $a$ describes the steepness of the potential, $P_{0}$ gives us the
position at which it is centered. Such SDEs are used in different applications
in the literature and are refered to as a description of an overdamped
Brownian particle in a double-well potential [71] in statistical physics. As
an illustration, we can think of the underlying dynamics as a particle that
would jump between the potential minima, driven by correlated fluctuations.
The latter consequently introduce a characteristic time scale for the
transitions between the fixed points. Often, such models are extended by a
driving periodical force that entails a biased occupation of one fixed point
with respect to a certain seasonality. Since power generation from wind energy
follows seasonal variations, we incorporate this into our model with the
simple approach
$\displaystyle F(t)\ =\ A\,\mathrm{cos}\,\omega t$ (7)
which is added to eq.4 as the driving force. We determine $\omega$ from
Fourier analysis as the leading frequency within the spectrum of $P(t)$. $A$
gives us the adaptable strength of the seasonal variations. With this
extension, we introduce another characteristic time scale into our approach.
The interplay of both this component and the stochastic fluctuations
determines the transition dynamics as described by the general phenomenon of
stochastic resonance [72]. Our approach effectively biases the bimodal PDF of
modelled power towards one of its peaks which reproduces the seasonal
variation of wind power generation on a rather basic level. In the parameter
estimation of $\omega$, we incorporate data from a specific month to calibrate
the seasonality according to the month in the data.
As we have seen in fig.1, the control of the WTs results in extremely narrow
peaks of the PDF of $P(t)$. This fact is caused by the intervening external
control of power output (curtailment). Our model approach already succeeds to
concentrate the power values around $P_{-}$ and $P_{+}$ as fixed points in a
similar manner but fails to narrow the peaks down as sharply as required due
to the simple analytical approach. Hence we have to incorporate the artificial
flattening of time series we observe in the data into our model sufficiently.
To do so, we cut off power values beyond the operating range $P_{-}\leq P\leq
P_{+}$ by setting them to the respective constant threshold value $P_{-}$ or
$P_{+}$. As we observe in the data, these limits are sometimes exceeded in the
empirical time series anyway. Consequently, this procedure is only applied
with a certain probability $p_{\gtrless}$. Note that the model can not be
regarded as a typical Langevin–equation driven model since the correlated
noise term $\xi_{H}(t)$ on the one hand and the artificial flattening due to
curtailment on the other hand differ strongly from the standard
delta–correlation of such models. Taking all explained considerations of our
model approach into account, the resulting model formula is
$\displaystyle\begin{split}P(t)\ &=\
\begin{cases}\tilde{P}(t)\qquad\quad&\text{if}\quad P_{-}\leq P(t)\leq
P_{+}\\\ P_{\pm}\qquad\quad&\text{if}\ \tilde{P}(t)\gtrless
P_{\pm}\quad\text{with}\quad p=1-p_{\gtrless}\\\
P_{\pm}+z\qquad\quad&\text{if}\ \tilde{P}(t)\gtrless
P_{\pm}\quad\text{with}\quad p_{\gtrless}\end{cases}\\\ \\\
\dfrac{\mathrm{d}\tilde{P}(t)}{\mathrm{d}t}\ &=\
\left[-\left(\frac{\tilde{P}(t)-P_{0}}{a}\right)^{3}+\frac{\tilde{P}(t)-P_{0}}{a}+A\,\mathrm{cos}\,\omega
t\right]\,+\,D\xi_{H}(t)\end{split}$ (8)
with the respective probabilities $p_{\gtrless}$ for having a power value
beyond the operation range and a $\mathcal{N}(0,\sigma)\,$–distributed random
number $z$. The model includes ten parameters in total. A detailed description
of the parameter calibration and all resulting values are given in app.
Appendix: Parameter Estimation. Six parameters
($P_{0},\,\omega,\,p_{\gtrless},\,\sigma_{\gtrless}$) are calibrated on the
data before a time series is to be modelled. $P_{0}$ determines around which
power value the distribution of values should be centered. The frequency
$\omega$ should include some limited degree of season-specific variation and
is fixed via Fourier analysis. The remaining four parameters $p_{\gtrless}$
and $\sigma_{\gtrless}$ calibrate the curtailment and variability of power
beyond the operating range. The three parameters $a\,,\,A$ and $D$ are
optimized such that the model reproduces the empirical statistical features
most effectively while avoiding overfitting. The Hurst exponent $H$ will be
varied in sec. 4.2 to observe how different autocorrelations affect the
generated power.
### 4.2 Results
We get a first glance of the model results by inspecting simulated time
series. We vary the Hurst exponent $H$ from diffusive ($H=0.5$) to persistent
($H=0.7$) up to strongly persistent ($H=0.9$) dynamics to account for
different degrees of correlation within the model. Thus, our approach does not
incorporate the uncovered crossover behaviour that would separate between
different autocorrelations below and above $s_{c}$ but only account for scales
$s<s_{c}$ in the stochastic component. Yet the deterministic component results
in strong ramps that are intended to resemble the observed power ramps that
lead to the Hurst exponent $H>1$ for $s>s_{c}$. We always compare the
simulated results to only one exemplary WT since the model aims at
characterizing the typical dynamics instead of a single specific WT.
Figure 5 shows three randomly chosen numerical time series $P(t)$ with the
different Hurst exponents and compares them to one empirical time series at
the bottom. A visual inspection gives a first indication of the similarity
between the empirical results from sec. 3 and the modelled time series. For
$H=0.9$ the modelled time series reproduces both the transitions between
$P_{-}$ and $P_{+}$ and the fluctuations around local trends most accurately.
The strong abrupt ramps on short time scales can also be observed for all
modelled time series. Note that if power time series had uncorrelated
fluctuations (red curve), they would apparently tend to change their local
trend more frequently. After this first visual comparison we next present a
more quantitative comparison of model and empirical data. Therefore we
consider the bimodality of power statistics, the intermittent behaviour of the
increment statistics and both the autocorrelation and power spectra of power
time series.
Figure 5: Comparison between empirical (bottom) and simulated power time
series for different Hurst exponents $H$. The black line represents a typical
empirical time series for one month. The simulated time series are generated
for the same month and chosen randomly.
The quality of the reproduced PDF shown in fig.6 for different values of $H$
is not as obvious. This time, we choose the sampled time series $P(t)$ whose
PDF reproduces the height of the bimodal peaks most sufficiently but still
reflects a typical result. While the PDF for $H=0.9$ reproduces the bimodality
of the power time series in a satisfactory manner, the average values (dashed
line) clearly differ. For $H=0.5$, peak heights can not be reproduced: a
sample of $P(t)$ with uncorrelated fluctuations tends to occupy both fixed
points $P_{-}$ and $P_{+}$ with the same frequency. This behaviour is entailed
by the enhanced frequency of transitions we observed in fig.5. Yet it fails to
give a convincing result for the mean value even though it should be more
stably centered for a balanced PDF.
The power increments $\Xi_{k}(t)$ are an essential quantity for the control of
WTs and grid stability. Besides the analysis in terms of autocorrelation and
power spectra of $P_{k}(t)$ (see below), higher statistical features of the
power fluctuations can be grasped by investigating the statistics of the power
increments, yielding higher order statistical features of power fluctuations.
A comparison of increment statistics between real and modelled data can
display strengths and weaknesses of the model to capture the intermittent
nature of power generation, expressed by strong correlated fluctuations. To
this extent, we compare the quantiles of the model results for the power
increments with the empirical data. This empirical test clarifies
qualitatively in which way the two distributions generally differ regarding
their quantiles.
Figure 6: Comparison between PDF of empirical (black) and simulated (orange)
power time series for one exemplary month. The Hurst exponent $H$ is varied
for every histogram. The dashed lines indicate the respective average power
values.
Since our model captures the effect of different degrees of autocorrelation
for the power time series with the Hurst parameter $H$, we can also study the
effect of autocorrelation in the power time series on the increment statistics
and possible non-Gaussian characteristics. In fig.7, the quantiles of three
simulated power time series with varied $H$ are plotted against the quantiles
of an empricial power time series. If they were equally distributed, they
would follow the diagonal black line. It becomes apparent that in the strongly
persistent regime of $H=0.9$, the larger increments which possess the non-
Gaussian property are reproduced most sufficiently. Increments up to
$|\Xi_{k}(t)|<4\sigma$ are almost equally distributed to the randomly chosen
empirical time series. The deviations mostly occur in the heavy tails of the
distributions for larger absolute increments. Small increments
$|\Xi_{k}(t)|<\sigma$ are not adequately described by any of the sampled time
series. We point out that the shape of the QQ–curve for persistent samples
$P(t)$ with $H=0.7$ differs only slightly from the one with $H=0.5$.
Figure 7: Direct comparison between empirical and simulated distributions of
power increments $\Xi(t)$ in a QQ–plot for different Hurst exponents $H$. The
ordinate shows the quantiles of the modelled distribution, the abscissa those
of the empirical distribution. The black diagonal line serves as a reference
for an equal distribution of both.
From this we conclude that a certain threshold of persistence has to be
exceeded so that big jumps between $P_{-}$ and $P_{+}$ dominate and generate
strong heavy tails. Thus, the persistence of power is closely related to its
intermittent behaviour regarding its fluctuations. The results we compared so
far demonstrated that the model succesfully reproduces important statistical
characteristics of power generation with respect to the statistical
distribution. The variation of Hurst exponents as a tool to incorporate the
temporal correlation of power time series should yet yield a strong impact on
the autocorrelation function. Thus, we analyse how well the autocorrelation
function $\Theta(\tau)$ and also the power spectrum is reproduced by our
model. Figure 8 shows in black the monthly averaged autocorrelation of
empirical power time series. We compare autocorrelations for the three
different Hurst exponents in the same average. The model in general succeeds
to reproduce the slowly decreasing autocorrelation with a decay on the same
time scale. Although we observe the expected relation that higher values of
$H$ lead to a more slowly decaying autocorrelation, none of the model results
yields a sufficient reproduction of the decay for $\tau\leq 1\,\mathrm{d}$.
Only for $H=0.9$ the simulated autocorrelation approaches the empirical one
within this interval. In general, the results for $\tau>1\,\mathrm{d}$ are
more applicable and even tend to reproduce the weak anticorrelation for
$\tau>5\,\mathrm{d}$. The discussed seasonal bumps are not clearly
distinguishable for any of the model results. The model autocorrelations show
stronger fluctuations which indicates that the monthly average does not cancel
out the impact of monthly varying time scales as much as for the empirical
data. A more sophisticated approach for the seasonal component in our model
could enhance this aspect.
Figure 8: Comparison between autocorrelation $\Theta(\tau)$ of empirical and
simulated power time series for different Hurst exponents $H$. The maximum
delay $\tau$ is one week. The dashed horizontal lines represent the averaged
level of significance.
Finally, fig.3(b) shows the three corresponding averaged power spectra for the
model time series (green). In contrast to the empirical spectrum, the slope
remains almost constant for all $H$ values even beyond the cut-off value
(dashed line), yielding lower power for the higher frequencies. Within the
fitted region, all three spectra show a qualitatively similar decay with
$\beta\approx 5/3$. In particular, the time series with the highest
persistence again gives the most convincing result ($H=0.5:\,\beta=1.71\pm
0.03\,,\ H=0.7:\,\beta=1.69\pm 0.04\,,$ $H=0.9:\,\beta=1.64\pm 0.03$).
## 5 Conclusion
We analysed a data set comprising time series of 30 WTs located at the German
offshore windfarm RIFFGAT over a total time period of one year. As important
basic features, we found power values of each WT to be bimodally distributed
with density peaks at zero power output $P_{-}$ and rated power output
$P_{+}$. The power output $P_{k}(T)$ of a WT turned out to have a slowly
decreasing autocorrelation which connects to general findings in the
literature. Beyond these basic features, we took into account the
non–stationarity of time series by applying DFA to both $P_{k}(T)$. In terms
of the Hurst exponent as a general classifier for autocorrelations of single
WTs, different characteristic time scales were identified on which
correlations vary between persistence and more involved nonstationarity,
separated by a crossover $s_{c}\approx 3\,\mathrm{d}$. Based on these
findings, we put forward a reduced form model for the power output $P_{k}(t)$.
With this model we intended to provide a tool for reproducing several
statistical properties and testing power generation scenarios with respect to
empirically motivated parameters. To this extent, we employed a nonlinear
stochastic differential equation with a double–well potential and a correlated
diffusion component. Most parameters were estimated empirically while some
were optimized to capture statistical properties of the data. The model
succeeded to qualitatively reproduce important statistical characteristics
such as a bimodal PDF, intermittent fluctuations and a slowly decreasing
autocorrelation. It thus enriches the modelling perspective on power
generation by focussing on a comprehensive set of statistical features only
while still producing characteristical results. Possible applications include
wind farm or large power network simulations that still aim at reflecting
single WT dynamics adequately. Even for such aggregated large-scale
approaches, taking correlations on a single WT scale into account is essential
since models with uncorrelated fluctuations will underestimate important
statistical effects such as heavy tailed distributions and persistence.
Anyway, the proposed model should be regarded as a first step towards a study
that quantitatively reflects on the model dynamics in greater detail and
undermines the significance of the obtained results. While a systematic
examination of parameter effects on the outcome remains to be done, some
parameter estimation methods could also be subject to further improvement. The
observed crossover which uncovers different correlations on time scales $t\leq
3\,\mathrm{d}$ could be of interest for possible model extensions. The strong
ramping behaviour on short time scales that is suspected to entail the found
crossover is an important component in the context of predictability despite
strong intermittent fluctuations. Finally, we only focused on single WTs. An
evident extension of the proposed model would be to aggregate several
correlated single WTs’ [73] and wind farms’ [74] model power outputs and study
characteristics of aggregated output [73], especially with regards to an
amplification of correlated fluctuations on the aggregated level.
## Acknowledgments
Parts of this work have been financially supported by the Ministry for Science
and Culture of the Federal State of Lower Saxony within the project ’Ventus
Efficiens’ under grant num- ber ZN3024. We further acknowledge support by the
Open Access Publication Fund of the University of Duisburg-Essen.
## Conflict of interest
The authors declare that they have no conflict of interest.
## Appendix: Parameter Estimation
We briefly provide an overview of the model parameters and explain the methods
of how they are calibrated. To solve the SDE, we use the
Euler-–Maruyama–method for SDEs with colored noise [75]. We sample the latter
by using the so called Harte–method, applying the Durbin–Levinson–algorithm
[76]. Averaged over 10000 samples, it takes us $0.095\pm 0.009\,\mathrm{s}$ to
simulate a power time series covering a period of one month with the hardware
setup used in this study. One advantage of our model lies in the fact that we
can already fix six of ten parameters before we simulate a single time series
only by adjusting them in a particular way to the empirical data.
Consequently, the estimation of these six fitted parameters is enhanced the
more data is included. This data based approach enables us to assign
interpretations to the parameters in context of the empirical data.
* •
Center of potential function $P_{0}$:
The most apparent value for the center of the symmetric double–well potential
$P_{0}$ is to fix it at the center of the empirical range of power values
$P_{0}=1800\,\mathrm{kW}$. This choice simply ensures that for an approriate
choice of $a$, the model reproduces the bimodal peaks at the correct power
output.
* •
Seasonal frequency $\omega$:
Since we introduced the frequency $\omega$ of the periodical driving force
into our model to include seasonal variations, we estimate it from the
empirical wind speed time series $u(t)$ by applying Fourier analysis. By
choosing $\omega$ to be the Fourier frequency of the highest Fourier
amplitude, we make sure that it at least resembles the most dominant seasonal
frequency in the data. We choose the wind speed time series $u(t)$ instead of
the power times $P(t)$ because we regard $u(t)$ to be the external drive for
the WTs analogously to an external driving force of a periodically driven
oscillator. This simple approach can only account for a rough estimation of
the seasonal variations.
* •
Probabilities for excess/negative power $p_{\gtrless}$:
Both parameters need to be set with respect to the likelihood of empirical
power values beyond the operating range. A power lower than $P_{-}$ or higher
than $P_{+}$ can only be generated if the corresponding wind speed $u$ is
lower than $u_{-}$ or higher than $u_{+}$ respectively. Hence we calculate the
probabilities for such wind speed values from the empirical data and use these
probabilities to fix the model probabilities $p_{\gtrless}$. Since these
probabilities show significant variations between different months, we obtain
24 values in total that can be interpreted as the monthly likelihood for the
WT to generate power beyond its operating range.
* •
Standard deviation for excess/negative power $\sigma_{\gtrless}$:
How strongly does power fluctuate around $P_{-}$ and $P_{+}$ when this range
is exceeded? — As we already stated, we choose a simple Gaussian distribution
for the power values around these two values. We fix $\sigma_{\gtrless}$
applying a Gaussian empirical fit around $P_{-}$ and $P_{+}$ respectively.
Since this empirical standard deviation hardly varies for different month, we
only obtain the two values $\sigma_{+}=68.93\,\mathrm{kW}$ and
$\sigma_{-}=4.47\,\mathrm{kW}$.
The remaining four parameters should be regarded as more dynamical parameters
that we try to estimate by optimizing a certain statistical feature with
respect to this parameter. One exception is the Hurst exponent $H$ which we
vary between the three values $H=0.5\,,\,0.7\,,\,0.9$ in sec. 4.2 towards a
better understanding of the impact of power law autocorrelations on power
generation. We briefly sketch the estimation methods for the parameter $a$ of
the potential function, the diffusion strength $D$ and the strength of the
seasonal variations $A$. For each parameter, we optimize it by sampling a
reasonable number of time series for each of the 12 months.
* •
Curtailment indicator $a$:
After we have set $P_{0}=1800\,\mathrm{kW}$, the most obvious choice for $a$
would also be $a=1800\,\mathrm{kW}$ to ensure that the bimodal peaks center
around the correct fixed points $P_{-}=0\,\mathrm{kW}$ and
$P_{+}=3600\,\mathrm{kW}$. Anyway, the artificial flattening procedure plays
an important role in this context: if we choose $a=1800\,\mathrm{kW}$, the
amount of values that are either set to one of the thresholds $P_{-}$ or
$P_{+}$ or scattered around them is too high compared to the empirical data.
This hints at choosing a value $a<1800\,\mathrm{kW}$. We optimize $a$ by
sampling power time series with different $a<1800\,\mathrm{kW}$ and
calculating the amount $p_{\gtrless}$ of power values above (below) or around
$P_{+}$ ($P_{-}$). The resulting value (for fixed $H$) minimizes the
difference between the respective empirical and simulated value. Consequently,
we regard the difference $\,(P_{0}-a)\,$ as the power difference that entails
how strongly the empirical power time series are subject to curtailment.
* •
Diffusion strength $D$:
The constant diffusion strength $D$ determines how strongly the stochastic
component dominates power generation. It is an important parameter in the
context of transition time scales between the two fixed points $P_{-}$ and
$P_{+}$. If we regard $D\xi_{H}(t)$ as the diffusion function of a Langevin
equation, we can apply the standard method to obtain this diffusion function
directly from the empirical data [77]. It can be shown [78] that the diffusion
function can be extracted by calculating
$\displaystyle D^{(2)}(P,t)\ $ $\displaystyle=\ \lim\limits_{\tau\rightarrow
0}{\frac{\langle(P(t+\tau)-P(t))^{2}\rangle}{\tau}\bigg{\rvert}_{P(t)=P_{0}}}\,.$
(9)
Since this method potentially yields errorenous results for correlated noise
processes, we can only expect an approximate estimate of $D$ [79].
Nevertheless, we follow this approach by minimizing the difference
$\mathrm{min}_{D}\left[|D_{2}^{\mathrm{emp}}(P(t))-D_{2}^{\mathrm{sim}}(P(t);D)|\right]$
of the empirical and the model diffusion function with respect to $D$.
* •
Seasonal variation strength $A$:
$A$ biases the power generation towards one of the two peaks of the bimodal
power distribution. In winter months, a high average power output is likely
whereas in spring, usually lower wind speeds occur which lead to a pronounced
peak around $P_{-}$. Thus we can calculate
$\displaystyle E(t)=\sum\limits_{t^{\prime}=t_{0}}^{t}P(t^{\prime})$ (10)
resembling an energy indicator for the empirical and simulated data to
minimize the difference between the respective time-averaged values $\langle
E(t)\rangle_{t}$. We consider $A$ to be optimized when this difference of
averaged accumulated power values is minimized for each month.
We summarize the parameters choices we obtain from the above mentioned
estimation methods:
Table 1: Estimated parameters for different Hurst exponents $H$. $\mathbf{H}$ | $\mathbf{a\ [\mathrm{kW}]}$ | $\mathbf{D\ [\mathrm{kW}/\sqrt{\mathrm{s}}]}$ | $\mathbf{A\ [\mathrm{kW}/\mathrm{s}]}$
---|---|---|---
0.5 | 1755 | 355 | 0.15
0.7 | 1445 | 360 | 0.17
0.9 | 1235 | 485 | 0.26
## References
* [1] B. E. D. G. B. K. A. M. H. E. M. E. M. L. R. J. L. S. K. S. J. S. F. S. Fabiani Appavou, Adam Brown, Renewables 2019 Global Status Report. Paris: REN21 Secretariat: REN21, (2019).
* [2] X. Lu, M. McElroy, and J. Kiviluoma, “Global potential for wind-generated electricity,” Proceedings of the National Academy of Sciences, vol. 106, no. 27, pp. 10933–10938, 2009.
* [3] P. Crossley and A. Beviz, “Smart energy systems: Transitioning renewables onto the grid,” Renewable Energy Focus, vol. 11, no. 5, pp. 54–59, 2010.
* [4] J. M. Carrasco, L. García Franquelo, J. T. Bialasiewicz, E. Galván, R. C. Portillo Guisado, M. d. l. Á. Martín Prats, J. I. León, and N. Moreno-Alfonso, “Power-electronic systems for the grid integration of renewable energy sources: A survey,” IEEE Transactions on Industrial Electronics, 53 (4), 1002-1016., 2006.
* [5] Z. Jiang, Q.-S. Jia, and X. Guan, “Review of wind power forecasting methods: From multi-spatial and temporal perspective,” in 2017 36th Chinese Control Conference (CCC), pp. 10576–10583, IEEE, 2017.
* [6] S. S. Soman, H. Zareipour, O. Malik, and P. Mandal, “A review of wind power and wind speed forecasting methods with different time horizons,” in North American Power Symposium 2010, pp. 1–8, IEEE, 2010.
* [7] G. Ren, J. Wan, J. Liu, D. Yu, and L. Söder, “Analysis of wind power intermittency based on historical wind power data,” Energy, vol. 150, pp. 482–492, 2018.
* [8] M. Anvari, G. Lohmann, M. Wächter, P. Milan, E. Lorenz, D. Heinemann, M. R. R. Tabar, and J. Peinke, “Short term fluctuations of wind and solar power systems,” New Journal of Physics, vol. 18, no. 6, p. 063027, 2016\.
* [9] C. M. S. Martin, J. K. Lundquist, and M. A. Handschy, “Variability of interconnected wind plants: correlation length and its dependence on variability time scale,” Environmental Research Letters, vol. 10, no. 4, p. 044004, 2015.
* [10] W. Katzenstein, E. Fertig, and J. Apt, “The variability of interconnected wind plants,” Energy policy, vol. 38, no. 8, pp. 4400–4410, 2010.
* [11] J. Heitzig, N. Fujiwara, K. Aihara, and J. Kurths, “Interdisciplinary challenges in the study of power grid resilience and stability and their relation to extreme weather events,” 2014.
* [12] L. R. Gorjão, M. Anvari, H. Kantz, C. Beck, D. Witthaut, M. Timme, and B. Schäfer, “Data-driven model of the power-grid frequency dynamics,” arXiv preprint arXiv:1909.08346, 2019.
* [13] Y. Stepanov, P. Rinn, T. Guhr, J. Peinke, and R. Schäfer, “Stability and hierarchy of quasi-stationary states: Financial markets as an example,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2015, no. 8, p. P08011, 2015.
* [14] J. Kantelhardt, E. Koscielny-Bunde, H. H. Rego, S. Havlin, and A. Bunde, “Detecting long-range correlations with detrended fluctuation analysis,” Physica A: Statistical Mechanics and its Applications, vol. 295, no. 3, pp. 441–454, 2001.
* [15] A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers,” Cr Acad. Sci. URSS, vol. 30, pp. 301–305, 1941.
* [16] U. Frisch and A. N. Kolmogorov, Turbulence: the legacy of AN Kolmogorov. Cambridge university press, 1995.
* [17] R. Calif, “Pdf models and synthetic model for the wind speed fluctuations based on the resolution of langevin equation,” Applied energy, vol. 99, pp. 173–182, 2012.
* [18] R. Calif and F. G. Schmitt, “Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 109, pp. 1–8, 2012.
* [19] R. Baıle, P. Poggi, and J.-F. Muzy, “Intermittency model for surface layer wind speed fluctuations. applications to short term forecasting and calibration of the wind resource,” in Proceedings of EWEC, 2010.
* [20] P. Milan, M. Wächter, and J. Peinke, “Stochastic modeling and performance monitoring of wind farm power production,” Journal of Renewable and Sustainable Energy, vol. 6, no. 3, p. 033119, 2014.
* [21] E. Fertig, J. Apt, P. Jaramillo, and W. Katzenstein, “The effect of long-distance interconnection on wind power variability,” Environmental research letters, vol. 7, no. 3, p. 034017, 2012.
* [22] M. S. Taqqu, V. Teverovsky, and W. Willinger, “Estimators for long-range dependence: an empirical study,” Fractals, vol. 3, no. 04, pp. 785–798, 1995.
* [23] R. Kavasseri and R. Nagarajan, “A multifractal description of wind speed records,” Chaos, Solitons & Fractals, vol. 24, no. 1, pp. 165–173, 2005\.
* [24] R. Govindan and H. Kantz, “Long-term correlations and multifractality in surface wind speed,” EPL (Europhysics Letters), vol. 68, no. 2, p. 184, 2004.
* [25] L. Telesca and M. Lovallo, “Analysis of the time dynamics in wind records by means of multifractal detrended fluctuation analysis and the fisher–shannon information plane,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2011, no. 07, p. P07001, 2011.
* [26] P. Wei, J. Liu, Q. Zhou, and X. Zhu, “Multidimensional wind power correlation analysis and modeling based on pair copula,” DEStech Transactions on Environment, Energy and Earth Sciences, no. appeec, 2018.
* [27] R. Calif and F. G. Schmitt, “Multiscaling and joint multiscaling description of the atmospheric wind speed and the aggregate power output from a wind farm,” Nonlinear Processes in Geophysics, vol. 21, no. 2, pp. 379–392, 2014.
* [28] P. Milan, M. Wächter, and J. Peinke, “Turbulent character of wind energy,” Physical Review Letters, vol. 110, no. 13, p. 138701, 2013.
* [29] L. McArthur, S. Mackenzie, and J. Boland, “Multifractal analysis of wind farm power output,” in Proceedings of the 20th International Congress on Modelling and Simulation (MODSIM 2013), MSSANZ, (2013).
* [30] X. Wang, P. Guo, and X. Huang, “A review of wind power forecasting models,” Energy Procedia, vol. 12, pp. 770–778, 2011.
* [31] S. Kolumbán, S. Kapodistria, and N. Nooraee, “Short and long-term wind turbine power output prediction,” arXiv preprint arXiv:1707.06497, 2017\.
* [32] J. Gottschall and J. Peinke, “How to improve the estimation of power curves for wind turbines,” Environmental Research Letters, vol. 3, no. 1, p. 015005, 2008.
* [33] M. Cui, V. Krishnan, B.-M. Hodge, and J. Zhang, “A copula-based conditional probabilistic forecast model for wind power ramps,” IEEE Transactions on Smart Grid, 2018.
* [34] J. Jeon and J. W. Taylor, “Using conditional kernel density estimation for wind power density forecasting,” Journal of the American Statistical Association, vol. 107, no. 497, pp. 66–79, 2012.
* [35] P. Sørensen, A. Hansen, and P. Rosas, “Wind models for simulation of power fluctuations from wind farms,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 90, no. 12, pp. 1381–1402, 2002.
* [36] M. Yang, X. Chen, and B. Huang, “Ultra-short-term multi-step wind power prediction based on fractal scaling factor transformation,” Journal of Renewable and Sustainable Energy, vol. 10, no. 5, p. 053310, 2018.
* [37] P. Chen, T. Pedersen, B. Bak-Jensen, and Z. Chen, “Arima-based time series model of stochastic wind power generation,” IEEE Transactions on Power Systems, vol. 25, no. 2, pp. 667–676, 2010.
* [38] P. Hänggi and P. Jung, “Colored noise in dynamical systems,” Advances in Chemical Physics, vol. 89, pp. 239–326, 1995.
* [39] B. Lotfi, M. Mourad, M. Najiba, and E. Mohamed, “Treatment methodology of erroneous and missing data in wind farm dataset,” in Systems, Signals and Devices (SSD), 2011 8th International Multi-Conference on, pp. 1–6, IEEE, (2011).
* [40] Y.-H. Wan, “Wind power plant behaviors: analyses of long-term wind power data,” tech. rep., National Renewable Energy Lab., Golden, CO (US), 2004.
* [41] J. F. Manwell, J. G. McGowan, and A. L. Rogers, Wind energy explained: theory, design and application. John Wiley & Sons, 2010.
* [42] H.-J. Wagner, “Introduction to wind energy systems,” in EPJ Web of Conferences, vol. 54, p. 01011, EDP Sciences, Springer, (2013).
* [43] T. Scholz, V. V. Lopes, P. Lind, and F. Raischel, “Modeling and Analysis of Cyclic Inhomogeneous Markov Processes: A Wind Turbine Case Study,” ArXiv e-prints, 2014.
* [44] A. Muñoz, E. Sánchez-Úbeda, A. Cruz, and J. Marín, “Short-term forecasting in power systems: A guided tour,” Handbook of power systems II, pp. 129–160, 2010.
* [45] W.-G. Früh, “From local wind energy resource to national wind power production,” AIMS Energy, vol. 3, no. 1, pp. 101–120, 2015.
* [46] N. Zhang, C. Kang, D. Kirschen, Q. Xia, W. Xi, J. Huang, and Q. Zhang, “Planning pumped storage capacity for wind power integration,” IEEE Transactions on Sustainable Energy, vol. 4, no. 2, pp. 393–401, 2013.
* [47] F. Ziel, C. Croonenbroeck, and D. Ambach, “Forecasting wind power–modeling periodic and non-linear effects under conditional heteroscedasticity,” Applied Energy, vol. 177, pp. 285–297, 2016.
* [48] G. Papaefthymiou and B. Klockl, “Mcmc for wind power simulation,” IEEE Transactions on Energy Conversion, vol. 23, no. 1, pp. 234–240, 2008.
* [49] B. Wen, S. Wei, K. Wei, W. Yang, Z. Peng, and F. Chu, “Power fluctuation and power loss of wind turbines due to wind shear and tower shadow,” Frontiers of Mechanical Engineering, vol. 12, no. 3, pp. 321–332, 2017.
* [50] O. D. Medina, F. G. Schmitt, and R. Calif, “Multiscale analysis of wind velocity, power output and rotation of a windmill,” Energy Procedia, vol. 76, pp. 193–199, 2015.
* [51] J. Gao, J. Hu, W.-W. Tung, Y. Cao, N. Sarshar, and V. Roychowdhury, “Assessment of long-range correlation in time series: How to avoid pitfalls,” Physical Review E, vol. 73, no. 1, p. 016117, 2006.
* [52] A. Bashan, R. Bartsch, J. Kantelhardt, and S. Havlin, “Comparison of detrending methods for fluctuation analysis,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 21, pp. 5080–5090, 2008.
* [53] H. Hurst, “The problem of long-term storage in reservoirs,” Hydrological Sciences Journal, vol. 1, no. 3, pp. 13–27, 1956.
* [54] R. Kavasseri and R. Nagarajan, “Evidence of crossover phenomena in wind-speed data,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 11, pp. 2255–2262, 2004.
* [55] R. Bryce and K. Sprague, “Revisiting detrended fluctuation analysis,” Scientific Reports, vol. 2, 2012.
* [56] O. Løvsletten, “Consistency of detrended fluctuation analysis,” Physical Review E, vol. 96, no. 1, p. 012141, 2017.
* [57] R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: A fractional dynamics approach,” Phys. Rep, vol. 339, no. 1, 2000.
* [58] C.-K. Peng, S. Havlin, H. Stanley, and A. Goldberger, “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 5, no. 1, pp. 82–87, 1995.
* [59] Q. Ma, R. Bartsch, P. Bernaola-Galván, M. Yoneyama, and P. Ivanov, “Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis,” Physical Review E, vol. 81, no. 3, p. 031101, 2010.
* [60] O. Løvsletten, “Consistency of detrended fluctuation analysis,” Physical Review E, vol. 96, no. 1, p. 012141, 2017.
* [61] K. Hu, P. Ivanov, Z. Chen, P. Carpena, and H. Stanley, “Effect of trends on detrended fluctuation analysis,” Physical Review E, vol. 64, no. 1, p. 011114, 2001.
* [62] Z. Chen, P. Ivanov, K. Hu, and H. Stanley, “Effect of nonstationarities on detrended fluctuation analysis,” Physical Review E, vol. 65, no. 4, p. 041107, 2002.
* [63] J. Olauson, M. Bergkvist, and J. Rydén, “Simulating intra-hourly wind power fluctuations on a power system level,” Wind Energy, vol. 20, no. 6, pp. 973–985, 2017.
* [64] E. Fertig, “Simulating subhourly variability of wind power output,” Wind Energy, vol. 22, no. 10, pp. 1275–1287, 2019.
* [65] P.-J. Trombe, P. Pinson, and H. Madsen, “A general probabilistic forecasting framework for offshore wind power fluctuations,” Energies, vol. 5, no. 3, pp. 621–657, 2012.
* [66] T. Pesch, S. Schröders, H. Allelein, and J. Hake, “A new markov-chain-related statistical approach for modelling synthetic wind power time series,” New Journal of Physics, vol. 17, no. 5, p. 055001, 2015.
* [67] D. Stroock, An Introduction to Markov Processes, vol. 230. Springer Science & Business Media, (2013).
* [68] C. Wang, H. Zhang, W. Fan, and X. Fan, “A new wind power prediction method based on chaotic theory and bernstein neural network,” Energy, vol. 117, pp. 259–271, 2016.
* [69] H. Akçay and T. Filik, “Wind speed forecasting with missing values,” in Information Science and Technology (ICIST), 2017 Seventh International Conference on, pp. 51–56, IEEE, (2017).
* [70] D. Koutsoyiannis, “The hurst phenomenon and fractional gaussian noise made easy,” Hydrological Sciences Journal, vol. 47, no. 4, pp. 573–595, 2002\.
* [71] Y. Kalmykov, W. Coffey, and S. Titov, “On the brownian motion in a double-well potential in the overdamped limit,” Physica A: Statistical Mechanics and its Applications, vol. 377, no. 2, pp. 412–420, 2007.
* [72] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Reviews of Modern Physics, vol. 70, no. 1, p. 223, 1998.
* [73] P. Chen, P. Siano, B. Bak-Jensen, and Z. Chen, “Stochastic optimization of wind turbine power factor using stochastic model of wind power,” IEEE transactions on Sustainable Energy, vol. 1, no. 1, pp. 19–29, 2010.
* [74] L. Guan, B. Wen, Y. Zhuo, L. Wu, and B. Zhou, “Multiple wind power time series modeling method considering correlation,” in 2018 International Conference on Power System Technology (POWERCON), pp. 1–7, IEEE, 2018.
* [75] G. Milshtein and M. Tret’yakov, “Numerical solution of differential equations with colored noise,” Journal of Statistical Physics, vol. 77, no. 3, pp. 691–715, 1994.
* [76] A. McLeod, H. Yu, Z. Krougly, et al., “Algorithms for linear time series analysis: With r package,” Journal of Statistical Software, vol. 23, no. 5, pp. 1–26, 2007.
* [77] R. Friedrich, S. Siegert, J. Peinke, M. Siefert, M. Lindemann, J. Raethjen, G. Deuschl, G. Pfister, et al., “Extracting model equations from experimental data,” Physics Letters A, vol. 271, no. 3, pp. 217–222, 2000\.
* [78] S. Siegert, R. Friedrich, and J. Peinke, “Analysis of data sets of stochastic systems,” Physics Letters A, vol. 243, no. 5-6, pp. 275–280, 1998.
* [79] F. Ghasemi, M. Sahimi, J. Peinke, and M. Tabar, “Analysis of non-stationary data for heart-rate fluctuations in terms of drift and diffusion coefficients,” Journal of Biological Physics, vol. 32, no. 2, pp. 117–128, 2006.
|
# A Gauss-Seidel projection method with the minimal number of updates for
stray field in micromagnetic simulations
Panchi Li<EMAIL_ADDRESS>Zetao Ma<EMAIL_ADDRESS>Rui Du
<EMAIL_ADDRESS>Jingrun Chen<EMAIL_ADDRESS>School of Mathematical
Sciences, Soochow University, Suzhou, 215006, China. Mathematical Center for
Interdisciplinary Research, Soochow University, Suzhou, 215006, China.
###### Abstract
Magnetization dynamics in magnetic materials is often modeled by the Landau-
Lifshitz equation, which is solved numerically in general. In micromagnetic
simulations, the computational cost relies heavily on the time-marching scheme
and the evaluation of stray field. Explicit marching schemes are efficient but
suffer from severe stability constraints, while nonlinear systems of equations
have to be solved in implicit schemes though they are unconditionally stable.
A better compromise between stability and efficiency is the semi-implicit
scheme, such as the Gauss-Seidel projection method (GSPM) and the second-order
backward differentiation formula scheme (BDF2). At each marching step, GSPM
solves several linear systems of equations with constant coefficients and
updates the stray field several times, while BDF2 updates the stray field only
once but solves a larger linear system of equations with variable coefficients
and a nonsymmetric structure. In this work, we propose a new method, dubbed as
GSPM-BDF2, by combing the advantages of both GSPM and BDF2. Like GSPM, this
method is first-order accurate in time and second-order accurate in space, and
is unconditionally stable with respect to the damping parameter. However,
GSPM-BDF2 updates the stray field only once per time step, leading to an
efficiency improvement of about $60\%$ than the state-of-the-art GSPM for
micromagnetic simulations. For Standard Problem #4 and #5 from National
Institute of Standards and Technology, GSPM-BDF2 reduces the computational
time over the popular software OOMMF by $82\%$ and $96\%$, respectively. Thus,
the proposed method provides a more efficient choice for micromagnetic
simulations.
###### keywords:
Micromagnetic simulation, Landau-Lifshitz equation, Gauss-Seidel projection
method, Backward differentiation formula , Stray field
###### MSC:
[2000] 35Q99 , 65Z05 , 65M06
## 1 Introduction
Due to their intrinsic magnetic properties, ferromagnets have been ideal
materials for magnetic recording devices in the past several decades [1]. The
basic quantity of interest in a ferromagnet is the magnetization, whose
dynamics is modeled by the Landau-Lifshitz (LL) equation [2, 3]
phenomenologically. The model and its generalizations in the presence of
external controls, such as spin current and temperature gradient, have been
successfully used to interpret many interesting experimental observations.
Numerically, micromagnetic simulation becomes increasingly important to study
the dynamics of magnetization, in addition to experiment and theory. In the LL
equation, magnetization dynamics is driven by the gyromagnetic term and the
damping term, which are both nonlinear with respect to magnetization.
Moreover, the length of magnetization does not change during its dynamic
evolution. These pose challenges in designing efficient and simple numerical
methods. Besides, the calculation of the stray field in micromagnetic
simulations is time-consuming since it involves a problem defined over the
entire space instead of the ferromagnetic body [4].
Explicit schemes such as Runge-Kutta method [5] are favored in the early days,
and the fifth-order Cash-Karp Runge-Kutta method is still used in OOMMF [6,
7]. These schemes are simple and efficient in the sense that no
linear/nonlinear systems of equations need to be solved at each marching step,
but the temporal stepsize is rather small due to the strong stability
restriction of explicit schemes. Implicit methods, such as [8, 9, 10, 11], are
proved to be unconditionally stable, and can preserve the length of
magnetization automatically. However, a nonlinear system of equations with
variable coefficients and a nonsymmetric structure has to be solved at each
step.
A nice compromise between efficiency and stability is the semi-implicit
scheme. One notable example is the Gauss-Seidel projection method (GSPM) [12,
13]. At each step, GSPM only solves linear systems of equations with constant
coefficients seven times and updates the stray field four times. GSPM is
tested to be unconditionally stable, and is first-order accurate in time and
second-order accurate in space. Recently, it has been found that the number of
linear systems to be solved and the number of updates for the stray field at
each step can be reduced without sacrificing the stability and accuracy [14,
15]. This leads to a reduction in computational cost of about $30\%$ in
micromagnetic simulations. Other semi-implicit schemes, such as [16, 17, 18],
are second-order accurate in time. In [17], the semi-implicit schemes are
constructed using the backward differentiation formula (BDF) and one-sided
interpolation, and the second-order BDF scheme (BDF2) is proved to converge
with second-order accuracy in both space and time [18]. Note that a projection
step is needed in semi-implicit schemes to preserve the length of
magnetization at each step.
To compare GSPM and BDF2 in details, let us consider a discrete problem with
$n$ degrees of freedom (dofs) and the dofs of magnetization is thus $3n$. At
each step, GSPM (referred to Scheme A in [14] throughout the paper) solves
linear systems of equations with constant coefficients and $n$ dofs five times
and updates the stray field with $3n$ dofs three times. BDF2 solves linear
systems of equations once with variable coefficients, a nonsymmetric
structure, and $3n$ dofs and updates the stray field once with $3n$ dofs. Both
schemes are second-order accurate in space. GSPM is first-order accurate in
time while BDF2 is second-order accurate in time. A natural question arises
here: can one design a more efficient method which combines the strengths of
both GSPM and BDF2?
In the work, we provide an affirmative answer to the aforementioned question.
The new method is dubbed as GSPM-BDF2, which solves linear systems of
equations with constant coefficients and $n$ dofs five times and updates the
stray field with $3n$ dofs only once. The method is tested to be
unconditionally stable with respect to the damping parameter, first-order
accurate in time, and second-order accurate in space. In micromagnetic
simulations, GSPM-BDF2 reduces the number of evaluations of the stray field
from $3$ to $1$, yielding about $60\%$ reduction of computational time on top
of GSPM [14]. For Standard Problem #4 and #5 from National Institute of
Standards and Technology (NIST) [19], GSPM-BDF2 reduces the computational time
over OOMMF [20] by $82\%$ and $96\%$, respectively.
The paper is organized as follows. The LL equation is introduced in Section 2.
The GSPM and the proposed method: GSPM-BDF2 are given in Section 3. In Section
4, the accuracy and stability of GSPM-BDF2 with respect to the damping
parameter are checked in both 1D and 3D, and two benchmark problems from NIST
are simulated. Conclusions are drawn in Section 5.
## 2 Landau-Lifshitz equation
The dynamics of magnetization $\mathbf{M}=(M_{1},M_{2},M_{3})^{T}$ in
ferromagnetic materials is modeled by the phenomenological Landau-Lifshitz
equation [2, 3]
$\mathbf{M}_{t}=-\gamma\mathbf{M}\times\mathbf{H}-\frac{\gamma\alpha}{M_{s}}\mathbf{M}\times(\mathbf{M}\times\mathbf{H}),$
(1)
where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the dimensionless
damping parameter and $|\mathbf{M}|=M_{s}$ in the point-wise sense with
$M_{s}$ the saturation magnetization. The local effective field
$\mathbf{H}=-\frac{\delta F}{\delta\mathbf{M}}$ is obtained from the Landau-
Lifshitz energy functional
$F[\mathbf{M}]=\frac{1}{2}\int_{\Omega}\left\\{\frac{A}{M_{s}^{2}}|\nabla\mathbf{M}|^{2}+\Phi\left(\frac{\mathbf{M}}{M_{s}}\right)-2\mu_{0}\mathbf{H}_{e}\cdot\mathbf{M}\right\\}\mathrm{d}\mbox{\boldmath$x$}+\frac{\mu_{0}}{2}\int_{\mathbb{R}^{3}}|\nabla
U|^{2}\mathrm{d}\mbox{\boldmath$x$}$ (2)
with $A$ the exchange constant and $\mu_{0}$ the permeability of vacuum.
$\mathbf{H}_{e}$ is the external magnetic field and $\Omega$ is the volume
occupied by the material. For a uniaxial material with $x$-axis the easy
direction, the anisotropy energy can be described by
$\Phi\left(\frac{\mathbf{M}}{M_{s}}\right)=\frac{K_{u}}{M_{s}^{2}}(M_{2}^{2}+M_{3}^{2})$
with $K_{u}$ the anisotropy constant. The last term in (2) is the self-
interacting energy induced by the magnetization distribution inside the
material with
$U(\mbox{\boldmath$x$})=\int_{\Omega}\nabla
N(\mbox{\boldmath$x$}-\mbox{\boldmath$y$})\cdot\mathbf{M}(\mbox{\boldmath$y$})\mathrm{d}\mbox{\boldmath$y$},$
(3)
where
$N(\mbox{\boldmath$x$}-\mbox{\boldmath$y$})=-\frac{1}{4\pi}\frac{1}{|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}|}$
is the Newtonian potential. Fast Fourier Transform (FFT) is employed for the
evaluation of the stray field $\boldsymbol{\mathbf{H}}_{s}=-\nabla U$ in
regular-shaped materials [21, 22].
For convenience, we nondimensionalize (1) by rescaling variables
$t\rightarrow(\mu_{0}\gamma M_{s})^{-1}t$ and $x\rightarrow Lx$ with $L$ the
diameter of $\Omega$. With $\mathbf{m}=\mathbf{M}/M_{s}$,
$\mathbf{h}_{e}=\mathbf{H}_{e}/M_{s}$, and
$\mathbf{h}_{s}=\mathbf{H}_{s}/M_{s}$, the dimensionless LL equation reads as
$\mathbf{m}_{t}=-\mathbf{m}\times(\epsilon\Delta\mathbf{m}+\mathbf{f}(\mathbf{m}))-\alpha\mathbf{m}\times(\mathbf{m}\times(\epsilon\Delta\mathbf{m}+\mathbf{f}(\mathbf{m}))),$
(4)
where
$\mathbf{f}(\mathbf{m})=-Q(m_{2}\mathbf{e}_{2}+m_{3}\mathbf{e}_{3})+\mathbf{h}_{e}+\mathbf{h}_{s}.$
(5)
Here dimensionless parameters $\epsilon=A/(\mu_{0}M_{s}^{2}L^{2})$,
$Q=K_{u}/(\mu_{0}M_{s}^{2})$, $\mathbf{e}_{2}=(0,1,0)^{T}$, and
$\mathbf{e}_{3}=(0,0,1)^{T}$. Neumann boundary condition is used
$\frac{\partial\mathbf{m}}{\partial\boldsymbol{\nu}}\Big{|}_{\partial\Omega}=0,$
(6)
where $\boldsymbol{\nu}$ is the outward unit normal vector on
$\partial\Omega$.
An equivalent form of (1) is the so-called Landau-Lifshitz-Gilbert (LLG)
equation
$\mathbf{M}_{t}=-\gamma\mathbf{M}\times\mathbf{H}+\frac{\alpha}{M_{s}}\mathbf{M}\times\mathbf{M}_{t}.$
(7)
In the presence of spin transfer torque (STT), the generalized LLG equation
reads as [23]
$\mathbf{M}_{t}=-\gamma\mathbf{M}\times\boldsymbol{\mathbf{H}}+\frac{\alpha}{M_{s}}\mathbf{M}\times\mathbf{M}_{t}-\frac{b}{M_{s}^{2}}\mathbf{M}\times(\mathbf{M}\times(\mathbf{j}\cdot\nabla)\mathbf{M})-\frac{b\xi}{M_{s}}\mathbf{M}\times(\mathbf{j}\cdot\nabla)\mathbf{M},$
(8)
where $\mathbf{j}$ is the spin polarization current density with magnitude
$J$, $b=P\mu_{B}\left(eM_{s}(1+\xi^{2})\right)^{-1}$ with $e$ the electron
charge, $\mu_{B}$ the Bohr magneton, and $P$ the spin current polarization.
After rescaling $t\rightarrow(1+\alpha^{2})(\mu_{0}\gamma M_{s})^{-1}t$ and
$x\rightarrow Lx$, the LLG equation can be rewritten into (4) with the
dimensionless local field $\mathbf{f}(\mathbf{m})$ of the following form
$\mathbf{f}(\mathbf{m})=-Q(m_{2}\mathbf{e}_{2}+m_{3}\mathbf{e}_{3})+\mathbf{h}_{e}+\mathbf{h}_{s}+\frac{b}{\gamma
M_{s}L}\mathbf{m}\times(\mathbf{j}\cdot\nabla)\mathbf{m}+\frac{b\xi}{\gamma
M_{s}L}(\mathbf{j}\cdot\nabla)\mathbf{m}.$
In particular, when an in-plane current is applied along the $x$-direction,
the local field reduces to
$\mathbf{f}(\mathbf{m})=-Q(m_{2}\mathbf{e}_{2}+m_{3}\mathbf{e}_{3})+\mathbf{h}_{e}+\mathbf{h}_{s}+\frac{bJ}{\gamma
M_{s}L}\mathbf{m}\times\mathbf{m}_{x}+\frac{bJ\xi}{\gamma
M_{s}L}\mathbf{m}_{x}.$ (9)
## 3 The proposed method
In this section, we shall describe the proposed method in details. The GSPM
[14] is introduced firstly for completeness and comparison. For the LL
equation (4) with field (5) and Neumann boundary condition (6), the fractional
step is applied
$\displaystyle\frac{\mathbf{m}^{*}-\mathbf{m}^{n}}{\Delta t}$
$\displaystyle=\epsilon\Delta_{h}\mathbf{m}^{*}+\mathbf{f}(\mathbf{m}^{n}),$
(10) $\displaystyle\frac{\mathbf{m}^{n+1}-\mathbf{m}^{n}}{\Delta t}$
$\displaystyle=-\mathbf{m}^{n}\times\frac{\mathbf{m}^{*}-\mathbf{m}^{n}}{\Delta
t}-\alpha\mathbf{m}^{n}\times\left(\mathbf{m}^{n}\times\frac{\mathbf{m}^{*}-\mathbf{m}^{n}}{\Delta
t}\right),$
where $\Delta_{h}$ represents a discrete approximation to the Laplacian
operator. In 3D, we use the second-order centered difference
$\displaystyle\Delta_{h}\mathbf{m}_{i,j,k}=$
$\displaystyle\frac{\mathbf{m}_{i+1,j,k}-2\mathbf{m}_{i,j,k}+\mathbf{m}_{i-1,j,k}}{\Delta
x^{2}}+$ (11)
$\displaystyle\frac{\mathbf{m}_{i,j+1,k}-2\mathbf{m}_{i,j,k}+\mathbf{m}_{i,j-1,k}}{\Delta
y^{2}}+$
$\displaystyle\frac{\mathbf{m}_{i,j,k+1}-2\mathbf{m}_{i,j,k}+\mathbf{m}_{i,j,k-1}}{\Delta
z^{2}},$
where $\mathbf{m}_{i,j,k}=\mathbf{m}((i-\frac{1}{2})\Delta
x,(j-\frac{1}{2})\Delta y,(k-\frac{1}{2})\Delta z)$ with $i=0,1,\cdots,M,M+1$,
$j=0,1,\cdots,N,N+1$, and $k=0,1,\cdots,K,K+1$ being the indices of grid
points in $x$-, $y$\- and $z$-directions, respectively. For the Neumann
boundary condition (6), a second-order approximation yields
$\displaystyle\mathbf{m}_{0,j,k}$
$\displaystyle=\mathbf{m}_{1,j,k},\quad\mathbf{m}_{M,j,k}=\mathbf{m}_{M+1,j,k},\quad
j=1,\cdots,N,k=1,\cdots,K,$ $\displaystyle\mathbf{m}_{i,0,k}$
$\displaystyle=\mathbf{m}_{i,1,k},\quad\mathbf{m}_{i,N,k}=\mathbf{m}_{i,N+1,k},\quad
i=1,\cdots,M,k=1,\cdots,K,$ $\displaystyle\mathbf{m}_{i,j,0}$
$\displaystyle=\mathbf{m}_{i,j,1},\quad\mathbf{m}_{i,j,K}=\mathbf{m}_{i,j,K+1},\quad
i=1,\cdots,M,j=1,\cdots,N.$
Denote $\mathcal{L}=(I-\epsilon\Delta t\Delta_{h})^{-1}$ with $I$ the identity
operator. (10) can be rewritten as
$\displaystyle m_{i}^{*}=\mathcal{L}(m_{i}^{n}+\Delta
tf_{i}(\mathbf{m}^{n})),\;\;i=1,2,3,$ (12)
$\displaystyle\mathbf{m}^{n+1}=\mathbf{m}^{n}-\mathbf{m}^{n}\times\mathbf{m}^{*}-\alpha\mathbf{m}^{n}\times\left(\mathbf{m}^{n}\times\mathbf{m}^{*}\right).$
(13)
Note that (13) is updated in a Gauss-Seidel manner to avoid the stability
issue [12]. It is nice that all linear systems in (12) have constant
coefficients with the symmetric and positive definite (spd) property, thus the
optimal computational cost of solving linear systems here is $\mathcal{O}(n)$
where $n$ is the dofs. Due to the homogeneous Neumann boundary condition, the
computational complexity is $\mathcal{O}(n\log(n))$ if discrete cosine
transform is used to solve the linear systems here [22]. Meanwhile, FFT has
also been used to calculate the stray field (3) to reduce the computational
complexity to $\mathcal{O}(n\log(n))$ [21]. Thus solving a linear system and
updating the stray field have the same computational complexity in the
asymptotic sense. However, the dofs of magnetization is actually $3n$. As a
consequence, updating the stray field is roughly three times costly compared
to solving a linear system in (12).
### 3.1 GSPM for LL equation
In [14], GSPM solves the LL equation in two steps:
* 1.
Implicit Gauss-Seidel update
$\displaystyle{g}_{i}^{n}$ $\displaystyle=\mathcal{L}({m}_{i}^{n}+\Delta
tf_{i}(\mathbf{m}^{n})),\quad i=1,2,3,$ (14) $\displaystyle{g}_{i}^{*}$
$\displaystyle=\mathcal{L}({m}_{i}^{*}+\Delta tf_{i}(\mathbf{m}^{*})),\quad
i=1,2,$ $\begin{pmatrix}{m}_{1}^{*}\\\ {m}_{2}^{*}\\\
{m}_{3}^{*}\end{pmatrix}=\begin{pmatrix}m_{1}^{n}-(m_{2}^{n}g_{3}^{n}-m_{3}^{n}g_{2}^{n})-\alpha(m_{1}^{n}g_{1}^{n}+m_{2}^{n}g_{2}^{n}+m_{3}^{n}g_{3}^{n})m_{1}^{n}+\alpha
g_{1}^{n}\\\
m_{2}^{n}-(m_{3}^{n}g_{1}^{*}-m_{1}^{*}g_{3}^{n})-\alpha(m_{1}^{*}g_{1}^{*}+m_{2}^{n}g_{2}^{n}+m_{3}^{n}g_{3}^{n})m_{2}^{n}+\alpha
g_{2}^{n}\\\
m_{3}^{n}-(m_{1}^{*}g_{2}^{*}-m_{2}^{*}g_{1}^{*})-\alpha(m_{1}^{*}g_{1}^{*}+m_{2}^{*}g_{2}^{*}+m_{3}^{n}g_{3}^{n})m_{3}^{n}+\alpha
g_{3}^{n}\end{pmatrix}.$ (15)
* 2.
Projection onto $S^{2}$
$\displaystyle\begin{pmatrix}{m}_{1}^{n+1}\\\ {m}_{2}^{n+1}\\\
{m}_{3}^{n+1}\end{pmatrix}=\frac{1}{|\mathbf{m}^{*}|}\begin{pmatrix}{m}_{1}^{*}\\\
{m}_{2}^{*}\\\ {m}_{3}^{*}\end{pmatrix}.$ (16)
###### Remark 1.
In [16], the semi-implicit BDF1 scheme for the LL equation reads as
$\frac{\mathbf{m}^{n+1}-\mathbf{m}^{n}}{\Delta
t}=-\mathbf{m}^{n}\times(\epsilon\Delta\mathbf{m}^{n+1}+\mathbf{f}(\mathbf{m}^{n}))-\alpha\mathbf{m}^{n}\times(\mathbf{m}^{n}\times(\epsilon\Delta\mathbf{m}^{n+1}+\mathbf{f}(\mathbf{m}^{n}))).$
(17)
A projection step is applied after (17) at each step. In fact, GSPM for the LL
equation (14)-(16) can be obtained by applying the splitting strategy
(12)-(13) to (17) and then updating (13) in the Gauss-Seidel manner. This
motivates the current work of applying the splitting strategy and updating in
the Gauss-Seidel manner for the semi-implicit BDF2 scheme.
###### Remark 2.
From (14), we know that at each time step GSPM solves linear systems of
equations with $n$ dofs five times and updates the stray field with $3n$ dofs
three times.
### 3.2 GSPM-BDF2 scheme for LL equation
In [17, 18], the semi-implicit BDF2 scheme for the LL equation reads as
$\frac{\frac{3}{2}\mathbf{\hat{m}}^{n+2}-2\mathbf{m}^{n+1}+\frac{1}{2}\mathbf{m}^{n}}{\Delta
t}=-\mathbf{\tilde{m}}^{n+2}\times(\epsilon\Delta\mathbf{\hat{m}}^{n+2}+\mathbf{f}(\mathbf{\tilde{m}}^{n+2})),\\\
-\alpha\mathbf{\tilde{m}}^{n+2}\times(\mathbf{\tilde{m}}^{n+2}\times(\epsilon\Delta\mathbf{\hat{m}}^{n+2}+\mathbf{f}(\mathbf{\tilde{m}}^{n+2})))$
(18)
$\mathbf{m}^{n+2}=\frac{1}{\left\lvert\mathbf{\hat{m}}^{n+2}\right\rvert}\mathbf{\hat{m}}^{n+2}$
(19)
with $\mathbf{\tilde{m}}^{n+2}=2\mathbf{m}^{n+1}-\mathbf{m}^{n}$. This scheme
has second-order accuracy in both space and time. At each step, it solves one
linear system of equations with variable coefficients, a nonsymmetric
structure, and $3n$ dofs. Note that only one update is needed for the stray
field per step in (18).
Similarly, we solve the semi-implicit BDF2 scheme (18) using the splitting
strategy and updating in the Gauss-Seidel manner. The Gauss-Seidel update is
applied for the following equation after splitting
$\frac{\frac{3}{2}\mathbf{\hat{m}}^{n+2}-2\mathbf{m}^{n+1}+\frac{1}{2}\mathbf{m}^{n}}{\Delta
t}=-\mathbf{\tilde{m}}^{n+2}\times\mathcal{L}\mathbf{\tilde{m}}^{n+2}-\alpha\mathbf{\tilde{m}}^{n+2}\times(\mathbf{\tilde{m}}^{n+2}\times\mathcal{L}\mathbf{\tilde{m}}^{n+2}).$
(20)
Therefore, in details, GSPM-BDF2 works as follows
$\displaystyle g_{i}^{n+2}=\mathcal{L}(\tilde{m}_{i}^{n+2}+\Delta
tf_{i}(\mathbf{\tilde{m}}^{n+2})),\;\;i=1,2,3,$
$\displaystyle\frac{3}{2}m_{1}^{*}=2m_{1}^{n+1}-\frac{1}{2}m_{1}^{n}-(\tilde{m}^{n+2}_{2}g^{n+2}_{3}-\tilde{m}_{3}^{n+2}g_{2}^{n+2})-\alpha(\tilde{m}_{1}^{n+2}g_{1}^{n+2}+\tilde{m}_{2}^{n+2}g_{2}^{n+2}+\tilde{m}_{3}^{n+2}g_{3}^{n+2})\tilde{m}_{1}^{n+2}+\alpha
g_{1}^{n+2},$ $\displaystyle\tilde{m}^{*}_{1}=2m_{1}^{*}-m_{1}^{n+1},\ \
g_{1}^{*}=\mathcal{L}(\tilde{m}^{*}_{1}+\Delta
tf_{1}(\mathbf{\tilde{m}}^{n+2})),$
$\displaystyle\frac{3}{2}m_{2}^{*}=2m_{2}^{n+1}-\frac{1}{2}m_{2}^{n}-(\tilde{m}^{n+2}_{3}g^{*}_{1}-\tilde{m}_{1}^{*}g_{3}^{n+2})-\alpha(\tilde{m}_{1}^{*}g_{1}^{*}+\tilde{m}_{2}^{n+2}g_{2}^{n+2}+\tilde{m}_{3}^{n+2}g_{3}^{n+2})\tilde{m}_{2}^{n+2}+\alpha
g_{2}^{n+2},$ $\displaystyle\tilde{m}^{*}_{2}=2m_{2}^{*}-m_{2}^{n+1},\ \
g_{2}^{*}=\mathcal{L}(\tilde{m}^{*}_{2}+\Delta
tf_{2}(\mathbf{\tilde{m}}^{n+2})),$
$\displaystyle\frac{3}{2}m_{3}^{*}=2m_{3}^{n+1}-\frac{1}{2}m_{3}^{n}-(\tilde{m}^{*}_{1}g^{*}_{2}-\tilde{m}_{2}^{*}g_{1}^{*})-\alpha(\tilde{m}_{1}^{*}g_{1}^{*}+\tilde{m}_{2}^{*}g_{2}^{*}+\tilde{m}_{3}^{n+2}g_{3}^{n+2})\tilde{m}_{3}^{n+2}+\alpha
g_{3}^{n+2},$ $\displaystyle\begin{pmatrix}m_{1}^{n+2}\\\ m_{2}^{n+2}\\\
m_{3}^{n+2}\end{pmatrix}=\frac{1}{\left\lvert\mathbf{m}^{*}\right\rvert}\begin{pmatrix}m_{1}^{*}\\\
m_{2}^{*}\\\ m_{3}^{*}\end{pmatrix}.$
In (18), one linear system with variable coefficients, a nonsymmetric
structure, and $3n$ dofs is solved at each step. This is avoided in GSPM-BDF2
thanks to the splitting strategy. Five linear systems with constant
coefficients, spd structure, and $n$ dofs are solved. Meanwhile, the Gauss-
Seidel update is tested to be unconditionally stable. GSPM-BDF2 inherits the
advantage of BDF2 that only one update of the stray field is needed with $3n$
dofs. We summarize the main computational costs of GSPM and GSPM-BDF2 in table
1 by counting the number of linear systems of equations and the number of
updates for the stray field per step. Simulations suggest that one update of
stray field with $3n$ dofs is computationally comparable to solving three
linear systems of equations with $n$ dofs, thus GSPM-BDF2 saves about $60\%$
computational time over GSPM; see Section 4 for details.
Table 1: Main computational costs of GSPM and GSPM-BDF2 in micromagnetic simulations. | #(linear systems of equations) (dofs) | #(stray field updates) (dofs)
---|---|---
GSPM | 5 ($n$) | 3 ($3n$)
GSPM-BDF2 | 5 ($n$) | 1 ($3n$)
GSPM-BDF2 is expected to be first-order accurate in time and second-order
accurate in space due to the splitting strategy and the Gauss-Seidel update,
both shall be verified numerically later. Compared to the semi-implicit BDF2
scheme, GSPM-BDF2 only solves linear systems of equations with constant
coefficients and a spd structure. Many efficient linear solvers can be
implemented directly, while the linear system in BDF2 has to be solved by
GMRES which is an iterative solver and is not so efficient as solvers for spd
matrices.
###### Remark 3.
There are two improved GSPMs proposed in [14]: Scheme A and Scheme B. The
current work proposes a more efficient method by combining Scheme A and BDF2.
Scheme B solves three linear systems of equations with $n$ dofs and updates
the stray field with $3n$ dofs three times. One may wonder how the combination
of Scheme B and BDF2 works. Unfortunately, we find that the stability of the
resulting scheme depends on the damping parameter $\alpha$ for micromagnetic
simulations. The underlying reason is that Scheme B updates the magnetization
in one step and does a projection step in a subsequent step and a delay of the
update for the stray field leads to the instability with respect to the
damping parameter.
## 4 Numerical simulations
### 4.1 Accuracy check
In 1D, we consider the following LL equation
$\mathbf{m}_{t}=-\mathbf{m}\times\mathbf{m}_{xx}-\alpha\mathbf{m}\times(\mathbf{m}\times\mathbf{m}_{xx}).$
(21)
Choose the exact solution
$\mathbf{m}_{e}=(\cos(\bar{x})\sin(t),\sin(\bar{x})\sin(t),\cos(t))$
with $\bar{x}=x^{2}(1-x)^{2}$. A forcing term will be added on the right-hand
side of (21) with
$\mathbf{\hat{f}}=\mathbf{m}_{et}+\mathbf{m}_{e}\times\mathbf{m}_{exx}+\alpha\mathbf{m}_{e}\times(\mathbf{m}_{e}\times\mathbf{m}_{exx})$.
The convergence rate with respect to the temporal step size and the spatial
step size is recorded in table 2.
Table 2: Convergence rates in time and space for the 1D example. The final time $T=1.0e-02$ and the damping parameter $\alpha=0.01$. $\Delta x=1.0e-03$ is used for the temporal accuracy and $\Delta t=1.0e-06$ is used for the spatial accuracy, respectively. Temporal accuracy | nt | 1000 | 2000 | 4000 | 8000 | order
---|---|---|---|---|---|---
GSPM | 6.01e-08 | 3.01e-08 | 1.52e-08 | 7.72e-09 | 0.99
GSPM-BDF2 | 6.01e-08 | 3.02e-08 | 1.52e-08 | 7.73e-09 | 0.99
Spatial accuracy | nx | 10 | 20 | 40 | 80 | order
GSPM | 1.24e-06 | 4.24e-07 | 1.27e-07 | 4.03e-08 | 1.66
GSPM-BDF2 | 1.24e-06 | 4.24e-07 | 1.27e-07 | 4.03e-08 | 1.66
In 3D, we choose the exact solution
$\mathbf{m}_{e}=(\cos(\bar{x}\bar{y}\bar{z})\sin(t),\sin(\bar{x}\bar{y}\bar{z})\sin(t),\cos(t)),$
where $\bar{x}=x^{2}(1-x)^{2}$, $\bar{y}=y^{2}(1-y)^{2}$ and
$\bar{z}=z^{2}(1-z)^{2}$. Uniform discretization over $\Omega=[0,1]^{3}$ is
used for the spatial accuracy. $\Omega=[0,2]\times[0,1]\times[0,0.2]$ with
$128\times 64\times 10$ cubes is used for the temporal accuracy. Errors are
recorded in table 3. From 1D and 3D results, the accuracy of GSPM-BDF2 is
$\mathcal{O}(\Delta t+(\Delta x)^{2})$. Moreover, the scheme is tested to be
unconditionally stable by varying the temporal step size.
Table 3: Convergence rates in time and space for the 3D example. The final time $T=1.0e-05$ and the damping parameter $\alpha=0.01$. $\Delta x=1.0e-03$ is used for the temporal accuracy and $\Delta t=1.0e-09$ is used for the spatial accuracy, respectively. Temporal accuracy | nt | 10 | 20 | 40 | 80 | order
---|---|---|---|---|---|---
GSPM | 1.00e-06 | 5.00e-07 | 2.50e-07 | 1.25e-07 | 1.00
GSPM-BDF2 | 1.00e-06 | 5.00e-07 | 2.50e-07 | 1.25e-07 | 1.00
Spatial accuracy | nx=ny=nz | 6 | 8 | 10 | 12 | order
GSPM | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88
GSPM-BDF2 | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88
### 4.2 Stability with respect to the damping parameter
The original GSPM is found to be conditionally stable with respect to the
damping parameter [13] if the stray field is updated only once per step [12].
Here we examine the performance of GSPM-BDF2 for small damping parameters.
Consider the magnetization dynamics of a ferromagnet over
$\Omega=1\;\mu\textrm{m}\times 1\;\mu\textrm{m}\times 0.02\;\mu\textrm{m}$ and
the final time $1.6$ nanoseconds. The spatial mesh size is
$4\;\textrm{nm}\times 4\;\textrm{nm}\times 4\;\textrm{nm}$, and the temporal
step size is $1$ picosecond.
For comparison, we first update the stray field in GSPM (14)-(16) only once,
i.e., only $\mathbf{f}(\mathbf{m}^{n})$ is used per step. Results are shown in
Figure 1 when $\alpha=0.01$. The whole simulation takes $493.75$ seconds. If
we use the GSPM (14)-(16) with three updates of the stray field, we obtain the
simulation results in Figure 2. The whole simulation takes $1225.92$ seconds.
Comparing these two results, we find that updating the stray field only once
leads to the numerical instability with respect to the damping parameter in
micromagnetic simulations, although the computational cost is reduced by
$60\%$.
(a) Angle profile (b) Magnetization profile
Fig. 1: Simulation results of the LL equation using GSPM with only one update
of stray field at each step. The magnetization on the centered slice of the
material in the $xy$ plane is used. Left: a color plot of the angle between
the in-plane magnetization and the $x$ axis; Right : an arrow plot of the in-
plane magnetization.
(a) Angle profile (b) Magnetization profile
Fig. 2: Simulation results of the LL equation using GSPM with three updates of
stray field at each step. The magnetization on the centered slice of the
material in the $xy$ plane is used. Left: a color plot of the angle between
the in-plane magnetization and the $x$ axis; Right : an arrow plot of the in-
plane magnetization.
Simulation results of GSPM-BDF2 are plotted in fig. 3 when $\alpha=0.01$. It
is clear that GSPM-BDF2 is able to produce correct magnetization dynamics
though the stray field is updated only once per step. The whole simulation
takes $511.39$ seconds and saves $58\%$ computational time over GSPM
(14)-(16). This study shows that the proposed method is stable with respect to
the damping parameter even the number of updates for the stray field is
minimized.
(a) Angle profile (b) Magnetization profile
Fig. 3: Simulation results of the LL equation using GSPM-BDF2. The
magnetization on the centered slice of the material in the $xy$ plane is used.
Left: a color plot of the angle between the in-plane magnetization and the $x$
axis; Right : an arrow plot of the in-plane magnetization.
### 4.3 Standard Problem #4
The setup of this benchmark problem is as follows: film size
$500\;\mathrm{nm}\times 125\;\mathrm{nm}\times 3\;\mathrm{nm}$; initial state:
an equilibrium s-state. A magnetic field with sufficient magnitude is applied
to reverse the magnetization of the s-state. Following the description of
Standard Problem #4, we first generate the initial s-state in fig. 4.
Fig. 4: Magnetization profile of the initial s-state. This is generated for a
random initial state under a magnetic field $100\;\mathrm{mT}$ along the
$[1,1,1]$ direction with a successive reduction to $0$.
Two different magnetic fields are applied:
* 1.
A field 25 $\mathrm{mT}$, directed 170 degrees counterclockwise from the
positive $\mathrm{x}$ axis;
* 2.
A field 36 $\mathrm{mT}$, directed 190 degrees counterclockwise from the
positive $\mathrm{x}$ axis.
Two different mesh strategies are used: a coarse mesh with cell size
$5\;\mathrm{nm}\times 5\;\mathrm{nm}\times 3\;\mathrm{nm}$ and a fine mesh
with cell size $2.5\;\mathrm{nm}\times 2.5\;\mathrm{nm}\times 3\;\mathrm{nm}$.
All the results shown here are independent of the mesh strategy.
Dynamics of the spatially averaged magnetization under the external field of
$25\;\mathrm{mT}$ is plotted in fig. 5.
(a) Magnetization dynamics
(b) Comparison of $<m_{y}>$
Fig. 5: Left: dynamics of the spatially averaged magnetization on the coarse
mesh under the external field of $25\;\mathrm{mT}$; Right: comparison of the
averaged $<m_{y}>$ on two different meshes.
Magnetization profile when the averaged $<m_{x}>=0$ under the external field
of $25\;\mathrm{mT}$ is visualized in fig. 6. The color map is given by the
$\mathrm{z}$-component of the magnetization.
Fig. 6: Magnetization profile when the averaged $<m_{x}>=0$ under the external
field of $25\;\mathrm{mT}$. The color map is given by the
$\mathrm{z}$-component of the magnetization.
In the presence of the $36\;\mathrm{mT}$ field, dynamics of the spatially
averaged magnetization is plotted in fig. 7 and the magnetization profile when
the averaged $<m_{x}>=0$ is visualized in fig. 8, respectively.
(a) Magnetization dynamics
(b) Comparison of $<m_{y}>$
Fig. 7: Left: dynamics of the spatially averaged magnetization on the coarse
mesh under the external field of $36\;\mathrm{mT}$; Right: comparison of the
averaged $<m_{y}>$ on two different meshes. Fig. 8: Magnetization profile when
the averaged $<m_{x}>=0$ under the external field of $36\;\mathrm{mT}$. The
color map is given by the $\mathrm{z}$-component of the magnetization.
When the field is applied at 170 degrees, the magnetization at the center of
the rectangle rotates in the same direction as that at the two ends during the
reversal process. When the field is applied at 190 degrees, the magnetization
at the center of the rectangle rotates in the opposite direction as that at
the two ends during the reversal process. These are in good agreements with
the reports listed in [19]. To check the efficiency, we record the
computational costs of the proposed method and OOMMF in table 4. The current
work saves $82\%$ computational time over OOMMF.
Table 4: Computational costs of the proposed method and OOMMF for Standard Problem #4 (unit: seconds) when the coarse mesh is used. Standard Problem #4 | The proposed method | OOMMF | Saving
---|---|---|---
field $25\;\mathrm{mT}$ | 20.47 | 115.32 | 82%
field $36\;\mathrm{mT}$ | 20.33 | 116.41 | 83%
### 4.4 Standard Problem #5
Standard Problem #5 considers the magnetization dynamics in the presence of
STT [23], which is modeled by the LL equation with (9). Since Neumann boundary
condition is used, zero-spin torques of the magnetization on the boundaries
where the spin current enters and leaves is naturally satisfied. The setup is
as follows: ferromagnet size $100\;\mathrm{nm}\times 100\;\mathrm{nm}\times
10\;\mathrm{nm}$; cell size $2\mathrm{nm}\;\times 2\mathrm{nm}\;\times
2\;\mathrm{nm}$. At each point $(x,y,z)\in\Omega$, the initial magnetization
is chosen as $\mathbf{m}=\mathbf{g}/\left\lvert\mathbf{g}\right\rvert$, where
$\mathbf{g}(x,y,z)=[-y,x,R]$ (22)
and $R=10\;\mathrm{nm}$. The equilibrium configuration after relaxation is
chosen as the initial state (vortex pattern) for all simulations in what
follows.
There are four sets of parameters used in Standard Problem #5:
* 1)
$bJ=72.35\mathrm{m}/\mathrm{s}$ and $\xi=0$;
(a) Magnetization dynamics (b) Final state
Fig. 9: Magnetization dynamics and the final state in the case of $\xi=0$ and
$bJ=72.35\;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially averaged
magnetization with the result of D. G. Porter as the reference; Right: the
final state colored by the $x$-component of magnetization.
* 2)
$bJ=72.17\mathrm{m}/\mathrm{s}$ and $\xi=0.05$;
(a) Magnetization dynamics (b) Final state
Fig. 10: Magnetization dynamics and the final state in the case of $\xi=0.05$
and $bJ=72.17\;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially
averaged magnetization with the result of D. G. Porter as the reference;
Right: the final state colored by the $x$-component of magnetization.
* 3)
$bJ=71.64\mathrm{m}/\mathrm{s}$ and $\xi=0.1$;
(a) Magnetization dynamics (b) Final state
Fig. 11: Magnetization dynamics and the final state in the case of $\xi=0.1$
and $bJ=71.64\;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially
averaged magnetization with the result of D. G. Porter as the reference;
Right: the final state colored by the $x$-component of magnetization.
* 4)
$bJ=57.88\mathrm{m}/\mathrm{s}$ and $\xi=0.5$.
(a) Magnetization dynamics (b) Final state
Fig. 12: Magnetization dynamics and the final state in the case of $\xi=0.5$
and $bJ=57.88\;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially
averaged magnetization with the result of D. G. Porter as the reference;
Right: the final state colored by the $x$-component of magnetization.
In the current work, the first component of the spatially averaged
magnetization $<M_{x}>$ at $10\mathrm{ns}$ is $-1.43\times
10^{5}\;\mathrm{A}/\mathrm{m}$, while the value is $-1.71\times
10^{5}\;\mathrm{A}/\mathrm{m}$ produced by OOMMF for case 1), 2), and 3). We
attribute this discrepancy to different calculations of the stray field. This
has been observed in [7] that the spatially averaged magnetization simulated
by OOMMF and M3S are different due to different evaluations of the stray
field. In our implementation, the stray field is evaluated in 3D without any
simplification. For case 4), a vortex state with different location is
observed in our simulation. Both $<M_{y}>$ and $<M_{x}>$ are different
compared to the result of D. G. Porter. However, if the result of G. Finocchio
et al. is used for comparison with the parameters $\xi=0.5$ and
$bJ=72.45\;\mathrm{m}/\mathrm{s}$, our result has a small difference in
$<M_{x}>$ but no difference in $<M_{y}>$; as shown in fig. 13. This state is
found to be stable and may be caused by different treatments of STT terms in
the LLG equation [19].
For each case, GSPM-BDF2 takes about $100$ seconds while softwares such as
NMAG [24], OOMMF [20], and the work in [7] take more than half an hour on the
same personal computer. Below we list the computational costs of GSPM-BDF2 and
OOMMF in table 5. The proposed method saves $96\%$ computational time over
OOMMF.
Table 5: Computational costs of the proposed method and OOMMF for Standard Problem #5 (unit: seconds). Standard Problem #5 | The proposed method | OOMMF | Saving
---|---|---|---
case 1) | 97.58 | 2216.85 | 96%
case 2) | 97.55 | 2226.45 | 96%
case 3) | 93.91 | 2229.22 | 96%
case 4) | 95.59 | 2246.40 | 96%
(a) Magnetization dynamics (b) Final state
Fig. 13: Magnetization dynamics and the final state in the case of $\xi=0.5$
and $bJ=72.45\;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially
averaged magnetization with the result of G. Finocchio et al. as the
reference; Right: the final state colored by the $x$-component of
magnetization. The first component of the spatially averaged magnetization
$<M_{x}>$ at $10\;\mathrm{ns}$ is $-1.43\times 10^{5}\;\mathrm{A}/\mathrm{m}$.
## 5 Conclusions
In this work, we propose a new method (GSPM-BDF2 in short) by combing the
advantages of the Gauss-Seidel projection method and the semi-implicit BDF2
scheme. The proposed method solves linear systems of equations with constant
coefficients and the spd structure five times and updates the stray field only
once. The method is tested to be unconditionally stable with respect to the
damping parameter, first-order accurate in time, and second-order accurate in
space. In micromagnetic simulations, the proposed method reduces the number of
evaluations of the stray field from $3$ to $1$, yielding about $60\%$
reduction of computational time on top of GSPM [14]. For Standard Problem #4
and #5 from National Institute of Standards and Technology [19], GSPM-BDF2
reduces the computational time over OOMMF [20] by $82\%$ and $96\%$,
respectively. Thus, the proposed method provides a more efficient choice for
micromagnetic simulations.
We shall mention that the number of updates for the stray field is minimized
in the current work and its evaluation is implemented using FFT which only
applies to regular geometries. How to effectively evaluate the stray field
over a general geometry is still a difficult task. Realizing this will
maximize the applicability of the proposed method for micromagnetic
simulations in general.
## Acknowledgments
P. Li is grateful to Kelong Cheng for helpful discussions during the 18th
CSIAM annual meeting and acknowledges the financial support from the
Postgraduate Research & Practice Innovation Program of Jiangsu Province via
grant KYCX20_2711. The work of R. Du was supported in part by NSFC via grant
11501399. The work of J. Chen was supported by NSFC via grant 11971021.
## References
* Žutić et al. [2004] I. Žutić, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Reviews of Modern Physics 76 (2004) 323–410.
* Landau and Lifshitz [1935] L. D. Landau, E. M. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies, “Zeitschrift fur Physik (Sowjetunion),” Physics 8 (1935) 153–169.
* Gilbert [1955] T. L. Gilbert, A lagrangian formulation of gyromagnetic equation of the magnetization field, Physical Review D 100 (1955) 1243–1255.
* Abert et al. [2013] C. Abert, L. Exl, G. Selke, A. Drews, T. Schrefl, Numerical methods for the stray field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials 326 (2013) 176–185.
* Romeo et al. [2008] A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, B. Azzerboni, A numerical solution of the magnetization reversal modeling in a permalloy thin film using fifth order Runge-Kutta method with adaptive step size control, Physica B: Condensed Matter 403 (2008) 1163–1194.
* Cash and Karp [1999] J. R. Cash, A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transaction on Mathematical Software 16 (1999).
* Massoud et al. [2009] N. Massoud, K. Benjamin, B. Stellan, F. Matteo, F. Hans, V. Antoine, A. Rolf, B. Markus, M. Ulrich, P. Daniela, M. D. P. F., M. Guido, Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, Journal of Applied Physics 105 (2009) 113914.
* Yamada and Hayashi [2004] H. Yamada, N. Hayashi, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, Journal of the Magnetics Society of Japan 28 (2004) 924–931.
* Sören and Andreas [2006] B. Sören, P. Andreas, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM Journal on Numerical Analysis 44 (2006) 1405–1419.
* Jeong and Kim [2010] D. Jeong, J. Kim, A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping, Journal of Computational and Applied Mathematics 234 (2010) 613–623.
* Atsushi et al. [2012] F. Atsushi, I. Tetsuya, T. Masayoshi, Finite difference scheme for the Landau-Lifshitz equation, Japan Journal of Industrial and Applied Mathematics 29 (2012) 83–110.
* Wang et al. [2001] X.-P. Wang, C. J. García-Cervera, W. E, A Gauss-Seidel projection method for micromagnetics simulations, Journal of Computational Physics 171 (2001) 357–372.
* García-Cervera and E [2003] C. J. García-Cervera, W. E, Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE Transactions on Magnetics 39 (2003) 1766–1770.
* Li et al. [2020a] P. Li, C. Xie, R. Du, J. Chen, X.-P. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation, Journal of Computational Physics 401 (2020a) 109046\.
* Li et al. [2020b] P. Li, J. Chen, R. Du, X.-P. Wang, Numerical methods for antiferrimagnets, IEEE Transactions on Magnetics 56 (2020b) 7200509\.
* Ivan [2005] C. Ivan, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA Journal of Numerical Analysis (2005) 611–634.
* Xie et al. [2020] C. Xie, C. J. García-Cervera, C. Wang, Z. Zhou, J. Chen, Second-order semi-implicit projection methods for micromagnetics simulations, Journal of Computational Physics (2020) 109104.
* Chen et al. [2019] J. Chen, C. Wang, C. Xie, Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, arXiv 1902.0974 (2019).
* Micromagnetic Modeling Activity Group [2000] Micromagnetic Modeling Activity Group, National institute of standards and technology, https://www.ctcms.nist.gov/~rdm/mumag.org.html, 2000\.
* Donahue and Porter [2019] M. J. Donahue, D. G. Porter, Oommf user’s guide (2019).
* García-Cervera [2007] C. J. García-Cervera, Numerical micromagnetics: a review, Boletín de la Sociedad Española de Matemática Aplicada. SeMA 39 (2007) 103–135.
* Yang [2008] L. Yang, Current induced domain wall motion: Analysis and simulation, Ph.D Thesis (2008).
* Zhang and Li [2004] S. Zhang, Z. Li, Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Physical Review Letter 93 (2004) 127204.
* Fangohr et al. [2012] H. Fangohr, T. Fischbacher, M. Franchin, G. Bordignon, J. Generowicz, A. Knittel, M. Walter, M. Albert, Nmag user manual (0.2.1) (2012).
|
<EMAIL_ADDRESS>Corresponding author. This work was partially carried
out at Caltech<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>11institutetext: National
Institute of Chemical Physics and Biophysics (NICPB), Rävala pst 10, 10143
Tallinn, Estonia 22institutetext: California Institute of Technology,
Pasadena, CA 91125, USA 33institutetext: University of California San Diego,
La Jolla, CA 92093, USA 44institutetext: European Center for Nuclear Research
(CERN), CH 1211, Geneva 23, Switzerland
# MLPF: Efficient machine-learned particle-flow reconstruction using graph
neural networks
Joosep Patae1,addr1,addr2 Javier Duartee2,addr3 Jean-Roch Vlimante3,addr2
Maurizio Pierinie4,addr4 Maria Spiropulue5,addr2
(Received: 5 February 2021 / Accepted: 19 April 2021)
###### Abstract
In general-purpose particle detectors, the particle-flow algorithm may be used
to reconstruct a comprehensive particle-level view of the event by combining
information from the calorimeters and the trackers, significantly improving
the detector resolution for jets and the missing transverse momentum. In view
of the planned high-luminosity upgrade of the CERN Large Hadron Collider
(LHC), it is necessary to revisit existing reconstruction algorithms and
ensure that both the physics and computational performance are sufficient in
an environment with many simultaneous proton-proton interactions (pileup).
Machine learning may offer a prospect for computationally efficient event
reconstruction that is well-suited to heterogeneous computing platforms, while
significantly improving the reconstruction quality over rule-based algorithms
for granular detectors. We introduce MLPF, a novel, end-to-end trainable,
machine-learned particle-flow algorithm based on parallelizable,
computationally efficient, and scalable graph neural networks optimized using
a multi-task objective on simulated events. We report the physics and
computational performance of the MLPF algorithm on a Monte Carlo dataset of
top quark-antiquark pairs produced in proton-proton collisions in conditions
similar to those expected for the high-luminosity LHC. The MLPF algorithm
improves the physics response with respect to a rule-based benchmark algorithm
and demonstrates computationally scalable particle-flow reconstruction in a
high-pileup environment.
††journal: Eur. Phys. J. C
## 1 Introduction
Reconstruction algorithms at general-purpose high-energy particle detectors
aim to provide a holistic, well-calibrated physics interpretation of the
collision event. Variants of the particle-flow (PF) algorithm have been used
at the CELLO Behrend:1982gk , ALEPH Buskulic:1994wz , H1 H1:2020zpd , ZEUS
Breitweg:1997aa ; Breitweg:1998gc , DELPHI Abreu:1995uz , CDF Bocci:2001zx ;
Connolly:2003gb ; Abulencia:2007iy , D0 Abazov:2008ff , CMS Sirunyan:2017ulk
and ATLAS Aaboud:2017aca experiments to reconstruct a particle-level
interpretation of high-multiplicity hadron collision events, given individual
detector elements such as tracks and calorimeter clusters from a multi-
layered, heterogeneous, irregular-geometry detector. The PF algorithm
generally correlates tracks and calorimeter clusters from detector layers such
as the electromagnetic calorimeter (ECAL), hadron calorimeter (HCAL) and
others to reconstruct charged and neutral hadron candidates as well as
photons, electrons, and muons with an optimized efficiency and resolution.
Existing PF reconstruction implementations are tuned using simulation for each
specific experiment because detailed detector characteristics and geometry are
critical for the best possible physics performance.
Recently, there has been significant interest in adapting the PF
reconstruction approach for future high-luminosity experimental conditions at
the LHC Chlebana:2203028 , as well as for proposed future collider experiments
such as the Future Circular Collider (FCC) Selvaggi:2715344 ; Benedikt:2018csr
. PF reconstruction is also a key driver in the detector design for future
lepton colliders Abada:2019zxq ; Behnke:2013xla ; CEPCStudyGroup:2018ghi .
While reconstruction algorithms are often based on an imperative, rule-based
approach, the use of supervised machine learning (ML) to define reconstruction
parametrically based on data and simulation samples may improve the physics
reach of the experiments by allowing a more detailed reconstruction to be
deployed given a fixed computing budget. Reconstruction algorithms based on ML
may be well-suited to irregular, high-granularity detector geometries and for
novel signal models, where it may not be feasible to encode the necessary
granularity in the ruleset. A fully probabilistic particle-level
interpretation of the event from an ML-based reconstruction may also improve
the physics performance of downstream algorithms such as jet tagging with more
granular inputs. At the same time, ML-solutions for computationally intensive
problems may offer a modern computing solution that may scale better with the
expected progress on ML-specific computing infrastructures, e.g., at high-
performance computing centers.
ML-based reconstruction approaches using graph neural networks gnn ;
gilmer2017neural ; pointnet ; Battaglia:2016jem ; DGCNN ; pointnet have been
proposed for various tasks in particle physics Shlomi:2020gdn , including
tracking Farrell:2018cjr ; Ju:2020xty ; Amrouche:2019wmx ; Amrouche:2019yxv ;
Choma:2020cry , jet finding Ju:2020tbo ; Li:2020grn ; Guo:2020vvt and tagging
Moreno:2019bmu ; Moreno:2019neq ; Qu:2019gqs ; Mikuni:2020wpr , calorimeter
reconstruction Qasim:2019otl , pileup mitigation Martinez:2018fwc , and PF
reconstruction Kieseler:2020wcq ; DiBello:2020bas ; Duarte:2020ngm . The
clustering of energy deposits in detectors with a realistic, irregular-
geometry detector using GNNs has been first proposed in Ref. Qasim:2019otl .
The ML-based reconstruction of overlapping signals without a regular grid was
further developed in Ref. Kieseler:2020wcq , where an optimization scheme for
reconstructing a variable number of particles based on a potential function
using an object condensation approach was proposed. The clustering of energy
deposits from particle decays with potential overlaps is an essential input to
PF reconstruction. In Ref. DiBello:2020bas , various ML models including GNNs
and computer-vision models have been studied for reconstructing neutral
hadrons from multi-layered granular calorimeter images and tracking
information. In particle gun samples, the ML-based approaches achieved a
significant improvement in neutral hadron energy resolution over the default
algorithm, which is an important step towards a fully parametric, simulation-
driven reconstruction using ML.
In this paper, we build on the previous ML-based reconstruction approaches by
extending the ML-based PF algorithm to reconstruct particle candidates in
events with a large number of simultaneous pileup (PU) collisions. In Section
2, we propose a benchmark dataset that has the main components for a particle-
level reconstruction of charged and neutral hadrons with PU. In Section 3, we
propose a GNN-based machine-learned particle-flow (MLPF) algorithm where the
runtime scales approximately linearly with the input size. Furthermore, in
Section 4, we characterize the performance of the MLPF model on the benchmark
dataset in terms of hadron reconstruction efficiency, fake rate and
resolution, comparing it to the baseline PF reconstruction, while also
demonstrating using synthetic data that MLPF reconstruction can be
computationally efficient and scalable. Finally, in Section 5 we discuss some
potential issues and next steps for ML-based PF reconstruction.
## 2 Physics simulation
We use pythia 8 Sjostrand:2006za ; Sjostrand:2007gs and delphes 3
deFavereau:2013fsa from the HepSim software repository Chekanov:2014fga to
generate a particle-level dataset of 50,000 top quark-antiquark
($\mathrm{t}\overline{\mathrm{t}}$) events produced in proton-proton
collisions at 14 TeV, overlaid with minimum bias events corresponding to a PU
of 200 on average. The $\mathrm{t}\overline{\mathrm{t}}$ dataset is used for
training the MLPF model. We additionally generate 5,000 events composed
uniquely of jets produced through the strong interaction, referred to as
quantum chromodynamics (QCD) multijet events, with the same PU conditions for
validation to evaluate the model in a different physics regime from the
training dataset. The dataset consists of detector hits as the input,
generator particles as the ground truth and reconstructed particles from
delphes for additional validation. The QCD sample uses a minimum invariant
$p_{\mathrm{T}}$ of 20 GeV, otherwise, the same generator settings are used as
for the $\mathrm{t}\overline{\mathrm{t}}$ sample. The delphes model
corresponds to a CMS-like detector with a multi-layered charged particle
tracker, an electromagnetic and hadron calorimeter. The full pythia 8 and
delphes data cards are available on Zenodo along with the dataset
pata_joosep_2021_4452283 .
Although this simplified simulation does not include important physics effects
such as pair production, Brehmsstrahlung, nuclear interactions,
electromagnetic showering or a detailed detector simulation, it allows the
study of overall per-particle reconstruction properties for charged and
neutral hadrons in a high-PU environment. Different reconstruction approaches
can be developed and compared on this simplified dataset, where the expected
performance is straightforward to assess, including from the aspect of
computational complexity.
The inputs to PF are charged particle tracks and calorimeter clusters. We use
these high-level detector inputs (elements), rather than low-level tracker
hits or unclustered calorimeter hits to closely follow how PF is implemented
in existing reconstruction chains, where successive reconstruction steps are
decoupled, such that each step can be optimized and characterized
individually. In this toy dataset, tracks are characterized by transverse
momentum ($p_{\mathrm{T}}$) 111As common for collider physics, we use a
Cartesian coordinate system with the $z$ axis oriented along the beam axis,
the $x$ axis on the horizontal plane, and the $y$ axis oriented upward. The
$x$ and $y$ axes define the transverse plane, while the $z$ axis identifies
the longitudinal direction. The azimuthal angle $\phi$ is computed with
respect to the $x$ axis. The polar angle $\theta$ is used to compute the
pseudorapidity $\eta=-\log(\tan(\theta/2))$. The transverse momentum
($p_{\mathrm{T}}$) is the projection of the particle momentum on the ($x$,
$y$) plane. We fix units such that $c=\hslash=1$., charge, and the
pseudorapidity and azimuthal angle coordinates ($\eta,\phi$), including
extrapolations to the tracker edge
($\eta_{\mathrm{outer}},\phi_{\mathrm{outer}}$).
The track $\eta$ and $\phi$ coordinates are additionally smeared with a 1%
Gaussian resolution to model a finite tracker resolution. Calorimeter clusters
are characterized by electromagnetic or hadron energy $E$ and $\eta,\phi$
coordinates. In this simulation, an event has $N=(4.9\pm 0.3)\times 10^{3}$
detector inputs on average.
The targets for PF reconstruction are stable generator-level particles that
are associated to at least one detector element, as particles that leave no
detector hits are generally not reconstructable. Generator particles are
characterized by a particle identification (PID) which may take one of the
following categorical values: charged hadron, neutral hadron, photon,
electron, or muon. In case multiple generator particles all deposit their
energy completely to a single calorimeter cluster, we treat them as
reconstructable only in aggregate. In this case, the generator particles are
merged by adding the momenta and assigning it the PID of the highest-energy
sub-particle. In addition, charged hadrons are indistinguishable outside the
tracker acceptance from neutral hadrons, therefore we label generated charged
hadrons with $|\eta|>2.5$ to neutral hadrons. We also set a lower energy
threshold on reconstructable neutral hadrons to $E>9.0\,\text{GeV}$ based on
the delphes rule-based PF reconstruction, ignoring neutral hadrons that do not
pass this threshold. A single event from the dataset is visualized in Fig. 1,
demonstrating the input multiplicity and particle distribution in the event.
The differential distributions of the generator-level particles in the
simulated dataset are shown in Fig. 2.
Figure 1: A simulated $\mathrm{t}\overline{\mathrm{t}}$ event from the MLPF
dataset with 200 PU interactions. The input tracks are shown in gray, with the
trajectory curvature being defined by the inner and outer $\eta,\phi$
coordinates. Electromagnetic (hadron) calorimeter clusters are shown in blue
(orange), with the size corresponding to cluster energy for visualization
purposes. We also show the locations of the generator particles (all types)
with red cross markers. The radii and thus the $x,y$-coordinates of the
tracker, ECAL and HCAL surfaces are arbitrary for visualization purposes.
Figure 2: The $p_{\mathrm{T}}$ (upper) and $\eta$ (lower) distributions of the
generator particles in the simulated $\mathrm{t}\overline{\mathrm{t}}$ dataset
with PU, split by particle type.
We also store the PF candidates reconstructed by delphes for comparison
purposes. The delphes rule-based PF algorithm is described in detail in Ref.
deFavereau:2013fsa . Charged and neutral hadrons are identified based on track
and hadron calorimeter cluster overlaps and energy subtraction. Photons are
identified based on electromagnetic calorimeter clusters not matched to
tracks. In addition, we note that electrons and muons are identified by
delphes based on the generator particle associated to the corresponding track,
therefore, for electron and muon tracks we add the corresponding generator-
level identification as an input feature to the MLPF training to demonstrate
that given the appropriate detector inputs, these less common particles can
also be identified by the algorithm.
Each event is now fully characterized by the set of generator particles
$Y=\\{y_{j}\\}$ (target vectors), the set of detector inputs $X=\\{x_{i}\\}$
(input vectors), with
$\displaystyle y_{j}$
$\displaystyle=[\mathrm{PID},p_{\mathrm{T}},E,\eta,\phi,q]\,,$ (1)
$\displaystyle x_{i}$
$\displaystyle=[\mathrm{type},p_{\mathrm{T}},E_{\mathrm{ECAL}},E_{\mathrm{HCAL}},\eta,\phi,\eta_{\mathrm{outer}},\phi_{\mathrm{outer}},q]\,,$
(2) $\displaystyle\mathrm{PID}$ $\displaystyle\in\\{\mathrm{charged\
hadron},\mathrm{neutral\
hadron},\mathrm{\gamma},\mathrm{e}^{\pm},\mathrm{\mu}^{\pm}\\}\,$ (3)
$\displaystyle\mathrm{type}$
$\displaystyle\in\\{\mathrm{track},\mathrm{cluster}\\}\,.$ (4)
For input tracks, only the type, $p_{\mathrm{T}}$, $\eta$, $\phi$,
$\eta_{\mathrm{outer}}$, $\phi_{\mathrm{outer}}$, and $q$ features are filled.
Similarly, for input clusters, only the type, $E_{\mathrm{ECAL}}$,
$E_{\mathrm{HCAL}}$, $\eta$ and $\phi$ entries are filled. Unfilled features
for both tracks and clusters are set to zero. In future iterations of MLPF, it
may be beneficial to represent input elements of different types with separate
data matrices to improve the computational efficiency of the model.
Precomputing additional features such as track trajectory intersection points
with the calorimeters may further improve the performance of PF reconstruction
based on machine learning.
Functionally, the detector is modelled in simulation by a function $S(Y)=X$
that produces a set of detector signals from the generator-level inputs for an
event. Reconstruction imperfectly approximates the inverse of that function
$R\simeq S^{-1}(X)=Y$. In the following section, we approximate the
reconstruction as set-to-set translation and implement a baseline MLPF
reconstruction using GNNs.
## 3 ML-based PF reconstruction
For a given set of detector inputs $X$, we want to predict a set of particle
candidates $Y^{\prime}$ that closely approximates the target generator
particle set $Y$. The target and predicted sets may have a different number of
elements, depending on the quality of the prediction. For use in ML using
gradient descent, this requires a computationally efficient, differentiable
set-to-set metric $||Y-Y^{\prime}||\in\mathbb{R}$ to be used as the loss
function.
We simplify the problem numerically by first zero-padding the target set $Y$
such that $|Y|=|X|$. This turns the problem of predicting a variable number of
particles into a multi-classification prediction by adding an additional “no
particle” to the classes already defined by the target PID and is based on
Ref. Kieseler:2020wcq . Furthermore, for PF reconstruction, the target
generator particles are often geometrically and energetically close to well-
identifiable detector inputs. In physics terms, a charged hadron is
reconstructed based on a track, while a neutral hadron candidate can always be
associated to at least one primary source cluster, with additional corrections
taken from other nearby detector inputs. Therefore, we choose to preprocess
the inputs such that for a given arbitrary ordering of the detector inputs
$X=[\dots,x_{i},\dots]$ (sets of vectors are represented as matrices with some
arbitrary ordering for ML training), the target set $Y$ is arranged such that
if a target particle can be associated to a detector input, it is arranged to
be in the same location in the sequence. This data preprocessing step speeds
up model convergence, but does not introduce any additional assumptions to the
model. Since the target set now has a predefined size, we may compute the loss
function which approximates reconstruction quality element-by-element:
$\displaystyle||Y-Y^{\prime}||$
$\displaystyle\equiv\sum_{j\in\mathrm{event}}L(y_{j},y^{\prime}_{j})\,,$ (5)
$\displaystyle L(y_{j},y^{\prime}_{j})$
$\displaystyle\equiv\mathrm{CLS}(c_{j},c^{\prime}_{j})+\alpha\mathrm{REG}(p_{j},p^{\prime}_{j})\,,$
(6)
where the target values and predictions $y_{j}=[c_{j};p_{j}]$ are decomposed
such that the multi-classification is encapsulated in the scores and one-hot
encoded classes $c_{j}$, while the momentum and charge regression values in
$p_{j}$. We use CLS to denote the multi-classification loss, while REG denotes
the regression loss for the momentum components weighted appropriately by a
coefficient $\alpha$. This combined per-particle loss function serves as a
baseline optimization target for the ML training. Further physics improvements
may be reached by extending the loss to take into account event-level
quantities, either by using an energy flow distance as proposed in Ref.
Komiske:2018cqr ; Komiske:2019fks ; Romao:2020ojy , or using a particle-based
Kansal:2020svm ; Bellagente:2019uyp ; Belayneh:2019vyx ; Butter:2019cae
generative adversarial network (GAN) goodfellow2014generative to optimize the
reconstruction network in tandem with an adversarial classifier that is
trained to distinguish between the target and reconstructed events, given the
detector inputs.
### 3.1 Graph neural network implementation
Figure 3: Functional overview of the end-to-end trainable MLPF setup with
GNNs. The event is represented as a set of detector elements $x_{i}$. The set
is transformed into a graph by the graph building step, which is implemented
here using an locality sensitive hashing (LSH) approximation of kNN. The graph
nodes are then encoded using a message passing step, implemented using graph
convolutional nets. The encoded elements are decoded to the output feature
vectors $y_{j}$ using elementwise feedforward networks.
Given the set of detector inputs for the event $X=\\{x_{i}\\}$, we adopt a
message passing approach for reconstructing the PF candidates $Y=\\{y_{j}\\}$.
First, we need to construct a trainable graph adjacency matrix
$\mathcal{F}(X|w)=A$ for the given set of input elements, represented with the
graph building block in Fig. 3. The input set is heterogeneous, containing
elements of different type (tracks, ECAL clusters, HCAL clusters) in different
feature spaces. Therefore, defining a static neighborhood graph in the feature
space in advance is not straightforward. A generic approach to learnable graph
construction using kNN in an embedding space, known as GravNet, has been
proposed in Ref. Qasim:2019otl , where the authors demonstrated that a
learnable, dynamically-generated graph structure significantly improves the
physics performance of an ML-based reconstruction algorithm for calorimeter
clustering. Similar dynamic graph approaches have also been proposed in Ref.
DGCNN .
However, naive kNN graph implementations in common ML packages such as
TensorFlow or Pytorch-Geometric have $\mathcal{O}(n^{2})$ time complexity: for
each set element out of $n=|X|$, we must order the other $n-1$ elements by
distance and pick the $k$ closest. More efficient kNN graph construction is
possible with, for example, k-dimensional trees rajani2015parallel , but so
far, we are not aware of an implementation that interfaces with common,
differentiable ML tools. For reconstruction, given equivalent physics
performance, both computational efficiency (a low overall runtime) and
scalability (subquadratic time and memory scaling with the input size) are
desirable.
We build on the GravNet approach Qasim:2019otl by using an approximate kNN
graph construction algorithm based on locality sensitive hashing (LSH) to
improve the time complexity of the graph building algorithm. The LSH approach
has been recently proposed kitaev2020reformer for approximating and thus
speeding up ML models that take into account element-to-element relations
using an optimizable $n\times n$ matrix known as self-attention
vaswani2017attention . The method divides the input into bins using a hash
function, such that nearby elements are likely to be assigned to the same bin.
The bins contain only a small number of elements, such that constructing a kNN
graph in the bin is significantly faster than for the full set of elements,
and thus not strongly affected by the quadratic scaling of the kNN algorithm.
In the kNN+LSH approach, the $n$ input elements $x_{i}$ are projected into a
$d_{K}$-dimensional embedding space by a trainable, elementwise feed-forward
network $\mathrm{FFN}(x_{i}|w)=z_{i}\in\mathbb{R}^{d_{K}}$. As in Ref.
kitaev2020reformer , we now assign each element into one of $d_{B}$ bins
indexed by integers $b_{i}$ using $h(z_{i})=b_{i}\in[1,\dots,d_{B}]$, where
$h(x)$ is a hash function that assigns nearby $x$ to the same bin with a high
probability. We define the hash function as
$h(x)=\operatorname*{arg\,max}[xP;-xP]$ where $[u;v]$ denotes the
concatenation of two vectors $u$ and $v$ and $P$ is a random projection matrix
of size $[d_{K},d_{B}/2]$ drawn from the normal distribution at
initialization.
We now build $d_{B}$ kNN graphs based on the embedded elements $z_{i}$ in each
of the LSH bins, such that the full sparse graph adjacency $A_{ij}$ in the
inputs set $X$ is defined by the sum of the subgraphs. The embedding function
can be optimized with backpropagation and gradient descent using the values of
the nonzero elements of $A_{ij}$. Overall, this graph building approach has
$\mathcal{O}(n\log{n})$ time complexity and does not require the allocation of
an $n^{2}$ matrix at any point. The LSH step generates $d_{B}$ disjoint
subgraphs in the full event graph. This is motivated by physics, as we expect
subregions of the detector to be reconstructable approximately independently.
The existing PF algorithm in the CMS detector employs a similar approach by
producing disjoint PF blocks as an intermediate step of the algorithm
Sirunyan:2017ulk .
Having built the graph dynamically, we now use a variant of message passing
4700287 ; Battaglia:2016jem ; gilmer2017neural ; battaglia2018relational to
create hidden encoded states $\mathcal{G}(x_{i},A_{ij}|w)=h_{i}$ of the input
elements taking into account the graph structure. As a first baseline, we use
a variant of graph convolutional network (GCN) that combines local and global
node-level information kipf2016semi ; wu2019simplifying ; xin2020graph . This
choice is motivated by implementation and evaluation efficiency in
establishing a baseline. This message passing step is represented in Fig. 3 by
the GCN block. Finally, we decode the encoded nodes $H=\\{h_{i}\\}$ to the
target outputs with an elementwise feed-forward network that combines the
hidden state with the original input element
$\mathcal{D}(x_{i},h_{i}|w)=y^{\prime}_{i}$ using a skip connection.
We have a joint graph building, but separate graph convolution and decoding
layers for the multi-classification and the momentum and charge regression
subtasks. This allows each subtask to be retrained separately in addition to a
combined end-to-end training should the need arise. The classification and
regression losses are combined with constant empirical weights such that they
have an approximately equal contribution to the full training loss. We use
categorical cross-entropy for the classification loss, which measures the
similarity between the true label distribution $c_{j}$ and the predicted
labels $c^{\prime}_{j}$. For the regression loss, we use componentwise mean-
squared error between the true and predicted momenta, where the losses for the
individual momentum components $(p_{\mathrm{T}},\eta,\sin{\phi},\cos{\phi},E)$
are scaled by normalization factors such that the components have
approximately equal contributions to the total loss. It may be beneficial to
use specific multi-task training strategies such as gradient surgery
yu2020gradient to further improve the performance across all subtasks and to
reduce the reliance on ad-hoc scale factors between the losses in a multi-task
setup.
The multi-classification prediction outputs for each node are converted to
particle probabilities with the softmax operation. We choose the PID with the
highest probability for the reconstructed particle candidate, while ensuring
that the probability meets a threshold that matches a fake rate working point
defined by the baseline delphes PF reconstruction algorithm.
The predicted graph structure is an intermediate step in the model and is not
used in the loss function explicitly—we only optimize the model with respect
to reconstruction quality. However, using the graph structure in the loss
function when a known ground truth is available may further improve the
optimization process. In addition, access to the predicted graph structure may
be helpful in evaluating the interpretability of the model.
The set of networks for graph building, message passing and decoding has been
implemented with TensorFlow 2.3 and can be trained end-to-end using gradient
descent. The inputs are zero-padded to $n=6,400$ elements. Additional elements
beyond 6,400 are truncated for efficient training and performance evaluation,
amounting to about 0.007% of the total number of elements in the
$\mathrm{t}\overline{\mathrm{t}}$ simulation sample. The truncated elements
are always calorimeter towers as the order of the elements is set by the
delphes simulation. For inference during data taking, truncation should be
avoided. The LSH bin size chosen to be $128$ such that the number of bins
$d_{B}=50$ and the number of nearest neighbors $k=16$. We use two hidden
layers for each encoding and decoding net with 256 units each, with two
successive graph convolutions between the encoding and decoding steps.
Exponential linear activations (ELU) clevert2016fast are used for the hidden
layers and linear activations are used for the outputs. Overall, the model has
approximately 1.5 million trainable weights and 25,000 constant weights for
the random projections. For optimization, we use the Adam kingma2017adam
algorithm with a learning rate of $5\times 10^{-6}$ for 300 epochs, training
over $4\times 10^{4}$ events, with $10^{4}$ events used for testing. The
events are processed in minibatches of five simultaneous events per graphics
processing unit (GPU), we train for approximately 48 hours using five RTX
2070S GPUs using data parallelism on 40,000 simulated
$\mathrm{t}\overline{\mathrm{t}}$ events. We report the results of the multi-
task learning problem in the next section. The code and dataset to reproduce
the training are made available on the Zenodo platform
joosep_pata_2021_4452542 ; pata_joosep_2021_4452283 .
## 4 Results
In Fig. 4, we show the $p_{\mathrm{T}}$ distributions for the MLPF
reconstruction and generator-level truth for both simulated QCD multijet and
$\mathrm{t}\overline{\mathrm{t}}$ events. Although the MLPF model was trained
on $\mathrm{t}\overline{\mathrm{t}}$, we observe a slight underprediction at
high transverse momentum for photons and neutral hadrons, which could arise
from the much greater numbers of low-$p_{\mathrm{T}}$ particles relative to
high-$p_{\mathrm{T}}$ particles in this unweighted sample. Further work is
needed to improve the performance in the high-$p_{\mathrm{T}}$ tail of the
distribution. We find that the model generalizes well to the QCD sample that
was not used in the training, demonstrating that the MLPF-based reconstruction
is transferable across different physics samples.
Figure 4: The MLPF reconstruction compared to the truth-level $p_{\mathrm{T}}$
distribution for the QCD validation sample and the
$\mathrm{t}\overline{\mathrm{t}}$ sample used for training. The differences
between the MLPF and truth distributions are a measure of the prediction
error. Charged hadrons, electrons, and muons are identified based on tracks
with no misidentification or loss of efficiency, hence the prediction error is
negligible for both samples. For neutral hadrons and photons, the tail is
reconstructed at a lower efficiency for $\mathrm{t}\overline{\mathrm{t}}$ as
compared to QCD, which could arise from overrepresentation of
low-$p_{\mathrm{T}}$ particles in the unweighted
$\mathrm{t}\overline{\mathrm{t}}$ training sample.
For the following results, we focus on the charged and neutral hadron
performance in QCD events, as hadrons make up the bulk of the energy content
of the jets and thus are the primary target for PF reconstruction. We do not
report detailed performance characteristics for photons, electrons, and muons
at this time because of the limitations of the delphes dataset and the rule-
based PF algorithm. A realistic study of photon and electron disambiguation,
in particular, requires a more detailed dataset that includes additional
physics effects, as discussed in Section 2. In Fig. 5, we present the charged
and neutral hadron multiplicities from both the baseline rule-based PF and
MLPF algorithms as a function of the target multiplicities. The particle
multiplicities from the MLPF model correlate better with the generator-level
target than the rule-based PF algorithm, demonstrating that the multi-
classification model successfully reconstructs variable-multiplicity events.
In general, we do not observe significant differences in the physics
performance of the MLPF algorithm between the QCD and
$\mathrm{t}\overline{\mathrm{t}}$ samples in the phase space where we have
validated it.
Figure 5: True and predicted particle multiplicity for MLPF and delphes PF for
charged (upper) and neutral hadrons (lower) in simulated QCD multijet events
with PU. Both models show a high degree of correlation ($r$) between the
generated and predicted particle multiplicity, with the MLPF model
reconstructing the neutral particle multiplicities with improved resolution
($\sigma$) and a lower bias ($\mu$). Figure 6: Particle identification
confusion matrices in simulated QCD multijet events with PU, with gen-level
particles as the ground truth, showing the baseline rule-based delphes PF
(upper) and the MLPF (lower) outputs. The rows have been normalized to unit
probability, corresponding to normalizing the dataset according to the
generated PID.
In Fig. 6, we compare the per-particle multi-classification confusion matrix
for both reconstruction methods. We see overall a similar classification
performance for both approaches. The charged hadron identification performance
is driven by track efficiency and is the same for MLPF and the rule-based PF.
The neutral hadron identification efficiency is slightly higher for MLPF (0.91
vs 0.88), since hadron calorimeter cluster energies that are not matched to
tracks must be determined algorithmically for neutral hadron reconstruction.
The electron-photon misidentification is driven by the parametrized tracking
efficiency, as electromagnetic calorimeter clusters without an associated
track are reconstructed as photons. Electron and muon identification
performance is shown simply for completeness, as it is driven by the use of
generator-level PID values for those tracks. Improved Monte Carlo generation,
subsampling, or weighting may further improve reconstruction performance for
particles or kinematic configurations that occur rarely in a physical
simulation. In this set of results, we apply no weighting on the events or
particles in the event.
Figure 7: The efficiency of reconstructing charged hadron candidates as a
function of the generator particle pseudorapidity $\eta$ in simulated QCD
multijet events with PU. Since the simulation does not contain fake tracks,
the charged hadron reconstruction is driven entirely by tracking efficiency
and is the same for MLPF and the rule-based PF. Figure 8: The efficiency
(upper) and fake rate (lower) of reconstructing neutral hadron candidates as a
function of the generator particle energy in simulated QCD multijet events
with PU. The MLPF model shows comparable performance to the delphes PF
benchmark, with a somewhat lower fake rate at a similar efficiency.
In Fig. 7, we see that the $\eta$-dependent charged hadron efficiency (true
positive rate) for the MLPF model is somewhat higher than for the rule-based
PF baseline, while the fake rate (false positive rate) is equivalently zero,
as the delphes simulation includes no fake tracks. From Fig. 8, we observe a
similar result for the energy-dependent efficiency and fake rate of neutral
hadrons. Both algorithms exhibit a turn-on at low energies and show a constant
behaviour at high energies, with MLPF being comparable or slightly better than
the rule-based PF baseline.
Furthermore, we see on Figs. 9 and 10 that the energy, energy
($p_{\mathrm{T}}$) and angular resolution of the MLPF algorithm are generally
comparable to the baseline for neutral (charged) hadrons.
Overall, these results demonstrate that formulating PF reconstruction as a
multi-task ML problem of simultaneously identifying charged and neutral
hadrons in a high-PU environment and predicting their momentum may offer
comparable or improved physics performance over hand-written algorithms in the
presence of sufficient simulation samples and careful optimization. The
performance characteristics for the baseline and the proposed MLPF model are
summarized in Table 1.
Figure 9: The $p_{\mathrm{T}}$ and $\eta$ resolution of the delphes PF
benchmark and the MLPF model for charged hadrons in simulated QCD multijet
events with PU. The $p_{\mathrm{T}}$ resolution is comparable for both
algorithms, with the angular resolution being driven by the smearing of the
track $(\eta,\phi)$ coordinates. Figure 10: The energy and $\eta$ resolution
of the delphes PF benchmark and the MLPF model for neutral hadrons in
simulated QCD multijet events with PU. Both reconstruction algorithms show
comparable performance.
We also characterize the computational performance of the GNN-based MLPF
algorithm. In Fig. 11, we see that the average inference time scales roughly
linearly with the input size, which is necessary for scalable reconstruction
at high PU. We also note that the GNN-based MLPF algorithm runs natively on a
GPU, with the current runtime at around 50 ms/event on a consumer-grade GPU
for a full 200 PU event. The algorithm is simple to port to computing
architectures that support common ML frameworks like TensorFlow without
significant investment. This includes GPUs and potentially even field-
programmable gate arrays or ML-specific processors such as the GraphCore
intelligence processing units Mohan:2020vvi through specialized ML compilers
Duarte:2018ite ; Iiyama:2020wap ; Heintz:2020soy . These coprocessing
accelerators can be integrated into existing CPU-based experimental software
frameworks as a scalable service that grows to meet the transient demand
Duarte:2019fta ; Krupa:2020bwg ; Rankin:2020usv .
Figure 11: Average runtime of the MLPF GNN model with a varying input event
size (upper) and the relative inference time when varying the number of events
evaluated simultaneously, i.e. batch size (lower), normalized to batch size 1.
For a simulated event equivalent to 200 PU collisions, we see a runtime of
around 50 ms, which scales approximately linearly with respect to the input
event size. We see a weak dependence on batch size, with batching having a
minor positive effect for low-pileup events. The runtime for each event size
is averaged over 100 randomly generated events over three independent runs.
The timing tests were done using an Nvidia RTX 2060S GPU and an Intel
<EMAIL_ADDRESS>CPU. We assume a linear scaling between PU and the number of
detector elements.
| Charged hadrons | Neutral hadrons
---|---|---
Metric | Rule-based PF | MLPF | Rule-based PF | MLPF
Efficiency | 0.953 | 0.953 | 0.883 | 0.908
Fake rate | 0.000 | 0.000 | 0.071 | 0.068
$p_{\mathrm{T}}$ ($E$) resolution | 0.213 | 0.137 | 0.350 | 0.323
$\eta$ resolution | 0.240 | 0.245 | 0.050 | 0.058
$N$ resolution | 0.004 | 0.004 | 0.014 | 0.013
Table 1: Particle reconstruction efficiency and fake rate, multiplicity $N$,
$p_{\mathrm{T}}$ ($E$) and $\eta$ resolutions for charged (neutral) hadrons,
comparing the rule-based PF baseline and the proposed MLPF method. Bolded
values indicate better performance.
## 5 Discussion and outlook
We have developed a ML algorithm for PF reconstruction in a high-pileup
environment for a general-purpose multilayered particle detector based on
transforming input sets of detector elements to the output set of
reconstructed particles. The MLPF implementation with GNNs is based on graph
building with a LSH approximation for kNN, dubbed LSH+kNN, and message passing
using graph convolutions. Based on benchmark particle-level
$\mathrm{t}\overline{\mathrm{t}}$ and QCD multijet datasets generated using
pythia 8 and delphes 3, the MLPF GNN reconstruction offers comparable
performance to the baseline rule-based PF algorithm in delphes, demonstrating
that a purely parametric ML-based PF reconstruction can reach or exceed the
physics performance of existing reconstruction algorithms, while allowing for
greater portability across various computing architectures at a possibly
reduced cost. The inference time empirically scales approximately linearly
with the input size, which is useful for efficient evaluation in the high-
luminosity phase of the LHC. In addition, the ML-based reconstruction model
may offer useful features for downstream physics analysis like per-particle
probabilities for different reconstruction interpretations, uncertainty
estimates, and optimizable particle-level reconstruction for rare processes
including displaced signatures.
The MLPF model can be further improved with a more physics-motivated
optimization criterion, i.e. a loss function that takes into account event-
level, in addition to particle-level differences. While we have shown that a
per-particle loss function already converges to an adequate physics
performance overall, improved event-based losses such as the object
condensation approach or energy flow may be useful. In addition, an event-
based loss may be defined using an adversarial classifier that is trained to
distinguish the target particles from the reconstructed particles.
Reconstruction algorithms need to adapt to changing experimental
conditions—this may be addressed in MLPF by a periodic retraining on
simulation that includes up-to-date running condition data such as the beam-
spot location, dead channels, and latest calibrations. In a realistic MLPF
training, care must be taken that the reconstruction qualities of rare
particles and particles in the low-probability tails of distributions are not
adversely affected and that the reconstruction performance remains uniform.
This may be addressed with detailed simulations and weighting schemes. In
addition, for a reliable physics result, the interpretability of the
reconstruction is essential. The reconstructed graph structure can provide
information about causal relations between the input detector elements and the
reconstructed particle candidates.
In order to develop a usable ML-based PF reconstruction algorithm, a realistic
high-pileup simulated dataset that includes detailed interactions with the
detector material needs to be used for the ML model optimization. The model
should be optimized and validated on a mix of realistic high-PU events to
learn global properties of reconstruction, as well as on a set of particle gun
samples to ensure that local properties of particle reconstruction are learned
in a generalizable way. To evaluate the reconstruction performance,
efficiencies, fake rates, and resolutions for all particle types need to be
studied in detail as a function of particle kinematics and detector
conditions. Furthermore, high-level derived quantities such as pileup-
dependent jet and missing transverse momentum resolutions must be assessed for
a more complete characterization of the reconstruction performance. With
ongoing work in ML-based track and calorimeter cluster reconstruction upstream
of PF CERN-LHCC-2017-023 ; ATL-PHYS-PUB-2020-018 ; deOliveira:2018lqd ;
Belayneh:2019vyx ; Ju:2020xty ; Choma:2020cry and ML-based reconstruction of
high-level objects including jets and jet classification probabilities
downstream of PF Sirunyan:2017ezt ; Moreno:2019bmu ; Moreno:2019neq ;
Qu:2019gqs ; Aad:2019uoz ; Aad:2019aic ; Bols:2020bkb ; Sirunyan:2020lcu ,
care must be taken that the various steps are optimized and interfaced
coherently.
Finally, the MLPF algorithm is inherently parallelizable and can take
advantage of hardware acceleration of GNNs via GPUs, FPGAs or emerging ML-
specific processors. Current experimental software frameworks can easily
integrate coprocessing accelerators as a scalable service. By harnessing
heterogeneous computing and parallelizable, efficient ML, the burgeoning
computing demand for event reconstruction tasks in the high-luminosity LHC era
can be met while maintaining or even surpassing the current physics
performance.
###### Acknowledgements.
We would like to thank Guenther Dissertori for suggesting the idea of ML-
driven PF reconstruction several years ago in private discussions. We thank
our colleagues in the CMS Collaboration, especially in the Particle Flow,
Physics Performance and Datasets, Offline and Computing, and Machine Learning
groups, in particular Josh Bendavid, Kenichi Hatakeyama, Lindsey Gray, Jan
Kieseler, Danilo Piparo, Gregor Kasieczka, Laurits Tani, and Juska Pekkanen,
for helpful feedback in the course of this work. J. P. was supported by the
Prime National Science Foundation (NSF) Tier2 award 1624356 and the U.S.
Department of Energy (DOE), Office of Science, Office of High Energy Physics
under Award No. DE-SC0011925 while at Caltech, and is currently supported by
the Mobilitas Pluss Grant No. MOBTP187 of the Estonian Research Council. J. D.
is supported by the DOE, Office of Science, Office of High Energy Physics
Early Career Research program under Award No. DE-SC0021187 and by the DOE,
Office of Advanced Scientific Computing Research under Award No. DE-SC0021396
(FAIR4HEP). M. P. is supported by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program (Grant
Agreement No. 772369). J-R. V. and M. S. are supported by the DOE, Office of
Science, Office of High Energy Physics under Award No. DE-SC0011925, DE-
SC0019227, and DE-AC02-07CH11359. J-R. V. was additionally partially supported
the same ERC grant as M. P. We are grateful to Caltech and the Kavli
Foundation for their support of undergraduate student research in cross-
cutting areas of machine learning and domain sciences. This work was conducted
at “iBanks,” the AI GPU cluster at Caltech, and on the NICPB GPU resources,
supported by European Regional Development Fund through the CoE program grant
TK133. We acknowledge Nvidia, SuperMicro and the Kavli Foundation for their
support of iBanks. Part of this work was also performed using the Pacific
Research Platform Nautilus HyperCluster supported by NSF awards CNS-1730158,
ACI-1540112, ACI-1541349, OAC-1826967, the University of California Office of
the President, and the University of California San Diego’s California
Institute for Telecommunications and Information Technology/Qualcomm
Institute. Thanks to CENIC for the 100 Gpbs networks.
## References
* (1) CELLO Collaboration, “An Analysis of the Charged and Neutral Energy Flow in $\mathrm{e^{+}e^{-}}$ Hadronic Annihilation at 34 GeV, and a Determination of the QCD Effective Coupling Constant”, Phys. Lett. B 113 (1982) 427, doi:10.1016/0370-2693(82)90778-X.
* (2) ALEPH Collaboration, “Performance of the ALEPH detector at LEP”, Nucl. Instrum. Meth. A 360 (1995) 481, doi:10.1016/0168-9002(95)00138-7.
* (3) H1 Collaboration, “Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement entropy of partons”, arXiv:2011.01812.
* (4) ZEUS Collaboration, “Measurement of the diffractive structure function F2(D(4)) at HERA”, Eur. Phys. J. C 1 (1998) 81–96, doi:10.1007/s100520050063, arXiv:hep-ex/9709021.
* (5) ZEUS Collaboration, “Measurement of the diffractive cross-section in deep inelastic scattering using ZEUS 1994 data”, Eur. Phys. J. C 6 (1999) 43–66, doi:10.1007/PL00021606, arXiv:hep-ex/9807010.
* (6) DELPHI Collaboration, “Performance of the DELPHI detector”, Nucl. Instrum. Meth. A 378 (1996) 57, doi:10.1016/0168-9002(96)00463-9.
* (7) A. Bocci, S. Lami, S. Kuhlmann, and G. Latino, “Study of jet energy resolution at CDF”, Int. J. Mod. Phys. A 16S1A (2001) 255, doi:10.1142/S0217751X01006632.
* (8) A. L. Connolly, “A Search for Supersymmetric Higgs Bosons in the Di-tau Decay Mode in $p\bar{p}$ Collisions at 1.8 TeV”. PhD thesis, UC Berkeley, 2003. doi:10.2172/15017134.
* (9) CDF Collaboration, “Measurement of $\sigma(p\bar{p}\to Z).{\rm Br}(Z\to 2\tau)$ in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV”, Phys. Rev. D 75 (2007) 092004, doi:10.1103/PhysRevD.75.092004.
* (10) D0 Collaboration, “Measurement of $\sigma(p\bar{p}\to Z+X)$ Br($Z\to\tau^{+}\tau^{-}$) at $\sqrt{s}=1.96\,\text{TeV}$”, Phys. Lett. B 670 (2009) 292, doi:10.1016/j.physletb.2008.11.010, arXiv:0808.1306.
* (11) CMS Collaboration, “Particle-flow reconstruction and global event description with the CMS detector”, JINST 12 (2017) P10003, doi:10.1088/1748-0221/12/10/P10003, arXiv:1706.04965.
* (12) ATLAS Collaboration, “Jet reconstruction and performance using particle flow with the ATLAS Detector”, Eur. Phys. J. C 77 (2017) 466, doi:10.1140/epjc/s10052-017-5031-2, arXiv:1703.10485.
* (13) CMS Collaboration Collaboration, “Challenges of particle flow reconstruction in the CMS High-Granularity Calorimeter at the High-Luminosity LHC”, Technical Report CMS-CR-2016-151. 1, CERN, Geneva, Jul, 2016. doi:10.1088/1742-6596/928/1/012027.
* (14) FCC-hh Collaboration, “Physics requirements for the FCC-hh calorimeter system”, J. Phys. Conf. Ser. 1162 (2019) 012010, doi:10.1088/1742-6596/1162/1/012010.
* (15) FCC Collaboration, “FCC-hh: The Hadron Collider”, Eur. Phys. J. ST 228 (2019) 755, doi:10.1140/epjst/e2019-900087-0.
* (16) FCC Collaboration, “FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2”, Eur. Phys. J. ST 228 (2019) 261, doi:10.1140/epjst/e2019-900045-4.
* (17) T. Behnke et al., “The International Linear Collider Technical Design Report - Volume 1: Executive Summary”, arXiv:1306.6327.
* (18) CEPC Study Group Collaboration, “CEPC Conceptual Design Report: Volume 2 - Physics \& Detector”, arXiv:1811.10545.
* (19) F. Scarselli et al., “The graph neural network model”, IEEE Trans. Neural Netw. 20 (2009), no. 1, 61, doi:10.1109/TNN.2008.2005605.
* (20) J. Gilmer et al., “Neural message passing for quantum chemistry”, in Proceedings of the 34th International Conference on Machine Learning, D. Precup and Y. W. Teh, eds., volume 70, p. 1263. PMLR, 2017. arXiv:1704.01212.
* (21) C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation”, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 2017\. arXiv:1612.00593. doi:10.1109/CVPR.2017.16.
* (22) P. W. Battaglia et al., “Interaction Networks for Learning about Objects, Relations and Physics”, in Advances in Neural Information Processing Systems, D. Lee et al., eds., volume 29, p. 4502. Curran Associates, Inc., 2016. arXiv:1612.00222.
* (23) Y. Wang et al., “Dynamic Graph CNN for Learning on Point Clouds”, ACM Trans. Graph. 38 (2019) doi:10.1145/3326362, arXiv:1801.07829.
* (24) J. Shlomi, P. Battaglia, and J.-R. Vlimant, “Graph Neural Networks in Particle Physics”, Mach. Learn.: Sci. Technol. 2 (2021) 021001, doi:10.1088/2632-2153/abbf9a, arXiv:2007.13681.
* (25) S. Farrell et al., “Novel deep learning methods for track reconstruction”, in 4th International Workshop Connecting the Dots. 2018\. arXiv:1810.06111.
* (26) X. Ju et al., “Graph Neural Networks for Particle Reconstruction in High Energy Physics detectors”, in 2nd Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems. 3, 2020. arXiv:2003.11603.
* (27) S. Amrouche et al., “The Tracking Machine Learning Challenge : Accuracy Phase”, in The NeurIPS ’18 Competition, p. 231. 2020\. arXiv:1904.06778. doi:10.1007/978-3-030-29135-8_9.
* (28) S. Amrouche et al., “Similarity hashing for charged particle tracking”, in IEEE International Conference on Big Data 2019, p. 1595. 2019\. doi:10.1109/BigData47090.2019.9006316.
* (29) N. Choma et al., “Track Seeding and Labelling with Embedded-space Graph Neural Networks”, in 6th International Workshop Connecting the Dots. 2020\. arXiv:2007.00149.
* (30) X. Ju and B. Nachman, “Supervised Jet Clustering with Graph Neural Networks for Lorentz Boosted Bosons”, Phys. Rev. D 102 (2020) 075014, doi:10.1103/PhysRevD.102.075014, arXiv:2008.06064.
* (31) J. Li, T. Li, and F.-Z. Xu, “Reconstructing boosted Higgs jets from event image segmentation”, arXiv:2008.13529.
* (32) J. Guo, J. Li, and T. Li, “The Boosted Higgs Jet Reconstruction via Graph Neural Network”, arXiv:2010.05464.
* (33) E. A. Moreno et al., “JEDI-net: a jet identification algorithm based on interaction networks”, Eur. Phys. J. C 80 (2020) 58, doi:10.1140/epjc/s10052-020-7608-4, arXiv:1908.05318.
* (34) E. A. Moreno et al., “Interaction networks for the identification of boosted $H\rightarrow b\overline{b}$ decays”, Phys. Rev. D 102 (2020) 012010, doi:10.1103/PhysRevD.102.012010, arXiv:1909.12285.
* (35) H. Qu and L. Gouskos, “ParticleNet: Jet Tagging via Particle Clouds”, Phys. Rev. D 101 (2020) 056019, doi:10.1103/PhysRevD.101.056019, arXiv:1902.08570.
* (36) V. Mikuni and F. Canelli, “ABCNet: An attention-based method for particle tagging”, Eur. Phys. J. Plus 135 (2020), no. 6, 463, doi:10.1140/epjp/s13360-020-00497-3, arXiv:2001.05311.
* (37) S. R. Qasim, J. Kieseler, Y. Iiyama, and M. Pierini, “Learning representations of irregular particle-detector geometry with distance-weighted graph networks”, Eur. Phys. J. C 79 (2019) 608, doi:10.1140/epjc/s10052-019-7113-9, arXiv:1902.07987.
* (38) J. Arjona Martínez et al., “Pileup mitigation at the Large Hadron Collider with graph neural networks”, Eur. Phys. J. Plus 134 (2019) 333, doi:10.1140/epjp/i2019-12710-3, arXiv:1810.07988.
* (39) J. Kieseler, “Object condensation: one-stage grid-free multi-object reconstruction in physics detectors, graph and image data”, Eur. Phys. J. C 80 (2020) 886, doi:10.1140/epjc/s10052-020-08461-2, arXiv:2002.03605.
* (40) F. A. Di Bello et al., “Towards a Computer Vision Particle Flow”, arXiv:2003.08863.
* (41) J. Duarte and J.-R. Vlimant, “Graph Neural Networks for Particle Tracking and Reconstruction”, in Artificial Intelligence for Particle Physics. World Scientific Publishing, 2020. arXiv:2012.01249. Submitted to _Int. J. Mod. Phys. A_. doi:10.1142/12200.
* (42) T. Sjöstrand, S. Mrenna, and P. Z. Skands, “pythia 6.4 Physics and Manual”, JHEP 05 (2006) 026, doi:10.1088/1126-6708/2006/05/026, arXiv:hep-ph/0603175.
* (43) T. Sjöstrand, S. Mrenna, and P. Z. Skands, “A Brief Introduction to pythia 8.1”, Comput. Phys. Commun. 178 (2008) 852, doi:10.1016/j.cpc.2008.01.036, arXiv:0710.3820.
* (44) DELPHES 3 Collaboration, “delphes 3, A modular framework for fast simulation of a generic collider experiment”, JHEP 02 (2014) 057, doi:10.1007/JHEP02(2014)057, arXiv:1307.6346.
* (45) S. Chekanov, “HepSim: a repository with predictions for high-energy physics experiments”, Adv. High Energy Phys. 2015 (2015) 136093, doi:10.1155/2015/136093, arXiv:1403.1886.
* (46) J. Pata et al., “Simulated particle-level events of $\mathrm{t}\overline{\mathrm{t}}$ and QCD with PU200 using pythia 8+delphes 3 for machine learned particle flow (MLPF)”, 2021. doi:10.5281/zenodo.4559324.
* (47) P. T. Komiske, E. M. Metodiev, and J. Thaler, “Energy Flow Networks: Deep Sets for Particle Jets”, JHEP 01 (2019) 121, doi:10.1007/JHEP01(2019)121, arXiv:1810.05165.
* (48) P. T. Komiske, E. M. Metodiev, and J. Thaler, “Metric Space of Collider Events”, Phys. Rev. Lett. 123 (2019) 041801, doi:10.1103/PhysRevLett.123.041801, arXiv:1902.02346.
* (49) M. C. Romao et al., “Use of a Generalized Energy Mover’s Distance in the Search for Rare Phenomena at Colliders”, arXiv:2004.09360.
* (50) R. Kansal et al., “Graph Generative Adversarial Networks for Sparse Data Generation in High Energy Physics”, in 3rd Machine Learning and the Physical Sciences Workshop at the 34th Conference on Neural Information Processing Systems. 2020\. arXiv:2012.00173.
* (51) M. Bellagente et al., “How to GAN away Detector Effects”, SciPost Phys. 8 (2020) 070, doi:10.21468/SciPostPhys.8.4.070, arXiv:1912.00477.
* (52) D. Belayneh et al., “Calorimetry with deep learning: Particle simulation and reconstruction for collider physics”, Eur. Phys. J. C 80 (2020) 688, doi:10.1140/epjc/s10052-020-8251-9, arXiv:1912.06794.
* (53) A. Butter, T. Plehn, and R. Winterhalder, “How to GAN LHC Events”, SciPost Phys. 7 (2019) 075, doi:10.21468/SciPostPhys.7.6.075, arXiv:1907.03764.
* (54) I. J. Goodfellow et al., “Generative Adversarial Nets”, in Advances in Neural Information Processing Systems, Z. Ghahramani et al., eds., volume 27. Curran Associates, Inc., 2014. arXiv:1406.2661.
* (55) N. Rajani, K. McArdle, and I. S. Dhillon, “Parallel k nearest neighbor graph construction using tree-based data structures”, in 1st High Performance Graph Mining workshop, volume 1, pp. 3–11. 2015\.
* (56) N. Kitaev, Ł. Kaiser, and A. Levskaya, “Reformer: The Efficient Transformer”, in 8th International Conference on Learning Representations. 2020\. arXiv:2001.04451.
* (57) A. Vaswani et al., “Attention Is All You Need”, in Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30, p. 5998. Curran Associates, Inc., 2017. arXiv:1706.03762.
* (58) F. Scarselli et al., “The graph neural network model”, IEEE Trans. Neural Netw. 20 (2009) 61, doi:10.1109/TNN.2008.2005605.
* (59) P. W. Battaglia et al., “Relational inductive biases, deep learning, and graph networks”, arXiv:1806.01261.
* (60) T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks”, in 5th International Conference on Learning Representations. 2017\. arXiv:1609.02907.
* (61) F. Wu et al., “Simplifying Graph Convolutional Networks”, in Proceedings of the 36th International Conference on Machine Learning, K. Chaudhuri and R. Salakhutdinov, eds., volume 97, p. 6861. PMLR, 2019. arXiv:1902.07153.
* (62) X. Xin, A. Karatzoglou, I. Arapakis, and J. M. Jose, “Graph Highway Networks”, arXiv:2004.04635.
* (63) T. Yu et al., “Gradient surgery for multi-task learning”, in Advances in Neural Information Processing Systems, H. Larochelle et al., eds., volume 33. 2020\. arXiv:2001.06782.
* (64) D.-A. Clevert, T. Unterthiner, and S. Hochreiter, “Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)”, in 4th International Conference on Learning Representations. 2016\. arXiv:1511.07289.
* (65) D. P. Kingma and J. Ba, “Adam: A Method for Stochastic Optimization”, in 3rd International Conference on Learning Representations, Y. Bengio and Y. LeCun, eds. 2015\. arXiv:1412.6980.
* (66) J. Pata, J. M. Duarte, and A. Tepper, “jpata/particleflow: MLPF delphes paper software release”. https://github.com/jpata/particleflow, 2021. doi:10.5281/zenodo.4559587.
* (67) L. R. M. Mohan et al., “Studying the potential of Graphcore IPUs for applications in Particle Physics”, arXiv:2008.09210.
* (68) J. Duarte et al., “Fast inference of deep neural networks in FPGAs for particle physics”, JINST 13 (2018) P07027, doi:10.1088/1748-0221/13/07/P07027, arXiv:1804.06913.
* (69) Y. Iiyama et al., “Distance-Weighted Graph Neural Networks on FPGAs for Real-Time Particle Reconstruction in High Energy Physics”, Front. Big Data 3 (2021) 44, doi:10.3389/fdata.2020.598927, arXiv:2008.03601.
* (70) A. Heintz et al., “Accelerated charged particle tracking with graph neural networks on FPGAs”, in 3rd Machine Learning and the Physical Sciences Workshop at the 34th Conference on Neural Information Processing Systems. 2020\. arXiv:2012.01563.
* (71) J. Duarte et al., “FPGA-accelerated machine learning inference as a service for particle physics computing”, Comput. Softw. Big Sci. 3 (2019) 13, doi:10.1007/s41781-019-0027-2, arXiv:1904.08986.
* (72) J. Krupa et al., “GPU coprocessors as a service for deep learning inference in high energy physics”, Machine Learning: Science and Technology 2 (April, 2021) 035005, doi:10.1088/2632-2153/abec21.
* (73) D. S. Rankin et al., “FPGAs-as-a-Service Toolkit (FaaST)”, in 2020 IEEE/ACM International Workshop on Heterogeneous High-performance Reconfigurable Computing (H2RC). 2020\. arXiv:2010.08556. doi:10.1109/H2RC51942.2020.00010.
* (74) CMS Collaboration, “The Phase-2 Upgrade of the CMS Endcap Calorimeter”, CMS Technical Design Report CERN-LHCC-2017-023. CMS-TDR-019, CERN, 2017.
* (75) ATLAS Collaboration, “Deep Learning for Pion Identification and Energy Calibration with the ATLAS Detector”, ATLAS Public Note ATL-PHYS-PUB-2020-018, CERN, 2020.
* (76) L. De Oliveira, B. Nachman, and M. Paganini, “Electromagnetic Showers Beyond Shower Shapes”, Nucl. Instrum. Meth. A 951 (2020) 162879, doi:10.1016/j.nima.2019.162879, arXiv:1806.05667.
* (77) CMS Collaboration, “Identification of heavy-flavour jets with the CMS detector in pp collisions at 13 TeV”, JINST 13 (2018) P05011, doi:10.1088/1748-0221/13/05/P05011, arXiv:1712.07158.
* (78) ATLAS Collaboration, “Identification of boosted Higgs bosons decaying into b-quark pairs with the ATLAS detector at 13 TeV”, Eur. Phys. J. C 79 (2019) 836, doi:10.1140/epjc/s10052-019-7335-x, arXiv:1906.11005.
* (79) ATLAS Collaboration, “ATLAS b-jet identification performance and efficiency measurement with $\mathrm{t}\overline{\mathrm{t}}$ events in pp collisions at $\sqrt{s}=13\,\text{TeV}$”, Eur. Phys. J. C 79 (2019) 970, doi:10.1140/epjc/s10052-019-7450-8, arXiv:1907.05120.
* (80) E. Bols et al., “Jet Flavour Classification Using DeepJet”, JINST 15 (2020) P12012, doi:10.1088/1748-0221/15/12/P12012, arXiv:2008.10519.
* (81) CMS Collaboration, “Identification of heavy, energetic, hadronically decaying particles using machine-learning techniques”, JINST 15 (2020) P06005, doi:10.1088/1748-0221/15/06/P06005, arXiv:2004.08262.
*[LHC]: CERN Large Hadron Collider
*[MLPF]: machine-learned particle-flow
*[PF]: particle-flow
*[ECAL]: electromagnetic calorimeter
*[HCAL]: hadron calorimeter
*[FCC]: Future Circular Collider
*[ML]: machine learning
*[GNNs]: graph neural network
*[PU]: pileup
*[GNN]: graph neural network
*[PID]: particle identification
*[GAN]: generative adversarial network
*[LSH]: locality sensitive hashing
*[kNN]: k-nearest neighbors
*[GCN]: graph convolutional network
*[GPU]: graphics processing unit
*[GPUs]: graphics processing unit
*[FPGAs]: field-programmable gate array
|
# Monitoring nonstationary processes based on recursive cointegration analysis
and elastic weight consolidation
Jingxin Zhang, Donghua Zhou, , and Maoyin Chen This work was supported by
National Natural Science Foundation of China [grant numbers 62033008,
61751307, 61873143]. (Corresponding authors: Donghua Zhou; Maoyin Chen)Jingxin
Zhang is with the Department of Automation, Tsinghua University, Beijing
100084, China (e-mail: zjx18@mails.tsinghua.edu.cn). Donghua Zhou is with
College of Electrical Engineering and Automation, Shandong University of
Science and Technology, Qingdao 266000, China and also with the Department of
Automation, Tsinghua University, Beijing 100084, China (e-mail:
zdh@mail.tsinghua.edu.cn).Maoyin Chen is with the Department of Automation,
Tsinghua University, Beijing 100001, China and also with School of Automation
and Electrical Engineering, Linyi University, Linyi 276005, China (e-mail:
mychen@tsinghua.edu.cn).This paper has been submitted to IEEE Transaction on
Cybernetics for potential publication.
###### Abstract
This paper considers the problem of nonstationary process monitoring under
frequently varying operating conditions. Traditional approaches generally
misidentify the normal dynamic deviations as faults and thus lead to high
false alarms. Besides, they generally consider single relatively steady
operating condition and suffer from the catastrophic forgetting issue when
learning successive operating conditions. In this paper, recursive
cointegration analysis (RCA) is first proposed to distinguish the real faults
from normal systems changes, where the model is updated once a new normal
sample arrives and can adapt to slow change of cointegration relationship.
Based on the long-term equilibrium information extracted by RCA, the remaining
short-term dynamic information is monitored by recursive principal component
analysis (RPCA). Thus a comprehensive monitoring framework is built. When the
system enters a new operating condition, the RCA-RPCA model is rebuilt to deal
with the new condition. Meanwhile, elastic weight consolidation (EWC) is
employed to settle the ‘catastrophic forgetting’ issue inherent in RPCA, where
significant information of influential parameters is enhanced to avoid the
abrupt performance degradation for similar modes. The effectiveness of the
proposed method is illustrated by a practical industrial system.
###### Index Terms:
Nonstationary process monitoring, recursive cointegration analysis, elastic
weight consolidation, recursive PCA
## I Background
Process monitoring is increasingly significant and essential to guarantee the
process safety [1, 2, 3, 4, 5]. Approaches for stationary processes have been
intensively investigated and considerable achievements have been obtained [6,
7, 8, 9, 10]. However, process data are generally nonstationary due to varying
load, changes of raw materials, aging of equipments and product grade
transitions, etc [11, 12]. This phenomenon is ubiquitous in industrial
systems, for instance, the power stations, oil explorations, chemical
processes, etc. It is urgent and challenging to investigate the monitoring
techniques for nonstationary processes under various potential operating
conditions [13].
Recently, several methods have been developed for nonstationary process
monitoring. Canonical variate analysis extracts dynamic latent information by
state space formulations, which aims to reduce the order of dynamics and is
generally applied to linear systems [14]. Dynamic latent variable models
(DLVMs) extract dynamic and static latent components simultaneously, which
extract the most predictable information first [15]. The switching
autoregressive DLVM was proposed for multimode processes in the probabilistic
framework [16] and it requires that the model covers all operating modes.
These methods aforementioned fail to distinguish the real faults from normal
deviations under varying operating conditions, thus delivering high false
alarm rates. To settle this issue, slow feature analysis (SFA) was proposed to
identify the real fault from the operating point deviation, by separating
dynamic information from steady state information [17]. SFA requires that the
system operates in a particular steady condition [18], which is unsuitable for
frequently varying operating conditions.
Cointegration analysis (CA) is an effective method to deal with nonstationary
data [19, 20] and able to distinguish the real faults from normal dynamic
changes under various operating conditions. It is based on the general
consesus that the long-term equilibrium relationship, i.e., cointegration
relationship, exists in physical and chemical processes because the
nonstationary variables are correlated to each other and governed by
specifical laws [21]. When the cointegration relationship is broken, the
system enters a new mode if the dynamic equilibrium relationship returns to
normal. Cointegration testing method was adopted primarily for nonstationary
process monitoring in [21]. Zhao _et al_. intensively investigated CA and
proposed several extensions of CA, including dynamic distributed strategy for
large-scale processes [22], CA with SFA to establish a full-condition
monitoring model [23].
However, these CA-based methods assume that the cointegration relationship
remains the same[22, 23], which is unrealistic in practical systems. Take the
coal pulverizing system of power plant as an instance. The compositions and
characteristics of one coal may change slowly because they are influenced by
environments and it is difficult to mix the coal quite evenly. Thus, the
cointegration relationship would change accordingly. Hansen _et al_. presented
a recursive form of cointegrated vector autoregressive models [24], which
could update the cointegration relationship to adapt to the new condition.
Another form of recursive CA was proposed to adapt to the slowly changing
cointegration relationship and the model was updated based on a block of data
[25]. However, the monitoring consequences are affected by the length of data
block and it is intractable to determine the optimal value. Only the dynamic
information that reflected the control performance was extracted and the
remaining information was neglected, thus causing insensitivity to detecting
the faults that are orthogonal to cointegration space [26].
It is also a universal phenomenon in practical industrial systems that the
cointegration relationship may change sharply and frequently. For instance,
the type of coal changes in power plants frequently owing to the environmental
requirements and economical benefits. The compositions and calorific value of
different coals vary greatly. Assume that there are various variables and can
be sorted into three blocks. One block of variables shares the similar trend
and one block represents the critical manipulated variables, while the
remaining variables are contained in another block. Thus the cointegration
relationship and some manipulated variables may be transited from one steady
state to another. It has been mentioned [25] that it is necessary to establish
a new CA model from scratch, to quickly adjust to the new cointegration
relationship based on the newly collected data. But the recursive CA failed to
provide excellent performance [25] because it requires abundant data.
Aimed at the issues mentioned above, this paper investigates the general
nonstationary process monitoring, where the cointegration relationship and the
manipulated variables would change from one steady state to another
frequently. First, in order to distinguish the real faults from normal dynamic
deviations, a novel version of recursive cointegration analysis (RCA) is
proposed to track the long-term equilibrium relationship, where the CA model
is updated once a new sample arrives. Based on RCA, we introduce recursive
principal component analysis (RPCA) to deal with the remaining information,
thus establishing a comprehensive monitoring framework. For convenience, RCA
with RPCA is denoted as RCA-RPCA.
When the cointegration relationship is recognized as broken by RCA and the
dynamic equilibrium relationship returns to normal quickly, the system enters
a new operating mode. We need to retrain the RCA and RPCA models from scratch
based on the new data. Here, we employ elastic weight consolidation (EWC) to
settle the ‘catastrophic forgetting’ issue of RPCA [27], where the significant
information that is influential in previous modes is preserved to avoid the
dramatic performance degradation for similar operating modes. For convenient
description, the proposed RCA-RPCA with EWC is referred to as RCA-RPCA-EWC. In
addition, test statistics are established based on the prior knowledge and CA
theory, which is more sensitive to normal change than recursive CA [25].
The rest of this paper is organized below. Section II introduces the problem
and reviews the basic theory of CA. Section III presents the detailed
procedure of the proposed RCA and summarizes the monitoring algorithm based on
RCA-RPCA. Then, the proposed RCA-RPCA is extended to multimode processes in
Section IV, where EWC is employed to overcome the ‘catastrophic forgetting’
issue of RPCA when a new mode appears. Section V summarizes the general
procedure for nonstationary process monitoring, analyzes the computational
complexity and compares with the state-of-the-art approaches. The
effectiveness is illustrated by a practical industrial system in Section VI.
The concluding remark is presented in Section VII.
## II Problem formulation and preliminary
### II-A Problem statement
Since there are various variables of multiple trends in practical
applications, how to deal with the variables appropriately affects the
monitoring performance severely. Take the practical coal pulverizing system of
power plant as an instance.
The variables are affected by load and types of coal. Partial variables are
described in Fig. 1. The variables are decomposed into three blocks based on
prior knowledge and augmented Dicky Fuller (ADF), where the final results rely
on the prior knowledge and ADF test is the auxiliary to enhance universality.
Variables in Fig. 1 are nonstationary and share the common trend, which are
normally influenced by varying load. For variables in Fig. 1, the uppermost
variable is regulated by controllers and the manipulated variable is expected
to vary from one steady state to another one if the type of coal changes. The
other three variables change slowly or irregularly. The appropriate method
needs to be investigated to deal with data of different characteristics, thus
delivering an optimal monitoring performance. Note that this variable grouping
makes it possible that this proposed method is sensitive to changes of
manipulated variables and mode identification.
() CA variables
() other variables
Figure 1: Practical data from the coal pulverizing system
This paper studies the general case of sequential nonstationary process
monitoring, where the stationary variables and the cointegration relationship
change from one steady state to another. RCA processes the data with common
trend to extract long-term equilibrium information and RPCA is adopted to deal
with other variables to extract short-term dynamics, thus constructing a
comprehensive monitoring framework for nonstationary processes. When the
cointegrated relationship or stationary variables change, the system enters a
new mode and EWC is adopted to preserve the significant information of
previous modes, thus delivering excellent performance for successive modes
based on single model.
### II-B Conventional CA algorithm
Given the nonstationary time series
$\boldsymbol{X}^{0}=\left\\{\boldsymbol{x}^{0}_{t}\right\\}_{t=1}^{N}$ with
$\boldsymbol{x}^{0}_{t}\in\mathbb{R}^{m_{1}}$. The reference mean
${\boldsymbol{\tilde{\mu}_{1}}}$ and reference standard deviation
$\tilde{\sigma}_{1},\cdots,\tilde{\sigma}_{m_{1}}$ are calculated as
$\boldsymbol{\tilde{\mu}_{1}}=\frac{1}{N}(\boldsymbol{X}^{0})^{T}\boldsymbol{1}$
(1)
${{\tilde{\sigma}}_{i}}=\frac{1}{{N-1}}\sum\limits_{t=1}^{N}{{{\left({x_{t,i}^{0}-{{\tilde{\mu}}_{i}}}\right)}^{2}}},i\in\left\\{{1,\ldots,{m_{1}}}\right\\}$
(2)
where ${x_{t,i}^{0}}$ is the $i$th variable at $t$th sampling instant,
$\boldsymbol{1}$ is the vector of all ones with appropriate dimension. Thus,
the original data $\boldsymbol{X}^{0}$ are normalized as
$\boldsymbol{X}=\left({{\boldsymbol{X}^{0}}-\boldsymbol{1}{\boldsymbol{\tilde{\mu}}^{T}}}\right){\boldsymbol{\tilde{\Sigma}}^{-1}}$
(3)
where
$\boldsymbol{\tilde{\Sigma}}=diag\left\\{\tilde{\sigma}_{1},\cdots,\tilde{\sigma}_{m_{1}}\right\\}$.
The vector error-correction (VEC) model is described as:
$\varDelta\boldsymbol{x}_{t}=\sum_{i=1}^{p-1}{\boldsymbol{\varOmega}_{i}\varDelta\boldsymbol{x}_{t-i}+\boldsymbol{\varGamma}\boldsymbol{x}_{t-1}+\boldsymbol{\varepsilon}_{t}}$
(4)
where ${\varDelta\boldsymbol{x}_{t}=\boldsymbol{x}_{t}-\boldsymbol{x}_{t-1}}$,
$p$ is the order of VEC model and determined by AIC.
$\boldsymbol{\varepsilon}_{t}$ is the Gaussian white noise with
${\boldsymbol{\varepsilon}\sim N\left(\textbf{0},\boldsymbol{\varXi}\right)}$.
${\boldsymbol{\varGamma}=\boldsymbol{\Upsilon}\boldsymbol{B}_{f}^{T}\in\mathbb{R}^{m_{1}\times
m_{1}}}$, where ${\boldsymbol{\Upsilon}\in\mathbb{R}^{m_{1}\times r}}$ and
${\boldsymbol{B}_{f}\in\mathbb{R}^{m_{1}\times r}}$ are of full rank $r$. The
columns in $\boldsymbol{B}_{f}$ are cointegration vectors. The objective of CA
is to determine $\boldsymbol{B}_{f}$ to make the equilibrium errors
$\boldsymbol{X}\boldsymbol{B}_{f}$ as stationary as possible.
Johansen _et al_. proved that (4) could be settled by optimizing the
likelihood function [20, 28]:
$\displaystyle
L\left(\boldsymbol{\varOmega}_{1},\cdots,\boldsymbol{\varOmega}_{p-1},\boldsymbol{\Upsilon},\boldsymbol{B}_{f},\boldsymbol{\varXi}\right)$
(5)
$\displaystyle=-\frac{Nm_{1}}{2}\ln\left(2\right)-\frac{N}{2}\ln\left|\boldsymbol{\varXi}\right|-\frac{1}{2}\sum_{t=1}^{N}{\boldsymbol{\varepsilon}_{t}^{T}\boldsymbol{\varXi}^{-1}\boldsymbol{\varepsilon}_{t}}$
The maximum likelihood estimation of cointegration vectors in
$\boldsymbol{B}_{f}$ is acquired by eigenvalue decomposition (EVD) [20]
$\left|\tilde{\lambda}\boldsymbol{S}_{11}-\boldsymbol{S}_{10}\boldsymbol{S}_{00}^{-1}\boldsymbol{S}_{01}\right|=0$
(6)
where
$\boldsymbol{S}_{i,j}=\frac{1}{N-p}\boldsymbol{E}_{i}^{T}\boldsymbol{E}_{j}$,
$\boldsymbol{E}_{i}$ ($i=0,1$) is the prediction error and calculated by
$\boldsymbol{E}_{0}=\varDelta\boldsymbol{X}_{p}-\varDelta\boldsymbol{X}^{p}\boldsymbol{\varTheta}$
(7)
$\boldsymbol{E}_{1}=\boldsymbol{X}_{p}-\varDelta\boldsymbol{X}^{p}\boldsymbol{\varPhi}$
(8)
where $\varDelta\boldsymbol{X}_{p}\in\mathbb{R}^{\left(N-p\right)\times
m_{1}}$ is the difference matrix, the vector
$\varDelta\boldsymbol{x}_{p+1}=\boldsymbol{x}_{p+1}-\boldsymbol{x}_{p}$ is the
temporal difference between two neighboring data points.
$\boldsymbol{X}_{p}\in\mathbb{R}^{\left(N-p\right)\times m_{1}}$ originates
from the observation matrix $\boldsymbol{X}$.
$\varDelta\boldsymbol{X}^{p}\in\mathbb{R}^{\left(N-p\right)\times pm_{1}}$ is
the augmented matrix which contains $p$ lagged observations. The specific
structures are described as
$\boldsymbol{X}_{p}=\left[\begin{array}[]{c}\boldsymbol{x}_{p}\\\
\boldsymbol{x}_{p+1}\\\ \vdots\\\ \boldsymbol{x}_{N-1}\\\
\end{array}\right],\quad\varDelta\boldsymbol{X}_{p}=\left[\begin{array}[]{c}\varDelta\boldsymbol{x}_{p+1}\\\
\varDelta\boldsymbol{x}_{p+2}\\\ \vdots\\\ \varDelta\boldsymbol{x}_{N}\\\
\end{array}\right]$ (9)
$\varDelta\boldsymbol{X}^{p}=\left[\begin{matrix}\varDelta\boldsymbol{x}_{1}&\cdots&\varDelta\boldsymbol{x}_{p}\\\
\vdots&\ddots&\vdots\\\
\varDelta\boldsymbol{x}_{N-p}&\cdots&\varDelta\boldsymbol{x}_{N-1}\\\
\end{matrix}\right]=\left[\begin{array}[]{c}\varDelta\boldsymbol{x}_{1}^{p}\\\
\vdots\\\ \varDelta\boldsymbol{x}_{N-p}^{p}\\\ \end{array}\right]$ (10)
The coefficients $\boldsymbol{\varTheta}$ and $\boldsymbol{\varPhi}$ are
obtained by ordinary least squares (OLS). Actually, (6) can be reformulated as
$\boldsymbol{A}\boldsymbol{w}={\lambda}\boldsymbol{B}\boldsymbol{w}$ (11)
where
$\boldsymbol{A}=\left[\begin{matrix}\boldsymbol{0}&\boldsymbol{S}_{01}\\\
\boldsymbol{S}_{10}&\boldsymbol{0}\\\ \end{matrix}\right]$,
$\boldsymbol{B}=\left[\begin{matrix}\boldsymbol{S}_{00}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{S}_{11}\\\ \end{matrix}\right]$, the generalized
eigenvalues are listed in the descending order.
$\boldsymbol{W}=\left[{{\boldsymbol{w}_{1}},\cdots,{\boldsymbol{w}_{r}}}\right]\in\mathbb{R}^{2m_{1}\times
r}$ contains the generalized principal eigenvectors corresponding to $r$
largest eigenvalues and $r$ is determined by the trace test [20]. The
cointegration matrix $\boldsymbol{B}_{f}$ and dynamic cointegration matrix
$\boldsymbol{B}_{e}$ are acquired from $\boldsymbol{W}$, namely,
$\boldsymbol{W}=\left[{{\boldsymbol{B}_{e}};{\boldsymbol{B}_{f}}}\right]$.
More information about CA can be found in [19, 20].
## III The proposed RCA-RPCA for process monitoring
In this section, we propose RCA to adapt to new cointegration relationship
once a new sample arrives. The RCA issue is formulated into a recursive
generalized EVD problem and settled by standard EVD. Besides, four test
statistics are constructed according to prior knowledge and RCA-RPCA theory.
According to the prior knowledge and ADF test, this paper divides the
variables into three blocks, one block represents the nonstationary variables
with common trend, which are conducted by RCA and labeled as
$\boldsymbol{x}_{1}$. One block indicates the stationary variables that are
sensitive to operating conditions and are denoted as $\boldsymbol{x}_{2}$.
Generally, $\boldsymbol{x}_{2}$ is the critical manipulated variables and
especially significant for industrial systems. The remaining block includes
the variables independent of working conditions, which are expected to be
stationary or change over the external environment and labeled as
$\boldsymbol{x}_{3}$. As a note, the variables are not necessarily divided
into three blocks for any industrial system. It depends on the system
characteristics and change regularities. However, the monitoring framework
proposed in this paper also applies to this situation equally.
### III-A Recursive cointegration analysis
We establish the initial CA model based on Section II-B. If the cointegration
relationship changes slowly, the collected data are preprocessed by fixed mean
and standard deviation, as described in (1-2). The procedure of RCA is
proposed below.
At $k+1$ instant, collect $\boldsymbol{x}_{k+1}^{0}$ and scale data as
$\boldsymbol{x}_{k+1}$. The sample is divided into three blocks, namely,
$\boldsymbol{x}_{k+1}=\left[{\begin{array}[]{*{20}{c}}\boldsymbol{x}_{1,k+1}&\boldsymbol{x}_{2,k+1}&\boldsymbol{x}_{3,k+1}\end{array}}\right]$.
Only $\boldsymbol{x}_{1,k+1}$ is utilized for RCA. Thus, the observations for
RCA are
$\boldsymbol{X}_{1,k+1}=\left[\begin{array}[]{c}\boldsymbol{X}_{1,k}\\\
\boldsymbol{x}_{1,k+1}\\\ \end{array}\right]$. Similar to (9-10),
$\boldsymbol{X}_{p,k+1}$, $\varDelta\boldsymbol{X}_{p,k+1}$ and
$\varDelta\boldsymbol{X}_{k+1}^{p}$ are generated from
$\boldsymbol{X}_{1,k+1}$.
The prediction errors are
$\boldsymbol{E}_{0,k+1}=\varDelta\boldsymbol{X}_{p,k+1}-\varDelta\boldsymbol{X}_{k+1}^{p}\boldsymbol{\varTheta}_{k+1}$
(12)
$\boldsymbol{E}_{1,k+1}=\boldsymbol{X}_{p,k+1}-\varDelta\boldsymbol{X}_{k+1}^{p}\boldsymbol{\varPhi}_{k+1}$
(13)
According to recursive OLS, $\boldsymbol{\varTheta}_{k+1}$ and
$\boldsymbol{\varPhi}_{k+1}$ are determined by:
$\boldsymbol{\varTheta}_{k+1}=\boldsymbol{\varTheta}_{k}+\boldsymbol{R}_{k+1}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\left(\varDelta\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varTheta}_{k}\right)$
(14)
$\boldsymbol{\varPhi}_{k+1}=\boldsymbol{\varPhi}_{k}+\boldsymbol{R}_{k+1}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\left(\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varPhi}_{k}\right)\\\
$ (15)
where
$\boldsymbol{R}_{k+1}=\boldsymbol{R}_{k}-\frac{\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{R}_{k}}{1+\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}}$
(16)
Bring (14,16) into (12), then
$\displaystyle\boldsymbol{E}_{0,k+1}$ (17) $\displaystyle=$
$\displaystyle\varDelta\boldsymbol{X}_{p,k+1}-\varDelta\boldsymbol{X}_{k+1}^{p}\boldsymbol{\varTheta}_{k+1}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\varDelta\boldsymbol{X}_{p,k}\\\
\varDelta\boldsymbol{x}_{p,k+1}\\\
\end{array}\right]-\left[\begin{array}[]{c}\varDelta\boldsymbol{X}_{k}^{p}\\\
\varDelta\boldsymbol{x}_{k+1}^{p}\\\ \end{array}\right]$
$\displaystyle\cdot\left(\boldsymbol{\varTheta}_{k}+\frac{\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}}{1+c_{k+1}}\left(\varDelta\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varTheta}_{k}\right)\right)$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\boldsymbol{E}_{0,k}-\frac{\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}}{1+c_{k+1}}\left(\varDelta\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varTheta}_{k}\right)\\\
\frac{1}{1+c_{k+1}}\left(\varDelta\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varTheta}_{k}\right)\\\
\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\boldsymbol{E}_{0,k}-\boldsymbol{D}_{k+1}\\\
\boldsymbol{d}_{k+1}\\\ \end{array}\right]$
where
$\boldsymbol{d}_{k+1}=\frac{1}{1+c_{k+1}}\left(\varDelta\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varTheta}_{k}\right)$,
$c_{k+1}=\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}$,
$\boldsymbol{D}_{k+1}=\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\boldsymbol{d}_{k+1}$.
Let $\boldsymbol{J}_{k}=\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}$,
the recursion of $\boldsymbol{J}_{k}$ is
$\displaystyle\boldsymbol{J}_{k}$
$\displaystyle=\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}$ (18)
$\displaystyle=\left[\begin{array}[]{c}\varDelta\boldsymbol{X}_{k-1}^{p}\\\
\varDelta\boldsymbol{x}_{k}^{p}\\\
\end{array}\right]\left(\boldsymbol{R}_{k-1}-\frac{\boldsymbol{R}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}}{1+c_{k}}\right)$
$\displaystyle=\left[\begin{array}[]{c}\boldsymbol{J}_{k-1}-\frac{\boldsymbol{J}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}}{1+c_{k}}\\\
\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}-\frac{\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}}{1+c_{k}}\\\
\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}\boldsymbol{J}_{k-1}\boldsymbol{\tilde{J}}_{k}\\\
\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}\boldsymbol{\tilde{J}}_{k}\\\
\end{array}\right]$
where
$\boldsymbol{\tilde{J}}_{k}=\boldsymbol{I}-\frac{\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}}{1+c_{k}}$,
$\boldsymbol{I}$ is the identity matrix with appropriate dimension. Thus,
$\boldsymbol{J}_{k}$ and $\boldsymbol{D}_{k+1}$ are calculated recursively.
Similarly,
$\displaystyle\boldsymbol{E}_{1,k+1}$ (19) $\displaystyle=$
$\displaystyle\boldsymbol{X}_{p,k+1}-\varDelta\boldsymbol{X}_{k+1}^{p}\boldsymbol{\varPhi}_{k+1}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\boldsymbol{E}_{1,k}-\frac{\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}}{1+c_{k+1}}\left(\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varPhi}_{k}\right)\\\
\frac{1}{1+c_{k+1}}\left(\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varPhi}_{k}\right)\\\
\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\boldsymbol{E}_{1,k}-\boldsymbol{H}_{k+1}\\\
\boldsymbol{h}_{k+1}\\\ \end{array}\right]$
where
$\boldsymbol{h}_{k+1}=\frac{1}{1+c_{k+1}}\left(\boldsymbol{x}_{p,k+1}-\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{\varPhi}_{k}\right)$,
$\boldsymbol{H}_{k+1}=\varDelta\boldsymbol{X}_{k}^{p}\boldsymbol{R}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\boldsymbol{h}_{k+1}=\boldsymbol{J}_{k}\left(\varDelta\boldsymbol{x}_{k+1}^{p}\right)^{T}\boldsymbol{h}_{k+1}$.
Combining (17-19), $\boldsymbol{A}$ and $\boldsymbol{B}$ are calculated
recursively as
$\displaystyle\boldsymbol{A}_{k+1}$ (20) $\displaystyle=$
$\displaystyle\frac{1}{k+1}\left[\begin{matrix}\boldsymbol{0}&\boldsymbol{E}_{0,k+1}^{T}\boldsymbol{E}_{1,k+1}\\\
\boldsymbol{E}_{1,k+1}^{T}\boldsymbol{E}_{0,k+1}&\boldsymbol{0}\\\
\end{matrix}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{k+1}\left(\left[\begin{matrix}\boldsymbol{0}&\boldsymbol{E}_{0,k}^{T}\boldsymbol{E}_{1,k}\\\
\boldsymbol{E}_{1,k}^{T}\boldsymbol{E}_{0,k}&\boldsymbol{0}\\\
\end{matrix}\right]+\left[\begin{matrix}\boldsymbol{0}&\varDelta{\boldsymbol{A}}_{1,k+1}\\\
\varDelta{\boldsymbol{A}}_{2,k+1}&\boldsymbol{0}\\\
\end{matrix}\right]\right)$ $\displaystyle=$
$\displaystyle\alpha_{k+1}\boldsymbol{A}_{k}+(1-\alpha_{k+1})\varDelta{\boldsymbol{A}}_{k+1}$
where $\alpha_{k+1}=\frac{k}{k+1}$,
$\varDelta{\boldsymbol{A}}_{1,k+1}=-\boldsymbol{D}_{k+1}^{T}\boldsymbol{E}_{1,k}-\boldsymbol{E}_{0,k}^{T}\boldsymbol{H}_{k+1}+\boldsymbol{D}_{k+1}^{T}\boldsymbol{H}_{k+1}+\boldsymbol{d}_{k+1}^{T}\boldsymbol{h}_{k+1}$,
$\varDelta{\boldsymbol{A}}_{2,k+1}=\varDelta{\boldsymbol{A}}_{1,k+1}^{T}$.
$\displaystyle\boldsymbol{B}_{k+1}$ (21) $\displaystyle=$
$\displaystyle\frac{1}{k+1}\left[\begin{matrix}\boldsymbol{E}_{0,k+1}^{T}\boldsymbol{E}_{0,k+1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{E}_{1,k+1}^{T}\boldsymbol{E}_{1,k+1}\\\
\end{matrix}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{k+1}\left(\left[\begin{matrix}\boldsymbol{E}_{0,k}^{T}\boldsymbol{E}_{0,k}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{E}_{1,k}^{T}\boldsymbol{E}_{1,k}\\\
\end{matrix}\right]+\left[\begin{matrix}\varDelta{\boldsymbol{B}}_{1,k+1}&\boldsymbol{0}\\\
\boldsymbol{0}&\varDelta{\boldsymbol{B}}_{2,k+1}\\\
\end{matrix}\right]\right)$ $\displaystyle=$
$\displaystyle\alpha_{k+1}\boldsymbol{B}_{k}+(1-\alpha_{k+1})\varDelta{\boldsymbol{B}}_{k+1}$
where
$\varDelta{\boldsymbol{B}}_{1,k+1}=\boldsymbol{D}_{k+1}^{T}\boldsymbol{D}_{k+1}-\boldsymbol{D}_{k+1}^{T}\boldsymbol{E}_{0,k}+\boldsymbol{d}_{k+1}^{T}\boldsymbol{d}_{k+1}-\boldsymbol{E}_{0,k}^{T}\boldsymbol{D}_{k+1}$,
$\varDelta{\boldsymbol{B}}_{2,k+1}=\boldsymbol{H}_{k+1}^{T}\boldsymbol{H}_{k+1}+\boldsymbol{h}_{k+1}^{T}\boldsymbol{h}_{k+1}-\boldsymbol{E}_{1,k}^{T}\boldsymbol{H}_{k+1}-\boldsymbol{H}_{k+1}^{T}\boldsymbol{E}_{1,k}$.
Obviously,
$rank\left(\boldsymbol{d}_{k+1}^{T}\boldsymbol{d}_{k+1}-\boldsymbol{E}_{0,k}^{T}\boldsymbol{D}_{k+1}\right)=1$,
$rank(\boldsymbol{D}_{k+1}^{T})=1$. Thus,
$rank(\varDelta{\boldsymbol{B}}_{1,k+1})\leqslant 2$. Similarly,
$rank(\varDelta{\boldsymbol{B}}_{2,k+1})\leqslant 2$.
$\varDelta{\boldsymbol{A}}$ and $\varDelta{\boldsymbol{B}}$ are also
calculated recursively, as described in Appendix -B.
The proposed RCA is reformulated into settling the following generalized EVD
problem:
$\boldsymbol{A}_{k+1}\boldsymbol{W}_{k+1}=\boldsymbol{B}_{k+1}\boldsymbol{W}_{k+1}\boldsymbol{\bar{\varLambda}}_{k+1}$
(22)
where $\boldsymbol{\bar{\varLambda}}_{k+1}$ is the diagonal matrix and
elements are generalized eigenvalues with descending order.
### III-B Solution for numerical efficient recursive CA
In this paper, we convert a generalized EVD issue to a standard symmetric EVD
problem. As $\boldsymbol{B}_{\boldsymbol{k}+1}$ is symmetric and positive
definite, let
$\boldsymbol{K}_{k+1}=\left(\boldsymbol{B}^{\frac{1}{2}}_{k+1}\right)^{-1}=\boldsymbol{B}^{-\frac{1}{2}}_{k+1}$,
(22) can be reformulated as
$\boldsymbol{K}_{k+1}\boldsymbol{A}_{k+1}\boldsymbol{K}^{T}_{k+1}\boldsymbol{\bar{W}}_{k+1}=\boldsymbol{\bar{W}}_{k+1}\boldsymbol{\bar{\varLambda}}_{k+1}$
(23)
where $\boldsymbol{K}_{k+1}$ is positive definite,
$\boldsymbol{\bar{W}}_{k+1}=\boldsymbol{K}^{-T}_{k+1}\boldsymbol{W}_{k+1}$.
Computing $\boldsymbol{K}$ directly may be ill-conditioning per update, thus
it is essential to acquire the recursion of $\boldsymbol{K}$ and avoid
inverting a matrix repeatedly. The detailed derivation procedure is presented
in Appendix -C. To further reduce the computational burden, the recursion of
$\boldsymbol{K}$ is obtained based on the rank of $\varDelta\boldsymbol{B}$.
The procedure for (22) is summarized in Algorithm 1.
Algorithm 1 Solution based on Cholesky decomposition
1: Calculate $\boldsymbol{A}_{k+1}$ by (20) and
$\varDelta{\boldsymbol{A}}_{k+1}$ by Appendix -B;
2: Calculate $\boldsymbol{B}_{k+1}$ by (21) and
$\varDelta{\boldsymbol{B}}_{k+1}$ by Appendix -B;
3: Compute $\boldsymbol{K}_{k+1}=\boldsymbol{B}^{-\frac{1}{2}}_{k+1}$, as
described in Appendix -C;
4: Compute
$\boldsymbol{C}_{k+1}=\boldsymbol{K}_{k+1}\boldsymbol{A}_{k+1}\boldsymbol{K}^{T}_{k+1}$;
5: Solve the eigenvalue problem of $\boldsymbol{C}_{k+1}$ by symmetric QR
algorithm, the eigenvectors and eigenvalues are denoted as
$\boldsymbol{\bar{W}}_{k+1}$ and $\boldsymbol{\bar{\varLambda}}_{k+1}$,
respectively;
6: Compute
$\boldsymbol{W}_{k+1}=\boldsymbol{K}^{T}_{k+1}\boldsymbol{\bar{W}}_{k+1}$.
### III-C Monitoring statistics
In this section, we construct the monitoring statistics to judge the operating
conditions. The proposed RCA is utilized to extract the long-term equilibrium
information and the short-term dynamic features are handled by RPCA. The key
steps of RPCA have been elaborated in Appendix -A.
At $k+1$ instant, a new sample is collected and preprocessed as
$\boldsymbol{x}_{k+1}=\left[{\begin{array}[]{*{20}{c}}\boldsymbol{x}_{1,k+1}&\boldsymbol{x}_{2,k+1}&\boldsymbol{x}_{3,k+1}\end{array}}\right]$.
Let
${\hat{\boldsymbol{x}}_{1,k+1}}=\left[{{\boldsymbol{x}_{1,k+1}}{\boldsymbol{B}_{f,k}}}\quad{{\boldsymbol{x}_{2,k+1}}}\right]$.
The cointegration matrix $\boldsymbol{B}_{f,k}$ and dynamic cointegration
matrix $\boldsymbol{B}_{e,k}$ are generated from generalized eigenvectors
$\boldsymbol{W}_{k}$ in Algorithm 1.
$T^{2}_{f}$ is designed to judge whether the long-term static equilibrium
relationship is still preserved.
$T^{2}_{f}={\hat{\boldsymbol{x}}_{1,k+1}}{\hat{\boldsymbol{x}}_{1,k+1}}^{T}$
(24)
$T^{2}_{e}$ is designed to monitor the long-term dynamic equilibrium
relationship.
$T^{2}_{e}={\boldsymbol{e}_{0,k+1}}\boldsymbol{B}_{e,k}\boldsymbol{B}_{e,k}^{T}{\boldsymbol{e}_{0,k+1}}^{T}$
(25)
where the prediction error ${\boldsymbol{e}_{0,k+1}}$ is the last sample of
$\boldsymbol{E}_{0,k+1}$.
Define
$\boldsymbol{B}_{f,k}^{\bot}=\boldsymbol{I}-\boldsymbol{B}_{f,k}\left(\boldsymbol{B}_{f,k}^{T}\boldsymbol{B}_{f,k}\right)^{-1}\boldsymbol{B}_{f,k}^{T}$,
${\hat{\boldsymbol{x}}_{2,k+1}}=\left[{{\boldsymbol{x}_{1,k+1}}\boldsymbol{B}_{f,k}^{\bot}}\quad{{\boldsymbol{x}_{3,k+1}}}\right]$.
The short-term dynamic information ${\hat{\boldsymbol{x}}_{2,k+1}}$ is
monitored by RPCA and two statistics are calculated by
$T^{2}=\hat{\boldsymbol{x}}_{2,k+1}\boldsymbol{P}_{k}\boldsymbol{\Lambda}_{k}^{-1}\boldsymbol{P}_{k}^{T}\hat{\boldsymbol{x}}^{T}_{2_{k}+1}$
(26)
$SPE=\hat{\boldsymbol{x}}_{2,k+1}\left(\boldsymbol{I}-\boldsymbol{P}_{k}\boldsymbol{P}_{k}^{T}\right)\hat{\boldsymbol{x}}^{T}_{2,k+1}$
(27)
where $\boldsymbol{\Lambda}_{k}$ and $\boldsymbol{P}_{k}$ represent
eigenvalues and eigenvectors of RPCA, which are updated by (32-35) in Appendix
-A.
## IV Multimode process monitoring with EWC
In this section, we extend the nonstationary monitoring technique to multimode
processes. Here, we define a mode where the stationary variables and the long-
term static equilibrium fluctuate within acceptable range, which can be
measured by $T^{2}_{f}$ and $T^{2}_{e}$ statistics. Actually, the data are
still nonstationary in one mode.
When the system operates from one steady operating condition to another, the
data distribution may change accordingly. Meanwhile, the cointegration
relationship and the stationary variables may also vary dramatically. It has
been illustrated that the recursive strategy of CA based on all collected data
is unreasonable and may lead to high false alarms [25]. RPCA also fails to
track the rapid changes accurately. It is essential to build the proposed RCA-
RPCA monitoring model from scratch. However, similar to most machine learning
approaches [29, 30, 31, 32], RPCA suffers from the ‘catastrophic forgetting’
issue and most information of the previous modes is overlapped when a new
model is rebuilt. To settle this issue, EWC [27] is employed at the initial
training phase of RPCA, where significant information from influential
variables in previous modes is enhanced to avoid drastic changes. Thus, the
proposed RCA-RPCA-EWC method can deliver outstanding monitoring performance
when similar or the existing operating modes reappear.
Here, we introduce the procedure of RPCA with EWC (RPCA-EWC), which is similar
to PCA with EWC in [32]. Let $\boldsymbol{P}^{*}_{0}$ denote the projection
matrix for the previous operating mode. When a new mode is detected by RCA,
the initial collected short-term dynamic data are denoted as
$\boldsymbol{X}_{2}$. The off-line training model of RPCA is built with EWC,
thus the objective is designed as
$\displaystyle\mathcal{J}(\boldsymbol{P})$
$\displaystyle=\mathcal{J}_{2}(\boldsymbol{P})+\zeta\mathcal{J}_{loss}(\boldsymbol{P},\boldsymbol{P}^{*}_{0})$
(28)
$\displaystyle=\|\boldsymbol{X}_{2}-\boldsymbol{X}_{2}\boldsymbol{P}\boldsymbol{P}^{T}\|_{F}^{2}+\|\boldsymbol{P}-\boldsymbol{P}^{*}_{0}\|_{{\boldsymbol{\Omega}}}^{2}$
where the hyperparameter $\zeta$ measures the importance of previous modes.
The matrix ${\boldsymbol{\Omega}}$ is positive semidefinite, which is
influenced by $\zeta$ and determined by [32, 33]. The constraint is
$\boldsymbol{P}^{T}\boldsymbol{P}=\boldsymbol{I}$ with
${\boldsymbol{P}\in\mathbb{R}^{m_{2}\times l}}$, $l$ is the number of
principal components and determined by cumulative percent variance (CPV)
approach. $\mathcal{J}_{2}(\boldsymbol{P})$ is the loss function of RPCA for
the current mode. $\mathcal{J}_{loss}(\boldsymbol{P},\boldsymbol{P}^{*}_{0})$
is the loss function which measures the deviation of key parameters between
two successive operating modes.
Algorithm 2 Solution of RPCA-EWC
1: Let $\boldsymbol{P}_{0}=\boldsymbol{P}_{0}^{*}$ be the initial solution,
error $\varepsilon$, set $i=0$;
2: Calculate
$\boldsymbol{Y}_{i}={\boldsymbol{\Omega}}\boldsymbol{P}_{0}^{*}+{\boldsymbol{X}_{2}^{T}}{\boldsymbol{X}_{2}}\boldsymbol{P}_{i}$;
3: Conduct singular vector decomposition on $\boldsymbol{Y}_{i}$, namely,
${\boldsymbol{Y}_{i}}={\boldsymbol{W}_{i}}{\boldsymbol{\Upsilon}_{i}}\boldsymbol{V}_{i}^{T}$;
4:
$\boldsymbol{P}_{i+1}=\boldsymbol{W}_{i}\boldsymbol{I}_{m,l}\boldsymbol{V}_{i}^{T}$;
5: Let $i=i+1$, go to 2 until
$\lVert\boldsymbol{P}_{i+1}-\boldsymbol{P}_{i}\rVert_{F}^{2}<\varepsilon$.
The objective function (28) is actually the difference of convex (DC)
functions programming problem [34, 35]. DC programming includes linearizing
the convex function and solving the convex function. The specific deviation
process has been described in [32] and some key steps are listed in Appendix
-D. The solution is summarized in Algorithm 2. Note that the matrix
${\boldsymbol{\Omega}}$ measures the importance of parameters and should be
updated before a new mode appears. The calculation method can refer to [33,
36, 32].
In summary, when a new mode is judged by RCA, RCA model is rebuilt from
scratch and the procedure is similar to Section III. RPCA-EWC is adopted at
the initial training phase and then parameters are updated by (32-35), thus
avoiding abrupt degradation of monitoring performance when similar modes
revisit.
Algorithm 3 Off-line training
1: Collect the initial data $\boldsymbol{X}_{N_{0}}^{0}$, and set $k=N_{0}$;
2: According to prior knowledge and ADF test, divide $\boldsymbol{X}_{k}^{0}$
into three blocks, namely, $\boldsymbol{X}_{1,k}^{0}$,
$\boldsymbol{X}_{2,k}^{0}$ and $\boldsymbol{X}_{3,k}^{0}$;
3: Calculate the mean values and standard deviations of
$\boldsymbol{X}_{1,k}^{0}$ and $\boldsymbol{X}_{2,k}^{0}$, i.e.,
$\hat{\boldsymbol{\mu}}_{1}$, $\hat{\boldsymbol{\Sigma}}_{1}$,
$\hat{\boldsymbol{\mu}}_{2}$, $\hat{\boldsymbol{\Sigma}}_{2}$. Scale data and
denote as $\boldsymbol{X}_{1,k}$ and $\boldsymbol{X}_{2,k}$;
4: Conduct CA on $\boldsymbol{X}_{1,k}$: a) Construct $\boldsymbol{X}_{p,k}$,
$\varDelta\boldsymbol{X}_{p,k}$ and $\varDelta\boldsymbol{X}_{k}^{p}$ by
(9-10);b) Calculate coefficients of (7-8), labeled by
$\boldsymbol{\Theta}_{k}$ and $\boldsymbol{\Phi}_{k}$;c) Calculate
$\boldsymbol{E}_{0,k}$ and $\boldsymbol{E}_{1,k}$ in (7-8), and compute
$\boldsymbol{A}_{k}$ and $\boldsymbol{B}_{k}$;d) Solve (11) and obtain
$\boldsymbol{B}_{f,k}$, $\boldsymbol{B}_{e,k}$;
5: Calculate $\boldsymbol{B}_{f,k}^{\bot}$ and construct
${\hat{\boldsymbol{X}}_{2,k}^{0}}=\left[{{\boldsymbol{X}_{1,k}}\boldsymbol{B}_{f,k}^{\bot}}\quad{{\boldsymbol{X}_{3,k}^{0}}}\right]$.
Calculate the mean $\boldsymbol{\mu}_{k}$ and standard deviation
$\boldsymbol{\Sigma}_{k}$, scale data and denote as
$\hat{\boldsymbol{X}}_{2,k}$;
6: Conduct PCA on ${\hat{\boldsymbol{X}}_{2,k}}$, and calculate
$\boldsymbol{P}_{k}$ and $\boldsymbol{\Lambda}_{k}$;
7: Calculate test statistics by (24-27) and the corresponding thresholds by
KDE.
## V Monitoring algorithm
State-of-the-art approaches explore the nonstationary processes for a single
mode [14, 26, 22, 23], where the stationary variables and the long-term static
equilibrium fluctuate within a certain range. When the operating mode changes,
the data distribution may vary accordingly and the original cointegration
relationship is broken. This section introduces the general monitoring
framework for multimode processes, which is also appropriate for a single
mode.
Similar to Section III, the normal data are divided into three blocks. At the
training phase, the long-term equilibrium information is extracted by CA and
PCA is utilized to monitor the remaining short-term dynamic information. Four
test statistics are calculated by (24-27), where $T_{f}^{2}$ and $T_{e}^{2}$
are employed to identify the operating status and $T^{2}$ and SPE are utilized
to monitor the short-term dynamics. The corresponding thresholds are
calculated by kernel density estimation (KDE) [6]. The off-line training
procedure is summarized in Algorithm 3.
For the practical industrial applications, when a new sample arrives, the
operating status is judged and the monitoring model is updated if normal, as
described in Algorithm 4. The thresholds are updated by KDE. Note that an
occasional anomaly is regarded as noise or disturbance. The fault is detected
if the anomaly lasts a short time.
The monitoring rule is summarized below:
1. 1.
All test statistics are within their thresholds, it is regarded that the
process operates normally in the same operating mode. The proposed RCA-RPCA is
still employed to update the parameters;
2. 2.
If $T^{2}_{e}$, $T^{2}$ and $SPE$ return to normal after $T^{2}_{f}$ is over
its threshold, it indicates that the system enters a new operating state and
then RCA-RPCA-EWC is adopted to monitor the system;
3. 3.
If $T^{2}_{f}$ and $T^{2}_{e}$ are within their thresholds, while $T^{2}$ or
$SPE$ is over its threshold, then a fault may occur and it is essential to
check the operation of the systems;
4. 4.
All test statistics exceed their thresholds, then the process is out of
control. A real fault is detected and the alarm is triggered.
Algorithm 4 Online monitoring
1: Collect $\boldsymbol{x}_{k+1}^{0}$, divide the sample into three blocks
based on step 2 in Algorithm 3, and scale data;
2: Construct and scale $\hat{\boldsymbol{x}}_{1,k+1}$ and
$\hat{\boldsymbol{x}}_{2,k+1}$, calculate test statistics by (24-27);
3: Judge the operating status: a) Normal, go to step 4; b) A new mode appears.
$n_{0}$ samples are collected, set $k=n_{0}$, go to step 2 in Algorithm 3. Set
$\boldsymbol{X}_{2}=\hat{\boldsymbol{X}}_{2,k}$ and the step 6 is replaced by
RPCA-EWC in Algorithm 2; c) Potential fault or real fault occurs, thus the
alarm is triggered;
4: Conduct RCA based on the current CA model and $\boldsymbol{x}_{1,k+1}$: a)
Construct $\boldsymbol{x}_{p,k+1}$, $\varDelta\boldsymbol{x}_{p,k+1}$ and
$\varDelta\boldsymbol{x}_{k+1}^{p}$;b) Calculate $\boldsymbol{\Theta}_{k+1}$
and $\boldsymbol{\Phi}_{k+1}$ by (15-16);c) Calculate $\boldsymbol{E}_{0,k+1}$
and $\boldsymbol{E}_{1,k+1}$ by (17-19), and compute $\boldsymbol{A}_{k+1}$
and $\boldsymbol{B}_{k+1}$ by (20-21);d) Solve (22) by Algorithm 1, and obtain
$\boldsymbol{B}_{f,k+1}$, $\boldsymbol{B}_{e,k+1}$;
5: Construct
$\tilde{\boldsymbol{x}}_{2,k+1}=\left[{{\boldsymbol{x}_{1,k+1}}\boldsymbol{B}_{f,k+1}^{\bot}}\quad{{\boldsymbol{x}_{3,k+1}}}\right]$,
conduct RPCA:a) Calculate $\boldsymbol{\mu}_{k+1}$ and
$\boldsymbol{\Sigma}_{k+1}$ by (29-30);b) Calculate $\boldsymbol{P}_{k+1}$ and
$\boldsymbol{\Lambda}_{k+1}$ by (32-35);c) Select $l$ based on CPV;
6: Set $k=k+1$ and return to step 1.
### V-A Computational complexity analysis
For online monitoring phase, the computational complexity contains the
computation of RCA and RPCA at each step, and RPCA-EWC when the operating mode
changes. The RCA and RPCA algorithms occupy the most computational source and
are considered in this paper.
For RCA, the computation focuses on Algorithm 1. The complexity of
$\varDelta\boldsymbol{A}_{k+1}$ and $\varDelta\boldsymbol{B}_{k+1}$ is
$O(m_{1}^{2})$, as illustrated in Appendix -B. The complexity of
$\boldsymbol{K}_{k+1}$ is $O(m_{1}^{3})$ in Appendix -C. Then, the calculation
of $\boldsymbol{C}_{k+1}$ needs $8m_{1}^{3}$ flops. The symmetric QR algorithm
requires at most $32m_{1}^{3}$ flops theoretically because
$\boldsymbol{C}_{k+1}$ is a block skew diagonal matrix. The calculation of
$\boldsymbol{W}_{k+1}$ in Algorithm 1 requires $8m_{1}^{3}$ flops. In summary,
the computational complexity of RCA is $O(m_{1}^{3})$ per update. For RPCA,
the complexity of $\boldsymbol{P}$ and $\boldsymbol{\Lambda}$ is
$O(m_{2}^{3}+m_{2}^{2})$. Obviously, ${m_{1}}<m$, ${m_{2}}\leq m$, $m$ is the
dimension of collected data. That is, the computation per update will not grow
as the number of samples $k$ increases.
### V-B Comparison and Discussion
We compare the recursive CA [25] with the proposed RCA-RPCA-EWC method below:
$\bullet$ Model tracking accuracy. The recursive CA model is updated based on
a block of data [25], and it is intractable to determine the data length to
deliver the optimal performance. However, the proposed RCA model is updated
once a new normal sample arrives. In the case that the cointegration
relationship changes sharply and frequently, the recursive CA [25] may fail to
track the normal change, while the proposed approach can establish the inexact
model based on just a few data and correct the model gradually.
$\bullet$ Sensitivity of mode switching identification. The operating status
is judged by $T_{f}^{2}$ and $T_{e}^{2}$. The construction of two statistics
is only based on nonstationary data that reflect the control performance [25].
The proposed statistics consider the prior knowledge and data simultaneously,
which is more sensitive to mode switching.
$\bullet$ Memory properties. The EWC technique is adopted to overcome the
‘catastrophic forgetting’ issue of RPCA and significant information of
previous operating conditions is enhanced to avoid dramatic changes of
influential parameters. The proposed RCA-RPCA can be updated accurately when
previous or similar operating modes appear, thus delivering optimal monitoring
performance.
$\bullet$ Algorithm complexity. The computational burden is highly related to
the number of current collected samples at each update step [25]. Although the
models stop to update to reduce complexity and false alarm, it is hard to
satisfy the criterion. For the proposed method, the computational cost is
$O(m^{3})$ per update, irrespective of the number of samples.
Here we make a further discussion about the variable decomposition. In this
paper, the variables are divided into three blocks based on prior knowledge
and data, as mentioned in Section III. Specifically, we first employ the
theory of industrial systems to partition variables. Then we adopt ADF test
and correlation analysis to verify and strengthen the rationality of variable
grouping. Thus, the changes of stationary variables would not be covered by
the normal variations of nonstationary variables. Note that it is not
necessary that the variables are decomposed into three blocks in any
industrial system. The number of blocks relies on the characteristics of
systems and selected variables. However, the monitoring framework in Section
III is also applied. This variable grouping method is sensitive to critical
manipulated variables and mode identification, which is beneficial to enhance
monitoring performance.
## VI Case study
This section adopts a practical industrial system to illustrate the
effectiveness of the proposed method. Besides, we make a comparative analysis
with the state-of-the-art methods to highlight the superiorities of the
proposed method.
Figure 2: Schematic diagram of the coal pulverizing system TABLE I: Data information of the pulverizing system Case number | Data original | Training samples | Testing samples | Fault time | Fault cause
---|---|---|---|---|---
Case 1 | Aomei-Aomeng-Aomei | 2000 | 8800 | 6734 | The opening of the regulating baffle of the primary air is abnormally large
Case 2 | Aomeng-Youhun | 2000 | 15280 | 8731 | Abnormality from cold primary air electric regulating baffle card
Case 3 | Fudong-Aomeng-Fudong | 2000 | 13120 | 9010 | The cooling fan motor trip
TABLE II: Evaluation indexes of the case study Case number | Indexes | Recursive CA [25] | RCA-RPCA | RCA-RPCA-EWC
---|---|---|---|---
$T^{2}$ | $S^{2}$ | $D^{2}$ | $T_{f}^{2}$ | $T_{e}^{2}$ | $T^{2}$ | SPE | $T_{f}^{2}$ | $T_{e}^{2}$ | $T^{2}$ | SPE
Case 1 | FDRs($\%$) | 59.31 | 0.19 | 0 | 67.49 | 99.85 | 59.75 | 66.81 | 75.91 | 99.85 | 86.31 | 69.04
FARs($\%$) | 0 | 0.16 | 0 | 2.52 | 2.85 | 3.19 | 2.75 | 2.61 | 2.85 | 4.72 | 1.65
DD | 841 | 847 | - | 65 | 3 | 832 | 686 | 128 | 3 | 283 | 630
Case 2 | FDRs($\%$) | 100 | 99.95 | 33.88 | 12.34 | 99.98 | 0 | 0 | 4.19 | 99.98 | 0 | 93.86
FARs($\%$) | 20.01 | 19.55 | 7.92 | 3.95 | 3.76 | 3.22 | 5.20 | 3.04 | 3.56 | 4.53 | 7.62
DD | 0 | 0 | 2 | 34 | 1 | - | - | 205 | 1 | - | 0
Case 3 | FDRs($\%$) | 86.21 | 0.05 | 0 | 93.07 | 99.81 | 84.41 | 89.42 | 95.26 | 99.46 | 97.20 | 86.72
FARs($\%$) | 0.71 | 0 | 0 | 1.58 | 3.65 | 1.42 | 1.15 | 1.43 | 3.51 | 1.64 | 1.40
DD | 0 | - | - | 43 | 2 | 622 | 353 | 19 | 6 | 113 | 544
### VI-A Description of the pulverizing system
The 1000-MW ultra-supercritical thermal power plant is increasingly popular
owing to economic benefits and environmental requirements. In this paper, we
investigate one important unit of boiler, namely, the coal pulverizing system
in Zhoushan Power Plant, Zhejiang Province, China [32]. It contains coal
feeder, coal mill, rotary separator, raw coal hopper and stone coal scuttle,
as depicted in Fig. 2. The coal pulverizing system grinds the raw coal into
pulverized coal with desired coal fineness and optimal temperature. The
operating conditions would change over the types of coal and varying unit
load. For different types of coal, the cointegration relationship may change
and the controlled variables may work at different stable points.
We choose 26 key variables and some typical variables have been depicted in
Fig. 1 to illustrate the data characteristics. Variables in Fig. 1 are
relevant to the unit load, which are also nonstationary by ADF test and prior
knowledge. Variables in Fig. 1 are little correlated with load. For instance,
the air powder mixture temperature is required to be stationary and may be
different for different types of raw coal. When the coal changes, the
temperature would vary from one stable value to another one. The bearing
temperatures are expected to remain at a stable level. The temperature of cold
air is closely related to the external environment.
We select three typical cases to illustrate the effectiveness of the proposed
method, namely, abnormality from outlet temperature (Cases 1 and 2) and rotary
separator (Case 3). According to the historical records, these two types of
faults occur frequently and affect the working safety. The sample interval is
20 seconds. The data information is listed in Table I. For each case, the
process data come from two types of coal and the original cointegration
relationship may be broken when the type of coal changes.
() Recursive CA
() RCA-RPCA
() RCA-RPCA-EWC
Figure 3: Monitoring charts of Case 1
() Recursive CA
() RCA-RPCA
() RCA-RPCA-EWC
Figure 4: Monitoring charts of Case 2
() Recursive CA
() RCA-RPCA
() RCA-RPCA-EWC
Figure 5: Monitoring charts of Case 3
### VI-B Simulation analysis
In this paper, we compare recursive CA [25] with the proposed RCA to
illustrate the virtues of real-time update. Then, the proposed RCA-RPCA is
compared with RCA-RPCA-EWC to illustrate the superiorities of EWC. Note that
RCA-RPCA and RCA-RPCA-EWC share the same RCA algorithm proposed in Section
III.
Three indicators are considered to evaluate the performance, namely, fault
detection rates (FDRs), false alarm rates (FARs) and detection delay (DD). The
calculation method can refer to [6]. DD refers to the number of samples that
the fault is detected later than the recorded fault time, which is valuable
and significant for practical industrial systems. The monitoring consequences
of three case are described in Figs. 3-5, respectively. Note that the pink
vertical line represents the practical fault time instant.
The monitoring charts of Case 1 are presented in Fig. 3. Recursive CA [25]
fails to detect the fault accurately and the FDR is $59.31\%$. Besides,
$S^{2}$ and $D^{2}$ can not distinguish novelty from normal dynamic changes in
Fig. 3. For the proposed RCA, $T_{e}^{2}$ can detect the fault precisely and
timely, and the detection delay is about 1 minute. In the time period, where
$T^{2}_{e}$ and $T^{2}_{f}$ change significantly, the type of coal changes and
the current cointegration relationship may be broken. Thus, the CA model needs
to be retrained from scratch based on the newly collected data. During this
time period, the process is monitored by the current CA model, and new data
are collected to build the initial CA model that is appropriate for the new
material. Thus, $T^{2}_{e}$ and $T^{2}_{f}$ recover to be stable quickly.
Compared with RPCA, RPCA-EWC provides better performance in Fig. 3 and the FDR
of $T^{2}$ is $86.31\%$. However, the FDR of RPCA is $66.81\%$, $19.5\%$ lower
than RPCA-EWC. According to $T^{2}_{e}$ and $T^{2}_{f}$, it is observed that
the type of coal varies and a new model is built before the fault occurs,
which indicates that RPCA is not trained enough and can not track the system
change actually.
For Case 2, the monitoring results are depicted in Fig. 4. For the proposed
RCA method, $T_{e}^{2}$ can detect the fault accurately and the FDR is
$99.98\%$. In Figs. 4 and 4, $T_{e}^{2}$ and $T_{f}^{2}$ change sharply twice.
According to the coal records and original data analysis, the first sudden
change of two statistics originates from the switch of coal type, while the
second abrupt change is attributed to the critical parameters adjusted
artificially. Compared with RPCA, SPE of RPCA-EWC enables to detect the fault
precisely and the FDR is $93.86\%$. The short-term dynamic of two types of
coal has a certain degree of similarity, and the significant information of
previous coal is preserved and beneficial to monitor other coal. The FARs of
SPE are relatively high because RPCA is not able to track the rapid system
change at the initial stage. The system is judged as normal because SPE
returns to normal quickly. Regardless of the false alarms caused by this
situation, the FARs of SPE are $5.20\%$ and $7.62\%$ in Figs. 4-4,
respectively. However, recursive CA [25] misidentifies the normal parameter
variations as anomalies in Fig. 4 and the FAR is up to $20\%$. It is
insensitive to faults that are orthogonal to cointegration space and only
dynamic information is not enough to monitor the process effectively.
For Case 3, the monitoring consequences are exhibited in Fig. 5. The recursive
CA [25] detects the fault inexactly and the FDR of $T^{2}$ is $86.21\%$. The
FDRs of $S^{2}$ and $D^{2}$ are 0, and thus it is meaningless to mention delay
detection. The proposed RCA can detect the fault accurately and the FDRs are
more than $99\%$ in Figs. 5-5. For RPCA-EWC in Fig. 5, the FDR of $T^{2}$ is
$97.20\%$, which indicates that the significant information of the previous
coal is preserved by EWC and beneficial to deliver excellent monitoring
performance. However, the FDR of RPCA is less than $90\%$ in Fig. 5.
The evaluation indexes of three cases are summarized in Table II. Compared
with recursive CA [25], the proposed RCA is more sensitive to normal changes
from human intervention and raw materials changing. This phenomenon occurs
owing to several factors: a) The variables are selected and divided based on
prior knowledge and ADF test, which is more universal and accurate than just
ADF; b) Critical stationary variables, which are sensitive to raw materials
changing, are utilized to establish the $T^{2}_{f}$ statistic; c) In [25], the
model is updated based on a block of data and the monitoring performance is
effected by the block length, while the proposed RCA model is updated in real
time and more compatible with the current operating system. In addition,
compared with RCA-RPCA, RCA-RPCA-EWC preserves significant information of
previous influential parameters and avoids dramatic performance degradation
when similar operating modes revisit.
## VII Conclusion
In this paper, RCA-RPCA-EWC was developed to monitor the general nonstationary
processes, where the proposed RCA is updated in real time and able to
distinguish the real faults from normal system deviations. To avoid potential
ill-conditioning issue of matrix inversion, several calculation techniques are
adopted and the RCA issue is settled with low computational burden. As RCA is
insensitive to faults that are orthogonal to cointegration space, the
remaining information of RCA together with other short-term dynamic
information is monitored by RPCA to establish a comprehensive monitoring
framework. When the system enters a new operating mode, EWC is employed to
strengthen the significant information of previous operating modes and avoid
the abrupt performance degradation for future similar operating modes.
Besides, the test statistics are constructed based on RCA and the prior
knowledge, which are sensitive to mode identification. Compared with recursive
CA [25] and RCA-RPCA, the effectiveness and superiorities of the proposed
method are illustrated by a practical industrial pulverizing system.
In future, we will investigate the quality-related nonstationary process
monitoring. Besides, graceful forgetting will be considered as forgetting
older modes is essential to make space for learning newer modes.
### -A RPCA for process monitoring
In this paper, RPCA is implemented based on on rank-1 modification with first-
order perturbation (FOP). Detailed information can refer to [37].
At $k+1$ instant, the sample $\boldsymbol{x}_{k+1}^{0}\in\mathbb{R}^{m_{2}}$
is collected. Then, the mean $\boldsymbol{\mu}$ and standard deviation
$\sigma_{1},\cdots,\sigma_{m_{2}}$ are updated as:
$\boldsymbol{\mu}_{k+1}=\alpha_{k+1}\boldsymbol{\mu}_{k}+(1-\alpha_{k+1})\boldsymbol{x}_{k+1}^{0}$
(29)
$\sigma_{i,k+1}^{2}=\alpha_{k+1}\sigma_{i,k+1}^{2}+(1-\alpha_{k+1})(x_{i,k+1}^{0}-\mu_{i,k+1})^{2}$
(30)
where $i=1,\cdots,m_{2}$, $\alpha_{k+1}=\frac{k}{{k+1}}$ is the forgetting
factor. The sample is normalized as
${\boldsymbol{x}}_{k+1}=(\boldsymbol{x}_{k+1}^{0}-\boldsymbol{\mu}_{k+1})\boldsymbol{\Sigma}_{k+1}^{-1}$
(31)
where
$\boldsymbol{\Sigma}_{k+1}=diag(\sigma_{1,k+1},\cdots,\sigma_{m_{2},k+1})$.
Based on rank-1 modification and FOP, the eigenvectors and eigenvalues are
updated as:
$\boldsymbol{P}_{k+1}=\boldsymbol{P}_{k}\left(\boldsymbol{I}+\boldsymbol{Q}_{V}\right)$
(32)
$\boldsymbol{\Lambda}_{k+1}=\alpha_{k+1}\boldsymbol{\Lambda}_{k}+\left(1-\alpha_{k+1}\right)\boldsymbol{Q}_{\Lambda}$
(33)
Define the rank-1 matrix
$\boldsymbol{\kappa}_{k+1}=\boldsymbol{P}_{k}^{T}\boldsymbol{x}_{k+1}$, the
diagonal matrix $\boldsymbol{Q}_{\varLambda}$ and $\boldsymbol{Q}_{V}$ are
calculated by:
$Q_{\varLambda}\left(i,i\right)=\kappa_{i}$ (34)
$\begin{cases}Q_{V}\left(i,j\right)=\frac{\kappa_{i}\kappa_{j}}{\tau_{j}+\kappa_{j}^{2}-\tau_{i}+\kappa_{i}^{2}},i\neq
j\\\ Q_{V}\left(i,i\right)=0\\\ \end{cases}$ (35)
where $\kappa_{i}$ is the $i$th element of $\boldsymbol{\kappa}_{k+1}$,
$\tau_{i}$ and $\tau_{j}$ are the $i$th and $j$th corresponding elements of
$k\boldsymbol{\varLambda}_{k}$.
### -B Recursive computation of $\varDelta\boldsymbol{A}$ and
$\varDelta\boldsymbol{B}$
We illustrate the computation of $\varDelta\boldsymbol{A}_{k+1}$ and
$\varDelta\boldsymbol{B}_{k+1}$. Take the component
$\boldsymbol{D}_{k+1}^{T}\boldsymbol{E}_{1,k}$ of
$\varDelta\boldsymbol{A}_{1,k+1}$ as an example, we show that the computation
of $\varDelta\boldsymbol{A}_{1,k+1}$ is $O(m_{1}^{2})$, which is independent
of the number of existing samples.
$\boldsymbol{D}_{k+1}^{T}\boldsymbol{E}_{1,k}=\boldsymbol{d}_{k+1}^{T}\varDelta\boldsymbol{x}_{k+1}^{p}\boldsymbol{J}_{k}^{T}\boldsymbol{E}_{1,k}$
(36)
If we compute (36) directly, the computation is $O(km_{1}^{2})$ and increases
linearly with the emerging samples. We need to get the recursive form of
$\boldsymbol{J}_{k}^{T}\boldsymbol{E}_{1,k}$.
$\displaystyle\boldsymbol{J}_{k}^{T}\boldsymbol{E}_{1,k}$ (37)
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\boldsymbol{J}_{k-1}\boldsymbol{\tilde{J}}_{k}\\\
\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}\boldsymbol{\tilde{J}}_{k}\\\
\end{array}\right]^{T}\left[\begin{array}[]{c}\boldsymbol{E}_{1,k-1}-\boldsymbol{J}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\boldsymbol{h}_{k}\\\
\boldsymbol{h}_{k}\\\ \end{array}\right]$ $\displaystyle=$
$\displaystyle\boldsymbol{\tilde{J}}_{k}^{T}\boldsymbol{J}_{k-1}^{T}\left(\boldsymbol{E}_{1,k-1}-\boldsymbol{J}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\boldsymbol{h}_{k}\right)+\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}\boldsymbol{\tilde{J}}_{k}\boldsymbol{h}_{k}$
$\displaystyle=$
$\displaystyle\boldsymbol{\tilde{J}}_{k}^{T}\boldsymbol{J}_{k-1}^{T}\boldsymbol{E}_{1,k-1}-\boldsymbol{\tilde{J}}_{k}^{T}\boldsymbol{J}_{k-1}^{T}\boldsymbol{J}_{k-1}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\boldsymbol{h}_{k}$
The recursion of $\boldsymbol{J}_{k}^{T}\boldsymbol{J}_{k}$ is:
$\boldsymbol{J}_{k}^{T}\boldsymbol{J}_{k}=\boldsymbol{\tilde{J}}_{k}^{T}\boldsymbol{J}_{k-1}^{T}\boldsymbol{J}_{k-1}\boldsymbol{\tilde{J}}_{k}+\boldsymbol{\tilde{J}}_{k}^{T}\boldsymbol{R}_{k-1}^{T}\left(\varDelta\boldsymbol{x}_{k}^{p}\right)^{T}\varDelta\boldsymbol{x}_{k}^{p}\boldsymbol{R}_{k-1}\boldsymbol{\tilde{J}}_{k}$
(38)
Based on (36-38), $\boldsymbol{D}_{k+1}^{T}\boldsymbol{E}_{1,k}$ can be
calculated recursively and the computational complexity is $O(m_{1}^{2})$ by
reasonable arrangement of matrix calculation order. It is obviously true
because each component contains at least one vector. Similarly, other
components of $\varDelta\boldsymbol{A}_{k+1}$ and
$\varDelta\boldsymbol{B}_{k+1}$ need $O(m_{1}^{2})$. In conclusion, the
computational complexity of $\varDelta\boldsymbol{A}_{k+1}$ and
$\varDelta\boldsymbol{B}_{k+1}$ is $O(m_{1}^{2})$.
### -C The recursive calculation of $\boldsymbol{K}$
$\boldsymbol{K}_{k+1}$ is calculated by:
$\boldsymbol{K}_{k+1}=\boldsymbol{B}_{k+1}^{-\frac{1}{2}}\,\,=\,\,\left({\alpha}_{k+1}\boldsymbol{B}_{k}+\left(1-{\alpha}_{k+1}\right)\boldsymbol{\varDelta
B}_{k+1}\right)^{-\frac{1}{2}}$ (39)
Considering that $\boldsymbol{B}$ is a block diagonal matrix with
$\boldsymbol{B}=\left[{\begin{array}[]{*{20}{c}}{{\boldsymbol{B}_{1}}}&\boldsymbol{0}\\\
\boldsymbol{0}&{{\boldsymbol{B}_{2}}}\end{array}}\right]$, we mainly introduce
the recursion of $\boldsymbol{B}_{1}$, and $\boldsymbol{B}_{2}$ can be updated
similarly.
$\displaystyle\boldsymbol{K}_{1,k+1}=\boldsymbol{B}_{1,{k}+1}^{-\frac{1}{2}}\,\,$
(40) $\displaystyle=$
$\displaystyle\,\,\left(\alpha_{{k}+1}\boldsymbol{B}_{1,{k}}+\left(1-\alpha_{{k}+1}\right){\varDelta\boldsymbol{B}}_{1,{k}+1}\right)^{-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left(\left(\sqrt{\alpha_{k+1}}\boldsymbol{B}_{1,k}^{\frac{1}{2}}\right)\left(\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{B}_{1,k}^{-\frac{1}{2}}{\varDelta\boldsymbol{B}}_{1,{k}+1}\right.\right.$
$\displaystyle\left.\left.\left(\boldsymbol{B}_{1,k}^{-\frac{1}{2}}\right)^{T}\right)\left(\sqrt{\alpha_{k+1}}\boldsymbol{B}_{1,k}^{\frac{1}{2}}\right)^{T}\right)^{-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha_{k+1}}}\left(\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{B}_{1,k}^{-\frac{1}{2}}{\varDelta\boldsymbol{B}}_{1,{k}+1}\boldsymbol{B}_{1,k}^{-\frac{1}{2}}\right)^{-\frac{1}{2}}\boldsymbol{B}_{1,k}^{-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha_{k+1}}}\left(\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{K}_{1,k}{\varDelta\boldsymbol{B}}_{1,{k}+1}\boldsymbol{K}^{T}_{1,k}\right)^{-\frac{1}{2}}\boldsymbol{K}_{1,k}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha_{k+1}}}\boldsymbol{\tilde{K}}_{1,k+1}^{-\frac{1}{2}}\boldsymbol{K}_{1,k}$
where
${\boldsymbol{\tilde{K}}_{1,k+1}}=\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{K}_{1,k}{\varDelta\boldsymbol{B}}_{1,{k}+1}\boldsymbol{K}^{T}_{1,k}$.
The key is to calculate $\boldsymbol{\tilde{K}}_{1,k+1}^{-\frac{1}{2}}$.
${\varDelta\boldsymbol{B}}_{1,{k}+1}$ is symmetric and the rank is no more
than 2. Thus, it can be reformulated into
$\displaystyle\varDelta\boldsymbol{B}_{1,k+1}$ (41) $\displaystyle=$
$\displaystyle\left[\begin{matrix}\boldsymbol{q}_{1,k+1}&\boldsymbol{q}_{2,k+1}\\\
\end{matrix}\right]\left[\begin{matrix}\beta_{1,k+1}&\\\ &\beta_{2,k+1}\\\
\end{matrix}\right]\left[\begin{matrix}\boldsymbol{q}_{1,k+1}&\boldsymbol{q}_{2,k+1}\\\
\end{matrix}\right]^{T}$ $\displaystyle=$
$\displaystyle\boldsymbol{Q}_{1,k+1}\boldsymbol{\varXi}_{1,k+1}\boldsymbol{Q}_{1,k+1}^{T}$
where $\beta_{1,k+1}$ and $\beta_{2,k+1}$ are non-zero eigenvalues if
$rank({\varDelta\boldsymbol{B}}_{1,{k}+1})=2$. $\boldsymbol{q}_{1,k+1}$ and
$\boldsymbol{q}_{2,k+1}$ are the corresponding eigenvectors. If the rank is
$1$, then $\beta_{1,k+1}$ or $\beta_{2,k+1}$ is $0$. Thus,
$\displaystyle{\boldsymbol{\tilde{K}}_{1,k+1}}$ (42) $\displaystyle=$
$\displaystyle\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{K}_{1,k}\boldsymbol{Q}_{1,k+1}\boldsymbol{\varXi}_{1,k+1}\boldsymbol{Q}_{1,k+1}^{T}\boldsymbol{K}^{T}_{1,k}$
$\displaystyle=$
$\displaystyle\boldsymbol{I}+\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\left(\boldsymbol{K}_{1,k}\boldsymbol{Q}_{1,k+1}\right)\boldsymbol{\varXi}_{1,k+1}\left(\boldsymbol{K}_{1,k}\boldsymbol{Q}_{1,k+1}\right)^{T}$
$\displaystyle=$
$\displaystyle\boldsymbol{I}+{\tilde{\boldsymbol{Q}}}_{1,k+1}{\tilde{\boldsymbol{\varXi}}}_{1,k+1}{\tilde{\boldsymbol{Q}}}_{1,k+1}^{T}$
where
${\tilde{\boldsymbol{Q}}}_{1,k+1}=\boldsymbol{K}_{1,k}\boldsymbol{Q}_{1,k+1}\in\mathbb{R}^{m\times
2}$,
${\tilde{\boldsymbol{\varXi}}}_{1,k+1}=\frac{1-\alpha_{k+1}}{\alpha_{k+1}}\boldsymbol{\varXi}_{1,k+1}$
and $rank({\tilde{\boldsymbol{\varXi}}}_{1,k+1})\leqslant 2$. As
$\boldsymbol{B}_{1,k+1}$ is obviously positive definite, then
$\boldsymbol{I}+{\tilde{\boldsymbol{Q}}}_{1,k+1}{\tilde{\boldsymbol{\varXi}}}_{1,k+1}{\tilde{\boldsymbol{Q}}}_{1,k+1}^{T}$
is also positive definite. For convenience, let
$\tilde{\boldsymbol{Q}}_{1,k+1}=\left[\begin{matrix}\tilde{\boldsymbol{q}}_{1,k+1}&\tilde{\boldsymbol{q}}_{2,k+1}\\\
\end{matrix}\right]$,
${\tilde{\boldsymbol{\varXi}}}_{1,k+1}=\left[\begin{matrix}\tilde{\beta}_{1,k+1}&\\\
&\tilde{\beta}_{2,k+1}\\\ \end{matrix}\right]$. We select the calculation
manner of $\boldsymbol{\tilde{K}}_{1,k+1}^{-\frac{1}{2}}$ based on the rank of
${\tilde{\boldsymbol{\varXi}}}_{1,k+1}$.
1) $rank({\tilde{\boldsymbol{\varXi}}}_{1,k+1})=1$.
Let $\beta_{1,k+1}\neq 0$ and $\beta_{2,k+1}=0$, here
${\tilde{\boldsymbol{\varXi}}}_{1,k+1}=\tilde{\beta}_{1,k+1}$,
${\tilde{\boldsymbol{Q}}}_{1,k+1}=\tilde{\boldsymbol{q}}_{1,k+1}$. Then,
$\displaystyle\left(\boldsymbol{I}+{\tilde{\boldsymbol{Q}}}_{1,k+1}{\tilde{\boldsymbol{\varXi}}}_{1,k+1}{\tilde{\boldsymbol{Q}}}_{1,k+1}^{T}\right)^{-\frac{1}{2}}$
(43) $\displaystyle=$
$\displaystyle\left(\boldsymbol{I}+\tilde{\beta}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}\right)^{-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\boldsymbol{I}+\frac{\tilde{\boldsymbol{q}}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}}{\tilde{\boldsymbol{q}}_{1,k+1}^{T}\tilde{\boldsymbol{q}}_{1,k+1}}\left(\frac{1}{\sqrt{1+\tilde{\beta}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}\tilde{\boldsymbol{q}}_{1,k+1}}}-1\right)$
$\displaystyle=$
$\displaystyle\boldsymbol{I}+\gamma_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}$
where
$\gamma_{1,k+1}=\frac{1}{\tilde{\boldsymbol{q}}_{1,k+1}^{T}\tilde{\boldsymbol{q}}_{1,k+1}}\left(\frac{1}{\sqrt{1+\tilde{\beta}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}\tilde{\boldsymbol{q}}_{1,k+1}}}-1\right)$.
Thus, the recursion of $\boldsymbol{K}_{1}$ is
$\boldsymbol{K}_{1,{k}+1}=\frac{1}{\sqrt{\alpha_{k+1}}}\left(\boldsymbol{I}+\gamma_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}\right)\boldsymbol{K}_{1,k}$
(44)
2) $rank({\tilde{\boldsymbol{\varXi}}}_{1,k+1})=2$.
The formula (42) can be further reformulated into
$\displaystyle\boldsymbol{I}+{\tilde{\boldsymbol{Q}}}_{1,k+1}{\tilde{\boldsymbol{\varXi}}}_{k+1}{\tilde{\boldsymbol{Q}}}_{1,k+1}^{T}$
(45) $\displaystyle=$
$\displaystyle\boldsymbol{I}+\gamma_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}\tilde{\boldsymbol{q}}_{1,k+1}^{T}+\gamma_{2,k+1}\tilde{\boldsymbol{q}}_{2,k+1}\tilde{\boldsymbol{q}}_{2,k+1}^{T}$
$\displaystyle=$
$\displaystyle\check{\boldsymbol{Q}}_{1,k+1}\boldsymbol{\check{\varLambda}}_{1,k+1}\check{\boldsymbol{Q}}_{1,k+1}^{T}$
where $\check{\boldsymbol{Q}}_{1,k+1}$ is the eigen matrix with
$\check{\boldsymbol{Q}}_{1,k+1}^{T}\check{\boldsymbol{Q}}_{1,k+1}=\boldsymbol{I}$,
$\boldsymbol{\check{\varLambda}}_{1,k+1}$ contains the eigenvalues. (42) is
realized by two successive rank-1 modification with FOP [37]. Thus, (40) is
further calculated:
$\boldsymbol{K}_{1,k+1}=\frac{1}{\sqrt{\alpha_{k+1}}}\boldsymbol{\check{\varLambda}}^{-\frac{1}{2}}_{1,k+1}{\check{\boldsymbol{Q}}}_{1,k+1}^{T}\boldsymbol{K}_{1,k}$
(46)
$\boldsymbol{K}_{2,k+1}$ can be calculated in the similar way. Thus,
$\boldsymbol{K}_{k+1}=\left[\begin{matrix}\boldsymbol{K}_{1,k+1}&\\\
&\boldsymbol{K}_{2,k+1}\\\ \end{matrix}\right]$.
### -D Solution of RPCA-EWC
The formula (28) can be reformulated as
$\displaystyle\mathcal{J}(\boldsymbol{P})=$ $\displaystyle
tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P})-tr(\boldsymbol{P}^{T}\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P})-2tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P}^{*}_{0})$
(47)
$\displaystyle+\underbrace{\\{tr(\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2})+tr({\boldsymbol{P}^{*}_{0}}^{T}{\boldsymbol{\Omega}}\boldsymbol{P}^{*}_{0})\\}}_{constant}$
Let
$G(\boldsymbol{P})=tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P})-2tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P}^{*}_{0})$,
$H(\boldsymbol{P})=tr(\boldsymbol{P}^{T}\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P})$.
Thus,
$\mathcal{J}(\boldsymbol{P})=G(\boldsymbol{P})-H(\boldsymbol{P})+constant$.
The minimization of (28) is equivalent to
$\displaystyle\mathop{min}\limits_{\boldsymbol{P}}\quad
G(\boldsymbol{P})-H(\boldsymbol{P})$ (48) $\displaystyle
s.t.\qquad\boldsymbol{P}^{T}\boldsymbol{P}=\boldsymbol{I}\in R^{l\times l}$
Since $G(\boldsymbol{P})$ and $H(\boldsymbol{P})$ are convex, the objective
function (48) is actually DC programming problem [34, 35]. DC programming
includes linearizing the convex function and solving the convex function as
follows [32].
Assume that $\boldsymbol{P}_{i}$ is the solution at $i$th iteration, we
approximate the second part $H(\boldsymbol{P})$ by linearizing
$H_{l}(\boldsymbol{P})=H(\boldsymbol{P}_{i})+\langle\boldsymbol{P}-\boldsymbol{P}_{i},\boldsymbol{U}_{i}\rangle$
(49)
Since the subgradient $\boldsymbol{U}\in\partial
H(\boldsymbol{P})=2\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}$,
let
$\boldsymbol{U}_{i}=2\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}_{i}$.
Then, (48) is approximated by
$\boldsymbol{P}_{i+1}\doteq\underset{\boldsymbol{P}^{T}\boldsymbol{P}=\boldsymbol{I}}{\arg\min}\quad
G(\boldsymbol{P})-<\boldsymbol{P},\boldsymbol{U}_{i}>$ (50)
Since ${\boldsymbol{\Omega}}$ is semidefinite, let
${\boldsymbol{\Omega}}=\boldsymbol{L}^{T}\boldsymbol{L}$ and $\boldsymbol{L}$
is the triangle matrix [32]. Thus, we can get
$\displaystyle G(\boldsymbol{P})-<\boldsymbol{P},\boldsymbol{U}_{i}>$ (51)
$\displaystyle=$ $\displaystyle
tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P})-2tr(\boldsymbol{P}^{T}{\boldsymbol{\Omega}}\boldsymbol{P}_{0}^{*})-2tr(\boldsymbol{P}^{T}\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}_{i})$
$\displaystyle=$
$\displaystyle{\langle\boldsymbol{L}\boldsymbol{P},\boldsymbol{L}\boldsymbol{P}\rangle}-2\langle\boldsymbol{L}\boldsymbol{P},\boldsymbol{L}\boldsymbol{P}_{0}^{*}+(\boldsymbol{L}^{T})^{-1}\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}_{i}\rangle$
$\displaystyle=$
$\displaystyle\lVert\boldsymbol{Z}_{i}-\boldsymbol{L}\boldsymbol{P}\rVert_{F}^{2}-\lVert\boldsymbol{Z}_{i}\rVert_{F}^{2}$
where
$\boldsymbol{Z}_{i}=\boldsymbol{L}\boldsymbol{P}_{0}^{*}+(\boldsymbol{L}^{T})^{-1}\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}_{i}$
is constant at $i+1$th iteration [32].
Then, (51) is equivalent to [38, 32]
$\boldsymbol{P}_{i+1}=\underset{\boldsymbol{P}^{T}\boldsymbol{P}=\boldsymbol{I}}{\arg\min}\quad\lVert\boldsymbol{P}-\boldsymbol{L}^{T}\boldsymbol{Z}_{i}\rVert_{F}^{2}$
(52)
Let
$\boldsymbol{Y}_{i}=\boldsymbol{L}^{T}\boldsymbol{Z}_{i}={\boldsymbol{\Omega}}\boldsymbol{P}_{0}^{*}+\boldsymbol{X}_{2}^{T}\boldsymbol{X}_{2}\boldsymbol{P}_{i}$.
According to the lemma in [38], we can obtain that
$\boldsymbol{P}_{i+1}=\boldsymbol{W}_{i}\boldsymbol{I}_{m,l}\boldsymbol{V}_{i}^{T}$,
where $\boldsymbol{W}_{i}\in\mathbb{R}^{m\times m}$ and
$\boldsymbol{V}_{i}\in\mathbb{R}^{l\times l}$ are left and right singular
values of the singular vector decomposition of $\boldsymbol{Y}_{i}$ [32]. The
procedure is summarized in Algorithm 2.
## References
* [1] I. B. Khediri, M. Limam, and C. Weihs, “Variable window adaptive kernel principal component analysis for nonlinear nonstationary process monitoring,” Computers & Industrial Engineering, vol. 61, no. 3, pp. 437–446, 2011.
* [2] S. Baek and D. Y. Kim, “Empirical sensitivity analysis of discretization parameters for fault pattern extraction from multivariate time series data,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 47, no. 5, pp. 1198–1209, 2017.
* [3] S. Yin, H. Gao, J. Qiu, and O. Kaynak, “Fault detection for nonlinear process with deterministic disturbances: A just-in-time learning based data driven method,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 47, no. 11, pp. 3649–3657, 2017.
* [4] A. Teixeira, I. Shames, H. Sandberg, and K. H. Johansson, “Distributed fault detection and isolation resilient to network model uncertainties,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 44, no. 11, pp. 2024–2037, 2014.
* [5] J. Zhang, M. Chen, H. Chen, X. Hong, and D. Zhou, “Process monitoring based on orthogonal locality preserving projection with maximum likelihood estimation,” Industrial & Engineering Chemistry Research, vol. 58, no. 14, pp. 5579–5587, 2019.
* [6] J. Zhang, H. Chen, S. Chen, and X. Hong, “An improved mixture of probabilistic PCA for nonlinear data-driven process monitoring,” IEEE Transactions on Cybernetics, vol. 49, no. 1, pp. 198–210, 2019.
* [7] T. Rato, M. Reis, E. Schmitt, M. Hubert, and B. D. Ketelaere, “A systematic comparison of PCA-based statistical process monitoring methods for high-dimensional, time-dependent processes,” AIChE Journal, vol. 62, no. 5, pp. 1478–1493, 2016.
* [8] J. Yu and X. Yan, “Whole process monitoring based on unstable neuron output information in hidden layers of deep belief network,” IEEE Transactions on Cybernetics, vol. 50, no. 9, pp. 3998–4007, 2020.
* [9] X. Yin and Z. Li, “Decentralized fault prognosis of discrete-event systems using state-estimate-based protocols,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 49, no. 4, pp. 1302–1313, 2019.
* [10] M. Wang, D. Zhou, M. Chen, and Y. Wang, “Anomaly detection in the fan system of a thermal power plant monitored by continuous and two-valued variables,” Control Engineering Practice, vol. 102, p. 104522, 2020.
* [11] C. Shang, F. Yang, B. Huang, and D. Huang, “Recursive slow feature analysis for adaptive monitoring of industrial processes,” IEEE Transactions on Industrial Electronics, vol. 65, no. 11, pp. 8895–8905, 2018\.
* [12] S. Yin, X. Xie, J. Lam, K. C. Cheung, and H. Gao, “An improved incremental learning approach for KPI prognosis of dynamic fuel cell system,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 46, no. 12, pp. 3135–3144, 2016.
* [13] R. Tan, J. R. Ottewill, and N. F. Thornhill, “Nonstationary discrete convolution kernel for multimodal process monitoring.,” IEEE Transactions on Neural Networks, vol. 31, no. 9, pp. 3670–3681, 2020.
* [14] K. E. S. Pilario and Y. Cao, “Canonical variate dissimilarity analysis for process incipient fault detection,” IEEE Transactions on Industrial Informatics, vol. 14, no. 12, pp. 5308–5315, 2018.
* [15] Y. Dong, Y. Liu, and S. J. Qin, “Efficient dynamic latent variable analysis for high-dimensional time series data,” IEEE Transactions on Industrial Informatics, vol. 16, no. 6, pp. 4068–4076, 2020.
* [16] L. Zhou, J. Zheng, Z. Ge, Z. Song, and S. Shan, “Multimode process monitoring based on switching autoregressive dynamic latent variable model,” IEEE Transactions on Industrial Electronics, vol. 65, no. 10, pp. 8184–8194, 2018\.
* [17] C. Shang, F. Yang, X. Gao, X. Huang, J. A. Suykens, and D. Huang, “Concurrent monitoring of operating condition deviations and process dynamics anomalies with slow feature analysis,” AIChE Journal, vol. 61, no. 11, pp. 3666–3682, 2015.
* [18] S. Zhang and C. Zhao, “Slow-feature-analysis-based batch process monitoring with comprehensive interpretation of operation condition deviation and dynamic anomaly,” IEEE Transactions on Industrial Electronics, vol. 66, no. 5, pp. 3773–3783, 2019.
* [19] R. F. E. W. J. Granger, “Co-integration and error correction: Representation, estimation, and testing,” Econometrica, vol. 55, no. 2, pp. 251–276, 1987\.
* [20] S. Johansen, “Statistical analysis of cointegration vectors,” Journal of Economic Dynamics and Control, vol. 12, no. 23, pp. 231–254, 1988.
* [21] Q. Chen, U. Kruger, and A. Y. T. Leung, “Cointegration testing method for monitoring nonstationary processes,” Industrial & Engineering Chemistry Research, vol. 48, no. 7, pp. 3533–3543, 2009.
* [22] C. Zhao and H. Sun, “Dynamic distributed monitoring strategy for large-scale nonstationary processes subject to frequently varying conditions under closed-loop control,” IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4749–4758, 2019.
* [23] C. Zhao and B. Huang, “A full-condition monitoring method for nonstationary dynamic chemical processes with cointegration and slow feature analysis,” AIChE Journal, vol. 64, no. 5, pp. 1662–1681, 2018.
* [24] H. Hansen and S. Johansen, “Recursive estimation in cointegrated VAR-models,” Discussion Papers, 1992.
* [25] W. Yu, C. Zhao, and B. Huang, “Recursive cointegration analytics for adaptive monitoring of nonstationary industrial processes with both static and dynamic variations,” Journal of Process Control, vol. 92, pp. 319–332, 2020.
* [26] Y. Lin, U. Kruger, F. Gu, A. Ball, and Q. Chen, “Monitoring nonstationary processes using stationary subspace analysis and fractional integration order estimation,” Industrial & Engineering Chemistry Research, vol. 58, no. 16, pp. 6486–6504, 2019.
* [27] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, and A. Grabska-Barwinska, “Overcoming catastrophic forgetting in neural networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 114, no. 13, pp. 3521–3526, 2017.
* [28] H. Hansen and S. Johansen, “Some tests for parameter constancy in cointegrated VAR-models,” Econometrics Journal, vol. 2, no. 2, pp. 306–333, 1999.
* [29] G. M. van de Ven, H. T. Siegelmann, and A. S. Tolias, “Brain-inspired replay for continual learning with artificial neural networks,” Nature Communications, vol. 11, no. 1, p. 4069, 2020.
* [30] G. Zeng, Y. Chen, B. Cui, and S. Yu, “Continual learning of context-dependent processing in neural networks,” Nature Machine Intelligence, vol. 1, no. 8, pp. 364–372, 2019.
* [31] N. Y. Masse, G. D. Grant, and D. J. Freedman, “Alleviating catastrophic forgetting using context-dependent gating and synaptic stabilization,” Proceedings of the National Academy of Sciences of the United States of America, vol. 115, no. 44, pp. 100467–10475, 2018.
* [32] J. Zhang, D. Zhou, and M. Chen, “Monitoring multimode processes: a modified PCA algorithm with continual learning ability,” arXiv:2012.07044, 2020\.
* [33] F. Huszár, “On quadratic penalties in elastic weight consolidation,” Proceedings of the National Academy of Sciences, 2017.
* [34] J. C. O. Souza, P. R. Oliveira, and A. Soubeyran, “Global convergence of a proximal linearized algorithm for difference of convex functions,” Optimization Letters, vol. 10, no. 7, pp. 1–11, 2015.
* [35] T. V. Voorhis and F. A. Al-Khayyal, “Difference of convex solution of quadratically constrained optimization problems,” European Journal of Operational Research, vol. 148, no. 2, pp. 349–362, 2003.
* [36] J. Schwarz, J. Luketina, W. M. Czarnecki, A. Grabska-Barwinska, Y. W. Teh, R. Pascanu, and R. Hadsell, “Progress & compress: A scalable framework for continual learning,” in International Conference on Machine Learning, pp. 4528–4537, 2018.
* [37] L. M. Elshenawy, S. Yin, A. S. Naik, and S. X. Ding, “Efficient recursive principal component analysis algorithms for process monitoring,” Industrial & Engineering Chemistry Research, vol. 49, no. 1, pp. 252–259, 2010.
* [38] J. Huang, F. Nie, H. Huang, and C. Ding, “Robust manifold nonnegative matrix factorization,” ACM Transactions on Knowledge Discovery from Data (TKDD), vol. 8, no. 3, pp. 1–21, 2014.
|
Hydroxide salts in the clouds of Venus: their effect on the sulfur cycle and cloud droplet pH
Paul B. Rimmer
0000-0002-7180-081X]Paul B. Rimmer
Department of Earth Sciences, University of Cambridge, Downing St, Cambridge CB2 3EQ, United Kingdom
Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge CB3 0HE, United Kingdom
MRC Laboratory of Molecular Biology, Francis Crick Ave, Cambridge CB2 0QH, United Kingdom
Sean Jordan
Institute of Astronoomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, United Kingdom
Tereza Constantinou
Institute of Astronoomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, United Kingdom
Peter Woitke
SUPA, School of Physics & Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK
Centre for Exoplanet Science, University of St Andrews, St Andrews, UK
0000-0002-8713-1446]Oliver Shorttle
Department of Earth Sciences, University of Cambridge, Downing St, Cambridge CB2 3EQ, United Kingdom
Institute of Astronomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, United Kingdom
Alessia Paschodimas
Earth and Environmental Sciences, University of St Andrews, Irvine Building, North Street, St Andrews, KY16 9AL, United Kingdom
Centre for Exoplanet Science, University of St Andrews, St Andrews, UK
Richard Hobbs
Institute of Astronomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, United Kingdom
The depletion of SO_2 and H_2O in and above the clouds of Venus (45 – 65 km) cannot be explained by known gas-phase chemistry and the observed composition of the atmosphere. We apply a full-atmosphere model of Venus to investigate three potential explanations for the SO_2 and H_2O depletion: (1) varying the below-cloud water vapor (H_2O), (2) varying the below-cloud sulfur dioxide (SO_2), and (3) the incorporation of chemical reactions inside the sulfuric acid cloud droplets. We find that increasing the below-cloud H_2O to explain the SO_2 depletion results in a cloud top that is 20 km too high, above-cloud O_2 three orders of magnitude greater than observational upper limits and no SO above 80 km. The SO_2 depletion can be explained by decreasing the below-cloud SO_2 to $20\,{\rm ppm}$. The depletion of SO_2 in the clouds can also be explained by the SO_2 dissolving into the clouds, if the droplets contain hydroxide salts. These salts buffer the cloud pH. The amount of salts sufficient to explain the SO_2 depletion entail a droplet pH of $\sim 1$ at 50 km. Since sulfuric acid is constantly condensing out into the cloud droplets, there must be a continuous and pervasive flux of salts of $\approx 10^{-13} \, {\rm mol \, cm^{-2} \, s^{-1}}$ driving the cloud droplet chemistry. An atmospheric probe can test both of these explanations by measuring the pH of the cloud droplets and the concentrations of gas-phase SO_2 below the clouds.
§ INTRODUCTION
Both sulfur dioxide (SO_2) and water vapor (H_2O) are known to be depleted in the cloud layer of Venus <cit.>, and to vary in abundance above the cloud top by an order of magnitude or more both spatially [Jessup et al., 2015, Encrenaz et al., 2019, Marcq et al., 2020], and temporally in years-long cycles [Marcq et al., 2013, Vandaele et al., 2017]. Both of these species participate in Venus's atmospheric sulfur cycle [Yung & Demore, 1982, Krasnopolsky, 1982, Krasnopolsky, 2007, Krasnopolsky, 2010, Krasnopolsky, 2012, Mills et al., 2007, Zhang et al., 2012, Bierson & Zhang, 2020]. Their photo-destruction in the upper cloud layer (60 – 70 km) leads to formation of sulfuric acid (H_2SO_4), that condenses out and forms the clouds in Venus's atmosphere [Yung & Demore, 1982]. The droplets rain out of the clouds at a height of $\lesssim 48$ km, where they evaporate [Yung & Demore, 1982, Krasnopolsky, 2007]. The H_2SO_4 then dissociates and replenishes SO_2 and H_2O in the lower atmosphere <cit.>. The behavior of all other known chemically reactive species in Venus's atmosphere is influenced by this cycle [Krasnopolsky, 2007, Krasnopolsky, 2010, Krasnopolsky, 2012], and many of these species participate in this cycle. The sulfur cycle in the atmosphere of Venus establishes a strong and persistent redox gradient through the atmosphere of Venus. Venus is more reduced above the clouds and more oxidized below the clouds.
Though this cycle is central to the atmospheric chemistry of Venus, it is not fully understood, and no self-consistent full atmospheric model of Venus yet accounts for this cycle.
There are several models of the lower atmosphere of Venus (0 – 40 km) that account for the efficient evaporation of H_2SO_4 and the effect of its dissociation products, SO_3 and H_2O, on the abundances of carbon monoxide (CO), carbonyl sulfide (OCS), and SO_2 [Krasnopolsky, 2007, Krasnopolsky, 2013]. Other models describe the middle atmosphere of Venus (60 – 120 km) <cit.>, investigating the chemistry above the clouds (60 – 80 km) where SO_2 is depleted and then re-appears between 85 and 105 km [Sandor et al., 2010, Belyaev et al., 2012]. Zhang et al., 2010 propose that night-side evaporation of H_2SO4 at 85 – 105 km, followed by rapid displacement to the day side by strong winds and subsequent photodissociation, can explain this behavior. The distribution of upper atmospheric aerosols at day and night side [Parkinson et al., 2015], could be applied as constraints for the proposed mechanism of Zhang et al., 2010. Pinto et al., 2021 suggest an alternative hypothesis, involving (SO)_2 chemistry. There has recently been a model of the upper clouds layer of Venus exploring the SO_2 depletion from $1\,\mathrm{ppm}$ to $\sim 10\,\mathrm{ppb}$, and the correlation with H_2O abundances in the clouds [Shao et al., 2020]. A diagram of the sulfur cycle and its connection to other trace atmospheric species in the atmosphere of Venus is shown in Figure <ref>.
At least three atmospheric models of Venus also exist, but they either do not predict the observed SO_2 depletion [Yung et al., 2009], do not couple the SO_2 depletion to the sulfur cycle [Greaves et al., 2020], or do not consider the H_2O and SO_2 depletion in concert. The best current full-atmospheric model that accounts for the SO_2 depletion, from Bierson & Zhang, 2020, reproduces the SO_2 reasonably well, though only by fixing the H_2O profile and by inhibiting the vertical transport within the clouds, as suggested by Marcq et al., 2018 and Bierson & Zhang, 2020.
To explain the depletion of sulfur dioxide and water in concert, either some unknown chemistry must take place within the cloud layer, or observations of lower atmospheric SO_2 and/or H_2O must be mistaken. We explore both of these possibilities in this paper.
In Section <ref> we show why the sulfur cycle cannot be explained without either decreasing the amount of sulfur in the lower atmosphere or increasing the amount of hydrogen in the clouds, either by increasing the water vapor in the clouds or by transporting the hydrogen into the clouds in a different form. We propose that hydrogen could be contained either in aerosols that are lifted by winds into the clouds from the surface, delivered exogenously, or contained within the clouds within some unknown chemical species. In Section <ref>, we discuss our full atmospheric model for Venus. We then show the results of our model if the observational constraints on SO_2 and H_2O in the lower atmosphere are wrong (Sections <ref> and <ref>) or if we introduce cloud chemistry (Section <ref>). In Section <ref> we also predict the effect of this source of hydrogen on the cloud chemistry, chiefly on how it would act as a pH buffer in the clouds. We discuss the implications of our results in Section <ref>, particularly about how rainout and replenishment of hydrogen is needed to sustain the SO_2 gradient. We also speculate about possible sources of hydrogen and their delivery into the clouds, and ways of reconciling the changing cloud chemistry with observations. Section <ref> contains our conclusions.
A scheme of the sulfur cycle on Venus. We include a hypothetical mechanism for cloud buffering by volcanic release of salts into the clouds, or levitation of dust particles with salts by winds into the cloud layer. For simplicity, this figure only shows the upward transport of species relevant for formation of the cloud and initiation of droplet chemistry, and not the settling, rainout and evaporation of cloud droplets, needed to complete the sulfur cycle.
§ THE PUZZLE OF SULFUR DEPLETION
There is good observational evidence that the concentration of SO_2, the dominant sulfur-bearing species in Venus's atmosphere, varies by several orders of magnitude between altitudes of 40 – 80 km (see Appendix <ref>. The concentration of SO_2 is above 100 ppm at 40 km and between 1 and 100 ppb at 80 km, and we call the decrease in SO_2 with height the “depletion” of SO_2. The standard explanation for this depletion predicts the major presumed constituent of the clouds of Venus: H2SO4. Near the top of the clouds:
\begin{align}
\ce{CO_2} + h\nu &\rightarrow \ce{CO} + \ce{O} \label{eqn:co2-pd}\\
\ce{SO_2} + h\nu &\rightarrow \ce{SO} + \ce{O} \label{eqn:so2-pd}\\
\ce{SO_2} + \ce{O} + \ce{M} &\rightarrow \ce{SO_3} + \ce{M} \label{eqn:so2-o}\\
\ce{SO_3} + 2\ce{H_2O} &\rightarrow \ce{H_2SO_4} + \ce{H_2O} \label{eqn:so3-h2o}
\end{align}
The H_2SO_4 condenses out to droplets that then drop down to $\sim 40$ km, where the droplets evaporate. These droplets are predicted to make up the cloud layer. This mechanism ends up reducing the upper atmosphere by replacing CO_2 with CO and SO_2 with SO. The excess oxygen is bound up in the sulfuric acid which has condensed out of the atmosphere.
An oxygen atom is needed to form SO_3, and this O must come from either CO_2 or SO_2 because they are by far the most abundant O-containing molecules. The formation of one molecule of H_2SO_4 is the destruction of one molecule of H_2O and at least one molecule of SO_2 (the sum of Reactions (<ref>) and (<ref>)), at most two molecules of SO_2 (the sum of Reactions (<ref>) and (<ref>)). Between one and two molecules of SO_2 is lost with every molecule of H_2O to make a molecule of H_2SO_4 this way, and so this mechanism predicts that the below-cloud H_2O and SO_2 concentrations be within a factor of two of each other. Therefore, the maximum depletion of SO_2 in the atmosphere by this mechanism is equal to $\big(\chi + 1\big)\ce{[H_2O]}$, where [H_2O] (cm$^{-3}$) is the atmospheric mixing ratio of H_2O and $\chi$ is the fraction of H2SO4-bound O that was produced by SO_2 dissociation.
The observational constraints, however, are $[\ce{SO_2}] \approx 150$ ppm and $[\ce{H_2O}] \approx 30$ ppm. Even if $\chi = 1$, and all water was converted to H_2SO_4, the SO_2 would only be depleted by $\sim 20\%$. This is insufficient to explain the several-orders-of-magnitude depletion of SO_2.
Therefore the SO_2 depletion is a puzzle for which there is no successful solution in the literature consistent with observations. This implies that either the observational constraints on H_2O and/or SO_2 in the lower atmosphere are wrong and their abundances below the clouds are within a factor of two of each other, or that some alternative chemistry explains the SO_2 depletion. We are not the first to notice the implications of this puzzle: a missing sulfur reservoir <cit.>.
One alternative mechanism to explain the SO_2 depletion is the formation of condensible sulfur allotropes out of SO_2 photodissociation or thermal dissociation products. The remaining sulfur cannot be in the form of SO, because the resulting concentrations of SO would be at least two orders of magnitude greater than indicated by above-cloud observations. This explanation requires photodissocation of both SO and SO_2 near the cloud top that is many orders of magnitude more efficient than predicted by any model, or similarly more efficient thermal dissociation of SO_2, and must explain 80% of the SO_2 depletion. 20% of the SO_2 will be converted into SO_3 and will react with H_2O to form H_2SO_4, condensing out. Another 20% will balance the reducing power of the SO_3 removal. The remaining 60% would have to go through the either or both of the total reactions:
\begin{align}
2\ce{SO_2} &\rightarrow \ce{S_2} + 2 \ce{O_2}, & {\rm Thermochemistry};\\
2\ce{SO_2} + 4h\nu &\rightarrow \ce{S_2} + 2 \ce{O_2} & {\rm Photochemistry}.
\end{align}
This would either predict 150 ppm additional O_2 in the upper atmosphere of Venus, exacerbating the O_2 overabundance problem discussed in the Introduction, or would lead to oxidation of CO. However, oxidation of CO would cause it to become depleted, whereas we see that CO increases above the clouds. It is not possible that the excess oxygen would remain in the form of atomic oxygen, because atomic oxygen is not chemically stable at above-cloud altitudes. It is possible that the oxygen is stored in some other chemical species that has not yet been identified, but thus far there is no known candidate species at several ppm concentrations needed to contain the excess oxygen. For these reasons we do not consider this explanation further here.
§ THE MODEL
For this work we use a photochemistry-diffusion model based on the model of Rimmer & Helling, 2016. The model is composed of a solver and a network. The solver, Argo, solves the time-dependent set of coupled non-linear differential equations:
\begin{equation}
\dfrac{dn_{\ce{X}}}{dt} = P_X - L_Xn_{\ce{X}} - \dfrac{\partial \Phi_X}{\partial z},
\end{equation}
where $n_{\ce{X}}$ (cm$^{-3}$) is the number density of species $\ce{X}$ at height $z$ (cm) and time $t$ (s), $P_X$ (cm$^{-3}$ s$^{-1}$) is the rate of production of X at height $z$ and time $t$, $L_X$ (s$^{-1}$) is the rate constant for destruction of species X at height $z$ and time $t$. The term $\partial \Phi_X/\partial z$ (cm$^{-3}$ s$^{-1}$) describes the divergence of the vertical diffusion flux.
As described by Rimmer & Helling, 2016, the chemistry is solved by following the motion of a parcel up and down through a one-dimensional atmosphere described by a grid of set temperature, $T$ (K), pressure, $p$ (bar) and other properties. A parcel starts at the surface with a particular set of initial chemical conditions, and then moves through the atmosphere at a velocity determined by the Eddy diffusion coefficient:
\begin{equation}
t_{\rm chem} = \dfrac{\big(\Delta z\big)^2}{2 K_{zz}},
\end{equation}
where $\Delta z$ (km) is the change in height from one part of the grid to the next, and $K_{zz}$ (cm$^2$ s$^{-1}$) is the Eddy diffusion coefficient. The volume mixing ratios are recorded for each species at each grid height, constructing chemical profiles for the atmosphere. There is a method to account for molecular diffusion, described by Rimmer & Helling, 2016 and corrected by Rimmer & Helling, 2019. We do not solve above Venus's homopause, so molecular diffusion, though included, is not significant and we do not describe it in detail here. After the parcel makes a single complete trip, a UV radiative transfer model is run on the recorded profiles, completing a single global iteration and recording the actinic flux[The actinic flux is the total (direct and diffusive) spectral irradiance integrated over a unit sphere.] $F_{\lambda}$ (photons cm$^{-2}$ s$^{-1}$; photons will be excluded from the units hereafter).
This method reproduces results for modern Earth and Jupiter [Rimmer & Helling, 2016] and agrees with Eulerian solvers for chemical quenching heights in Hot Jupiter atmospheres [Tsai et al., 2017, Hobbs et al., 2019, Hobbs et al., 2020].
Some of the values of $L_X$ (and corresponding production coefficients) are the result of direct photodissociation and photoionization. These are all set to zero for the first global iteration, and then are calculated for global iteration $I$ using the chemical profiles of global iteration $I-1$. The photodissociation and photoionization rate constant for species X are:
\begin{equation}
k_{\lambda}(\ce{X}) = \dfrac{1}{2}\int F_{\lambda} \, \sigma_{\lambda}(\ce{X}) \; d\lambda
\end{equation}
where $\lambda$ (Å) is the wavelength, $z$ (km) is the atmospheric height, $\sigma_{\lambda}(\ce{X})$ (cm$^2$) are the photochemical cross-sections (see Appendix <ref>). The factor of 1/2 is typically included to account for rotation of the planet. Venus rotates too slowly to include this factor for the same reason, and would typically be treated as having a dayside and nightside chemistry. However, the zonal winds of Venus are very strong, horizontal mixing is fast, and for simplicity we consider the atmosphere to be a well-mixed average of day-side and night-side chemistries, which will be true for the long-lifetime and medium-lifetime species. The flux at height $z$ is given by:
\begin{equation}
F_{\lambda}(z) = F_{\lambda}(z_0) e^{-(\tau + \tau_a)/\mu_0} + F_{\rm diff}.
\label{eqn:actinic-flux}
\end{equation}
Here $F_{\lambda}(z_0)$ (cm$^{-2}$ s$^{-1}$ Å$^{-1}$) is the top-of-atmosphere (TOA) flux as described in Section <ref>, $F_{\rm diff}$ (cm$^{-2}$ s$^{-1}$ Å$^{-1}$) is the diffuse flux from scattering <cit.>, the cosine of the average zenith angle $\mu_0 = 0.54$ <cit.>, $\tau$ is the optical depth from molecular absorption calculated from the chemical profiles using the prior global solution as well as the photochemical cross-sections (see Appendix <ref> for details). In addition, $\tau_a$ is the additional optical depth due to Venus's mysterious UV absorber, described in Section <ref>.
Beyond these photodissociation and photoionization rate constants, the coefficients used to construct $P_X$ and $L_X$ are provided by the chemical network, Stand2020, which we introduce here. We start with the sulfur network of Hobbs et al., 2020 and add all relevant reactions involving species without thermochemical data from Greaves et al., 2020. The network includes 2901 reversible reactions and 537 irreversible reactions involving 480 species comprised of H/C/N/O/S/Cl and a handful of other elements, including condensation of H_2O, and a host of other species, most of which condense at temperatures far lower than achieved in Venus's atmosphere, see Appendix <ref>. The network, including added condensation chemistry for sulfuric acid and sulfur allotropes, is described in detail in Appendix <ref>.
In addition, we track the dissolution of SO_2 into the cloud droplets and subsequent liquid-phase chemistry for the cloud chemistry model described in Section <ref>.
The chemical profiles from the most recent global iteration, $I$ are compared to the profiles from the next most recent global iteration, $I-1$ in order to determine convergence. Convergence criteria are the same as for Greaves et al., 2020.
We give the parameters and initial conditions for our model in Section <ref>, including the temperature profile, Eddy diffusion, stellar irradiation, and surface boundary conditions for the chemistry. In Section <ref>, we discuss the consistency of these initial conditions compared to chemical equilibrium at the surface.
§.§ Parameters and Initial Conditions
The initial conditions and parameters we use are are similar to those used for Greaves et al., 2020. We use the same fixed temperature profile as Greaves et al., 2020, which was initially taken from Krasnopolsky, 2007 and Krasnopolsky, 2012. We use Eddy diffusion profiles from the same sources, though we also explore the effect of using the in-cloud Eddy diffusion coefficients of Bierson & Zhang, 2020 in Section <ref>. The profiles we use are shown in Figure <ref>.
We use a scaled top-of-atmosphere (TOA) solar spectrum from 1 – 10000 Å, compiled by Granville-Willett, 2017, for our TOA boundary condition. This data was compiled using Matthes et al., 2017 for the 401 – 1149 Å spectral region and Coddington et al., 2015 for the other wavelengths. The actinic flux is then multiplied by 1.913 to account for the difference in average distances of Earth and Venus from the Sun. The TOA spectrum is included in the Supplementary Materials.
We also include, in addition to molecular absorption and scattering as described above in Section <ref>, a parameterization of the mysterious UV absorber present within Venus's cloud layer. The parameterization originates with Krasnopolsky, 2012 and is the same as that used by Greaves et al., 2020, it takes the form:
\begin{align}
\dfrac{d\tau_a}{dz} &= 0.056 \; \mathrm{km}^{-1} \exp\Big\{-\dfrac{z-67 \, \mathrm{km}}{3 \, \mathrm{km}} - \dfrac{\lambda - 3600 \mathrm{\AA}}{1000 \mathrm{\AA}}\Big\}, & z > 67 \; \mathrm{km}; \notag\\
\dfrac{d\tau_a}{dz} &= 0.056 \; \mathrm{km}^{-1} \exp\Big\{- \dfrac{\lambda - 3600 \mathrm{\AA}}{1000 \mathrm{\AA}}\Big\}, & 58 \; \mathrm{km} < z \leq 67 \; \mathrm{km}; \label{eqn:mysterious-flux}\\
\dfrac{d\tau_a}{dz} &= 0.0 \; \mathrm{km}^{-1}, & z \leq 58 \; \mathrm{km}. \notag
\end{align}
Where $\tau_a$ is added to the optical depth in Eq. (<ref>).
The initial surface mixing ratios we set for Venus's atmosphere are given in Table <ref>, defining the bulk atmosphere. For all of the models presented outside of Section <ref>, we do not include PH_3. Now that we've defined our model and parameters, we will lay out the different scenarios that we consider for explaining the observed SO_2 depletion.
Initial Surface Abundances used for Model Atmospheres of Venus
CO_2 N_2
SO_2 H_2O
OCS CO HCl H_2 H_2S NO
0.96 0.03 150 ppm$^*$ 30 ppm$^*$ 5 ppm 20 ppm 500 ppb 3 ppb 10 ppb 5.5 ppb
$^*$The mixing ratios of SO_2 and H_2O are varied for some of the models.
Atmospheric temperature, $T$ (K, left), and $K_{zz}$ (cm$^{2}$ s$^{-1}$, right), as a function of height $h$ (km) <cit.>. The two values of $K_{zz}$ are for the cloud chemistry and the low-sulfur models (Sections <ref>, <ref>, <ref> and <ref>, solid) and high-water models (Sections <ref> and <ref>, dashed and dotted), from Krasnopolsky, 2007, Krasnopolsky, 2012 and Bierson & Zhang, 2020 respectively.
§.§ Equilibrium Surface Composition
Here we discuss the consistency of the chemical boundary conditions (Table <ref>) compared to chemical equilibrium, and the implications these conditions may have on surface mineralogy. We find there are solutions where the gas-phase chemistry is consistent with our chosen boundary conditions and the condensed-phase chemistry is broadly consistent with observed surface mineral compositions. We will explore these conditions and their implications for hypothetical cloud chemistry in Section <ref>.
In June 1985, the Vega 2 lander determined the composition of the Venusian surface rock in the northern region of Aphrodite Terra <cit.>. The rock was analysed by X-ray fluorescence employing instrumentation that had been improved based on the experience with the previous Venera 13 and 14 missions [Surkov et al., 1982, Surkov et al., 1984]. The measured oxide ratios are listed in Table <ref>. We have used these data, in combination with the observed gas composition at the surface in Table <ref>, to investigate the question in how far the gas at the bottom of the Venusian atmosphere is in chemical equilibrium and in phase equilibrium with the surface rock.
Oxide mass fractions [%] of the surface rock measured by the
Vega 2 lander [Surkov et al., 1986].
SiO2 TiO2 Al2O3 FeO MnO MgO
CaO Na2O K2O SO3
$45.6 \pm 3.2$ $0.2 \pm 0.1$ $16 \pm 1.8$ $7.7 \pm 1.1$
$0.14 \pm 0.12$ $11.5 \pm 3.7$ $7.5 \pm 0.7$ $2.0$
$0.1\pm 0.08$ $4.7 \pm 1.5$
The following elements were not detected:
Cl, Cu and Pb $<$0.3% , Zn$<$0.2%, Pb, As, Se, Br, Sr, Y, Zr, Nb, Mo $<$0.1%.
These investigations were carried out by means of the chemical and phase equilibrium code GGchem [Woitke et al., 2018] taking into account the following 16 elements: H, C, N, O, F, S, Cl, Fe, Mn, Si, Mg, Ca, Al, Na, K, and Ti. No information is available about phosphorous at the Venusian surface, so we have excluded that element from this investigation. GGchem finds 442 gas phase species (atoms, ions, molecules and molecular ions) and 190 condensed species in its databases for this selection of elements. The thermochemical data for the molecules are based on the NIST-Janaf tables [Chase et al., 1982, Chase, 1986, Chase, 1998], as fitted by Stock, 2008, with some additions for diatomic molecules from Barklem & Collet, 2016. Condensed phase data are taken from the SUPCRTBL database [Zimmer et al., 2016] and from the NIST-Janaf database. Some additional vapour pressure data are taken from Yaws, 1999, Weast, 1971, Ackerman & Marley, 2001, and Zahnle et al., 2016.
We consider a mixture of gas and condensed species at $T\!=\!735\,$K and $p\!=\!90\,$bars with total (gas $+$ condensed) element abundances $\epsilon^0$, see <cit.>. In order to find these element abundances we first convert the solid oxide mass ratios given in Table <ref> into element particle ratios. Second, we multiply by an arbitrary factor of 1000 and then add the observed gas phase element abundances from Table <ref>. Third, we carefully adjust the total oxygen abundance $\epsilon^0_{\rm O}$ until the gas over the condensates has a SO2 concentration of 150 ppm in the model. The arbitrary factor in preparation step two causes the model to produce mostly condensed phases with only small amounts of gas above it. That factor has little influence on the results as long as it is large. The reason for this behaviour is that once GGchem has determined the solid composition in form of active condensates, which have supersaturation ratio $S\!=\!1$ (all other condensates have $S\!<\!1$), one can add arbitrary amounts of those condensates to $\epsilon^0_{\rm O}$, they will just fall out again without changing the resulting gas composition.
The results of this model are shown in Table <ref>. The resultant solid composition of the Venusian surface rock is a felsic mixture of enstatite (MgSiO3[s]) and quartz (SiO2[s]). The condensates anorthite (CaAl2Si2O8[s]), albite (NaAlSi3O3[s]), and microcline (KAlSi3O8[s]) are the three main minerals forming feldspar which is found e.g. in basaltic rock on Earth. Iron is found to be entirely bound in magnetite (Fe2O3[s]). No carbonates and no phyllosilicates are found to be stable under the assumed conditions, nor any minerals containing chlorine. The only halide found to be stable is magnesium fluoride (MgF2[s]). Therefore, all carbon, nitrogen, hydrogen and chlorine assumed in the model is present in the gas, which allows us to directly fit the observed gas phase concentrations of CO2, N2, H2O and HCl.
Fitting the SO2 concentration is more difficult, because sulphur is mostly contained in anhydrite (CaSO4). In the close vicinity of the equilibrium solution outlined in Table <ref>, we see that additional oxygen is used to form more anhydrite in the model on the expense of gaseous SO2 and anorthite (CaAl2Si2O8[s]) via
\begin{equation}
\ce{O} \,+\, \ce{SO2} \,+\, \ce{CaAl2Si2O8[s]} ~\longleftrightarrow~
\ce{CaSO4[s]} \,+\, \ce{Al2O3[s]} \,+\, 2 \ce{SiO2[s]} \ ,
\label{eq:regulation}
\end{equation}
which is a potentially important buffer mechanism to understand the SO2 concentration in the lower Venus atmosphere. It allows us in the model to adjust the total oxygen abundance $\epsilon^0_{\rm O}$ to find the desired SO2 concentration (more oxygen means less SO2).
Table <ref> shows that it is possible to explain both the observed composition of the near-crust Venusian atmosphere and the solid composition of the surface rock by a simple consistent phase equilibrium model. All molecules that are predicted to be abundant in our model (those with percent or ppm concentrations) have observed counterparts. The abundance hierarchy matches between model and observations. Other common molecules like CH4 and NH3, which have extremely small abundances ($<\!10^{-15}$) in our model, are not observed. The Venus atmosphere can hence be classified as type B atmosphere according to Woitke et al., 2020, along with the atmospheres of Earth and Mars. All predicted molecular concentrations are in reasonable agreement with the observed values, in particular when taking into account the large measurement uncertainties (see Table <ref>). We note, however that this does not prove that the Venusian atmosphere is in chemical and phase equilibrium with the surface rock, it only shows that the data can be interpreted that way.
For the water-rich scenario, see Sect. <ref>, we can increase the hydrogen abundance to find a model with 200 ppm H2O, which has little effect on CO and OCS, but results in slightly increased abundances of the ppb molecules,
149 ppm SO2,
12 ppm CO,
9 ppm OCS,
505 ppb HCl,
505 ppb HF,
110 ppb S2,
370 ppb H2S,
110 ppb S2O,
and 18 ppb H2, which is arguably still consistent with the observations.
For the sulphur-poor scenario, we can increase the oxygen abundance to find a model with 20 ppm SO2. In that case, the atmosphere is found to have a purer, more oxidising character with
30 ppm H2O,
2 ppm CO,
4 ppb OCS,
505 ppb HCl,
355 ppb HF,
and S2, H2S, S2O, H2 all $<$ 1ppb,
which seems inconsistent with the observations.
Results of the GGchem model for the bottom of the
Venusian atmosphere assuming the gas to be in chemical equilibrium
and in phase equilibrium with the surface rock at $T\!=\!735\,$K and
Gas phase composition
CO2 N2 SO2 H2O CO OCS
HCL HF S2 H2S S2O H2
96 % 3 % 150 ppm 30 ppm 20 ppm 5 ppm
500 ppb 500 ppb - 10 ppb - 3 ppb
97 % 3 % 150 ppm 30 ppm 12 ppm 9 ppm
505 ppb 355 ppb 114 ppb 57 ppb 17 ppb 3 ppb
All other molecules have concentrations $<$ 1 ppm in
the model.
Solid composition (results in mass fractions)
MgSiO3 SiO2 CaAl2Si2O8 NaAlSi3O8
CaSO4 Fe2O3
Al2O3 TiO2 KAlSi3O8 Mn3Al2Si3O12 MgF2
29.7 % 7.6 % 21.7 % 17.6 % 8.3 % 8.9 %
5.1 % 0.2 % 0.6 % 0.3 % (trace)
MgSiO3 $=$ enstatite,
SiO2 $=$ quartz,
CaAl2Si2O8 $=$ anorthite,
NaAlSi3O8 $=$ albite,
CaSO4 $=$ anhydrite,
Fe2O3 $=$ magnetite,
Al2O3 $=$ corundum,
TiO2 $=$ rutile,
KAlSi3O8 $=$ microcline,
Mn3Al2Si3O12 $=$ spessartine,
MgF2 $=$ magnesium fluoride.
All other condensates are under-saturated in the model and have zero mass fractions.
§ HYPOTHESIS: THE OBSERVATIONAL CONSTRAINTS ARE WRONG
One possibility to consider is that the below-cloud observational constraints are incorrect. Possibly the below-cloud water vapor is much higher than observations suggest, a possibility considered by Yung & Demore, 1982. Or the below-cloud sulfur dioxide is much lower than most observations suggest.
In order to explore the sulfur-poor hypothesis we vary the below-cloud SO_2 abundance from 80 ppm down to 6 ppm with below-cloud H_2O kept at the nominal value (30 ppm). To explore the water-rich hypothesis we vary the below-cloud H_2O abundance from 30 ppm up to 200 ppm with SO2 kept at the nominal value (150 ppm). In this case we find that the observed above-cloud SO2 depletion is not achieved for any value of below-cloud H_2O due to H_2O self-shielding effects. Some support for the water-rich hypothesis is the $>100 \, {\rm ppm}$ abundances of water vapor observed in and directly below the clouds of Venus [Mukhin et al., 1982, Surkov et al., 1982, Bell et al., 1991], though these are inconsistent with $<100 \, {\rm ppm}$ abundances close to the surface [Bertaux et al., 1996].
To explore the possibility of the water-rich hypothesis further we test the effect of introducing a trap in the eddy diffusion profile within the cloud layer alongside varying the H_2O abundance below the clouds. Observational constraints on the eddy diffusivity as a function of altitude in the atmosphere are sparse. Marcq et al., 2018 have suggested that the existence of statically stable layers in the cloud region may inhibit dynamical exchange between the upper and lower regions of the atmosphere, a possibility explored by Bierson & Zhang, 2020. In the present work we test this possibility by modifying the nominal eddy diffusion profile taken from Krasnopolsky, 2007 and Krasnopolsky, 2012 to include a trap of constant lower $K_{zz}$ across the extent of the cloud layer. The range of values that we explore for $K_{zz}$ in the trap are 5000, 1000, 500 and $100 {\rm \, cm^2 \, s^{-1}}$, shown in figure <ref>. Such a $K_{zz}$ trap, if it exists in the Venus cloud layer, is particularly relevant to the water-rich hypothesis as enhanced water abundance below the clouds would result in lesser thermal heating flux at the cloud base due to H_2O IR absorption, which in turn increases the convective stability and decreases the eddy diffusivity, shown by Yamamoto, 2014. We investigate the results from combining a $K_{zz}$ trap and enhanced below-cloud water abundance.
§ HYPOTHESIS: ANOTHER SOURCE OF HYDROGEN IN THE CLOUDS
The SO_2 depletion can be explained if there is another source of hydrogen in the clouds. Here we will use NaOH as that source of hydrogen for the sake of convenience when calculating the cloud droplet chemistry. We are not claiming that this is the source of hydrogen. We will discuss possible sources of this excess hydrogen in Section <ref>.
Some SO_2 will dissolve into the cloud droplets directly, with a concentration in the droplet, $c {\rm \,(mol/L)}$, linearly proportional to the partial pressure of SO_2, with the Henry's Law constant, $H_{\ce{SO_2}} {\rm \big(mol/(L bar)\big)}$, as the constant of proportionality:
\begin{equation}
c(\ce{SO_2}) = H_{\ce{SO_2}} \; p f_{\ce{SO_2}}.
\label{eqn:henrys-law}
\end{equation}
Here $p f_{\ce{SO_2}}$ is the partial pressure of SO_2, with the total gas pressure $p$ (bar) and $f_{\ce{SO_2}}$ as the volume mixing ratio of SO_2, equal to $n_{\ce{SO_2}}/\big(\Sigma_{\ce{X}} n_{\ce{X}}\big)$.
For SO_2 dissolved in water, the Henry's Law constant is $\approx 10^{-2}$ mol/(L bar) [Burkholder et al., 2020]. For SO_2 dissolved in sulfuric acid, the constant increases with the sulfuric acid concentration between 0.1 and 1 mol/(L bar) [Zhang et al., 1998]. The SO_2 then participates in the following reactions (here g is gas-phase, $\ell$ is in the droplet). We first consider the dissolution of SO_2 and H_2O into the droplets by Henry's law.
\begin{align}
\ce{SO_2(g)} &\rightleftharpoons \ce{SO_2($\ell$)}, \label{eqn:liquid-first}\\
\ce{H_2O(g)} &\rightleftharpoons \ce{H_2O($\ell$)}. \label{eqn:liquid-second}
\end{align}
The rate constants for this reaction are balanced such that the concentration of SO_2 at any point agrees with Eq. (<ref>) when accounting for the droplet volume (see below). Also, in reality, the Henry's law constant for both water vapor and sulfur dioxide will vary with the composition of the droplet, and the composition of the droplet will vary as more water vapor and sulfur dioxide dissolves in the droplet. Our model does not account for these variations, and thus Henry's law as treated here is only an approximation. The rest of the reactions are dissociation reactions for which bimolecular rate constants are set to $k_f = 5 \times 10^{10}\,{\rm mol^{-1} \, L \, s^{-1}}$ in order to rapidly achieve equilibrium and avoid any dynamic effects, and the unimolecular rate constant is assigned a value that preserves equilibrium set by the $pK_a$ or $pK_b$. The rate constant for the reverse reaction is (with units s$^{-1}$):
\begin{align}
k_r &= \dfrac{\rho k_f}{\mu} \, 10^{-pK_a}, \\
k_r &= \dfrac{\rho k_f}{\mu} \, 10^{-pK_b},
\end{align}
where $\rho\,{\rm (g\,cm^{-3})}$ is the density of the liquid and $\mu\,({\rm g\, mol^{-1}})$ is the molar weight of the species. The reactions for the sulfates are:
\begin{align}
\ce{H_2SO_4($\ell$)} &\rightleftharpoons \ce{HSO_4^-($\ell$)} + \ce{H^+($\ell$)}, & pK_{a,1} = -2.8;\\
\ce{HSO_4^-($\ell$)} &\rightleftharpoons \ce{SO_4^{2-}($\ell$)} + \ce{H^+($\ell$)}, & pK_{a,2} = 1.99;
\end{align}
and the reactions involving sulfurous acid and sulfites:
\begin{align}
\ce{SO_2($\ell$)} + \ce{H_2O($\ell$)} &\rightleftharpoons \ce{HSO_3^-($\ell$)} + \ce{H^+($\ell$)}, & {\rm see \; below};\\
\ce{H_2SO_3($\ell$)} &\rightleftharpoons \ce{SO_2($\ell$)} + \ce{H_2O($\ell$)}, & {\rm see \; below};\\
\ce{H_2SO_3($\ell$)} &\rightleftharpoons \ce{HSO_3^-($\ell$)} + \ce{H^+($\ell$)}, & pK_{a,1} = 1.857;\\
\ce{HSO_3^-($\ell$)} &\rightleftharpoons \ce{SO_3^{2-}($\ell$)} + \ce{H^+($\ell$)} & pK_{a,2} = 7.172.
\end{align}
The rate constant for SO_2 to react with H_2O is $2 \times 10^8 \, {\rm mol^{-1} \, L \, s^{-1}}$ [Eigen et al., 1961, Brandt & Van Eldik, 1995] and the rate constant for H_2SO_3 dissociation is $10^8$ s$^{-1}$ [Eigen et al., 1961, Brandt & Van Eldik, 1995]. These reactions are sufficiently fast to draw the gas-droplet chemistry into equilibrium at all atmospheric heights. It was for this reason that we chose NaOH as our candidate salt. Finally, we consider the self-dissociation of water and the dissociation of sodium hydroxide:
\begin{align}
\ce{H^+($\ell$)} + \ce{OH^-($\ell$)} &\rightleftharpoons \ce{H_2O($\ell$)}, & pK_a = 14\\
\ce{NaOH($\ell$)} &\rightleftharpoons \ce{Na^+($\ell$)} + \ce{OH^-($\ell$)}, & pK_b = 0.2 \label{eqn:liquid-last}
\end{align}
Sodium hydroxide is the example species we will use to buffer the clouds of Venus, freeing up more water in the droplet to react with SO_3, forming sulfuric acid, or with SO_2 to form sulfurous acid.
If the excess source of hydrogen is a salt, as is the case with NaOH, H^+ is replaced by Na^+. We calculate the pH where $\ce{pH} = - \log_{10}\big(a_{\mathrm{H^+}}\big)$, where $a_\mathrm{H^+}$ is the H^+ activity. The concentration of NaOH needed in the droplets to sequester SO_2 is determined by considering the cloud droplet volume as a function of height:
\begin{equation}
V_d(h) = \int \dfrac{4}{3}\pi r_d^3 \dfrac{\partial N_d}{\partial r_d} \, dr_d,
\label{eqn:droplet-volume}
\end{equation}
where $r_d$ ($\mu$m) is the droplet radius, and the function $r_d \big(\partial N_d/\partial r_d\big)$ is the droplet size distribution, which we take from Gao et al., 2014:
\begin{equation}
\dfrac{\partial N_d}{\partial r_d} = \dfrac{\partial n_d}{\partial r_d} V_{\rm atm}
\end{equation}
where $n_d$ (cm$^{-3}$) is the droplet number density (across all sizes) and $V_{\rm atm} = 4\pi R_p^2 \Delta z$, with $R_p = 6052$ km as the radius of Venus and $\Delta z$ (km) is the model height step. We can calculate the amount of NaOH needed to deplete the SO2 to the observed levels. We do this by dividing the total number of SO_2 molecules by the total droplet volume at height $z$:
\begin{equation}
c(z) = \dfrac{p f_{\ce{SO_2}}(z)}{N_AkT(z)} \, \Bigg[\int \dfrac{4}{3}\pi r_d^2 \Bigg(r_d \dfrac{\partial n_d(z)}{\partial r_d}\Bigg)\, dr_d\Bigg]^{-1},
\end{equation}
where $N_A = 6.022 \times 10^{23}$ is Avogadro's Number and $k = 1.38065 \times 10^{-16}$ erg/K is Boltzmann's constant. The above equation is used to prescribe the initial concentration of NaOH (in units of mol/cm$^3$) for our solver. The NaOH will rapidly dissociate and the OH^- will then recombine with H^+ to form H_2O, and this, along with dissolved H_2O from the gas phase will react with SO_2 to form H_2SO_3. This will rapidly dissociate to form HSO_3^- and H^+, and will buffer the solution, meaning that if the H^+ activity is perturbed, the reactions between the anions and H^+ will bring the activity of H^+ back to a given value determined by the $pK_a$. It should be emphasized that Eq's (<ref>) and (<ref>) do not describe condensation of either of these species, who generally have partial pressures throughout Venus's atmosphere that are well below the vapor pressure. Rather, these are the equilibrium concentrations in the droplets due to dissolution, balanced with the partial pressures in the gas.
Prescribing the NaOH this way introduces extra hydrogen into the model in a way not accounted by mass balance or atmospheric redox balance. A full solution of the atmospheric chemistry coupled with surface chemistry would preserve this balance, but that would require us to identify the source of hydrogen. Though we speculate on possible sources of hydrogen, from the surface (delivered via volcanism or winds) or exogenous, in Section <ref>, we do not know enough about Venus's atmosphere, surface or clouds to confidently identify a source, and therefore do not self-consistently include this source in our model.
§ RESULTS
The depletion of SO_2 can be explained by the SO_2 dissolving into the clouds, if the cloud pH is higher than previously believed. It can also be explained by varying the below-cloud SO_2 and H_2O. We discuss the consequences of varying the SO_2 in Section <ref> and H_2O in Section <ref>. The results of incorporating cloud chemistry are presented in Section <ref>.
§.§ Sulfur-Poor Venus
SO_2 begins to significantly decrease through and above the clouds when surface $f_{\ce{SO_2}} \sim 50 {\rm \, ppm}$, and achieves best results for below-cloud concentrations of 15 ppm. See Figure <ref>. The depletion occurs at 70-80 km unless the surface SO_2 is lower, around 15 ppm. Besides being inconsistent with most below-cloud observational constraints on $f_{\ce{SO_2}}$ by almost an order of magnitude, the sulfur-poor model predictions agree with observations for all the species we consider reasonably well. The results agree with observations as closely as when we consider cloud droplet chemistry below with one possible exception of SO, which is below 1 ppb at between 80 and 100 km. This may be brought into better agreement by including a mesospheric source of sulfur acid vapor, or by adjusting the below-cloud concentrations of SO_2, since we have found that the above-cloud SO_2 is very sensitive to the below-cloud SO_2 and the Eddy diffusion profile when $f_{\ce{SO_2}} \approx $ 15 ppm. The 15 ppm below-cloud SO_2 model also predicts H_2 concentrations of $\sim 10 \, {\rm ppm}$ above 70 km.
Mixing ratios as a function of height for SO_2 (a), H_2O (b), O_2 (c) and CO (d) by varying the below-cloud abundance of SO_2 from 6 ppm to 80 ppm. No droplet chemistry is included.
§.§ Water-Rich Venus
For the water-rich case, SO_2 does not deplete for any value of below-cloud H_2O unless a $K_{zz}$ trap extending to 85 km in altitude is introduced in the eddy diffusion profile. Upon introduction of this trap the SO_2 begins to significantly deplete at the top of the atmosphere, and this depletion height then lowers with increasing below-cloud water abundance, achieving best results around surface $f_{\ce{H_2O}} = {\rm 200 \, ppm}$. See Figure <ref>. For in-cloud $K_{zz} = {\rm 5000 \, cm^2 \, s^{-1}}$ the depletion height is higher than observations suggest [Encrenaz et al., 2019], dropping off at $\gtrsim 75{\rm \, km}$, and the O_2 concentration in the upper atmosphere exceeds 100 ppm. The depletion height can be lowered further to $\sim {\rm \, 70 \, km}$ by decreasing the $K_{zz}$ value in the trap, however this exacerbates the overabundance of O_2 and causes CO to deviate from smooth monotonic growth with altitude. The CO profile has a downward spike at $\sim 70 {\rm \, km}$, consistent the profile of Pollack et al., 1993, as scaled by Marcq et al., 2005. This is likely due to the change of $K_{zz}$, see Section <ref>. Neither the SO_2 depletion or O_2 abundances agree with observations as well as in the sulfur-poor and cloud-chemistry cases. The reason for this is the self-shielding of the excess water, which is not as effectively removed as SO_2 within the clouds. The 200 ppm below-cloud H_2O model also predicts H_2 concentrations of $\sim 100 \, {\rm ppm}$ above 70 km, inconsistent with observations (see Appendix <ref>).
Mixing ratios as a function of height for SO_2 (a), H_2O (b), O_2 (c) and CO (d) by varying $K_{zz}$ (see Figure <ref>) with the abundance of below-cloud H_2O of 200 ppm. The divot in the water abundance for nominal K_zz is due to condensation. The CO profile when $K_{zz} = 1000 {\rm \, cm^{2} s^{-1}}$ bears a remarkable similarity to the CO profile from Marcq et al., 2005. No droplet chemistry is included.
§.§ Cloud Chemistry
In this scenario, described in Section <ref>, SO_2 depletion through the clouds is accomplished by removing the SO_2 into H_2SO_3 and H_2SO_4 via droplet chemistry. In our model, the droplet chemistry is driven specifically by sodium hydroxide (NaOH), and the NaOH itself buffers the cloud pH. The NaOH is a proxy used for convenience in modelling, and represents other plausible sources of delivered hydrogen, discussed in Section <ref>. We adjust the amount of NaOH as a function of height to reproduce the observationally constrained SO_2 profile (see Figure <ref>). This initial NaOH is prescribed for each height and is not solved for within the model. The function that reproduces the SO_2 depletion, given the estimated droplet volume, was determined by solving the aqueous chemistry and cloud chemistry for different amounts of NaOH, and the concentration of NaOH and the speciation of sulfuric and sulfurous acid as a function of height that results in the observed SO_2 profile is shown in Figure <ref>. The prediction that the bulk of the clouds is HSO_3^- only holds if there is no mechanism for oxidizing the sulfur in the clouds. Sulfite aerosols are rapidly oxidized in Earth's atmosphere [Townsend et al., 2012], and this may be the case for the atmosphere of Venus as well. We show the predicted mass loading in Figure <ref>. The mass of the lower cloud is largely in the form of sulfite, and this region overlaps with the larger mode 3 aerosols [Knollenberg & Hunten, 1980]. Profiles are similar for Ca(OH)_2, assuming the kinetics are identical between Ca(OH)_2 and NaOH. The similarity is due to the large $pK_a$ which results in all the Ca(OH)_2 becoming fully dissociated, along with the similar mass to basicity potential: the mass of a calcium atom is almost double that of a sodium atom, and each calcium carries two hydrides, which replace two protons.
We can then consider our solution of the droplet chemistry, set out in Equations (<ref>) – (<ref>), which predicts the H^+ activity, from which we can calculate the cloud droplet pH. As described in Section <ref>, the SO_2 depletion is set by the capacity of the liquid to hold SO_2, which is controlled by the amount of NaOH, and by the total volume of the liquid, which is determined by the cloud droplet size distribution. The predicted droplet pH is plotted as a function of height in Figure <ref>.
In this model, throughout the clouds the gas-phase SO_2 is in equilibrium with the concentration of sulfur in the droplet that is specifically in the form of SO_2; i.e., $\mathrm{SO_2(g)}\propto\mathrm{SO_2}(\ell)$ and adding or removing gas-phase SO_2 results in proportionally changing the droplet SO_2, and the balance of the other sulfur species. This equilibrium allows us to write out effective rate constants for SO_2 and H_2O dissolution into the droplets with rate constants tuned to reproduce the results from solving Eq's (<ref>) – (<ref>). For SO_2 the effective reaction is:
\begin{equation}
\ce{SO_2(g)} + \ce{H_2SO_4($\ell$)} \rightleftharpoons \ce{SO_2($\ell$)} + \ce{H_2SO_4($\ell$)}.
\end{equation}
The rate constants for the forward reaction, $k_f$ (cm$^3$ s$^{-1}$), and reverse reaction $k_r$ (cm$^3$ s$^{-1}$) are:
\begin{align}
k_f &= 10^{-32} \; \mathrm{cm^3 \, s^{-1}} \; e^{9000 \, {\rm K}/T}, \\
k_r &= 6.67 \times 10^{-36} \; \mathrm{cm^3 \, s^{-1}} \; e^{9000 \, {\rm K}/T}.
\end{align}
For H_2O the effective reaction is:
\begin{equation}
\ce{H_2O(g)} + 2\ce{SO_2($\ell$)} \rightleftharpoons \ce{H_2SO_3($\ell$)} + \ce{SO_2($\ell$)}.
\end{equation}
Neither of these reactions account for condensation. We determine the effective rate constants for the forward reaction, $k_f$ (cm$^3$ s$^{-1}$), and reverse reaction $k_r$ (cm$^3$ s$^{-1}$) to be:
\begin{align}
k_f &= 2.53 \times 10^{-36} \; \mathrm{cm^3 \, s^{-1}} \; e^{9000 \, {\rm K}/T}, \\
k_r &= 8.43 \times 10^{-38} \; \mathrm{cm^3 \, s^{-1}} \; e^{9000 \, {\rm K}/T}.
\end{align}
Finally, an effective reaction needs to be included to encapsulate the release of SO_2 when the droplet rains out and evaporates:
\begin{equation}
\ce{SO_2($\ell$)} + \ce{H_2SO_4(g)} \rightarrow \ce{SO_2(g)} + \ce{H_2SO_4(g)},
\end{equation}
with rate constant of $2.2 \times 10^{-4}$ cm$^3$ s$^{-1}$ $e^{-10000 \, {\rm K}/T}$.
The results of our model agree within an order of magnitude for all species considered, and within a factor of 3 for all species except for OCS and the sulfur allotropes. Our model underestimates S_4 and does not predict the steep below-cloud gradient of OCS. In addition, it over-predicts O_2 in the upper atmosphere by a factor of 2-3. Comparison of this model and the best sulfur-poor model, with $f(\ce{SO_2}) = 20$ ppm and Krasnopolsky's Eddy diffusion profile (from Section <ref>) is shown in Figure <ref>.
Predicted cloud droplet pH (bottom axis, solid black line) initial NaOH concentrations (mol/L, top axis, dashed green line), and the speciation of the sulfuric and sulfurous acid (red and blue lines), as a function of height (km) based on SO_2 depletion. The initial NaOH is an input that we use to reproduce the SO_2 depletion. This solution for SO_2 is not unique. Changing the pH by changing the initial NaOH will affect the SO_2 depletion. Alternatively, the pH could be higher than plotted, and the depletion could be limited by kinetics. The profiles with Ca(OH)_2 are similar.
Mass composition of the droplets (mass loading, mg m$^{-3}$) as a function of altitude (km). The total mass loading is based on Pioneer Venus observations summed over the three aerosol modes [Knollenberg & Hunten, 1980, Krasnopolsky, 2016]. The species mass loading is equal to the mass fraction multiplied by the total mass loading. The profiles with Ca(OH)_2 are similar.
Predicted volume mixing ratios of SO_2, H_2O, CO, OCS, O_2, HCl, H_2SO_4, H_2, SO, S_3, S_4 and H_2S as a function of height (km), and compared to data (see Appendix <ref>), for four models: the cloud-chemistry model (yellow dashed, Sections <ref> and <ref>), the sulfur-poor model with $f(\ce{SO_2}) = 20\,{\rm ppm}$ (red dash-dot, Section <ref>), the water-rich model with $f(\ce{H_2O}) = 200\,{\rm ppm}$ and in-cloud $K_{zz} = 5000\,{\rm cm^{2}\,s^{-1}}$ (blue dotted, Section <ref>), and the fiducial model with $f(\ce{SO_2}) = 150\,{\rm ppm}$, $f(\ce{H_2O}) = 30\,{\rm ppm}$ and no cloud chemistry (black solid). Observations and upper limits of these species (from Table <ref>) are also plotted in gray to compare. H_2SO_4 includes both condensed and gas-phase H_2SO_4.
§ DISCUSSION
Below-cloud SO_2 $<50\,\mathrm{ppm}$ is inconsistent with most observations but not all. Vega 1 and 2 observed $\ge100\,\mathrm{ppm}$ concentrations of SO_2 directly below the clouds, though the error bars are large and $50\,\mathrm{ppm}$ abundances would be within $2\sigma$ of the measurements [Bertaux et al., 1996]. Even lower SO_2 was measured within 20 km of the surface [Bertaux et al., 1996], which could indicate rapid surface depletion of SO_2 [Yung & Demore, 1982].
The reported observations of below-cloud SO2 at 100–$200\,\mathrm{ppm}$ also have large uncertainties, typically on the order of 50 ppm <cit.>, so true values below 50 ppm would amount to discrepancies of 2 – 3 $\sigma$. Exploring the hypothesis of low below-cloud SO_2 as an explanation for the above-cloud SO_2 depletion will require both more precise and more frequent observations of the below-cloud SO_2, to see whether it varies and by how much. Such data may only be obtainable with in situ measurements.
Below-cloud H_2O is better constrained, so it is less likely that there is an undetected large source of water vapor beneath the clouds. There are some variations in the measurements, from less than 10 ppm to 60 ppm, but with relatively small error bars. There is some observational support for a large reservoir of water within the clouds, with 700 – 2000 ppm concentrations observed in the cloud layers by Vega 1,2 [Surkov et al., 1987] and Venera 13,14 [Mukhin et al., 1982, Surkov et al., 1982], and 100 – 200 ppm concentrations observed from the ground [Bell et al., 1991]. These high quantities may have been due to incidental sampling of cloud droplets which are expected to be composed of $\sim 15-25 \%$ water, even without considering the cloud pH buffer hypothesis proposed here. Even if these constraints were not so tight, the model where we increase the below-cloud water vapor predicts that the clouds to extend up to $\sim 80$ km, and the depletion would be more gradual and higher in the atmosphere. Further to this, the water-rich model overproduces O_2 above $\sim 70$ km pushing it beyond the observed upper limit by two orders of magnitude, overproduces H_2, and does not allow for SO_2 to return at the $\sim 1 {\rm \, ppb}$ level at 90 km, problems which are not shared by the other best fitting models.
In addition to enhancing the cloud layers, we had to vary the above-cloud $K_{zz}$ in order to induce sulfur and water vapor depletion. Decreasing the $K_{zz}$ to $\sim 1000 \, {\rm cm^{2} s^{-1}}$ creates a negative spike in the CO at $\sim 70 \, {\rm km}$, reproducing a feature seen in suggested mesospheric profiles [Marcq et al., 2005]. This is a consequence of varying the $K_{zz}$ and not increasing the below-cloud water vapor. Our model suggests that this CO negative spike may trace changes in the eddy diffusion, a hypothesis that is worth further investigation but is outside the scope of this paper.
Both the sulfur-poor and water-rich models predict $> 10 \, {\rm ppm}$ H_2 concentrations in the mesosphere, above 70 km, where the cloud chemistry model predicts $\sim 0.1 \, {\rm ppm}$ H_2 concentrations. Better observational constraints of H_2 may be useful for distinguishing these models.
As we have shown, cloud chemistry is a possible explanation for the depletion of SO_2. Aerosols could provide the excess hydrogen capable of depleting gas-phase SO_2. This hydrogen could be bound in salts, or could be in some other form, such as hydrocarbons. It is important that whatever the source of hydrogen, it is replenished to sustain the SO_2 gradient. Otherwise the hydrogen will be consumed, the clouds will be saturated with sulfur, and the gradient of SO_2 will disappear. It is also worth mentioning that three phases can participate in this chemistry: the gas phase, the liquid of the droplet (at atmospheric heights where the droplet has not frozen), and the solid aerosol material, either at the core, surrounding the droplet, or suspended within the droplet. This provides a rich and complex multi-phase chemistry worth exploring in the lab.
What follows in this section is a discussion of the requirements for and implications of cloud chemistry. In Section <ref>, we discuss possible sources of hydrogen within the clouds. The measured optical constants and spectral features of the clouds of Venus are consistent with cloud droplets composed of a large percentage of sulfuric acid. Any proposed cloud chemistry must either preserve sulfuric acid as the dominant species in the clouds, or must propose a species with similar optical properties and spectral features. We discuss the observable implications of our cloud chemistry in Section <ref>. The dissolution of hydrogen halides into the clouds is discussed in Section <ref>. The cloud chemistry also has implications for above-cloud radical concentrations, which affects the lifetime of hypothetical PH_3 within the clouds. We discuss the status of PH_3 and its lifetime within the clouds in Section <ref>. Finally, in Section <ref>, we briefly discuss the implications of different cloud chemistry for hypothetical life within the cloud droplets of Venus.
§.§ Possible Sources of Hydrogen in the Clouds
If a buffer explains the sulfur depletion, it is possibly a salt. Salts will dissociate quickly, and some will provide efficient buffers. The salts must get into the clouds in order to buffer them. This can be accomplished either by transport from the surface or exogenous delivery.
Exogenous Delivery: Exogenous delivery, meaning delivery of material from the interplanetary medium, is unlikely to provide significant material to buffer the clouds of Venus based on the estimated incoming flux of interplanetary dust. The clouds must be able to retain virtually all of the SO_2 over the timescale of transport through the clouds, requiring a flux of salts, $\Phi_s$ (mol cm$^{-2}$ s$^{-1}$), of:
\begin{equation}
\Phi_s = \dfrac{n_{\ce{SO_2}} \, H_0}{\tau_{\rm dyn}} = \dfrac{f_{\ce{SO_2}} \, nkT \, K_{zz}}{\mu_{\rm av} m_p \, N_A \, g \, (\Delta h)^2},
\end{equation}
where $n_{\ce{SO_2}} \, ({\rm cm^{-3}})$ is the number density of SO_2 at the height where SO_2 begins to deplete (50 km), $H_0 \, {\rm (km)}$ is the scale height, $R = 6052 \, {\rm km}$ is the radius of Venus, $\tau_{\rm dyn} \, {\rm (s)}$ is the dynamic timescale of the atmosphere at 40 km, $f_{\ce{SO_2}} = 150$ ppm is the volume mixing ratio of SO_2, $n = \Sigma_{\ce{X}} n_{\ce{X}} = 2.189 \times 10^{19}$ cm$^{-3}$ is the gas density at 50 km, the height where the depletion begins, $k = 1.38065 \times 10^{-16}$ erg/K is Boltzmann's constant, $T = 349.7$ K is the temperature at 40 km, $K_{zz} \approx 100$ cm$^2$ s$^{-1}$ is the Eddy diffusion of the droplets within the cloud, a low estimate more favorable for exogenous delivery. $\mu_{\rm av} = 44$ is the mean molecular weight of the atmosphere, $m_p = 1.6726 \times 10^{-24}$ g is the mass of a proton, $N_A = 6.022 \times 10^{23}$ is Avogadro's Number, $g = 8.87$ m s$^{-2}$ is the surface gravity of Venus and $\Delta h = 20$ km is the thickness of the cloud layer. Applying all these estimates yields:
\begin{equation}
\Phi_s \approx 10^{-13} \; \mathrm{mol \, cm^{-2} \, s^{-1}}.
\end{equation}
Continuous exogenous delivery of material is insufficient to match this flux even assuming that 100% of the material is in the form of hydrated minerals that will deplete SO_2. If we assume that delivery of exogenous material to Venus is comparable to delivery to Earth, 20 – 70 ktonnes/year [Peucker-Ehrenbrink, 1996, Greaves et al., 2020], which translates to $\sim 10^{-17}$ mol cm$^{-2}$ s$^{-1}$, or four orders of magnitude too little to account for the missing hydrogen.
One other possibility is stochastic delivery. If a recent airburst, a large impact that breaks up in the atmosphere, occurred in Venus's atmosphere, the metals released could permeate the clouds, providing a transient in-cloud source of hydrogen. If this is the case, then the depletion of SO_2 will be temporary, lasting as long as the below-cloud store of these elements persists, on the order of the diffusion timescale or $\sim 1000$ years.
Dust from the Surface: Transport from the surface also struggles to meet the required flux of hydrated material. Although calculations of dust transport favor a more dusty atmosphere for Venus than Earth [Sagan, 1975], Venera measurements suggest that the lower atmosphere is clear, placing upper limits on dust transport to the clouds. It is possible that there is heterogeneous low atmospheric weather, with dust rapidly transported to tens of km above the surface, and that Venera happened to land in a region where the vertical diffusion and winds were insufficient to change the atmospheric opacity. A low level haze inferred from Venera probe data [Grieger et al., 2004], and consistent with heavy metal frost at higher elevations on Venus's surface [Schaefer & Fegley, 2004], may itself be the suspended dust [Titov et al., 2018], and winds may cause that dust to periodically move into the clouds. The reaction products of the salt with SO_2 in and below the clouds could result in transient high abundances of water, which may then be removed by the left-over oxides. This could reconcile the transient observations of $>100\,{\rm ppm}$ of water vapor in and below the clouds [Mukhin et al., 1982, Surkov et al., 1982, Bell et al., 1991], with the $\sim 25\,{\rm ppm}$ abundances near the surface [Bertaux et al., 1996].
To explain the sulfur depletion, with dust containing 5 wt.% salt, requires a dust flux to the clouds of $\approx 16$ Gt/year, well within the estimates of surface dust fluxes estimated from analogue experiments [Greeley et al., 1984]. The composition of the dust and the form of the salts is unknown. Here we will speculate on some possible candidates. Our speculation is based on chemical and physical stability of the salts:
* NaOH: Sodium hydroxide is the example salt we use for our calculations, but is an unrealistic candidate salt. It is sufficiently stable to heat, persisting as a liquid up to 1661 K [Haynes, 2014]. However, it is known to react rapidly with SO_2 to form sodium sulfite and water vapor. Given the concentrations of below-cloud SO_2, NaOH cannot plausibly survive to reach the clouds unless it is injected rapidly, e.g. via a volcanic plume.
* Ca(OH)_2: Calcium hydroxide is stable as a solid at the surface of Venus. Strictly speaking, calcium hydroxide has no melting point. Instead, at 93 bar and $\sim 1000$ K it is expected to decompose into CaO and H_2O, based on extrapolations of the vapor pressure curve of Halstead & Moore, 1957. Ca(OH)_2 will undergo carbonation, and there is ample CO_2, but this reaction is very slow, and Ca(OH)_2 is kinetically stable at temperatures above 723 K [Materic & Smedley, 2011]. Ca(OH)_2 will also react rapidly with SO_2, but only in the presence of water vapor at concentrations of $>3000$ ppm [Liu et al., 2010]. The kinetic stability vs. the dynamic timescale for Ca(OH)_2 aerosols is unknown, but presently Ca(OH)_2 cannot be ruled out as a candidate.
* Mg(OH)_2 and Fe(OH)_2: Neither magnesium hydroxide nor iron hydroxide (either the Fe(II) hydroxide or Fe(III) hydroxides) are stable at Venus's surface temperature and pressure [Wang et al., 1998, Haynes, 2014].
* Other Hydroxides: It may be that more exotic hydroxides, such as Al(OH)_3, could deliver hydrogen to Venus's cloud layer. The requirements are sufficient concentrations to satisfy the required fluxes, and the thermochemical and kinetic stability of the salt in the presence of major atmospheric constituents.
* Oxides: Oxides, either resident surface oxides or oxides produced by the dehydration of hydroxides, may participate in cloud chemistry in unknown ways, sequestering SO_3 directly, for example, {Mg,Fe}O + SO_3 may be converted into {Mg,Fe}SO_4 directly. The subsequent dissociation in sulfuric acid will buffer the cloud droplet pH. There is some indirect evidence of the presence of oxides from the near-surface haze, since these oxides can react with hydrochloric acid vapor to form FeCl_3, which has been observed in the clouds and is a candidate for the mysterious UV absorber [Krasnopolsky, 2017].
* Sulfates: It is possible hydrogen-bearing sulfates could find their way into the clouds, but no known hydrogen-bearing sulfate is thermally stable at Venus's surface pressure and temperature. It is possible that they are produced from gas-phase reactions, e.g., the possible production of ammonium sulfate from reaction with NH_3 and SO_2 [Titov, 1983]. Sulfates that do not contain hydrogen are plausible aerosols. Indeed, we predict they are produced via cloud chemistry. But these aerosols will not deliver hydrogen and cannot directly participate in the depletion of gas-phase SO_2.
The above list is not intended to be exhaustive, and does not consider whether these salts are expected at the temperatures and pressures at the surface of Venus, where we would generally expect chemistry to tend toward thermodynamic equilibrium. The results of comparing our surface boundary conditions (Section <ref>) to chemical equilibrium (Section <ref>) predicts that no phyllosilicates or hydrated minerals are present at the surface of Venus if the surface and gas are in equilibrium.
GGchem model for the bottom of the Venus atmosphere, assuming chemical and phase equilibrium, as a function of surface temperature. The temperature of the current Venus surface (735 K) is marked as dotted vertical lines. A constant gas pressure of $90$ bar and constant total element abundances $\epsilon^0$ are assumed. $n_{\rm mol}/n_{\rm tot}$ are the molecular concentrations in the gas, and $n_{\rm solid}/n_{\langle\rm H\rangle}$ are the densities of solid species per hydrogen nuclei density. Only solid species which change substantially with temperature are shown. Other stable solids are MgSiO3[s], SiO2[s], CaAl2Si2O8[s], NaAlSi3O8[s], KAlSi3O8[s], CaSO4[s], Fe2O3[s], TiO2[s] and Mn3Al2Si3O12[s], which have about constant concentrations as listed in Table <ref>.
The model presented in Sect. <ref> can be used to explore the sensitivity of the mineral composition to surface temperature. Figure <ref> shows the results of our GGchem model at the same constant pressure and total element abundances when varying the surface temperature between 450 K and 850 K. Venus is just about 15 K too warm to have pyrite (FeS2[s]) as a stable condensate on the surface according to this model. For surface temperatures lower than about 720 K, the formation of FeS2[s] would start to remove the SO2 from the atmosphere according to the following complex net reaction
\begin{equation}
\frac{15}{11}\,\ce{SO2}
\,+\, \frac{1}{11}\,\ce{Fe2O3[s]}
\,+\, \ce{CaAl2Si2O8[s]}
\frac{2}{11}\,\ce{FeS2[s]}
\,+\, \ce{CaSO4[s]}
\,+\, \ce{Al2O3[s]}
\,+\, 2\,\ce{SiO2[s]} \ ,
\end{equation}
which is a variant of Eq. (<ref>) where the oxygen
on the left side is provided by Fe2O3[s]. At temperatures below about 580 K in this model, the first carbonate magnesite (MgCO3[s]) becomes stable, which could initiate a dramatic change of the atmosphere as the main atmospheric molecule CO2 could deposit at the surface to form MgCO3[s]:
\begin{equation}
\ce{CO2} \,+\, \ce{MgSiO3[s]} ~\longrightarrow~
\ce{MgCO3[s]} \,+\, \ce{SiO2[s]} \ ,
\end{equation}
leaving an atmosphere that is dominated by N2 with more H2O. Finally, below a temperature of about 520 K, the first phyllosilicate, talc (Mg3Si4O12H2[s]) becomes stable, which could partly remove the water from the atmosphere. Only for temperatures below 520 K our model predicts the presence of hydrogen-containing minerals.
Whether or not this means that the gas at the bottom of the Venusian atmosphere is in fact in chemical equilibrium, and whether the element abundances in the gas are regulated by outgassing/deposition via the contact with the hot rock at the surface are yet unsolved questions. Disequilibrium processes might supply phyllosilicates or hydrated salts. For example, geological processes such as volcanism may resurface Venus's crust with hydrated components (see also discussion below). Our phase-equilibrium model suggests that hydrogenated rock and salts are not stable on the surface of Venus and will sublimate or react with the atmospheric gases to form other chemicals. However, the salts may be stable enough to be swept up to greater altitudes and cooler temperatures, before they react away and equilibrium is restored.
More research in the lab and in situ observations of the clouds of Venus will be needed to determine if salts are present and, if so, what their chemistry is.
Volcanic Delivery: A variant of the dust delivery mechanism to achieve a hydrogen flux to the clouds is volcanism. The presence of active volcanism on Venus has long been speculated upon, motivated both by the transient SO2 detected at 40 mbar by Pioneer Venus <cit.> and the recognition that SO2 may react with surface minerals and require continual replenishment [Fegley & Prinn, 1989].
Volcanism could deliver material to Venus's clouds in three ways: 1) as solid material deposited at the surface, which is subsequently lofted by winds; 2) as an explosive eruption introducing material into the below-cloud atmosphere, where vertical mixing slowly raises it into the cloud layer; and 3) as a large explosive eruption injecting material directly into the cloud layer.
Scenario (1) is effectively the `dust from the surface' mechanism described above. Albeit, by explicitly considering volcanism a source of juvenile OH-bearing material is introduced, which could help overcome the short surface lifetime of some OH-bearing phases. The second and third possibilities take advantage of the dynamics of volcanism to shorten the distance, and thereby potentially increase the flux, of mineral sources of OH to the clouds by reducing the time the material spends at high temperature near Venus's surface. Volcanism may also help loft material above the sluggish surface winds to an altitude where winds more readily carry dust higher <cit.>.
Modelling work by Glaze, 1999 and Airey et al., 2015 suggests that it is possible for volcanic eruption columns on Venus to reach the cloud base, however it requires particular circumstances that may or may not be frequently met: in particular, a high elevation vent and a magma containing several wt% water. Whether such water-rich magmas exist on Venus is unknown; taking Earth as an analogue, magmas with $> 3\,\mathrm{wt}\%$ water occur only where subduction is introducing surface water back into the mantle, a tectonic mode that cannot have prevailed on Venus for hundreds of millions of years, if ever. Voluminous water-rich explosive volcanism is also problematic given the tight constraints on the below-cloud the H2O mixing ratio of $<60\,\mathrm{ppm}$. At best therefore, these constraints would imply a highly stochastic delivery of volcanic material directly into the clouds.
The best case for a volcanic contribution to mineral buffering of Venus's cloud chemistry is therefore by enhancing background dust levels in the below-cloud atmosphere. Fegley & Prinn, 1989 estimate that a volcanic flux of $\sim1\,\mathrm{km^3\,yr^{-1}}$ is required to sustain atmospheric SO_2 at the levels observed. This translates to a mass flux of $\sim{3\times10^{12}\,\mathrm{kg\,yr^{-1}}}$ of magma, or $\sim6\times10^{11}\,\mathrm{mol\,yr^{-1}}$ of hydrated phases assuming terrestrial-levels of water in Venus's magmas. As an upper bound, this volcanic flux could provide a potential $4\times10^{-15}\,\mathrm{mol\,cm^{-2}\,s^{-1}}$ of hydrated phases to Venus's clouds if the entire mass was mobilised as dust in the atmosphere. Being below the $\sim{10^{-13}\,\mathrm{mol\,cm^{-2}\,s^{-1}}}$ of salt delivery estimated above, rates of volcanism either need to be (at least) an order of magnitude larger than assumed here, or the magmas correspondingly more volatile rich, for volcanism to be contributing to chemical buffering of Venus's clouds by water delivery. We note that although seeming unlikely, given all the current uncertainties on the composition and dynamics of Venus's interior, this possibility cannot be entirely ruled out.
§.§ Reconciling the Cloud Chemistry with Cloud Observations
There is considerable evidence that the clouds of Venus are mostly sulfuric acid or something very like sulfuric acid. The classic paper by Young, 1973 identifies most of the lines of evidence. The refractive index of the clouds obtained from infrared polarimetry, constrained to within 1.425 and 1.455 at the time, is best explained by droplets of $\sim 75\%$ sulfuric acid. In addition, the bottom of the cloud layer matches the condensation temperature of sulfuric acid, and specific spectral features between 8 – 13 $\mu$m are very similar to spectra of condensed sulfuric acid. Sulfuric acid is also expected at concentrations of $\sim 75\%$ based on models of H_2SO_4 and H_2O condensation [Krasnopolsky, 2015]. Subsequent studies have further refined the estimated concentrations of sulfuric acid in the clouds. Barstow et al., 2012 perform a retrieval on VIRTIS data of the atmosphere from Venus Express, and find that the 2.2:1.74 $\mu$m radiance ratio is sensitive only to the imaginary index of refraction, and therefore the sulfuric acid concentrations, and that most of the retrieved sulfuric acid concentrations in the lower clouds are between 85 – 96 wt%, or between 16 – 18 mol/L. Arney et al., 2014 used the same 2.2:1.74 $\mu$m radiance ratio and found that the sulfuric acid concentrations vary by time and latitude between 73 – 87 wt% in the upper clouds, or between 14 – 16 mol/L sulfuric acid concentration, lower than Barstow et al., 2012, suggesting that the concentration of sulfuric acid changes as a function of height. Titov et al., 2018 provide a comprehensive review of the research into Venus's clouds. None of the UV absorption due to our predicted profiles of gas-phase species, for the cloud model or any other model, explains the mysterious UV absorber, at $\lambda > 200$ nm. However, we do not consider the absorption properties of the aerosol particles themselves. It would be useful to determine the UV optical constants of these aerosols, or of sulfuric acid droplets themselves, to see whether a feasible candidate for the mysterious UV absorber is already in our midst.
Recent retrievals rely on infrared bands that constrain the index of refraction. If our proposed cloud chemistry is accurate and if all droplets have the same chemistry, hydrated sulfites and sulfates compose a large fraction of Venus's clouds. Sulfate achieves an index of refraction of $>1.44$ at $<20$ wt% H_2O [Cotterell et al., 2017], sufficient to explain the observed index of refraction. In addition, the S-H and S-O bonds are all similar, and so similar spectral features are expected. We predict a significant fraction of the clouds is made up of sulfites, in line with terrestrial SO_2 + H_2O chemistry [Terraglio & Manganelli, 1967]. The refractive index of sulfites is not as well known, though it has been measured for cyclic sulfites to be $\sim 1.5$ [Pritchard & Lauterbur, 1961], which is consistent with Venus cloud observations, particularly those that favor values higher than can be achieved by pure sulfuric acid [Markiewicz et al., 2018, Petrova, 2018], but may be achievable by sulfites and sulfates.
One other possibility is that the droplets of Venus do not all have the same chemistry. The salt content of the aerosols could be heterogeneous, either because they derive from different surface, volcanic or delivered materials, or because some cloud nuclei are produced photochemically, as sulfur allotropes and sulfuric acid. The observed bimodal distribution of cloud droplet sizes [Wilquet et al., 2009, Wilquet et al., 2012], is consistent with this hypothesis. It may be that the smaller more numerous cloud droplets lack salts, while less numerous droplets have salts. The salts will afford those droplets a far greater capacity for SO_2, and this may explain their larger size. If this is the case, we would predict the small mode droplets have a pH $< 0$ and the large mode droplets a pH $> 1$.
§.§ The Effect of Cloud Chemistry on Gas-Phase Halogens
Hydrogen halides have been observed on Venus, namely HCl and HF (see Appendix <ref>). Attempts have been made to observe HBr, but thus far only upper limits have been established [Krasnopolsky & Belyaev , 2016]. These hydrogen halides are likely to dissolve into the droplets, especially if the pH is higher than previously thought, and this may affect both the droplet chemistry and the atmospheric profiles of these halides. We investigate this possibility, considering only HCl, the sole hydrogen halide included in our model.
First, we have to determine the Henry's law constant for HCl dissolved into water/sulfuric acid mixtures. Williams & Golden, 1993 find that the effective Henry's Law of HCl in a solution of 50 wt% H2SO4 is:
\begin{equation}
H^* \approx 10^5 \, {\rm M/bar},
\end{equation}
at room temperature (which we will use as a proxy for the cloud layer). We use the relation (where $H^*$ (mol/(L bar)) is the effective Henry's law constant and $K_a$ is the acid dissociation constant):
\begin{equation}
H^* = H \Big(1 + \dfrac{K_a}{a_{H^+}}\Big),
\end{equation}
to solve for the hydrogen activity, $a_{H^+}$. The concentration of 50 wt% H2SO4 is approximately 2 mol/L, which will completely dissociate, and result in a pH of roughly -0.5, or an $a_{H^+} \approx 2$. For HCl, the $K_a \approx 2 \times 10^6$. This means that $H \approx 0.1$ for HCl in sulfuric acid.
We can now predict the depletion of HCl for the sulfuric acid case, where pH $\sim -3$. The equation for the capacity, $\kappa$, of the cloud droplets for a particular species, assuming that the dissociation products are not removed and so a simple equilibrium is achieved, is:
\begin{align}
\kappa &= \dfrac{H^* p(\ce{HCl}) N_A}{n(\ce{HCl})} \, \int 4 \pi r_d^2 \, \Bigg(r_d \dfrac{dn_d}{dr_d}\Bigg) \, dr_d\\
&= H \Bigg(1 + \dfrac{K_a}{a_{H^+}}\Bigg) \, kTN_A \, \int \dfrac{4}{3} \pi r_d^2 \, \Bigg(r_d \dfrac{dn_d}{dr_d}\Bigg) \, dr_d. \label{eqn:capacity}
\end{align}
Though it is true that, if there is any Na^+ or other cation, it will form a salt with Cl^-, this will act in solution as though entirely dissociated, and so effectively the amount of Cl- and H^+ due to HCl remains unchanged. If $\kappa < 1$, then the capacity of the droplets is not great enough to appreciably affect the composition of the gas-phase species. In the pure sulfuric acid case, $c \approx 1.6 \times 10^{-5}$, and so very little HCl will be in the droplets, compared to the amount in the gas-phase.
At higher pH, the situation changes, but still, at pH $\sim 1$, $\kappa \approx 0.16$, and so there should not be significant depletion. The value of $\kappa$ in Equation (<ref>) can be set to one, and $a_{H^+}$ solved for to predict the pH at which HCl should be significantly depleted, pH $\approx 1.8$. Though the profiles of gas-phase hydrogen halides should not not be affected at lower pH's, the concentrations of halogens in the droplets could be significant, and it would be informative to measure these concentrations in situ.
§.§ The Presence and Lifetime of Phosphine
A broad time-variable feature has been observed at 267 GHz by both ALMA and JCMT [Greaves et al., 2020, Greaves et al., 2020]. The existence and source of the feature has been disputed [Snellen et al., 2020, Thompson, 2020], though see the reply by Greaves et al., 2020. This feature has been attributed to phosphine [Greaves et al., 2020, Greaves et al., 2020], consistent with possible in situ detection of phosphine discovered during re-analysis of Venus Pioneer data [Mogul et al., 2021]. The updated ALMA data indicates that the 267 GHz feature is now consistent either with 1.5-7 ppb phosphine (PH_3) or $\sim 50$ ppb mesospheric SO_2 [Greaves et al., 2020, Lincowski et al., 2021], through this amount of SO_2 is not consistent with the non-detection of the 267.5 GHz SO_2 feature, from the same ALMA data, providing a 10 ppb upper limit [Greaves et al., 2020]. Careful modelling of different PH_3 profiles by Lincowski et al., 2021 demonstrate that the proposed in-cloud profile for PH_3 from Greaves et al., 2020 is insufficient to explain the observed feature. This leaves open the possibility that the signal is due to $\sim 50$ ppb mesospheric SO_2, with an anomalous velocity shift, or mesospheric PH_3 in quantities difficult to reconcile with the PH_3 lifetime, unless there is an unknown mesospheric source of PH_3. Further observations will be needed to determine which if either PH_3 or SO_2, or if some other unidentified molecule is the source of the 267 GHz feature.
Greaves et al., 2020 and Bains et al., 2020 estimated the lifetime and required flux of 20 ppb PH_3. The amount inferred from the ALMA observations has decreased to 1 ppb [Greaves et al., 2020], and we can apply our new model results to estimate that a flux of $10^7$ cm$^{-2}$ s$^{-1}$ PH_3 is required to explain PH_3 at these abundances, assuming the 267 GHz feature probed the clouds (this also includes effective dry deposition of PH_2 set to fix the abundance of PH_2 at the surface equal to zero, and dry deposition of PH_3 fixed to $10^{-4} {\rm \, cm \, s^{-1}}$). We also find that PH_3 is efficiently destroyed above 60 km, and the steep gradient in its profile is consistent with the non-detection above 61 km [Encrenaz et al., 2020]. The in-cloud PH_3 profile cannot explain the observed feature [Lincowski et al., 2021], but this profile is expected if there were a source of PH_3 in the clouds.
§.§ Implications for Hypothetical Life in the Clouds of Venus
The implications of this cloud chemistry on hypothetical Venusian life depends on how and in what form the sulfur is depleted. If surface, volcanic or exogenous delivery is responsible, this would explain the provision of various alkaline salts and/or other metals, essential for life as we know it. The higher pH is within the range where known acidophiles can thrive [Messerli et al., 2005]. However, although acidophilic and halophilic extremophiles exist on Earth, there are, as far as we know, no known extremophiles that are both acidophilic and halophilic enough to thrive in these conditions [Belilla et al., 2019].
If on the other hand the pH is being buffered within the clouds by the organisms themselves, either by use of phosphine or by burning hydrocarbons, sacrificing themselves so that others may live, then the available water will be produced and scavenged by means of the reaction. The remaining question is whether the biomass is sufficient to explain the sulfur depletion, and there is no reliable estimate of the biomass in the clouds of Venus. Early estimates based on tentative detections of phosphine have been made [Lingam & Loeb, 2020], but there is significant work left to better constrain the possibility that life is making use of the in-cloud SO_2.
§ CONCLUSION
In this paper, we discussed the puzzle of SO_2 depletion in the cloud layer. If the below-cloud observations of SO_2 and H_2O are correct, then there is too little H_2O to explain this depletion. We found that increasing the amount of below-cloud H_2O predicts chemistry above the clouds that does not agree with observations, but decreasing the below-cloud SO_2 results in above-cloud chemistry that generally agrees with observations. We also explored the possibility that hydrogen is delivered into the clouds in the form of aerosols, salts or metals, either from an exogenous source, from dust rising up from the surface, from volcanism, and from processes occurring within the clouds themselves. These processes buffer the pH of the clouds to values of 1-2. We discuss the implication of these predictions for observations of other trace gas-phase species and the optical properties of the clouds themselves.
Probes into the clouds of Venus will be necessary to determine what is happening within the clouds: the depletion of SO_2, the droplet chemistry (whether or not this chemistry has anything to do with the SO_2 depletion), the mysterious UV absorber, the known presence of heavy metals such as iron, the plausible presence of several reduced species in surprisingly large quantities [Greaves et al., 2020, Mogul et al., 2021]. In particular, the DAVINCI+ mission concept is planned to travel into and below the clouds to measure atmospheric redox and better constrain the chemical cycles that are thought to sustain the clouds [Garvin et al., 2020]. The cloud chemistry itself can be examined by including a design like the “JPL Venus Aerosol Mass Spectrometer Concept”, where a nebulizer is incorporated onto a mass spectrometer to separate gas and cloud particles and analyze the aerosol chemistry directly [Baines et al., 2018]. This would be the most straight-forward way to test this hypothesis, and other hypotheses, e.g. Krasnopolsky, 2017, that involve cloud chemistry.
Probes to the surface will be relevant for constraining the surface mineralogy and determining whether Venus's surface composition is in chemical equilibrium. These observations can be combined with a climate history of Venus, based on observations and models, to discover more about Venus's atmospheric evolution. Specifically, Figure <ref> can also be used to speculate about the possible cause for the origin of the thick Venusian atmosphere that we observe today. If Venus once was a cooler planet with a thinner N2 dominated atmosphere, just like Earth, but for some reason it warmed up to temperatures above 580 K, possibly due to large amounts of CO_2 released during global resurfacing [Strom et al., 1994], all carbonates in the surface rock would decompose and liberate even more CO2 into the atmosphere. Not only would this increase the greenhouse effect, but it would also make the Venusian atmosphere thicker. Both effects would have increased the surface temperature, leading to a run-away build-up of the thick CO2 Venusian atmosphere that we find today. Observations of surface minerals, possible with VERITAS [Smrekar et al., 2020], and EnVision [de Oliveira et al., 2018], would allow us to test the predictions of this and other hypotheses for the present state of Venus's atmosphere and climate, in particular, whether the surface and atmosphere of Venus are at thermochemical equilibrium.
To prepare for these missions, experiments are needed in Venus analogue environments to predict the chemistry that takes place, especially the largely unexplored chemistry that may take place within high concentrations of sulfuric acid, and the surface chemistry that may take place on efflorescent sulfates. In the meantime, a detailed cloud model of the form published by Gao et al., 2014, but that incorporates this chemistry, would be of value in order to see if any remote predictions would distinguish between sulfuric acid/water droplets and sulfate/sulfite/water droplets. It may be possible to falsify the cloud chemistry hypothesis based on a combination of cloud formation and radiative transfer models and observations. It is unlikely the puzzles addressed in our paper are likely to be resolved without returning to the clouds of Venus. If other missions to other planets are any indication, what we find will be entirely unexpected.
The authors thank Joanna Petkowska-Hankel for the preparation of Fig. <ref>. P. B. R. thanks the Simons Foundation for funding (SCOL awards 599634). P. W. acknowledges funding from the European Union H2020-MSCA-ITN-2019 under Grant Agreement no. 860470 (CHAMELEON). We thank David Grinspoon, Stephen Mojzsis and Kevin Zahnle for helpful discussions, Alex T. Archibald for his advice for improving the Earth-relevant reactions for our network, and the entire team involved with Greaves et al., 2020 for initiating this investigation into the atmospheric chemistry of Venus. We also thank the two anonymous referees for helpful and insightful comments that helped improve the quality of this manuscript. A. P. thanks Sami Mikhail for providing Venus and Venus II books and Christiane Helling for the support.
Argo [Rimmer & Helling, 2016], GGchem [Woitke et al., 2018]
§ OBSERVATIONAL CONSTRAINTS ON THE ATMOSPHERIC COMPOSITION OF VENUS
The species we included in our network robustly, and that are also observationally constrained in Venus's atmosphere are carbon dioxide (CO_2), molecular nitrogen (N_2), sulfur dioxide (SO_2), water vapor (H_2O), carbon monoxide (CO), molecular oxygen (O_2), carbonyl sulfide (OCS), sulfuric acid vapor (H_2SO_4), hydrogen chloride (HCl), sulfur monoxide (SO), trisulfur (S_3), tetrasulfur (S_4), hydrogen sulfide (H_2S) and molecular hydrogen (H_2). Phosphine (PH_3) is also included in our network but not in a robust way. PH_3 may have been observed in the atmosphere of Venus (see Section <ref> for details). There are also many remote observations that constrain cloud properties such as average particle size and indirectly infer that the clouds are made of droplets of high concentration sulfuric acid (See Section <ref>). To this date no definitive in situ measurements of the cloud droplet chemistry have been made, and so virtually nothing is directly known about the cloud droplet chemistry. Observations have been made by a variety of instruments on the ground, by orbital probes, and by in situ probes. We compiled this data ourselves from a variety of sources, and a more complete compilation has been made by Johnson & de Oliveira, 2019, which includes several reactive species not incorporated into our network, such as HF, as well as unreactive species. Our compilation is given in Table <ref>.
Observational Constraints of Chemical Species in the Atmosphere of Venus
Species $h_{\rm min}$
$h_{\rm max}$ Mixing Ratio$^*$
Reference Obs Type
(km) (km)
SO_2 ppm
30.0 40.0 $130. \pm 50. $ Marcq et al., 2008 Venus Express
0. 22.0 $130. \pm 35. $ Gelman et al., 1979 Venera 12
22 22 $185. \pm 43. $ Oyama et al., 1979 Pioneer Venus
35 45 $130. \pm 40. $ Bézard & de Bergh, 2007 Ground
12 12 $22.5 \pm 2.5 $ Bertaux et al., 1996 Vega 1, Vega 2
22 22 $38.0 \pm 3.8 $ Bertaux et al., 1996 Vega 1, Vega 2
42 42 $132.5 \pm 14. $ Bertaux et al., 1996 Vega 1, Vega 2
52 52 $107.5 \pm 42.5 $ Bertaux et al., 1996 Vega 1, Vega 2
35 45 $130. \pm 40. $ Bézard & de Bergh, 2007 Ground
42 42 $180. \pm 70. $ Pollack et al., 1993 Ground
62 62 $0.5 \pm 0.1 $ Zasova et al., 1993 Ground
62 62 $1.5 \pm 0.5 $ Zasova et al., 1993 Ground
22 22 $38. \pm 10. $ Bertaux et al., 1996 Vega 1
12 12 $25. \pm 10. $ Bertaux et al., 1996 Vega 1
52 52 $150. \pm 70. $ Bertaux et al., 1996 Vega 1
52 52 $65. \pm 30. $ Bertaux et al., 1996 Vega 2
42 42 $125. \pm 50. $ Bertaux et al., 1996 Vega 1
42 42 $200. \pm 100. $ Bertaux et al., 1996 Vega 2
42 42 $< 176. $ Oyama et al., 1980 Pioneer Venus
22 22 $185 \pm 43.1$
Oyama et al., 1980 Pioneer Venus
42 42 $324 \pm 148$ Oyama et al., 1980 Pioneer Venus
52 52 $< 600. $ Oyama et al., 1980 Pioneer Venus
22 22 $< 300. $ Hoffman et al., 1980 Pioneer Venus
52 52 $< 10. $ Hoffman et al., 1980 Pioneer Venus
30 40 $130. \pm 50. $ Marcq et al., 2008 Venus Express
88 88 $0.012 \pm 0.003 $ Encrenaz et al., 2012 Ground
70 70 $0.075 \pm 0.025 $ Encrenaz et al., 2012 Ground
70 70 $0.125 \pm 0.050 $ Na et al., 1990 Ground
45 55 $140. \pm 37. $ Arney et al., 2014 Ground
45 55 $126. \pm 32. $ Arney et al., 2014 Ground
65 75 $0.38 \pm 0.07 $ Na et al., 1990 IUE
65 75 $0.05 \pm 0.02 $ Na et al., 1990 IUE
75 85 $50.0 \pm 10 $ Greaves et al., 2020 Ground$^{**}$
SO_2 ppm
H_2O ppm
30 40 $31 \pm 2 $ Marcq et al., 2008 Venus Express
30 40 $30 \pm 10 $ de Bergh et al., 1995 Venus Express
5 40 $30 \pm 10 $ Bézard & de Bergh, 2007 Venus Express
51.3 51.3 $6.3 \pm 4. $ Donahue et al., 1997 Pioneer Venus
55.3 55.3 $4.2 \pm 3. $ Donahue et al., 1997 Pioneer Venus
69.5 69.5 $3. \pm 1. $ Cottini et al., 2012 Venus Express
15 30 $10. \pm 1. $ Evans & Ingalls, 1969 Ground
49 58 $700. \pm 300. $ Mukhin et al., 1982 Venera 13, 14
46 50 $2000.\pm 400. $ Surkov et al., 1982 Venera 13, 14
26 45 $30. \pm 10. $ de Bergh et al., 1995 Ground
15 30 $30. \pm 10. $ de Bergh et al., 1995 Ground
0 15 $30. \pm 10. $ de Bergh et al., 1995 Ground
10 10 $45. \pm 10. $ Meadows & Crisp, 1996 Ground
0 10 $7. \pm 3. $ Donahue & Hodges, 1992 Pioneer Venus
10 26 $28. \pm 18. $ Donahue & Hodges, 1992 Pioneer Venus
58 60 $20. \pm 10. $ Moroz et al., 1990 Ground
52 60 $1000.\pm 500. $ Surkov et al., 1987 Vega 1, Vega 2
42 42 $150. \pm 50. $ Moroz et al., 1979 Venera 11, 12
22 22 $60. \pm 30. $ Moroz et al., 1979 Venera 11, 12
35 45 $40. \pm 20. $ Bézard et al., 1990 Ground
10 40 $30. \pm 10. $ Pollack et al., 1993 Ground
15 25 $30. \pm 10. $ de Bergh et al., 1995 Ground
0 15 $30. \pm 15. $ de Bergh et al., 1995 Ground
0 0 $20. \pm 10. $ Moroz et al., 1979 Venera 11, 12
30 40 $26. \pm 4. $ Marcq et al., 2006 Ground
30 40 $35. \pm 4. $ Tsang et al., 2008 Venus Express
30 40 $30. \pm 4. $ Tsang et al., 2008 Venus Express
30 40 $34. \pm 2. $ Arney et al., 2014 Ground
30 40 $33. \pm 3. $ Arney et al., 2014 Ground
15 30 $33. \pm 2. $ Arney et al., 2014 Ground
15 30 $32. \pm 2. $ Arney et al., 2014 Ground
0 15 $44. \pm 9. $ Bézard & de Bergh, 2007 Venus Express
0 15 $30. \pm 10. $ Bézard et al., 2011 Venus Express
0 15 $31. \pm 9. $ Chamberlain et al., 2013 AAT
0 15 $29. \pm 2. $ Arney et al., 2014 Ground
0 15 $27. \pm 2. $ Arney et al., 2014 Ground
0 15 $25.7 \pm 1.4 $ Fedorova et al., 2015 Venus Express
0 15 $29.4 \pm 1.6 $ Fedorova et al., 2015 Venus Express
65 74 $6.0 \pm 4.0 $ Fedorova et al., 2016 Pioneer Venus
70 110 $1.0 \pm 0.9 $ Fedorova et al., 2008 Venus Express
45 55 $20.0 \pm 10. $ Meadows & Crisp, 1996 Ground
0 15 $45. \pm 15. $ Meadows & Crisp, 1996 Ground
65 100 $1.8 \pm 1.8 $ Sandor & Clancy, 2005 Ground
65 120 $2.5 \pm 0.6 $ Encrenaz et al., 2015 Ground
65 75 $3.0 \pm 1.0 $ Cottini et al., 2012 Venus Express
65 75 $5.0 \pm 2.0 $ Cottini et al., 2012 Venus Express
65 65 $0.8 $ Bell et al., 1991 Ground
55 55 $100.0 $ Bell et al., 1991 Ground
40 40 $200.0 $ Bell et al., 1991 Ground
H_2O ppm
CO ppm
36 36 $27.5 \pm 3.5 $ Marcq et al., 2008 Venus Express
35 35 $23.0 \pm 2.0 $ Tsang et al., 2008 Venus Express
35 35 $32.0 \pm 2.0 $ Tsang et al., 2008 Venus Express
22 22 $20.0 \pm 0.4 $ Oyama et al., 1979 Pioneer Venus
64 64 $45.0 \pm 10.0 $ Fegley, 2014 Venera 13, 14
0 0 $3.8 \pm 3.2 $ Fegley, 2014 Venera 13, 14
52 52 $32.0 \pm 22.0 $ Oyama et al., 1980 Pioneer Venus
42 42 $30.0 \pm 18.0 $ Oyama et al., 1980 Pioneer Venus
22 22 $20.0 \pm 3.0 $ Oyama et al., 1980 Pioneer Venus
0 42 $28.0 \pm 7.0 $ Gelman et al., 1979 Venera 12
64. 64. $45.0 \pm 10.0 $ Connes et al., 1968 Ground
64. 64. $51.0 \pm 1.0 $ Young, 1972 Ground
90 90 $180. \pm 90.0 $ Wilson et al., 1981 Ground
36 36 $23.0 \pm 5.0 $ Pollack et al., 1993 Ground
40 40 $29.0 \pm 7.0 $ Pollack et al., 1993 Ground
35 45 $45.0 \pm 10.0 $ Bézard et al., 1990 Ground
36 36 $23.0 \pm 10.0 $ Pollack et al., 1993 Ground
28 28 $23.0 \pm 10.0 $ Bézard & de Bergh, 2007 Ground
42 42 $30.0 \pm 15.0 $ Bézard & de Bergh, 2007 Ground
36 36 $24.0 \pm 2.0 $ Marcq et al., 2006 Ground
36 36 $27.0 \pm 3.0 $ Cotton et al., 2012 Ground
68 71 $70.0 \pm 8.0 $ Krasnopolsky, 2008 Ground
68 68 $51.0 \pm 4.0 $ Krasnopolsky, 2010 Ground
68 68 $40.0 \pm 4.0 $ Krasnopolsky, 2010 Ground
104 104 $560.0 \pm 100.0 $ Krasnopolsky, 2014 Ground
70 70 $35.0 \pm 10.0 $ Marcq et al., 2015 Ground
30 30 $30.0 \pm 5.0 $ Collard et al., 1993 Galileo
30 30 $40.0 \pm 5.0 $ Collard et al., 1993 Galileo
35 35 $30.0 \pm 15.0 $ Marcq et al., 2005 Ground
104 104 $560.0 \pm 100.0 $ Krasnopolsky, 2014 Ground
65 75 $70.0 \pm 10.0 $ Grassi et al., 2014 Venus Express
65 75 $60.0 \pm 5.0 $ Grassi et al., 2014 Venus Express
CO ppm
O_2 ppm
52 52 $43. \pm 25. $ Oyama et al., 1979 Pioneer Venus
42 42 $16. \pm 8. $ Oyama et al., 1979 Pioneer Venus
52 52 $44. \pm 25. $ Oyama et al., 1980 Pioneer Venus
42 42 $16. \pm 7. $ Oyama et al., 1980 Pioneer Venus
35 58 $18. \pm 4. $ Mukhin et al., 1982 Venera 13, 14
60 100 $<2.8$ Marcq et al., 2018 Ground
O_2 ppm
OCS ppm
30 30 $14.0 \pm 6.0 $ Pollack et al., 1993 Ground
33 33 $3.25 \pm 0.75 $ Marcq et al., 2008 Venus Express
29 37 $40.0 \pm 20.0 $ Mukhin et al., 1982 Venera 13, 14
33 33 $4.4 \pm 1.0 $ Pollack et al., 1993 Ground
28 28 $30.0 \pm 10.0 $ Pollack et al., 1993 Ground
38 38 $0.35 \pm 0.1 $ Marcq et al., 2005 Ground
0 30 $15.0 \pm 5.0 $ Bézard & de Bergh, 2007 Ground
30 30 $14.0 \pm 6.0 $ Pollack et al., 1993 Ground
29 37 $40.0 \pm 20.0 $ Mukhin et al., 1983 Venera 13, 14
36 36 $0.52 \pm 0.05 $ Marcq et al., 2006 Ground
30 30 $16.0 \pm 8.0 $ Bézard & de Bergh, 2007 Ground
38 38 $0.35 \pm 0.1 $ Bézard & de Bergh, 2007 Ground
36 36 $0.44 \pm 0.1 $ Arney et al., 2014 Ground
36 36 $0.57 \pm 0.12 $ Arney et al., 2014 Ground
36 36 $0.5 \pm 0.02 $ Marcq et al., 2005 Ground
36 36 $0.46 \pm 0.01 $ Marcq et al., 2005 Ground
36 36 $0.54 \pm 0.13 $ Arney et al., 2014 Ground
36 36 $0.61 \pm 0.12 $ Arney et al., 2014 Ground
65 65 $<0.004 $ Krasnopolsky, 2010 Ground
65 65 $0.005 \pm 0.003 $ Krasnopolsky, 2010 Ground
OCS ppm
H_2SO_4 ppm
50 52 $3.0 \pm 2.0$ Oschlisniok et al., 2012 Venus Express
H_2SO_4 ppm
HCl ppm
15 25 $0.41 \pm 0.04 $ Arney et al., 2014 Ground
15 25 $0.42 \pm 0.055 $ Arney et al., 2014 Ground
74 74 $0.4 \pm 0.03 $ Krasnopolsky, 2010 Ground
61 67 $0.76 \pm 0.1 $ Iwagami et al., 2008 Ground
74 74 $0.4 \pm 0.03 $ Krasnopolsky, 2010 Ground
70 70 $0.4 \pm 0.04 $ Sandor & Clancy, 2012 Ground
90 90 $0.2 \pm 0.02 $ Sandor & Clancy, 2012 Ground
35 70 $0.5 \pm 0.12 $ Connes et al., 1967 Ground
74 74 $0.4 \pm 0.04 $ Krasnopolsky, 2010 Ground
36 36 $0.42 \pm 0.07 $ Young, 1972 Ground
50 50 $0.61 \pm 0.06 $ Young, 1972 Ground
30 40 $0.005 \pm 0.002 $ Bézard et al., 1990 Ground
70 70 $0.4 \pm 0.04 $ Bézard et al., 1990 Ground
15 30 $0.5 \pm 0.05 $ Bézard et al., 1990 Ground
45 55 $0.5 \pm 0.05 $ Iwagami et al., 2008 Ground
65 75 $0.6 \pm 0.1 $ Connes et al., 1967 Ground
HCl ppm
PH_3 ppb
55.0 60.0 $20.0 \pm 10.0$ Greaves et al., 2020 Ground$^{\dagger}$
55.0 60.0 $1.5 \pm 1.0$ Greaves et al., 2020 Ground$^{\dagger}$
60.0 65.0 $< 5.0$ Encrenaz et al., 2020
PH_3 ppb
SO ppb
84.0 90.0 $ 8.0 \pm 2.0 $ Encrenaz et al., 2015 Ground
84.0 90.0 $ 8.0 \pm 2.0 $ Encrenaz et al., 2015 Ground
75.0 85.0 $ 5.0 \pm 4.0 $ Na et al., 1990 Ground
SO ppb
S_3 ppb
23.0 23.0 $0.04 \pm 0.01 $ Bézard & de Bergh, 2007 Venera 11-14
3.0 19.0 $0.065 \pm 0.035 $ Maiorov et al., 2005 Venera 11
3.0 19.0 $11. \pm 3.0 $ Krasnopolsky, 2013 Venera 11
10.0 19.0 $18. \pm 3.0 $ Krasnopolsky, 2013 Venera 11
S_3 ppb
S_4 ppb
3.0 10.0 $4. \pm 3.9 $ Krasnopolsky, 2013 Venera 11
10.0 19.0 $6. \pm 2.0 $ Krasnopolsky, 2013 Venera 11
S_4 ppb
H_2S ppb
70.0 70.0 $<13.0$ Krasnopolsky, 2008 Ground
H_2S ppb
H_2 ppm
0.0 25.0 $<10.$ Donahue et al., 1997 Pioneer Venus
22.0 64.0 $<10.$ Oyama et al., 1980 Pioneer Venus
49.0 58.0 $25. \pm 10$ Mukhin et al., 1982 Venera 13,14
H_2 ppm
$^*$ Values and errors as found in the cited literature.
$^{**}$ If interpreted as mesospheric SO_2, see Section <ref>
$^{\dagger}$ If interpreted as mesospheric PH_3, see Section <ref>
$^{\ddagger}$ Based on their wet profile, which fits their data best.
§ THE CHEMICAL NETWORK: STAND2020
[Ackerman & Marley, 2001]
Ackerman, A. S., & Marley, M. S. 2001, , 556, 872,
[Ackerman et al., 1970]
Ackerman, M., Biaumé, F., & Kockarts, G. 1970, , 18, 1639,
[Airey et al., 2015]
Airey, M. W., Mather, T., Pyle, D., et al. 2015, Planetary and Space Science,
113, 33
[Amano et al., 1983]
Amano, A., Yamada, M., Hashimoto, K., & Sugiura, K. 1983, Nippon Kagaku
Kaishi, 385
[Ambrose, 1956]
Ambrose, D. 1956, Transactions of the Faraday Society, 52, 772
[Ambrose & Townsend, 1964]
Ambrose, D., & Townsend, R. 1964, Transactions of the Faraday Society, 60,
[Andersson et al., 2003]
Andersson, S., Marković, N., & Nyman, G. 2003, Journal of Physical
Chemistry A, 107, 5439, 10.1021/jp0222604
[Anglada et al., 2017]
Anglada, J. M., Crehuet, R., Martins-Costa, M., Francisco, J. S., &
Ruiz-López, M. 2017, Physical Chemistry Chemical Physics (Incorporating
Faraday Transactions), 19, 12331, 10.1039/C7CP01976A
[Archer et al., 2013]
Archer, L., Stark, G., Smith, P., et al. 2013, Journal of Quantitative
Spectroscopy and Radiative Transfer, 117, 88
[Arney et al., 2014]
Arney, G., Meadows, V., Crisp, D., et al. 2014, Journal of Geophysical
Research (Planets), 119, 1860, 10.1002/2014JE004662
[Arthur & Cooper, 1997]
Arthur, N. L., & Cooper, I. A. 1997, Journal of the Chemical Society, Faraday
Transactions, 93, 521
[Ashmore & Burnett, 1962]
Ashmore, P., & Burnett, M. 1962, Transactions of the Faraday Society, 58, 253
[Aston et al., 1937]
Aston, J., Siller, C., & Messerly, G. 1937, Journal of the American Chemical
Society, 59, 1743
[Atkinson et al., 1989]
Atkinson, R., Baulch, D. L., Cox, R. A., et al. 1989, Journal of
Physical and Chemical Reference Data, 18, 881, 10.1063/1.555832
[Atkinson et al., 2004]
—. 2004, Atmospheric Chemistry & Physics, 4, 1461
[Back & Griffiths, 1967]
Back, R. A., & Griffiths, D. W. L. 1967, , 46, 4839,
[Baines et al., 2018]
Baines, K. H., Cutts, J. A., Nikolić, D.,
Madzunkov, S. M., Delitsky, M. L., Limaye, S. S.,
& McGouldrick, K. 2018, The Venus Aerosol Mass Spectrometer Concept. In 16th Meeting of the Venus Exploration and Analysis Group (VEXAG), 16, 8031
[Bains et al., 2020]
Bains, W., Petkowski, J. J., Seager, S., et al. 2020, Accepted for Publication in Astrobiology, arXiv e-prints, arXiv:2009.06499.
[Barfield et al., 1972]
Barfield, W., Koontz, G. D., & Huebner, W. F. 1972, , 12, 1409,
[Barklem & Collet, 2016]
Barklem, P. S., & Collet, R. 2016, , 588, A96,
[Barnes & Simpson, 1963]
Barnes, E. E., & Simpson, W. T. 1963, , 39, 670,
[Barstow et al., 2012]
Barstow, J. K., Tsang, C. C. C., Wilson, C. F., et al. 2012, ,
217, 542, 10.1016/j.icarus.2011.05.018
[Barsuhn & Nesbet, 1978]
Barsuhn, J., & Nesbet, R. K. 1978, , 68, 2783,
[Barth et al., 2020]
Barth, P., Helling, C., Stüeken, E., et al. 2020, , submitted
[Baulch et al., 1981]
Baulch, D., Duxbury, J., Grant, S., & Montague, D. 1981, Evaluated kinetic
data for high temperature reactions. Volume 4. Homogeneous gas phase
reactions of halogen-and cyanide-containing species, Tech. rep., National
Standard Reference Data System
[Baulch et al., 1992]
Baulch, D. L., Cobos, C. J., Cox, R. A., et al. 1992, Journal of
Physical and Chemical Reference Data, 21, 411, 10.1063/1.555908
[Baulch et al., 1994]
—. 1994, Journal of Physical and Chemical Reference Data, 23, 847,
[Baulch et al., 2005]
Baulch, D. L., Bowman, C. T., Cobos, C. J., et al. 2005, Journal of
Physical and Chemical Reference Data, 34, 757, 10.1063/1.1748524
[Becker et al., 1991]
Becker, E., Wille, U., Rahman, M., & Schindler, R. 1991, Berichte der
Bunsengesellschaft für physikalische Chemie, 95, 1173
[Belilla et al., 2019]
Belilla, J., Moreira, D., Jardillier, L., Reboul, G., Benzerara, K., López-García, J. M., Bertolino, P., López-Archilla, A. I., & López-García, P., 2019, Nature ecology & evolution, 3, 1552
[Bell et al., 1991]
Bell, J. F., Crisp, D., Lucey, P. G., Ozoroski, T. A., Sinton, W. M., Willis, S. C., & Campbell, B. A. 1991, Science, 252, 1293
[Belyaev et al., 2012]
Belyaev, D. A., Montmessin, F., Bertaux, J.-L., et al. 2012, ,
217, 740, 10.1016/j.icarus.2011.09.025
[Benson, 1989]
Benson, S. W. 1989, International journal of chemical kinetics, 21, 233
[Bertaux et al., 1996]
Bertaux, J.-L., Widemann, T., Hauchecorne, A., Moroz, V. I., &
Ekonomov, A. P. 1996, , 101, 12709, 10.1029/96JE00466
[Bethe & Salpeter, 1957]
Bethe, H., & Salpeter, E. 1957, Quantum mechanics of one-and two-electron
atoms, Springer-Verlag, Berlin
[Bézard & de Bergh, 2007]
Bézard, B., & de Bergh, C. 2007, Journal of Geophysical Research
(Planets), 112, E04S07, 10.1029/2006JE002794
[Bézard et al., 1990]
Bézard, B., de Bergh, C., Crisp, D., & Maillard, J. P. 1990, ,
345, 508, 10.1038/345508a0
[Bézard et al., 2011]
Bézard, B., Fedorova, A., Bertaux, J.-L., Rodin, A., & Korablev,
O. 2011, , 216, 173, 10.1016/j.icarus.2011.08.025
[Bierson & Zhang, 2020]
Bierson, C., & Zhang, X. 2020, Journal of Geophysical Research: Planets, 125,
[Black et al., 1975]
Black, G., Sharpless, R. L., Slanger, T. G., & Lorents, D. C. 1975,
, 62, 4274, 10.1063/1.430348
[Blitz et al., 2006]
Blitz, M. A., Hughes, K. J., Pilling, M. J., & Robertson, S. H. 2006,
Journal of Physical Chemistry A, 110, 2996, 10.1021/jp054722u
[Bogdanchikov et al., 2004]
Bogdanchikov, G., Baklanov, A., & Parker, D. 2004, Chemical Physics Letters,
385, 486 , https://doi.org/10.1016/j.cplett.2004.01.015
[Bogumil et al., 2003]
Bogumil, K., Orphal, J., Homann, T., et al. 2003, Journal of Photochemistry
and Photobiology A: Chemistry, 157, 167
[Bozzelli & Dean, 1989]
Bozzelli, J. W., & Dean, A. M. 1989, The Journal of Physical Chemistry, 93,
[Bozzelli & Dean, 1995]
—. 1995, International journal of chemical kinetics, 27, 1097
[Brandt & Van Eldik, 1995]
Brandt, C., & Van Eldik, R. 1995, Chemical Reviews, 95, 119
[Brion et al., 1979]
Brion, C., Tan, K., Van der Wiel, M., & Van der Leeuw, P. E. 1979, Journal of
Electron Spectroscopy and Related Phenomena, 17, 101
[Broad & Reinhardt, 1976]
Broad, J. T., & Reinhardt, W. P. 1976, , 14, 2159,
[Brolley et al., 1973]
Brolley, J. E., Porter, L. E., Sherman, R. H., Theobald, J. K., &
Fong, J. C. 1973, , 78, 1627, 10.1029/JA078i010p01627
[Brown, 1973]
Brown, R. L. 1973, International Journal of Chemical Kinetics, 5, 663
[Browning & Fryar, 1973]
Browning, R., & Fryar, J. 1973, Journal of Physics B Atomic Molecular
Physics, 6, 364, 10.1088/0022-3700/6/2/019
[Brudnik et al., 2009]
Brudnik, K., Gola, A. A., & Jodkowski, J. T. 2009, Journal of molecular
modeling, 15, 1061
[Bulatov et al., 1990]
Bulatov, V., Vereshchuk, S., Dzegilenko, F., Sarkisov, O., & Khabarov, V.
1990, Khimicheskaya Fizika, 9, 1214
[Burkholder et al., 2020]
Burkholder, J., Sander, S., Abbatt, J., et al. 2020, Chemical kinetics and
photochemical data for use in atmospheric studies; evaluation number 19,
Tech. rep., Pasadena, CA: Jet Propulsion Laboratory, National Aeronautics and
Space …
[Burkholder & McKeen, 1997]
Burkholder, J. B., & McKeen, S. 1997, , 24, 3201,
[Cairns & Samson, 1965]
Cairns, R. B., & Samson, J. A. R. 1965, , 70, 99,
[Campbell & Gray, 1973]
Campbell, I. M., & Gray, C. N. 1973, Chemical Physics Letters, 18, 607,
[Cardelino et al., 2003]
Cardelino, B. H., Moore, C. E., Cardelino, C. A., et al. 2003, Journal
of Physical Chemistry A, 107, 3708, 10.1021/jp026289j
[Caridade et al., 2005]
Caridade, P. J. S. B., Rodrigues, S. P. J., Sousa, F., & Varandas,
A. J. C. 2005, Journal of Physical Chemistry A, 109, 2356,
[Carl et al., 2005]
Carl, S. A., Minh Thi Nguyen, H., Elsamra, R. M. I., Tho Nguyen, M., &
Peeters, J. 2005, , 122, 114307, 10.1063/1.1861887
[Carnovale et al., 1982]
Carnovale, F., Hitchcock, A. P., Cook, J. P. D., & Brion, C. E. 1982,
Chemical Physics, 66, 249, 10.1016/0301-0104(82)88025-7
[Chamberlain et al., 2013]
Chamberlain, S., Bailey, J., Crisp, D., & Meadows, V. 2013, ,
222, 364, 10.1016/j.icarus.2012.11.014
[Chase, 1998]
Chase, Jr., M. 1998, Journal of Physical and Chemical Reference Data, Monograph
[Chase, 1986]
Chase, M. W. 1986, JANAF thermochemical tables (American Chemical Society;
New York)
[Chase et al., 1982]
Chase, M. W., Curnutt, J. L., Downey, J. R., et al. 1982, in Journal of
Physics Conference Series, Vol. 11, Journal of Physics Conference Series,
[Chen et al., 1991]
Chen, F., Judge, D. L., Wu, C. Y. R., et al. 1991, , 96, 17519,
[Chen & Wu, 2004]
Chen, F. Z., & Wu, C. Y. R. 2004, , 85, 195,
[Chen et al., 2002]
Chen, Y., Zhu, L., & Francisco, J. S. 2002, Journal of Physical
Chemistry A, 106, 7755, 10.1021/jp014544e
[Cheng et al., 2006]
Cheng, B.-M., Lu, H.-C., Chen, H.-K., et al. 2006, , 647, 1535,
[Choi & Lin, 2005]
Choi, Y., & Lin, M.-C. 2005, International journal of chemical kinetics, 37,
[Chung et al., 1975]
Chung, K., Calvert, J. G., & Bottenheim, J. W. 1975, International Journal of
Chemical Kinetics, 7, 161
[Clark et al., 1978]
Clark, J. H., Moore, C. B., & Nogar, N. S. 1978, , 68, 1264,
[Clyne & Townsend, 1975]
Clyne, M., & Townsend, L. 1975, Int. J. Chem. Kinet.;(United States), 7
[Clyne et al., 1984]
Clyne, M. A., MacRobert, A. J., Murrells, T. P., & Stief, L. J. 1984, Journal
of the Chemical Society, Faraday Transactions 2: Molecular and Chemical
Physics, 80, 877
[Clyne & Watson, 1974]
Clyne, M. A., & Watson, R. T. 1974, Journal of the Chemical Society, Faraday
Transactions 1: Physical Chemistry in Condensed Phases, 70, 2250
[Cobos et al., 1985]
Cobos, C., Hippler, H., & Troe, J. 1985, The Journal of Physical Chemistry,
89, 342
[Coddington et al., 2015]
Coddington, O., Lean, J., Lindholm, D., Pilewskie, P., & Snow, M. 2015, NOAA
National Centers for Environmental Information, 10.7289/V51J97P6
[Cohen & Westberg, 1991]
Cohen, N., & Westberg, K. R. 1991, Journal of Physical and Chemical
Reference Data, 20, 1211, 10.1063/1.555901
[Collard et al., 1993]
Collard, A. D., Taylor, F. W., Calcutt, S. B., et al. 1993, ,
41, 487, 10.1016/0032-0633(93)90031-V
[Connes et al., 1967]
Connes, P., Connes, J., Benedict, W. S., & Kaplan, L. D. 1967, ,
147, 1230, 10.1086/149124
[Connes et al., 1968]
Connes, P., Connes, J., Kaplan, L. D., & Benedict, W. S. 1968, ,
152, 731, 10.1086/149590
[Cook & Metzger, 1964]
Cook, G., & Metzger, P. 1964, The Journal of Chemical Physics, 41, 321
[Cook et al., 1965]
Cook, G., Metzger, P., & Ogawa, M. 1965, Canadian Journal of Physics, 43, 1706
[Cook & Metzger, 1964]
Cook, G. R., & Metzger, P. H. 1964, Journal of the Optical Society of
America (1917-1983), 54, 968
[Cook et al., 1981]
Cook, J.-E. L., Ennis, C. A., Leck, T. J., & Birks, J. W. 1981, ,
74, 545, 10.1063/1.440807
[Cooper & Hershberger, 1992]
Cooper, W. F., & Hershberger, J. F. 1992, The Journal of Physical Chemistry,
96, 5405
[Cotterell et al., 2017]
Cotterell, M. I., Willoughby, R. E., Bzdek, B. R., Orr-Ewing, A. J., & Reid,
J. P. 2017, Atmos. Chem. Phys, 17, 9837
[Cottini et al., 2012]
Cottini, V., Ignatiev, N. I., Piccioni, G., et al. 2012, , 217,
561, 10.1016/j.icarus.2011.06.018
[Cotton et al., 2012]
Cotton, D. V., Bailey, J., Crisp, D., & Meadows, V. S. 2012, ,
217, 570, 10.1016/j.icarus.2011.05.020
[Cox & Derwent, 1976]
Cox, R., & Derwent, R. 1976, Journal of Photochemistry, 6, 23
[Craven & Murrell, 1987]
Craven, W., & Murrell, J. N. 1987, Journal of the Chemical Society, Faraday
Transactions 2: Molecular and Chemical Physics, 83, 1733
[Dannhauser & Flueckinger, 1963]
Dannhauser, W., & Flueckinger, A. F. 1963, , 38, 69,
[de Bergh et al., 1995]
de Bergh, C., Bezard, B., Crisp, D., et al. 1995, Advances in Space
Research, 15, 79, 10.1016/0273-1177(94)00067-B
[DeMore et al., 1985]
DeMore, W. B., Leu, M.-T., Smith, R. H., & Yung, Y. L. 1985, ,
63, 347, 10.1016/0019-1035(85)90051-X
[de Oliveira et al., 2018]
de Oliveira, M. R., Gil, P. J., & Ghail, R. 2018, Acta Astronautica, 148, 260
[Ditchburn, 1955]
Ditchburn, R. W. 1955, Proceedings of the Royal Society of London Series A,
229, 44, 10.1098/rspa.1955.0073
[Dixon & Kirby, 1968]
Dixon, R., & Kirby, G. 1968, Transactions of the Faraday Society, 64, 2002
[Donahue et al., 1997]
Donahue, T. M., Grinspoon, D. H., Hartle, R. E., & Hodges, R. R., J.
1997, in Venus II: Geology, Geophysics, Atmosphere, and Solar Wind
Environment, ed. S. W. Bougher, D. M. Hunten, & R. J. Phillips, 385
[Donahue & Hodges, 1992]
Donahue, T. M., & Hodges, R. R., J. 1992, , 97, 6083,
[Dorthe et al., 1991]
Dorthe, G., Caubet, P., Vias, T., Barrere, B., & Marchais, J. 1991, The
Journal of Physical Chemistry, 95, 5109
[Du et al., 2008]
Du, S., Francisco, J. S., Shepler, B. C., & Peterson, K. A. 2008,
, 128, 204306, 10.1063/1.2919569
[Duran et al., 1989]
Duran, R., Amorebieta, V., & Colussi, A. 1989, International journal of
chemical kinetics, 21, 847
[Eibling & Kaufman, 1983]
Eibling, R. E., & Kaufman, M. 1983, Atmospheric Environment, 17, 429,
[Eigen et al., 1961]
Eigen, M., Kustin, K., & Maass, G. 1961, Zeitschrift für Physikalische
Chemie, 30, 130
[Eiteneer et al., 1998]
Eiteneer, B., Yu, C. L., Goldenberg, M., & Frenklach, M. 1998, Journal
of Physical Chemistry A, 102, 5196, 10.1021/jp981184v
[Encrenaz et al., 2012]
Encrenaz, T., Greathouse, T. K., Roe, H., et al. 2012, , 543, A153,
[Encrenaz et al., 2015]
Encrenaz, T., Moreno, R., Moullet, A., Lellouch, E., & Fouchet, T.
2015, , 113, 275, 10.1016/j.pss.2015.01.011
[Encrenaz et al., 2019]
Encrenaz, T., Greathouse, T. K., Marcq, E., et al. 2019, , 623,
A70, 10.1051/0004-6361/201833511
[Encrenaz et al., 2020]
—. 2020, , 643, L5, 10.1051/0004-6361/202039559
[Ennis & Birks, 1988]
Ennis, C., & Birks, J. 1988, Journal of physical chemistry (1952), 92, 1119
[Esposito, 1984]
Esposito, L. W. 1984, Science, 223, 1072
[Estupiñán et al., 2001]
Estupiñán, E. G., Nicovich, J. M., & Wine, P. H. 2001, Journal
of Physical Chemistry A, 105, 9697, 10.1021/jp011940o
[Evans & Ingalls, 1969]
Evans, J. V., & Ingalls, R. P. 1969, in The Venus Atmosphere, ed.
R. Jastrow & S. I. Rasool, 169
[Fairbairn, 1969]
Fairbairn, A. R. 1969, Proceedings of the Royal Society of London Series A,
312, 207, 10.1098/rspa.1969.0149
[Farahani et al., 2019]
Farahani, S., Frandsen, B. N., Kjaergaard, H. G., & Lane, J. R. 2019, The
Journal of Physical Chemistry A, 123, 6605
[Faravelli et al., 2000]
Faravelli, T., Goldaniga, A., Zappella, L., et al. 2000, Proceedings of the
Combustion Institute, 28, 2601
[Fedorova et al., 2015]
Fedorova, A., Bézard, B., Bertaux, J.-L., Korablev, O., &
Wilson, C. 2015, , 113, 66, 10.1016/j.pss.2014.08.010
[Fedorova et al., 2016]
Fedorova, A., Marcq, E., Luginin, M., et al. 2016, , 275, 143,
[Fedorova et al., 2008]
Fedorova, A., Korablev, O., Vandaele, A. C., et al. 2008, Journal of
Geophysical Research (Planets), 113, E00B22, 10.1029/2008JE003146
[Fegley & Prinn, 1989]
Fegley, B., & Prinn, R. G. 1989, Nature, 337, 55
[Fegley, 2014]
Fegley, B., J. 2014, Venus, Vol. 2 (Elsevier), 127–148
[Fernandes et al., 2006]
Fernandes, R. X., Luther, K., & Troe, J. 2006, Journal of Physical
Chemistry A, 110, 4442, 10.1021/jp056850o
[Ferradaz et al., 2009]
Ferradaz, T., Bénilan, Y., Fray, N., et al. 2009, , 57, 10,
[Freeman et al., 1984]
Freeman, D. E., Yoshino, K., Esmond, J. R., & Parkinson, W. H. 1984,
, 32, 1125, 10.1016/0032-0633(84)90139-9
[Fritz et al., 1982]
Fritz, B., Lorenz, K., Steinert, W., & Zellner, R. 1982, Phys. Chem. Behav.
Atmos. Pollut. Proc. Eur. Symp.
[Fujii et al., 1985]
Fujii, N., Kakuda, T., Sugiyama, T., & Miyama, H. 1985, Chemical
Physics Letters, 122, 489, 10.1016/0009-2614(85)87251-1
[Fulle et al., 1996]
Fulle, D., Hamann, H. F., Hippler, H., & Troe, J. 1996, , 105,
1001, 10.1063/1.471944
[Fulle et al., 1998]
—. 1998, , 108, 5391, 10.1063/1.475971
[Gao et al., 2014]
Gao, P., Zhang, X., Crisp, D., Bardeen, C. G., & Yung, Y. L. 2014,
, 231, 83, 10.1016/j.icarus.2013.10.013
[Gao et al., 2015]
Gao, Y., Alecu, I. M., Goumri, A., & Marshall, P. 2015, Chemical
Physics Letters, 624, 83, 10.1016/j.cplett.2015.02.011
[Garland, 1998]
Garland, N. L. 1998, Chemical Physics Letters, 290, 385,
[Garvin et al., 2020]
Garvin, J. B., Arney, G., Getty, S., Johnson, N., Kiefer, W., Lorenz, R., Ravine, M., Malespin, C., Webster, C., Campbell, B., & Izenberg, N., 2020, DAVINCI+: Deep Atmosphere of Venus Investigation of Noble Gases, Chemistry, and Imaging Plus. In Lunar and Planetary Science Conference, 2326, 2599
[Gelman et al., 1979]
Gelman, B. G., Zolotukhin, V. G., Lamonov, N. I., et al. 1979, An
analysis of the chemical composition of the atmosphere of Venus on an AMS of
the Venera-12 using a gas chromatograph, NASA STI/Recon Technical Report N
[Geltman, 1962]
Geltman, S. 1962, , 136, 935, 10.1086/147447
[Gentieu & Mentall, 1970]
Gentieu, E. P., & Mentall, J. E. 1970, Science, 169, 681,
[Glaze, 1999]
Glaze, L. S. 1999, Journal of Geophysical Research: Planets, 104, 18899
[Glicker & Stief, 1971]
Glicker, S., & Stief, L. J. 1971, , 54, 2852,
[Golden et al., 2003]
Golden, D. M., Barker, J. R., & Lohr, L. L. 2003, Journal of Physical
Chemistry A, 107, 11057, 10.1021/jp0353183
[Golomb et al., 1962]
Golomb, D., Watanabe, K., & Marmo, F. F. 1962, , 36, 958,
[Goodeve & Stein, 1931]
Goodeve, C., & Stein, N. 1931, Transactions of the Faraday Society, 27, 393
[Gordon & Ausloos, 1967]
Gordon, R., & Ausloos, P. 1967, J. Chem. Phys, 46, 4823
[Gordon et al., 1971]
Gordon, S., Mulac, W., & Nangia, P. 1971, Journal of Physical Chemistry,
75, 2087
[Goumri et al., 1999]
Goumri, A., Rocha, J.-D. R., Laakso, D., Smith, C. E., & Marshall,
P. 1999, Journal of Physical Chemistry A, 103, 11328,
[Goumri et al., 1995]
Goumri, A., Rocha, J.-D. R., & Marshall, P. 1995, The Journal of Physical
Chemistry, 99, 10834
[Graham & Johnston, 1978]
Graham, R. A., & Johnston, H. S. 1978, The Journal of Physical Chemistry, 82,
[Granville-Willett, 2017]
Granville-Willett, A. 2017, Master's thesis, University of Cambridge,
Cambridge, UK
[Grassi et al., 2014]
Grassi, D., Politi, R., Ignatiev, N. I., et al. 2014, Journal of
Geophysical Research (Planets), 119, 837, 10.1002/2013JE004586
[Greaves et al., 2020]
Greaves, J. S., Richards, A. M. S., Bains, W., et al.
2020a, Nature Astronomy, 10.1038/s41550-020-1174-4
[Greaves et al., 2020]
—. 2020b, Matters Arising
[Greeley et al., 1984]
Greeley, R., Iversen, J., Leach, R., et al. 1984, , 57, 112,
[Grieger et al., 2004]
Grieger, B., Ignatiev, N. I., Hoekzema, N. M., & Keller, H. U. 2004,
in ESA Special Publication, Vol. 544, Planetary Probe Atmospheric Entry and
Descent Trajectory Analysis and Science, ed. A. Wilson, 63–70
[Griggs, 1968]
Griggs, M. 1968, , 49, 857, 10.1063/1.1670152
[Guyon et al., 1976]
Guyon, P. M., Chupka, W. A., & Berkowitz, J. 1976, , 64, 1419,
[Halstead & Moore, 1957]
Halstead, P., & Moore, A. 1957, Journal of the Chemical Society (Resumed),
[Harada et al., 2010]
Harada, N., Herbst, E., & Wakelam, V. 2010, , 721, 1570,
[Harding et al., 2007]
Harding, L. B., Klippenstein, S. J., & Georgievskii, Y. 2007, Journal of
Physical Chemistry A, 111, 3789, 10.1021/jp0682309
[Hayden et al., 1982]
Hayden, C. C., Neumark, D. M., Shobatake, K., Sparks, R. K., & Lee,
Y. T. 1982, , 76, 3607, 10.1063/1.443397
[Haynes, 2014]
Haynes, W. M. 2014, CRC Handbook of Chemistry and Physics (CRC Press)
[Henry & McElroy, 1968]
Henry, R. J., & McElroy, M. B. 1968, avm, 251
[Henry, 1970]
Henry, R. J. W. 1970, , 161, 1153, 10.1086/150615
[Herbort et al., 2020]
Herbort, O., Woitke, P., Helling, C., & Zerkle, A. 2020, , 636,
A71, 10.1051/0004-6361/201936614
[Herman & Mentall, 1982]
Herman, J. R., & Mentall, J. E. 1982, , 87, 8967,
[Herndon et al., 1999]
Herndon, S. C., Froyd, K. D., Lovejoy, E. R., & Ravishankara, A. R.
1999, Journal of Physical Chemistry A, 103, 6778, 10.1021/jp9911853
[Hickson & Keyser, 2005]
Hickson, K. M., & Keyser, L. F. 2005, Journal of Physical Chemistry A,
109, 6887, 10.1021/jp051176w
[Hindiyarti et al., 2007]
Hindiyarti, L., Glarborg, P., & Marshall, P. 2007, Journal of Physical
Chemistry A, 111, 3984, 10.1021/jp067499p
[Hintze et al., 2003]
Hintze, P. E., Kjaergaard, H. G., Vaida, V., & Burkholder, J. B. 2003,
Journal of Physical Chemistry A, 107, 1112, 10.1021/jp0263626
[Hobbs et al., 2019]
Hobbs, R., Shorttle, O., Madhusudhan, N., & Rimmer, P. 2019, ,
487, 2242, 10.1093/mnras/stz1333
[Hobbs et al., 2020]
Hobbs, R., Shorttle, O., Rimmer, P. B., & Madhusudhan, N. 2020,
, submitted
[Hoffman et al., 1980]
Hoffman, J. H., Hodges, R. R., Donahue, T. M., & McElroy, M. M. 1980,
, 85, 7882, 10.1029/JA085iA13p07882
[Hong et al., 2012]
Hong, Z., Davidson, D. F., Lam, K.-Y., & Hanson, R. K. 2012, Combustion and
flame, 159, 3007
[Hu et al., 2012]
Hu, R., Seager, S., & Bains, W. 2012, , 761, 166,
[Hubert & Herzberg, 1979]
Hubert, K., & Herzberg, G. 1979, Volume IV: Constants of Diatomic Molecules
[Huebner & Mukherjee, 2015]
Huebner, W. F., & Mukherjee, J. 2015, , 106, 11,
[Huffman et al., 1963]
Huffman, R., Tanaka, Y., & Larrabee, J. 1963, The Journal of Chemical Physics,
39, 910
[Huffman, 1969]
Huffman, R. E. 1969, Canadian Journal of Chemistry, 47, 1823
[Huynh & Violi, 2008]
Huynh, L. K., & Violi, A. 2008, The Journal of organic chemistry, 73, 94
[Hwang et al., 2010]
Hwang, S., Cooke, J., De Witt, K., & Rabinowitz, M. 2010, International
Journal of Chemical Kinetics, 42, 168
[Inn, 1975]
Inn, E. C. Y. 1975, Journal of Atmospheric Sciences, 32, 2375,
[Ityaksov et al., 2008]
Ityaksov, D., Linnartz, H., & Ubachs, W. 2008, Chemical Physics Letters,
462, 31, 10.1016/j.cplett.2008.07.049
[Iwagami et al., 2008]
Iwagami, N., Ohtsuki, S., Tokuda, K., et al. 2008, , 56, 1424,
[Iyer et al., 1983]
Iyer, R. S., Rogers, P. J., & Rowland, F. 1983, The Journal of Physical
Chemistry, 87, 3799
[Jacobs et al., 1989]
Jacobs, A., Wahl, M., Weller, R., & Wolfrum, J. 1989, Symposium
(International) on Combustion, 22, 1093
[Jasper et al., 2009]
Jasper, A. W., Klippenstein, S. J., & Harding, L. B. 2009, Proceedings of the
combustion institute, 32, 279
[Jasper et al., 2007]
Jasper, A. W., Klippenstein, S. J., Harding, L. B., & Ruscic, B. 2007,
Journal of Physical Chemistry A, 111, 3932, 10.1021/jp067585p
[Javoy et al., 2003]
Javoy, S., Naudet, V., Abid, S., & Paillard, C. 2003, Experimental thermal and
fluid science, 27, 371
[Jessup et al., 2015]
Jessup, K. L., Marcq, E., Mills, F., et al. 2015, , 258, 309,
[Jodkowski et al., 1995]
Jodkowski, J. T., Ratajczak, E., Fagerström, K., et al. 1995,
Chemical Physics Letters, 240, 63, 10.1016/0009-2614(95)00470-O
[Johnson & de Oliveira, 2019]
Johnson, N. M., & de Oliveira, M. R. R. 2019, Earth and Space Science, 6,
1299, 10.1029/2018EA000536
[Jourdain et al., 1979]
Jourdain, J., Bras, G. L., & Combourieu, J. 1979, International Journal of
Chemical Kinetics, 11, 569
[Ju & Wang, 2010]
Ju, L., & Wang, D. 2010, International Journal of Chemical Kinetics, 42, 289
[Karkach & Osherov, 1999]
Karkach, S. P., & Osherov, V. I. 1999, , 110, 11918,
[Katayama et al., 1973]
Katayama, D. H., Huffman, R. E., & O'Bryan, C. L. 1973, , 59, 4309,
[Keller-Rudek et al., 2013]
Keller-Rudek, H., Moortgat, G. K., Sander, R., & Sörensen, R.
2013, Earth System Science Data Discussions, 6, 411,
[Keyser, 1980]
Keyser, L. 1980, The Journal of Physical Chemistry, 84, 11
[Kley & Welge, 1965]
Kley, D., & Welge, K. H. 1965, Zeitschrift Naturforschung Teil A, 20, 124,
[Klippenstein et al., 2009]
Klippenstein, S. J., Harding, L., Ruscic, B., et al. 2009, The Journal of
Physical Chemistry A, 113, 10241
[Klippenstein & Kim, 1993]
Klippenstein, S. J., & Kim, Y.-W. 1993, , 99, 5790,
[Knipovich et al., 1988]
Knipovich, O., Rubtsova, E., & Nekrasov, L. 1988, Volume Recombination Of
Nitrogen-Atoms In Afterglow Of Condensed Discharges, MEZHDUNARODNAYA KNIGA
39 DIMITROVA UL., 113095 MOSCOW, RUSSIA
[Knollenberg & Hunten, 1980]
Knollenberg, R. G., & Hunten, D. M. 1980, ournal of Geophysical Research: Space Physics, 85, 8039
[Koch & Skibowski, 1971]
Koch, E. E., & Skibowski, M. 1971, Chemical Physics Letters, 9, 429,
[Krasnoperov et al., 1984]
Krasnoperov, L. N., Chesnokov, E. N., & Panfilov, V. N. 1984, Chemical
Physics, 89, 297, 10.1016/0301-0104(84)85317-3
[Krasnoperov & Michael, 2004]
Krasnoperov, L. N., & Michael, J. V. 2004, Journal of Physical Chemistry
A, 108, 5643, 10.1021/jp040186e
[Krasnopolsky, 1982]
Krasnopolsky, V. A. 1982, Moscow Izdatel Nauka
[Krasnopolsky, 2007]
—. 2007, , 191, 25, 10.1016/j.icarus.2007.04.028
[Krasnopolsky, 2008]
—. 2008, , 197, 377, 10.1016/j.icarus.2008.05.020
[Krasnopolsky, 2010]
—. 2010a, , 209, 314,
[Krasnopolsky, 2010]
—. 2010b, , 208, 539,
[Krasnopolsky, 2012]
—. 2012, , 218, 230, 10.1016/j.icarus.2011.11.012
[Krasnopolsky, 2013]
—. 2013, , 225, 570, 10.1016/j.icarus.2013.04.026
[Krasnopolsky, 2014]
—. 2014, , 237, 340, 10.1016/j.icarus.2014.04.043
[Krasnopolsky, 2015]
—. 2015, , 252, 327, 10.1016/j.icarus.2015.01.024
[Krasnopolsky, 2016]
—. 2016, , 274, 33, 10.1016/j.icarus.2016.03.010
[Krasnopolsky, 2017]
—. 2017, , 286, 134, 10.1016/j.icarus.2016.10.003
[Krasnopolsky & Pollack, 1994]
Krasnopolsky, V. A., & Pollack, J. B. 1994, , 109, 58,
[Krasnopolsky & Belyaev , 2016]
Krasnopolsky, V. A., & Belyaev, D. A. 2016, , 293, 114, 10.1016/j.icarus.2017.04.016
[Kronebusch & Berkowitz, 1976]
Kronebusch, P. L., & Berkowitz, J. 1976, International Journal of Mass
Spectrometry and Ion Processes, 22, 283, 10.1016/0020-7381(76)80088-5
[Kurbanov & Mamedov, 1995]
Kurbanov, M., & Mamedov, K. F. 1995, Kinetics and catalysis, 36
[Kurtén et al., 2010]
Kurtén, T., Lane, J. R., Jørgensen, S., & Kjaergaard, H. G.
2010, Physical Chemistry Chemical Physics (Incorporating Faraday
Transactions), 12, 12833, 10.1039/c0cp00383b
[Laidler & Wojciechowski, 1961]
Laidler, K. J., & Wojciechowski, B. W. 1961, Proceedings of the Royal
Society of London Series A, 260, 103, 10.1098/rspa.1961.0016
[Lamb et al., 1984]
Lamb, J., Mozurkewich, M., & Benson, S. 1984, Journal of physical chemistry
(1952), 88, 6441
[Lane & Kjaergaard, 2008]
Lane, J. R., & Kjaergaard, H. G. 2008, Journal of Physical Chemistry A,
112, 4958, 10.1021/jp710863r
[Lavendy et al., 1984]
Lavendy, H., Gandara, G., & Robbe, J. M. 1984, Journal of Molecular
Spectroscopy, 106, 395, 10.1016/0022-2852(84)90170-X
[Lavendy et al., 1987]
Lavendy, H., Robbe, J. M., & Gandara, G. 1987, Journal of Physics B
Atomic Molecular Physics, 20, 3067, 10.1088/0022-3700/20/13/018
[Lawrence, 1972]
Lawrence, G. M. 1972a, , 56, 3435, 10.1063/1.1677717
[Lawrence, 1972]
—. 1972b, , 57, 5616, 10.1063/1.1678271
[Lee & Chiang, 1982]
Lee, L., & Chiang, C. 1982, Chemical Physics Letters, 19, 3
[Lee et al., 1973]
Lee, L. C., Carlson, R. W., Judge, D. L., & Ogawa, M. 1973, ,
13, 1023, 10.1016/0022-4073(73)90075-7
[Lee et al., 1977]
Lee, L. C., Slanger, T. G., Black, G., & Sharpless, R. L. 1977, ,
67, 5602, 10.1063/1.434759
[Li et al., 2004]
Li, H.-Y., Pu, M., Ji, Y.-Q., Xu, Z.-F., & Feng, W.-L. 2004,
Chemical Physics, 307, 35, 10.1016/j.chemphys.2004.07.014
[Lias et al., 1970]
Lias, S. G., Collin, G. J., Rebbert, R. E., & Ausloos, P. 1970, ,
52, 1841, 10.1063/1.1673226
[Lilenfeld & Richardson, 1977]
Lilenfeld, H. V., & Richardson, R. J. 1977, , 67, 3991,
[Lin et al., 1978]
Lin, C. L., Rohatgi, N. K., & Demore, W. B. 1978, , 5, 113,
[Lincowski et al., 2021]
Lincowski, A. P., Meadows, V. S., Crisp, D., Akins, A. B., Schwieterman, E. W., Arney, G. N., Wong, M. L., Steffes, P. G., Parenteau, M. N., & Domagal-Goldman, S. 2021, , 908, L44
[Lingam & Loeb, 2020]
Lingam, M., & Loeb, A. 2020, arXiv e-prints, arXiv:2009.07835.
[Linkin et al., 1986]
Linkin, V., Kerzhanovich, V., Lipatov, A., et al. 1986, Science, 231, 1417
[Lissianski et al., 1995]
Lissianski, V., Yang, H., Qin, Z., et al. 1995, Chemical Physics
Letters, 240, 57, 10.1016/0009-2614(95)00496-Q
[Liu et al., 2010]
Liu, C.-F., Shih, S.-M., & Huang, T.-B. 2010, Industrial & engineering
chemistry research, 49, 9052
[Lofthus & Krupenie, 1977]
Lofthus, A., & Krupenie, P. H. 1977, Journal of Physical and Chemical
Reference Data, 6, 113, 10.1063/1.555546
[Lombos et al., 1967]
Lombos, B. A., Sauvageau, P., & Sandorfy, C. 1967, Journal of Molecular
Spectroscopy, 24, 253, 10.1016/0022-2852(67)90089-6
[Loomis & Walters, 1927]
Loomis, A., & Walters, J. 1927, Journal of the Franklin Institute, 203, 333
[Lovejoy et al., 1996]
Lovejoy, E. R., Hanson, D. R., & Huey, L. G. 1996, The Journal of Physical
Chemistry, 100, 19911
[Lovejoy et al., 1987]
Lovejoy, E. R., Wang, N.-S., & Howard, C. J. 1987, Journal of physical
chemistry, 91, 5749
[Lu et al., 2003]
Lu, C.-W., Wu, Y.-J., Lee, Y.-P., Zhu, R. S., & Lin, M. C. 2003,
Journal of Physical Chemistry A, 107, 11020, 10.1021/jp036025c
[Lukirskii et al., 1964]
Lukirskii, A., Brytov, I., & Zimkina, T. 1964, OptSp, 17, 234
[Lyons, 2008]
Lyons, J. R. 2008, Journal of Sulfur Chemistry, 29, 269
[Magnotta & Johnston, 1980]
Magnotta, F., & Johnston, H. S. 1980, , 7, 769,
[Maiorov et al., 2005]
Maiorov, B. S., Ignat'ev, N. I., Moroz, V. I., et al. 2005, Solar
System Research, 39, 267, 10.1007/s11208-005-0042-1
[Manion et al., 2015]
Manion, J., Huie, R., Levin, R., et al. 2015, URL: http://kinetics.nist.gov
(Date: 27.01. 2018)
[Marcq et al., 2013]
Marcq, E., Bertaux, J.-L., Montmessin, F., & Belyaev, D. 2013, Nature
Geoscience, 6, 25, 10.1038/ngeo1650
[Marcq et al., 2008]
Marcq, E., Bézard, B., Drossart, P., et al. 2008, Journal of
Geophysical Research (Planets), 113, E00B07, 10.1029/2008JE003074
[Marcq et al., 2005]
Marcq, E., Bézard, B., Encrenaz, T., & Birlan, M. 2005, ,
179, 375, 10.1016/j.icarus.2005.06.018
[Marcq et al., 2006]
Marcq, E., Encrenaz, T., Bézard, B., & Birlan, M. 2006, ,
54, 1360, 10.1016/j.pss.2006.04.024
[Marcq et al., 2015]
Marcq, E., Lellouch, E., Encrenaz, T., et al. 2015, , 113, 256,
[Marcq et al., 2018]
Marcq, E., Mills, F. P., Parkinson, C. D., & Vandaele, A. C. 2018,
, 214, 10, 10.1007/s11214-017-0438-5
[Marcq et al., 2020]
Marcq, E., Lea Jessup, K., Baggio, L., et al. 2020, , 335,
113368, 10.1016/j.icarus.2019.07.002
[Markiewicz et al., 2018]
Markiewicz, W. J., Petrova, E. V. & Shalygina, O. S. 2018, , 299, 272
[Markwalder et al., 1993]
Markwalder, B., Gozel, P., & Van den Bergh, H. 1993, The Journal of Physical
Chemistry, 97, 5260
[Marmo, 1953]
Marmo, F. F. 1953, Journal of the Optical Society of America (1917-1983), 43,
[Martínez et al., 2000]
Martínez, E., Albaladejo, J., Jiménez, E., Notario, A., &
Díaz de Mera, Y. 2000, Chemical Physics Letters, 329, 191,
[Martinotti et al., 1968]
Martinotti, F. F., Welch, M. J., & Wolf, A. P. 1968, Chemical Communications
(London), 115
[Masuoka & Samson, 1981]
Masuoka, T., & Samson, J. A. R. 1981, , 74, 1093,
[Materic & Smedley, 2011]
Materic, V., & Smedley, S. I. 2011, Industrial & engineering chemistry
research, 50, 5927
[Matsumi et al., 2002]
Matsumi, Y., Comes, F. J., Hancock, G., et al. 2002, Journal of
Geophysical Research (Atmospheres), 107, 4024, 10.1029/2001JD000510
[Matsunaga & Watanabe, 1967]
Matsunaga, F., & Watanabe, K. 1967, Sci. Light, 16, 31
[Matsunaga & Watanabe, 1967]
Matsunaga, F. M., & Watanabe, K. 1967, , 46, 4457,
[Matthes et al., 2017]
Matthes, K., Funke, B., Andersson, M. E., et al. 2017, Geoscientific
Model Development, 10, 2247, 10.5194/gmd-10-2247-2017
[McNesby & Okabe, 1964]
McNesby, J., & Okabe, H. 1964, Advances in Photochemistry, 3, 166
[McNesby et al., 1962]
McNesby, J. R., Tanaka, I., & Okabe, H. 1962, , 36, 605,
[Meadows & Crisp, 1996]
Meadows, V. S., & Crisp, D. 1996, , 101, 4595,
[Mentall et al., 1971]
Mentall, J. E., Gentieu, E. P., Krauss, M., & Neumann, D. 1971, ,
55, 5471, 10.1063/1.1675711
[Messerli et al., 2005]
Messerli, M. A., Amaral-Zettler, L. A., Zettler, E., et al. 2005, Journal of
experimental biology, 208, 2569
[Metzger & Cook, 1964]
Metzger, P. H., & Cook, G. R. 1964, , 41, 642,
[Michels & Wassenaar, 1950]
Michels, A., & Wassenaar, T. 1950, Physica, 16, 221,
[Miller et al., 2003]
Miller, J. A., Klippenstein, S. J., & Glarborg, P. 2003, Combustion and flame,
135, 357
[Mills, 1998]
Mills, F. P. 1998, PhD thesis, CALIFORNIA INSTITUTE OF TECHNOLOGY
[Mills et al., 2007]
Mills, F. P., Esposito, L. W., & Yung, Y. L. 2007, Washington DC
American Geophysical Union Geophysical Monograph Series, 176, 73,
[Mogul et al., 2021]
Mogul, R., Limaye, S. S., Way, M., & Cordova Jr, J. A. 2021, , e2020GL091327
[Molina et al., 1981]
Molina, L. T., Lamb, J. J., & Molina, M. J. 1981, , 8, 1008,
[Molina & Arguello, 1979]
Molina, M. J., & Arguello, G. 1979, , 6, 953,
[Monks et al., 1993]
Monks, P. S., Romani, P. N., Nesbitt, F. L., Scanlon, M., & Stief,
L. J. 1993, , 98, 17115, 10.1029/93JE01789
[Montoya et al., 2005]
Montoya, A., Sendt, K., & Haynes, B. S. 2005, Journal of Physical
Chemistry A, 109, 1057, 10.1021/jp047903p
[Moortgat & Warneck, 1975]
Moortgat, G. K., & Warneck, P. 1975, Zeitschrift Naturforschung Teil A,
30, 835, 10.1515/zna-1975-6-720
[Moroz et al., 1979]
Moroz, V. I., Moshkin, B. E., Ekonomov, A. P., et al. 1979, Soviet
Astronomy Letters, 5, 118
[Moroz et al., 1990]
Moroz, V. I., Spankuch, D., Titov, D. V., et al. 1990, Advances in
Space Research, 10, 77, 10.1016/0273-1177(90)90168-Y
[Moses et al., 1992]
Moses, J. I., Allen, M., & Yung, Y. L. 1992, , 99, 318,
[Moses et al., 2002]
Moses, J. I., Zolotov, M. Y., & Fegley, B. 2002, , 156, 76,
[Moule & Foo, 1971]
Moule, D. C., & Foo, P. D. 1971, , 55, 1262, 10.1063/1.1676214
[Mount & Moos, 1978]
Mount, G. H., & Moos, H. W. 1978, , 224, L35, 10.1086/182753
[Mousavipour et al., 2003]
Mousavipour, S. H., Namdar-Ghanbari, M. A., & Sadeghian, L. 2003,
Journal of Physical Chemistry A, 107, 3752, 10.1021/jp022291z
[Mousavipour et al., 2009]
Mousavipour, S. H., Pirhadi, F., & Habibagahi, A. 2009, Journal of
Physical Chemistry A, 113, 12961, 10.1021/jp905197h
[Mukhin et al., 1983]
Mukhin, L., Gel'man, B., Lamonov, N., et al. 1983, Kosmicheskie
Issledovaniya, 21, 225
[Mukhin et al., 1982]
Mukhin, L. M., Gelman, B. G., Lamonov, N. I., et al. 1982, Soviet
Astronomy Letters, 8, 216
[Murakami et al., 2003]
Murakami, Y., Onishi, S., Kobayashi, T., et al. 2003, Journal of
Physical Chemistry A, 107, 10996, 10.1021/jp030471i
[Myer & Samson, 1970]
Myer, J. A., & Samson, J. A. R. 1970, , 52, 266,
[Na et al., 1990]
Na, C. Y., Esposito, L. W., & Skinner, T. E. 1990, , 95, 7485,
[Nakata et al., 1965]
Nakata, R., Watanabe, K., & Matsunaga, F. 1965, Sci. Light Tokyo, 14, 54V71
[Nakayama et al., 1959]
Nakayama, T., Kitamura, M. Y., & Watanabe, K. 1959, , 30, 1180,
[Nakayama & Watanabe, 1964]
Nakayama, T., & Watanabe, K. 1964, , 40, 558,
[Natarajan et al., 1997]
Natarajan, K., Woiki, D., & Roth, P. 1997, International Journal of Chemical
Kinetics, 29, 35
[Nava & Stief, 1989]
Nava, D. F., & Stief, L. J. 1989, The Journal of Physical Chemistry, 93, 4044
[Nee & Lee, 1984]
Nee, J. B., & Lee, L. C. 1984, , 81, 31, 10.1063/1.447387
[Nicholas et al., 1979]
Nicholas, J. E., Amodio, C. A., & Baker, M. J. 1979, Journal of the Chemical
Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases, 75,
[Nickolaisen et al., 1994]
Nickolaisen, S. L., Friedl, R. R., & Sander, S. P. 1994, The Journal of
Physical Chemistry, 98, 155
[Nuth & Glicker, 1982]
Nuth, J. A., & Glicker, S. 1982, , 28, 223,
[Okabe, 1970]
Okabe, H. 1970, , 53, 3507, 10.1063/1.1674525
[Okabe, 1978]
—. 1978, Photochemistry of small molecules (John Wiley &. Sons)
[Okabe, 1980]
—. 1980, , 72, 6642, 10.1063/1.439123
[Okabe, 1981]
—. 1981, , 75, 2772, 10.1063/1.442348
[Okabe, 1983]
—. 1983, , 78, 1312, 10.1063/1.444868
[Okabe & Becker, 1963]
Okabe, H., & Becker, D. A. 1963, , 39, 2549, 10.1063/1.1734060
[Okabe et al., 1971]
Okabe, H., Laufer, A. H., & Ball, J. J. 1971, , 55, 373,
[Oschlisniok et al., 2012]
Oschlisniok, J., Häusler, B., Pätzold, M., et al. 2012,
, 221, 940, 10.1016/j.icarus.2012.09.029
[Oyama et al., 1979]
Oyama, V. I., Carle, G. C., Woeller, F., & Pollack, J. B. 1979,
Science, 203, 802, 10.1126/science.203.4382.802
[Oyama et al., 1980]
Oyama, V. I., Carle, G. C., Woeller, F., et al. 1980, , 85, 7891,
[Padial et al., 1985]
Padial, N. T., Collins, L. A., & Schneider, B. I. 1985, , 298, 369,
[Pagsberg et al., 1979]
Pagsberg, P. B., Eriksen, J., & Christensen, H. 1979, Journal of Physical
Chemistry, 83, 582
[Parkinson et al., 2015]
Parkinson, C. D, Gao, P., Schulte, R., Bougher, S.W., Yung, Y.L., Bardeen, C.G., Wilquet, V., Vandaele, A.C., Mahieux, A., Tellmann, S., & Pätzold, M.. 2015, Planetary and Space Science, 113, 205
[Peralta et al., 2017]
Peralta, J., Lee, Y., Hueso, R., et al. 2017, Geophysical Research Letters,
44, 3907
[Petrova, 2018]
Petrova, E. V. 2018, , 306, 163
[Peucker-Ehrenbrink, 1996]
Peucker-Ehrenbrink, B. 1996, Geochimica et Cosmochimica Acta, 60, 3187
[Phillips et al., 1977]
Phillips, E., Lee, L. C., & Judge, D. L. 1977, , 18, 309,
[Phillips, 1981]
Phillips, L. F. 1981, Journal of Physical Chemistry, 85, 3994
[Pinto et al., 2021]
Pinto, J.P., Li, J., Mills, F.P., Marcq, E., Evdokimova, D., Belyaev, D. and Yung, Y.L.. 2021, Nature Communications, 12, 1
[Pollack et al., 1993]
Pollack, J. B., Dalton, J. B., Grinspoon, D., et al. 1993, ,
103, 1, 10.1006/icar.1993.1055
[Porter & Noyes Jr, 1959]
Porter, R. P., & Noyes Jr, W. A. 1959, Journal of the American Chemical
Society, 81, 2307
[Pouilly et al., 1983]
Pouilly, B., Robbe, J. M., Schamps, J., & Roueff, E. 1983, Journal of
Physics B Atomic Molecular Physics, 16, 437,
[Pritchard & Lauterbur, 1961]
Pritchard, J., & Lauterbur, P. 1961, Journal of the American Chemical Society,
83, 2105
[Ranjan et al., 2020]
Ranjan, S., Schwieterman, E. W., Harman, C., et al. 2020, , 896,
148, 10.3847/1538-4357/ab9363
[Rimmer & Helling, 2016]
Rimmer, P. B., & Helling, C. 2016, , 224, 9,
[Rimmer & Helling, 2019]
—. 2019, , 245, 20, 10.3847/1538-4365/ab5192
[Rimmer & Rugheimer, 2019]
Rimmer, P. B., & Rugheimer, S. 2019, , 329, 124,
[Rimmer et al., 2020]
Rimmer, P. B., Ferus, M., Waldmann, I. P., et al. 2020, , 888, 21,
[Robertson & Smith, 2006]
Robertson, R., & Smith, G. P. 2006, Journal of Physical Chemistry A, 110,
6673, 10.1021/jp055863z
[Roose et al., 1978]
Roose, T., Hanson, R., & Kruger, C. 1978, Shock tube and shock wave research,
[Rowe et al., 1998]
Rowe, B. R., Brownsword, R. A., Smith, I. W., et al. 1998, Journal of the
Chemical Society, Faraday Transactions, 94, 2889
[Roxlo & Mandl, 1980]
Roxlo, C., & Mandl, A. 1980, Journal of Applied Physics, 51, 2969,
[Sagan, 1975]
Sagan, C. 1975, Journal of Atmospheric Sciences, 32, 1079,
[Salahub & Sandorfy, 1971]
Salahub, D. R., & Sandorfy, C. 1971, Chemical Physics Letters, 8, 71,
[Samson & Cairns, 1964]
Samson, J. A., & Cairns, R. 1964, Journal of Geophysical Research, 69, 4583
[Samson & Cairns, 1965]
—. 1965, JOSA, 55, 1035
[Sandor & Clancy, 2005]
Sandor, B. J., & Clancy, R. T. 2005, , 177, 129,
[Sandor & Clancy, 2012]
—. 2012, , 220, 618, 10.1016/j.icarus.2012.05.016
[Sandor et al., 2010]
Sandor, B. J., Todd Clancy, R., Moriarty-Schieven, G., & Mills, F. P.
2010, , 208, 49, 10.1016/j.icarus.2010.02.013
[Sauter, 1931]
Sauter, F. 1931, Ann. Phys.(Leipzig), 11, 454
[Saxon et al., 1983]
Saxon, R. P., Lengsfield, B. H., I., & Liu, B. 1983, , 78, 312,
[Schaefer & Fegley, 2004]
Schaefer, L., & Fegley, B. 2004, , 168, 215,
[Schindler et al., 1996]
Schindler, R. N., Dethlefs, J., & Schmidt, M. 1996, Berichte der
Bunsengesellschaft für physikalische Chemie, 100, 1242
[Schoen, 1962]
Schoen, R. I. 1962, , 37, 2032, 10.1063/1.1733423
[Schurath et al., 1969]
Schurath, U., Tiedemann, P., & Schindler, R. N. 1969, The Journal of Physical
Chemistry, 73, 456
[Schürgers & Welge, 1968]
Schürgers, M., & Welge, K. H. 1968, Zeitschrift Naturforschung Teil A,
23, 1508, 10.1515/zna-1968-1011
[Seinfeld & Pandis, 2016]
Seinfeld, J. H., & Pandis, S. N. 2016, Atmospheric chemistry and physics: from
air pollution to climate change (John Wiley & Sons)
[Sellevåg et al., 2009]
Sellevåg, S. R., Georgievskii, Y., & Miller, J. A. 2009, Journal
of Physical Chemistry A, 113, 4457, 10.1021/jp8110524
[Selwyn et al., 1977]
Selwyn, G., Podolske, J., & Johnston, H. S. 1977, , 4, 427,
[Sendt & Haynes, 2005]
Sendt, K., & Haynes, B. S. 2005, Journal of Physical Chemistry A, 109,
8180, 10.1021/jp052622i
[Sendt et al., 2002]
Sendt, K., Jazbec, M., & Haynes, B. 2002, Proceedings of the Combustion
Institute, 29, 2439
[Senosiain et al., 2006]
Senosiain, J. P., Klippenstein, S. J., & Miller, J. A. 2006, Journal of
Physical Chemistry A, 110, 5772, 10.1021/jp054934r
[Shao et al., 2020]
Shao, W. D., Zhang, X., Bierson, C. J., & Encrenaz, T. 2020, Journal
of Geophysical Research (Planets), 125, e06195, 10.1029/2019JE006195
[Sharkey et al., 1994]
Sharkey, P., Sims, I. R., Smith, I. W., Bocherel, P., & Rowe, B. R. 1994,
Journal of the Chemical Society, Faraday Transactions, 90, 3609
[Shum & Benson, 1985]
Shum, L. G., & Benson, S. W. 1985, International journal of chemical kinetics,
17, 749
[Singleton & Cvetanović, 1988]
Singleton, D. L., & Cvetanović, R. J. 1988, Journal of Physical and
Chemical Reference Data, 17, 1377, 10.1063/1.555811
[Sinozaki et al., 1926]
Sinozaki, H., Hara, R., & Mitsukuri, S. 1926, Bulletin of the Chemical Society
of Japan, 1, 59
[Slack, 1976]
Slack, M. W. 1976, , 64, 228, 10.1063/1.431955
[Slanger & Black, 1982]
Slanger, T. G., & Black, G. 1982, , 77, 2432, 10.1063/1.444111
[Smith et al., 2006]
Smith, C. A., Pope, F. D., Cronin, B., Parkes, C. B., & Orr-Ewing,
A. J. 2006, Journal of Physical Chemistry A, 110, 11645,
[Smith et al., 1998]
Smith, N. S., Bénilan, Y., & Bruston, P. 1998, , 46, 1215,
[Smrekar et al., 2020]
Smrekar, S., Dyar, D., Helbert, J., Hensley, S., Nunes, D., & Whitten, J. 2020, VERITAS (Venus Emissivity, Radio Science, InSAR, Topography and Spectroscopy): A Proposed Discovery Mission. In European Planetary Science Congress EPSC2020-447
[Snellen et al., 2020]
Snellen, I. A. G., Guzman-Ramirez, L., Hogerheijde, M. R., Hygate,
A. P. S., & van der Tak, F. F. S. 2020, arXiv e-prints, arXiv:2010.09761.
[Steer & Knight, 1968]
Steer, R., & Knight, A. 1968, The Journal of Physical Chemistry, 72, 2145
[Stief et al., 1972]
Stief, L. J., Donn, B., Glicker, S., Gentieu, E. P., & Mentall,
J. E. 1972, , 171, 21, 10.1086/151253
[Stobbe, 1930]
Stobbe, M. 1930, Annalen der Physik, 399, 661
[Stock, 2008]
Stock, J. 2008, diploma thesis, Technische Universität Berlin, Straße
des 17. Juni 135, 10623 Berlin, Germany
[Stockwell & Calvert, 1978]
Stockwell, W. R., & Calvert, J. G. 1978, Journal of Photochemistry, 8, 193
[Stoeckel et al., 1985]
Stoeckel, F., Schuh, M. D., Goldstein, N., & Atkinson, G. H. 1985,
Chemical Physics, 95, 135, 10.1016/0301-0104(85)80154-3
[Stone & Rowley, 2005]
Stone, D., & Rowley, D. M. 2005, Physical Chemistry Chemical Physics
(Incorporating Faraday Transactions), 7, 2156, 10.1039/B502673C
[Strausz et al., 1968]
Strausz, O., Donovan, R., & De Sorgo, M. 1968, Berichte der Bunsengesellschaft
für physikalische Chemie, 72, 253
[Streit et al., 1976]
Streit, G. E., Howard, C. J., Schmeltekopf, A. L., Davidson, J. A., &
Schiff, H. I. 1976, , 65, 4761, 10.1063/1.432930
[Strom et al., 1994]
Strom, R. G., Schaber, G. G., & Dawsow, D. D. 1994, , 99, 10899,
[Stull, 1947]
Stull, D. R. 1947, Industrial & Engineering Chemistry, 39, 517
[Sun & Weissler, 1955]
Sun, H., & Weissler, G. L. 1955, , 23, 1160, 10.1063/1.1742205
[Surkov et al., 1987]
Surkov, I. A., Shcheglov, O. P., Ryvkin, M. L., Sheinin, D. M., &
Davydov, N. A. 1987, Kosmicheskie Issledovaniia, 25, 678
[Surkov et al., 1984]
Surkov, Y. A., Barsukov, V. L., Moskalyeva, L. P., Kharyukova, V. P.,
& Kemurdzhian, A. L. 1984, Lunar and Planetary Science Conference
Proceedings, 89, B393, 10.1029/JB089iS02p0B393
[Surkov et al., 1982]
Surkov, Y. A., Ivanova, V. F., Pudov, A. N., et al. 1982, Pisma v
Astronomicheskii Zhurnal, 8, 411
[Surkov et al., 1986]
Surkov, Y. A., Moskalyova, L. P., Kharyukova, V. P., et al. 1986, ,
91, E215, 10.1029/JB091iB13p0E215
[Takahashi & Miyazaki, 1977]
Takahashi, S., & Miyazaki, S. 1977, Bulletin of the Chemical Society of Japan,
50, 1627
[Tanaka et al., 1953]
Tanaka, Y., Inn, E. C., & Watanabe, K. 1953, The Journal of Chemical Physics,
21, 1651
[Tellinghuisen, 2003]
Tellinghuisen, J. 2003, Journal of Physical Chemistry A, 107, 753,
[Terraglio & Manganelli, 1967]
Terraglio, F. P., & Manganelli, R. M. 1967, Journal of the Air Pollution
Control Association, 17, 403
[Thompson et al., 1963]
Thompson, B. A., Harteck, P., & Reeves, R. R. 1963, , 68, 6431,
[Thompson, 2020]
Thompson, M. A. 2020, arXiv e-prints, arXiv:2010.15188.
[Titov, 1983]
Titov, D. V. 1983, Kosmicheskie Issledovaniia, 21, 401
[Titov et al., 2018]
Titov, D. V., Ignatiev, N. I., McGouldrick, K., Wilquet, V., &
Wilson, C. F. 2018, , 214, 126, 10.1007/s11214-018-0552-z
[Tizniti et al., 2010]
Tizniti, M., Le Picard, S. D., Canosa, A., Sims, I. R., & Smith, I.
W. M. 2010, Physical Chemistry Chemical Physics (Incorporating Faraday
Transactions), 12, 12702, 10.1039/C0CP00591F
[Toby et al., 1984]
Toby, S., Sheth, S., & Toby, F. S. 1984, International journal of chemical
kinetics, 16, 149
[Townsend et al., 2012]
Townsend, T. M., Allanic, A., Noonan, C. & Sodeau, J. R. 2012, Journal of Physical Chemistry A, 116, 4035
[Tsai et al., 2017]
Tsai, S.-M., Lyons, J. R., Grosheintz, L., et al. 2017, , 228, 20,
[Tsang et al., 2008]
Tsang, C. C. C., Irwin, P. G. J., Wilson, C. F., et al. 2008, Journal
of Geophysical Research (Planets), 113, E00B08, 10.1029/2008JE003089
[Tsang, 1987]
Tsang, W. 1987, Journal of physical and chemical reference data, 16, 471
[Tsang, 1992]
Tsang, W. 1992, Journal of Physical and Chemical Reference Data, 21, 753,
[Tsang & Hampson, 1986]
Tsang, W., & Hampson, R. F. 1986, Journal of Physical and Chemical
Reference Data, 15, 1087, 10.1063/1.555759
[Tsang & Herron, 1991]
Tsang, W., & Herron, J. T. 1991, Journal of Physical and Chemical
Reference Data, 20, 609, 10.1063/1.555890
[Tsang & Walker, 1992]
Tsang, W., & Walker, J. A. 1992, The Journal of Physical Chemistry, 96, 8378
[Tsuboi & Hashimoto, 1981]
Tsuboi, T., & Hashimoto, K. 1981, Combustion and Flame, 42, 61
[Tsuboi et al., 1981]
Tsuboi, T., Katoh, M., Kikuchi, S., & Hashimoto, K. 1981, Japanese
Journal of Applied Physics, 20, 985, 10.1143/JJAP.20.985
[Tsuchiya et al., 1997]
Tsuchiya, K., Kamiya, K., & Matsui, H. 1997, International Journal of Chemical
Kinetics, 29, 57
[Tsuchiya et al., 1996]
Tsuchiya, K., Yamashita, K., Miyoshi, A., & Matsui, H. 1996, The Journal of
Physical Chemistry, 100, 17202
[Tsuchiya et al., 1994]
Tsuchiya, K., Yokoyama, K., Matsui, H., Oya, M., & Dupre, G. 1994, The Journal
of Physical Chemistry, 98, 8419
[Turányi et al., 2012]
Turányi, T., Nagy, T., Zsély, I. G., et al. 2012, International
Journal of Chemical Kinetics, 44, 284
[Vaghjiani & Ghanshyam, 1993]
Vaghjiani, & Ghanshyam, L. 1993, , 99, 5936, 10.1063/1.465917
[van Dishoeck, 1987]
van Dishoeck, E. F. 1987, , 86, 196, 10.1063/1.452610
[van Dishoeck & Dalgarno, 1984]
van Dishoeck, E. F., & Dalgarno, A. 1984, , 59, 305,
[Vandaele et al., 2017]
Vandaele, A. C., Korablev, O., Belyaev, D., et al. 2017a,
, 295, 16, 10.1016/j.icarus.2017.05.003
[Vandaele et al., 2017]
—. 2017b, , 295, 1, 10.1016/j.icarus.2017.05.001
[Vandeputte et al., 2010]
Vandeputte, A. G., Reyniers, M.-F., & Marin, G. B. 2010, Journal of
Physical Chemistry A, 114, 10531, 10.1021/jp103357z
[Verner et al., 1996]
Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996,
, 465, 487, 10.1086/177435
[Verner & Yakovlev, 1995]
Verner, D. A., & Yakovlev, D. G. 1995, , 109, 125
[Vidal et al., 2017]
Vidal, T. H. G., Loison, J.-C., Jaziri, A. Y., et al. 2017, ,
469, 435, 10.1093/mnras/stx828
[Wakelam et al., 2012]
Wakelam, V., Herbst, E., Loison, J. C., et al. 2012, , 199, 21,
[Walker & Kelly, 1972]
Walker, T. E. H., & Kelly, H. P. 1972, Chemical Physics Letters, 16, 511,
[Wang & Hou, 2005]
Wang, B., & Hou, H. 2005, Chemical Physics Letters, 410, 235,
[Wang et al., 1998]
Wang, J., Novaro, O., Bokhimi, X., et al. 1998, Materials letters, 35, 317
[Wang & Howard, 1990]
Wang, N. S., & Howard, C. J. 1990, Journal of physical chemistry, 94, 8787
[Watanabe, 1954]
Watanabe, K. 1954, , 22, 1564, 10.1063/1.1740459
[Watanabe, 1958]
—. 1958, Advances in Geophysics, 5, 153,
[Watanabe & Jursa, 1964]
Watanabe, K., & Jursa, A. S. 1964, , 41, 1650,
[Watanabe et al., 1967]
Watanabe, K., Matsunaga, F. M., & Sakai, H. 1967, , 6, 391,
[Watanabe & Sood, 1965]
Watanabe, K., & Sood, S. 1965, Sci. Light, 14, 36
[Watanabe & Zelikoff, 1953]
Watanabe, K., & Zelikoff, M. 1953, Journal of the Optical Society of
America (1917-1983), 43, 753
[Watson et al., 1976]
Watson, R., Machado, G., Fischer, S., & Davis, D. D. 1976, , 65,
2126, 10.1063/1.433369
[Weast, 1971]
Weast, R. 1971, CRC Handbook of Chemistry and Physics: A Ready-reference Book
of Chemical and Physical Data, CRC handbook series (Chemical Rubber Company).
[Weaver et al., 1976]
Weaver, J., Meagher, J., & Heicklen, J. 1976, Journal of Photochemistry, 6,
[Wei & Timmons, 1975]
Wei, C.-N., & Timmons, R. B. 1975, , 62, 3240,
[West & Berry, 1974]
West, G. A., & Berry, M. J. 1974, , 61, 4700,
[Williams & Golden, 1993]
Williams, L. R., & Golden, D. M. 1993, , 20, 2227
[Wilquet et al., 2012]
Wilquet, V., Drummond, R., Mahieux, A., et al. 2012, , 217, 875,
[Wilquet et al., 2009]
Wilquet, V., Fedorova, A., Montmessin, F., et al. 2009, Journal of
Geophysical Research (Planets), 114, E00B42, 10.1029/2008JE003186
[Wilson et al., 1981]
Wilson, W. J., Klein, M. J., Kahar, R. K., et al. 1981, , 45,
624, 10.1016/0019-1035(81)90029-4
[Wine et al., 1988]
Wine, P., Wells, J., & Nicovich, J. 1988, The Journal of Physical Chemistry,
92, 2223
[Woiki & Roth, 1995]
Woiki, D., & Roth, P. 1995, International journal of chemical kinetics, 27, 59
[Woitke et al., 2018]
Woitke, P., Helling, C., Hunter, G. H., et al. 2018, , 614, A1,
[Woitke et al., 2020]
Woitke, P., Herbort, O., Helling, C., et al. 2020, arXiv e-prints,
[Wooldridge et al., 1996]
Wooldridge, S. T., Hanson, R. K., & Bowman, C. T. 1996, International Journal
of Chemical Kinetics, 28, 245
[Wu & Judge, 1981]
Wu, C. Y. R., & Judge, D. L. 1981, , 74, 3804,
[Wu et al., 2007]
Wu, Y.-J., Lu, H.-C., Chen, H.-K., et al. 2007, , 127, 154311,
[Xu & Lin, 2009]
Xu, S., & Lin, M.-C. 2009, International Journal of Chemical Kinetics, 41, 667
[Xu & Sun, 1999]
Xu, Z.-F., & Sun, C.-C. 1999, Journal of Molecular Structure: THEOCHEM, 459,
[Xu & Sun, 1998]
Xu, Z.-F., & Sun, J.-Z. 1998, Journal of Physical Chemistry A, 102, 1194,
[Yamamoto, 2014]
Yamamoto, M. 2014, Earth, Planets, and Space, 66, 27,
[Yasunaga et al., 2008]
Yasunaga, K., Kubo, S., Hoshikawa, H., Kamesawa, T., & Hidaka, Y. 2008,
International Journal of Chemical Kinetics, 40, 73
[Yaws, 1999]
Yaws, C. 1999, Chemical Properties Handbook, Chemical engineering books
(McGraw-Hill Education).
[Yoshimura et al., 1992]
Yoshimura, M., Koshi, M., Matsui, H., Kamiya, K., & Umeyama, H.
1992, Chemical Physics Letters, 189, 199,
[Yoshino et al., 1993]
Yoshino, K., Esmond, J. R., Freeman, D. E., & Parkinson, W. H. 1993,
, 98, 5205, 10.1029/93JD00028
[Young, 1973]
Young, A. T. 1973, , 18, 564, 10.1016/0019-1035(73)90059-6
[Young, 1972]
Young, L. D. G. 1972, , 17, 632, 10.1016/0019-1035(72)90029-2
[Yung & Demore, 1982]
Yung, Y. L., & Demore, W. B. 1982, , 51, 199,
[Yung et al., 2009]
Yung, Y. L., Liang, M. C., Jiang, X., et al. 2009, Journal of
Geophysical Research (Planets), 114, E00B34, 10.1029/2008JE003094
[Zahnle et al., 2016]
Zahnle, K., Marley, M. S., Morley, C. V., & Moses, J. I. 2016, ,
824, 137, 10.3847/0004-637X/824/2/137
[Zanchet et al., 2018]
Zanchet, A., del Mazo, P., Aguado, A., et al. 2018, Physical Chemistry
Chemical Physics (Incorporating Faraday Transactions), 20, 5415,
[Zasova et al., 2007]
Zasova, L., Ignatiev, N., Khatuntsev, I., & Linkin, V. 2007, Planetary and
Space Science, 55, 1712
[Zasova et al., 1993]
Zasova, L. V., Moroz, V. I., Esposito, L. W., & Na, C. Y. 1993,
, 105, 92, 10.1006/icar.1993.1113
[Zelikoff & Watanabe, 1953]
Zelikoff, M., & Watanabe, K. 1953, Journal of the Optical Society of
America (1917-1983), 43, 756
[Zelikoff et al., 1953]
Zelikoff, M., Watanabe, K., & Inn, E. C. Y. 1953, , 21, 1643,
[Zhang et al., 1998]
Zhang, Q., Wang, H., Dalla Lana, I. G., & Chuang, K. T. 1998, Industrial &
engineering chemistry research, 37, 1167
[Zhang et al., 2012]
Zhang, X., Liang, M. C., Mills, F. P., Belyaev, D. A., & Yung, Y. L.
2012, , 217, 714, 10.1016/j.icarus.2011.06.016
[Zhang et al., 2010]
Zhang, X., Liang, M.-C., Montmessin, F., et al. 2010, Nature
Geoscience, 3, 834, 10.1038/ngeo989
[Zhang & Bauer, 1997]
Zhang, Y.-X., & Bauer, S. 1997, The Journal of Physical Chemistry B, 101, 8717
[Zimmer et al., 2016]
Zimmer, K., Zhang, Y., Lu, P., et al. 2016, Computers and Geosciences,
90, 97, 10.1016/j.cageo.2016.02.013
|
# Ultrafast photomechanical transduction through thermophoretic implosion
Nikita Kavokine† Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne
Paris Cité, Paris, France Shuangyang Zou† Beijing Key Lab of nanophotonics
and ultrafine optoelectronic systems, Beijing Institute of Technology, Beijing
100081, China Ruibin Liu Beijing Key Lab of nanophotonics and ultrafine
optoelectronic systems, Beijing Institute of Technology, Beijing 100081, China
Antoine Niguès Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne
Paris Cité, Paris, France Bingsuo Zou∗ Beijing Key Lab of nanophotonics and
ultrafine optoelectronic systems, Beijing Institute of Technology, Beijing
100081, China Key Lab of Featured Metal Resources Utilization and Advanced
Materials, Guangxi University, Nanning 530004, China Lydéric Bocquet∗
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité,
Paris, France
∗ corresponding authors. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
† equal contribution.
## Abstract
Since the historical experiments of Crookes, the direct manipulation of matter
by light has been both a challenge and a source of scientific debate. Here we
show that laser illumination allows to displace a vial of nanoparticle
solution over centimetre-scale distances. Cantilever-based force measurements
show that the movement is due to millisecond long force spikes, which are
synchronised with a sound emission. We observe that the nanoparticles undergo
negative thermophoresis, while ultrafast imaging reveals that the force spikes
are followed by the explosive growth of a bubble in the solution. We propose a
mechanism accounting for the propulsion based on a thermophoretic instability
of the nanoparticle cloud, analogous to the Jeans instability that occurs in
gravitational systems. Our experiments demonstrate a new type of laser
propulsion, and a remarkably violent actuation of soft matter, reminiscent of
the strategy used by certain plants to propel their spores.
## Introduction
In 1874, William Crookes observed that a light-absorbing vane placed in a
vacuum-filled glass bulb would rotate when exposed to sunlight, and
interpreted the results as the effect of radiation pressure Crookes:1874wx .
Crookes’ interpretation was the cause of much debate at the time, and it is
five years later that Maxwell proposed the currently accepted explanation: the
vane actually rotates due to thermophoresis of the residual gas molecules in
the bulb ClerkMaxwell:1878ub . Since then, a variety of methods for propelling
macroscopic objects with light have been proposed White:2017ep , involving, if
not radiation pressure Tsuda:2011hy ; Tsu:1959ju , the light-induced ejection
of matter, resulting in propulsion through momentum conservation Phipps:2010jk
; Dogariu:2013bb ; Zhang:2015jo . However, the potential of thermophoresis –
the driving mechanism in Crookes’ experiment – for macroscopic light-induced
propulsion has hardly been explored since the nineteenth century. While light-
induced self-thermophoresis has been highlighted as a means of controlled
optical manipulation of individual colloids Schermer:2011bz ; Jiang:2010el ,
light-induced thermal gradients have been overlooked as a means of macroscopic
actuation.
We describe in this Article a macroscopic system that is propelled over
centimetre-scale distances by the sole action of light, yet without any
exchange of matter with the surrounding medium; we show that the propulsion
mechanism is based on light-induced thermophoresis.
## Results
### The photomechanical effect
Our system consists of a closed vial containing 1 mL of a concentrated
solution of lead sulphide (PbS) nanoparticles in cyclohexane. The particles
have an average diameter of 8 nm and strong absorption in the near-infrared
(supplementary figure 1). We observe that when illuminated with a $\sim 1.5$
W, 975 nm laser, the vial suddenly "jumps" away from the laser source (figure
1a, supplementary movie 1), over a few millimetres. The typical velocity of
the vial during a jump is about 1$~{}\rm cm\cdot s^{-1}$, corresponding to a
mechanical work around 30 $\rm\mu J$. After one such jump, the vial falls out
of range of the laser, which diverges from a fibre tip. Moving the fibre tip
closer to the vial results in another jump, and the process can thus go on,
yielding propulsion of the vial over several centimetres. In the experiment
shown in figure 1, an average speed of 1 $\rm mm\cdot s^{-1}$ was obtained
(figures 1b and c, supplementary movie 2), which was essentially limited by
the rate of laser repositioning.
Figure 1: A vial filled with a lead sulphide nanoparticle solution is
propelled by a near-infrared laser. a. Snapshots of the vial motion during one
laser-induced jump. The red arrow indicates the propagation direction of the
laser. b. Temporal colour-code projection of the large scale vial motion
obtained when the fiber tip is kept at an approximately constant distance from
the vial. The image is a superposition of ten snapshots of the vial motion,
with the colour encoding time. c. Vial and fibre tip position as a function of
time corresponding to the large-scale motion shown in panel b.
### Macroscopic characterisation
In order to characterise this intriguing phenomenon, we built a force-
measurement setup based on a cantilever, whose deflection is monitored using a
quadrant photodiode (figure 2a). The vial was glued on the cantilever and its
deflection was recorded as a function of time upon application of the laser
light. Then, knowing the cantilever stiffness (supplementary figure 2), we
could translate the observed deflection into the horizontal component of the
force exerted on the vial-cantilever system. A typical measurement is shown in
figure 2b: the cantilever registers a series of force spikes which starts when
the laser is switched on; it ceases as soon as it is switched off. Each force
spike is accompanied by the emission of audible sound (figure 2b), and $\sim$
97% of the recorded sound spikes match a force spike. A typical force spike
lasts around 5 ms and contains several oscillations of the force between
positive and negative values (figure 2b, inset). Note that a positive force is
oriented here along the direction of propagation of the laser and the initial
increase of the force is always towards positive values. Varying the laser
power, one observes that the force spikes are triggered only above a threshold
power, as shown in figure 2c; above the threshold, the average spike frequency
(that is the number of spikes per unit time) increases with increasing power.
The threshold laser power depends on particle concentration: the lower the
concentration the higher the threshold. For a 2 % in weight particle
concentration, no spikes were observed up to 7 W laser power. The average
amplitude of a spike is around 6 mN; remarkably, it does not have any
appreciable dependence on laser power or particle concentration (figure 2d).
Figure 2: Cantilever-based measurements of the laser-induced force. a.
Schematic of the experimental setup. The vial is suspended on a cantilever; a
low power laser and a quadrant photodiode are used to monitor the cantilever
position as a function of time. b. Typical force and sound pressure versus
time measurement. The particle concentration is 6.4 wt % and the laser power
is 1.5 W. Inset: zoom on a single force spike and the corresponding spike in
sound pressure. c. Frequency of the force spikes as a function of laser power,
for different particle concentrations. Spikes are observed above a critical
laser power, which increases with decreasing concentration. d. Average peak
force as a function of laser power, for two different particle concentrations.
Error bars represent the standard deviation.
### Microscopic mechanism
We investigated the origin of the photomechanical effect in light of the
macroscopic characterisation described above. Radiation pressure could clearly
be ruled out since it acts continuously and not in spikes; moreover, one can
estimate the radiation pressure in our system at a value around 5 nano-Newton,
which is orders of magnitude below the forces that are measured, in the
several milli-Newton range. Furthermore, the vial is sealed and can hardly be
expected to exchange matter with the surrounding medium. Thus, the momentum of
the vial and solution it contains must be conserved, and if the vial moves
away from the laser, then the fluid inside must acquire momentum towards the
laser. In order to understand how the fluid acquires this momentum, we placed
the nanoparticle solution in a 1 mm thick spectroscopic cuvette, and
synchronised the force measurement with high-speed imaging of the laser-
illuminated region. The results are presented in figure 3.
Figure 3: Microscopic investigation of the origin of the photomechanical
effect. The four panels show results of simultaneous measurement of force and
sound pressure, coupled to ultrafast imaging (11 000 fps), in a 1 mm thick
spectroscopic cuvette. a. High speed photography snapshots of the dynamics of
a bubble growing and collapsing upon laser illumination. The red arrow
indicates the laser impact point. b. Typical recording of force and sound
versus time, and of the volume of the bubble in solution as determined by
optical imaging. c. Image of the particle aggregate that becomes visible after
about 10 s illumination. The red arrow shows the impact point of the laser,
the grey striped rectangle emphasises the cuvette wall, and the dashed white
line highlights the aggregate in the image. d. Bubble volume, force, and sound
pressure versus time, averaged over the first six spikes shown in panel b. The
grey rectangle is a guide to the eye highlighting that the onset of the force
spike precedes bubble growth. Inset: aligned time traces of six force spikes,
showing very reproducible synchronisation with the bubble dynamics.
As opposed to the large vial, where force spikes could be observed
indefinitely (we observed no change in behaviour for up to 30 seconds), in the
thin spectroscopic cuvette the spiking ceased after less than a second of
illumination (figure 3b). After about 10 seconds, imaging of the illuminated
region revealed the formation of a solid aggregate (figure 3c and
supplementary movies 3, 4), due to the PbS particles accumulating next to the
cuvette wall, somehow "jamming" the spiking mechanism. Therefore, there is a
particle-laser interaction that causes the particles to migrate towards the
laser. Such migration could be due to the direct interaction of the particles
with the laser electric field: the phenomenon in question would be
dielectrophoresis dielectro . However, one can estimate the dielectrophoretic
migration velocity in our system as less than 1 nm per hour (see supplementary
discussion), hence the contribution of dielectrophoretic driving is
negligible. On the other hand, one could expect the particle migration to be
the result of temperature driving, since the particles have strong absorption
at the laser wavelength. Indeed, when performing infrared imaging (see
supplementary movie 5) we observe temperature differences of up to 30 K across
the system, and the temperature gradient reaches up to 2 $\rm K\cdot mm^{-1}$.
We expect such temperature differences to drive particle motion through
thermophoresis; here negative thermophoresis since particles move towards
higher temperatures. The particle flux $j(\mathbf{r})$ due to thermophoresis
is characterised by the Soret coefficient $S$, defined by:
$j(\mathbf{r})=-DS\rho(\mathbf{r})\nabla T(\mathbf{r}),$ (1)
where $D$ is the particle diffusion coefficient and $\rho(\mathbf{r})$ is the
particle concentration Wurger:2010dj ; Piazza:2008bq . Thermal imaging
combined with monitoring of the aggregate size as a function of time
(supplementary figure 3) allowed us to make a rough estimate of the Soret
coefficient of the PbS particles, yielding $S\approx-4~{}\rm K^{-1}$. To our
knowledge there have been no reported measurements of the Soret coefficient
for PbS particles in cyclohexane. For a more common experimental system,
polystyrene particles in water, Soret coefficient values ranging from $-1.5$
to $+40~{}\rm K^{-1}$ have been reported Wurger:2010dj ; Piazza:2008bq ;
Duhr:2006fwa ; Putnam2005 ; Braibanti2008 , depending on conditions and
particle size. Our estimate is therefore not unreasonable with regard to this
range, and corresponds to quite strong negative thermophoresis.
Furthermore, high speed imaging of the solution revealed another striking
phenomenon. We observed that every force spike is accompanied by the explosive
formation of a bubble next to the cuvette wall (figure 3a, supplementary movie
3). The bubble grows in about 0.5 ms and collapses as rapidly, which could be
a signature of either cavitation Lauterborn:2010cd ; BORKENT:2008fb or
explosive boiling Hou:2015ex ; Jollans:2019uu ; in that case our bubble would
be analogous to a plasmonic bubble Wang2018 . One could expect these violent
bubble dynamics to be the cause of the observed photomechanical effect.
However, careful time-resolution of the dynamics contradicts this hypothesis.
Indeed, the bubble growth results in fluid moving away from the laser, which
should trigger macroscopic motion of the vial towards the laser through
momentum conservation. This is in contradiction with the macroscopic
observation that the vial always moves away from the laser source. Now, one
could argue that the bubble dynamics may still be responsible for the vial
motion if the necessary momentum was released during bubble collapse. However,
this scenario is again contradicted by the experimental observations, since
high speed imaging shows that the vial takes off while the bubble is still
growing (supplementary movie 1). A further confirmation of this macroscopic
observation is provided by the parallel high-speed imaging and force
measurement (figure 3d), whose synchronisation we ensured down to one camera
frame (90 $\mu$s, see supplementary figure 5). The time traces of the various
measured quantities were found to be very reproducible over several spikes
(see figure 3d, inset), in terms of shape, time delay and order of occurence.
Thus, the measurement unambiguously shows that the onset of the force spike
precedes bubble collapse, and even precedes bubble growth; moreover, the spike
contains oscillations on a timescale which is about three times faster than
the bubble dynamics. Hence the bubble seems to arise as a collateral
consequence of another phenomenon that is actually responsible for the
photomechanical effect.
### The thermophoretic instability
Based on these various observations – and the dismissal of several scenarios
–, we propose a physical origin for the photomechanical effect, which is based
on a collapsing instability of the colloidal suspension. The key point in our
reasoning is that the thermophoretic force that drives the particles towards
the laser also induces their mutual attraction. It was proposed theoretically
that such a thermophoretic attraction could lead to an instability, analogous
to the Jeans instability observed in gravitational systems Golestanian:2012kp
; Saha:2014fr , as we detail in the following.
Assuming for simplicity that all the particles at positions $\mathbf{r}_{j}$
are illuminated with intensity $I$, each particle behaves as a point heat
source and the temperature field satisfies $\kappa\Delta
T(\mathbf{r})=4\pi\sigma I\sum_{j}\delta(\mathbf{r}-\mathbf{r}_{j})$, where
$\sigma$ is the particle absorption cross-section and $\kappa$ the thermal
conductivity of the solvent. This leads immediately to a temperature field
which depends on the colloidal structure according to
$T(\mathbf{r})=T_{0}+\frac{\sigma
I}{\kappa}\sum_{j}\frac{1}{\mathbf{|r-r}_{j}|},$ (2)
where $T_{0}$ is the temperature far away from the laser. Now if the particles
undergo thermophoresis characterised by the Soret coefficient $S$, then the
particle flux, given by Eq. (1), can be rewritten introducing the effective
potential ${\cal V}_{\rm eff}(\mathbf{r})$ which represents the
"thermophoretic interaction":
$\mathbf{j(r)}=-DS\rho(\mathbf{r})\nabla T(\mathbf{r})\equiv\frac{D}{k_{\rm
B}T_{0}}\rho(\mathbf{r})(-\nabla{\cal V}_{\rm eff})(\mathbf{r}),$ (3)
with
${{\cal V}_{\rm eff}}(\mathbf{r})=-{\cal
G}\sum_{j}\frac{1}{\mathbf{|r-r}_{j}|},$ (4)
and ${\cal G}\equiv k_{\rm B}T_{0}|S|\sigma I/\kappa$. Therefore, the laser-
induced thermophoresis results in the particles interacting with a $1/r$
attractive potential ($S$ is negative), which is analogous to a gravitational
potential, and ${\cal G}$ plays the role of a gravitational constant. Now, an
ensemble of particles – say a cloud of radius $R$ – with gravitational
interactions is known to undergo a Jeans instability when it exceeds a
critical mass Jeans1902 : quantitatively, the cloud collapses to a point when
the gravitational energy per particle exceeds the thermal energy:
$\rho R^{2}{\cal G}\sim k_{\rm B}T_{0}.$ (5)
Such a collapsing instability explains the formation of stars from clouds of
interstellar dust astro . The profound analogy between gravitational and
certain types of colloidal interactions was first noted by Keller and Segel in
the case of chemotactic bacteria Keller:1970kv ; Brenner:1998jc : when
bacteria are attracted by a molecule that they themselves produce, they may
collapse on each other to form dense aggregates. Similar behaviour was later
highlighted with diffusiophoretic Theurkauff:2012jo ; Marbach:2019jg , or even
capillary Bleibel:2011fw interactions, and more recently predicted
theoretically for thermophoretic interactions such as the ones considered here
Golestanian:2012kp ; Saha:2014fr . It is therefore likely that the analog of
Jeans’ instability with thermophoretic interactions explains the brutal force
release that we observe. This is further supported by the scaling law argument
we develop in the following.
The dynamics of the thermophoretic collapse couple the particle transport in
the pseudo-gravitational field to the fluid dynamics. The latter is described
with a Navier-Stokes equation:
$\mu_{\rm s}[\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}]=-\nabla
p+\rho(-\nabla{\cal V}_{\rm eff})+\eta\Delta\mathbf{v},$ (6)
where $\mathbf{v}$ is the velocity field, $p$ is the pressure, and $\mu_{\rm
s}$ and $\eta$ the suspension mass density and viscosity, respectively. The
driving term $\rho(-\nabla{\cal V}_{\rm eff})$ takes its origin in the
thermophoretic interaction introduced above. Solving this equation in the
presence of the thermophoretic interaction represents a formidable challenge,
but one may propose some scaling relations for the collapse dynamics. First,
the observations indicate that the collapse occurs over a short time-scale, of
the order of 100 $\mu$s (figure 3b). This suggests that the Reynolds number
associated with the collapse is relatively large: indeed, using a millimetric
size for the collapsing region, one can estimate $\mathcal{R}e\approx 10$;
thus the dynamics are dominated by the transient (inertial) terms in the
Navier-Stokes equation, while viscous terms should be small. As a further
note, one may remark that the viscous (shear) term cancels for an
incompressible flow with spherical symmetry as expected in the present
geometry, hence discarding viscosity effects on a more general ground. The
collapse thus results from the balance between the transient inertial term
$\mu_{\rm s}R^{3}\mathbf{\dot{v}}\sim\mu_{\rm s}R^{4}/\tau^{2}$ (integrated
over the size $R\sim 1~{}\rm mm$ of the collapsing cloud) and the total
thermophoretic force $\sim\rho R^{3}(-\nabla{\cal V}_{\rm eff})$, with ${\cal
V}_{\rm eff}$ the thermophoretic interaction potential defined in Eq. (3).
Since the pseudo-gravitational potential results from the sum of the
interactions of particles within the sphere of radius $R$, one estimates
${\cal V}_{\rm eff}\approx-\rho R^{3}{{\cal G}\over R}$. This leads to a
scaling law for the timescale $\tau$ of the collapse, as
$\mu_{\rm s}\cdot\frac{R^{4}}{\tau^{2}}\sim\rho^{2}{\cal G}R^{5}\sim\rho
R^{2}k_{\rm B}T_{0}$ (7)
hence
$\tau\sim\sqrt{\frac{\mu_{\rm s}R^{2}}{\rho k_{\rm B}T_{0}}},$ (8)
Due to the thermophoretic attraction, the density in the illuminated region is
expected to be close to the close packing density of the PbS particles, so
that assuming $\rho\sim 10^{24}~{}\rm m^{-3}$, we find $\tau\sim 100~{}\mu$s.
This estimate is indeed in good agreement with the experimental result and
justifies a posteriori the inertial assumptions on the dynamics. As a further
prediction, the force generated in the collapse can be estimated as the
collapsing mass times its acceleration:
$F\sim m\rho R^{3}\frac{R}{\tau^{2}}=\frac{m\rho^{2}}{\mu_{\rm s}}k_{\rm
B}T_{0}R^{2}.$ (9)
We find $F\sim 40~{}\rm mN$, which is again compatible with the experimentally
observed range.
Figure 4: Schematic mechanism of the photomechanical effect. a. Force versus
time signal corresponding to a force spike. The four dashed rectangles
correspond to the four steps in the proposed mechanism for the generation of a
spike. b. Schematics for the four steps highlighted in panel a. The black
circles correspond to PbS particles and the red colour represents a
temperature gradient. (1) Phoretic motion of the particles towards the laser
source. (2) Jeans instability, resulting in brutal force release. (3)
Explosive growth of a bubble, which disperses the accumulated particles. (4)
Collapse of the bubble and return to the initial state.
## Discussion
The above considerations allow us to propose a complete scenario for the
origin of the photomechanical effect, which is summarised in figure 4. The PbS
particles absorb the laser light and become point heat sources. They undergo
negative thermophoresis and thus migrate towards the laser source. The
particle density increases in the illuminated region, up to the point where
the Jeans instability occurs, resulting in collective motion of the particles
towards the laser, giving the whole vial momentum away from the laser. The
"gravitational" collapse is expected to result in a temperature increase and
pressure decrease near the wall of the vial, resulting from the increase in
the particle concentration and their rapid motion, respectively. This triggers
a cavitation event, with a bubble explosively growing and then collapsing next
to the wall. The resulting dynamics disperse the particles that have
accumulated near the wall BORKENT:2008fb ; Leveugle:2007cd , so that the
process can start again and result in a new propulsion event. We expect that
the momentum of the particles is transmitted to the vial wall, resulting in
the propagation of a shock wave through the glass, which would be responsible
for the oscillations observed in the force spikes and the sound: the observed
oscillation frequencies were different in the vial and the spectroscopic
cuvette (supplementary figure 4). When the vial stands freely on a substrate,
the momentum carried by the shock wave can be transmitted to the substrate and
then back to the vial so that it takes off: this momentum transfer is similar
to what occurs in the propulsion of the Mexican jumping beans hu . We thus
expect only the first positive force peak to matter: the subsequent
oscillations contained in the spike (figure 4a) occur when the vial is already
in the air and they cannot therefore contribute to propulsion.
We have demonstrated the propulsion of a macroscopic object by the sole action
of light through a novel mechanism involving negative thermophoresis as a key
ingredient. A 1.5 W laser illumination is sufficient to propel a vial weighing
3.5 grams at a 1 $\rm mm\cdot s^{-1}$ average speed. Remarkably, it is the
interplay between three different phenomena – thermophoretic migration, the
Jeans instability and bubble cavitation –, that results in the system
propelling itself through repeatable, discrete force spikes. We believe it is
also remarkable that a "soft" system, whose dynamics are usually viscous,
actually exhibits ultrafast transport, way beyond the viscous timescale. The
slow energy accumulation followed by a fast release is reminiscent of the
process used by certain plants, such as the fern, to propel their spores
Noblin:2012ie . Our novel mechanism for soft matter actuation could therefore
be of interest for mimicking certain bio-inspired functionalities.
## Acknowledgements
The authors thank A. Kavokin for initiating the collaboration, and L. Canale,
A. Lainé, T. Mouterde, P. Robin, D. Baigl and D. Lohse for discussions. The
authors are indebted to J. Delannoy, A. Bouillant, C. Cohen and D. Quéré for
lending the high speed and infrared cameras. LB acknowledges funding from the
EU H2020 Framework Programme/ERC Advanced Grant agreement number
785911-Shadoks. The work was partially funded by the Horizon 2020 programme
through 766972-FET-OPEN-NANOPHLOW and by Agence Nationale de la Recherche
(France) through the project ILLIAD.
## Methods
### Nanoparticle synthesis and characterisation
The PbS nanoparticles used in all experiments were synthesised following a
typical hot-injection method, involving injection of a sulfur precursor into a
lead precursor in an organic solvent. All the syntheses were carried out under
air-free conditions using a standard Schlenk-line setup. PbS quantum dots
purchased from Merck exhibited the same behaviour as the in-house-synthesised
ones.
Transmission electron microscopy (TEM, JEOL-2100F) was used to visualise the
nanoparticles. Their crystal structure was obtained by the X-ray diffraction
(XRD, D8-Advance X-ray diffractometer). The absorption spectra in cyclohexane
were recorded using a Shimadzu UV-3600 spectrometer.
### Force measurement setup
A schematic of our force measurement setup is given in figure 2a. A Sheaumann
HF-975-7500-25C fibre-coupled laser diode, powered by an Arroyo LaserSource
controller, was used for actuating the system. The vial or spectroscopic
cuvette were glued to a 3 cm long metal cantilever (made from a metal ruler).
A small mirror was glued to the other side of the cantilever. A low power red
laser was reflected on the small mirror, and directed onto a quadrant
photodiode (Thorlabs, PDQ80A). Laser illumination above the threshold power
(see figure 2c) resulted in force spikes which induced deflection of the
cantilever in the 10 $\mu$m range. The resolution in cantilever deflection was
below 100 nm, and the time resolution was given by the photodiode bandwidth
(150 kHz). In all experiments, the excitation laser fibre tip was placed at
exactly 1 mm from the vial wall using a stepper motor, and 2 mm above the vial
bottom. The analog control of the laser power, camera triggers and force and
sound measurements were synchronised using a National Instruments DAQ card (NI
USB-6363) and a custom Labview software.
### Imaging
High speed imaging was performed using a Phantom v642 or a Hamamatsu Orca
Flash 4.0 camera, and either a Nikon 50 mm macro lens, or a Thorlabs MVL12X3Z
12X zoom lens, with coaxial illumination from an Olympus U-HGLGPS (130 W)
light source. Thermal imaging was performed using a FLIR Q655sc infrared
camera.
## Data availability
All data associated with the plots is available from LB upon reasonable
request.
## Competing interests
The authors declare no competing interests.
## Author contributions
RL observed the photomechanical effect; SZ synthesised the nanoparticles and
performed initial experiments with supervision from BZ. NK performed all
experiments reported in the paper with contributions from AN. NK and LB
analysed the data and wrote the paper. All authors discussed the results and
commented on the manuscript. BZ and LB supervised all aspects of the project.
## References
## References
* (1) Crookes, W. On attraction and repulsion resulting from radiation. _Philosophical transactions of the Royal Society_ 164, 501–527 (1874).
* (2) Clerk Maxwell, J. VII. On stresses in rarified gases arising from inequalities of temperature. _Philosophical transactions of the Royal Society_ 170, 231–256 (1878).
* (3) White, T. J. (ed.) _Photomechanical Materials, Composites, and Systems_. Wireless Transduction of Light into Work (John Wiley & Sons, Ltd, Chichester, UK, 2017).
* (4) Tsuda, Y. _et al._ Flight status of IKAROS deep space solar sail demonstrator. _Acta Astronautica_ 69, 833–840 (2011).
* (5) Tsu, T. C. Interplanetary Travel by Solar Sail. _ARS Journal_ 29, 422–427 (1959).
* (6) Phipps, C. _et al._ Review: Laser-Ablation Propulsion. _Journal of Propulsion and Power_ 26, 609–637 (2010).
* (7) Dogariu, A., Sukhov, S. & Sáenz, J. Optically induced ’negative forces’. _Nature Photonics_ 7, 24–27 (2013).
* (8) Zhang, T. _et al._ Macroscopic and direct light propulsion of bulk graphene material. _Nature Photonics_ 9, 471–476 (2015).
* (9) Schermer, R. T., Olson, C. C., Coleman, J. P. & Bucholtz, F. Laser-induced thermophoresis of individual particles in a viscous liquid. _Optics Express_ 19, 10571–10586 (2011).
* (10) Jiang, H.-R., Yoshinaga, N. & Sano, M. Active Motion of a Janus Particle by Self-Thermophoresis in a Defocused Laser Beam. _Physical Review Letters_ 105, 268302 (2010).
* (11) Irimajiri, A., Hanai, T. & Inouye, A. A dielectric theory of "multi-stratified shell" model with its application to a lymphoma cell. _Journal of Theoretical Biology_ 78, 251–269 (1979).
* (12) Würger, A. Thermal non-equilibrium transport in colloids. _Reports on Progress in Physics_ 73, 126601 (2010).
* (13) Piazza, R. & Parola, A. Thermophoresis in colloidal suspensions. _Journal of Physics: Condensed Matter_ 20, 153102 (2008).
* (14) Duhr, S. & Braun, D. Why molecules move along a temperature gradient. _Proceedings of the National Academy of Sciences_ 103, 19678–19682 (2006).
* (15) Putnam, S. A. & Cahill, D. G. Transport of nanoscale latex spheres in a temperature gradient. _Langmuir_ 21, 5317–5323 (2005).
* (16) Braibanti, M., Vigolo, D. & Piazza, R. Does thermophoretic mobility depend on particle size? _Physical Review Letters_ 100, 1–4 (2008).
* (17) Lauterborn, W. & Kurz, T. Physics of bubble oscillations. _Reports on Progress in Physics_ 73, 106501 (2010).
* (18) Borkent, B. M. _et al._ The acceleration of solid particles subjected to cavitation nucleation. _Journal of Fluid Mechanics_ 610, 157–182 (2008).
* (19) Hou, L., Yorulmaz, M., Verhart, N. R. & Orrit, M. Explosive formation and dynamics of vapor nanobubbles around a continuously heated gold nanosphere. _New Journal of Physics_ 17, 013050 (2015).
* (20) Jollans, T. & Orrit, M. Explosive, oscillatory, and leidenfrost boiling at the nanoscale. _Phys. Rev. E_ 99, 063110 (2019).
* (21) Wang, Y. _et al._ Giant and explosive plasmonic bubbles by delayed nucleation. _Proceedings of the National Academy of Sciences of the United States of America_ 115, 7676–7681 (2018).
* (22) Golestanian, R. Collective Behavior of Thermally Active Colloids. _Physical Review Letters_ 108, 539 (2012).
* (23) Saha, S., Golestanian, R. & Ramaswamy, S. Clusters, asters, and collective oscillations in chemotactic colloids. _Physical Review E_ 89, 062316 (2014).
* (24) Jeans, J. H. The stability of a spherical nebula. _Philos. Trans. Royal Soc. A_ 199, 312–320 (1902).
* (25) Battaner, E. _Astrophysical Fluid Dynamics_ , 166–167 (Cambridge University Press, 1996).
* (26) Keller, E. F. & Segel, L. A. Initiation of slime mold aggregation viewed as an instability. _Journal of Theoretical Biology_ 26, 399–415 (1970).
* (27) Brenner, M. P., Levitov, L. S. & Budrene, E. O. Physical Mechanisms for Chemotactic Pattern Formation by Bacteria. _Biophysical Journal_ 74, 1677–1693 (1998).
* (28) Theurkauff, I., Cottin-Bizonne, C., Palacci, J., Ybert, C. & Bocquet, L. Dynamic Clustering in Active Colloidal Suspensions with Chemical Signaling. _Physical Review Letters_ 108, 268303 (2012).
* (29) Marbach, S. & Bocquet, L. Osmosis, from molecular insights to large-scale applications. _Chem. Soc. Rev._ 48, 3102–3144 (2019).
* (30) Bleibel, J., Dietrich, S., Domínguez, A. & Oettel, M. Shock Waves in Capillary Collapse of Colloids: A Model System for Two-Dimensional Screened Newtonian Gravity. _Physical Review Letters_ 107, 128302 (2011).
* (31) Leveugle, E., Sellinger, A., Fitz-Gerald, J. M. & Zhigilei, L. V. Making Molecular Balloons in Laser-Induced Explosive Boiling of Polymer Solutions. _Physical Review Letters_ 98, 216101 (2007).
* (32) West, D. M., Lal, I. K., Leamy, M. J. & Hu, D. L. Locomotion of mexican jumping beans. _Bioinspiration & Biomimetics_ 7, 036014 (2012).
* (33) Noblin, X. _et al._ The Fern Sporangium: A Unique Catapult. _Science_ 335, 1322–1322 (2012).
|
# Positive Geometries for Barycentric Interpolation
Márton Vaitkus Budapest University of Technology and Economics
###### Abstract
We propose a novel theoretical framework for barycentric interpolation, using
concepts recently developed in mathematical physics. Generalized barycentric
coordinates are defined similarly to Shepard’s method, using positive
geometries – subsets which possess a rational function naturally associated to
their boundaries. Positive geometries generalize certain properties of
simplices and convex polytopes to a large variety of geometric objects. Our
framework unifies several previous constructions, including the definition of
Wachspress coordinates over polytopes in terms of adjoints and dual polytopes.
We also discuss potential applications to interpolation in 3D line space,
mean-value coordinates and splines.
## 1 Overview
The interpolation of scalar- or vector-valued data is an important task in
many fields, including numerical analysis, geometric modeling and computer
graphics. _Barycentric coordinates_ can be defined over segments, triangles
and simplices as well as more complex shapes, such as polytopes [4]. These
include _Wachspress coordinates_ [17] which are rational functions defined for
convex polytopes. Wachspress coordinates were also generalized to subsets
bounded by non-linear hypersurfaces [2] and can be defined in several
equivalent ways [18, 11, 9]. We describe a theoretical framework that
clarifies the relationship between these different formulations, and provides
opportunities for novel generalizations.
We propose a set of basic building blocks for barycentric interpolation
methods: _positive geometries_. Positive geometries have been recently
introduced in the theoretical physics literature [1], but their application to
interpolation problems has not been explored before. Besides simplices and
polytopes, the category of positive geometries includes objects with similar
combinatorial properties, such as polycons [17] or “positive” parts of toric
varieties [15] and Grassmannians [12, 6]. A positive geometry carries a
“canonical” differential form: a rational function with the properties of a
signed volume, that has its poles (where the denominator vanishes) along the
boundaries of the “positive” region. Crucially, these poles have a recursive
structure, i.e. restricting a canonical form to a boundary component (via a
generalization of taking complex residues) results in another canonical form.
Thus, positive geometries share many properties with polytopes – most
importantly, that complicated objects can be constructed by adding together
simpler ones. To define interpolation in terms of canonical forms, we use a
variant of Shepard’s method [4, 13]: barycentric coordinates are ratios
tending to $\frac{\infty}{\infty}$ along the interpolated boundaries. For
polytopes, our construction recovers the definition of Wachspress coordinates
in terms of dual volumes [9].
In this preliminary work, our goal is to introduce the theory of positive
geometries to our audience, and demonstrate how they generalize earlier
constructions for barycentric interpolation. After some motivating
observations (Section 2), we give a definition of positive geometries and
their canonical forms, also giving some examples (Section 3). We then describe
how to use the canonical forms of positive geometries for barycentric
interpolation (Section 4). Finally, we discuss potential applications of this
framework to interpolation in line space, and possible extensions to non-
rational barycentric coordinates and splines (Section 5).
## 2 Motivation
Let us consider the basic example of linear interpolation over a segment
$[a,b]\subset\mathbb{R}$. The standard formula
$\displaystyle f(x)=\frac{b-x}{b-a}f(a)+\frac{x-a}{b-a}f(b),$ (2.1)
can be written in an alternative _barycentric form_ [8]:
$\displaystyle
f(x)=\frac{\frac{1}{x-a}f(a)-\frac{1}{x-b}f(b)}{-\frac{b-a}{(x-a)(x-b)}},$
(2.2)
as a rational function with both the numerator and the denominator having
poles at the boundaries $a$ and $b$.
Consider next linear interpolation over triangles
$(\mathbf{p}_{0},\mathbf{p}_{1},\mathbf{p}_{2})\subset\mathbb{R}^{2}$ in the
plane. The usual formula for barycentric interpolation gives
$\displaystyle
f(\mathbf{x})=\frac{A_{0}f(\mathbf{p}_{0})+A_{1}f(\mathbf{p}_{1})+A_{2}f(\mathbf{p}_{2})}{A},$
(2.3)
which can be reorganized into the equivalent form
$\displaystyle
f(\mathbf{x})=\frac{\frac{1}{A_{1}A_{2}}f(\mathbf{p}_{0})+\frac{1}{A_{2}A_{0}}f(\mathbf{p}_{1})+\frac{1}{A_{0}A_{1}}f(\mathbf{p}_{2})}{\frac{A}{A_{0}A_{1}A_{2}}},$
(2.4)
with the notations shown in Figure 1.
Figure 1: Notations for barycentric coordinates. Figure 2: Canonical rational
function of a triangle.
Wachspress proposed generalized barycentric coordinates over convex polytopes
[17]. For a planar $n$-gon with vertices
$\mathbf{p}_{0},\ldots\mathbf{p}_{n-1}$ these coordinates are defined as:
$\displaystyle
f(\mathbf{x})=\sum_{i=0}^{n}\frac{\frac{C_{i}}{A_{i-1,i}A_{i,i+1}}}{\frac{\sum_{k}C_{k}A_{k-3,k-2}\cdots
A_{k+1,k+2}}{\prod_{k}A_{k,k+1}}}f(\mathbf{p}_{i}),$ (2.5)
where the indices are cyclical modulo $n$ – see Figure 1.
Figure 3: Canonical rational function of a pentagon.
In each of these cases, some terms in the numerator and denominator approach
infinity along an interpolated subset, which is reminiscent of Shepard’s
method [8, 13]. The rational functions in the denominators are illustrated in
Figure 2 and Figure 3. The common pattern involves functions with poles along
the boundaries of some shape. Our goal is to make this idea rigorous using the
notion of a positive geometry.
## 3 Positive Geometries and Canonical Differential Forms
The central concept of our work is that of a positive geometry – a relatively
new concept originating from mathematical physics. In this chapter, we give an
informal overview, and refer to [1] for technical details. For the necessary
background in projective and algebraic geometry, see e.g. [12, 7].
### 3.1 Definition of Positive Geometries
Positive geometries were introduced as generalizations of shapes with a
recursive structure similar to polytopes, defined by some sort of “positivity”
criterion (e.g. the interior of polytopes, or totally positive matrices [5]).
The definition was motivated by recent developments in quantum physics, where
the solution of differential equations in Fourier space led to rational
functions with poles at the boundaries of certain regions [1].
###### Definition 1.
A positive geometry is a pair $(X,X_{\geq 0})$, where $X$ is a $d$-dimensional
complex projective algebraic variety, and $X_{\geq 0}$ is an oriented semi-
algebraic subset of its real part, with a unique differential $d$-form
$\Omega(X,X_{\geq 0})$ called its canonical form, defined by recursion on the
boundary dimension:
* •
If $d=0$, then $X$ is an (oriented) point, and $\Omega(X,X_{\geq 0})=\pm 1$
depending on the orientation.
* •
If $d>0$, then the boundary components of $X_{\geq 0}$ are themselves positive
geometries, and the multivariate residue (see [1, A.3] or [7, Ch. 5]) along a
component is the canonical form for the associated positive geometry.
We stress that a positive geometry is determined by an ambient complex
manifold (most often a projective space $\mathbb{P}^{n}$), together with a
“positive” real subset, thus many different positive geometries could be
associated to the same ambient space.
For positive geometries over projective spaces with homogeneous coordinates
$\mathbf{x}=\left[x_{0}:x_{1}:\ldots x_{d}\right]$, the canonical form can
always be written in terms of a rational function [1, C.1]:
$\displaystyle\Omega(X,X_{\geq 0})=C(\mathbf{x})\omega(\mathbf{x}),$ (3.6)
where $C$ is called the canonical rational function of the positive geometry
and
$\displaystyle\omega(\mathbf{x})=\frac{1}{d!}x_{0}dx_{1}\wedge\ldots\wedge
dx_{d}+\ldots+(-1)^{d}x_{d}dx_{0}\wedge\ldots\wedge dx_{d-1}$
is the _standard measure_ on projective $d$-space.
### 3.2 Examples of Positive Geometries
Some elementary examples of positive geometries are the following:
* •
Segments, bounded by two points $\mathbf{a}=\left[a:1\right]$ and
$\mathbf{b}=\left[b:1\right]$ in the projective line $\mathbb{P}^{1}$
parameterized using homogeneous coordinates. The canonical form at the point
$\mathbf{x}=\left[x:y\right]$ is
$\displaystyle\Omega(\mathbb{P}^{1},\left[a,b\right])=\frac{\left\langle\mathbf{b},\mathbf{a}\right\rangle}{\left\langle\mathbf{x},\mathbf{a}\right\rangle\left\langle\mathbf{x},\mathbf{b}\right\rangle}\omega(\mathbf{x}),$
(3.7)
where
$\left\langle\mathbf{v},\mathbf{w}\right\rangle:=\det(\mathbf{v},\mathbf{w})$
denotes the determinant of vectors of homogeneous coordinates.
* •
Simplices in projective spaces. For triangles formed by three points
$\mathbf{p}_{i}=\left[x_{i}:y_{i}:1\right],i=0,1,2$ in the projective plane
$\Delta\subset\mathbb{P}^{2}$ parameterized using homogeneous coordinates
$\mathbf{x}=\left[x:y:z\right]$, the canonical form is
$\displaystyle\Omega(\mathbb{P}^{2},\Delta)$
$\displaystyle=\frac{1}{2}\frac{\left\langle\mathbf{p}_{0},\mathbf{p}_{1},\mathbf{p}_{2}\right\rangle^{2}}{\left\langle\mathbf{x},\mathbf{p}_{0},\mathbf{p}_{1}\right\rangle\left\langle\mathbf{x},\mathbf{p}_{1},\mathbf{p}_{2}\right\rangle\left\langle\mathbf{x},\mathbf{p}_{2},\mathbf{p}_{0}\right\rangle}\omega(\mathbf{x}).$
(3.8)
Both of these forms are invariant under independent scaling of the homogeneous
coordinates of the vertices, as well as the point of evaluation, and are thus
well-defined functions over projective spaces.
Many other examples of positive geometries were identified [1, Ch. 5]:
* •
Planar regions bounded by a conic section and a line.
* •
Regions in projective 3-space bounded by a quadric or cubic surface and a
plane.
* •
Positive, real parts of _Grassmannians_ (manifolds of $k$-planes in
$n$-dimensional space, denoted $G(k,n)$).
* •
Positive, real parts of _toric varieties_ [15].
These examples – called generalized simplices – all have canonical forms with
constant numerators.
The canonical forms of positive geometries are additive, i.e. the canonical
form of a union is the sum of the canonical forms of its parts. The poles
along boundaries meeting with opposite orientation will cancel, so that only
poles on the exterior boundary remain (technically, this is true for a “signed
triangulation of the empty set” – see [1, Ch. 3]). This implies that the
canonical forms of more complicated regions can be determined by
“triangulating” them. It follows that _convex polytopes_ – which can be
triangulated in the usual sense – are also positive geometries.
In analogy with generalized simplices, there are also various generalized
polytopes:
* •
Convex regions of the projective plane bounded by straight lines and conics,
which are examples of _polycons_ , as defined by Wachspress [17].
* •
_Grassmann polytopes_ , in particular _Amplituhedra_ , which are
generalizations of cyclic polytopes into Grassmannians.
Generalized polytopes have canonical forms with a non-constant numerator. In
fact, for polytopes and polycons the numerator is known as the _adjoint
polynomial_ [10, 2].
### 3.3 Relation to dual polytopes
A canonical form often has a natural geometric interpretation. In particular,
the canonical rational function of a convex projective polytope
$P\subset\mathbb{P}^{d}$ is the signed volume of its _polar dual_
$P_{\mathbf{x}}^{\ast}\subset(\mathbb{P}^{d})^{\ast}$, as shown in [1, Ch.
7.4.1]. The polar dual is the intersection of the dual projective cone of the
polytope, with the dual hyperplane of the point $\mathbf{x}\in P$, and its
volume is a rational function over $P$ computed by the integral formula
$\displaystyle\mathrm{Vol}(P_{\mathbf{x}}^{\ast})=\frac{1}{d!}\int_{\mathbf{y}\in
P_{\mathbf{x}^{\ast}}}\frac{1}{(\mathbf{x}^{T}\mathbf{y})^{d+1}}\omega(\mathbf{y}).$
(3.9)
## 4 Barycentric Interpolation over Positive Geometries
We claim that the canonical forms of positive geometries can be used to define
barycentric interpolation in a way similar to Shepard’s method.
Consider barycentric (linear) interpolation over a _segment_. Define the
weight function for the endpoint $a$ as the ratio of canonical forms for
positive geometries over the projective line
$\displaystyle\lambda_{a}(x)=\frac{\Omega(\mathbb{P}^{1},\left[a,x^{\ast}\right])}{\Omega(\mathbb{P}^{1},\left[a,b\right])},$
(4.10)
where $x^{\ast}$ is the _projective dual_ of the point $x$. If we choose
coordinates so that $x$ is the origin and $x^{\ast}$ is the point at infinity,
we get the barycentric formula (2.2) for linear interpolation.
The same construction applies to a _triangle_ as well. For the the vertex
$\mathbf{p}_{i}$, we take the ratio of two canonical forms over the projective
plane – that of the original triangle, and the triangle bounded by the two
sides meeting at $\mathbf{p}_{i}$ together with the dual line of the current
point $\mathbf{x}$. Using the notations of Figure 4:
$\displaystyle\lambda_{i}^{\Delta}(\mathbf{x})=\frac{\Omega(\mathbb{P}^{2},\Delta(l_{ki},l_{ij},\mathbf{x}^{\ast}))}{\Omega(\mathbb{P}^{2},\Delta(l_{ij},l_{jk},l_{ki}))}.$
(4.11)
This is simply the usual formula (2.4) for barycentric interpolation. Along
each of the sides, the forms (technically, their residues) restrict to linear
interpolation over a segment, as expected.
Figure 4: Triangles used for barycentric coordinates.
For a _convex polytope_ $P\subset\mathbb{P}^{2}$ we follow the same recipe –
the denominator for the vertex $\mathbf{p}_{i}$ will be the canonical form of
$P$, while the numerator is the canonical form defined by the sides meeting at
the vertex, with the dual line of the current point:
$\displaystyle\lambda_{i}^{P}(\mathbf{x})=\frac{\Omega(\mathbb{P}^{2},\Delta(l_{\left(i-1\right)i},l_{i\left(i+1\right)},\mathbf{x}^{\ast}))}{\Omega(\mathbb{P}^{2},P)}.$
(4.12)
These weight functions are the Wachpress coordinates (2.5) over $P$. The
generalization to higher-dimensional simplices and (simplicial) polytopes is
straightforward.
In each case, the numerators are defined by positive geometries that form a
signed triangulation of the domain. The additivity of canonical forms under
unions then implies that their sum is the canonical form of the original
polytope, and thus these functions form a _partition of unity_.
While these triangulations are not the most natural with respect to the
original polytope, they correspond to a natural triangulation of the polar
dual polytope including the origin (i.e. the current point $\mathbf{x}$).
Recalling the interpretation of canonical forms as dual volumes, the numerator
is seen as the volume of the dual pyramid formed by the current point and the
dual face of the vertex. Thus, we connect to the earlier work of [9], who
characterized Wachspress coordinates as ratios of polar dual volumes. Note
that a triangulation of the polar dual through its vertices is analogous to
Warren’s triangulation-based definition of the adjoint polynomial for a
polytope [18].
Wachspress coordinates can be generalized also to regions bounded by subsets
bounded by higher-order algebraic varieties, such as _polycons_ [17]. We omit
the discussion of these cases to conform to spatial limitations.
## 5 Discussion and Future Work
Our approach to barycentric interpolation with canonical forms of positive
geometries unifies earlier approaches using adjoint polynomials, dual volumes
and Shepard-like interpolation. An advantage of this framework is that it
extends to positive geometries other than polytopes, such as Grassmannians and
toric varieties.
We also mention that the definition of a positive geometry embeds the domain
into a higher-dimensional complex manifold, thus our approach can be viewed as
a multivariate generalization of complex analytical methods (contour
integrals, residues) used in univariate approximation theory [16].
### 5.1 Open Problem: Interpolation in Grassmannians
As was mentioned previously, certain “positive” subsets of Grassmannians are
also examples of positive geometries. Points in a Grassmannian $G(k,n)$ can be
represented by $k\times n$ matrices, and the positive geometry known as the
_positive Grassmannian_ corresponds to _totally positive_ matrices, which are
of great interest for approximation theory and geometric modeling [5].
The Grassmannian of 2-planes in 4-space, $G(2,4)$ – which is also the manifold
of lines within 3-space – is particularly interesting for many applications
[12]. $G(2,4)$ is a 4-dimensional hypersurface in $\mathbb{P}^{5}$ cut out by
a quadratic equation in Plücker coordinates, and the positive Grassmannian is
a semi-algebraic subset with non-linear boundaries. The line-geometric
analogue of a simplex might be related to the _tetrahedral line complex_ , the
boundary of which is the set of lines defined by the edges of a tetrahedron.
The combinatorial structure of this boundary is that of an octahedron – an
example of a _hypersimplex_ – as shown in Figure 5, where each vertex
represents a line along an edge of the tetrahedron, while each face represents
lines through either a vertex or a face [6].
Figure 5: Tetrahedral line complex and corresponding hypersimplex. Shaded
faces correspond to lines through vertices.
Being a positive geometry, the positive Grassmannian has a canonical form,
i.e. a rational function of Plücker coordinates with poles along its
boundaries [1, Ch. 5.5]. This suggests that generalizations of barycentric
coordinates into line space might be possible using our framework.
### 5.2 Open Problem: Generalized Positive Geometries for Mean-Value
Coordinates
Canonical forms are defined by rational functions, so Mean-Value Coordinates
(MVCs) [4] – defined by transcendental functions – are apparently incompatible
with the proposed framework. We could adapt the approach of [14], where MVCs
are expressed as dual Shepard interpolants, after deforming the original
boundary to a unit circle around the current point. A disc is not a positive
geometry in the usual sense – it lacks zero-dimensional boundary components,
for example. Nevertheless, we can easily find a projectively well-defined
function with singularities along a projective conic $\mathcal{C}$ given by
the quadratic equation $\mathbf{x}^{T}\mathbf{Q}\mathbf{x}=0$ (see [1, Ch. 10]
for a lengthy discussion):
$\displaystyle\Omega(\mathbb{P}^{2},\mathcal{C})=\frac{\pi\det(\mathbf{Q})^{\frac{3}{4}}}{(\mathbf{x}^{T}\mathbf{Q}\mathbf{x})^{\frac{3}{2}}}\omega(\mathbf{x}).$
(5.13)
Observe the similarity with the transfinite form of MVCs [4, Ch. 10], when
$\mathcal{C}$ is a circle. This kind of canonical function is not rational and
its singularities along $\mathcal{C}$ are not simple poles, but _branch
points_ (similar to the origin of the complex plane for fractional powers and
logarithms). The authors of [1] also identified such transcendental
generalizations of positive geometries as promising subjects for future
research.
### 5.3 Open Problem: Relation to Splines and Integral Geometry
Formulas such as the dual volume (3.9) also appear in the context of
multivariate (box/simplex/cone) splines [3], as Laplace transforms of
indicator functions. This suggests that splines and barycentric interpolants
could both fit within an even more general theoretical framework related to
integral geometry.
## Acknowledgements
This project has been supported by the Hungarian Scientific Research Fund
(OTKA, No.124727). The author thanks Tamás Várady for support and Péter Salvi
for interesting discussions.
## References
* [1] N. Arkani-Hamed, Y. Bai, and T. Lam. Positive geometries and canonical forms. Journal of High Energy Physics, 2017(11):39, 2017.
* [2] G. Dasgupta and E. Wachspress. The adjoint for an algebraic finite element. Computers & Mathematics with Applications, 55(9):1988–1997, 2008\.
* [3] C. De Concini and C. Procesi. Topics in Hyperplane Arrangements, Polytopes and Box-Splines. Springer, 2010.
* [4] M.S. Floater. Generalized barycentric coordinates and applications. Acta Numerica, 24:161–214, 2015.
* [5] M. Gasca and C.A. Micchelli, editors. Total Positivity and Its Applications. Kluwer, 1996.
* [6] I.M. Gelfand and R.D. MacPherson. Geometry in Grassmannians and a generalization of the dilogarithm. Advances in mathematics, 44(3):279–312, 1982.
* [7] P. Griffiths and J. Harris. Principles of Algebraic Geometry. John Wiley & Sons, 1978.
* [8] K. Hormann. Barycentric interpolation. In Approximation Theory XIV: San Antonio 2013. Springer, 2014.
* [9] T. Ju, S. Schaefer, J. Warren, and M. Desbrun. A geometric construction of coordinates for convex polyhedra using polar duals. In Proceedings of the 3rd Eurographics Symposium on Geometry Processing. Eurographics, 2005.
* [10] K. Kohn and K. Ranestad. Projective geometry of Wachspress coordinates. arXiv preprint arXiv:1904.02123, 2019.
* [11] M. Meyer, A. Barr, H. Lee, and M. Desbrun. Generalized barycentric coordinates on irregular polygons. Journal of graphics tools, 7(1):13–22, 2002.
* [12] H. Pottmann and J. Wallner. Computational Line Geometry. Mathematics and Visualization. Springer, 2001.
* [13] V.L. Rvachev, T.I. Sheiko, V. Shapiro, and I. Tsukanov. Transfinite interpolation over implicitly defined sets. Computer Aided Geometric Design, 18(3):195–220, 2001.
* [14] S. Schaefer, T. Ju, and J. Warren. A unified, integral construction for coordinates over closed curves. Computer Aided Geometric Design, 24(8-9):481–493, 2007.
* [15] F. Sottile. Toric ideals, real toric varieties, and the algebraic moment map. In Topics in Algebraic Geometry and Geometric Modeling. AMS, 2003\.
* [16] L.N. Trefethen. Approximation Theory and Approximation Practice. SIAM, 2013.
* [17] E. Wachspress. Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics. Springer, 2016.
* [18] J. Warren. Barycentric coordinates for convex polytopes. Advances in Computational Mathematics, 6(1):97–108, 1996.
|
# When the ends don’t justify the means: Learning a treatment strategy to
prevent harmful indirect effects
Kara E. Rudolph
Department of Epidemiology The authors gratefully acknowledge R00DA042127
from the National Institute on Drug Abuse Columbia University
and
Iván Díaz
Department of Population Health Sciences Weill Cornell Medicine
###### Abstract
There is a growing literature on finding so-called optimal treatment rules,
which are rules by which to assign treatment to individuals based on an
individual’s characteristics, such that a desired outcome is maximized. A
related goal entails identifying individuals who are predicted to have a
harmful indirect effect (the effect of treatment on an outcome through
mediators) even in the presence of an overall beneficial effect of the
treatment on the outcome. In some cases, the likelihood of a harmful indirect
effect may outweigh a likely beneficial overall effect, and would be reason to
caution against treatment for indicated individuals. We build on both the
current mediation and optimal treatment rule literature to propose a method of
identifying a subgroup for which the treatment effect through the mediator is
harmful. Our approach is nonparametric, incorporates post-treatment variables
that may confound the mediator-outcome relationship, and does not make
restrictions on the distribution of baseline covariates, mediating variables
(considered jointly), or outcomes. We apply the proposed approach to identify
a subgroup of boys in the Moving to Opportunity housing voucher experiment who
are predicted to have harmful indirect effects, though the average total
effect is beneficial.
Keywords: dynamic treatment regime, optimal individualized treatment regime,
cross-validation, causal inference, optimal rule, mediation, interventional
indirect effect
## 1 Introduction
When individuals differ in their responses to a treatment or intervention, it
may be desirable to use individual-level information to predict for whom
treatments/interventions may work well versus not. This goal has fueled the
growing literature on finding so-called optimal individualized treatment
rules/regimes, which assign treatment based on an individual’s characteristics
such that a desired outcome is maximized (Murphy, 2003). For a single point in
time and a binary treatment, the optimal treatment rule is often defined as
the maximizer over all possible treatment rules of the counterfactual mean
outcome observed under a hypothetical implementation of the rule (Murphy,
2003; Robins, 2003).
Nabi et al. (2018) and Shpitser and Sherman (2018) recently extended these
methods to finding a treatment rule that optimizes a path-specific effect.
Nabi et al. (2018) focus, in particular, on optimizing a direct effect (an
effect of treatment on an outcome, not through a mediator); in their
application, the direct effect of interest is the effect of a treatment not
operating through adherence. A related goal entails identifying individuals
who are predicted to have a harmful indirect effect (the effect of treatment
on an outcome through mediators). Such a goal would be of interest if the
process by which an effect of a treatment or intervention might occur may be
traumatic or otherwise harmful, or if the harmful indirect effect is
indicative of more general harmful social processes, and consequently
desirable to avoid. It is possible for harmful indirect effects to be present
even if the overall effect of treatment on the outcome is beneficial,
sometimes referred to as inconsistent mediation (MacKinnon et al., 2000). In
some cases, the likelihood of a harmful indirect effect may outweigh a likely
beneficial overall effect and would be reason to caution against treatment for
indicated individuals.
The Moving to Opportunity study (MTO) presents an example where it may be of
interest to identify those with potentially harmful indirect effects. In MTO,
families were randomized to receive a Section 8 housing voucher that would
allow them to move out of public housing and into a rental on the private
market (Kling et al., 2007). Participants were followed for 10-15 years and a
broad array of outcomes related to health, education and income were assessed.
Among the children, boys in the voucher group were generally found to have
negative effects on mental health and risk behaviors (though this was not true
for all such outcomes) whereas girls were generally found to have positive
effects (Clampet-Lundquist et al., 2011; Sanbonmatsu et al., 2011; Schmidt et
al., 2017; Kessler et al., 2014; Rudolph et al., 2018; Kling et al., 2007;
Rudolph et al., 2020). Much of the negative effects were explained by
mediators related to the neighborhood and school environments and instability
of the social environment (Rudolph et al., 2020). However, measures of the
school and neighborhood environments appeared to generally improve among those
receiving the intervention, indicating that the “objective” mediator measures
of a school’s academic performance, poverty level, etc., may in fact be
surrogates for harmful processes like discrimination and increased risk of
suspensions/expulsions for some (Rudolph et al., 2020). In the context of MTO,
one could argue that a slightly beneficial long-term overall effect of housing
voucher receipt on reduced risk of a particular outcome, e.g., adolescent
alcohol use, may be outweighed by harmful indirect effects, as these may
indicate the occurrence of harmful social processes more generally.
Such scenarios motivate identifying those predicted to have harmful indirect
effects, and for that subgroup, treatment may be reconsidered. In this sense,
such subgroup identification is descriptive rather than prescriptive. Although
the approach we propose is mathematically grounded in the optimal
individualized treatment rule literature, we avoid using the phrase “treatment
rule” in describing it, because our subgroup identification is not a treatment
recommendation (in part because the overall treatment effect may still be
positive). A decision to treat (or not) individuals in this subset would
likely be based on a comprehensive assessment involving aspects like the
predicted direct effect as well as ethics, costs, and qualitative measures.
In this paper, we build on both the current mediation and optimal
individualized treatment rule literature to propose a method of identifying a
subgroup who may have an unintended harmful indirect effect, and then estimate
how the population path-specific effects would hypothetically change if the
identified subgroup were not treated. Our approach is nonparametric,
incorporates an ensemble of machine learning algorithms in model fitting,
incorporates post-treatment variables that may confound the mediator-outcome
relationship, and does not make restrictions on the distribution of baseline
covariates, mediating variables (considered jointly), or outcomes. We apply
the proposed approach to identify a subgroup of boys in the MTO housing
voucher experiment who are predicted to have harmful indirect effects, though
the average total effect on adolescent alcohol use is beneficial.
## 2 Notation
Let $O=(W,A,Z,M,Y)$ represent observed data, where $W$ is a vector of baseline
variables; $A$ is a binary treatment representing housing voucher receipt; $M$
represents mediating variables related to the school, neighborhood, and social
environments; $Y$ represents the outcome of alcohol use in adolescence, and
$Z$ represents post-treatment variables that may confound the $M\rightarrow Y$
relationship (in the case of MTO, moving with the housing voucher). Let
$O_{1},...,O_{n}$ represent the sample of $n$ i.i.d. observations of $O$. We
use capital letters to denote random variables and lowercase letters to denote
realizations of those variables. Let $\mathsf{P}$ represent the distribution
of $O$. We assume the data are generated according to the following
nonparametric structural equation model (Pearl, 2000):
$W=f_{W}(U_{W});A=f_{A}(W,U_{A});Z=f_{Z}(A,W,U_{Z});\\\
M=f_{M}(Z,A,W,U_{M});Y=f_{Y}(M,Z,A,W,U_{Y}),$ (1)
where each $U$ represents unmeasured, exogenous factors and each $f$
represents an unknown, deterministic function.
Our method to identify subsets of individuals at risk of a harmful indirect
effect are defined in terms of counterfactual outcomes under interventions on
the treatment and mediator. Specifically, we define a counterfactual outcome
under interventions that set the treatment and mediator to $(A,M)=(a,m)$ as
$Y_{a,m}=f_{Y}(m,Z_{a},a,W,U_{Y})$, where $Z_{a}=f_{Z}(a,W,U_{Z})$ is the
counterfactual variable $Z$ observed under an intervention setting $A=a$.
Direct and indirect effects are not generally point-identified in the presence
of a variable $Z$ that confounds the mediator-outcome relation and that is
affected by treatment (Avin et al., 2005). We overcome this problem using so-
called interventional effects (VanderWeele et al., 2014; Vansteelandt and
Daniel, 2017; Zheng and van der Laan, 2017; Lok, 2016, 2019; Rudolph et al.,
2017; Didelez et al., 2006), which rely on interventions on that set the
mediator to a random draw from its counterfactual distribution conditional on
baseline variables. Specifically, we let $M_{a}=f_{M}(Z_{a},a,W,U_{M})$, and
use $G_{a}$ to denote a random draw from the distribution of $M_{a}$
conditional on $W$.
We define $\mathsf{P}f=\int f(o)\mathrm{d}\mathsf{P}(o)$ for a given function
$f(o)$. We assume $\mathsf{P}$ is continuous with respect to some dominating
measure $\nu$ and let $\mathsf{p}$ denote the corresponding probability
density function. We will also use the following notation:
$\mathsf{b}(a,z,m,w)$ denotes $\mathsf{E}(Y\mid A=a,Z=z,M=m,W=w)$;
$\mathsf{g}(a\mid w)$ denotes $\mathsf{P}(A=a\mid W=w)$; $\mathsf{e}(a\mid
m,w)$ denotes $\mathsf{P}(A=a\mid M=m,W=w)$; $\mathsf{q}(z\mid a,w)$ denotes
$\mathsf{P}(Z=z\mid A=a,W=w)$; and $\mathsf{r}(z\mid a,m,w)$ denotes
$\mathsf{P}(Z=z\mid A=a,M=m,W=w)$.
## 3 Identification of the subgroup with a predicted harmful indirect effect
### 3.1 Estimand and identification
Throughout, we use the example from MTO in which a reduction in risk of
subsequent adolescent alcohol use is the outcome of interest; thus, negative
risk differences are considered “beneficial” and positive risk differences are
considered “harmful”. The definition of “harmful” for what follows should be
changed to reflect the particular research question.
As noted above, we use population interventional direct and indirect effects.
We only provide their definition here, as they have have been described
extensively elsewhere (Petersen et al., 2006; van der Laan and Petersen, 2008;
Zheng and van der Laan, 2012; VanderWeele et al., 2014; Rudolph et al., 2017;
VanderWeele and Tchetgen Tchetgen, 2017; Díaz et al., 2019). The population
interventional indirect effect,
$\mathsf{E}(Y_{1,G_{1}})$-$\mathsf{E}(Y_{1,G_{0}})$, compares the expected
average counterfactual outcomes under hypothetical interventions in which the
treatment is fixed but the mediator is changed from $G_{1}$ to $G_{0}$. This
corresponds to the effect of treatment on the outcome operating through $M$,
and is discussed further in VanderWeele and Tchetgen Tchetgen (2017). In terms
of our MTO research question, the population interventional indirect effect is
interpreted as the average difference in predicted risk of adolescent alcohol
use setting voucher to be received and drawing mediator values from their
counterfactual joint distribution if the voucher had been received versus had
it not been received; or more simply, the effect of voucher receipt on the
long-term risk of adolescent alcohol use that operates through aspects of the
school, neighborhood, and social environments.
The subgroup predicted to not have a harmful indirect effect (where “harmful”
is indicated by a positive effect, as described above) can be identified by
the target parameter
$\mathsf{d}(v)=\mathds{1}(\mathsf{B}(v)\leq 0),$
where $\mathsf{B}(v)$ is the average indirect effect conditional on a subset
of baseline variables $V\subseteq W$, and is defined as
$\mathsf{B}(v)=\mathsf{E}(Y_{1,G_{1}}-Y_{1,G_{0}}\mid V=v).$
In what follows we let $W^{c}=W\backslash V$. The function $\mathsf{B}$ is
often referred to as a “blip” function (Robins, 2004), and is a predictive
function that takes covariates $V$ as input and outputs the conditional
indirect effect of treatment on an additive scale. The function
$\mathsf{d}(v)$ is then an indicator of a predicted nonharmful indirect effect
for covariate profile $V=v$. In this case, if $\mathsf{B}(v)\leq 0,$ then
individuals in strata $V=v$ are indicated as not having a predicted harmful
effect; if $\mathsf{B}(v)>0,$ then individuals are indicated as having a
predicted harmful effect.
We introduce the following assumptions, which will allow us to identify the
causal parameters $\mathsf{B}(v)$ and $\mathsf{d}(v)$:
###### Assumption 1 (No unmeasured confounders of the $A\rightarrow Y$
relation).
$Y_{1,m}\mbox{$\perp\\!\\!\\!\perp$}A\mid W$
###### Assumption 2 (No unmeasured confounders of the $A\rightarrow M$
relation).
$M_{a}\mbox{$\perp\\!\\!\\!\perp$}A\mid W$, for $a\in\\{0,1\\}$.
###### Assumption 3 (No unmeasured confounders of the $M\rightarrow Y$
relation).
$Y_{1,m}\mbox{$\perp\\!\\!\\!\perp$}M\mid(W,A,Z)$.
###### Assumption 4 (Positivity).
$\mathsf{p}(W)>0$ implies $\mathsf{p}(a\mid W)>0$, $\mathsf{p}(M\mid
a^{\star},W)>0$ and $\mathsf{p}(Z\mid a^{\prime},W)>0$ imply $\mathsf{p}(M\mid
a^{\prime},Z,W)>0$ with probability one for $(a^{\prime},a^{\star})=(1,1)$ and
$(a^{\prime},a^{\star})=(1,0)$.
In the context of the conditional indirect effect of MTO housing voucher
receipt on subsequent risk of adolescent alcohol use through the school,
neighborhood and social environments, Assumptions 1 and 2 are expected to
hold, as the intervention is randomized. Assumption 3 may not hold, most
likely due to unmeasured post-randomization confounding variables, such as
those related to changes in parental employment, income, parent-child
dynamics, etc. However, we do include a large number of baseline covariates
related parental socioeconomic status, motivations for enrolling in MTO, and
relationships; child characteristics; and neighborhood characteristics. We
also include an indicator representing moving with the housing voucher, which
could be one such post-treatment confounder. Lastly, we include numerous
mediating variables capturing aspects of the school, neighborhood and social
environments. Any unmeasured variables would need to contribute to mediator-
outcome confounding independently of the aforementioned variables to violate
Assumption 3.
Under Assumptions 1-4, the conditional indirect effect is identified from the
observed data as follows. First, VanderWeele and Tchetgen Tchetgen (2017) show
that for any values $(a^{\prime},a^{*})\in\\{0,1\\}^{2}$,
$\mathsf{E}(Y_{a^{\prime},G_{a^{*}}}\mid V=v)$ is identified as
$\mathsf{E}(Y_{a^{\prime},G_{a^{*}}}\mid
V=v)=\int\mathsf{b}(a^{\prime},z,m,w)\mathsf{q}(z\mid
a^{\prime},w)\mathsf{p}(m\mid a^{*},w)\mathsf{p}(w^{c}\mid
v)\mathrm{d}\nu(w^{c},z,m).$
And thus we have
$\mathsf{B}(v)=\int\mathsf{b}(1,z,m,w)\mathsf{q}(z\mid 1,w)\\{\mathsf{p}(m\mid
1,w)-\mathsf{p}(m\mid 0,w)\\}\mathsf{p}(w^{c}\mid v)\mathrm{d}\nu(w^{c},z,m).$
(2)
#### Alternative interpretation in terms of the optimal dynamic treatment
rule.
Although in the context of identifying boys in MTO who are predicted to have
harmful indirect effects, we believe it would be inappropriate to interpret
$\mathsf{d}(v)$ as a treatment rule, we recognize that other contexts exist in
which such an interpretation may be justified. In these cases, $\mathsf{d}(v)$
may be alternatively defined in terms of an optimal individualized dynamic
treatment rule:
$\mathsf{d}(v)\in\operatorname*{arg\,min}_{\mathsf{d}^{\prime}\in\mathcal{D}}\mathsf{E}(Y_{\mathsf{d}^{\prime},G_{\mathsf{d}^{\prime}}})-\mathsf{E}(Y_{\mathsf{d}^{\prime},G_{0}}),$
where, for any $\mathsf{d}^{\prime}$ and $\mathsf{d}^{\star}$,
$Y_{\mathsf{d}^{\prime},G_{\mathsf{d}^{\star}}}$ is the counterfactual outcome
in a hypothetical world where treatment is set to the rule
$\mathsf{d}^{\prime}(V)\in\\{0,1\\}$ and the mediator is set to a random draw
$G_{\mathsf{d}^{\star}}$ from the distribution of $M$ conditional on
$A=\mathsf{d}^{\star}(V)$. Here $\mathcal{D}$ is the space of all functions
that map the covariates $V$ to a treatment decision rule in $\\{0,1\\}$.
### 3.2 Estimation
As pointed out in the Introduction, estimation of $\mathsf{B}(v)$ and
$\mathsf{d}(v)$ are equivalent to estimation of conditional average effects
(here, the conditional interventional indirect effect) and estimation of
optimal treatment rules. Consequently, estimation techniques such as
Q-learning (Murphy, 2003; Qian and Murphy, 2011; Moodie et al., 2012; Laber et
al., 2014), outcome-weighted learning (OWL) (Zhang et al., 2012; Zhao et al.,
2012, 2015), and doubly robust techniques (Luedtke and van der Laan, 2016;
Díaz et al., 2018; Kennedy, 2020) may be used. Q-learning is a regression-
based approach related to g-computation that relies on sequential regression
formulas. Q-learning is not directly applicable to our problem because
Equation 2 does not readily yield a sequential regression representation. OWL
uses inverse probability weights to recast the problem of estimating
$\mathsf{d}(v)$ as a weighted classification problem. Although we could use
OWL to estimate $\mathsf{B}(v)$ and $\mathsf{d}(v)$ here, we instead choose to
use a doubly robust approach that combines regressions with inverse
probability weights to obtain an estimator that remains consistent under
certain configurations of inconsistent estimation of nuisance parameters. This
approach relies on so-called unbiased transformations, which we define below.
###### Definition 1 (Unbiased transformation).
The function $D(o)$ is an unbiased transformation for $\mathsf{B}(v)$ if it
satisfies $\mathsf{E}[D(O)\mid V=v]=\mathsf{B}(v)$.
In particular, we use a multiply robust unbiased transformation (Rubin and van
der Laan, 2007). For the case of optimal treatment rules, multiply robust
unbiased transformations are constructed using the efficient influence
function (EIF) of the average treatment effect (Luedtke and van der Laan,
2016; Díaz et al., 2018; Kennedy, 2020). We take a similar approach here.
Specifically, the EIF for the counterfactual mean
$\mathsf{E}(Y_{a^{\prime},G_{a}^{*}})$ under Assumptions 1-4 was derived by
Díaz et al. (2019) as follows. Define
$\displaystyle\mathsf{h}(z,m,w)$
$\displaystyle=\frac{\mathsf{g}(a^{\prime}\mid w)}{\mathsf{g}(a^{\star}\mid
w)}\frac{\mathsf{q}(z\mid a^{\prime},w)}{\mathsf{r}(z\mid
a^{\prime},m,w)}\frac{\mathsf{e}(a^{\star}\mid m,w)}{\mathsf{e}(a^{\prime}\mid
m,w)}$ $\displaystyle\ \mathsf{u}(z,w)$
$\displaystyle=\mathsf{E}\left\\{\mathsf{b}(a^{\prime},Z,M,W)\mathsf{h}(Z,M,W),\mid\,Z=z,A=a^{\prime},W=w\right\\},$
$\displaystyle\mathsf{v}(w)$
$\displaystyle=\mathsf{E}\left\\{\int_{\mathcal{Z}}\mathsf{b}(a^{\prime},z,M,W)\mathsf{q}(z\mid
a^{\prime},W)\mathrm{d}\nu(z)\,\mid\,A=a^{\star},W=w\right\\}.$
Let
$\eta=(\mathsf{g},\mathsf{e},\mathsf{q},\mathsf{r},\mathsf{b},\mathsf{u},\mathsf{v})$
denote a vector of nuisance parameters. Then the EIF for
$\mathsf{E}(Y_{a^{\prime},G_{a}^{*}})$ under Assumptions 1-4 is equal to
$\displaystyle D_{\eta}^{(a^{\prime},a^{\star})}(o)$
$\displaystyle=\frac{\mathds{1}\\{a=a^{\prime}\\}}{\mathsf{g}(a^{\prime}\mid
w)}\mathsf{h}(z,m,w)\\{y-\mathsf{b}(a^{\prime},z,m,w)\\}$
$\displaystyle+\frac{\mathds{1}\\{a=a^{\prime}\\}}{\mathsf{g}(a^{\prime}\mid
w)}\\{\mathsf{u}(1,w)-\mathsf{u}(0,w)\\}\left\\{z-\mathsf{q}(1\mid
a^{\prime},w)\right\\}$
$\displaystyle+\frac{\mathds{1}\\{a=a^{\star}\\}}{\mathsf{g}(a^{\star}\mid
w)}\left\\{\int\mathsf{b}(a^{\prime},z,m,w)\mathsf{q}(z\mid
a^{\prime},w)\mathrm{d}\nu(z)-\mathsf{v}(w)\right\\},$
$\displaystyle+\mathsf{v}(w).$
Thus, we use $D_{\eta}(o)=D_{\eta}^{(1,1)}(o)-D_{\eta}^{(1,0)}(o)$ as an
unbiased transformation for $\mathsf{B}(v)$. For a fixed
$\eta_{1}=(\mathsf{g}_{1},\mathsf{e}_{1},\mathsf{q}_{1},\mathsf{r}_{1},\mathsf{b}_{1},\mathsf{u}_{1},\mathsf{v}_{1})$,
Díaz et al. (2019) show that $D_{\eta_{1}}(o)$ satisfies Definition 1 as long
as
1. 1.
$\mathsf{v}_{1}=\mathsf{v}$ and either
$(\mathsf{q}_{1},\mathsf{e}_{1},\mathsf{r}_{1})=(\mathsf{q},\mathsf{e},\mathsf{r})$
or $(\mathsf{b}_{1},\mathsf{q}_{1})=(\mathsf{b},\mathsf{q})$, or
2. 2.
$\mathsf{g}_{1}=\mathsf{g}$ and either
$(\mathsf{q}_{1},\mathsf{e}_{1},\mathsf{r}_{1})=(\mathsf{q},\mathsf{e},\mathsf{r})$
or $(\mathsf{b}_{1},\mathsf{q}_{1})=(\mathsf{b},\mathsf{q}).$
In this sense, using a multiply robust unbiased transformation provides
robustness to misspecification of some models, hence the name.
Our estimator proceeds by obtaining an estimate, $\hat{\eta}$, of $\eta$. All
nuisance parameters are estimated using an ensemble regression approach known
as the Super Learner (Van der Laan et al., 2007; van der Laan & S. Dudoit &
A.W. van der Vaart, 2006), which creates an optimally weighted combination of
algorithms. Then, the pseudo-outcome $D_{\hat{\eta}}(O_{i})$ is computed for
all individuals in the sample, and a regression of the pseudo-outcome on
baseline covariates $V$ is fitted. Following the approach of Luedtke and van
der Laan (2016), we also use Super Learner in fitting the regression of
$D_{\hat{\eta}}(O)$ on $V$ and use the fitted values to obtain predictions
$\hat{\mathsf{B}}(v)$. The group of individuals at risk for a harmful indirect
effect is identified as those indices $i$ with
$\hat{\mathsf{d}}(v_{i})=\mathds{1}\\{\hat{\mathsf{B}}(v_{i})>0\\}$.
We use cross-fitting (Klaassen, 1987; Zheng and van der Laan, 2011;
Chernozhukov et al., 2016) throughout the estimation process in all fits. Let
${\cal V}_{1},\ldots,{\cal V}_{J}$ denote a random partition of data with
indices $i\in\\{1,\ldots,n\\}$ into $J$ prediction sets of approximately the
same size such that $\bigcup_{j=1}^{J}{\cal V}_{j}=\\{1,\ldots,n\\}$. For each
$j$, the training sample is given by ${\cal
T}_{j}=\\{1,\ldots,n\\}\setminus{\cal V}_{j}$. $\hat{\eta}_{j}$ denotes the
estimator of $\eta$, obtained by training the corresponding prediction
algorithm using only data in the sample ${\cal T}_{j}$, and $j(i)$ denotes the
index of the validation set which contains observation $i$. We then use these
fits, $\hat{\eta}_{j(i)}(O_{i})$ in computing $D_{\hat{\eta}_{j(i)}}(O_{i})$.
Likewise, regressions of the pseudo-outcome $D_{\hat{\eta}_{j(i)}}(O_{i})$ on
$V_{i}$ are trained within each training sample, and the final estimate is
computed by predicting in the corresponding validation data set.
## 4 Estimating the indirect effect under a hypothetical treatment decision
$\mathsf{d}(v)$
As stated in the Introduction, our goal is to identify a subset of the
population that is predicted to have a harmful indirect effect, and for whom
treatment may be carefully considered or reconsidered. Even though our goal is
not to develop a treatment rule, in some situations it may be important to
assess the population effects that would be observed if the function
$\mathsf{d}$ were used as a treatment rule. To that end, in this section we
estimate the hypothetical population interventional indirect effect if we were
to use $\mathsf{d}(V)$ to assign treatment to each individual. Specifically,
we define the total effect of implementing $\mathsf{d}(V)$ as
$\mathsf{E}(Y_{\mathsf{d}}-Y_{0})$, and decompose it in terms of direct and
indirect effects as
$\mathsf{E}(Y_{\mathsf{d}}-Y_{0})=\underbrace{\mathsf{E}(Y_{\mathsf{d},G_{\mathsf{d}}}-Y_{\mathsf{d},G_{0}})}_{\text{indirect
effect}}+\underbrace{\mathsf{E}(Y_{\mathsf{d},G_{0}}-Y_{0,G_{0}})}_{\text{direct
effect}}.$
The population interventional indirect effect can be identified under the
sequential randomization assumptions and positivity in Section 3.1 as:
$\mathsf{E}(Y_{\mathsf{d},G_{\mathsf{d}}}-Y_{\mathsf{d},G_{0}})=\\\
\int\mathsf{b}(\mathsf{d}(v),z,m,w)\mathsf{q}(z\mid\mathsf{d}(v),w)\\{\mathsf{p}(m\mid\mathsf{d}(v),w)-\mathsf{p}(m\mid
0,w)\\}\mathsf{p}(w)\mathrm{d}\nu(w,z,m).$ (3)
Because we do not have the true, unknown $\mathsf{d}$, but instead have an
estimate of it, $\hat{\mathsf{d}}$, we estimate the population indirect effect
$\mathsf{E}(Y_{\hat{\mathsf{d}},G_{\hat{\mathsf{d}}}}-Y_{\hat{\mathsf{d}},G_{0}})$
that would be observed if our estimate were implemented. We use the one-step
estimation approach described in Díaz et al. (2019), and which is based on
solving the EIF estimating equation. Specifically, we let our estimator be
defined as
$\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}\\{D_{\hat{\eta}}^{\hat{\mathsf{d}},\hat{\mathsf{d}}}(O_{i})-D_{\hat{\eta}}^{\hat{\mathsf{d}},0}(O_{i})\\},$
We use a cross-fitted version of this estimator, as described in Section 3.2.
The variance of this estimator can be estimated as the sample variance of the
EIF. Theorem 5 of van der Laan and Luedtke (2015) proves that plugging in
$\hat{\mathsf{d}}$ estimated from the data results in asymptotically linear
estimation of its effect, even though the same data were used to estimate
$\hat{\mathsf{d}}$ and to assess its effect.
The R code to implement this cross-fitted estimator is available:
blindedforreview.
## 5 Identifying those with predicted harmful indirect effects
We now apply our proposed approach to identify the subgroup of boys with a
predicted harmful indirect effect of voucher receipt on adolescent alcohol use
through aspects of the school, neighborhood, and social environments. In this
case, the average treatment effect of Section 8 housing voucher receipt on
long-term risk of adolescent alcohol use among boys is beneficial,
contributing to decreased risk, whereas the indirect effect through mediators
related to the school and neighborhood environments and instability of the
social environment is estimated to contribute to an increased risk of
subsequent adolescent alcohol use. There is evidence that improvements in
measures of the school and neighborhood environment (e.g., neighborhood
poverty, school rank, school poverty) may actually negatively impact a
subgroup of boys, possibly via increased risk of suspensions/expulsions and
less social support (Rudolph et al., 2020), which in turn, may increase risk
of a variety of negative mental health and substance use outcomes (Rudolph et
al., 2020).
If one is interested in optimizing the overall effect, one could apply
existing methods to identify the subgroup who may be harmed by the
intervention (Luedtke and van der Laan, 2016; Robins, 2004; Zhao et al.,
2012). However, instances where the overall effect is beneficial, but the
indirect effect is harmful—like this one—motivate our proposed approach to
identify a subgroup predicted to experience an unintended, harmful indirect
effect, even in the presence of a predicted beneficial overall effect. In this
application, we 1) identify the subgroup of boys who would be predicted to
have a harmful indirect effect and thus, for whom voucher receipt may not be
recommended, and 2) estimate the hypothetical indirect effect and hypothetical
total effect had the identified subgroup not received the intervention.
A limitation of using so-called “black-box” algorithms proposed in Section 3.2
for this task is that we are left without knowing which characteristics are
important in predicting whether an indirect effect will be positive or
negative. Consequently, we also use a recently proposed adaptive lasso
(Bahamyirou et al., 2020) to learn less-than-optimal, but interpretable
$\mathsf{d}$.
### 5.1 Data and Analysis
MTO was a large-scale experiment in which families living in low-income public
housing could sign-up to be randomized to receive a Section 8 housing voucher,
which is a form of housing assistance that subsidizes rent on the private
market, allowing families to move out of public housing. We restrict to
adolescent boys who were surveyed at the final follow-up timepoint in the
Boston, Chicago, New York City, and Los Angeles sites (N=2,100, rounded sample
size). We consider baseline covariates at the individual, family, and
neighborhood level ($W$, study site, age, race/ethnicity, number of family
members, previous problems in school, enrolled in special class for gifted and
talented students, parent is high school graduate, parent marital status,
parent work status, receipt of AFDC/TANF, whether any family member has a
disability, perceived neighborhood safety, neighborhood satisfaction,
neighborhood poverty level, reported reasons for participating in MTO,
previous number of moves, previous application for Section 8 voucher),
baseline randomized intervention status of received voucher or not ($A$,
binary 1/0), whether or not the family used the voucher to move ($Z$, binary
1/0), mediating variables representing aspects of the school and neighborhood
environments and instability of the social environment over the 10-15 years of
follow-up ($M$, school rank, student-to-teacher ratio, % students receiving
free or reduced-price lunch, % schools attended that were Title I, number of
schools attended, number of school changes within the year, whether or not the
most recent school was in the same district as the baseline school, number of
moves, neighborhood poverty; all weighted over the duration of follow-up), and
the long-term outcome of past-month alcohol use at the final timepoint, when
the children were adolescents ($Y$, binary 1/0). For the purposes of this
illustration, we use just one imputed dataset. As recommended for all MTO
analyses, we use individual-level weights that account for the sampling of
children within families, assignment ratios, and loss-to-follow-up
(Sanbonmatsu et al., 2011).
We first apply the estimation approach in Section 3.2 to identify the subgroup
$\\{i\in\\{1,\ldots,n\\}:\hat{\mathsf{d}}(V_{i})=0\\}$ of boys with a
predicted harmful indirect effect of voucher receipt on adolescent alcohol use
through aspects of the school, neighborhood, and social environments. We then
apply the estimation procedure in Section 4 to estimate indirect effects:
$\mathsf{E}(Y_{\hat{\mathsf{d}},G_{\hat{\mathsf{d}}}}-Y_{\hat{\mathsf{d}},G_{0}})$,
$\mathsf{E}(Y_{1,G_{1}}-Y_{1,G_{0}})$, and total effects:
$\mathsf{E}(Y_{\hat{\mathsf{d}},G_{\hat{\mathsf{d}}}}-Y_{0,G_{0}})$,
$\mathsf{E}(Y_{1,G_{1}}-Y_{0,G_{0}})$. We use five folds in cross-fitting, and
include: a simple mean model, main-terms generalized linear model, lasso,
multivariate adaptive regression splines, and extreme gradient boosted
machines in our Super Learner ensemble. Finally, we report the estimated
indirect effects and total effects using the alternative, interpretable
$\hat{\mathsf{d}}$ estimated using adaptive lasso.
### 5.2 Results
Figure 1 shows: the typical population interventional indirect effect (PIIE,
$\mathsf{E}(Y_{1,G_{1}}-Y_{1,G_{0}})$) estimates and total effect (PITE,
$\mathsf{E}(Y_{1,G_{1}}-Y_{0,G_{0}})$) estimates not using any
individualization, which are labeled as “no individualization”, and the PIIE
and PITE estimates using $\hat{\mathsf{d}}$ to hypothetically assign boys to
receive the voucher who were not predicted to have a harmful experience
through mediators of the school, neighborhood, and social environments
(learned via Super Learner), denoted “superlearner estimation”. Using Super
Learner to identify the subgroup predicted to experience a harmful indirect
effect and then not giving the voucher to that subgroup would be expected to
reduce the otherwise harmful indirect effect down to 0.0004 increased risk of
subsequent adolescent alcohol use (95% CI: -0.0690, 0.0698) as opposed to
0.0432 increased risk (95% CI: -0.1009, 0.1873). It would also be expected to
slightly decrease the beneficial total average effect of voucher assignment to
0.0224 reduced risk of subsequent adolescent alcohol use (-0.0224, 95% CI:
-0.1500, 0.1053) as opposed to 0.0683 reduced risk (95% CI: -0.2882, 0.1053).
Applying the adaptive lasso to estimate $\hat{d}$, we find that those whose
parent graduated from high school and whose race is not white, black, nor
Hispanic/Latino would be predicted to not have a harmful indirect effect. The
PIIE and PITE estimates using this $\hat{\mathsf{d}}$ are denoted “lasso
estimation”.
Figure 1: Population interventional indirect effects and total effects by rule
type. All results were approved for release by the U.S. Census Bureau,
authorization number CBDRB-FY21-ERD003-004.
## 6 Conclusions
We proposed a nonparametric approach to identify a subgroup who may have
unintended harmful indirect effects even in the presence of beneficial overall
effects. We then proposed a nonparametric estimator to estimate the
hypothetical population interventional indirect and total effects were one to
make the decision not to treat the subgroup predicted to have harmful indirect
effects. Our cross-fitted estimators solve the efficient influence function,
so are robust and can incorporate machine learning in model fitting.
This work was motivated by surprising results from a large, housing policy
intervention (the Moving to Opportunity Study) in which boys whose families
were randomized to receive a housing voucher had slightly reduced risk of
subsequent alcohol use as adolescents (total effect) but the indirect pathway
from voucher receipt to that positive effect operating through aspects of the
school and social environments was harmful. This harmful indirect effect could
reflect processes like boys whose families moved with the vouchers going to
schools with higher academic performance but where they were more likely to
face discriminatory practices, like suspension and expulsion, and more social
isolation (Rudolph et al., 2020). It is possible that some may wish to avoid
such negative processes, prompting the desire to identify those at risk for
them. In this example, we do so and find that not giving the intervention to
this subgroup would have indeed reduced the harmful predicted indirect effect
point estimate, though the confidence intervals are wide and overlapping.
In terms of identification and estimation, our proposal builds on both the
causal mediation and individualized optimal treatment regime literatures.
However, we caution against interpreting $\hat{\mathsf{d}}$ as a decision
rule, understanding that, at least in some instances, deciding whether or not
someone would benefit from an intervention/treatment is necessarily more
nuanced than the mediation mechanisms and total effects we consider. Future
work could focus on incorporating additional complexities and nuances into the
existing tools used in personalized medicine to reflect the multiple
mechanisms by which treatments or interventions may affect numerous, and
sometimes competing, outcomes among individuals, recognizing heterogeneities
(anticipated or not) at each stage in the process.
## References
* Avin et al. (2005) Avin, C., I. Shpitser, and J. Pearl (2005). Identifiability of path-specific effects.
* Bahamyirou et al. (2020) Bahamyirou, A., M. E. Schnitzer, E. H. Kennedy, L. Blais, and Y. Yang (2020). Doubly robust adaptive lasso for effect modifier discovery. arXiv preprint arXiv:2011.12746.
* Chernozhukov et al. (2016) Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, et al. (2016). Double machine learning for treatment and causal parameters. arXiv preprint arXiv:1608.00060.
* Clampet-Lundquist et al. (2011) Clampet-Lundquist, S., K. Edin, J. R. Kling, and G. J. Duncan (2011). Moving teenagers out of high-risk neighborhoods: How girls fare better than boys. American Journal of Sociology 116(4), 1154–89.
* Díaz et al. (2019) Díaz, I., N. S. Hejazi, K. E. Rudolph, and M. J. van der Laan (2019). Non-parametric efficient causal mediation with intermediate confounders. arXiv preprint arXiv:1912.09936.
* Díaz et al. (2018) Díaz, I., O. Savenkov, and K. Ballman (2018). Targeted learning ensembles for optimal individualized treatment rules with time-to-event outcomes. Biometrika 105(3), 723–738.
* Didelez et al. (2006) Didelez, V., A. P. Dawid, and S. Geneletti (2006). Direct and indirect effects of sequential treatments. In Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, pp. 138–146. AUAI Press.
* Kennedy (2020) Kennedy, E. H. (2020). Optimal doubly robust estimation of heterogeneous causal effects. arXiv preprint arXiv:2004.14497.
* Kessler et al. (2014) Kessler, R. C., G. J. Duncan, L. A. Gennetian, L. F. Katz, J. R. Kling, N. A. Sampson, L. Sanbonmatsu, A. M. Zaslavsky, and J. Ludwig (2014). Associations of housing mobility interventions for children in high-poverty neighborhoods with subsequent mental disorders during adolescence. JAMA 311(9), 937–948.
* Klaassen (1987) Klaassen, C. A. (1987). Consistent estimation of the influence function of locally asymptotically linear estimators. The Annals of Statistics, 1548–1562.
* Kling et al. (2007) Kling, J. R., J. B. Liebman, and L. F. Katz (2007). Experimental analysis of neighborhood effects. Econometrica 75(1), 83–119.
* Laber et al. (2014) Laber, E. B., K. A. Linn, and L. A. Stefanski (2014). Interactive model building for q-learning. Biometrika 101(4), 831–847.
* Lok (2016) Lok, J. J. (2016). Defining and estimating causal direct and indirect effects when setting the mediator to specific values is not feasible. Statistics in medicine 35(22), 4008–4020.
* Lok (2019) Lok, J. J. (2019). Causal organic direct and indirect effects: closer to baron and kenny. arXiv preprint arXiv:1903.04697.
* Luedtke and van der Laan (2016) Luedtke, A. R. and M. J. van der Laan (2016). Super-learning of an optimal dynamic treatment rule. The international journal of biostatistics 12(1), 305–332.
* MacKinnon et al. (2000) MacKinnon, D. P., J. L. Krull, and C. M. Lockwood (2000). Equivalence of the mediation, confounding and suppression effect. Prevention science 1(4), 173–181.
* Moodie et al. (2012) Moodie, E. E., B. Chakraborty, and M. S. Kramer (2012). Q-learning for estimating optimal dynamic treatment rules from observational data. Canadian Journal of Statistics 40(4), 629–645.
* Murphy (2003) Murphy, S. (2003). Optimal dynamic treatment regimes. Journal of the Royal Statistical Society: Series B 65(2), 331–66.
* Nabi et al. (2018) Nabi, R., P. Kanki, and I. Shpitser (2018). Estimation of personalized effects associated with causal pathways. In Uncertainty in artificial intelligence: proceedings of the… conference. Conference on Uncertainty in Artificial Intelligence, Volume 2018\. NIH Public Access.
* Pearl (2000) Pearl, J. (2000). Causality: models, reasoning, and inference. Cambridge University Press. ISBN 0 521(77362), 8.
* Petersen et al. (2006) Petersen, M. L., S. E. Sinisi, and M. J. van der Laan (2006). Estimation of direct causal effects. Epidemiology, 276–284.
* Qian and Murphy (2011) Qian, M. and S. A. Murphy (2011). Performance guarantees for individualized treatment rules. Annals of statistics 39(2), 1180.
* Robins (2003) Robins, J. (2003). Discussion of “optimal dynamic treatment regimes" by susan a. murphy. Journal of the Royal Statistical Society: Series B 65(2), 355–66.
* Robins (2004) Robins, J. M. (2004). Optimal structural nested models for optimal sequential decisions. In D. Y. Lin and P. J. Heagerty (Eds.), Proceedings of the second seattle Symposium in Biostatistics, pp. 189–326. Seattle, Washington: Springer Science+ Business Media.
* Rubin and van der Laan (2007) Rubin, D. and M. J. van der Laan (2007). A doubly robust censoring unbiased transformation. The international journal of biostatistics 3(1).
* Rudolph et al. (2020) Rudolph, K. E., C. Gimbrone, and I. Diaz (2020). Helped into harm: Mediation of the effect of a housing voucher intervention on long-term mental health and substance use outcomes in adolescent boys. arXiv preprint arXiv:2006.07708.
* Rudolph et al. (2018) Rudolph, K. E., O. Sofrygin, N. M. Schmidt, R. Crowder, M. M. Glymour, J. Ahern, and T. L. Osypuk (2018). Mediation of neighborhood effects on adolescent substance use by the school and peer environments. Epidemiology 29(4), 590–598. PMCID: PMC5987191.
* Rudolph et al. (2017) Rudolph, K. E., O. Sofrygin, W. Zheng, and M. J. Van Der Laan (2017). Robust and flexible estimation of stochastic mediation effects: A proposed method and example in a randomized trial setting. Epidemiologic Methods 7(1).
* Sanbonmatsu et al. (2011) Sanbonmatsu, L., L. F. Katz, J. Ludwig, L. A. Gennetian, G. J. Duncan, R. C. Kessler, E. K. Adam, T. McDade, and S. T. Lindau (2011). Moving to opportunity for fair housing demonstration program: Final impacts evaluation.
* Schmidt et al. (2017) Schmidt, N. M., M. M. Glymour, and T. L. Osypuk (2017). Adolescence is a sensitive period for housing mobility to influence risky behaviors: An experimental design. Journal of Adolescent Health 60(4), 431–437.
* Shpitser and Sherman (2018) Shpitser, I. and E. Sherman (2018). Identification of personalized effects associated with causal pathways. In Uncertainty in artificial intelligence: proceedings of the… conference. Conference on Uncertainty in Artificial Intelligence, Volume 2018\. NIH Public Access.
* van der Laan and Luedtke (2015) van der Laan, M. J. and A. R. Luedtke (2015). Targeted learning of the mean outcome under an optimal dynamic treatment rule. Journal of causal inference 3(1), 61–95.
* van der Laan and Petersen (2008) van der Laan, M. J. and M. L. Petersen (2008). Direct effect models. The international journal of biostatistics 4(1).
* Van der Laan et al. (2007) Van der Laan, M. J., E. C. Polley, and A. E. Hubbard (2007). Super learner. Statistical applications in genetics and molecular biology 6(1).
* van der Laan & S. Dudoit & A.W. van der Vaart (2006) van der Laan & S. Dudoit & A.W. van der Vaart, M. (2006). The cross-validated adaptive epsilon-net estimator. Statistics & Decisions 24(3), 373–395.
* VanderWeele and Tchetgen Tchetgen (2017) VanderWeele, T. J. and E. J. Tchetgen Tchetgen (2017). Mediation analysis with time varying exposures and mediators. Journal of the Royal Statistical Society. Series B, Statistical Methodology 79(3), 917.
* VanderWeele et al. (2014) VanderWeele, T. J., S. Vansteelandt, and J. M. Robins (2014). Effect decomposition in the presence of an exposure-induced mediator-outcome confounder. Epidemiology (Cambridge, Mass.) 25(2), 300.
* Vansteelandt and Daniel (2017) Vansteelandt, S. and R. M. Daniel (2017). Interventional effects for mediation analysis with multiple mediators. Epidemiology (Cambridge, Mass.) 28(2), 258.
* Zhang et al. (2012) Zhang, B., A. A. Tsiatis, M. Davidian, M. Zhang, and E. Laber (2012). Estimating optimal treatment regimes from a classification perspective. Stat 1(1), 103–114.
* Zhao et al. (2012) Zhao, Y., D. Zeng, A. J. Rush, and M. R. Kosorok (2012). Estimating individualized treatment rules using outcome weighted learning. Journal of the American Statistical Association 107(499), 1106–1118.
* Zhao et al. (2015) Zhao, Y.-Q., D. Zeng, E. B. Laber, and M. R. Kosorok (2015). New statistical learning methods for estimating optimal dynamic treatment regimes. Journal of the American Statistical Association 110(510), 583–598.
* Zheng and van der Laan (2017) Zheng, W. and M. van der Laan (2017). Longitudinal mediation analysis with time-varying mediators and exposures, with application to survival outcomes. Journal of causal inference 5(2).
* Zheng and van der Laan (2011) Zheng, W. and M. J. van der Laan (2011). Cross-validated targeted minimum-loss-based estimation. In Targeted Learning, pp. 459–474. Springer.
* Zheng and van der Laan (2012) Zheng, W. and M. J. van der Laan (2012). Targeted maximum likelihood estimation of natural direct effects. The international journal of biostatistics 8(1), 1–40.
|
# Machine learning time-local generators of open quantum dynamics
Paolo P. Mazza Institut für Theoretische Physik and Center for Quantum
Science, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen,
Germany Dominik Zietlow Max Planck Institute for Intelligent Systems, Max-
Planck-Ring 4, 72076 Tübingen, Germany Federico Carollo Institut für
Theoretische Physik and Center for Quantum Science, Universität Tübingen, Auf
der Morgenstelle 14, 72076 Tübingen, Germany Sabine Andergassen Institut für
Theoretische Physik and Center for Quantum Science, Universität Tübingen, Auf
der Morgenstelle 14, 72076 Tübingen, Germany Georg Martius Max Planck
Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany
Igor Lesanovsky Institut für Theoretische Physik and Center for Quantum
Science, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen,
Germany School of Physics and Astronomy, University of Nottingham,
Nottingham, NG7 2RD, UK Centre for the Mathematics and Theoretical Physics of
Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7
2RD, UK
###### Abstract
In the study of closed many-body quantum systems one is often interested in
the evolution of a subset of degrees of freedom. On many occasions it is
possible to approach the problem by performing an appropriate decomposition
into a bath and a system. In the simplest case the evolution of the reduced
state of the system is governed by a quantum master equation with a time-
independent, i.e. Markovian, generator. Such evolution is typically emerging
under the assumption of a weak coupling between the system and an infinitely
large bath. Here, we are interested in understanding to which extent a neural
network function approximator can predict open quantum dynamics — described by
time-local generators — from an underlying unitary dynamics. We investigate
this question using a class of spin models, which is inspired by recent
experimental setups. We find that indeed time-local generators can be learned.
In certain situations they are even time-independent and allow to extrapolate
the dynamics to unseen times. This might be useful for situations in which
experiments or numerical simulations do not allow to capture long-time
dynamics and for exploring thermalization occurring in closed quantum systems.
_Introduction_ —The investigation of isolated quantum systems out of
equilibrium is a central problem in physics D’Alessio _et al._ (2016);
Polkovnikov _et al._ (2011). Often, partial information that concerns a
spatially localized subsystem ${S}$ is sufficient to study dynamical effects
related to relaxation or thermalization. This information is encoded in the
so-called reduced quantum state $\rho_{S}$, [c.f. Fig. 1(a,b)], whose time
dependence is obtained by “integrating out” the remainder of the evolving
many-body system, which is often referred to as bath ${B}$ [see Fig. 1(c)]. In
certain settings, e.g. for systems which are weakly interacting with a large
environment Breuer _et al._ (2002); Gardiner and Zoller (2004), the dynamics
of the reduced state is effectively described by a quantum master equation
Lindblad (1976); Gorini _et al._ (1976); Alicki and Lendi (2007). In the
simplest manifestation, the dynamics of the reduced quantum state $\rho_{S}$
is then governed by a time-independent generator $\mathcal{L}$ acting only on
$S$:
$\displaystyle\rho_{S}(t):=\textnormal{Tr}_{B}\left(U_{t}\left(\rho_{S}\otimes\rho_{B}\right)U_{t}^{\dagger}\right)\approx
e^{t\,\mathcal{L}}\left[\rho_{S}\right].$ (1)
Here, $U_{t}$ is the unitary evolution operator of the full many-body system,
the state $\rho_{S}\otimes\rho_{B}$ is the initial state of the many-body
system in product form, and $\textnormal{Tr}_{B}$ indicates the average over
the bath ${B}$. However, the generator does not need to be time-independent
and, in most generic instances Chruściński and Kossakowski (2010), it may
depend locally on time, $\mathcal{L}_{t}$, as sketched in Fig. 1(c).
Figure 1: Reduced local dynamics of a many-body quantum system. The reduced
state $\rho_{S}$ describes local degrees of freedom of a quantum spin system.
The remainder of the system, $B$, takes the role of a “bath”, with initial
state $\rho_{B}$. We consider two scenarios: (a) Model I: spin chain with open
boundary conditions, where the interaction strength $V$ and range $\alpha$ are
varied [see Eq. (2)]; (b) Model II: spin chain with closed boundaries and
nearest-neighbor interaction, where the system-bath coupling $V^{\prime}$ and
the initial temperature $\beta$ of the bath are modified. (c) The dynamics of
the reduced state, $\rho_{S}(t)$, is obtained by tracing out the degrees of
freedom of $B$ and to be learned by a neural network. The learned generator
may be either Markovian (time-independent) or time-dependent.
In this paper, we show how simple neural network architectures can learn time-
local generators that govern the local dynamics of a closed quantum spin
system. The input of the networks consists of the time-dependent average
values of the reduced system observables from which the dynamical generator is
estimated. Firstly, we consider an architecture which provides a time-averaged
generator $\overline{\mathcal{L}}$ corresponding to an effectively Markovian
description of the system dynamics. Our findings indicate that this can yield
indeed a valid approximation for the generator even beyond typical scenarios
justifying a Markovian weak-coupling assumption Breuer _et al._ (2002);
Gardiner and Zoller (2004). Once the generator is known, the network can
further be exploited to make predictions for times which go beyond the
previously analyzed time frame, as is sketched in Fig. 2. To study settings
with a time-dependent generator, $\mathcal{L}_{t}$, we use a different neural
network architecture which is based on so-called hypermodels and allows to
assess the time-dependence of the generator of the reduced dynamics. Our work
links to recent efforts aiming at understanding quantum dynamics with neural
networks Yoshioka and Hamazaki (2019); Hartmann and Carleo (2019a); Nagy and
Savona (2019); Krastanov _et al._ (2020); Carleo _et al._ (2019); Krastanov
_et al._ (2019); Carleo and Troyer (2017); Hartmann and Carleo (2019b); Liu
_et al._ (2020); Luchnikov _et al._ (2020); Martina _et al._ (2021); Miles
_et al._ (2020). Our approach, which uses machine learning tools and provides
a directly interpretable object such as the physical generator of the
dynamics, highlights a possible route for the application of neural networks
in the study of the long-time dynamics of local observables in closed quantum
systems. Moreover, it may find applications in the context of quantifying the
degree of non-Markovianity in reduced quantum dynamics Chruściński and
Kossakowski (2010).
Figure 2: Effective time-independent generator: Training and extrapolation. To
obtain a time-independent generator we use a multi-layer perceptron (MLP). The
inputs are the time-dependent expectation values of the reduced system
observables — for a single spin contained in the Bloch vector $\mathbf{v}_{\rm
b}$– in a given training time window (shaded blue region) and for several
initial Bloch vectors (top to bottom). The MLP learns the “propagator”
$M[\theta^{*}]$, which depends on the optimized network parameters
$\theta^{*}$ but not on the actual time. Considering an Euler integration step
of length $dt$, this provides a matrix $\bar{{L}}$ representing the action of
an “averaged” generator on the Bloch vector. When the reduced dynamics is
Markovian, this learned generator allows for the network to make predictions
for unseen future times (red shaded region).
_Many-body spin models_ —For illustrating our ideas, we consider two one-
dimensional systems (in the following referred to as Model I and II)
consisting of $N$ interacting spins, which are inspired by recent state-of-
the-art quantum simulator experiments with Rydberg atoms Bloch _et al._
(2012); Kim _et al._ (2018); Ebadi _et al._ (2020); Browaeys and Lahaye
(2020); Bloch _et al._ (2008) or trapped ions Islam _et al._ (2011); Bohnet
_et al._ (2016); Joshi _et al._ (2020). Model I has open boundary conditions
[see Fig. 1(a)]. It features power-law interactions and is described by the
Hamiltonian
$H_{\mathrm{I}}=\Omega\sum_{i=1}^{N}\sigma_{i}^{x}+V\sum^{N}_{i<j}\frac{n_{i}n_{j}}{|i-j|^{\alpha}}.$
(2)
Here, $n_{i}=(1+\sigma^{z}_{i})/2$ is a projector on the “up”-state of the
$i$-th spin and $\sigma^{x,y,z}_{i}$ are Pauli matrices. The parameter
$\alpha$ accounts for the interaction range, i.e. the power-law decay of the
interaction potential, and $V$ controls the overall interaction strength with
respect to the strength $\Omega$ of a transverse field term. The reduced
system $S$, for this model, is the middle spin of the chain, as shown in Fig.
1(a). Model II is a closed spin chain [sketched in Fig. 1(b)], with nearest-
neighbor interactions and Hamiltonian
$H_{\mathrm{II}}=\Omega\sum_{i=2}^{N}\sigma_{i}^{x}+V\sum_{j=2}^{N-1}n_{j}n_{j+1}+\Omega^{\prime}\sigma_{1}^{x}+V^{{}^{\prime}}n_{1}(n_{2}+n_{N}).$
(3)
Here, the spin with label $1$ is singled out as reference spin (forming the
system $S$) and the interaction strength with its neighbors is given by
$V^{\prime}$, while the interaction strength among all other spins is $V$.
Moreover, we assume that the strength $\Omega^{\prime}$ of the transverse
field acting on the reference spin can be controlled. Both models capture a
whole variety of different scenarios, including short- and long-range
interactions, absence and presence of translational invariance as well as weak
and strong coupling between local and bath degrees of freedom. They moreover
encompass a number of standard scenarios often explored in quantum many-body
physics, such as the so-called PXP-model Sun and Robicheaux (2008); Ates _et
al._ (2012); Turner _et al._ (2018), the Ising model in the presence of
longitudinal and transverse fields, and all-to-all connected spin systems with
resemblance to the Dicke model Emary and Brandes (2003) or the Lipkin-Meshkov-
Glick model Dusuel and Vidal (2004).
Our aim is to investigate the reduced dynamics of local degrees of freedom, as
for instance a single spin $S$, as depicted in Fig. 1(a,b). To this end, we
assume that the system is initialized at time $t=0$ in the state
$\rho=\rho_{S}\otimes\rho_{B}$. In our studies concerning Model I, we assume
the bath to be in the infinite temperature state,
$\rho_{B}\propto\mathds{1}_{B}$. When considering Model II, instead, we assume
a finite temperature situation with $\rho_{B}\propto
e^{-\beta\tilde{H}_{\mathrm{II}}}$, where the Hamiltonian
$\tilde{H}_{\mathrm{II}}$ is the one of Eq. (3) with
$\Omega^{\prime}=V^{\prime}=0$, and $\beta$ is the inverse temperature. We let
the state $\rho$ evolve according to the unitary dynamics $\rho(t)=U_{t}\rho
U^{\dagger}_{t}$, with $U_{t}=\textnormal{e}^{-iHt}$ from which we obtain the
full dynamical information about the subsystem $S$. Throughout we will make
use of the Bloch vector representation of the subsystem’s density matrix:
$\rho_{S}(t)=(\mathbf{v}_{\rm b}(t)\cdot\mathbf{\sigma})/2$ with
$\mathbf{\sigma}=(\mathds{1},\sigma^{x},\sigma^{y},\sigma^{z})$. Here the
Bloch vector $\mathbf{v}_{\rm
b}(t)=\left(1,\braket{\sigma_{k}^{x}}_{t},\braket{\sigma_{k}^{y}}_{t},\braket{\sigma_{k}^{z}}_{t}\right)$
is given in terms of the expectation $\braket{\cdot}_{t}$ with respect to the
full quantum state at time $t$, and $k$ labels the system spin [see Fig.
1(a,b)].
In the following, we will characterize the generator of this reduced dynamics;
that is, we want to find a generator acting solely on $S$ that reproduces (or
approximates) the dynamics of $\mathbf{v}_{\rm b}(t)$ for the two models under
consideration [Fig. 1(a,b)]. For Model I we explore for which parameter
combinations $(V,\alpha)$, the local generator, which is in principle time-
dependent $\mathcal{L}_{t}$, can be approximated by an effective “averaged”
time-independent one, $\mathcal{L}_{t}\approx\bar{\mathcal{L}}$. We remark
here that, while $B$ acts as a fictitious bath for subsystem $S$, the
situation described here is far from that of typical weak-coupling limits.
Indeed, $B$ is a finite quantum system and its Hamiltonian has a discrete
spectrum. As such, $B$ can hardly be thought of as an infinite Markovian bath
of bosonic oscillators, whose state is unaffected by the interaction with $S$.
The only aspect in common with a weak-coupling limit is that the interaction
strength can be made small. However, this is not sufficient to argue that the
local dynamics may be described by time-independent generators.
For Model II we consider the case in which an “isolated” spin (on site $1$)
interacts with $B$ with an independently tunable strength $V^{\prime}$.
Experimentally such a situation can be realized, e.g., by exploiting the
distance dependence of dipolar interactions of atoms in Rydberg states
Browaeys and Lahaye (2020). Also for Model II we investigate how well the
generator of the reduced dynamics is approximated by a time-independent model
when the initial state of $B$ (parametrized by the inverse temperature
$\beta$) and the interaction strength $V^{{}^{\prime}}$ is varied. In
addition, for this scenario, we use a hypermodel in order to analyze regimes
that necessitate a time-dependent generator $\mathcal{L}_{t}$.
_Network architectures and training_ —In order to learn dynamical generators
for the reduced dynamics of subsystem $S$, we employ machine learning
algorithms. Our approach is completely data-driven, i.e. the network has no
prior information about the physical system. For learning time-independent
generators, we use a linear multi-layer perceptron (MLP) architecture (see
Fig. 2) which turns out to be the simplest possible one. Every data-point of
our data-set $\mathcal{D}$ contains the triple $(\mathbf{v}_{\rm
b},\mathbf{v}_{\rm b}^{\prime},t)$ where $\mathbf{v}_{\rm b}:=\mathbf{v}_{\rm
b}(t)$ is the value of the Bloch vector at a given time-point $t$, and
$\mathbf{v}_{\rm b}^{\prime}$ is the value of the Bloch vector after a further
discrete time step $dt$, i.e. $\mathbf{v}_{\rm b}^{\prime}:=\mathbf{v}_{\rm
b}(t+dt)$.
The input of our model is $\mathbf{v}_{\rm b}$, and the output is the vector
$\mathbf{o}(t)=M[{\theta}]\mathbf{v}_{\rm b}$, where $M[{\theta}]$ is a
$4\times 4$-matrix (matching the dimension of the Bloch vector), that depends
on the parameters ${\theta}$ of the network. To optimize the network, we
introduce the following loss function, which is given by the norm of the
difference between the next time step $\mathbf{v}_{\rm b}^{\prime}$ and the
output of the network:
$C(\theta)=\mathbb{E}_{(\mathbf{v}_{\rm b},\mathbf{v}_{\rm
b}^{\prime})\sim\mathcal{D}}\left[\left\lVert M[{\theta}]\mathbf{v}_{\rm
b}-\mathbf{v}_{\rm b}^{\prime}\right\rVert\right],$ (4)
where the expectation is taken over the training data $\mathcal{D}$.
Minimizing the above function thus provides a model $M[{\theta}^{\ast}]$ for
the propagation of the Bloch vector over an infinitesimal time step $dt$. This
model is related to the generator of the local dynamics: indeed, we have
$M[{\theta}^{\ast}]=\mathds{1}+\overline{{L}}dt$, where $\bar{L}$ is a
representation of the time-averaged generator $\overline{\mathcal{L}}$ acting
on Bloch vectors. The training data-set is constructed by evolving the Bloch
vector $\mathbf{v}_{\rm b}$ over a time interval $[0,T_{\mathrm{train}}]$. The
trajectories are generated evolving $100$ randomly chosen initial states of
the form
$\rho_{0}=\frac{1}{2}\left(\begin{array}[]{@{}cc@{}}1+z&x-iy\\\
x+iy&1-z\end{array}\right)\otimes\rho_{B},$ (5)
with $x^{2}+y^{2}+z^{2}<1$, i.e. we consider non-pure initial states. We split
our data-set such that we consider the $80\%$ of it for training, i.e. finding
the best parameters of our model. The remaining $20\%$ form the validation set
to assess the performance of the model built in the training phase.
To encode and quantify the time-dependence of dynamical generators,
$\mathcal{L}_{t}$, we use a different architecture and consider a so-called
hypermodel, which computes network weights based on a context input. In our
case, the context input is the time $t$ and a multi-layer perceptron (MLP) $F$
with non-linear activation functions transforms this input to the $4\times 4$
matrix of the generator $M[F(t;\theta)]$.
The data-set is organized in the same way as for the time-independent model.
The only difference with respect to the previous case is that we also pass the
information about the actual time step $t$. The network parameters $\theta$
are optimized by minimizing the loss function
$C_{\mathrm{H}}(\theta)=\mathbb{E}_{(\mathbf{v}_{\rm b},\mathbf{v}_{\rm
b}^{\prime},t)\sim\mathcal{D}}\left[\left\lVert M[F(t;\theta)]\mathbf{v}_{\rm
b}-\mathbf{v}_{\rm b}^{\prime}\right\rVert\right].$ (6)
By training the model in this way we obtain a time-dependent representation of
the propagator of the local dynamics, i.e.
$M[F(t;\theta^{*})]=\mathds{1}+L_{t}dt$. For the training set, we choose
quantum states as in the previous case. For the evaluation data-set, instead,
we consider
$\rho_{0}=\left(\begin{array}[]{@{}cc@{}}1-c&0\\\
0&c\end{array}\right)\otimes\rho_{B},$ (7)
with $c$ being a random number, $c\in(0.01,0.7)$111See supplemental material
for details.
Figure 3: Time-independent generator. (a) Model I: Comparison between the
exact dynamics of the Bloch vector and the one obtained by learning an
effective time-independent generator with neural networks. The density plot
shows the time-averaged norm difference, $\overline{\epsilon}$ [Eq. (8)
averaged over $5$ randomly selected values $c$ of the initial conditions, see
Eq. (7)], for various parameter choices $(V,\alpha)$. The data is calculated
for $N=9$ spins with a training time of $T_{\mathrm{train}}=10/\Omega$,
$dt=0.01/\Omega$ and a total time $T_{\mathrm{tot}}=20/\Omega$ (see main text
for details). We also show two examples for the Bloch vector evolution, where
we choose $c=1$ and $T_{\mathrm{tot}}=20/\Omega$ for illustration. Here solid
curves correspond to the components of the Bloch vector propagated with the
learned time-averaged generator. The dashed lines correspond to the
numerically exact solution. (b) Same as panel (a), but for Model II and the
parameter set $(V^{\prime},\beta)$.
_Time-independent generators_ —With the above numerical approach, we can now
investigate under which circumstances the generator of the dynamics of the
subsystem $S$ is time-independent. We consider first the open spin chain [see
Fig. 1(a) and Eq. (2)] and explore different values of $V$ and $\alpha$. To
test the performance of the model after the training, we study the _time-
averaged_ norm difference between the exact Bloch vector $\mathbf{v}^{\rm
ex}_{\rm b}$ — obtained by simulating the full many-body spin chain dynamics —
and the time-evolution of the same quantity as predicted by our model
$\mathbf{v}^{\rm mod}_{\rm b}$:
$\epsilon=\frac{1}{T_{\rm tot}}\int_{0}^{T_{\rm
tot}}\\!\\!dt\left\lVert\mathbf{v}^{\rm mod}_{\rm b}(t)-\mathbf{v}_{\rm
b}^{\rm ex}(t)\right\rVert\,,$ (8)
computed by numerical integration.
As shown in Fig. 3(a), the error $\epsilon$ is small for short-range (large
$\alpha$) and weakly interacting (with small ratio $V/\Omega$) spin chains.
This means that our time-independent model captures the relevant features of
the reduced dynamics, and suggests that for finite systems with sufficiently
weak interactions, the dynamics of local degrees of freedom can indeed be
effectively described by a _time-independent_ generator $\bar{\mathcal{L}}$.
Furthermore, for this Markovian regime of the dynamics, our model allows to
make predictions for times exceeding the training time.
We now turn to the discussion of Model II [see Fig. 1(b) and Eq. (3)]. Using
the same procedure explained above, we can learn the time-averaged dynamical
generator for subsystem $S$. In Fig. 3(b) we summarize the results of this
analysis. We display the time-averaged error, as defined in Eq. (8), for
different parameter choices, which in this case are the (inverse) temperature
$\beta$ of the initial state $\rho_{B}$ and the interaction strength
$V^{{}^{\prime}}$. We furthermore take $\Omega^{\prime}=\Omega$. As shown in
the figure the dynamics is well approximated by a time-independent generator,
$\bar{\mathcal{L}}$, for weak interactions $V^{\prime}$ and for small $\beta$.
Here also the extrapolation to times that exceed $T_{\mathrm{train}}$ is
possible. The data in Fig. 3 shows that in certain parameter regimes the
approximation of the dynamics through a time-independent generator does not
work well. Here significant deviations are even observable during the training
period.
Figure 4: Hypermodel and time-dependent generator. Hypermodels allow to encode
time-dependent generators $\mathcal{L}_{t}$. The time-dependence is quantified
through the derivative of $L_{t}$ with respect to time, expressed by the
positive quantity $\xi(t)$ (see text for definition). Panel (a) shows three
example curves corresponding to Model II ($N=9$). The function $\xi(t)$
depends only on the parameters of the model and is independent from the
initial condition chosen. Generally, the stronger the interaction
$V^{\prime}$, the more pronounced the time dependence. This trend is clearly
visible in panel (b), where we show the averaged value of $\xi(t)$ in the
interval $0$ to $T=10/\Omega$, denoted as $\Xi$. In order to obtain the
density plot we averaged, also in this case, over $5$ different values of $c$
in Eq. (7), also in this case $dt=0.01\Omega$.
_Time-dependent time-local generators_ —We are now interested in analyzing how
strongly the dynamical generator depends on time, in those instances where the
model introduced in the previous section fails. To this end, we adopt a
hypermodel which can take as input the information about the running time. In
this way, the neural network is able to learn an optimal parametrization of
the “propagator” $M[F(t;\theta^{*})]=\mathds{1}+{{L}_{t}}dt$ for the Bloch
vector dynamics which explicitly depends on time. Here, the matrix $L_{t}$
encodes the action of the generator on the Bloch vector and, essentially,
implements the differential equations obeyed by the entries of the Bloch
vector $\mathbf{v}_{\rm b}$. This architecture allows to learn and accurately
reproduce the dynamics of Model II for all studied parameter regimes (see
Supplemental Material for examples). This is not surprising, but gives us a
handle for analyzing the time-dependence of the dynamical generator $L_{t}$.
To this end we consider the positive quantity $\xi(t)=\sqrt{{\rm
Tr}(\dot{L}_{t}^{\dagger}\dot{L}_{t})}$, where the dot denotes the derivative
with respect to time. This can only be zero when $\dot{L}_{t}=0$, i.e. when
the action of the generator is not depending on the running time. To quantify
an overall time-dependence within a time window $[0,T]$, we define the time-
averaged value $\Xi=\frac{1}{T}\int_{0}^{T}dt\,\xi(t)$. In Fig. 4 we show the
corresponding data. For strong interactions $V^{\prime}$, $\xi(t)$ is non-zero
throughout which indicates a strong explicit time-dependence. This is also
reflected in a large $\Xi$. For weak interactions, on the other hand, $\xi(t)$
remains small for all times considered. The oscillation we attribute to the
finite size of the system. This confirms that, in such parameter regime, it is
indeed possible to approximate the dynamics of the Bloch vector by means of a
time-independent matrix, as considered in the previous section.
_Conclusions_ —We have presented two simple examples of neural network
architectures that can learn the dynamical features of reduced quantum states.
When such time evolution is effectively Markovian, the network can find a
suitable approximation for the generator of the local dynamics. Here one can
extrapolate the dynamics of reduced degrees of freedom to times that were
unexplored during the training procedure. This possibility is particularly
promising for applications in combination with tensor networks, which perform
extremely well for short times. A neural network could learn the time-
independent generator during this time interval and then possibly extrapolate
to long times. In principle, this might enable the exploration of the onset of
stationary or thermalization regimes. When the dynamics is not of Markovian
type, we have shown that hypermodels can recover the generator of the reduced
quantum dynamics. This allows to quantify non-Markovian effects Benatti _et
al._ (2016), which manifest in an explicit time-dependence of the dynamical
generator Ref. Chruściński and Kossakowski (2010). Furthermore, being able to
reconstruct the generator of the reduced dynamics, as we have done here with
hypermodels, makes it possible to explore different measures of non-
Markovianity based on non-divisibility criteria for quantum dynamical time-
evolutions Breuer _et al._ (2009); Laine _et al._ (2010); Breuer (2012);
Rivas _et al._ (2014).
A possible future development in this regard could be the application of more
advanced machine learning methods for learning time correlations in time
series. This can be achieved with models borrowed from language studies (e.g.
long-short term memory recurrent neural networks (LSTM) Hochreiter and
Schmidhuber (1997) and transformers Vaswani _et al._ (2017)) or with
algorithms geared towards more interpretable models by either learning
analytical expressions of the differential equation Sahoo _et al._ (2018) or
by modelling the time-evolution dynamics directly Chen _et al._ (2018).
###### Acknowledgements.
_Acknowledgments_ —We acknowledge financial support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-Number 2064/1 – Project number 390727645. IL
acknowledges funding from the “Wissenschaftler Rückkehrprogramm GSO/CZS” of
the Carl-Zeiss-Stiftung and the German Scholars Organization e.V., and through
the DFG projects number 428276754 (SPP GiRyd) and 435696605. The authors thank
the International Max Planck Research School for Intelligent Systems (IMPRS-
IS) for supporting DZ.
## References
* D’Alessio _et al._ (2016) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
* Polkovnikov _et al._ (2011) A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
* Breuer _et al._ (2002) H.-P. Breuer, F. Petruccione, _et al._ , _The theory of open quantum systems_ (Oxford University Press on Demand, 2002).
* Gardiner and Zoller (2004) C. Gardiner and P. Zoller, _Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics_ (Springer Science & Business Media, 2004).
* Lindblad (1976) G. Lindblad, Commun. Math. Phys. 48, 119 (1976).
* Gorini _et al._ (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics 17, 821 (1976).
* Alicki and Lendi (2007) R. Alicki and K. Lendi, _Quantum dynamical semigroups and applications_ , Vol. 717 (Springer, 2007).
* Chruściński and Kossakowski (2010) D. Chruściński and A. Kossakowski, Phys. Rev. Lett. 104, 070406 (2010).
* Yoshioka and Hamazaki (2019) N. Yoshioka and R. Hamazaki, Phys. Rev. B 99, 214306 (2019).
* Hartmann and Carleo (2019a) M. J. Hartmann and G. Carleo, Phys. Rev. Lett. 122, 250502 (2019a).
* Nagy and Savona (2019) A. Nagy and V. Savona, Phys. Rev. Lett. 122, 250501 (2019).
* Krastanov _et al._ (2020) S. Krastanov, K. Head-Marsden, S. Zhou, S. T. Flammia, L. Jiang, and P. Narang, arXiv:2009.03902 (2020).
* Carleo _et al._ (2019) G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Rev. Mod. Phys. 91, 045002 (2019).
* Krastanov _et al._ (2019) S. Krastanov, S. Zhou, S. T. Flammia, and L. Jiang, Quantum Sci. Technol. 4, 035003 (2019).
* Carleo and Troyer (2017) G. Carleo and M. Troyer, Science 355, 602 (2017).
* Hartmann and Carleo (2019b) M. J. Hartmann and G. Carleo, Phys. Rev. Lett. 122, 250502 (2019b).
* Liu _et al._ (2020) Z. Liu, L.-M. Duan, and D.-L. Deng, arXiv:2008.05488 (2020).
* Luchnikov _et al._ (2020) I. A. Luchnikov, S. V. Vintskevich, D. A. Grigoriev, and S. N. Filippov, Phys. Rev. Lett. 124, 140502 (2020).
* Martina _et al._ (2021) S. Martina, S. Gherardini, and F. Caruso, arXiv:2101.03221 (2021).
* Miles _et al._ (2020) C. Miles, A. Bohrdt, R. Wu, C. Chiu, M. Xu, G. Ji, M. Greiner, K. Q. Weinberger, E. Demler, and E.-A. Kim, arXiv:2011.03474 (2020).
* Bloch _et al._ (2012) I. Bloch, J. Dalibard, and S. Nascimbène, Nat. Phys. 8, 267 (2012).
* Kim _et al._ (2018) H. Kim, Y. Park, K. Kim, H.-S. Sim, and J. Ahn, Phys. Rev. Lett. 120, 180502 (2018).
* Ebadi _et al._ (2020) S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, _et al._ , arXiv:2012.12281 (2020).
* Browaeys and Lahaye (2020) A. Browaeys and T. Lahaye, Nat. Phys. 16, 132–142 (2020).
* Bloch _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
* Islam _et al._ (2011) R. Islam, E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. J. Wang, J. Freericks, _et al._ , Nat. Comm. 2, 1 (2011).
* Bohnet _et al._ (2016) J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Science 352, 1297 (2016).
* Joshi _et al._ (2020) M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Phys. Rev. Lett. 124, 240505 (2020).
* Sun and Robicheaux (2008) B. Sun and F. Robicheaux, New J. Phys. 10, 045032 (2008).
* Ates _et al._ (2012) C. Ates, J. P. Garrahan, and I. Lesanovsky, Phys. Rev. Lett. 108, 110603 (2012).
* Turner _et al._ (2018) C. Turner, A. Michailidis, D. Abanin, M. Serbyn, and Z. Papić, Nat. Phys. 14, 745 (2018).
* Emary and Brandes (2003) C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003).
* Dusuel and Vidal (2004) S. Dusuel and J. Vidal, Phys. Rev. Lett. 93, 237204 (2004).
* Note (1) See supplemental material for details.
* Benatti _et al._ (2016) F. Benatti, F. Carollo, R. Floreanini, and H. Narnhofer, Phys. Lett. A 380, 381 (2016).
* Breuer _et al._ (2009) H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).
* Laine _et al._ (2010) E.-M. Laine, J. Piilo, and H.-P. Breuer, Phys. Rev. A 81, 062115 (2010).
* Breuer (2012) H.-P. Breuer, J. Phys. B 45, 154001 (2012).
* Rivas _et al._ (2014) Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014).
* Hochreiter and Schmidhuber (1997) S. Hochreiter and J. Schmidhuber, Neural Comput. 9, 1735 (1997).
* Vaswani _et al._ (2017) A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, arXiv:1706.03762 (2017).
* Sahoo _et al._ (2018) S. S. Sahoo, C. H. Lampert, and G. Martius, in _Proc. 35th International Conference on Machine Learning, ICML 2018, Stockholm, Sweden, 2018_, Vol. 80 (PMLR, 2018) pp. 4442–4450.
* Chen _et al._ (2018) R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, in _Advances in Neural Information Processing Systems_, Vol. 31, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, Inc., 2018) pp. 6571–6583.
Supplemental material: Machine learning time-local generators of open quantum
dynamics
In this supplemental material we report a more detailed analysis concerning
the effectiveness of the hypermodel in predicting the dynamics of the Bloch
vector within the training time-window. In addition, we provide the hyper-
parameters of the used models.
## I Hypermodel vs. exact dynamics
In the main text we analyze the time-dependence of time-local generators
parametrizing the “propagator” $M[F(t;\theta^{\ast})]=\mathds{1}+dtL_{t}$ for
the Bloch vector, using a hypermodel architecture. The matrix $L_{t}$ encodes
all the information concerning the dynamics of local (one-site) observables.
As stated in the main text, a suitable measure to quantify the dependence on
time of this matrix is given by $\Xi=\frac{1}{T}\int_{0}^{T}dt\,\xi(t)$, where
$\xi(t)=\sqrt{{\rm Tr}(\dot{L}_{t}^{\dagger}\dot{L}_{t})}$. However, this can
be a faithful measure of the time-dependence of the physical generator, only
if $L_{t}$ correctly implements the dynamics of the Bloch vector. It is thus
important to check that the prediction of the hypermodel matches the exact
dynamics, in the time window $[0,T]$ in which we want to analyze the generator
$L_{t}$. To this end, we report in Fig. 1 a detailed comparison between the
predicted time evolution of the Bloch vector components and the exact ones for
six different parameter regimes.
Figure 1: Hypermodel accuracy: Comparison between the exact dynamics and the
dynamics predicted by the hypermodel for initial state $\rho_{0}$ reported in
Eq. (1) and for $N=7$. As it can be observed the agreement between the model
prediction (dotted line) and the exact dynamics (full line) is excellent. This
fact enables us to use the local operator learned using the hypermodel for
studying the time-dependence of $L_{t}$.
We start from a generic initial state of the form
$\rho_{0}=\frac{1}{2}\left(\begin{array}[]{@{}cc@{}}1+z&x-iy\\\
x+iy&1-z\end{array}\right)\otimes\rho_{B},$ (1)
with $x^{2}+y^{2}+z^{2}<1$, in particular we choose $x=0.6$, $y=0.3$ and
$z=0.4$, and we consider different interactions and initial bath temperatures.
In the training time-window $[0,T]$, we observe an excellent agreement between
the two curves for $T=10/\Omega$. In Fig. 2 we report the time evolution of
the quantity $\xi(t)$ for the same parameter regimes.
Figure 2: Time-dependence: Behavior of the quantity $\xi(t)$ for the same
parameter regimes of Fig. 1. The quantity $\Xi$ grows with the interaction,
meaning that the generator of the local dynamics $L_{t}$ becomes more time-
dependent.
As it can be observed for weakly interacting systems the generator $L_{t}$ is
almost time-independent. In this regime, as discussed in the main text, it is
possible to predict the local dynamics using simple time-independent
architectures. Increasing the interaction, the parameter $\Xi$ grows
accordingly signaling that the dependence on time of $L_{t}$ becomes stronger.
The training procedure for the hyper-model is slightly different with respect
what we have done for the time-independent case, in particular it is different
the differentiation between training and evaluation set. In this case, in
fact, we generate two different types of data-sets, the first data-set for
training consists in $100$ trajectories with randomly chosen initial state
$\rho_{0}$ of the the same form reported in Eq. (1). For evaluation we select,
instead, randomly data-points from trajectories generated from initial states
$\rho_{0}^{\mathrm{ev}}$ of type
$\rho_{0}^{\mathrm{ev}}=\left(\begin{array}[]{@{}cc@{}}1-c&0\\\
0&c\end{array}\right)\otimes\rho_{B},$ (2)
with $0.01<c<0.7$. In this way we are performing a separation of the data-sets
in the Hilbert space. This is contrary to what we had for the time-independent
case where the data-set was given by separated time-intervals of the same
trajectory.
## II Network architectures
Setting | Architecture | Optimizer | Dataset
---|---|---|---
Time-independent | Linear perceptron (no hidden layers) | Adam Learning rate: $10^{-3}$ Betas: $0.9$, $0.999$ Epsilon: $10^{-8}$ Batch size: $256$ Batches per epoch: $512$ Epochs: $5$ | Trajectories (train / val): $100$ / $20$ Total time ($\Omega\,T$): $10$ Time step ($\Omega\,dt$): $0.01$
Time-dependent (hyper-model) | Hidden layers: $3$ Nonlinearity: tanh Output nonlinearity: none | Adam Learning rate: $10^{-3}$ Betas: $0.9$, $0.999$ Epsilon: $10^{-8}$ Batch size: $256$ Batches per epoch: $256$ Epochs: $500$ | Trajectories (train / val): $100$ / $20$ Total time ($\Omega\,T$): $10$ Time step ($\Omega\,dt$): $0.01$
Table 1: Overview over the architecture, optimizers, training- and system-
parameters for Model I and Model II (top) and the hyper-model (bottom)
|
# Fast Clustering of Short Text Streams Using Efficient Cluster Indexing and
Dynamic Similarity Thresholds
Md Rashadul Hasan Rakib Dalhousie UniversityNova ScotiaCanada<EMAIL_ADDRESS>and Muhammad Asaduzzaman Queen’s UniversityOntarioCanada
<EMAIL_ADDRESS>
(2020)
###### Abstract.
Short text stream clustering is an important but challenging task since
massive amount of text is generated from different sources such as micro-
blogging, question-answering, and social news aggregation websites. One of the
major challenges of clustering such massive amount of text is to cluster them
within a reasonable amount of time. The existing state-of-the-art short text
stream clustering methods can not cluster such massive amount of text within a
reasonable amount of time as they compute similarities between a text and all
the existing clusters to assign that text to a cluster. To overcome this
challenge, we propose a fast short text stream clustering method (called
FastStream) that efficiently index the clusters using inverted index and
compute similarity between a text and a selected number of clusters while
assigning a text to a cluster. In this way, we not only reduce the running
time of our proposed method but also reduce the running time of several state-
of-the-art short text stream clustering methods. FastStream assigns a text to
a cluster (new or existing) using the dynamically computed similarity
thresholds based on statistical measure. Thus our method efficiently deals
with the concept drift problem. Experimental results demonstrate that
FastStream outperforms the state-of-the-art short text stream clustering
methods by a significant margin on several short text datasets. In addition,
the running time of FastStream is several orders of magnitude faster than that
of the state-of-the-art methods.
text stream clustering, inverted index, dynamic similarity threshold
††copyright: acmcopyright††journalyear: 2020††conference: aa ’20: aa; 2020;
USA††booktitle: aaa ’20: aa, June 03–05, 2020, USA††price: 15.00††journalyear:
2020††copyright: acmcopyright††conference: aa 2020; September 29-October 2,
2020; Virtual Event, CA, USA††booktitle: aa 2020 (aa ’20), September
29-October 2, 2020, Virtual Event, CA, USA††price: 15.00††doi:
10.1145/3395027.3419589††isbn: 978-1-4503-8000-3/20/09††ccs: Computing
methodologies Natural language processing††ccs: Computing methodologies
Cluster analysis
## 1\. Introduction
Due to technological advances short text streams are generated at large
volumes from different sources. Organizing these text streams within
reasonable time-frame is an important step towards discovering real-time
trends (e.g., political, economic) in conversations, finding groups of users
sharing similar topics of interest (Zhao et al., 2016), and monitoring the
activity of individuals over time. The task of clustering short text streams
is to assign a text to a new cluster or to one of the existing clusters as it
arrives (Yin et al., 2018). The recent short text stream clustering methods
were proposed based on dirichlet process multinational mixture model as
described in (Yin et al., 2018; Kumar et al., 2020; Chen et al., 2020).
Generally, these algorithms used Gibbs Sampling (Ishwaran and James, 2001) to
estimate the parameters of the mixture model so as to obtain the clustering of
text streams. These algorithms assign a text to a cluster by computing
similarities between a text and the clusters based on the common features
between them (e.g., word, biterm). However, there may be some clusters with
which a text may not share any common feature, thus the similarities between
that text and those clusters are zero. Therefore, those similarity
computations can be ignored.
Motivated by this, FastStream selects a specific set of clusters while
computing similarities between a text and the clusters. These selected
clusters share common features with a particular text. By limiting the number
of similarity computations, we significantly reduce the running time of our
proposed method and thus the running time of our method is several orders of
magnitude faster than that of the state-of-the-art methods.
In general, similarity-based text stream clustering methods use user defined
similarity threshold to assign a text to a new or one of the existing clusters
(Yin et al., 2018). Rakib et al. (2020) proposed a short text stream
clustering method that removes outliers from the clusters and reassigns them
to the clusters using dynamically computed similarity thresholds (Rakib et
al., 2020). This motivates us to dynamically calculate the similarity
threshold for each text based on statistical measure and use this similarity
threshold to assign the text to a new or one of the existing clusters. Thus
our method efficiently handles the _concept drift_ problem (Yin et al., 2018)
(the problem that topics of the text streams may change over time).
The three major contributions of our work are as follows.
* •
First, we improve the running time of our
method111https://github.com/rashadulrakib/short-text-stream-
clustering/tree/master/OnlineClustering by computing the similarity between a
text and a selected number of clusters (obtained by inverted index (Ilic et
al., 2014)) instead of all clusters. Therefore, the running time of our method
is several orders of magnitude faster than that of the state-of-the-art short
text stream clustering methods. Moreover, the experimental results demonstrate
that by applying inverted index, we are able to reduce the running time of
several state-of-the-art methods.
* •
Second, our method utilizes dynamically calculated similarity thresholds based
on the statistical measure to assign a text to a new or an existing cluster.
The use of dynamic thresholds helps to avoid the concept drift problem (Yin et
al., 2018).
* •
Third, we perform an extensive comparison of our proposed method using four
different datasets and our method outperforms the state-of-the-art short text
stream clustering methods by a significant margin on several datasets.
## 2\. Related Work
### 2.1. Similarity-based Stream Clustering
In general, similarity-based text stream clustering methods use the vector
space model (Erk, 2012) to represent the documents. A document is assigned to
a new or one of the existing clusters based on the similarity threshold which
needs to be manually determined by the user (Yin et al., 2018). A detailed
survey of clustering data streams can be found in (Carnein and Trautmann,
2019). In our method, we dynamically calculate the similarity threshold for
each text and use this similarity threshold to assign the text to a new or one
of the existing clusters.
CluStream (Aggarwal et al., 2003) is a stream clustering method consisting of
an online micro-clustering phase and an offline macro-clustering phase. In
online phase, it assigns a data point to a new or an existing micro-cluster
based on a similarity threshold. In offline phase, it applies k-means
clustering on the micro-clusters and obtain k user specified macro-clusters.
Sumblr (Shou et al., 2013) is a tweet stream summarization prototype. It
consists of a text stream clustering module that compresses tweets into tweet
feature vectors (TCVs) and assigns future tweets to the clusters based on the
statistics of TCVs.
An efficient text stream clustering method using term burst information was
proposed in (Kalogeratos et al., 2016). Bursty terms are the terms that appear
in many documents during a short period of time. This approach considers the
fact that the documents that are published on a particular topic within a
certain time period contain a particular set of bursty terms. An user-defined
threshold for the number of occurrences of terms in documents is used to
identify bursty terms.
### 2.2. Model-based Stream Clustering
A recent short text stream clustering algorithm based on dirichlet process
multinational mixture model was proposed in (Yin et al., 2018) which uses two
dirichlet priors $\alpha$ and $\beta$. $\alpha$ refers to the prior
probability of a text choosing a new cluster and $\beta$ corresponds to the
prior probability of a text choosing a cluster with which the text shares more
similar content than other clusters. This algorithm has two variants: one is
by retaining all previous clusters (called MStream) and other one is by
removing old clusters (called MStreamF).
Rakib et al. (2020) proposed a short text stream clustering method that
clusters a fraction of texts in each batch2221A Batch (or stream) is defined
as a collection of texts (Yin et al., 2018). The terms “Batch” and “Stream”
are interchangeably used in our paper using the frequent biterms in texts,
then it populates the clustering model of MStream algorithm using the cluster
assignments obtained by the frequently occurred biterms in texts. After that
it clusters rest of the texts in the batch using the populated clustering
model of MStream algorithm. Following that it removes outliers from the
clusters and reassigns them to the clusters (new or existing) using the
dynamically computed similarity thresholds.
A biterm based mixture model for short text stream clustering was proposed in
(Chen et al., 2020). Similar to MStream(F) algorithm (Yin et al., 2018), the
biterm based clustering method developed two variants: one is by retaining the
clusters obtained in previous batches (called DP-BMM) and other is by
discarding the clusters obtained in previous batches (called DP-BMM-FP). The
main difference between MStream(F) and DP-BMM(-FP) is that DP-BMM(-FP)
represents the texts using biterm features instead of unigrams. In particular,
DP-BMM(-FP) represents a text of $n$ words using $n*(n-1)/2$ biterm features.
OSDM (Kumar et al., 2020) is a semantic-enhanced dirichlet model for short
text stream clustering. OSDM extends the MStream (Yin et al., 2018) algorithm
by integrating the word to word co-occurrence based semantic information
obtained from the common words between a text and a cluster and uses this
semantic information to compute similarity between a text and a cluster.
In general, a Dirichlet process mixture model based clustering algorithm
requires tuning the parameters (i.e., $\alpha$ and $\beta$) to obtain the
desired clustering performance. For example, MStream(F) uses $\alpha=0.03$ and
$\beta=0.03$ to obtain optimal clustering performance. The DP-BMM(-FP) and
OSDM performed grid search to obtain the optimal values for $\alpha$ and
$\beta$ and set ($\alpha=0.6$ and $\beta=0.02$) and ($\alpha=2e^{-3}$ and
$\beta=4e^{-5}$) respectively. On the contrary, our proposed method
(FastStream) does not require this kind of parameter tuning, instead it uses
the dynamically computed similarity threshold (based on statistical measure)
to assign a text to a new or an existing cluster.
DCT-L (Liang et al., 2016) is a dynamic clustering topic model for short text
streams based on dirichlet process multinomial mixture model. It assigns a
single topic (i.e., cluster) to each short text at a particular timestamp and
uses the resulting topic distribution as priors for inferring the topics of
subsequent documents.
DTM (Blei and Lafferty, 2006) is a dynamic topic model that analyzes the
topics of a collection of documents over time. This method assumes that a
document is rich enough to contain multiple topics. However, this assumption
does not work well for short texts, which results low quality performance on
short text streams.
## 3\. Background
### 3.1. Text Representation
We represent each text using three different kinds of features ($f$) which are
unigram, bigram and biterm and use each representation separately to cluster
the streams of texts. The words of a text are considered as the unigram
features of that text. The two consecutive words of a text are considered as
the bigram features of that text. The biterm feature is defined as an
unordered _word pair_ constructed using the words contained in a text (Chen et
al., 2020). For example, the text “ai improves healthcare system” will be
represented by the following bigrams: “ai improves”, “improves healthcare”,
and “healthcare system”. The same text will be represented by the following
biterms: “ai improves”, “ai healthcare”, “ai system”, “improves healthcare”,
“improves system”, and “healthcare system”.
### 3.2. Cluster Representation
In our method, each cluster is represented by a cluster feature ($CF$) vector
(Yin et al., 2018) consisting of 4 tuples {$n_{z}^{f}$, $n_{z}$, $m_{z}$,
$id_{z}$ } where $n_{z}^{f}$ refers to the features (unigram, bigram or
biterm) along with frequencies in cluster $z$, $n_{z}$ refers to the number of
features in cluster $z$, $m_{z}$ refers to the number of texts in cluster $z$,
and $id_{z}$ refers to the unique _id_ for cluster $z$.
### 3.3. Inverted Index
Inverted Index is a hashmap like data structure that creates mapping from
document-features (unigram, bigram or biterm) to documents (Ilic et al.,
2014). To keep track of which clusters are associated with which features, we
adopt the inverted index based searching technique and create a Feature vector
$F$ for each feature, defined as a tuple of {$l_{f}^{id}$} where $l_{f}^{id}$
refers to the list of cluster $ids$ associated with a feature $f$.
## 4\. Proposed Method
The proposed method (FastStream) clusters each short text one by one as it
arrives. At first, the features are extracted from the text. At a time, we use
only one type of feature to cluster the streams of texts. That means, we
extract only either unigram, bigram or biterm features from the text. After
that we select the clusters that contain the features of the text. Then our
method computes similarities between the text and the selected clusters using
common features. Following that it assigns the text to an appropriate cluster
(new or existing) using the dynamically computed similarity thresholds based
on statistical measure. After that it builds a clustering model using the
cluster assignment of the text to reflect the addition of this text to a new
or an existing cluster; and uses the current clustering model to cluster the
subsequent text. The details are shown in Algorithm 1 and are described next.
Algorithm 1 Proposed Text Stream Clustering
Input: Texts: $t_{1}...t_{\infty}$, $DI$: Delete-interval
Output: Cluster assignments: $z_{t_{1}...t_{\infty}}$
1: for $t_{i}$ in $t_{1}...t_{\infty}$ do
2: Extract features ($f$) from $t_{i}$
3: Select $L$ clusters that share common features with $t_{i}$ (described in
Section 4.1)
4: Compute similarities ($s_{l}$) between $t_{i}$ and the $L$ selected
clusters
5: Compute the maximum ($\max_{l}$), mean ($\mu_{l}$) and standard deviation
($\sigma_{l}$) of the $s_{l}$ similarities
6: if $\max_{l}>\mu_{l}+\sigma_{l}$ then
7: $j=\,$ cluster index for $\max_{l}$
8: Assign $t_{i}$ to $j^{th}$ cluster
9: else
10: Assign $t_{i}$ to a new cluster
11: end if
12: Build clustering model (described in Section 4.4)
13: if $i\mod DI=0$ then
14: Delete outdated clusters (described in Section 4.5)
15: end if
16: end for
### 4.1. Selecting Clusters for the Text
For each short text, we select a specific set of clusters that share common
features with the text. For each feature in a text, we obtain the cluster
$ids$ from the corresponding feature vector $F$. We combine the cluster $ids$
obtained using the features in a text. These are the selected clusters that
share common features with the text.
### 4.2. Computing Similarities between the Text and Selected Clusters
FastStream computes similarities between the text and the selected clusters
based on the common features as shown in Equation 1.
(1) $similarity(t,z)=\frac{2\times\sum_{f\in
t}\min(n_{z}^{f},N_{t}^{f})}{N_{t}+n_{z}}$
To compute similarity between a text $t$ and a cluster $z$, we sum the
occurrences of the common features between $t$ and $z$ which is then
normalized by the summation of the total number of features in text $t$ and
cluster $z$ denoted by $N_{t}$ and $n_{z}$ respectively. Here $n_{z}^{f}$,
$N_{t}^{f}$ refer to the features ($f$) along with frequencies in cluster $z$
and text $t$ respectively.
### 4.3. Assigning Text to Cluster
To assign a text to a cluster, we compute the maximum ($max$), mean ($\mu$)
and standard deviation ($\sigma$) of the similarities between the text $t$ and
the selected clusters. We assign the text to the cluster with the maximum
similarity if the maximum similarity is greater than the $\mu+\sigma$ of the
similarities. Otherwise, we create a new cluster containing this text. Thus
the texts are assigned to clusters based on the dynamically computed
similarity thresholds. The intuition behind using maximum similarity greater
than $\mu+\sigma$ is that, this maximum similarity is above the average
similarities reflecting that both the text and the target cluster share highly
similar content.
### 4.4. Building Clustering Model
We build clustering model using the cluster assignment of the text to reflect
the addition of this text to an existing or a new cluster. When a text $t$ is
added to a cluster $z$, we update the corresponding $CF$ vector by updating
its features with frequencies ($n_{z}^{f}$), number of features ($n_{z}$), and
number of texts ($m_{z}$). The addible property of the $CF$ vector is
described in the following.
##### Addible Property of Text to Cluster:
$\\\ n_{z}^{f}=n_{z}^{f}+N_{t}^{f}\;\;\;\forall f\in\mathbb{\,}t\\\
n_{z}=n_{z}+N_{t}\\\ m_{z}=m_{z}+1\\\ id_{z}=\begin{cases}id_{z},\;\text{if
successive texts are in same cluster}\\\ max(id_{z})+1\;\;\;\forall
z\in\mathbb{\,}Z,\;\text{otherwise}\end{cases}\\\ \\\ $
Here, $N_{t}^{f}$ and $N_{t}$ refer to the features with frequencies in text
$t$ and the total number of features in text $t$ respectively. $Z$ is the set
of all $CF$ vectors (i.e., $Z=\\{CF\\}$).
A new $id$ is assigned to the cluster $z$ (new or existing) if the cluster
assignment of the current text is different from that of the previous text,
otherwise the cluster $id$ of the current text remains same as that of the
previous text. This implies that the recently created or updated cluster will
have the highest cluster $id$. For each feature in the text, we append the
cluster $id$ to the corresponding feature vector $F$.
### 4.5. Deleting Outdated Clusters
We remove the outdated clusters based on their update-timestamps (represented
by cluster $id$) and cluster-sizes (i.e., number of texts in clusters). The
recently created or updated clusters will have higher cluster $ids$.
We remove outdated clusters in every Delete-interval. Delete-interval is the
interval when we remove outdated clusters after clustering a certain number of
texts. For example, the Delete-interval equal to 500 implies that we delete
outdated clusters after we cluster every 500 texts. To obtain outdated
clusters, we calculate the $\mu$ and $\sigma$ of the cluster $ids$ and
cluster-sizes in every Delete-interval. If the _cluster id_ is less than the
$\mu-\sigma$ of cluster $ids$ and the cluster-size is less than the
$\mu-\sigma$ of cluster-sizes, then we delete the cluster by deleting the
corresponding $CF$ vector and remove the corresponding cluster $id$ from the
feature vectors ($F$) that contain that particular cluster $id$.
## 5\. Experimental Study
### 5.1. Datasets
We used four different datasets of short texts in our experiments. The basic
properties of these datasets are shown in Table 1.
Table 1. Summary of the short text datasets Dataset | #Clusters | #Texts | Avg. #words/text
---|---|---|---
Ns-T | 152 | 11,109 | 6.23
Ts-T | 269 | 30,322 | 7.97
SO-T | 10,573 | 1,23,342 | 5.57
NTSO-T | 10,994 | 1,64,773 | 6.02
The datasets Ns-T (Yin et al., 2018) and Ts-T (Yin et al., 2018) consist of
11,109 news titles and 30,322 tweets and are distributed into 152 and 269
groups respectively. We create a new dataset called SO-T using the titles of
the StackoverFlow questions consisting of 1,23,342 question titles distributed
into 10,573 clusters. We combined the texts of the datasets Ns-T, Ts-T, and
SO-T and created a combined dataset NTSO-T consisting of 1,64,773 texts
distributed into 10,994 clusters. The average number of words per text in the
datasets Ns-T, Ts-T, SO-T, and NTSO-T are 6.23, 7.97, 5.57, and 6.02
respectively. The texts in these four datasets were randomly shuffled to
examine how FastStream and the state-of-the-art methods (MStream(F) (Yin et
al., 2018), DP-BMM(-FP) (Chen et al., 2020), and OSDM (Kumar et al., 2020))
perform when dealing with the texts from different domains arriving in random
order.
#### 5.1.1. Construction of the Dataset SO-T
We create a dataset SO-T using the titles of the duplicate questions posted in
StackOverflow website333https://stackoverflow.com/ on various topics such as
_Java_ , _Python_ , _JQuery_ , _R_ , and so on. We consider that duplicate
questions are similar to each other and a group of similar questions can form
a cluster.
We obtain the question titles from the file
Posts.xml444https://meta.stackexchange.com/questions/2677/database-schema-
documentation-for-the-public-data-dump-and-sede and obtain the information
about the duplicate questions from PostLinks.xml. Each item in Posts.xml
represents a single post which can be of different types (e.g., question,
answer, and so on). Each item of the PostLinks.xml contains the information
about a pair of duplicate questions. For instance, the PostLinks.xml contains
the questions A and B if they are duplicate. There are 20,094,655 questions in
Posts.xml and 1,009,249 pair of duplicate questions in PostLinks.xml. Among
the 1,009,249 pair of duplicate questions, we randomly select 400,000
pairs555We select this specific number of pairs (400,000) because of the
maximum capacity of the computer (Core i5-4200U and 8GB memory) where the
experiments were carried out..
Using the duplicate information in PostLinks.xml, we create a list of directed
edges (e.g., $A\rightarrow B$, $B\rightarrow C$) which are then used to create
a graph. To obtain the clusters of duplicate questions, we compute connected
components666We use the library in https://networkx.github.io/ to compute
connected components. using the representation of the graph. In particular, If
A and B are duplicate, and B and C are duplicate, then we obtain the connected
component $A\rightarrow B\rightarrow C$ which is considered as a cluster of
the duplicate questions of A, B, and C.
We compute the length of the connected components defined as the number of
questions in the component. After that, we compute the mean ($\mu$) and
standard deviation ($\sigma$) of the lengths of the connected components and
select the components whose lengths are between the $\mu\pm\sigma$ which in
turn produces 10,573 connected components (i.e., clusters) consisting of
1,23,342 question titles. Sample StackOverflow question titles (with _PostId_)
of a cluster in the dataset SO-T are shown in Figure 1.
– Python Video Framework (1003376)
– Best video manipulation library for Python? (220866)
– Trim (remove frames from) a video using Python (7291653)
Figure 1. Sample StackOverflow question titles (with _PostId_) of a cluster in
the dataset SO-T.
### 5.2. Proposed Clustering Method with Different Types of Features
We represent each text using three different types of features which are
unigram, bigram and biterm and use each representation separately to cluster
the streams of texts. When we use unigram, bigram or biterm feature, our
method is called as FastStream-unigram, FastStream-bigram, and FastStream-
biterm respectively.
### 5.3. Optimal Delete-interval for Proposed Method
Our proposed method (FastStream) requires only one parameter called Delete-
interval ($DI$) to delete the outdated clusters. We set Delete-interval to 500
for all the datasets used in our experiments. How we choose this value is
discussed in the following.
Delete-interval is the interval when we remove outdated clusters after
clustering a certain number of texts. To determine the optimal value of
Delete-interval, we choose the dataset Ns-T and represent the text using
biterm features. We determine the value of Delete-interval based on the
optimal clustering performance of our method for the dataset Ns-T; and use
this value to delete outdated clusters for other datasets. The clustering
performance (in terms of NMI) of our method for different Delete-intervals on
the dataset Ns-T is shown in Figure 2.
$100$$300$$500$$700$$900$$70$$75$$80$$85$$90$Delete-intervals ($DI$)NMI (%)NMI
for Ns-T Figure 2. NMI results of our short text stream clustering method for
different Delete-intervals ($DI$) on dataset Ns-T.
Based on the different values of $DI$, we observe that we achieve the highest
NMI for Ns-T when $DI$=500. Therefore, we choose $DI$=500 for all datasets
used in our experiments to delete outdated clusters.
### 5.4. Baseline Clustering Methods
We compare the performance of FastStream with recent state-of-the-art short
text stream clustering methods as described in the following.
* •
MStream (Yin et al., 2018) algorithm clusters each batch of short texts at a
time. It stores all the clusters produced over the course of time. MStream has
one pass clustering process and update clustering process of each batch. In
update clustering process, it applies gibbs sampling to the same batch of
texts multiple times to improve the initial clustering result obtained in one
pass clustering process.
* •
MStreamF (Yin et al., 2018) is a variant of MStream algorithm that deletes the
outdated clusters of previous batches and only stores the clusters of the
current batch.
* •
DP-BMM (Chen et al., 2020) is a short text stream clustering algorithm that
adopts the similar approach as MStream and clusters each batch of short texts
at a time as it arrives. The principal difference between DP-BMM and MStream
is that DP-BMM represents the texts using biterm features instead of unigrams
(i.e., words). Therefore, DP-BMM stores the clusters of texts using biterm
features.
* •
DP-BMM-FP (Chen et al., 2020) is a variant of DP-BMM algorithm that deletes
the outdated clusters of previous batches and only stores the clusters of the
current batch similar to MStreamF.
* •
Rakib et al. (2020) proposed a short text stream clustering method by adopting
the clustering model of MStream algorithm (Yin et al., 2018). The main
difference between this method and MStream algorithm is that this method
removes outliers from the clusters obtained by MStream algorithm and reassigns
the outliers to the clusters using dynamically computed similarity thresholds.
* •
OSDM (Kumar et al., 2020) is a short text stream clustering algorithm that
clusters each short text one by one as it arrives. OSDM deletes a cluster if a
cluster becomes outdated, that is, the cluster is not being updated for a
while over time.
For MStream(F), we set the parameters $\alpha=0.03$ and $\beta=0.03$ for all
the datasets as defined in (Yin et al., 2018). Likewise, for DP-BMM(-FP), we
set $\alpha=0.6$ and $\beta=0.02$ as mentioned in (Chen et al., 2020).
MStream(F) and DP-BMM(-FP) cluster each batch of texts at a time. We set the
batch size777The batch size equal to 2000 was chosen for MStream(F) and DP-
BMM(-FP) based on their optimal performance on the datasets used in this
paper. to 2000 for MStream(F) and DP-BMM(-FP) for all datasets. We set the
number of iterations to 10 for MStream(F) and DP-BMM(-FP), and number of saved
batches to one for MStreamF and DP-BMM-FP as mentioned in (Chen et al., 2020).
For OSDM, we set $\alpha=2e^{-3}$, $\beta=4e^{-5}$, and an additional
parameter $\lambda=6e^{-6}$ for all the datasets as defined in (Kumar et al.,
2020). OSDM used $\lambda$ as the decay rate which is used to set lower
weights to the clusters which are not being updated over time and need to be
deleted in the future.
### 5.5. Faster Version of Baseline Clustering Methods
* •
Fast-MStream888https://github.com/rashadulrakib/short-text-stream-
clustering/tree/master/FastBatchClustering and Fast-MStreamF are the faster
versions of MStream and MStreamF algorithms that we developed respectively. We
apply the inverted index (Ilic et al., 2014) based searching technique using
the words of the clusters produced by the MStream and MStreamF algorithms.
* •
Fast-Rakib et al. (2020) is the faster version of Rakib et al. (2020). We
index the clusters produced by Rakib et al. (2020) using unigrams following
the notion of Fast-MStream.
### 5.6. Comparison with State-of-the-art Methods
#### 5.6.1. Comparison of Clustering Results
We compare the performance of our proposed method (FastStream) and the
proposed faster versions of the state-of-the-art short text stream clustering
methods with the state-of-the-art methods. We apply different types of text
representations (unigram, bigram, and biterm) to FastStream denoted as
FastStream-unigram, FastStream-bigram, and FastStream-biterm. We use
normalized mutual information (NMI) as the evaluation measures for evaluating
the performance of different clustering methods. We randomly shuffle each
dataset 20 times. Then we perform 20 independent trials for each of the
methods on each dataset. The average NMI results999We could not run DP-
BMM(-FP) on the datasets SO-T and NTSO-T because of their longer running time
that exceeds the capacity of the experimental computer. of these runs are
shown in Table 2.
Table 2. Normalized Mutual Information (NMI) score of different clustering methods. The highest result for a particular dataset is denoted bold. Clustering | Eva. | Data Sets
---|---|---
Methods | | Ns-T | Ts-T | SO-T | NTSO-T
FastStream-unigram | | 0.829 | 0.831 | 0.732 | 0.725
FastStream-bigram | | 0.837 | 0.826 | 0.754 | 0.729
FastStream-biterm | | 0.844 | 0.848 | 0.777 | 0.774
Fast-MStream | | 0.857 | 0.868 | 0.615 | 0.611
Fast-MStreamF | | 0.877 | 0.878 | 0.651 | 0.613
Fast-Rakib et al. (2020) | NMI | 0.864 | 0.901 | 0.654 | 0.639
MStream | | 0.859 | 0.867 | 0.618 | 0.608
MStreamF | | 0.879 | 0.877 | 0.652 | 0.619
DP-BMM | | 0.883 | 0.862 | – | –
DP-BMM-FP | | 0.838 | 0.875 | – | –
OSDM | | 0.858 | 0.842 | 0.463 | 0.441
Rakib et al. (2020) | | 0.862 | 0.892 | 0.658 | 0.641
Our experimental results show that FastStream (-unigram,-bigram, and -biterm)
perform significantly better than the state-of-the methods in terms of NMI on
the dataset SO-T and the combined dataset NTSO-T. Among the three variants of
FastStream, FastStream-biterm performs better than FastStream-unigram and
FastStream-bigram on all the datasets. The reason is that, by using biterm
features, we can extract sufficient contexts for short texts and the biterm
features are more distinctive which in turn more likely to cluster the texts
into their proper clusters (Chen et al., 2020).
The state-of-the-art methods MStream(F), DP-BMM(-FP), and OSDM perform better
than FastStream on the datasets Ns-T and Ts-T since these methods tune the
hyper parameters (e.g., $\alpha$, $\beta$) on these datasets to obtain optimal
clustering performance. On the contrary, our method uses dynamic similarity
thresholds to assign texts to the clusters (new or existing) and does not
require this kind of hyper parameter tuning on a particular dataset.
The overall performance of FastStream is comparable to the performance of the
state-of-the-art methods on the datasets Ns-T and Ts-T. In addition, our
method significantly outperforms the state-of-the-art methods on the datasets
SO-T and NTSO-T as the hyper parameters of the state-of-the-art methods were
not tuned on these two datasets.
#### 5.6.2. Comparison of Running Time
The average running times (in seconds) of FastStream and the state-of-the-art
methods are shown in Table 3.
Table 3. Average running times (in seconds) of different methods. Clustering | Data Sets
---|---
Methods | Ns-T | Ts-T | SO-T | NTSO-T
FastStream-unigram | 3 | 20 | 85 | 168
FastStream-bigram | 3 | 18 | 86 | 167
FastStream-biterm | 4 | 24 | 101 | 183
Fast-MStream | 115 | 219 | 1318 | 1790
Fast-MStreamF | 83 | 193 | 811 | 1198
Fast-Rakib et al. (2020) | 32 | 79 | 479 | 613
MStream | 227 | 541 | 3319 | 4289
MStreamF | 173 | 295 | 1481 | 2137
DP-BMM | 5016 | 12307 | – | –
DP-BMM-FP | 880 | 2854 | – | –
OSDM | 31 | 196 | 1103 | 1882
Rakib et al. (2020) | 80 | 165 | 889 | 1389
The running time of FastStream is several orders of magnitude faster than that
of MStream(F), DP-BMM(-FP), and OSDM on all datasets. In addition, we improve
the running time of the existing methods (e.g., Fast-MStream, Fast-MStreamF,
and Fast-Rakib et al. (2020)) by using inverted index (Ilic et al., 2014). The
reason is that we do not compute similarity between a text and all clusters
while assigning a text to a cluster. Instead we select a specific set of
clusters using the biterms of the text based on inverted index and compute
similarities between the text and the selected clusters. We store almost twice
the number of biterms than other methods as we use inverted index to select a
specific set of clusters for a particular text. We consider this as a small
price to pay for the significant improvement in running time of our proposed
method (FastStream) and the existing state-of-the-art methods.
## 6\. Conclusion and Future Work
We have demonstrated that building an efficient clustering model based on
inverted index and by assigning texts to the clusters using dynamic similarity
thresholds improves the clustering quality of our proposed method and
outperforms the state-of-the-art short text stream clustering methods on
larger datasets.
We also demonstrated that by computing similarity between a text and a
specific set of clusters instead of all clusters, we significantly reduce the
running time of our method (FastStream) and the faster versions of the state-
of-the-art methods than that of the existing state-of-the-art methods. We
contribute two new datasets SO-T and NTSO-T comprising of texts from various
domains to examine how the FastStream and other state-of-the-art methods
perform when texts from different domains arrive in random order.
In the future, we plan to cluster significantly larger short text streams
(composed of a few million texts). We also plan to apply our improved text
stream clustering algorithm to better identify groups of users of social media
platforms interested in similar topics.
## References
* (1)
* Aggarwal et al. (2003) Charu C. Aggarwal, Jiawei Han, Jianyong Wang, and Philip S. Yu. 2003\. A Framework for Clustering Evolving Data Streams. In _Proceedings of the 29th International Conference on Very Large Data Bases_ (Berlin, Germany). 81–92.
* Blei and Lafferty (2006) David M. Blei and John D. Lafferty. 2006. Dynamic Topic Models. In _Proceedings of the 23rd International Conference on Machine Learning_ (Pittsburgh, Pennsylvania, USA). ACM, New York, NY, USA, 113–120.
* Carnein and Trautmann (2019) Matthias Carnein and Heike Trautmann. 2019. Optimizing Data Stream Representation: An Extensive Survey on Stream Clustering Algorithms. _Business & Information Systems Engineering_ 61, 3 (01 Jun 2019), 277–297.
* Chen et al. (2020) Junyang Chen, Zhiguo Gong, and Weiwen Liu. 2020. A Dirichlet process biterm-based mixture model for short text stream clustering. _Applied Intelligence_ 50, 5 (2020), 1609–1619.
* Erk (2012) Katrin Erk. 2012\. Vector Space Models of Word Meaning and Phrase Meaning: A Survey. _Language and Linguistics Compass_ 6 (2012), 635–653.
* Ilic et al. (2014) M. Ilic, P. Spalevic, and M. Veinovic. 2014. Inverted index search in data mining. In _2014 22nd Telecommunications Forum Telfor (TELFOR)_. 943–946.
* Ishwaran and James (2001) Hemant Ishwaran and Lancelot F. James. 2001. Gibbs Sampling Methods for Stick-Breaking Priors. _J. Amer. Statist. Assoc._ 96, 453 (1 3 2001), 161–173.
* Kalogeratos et al. (2016) Argyris Kalogeratos, Panagiotis Zagorisios, and Aristidis Likas. 2016. Improving Text Stream Clustering Using Term Burstiness and Co-Burstiness. In _Proceedings of the 9th Hellenic Conference on Artificial Intelligence_ (Thessaloniki, Greece) _(SETN ’16)_. Association for Computing Machinery, New York, NY, USA, Article 16, 9 pages.
* Kumar et al. (2020) Jay Kumar, Junming Shao, Salah Uddin, and Wazir Ali. 2020\. An Online Semantic-enhanced Dirichlet Model for Short Text Stream Clustering. In _Proc. of the 58th Annual Meeting of the Association for Computational Linguistics_. Online, 766–776.
* Liang et al. (2016) Shangsong Liang, Emine Yilmaz, and Evangelos Kanoulas. 2016\. Dynamic Clustering of Streaming Short Documents. In _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ (San Francisco, California, USA). 995–1004.
* Rakib et al. (2020) Md Rashadul Hasan Rakib, Norbert Zeh, and Evangelos Milios. 2020\. Short Text Stream Clustering via Frequent Word Pairs and Reassignment of Outliers to Clusters _(DocEng ’20)_. Association for Computing Machinery, New York, NY, USA, Article 13, 4 pages.
* Shou et al. (2013) Lidan Shou, Zhenhua Wang, Ke Chen, and Gang Chen. 2013\. Sumblr: Continuous Summarization of Evolving Tweet Streams. In _Proceedings of the 36th International ACM SIGIR Conference on Information Retrieval_ (Dublin, Ireland). 533–542.
* Yin et al. (2018) Jianhua Yin, Daren Chao, Zhongkun Liu, Wei Zhang, Xiaohui Yu, and Jianyong Wang. 2018\. Model-based Clustering of Short Text Streams. In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ (London, United Kingdom). 2634–2642.
* Zhao et al. (2016) Yukun Zhao, Shangsong Liang, Zhaochun Ren, Jun Ma, Emine Yilmaz, and Maarten de Rijke. 2016\. Explainable User Clustering in Short Text Streams. In _Proceedings of the 39th International ACM SIGIR Conference on Research and Development in Information Retrieval_ (Pisa, Italy). ACM, 155–164.
|
# A Family of Supercongruences Involving Multiple Harmonic Sums
Megan McCoy∗, Kevin Thielen∗, Liuquan Wang†, and Jianqiang Zhao⋆
<EMAIL_ADDRESS><EMAIL_ADDRESS>∗Department of Mathematics, Eckerd College,
St. Petersburg, FL 33711, USA †Department of Mathematics, National University
of Singapore, Singapore, 119076, Singapore ⋆Department of Mathematics, The
Bishop’s School, San Diego, CA 92037
###### Abstract.
In recent years, the congruence
$\sum_{\begin{subarray}{c}i+j+k=p\\\
i,j,k>0\end{subarray}}\frac{1}{ijk}\equiv-2B_{p-3}\pmod{p},$
first discovered by the last author have been generalized by either increasing
the number of indices and considering the corresponding supercongruences, or
by considering the alternating version of multiple harmonic sums. In this
paper, we prove a family of similar supercongruences modulo prime powers
$p^{r}$ with the indexes summing up to $mp^{r}$ where $m$ is coprime to $p$,
where all the indexes are also coprime to $p$.
###### Key words and phrases:
Multiple harmonic sums, Bernoulli numbers, Supercongruences
###### 2010 Mathematics Subject Classification:
11A07, 11B68
## 1\. Introduction
Multiple harmonic sums are multiple variable generalization of harmonic
numbers. Let ${\mathbb{N}}$ be the set of natural numbers. For ${\bf
s}=(s_{1},\dots,s_{d})\in{\mathbb{N}}^{d}$ and any $N\in{\mathbb{N}}$, we
define the multiple harmonic sums (MHS) by
$H_{N}({\bf s}):=\sum_{N\geq
k_{1}>\dots>k_{d}>0}\prod_{i=1}^{d}\frac{1}{k_{i}^{s_{i}}}.$
Since mid 1980s these sums have appeared in a few diverse areas of mathematics
as well as theoretical physics such as multiple zeta values [4, 5, 7], Feynman
integrals [1, 3], quantum electrodynamics and quantum chromodynamics [2, 10].
In [17] the last author started to investigate congruence properties of MHSs,
which were also considered by Hoffman [5] independently. As a byproduct, the
following intriguing congruence was noticed: for all primes $p\geq 3$
$\sum_{\begin{subarray}{c}i+j+k=p\\\
i,j,k>0\end{subarray}}\frac{1}{ijk}\equiv-2B_{p-3}\pmod{p},$ (1)
where $B_{k}$ are Bernoulli numbers defined by the generating series
$\frac{t}{e^{t}-1}=\sum_{k=0}^{\infty}B_{k}\frac{t^{k}}{k!}.$
This was proved by the last author using MHSs in [16], and by Ji using some
combinatorial identities in [6]. Later on, a few generalizations and analogs
were obtained by either increasing the number of indices and considering the
corresponding supercongruences (see [11, 13, 15, 21]), or considering the
alternating version of MHSs (see [9, 12]).
Let $\mathcal{P}_{n}$ be the set of positive integers not divisible by $n$. To
generalize the congruence in (1), we wonder if for every _odd_ integer $d\geq
3$ there exists a rational number $q_{d}$ such that
$\sum_{\begin{subarray}{c}l_{1}+l_{2}+\dots+l_{d}=p^{r}\\\
l_{1},\dots,l_{d}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{d}}\equiv q_{d}\cdot p^{r-1}B_{p-d}\pmod{p^{r}}$ (2)
for any prime $p>d$ and integer $r\geq 2$. In [11, 13] it is shown that
$q_{3}=-2$ and $q_{5}=-5!/6$. We should point it out that when $d$ is even,
the congruence pattern is quite different, see [14, 19]. In this paper, we
shall prove the following main result when $d=7$.
###### Theorem 1.1.
Let $r$ and $m$ be positive integers and $p>7$ be a prime such that $p\nmid
m$.
1. (i)
If $r=1$, then
$\sum_{\begin{subarray}{c}l_{1}+l_{2}+\dots+l_{7}=mp\\\
l_{1},\dots,l_{7}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{7}}\equiv-(504m+210m^{3}+6m^{5})B_{p-7}\pmod{p}.$
2. (ii)
If $r\geq 2$, then
$\sum_{\begin{subarray}{c}l_{1}+l_{2}+\dots+l_{7}=mp^{r}\\\
l_{1},\dots,l_{7}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{7}}\equiv-\frac{7!}{10}\cdot mp^{r-1}B_{p-7}\pmod{p^{r}}.$
To establish this result, for all positive integers $n$, $m$, $r$ and primes
$p$, following the notation in [13], we define
$S_{n}^{(m)}(p^{r}):=\sum_{\begin{subarray}{c}l_{1}+l_{2}+\dots+l_{n}=mp^{r}\\\
p^{r}>l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{n}}.$
Notice that the sum in the theorem is not exactly the same type as that
appearing in $S_{n}^{(m)}$ since the condition $p^{r}>l_{i}$ for all $i$ is
not present. The main idea of our proof is to show the special case when $m=1$
first. In order to do this we will first prove the relation
$S_{n}^{(1)}(p^{r+1})\equiv
pS_{n}^{(1)}(p^{r})\quad\pmod{p^{r+1}},\quad\forall r\geq 2,$ (3)
and then use induction. Notice that when $r=1$ the above congruence usually
does not hold anymore. So we will compute the congruence of
$S_{n}^{(1)}(p^{2})$ and $S_{n}^{(1)}(p)$ separately by relating them to the
following quantities:
$R_{n}^{(m)}(p):=\sum_{\begin{subarray}{c}l_{1}+l_{2}+\dots+l_{n}=mp\\\
l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{n}}.$
To save space, throughout the paper when the prime $p$ is fixed we often use
the shorthand $H({\bf s})=H_{p-1}({\bf s})$. Moreover, we shall also need the
modified sum
$H^{(p)}_{N}({\bf s}):=\sum_{\begin{subarray}{c}N\geq k_{1}>\dots>k_{d}>0\\\
k_{1},\dots,k_{d}\in{\mathcal{P}}_{p}\end{subarray}}\
\prod_{i=1}^{d}\frac{1}{k_{i}^{s_{i}}}.$
## 2\. Preliminary lemmas
Let $C^{(m)}_{a,p}(n)$ denote the number of solutions $(x_{1},\dots,x_{n})$ of
the equation
$x_{1}+\dots+x_{n}=mp-a,\quad 0\leq x_{i}<p\ \forall i=1,\dots,n.$
For all $b\geq 1$ set
${\beta}_{n}(a,b):=\binom{bp-a+n-1}{n-1}\qquad\text{and}\qquad{\gamma}_{n}(a):=\frac{(-1)^{a-1}}{a\binom{n-1}{a}}.$
It is not hard to see that
${\beta}_{n}(a,b)\equiv\frac{b(-1)^{a-1}(n-a-1)!(a-1)!}{(n-1)!}p\
\equiv\frac{b(-1)^{a-1}}{a\binom{n-1}{a}}p\equiv
b{\gamma}_{n}(a)p\pmod{p^{2}}.$ (4)
###### Lemma 2.1.
For all $m,n,a\in{\mathbb{N}}$ and primes $p$, we have
$C^{(m)}_{a,p}(n)\equiv(-1)^{m-1}\binom{n-2}{m-1}{\gamma}_{n}(a)p\equiv(-1)^{m-1}\binom{n-2}{m-1}C^{(1)}_{a,p}(n)\pmod{p^{2}}.$
###### Proof.
The coefficient of $x^{mp-a}$ in the expansion of
$\big{(}1+x+\cdots+x^{p-1}\big{)}^{n}=(x^{p}-1)^{n}(x-1)^{-n}$ is
$\begin{split}C^{(m)}_{a,p}(n)&=\sum_{i=0}^{m}\binom{n}{i}\binom{-n}{mp-
ip-a}(-1)^{mp-a}\\\ &=\sum_{i=0}^{m}\binom{n}{i}(-1)^{ip}\binom{n+mp-
ip-a-1}{n-1}\\\ &=\sum_{i=0}^{m}(-1)^{i}\binom{n}{i}\binom{n+mp-
ip-a-1}{n-1}\\\
&\equiv\sum_{i=0}^{m}(-1)^{i}\binom{n}{i}(m-i){\gamma}_{n}(n-a)p\pmod{p^{2}}\end{split}$
by (4). Now we calculate the sum
$A(m)=\sum_{i=0}^{m}{(-1)^{i}\binom{n}{i}(m-i)}.$
It is easy to see that $A(m)$ is the coefficient of $x^{m}$ in the expansion
of
$(1-x)^{n}\cdot\sum_{i=0}^{\infty}{ix^{i}}=(1-x)^{n}\cdot\frac{x}{(1-x)^{2}}=x(1-x)^{n-2}=\sum_{m=1}^{n-1}(-1)^{m}\binom{n-2}{m-1}x^{m},$
as desired. ∎
###### Corollary 2.2.
When $n=7$, we have
$\displaystyle C^{(2)}_{1,p}(7)-C^{(2)}_{6,p}(7)$
$\displaystyle\equiv-(5/3)p,\quad
C^{(3)}_{1,p}(7)-C^{(3)}_{6,p}(7)\equiv\phantom{,}(10/3)p\pmod{p^{2}},$
$\displaystyle C^{(3)}_{2,p}(7)-C^{(3)}_{5,p}(7)$
$\displaystyle\equiv-(2/3)p,\quad
C^{(2)}_{2,p}(7)-C^{(2)}_{5,p}(7)\equiv\phantom{-}(1/3)p\pmod{p^{2}},$
$\displaystyle C^{(3)}_{3,p}(7)-C^{(3)}_{4,p}(7)$
$\displaystyle\equiv\phantom{-}(1/3)p,\quad
C^{(2)}_{3,p}(7)-C^{(2)}_{4,p}(7)\equiv-(1/6)p\pmod{p^{2}}.$
Part (ii) of the following lemma generalizes [13, Lemma 1(ii)].
###### Lemma 2.3.
Let $1\leq k\leq n-1$ and $p>n$ a prime. For all $r\geq 1$, we have
1. (i)
$S_{n}^{(k)}(p^{r})\equiv(-1)^{n}S_{n}^{(n-k)}(p^{r})$ (mod $p^{r}$).
2. (ii)
$\displaystyle
S_{n}^{(m)}(p^{r+1})\equiv\sum_{a=1}^{n-1}C^{(m)}_{a,p}(n)S_{n}^{(a)}(p^{r})\pmod{p^{r+1}}.$
###### Proof.
(i) can be found in [13]. We now prove (ii). For any $n$-tuples
$(l_{1},\cdots,l_{n})$ of integers satisfying $l_{1}+\cdots+l_{n}=mp^{r+1}$,
$p^{r+1}>l_{i}\in{\mathcal{P}}_{p}$, $1\leq i\leq n$, we rewrite them as
$l_{i}=x_{i}p^{r}+y_{i},\quad 0\leq x_{i}<p,\quad 1\leq y_{i}<p^{r},\quad
y_{i}\in{{\mathcal{P}}_{p}},\quad 1\leq i\leq n.$
Since
$\Big{(}\sum_{i=1}^{n}{x_{i}}\Big{)}p^{r}+\sum_{i=1}^{n}{y_{i}}=mp^{r+1}$
and $n<p$, we know there exists $1\leq a<n$ such that
$\left\\{\begin{array}[]{ll}x_{1}+\cdots+x_{n}=mp-a,\quad 0\leq x_{i}<p,\\\
y_{1}+\cdots+y_{n}=ap^{r}.\\\ \end{array}\right.$
For $1\leq a<n$, the equation $x_{1}+\cdots+x_{n}=mp-a$ has $C^{(m)}_{a,p}(n)$
integer solutions with $0\leq x_{i}<p$. Hence by Lemma 2.1
$\begin{split}S_{n}^{(m)}(p^{r+1})&=\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{n}=mp^{r+1}\\\
l_{1},\cdots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\cdots
l_{n}}\\\
&=\sum_{a=1}^{n-1}\sum_{\begin{subarray}{c}x_{1}+\cdots+x_{n}=mp-a\\\ 0\leq
x_{i}<p\end{subarray}}\ \sum_{\begin{subarray}{c}y_{1}+\cdots+y_{n}=ap^{r}\\\
y_{i}\in{\mathcal{P}}_{p},\,y_{i}<p^{r}\end{subarray}}\frac{1}{(x_{1}p^{r}+y_{1})\cdots(x_{n}p^{r}+y_{n})}\\\
&\equiv\sum_{a=1}^{n-1}\sum_{\begin{subarray}{c}x_{1}+\cdots+x_{n}=mp-a\\\
0\leq x_{i}<p\end{subarray}}\
\sum_{\begin{subarray}{c}y_{1}+\cdots+y_{n}=ap^{r}\\\
y_{i}\in{\mathcal{P}}_{p},\,y_{i}<p^{r}\end{subarray}}\left(1-\frac{x_{1}}{y_{1}}p^{r}-\cdots-\frac{x_{n}}{y_{n}}p^{r}\right)\frac{1}{y_{1}\cdots
y_{n}}\pmod{p^{r+1}}\\\
&\equiv\sum_{a=1}^{n-1}C^{(m)}_{a,p}(n)S_{n}^{(a)}(p^{r})\pmod{p^{r+1}}\end{split}$
since for each $x_{j}$ ($j=1,\dots,n$), we have
$\sum_{\begin{subarray}{c}x_{1}+\dots+x_{n}=mp-a\\\ 0\leq
x_{i}<p\end{subarray}}x_{j}=\frac{1}{n}\sum_{x_{1}+\dots+x_{n}=mp-a}(x_{1}+x_{2}+\dots+x_{n})=\frac{mp-a}{n}C^{(m)}_{a,p}(n)\equiv
0\pmod{p}$
by Lemma 2.1. ∎
## 3\. Congruences involving multiple harmonic sums
We first consider some un-ordered sums. Lemmas 3.1 and 3.3 were proved by Zhou
and Cai [21].
###### Lemma 3.1.
Let $p$ be a prime and ${\alpha}_{1},\dots,{\alpha}_{n}$ be positive integers,
$r={\alpha}_{1}+\dots+{\alpha}_{n}\leq p-3$. Define the un-ordered sum
$U_{b}({\alpha}_{1},\dots,{\alpha}_{n})=\sum_{\begin{subarray}{c}0<l_{1},\dots,l_{n}<bp\\\
l_{i}\neq l_{j}\forall i\neq
j,l_{i}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}^{{\alpha}_{1}}\cdots
l_{n}^{{\alpha}_{n}}}.$
Then
$U_{1}({\alpha}_{1},\dots,{\alpha}_{n})\equiv\left\\{\begin{array}[]{ll}\displaystyle(-1)^{n}(n-1)!\frac{r(r+1)}{2(r+2)}B_{p-r-2}\cdot
p^{2}&\pmod{p^{3}},\quad\hbox{if $r$ is odd;}\\\
\displaystyle(-1)^{n-1}(n-1)!\frac{r}{r+1}B_{p-r-1}\cdot
p&\pmod{p^{2}},\quad\hbox{if $r$ is even.}\end{array}\right.$
This easily leads to the following corollary (see also [17]).
###### Corollary 3.2.
Let $p$ be a prime and ${\alpha}$ be positive integer. Then
$H(\\{{\alpha}\\}^{n})\equiv\left\\{\begin{array}[]{ll}\displaystyle(-1)^{n}\frac{{\alpha}(n{\alpha}+1)}{2(n{\alpha}+2)}B_{p-n{\alpha}-2}\cdot
p^{2}&\pmod{p^{3}},\quad\hbox{if $n{\alpha}$ is odd;}\\\
\displaystyle(-1)^{n-1}\frac{{\alpha}}{n{\alpha}+1}B_{p-n{\alpha}-1}\cdot
p&\pmod{p^{2}},\quad\hbox{if $n{\alpha}$ is even.}\end{array}\right.$
###### Lemma 3.3.
Let $n>1$ be positive integer and let $p>n+1$ be a prime. Then
$R_{n}^{(1)}(p)=\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=p\\\
l_{1},\dots,l_{n}>0\end{subarray}}\frac{1}{l_{1}\cdots
l_{n}}\equiv\left\\{\begin{array}[]{ll}\displaystyle-(n-1)!B_{p-n}&\pmod{p},\quad\hbox{if
$n$ is odd;}\\\ \displaystyle-\frac{n\cdot
n!}{n+1}B_{p-n-1}p&\pmod{p^{2}},\quad\hbox{if $n$ is even.}\end{array}\right.$
The next result generalizes Lemma 3.1.
###### Lemma 3.4.
Let $p$ be a prime and ${\alpha}_{1},\dots,{\alpha}_{n}$ be positive integers,
$r={\alpha}_{1}+\dots+{\alpha}_{n}\leq p-3$. Then
$U_{b}({\alpha}_{1},\dots,{\alpha}_{n})\equiv\left\\{\begin{array}[]{ll}\displaystyle(-1)^{n}(n-1)!\frac{b^{2}r(r+1)}{2(r+2)}B_{p-r-2}\cdot
p^{2}&\pmod{p^{3}},\quad\hbox{if $r$ is odd;}\\\
\displaystyle(-1)^{n-1}(n-1)!\frac{br}{r+1}B_{p-r-1}\cdot
p&\pmod{p^{2}},\quad\hbox{if $r$ is even.}\end{array}\right.$
###### Proof.
For all $k\geq 1$, we have
$\displaystyle\sum_{kp<l<(k+1)p}\frac{1}{l^{\alpha}}=$ $\displaystyle\
\sum_{l=1}^{p-1}\frac{1}{(l+kp)^{\alpha}}=\sum_{l=1}^{p-1}\frac{1}{(1+kp/l)^{\alpha}}\frac{1}{l^{\alpha}}$
$\displaystyle\equiv$ $\displaystyle\
\sum_{l=1}^{p-1}\left(1-\frac{{\alpha}kp}{l}+\frac{{\alpha}({\alpha}+1)}{2l^{2}}k^{2}p^{2}\right)\frac{1}{l^{\alpha}}\pmod{p^{3}}$
$\displaystyle\equiv$ $\displaystyle\
\sum_{l=1}^{p-1}\frac{1}{l^{\alpha}}-{\alpha}kp\sum_{l=1}^{p-1}\frac{1}{l^{{\alpha}+1}}\pmod{p^{3}}.$
By Lemma 3.1 we see that
$\sum_{kp<l<(k+1)p}\frac{1}{l^{\alpha}}\equiv\left\\{\begin{array}[]{ll}\displaystyle-\frac{{\alpha}({\alpha}+1)}{{\alpha}+2}\left(\frac{1}{2}+k\right)B_{p-{\alpha}-2}p^{2}&\pmod{p^{3}},\quad\hbox{if
${\alpha}$ is odd;}\\\
\displaystyle\frac{{\alpha}}{{\alpha}+1}B_{p-{\alpha}-1}p&\pmod{p^{2}},\quad\hbox{if
${\alpha}$ is even.}\end{array}\right.$
Therefore for any positive integer $b$, we have
$\sum_{0<l<bp,\,p\nmid
l}\frac{1}{l^{\alpha}}\equiv\left\\{\begin{array}[]{ll}\displaystyle-\frac{b^{2}{\alpha}({\alpha}+1)}{2({\alpha}+2)}B_{p-{\alpha}-2}p^{2}&\pmod{p^{3}},\quad\hbox{if
${\alpha}$ is odd;}\\\
\displaystyle\frac{b{\alpha}}{{\alpha}+1}B_{p-{\alpha}-1}p&\pmod{p^{2}},\quad\hbox{if
${\alpha}$ is even.}\end{array}\right.$
This proves the lemma in the case $n=1$. Now assume the lemma holds when the
number of variables is less than $n$. Then
$U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{n})=\sum_{\begin{subarray}{c}1\leq
l_{1},\dots,l_{n-1}<bp\\\ l_{i}\neq
l_{j},\,l_{i}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}^{{\alpha}_{1}}\dots
l_{n-1}^{{\alpha}_{n-1}}}\left(\sum_{1\leq
l_{n}<bp,\,l_{n}\in{\mathcal{P}}_{p}}\frac{1}{l_{n}^{{\alpha}_{n}}}-\sum_{i=1}^{n-1}\frac{1}{l_{i}^{{\alpha}_{n}}}\right)\\\
\equiv U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{n-1})\left(\sum_{1\leq
l_{n}<bp,\,l_{n}\in{\mathcal{P}}_{p}}\frac{1}{l_{n}^{{\alpha}_{n}}}\right)-\sum_{i=1}^{n-1}U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{i-1},{\alpha}_{i}+{\alpha}_{n},{\alpha}_{i+1},\dots,{\alpha}_{n-1}\big{)}.$
By the induction assumption, we have
$U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{n-1})\sum_{1\leq
l_{n}<bp,\,l_{n}\in{\mathcal{P}}_{p}}\frac{1}{l_{n}^{{\alpha}_{n}}}\equiv\left\\{\begin{array}[]{ll}0&\pmod{p^{3}},\quad\hbox{if
$r$ is odd;}\\\ 0&\pmod{p^{2}},\quad\hbox{if $r$ is even.}\end{array}\right.$
Thus if $r$ is odd, we have
$\displaystyle U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{n})$ $\displaystyle\
\equiv-(n-1)U_{b}\big{(}{\beta}_{1},\dots,{\beta}_{n-1})\qquad\Big{(}\text{here
}\sum_{j=1}^{n-1}{\beta}_{j}=r\Big{)}$ $\displaystyle\
\equiv-(n-1)(-1)^{n-1}(n-2)!\frac{b^{2}r(r+1)}{2(r+2)}p^{2}B_{p-r-2}\pmod{p^{3}}$
$\displaystyle\
\equiv(-1)^{n}(n-1)!\frac{b^{2}r(r+1)}{2(r+2)}p^{2}B_{p-r-2}\pmod{p^{3}}.$
Similarly, if $r$ is even, we can derive
$U_{b}\big{(}{\alpha}_{1},\dots,{\alpha}_{n})\equiv(-1)^{n-1}(n-1)!\frac{br}{r+1}pB_{p-r-1}\pmod{p^{2}}.$
∎
###### Lemma 3.5.
Let $n$ be an odd positive integer and $p$ be a prime. Then
$R_{n}^{(2)}(p)=\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}\equiv-\frac{n+1}{2}\cdot(n-1)!B_{p-n}\pmod{p}.$
###### Proof.
We have
$\displaystyle\ \sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}$ $\displaystyle=$ $\displaystyle\
\sum_{l_{1}+\dots+l_{n}=2p}\frac{1}{l_{1}\dots
l_{n}}-\frac{n}{p}\sum_{l_{1}+\dots+l_{n-1}=p}\frac{1}{l_{1}\dots l_{n-1}}$
$\displaystyle=$ $\displaystyle\
\frac{n!}{2p}\sum_{0<u_{1}<\dots<u_{n-1}<2p}\frac{1}{u_{1}\dots
u_{n-1}}-\frac{n!}{p^{2}}\sum_{0<u_{1}<\dots<u_{n-2}<p}\frac{1}{u_{1}\dots
u_{n-2}}$ $\displaystyle=$ $\displaystyle\
\frac{n!}{2p}H^{(p)}_{2p-1}(\\{1\\}^{n-1})+\frac{n!}{2p^{2}}\sum_{j=1}^{n-1}\sum_{\begin{subarray}{c}0<u_{1}<\dots<u_{j-1}<p\\\
p<u_{j+1}<\dots<u_{n-1}<2p\end{subarray}}\frac{1}{u_{1}\dots
u_{j-1}u_{j+1}\dots u_{n-1}}-$ $\displaystyle\frac{n!}{p^{2}}H(\\{1\\}^{n-2})$
$\displaystyle\equiv$ $\displaystyle\
\frac{n!}{2p}\cdot\frac{U_{2}(\\{1\\}^{n-1})}{(n-1)!}+\frac{n!}{2p^{2}}\left(2H(\\{1\\}^{n-2})-p\frac{U_{1}(2,\\{1\\}^{n-3})}{(n-3)!}\right)$
$\displaystyle+$ $\displaystyle\
\frac{n!}{2p^{2}}\sum_{j=2}^{n-2}H(\\{1\\}^{j-1})\left(H(\\{1\\}^{n-j-1})-p\frac{U_{1}(2,\\{1\\}^{n-j-2})}{(n-j-2)!}\right)-\frac{n!}{p^{2}}H(\\{1\\}^{n-2})$
$\displaystyle\pmod{p}$ $\displaystyle\equiv$ $\displaystyle\
\frac{n}{2p}U_{2}(\\{1\\}^{n-1})-\frac{n!}{2p}\frac{U_{1}(2,\\{1\\}^{n-3})}{(n-3)!}$
$\displaystyle\pmod{p}$ $\displaystyle\equiv$ $\displaystyle\
-\frac{n}{p}(n-2)!\frac{n-1}{n}B_{p-n}p-(-1)^{n-3}\frac{n!}{2p}(n-3)!\frac{n-1}{n(n-3)!}B_{p-n}p$
$\displaystyle\pmod{p}$ $\displaystyle\equiv$ $\displaystyle\
-\frac{n+1}{2}(n-1)!B_{p-n}$ $\displaystyle\pmod{p}$ $\displaystyle,\ $
as desired. ∎
###### Corollary 3.6.
Let $n$ be an odd positive integer with $n\geq 5$. Then for all prime $p>n$,
we have
$S_{n}^{(2)}(p)\equiv\frac{n-1}{2}\cdot(n-1)!B_{p-n}\pmod{p}.$
###### Proof.
We observe that
$\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{j}\in{\mathcal{P}}_{p}\,\forall j\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}\equiv\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{1},\dots,l_{n}<p\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}+n\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=p\\\
l_{1},\dots,l_{n}<p\end{subarray}}\frac{1}{(l_{1}+p)l_{2}\dots
l_{n}}\pmod{p}.$
By Lemma 3.3, we have $S_{n}^{(1)}(p)\equiv-(n-1)!B_{p-n}\pmod{p}$. So we
deduce
$S_{n}^{(2)}(p)\equiv\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{j}\in{\mathcal{P}}_{p}\,\forall j\end{subarray}}\frac{1}{l_{1}l_{2}\dots
l_{n}}-nS_{n}^{(1)}(p)\equiv\frac{n-1}{2}\cdot(n-1)!B_{p-n}\pmod{p}.$
∎
###### Lemma 3.7.
Let $n\geq 3$ be an odd positive integer. Then for all prime
$p\geq\max\\{n,5\\}$, we have
$\displaystyle R_{n}^{(3)}(p)=\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=3p\\\
l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}$ $\displaystyle\equiv$ $\displaystyle-\frac{1}{n}{n+2\choose
3}\cdot(n-1)!B_{p-n}-\frac{n!}{6}\sum\limits_{\begin{subarray}{c}a+b+c=\frac{n-3}{2}\\\
a,b,c\geq
1\end{subarray}}\frac{B_{p-2a-1}B_{p-2b-1}B_{p-2c-1}}{(2a+1)(2b+1)(2c+1)}\pmod{p}.$
###### Proof.
Let $\nu=n-1$ throughout the proof. Let $u_{i}=l_{1}+\dots+l_{i}$, $1\leq
i\leq\nu$. We have
$\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=3p\\\
l_{1},\dots,l_{n}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}=\frac{n!}{3p}\sum_{\begin{subarray}{c}1\leq u_{1}<\dots<u_{n-1}<3p\\\
u_{1},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2},u_{\nu}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}.$ (5)
Evidently
$\displaystyle\ \ \ \sum_{\begin{subarray}{c}1\leq u_{1}<\dots<u_{\nu}<3p\\\
u_{1},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2},u_{\nu}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle\ =\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p\\\ u_{1},u_{2},\dots,u_{\nu}\in{\mathcal{P}}_{p}\\\
u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}+\sum_{i=2}^{n-4}\sum_{j=i+2}^{n-2}\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{i}=p,u_{j}=2p\\\ \forall k\neq i,k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle\ +\sum_{j=2}^{n-2}\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{j}=p\\\ \forall k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}+\sum_{j=2}^{n-2}\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{j}=2p\\\ \forall k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}.$ (6)
Now we deal with the sums in (6) one by one. For $2\leq j\leq n-2$,
$\displaystyle\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{j}=p\\\ \forall k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle=$ $\displaystyle\ \frac{1}{p}\sum_{1\leq
u_{1}<\cdots<u_{j-1}<p}\frac{1}{u_{1}\cdots
u_{j-1}}\sum_{\begin{subarray}{c}p<u_{j+1}<\cdots<u_{\nu}<3p,\\\ \forall
k>j,u_{k},u_{j+2}-u_{j+1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{j+1}\cdots
u_{\nu}}$ $\displaystyle=$ $\displaystyle\
\frac{1}{p}H(\\{1\\}^{j-1})\sum_{\begin{subarray}{c}0<u_{1}<\cdots<u_{\nu-j}<2p\\\
\forall
k,u_{k},u_{2}-u_{1},\dots,u_{\nu-j}-u_{\nu-j-1}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{(u_{1}+p)\cdots(u_{\nu-j}+p)}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{p}H(\\{1\\}^{j-1})\bigg{(}H^{(p)}_{2p-1}(\\{1\\}^{n-j-1})-p\sum_{i=0}^{n-j-2}H^{(p)}_{2p-1}(\\{1\\}^{i},2,\\{1\\}^{n-i-j-2})$
$\displaystyle\ \hskip
56.9055pt-\sum_{i=1}^{\nu-j-2}\sum_{0<u_{1}<\cdots<u_{\nu-j-1}<p}\frac{1}{(u_{1}+p)\cdots(u_{i}+p)(u_{i}+2p)\cdots(u_{\nu-j-1}+2p)}\bigg{)}\pmod{p^{2}}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{p}H(\\{1\\}^{j-1})\bigg{(}\frac{U_{2}(\\{1\\}^{n-j-1})}{(n-j-1)!}-\frac{U_{2}(2,\\{1\\}^{n-j-2})}{(n-j-2)!}p-\frac{U_{2}(2,\\{1\\}^{n-j-2})}{(n-j-2)!}$
$\displaystyle\
+p\sum_{i=1}^{\nu-j-2}H(\\{1\\}^{i-1},3,\\{1\\}^{n-i-j-2})+p\sum_{i=1}^{\nu-j-2}\sum_{k=0}^{i-1}H(\\{1\\}^{k},2,\\{1\\}^{i-k-2},2,\\{1\\}^{n-i-j-2})$
$\displaystyle\
+2p\sum_{i=1}^{\nu-j-2}H(\\{1\\}^{i-1},3,\\{1\\}^{n-i-j-2})+2p\sum_{i=1}^{\nu-j-2}\sum_{k=0}^{n-i-j-2}H(\\{1\\}^{i-1},2,\\{1\\}^{k},2,\\{1\\}^{n-i-
j-k-3})\bigg{)}$ $\displaystyle\equiv$ $\displaystyle\ 0\pmod{p^{2}}$ (7)
by Lemma 3.1 and Corollary 3.2 since one of $j-1$ and $n-j-1$ is even and the
other is odd.
Similarly, for $2\leq j\leq n-2$,
$\displaystyle\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{j}=2p\\\ \forall
k<j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle=$ $\displaystyle\
\frac{1}{2p}\sum_{\begin{subarray}{c}1\leq u_{1}<\cdots<u_{j-1}<2p\\\ \forall
k<j,u_{k},u_{2}-u_{1},\dots,u_{j-1}-u_{j-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\cdots
u_{j-1}}\sum_{2p<u_{j+1}<\cdots<u_{\nu}<3p}\frac{1}{u_{j+1}\cdots u_{\nu}}$
$\displaystyle=$ $\displaystyle\
\frac{1}{2p}\left(H^{(p)}_{2p-1}(\\{1\\}^{j-1})-\sum_{i=1}^{j-2}\sum_{1\leq
u_{1}<\cdots<u_{j-2}<p}\frac{1}{u_{1}\cdots
u_{i}(u_{i}+p)\cdots(u_{j-2}+p)}\right)$ $\displaystyle\ \hskip
85.35826pt\times\sum_{0<u_{1}<\cdots<u_{n-j-1}<p}\frac{1}{(u_{1}+2p)\cdots(u_{n-j-1}+2p)}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{2p}\bigg{(}H^{(p)}_{2p-1}(\\{1\\}^{j-1})-\sum_{i=1}^{j-2}H(\\{1\\}^{i-1},2,\\{1\\}^{j-i-2})+p\sum_{i=1}^{j-2}\sum_{k=0}^{j-i-2}H(\\{1\\}^{i-1},2,\\{1\\}^{k},2,\\{1\\}^{j-i-k-3})\bigg{)}$
$\displaystyle\ \hskip
85.35826pt\times\bigg{(}H(\\{1\\}^{n-j-1})-2p\sum_{i=0}^{n-j-2}H(\\{1\\}^{i},2,\\{1\\}^{n-i-j-2})\bigg{)}\pmod{p^{2}}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{2p}\left(\frac{U_{2}(\\{1\\}^{j-1})}{(j-1)!}-\frac{U_{1}(\\{1\\}^{j-3})}{(j-3)!}+\frac{U_{1}(2,2,\\{1\\}^{j-4})}{2!(j-4)!}p\right)\left(H(\\{1\\}^{n-j-1})-\frac{2U_{1}(2,\\{1\\}^{n-j-2})}{(n-j-2)!}p\right)$
$\displaystyle\equiv$ $\displaystyle\ 0\pmod{p^{2}}.$ (8)
Further, for all $2\leq i\leq j-2\leq n-4$, we obtain
$\displaystyle\ \sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{i}=p,u_{j}=2p\\\ \forall k\neq i,k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle=$ $\displaystyle\ \frac{1}{2p^{2}}\sum_{1\leq
u_{1}<\cdots<u_{i-1}<p}\frac{1}{u_{1}\cdots<u_{i-1}}\sum_{p<u_{i+1}<\cdots<u_{j-1}<2p}\frac{1}{u_{i+1}\cdots
u_{j-1}}\sum_{2p<u_{j+1}<\cdots<u_{\nu}<3p}\frac{1}{u_{j+1}\cdots u_{\nu}}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{2p^{2}}H(\\{1\\}^{i-1})\Big{(}H(\\{1\\}^{j-i-1})-p\sum_{\ell=1}^{j-i-1}H(\\{1\\}^{\ell},2,\\{1\\}^{j-i-\ell-2})\Big{)}$
$\displaystyle\ \hskip
85.35826pt\times\Big{(}H(\\{1\\}^{n-j-1})-2p\sum_{\ell=1}^{n-j-1}H(\\{1\\}^{\ell},2,\\{1\\}^{n-j-\ell-2})\Big{)}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{2p^{2}}H(\\{1\\}^{i-1})H(\\{1\\}^{j-i-1})H(\\{1\\}^{n-j-1})$
$\displaystyle-$ $\displaystyle\
\frac{1}{2p}H(\\{1\\}^{i-1})H(\\{1\\}^{n-j-1})\frac{U_{1}(2,\\{1\\}^{j-i-2})}{(j-i-2)!}-\frac{1}{p}H(\\{1\\}^{i-1})H(\\{1\\}^{j-i-1})\frac{U_{1}(2,\\{1\\}^{n-j-2})}{(n-j-2)!}$
$\displaystyle\equiv$ $\displaystyle\
\frac{1}{2p^{2}}H(\\{1\\}^{i-1})H(\\{1\\}^{j-i-1})H(\\{1\\}^{n-j-1})\pmod{p^{2}}.$
Thus by Corollary 3.2 we deduce that
$\displaystyle\sum_{i=2}^{n-4}\sum_{j=i+2}^{n-2}\sum_{\begin{subarray}{c}1\leq
u_{1}<\dots<u_{\nu}<3p,\>u_{i}=p,u_{j}=2p\\\ \forall k\neq i,k\neq
j,u_{k},u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ (9) $\displaystyle\equiv$
$\displaystyle\frac{1}{2p^{2}}\sum_{i=2}^{n-4}\sum_{j=i+2}^{n-2}H(\\{1\\}^{i-1})H(\\{1\\}^{j-i-1})H(\\{1\\}^{n-j-1})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2p^{2}}\sum\limits_{\begin{subarray}{c}a+b+c=n-3\\\
a,b,c\geq 1\end{subarray}}H(\\{1\\}^{a})H(\\{1\\}^{b})H(\\{1\\}^{c})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2p^{2}}\sum\limits_{\begin{subarray}{c}a+b+c=\frac{n-3}{2}\\\
a,b,c\geq 1\end{subarray}}H(\\{1\\}^{2a})H(\\{1\\}^{2b})H(\\{1\\}^{2c})$
$\displaystyle\equiv$
$\displaystyle-\frac{p}{2}\sum\limits_{\begin{subarray}{c}a+b+c=\frac{n-3}{2}\\\
a,b,c\geq
1\end{subarray}}\frac{B_{p-2a-1}B_{p-2b-1}B_{p-2c-1}}{(2a+1)(2b+1)(2c+1)}\pmod{p^{2}}.$
For the first sum in (6), by the inclusion-exclusion principle,
$\displaystyle\sum_{\begin{subarray}{c}1\leq u_{1}<\dots<u_{\nu}<3p\\\
u_{1},\dots,u_{\nu}\in{\mathcal{P}}_{p}\\\
u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ (10) $\displaystyle\equiv$
$\displaystyle\sum_{\begin{subarray}{c}1\leq u_{1}<\dots<u_{\nu}<3p\\\
u_{1},\dots,u_{\nu}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}-\sum_{j=1}^{n-2}D_{j}-\sum_{j=1}^{n-2}T_{j}+\sum_{j=1}^{n-3}\sum_{k=j+2}^{n-2}T_{j,k}+\sum_{j=1}^{n-3}W_{j}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{(n-1)!}U_{3}(\\{1\\}^{n-1})-\sum_{j=1}^{n-2}D_{j}-\sum_{j=1}^{n-2}T_{j}+\sum_{j=1}^{n-3}\sum_{k=j+2}^{n-2}T_{j,k}+\sum_{j=1}^{n-3}W_{j}\pmod{p^{2}},$
where (setting $v_{n-1}=3p$)
$\displaystyle D_{j}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\dots<v_{j}<v_{j}+2p<v_{j+1}<\dots<3p\\\
v_{1},\dots,v_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\dots
v_{j}(v_{j}+2p)v_{j+1}\dots v_{n-2}},$ $\displaystyle T_{j}=$ $\displaystyle\
\sum_{\begin{subarray}{c}1\leq v_{1}<\dots<v_{j}<v_{j}+p<v_{j+1}<\dots<3p\\\
v_{1},\dots,v_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\dots
v_{j}(v_{j}+p)v_{j+1}\dots v_{n-2}},$ $\displaystyle T_{j,k}=$ $\displaystyle\
\sum_{\begin{subarray}{c}1\leq
v_{1}<\cdots<v_{j}<v_{j}+p<v_{j+1}<\cdots<v_{k}<v_{k}+p<\cdots<3p\\\
v_{1},\cdots,v_{n-3}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\cdots
v_{j}(v_{j}+p)v_{j+1}\cdots v_{k}(v_{k}+p)v_{k+1}\cdots v_{n-3}},$
$\displaystyle W_{j}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\cdots<v_{j}<v_{j}+p<v_{j}+2p<v_{j+1}<\cdots<3p\\\
v_{1},\cdots,v_{n-3}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\cdots
v_{j}(v_{j}+p)(v_{j}+2p)v_{j+1}\cdots v_{n-3}}.$
We have
$\displaystyle D_{j}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\dots<v_{n-2}<p\\\
v_{1},\dots,v_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\dots
v_{j}(v_{j}+2p)(v_{j+1}+2p)\dots(v_{n-2}+2p)}$ $\displaystyle\equiv$
$\displaystyle\
H(\\{1\\}^{j-1},2,\\{1\\}^{n-j-2})-2p\bigg{(}H(\\{1\\}^{j-1},3,\\{1\\}^{n-j-2})$
$\displaystyle+$ $\displaystyle\
\sum_{i=0}^{n-j-3}H(\\{1\\}^{j-1},2,\\{1\\}^{i},2,\\{1\\}^{n-j-i-3})\bigg{)}\pmod{p^{2}}.$
So by Lemma 3.1 we have
$\displaystyle\sum_{j=1}^{n-2}D_{j}=$ $\displaystyle\
\frac{U_{1}(2,\\{1\\}^{n-3})}{(n-3)!}-\frac{2U_{1}(3,\\{1\\}^{n-3})}{(n-3)!}p-\frac{2U_{1}(2,2,\\{1\\}^{n-4})}{(n-4)!}p\equiv\frac{n-1}{n}B_{p-n}\cdot
p\pmod{p^{2}}.$
Similarly,
$\displaystyle T_{j}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\dots<v_{n-2}<2p\\\
v_{1},\dots,v_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\dots
v_{j}(v_{j}+p)(v_{j+1}+p)\dots(v_{n-2}+p)}$ $\displaystyle\equiv$
$\displaystyle\
H^{(p)}_{2p-1}(\\{1\\}^{j-1},2,\\{1\\}^{n-j-2})-p\bigg{(}H^{(p)}_{2p-1}(\\{1\\}^{j-1},3,\\{1\\}^{n-j-2})$
$\displaystyle+$ $\displaystyle\
\sum_{i=0}^{n-j-3}H^{(p)}_{2p-1}(\\{1\\}^{j-1},2,\\{1\\}^{i},2,\\{1\\}^{n-j-i-3})\bigg{)}\pmod{p^{2}}.$
So by Lemma 3.1 we have
$\displaystyle\sum_{j=1}^{n-2}T_{j}=$ $\displaystyle\
\frac{U_{2}(2,\\{1\\}^{n-3})}{(n-3)!}-\frac{U_{2}(3,\\{1\\}^{n-3})}{(n-3)!}p-\frac{U_{2}(2,2,\\{1\\}^{n-4})}{(n-4)!}p\equiv\frac{2(n-1)}{n}B_{p-n}\cdot
p\pmod{p^{2}}.$
Moreover,
$\displaystyle T_{j,k}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\cdots<v_{n-3}<p\\\
v_{1},\cdots,v_{n-3}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\cdots
v_{j}(v_{j}+p)(v_{j+1}+p)\cdots(v_{k}+p)(v_{k}+2p)\cdots(v_{n-3}+2p)}$
$\displaystyle\equiv$ $\displaystyle\
H(\\{1\\}^{j-1},2,\\{1\\}^{k-j-1},2,\\{1\\}^{n-k-3})-p\bigg{(}H(\\{1\\}^{j-1},3,\\{1\\}^{k-j-1},2,\\{1\\}^{n-k-3})$
$\displaystyle+$ $\displaystyle\
\sum_{i=0}^{k-j-2}H(\\{1\\}^{j-1},2,\\{1\\}^{i},2,\\{1\\}^{k-i-j-2},2,\\{1\\}^{n-k-3})+H(\\{1\\}^{j-1},2,\\{1\\}^{k-j-1},3,\\{1\\}^{n-k-3})\bigg{)}$
$\displaystyle-$ $\displaystyle\
2p\bigg{(}H(\\{1\\}^{j-1},2,\\{1\\}^{k-j-1},3,\\{1\\}^{n-k-3})$
$\displaystyle+$ $\displaystyle\
\sum_{i=0}^{n-k-4}H(\\{1\\}^{j-1},2,\\{1\\}^{k-j-1},2,\\{1\\}^{i},2,\\{1\\}^{n-i-k-4})\bigg{)}\pmod{p^{2}}.$
Finally
$\displaystyle W_{j}=$ $\displaystyle\ \sum_{\begin{subarray}{c}1\leq
v_{1}<\cdots<v_{n-3}<p\\\
v_{1},\cdots,v_{n-3}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{v_{1}\cdots
v_{j}(v_{j}+p)(v_{j}+2p)(v_{j+1}+2p)\cdots(v_{n-3}+2p)}$ $\displaystyle\equiv$
$\displaystyle\
H(\\{1\\}^{j-1},3,\\{1\\}^{n-j-3})-pH(\\{1\\}^{j-1},4,\\{1\\}^{n-j-3})$
$\displaystyle-$ $\displaystyle\
2p\bigg{(}H(\\{1\\}^{j-1},4,\\{1\\}^{n-j-3})+\sum_{i=0}^{n-k-4}H(\\{1\\}^{j-1},3,\\{1\\}^{i},2,\\{1\\}^{n-i-j-4})\bigg{)}\pmod{p^{2}}.$
Thus
$\sum_{j=1}^{n-3}\sum_{k=j+1}^{n-2}T_{j,k}+\sum_{j=1}^{n-3}W_{j}\equiv\frac{U_{1}(2,2,\\{1\\}^{n-5})}{2(n-5)!}+\frac{U_{1}(3,\\{1\\}^{n-4})}{(n-4)!}\\\
-3p\left(\frac{U_{1}(4,\\{1\\}^{n-4})}{(n-4)!}+\frac{U_{1}(3,2,\\{1\\}^{n-5})}{(n-5)!}+\frac{U_{1}(\\{2\\}^{3},\\{1\\}^{n-6})}{3!(n-6)!}\right)\pmod{p^{2}}\\\
\equiv-(n-4)!\frac{(n-1)}{n}\left(\frac{1}{2(n-5)!}+\frac{1}{(n-4)!}\right)B_{p-n}p\equiv-\frac{(n-1)(n-2)}{2n}B_{p-n}p\pmod{p^{2}}.$
Plugging this into (10), we have
$\displaystyle\sum_{\begin{subarray}{c}1\leq u_{1}<\dots<u_{\nu}<3p\\\
u_{1},\dots,u_{\nu}\in{\mathcal{P}}_{p}\\\
u_{2}-u_{1},\dots,u_{\nu}-u_{n-2}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{u_{1}\dots
u_{\nu}}$ $\displaystyle\equiv$
$\displaystyle-\frac{n!}{3}\frac{(n^{2}+3n+2)}{2n}B_{p-n}$ (11)
$\displaystyle\equiv$
$\displaystyle-\frac{(n+1)(n+2)}{6}(n-1)!B_{p-n}\pmod{p}$
by Lemma 3.1 again.
Now plugging (3), (3), (9) and (11) into (6), and then combining with (5), we
get the desired result. ∎
###### Corollary 3.8.
Let $n\geq 3$ be an odd positive integer. Then for all prime
$p\geq\max\\{n,5\\}$, we have
$S_{n}^{(3)}(p)\equiv-\frac{1}{n}\binom{n}{3}\cdot(n-1)!B_{p-n}-\frac{n!}{6}\sum\limits_{\begin{subarray}{c}a+b+c=\frac{n-3}{2}\\\
a,b,c\geq
1\end{subarray}}\frac{B_{p-2a-1}B_{p-2b-1}B_{p-2c-1}}{(2a+1)(2b+1)(2c+1)}\pmod{p}.$
###### Proof.
We observe that
$\displaystyle\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=3p\\\
l_{j}\in{\mathcal{P}}_{p}\,\forall j\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}\equiv$ $\displaystyle\ \sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=3p\\\
l_{j}<p,\,l_{j}\in{\mathcal{P}}_{p}\,\forall
j\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}+\binom{n}{2}\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=p\\\
l_{1},\dots,l_{n}<p\end{subarray}}\frac{1}{(l_{1}+p)(l_{2}+p)l_{3}\dots
l_{n}}$ $\displaystyle+$ $\displaystyle\
n\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=p\\\
l_{1},\dots,l_{n}<p\end{subarray}}\frac{1}{(l_{1}+2p)l_{2}\dots
l_{n}}+n\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=2p\\\
l_{1},\dots,l_{n}<p\end{subarray}}\frac{1}{(l_{1}+p)l_{2}\dots
l_{n}}\pmod{p}.$
So we deduce
$\displaystyle S_{n}^{(3)}(p)$
$\displaystyle\,\equiv\sum_{\begin{subarray}{c}l_{1}+\dots+l_{n}=3p\\\
l_{j}\in{\mathcal{P}}_{p}\,\forall j\end{subarray}}\frac{1}{l_{1}\dots
l_{n}}-\binom{n+1}{2}S_{n}^{(1)}(p)-nS_{n}^{(2)}(p)$
$\displaystyle\,\equiv-\frac{1}{n}\binom{n}{3}\cdot(n-1)!B_{p-n}-\frac{n!}{6}\sum\limits_{\begin{subarray}{c}a+b+c=\frac{n-3}{2}\\\
a,b,c\geq
1\end{subarray}}\frac{B_{p-2a-1}B_{p-2b-1}B_{p-2c-1}}{(2a+1)(2b+1)(2c+1)}\pmod{p}$
by Lemma 3.7, since $S_{n}^{(1)}(p)\equiv-(n-1)!B_{p-n}\pmod{p}$ by Lemma 3.3
and $S_{n}^{(2)}(p)\equiv-\frac{n-1}{2}(n-1)!B_{p-n}\pmod{p}$ by Corollary
3.6. ∎
## 4\. Proof of the main theorem
First, we prove a special case of Theorem 1.1.
###### Proposition 4.1.
For all $r\geq 1$ and prime $p>7$ we have
$S_{7}^{(1)}(p^{r+1})\equiv-\frac{7!}{10}B_{p-7}p^{r}\pmod{p^{r+1}}.$
###### Proof.
By Lemma 2.3, for all $r\geq 1$, we have
$\displaystyle
S_{n}^{(m)}(p^{r+1})\equiv\sum_{a=1}^{n-1}\big{(}(-1)^{m-1}{n-2\choose
m-1}{\gamma}_{n}(a)p+O(p^{2})\big{)}S_{n}^{(a)}(p^{r})\pmod{p^{r+1}}.$
Here the $O(p^{2})$ means a quantity which remains a $p$-adic integer after
dividing by the $p^{2}$. By induction on $r$ it is not hard to see that for
all $m=1,\dots,n-1$, we have
$S_{n}^{(m)}(p^{r+1})\equiv 0\pmod{p^{r}},\quad\text{ for all }r\geq 1.$
Thus for all $m=1,\dots,n-1$, by Lemmas 2.1 and 2.3, we have
$\displaystyle S_{n}^{(m)}(p^{r+1})$
$\displaystyle\,\equiv\sum_{a=1}^{n-1}(-1)^{m-1}\binom{n-2}{m-1}{\gamma}_{n}(a)pS_{n}^{(a)}(p^{r})$
$\displaystyle\pmod{p^{r+1}}\,$
$\displaystyle\,\equiv(-1)^{m-1}\binom{n-2}{m-1}S_{n}^{(1)}(p^{r+1})$
$\displaystyle\pmod{p^{r+1}}.$
Thus by Lemmas 2.1 and 2.3, for all $r\geq 2$
$\displaystyle S_{n}^{(1)}(p^{r+1})\equiv$ $\displaystyle\
\sum_{m=1}^{n-1}C_{a,p}^{m}(n)S_{n}^{(m)}(p^{r})$
$\displaystyle\pmod{p^{r+1}}\,$ $\displaystyle\equiv$ $\displaystyle\
\sum_{m=1}^{n-1}(-1)^{m-1}\binom{n-2}{m-1}p{\gamma}_{n}(m)S_{n}^{(1)}(p^{r})$
$\displaystyle\pmod{p^{r+1}}\,$ $\displaystyle\equiv$ $\displaystyle\
\sum_{m=1}^{n-1}\frac{(n-m-1)!(m-1)!p}{(n-1)!}\binom{n-2}{m-1}S_{n}^{(1)}(p^{r})$
$\displaystyle\pmod{p^{r+1}}\,$ $\displaystyle\equiv$ $\displaystyle\
pS_{n}^{(1)}(p^{r})$ $\displaystyle\pmod{p^{r+1}},$
which proves (3). Finally, by applying Lemma 2.3 when $n=7$, we get
$\displaystyle S_{7}^{(1)}(p^{2})\equiv$
$\displaystyle\,\frac{p}{3}S_{7}^{(1)}(p)-\frac{p}{15}S_{7}^{(2)}(p)+\frac{p}{30}S_{7}^{(3)}(p^{r})$
$\displaystyle\pmod{p^{2}}$ $\displaystyle\equiv$
$\displaystyle\,\left(-\frac{p}{3}-\frac{3p}{15}-\frac{5p}{30}\right)6!B_{p-7}\equiv-\frac{7!}{10}B_{p-7}p$
$\displaystyle\pmod{p^{2}}$
by Lemma 3.3, Corollary 3.6 and Corollary 3.8. ∎
We are now ready to prove Theorem 1.1.
Let $n=mp^{r}$, where $p$ does not divide $m$. For any 7-tuples
$(l_{1},\cdots,l_{7})$ of integers satisfying $l_{1}+\cdots+l_{7}=n$,
$l_{i}\in{\mathcal{P}}_{p}$, $1\leq i\leq 7$, we rewrite them as
$l_{i}=x_{i}p^{r}+y_{i},\quad x_{i}\geq 0,\quad 1\leq y_{i}<p^{r},\quad
y_{i}\in{\mathcal{P}}_{p},\quad 1\leq i\leq 7.$
Since
$\Big{(}\sum_{i=1}^{7}x_{i}\Big{)}p^{r}+\sum_{i=1}^{7}{y_{i}}=mp^{r},$
we know there exists $1\leq a\leq 6$ such that
$\left\\{\begin{array}[]{ll}x_{1}+\cdots+x_{7}=m-a,\\\
y_{1}+\cdots+y_{7}=ap^{r}.\\\ \end{array}\right.$
For $1\leq a\leq 6$, the equation $x_{1}+\cdots+x_{7}=m-a$ has
$\binom{m+6-a}{6}$ nonnegative integer solutions. Hence
$\displaystyle\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{7}=mp^{r}\\\
l_{1},\cdots,l_{7}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\cdots
l_{7}}$ $\displaystyle=\sum_{a=1}^{6}\ \sum_{x_{1}+\cdots+x_{7}=m-a}\
\sum_{\begin{subarray}{c}y_{1}+\cdots+y_{7}=ap^{r}\\\
y_{i}\in{\mathcal{P}}_{p},y_{i}<p^{r}\end{subarray}}\frac{1}{(x_{1}p^{r}+y_{1})\cdots(x_{7}p^{r}+y_{7})}$
$\displaystyle\equiv\sum_{a=1}^{6}\binom{m+6-a}{6}S_{7}^{(a)}(p^{r})\pmod{p^{r}}.$
(12)
(i) If $r=1$, then since $S_{7}^{(1)}(p)\equiv-6!B_{p-7}\pmod{p}$. We also
have $S_{7}^{(2)}(p)\equiv 3\cdot 6!B_{p-7}\pmod{p}$,
$S_{7}^{(3)}(p)\equiv-5\cdot 6!B_{p-5}\pmod{p}$ and $S_{7}^{(a)}(p)\equiv-
S_{7}^{(7-a)}(p)\pmod{p}$ for $4\leq a\leq 6$. Hence from (4) we have
$\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{7}=n\\\
l_{1},\cdots,l_{7}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\cdots
l_{7}}\equiv\frac{1}{6!}\Big{(}504m+210m^{3}+6m^{5}\Big{)}S_{7}^{(1)}(p)\pmod{p}.$
Since $S_{7}^{(1)}(p)\equiv-6!B_{p-7}\pmod{p}$ we complete the proof of (i).
(ii) If $r\geq 2$, then we have
$S_{7}^{(2)}(p^{r})\equiv-5S_{7}^{(1)}(p^{r})\pmod{p^{r}}$ and
$S_{7}^{(3)}(p^{r})\equiv 10S_{7}^{(1)}(p^{r})\pmod{p^{r}}$. Meanwhile, we
have $S_{7}^{(a)}(p)\equiv-S_{7}^{(7-a)}(p)\pmod{p^{r}}$ for $4\leq a\leq 6$.
Hence from (4) we obtain
$\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{7}=n\\\
l_{1},\cdots,l_{7}\in{\mathcal{P}}_{p}\end{subarray}}\frac{1}{l_{1}l_{2}\cdots
l_{7}}\equiv\sum_{a=0}^{5}(-1)^{a}\binom{5}{a}\binom{m+5-a}{6}S_{7}^{(1)}(p^{r})\equiv
mS_{7}^{(1)}(p^{r})\pmod{p^{r}}.$
Since $S_{7}^{(1)}(p^{r})\equiv-\frac{7!}{10}p^{r-1}B_{p-7}\pmod{p^{r}}$ by
Proposition 4.1, we complete the proof of (ii).
## 5\. Concluding remarks
Using similar ideas from [19] we find that it is unlikely to further
generalize our main result to congruence (2) for $r\geq 2$, odd integer $d\geq
9$, and $q_{d}\in{\mathbb{Q}}$ depending only on $d$. By using PSLQ algorithm
we find that both the numerator and the denominator of $q_{9}$ would have at
least 60 digits if the congruence (2) holds for every prime $p\geq 11$.
However, when $r=1$ we have obtained a few general congruences in Lemma 3.3,
Lemma 3.5 and Corollary 3.6, which can be rephrased as follows. Let $m=1,2$
and $d$ be any odd integer greater than $2$. Then for any prime $p>d$, we have
$S_{d}^{(m)}(p)\equiv c_{d,m}\cdot(d-1)!B_{p-d}\pmod{p},$ (13)
where $c_{d,1}=-1$ and $c_{d,2}=(d-1)/2$, and
$R_{d}^{(m)}(p)\equiv c^{\prime}_{d,m}\cdot(d-1)!B_{p-d}\pmod{p},$ (14)
where $c^{\prime}_{d,1}=-1$ and $c^{\prime}_{d,2}=-(d+1)/2$. Unfortunately,
Lemma 3.7 and Corollary 3.8 imply that these do not generalize to $m\geq 3$.
Computation with PSLQ algorithm suggests that if (13) and (14) hold for
$d=9,11,13,15$, $m=3,4$ then both the numerators and the denominators of
$c_{d,m}$ and $c^{\prime}_{d,m}$ would have at least 60 digits. In fact,
numerical evidence suggests the following conjecture.
###### Conjecture 5.1.
For any prime $p\geq 11$, we have
$\displaystyle R_{8}^{(m)}(p)\equiv$
$\displaystyle\,\frac{112}{5}m(m^{2}+16)(m^{2}-1)B_{p-3}B_{p-5}\pmod{p},$
$\displaystyle R_{9}^{(m)}(p)\equiv$
$\displaystyle\,-\frac{8!}{18}\binom{m+2}{5}B_{p-3}^{3}-8m(m^{6}+126m^{4}+1869m^{2}+3044)B_{p-9}\pmod{p},$
$\displaystyle R_{10}^{(m)}(p)\equiv$
$\displaystyle\,-\frac{24}{35}m(m^{4}+71m^{2}+540)(m^{2}-1)\left(50B_{p-3}B_{p-7}+21B_{p-5}^{2}\right)\pmod{p}.$
This conjecture is consistent with the general philosophy we have observed for
the finite multiple zeta values (FMZVs). See, for example, [18, 20] for the
definition of FMZVs and the relevant results. Note that according to the
dimension conjecture of FMZVs discovered by Zagier and independently by the
last author (see [20]) the weight 8 (resp. weight 10) piece of FMZVs has
conjectural dimension 2 (resp. 3). Theorem 1.1 (i), Conjecture 5.1 and all the
previous works in lower weights imply that $R_{d}^{(m)}(p)$ ($d\leq 10$ and
$m\geq 2$) should lie in the proper subalgebra generated by the so-called
${\mathcal{A}}_{1}$-Bernoulli numbers defined in [20]. According to the
analogy between FMZVs and MZVs, this subalgebra is the FMZV analog of the MZV
subalgebra generated by the Riemann zeta values. It would be interesting to
see if this phenomenon holds in every weight.
Acknowledgements. JZ is partially supported by the NSF grant DMS 1162116. Part
of this work was done while he was visiting the Max Planck Institute for
Mathematics, IHES and ICMAT at Madrid, Spain, whose supports are gratefully
acknowledged. The authors also thank the anonymous referee for a number of
valuable suggestions which improved the paper greatly.
## References
* [1] J. Blümlein, Harmonic sums and Mellin transforms, _Nucl. Phys. Proc. Suppl._ 79 (1999), pp. 166–168.
* [2] J. Blümlein, Relations between harmonic sums in massless QCD calculations, _Few-Body Systems_ 36 (2005), pp. 29–34.
* [3] A. Devoto and D.W. Duke, Table of integrals and formulae for Feynman diagram calculations, _Riv. Nuovo Cim._ 7(6) (1984), pp. 1–39.
* [4] Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso, New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner’s series, _Trans. Amer. Math. Soc._ , 366 (6) (2014), pp. 3131–3159.
* [5] M.E. Hoffman, Algebraic aspects of multiple zeta values, in: Zeta Functions, Topology and Quantum Physics, T. Aoki et. al. (eds.), _Dev. Math._ 14, Springer, New York, 2005, pp. 51–74.
* [6] C. Ji, A simple proof of a curious congruence by Zhao, Proc. Amer. Math. Soc. 133 (2005), pp. 3469–3472.
* [7] M. Kaneko and D. Zagier, Finite multiple zeta values, in preparation.
* [8] M. Petkovsek, H. Wilf and D. Zeilberger, A=B, A K Peters/CRC Press, 1996.
* [9] Z. Shen and T. Cai, Congruences for alternating triple harmonic sums, _Acta Math. Sinica (Chin. Ser.)_ , 55 (4) (2012), pp. 737–748.
* [10] J. A. M. Vermaseren, Harmonic sums, Mellin transforms and integrals, _Internat. J. Modern Phys. A_ 14(13) (1999), pp. 2037–2076.
* [11] L. Wang and T. Cai, A curious congruence modulo prime powers, _J. Number Theory_ , 144 (2014), pp. 15–24.
* [12] L. Wang, A curious congruence involving alternating harmonic sums, _J. Comb. Number Theory_ 6 (2014), pp. 209–214.
* [13] L. Wang, A new curious congruence involving multiple harmonic sums, _J. Number Theory_ 154 (2015), pp. 16–31.
* [14] L. Wang, A new congruence on multiple harmonic sums and Bernoulli numbers. arXiv:1504.03227.
* [15] B. Xia and T. Cai, Bernoulli numbers and congruences for harmonic sums, _Int. J. Number Theory_ 6 (4) (2010), pp. 849–855.
* [16] J. Zhao, Bernoulli numbers, Wolstenholme’s Theorem, and $p^{5}$ variations of Lucas’ Theorem, _J. Number Theory_ 123 (2007), pp. 18–26.
* [17] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, _Int. J. Number Theory_ 4 (1)(2008), pp. 73–106.
* [18] J. Zhao, Mod $p$ structure of alternating and non-alternating multiple harmonic sums. _J. Théor. Nombres Bordeaux_ 23 (1) (2011), pp. 259–268. (MR 2780631)
* [19] J. Zhao, A super congruence involving multiple harmonic sums. arXiv:1404.3549.
* [20] J. Zhao, Finite multiple zeta values and finite Euler sums. arXiv:1507.04917.
* [21] X. Zhou and T. Cai, A generalization of a curious congruence on harmonic sums, _Proc. Amer. Math. Soc._ 135 (2007), pp. 1329–1333.
|
11institutetext: HSE University, Russian Federation
11email<EMAIL_ADDRESS>
# On Separation between the Degree of a Boolean Function and the Block
Sensitivity††thanks: The article was prepared within the framework of the HSE
University Basic Research Program
Nikolay V. Proskurin
###### Abstract
In this paper we study the separation between two complexity measures: the
degree of a Boolean function as a polynomial over the reals and the block
sensitivity. We show that the upper bound on the largest possible separation
between these two measures can be improved from $d^{2}(f)\geq bs(f)$,
established by Tal [19], to $d^{2}(f)\geq(\sqrt{10}-2)bs(f)$. As a corollary,
we show that the similar upper bounds between some other complexity measures
are not tight as well, for instance, we can improve the recent sensitivity
conjecture result by Huang [10] $s^{4}(f)\geq bs(f)$ to
$s^{4}(f)\geq(\sqrt{10}-2)bs(f)$. Our techniques are based on the paper by
Nisan and Szegedy [14] and include more detailed analysis of a symmetrization
polynomial.
In our next result we show the same type of improvement for the separation
between the approximate degree of a Boolean function and the block
sensitivity: we show that $\deg_{1/3}^{2}(f)\geq\sqrt{6/101}bs(f)$ and improve
the previous result by Nisan and Szegedy [14]
$\deg_{1/3}(f)\geq\sqrt{bs(f)/6}$. In addition, we construct an example
showing that the gap between the constants in the lower bound and in the known
upper bound is less than $0.2$.
In our last result we study the properties of a conjectured fully sensitive
function on 10 variables of degree 4, existence of which would lead to
improvement of the biggest known gap between these two measures. We prove that
there is the only univariate polynomial that can be achieved by symmetrization
of such a function by using the combination of interpolation and linear
programming techniques.
###### Keywords:
degree of a Boolean function approximate degree block sensitivity
## 1 Introduction
Let $f$: $\\{0,1\\}^{n}\to\\{0,1\\}$ be a Boolean function. We can represent
$f$ in many ways, for example, as a polynomial over the reals. It is easy to
show that every Boolean function can be uniquely represented by such a
polynomial (see [11, exercise 2.23]), so we can introduce a complexity measure
that is the degree of the polynomial that represents $f$, denoted by $d(f)$.
Another representation of $f$ related to polynomials is the approximating one:
a polynomial is called a $\varepsilon$-approximation of $f$ if for any
$x\in\\{0,1\\}^{n}$ we have $|f(x)-p(x)|\leq\varepsilon$. Such polynomials
make sense for any $0<\varepsilon<\frac{1}{2}$, and it is often assumed that
$\varepsilon=\frac{1}{3}$. By $\deg_{\varepsilon}(f)$ we denote the minimum
degree among the polynomials that $\varepsilon$-approximates $f$.
Exact and approximation degrees are closely related to the model called
decision trees. The main measure in this model is a decision tree complexity
$D(f)$, which is equal to the amount of bits in input we need to ask in order
to give the value of $f$ on such input. Other complexity measures include a
sensitivity $s(f)$ and a block sensitivity $bs(f)$. If we denote
$x^{(R)}=\begin{cases}1-x_{i}&i\in R\\\ x_{i}&i\notin R\end{cases}$
then a local block sensitivity $bs(f,x)$ is the largest amount of disjoint
blocks $R_{1},\ldots,R_{t}$ such that $f(x)\neq f(x^{(R_{i})})$ for every
$i=1,\ldots,t$. A block sensitivity in general is the maximum over the local
block sensitivities for $x\in\\{0,1\\}^{n}$. A local sensitivity and
sensitivity defined similarly with a restriction that all the blocks must be
of size 1. See the [4] for an overview of these and other complexity measures
in the decision tree model.
One of the questions involving various complexity measures is determining the
relations between them. For example, recently Huang resolved [10] the well-
known sensitivity conjecture and established that $s^{4}(f)\geq bs(f)$. As for
polynomials, the first result of this kind was made by Nisan and Szegedy: they
analyzed symmetrizations of Boolean functions and showed that $2d^{2}(f)\geq
bs(f)$ [14]. Later, Tal improved this bound by a constant factor by studying a
function composition, proving that $d^{2}(f)\geq bs(f)$ [19]. However, the
best known example with low degree and high block sensitivity is due to
Kushilevitz [9, Example 6.3.2], in which $bs(f)=n=6^{k}$ while
$d(f)=3^{k}=n^{\log_{6}{3}}\simeq n^{0.61}$. That means there is still a large
gap between the upper and lower bounds in this separation. Our result is the
next constant factor improving:
###### Theorem 1.1
For all Boolean functions $\\{0,1\\}^{n}\to\\{0,1\\}$, we have
$d^{2}(f)\geq(\sqrt{10}-2)bs(f)\simeq 1.16bs(f)$ (1)
As a corollary of this result, we also improve some other relations between
complexity measures, including the Huang’s result: we prove that
$s^{4}(f)\geq(\sqrt{10}-2)bs(f)$.
As for approximating polynomials, Nisan and Szegedy proved that
$\deg_{1/3}(f)\geq\sqrt{bs(f)/6}$ and provided an example (namely, the
$OR_{n}$ function), for which the constrain is tight up to a constant factor.
Later, similar results were archived for this and other Boolean functions [18,
5, 2]. In presented papers, authors were not interested in a constant factor
in bounds. In our result, we improve the constant in the lower bound and prove
that
###### Theorem 1.2
For all Boolean functions $\\{0,1\\}^{n}\to\\{0,1\\}$, we have
$\deg_{1/3}^{2}(f)\geq\sqrt{\frac{6}{101}}bs(f)\simeq 0.24bs(f)$ (2)
We also provide an example of a Boolean function (namely, the $NAE_{n}$) that
can be approximated with a polynomial of degree asymptotically tight to our
bound and with a low constant factor in it; in fact, the difference is less
than $0.2$, which shows that the lower bound is not far from optimal.
Another way to approach the problem of the separation between $d(f)$ and
$bs(f)$ is to provide examples of functions of low degree and high
sensitivity. The first known example was given by Nisan and Szegedy: like in
the Kushilevitz’s function, $f$ is fully sensitive, depends on $n=3^{k}$
variables and $d(f)=2^{k}=n^{\log_{3}{2}}\simeq n^{0.63}$. Both examples
achieved by composing the base function with itself arbitrary amount of times,
and one can show that in the fully sensitive case in such a composition both
$d(f)$ and $bs(f)$ remain the same in terms of $n$. This technique was later
studied by Tal in [19]. In [14] the base polynomial consists of 3 variables
and has the degree of 2, while in the [9, Example 6.3.2] it has 6 variables
and the degree of 3. In both examples $2n=d(d+1)$, so the next natural step is
the fully sensitive $\tilde{f}$ on 10 variables with $d(\tilde{f})=4$.
Existence of such a function would lead to the new best example of the
separation with $bs(\tilde{f})=n$ and $d(\tilde{f})=n^{\log_{10}{4}}\simeq
n^{0.60}$. While we do not provide an example of $\tilde{f}$, we prove that
the only polynomial that can be achieved by symmetrization of it is:
$\tilde{p}(x)=-\frac{x^{4}}{144}+\frac{5x^{3}}{36}-\frac{125x^{2}}{144}+\frac{125x}{72}$
(3)
Our techniques for the lower bounds are based on Nisan and Szegedys’ paper. We
use the same symmetrization approach but with the more detailed analysis of a
symmetrization polynomial: we apply better bounds and study higher order
derivatives. As for the upper bounds, we analyze the Chebyshev polynomials of
the first kind for approximating polynomials and use the combination of
interpolation and linear programming for exact polynomials.
In Section 2 we provide necessary definitions and theorems. In Sections 3, 4
and 5 we prove the lower bound for exact polynomials, the result for
approximating polynomials and the property of the low degree function
$\tilde{f}$ respectively.
## 2 Preliminaries
In this paper we assume that in input of a Boolean function 1 corresponds to
the logical true while 0 corresponds to the logical false. The weight of an
input is the amount of the positive bits in it. We use the notation
$||P||=\sup_{x\in[-1;1]}|P(x)|$ and denote the set of polynomials of degree at
most $d$ by $\mathcal{P}_{d}$.
A symmetrization of a polynomial $p$: $\mathbb{R}^{n}\to\mathbb{R}$ defined as
follows:
$p^{sym}(x)=\frac{1}{n!}\sum_{\pi\in S_{n}}p(\pi(x))$ (4)
$S_{n}$ denotes the group of permutations of size $n$ and $\pi(x)$ denotes the
new input, with bits from $x$ moved according to the permutation $\pi$.
The following lemma allows us to represent $p^{sym}$ as a univariate
polynomial of small degree:
###### Lemma 1 (Symmetrization lemma [13])
If $p$: $\mathbb{R}^{n}\to\mathbb{R}$ is a multilinear polynomial, then there
exists a univariate polynomial $\tilde{p}$: $\mathbb{R}\to\mathbb{R}$ of
degree at most the degree of $p$ such that:
$p^{sym}(x_{1},\ldots,x_{n})=\tilde{p}(x_{1}+\ldots+x_{n}),\quad\forall
x\in\\{0,1\\}^{n}$
Note that the value of $\tilde{p}(k)$ for $k=0,1,\ldots,n$ is equal to the
fraction of inputs such that the weight of $x$ is equal to $k$ and $p(x)=1$.
From the proof of this lemma we can also get the explicit formula for the
$\tilde{p}$:
$\tilde{p}(x)=c_{0}+c_{1}\binom{x}{1}+c_{2}\binom{x}{2}+\ldots+c_{d}\binom{x}{d},\quad
d\leq\deg{p}$ (5)
By definition for the binomial coefficients. put:
$\binom{x}{k}=\frac{x\cdot(x-1)\cdot\ldots\cdot(x-k+1)}{k!}$
The original work of Nisan and Szegedy used the following theorem to bound the
degree of a polynomial:
###### Theorem 2.1 ([7, 16])
Let $p$: $\mathbb{R}\to\mathbb{R}$ be a polynomial such that $b_{1}\leq
p(k)\leq b_{2}$ for every integer $0\leq k\leq n$ and a derivative satisfies
$|p^{\prime}(\eta)|\geq c$ for some real $0\leq\eta\leq n$; then
$\deg(p)\geq\sqrt{\dfrac{nc}{c+b_{2}-b_{1}}}.$
However, it is obvious that $\sqrt{\dfrac{c}{c+b_{2}-b_{1}}}<1$, so any bound
achieved using this theorem would be weaker than Tal’s $d^{2}(f)\geq bs(f)$.
In order to make progress, we are going to use the following theorem by Ehlich
and Zeller, as well as the Markov brothers’ inequality:
###### Theorem 2.2 ([7])
Let $p$: $\mathbb{R}\to\mathbb{R}$ be a polynomial of degree $d$. Suppose
$n\in\mathbb{N}$ satisfies:
1. 1.
$\rho=\dfrac{d^{2}(d^{2}-1)}{6n^{2}}<1$, and
2. 2.
$\forall k=0,1,\ldots,n$: $x_{k}=-1+\dfrac{2k}{n}$, $|p(x_{k})|<1$
then $||p||\leq\dfrac{1}{1-\rho}$.
###### Theorem 2.3 (Markov brothers’ inequality)
For any $p\in\mathcal{P}_{d}$ and $k<d$:
$||p^{(k)}||\leq\frac{d^{2}\cdot(d^{2}-1)\cdot\ldots\cdot(d^{2}-k+1)}{1\cdot
3\cdot\ldots\cdot(2k-1)}||p||$ (6)
A Boolean function $f$ is called fully sensitive at 0 iff $f(0)=0$ and
$s(f,0)=n$. The next theorem by Nisan and Szegedy explains why it is enough
for us to focus only on fully sensitive functions.
###### Theorem 2.4 ([14])
For every Boolean function $f$ there exists the fully sensitive at 0 function
$\tilde{f}$ that depends on $bs(f)$ variables and $d(\tilde{f})\leq d(f)$.
The simple proofs of Theorems 1, 2.2 and 2.4 are given in Appendix 0.A. While
the proof of the Markov brothers’ inequality is significantly harder, two
“book-proof”s of it are given in [17].
## 3 Exact Polynomials
In this section we study the separation between the degree of a Boolean
function as an exact polynomial and the block sensitivity. The result
organized as follows. Firstly, we will prove the warm-up result, in which we
introduce a new approach for bounding degree of a polynomial. Secondly, we
will prove a series of lemmas for the main result and prove Theorem 1.1.
### 3.1 Warm-up
In order to show new techniques, first we are going to prove a simpler result,
which, however, is still better than the previously known upper bound for this
separation:
###### Theorem 3.1
For all Boolean functions $\\{0,1\\}^{n}\to\\{0,1\\}$, we have
$d^{2}(f)\geq\sqrt{6/5}bs(f)\simeq 1.09bs(f)$ (7)
First of all, we need to derive a new approach to bound the degree of a
polynomial that would be stronger than Theorem 2.1.
###### Theorem 3.2
Let $p$: $\mathbb{R}\to\mathbb{R}$ be a polynomial of degree $d$ such that:
1. 1.
$\forall i=0,1,\ldots,n$: $0\leq p(i)\leq 1$, and
2. 2.
$\sup_{x\in[0;n]}|p^{(k)}(x)|\geq c$.
Then either $d^{2}\geq\sqrt{6}n$ or the ratio $x=\dfrac{d^{2}}{n}$ satisfies
the following inequality:
$\left(1-\frac{x^{2}}{6}\right)\frac{(2k-1)!!}{2^{k-1}}c<x^{k}$ (8)
$(2k-1)!!$ denotes the double factorial:
$(2k-1)!!=(2k-1)\cdot(2k-3)\cdot\ldots\cdot 3\cdot 1$.
###### Proof
Suppose $d^{2}<\sqrt{6}n$, otherwise the first statement holds. In terms of
Theorem 2.2:
$\rho=\frac{d^{2}(d^{2}-1)}{6n^{2}}<\frac{d^{4}}{6n^{2}}<1$ (9)
Let $P(x)$ be defined as follows:
$P(x)=p\left(\frac{n}{2}(x+1)\right)-\frac{1}{2}$
By Theorem 2.2, $||P||\leq\dfrac{1}{2}\cdot\dfrac{1}{1-\rho}$. On the other
hand, $||P^{(k)}||\geq\dfrac{n^{k}}{2^{k}}\cdot c$. Combined with Inequality
2.3, we get
$c\left(\frac{n}{2}\right)^{k}\leq||P^{(k)}||\leq\frac{d^{2}\cdot(d^{2}-1)\cdot\ldots\cdot(d^{2}-k+1)}{1\cdot
3\cdot\ldots\cdot(2k-1)}||P||<\frac{d^{2k}}{(2k-1)!!}\cdot\frac{1}{2(1-\rho)}$
$(1-\rho)\frac{(2k-1)!!}{2^{k-1}}c\leq\left(\frac{d^{2}}{n}\right)^{k}$ (10)
By substituting (9) and $x=\dfrac{d^{2}}{n}$ into (10), we obtain exactly the
inequality (8).
The same approach was used by Beigel [1, lemma 3.2], however, he didn’t
parameterized his result and proved it only for the first derivative. If we
apply his result with a trivial bound $\sup_{x\in[0;n]}|p^{\prime}(x)|\geq 1$,
it follows that $1-\dfrac{x^{2}}{6}<x$ for $x=\dfrac{d^{2}}{n}$. As $x>0$, the
solution is $x\geq\sqrt{15}-3\simeq 0.87$, which is stronger than the original
bound, but weaker than the Tal’s result.
Now we are ready to prove the warm-up result.
###### Proof (Theorem 3.1)
Because of reduction 2.4, we can assume without loss of generality that $f$ is
fully sensitive at 0, and so a polynomial $p$ derived from Lemma 1 satisfies
$p(0)=0$ and $p(1)=1$. Also, it is obvious that if $\rho$ from Theorem 2.2 is
not less than $1$, then $d^{2}\geq\sqrt{6}n>\sqrt{6/5}n$, so we assume that
$\rho<1$.
Suppose $p\in\mathcal{P}_{2}$, i.e. $p(x)=ax^{2}+bx+c$. From the values of
$p(0),\ p(1)$ and $p(2)$ we obtain the following constrains:
1. 1.
$p(0)=c=0\Rightarrow c=0$.
2. 2.
$p(1)=a+b+c=1\Rightarrow a+b=1$.
3. 3.
$p(2)=4a+2b\Rightarrow-2\leq 2a\leq-1$.
As a result, $|p^{\prime\prime}(x)|=|2a|\geq 1$.
In general case, let $q(x)$ be the quadratic polynomial that equals to $p(x)$
at $x\in\\{0,1,2\\}$, and $\tilde{p}(x)=p(x)-q(x)$. From our definition it
follows that $\tilde{p}(0)=\tilde{p}(1)=\tilde{p}(2)=0$, so there exists
$\xi\in[0;2]$: $\tilde{p}^{\prime\prime}(\xi)=0$. Then
$|p^{\prime\prime}(\xi)|=|q^{\prime\prime}(\xi)+\tilde{p}^{\prime\prime}(\xi)|\geq
1$, and as a direct consequence $\sup_{x\in[0;n]}|p^{\prime\prime}(\xi)|\geq
1$. Substituting $c=1$ and $k=2$ in (8), we obtain the following inequality
for $x=\dfrac{d^{2}}{n}$:
$\left(1-\frac{x^{2}}{6}\right)\frac{3}{2}<x^{2}$ (11)
Since $x\geq 0$, we get $x\geq\sqrt{6/5}$ and $d^{2}\geq\sqrt{6/5}n$.
### 3.2 Proof of Theorem 1.1
To increase the constant factor from $\sqrt{6/5}$ to $\sqrt{10}-2$, we should
analyze higher order derivatives. In order to do so, we are going to use
representation (5). We also need to establish a series of lemmas.
###### Lemma 2
Suppose $f$: $\\{0,1\\}^{n}\to\\{0,1\\}$ is fully sensitive at 0 and $n\geq
4$; then for symmetrization polynomial $p$ we have
$\sup_{x\in[0;n]}|p^{\prime\prime\prime}(x)|\geq 1-p(3)$.
###### Lemma 3
$\sum_{k=4}^{\infty}\frac{1}{k2^{k-2}}<\frac{1}{8}$ (12)
The proofs are omitted to Appendix 0.A. With this lemmas we are ready for the
main proof.
###### Proof (Theorem 1.1)
If $n<4$, then the theorem follows from Tal’s bound $d^{2}(f)\geq bs(f)$. As
in Theorem 3.1, we assume that $f$ is fully sensitive at $0$ and $\rho<1$.
It is easy to show that
$\binom{x}{k}^{\prime}\bigg{|}_{x=0}=\frac{(-1)^{k+1}}{k},\quad
k\in\mathbb{N}$
If we combine this with representation (5), we get the formula for the first
derivative of the symmetrization polynomial at $x=0$:
$p^{\prime}(0)=\sum_{k=1}^{d}(-1)^{k+1}\frac{c_{k}}{k}$ (13)
We can bound the first three coefficients:
1. 1.
$p(1)=c_{1}=1\Rightarrow c_{1}=1$.
2. 2.
$p(2)=2c_{1}+c_{2}\leq 1\Rightarrow c_{2}\leq-1$.
3. 3.
$p(3)=3c_{1}+3c_{2}+c_{3}\geq 0\Rightarrow c_{3}\geq p(3)$.
If $p(3)<\dfrac{3}{8}$, then by Lemma 2
$\sup_{x\in[0;n]}|p^{\prime\prime\prime}(x)|>\dfrac{5}{8}$. Substituting
$c=\dfrac{5}{8}$ and $k=3$ in (8), we get the inequality for
$x=\dfrac{d^{2}}{n}$:
$\left(1-\frac{x^{2}}{6}\right)\frac{75}{32}<x^{3}$
Inequality implies that $x>1.2$, which satisfies the statement of the theorem.
In remaining case, we have $c_{3}\geq\dfrac{3}{8}$. Substituting all the
constrains in (13), we obtain:
$p^{\prime}(0)\geq\frac{3}{2}+\frac{1}{8}+\sum_{k=4}^{d}(-1)^{k+1}\frac{c_{k}}{k}$
(14)
Suppose $|c_{k}|<\dfrac{1}{2^{k-2}}$ for $k>3$, then
$p^{\prime}(0)\geq\frac{3}{2}+\frac{1}{8}-\sum_{k=4}^{d}\frac{1}{k2^{k-2}}>\frac{3}{2}+\frac{1}{8}-\sum_{k=4}^{\infty}\frac{1}{k2^{k-2}}>\frac{3}{2}+\frac{1}{8}-\frac{1}{8}=\frac{3}{2}$
Inequality (8) with $c=\dfrac{3}{2}$ and $k=1$ implies
$\left(1-\dfrac{x^{2}}{6}\right)\dfrac{3}{2}\leq x$ for $x=\dfrac{d^{2}}{n}$.
The solution is $x\geq(\sqrt{10}-2)$, which means that
$d^{2}\geq(\sqrt{10}-2)n$.
The only case left to consider is if there exists such $k>3$ that
$|c_{k}|\geq\dfrac{1}{2^{k-2}}$. To deal with it, we first need to show that
$\sup_{x\in[0;n]}|p^{(k)}(x)|\geq c_{k}$. If $d=k$, then the derivative is a
constant and equals to $c_{k}$ because $\binom{x}{k}^{(k)}=1$. In other case,
let $q(x)$ be the polynomial that consists of all the terms from (5) up to one
with the $c_{k}$. Then $\tilde{p}(x)=p(x)-q(x)$ equals to 0 for
$x=0,1,\ldots,k$, so $\exists\xi\in[0;k]$: $\tilde{p}^{(k)}(\xi)=0$ and
$|p^{(k)}(\xi)|=|c_{k}|$.
Now, for $k=4,5$ we obtain the following inequalities from (8):
$\left(1-\frac{x^{2}}{6}\right)\frac{105}{32}\leq x^{4}\quad\Rightarrow\quad
x>1.24>(\sqrt{10}-2)$ $\left(1-\frac{x^{2}}{6}\right)\frac{945}{128}\leq
x^{5}\quad\Rightarrow\quad x>1.3>(\sqrt{10}-2)$
For $k>5$, we need to show that the solution from the inequality would not be
worse than for $k=5$. Notice that every time we increase $k$ in inequality (8)
we multiply the left hand side by $\dfrac{2k+1}{4}>\sqrt{6}$ and the right
hand side by $x$. But if we recall that $\rho<1$, we get $x<\sqrt{6}$, and
thus the inequality becomes tighter. As a result, the statement holds for
$k>5$ as well.
### 3.3 Corollaries
While our improvement may seem insignificant, it shows that the currently
known bound between $d(f)$ and $bs(f)$ is not tight. Next two corollaries
shows that the same holds for some other pairs of complexity measures.
###### Corollary 1
For all Boolean functions $\\{0,1\\}^{n}\to\\{0,1\\}$, we have
$s^{4}(f)\geq(\sqrt{10}-2)bs(f)$ (15)
###### Proof
In the proof of the sensitivity conjecture [10], Huang established the
following bound for $d(f)$:
$s^{2}(f)\geq d(f)$ (16)
Combined with Theorem 1.1, it follows that
$s^{4}(f)\geq d^{2}(f)\geq(\sqrt{10}-2)bs(f)$
###### Corollary 2
For all Boolean functions $\\{0,1\\}^{n}\to\\{0,1\\}$, we have
$d^{3}(f)\geq(\sqrt{10}-2)D(f)$ (17)
###### Proof
Combining Theorem 1.1 with the bound $D(f)\leq bs(f)\cdot d(f)$ from the paper
[12], we get
$d^{3}(f)\geq d(f)\cdot(\sqrt{10}-2)bs(f)\geq(\sqrt{10}-2)D(f)$
## 4 Approximating Polynomials
In this section, we improve the constant factor in the separation between the
degree of an approximating polynomial and the block sensitivity and provide an
example of a polynomial for $NAE_{n}$ function that shows that not only our
bound is asymptotically tight, but the difference between the best known
constant in the lower bound and the constant in our example is relatively
small as well.
### 4.1 Lower bound
Before the proof we need to derive a similar to 3.2 lemma, but this time for
approximating polynomials.
###### Lemma 4
Let $p$: $\mathbb{R}\to\mathbb{R}$ be a polynomial of degree $d$ such that:
1. 1.
$\forall i=0,1,\ldots,n$: $-\dfrac{1}{3}\leq p(i)\leq\dfrac{4}{3}$, and
2. 2.
$\sup_{x\in[0;n]}|p^{(k)}(x)|\geq c$.
Then either $d^{2}\geq\sqrt{6}n$ or the ratio $x=\dfrac{d^{2}}{n}$ satisfies
the following inequality:
$\left(1-\frac{x^{2}}{6}\right)\frac{(2k-1)!!}{2^{k}}\cdot\frac{6c}{5}<x^{k}$
(18)
###### Proof
The only difference between this lemma and Theorem 3.2 is the bounds for
$p(k)$, therefore if we define $P(x)$ the same as earlier, we get the weaker
upper bound: $||P||\leq\dfrac{5}{6}\cdot\dfrac{1}{1-\rho}$. The remaining part
of the proof is the same as in 3.2.
###### Proof (Theorem 1.2)
Using reduction 2.4, we can assume that the symmetrization polynomial of $f$
satisfies $-\dfrac{1}{3}\leq p(0)\leq\dfrac{1}{3}$ and $\dfrac{2}{3}\leq
p(1)\leq\dfrac{4}{3}$. As always, we can only consider the case $\rho<1$.
Also, if $n<5$, then the theorem follows from the original bound
$6\deg^{2}_{1/3}(f)\geq bs(f)$.
Suppose that $p\in\mathcal{P}_{3}$. By Lagrange’s interpolation formula for
$x\in\\{0,1,2\\}$ and $x\in\\{0,2,5\\}$:
$p(x)=\frac{(x-1)(x-2)}{2}p(0)+x(2-x)p(1)+\frac{x(x-1)}{2}p(2)$ (19)
$p(x)=\frac{(x-2)(x-5)}{10}p(0)-\frac{x(x-5)}{6}p(2)+\frac{x(x-2)}{15}p(5)$
(20)
From (19) we get $\forall x\quad
p^{\prime\prime}(x)=p(0)-2p(1)+p(2)\leq-1+p(2)$, and if
$p(2)\leq\dfrac{14}{15}$, then $p^{\prime\prime}(x)\leq-\dfrac{1}{15}$.
Otherwise, from (20) we get $\forall x\quad
p(x)=\dfrac{1}{5}p(0)-\dfrac{1}{3}p(2)+\dfrac{2}{15}p(5)\leq\dfrac{11}{45}-\dfrac{1}{3}p(2)\leq-\dfrac{1}{15}$.
If $\deg{p}>3$, we can use the same reduction as in the proof of Theorem 3.1.
Applying (18) with $c=\dfrac{1}{15}$ and $k=2$, we obtain the following
inequality:
$\left(1-\frac{x^{2}}{6}\right)\frac{3}{50}<x^{2}$ (21)
It now follows that $x\geq\sqrt{\dfrac{6}{101}}$ and
$\deg_{1/3}^{2}(f)\geq\sqrt{\dfrac{6}{101}}bs(f)$.
### 4.2 Upper bound
A function $NAE_{n}$: $\\{0,1\\}^{n}\to\\{0,1\\}$ equals to 1 iff
$x\in\\{0^{n},1^{n}\\}$, i.e. all the bits in the input are the same. The next
theorem provides a polynomial that approximates $NAE_{n}$ and gives the upper
bound for $\deg_{1/3}(f)$ in terms of the block sensitivity.
###### Theorem 4.1
Define $d=\lceil\sqrt{c(n-2)}\rceil$ with a constant $c$ satisfying the
following inequality:
$2c+\frac{2}{3}c^{2}-\frac{2c}{3(n-2)}>1$ (22)
Then there exists a polynomial of degree $d$ if $d$ is even and $d+1$
otherwise that is a $\frac{1}{3}$-approximation of $NAE_{n}$.
###### Proof
In our construction, we use the Chebyshev polynomials of the first kind,
defined as $T_{k}(x)=\cos{(k\arccos{x})}$. We need the following properties of
them; proof of property 3 is omitted to Appendix 0.A, and property 4 is [17,
lemma 5.17] for $k=1$ and $k=2$.
1. 1.
If $k$ is even, then $T_{k}(x)=T_{k}(-x)$.
2. 2.
$\forall x\in[-1;1]\quad|T_{k}(x)|\leq 1$.
3. 3.
$T_{k}^{\prime\prime}(\theta)\geq T_{k}^{\prime\prime}(1)$ for $\theta\geq 1$.
4. 4.
$T_{k}^{\prime}(1)=k^{2}$ and
$T_{k}^{\prime\prime}(1)=\dfrac{k^{4}-k^{2}}{3}$.
By definition, put
$p(x)=1-\dfrac{2T_{k}(\frac{2x-n}{n-2})}{3T_{k}(\frac{n}{n-2})}$ (23)
It is clear from property 1 that $p(0)=p(n)=\dfrac{1}{3}$. If we show that
$T_{k}\left(\dfrac{n}{n-2}\right)\geq 2$, then by property 2 for all $1\leq
k\leq n-1$ we have
$\left|\frac{2T_{k}(\frac{2x-n}{n-2})}{3T_{k}(\frac{n}{n-2})}\right|\leq\frac{2\cdot
1}{3\cdot 2}=\frac{1}{3}\quad\Rightarrow\quad\frac{2}{3}\leq
p(k)\leq\frac{4}{3}$
and $q(x_{1},\ldots,x_{n})=p(n-x_{1}-\ldots-x_{n})$ is indeed the
$\frac{1}{3}$-approximation of $NAE_{n}$.
Substituting $x=1$ in the Taylor series for $T_{k}(\dfrac{n}{n-2})$, we obtain
$\begin{array}[]{c}T_{k}\left(\dfrac{n}{n-2}\right)=T_{k}(1)+\dfrac{2}{n-2}T_{k}^{\prime}(1)+\dfrac{2}{(n-2)^{2}}T_{k}^{\prime\prime}(\theta)\geq\\\
\\\ \geq
T_{k}(1)+\dfrac{2}{n-2}T_{k}^{\prime}(1)+\dfrac{2}{(n-2)^{2}}T_{k}^{\prime\prime}(1)\end{array}$
(24)
The last inequality holds because of property 3. Combining property 4 and
(22), we get:
$\begin{array}[]{c}T_{k}\left(\dfrac{n}{n-2}\right)\geq
1+2c+\dfrac{2}{3(n-2)^{2}}(c^{2}(n-2)^{2}-c(n-2))=\\\ \\\
=1+2c+\dfrac{2}{3}c^{2}-\dfrac{2c}{3(n-2)}\geq 2\end{array}$ (25)
Because the last term in the left hand side of (22) tends to zero as $n$ tends
to infinity, the optimal $c$ tends to the solution of the following
inequality: $2x+\dfrac{2}{3}x^{2}>1$. The solution is
$x>\dfrac{1}{2}(\sqrt{15}-3)\simeq 0.43$, so the difference between $c$ and
the best known lower bound is less than $0.2$, which shows that the bound 1.2
is close to be tight.
## 5 Fully Sensitive Function of Small Degree
The last result of this paper is about an example of a function with low
degree and high block sensitivity. We study properties of conjectured function
$\tilde{f}$ on $10$ variables with $d(\tilde{f})=4$. By applying the same
composition scheme as in the previous examples, we can generalize $\tilde{f}$
for the arbitrary large $n$. While we do not provide an example of
$\tilde{f}$, we prove that if $\tilde{f}$ is fully sensitive at 0, then by
applying Lemma 1 to it, the only univariate polynomial we can get is (3).
We prove this statement in two steps. Firstly, we achieve such a polynomial by
interpolation. Secondly, we prove the uniqueness using the linear programming.
### 5.1 Interpolation
The first part of the proof is to construct polynomial (3). We do this by
establishing the extremal property of all the symmetrizations of degree 4 for
$n\geq 8$.
###### Theorem 5.1
Let $f$: $\\{0,1\\}^{n}\to\\{0,1\\}$ be fully sensitive at 0 and $n\geq 8$;
then for symmetrization polynomial $p$ we have
$\sup_{x\in[0;n]}|p^{(4)}(x)|\geq\frac{1}{6}$. Moreover, the only polynomial
for which inequality is tight is (3).
###### Proof
Using the Lagrange’s interpolation formula for $x\in\\{0,1,2,7,8\\}$ and
$x\in\\{0,1,2,5,7\\}$, we get the following representations:
1. 1.
$\begin{array}[]{c}\forall x\quad p^{(4)}(x)=-\dfrac{24}{1\cdot 1\cdot 6\cdot
7}+\dfrac{24}{2\cdot 1\cdot 5\cdot 6}p(2)-\dfrac{24}{7\cdot 6\cdot 5\cdot
1}p(7)+\\\ \\\ +\dfrac{24}{8\cdot 7\cdot 6\cdot
1}p(8)=-\dfrac{4}{7}+\dfrac{2}{5}p(2)-\dfrac{4}{35}p(7)+\dfrac{1}{14}p(8)\leq-\dfrac{1}{10}-\dfrac{4}{35}p(7)\end{array}$
(26)
2. 2.
$\begin{array}[]{c}\forall x\quad p^{(4)}(x)=-\dfrac{24}{1\cdot 1\cdot 4\cdot
6}+\dfrac{24}{2\cdot 1\cdot 3\cdot 5}p(2)-\dfrac{24}{5\cdot 4\cdot 3\cdot
2}p(5)+\\\ \\\ +\dfrac{24}{7\cdot 6\cdot 5\cdot
2}p(7)=-1+\dfrac{4}{5}p(2)-\dfrac{1}{5}p(5)+\dfrac{2}{35}p(7)\leq-\dfrac{1}{5}+\dfrac{2}{35}p(7)\end{array}$
(27)
If $p(7)\leq\dfrac{7}{12}$, then from (26) we get
$p^{(4)}(x)\leq-\dfrac{1}{6}$, otherwise we get the same result from (27).
Inequality is tight iff $p(7)=\dfrac{7}{12},\ p(2)=1,\ p(5)=0$ and $p(8)=1$.
Combined with $p(0)=0$ and $p(1)=1$, we obtain that there is the only
polynomial of degree at most 5 that satisfies all the constrains. By applying
the Lagrange’s interpolation formula, we get polynomial (3).
Note that (3) is indeed a symmetrization polynomial for some function, because
for $k=0,\ldots,10$ the values $p(k)$ represent the fraction of inputs for
corresponding weights (i.e. $\binom{n}{k}\cdot p(k)\in\mathbb{N}_{0}$).
### 5.2 Linear programming
The second part of the proof is to show that (3) is the only symmetrization
polynomial for $n=10$ and $d(f)\leq 4$. This part is done with a linear
programming solver. In our case we are going to use the
scipy.optimize.linprog, the full code for the problem is available at Google
Colab [15].
###### Theorem 5.2
The only symmetrization polynomial for the fully sensitive at 0 function of 10
variables with degree at most 4 is (3).
###### Proof
Suppose $p(x)=c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}$ is the needed
polynomial. We use the necessary (but not sufficient) conditions for $p(x)$ to
create a linear programming task. Namely, we require $p(1)=1,\ \forall
k=2,\ldots,10$: $0\leq p(k)\leq 1$ and some additional constrain for $c_{4}$.
We require a solver to minimize $c_{4}$.
If we add a constrain $c_{4}>0$, solver proves that problem is infeasible.
Without any constrains for $c_{4}$, solver states that the solution is
$-\dfrac{1}{144}$, i.e. the minimum value for $p^{(4)}(x)$ for which solver
can find the solution is $-\dfrac{1}{144}\cdot 24=-\dfrac{1}{6}$. But we know
that the only polynomial for this value is (3), thus we obtain it’s
uniqueness.
## 6 Conclusion
Although we made improvements in relations between some complexity measures,
we strongly suspect that the current results are still not tight. For example,
the choice of points for interpolation in many theorems was not really
justified, so we think that finding a pattern for the choice of interpolation
set is one of the keys for the further improvements. Also, we suspect that by
using Bernstein’s inequality (see [3, theorem 5.1.7]) in Theorem 1.1 with or
instead of the Markov brothers’ inequality for the first derivative one might
improve the result as well.
Another open question occurs if we add further restrictions for the function
$f$, for example, if we want $f$ to be symmetrical. It was proved by von zur
Gathen and Roche that if $n=p-1$ for prime $p$, then $d(f)=n$, and as a
corollary in general $d(f)=n-\mathcal{O}(n^{0.525})$ [8]. It is conjectured
that $d(f)\geq n-3$, but very little progress was made since. For instance,
the result by Cohen and Shpilka [6] states that if $n=p^{2}-1$, then $d(f)\geq
n-\sqrt{n}$.
#### 6.0.1 Acknowledgments.
Author would like to thank Vladimir V. Podolskii for the proof idea for
Theorem 3.1.
## References
* [1] Beigel, R.: Perceptrons, pp, and the polynomial hierarchy. Comput. Complex. 4, 339–349 (1994). https://doi.org/10.1007/BF01263422, https://doi.org/10.1007/BF01263422
* [2] Bogdanov, A., Mande, N.S., Thaler, J., Williamson, C.: Approximate degree, secret sharing, and concentration phenomena. In: Achlioptas, D., Végh, L.A. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019, September 20-22, 2019, Massachusetts Institute of Technology, Cambridge, MA, USA. LIPIcs, vol. 145, pp. 71:1–71:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.71, https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.71
* [3] Borwein, P., Erdelyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, Springer New York (1995), https://books.google.ru/books?id=386CC7JnuuwC
* [4] Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002). https://doi.org/10.1016/S0304-3975(01)00144-X, https://doi.org/10.1016/S0304-3975(01)00144-X
* [5] Bun, M., Thaler, J.: Dual lower bounds for approximate degree and markov-bernstein inequalities. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M.Z., Peleg, D. (eds.) Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I. Lecture Notes in Computer Science, vol. 7965, pp. 303–314. Springer (2013). https://doi.org/10.1007/978-3-642-39206-1_26, https://doi.org/10.1007/978-3-642-39206-1_26
* [6] Cohen, G., Shpilka, A.: On the degree of symmetric functions on the boolean cube. Electron. Colloquium Comput. Complex. 17, 39 (2010), http://eccc.hpi-web.de/report/2010/039
* [7] Ehlich, H., Zeller, K.: Schwankung von polynomen zwischen gitterpunkten. Mathematische Zeitschrift pp. 41–44 (1964)
* [8] von zur Gathen, J., Roche, J.R.: Polynomials with two values. Comb. 17(3), 345–362 (1997). https://doi.org/10.1007/BF01215917, https://doi.org/10.1007/BF01215917
* [9] Hatami, P., Kulkarni, R., Pankratov, D.: Variations on the sensitivity conjecture. Theory Comput. 4, 1–27 (2011). https://doi.org/10.4086/toc.gs.2011.004, https://doi.org/10.4086/toc.gs.2011.004
* [10] Huang, H.: Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. Annals of Mathematics 190(3), 949–955 (2019), https://www.jstor.org/stable/10.4007/annals.2019.190.3.6
* [11] Jukna, S.: Boolean Function Complexity - Advances and Frontiers, Algorithms and combinatorics, vol. 27. Springer (2012). https://doi.org/10.1007/978-3-642-24508-4, https://doi.org/10.1007/978-3-642-24508-4
* [12] Midrijanis, G.: Exact quantum query complexity for total boolean functions. arXiv preprint quant-ph/0403168 (2004)
* [13] Minsky, M., Papert, S.: Perceptrons - an introduction to computational geometry. MIT Press (1987)
* [14] Nisan, N., Szegedy, M.: On the degree of boolean functions as real polynomials. Comput. Complex. 4, 301–313 (1994). https://doi.org/10.1007/BF01263419, https://doi.org/10.1007/BF01263419
* [15] Proskurin, N.: Symmetrization linprog. https://colab.research.google.com/drive/1XKJSYLElVxGgZuwHaFy4BdoTN4JIgKXJ?usp=sharing (2020)
* [16] Rivlin, T.J., Cheney, E.W.: A comparison of uniform approximations on an interval and a finite subset thereof. SIAM Journal on Numerical Analysis 3(2), 311–320 (1966), http://www.jstor.org/stable/2949624
* [17] Shadrin, A.: Twelve proofs of the markov inequality. Approximation theory: a volume dedicated to Borislav Bojanov pp. 233–298 (2004)
* [18] Spalek, R.: A dual polynomial for OR. CoRR abs/0803.4516 (2008), http://arxiv.org/abs/0803.4516
* [19] Tal, A.: Properties and applications of boolean function composition. In: Kleinberg, R.D. (ed.) Innovations in Theoretical Computer Science, ITCS ’13, Berkeley, CA, USA, January 9-12, 2013. pp. 441–454. ACM (2013). https://doi.org/10.1145/2422436.2422485, https://doi.org/10.1145/2422436.2422485
## Appendix 0.A Omitted Proofs
###### Proof (Lemma 1)
Define $d=\deg{p^{sym}}$ and $P_{k}=\sum_{|S|=k}\prod_{i\in S}x_{i}$ where $S$
are chosen from the subsets of $[n]=\\{1,2,\ldots,n\\}$. Suppose that $S$ is a
monomial in $p$; then by definition symmetrization adds up all the monomials
of size $|S|$ to $p^{sym}$ equal amount of times: in order to get a specific
monomial $S^{\prime}$ one should fix the permutation of variables in
$S^{\prime}$ and in $[n]\setminus S^{\prime}$, thus amount of every monomial
is equal. Therefore, we can rewrite $p^{sym}$ as
$p^{sym}(x)=c_{0}+c_{1}P_{1}(x)+\ldots+c_{d}P_{d}(x)$
Note that we only interested in $x\in\\{0,1\\}^{n}$; therefore, every term in
$P_{k}$ is equal to 1 iff every variable in it is equal to 1. Thus it is
obvious that if $z=x_{1}+x_{2}+\ldots+x_{n}$, then
$p^{sym}(x)=c_{0}+c_{1}\binom{z}{1}+\ldots+c_{d}\binom{z}{d}=\tilde{p}(z)$
$\deg{\tilde{p}}\leq\deg{p}$ because $\deg{p^{sym}}\leq\deg{p}$.
###### Proof (Theorem 2.2)
Define $K=\inf\\{k:||p||\leq 1+k\\}$. From Inequality 2.3 we get
$||p^{\prime\prime}(x)||\leq\frac{d^{2}(d^{2}-1)}{3}(1+K)$ (28)
Let $\xi$ be the point of maximum on $[-1;1]$, i.e. $||p||=|p(\xi)|$. The
cases $\xi=\pm 1$ are trivial because $|p(\xi)|\leq 1$, so we can assume that
$\xi$ is an inner point and $p^{\prime}(\xi)=0$. Also, because $\forall
k=0,1,\ldots,n-1\quad x_{k+1}-x_{k}=\dfrac{2}{n}$ there exists such $k$ that
$\left|x_{k}-\xi\right|\leq\dfrac{1}{n}$. Applying Taylor series for $p(x)$,
we obtain
$p(x_{k})=p(\xi)+(x_{k}-\xi)p^{\prime}(\xi)+\frac{(x_{k}-\xi)^{2}}{2}p^{\prime\prime}(\theta)=p(\xi)+\frac{(x_{k}-\xi)^{2}}{2}p^{\prime\prime}(\theta),\quad\theta\in[-1;1]$
Substituting (28) in the last equality, we get another bound for $||p||$:
$\begin{array}[]{c}||p||=|p(\xi)|=\left|p(x_{k})-\dfrac{(x_{k}-\xi)^{2}}{2}p^{\prime\prime}(\theta)\right|\leq|p(x_{k})|+\left|\dfrac{(x_{k}-\xi)^{2}}{2}p^{\prime\prime}(\theta)\right|\leq\\\
\\\
1+\dfrac{1}{2n^{2}}\cdot\dfrac{d^{2}(d^{2}-1)}{3}(1+K)=1+\rho(1+K)\end{array}$
(29)
By definition $K\leq\rho(1+K)$ and as a corollary $K\leq\dfrac{\rho}{1-\rho}$
and $||p||\leq 1+K\leq\dfrac{1}{1-\rho}$.
###### Proof (Theorem 2.4)
Let $x$ be the input such that $bs(f)=bs(f,x)$, and let
$S_{1},S_{2},\ldots,S_{t}$ be the blocks on which we achieve such block
sensitivity. Without loss of generality we can assume that $f(0)=0$, otherwise
we can introduce the new function $g(x)=1-f(x)$.
Let $\tilde{f}$ be defined as follows
$\tilde{f}(y_{1},\ldots,y_{t})=f(x\oplus y_{1}S_{1}\oplus\ldots\oplus
y_{t}S_{t})$ (30)
i.e. we create a new input for $f$ such that every bit $x_{j}$ is equal to
$x_{j}\oplus y_{i}$ if $x_{j}\in S_{i}$ or $x_{j}$ is left unchanged
otherwise.
$d(\tilde{f})\leq d(f)$ because $\tilde{f}$ is a linear substitution in $f$.
On the other hand, $\tilde{f}$ is fully sensitive at 0 as
$\tilde{f}(0)=f(0)=0$ and $f(e_{j})=f(x^{(S_{i})})=1$. Thus $\tilde{f}$
satisfies the statement as $t=bs(f)$.
###### Proof (Lemma 2)
Suppose that $p\in\mathcal{P}_{3}$. Using the Lagrange’s interpolation formula
for $x\in\\{0,1,3,4\\}$, we get the following representation:
$p(x)=\frac{x(x-3)(x-4)}{6}-\frac{x(x-1)(x-4)}{6}p(3)+\frac{x(x-1)(x-3)}{12}p(4)$
$\forall x\quad p^{\prime\prime\prime}(x)=1-p(3)+\frac{p(4)}{2}\geq 1-p(3)$
In general case, let $q\in\mathcal{P}_{3}$ be equal to $p$ on the same set of
points. Similarly to Theorem 3.1, if $\tilde{p}(x)=p(x)-q(x)$, then
$\exists\xi\in[0;4]$: $\tilde{p}^{\prime\prime\prime}(\xi)=0$ and
$|p^{\prime\prime\prime}(\xi)|\geq 1-p(3)$.
###### Proof (Lemma 3)
The Maclaurin series for the natural logarithm converges for $-1\leq x<1$.
Substituting $x=-\dfrac{1}{2}$, we can calculate and bound our sum:
$\ln(1+x)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}x^{k}$
$\ln{\frac{1}{2}}=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\left(-\frac{1}{2}\right)^{k}=-\sum_{k=1}^{\infty}\frac{1}{k2^{k}}\quad\Rightarrow\quad\sum_{k=1}^{\infty}\frac{1}{k2^{k}}=\ln{2}$
$\sum_{k=4}^{\infty}\frac{1}{k2^{k-2}}=4\sum_{k=4}^{\infty}\frac{1}{k2^{k}}=4\left(\sum_{k=1}^{\infty}\frac{1}{k2^{k}}-\frac{2}{3}\right)=4\left(\ln{2}-\frac{2}{3}\right)<\frac{1}{8}$
###### Proof (Theorem 4.1, property 3)
As $T_{k}(\cos{x})=\cos{kx}$, we can see that all the roots of the Chebyshev
polynomial lie on $[-1;1]$. By the Rolle’s theorem, all the roots of any
derivative of the Chebyshev polynomial also lie on $[-1;1]$. From [17, lemma
5.17] we get that
$T_{k}^{\prime\prime\prime}(1)=\dfrac{k^{2}(k^{2}-1)(k^{2}-2)}{15}>0$, so
$T_{k}^{\prime\prime\prime}(x)>0$ for $x\geq 1$ and
$T_{k}^{\prime\prime}(\theta)\geq T_{k}^{\prime\prime}(1)$ for $\theta\geq 0$.
|
∎ [table]capposition=above [table]captionskip=0cm
11institutetext: Authors 22institutetext: Department of Aerospace and
Mechanical Engineering,
University of Liège, 4000 Liège, Belgium.
Denis Trillet
22email<EMAIL_ADDRESS>
# Note on the minimum length scale and its defining parameters
Analytical relationships for Topology Optimization based on uniform
manufacturing uncertainties.
Denis Trillet Pierre Duysinx Eduardo Fernández
(Received: date / Accepted: date)
###### Abstract
The robust topology optimization formulation that introduces the eroded and
dilated versions of the design has gained increasing popularity in recent
years, mainly because of its ability to produce designs satisfying a minimum
length scale. Despite its success in various topology optimization fields, the
robust formulation presents some drawbacks. This paper addresses one in
particular, which concerns the imposition of the minimum length scale. In the
density framework, the minimum size of the solid and void phases must be
imposed implicitly through the parameters that define the density filter and
the smoothed Heaviside projection. Finding these parameters can be time
consuming and cumbersome, hindering a general code implementation of the
robust formulation. Motivated by this issue, in this article we provide
analytical expressions that explicitly relate the minimum length scale and the
parameters that define it. The expressions are validated on a density-based
framework. To facilitate the reproduction of results, MATLAB codes are
provided.
As a side finding, this paper shows that to obtain simultaneous control over
the minimum size of the solid and void phases, it is necessary to involve the
3 fields (eroded, intermediate and dilated) in the topology optimization
problem. Therefore, for the compliance minimization problem subject to a
volume restriction, the intermediate and dilated designs can be excluded from
the objective function, but the volume restriction has to be applied to the
dilated design in order to involve all 3 designs in the formulation.
###### Keywords:
Length Scale Robust Design SIMP
††journal: PREPRINT
## 1 Introduction
Since the seminal work of Bendsøe and Kikuchi (1988), topology optimization
has experienced huge advances and nowadays is being massively adopted in the
industry (Pedersen and Allinger, 2006; Zhou et al, 2011; Zhu et al, 2016).
Among the successful advancements, one can mention the famous density method
under the SIMP interpolation scheme (Bendsøe, 1989). The well-known
shortcomings of SIMP led to a succession of improvements seeking to avoid the
mesh dependency, the checkerboard patterns and the presence of intermediate
densities. To date, one of the most effective approaches to dealing with the
ill-effects of SIMP is the robust design approach that brings the eroded and
dilated versions of the design (Sigmund, 2009). This method considers
manufacturing errors that may incur in a uniformly thinner or uniformly
thicker component compared to the blueprint layout. Sigmund (2009) proposed a
robust formulation that maximizes the performance of the worst performing
design among the eroded, dilated and reference (intermediate) designs. This
means, the formulation guarantees a design with good performance even if it is
eventually eroded or dilated during the manufacturing process. Interestingly,
the robust formulation yields a reference (intermediate) design that features
minimum member size and minimum cavity size (Wang et al, 2011).
In addition to imposing minimum length scale in topology optimization, it has
been seen that the robust formulation provides a more stable convergence than
other projection or filtering strategies known so far, which allows to reach
almost discrete solutions in the density method (Wang et al, 2011). The
formulation has been recently applied in combination with existing methods
that allow to impose stress limits (da Silva et al, 2019; Silva et al, 2020),
overhang angle constraints (Pellens et al, 2018), maximum size restrictions
(Fernández et al, 2020), and geometric non-linearities (Lazarov and Sigmund,
2011; Silva et al, 2020), not only in the density method but also in the level
set method (Chen and Chen, 2011; Andreasen et al, 2020).
Despite proving its effectiveness in several fields of application (Wang et
al, 2011b; Christiansen et al, 2015), the robust design approach presents some
drawbacks with regard to its implementation. For example, the method requires
boundary treatments with respect to the density filter, since the filtering
region can be split at the edges of the design domain affecting the imposed
minimum size (Clausen and Andreassen, 2017; Kumar and Fernández, 2021). In
addition, due to the erosion and dilation distances with respect to the
intermediate design, boundary conditions can be disconnected from the designs
involved in the formulation (Clausen and Andreassen, 2017). Another difficulty
posed by the method is that the minimum length scale must be implicitly
imposed through the parameters defining the density filter and the Heaviside
projection. This drawback has been addressed in the literature using numerical
(Wang et al, 2011) and analytical (Qian and Sigmund, 2013) approaches. The
numerical approach consists of applying the density filter and the Heaviside
projection to a 1D design, measuring the resulting length scale, and repeating
the process several times with different filter and projection parameters to
subsequently construct a graph that relates the involved parameters. The
analytical method consists of applying the density filter as a convolution
integral in a continuous 1D design. The solution of the integration yields
explicit relationships of the projection/filtering parameters with the
obtained length scale (Qian and Sigmund, 2013). As the three field scheme is a
scalar function, the relationships obtained from the 1D designs remain valid
for 2D and 3D (Wang et al, 2011).
The numerical and analytical relationships reported by Wang et al (2011) and
Qian and Sigmund (2013) are intended for the particular case where the minimum
size of the void phase is set equal to that of the solid phase. This
simplifies and facilitates the procedures to explicitly relate the desired
minimum length scale to the filter and projection parameters, but in this way,
the minimum size of the solid phase cannot be different from the minimum size
of the void phase. Nonetheless, due to the rapid popularization of the robust
formulation, it becomes necessary to find a method that allows to impose
different minimum sizes for each phase, in addition to provide access to other
geometric information. For example, the erosion and dilation distances with
respect to the intermediate design have been required to impose a maximum
member size (Fernández et al, 2020) and to obtain designs tailored to the size
of a deposition nozzle (Fernández et al, 2021). To date, the erosion and
dilation distances have been reported for specific minimum length scales,
which hinders widespread applicability of the above methods.
The aim of this work is to provide a method to obtain the filter and
projection parameters that impose user-defined minimum length scales, where
the size of the solid phase can be defined differently from that of the void
phase. In addition, for the desired minimum length scales, the erosion and
dilation distances are provided. To this end, we extend the applicability of
the analytical method proposed by Qian and Sigmund (2013) and the resulting
relationships are validated using the numerical method proposed by Wang et al
(2011) and a set of 2D topology-optimized designs. To facilitate the
replication of results and the application of the methods discussed in this
paper, MATLAB codes are provided.
The remainder of the document is developed as follows. Section 2 introduces
the robust topology optimization formulation and the case studies used to
asses the analytical expressions that relate the minimum len-gth scale to the
parameters that impose it. Section 3 presents the analytical method proposed
by Qian and Sigmund (2013) and describes the main contribution of our work,
which is the extension of the analytical approach to allow choosing
independent minimum sizes for the solid phase and the void phase. Section 4
validates the analytical expressions using the numerical method proposed by
Wang et al (2011). Section 5 discusses the sources of error that are inherent
to the analytical method. Section 6 assesses the analytical expressions on 2D
topology-optimized designs. Section 7 gathers the final conclusions of this
work while Section 8 provides the codes that allow for the replication of
results.
## 2 Problem Definition
The analytical expressions that relate the minimum length scale to the
parameters that define it are developed for the robust topology optimization
formulation based on the eroded, intermediate and dilated designs. This work
considers the density approach based on the SIMP interpolation scheme
(Bendsøe, 1989), even though the proposed methodology can be applied to other
topology optimization approaches with little efforts.
Like most works in the literature, the eroded, intermediate and dilated
designs that constitute the robust formulation are built using a three-field
scheme (Sigmund and Maute, 2013). The first field, denoted by $\bm{\rho}$,
corresponds to the design variables. The second field, denoted by
$\bm{\tilde{\rho}}$, is obtained by a weighted average of the design variables
within a circle of radius $r_{\mathrm{fil}}$. The third field, denoted by
$\bm{\bar{\rho}}$, is obtained by projecting the components of the filtered
field towards 0 or 1. The filter and projection functions are identical to
those provided by Wang et al (2011), however, these are reminded herein for
the sake of clarity as they define the minimum length scale obtained in the
optimized design.
The filter of design variables, or the density filter (Bruns and Tortorelli,
2001; Bourdin, 2001), is defined as follows:
$\tilde{\rho}_{i}=\frac{\displaystyle\sum_{j=1}^{N}\rho_{j}\mathrm{v}_{j}w(\mathbf{x}_{i}-\mathbf{x}_{j})}{\displaystyle\sum_{j=1}^{N}{\mathrm{v}}_{j}w(\mathbf{x}_{i}-\mathbf{x}_{j})}\;\;,$
(1)
where $\rho_{j}$ is the design variable associated to the element $j$ and
$\tilde{\rho}_{i}$ is the filtered variable associated to the element $i$.
$\mathrm{v}_{j}$ is the volume of the element $j$ and
$w(\mathbf{x}_{i}-\mathbf{x}_{j})$ is the weigh of $\rho_{j}$ in the
definition of $\tilde{\rho}_{i}$. As is common practice in the literature, the
weighting function $w(\mathbf{x}_{i}-\mathbf{x}_{j})$ is defined as a linear
and decreasing function with respect to the distance between the elements $i$
and $j$, as follows:
$w(\mathbf{x}_{i}-\mathbf{x}_{j})=\mathrm{max}\left(0,\>1-\frac{\|\mathbf{x}_{i}-\mathbf{x}_{j}\|}{r_{\mathrm{fil}}}\right)\;\;,$
(2)
where $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ represent the centroid of the
elements $i$ and $j$, respectively. It is recalled that $r_{\mathrm{fil}}$ is
the radius of the density filter.
To reduce the amount of intermediate densities pre-sent in the filtered field
and to build the designs that constitute the robust formulation, the projected
field is obtained with the following smoothed Heaviside function (Wang et al,
2011):
$\bar{\rho}_{i}=\frac{\tanh{(\beta\eta)}\tanh{(\beta(\tilde{\rho}_{i}-\eta))}}{\tanh{(\beta\eta)}\tanh{(\beta(1-\eta))}}$
(3)
where $\beta$ and $\eta$ control the steepness and the threshold of the
projection, respectively. The eroded, intermediate and dilated designs,
denoted by ${\bm{\bar{\rho}}}_{\mathrm{ero}}$,
${\bm{\bar{\rho}}}_{\mathrm{int}}$ and ${\bm{\bar{\rho}}}_{\mathrm{dil}}$, are
obtained from the smoothed Heaviside function of Eq. (3) by using the same
$\beta$ but with different thresholds ${\eta_{\mathrm{ero}}}$,
$\eta_{\mathrm{int}}$ and $\eta_{\mathrm{dil}}$, thus leading to
${{\bm{\bar{\rho}}}_{\mathrm{ero}}}(\bm{\tilde{\rho}},\beta,{\eta_{\mathrm{ero}}})$,
${{\bm{\bar{\rho}}}_{\mathrm{int}}}(\bm{\tilde{\rho}},\beta,{\eta_{\mathrm{int}}})$
and
${{\bm{\bar{\rho}}}_{\mathrm{dil}}}(\bm{\tilde{\rho}},\beta,{\eta_{\mathrm{dil}}})$.
(a)
(b)
Figure 1: (a) Heat sink and (b) force inverter design domains considered in
this study.
To assess the scope of the equations that provide the necessary parameters to
impose the minimum length scale, different topology optimization problems are
sol-ved with variations in the desired minimum length scale. The minimum size
of the optimized designs is measured and compared against the intended values.
In this work, the problems that are chosen to illustrate the developments are
the heat conduction and the non-linear compliant mechanism design. The former
aims at minimizing the thermal compliance subject to a volume restriction
(Wang et al, 2011), whilst in the latter, the formulation of a force inverter
is considered where an output displacement is maximized for a given input
force (Sigmund, 1997). The design domains are shown in Fig. 1. According to
the robust design approach, the topology optimization problems can be written
as follows:
$\begin{split}\text{min}\quad&\mathrm{max}\left(c({{\bm{\bar{\rho}}}_{\mathrm{ero}}}),\;c({{\bm{\bar{\rho}}}_{\mathrm{int}}})\;,c({{\bm{\bar{\rho}}}_{\mathrm{dil}}})\right)\\\
\text{s.t.:}\quad&\mathbf{v}^{\intercal}\>{{\bm{\bar{\rho}}}_{\mathrm{dil}}}\leq
V^{*}_{\mathrm{dil}}(V^{*}_{\mathrm{int}})\\\ &0\leq\rho_{i}\leq
1\;\;,\;\;i=1,...,N\;\;,\end{split}$ (4)
where $c$ represents the thermal compliance of the heat sink or the output
displacement of the force inverter, $V^{*}_{\mathrm{dil}}$ is the upper bound
of the volume restriction and N is the total amount of design variables. As
the design intended for manufacturing is the intermediate design, the upper
bound of the volume constraint is scaled according to the user-defined limit
$V^{*}_{\mathrm{int}}$. The optimization problem and the scaling process of
the volume restriction are the same as described by Wang et al (2011) for the
heat conduction problem. In the nonlinear force inverter, the method to deal
with the mesh distorsion appearing in low density elements is the same as
explained by Wang et al (2014). To avoid overextending the contents of the
manuscript, the interested reader is refered to the cited articles.
## 3 Analytical minimum length scale
Comprehensive numerical tests have shown that the robust formulation in Eq.
(4) yields intermediate designs that have the same topology as their eroded
and dilated projections (Wang et al, 2011). In other words, the erosion
process does not destroy the solid members of the optimized topology
(intermediate), it just makes them thinner. This implies that a minimum size
of the solid members in the intermediate design is imposed according to the
erosion distance. Similarly, the dilation process does not close the cavities
of the optimized topology (intermediate), it only makes them narrower. This
implies that the minimum size of the void phase is imposed according to the
dilation distance.
Considering a two-dimensional design domain, the minimum size of the solid
phase is defined by the radius $r_{\mathrm{min.Solid}}$ of the largest circle
that can be circumscribed into the smallest solid member of the topology,
while the minimum size of the void phase is defined by the radius
$r_{\mathrm{min.Void}}$ of the largest circle that can be circumscribed into
the smallest cavity of the topology. It has been shown that the minimum length
scale, $r_{\mathrm{min.Solid}}$ and $r_{\mathrm{min.Void}}$, are defined by
the filter radius $r_{\mathrm{fil}}$ and the projection thresholds
$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{int}}$ and $\eta_{\mathrm{dil}}$.
Therefore, if specific values are to be imposed for the minimum length scale,
these must be implicitly imposed through the 4 parameters that define the
density filter and the Heaviside projection.
The dependence of the minimum length scale on the projection and filtering
parameters hurdles a general implementation of the robust formulation,
specially in commercial codes, as there is no prompt approach to find the
relationship between the involved parameters and the desired length scale.
Encouraged by this shortcoming, in this section we present and extend the
applicability of the analytical procedure proposed by Qian and Sigmund (2013),
which is described below.
To obtain an explicit relation between the minimum length scale and the
parameters that define it, Qian and Sigmund (2013) proposed to apply the
three-field scheme over a uni-dimensional and continuous design domain. For
instance, to obtain the minimum size of the solid phase, a design domain
$\bm{\rho}(x)$ centered at the coordinate $x_{m}$ and containing solid
elements on a stretch of size $h$ is assumed, as shown in Fig. 2(a). The
continuous form of the density filter reads as follows:
$\begin{matrix}{\tilde{\rho}}(x_{i})&=\frac{\displaystyle\int_{x_{i}-r_{\mathrm{fil}}}^{x_{i}+r_{\mathrm{fil}}}\bm{\rho}(x)\left(1-\frac{\left|x_{i}-x\right|}{r_{\mathrm{fil}}}\right)dx}{\displaystyle\int_{x_{i}-r_{\mathrm{fil}}}^{x_{i}+r_{\mathrm{fil}}}\left(1-\frac{\left|x_{i}-x\right|}{r_{\mathrm{fil}}}\right)dx}\vspace{4mm}\\\
&=\displaystyle\int_{x_{i}-r_{\mathrm{fil}}}^{x_{i}+r_{\mathrm{fil}}}\frac{\bm{\rho}(x)}{r_{\mathrm{fil}}}\left(1-\frac{\left|x_{i}-x\right|}{r_{\mathrm{fil}}}\right)dx\end{matrix}$
(5)
where $x_{i}$ is any coordinate of $x$. For example, by choosing a filter
radius greater than $h/2$, the one-dimensional filtered field in Fig. 2(b) is
obtained. The filtered field can be projected using the smoothed Heaviside
function of Eq. (3). To simplify the analysis, an infinite steepness parameter
($\beta\to\infty$) is assumed. For example, for an Heaviside threshold
$\eta_{i}=0.2$, the design of Fig. 2(c) is obtained. The size of the solid
phase in the projected design is defined by the length $L_{i}$, which can be
obtained by finding $x_{i}$ from the equation $\tilde{\rho}(x_{i})=\eta_{i}$.
However, to this extent, the size of the solid phase is a function
$L_{i}(r_{\mathrm{fil}},\eta_{i},h)$ that depends on the assumed $h$ value. To
determine the value of $h$ and to discover the explicit relation between the
size $L$ of the projected field and the filter and projection parameters, it
is necessary to refer to the foundation of the robust formulation.
(a) 1D domain
(b) Filtered Field
(c) Projected Field
(d) Filtered field
Figure 2: The three field scheme applied to a one-dimensional design domain in
order to obtain the analytical minimum length scale.
An erosion projection removes solid material from the surface of the reference
design. This operation enlarges the cavities and thins the structural members
present in the intermediate reference design (Sigmund, 2009). When robustness
with respect to erosion is desired, the structural members must be present in
both the eroded and the reference designs. It has been shown that a sufficient
condition to preserve robustness is that the solid member in the eroded design
has to be projected by at least an infinitesimal size, i.e.
$L({\eta_{\mathrm{ero}}})\approx 0$, as shown in Fig 2(d). This condition
allows determining the assumed size $h$ by solving the following equation:
$\tilde{\rho}(x_{\mathrm{ero}})=\int_{-h/2}^{h/2}\frac{1}{r_{\mathrm{fil}}}\left(1-\frac{|x|}{r_{\mathrm{fil}}}\right)dx={\eta_{\mathrm{ero}}}$
(6)
where the reference system is placed at $x_{\mathrm{ero}}$ for integration. By
solving the expression in Eq. (6), the size $h$ is obtained as:
$h=2r_{\mathrm{fil}}(1-\sqrt{1-{\eta_{\mathrm{ero}}}})$ (7)
By obtaining the distance $h$ that produces an eroded projection of
infinitesimal size, it is possible to relate the minimum size to the filter
radius for any projection whose threshold meets
$\eta_{i}<{\eta_{\mathrm{ero}}}$. A particular case is to choose
$\eta_{i}=0.5$, which is the value chosen by Qian and Sigmund (2013) to define
the intermediate design. However, to extend the analytical method we use an
arbitrary $\eta_{i}$ such that ${\eta_{\mathrm{i}}}<{\eta_{\mathrm{ero}}}$.
As mentioned above, the size of the solid phase $L$ in the projected field can
be obtained by equating $\tilde{\rho}(x_{i})$ to $\eta_{i}$. Since the
condition of robustness is imposed on $h$ (Eq. 7), the length corresponds to
the minimum size, i.e. $L=2r_{\mathrm{min.Solid}}$. The expression that allows
to relate the minimum size to the filter radius is as follows:
${\tilde{\rho}}(x_{i})=\displaystyle\int_{x_{i}-r_{\mathrm{fil}}}^{x_{i}+r_{\mathrm{fil}}}\frac{\bm{\rho}(x)}{r_{\mathrm{fil}}}\left(1-\frac{\left|x_{i}-x\right|}{r_{\mathrm{fil}}}\right)dx=\eta_{i}$
(8)
For the integration, it is convenient to place the origin of the reference
system at $x_{i}$. To this end, 4 situations must be considered to define the
integration limits, which are summarized in Fig. 3. For example, if the length
of the projected field $L$ is greater than $h$ and the filter radius is
greater than $(L+h)/2$ (Fig. 3(a)), then Eq. (8) becomes:
${\tilde{\rho}}(x_{i})=\displaystyle\int_{\frac{L-h}{2}}^{L-\frac{L-h}{2}}\frac{1}{r_{\mathrm{fil}}}\left(1-\frac{\left|x\right|}{r_{\mathrm{fil}}}\right)dx=\eta_{i}$
(9)
(a) $L\geq h$ and $r_{\mathrm{fil}}\geq\frac{L+h}{2}$.
(b) $L<h$ and $r_{\mathrm{fil}}\geq\frac{L+h}{2}$.
(c) $L\geq h$ and $r_{\mathrm{fil}}<\frac{L+h}{2}$.
(d) $L<h$ and $r_{\mathrm{fil}}<\frac{L+h}{2}$.
Figure 3: Situations to be considered when defining the limits of integration. Phase | Conditions | | Minimum size
---|---|---|---
Solid | $\eta_{i}\leq 2{\eta_{\mathrm{ero}}}-1$ | and | $\eta_{i}\geq 0.5$ | | $\displaystyle\frac{2r_{\mathrm{min.Solid}}}{r_{\mathrm{fil}}}=2\sqrt{2-2\eta_{i}}-2\sqrt{1-{\eta_{\mathrm{ero}}}}$
$\eta_{i}\geq 2{\eta_{\mathrm{ero}}}-2+2\sqrt{1-{\eta_{\mathrm{ero}}}}$ | and | $\eta_{i}>2{\eta_{\mathrm{ero}}}-1$ | | $\displaystyle\frac{2r_{\mathrm{min.Solid}}}{r_{\mathrm{fil}}}=2\sqrt{{\eta_{\mathrm{ero}}}-\eta_{i}}$
$\eta_{i}<4-4\sqrt{1-{\eta_{\mathrm{ero}}}}-2{\eta_{\mathrm{ero}}}$ | and | $\eta_{i}<0.5$ | | $\displaystyle\frac{2r_{\mathrm{min.Solid}}}{r_{\mathrm{fil}}}=4-2\sqrt{1-{\eta_{\mathrm{ero}}}}-2\sqrt{2\eta_{i}}$
$\eta_{i}\geq 4-4\sqrt{1-{\eta_{\mathrm{ero}}}}-2{\eta_{\mathrm{ero}}}$ | and | $\eta_{i}<2\eta_{\mathrm{ero}}-2+2\sqrt{1-{\eta_{\mathrm{ero}}}}$ | | $\displaystyle\frac{2r_{\mathrm{min.Solid}}}{r_{\mathrm{fil}}}=2-\frac{\eta_{i}}{1-\sqrt{1-{\eta_{\mathrm{ero}}}}}$
Void | $\eta_{i}\geq 2{\eta_{\mathrm{dil}}}$ | and | $\eta_{i}\leq 0.5$ | | $\displaystyle\frac{2r_{\mathrm{min.Void}}}{r_{\mathrm{fil}}}=2\sqrt{2\eta_{i}}-2\sqrt{{\eta_{\mathrm{dil}}}}$
$\eta_{i}\leq 2\eta_{d}+1-2\sqrt{{\eta_{\mathrm{dil}}}}$ | and | $\eta_{i}<2{\eta_{\mathrm{dil}}}$ | | $\displaystyle\frac{2r_{\mathrm{min.Void}}}{r_{\mathrm{fil}}}=2\sqrt{\eta_{i}-{\eta_{\mathrm{dil}}}}$
$\eta_{i}\geq 4\sqrt{{\eta_{\mathrm{dil}}}}-2{\eta_{\mathrm{dil}}}-1$ | and | $\eta_{i}>0.5$ | | $\displaystyle\frac{2r_{\mathrm{min.Void}}}{r_{\mathrm{fil}}}=4-2\sqrt{{\eta_{\mathrm{dil}}}}-2\sqrt{2-2\eta_{i}}$
$\eta_{i}<4\sqrt{{\eta_{\mathrm{dil}}}}-2{\eta_{\mathrm{dil}}}-1$ | and | $\eta_{i}>2{\eta_{\mathrm{dil}}}+1-2\sqrt{{\eta_{\mathrm{dil}}}}$ | | $\displaystyle\frac{2r_{\mathrm{min.Void}}}{r_{\mathrm{fil}}}=2-\frac{1-\eta_{i}}{1-\sqrt{{\eta_{\mathrm{dil}}}}}$
Table 1: The explicit relationship between the minimum length scale and the
filter and projection parameters.
Solving the integral of Eq. (9) leads to the following expression:
${\tilde{\rho}}(x_{i})=\frac{h}{r_{\mathrm{fil}}}(1-\frac{L}{2r_{\mathrm{fil}}})=\eta_{i}$
(10)
Finally, replacing Eq. (7) in (10):
$\frac{2r_{\mathrm{min.Solid}}}{r_{\mathrm{fil}}}=2-\frac{\eta_{i}}{1-\sqrt{1-{\eta_{\mathrm{ero}}}}}$
(11)
Eq. (11) explicitly relates the minimum size of the solid phase
($r_{\mathrm{min.Solid}}$), of a projected field defined by $\eta_{i}$, with
the filter radius ($r_{\mathrm{fil}}$) and the erosion threshold
($\eta_{\mathrm{ero}}$). However, Eq. (11) is only valid for $L>h$ and
$r_{\mathrm{fil}}\geq(L+h)/2$. For implementation purposes, it is more
convenient to express the range of application in terms of the projection
thresholds. To this end, $h$ can be replaced from Eq. (7) and $L$ from (11),
which leads to conditions depending only on $\eta_{i}$ and
${\eta_{\mathrm{ero}}}$, as follows:
$\begin{matrix}L>h&\implies\eta_{i}<2\eta_{\mathrm{ero}}-2+2\sqrt{1-{\eta_{\mathrm{ero}}}}\\\\[4.30554pt]
r_{\mathrm{fil}}\geq(L+h)/2&\implies\eta_{i}\geq
4-4\sqrt{1-{\eta_{\mathrm{ero}}}}-2{\eta_{\mathrm{ero}}}\end{matrix}$ (12)
By repeating the procedure from Eq. (9) to (12) for the 4 integration
conditions shown in Fig. 3, a set of equations is obtained which relate the
filter and projection parameters with the minimum size for any projection
threshold $\eta_{i}$, provided that $\eta_{i}<{\eta_{\mathrm{ero}}}$. The set
of equations are summarized in the four first rows of Table 1.
To obtain the relationships that define the minimum size of the void phase,
the same procedure must be used as for the solid phase, however, now starting
from a one-dimensional design domain containing a cavity of size $h$. To avoid
overextending the document with redundant information, this section is limited
to presenting the final equations that define the minimum size of the void
phase. The expressions are summarized in the last 4 rows of Table 1.
Having delivered the set of equations that expand the scope of the method
proposed by Qian and Sigmund (2013), the following section presents a
methodology to use these equations.
## 4 Imposing the desired minimum length scale
In structural design, the minimum length scale control is usually desired
because of design requirements or manufacturing limitations, hence in most
cases, the minimum size of the solid and void phases are known values
established for the intermediate design. Therefore, for the set of equations
presented in Table 1, the radii $r_{\mathrm{min.Solid}}^{\mathrm{int}}$ and
$r_{\mathrm{min.Void}}^{\mathrm{int}}$ are assumed user-defined input values.
In this case, the projection threshold $\eta_{i}$ corresponds to the
projection threshold ${\eta_{\mathrm{int}}}$, hence the desired length scale
for the intermediate design is a function of the projection thresholds and of
the size of the filter, namely,
$r_{\mathrm{min.Solid}}^{\mathrm{int}}({\eta_{\mathrm{int}}},{\eta_{\mathrm{ero}}},r_{\mathrm{fil}})$
and
$r_{\mathrm{min.Void}}^{\mathrm{int}}({\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}},r_{\mathrm{fil}})$.
Given the number of unknowns
$(r_{\mathrm{fil}},{\eta_{\mathrm{ero}}},\eta_{\mathrm{int}},\eta_{\mathrm{dil}})$,
the system of equations in Table 1 becomes indeterminate and the desired
minimum length scale can be imposed through multiple combinations of
parameters. Nevertheless, such freedom of parameters selection can be reduced
by considering the following three recommendations.
Firstly, a number of advantages have been observed when defining the
intermediate design with a threshold ${\eta_{\mathrm{int}}}=0.5$. For
instance, the projection features lower amounts of intermediate densities
compared to those projections that use a threshold other than 0.5 (Xu et al,
2010; Wang et al, 2011; da Silva et al, 2019), and a threshold
${\eta_{\mathrm{int}}}$ set to 0.5 provides the same size ranges (0.5) for the
erosion and dilatation thresholds, which is convenient for reducing rounding
errors, since small differences in projection thresholds could be insensitive
to the minimum size when using a coarse discretization of the design domain
(Qian and Sigmund, 2013). Secondly, for a particular combination of
thresholds, the filter size ($r_{\mathrm{fil}}$) can become considerably
larger than the desired minimum length scale, which could significantly
increase computational requirements (Lazarov and Sigmund, 2011). This can be
seen in Figs. 4 and 6(b). These figures show graphs that relate the filter
size ($r_{\mathrm{fil}}$) to the minimum size of solid or void phase and to
the erosion or dilation threshold. These graphs show that the closer
${\eta_{\mathrm{ero}}}$ and ${\eta_{\mathrm{dil}}}$ are to
${\eta_{\mathrm{int}}}$, the larger the filter radius, which inevitably
increases computational requirements. Under this observation, we recommend
choosing ${\eta_{\mathrm{ero}}}\geq 0.75$ and ${\eta_{\mathrm{dil}}}\leq
0.25$, thus it is ensured that $r_{\mathrm{fil}}\leq
2r_{\mathrm{min.Solid}}^{\mathrm{int}}$ and $r_{\mathrm{fil}}\leq
2r_{\mathrm{min.Void}}^{\mathrm{int}}$. Thirdly, erosion and dilation
thresholds too distant or too close to the intermediate threshold increases
oscillations of design variables during the optimization process. In general,
a good compromise is to choose $0.10\leq{\eta_{\mathrm{dil}}}\leq 0.4$ and
$0.60\leq{\eta_{\mathrm{ero}}}\leq 0.9$.
Figure 4: Graphical relationship between the minimum size of the void phase,
the filter radius and the dilation projection.
The first observation removes an unknown from the system of equations, since
${\eta_{\mathrm{int}}}$ is set to 0.5, while the second and third observations
limit the range of the erosion and dilation thresholds. Thus, for a user-
defined minimum length scale, it is possible to develop an algorithm that
solves the system of equations considering the three observations. Here we
propose an algorithm based on graphic relationships, so that the reader can
easily find the desired parameters without the need to resort to a
computational algorithm. Nonetheless, we also provide as supplementary
material a code written in MATLAB named SizeSolution.m that performs the
procedure described below. It is important to note that the above observations
are based on numerical tests considered for specific optimization problems
formulated in the density approach. Therefore, it is possible that under other
topology optimization approaches or formulations the above observations are no
longer valid. However, the proposed procedure can be applied for any other
value of ${\eta_{\mathrm{int}}}$, or any other combination of parameters that
the user may consider convenient.
Given that the ranges of application of the equations in Table 1 are defined
as a function of the projection thresholds ($\eta_{i}$,
${\eta_{\mathrm{ero}}}$ and ${\eta_{\mathrm{dil}}}$), it is rather simple to
construct graphs with respect to them. For instance, as shown in Fig. 5, to
find the range of application of the equations defining the minimum size of
the solid phase, it is simply necessary to know in which region of Fig. 5 the
projection $\eta_{i}$ falls.
The first graph proposed in this work is shown in Fig. 6(a) and gathers 4
parameters, the minimum length scale ($r_{\mathrm{min.Solid}}^{\mathrm{int}}$
and $r_{\mathrm{min.Void}}^{\mathrm{int}}$) and the projection thresholds
(${\eta_{\mathrm{ero}}}$ and ${\eta_{\mathrm{dil}}}$), provided that
${\eta_{\mathrm{int}}}=0.5$. In this graph, the user can easily find the set
of erosion and dilation threshold that leads to the desired length scale.
Then, the user can access the graph in Fig. 6(b) to obtain the filter radius.
For example, for the following minimum length scale,
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=3$ elements and
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=3$ elements, the graph in Fig. 6(a) is
accessed with a value of 1.0 for the ordinate. According to the aforementioned
observations, the combination of thresholds [${\eta_{\mathrm{ero}}}$,
${\eta_{\mathrm{dil}}}$] that can be chosen among others are:
$\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.75,\;0.25]\;,$ (13a)
$\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.80,\;0.20]\;,$ (13b)
$\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.85,\;0.15]\;,$ (13c)
$\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.90,\;0.10]\;.$ (13d)
Arbitrarily, [0.75, 0.25] is selected, and from the graph in Fig. 6(b), it is
obtained that $r_{\mathrm{fil}}=2r_{\mathrm{min.Solid}}^{\mathrm{int}}=6$
elements.
It should be noted that the graph in Fig. 6(a) is constructed considering a
resolution of 0.05 in the projection thresholds. This is due to the fact that
the discretization of the filter radius in topology optimization is generally
coarse, and decimal numbers smaller than 0.05 in the threshold value have
usually a negligible effect on the minimum length scale of the optimized
design. However, if a better resolution is required, the user can resort to
the attached code SizeSolution.m.
Figure 5: Graphic representation of the applicability of the set of equations
in Table 1 proposed for the solid phase. The denomination of the zone
corresponds to the row number of Table 1.
(a)
(b)
(c)
(d)
(a) (b)
(c) (d)
Figure 6: Graphical relationships between the minimum length scale and
$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{int}}$, $\eta_{\mathrm{dil}}$, and
$r_{\mathrm{fil}}$. The minimum length scale is defined for the intermediate
design. The graphs are built using both, the analytical and the numerical
method. Graphs (a) and (b) are designed to obtain the projection thresholds
and the filter radius, respectively. Graphs (c) and (d) are designed to obtain
the dilation and erosion distances, respectively.
As previously mentioned, the erosion and dilation distances could be required,
for instance, when implementing maximum size restrictions (Fernández et al,
2020). These distances can be easily obtained from the equations of Table 1.
As shown in Fig. 2(d), the dilation distance, denoted by $t_{\mathrm{dil}}$,
is the offset distance between the intermediate and dilated designs.
Analogously, the erosion distance, denoted by $t_{\mathrm{ero}}$, is the
offset distance between the intermediate and eroded designs. Therefore:
$\begin{matrix}t_{\mathrm{dil}}&=r_{\mathrm{min.Solid}}^{\mathrm{dil}}{(\eta_{\mathrm{dil}},\eta_{\mathrm{ero}},r_{\mathrm{fil}})}-r_{\mathrm{min.Solid}}^{\mathrm{int}}{(\eta_{\mathrm{int}},\eta_{\mathrm{ero}},r_{\mathrm{fil}})}\vspace{2mm}\\\
t_{\mathrm{ero}}&=r_{\mathrm{min.Void}}^{\mathrm{ero}}{(\eta_{\mathrm{ero}},\eta_{\mathrm{dil}},r_{\mathrm{fil}})}-r_{\mathrm{min.Void}}^{\mathrm{int}}{(\eta_{\mathrm{int}},\eta_{\mathrm{dil}},r_{\mathrm{fil}})}\end{matrix}$
(14)
where the minimum size of the solid phase in the dilated design
($r_{\mathrm{min.Solid}}^{\mathrm{dil}}$) is obtained by choosing
$\eta_{i}={\eta_{\mathrm{dil}}}$ in the equations of Table 1. Analogously, the
minimum size of the void phase in the eroded design
($r_{\mathrm{min.Void}}^{\mathrm{ero}}$) is obtained by choosing
$\eta_{i}={\eta_{\mathrm{ero}}}$. In this way, analytical expressions can be
obtained for the dilation and erosion distances.
It is noted from Eq. (14) that the erosion and dilation depend on the 3
projection thresholds and on the size of the filter, namely,
$t_{\mathrm{ero}}(r_{\mathrm{fil}},{\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})$
and
$t_{\mathrm{dil}}(r_{\mathrm{fil}},{\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})$.
Therefore, if for some particular reason the user needs to impose the minimum
length scale, and the erosion and dilation distances, then the following
system of equations must be solved:
$\begin{split}r_{\mathrm{min.Solid}}^{\mathrm{int}}&=r_{\mathrm{min.Solid}}^{\mathrm{int}}(r_{\mathrm{fil}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{ero}}})\vspace{1mm}\\\
r_{\mathrm{min.Void}}^{\mathrm{int}}&=r_{\mathrm{min.Solid}}^{\mathrm{int}}(r_{\mathrm{fil}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})\vspace{2mm}\\\
t_{\mathrm{ero}}&=r_{\mathrm{min.Solid}}^{\mathrm{int}}(r_{\mathrm{fil}},{\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})\vspace{1mm}\\\
t_{\mathrm{dil}}&=r_{\mathrm{min.Solid}}^{\mathrm{int}}(r_{\mathrm{fil}},{\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})\\\
\end{split}$ (15)
The system in Eq. (15) is determined and there is only one combination of
parameters [$r_{\mathrm{fil}}$, ${\eta_{\mathrm{ero}}}$,
${\eta_{\mathrm{int}}}$, ${\eta_{\mathrm{dil}}}$] that leads to the desired
length scale [$r_{\mathrm{min.Solid}}^{\mathrm{int}}$,
$r_{\mathrm{min.Void}}^{\mathrm{int}}$, $t_{\mathrm{ero}}$,
$t_{\mathrm{dil}}$]. This can be illustrated with the following example. For
each combination of thresholds [$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{dil}}$]
given in Eq. (13), the erosion and dilation distances [${t_{\mathrm{ero}}}$ ,
${t_{\mathrm{dil}}}$ ] are provided, as follows:
$\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.75,\;0.25]\;,[{t_{\mathrm{ero}}},{t_{\mathrm{dil}}}]=[1.76,\;1.76]$
(16a) $\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.80,\;0.20]\;,[{t_{\mathrm{ero}}},{t_{\mathrm{dil}}}]=[1.99,\;1.99]$
(16b) $\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.85,\;0.15]\;,[{t_{\mathrm{ero}}},{t_{\mathrm{dil}}}]=[2.21,\;2.21]$
(16c) $\displaystyle[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$
$\displaystyle=[0.90,\;0.10]\;,[{t_{\mathrm{ero}}},{t_{\mathrm{dil}}}]=[2.43,\;2.43]$
(16d)
It is recalled that all combinations of thresholds [$\eta_{\mathrm{ero}}$,
$\eta_{\mathrm{dil}}$] in Eq. (16) lead to the minimum length scale
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=r_{\mathrm{min.Void}}^{\mathrm{int}}=3$
elements. However, they all result in different erosion and dilation
distances.
To the authors’ knowledge, the need to impose specific values for the erosion
and dilation distances has not yet been claimed, so this manuscript is limited
to providing these values for a given combination of parameters. The proposed
graphs are shown in Fig. 6(c) and 6(d), which depend on the erosion and
dilation thresholds and on the corresponding minimum size. For instance,
considering $[{\eta_{\mathrm{ero}}},{\eta_{\mathrm{dil}}}]$ = [0.75, 0.25],
the erosion and dilation distances are $t_{\mathrm{ero}}=0.58\times 3$
elements and $t_{\mathrm{dil}}=0.58\times 3$ elements, where 0.58 is obtained
from the graphs.
## 5 Sources of error
The analytical procedure developed in this article allows to quickly obtain
the set of filtering and projection parameters that imposes the desired
minimum length scale. However, in practice, the length scale of the optimized
design often differs from the desired values (Qian and Sigmund, 2013). This is
mainly due to the fact that the analytical method assumes (i) a continuous
domain, (ii) a perfect Heaviside projection ($\beta\to\infty$), and (iii) that
the eroded and dilated fields project an infinitesimal minimum size in the
solid and void phases, respectively. The assumptions (i), (ii), and (iii) are
not met in topology optimization mainly because the design domains are
discretized into finite elements and because the Heaviside projection is
smoothed. To assess the error introduced by these assumptions, in the
following we compare the analytical method with the numerical method proposed
by Wang et al (2011), which considers a discrete domain and a smoothed
Heaviside function.
The procedure for obtaining the minimum size using the numerical method is
analogous to the analytical method but now using a one-dimensional domain
discretized into $N$ elements. For example, to obtain the minimum size of the
solid phase the three-field scheme is applied to a one-dimensional design
domain $\bm{\rho}$ containing solid elements in a length $h$. Then, the size
of the eroded, intermediate and dilated fields are measured, and the length
$h$ is adjusted so that the resulting eroded field has an infinitesimal size
(1 solid element). This process is repeated several times for different values
of ${\eta_{\mathrm{ero}}}$, and the resulting minimum size is normalized with
respect to the size of the chosen filter. For implementation details regarding
the numerical method, the reader is referred to the works of Wang et al (2011)
and Fernández et al (2020), and to the attached MATLAB code named
NumericalSolution.m.
To validate the analytical and numerical methods, the latter is implemented
using a Heaviside function at $\beta=500$, a filter radius of 1000 elements
and a design domain discretized into 10 thousand elements, so the numerical
method can be considered continuous and under similar assumptions than the
analytical one. The relationships obtained with the numerical method can be
seen in the graphs of Fig. 6. The agreement of results from both methods
allows us to validate the set of equations provided in Table 1.
The three sources of error that affect the analytical method are discussed
below.
### 5.1 Continuous design domain
When the design domain is discretized using finite elements, the radii that
define the minimum sizes ($r_{\mathrm{fil}}$,
$r_{\mathrm{min.Solid}}^{\mathrm{int}}$,
$r_{\mathrm{min.Solid}}^{\mathrm{dil}}$,
$r_{\mathrm{min.Void}}^{\mathrm{int}}$ and
$r_{\mathrm{min.Void}}^{\mathrm{ero}}$) are defined by a discrete number of
elements. In this case, the rounding error is $\pm$ 1 finite element in the
radius that defines the minimum size. Therefore, to reduce this error, it is
sufficient to reduce the size of the elements by mesh refinement. To
illustrate this remark graphically, we consider the following two
discretizations of a one-dimensional domain, one containing 100 elements and
the other containing 200 elements. In our implementation, the size of the
filter is chosen equal to $10\%$ of the domain size. Therefore, for the
discretizations containing 100 and 200 elements, the filter radius contains 10
and 20 elements, respectively. For each discretization, the relationship
between the minimum size of the solid phase, the size of the filter and the
erosion threshold is plotted in Fig. 7, using both the analytical and the
numerical method.
As the analytical method does not depend on the discretization, it provides
identical relationships on both discretizations. However, the rounding error
associated to the analytical method does depend on the discretization and can
be plotted as an offset of the analytical curve, as shown by the dashed curves
in Fig. 7. The rounding error, denoted as $\delta r$, corresponds to 1 element
in the radius, i.e. 2 elements in the diameter. That is, the vertical offset
of the analytical curve due to the rounding error is equal to $2\delta r=2/10$
and to $2\delta r=2/20$ in Figs. 7(a) and 7(b), respectively.
(a) $N=100$, $r_{\mathrm{fil}}=10$.
(b) $N=200$, $r_{\mathrm{fil}}=20$
Figure 7: Relationship between the minimum size of the solid phase, the eroded
threshold, and the filter radius. The dotted line represents the analytical
relationship including the rounding error.
The error curves agree with the results obtained from the numerical method,
which shows that the error of the analytical method coming from the assumption
of a continuous domain can be easily estimated. This estimation allows to know
either the margin of error of the minimum size ($\delta r$) for a given
analytical threshold (${\eta_{\mathrm{ero}}}$ or ${\eta_{\mathrm{dil}}}$), or
the margin of error of the analytical threshold ($\delta\eta$) for a given
minimum size, as shown in Fig. 7(a).
If the user of the robust approach needs to impose precisely the minimum size,
he can choose the projection thresholds (${\eta_{\mathrm{ero}}}$ and
${\eta_{\mathrm{dil}}}$) that lead to a larger filter radius, or he can use
the numerical method (Wang et al, 2011) using a discretization representative
of the design domain to be optimized.
### 5.2 Perfect Heaviside projection
The analytical method developed in this work assumes a perfect Heaviside
function, which results in a projected field $\bm{\bar{\rho}}$ containing
discrete densities (0 and 1). However, in topology optimization, the Heaviside
function is smoothed resulting in regions of intermediate densities that lie
on the surface of the optimized structure. For this reason, in practice, once
the optimized solution ${{\bm{\bar{\rho}}}_{\mathrm{int}}}$ is obtained, a
cut-off value $\varepsilon$ is usually added to the projected densities in
order to get the optimized structure. Therefore, the final design intended for
manufacturing is obtained by projecting the design
${{\bm{\bar{\rho}}}_{\mathrm{int}}}$ as follows:
${\bar{\rho}^{\varepsilon}}_{i}=\left\\{\begin{matrix}1\;\;,&\text{if}\;\;\;\;\bar{\rho}_{\mathrm{int}(i)}\geq\varepsilon\vspace{1mm}\\\
0\;\;,&\text{otherwise}\end{matrix}\right.$ (17)
Since the post-processed design $\bm{\bar{\rho}^{\varepsilon}}$ is the one
intended for manufacturing, the analytical expressions relating the minimum
length scale and the parameters that define it must be elaborated for the
field $\bm{\bar{\rho}^{\varepsilon}}$. As will be explained hereafter, the
equations provided in Table 1 can be easily adapted to consider the cut-off
value $\varepsilon$.
Figure 8: Illustration of a transition region where projected densities reach
intermediate values.
Consider Fig. 8 illustrating a transition zone between the solid structure and
the void region. In blue it is shown the filtered field $\bm{\tilde{\rho}}$
and in red the projected field $\bm{\bar{\rho}}$ obtained with $\beta=32$. In
this illustration, the density cut-off is defined as $\varepsilon=0.95$. A
perfect Heaviside projection ($\beta\to\infty$) would define the size of the
solid zone at the coordinate $x_{i}$. However, the smoothed Heaviside
projection defines the solid/void transition in the coordinate
$x_{\varepsilon}$. The idea then is to find the projection threshold
$\eta_{\varepsilon}$ for which a perfect Heaviside projection produces the
same solid/void transition coordinate than the one that is obtained with a
smoothed Heaviside projection with a threshold $\eta_{i}$. Thus, the value of
the filtered density present at the coordinate $x_{\varepsilon}$ must be
found. To this end, from Fig. 8 it is observed that:
$\varepsilon=\frac{\tanh{(\beta\eta_{i})}\tanh{(\beta(\eta_{\varepsilon}-\eta_{i}))}}{\tanh{(\beta\eta_{i})}\tanh{(\beta(1-\eta_{i}))}}$
(18)
Assuming $\beta>10$ at the end of the optimization process, which is a common
practice when dealing with Heaviside projection, Eq. (18) can be simplified
to:
$\tanh{(\beta(\eta_{\varepsilon}-\eta_{i}))}=2\varepsilon-1$ (19)
From Eq. (19), the filtered density $\eta_{\varepsilon}$ can be obtained:
$\eta_{\varepsilon}=\eta_{i}+\frac{1}{\beta}\mathrm{atanh}(2\varepsilon-1)$
(20)
We recall that the filtered density $\eta_{\varepsilon}$ represents the value
for which a perfect Heaviside function with threshold $\eta_{\varepsilon}$ and
a smoothed Heaviside function with threshold $\eta_{i}$ result in the same
size for the solid and void phases. Therefore, to take into account the
intermediate densities resulting from a smoothed Heaviside function, the
shifting term $\mathrm{atanh}(2\varepsilon-1)/\beta$ must be added to the
thresholds involved in the analytical equations (Table 1). Note that if
$\beta\to\infty$ or $\varepsilon=0.50$, then the shifting term is zero and
$\eta_{\varepsilon}=\eta_{i}$.
(a)
(b)
Figure 9: (a) Minimum length scale obtained with the analytical method
assuming a perfect Heaviside projection ($\beta\to\infty$), and with the
numerical method using a smoothed Heaviside projection ($\beta=32$) and three
different cut-off values $\varepsilon$. (b) The analytical curves obtained by
adding the shifting term $\mathrm{atanh}(2\varepsilon-1)/\beta$ to the
thresholds.
To assess the error introduced by a smoothed Heaviside projection, the minimum
length scale obtained with the analytical and the numerical methods are
compared. The analytical method is applied with $\beta\to\infty$, so it does
not consider the shifting term on the thresholds. For the numerical method we
use $\beta=30$, a filter radius equal to 200 and a one-dimensional domain
discretized into 2000 elements in order to avoid rounding errors and isolate
the effect of using a smoothed Heaviside projection. In the numerical method,
three cut-off values are used, $\varepsilon=0.01$, $\varepsilon=0.50$ and
$\varepsilon=0.99$. The results are shown in Fig. 9(a).
The results show that the influence of using a smoothed Heaviside function is
observed when $\varepsilon\neq 0.5$ and when ${\eta_{\mathrm{ero}}}>0.75$. For
this reason, we recommend to use a density cut-off value equal to 0.5, as this
avoids the need to add the shifting term to the projection thresholds.
Nevertheless, the user can easily adjust the analytical curves for
$\varepsilon\neq 0.5$, since it is only required to add
$\mathrm{atanh}(2\varepsilon-1)/\beta$ to the projection thresholds, as shown
in the corrected curves in Fig. 9(b).
It is worth mentioning that probably the effect of using a smoothed Heaviside
function is only seen for ${\eta_{\mathrm{ero}}}\geq 0.75$ because the
projected field becomes more discrete as the projection threshold is closer to
0.5 (Wang et al, 2011).
### 5.3 Infinitesimal size
It is recalled that the analytical and numerical methods are developed under
the condition of robustness which ensures that the manufactured design will
feature a good performance even if the blueprint design is uniformly thinned
or thickened during the manufacturing process. This condition dictates that
the structural members/cavities must be projected by at least one solid/void
element in the case of an eroded/dilated design.
To obtain the minimum size, the analytical method assumes that the size of the
structural members/cavities is infinitesimal ($\approx 0$) in the
eroded/dilated design, but in practice this does not occur. In topology
optimization, it is observed that the smallest structural members/cavities in
the eroded/dilated design are composed of one or a few elements, generally
described with intermediate densities. Thus, in practice, the minimum size of
the solid phase in the eroded design ($r_{\mathrm{min.Solid}}^{\mathrm{ero}}$)
results in a discrete number of elements. This can be seen in Fig. 10, which
is build using the numerical method. There, a filtered field and its eroded
projection under the robust condition are shown. It can be seen that the
minimum size of the eroded design is not infinitesimal, and therefore the
minimum size of the solid phase in the intermediate and dilated design would
be larger than the value predicted by the analytical method.
Figure 10: Illustration in which an infinitesimal size is not reached for the
eroded projection.
The minimum size of the solid and void phases in the eroded and dilated
designs ($r_{\mathrm{min.Solid}}^{\mathrm{ero}}$ and
$r_{\mathrm{min.Void}}^{\mathrm{dil}}$) are defined by the size of the
elements that discretize the design space. In addition, the amount of elements
with intermediate densities defining the minimum size depends on the steepness
of the smoothed Heaviside function ($\beta$), therefore, the error introduced
by assuming an infinitesimal size in the robust condition is related to the
two sources of error mentioned previously. To isolate the effect of the
infinitesimal size in the robust condition and illustrate the error introduced
by the assumption of infinitesimal size, we make use of the numerical method.
The numerical method is implemented in a discretized domain containing a large
number of elements ($10^{4}$ elements) and using a large filter size ($10^{3}$
elements) to simulate a continuous domain. The steepness parameter of the
smoothed Heaviside projection is set as $\beta=512$. Thus, the size and
density of the elements are excluded from the analysis. The effect of not
achieving an infinitesimal size in the condition of robustness is
intentionally imposed in the numerical code. For this, the robustness
condition is considered satisfied if
$r_{\mathrm{min.Solid}}^{\mathrm{ero}}=\alpha\>r_{\mathrm{fil}}$. Considering
that $r_{\mathrm{min.Solid}}^{\mathrm{ero}}$ represents one finite element in
topology optimization, and that representative values for $r_{\mathrm{fil}}$
are between 2 and 10 elements, it is reasonable to consider values for
$\alpha$ ranging from 0.1 to 0.5. Taking into consideration the above, we
build the graphical solutions that relate the minimum size in the solid phase
($r_{\mathrm{min.Solid}}^{\mathrm{int}}$), the filter radius and the erosion
threshold. The graph is shown in Fig. 11.
Figure 11: Effect of not reaching an infinitesimal size in the condition of
robustness $r_{\mathrm{min.Solid}}^{\mathrm{ero}}=0$. This condition is
assessed by imposing
$r_{\mathrm{min.Solid}}^{\mathrm{ero}}=\alpha\>r_{\mathrm{fil}}$.
The graph shows that the error of not achieving an infinitesimal size in the
eroded design produces a minimum size of the solid phase bigger than the value
predicted by the analytical method. The error related to the infinitesimal
size is low in comparison to the rounding error, so the latter would be the
most relevant source of error from the analytical method that assumes a
continuous design domain.
In summary, this section discussed the scope of the analytical method (Qian
and Sigmund, 2013) by comparing it with the numerical one (Wang et al, 2011),
both developed for a one-dimensional design domain. To this end, different
sources of error were examined. In general, the errors can be controlled by
mesh refinement or by correcting the projection thresholds according to the
cut-off $\varepsilon$ value. In the following section, the analytical method
is assessed using 2D-topology optimization problems.
## 6 Numerical examples and discussion
This section examines the reliability of the analytical expressions provided
in Table 1. To this end, a set of 2D topology optimization problems are
solved, from which the length scale is measured graphically and compared with
the imposed values. Then, some designs obtained with maximum size constraints
are provided to illustrate the use of the erosion and dilation distances.
Finally, this section provides a remark regarding the simplified robust
formulation, where the intermediate and dilated designs are removed from the
objective function.
### 6.1 Minimum length scale
The minimum length scale is assessed using the heat exchanger design problem
described in Section 2. A set of results is generated from this design
problem, which differ in the desired minimum length scale and in the
discretization used. Specifically, three sets of solutions are obtained by
discretizing the design domain into $100\times 100$, $200\times 200$ and
$400\times 400$ quadrilateral elements. In addition, three different length
scales are prescribed for each discretization, which are reported as the ratio
between the minimum size of the solid phase and the minimum size of the void
phase, i.e.
$r_{\mathrm{min.Solid}}^{\mathrm{int}}/r_{\mathrm{min.Volid}}^{\mathrm{int}}$.
The chosen ratios are $1/2$, $1/1$ and $2/1$. The minimum size in the solid
phase is the same in all scenarios and is defined as a physical dimension. In
number of finite elements, the radius that defines the minimum size of the
solid phase ($r_{\mathrm{min}}^{\mathrm{int}}$) is equal to 1, 2 and 4, for
the discretizations that use $100^{2}$, $200^{2}$ and $400^{2}$ elements,
respectively. It is well known that the initial values of design variables
have a huge influence on the resulting topology when it comes to thermal
compliance minimization (Yan et al, 2018), hence, in order to facilitate the
comparison of results, we impose a base structure as a starting point, which
is shown in Fig. 12. Before presenting the results, the procedure to obtain
the minimum length scale from the optimized designs is detailed.
Figure 12: Initial distribution of design variables considered for the thermal
compliance minimization problem.
The minimum size of the solid phase is measured graphically by counting the
number of finite elements that define the size of the thinnest structural
branch. Similarly, the minimum size of the void phase is measured by counting
the elements in the radius of the largest circumference that can be inscribed
at the re-entrant corners of the design. For example, consider the design of
Fig. 13 where the minimum length scales
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=\mathrm{min.Void}^{\mathrm{int}}=2$
elements are imposed. To determine the real minimum size of the void phase,
the largest circle in Fig. 13 that falls into the re-entrant corners of the
design is identified. The corners analyzed are those that form a sharp angle
between two structural branches. Fig. 13 shows three representative re-entrant
corners of the design depicted in Fig. 13. From there it is observed that the
minimum size is given by a circle of radius 2 elements (zone C). Similarly,
the largest region that fits into the thinnest structural members is
determined, as shown in Fig. 13. There, the minimum size of the solid phase is
given by a circle of radius 1.5 elements (zone D).
After describing the test case, the obtained results are presented. Table 2
contains the nine results generated in this example (3 length scales $\times$
3 discretizations). The imposed minimum length scales are reported graphically
next to each solution. The minimum size of the void phase is indicated in
blue, while the minimum size of the solid phase in magenta. The table also
reports the 3 parameters required to impose the desired minimum length scales,
i.e. $\eta_{\mathrm{ero}}$, $\eta_{\mathrm{dil}}$, and $r_{\mathrm{fil}}$.
These parameters have been obtained using the analytical method implemented in
the MATLAB code provided with this paper (SizeSolution.m). It is recalled that
all the examples assume $\eta_{\mathrm{int}}=0.5$.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Figure 13: Illustration of the minimum length scale measurement. In (a) the test regions and their radius sizes given in number of finite elements. In (b) the minimum size of the void phase, (c) the optimized heat exchanger, and (d) the minimum size of the solid phase. The imposed length scales are graphically shown in (c), at the upper left corner of the design. Mesh | $r_{\mathrm{min.Solid}}^{\mathrm{int}}/r_{\mathrm{min.Void}}^{\mathrm{int}}$
---|---
$1/2$ | $1/1$ | $2/1$
using [$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{dil}}$] = [0.65, 0.05] | using [$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{dil}}$] = [0.70, 0.30] | using [$\eta_{\mathrm{ero}}$, $\eta_{\mathrm{dil}}$] = [0.80, 0.42]
$100\times 100$ | | |
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=1$, $r_{\mathrm{fil}}=2.58$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=1$, $r_{\mathrm{fil}}=2.24$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=1$, $r_{\mathrm{fil}}=1.81$.
$200\times 200$ | | |
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=2$, $r_{\mathrm{fil}}=5.16$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=2$, $r_{\mathrm{fil}}=4.47$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=2$, $r_{\mathrm{fil}}=3.62$.
$400\times 400$ | | |
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=4$, $r_{\mathrm{fil}}=10.32$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=4$, $r_{\mathrm{fil}}=9.94$. | $r_{\mathrm{min.Solid}}^{\mathrm{int}}=4$, $r_{\mathrm{fil}}=7.24$.
Table 2: Optimized designs for the heat exchanger problem. The imposed volume
constraint is $V^{*}_{\mathrm{min}}=20\%$.
The measured minimum size of the solid and void phases are reported on Figs.
14 and 14, respectively. To construct these graphs, the measured minimum sizes
are normalized with respect to the desired minimum sizes, and the values are
placed in the ordinate coordinate. This is done for each discretization (which
determines the abscissa) and for each length scale. This procedure is also
carried out with results obtained from the numerical method (Wang et al,
2011), which is executed using representative discretizations for each case.
The values obtained from the numerical method are labeled as 1D in the graphs
of Fig. 14, while the values measured from the optimized designs are labeled
as 2D. The error bars in Fig. 14 illustrate the rounding error present in the
analytical method (see Section 5.1 for definition). Rounding error lines have
not been plotted in Fig. 14 because the length scale ratios have different
rounding errors.
As a general observation, we can point out from Fig. 14 that mesh refinement
reduces the error between the measured minimum size and the desired minimum
size (which is imposed through the set of parameters provided by the
analytical method). This is consistent with the observations made in the
previous section, where a continuous and uni-dimensional design domain was
used. In addition, the predicted error for the analytical method (the error
bars) are also consistent with the measured sizes, which validates the scope
and limitations of the analytical equations discussed in Section 5.
Regarding the graph in Fig. 14, it can be mentioned that in the nine 2D
designs, the error between the measured and desired minimum sizes is half a
finite element. This error is relatively large in the coarse discretization
(50$\%$ error), and small in the fine discretization (12.5$\%$ error).
However, despite the fact that the analytical method is not exact, it seems to
be accurate enough in the examples examined, since in all cases the error is
the same, half a finite element. It is interesting to note that the numerical
method that assumes a discrete 1D domain, estimates a minimum size that
differs by half a finite element with respect to the imposed value, which
matches the measured error. However, the numerical method provides a minimum
size of the solid phase larger than the measured one. This is probably due to
the fact that the condition of robustness refers to one finite element and not
to an infinitesimal size, as discussed in Section 5.3.
On the other hand, the graph built for the void phase (Fig. 14) shows a
different pattern. The measured minimum size of the void phase is often equal
to the desired one and even bigger for some designs. This could be explained
by the simple fact that the chosen test case does not present a geometric
singularity over the minimum size of the void phase, therefore it is not
possible to guarantee that the smallest re-entrant corner of the design will
indeed correspond to the minimum size imposed by the robust formulation. The
representative case might be that where $r_{\mathrm{min.Void}}^{\mathrm{int}}$
is chosen twice the size $r_{\mathrm{min.Solid}}^{\mathrm{int}}$ (ratio 1/2).
In such a case, the measured error corresponds to half a finite element
smaller than the desired value, that is, the same result as for the solid
phase.
(a)
(b)
(a)
(b)
Figure 14: The graphs summarize the measured minimum size of (a) the solid
phase and (b) the void phase from the designs in Table 2. For each
discretization, two set of results are provided, labeled as $2D$ and $1D$. The
$2D$ represents the graphical measurement normalized with respect to the
imposed value. The $1D$ represents the minimum size obtained from the
numerical method normalized with respect to the imposed value.
### 6.2 Erosion and dilation distances
In the following example we illustrate the use of the erosion and dilation
distances that are provided for a desired minimum length scale. To this end,
we use maximum size constraints, where the erosion and dilation distances are
essential information to impose a consistent length scale in the robust
formulation (Fernández et al, 2020). For the sake of completeness of the
manuscript, the non-linear force inverter is considered in this illustrative
example. The topology optimization problem of the non-linear force inverter
including maximum size constraint reads as follow :
$\begin{split}\text{min}\quad&\text{max}\left(\mathrm{c}({{\bm{\bar{\rho}}}_{\mathrm{ero}}}),\;\mathrm{c}({{\bm{\bar{\rho}}}_{\mathrm{int}}})\;,\mathrm{c}({{\bm{\bar{\rho}}}_{\mathrm{dil}}})\right)\\\
\text{s.t.:}\quad&\mathbf{v}^{\intercal}\>{{\bm{\bar{\rho}}}_{\mathrm{dil}}}\leq
V^{*}_{\mathrm{dil}}(V^{*}_{\mathrm{int}})\\\
&\mathrm{G_{ms}}({\bm{\bar{\rho}}}_{\mathrm{dil}})\leq 0\\\ &0\leq\rho_{i}\leq
1\;\;,\;\;i=1,...,N\;\;,\end{split}$ (21)
where $\mathrm{G_{ms}}$ is the maximum size constraint defined exactly as in
(Fernández et al, 2020). Therefore, $\mathrm{G_{ms}}$ represents a p-mean
aggregation function that gathers local maximum size restrictions. As in the
cited work, the $p$ aggregation exponent is set at 100.
Regarding the optimization parameters, these represent the implementation of
Wang et al (2014), i.e. the Heaviside parameter $\beta$ is initialized at 1
and is increased by 1 every 20 iterations until a value $\beta=16$ is reached.
Then, 20 more iterations are carried out with a $\beta=32$. The SIMP penalty
parameter is set at 3. We have found that this setting of parameters works
well for introducing maximum size restrictions into the non-linear force
inverter formulated under the robust design approach.
To impose minimum and maximum length scales, the maximum size constraint
$\mathrm{G_{ms}}$ is applied on the dilated design (Fernández et al, 2020). To
do so, the regions where the local maximum size constraints are applied must
be scaled according to the dilation distance. For instance, if the desired
maximum size in the intermediate design is defined by a circle of radius
$r_{\mathrm{max.Solid}}^{\mathrm{int}}$, the maximum size constraint should be
formulated for the dilated design using a circle of radius:
$r_{\mathrm{max.Solid}}^{\mathrm{dil}}=r_{\mathrm{max.Solid}}^{\mathrm{int}}+t_{\mathrm{dil}}$
(22)
Equation (22) explains the need of knowing the dilation distance when imposing
maximum size restrictions in the robust formulation. As mentioned previously,
this information can be easily obtained from the graphs generated by the
analytical equations. For example, in the following, the half force inverter
depicted in Fig. 1 is solved. The design domain is discretized into $200\times
100$ quadrilateral finite elements. The minimum size of the solid phase is set
as $r_{\mathrm{min.Solid}}^{\mathrm{int}}=2$ elements, while the maximum size
is set as $r_{\mathrm{max.Solid}}^{\mathrm{int}}=3$ elements. Two different
values for the minimum size of the void phase are chosen,
$r_{\mathrm{min.Void}}^{\mathrm{int}}=2$ and
$r_{\mathrm{min.Void}}^{\mathrm{int}}=3$ elements. The filter and projection
parameters used to impose the desired length scale are reported in Table 3. In
all cases, the volume constraint is set to $25\%$.
$r_{\mathrm{min.Void}}^{\mathrm{int}}$ | $r_{\mathrm{fil}}$ | $\eta_{\mathrm{ero}}$ | $\eta_{\mathrm{dil}}$ | $t_{\mathrm{dil}}$ | $t_{\mathrm{ero}}$
---|---|---|---|---|---
$2$ | $4.47$ | $0.70$ | $0.30$ | $1.03$ | $1.03$
$3$ | $4.47$ | $0.70$ | $0.11$ | $2.41$ | $1.03$
Table 3: Parameters to impose $r_{\mathrm{min.Solid}}^{\mathrm{int}}=2$ on the
design with $\beta=32$.
To obtain the dilation distance $t_{\mathrm{dil}}$, the projection thresholds
that define the minimum length scale must be defined. These values can be
graphically obtained from Fig. 6, or from the attached MATLAB code
(SizeSolution.m). For example, using this code, the sets of parameters in
Table 3 are obtained. The dilation distance is also reported there. For the
minimum sizes $r_{\mathrm{min.Void}}^{\mathrm{int}}=2$ and
$r_{\mathrm{min.Void}}^{\mathrm{int}}=3$ elements, the dilation distances are
$t_{\mathrm{dil}}=1$ and $t_{\mathrm{dil}}=2$ elements, respectively (the
numbers have been rounded to the nearest integer).
The results are shown in Figs. 15 and 16. Each figure reports 2 optimized
designs, which are obtained with and without maximum size restrictions. To
facilitate the visual comparison between results, the designs are placed with
respect to the symmetry axis and are reported in their deformed configuration.
The imposed length scale is also reported graphically next to each design. As
in the previous examples, the blue and magenta circles represent the minimum
size of the void and solid phase, respectively. The black circle represents
the maximum size desired for the intermediate design (which is imposed through
the dilated design)111The maximum size is actually imposed using an annular
region (Fernández et al, 2020), but for illustrative purposes, a circular
region is drawn..
Figure 15: Nonlinear force inverter without (upper half) and with (lower half)
maximum size constraints. The minimum length scales are
$r_{\mathrm{min.Solid}}^{\mathrm{int}}$ =
$r_{\mathrm{min.Void}}^{\mathrm{int}}$ = 2 elements. The maximum size is
$r_{\mathrm{max.Solid}}^{\mathrm{int}}=3$ elements. Figure 16: Nonlinear force
inverter without (upper half) and with (lower half) maximum size constraints.
The minimum length scales are $r_{\mathrm{min.Solid}}^{\mathrm{int}}$ = 2
elements and $r_{\mathrm{min.Void}}^{\mathrm{int}}$ = 3 elements. The
prescribed maximum size is $r_{\mathrm{max.Solid}}^{\mathrm{int}}=3$ elements.
The results are consistent with the imposed length scales. In the 4 designs
reported in Figs. 15 and 16, the minimum size of the solid and void phases are
met with a half-finite element of error (which agrees with the analytical
rounding error). Regarding the maximum size, this also matches the imposed
value. However, due to the inherent drawbacks of the aggregation function and
the strong non-linearity of the force inverter design problem, there are local
regions where the imposed maximum size restriction is not met, such as the
horizontal bar in the design of Fig. 16.
This force inverter test case shows the usefulness of the proposed equations
and provided codes, since they allow to quickly obtain the filter and
projection parameters, and the dilation distance that allow to impose the
desired length scales.
### 6.3 Remark on the simplified robust formulation
Before concluding this manuscript we devote a discussion concerning the robust
formulation. It is widely known that the robust formulation of the
optimization problem can be simplified when the objective function is
monotonically dependent on the volume fraction. The most widespread case in
the literature is the compliance minimization problem, where the intermediate
and dilated fields have less compliance than the eroded design and therefore
can be removed from the objective function without compromising the robustness
of the formulation. In this case, the objective function is evaluated only for
the eroded design, with the intermediate and dilated fields remaining solely
for formulating design constraints. When the volume restriction is the only
constraint included in the optimization problem, the constraint can be
formulated using the intermediate design, assuming that this design field is
the one intended for manufacturing. However, it has been mentioned in the
literature that evaluating the volume restriction in the dilated design
promotes convergence to better optimums since numerical instabilities are
prevented. In this section we add another reason that has not been mentioned
so far (to the best of the authors’ knowledge). If the dilated design is not
included in the volume restriction, then it is not possible to ensure the
control over the minimum size of the void phase. To explain this statement,
consider the following two robust topology optimization formulations for the
thermal compliance minimization problem:
P.IP.II $\overbrace{\hskip 85.35826pt}\hskip 11.38109pt\overbrace{\hskip
105.27519pt}$ $\displaystyle{\min_{\bm{\rho}}}$ $\displaystyle\quad
c({{\bm{\bar{\rho}}}_{\mathrm{ero}}})$ $\displaystyle{\min_{\bm{\rho}}}$
$\displaystyle\quad c({{\bm{\bar{\rho}}}_{\mathrm{ero}}})$
$\displaystyle\mathrm{s.t.:}$
$\displaystyle\quad\mathbf{v}^{\intercal}{{\bm{\bar{\rho}}}_{\mathrm{int}}}\leq
V^{*}_{\mathrm{int}}\>,$ $\displaystyle\mathrm{s.t.:}$
$\displaystyle\quad\mathbf{v}^{\intercal}{{\bm{\bar{\rho}}}_{\mathrm{dil}}}\leq
V^{*}_{\mathrm{dil}}(V^{*}_{\mathrm{int}})$ (23) $\displaystyle\quad
0\leq{\rho_{i}}\leq 1$ $\displaystyle\quad 0\leq{\rho_{i}}\leq 1$
The two optimization problems, P.I and P.II in Eq. (6.3), are formulated under
the robust design approach, but P.I evaluates the volume constraint directly
in the intermediate design while P.II does it through the dilated design,
whose upper bound $V^{*}_{\mathrm{dil}}$ is scaled according to
$V^{*}_{\mathrm{int}}$. From Eq. (6.3) it is clear that
P.I$({{\bm{\bar{\rho}}}_{\mathrm{ero}}},{{\bm{\bar{\rho}}}_{\mathrm{int}}})$
and
P.II$({{\bm{\bar{\rho}}}_{\mathrm{ero}}},{{\bm{\bar{\rho}}}_{\mathrm{int}}},{{\bm{\bar{\rho}}}_{\mathrm{dil}}})$,
or alternatively, P.I$({\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}})$ and
P.II$({\eta_{\mathrm{ero}}},{\eta_{\mathrm{int}}},{\eta_{\mathrm{dil}}})$.
Therefore, P.I is not influenced by the dilation threshold nor the dilated
design.
We recall that the condition of robustness imposed for the void phase involves
the dilated design, which has to project a cavity with at least one void
element to be present in the 3 fields that constitute the robust formulation.
The influence of the dilated design on the minimum size of the void phase can
be seen graphically in Fig. 6(a) by considering a fixed erosion threshold and
different dilation thresholds. For example, for the set
$[{\eta_{\mathrm{dil}}}$, ${\eta_{\mathrm{ero}}}]$ consider the points
$[0.14\>,\>0.60]$ and $[0.40\;,\;0.60]$. The first point corresponds to a
length scale where the minimum size of the void phase is equal to that of the
solid phase, while the second point imposes the size of the void phase twice
that of the solid phase. In the following, problems P.I and P.II are solved
using the two sets of thresholds indicated previously. To help to see the
differences, the volume fraction ratio has been increased to $0.3$. These sets
of parameters are resumed in Table 4. The results are summarized in Fig. 17.
Clearly, the results obtained using P.II show a length scale consistent with
the expected one, as the design in Fig. 17(a) features bigger reentrant
corners than the design in Fig. 17(b). However, for any value of
${\eta_{\mathrm{dil}}}$, the result from P.I is always the same and
corresponds to that shown in Fig. 17(d). The interpretation that can be made
of the P.I problem is that it uses a dilation threshold equal to the
intermediate one, i.e. ${\eta_{\mathrm{dil}}}={\eta_{\mathrm{int}}}=0.5$. This
can be corroborated by solving P.II with the set
$[{\eta_{\mathrm{dil}}}\>,\>{\eta_{\mathrm{ero}}}]$ equal to $[0.50,0.60]$,
whose result is shown in Fig. 17(c).
$r_{\mathrm{min.Solid}}^{\mathrm{int}}$ | $r_{\mathrm{min.Void}}^{\mathrm{int}}$ | $r_{\mathrm{fil}}$ | $\eta_{e}$ | $\eta_{d}$
---|---|---|---|---
$4$ | $0$ | $12.65$ | $0.60$ | $0.50$
$4$ | $4$ | $12.65$ | $0.60$ | $0.40$
$4$ | $8$ | $12.65$ | $0.60$ | $0.14$
Table 4: Parameters to compare the robust formulation and its simplified form.
(a) P.II, ${\eta_{\mathrm{dil}}}=0.40$.
(b) P.II, ${\eta_{\mathrm{dil}}}=0.14$.
(c) P.II, ${\eta_{\mathrm{dil}}}=0.50$.
(d) P.I for any ${\eta_{\mathrm{dil}}}$.
Figure 17: Heat exchanger design problem using two different variations of the
robust formulation. P.I evaluates the volume constraint directly in the
intermediate design, while P.II does it through the dilated design. Here,
${\eta_{\mathrm{ero}}}=0.60$ and $r_{\mathrm{min.Solid}}^{\mathrm{int}}=4$
elements.
## 7 Conclusion
The robust topology optimization formulation based on uniform manufacturing
errors has gained increasing acceptance in the topology optimization
community. This is mainly due to its ability to control the minimum size of
both the solid and void phases, and its potential to be combined with other
topology optimization approaches. Despite the increasing popularity of the
formulation, no method was yet available to easily obtain the filter and
projection parameters that produce the desired minimum length scales. This
need encouraged us to further develop the analytical method proposed by Qian
and Sigmund (2013). The scope and limitations of this method were assessed
using the numerical method of Wang et al (2011) and a set of 2D design results
from two topology optimization problems, the thermal compliance minimization
problem and the non-linear force inverter.
In addition to providing a fast and effective way to obtain the parameters
that impose the desired minimum length scale, this work shows that to obtain
simultaneous control over the minimum sizes of the solid and void phases, it
is necessary to involve the 3 fields (eroded, intermediate and dilated) in the
robust topology optimization problem. For example, for the compliance
minimization problem subject to a volume restriction, it is known that
intermediate and dilated designs can be excluded from the objective function,
but the volume restriction has to be applied to the dilated design in order to
involve all 3 designs in the formulation.
## 8 Replication of results
This manuscript contains two MATLAB codes as supplementary material. The first
is called SizeSolution.m and provides a list of filter and projection
parameters that impose user defined minimum length scales. The second is
called NumericalSolution.m and builds the graphs in Fig. 6 using the numerical
method proposed by Wang et al (2011).
###### Acknowledgements.
The authors acknowledge the research project FAFIL (Fabrication Additive par
Dépôt de Fil), funded by INTERREG and the European Regional Development Fund
(ERDF).
## Conflict of interest
On behalf of all authors, the corresponding author states that there is no
conflict of interest.
## References
* Andreasen et al (2020) Andreasen CS, Elingaard MO, Aage N (2020) Level set topology and shape optimization by density methods using cut elements with length scale control. Structural and Multidisciplinary Optimization pp 1–23
* Bendsøe and Kikuchi (1988) Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2):197–224
* Bendsøe (1989) Bendsøe M (1989) Bendsoe, m.p.: Optimal shape design as a material distribution problem. structural optimization 1, 193-202. Structural Optimization 1:193–202
* Bourdin (2001) Bourdin B (2001) Filters in topology optimization. International journal for numerical methods in engineering 50(9):2143–2158
* Bruns and Tortorelli (2001) Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Computer methods in applied mechanics and engineering 190(26-27):3443–3459
* Chen and Chen (2011) Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Structural and Multidisciplinary Optimization 44(1):1–18
* Christiansen et al (2015) Christiansen R, Lazarov B, Jensen J, Sigmund O (2015) Creating geometrically robust designs for highly sensitive problems using topology optimization: Acoustic cavity design. Structural and Multidisciplinary Optimization 52:737–754
* Clausen and Andreassen (2017) Clausen A, Andreassen E (2017) On filter boundary conditions in topology optimization. Structural and Multidisciplinary Optimization 56(5):1147–1155
* da Silva et al (2019) da Silva GA, Beck AT, Sigmund O (2019) Topology optimization of compliant mechanisms with stress constraints and manufacturing error robustness. Computer Methods in Applied Mechanics and Engineering 354:397 – 421
* Fernández et al (2020) Fernández E, Yang Kk, Koppen S, Alarcón P, Bauduin S, Duysinx P (2020) Imposing minimum and maximum member size, minimum cavity size, and minimum separation distance between solid members in topology optimization. Computer Methods in Applied Mechanics and Engineering 368:113,157
* Fernández et al (2021) Fernández E, Ayas C, Langelaar M, Duysinx P (2021) Topology optimization for large-scale additive manufacturing: Generating designs tailored to the deposition nozzle size (Under Review.)
* Kumar and Fernández (2021) Kumar P, Fernández E (2021) A numerical scheme for filter boundary conditions in topology optimization on regular and irregular meshes (Under Review, arXiv:2101.01122v1)
* Lazarov and Sigmund (2011) Lazarov BS, Sigmund O (2011) Filters in topology optimization based on helmholtz-type differential equations. International Journal for Numerical Methods in Engineering 86(6):765–781
* Pedersen and Allinger (2006) Pedersen C, Allinger P (2006) Industrial Implementation and Applications of Topology Optimization and Future Needs, vol 137, Springer, pp 229–238
* Pellens et al (2018) Pellens J, Lombaert G, Lazarov B, Schevenels M (2018) Combined length scale and overhang angle control in minimum compliance topology optimization for additive manufacturing. Structural and Multidisciplinary Optimization
* Qian and Sigmund (2013) Qian X, Sigmund O (2013) Topological design of electromechanical actuators with robustness toward over-and under-etching. Computer Methods in Applied Mechanics and Engineering 253:237–251
* Sigmund (1997) Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Journal of Structural Mechanics 25(4):493–524
* Sigmund (2009) Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mechanica Sinica 25(2):227–239
* Sigmund and Maute (2013) Sigmund O, Maute K (2013) Topology optimization approaches. Structural and Multidisciplinary Optimization 48(6):1031–1055
* Silva et al (2020) Silva G, Beck A, Sigmund O (2020) Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. Computer Methods in Applied Mechanics and Engineering 365:112,972
* Wang et al (2011) Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization 43(6):767–784
* Wang et al (2011b) Wang F, Jensen J, Sigmund O (2011b) Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. JOSA B 28:387–397
* Wang et al (2014) Wang F, Lazarov BS, Sigmund O, Jensen JS (2014) Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Computer Methods in Applied Mechanics and Engineering 276:453 – 472
* Xu et al (2010) Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Structural and Multidisciplinary Optimization 41(4):495–505
* Yan et al (2018) Yan S, Wang F, Sigmund O (2018) On the non-optimality of tree structures for heat conduction. International Journal of Heat and Mass Transfer 122:660–680
* Zhou et al (2011) Zhou M, Fleury R, Patten S, Stannard N, Mylett D, Gardner S (2011) Topology optimization-practical aspects for industrial applications. In: 9th World Congress on Structural and Multidisciplinary Optimization
* Zhu et al (2016) Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Archives of Computational Methods in Engineering 23:595–622
|
# Improving D-Optimality in Nonlinear Situations
Hana Sulieman
Department of Mathematics and Statistics
American University of Sharjah, P.O.Box 26666, Sharjah, U.A.E.
<EMAIL_ADDRESS>
###### Abstract
Experimental designs based on the classical D-optimal criterion minimize the
volume of the linear-approximation inference regions for the parameters using
local sensitivity coefficients. For nonlinear models, these designs can be
unreliable because the linearized inference regions do not always provide a
true indication of the exact parameter inference regions. In this article, we
apply the profile-based sensitivity coefficients developed by Sulieman et.al.
[12] in designing D-optimal experiments for parameter estimation in some
selected nonlinear models. Profile-based sensitivity coefficients are defined
by the total derivative of the model function with respect to the parameters.
They have been shown to account for both parameter co-dependencies and model
nonlinearity up to second order-derivative. This work represents a first
attempt to construct experiments using profile-based sensitivity coefficients.
Two common nonlinear models are used to illustrate the computational aspects
of the profile-based designs and simulation studies are conducted to
demonstrate the efficiency of the constructed experiments.
Keywords: D-optimality; Local sensitivity coefficient; Profile-based
sensitivity coefficient; Sequential design.
## 1 INTRODUCTION
Design of experiments has been a very active research area in many scientific
fields for the past two decades. For linear models, the theory of optimal
designs is well established in the literature and the properties of these
designs are fairly understood and used in various applications. On the other
hand, for nonlinear models, the theory of design optimality is still emerging
in the literature. The major difficulty when the underlying model is nonlinear
is that the optimal designs depend on the true value of the parameters. Hence,
poor estimates of the unknown parameter values generate poor designs.
Several design of experiment techniques have been developed and applied
successfully to wide range of model systems (Franceschini and Macchietto [8];
Berger and Wong [4]). The objectives of these techniques typically focus on
model precision or/and model discrimination. D-optimality is one of the most
popular design criteria used. The criterion minimizes the volume of the
linear-approximation inference regions for the parameters. The measure of
information content used in D-optimal designs involves local sensitivity
coefficients defined by the first-order derivative matrix of the model
function with respect to the parameters. Hence, the resulting designs are
termed locally optimal. Locally optimal designs can be unreliable for highly
nonlinear model functions. The works by Hamilton and Watts [10] and Vila and
Gauchi [17] represent examples of successful attempts to take into account the
model nonlinearity in the design formulation.
Sulieman et.al. [12, 13] proposed profile-based nonlinear sensitivity measure
which simultaneously accounts for model nonlinearity and parameter estimates
correlations. Applications of the measure to different models by Sulieman
et.al.[15] have shown that the measure gives more reliable reflection of the
sensitivity behavior of the model to the parameters than that given by the
local sensitivity measures. The profile-based sensitivity measure is defined
by the total derivative of the model function with respect to the parameter of
interest. Hence and like any derivative measure, it is inherently local, it
provides however, a broader picture of the model sensitivity in the presence
of parameter co-dependencies and model nonlinearity.
The primary goal of this article is to employ the profile-based sensitivity
information in the construction of D-optimal designs. The resulting designs
are compared with the classical local D-optimal designs in which the
conventional local sensitivity coefficients are used. In section 2 we give a
brief review of profile-based sensitivity measure and discuss its
characteristics. In Section 3 we construct the profile-based D-optimal design
and discuss its relations to the classical D-optimal design. Illustrative
model cases are presented in Section 4, and conclusions are summarized in
Section 5.
## 2 A brief Overview of Profile-based Sensitivity
Let us consider the general mathematical form of a single response nonlinear
regression model
${\bf y}={\bf f}({\bf X},\Theta)+\mbox{\boldmath$\epsilon$}$ (1)
where ${\bf y}$ is an $n$-element vector of observed values of the response
variable for particular values of the $m$-regressor variables ${\bf X}=\\{{\bf
x}_{1},{\bf x}_{2},\ldots,{\bf x}_{m}\\}$, each ${\bf x}_{i}$ is $n$-element
vector of experimental settings. $\Theta$ is a $k$-element vector of unknown
parameters, ${\bf f}$ is an $n$-element vector of predicted values of the
response variable for given ${\bf X}$ and $\Theta$, ${\bf f}({\bf
X},\Theta)=\\{f({\bf x}_{1},\Theta),f({\bf x}_{2},\Theta),\ldots f({\bf
x}_{m},\Theta)\\}$, and $\epsilon$ is an $n$-element vector of independent
random errors with a specified joint distribution. In most cases, including
the case here, $\epsilon$ is assumed to have a spherical normal distribution,
with $E(\mbox{\boldmath$\epsilon$})={\bf 0}$ and
$var(\mbox{\boldmath$\epsilon$})=E(\mbox{\boldmath$\epsilon$}\mbox{\boldmath$\epsilon$}^{\prime})=\sigma^{2}{\bf
I}$.
To emphasizes the dependence of the predicted response values on the
parameters $\Theta$, the above model is expressed as:
${\bf y}=\mbox{\boldmath$\eta$}(\Theta)+\mbox{\boldmath$\epsilon$}$ (2)
where the $j^{th}$ element of the $n$-dimensional vector
$\mbox{\boldmath$\eta$}(\Theta)=(\eta_{1}(\Theta),\eta_{2}(\Theta)\\\
,\ldots,\eta_{n}(\Theta))^{\prime}$ is given by
$\eta_{j}(\Theta)=f({\bf x}_{j},\Theta)\ \ \ \ \ \ j=1,2,\ldots,n.$ (3)
Conventionally, sensitivity of model predictions to variation in parameter
values is measured by the first-order partial derivative of predicted response
function, $\mbox{\boldmath$\eta$}(\Theta)$, with respect to the parameters.
Sulieman et al.[12] proposed partitioning the $p$-element parameter vector
$\Theta$ into $\Theta=(\theta_{i},\Theta_{-i})$ and
$\mbox{\boldmath$\eta$}(\Theta)$ into
$\mbox{\boldmath$\eta$}(\theta_{i},\Theta_{-i})$ , where $\theta_{i}$ is the
parameter of interest for which sensitivity is measured. $\theta_{i}$ is
varied across a specified range of uncertainty so that for each fixed value of
$\theta_{i}$, the conditional estimates of the remaining parameters
$\Theta_{-i}$ are obtained. Accordingly a sensitivity measure is defined by
the total derivative of the predicted response with respect to $\theta_{i}$.
It is given by:
$p_{i}({\bf
x}_{0})=\frac{D\eta_{0}(\theta_{i},\Theta_{-i}(\theta_{i}))}{D\theta_{i}}$ (4)
where $\eta_{0}=f({\bf x}_{0},\Theta)$ is the predicted response at a selected
point ${\bf x}_{0}$ and the notation $\displaystyle\frac{D}{D\theta_{i}}$
means total derivative with respect to $\theta_{i}$. Based on the least
squares estimation criterion, equation (4) can be expressed as:
$p_{i}({\bf
x}_{0})=\frac{\partial\eta_{0}}{\partial\theta_{i}}-\frac{\partial\eta_{0}}{\partial\Theta_{-i}}(\frac{\partial^{2}S}{\partial\Theta_{-i}\partial\Theta_{-i}^{\prime}})^{-1}\frac{\partial^{2}S}{\partial\theta_{i}\partial\Theta_{-i}}\Bigr{|}_{\tilde{\Theta}_{-i}}$
(5)
where the function $S$ is the sum of squares function given by
$S(\Theta)=\sum_{j=1}^{n}(y_{j}-\eta_{j}(\Theta))^{2}$ and
$\tilde{\Theta}_{-i}$ is the conditional least squares estimate of
$\Theta_{-i}$ given $\theta_{i}$.
The first term in equation (5),
$\displaystyle\frac{\partial\eta_{0}}{\partial\theta_{i}}$ gives the
conventional sensitivity coefficient of the predicted response with respect to
$\theta_{i}$. The second term is a correction term that involves the marginal
effects of $\Theta_{-i}$ on $\eta_{0}$ weighted by the correlations among the
elements of $\tilde{\Theta}_{-i}$, and correlations between $\hat{\theta}_{i}$
and $\tilde{\Theta}_{-i}$. These parameter correlations are based on the
observed value of the Hessian matrix, $\displaystyle
H(\Theta)=\frac{\partial^{2}S(\Theta)}{\partial\Theta\partial\Theta^{\prime}}$.
$p_{i}({\bf x}_{0})$ is called Profile-based Sensitivity Coefficient after the
profiling algorithm used to assess the extent of nonlinearity in the model and
construct likelihood regions for parameter estimates (Bates and Watts[1]).
While $p_{i}({\bf x}_{0})$ is a local derivative-based measure, the
incorporation of the correlation structure, based on the Hessian terms above,
makes it a more reliable measure of sensitivity as it accounts for
simultaneous changes in the parameter values and nonlinearity of the model.
$p_{i}({\bf x}_{0})$ can be expressed in terms of the first and second order
derivative information of the model function $\mbox{\boldmath$\eta$}(\Theta)$
as:
$p_{i}({\bf x}_{0})={v}_{0_{i}}-{{\bf
v}}^{\prime}_{0_{-i}}({V}_{-i}^{\prime}{V}_{-i}-[{e}^{\prime}][{V}_{-i-i}])^{-1}({V}_{-i}^{\prime}{{\bf
v}}_{i}-{V_{-ii}}^{\prime}{e})$ (6)
where ${v}_{0_{i}}$ is the $i^{th}$ component of the $k$-element first order
derivative vector ${\bf
v}_{0}=\displaystyle\frac{{\partial\eta_{0}}(\Theta)}{\partial\Theta}$
evaluated at ${\bf x}_{0}$; $V_{-i}$ is an $n\times(k-1)$ matrix consisting of
first derivative vectors of $\mbox{\boldmath$\eta$}(\Theta)$ with respect to
$\Theta_{-i}$; ${{\bf v}}_{0_{-i}}$ is a $(k-1)$ dimensional vector consisting
of the elements in the row of ${V}_{-i}$ that corresponds to ${\bf x}_{0}$;
$V_{-i-i}$ is the $n\times(k-1)\times(k-1)$ array of the second order
derivatives of $\mbox{\boldmath$\eta$}(\Theta)$ with respect to $\Theta_{-i}$;
$V_{-ii}$ is the $n\times(k-1)$ matrix of the second derivatives of
$\mbox{\boldmath$\eta$}(\Theta)$ with respect to $\Theta_{-i}$ and
$\theta_{i}$, and $e$ is the $n$-element residuals vector. In the $k-1$
parameter subspace, the entries of $[{e}^{\prime}][{V}_{-i-i}]$ give the
projections of the second-order derivative vectors on the residual vector.
Similarly the matrix ${V_{-ii}}^{\prime}{e}$ gives the model function
curvature in $\theta_{i}$ direction projected on the residual vector. Since
the residual vector is orthogonal to the tangent plane spanned by the $k-1$
local sensitivity vectors, $[{e}^{\prime}][{V}_{-i-i}]$ is a function of only
the intrinsic curvature portion of the second-order derivative array. For
significant intrinsic model nonlinearity these projections play major role in
the extent to which profile-based nonlinear sensitivities differ from the
local linear sensitivities. The quantities in equation (6) are evaluated at
$(\hat{\theta}_{i},\tilde{\Theta}_{-i}(\hat{\theta}_{i}))$.
In vector notations, equation (6) can be expressed as:
${\bf p}_{i}={\bf
v}_{i}-V_{-i}({V}_{-i}^{\prime}{V}_{-i}-[{e}^{\prime}][{V}_{-i-i}])^{-1}({V}_{-i}^{\prime}{\bf
v}_{i}-{{V^{\prime}}_{-ii}}{e})$ (7)
where ${\bf p}_{i}$ is $n\times 1$ vector containing profile-based sensitivity
coefficients for $\theta_{i}$ evaluated at the $n$ prediction points, ${\bf
v}_{i}$ is the corresponding vector of local sensitivity coefficients. If the
linear approximation to the model function is adequate, i.e., Hessian terms
can be set to zero or when the model fits data exactly ($e=0$), the vector of
profile-based sensitivity coefficients in equation (7) reduces to the
following:
${\bf p}_{i}={\bf
v}_{i}-V_{-i}({V}_{-i}^{\prime}{V}_{-i})^{-1}{V}_{-i}^{\prime}{\bf v}_{i}$ (8)
which can be expressed as:
${\bf p}_{i}=[\bf{I}_{n}-{\cal P}_{V_{-i}}]{\bf v}_{i}$ (9)
where $\bf{I}_{n}$ is the $n$-dimensional identity matrix and ${\cal
P}_{V_{-i}}=V_{-i}({V}_{-i}^{\prime}{V}_{-i})^{-1}{V}_{-i}^{\prime}$ is the
projection matrix that orthogonally projects the columns of $V_{-i}$ onto
themselves. The projection matrix $\bf{I}_{n}-{\cal P}_{V_{-i}}$ projects the
columns of $V_{-i}$ onto the orthogonal complement of the space spanned by the
columns of $V_{-i}$. A close examination of the formulation given in equation
(9) reveals that ${\bf p}_{i}$ is the vector of least squares residuals
obtained from linearly regressing ${\bf v}_{i}$ on regressor variables given
by $V_{-i}$. Note that the vector ${\bf v}_{i}$ can be orthogonally decomposed
as:
${\bf v}_{i}={\cal P}_{V_{-i}}{\bf v}_{i}+[\bf{I}_{n}-{\cal P}_{V_{-i}}]{\bf
v}_{i}$ (10)
If the local effect of $\theta_{i}$ on the predicted response is highly
correlated with the effects of the remaining parameters, $\Theta_{-i}$, the
first term in equation (10) will have large magnitude compared to the second
term indicating small magnitude of profile-based sensitivities. On the other
hand, weak correlations among the effects of parameters on the predicted
response indicate large magnitude of profile-based sensitivities. This is to
say that $\bf{p}_{i}$ measures the influence that $\theta_{i}$ exerts on the
predicted response after the removal of its co-dependencies with the remaining
parameters. With this understanding, profile-based sensitivities represent a
particular orthogonalization of parameter space so that the individual impacts
of the resulting parameters are more independent from each other than those of
original parameters $\Theta$. Inclusion of the second order derivatives in
Equation (7) suggests that model nonlinearity is accounted for in this
particular orthogonalization.
Sulieman et. al.[16] described profile-based sensitivity procedure using the
notion of model re-parametrization. They showed that the slopes of the profile
traces in the re-parametrized model represent the foundation for the
underlying definition of the profile-based sensitivity measure. Sulieman et
al.[13] extended the profile-based sensitivity assessment to the parameter
estimation in multi-response regression models. Sulieman et al.[15] presented
a comparative analysis of the profile-based sensitivity and Fourier Amplitude
Sensitivity Test (FAST). They showed that while FAST accounts for model
nonlinearity to all orders it fails to account for parameter co-dependencies
which are considered in the profile-based sensitivity measure.
## 3 PROFILE-BASED D-OPTIMAL DESIGN
D-optimal designs are one of the most commonly used alphabet designs. A
D-optimal design minimizes the volume of the parameter joint inference region
or equivalently maximizing the determinant of the Fisher Information matrix
with respect to the design settings. Box and Lucas [5] gave the first
formulation and geometric interpretation of the D-optimal design for nonlinear
models. They defined the D-optimality objective function, using the local
sensitivity coefficients, as:
$maxD=|V_{0}^{\prime}V_{0}|$ (11)
with respect to design settings, ${\bf x}$, where the matrix of local
sensitivity coefficients $V_{0}$ is evaluated at an initial parameter
estimates $\Theta_{0}$. Under the linear approximation, the model response
surface is replaced by its tangent plane and the usual ellipsoidal joint
inference region for $\Theta$ is the image in parameter space of a spherical
region on the tangent plane. The volume of the ellipsoidal region evaluated at
$\Theta_{0}$ is given by $|V_{0}^{\prime}V_{0}|^{-1/2}$. By maximizing $D$,
this volume is minimized. Thus, for a given nonlinear model response, the
D-optimal criterion ensures that the design is such that large regions on the
tangent plane map into small regions in the parameter space.When model
nonlinearity is pronounced, the local D-optimality can produce designs with
poor performance and little information about parameters. Hamilton and Watts
[10] introduced quadratic designs based on second-order approximation to the
volume of the inference region of $\Theta$. Quadratic designs have the
distinct advantage of taking into account the nonlinearity of response
function. Benabbas et al.[3] proposed a curvature-based method for optimal
experimental design for parameter estimation in multi-response nonlinear
dynamic models. Vila and Gauchi[17] constructed non-sequential optimal designs
based on the expected volume of exact parameter confidence regions. These
designs generally result in repeated experiments on $k$-support points for
models with $k$ parameters and tend to reduce parameter nonlinearities. Gao
and Zhou[9] developed a nonlinear D-optimal design criterion based on the
second-order least squares estimator for regression models with asymmetric
error distribution.
Using the profile-based sensitivity coefficients defined in equation (7), the
profile-based D-optimality can be defined as maximizing:
$maxD_{P}=|P_{0}^{\prime}P_{0}|$ (12)
with respect to ${\bf x}$, where the matrix $P=\left[{\bf p}_{1}{\bf
p}_{2}\ldots{\bf p}_{k}\right]$ is evaluated at $\Theta_{0}$, i.e., each
element ${\bf p}_{i}$ is evaluated at $\Theta_{0}$.
As discussed in the previous section, profile-based sensitivity coefficients
orthogonalize the parameter space so that the resulting parameters are less
correlated than the original parameters. By maximizing $D_{p}$, the volume of
the inference region in the less-correlated parameter space is minimized.
Hence, the resulting design produce more precise and less correlated parameter
estimates than the corresponding local $D$-optimal design.
The $(i,j)^{th}$ element of the $k\times k$ matrix $P^{\prime}P$ is given by:
${\bf p}_{i}^{\prime}{\bf p}_{j}={\bf v}_{i}^{\prime}{\bf v}_{j}-{\bf
v}_{i}^{{}^{\prime}}V_{-j}H^{-1}_{-j-j}{\bf h}_{-jj}-{\bf
h}^{{}^{\prime}}_{-ii}H^{-1}_{-i-i}V^{{}^{\prime}}_{-i}{\bf v}_{j}+{\bf
h}^{{}^{\prime}}_{-ii}H^{-1}_{-i-i}V^{{}^{\prime}}_{-i}V_{-j}H^{-1}_{-j-j}{\bf
h}_{-jj}$ (13)
where
$H_{-i-i}={V}_{-i}^{{}^{\prime}}{V}_{-i}-[{e}^{{}^{\prime}}][{V}_{-i-i}]$ is
$k-1$ square matrix containing Hessian terms corresponding to the parameters
$\Theta_{-i}$, ${\bf h}_{-ii}={V}_{-i}^{\prime}{\bf
v}_{i}-{{V^{\prime}}_{-ii}}{e}$ is $k-1$-element vector containing Hessian
terms corresponding to $\theta_{i}$ and $\Theta_{-i}$. Similarly, the matrix
$H_{-j-j}$ and vector ${\bf h}_{-jj}$ contain the corresponding Hessian terms
for $\Theta_{-j}$ and $\theta_{j}$. The first term in equation (13) is equal
to the $(i,j)^{th}$ element of the matrix $V_{0}^{\prime}V_{0}$ used in the
local D-optimality criterion. The remaining terms are proportionate to
pairwise products of various co-dependency structures among the conditional
parameter estimates in $\Theta_{-i}$ and those in $\Theta_{-j}$ in addition to
the corresponding co-dependencies with the conditioning parameters
$\theta_{i}$ and $\theta_{j}$. Thus, the objective function in
$D_{P}$-optimality is equal to the objective function in the conventional
$D$-optimality corrected for the correlation structures among parameters.
Equation (13) suggests that maximum $D_{P}$ is obtained by maximizing $D$
(first term) and minimizing the magnitudes of the middle two terms
representing correlation structures. This suggests that $D_{P}$ maximizes the
information content of the design while accounting for the correlations among
parameter estimates that result from an existing design or/and model
formulations. These correlations are ignored by the $D$-optimal criterion.
Similar to the $D$-optimal designs, the $D_{P}$-optimal designs are invariant
under nonsingular transformation of parameters.
The ability of the $D$-optimal design to estimate model parameters relative to
the $D_{P}$-optimal design is measured by its $D$-efficiency. The
$D$-efficiency is defined by:
$D_{eff}=(\frac{|V_{0}^{\prime}V_{0}|}{|P_{0}^{\prime}P_{0}|})^{\frac{1}{k}}\times
100\%$ (14)
The $D_{eff}$ gives the percentage of the experimental effort of the
$D$-optimal design required by $D_{p}$-optimal design in order to produce
parameter estimates of the same precision.
## 4 Model Examples and Simulations
To carry out the computations for constructing the optimal $D$ and $D_{p}$
designs, we initially evaluate the two criteria at a selected set of fine grid
points in the design region using initial parameter values, $\Theta_{0}$. Then
we performed a minimization algorithm for each of $D^{-1}$ and $D_{p}^{-1}$
using one of the two multivariable nonlinear minimization routines found in
MATLAB’s Optimization Toolbox: fminsearch for unconstrained optimization and
fmincon for constrained optimization. The starting point ${\bf x}_{0}$ used
for the minimization is the point at which $D$ and $D_{p}$ are maximized in
the initial grid search. To ensure that the resulting optimal design point is
global one, a further grid search exploring the interior of the design region
is carried out. A simulation study is conducted for each model example in
order to evaluate the performance of the constructed designs.
### 4.1 Michaelis-Menten Enzyme Kinetic Model
The Michaelis-Menten model is one of the most widely used models in the
biological sciences. It is commonly used in enzymatic kinetics with well-known
formulation:
${\bf y}=\frac{\theta_{1}x}{\theta_{2}+x}+\mbox{\boldmath$\epsilon$}$ (15)
where $y$ is the measured initial velocity of an enzymatic reaction and $x$ is
the substrate concentration. The unknown parameters $\theta_{1}$ and
$\theta_{2}$ represent maximum conversion rate and Michaelis-Menten constant,
respectively.
One of the most widely data sets used to fit the model is published in Bates
and Watts[1] representing reaction velocity measurements with enzyme treated
with Puromycin and with untreated enzyme. Table 1 depicts the design and
velocity values for the treated enzyme.
Table 1: Data set for the Michaelis-Menten model reported in Bates & Watts [1]
Observation no. Substrate Concentration(ppm) Velocity (Treated)(counts/min2) 1
0.02 76 2 47 3 0.06 97 4 107 5 0.11 123 6 139 7 0.22 159 8 152 9 0.56 191 10
201 11 1.10 207 12 200
${\theta}_{2}$${\theta}_{1}$
Figure 1: $90\%$ joint likelihood regions for the parameter in the Michaelis-
Menten model based on unconditional least squares estimation (solid line) and
conditional least squares estimations (dashed-dotted line). The unconditional
least squares estimate ($\hat{\theta}_{1}=212.68$, $\hat{\theta}_{2}=0.064$)
is indicated by x.
This original design consists of six different experimental settings, each
with two replications. We used the data to construct the joint inference
regions for the parameter estimates of $\theta_{1}$ and $\theta_{2}$,
$\hat{\theta}_{1}$ and $\hat{\theta}_{2}$. Figure 1 depicts the 90% likelihood
inference regions for the two parameters based on unconditional and
conditional least squares estimation of the model. The unconditional
likelihood contour (solid line) was determined by evaluating the sum of
squares function for an arbitrary grid of ($\hat{\theta}_{1}$,
$\hat{\theta}_{2}$) values in their respective ranges of uncertainty. Whereas,
for the conditional likelihood contour (dashed-dotted line) the sum of squares
function was evaluated using a grid points ($\tilde{\theta}_{1}(\theta_{2})$,
$\tilde{\theta}_{2}(\theta_{1})$) where $\tilde{\theta}_{1}(\theta_{2})$ is
the estimate of $\theta_{1}$ conditional on selected values of ${\theta}_{2}$
in its range of variation and $\tilde{\theta}_{2}(\theta_{1})$ is the least
squares estimate of $\theta_{2}$ conditional on selected values of
${\theta}_{1}$ in its range of variation.
It is evident from Figure 1 that the 90% conditional likelihood inference
region is less elliptical with reduced inclination reflecting decreased
correlation between the two parameters. The conditional estimation procedure
reduces the correlations induced by the simultaneous search for optimal least
squares estimates in $\theta_{1}$ and $\theta_{2}$ directions. Hence, the
$D_{p}$-optimality provides a basis for designing experimental settings that
produce less correlated parameter estimates.
#### 4.1.1 Starting Design
Using the local D-optimal criterion in equation (11), Bates and Watts[1]
constructed a starting design for the model in equation (15). They showed that
the design does not depend on the conditionally linear parameter $\theta_{1}$
and used $\theta_{2}^{0}=0.1$ as starting value of $\theta_{2}$. The maximum
$D$ occurred at $x_{1}=1.1$ and $x_{2}=0.085$ where the value 1.1 is the
maximum concentration reported in the original design given in Table 1 and
0.085 is nearly the half-concentration in the same design. They compared their
design to the original design and concluded that their design produced smaller
linear approximation inference region for $\theta_{1}$ and $\theta_{2}$ with
lower correlation between the parameter estimates.
${\theta}_{2}$${\theta}_{1}$
Figure 2: $95\%$ approximate confidence region for three designs for
estimating the Michaelis-Menten model parameters ${\theta}_{1}$ and
${\theta}_{2}$. The largest region (solid line) represents the original design
(Table 1), the middle region (dotted line) represents the $D$\- optimal design
and the smallest region (dashed line) represents the $D_{p}$-optimal design
In constructing the starting $D_{P}$ design, we set the unknown residual
vector in equation (7) to zero and so the ${\bf p}_{i}$ vectors are reduced to
the form given in equation (9). We write the model equation (15) as
$f(x,\Theta)=\theta_{1}g(x,\theta_{2})$ where
$g(x,\theta_{2})=\displaystyle\frac{x}{\theta_{2}+x}$. For notational
convenience, we write $g(x,\theta_{2})=g$ and so $f(x,\Theta)=\theta_{1}{\bf
g}$. It can be shown that the $2$-element vectors of profile-based sensitivity
of $\theta_{1}$ and $\theta_{2}$ are given by:
${\bf p}_{1}=[I_{2}-{\bf g}_{\theta_{2}}({\bf g}^{\prime}_{\theta_{2}}{\bf
g}_{\theta_{2}})^{-1}{\bf g}^{\prime}_{\theta_{2}}]{\bf g}_{\theta_{2}}$ (16)
${\bf p}_{2}=\theta_{1}[I_{2}-{\bf g}({\bf g}^{\prime}{\bf g})^{-1}{\bf
g}^{\prime}]{\bf g}\propto[I_{2}-{\bf g}({\bf g}^{\prime}{\bf g})^{-1}{\bf
g}^{\prime}]{\bf g}$ (17)
where ${\bf g}=(g(x_{1},\theta_{2}),g(x_{2},\theta_{2}))^{\prime}$ and ${\bf
g}_{\theta_{2}}=\displaystyle(\frac{\partial{g(x_{1},\theta_{2})}}{\partial\theta_{2}},\frac{\partial{g(x_{2},\theta_{2})}}{\partial\theta_{2}})^{\prime}$.
It is obvious that ${\bf p}_{1}$ is free of the conditionally linear parameter
$\theta_{1}$ and that $\theta_{1}$ appears linearly in ${\bf p}_{2}$ as a
proportion constant to a term that depends on $\theta_{2}$ only. This implies
that the optimum value of $D_{P}$ is independent of $\theta_{1}$. As shown by
Bates and Watts[1],the classical $D$-optimal designs are independent of the
conditionally linear parameters for most nonlinear models. This property holds
also true for the $D_{P}$-optimal designs only when models are intrinsically
linear (${V}_{-i-i}$ and ${V}_{-ii}$ can be set to zero) or when data fits the
model perfectly ($e$ is zero).
Using MATLAB 8.2.0 optimizer, the locations of maximum $D_{P}$ were found at
$x_{1}=1.1$ and $x_{2}=0.056$. In comparing the resulting $D_{P}$-optimal
design to the $D$-optimal design constructed by Bates and Watts[1], we places
six replications at $x_{1}=1.1$ and six replications at $x_{2}=0.056$. The
corresponding $D$-optimal design consists of six replications of $x_{1}=1.1$
and six replications of $x_{2}=0.085$. As shown in Table 1, the original
design consists of six design points, each has two replications. Figure 2
shows the $95\%$ linear approximation confidence regions for $\theta_{1}$ and
$\theta_{2}$ using the three designs assuming that all designs produced same
parameter estimates and residual variance.
We can clearly see that the $D_{P}$-optimal design produces the smallest joint
confidence region (dashed line) and smaller confidence intervals. Also the
correlation between the two parameter estimates is the lowest (0.65) for the
$D_{P}$-optimal design compared to 0.68 for the $D$-optimal design and 0.76
for the original design. The $D$-efficiency is $95\%$ indicating that $95\%$
of the optimal $D$ experimental effort is needed by the optimal $D_{P}$ design
in order to produce similar accuracy of parameter estimates. In other words,
the $D$ optimal design requires $5\%$ more of experimental effort in order to
obtain as accurate parameter estimates as that given by the $D_{P}$ design.
#### 4.1.2 Sequential Design
When some experiments are already done, a natural way of designing an
experiment is to use a sequential method. Sequential designs are appealing
because they offer the chance to change strategy after the first round of
experiments has been completed and new information is available. Of particular
interest is the case when one additional design point is desired. In
$D$-optimality, this is achieved by maximizing $|V_{n+1}^{\prime}V_{n+1}|$
with respect to the $(n+1)^{th}$ design point, $\bf{x}_{n+1}$, where
$V_{n+1}=\left[\begin{array}[]{c}V_{n}\\\ {\bf v}_{n+1}\end{array}\right]$
(18)
where $V_{n}$ is the design matrix consisting of the local sensitivity
measures for the pre-existing experimental settings and ${\bf v}_{n+1}$ is
$k$-element vector of sensitivity coefficients that correspond to the new
experimental setting being selected. The corresponding profile-based
sequential design strategy maximizes $|P_{n+1}^{\prime}P_{n+1}|$ where
$P_{n+1}=\left[\begin{array}[]{c}P_{n}\\\ {\bf p}_{n+1}\end{array}\right]$
(19)
and ${\bf p}_{n+1}$ is $k$-element vector of profile-based sensitivity
coefficients corresponding to the $(n+1)^{th}$ experimental setting. $V_{n+1}$
and $P_{n+1}$ are evaluated at the least squares parameter estimates from the
already existing $n$ experiments.
The original design for the Michaelis-Menten model given in Table 1 was used
to obtain the least squares estimates of $\theta_{1}$ and $\theta_{2}$. These
estimates were used to evaluate $V_{n+1}$ and $P_{n+1}$ above. The $13^{th}$
concentration point is generated using MATLAB 8.2.0 optimizer for restricted
$x$, $0<x\leq x_{max}=1.1$. The optimal value for the additional concentration
point is $x=0.0747$ when $D$ is maximized and $x=0.05116$ when $D_{P}$ is
maximized. With one additional design point, the efficiency for the new
$D_{P}$ design is $98\%$.
Simulation Study
In a simple attempt to evaluate the information content of the 13-point
design, formed by adding the new optimal value of $x$ to the existing design
in Table 1, the parameters $\theta_{1}$ and $\theta_{2}$ are re-estimated
twice: one time using the new design derived by $D$ criterion and the other
using the design derived by $D_{P}$ criterion. The response variable for the
$13^{th}$ optimal concentration point is simulated using the fitted model
obtained from the original 12-point design and adding normally distributed
random noise. The estimation procedure for each design was carried out for
2000 simulations. For each simulation, the estimated linear-approximation
based variance-covariance matrix, $s^{2}(V^{\prime}V)^{-1}$ is evaluated at
the least squares estimate $\hat{\Theta}$ and recorded.
Index$corr_{D}$$corr_{D_{P}}$
Figure 3: Simulated correlation coefficients between $\hat{\theta}_{1}$ and
$\hat{\theta}_{2}$ in Michaelis-Menten Model using 13-points sequential design
resulted from $D$-optimality (solid line) and $D_{P}$-optimality (dotted
line). The dashed line gives the correlation coefficient (0.77) from the
12-point original design
$se_{D}(\hat{\theta}_{2})$$se_{D}(\hat{\theta}_{1})$$se_{D_{P}}(\hat{\theta}_{1})$$se_{D_{P}}(\hat{\theta}_{2})$
Figure 4: Simulated standard errors of $\hat{\theta}_{1}$ and
$\hat{\theta}_{2}$ in Michaelis-Menten model using the 13-point sequential
design resulted from $D$-optimality and $D_{P}$-optimality (asterisk points).
The solid line is the line of equality
Figure 3 shows the resulting correlation coefficients between
$\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ plotted against the simulation
number. The dotted line represents the correlation coefficient resulting from
the $D_{P}$ design while the solid line represents the corresponding
correlation resulting from the $D$ design. The horizontal dashed line gives
the correlation coefficient (0.77) resulted from the original design given in
Table 1. The line charts of Figure 3 clearly demonstrate that, for vast
majority of the simulations, the $D_{P}$ design gives reduced correlation
coefficient from that given by the original design and these correlations are
remarkably lower than correlations generated from the $D$ design.
Furthermore, in Figure 4 we present scatter plots of the estimated standard
errors of $\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ from the 2000 simulations.
The parameter standard errors resulted from $D$ design ($se_{D}$) are plotted
on the horizontal axis while the $D_{P}$ design standard errors ($se_{D_{p}}$)
are plotted (asterisk points) on the vertical axis. The solid line is the line
of equality of $se_{D}$ and $se_{D_{P}}$ so that it is easily seen that in a
significant portion of the simulations the $se_{D_{P}}$ is less than $se_{D}$,
reflecting higher accuracy of parameter estimates. The average value of the
simulated D-efficiency scores is $97.6\%$ which suggests that the two designs
are of comparable efficiency in estimating the two parameters. The $D_{P}$
design, however, has the distinctive ability to increase parameter estimate
precision while reducing their correlations.
### 4.2 Hougen-Watson Model
The Hougen-Watson model is common in chemical kinetics of catalyzed reactions.
It expresses the reaction rate in terms of the catalyst variables, the
temperature and concentrations of reactants. One of the expressions the model
takes is:
$f({\bf
x},\Theta)=\frac{\theta_{1}\theta_{3}(x_{2}-x_{3}/1.632)}{1+\theta_{2}x_{1}+\theta_{3}x_{2}+\theta_{4}x_{3}}$
(20)
where $x_{1}$, $x_{2}$ and $x_{3}$ represent the partial pressures of the
reactants. The data set, reported in Bates and Watts[1], consists of 24 runs
of the design variables $(x_{1},x_{2},x_{3})$. The model was fitted to this
initial 24-point design. Table 2 below shows the results.
Table 2: Summary of parameter estimates for the Hougen-Watson model Parameter
Estimate St.error Correlation $\theta_{1}$ 35.92 8.21 1.00 $\theta_{2}$ 0.071
0.178 -0.805 1.00 $\theta_{3}$ 0.038 0.099 -0.840 0.998 1.00 $\theta_{4}$
0.167 0.415 -0.790 0.998 0.995 1.00
The correlations between the parameter estimates are clearly very high and the
model suffers from pronounced nonlinearity, Bates and Watts[1].
To generate an additional design point to improve parameter precision, Bates
and Watts[1] used the $D$-optimality with design variables restricted to the
following ranges: $100\leq x_{1}\leq 400$, $75\leq x_{2}\leq 350$ and $30\leq
x_{3}\leq 150$. They searched for the maximum $D$ at the corner points of the
restricted design region and inside its interior. They found the combination
$(x_{1}=100,x_{2}=350,x_{3}=30)$ to be the point at which the optimum $D$
occurred and hence they recommended this corner point for the next
experimental run.
We implemented the same strategy using MATLAB 8.2.0 optimizer. At first, we
evaluated the $D_{P}$ criterion using equation (7) at the corner points of the
design region above and at the original 24 design points. The maximum $D_{P}$
occurred at the design point $(x_{1}=251,x_{2}=294,x_{3}=41.5)$. We used this
point as starting point for the maximization of $D_{P}$. The $25^{th}$ design
point at which the optimum value was found is
$(x_{1}=245,x_{2}=300,x_{3}=40)$.
Simulation Study
Similar to the previous example, we ran 2000 simulations of the Hougen-Watson
model estimation in order to evaluate the information content of each of the
two 25-point designs constructed by $D$ and $D_{P}$ criteria. In each
simulation, the response variable for the additional point;
$(x_{1}=100,x_{2}=350,x_{3}=30)$ for $D$ and $(x_{1}=245,x_{2}=300,x_{3}=40)$
for $D_{P}$; was estimated by using the fitted model in Table 2 and adding
randomly generated noise from a normal distribution. The results are depicted
in Figures 4 and 5.
Index$corr(\hat{\theta}_{1},\hat{\theta}_{2})$$corr(\hat{\theta}_{1},\hat{\theta}_{3})$$corr(\hat{\theta}_{1},\hat{\theta}_{4})$$corr(\hat{\theta}_{2},\hat{\theta}_{3})$$corr(\hat{\theta}_{2},\hat{\theta}_{4})$$corr(\hat{\theta}_{3},\hat{\theta}_{4})$
Figure 5: Simulated correlation coefficients between the four parameter
estimates in the Hougen-Watson model. The solid line represent correlations
from the $D$-optimal design, the dotted line represent correlations from the
$D_{P}$-optimal design and the horizontal dashed line gives the correlation
coefficient from the 24-point original design
$\hat{se}_{1_{D}}$$\hat{se}_{1_{D_{P}}}$$\hat{se}_{2_{D}}$$\hat{se}_{2_{D_{P}}}$$\hat{se}_{3_{D}}$$\hat{se}_{3_{D_{P}}}$$\hat{se}_{4_{D}}$$\hat{se}_{4_{D_{P}}}$
Figure 6: Simulated standard errors of $\hat{\theta}_{1}$, $\hat{\theta}_{2}$,
$\hat{\theta}_{3}$ and $\hat{\theta}_{4}$ in the Hougen-Watson model using the
25-points sequential design resulted from $D$-optimality and
$D_{P}$-optimality (asterisk points). The solid line is the line of equality
Figure 4 clearly demonstrates that, for vast majority of the simulations, the
magnitudes of the correlations among parameter estimates are lower for the
$D_{P}$ (dotted line) than the $D$ design (solid line). The reduction in
correlations is most pronounced for relationships involving
$\hat{\theta}_{1}$. This is to say that the location of the additional design
point generated by the $D_{P}$ criterion provides informative experimental
setting to reduce dependencies between $\theta_{1}$ from the other parameter
estimates, thereby, reducing the overall ill-conditioning of the model
estimation. The additional design point generated by the classical $D$
criterion gave rise to higher magnitudes of all correlations involving
$\hat{\theta}_{1}$. As for the correlations involving $\hat{\theta}_{2}$,
$\hat{\theta}_{3}$ and $\hat{\theta}_{4}$, the $D_{P}$ continued to produce
lower values than those produced by the $D$ design. Because the $D_{P}$
criterion accounts for the correlation structure among parameters based on
second-order derivative information, the resulting correlations from $D_{P}$
design are seen to have more volatility that the correlations produced by the
classical $D$ design.
Figure 5 shows the scatter plots of the estimated standard errors of of the
parameter estimates. The solid straight is the line of equality of the the
estimated standard errors produced by the $D_{P}$ design,$se_{D_{P}}$, and the
corresponding standard errors produced by the $D$ design, $se_{D}$. Clearly,
the simulated standard errors of the four parameter estimates are lower for
the $D_{P}$ design (asterisk points) than that for the $D$ design in
substantially all the simulations. It should be pointed out that, except for
$\hat{\theta}_{1}$, the simulated standard errors of the other three parameter
estimates by the $D$ and $D_{P}$ designs are in general larger than those
reported in Table 2 above. This is to say that the additional experimental
setting in each design was not informative enough to improve the precision of
$\hat{\theta}_{2}$, $\hat{\theta}_{3}$ or $\hat{\theta}_{4}$. However, the
additional design point in both $D$ and $D_{P}$ produced a significant
improvement in the precision of $\hat{\theta}_{1}$ in nearly all of the
simulations. The additional design point for the $D_{P}$ optimal design was
significantly informative in reducing the correlations involving
$\hat{\theta}_{1}$ and improving its precision.
## 5 CONCLUSIONS
In this article we have applied the profile-based sensitivity coefficients
developed by Sulieman et.al. [12] in designing experiments for nonlinear
models. Given that the profile-based sensitivity coefficients account for both
parameter correlations and model nonlinearity, it has been shown that
utilizing them in the $D$-optimal criterion generates more informative
experimental settings. Two model examples have been used to demonstrate the
computational aspects of the profile-based $D$-optimal criterion. Furthermore,
simulation studies have shown that the constructed profile-based optimal
designs are more efficient and informative than the classical $D$-optimal
designs. Future work will include establishing more detailed theoretical
framework for the proposed design criterion and conducting further comparisons
with existing nonlinear design criteria including that of Hamilton and Watts
[10] and Vila and Gauchi [17].
## 6 ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support of the American
University of Sharjah, United Arab Emirates.
## References
* [1] Bates, D.M. and Watts, D.G. (1988) Nonlinear Regression Analysis and Its Applications. Wiley, New York.
* [2] Bates, D.M. and Watts, D.G. (1981) Parameter transformation for improved approximate confidence regions in nonlinear least squares. Annals of Statistics 9: 1152- 1167.
* [3] Benabbas, L., Asprey, S. P. and Macchietto, S. (2005) Curvature-based methods for designing optimally informative experiments in multiresponse nonlinear dynamic situations. Industrial and Engineering Chemistry Research44: 7120-7131.
* [4] Berger M.P.F. and Wong W.K. (2009) An introduction to optimal designs for social and biomedical research.Wiley, Chichester
* [5] Box, G.E.P. and Lucas, H.L. (1959) Design of experiments in non-linear situations. Biometrika46: 77-90.
* [6] Buzzi Ferraris, G.and Donati, G. (1974) A powerful method for Hougen Watson model parameter estimation with integral conversion data. Chemical Engineering Science 29: 1504-1509.
* [7] Clarke, G.P.Y. (1987) Approximate confidence limits for a parameter function in Nonlinear Regression. JASA82(397): 221-230.
* [8] Franceschini, G. and Macchietto, S. (2008) Model-based design of experiments for parameter precision: State of the art. Chemical Engineering Science 63:4846-4872.
* [9] Gao, L.L. and Zhou, J. (2017) D-optimal designs based on the second-order least squares estimator. Stat Papers 58: 77-94.
* [10] Hamilton, D.C. and Watts, D.G. (1985) A quadratic design criterion for precise estimation in nonlinear regression models. Technometrics 27(3): 241-250.
* [11] Ryan, T.P. (2007) Modern Experimental Design. John Wiley & Sons, Inc., Hoboken, New Jersey.
* [12] Sulieman, H., McLellan, P.J. and Bacon, D.W. (2001) A Profile-Based Approach to Parametric Sensitivity Analysis of Nonlinear Regression Models. Technometrics 43(4): 425-33.
* [13] Sulieman, H., McLellan, P.J. and Bacon, D.W. (2004) A Profile-based approach to parametric sensitivity in multiresponse regression models. Computational Statistics & Data Analysis 45: 721-740.
* [14] Sulieman H. (2008)Improved local sensitivity measures for regression models with correlated parameters. Proceedings in Computational Statistics 18th Symposium, Porto, Portugal.
* [15] Sulieman, H., Kucuk, I. and McLellan J.(2009) Parametric sensitivity: A case study comparison. Computational Statistics & Data Analysis 53(7): 2640-2652.
* [16] Sulieman, H., Kucuk, I. (2011) Global derivative based sensitivity method for parameter estimation. Journal of the Franklin Institute348(7): 1556–1573.
* [17] Vila, J.P. and Gauchi, J.P. (2010) Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model. Journal of Statistical Planning and Inference 137(9): 2935 2953.
|
1842 LABEL:LastPageJun. 20, 2021Oct. 21, 2022
An abridged version of this paper appeared in TACAS 2021; missing proofs have
been added to create this version.
# General Decidability Results
for Asynchronous Shared-Memory Programs:
Higher-Order and Beyond
Rupak Majumdar Max Planck Institute for Software Systems (MPI-SWS), Paul-
Ehrlich-Straße, Building G26, Kaiserslautern, Germany 67663.
<EMAIL_ADDRESS>, Ramanathan S. Thinniyam and Georg
Zetzsche
###### Abstract.
The model of asynchronous programming arises in many contexts, from low-level
systems software to high-level web programming. We take a language-theoretic
perspective and show general decidability and undecidability results for
asynchronous programs that capture all known results as well as show
decidability of new and important classes. As a main consequence, we show
decidability of safety, termination and boundedness verification for _higher-
order_ asynchronous programs—such as OCaml programs using Lwt—and
undecidability of liveness verification already for order-2 asynchronous
programs. We show that under mild assumptions, surprisingly, safety and
termination verification of asynchronous programs with handlers from a
language class are decidable _iff_ emptiness is decidable for the underlying
language class. Moreover, we show that configuration reachability and liveness
(fair termination) verification are equivalent, and decidability of these
problems implies decidability of the well-known “equal-letters” problem on
languages. Our results close the decidability frontier for asynchronous
programs.
###### Key words and phrases:
Higher-order asynchronous programs, Higher-order recursion schemes and
Decidability
## 1\. Introduction
_Asynchronous programming_ is a common way to manage concurrent requests in a
system. In this style of programming, rather than waiting for a time-consuming
operation to complete, the programmer can make _asynchronous_ procedure calls
which are stored in a _task buffer_ pending later execution. Each asynchronous
procedure, or _handler_ , is a sequential program. When run, it can change the
_global shared state_ of the program, make internal synchronous procedure
calls, and post further instances of handlers to the task buffer. A scheduler
repeatedly and non-deterministically picks pending handler instances from the
task buffer and executes their code _atomically_ to completion. Asynchronous
programs appear in many domains, such as operating system kernel code, web
programming, or user applications on mobile platforms. This style of
programming is supported natively or through libraries for most programming
environments. The interleaving of different handlers hides latencies of long-
running operations: the program can process a different handler while waiting
for an external operation to finish. However, asynchronous scheduling of tasks
introduces non-determinism in the system, making it difficult to reason about
correctness.
An asynchronous program is _finite-data_ if all program variables range over
finite domains. Finite-data programs are still infinite state transition
systems: the task buffer can contain an unbounded number of pending instances
and the sequential machine implementing an individual handler can have
unboundedly large state (e.g., if the handler is given as a recursive program,
the stack can grow unboundedly). Nevertheless, verification problems for
finite-data programs have been shown to be decidable for several kinds of
handlers [SV06, JM07, CV07, GM12]. Several algorithmic approaches have been
studied, which tailor to (i) the kinds of permitted handler programs and (ii)
the properties that are checked.
#### State of the art
We briefly survey the existing approaches and what is known about the
decidability frontier. The _Parikh approach_ applies to (first-order)
recursive handler programs. Here, the decision problems for asynchronous
programs are reduced to decision problems over Petri nets [GM12]. The key
insight is that since handlers are executed atomically, the order in which a
handler posts tasks to the buffer is irrelevant. Therefore, instead of
considering the sequential order of posted tasks along an execution, one can
equivalently consider its Parikh image. Thus, when handlers are given as
pushdown systems, the behaviors of an asynchronous program can be represented
by a (polynomial sized) Petri net. Using the Parikh approach, safety
(formulated as reachability of a global state), termination (whether all
executions terminate), and boundedness (whether there is an a priori upper
bound on the task buffer) are all decidable for asynchronous programs with
recursive handlers, by reduction to corresponding problems on Petri nets
[SV06, GM12]. Configuration reachability (reachability of a specific global
state and task buffer configuration), fair termination (termination under a
fair scheduler), and fair non-starvation (every pending handler instance is
eventually executed) are also decidable, by separate ad hoc reductions to
Petri net reachability [GM12]. A “reverse reduction” shows that Petri nets can
be simulated by polynomial-sized asynchronous programs (already with finite-
data handlers).
In the _downclosure approach_ , one replaces each handler with a finite-data
program that is equivalent up to “losing” handlers in the task buffer. Of
course, this requires that one can compute equivalent finite-data programs for
a given class of handler programs. This has been applied to checking safety
for recursive handler programs [ABQ09]. Finally, a bespoke _rank-based
approach_ has been applied to checking safety when handlers can perform
restricted higher-order recursion [CV07].
#### Contribution
Instead of studying individual kinds of handler programs, we consider
asynchronous programs in a general language-theoretic framework. The class of
handler programs is given as a language class $\mathcal{C}$: An asynchronous
program over a language class $\mathcal{C}$ is one where each handler defines
a language from $\mathcal{C}$ over the alphabet of handler names, as well as a
transformer over the global state. This view leads to general results: we can
obtain simple characterizations of which classes of handler programs permit
decidability. For example, we do not need the technical assumptions of
computability of equivalent finite-data programs from the Parikh and the
downclosure approach.
Our first result shows that, under a mild language-theoretic assumption,
safety and termination are decidable if and only if the underlying language
class $\mathcal{C}$ has a decidable emptiness problem.111 The “mild language-
theoretic assumption” is that the class of languages forms an effective full
trio: it is closed under intersections with regular languages, homomorphisms,
and inverse homomorphisms. Many language classes studied in formal language
theory and verification satisfy these conditions. Similarly, we show that
boundedness is decidable iff _finiteness_ is decidable for the language class
$\mathcal{C}$. These results are the best possible: decidability of emptiness
(resp., finiteness) is a requirement for safety and termination verification
already for verifying the safety or termination (resp., boundedness) of one
_sequential_ handler call. As corollaries, we get new decidability results for
all these problems for asynchronous programs over _higher-order recursion
schemes_ , which form the language-theoretic basis for programming in higher-
order functional languages such as OCaml [Kob09, Ong15], as well as other
language classes (lossy channel languages, Petri net languages, etc.).
Second, we show that configuration reachability, fair termination, and fair
starvation are mutually reducible; thus, decidability of any one of them
implies decidability of all of them. We also show decidability of these
problems implies the decidability of a well-known combinatorial problem on
languages: given a language over the alphabet ${\\{\mathtt{a},\mathtt{b}\\}}$,
decide if it contains a word with an equal number of $\mathtt{a}$s and
$\mathtt{b}$s. Viewed contrapositively, we conclude that all these decision
problems are undecidable already for asynchronous programs over order-2
pushdown languages, since the equal-letters problem is undecidable for this
class.
Together, our results “close” the decidability frontier for asynchronous
programs, by demonstrating reducibilities between decision problems heretofore
studied separately and connecting decision problems on asynchronous programs
with decision problems on the underlying language classes of their handlers.
While our algorithms do not assume that downclosures are effectively
computable, we use downclosures to prove their correctness. We show that the
safety, termination, and boundedness problems are invariant under taking
downclosures of runs; this corresponds to taking downclosures of the languages
of handlers.
The observation that safety, termination, and boundedness depend only on the
downclosure suggests a possible route to implementation. If there is an
effective procedure to compute the downclosure for class $\mathcal{C}$, then a
direct verification algorithm would replace all handlers by their (regular)
downclosures, and invoke existing decision procedures for this class. Thus, we
get a direct algorithm based on downclosure constructions for higher-order
recursion schemes, using the string of celebrated recent results on
effectively computing the downclosure of _word schemes_ [Zet15, HKO16,
CPSW16].
We find our general decidability result for asynchronous programs to be
surprising. Already for regular languages, the complexity of safety
verification jumps from ${\mathsf{NL}}$ (NFA emptiness) to
${\mathsf{EXPSPACE}}$ (Petri net coverability): asynchronous programs are far
more expressive than individual handler languages. It is therefore unexpected
that safety and termination remain decidable whenever they are decidable for
individual handler languages.
## 2\. Preliminaries
#### Basic Definitions
We assume familiarity with basic definitions of automata theory (see, e.g.,
[HMU07, Sip97]). If $w$ is a word over $\Sigma$ and $\Gamma\subseteq\Sigma$ is
a subalphabet, then the _projection of $w$ onto $\Gamma$_, denoted
$\mathrm{Proj}_{\Gamma}(w)$, is the word obtained by erasing from $w$ each
symbol that does not belong to $\Gamma$. For a language
$L\subseteq\Sigma^{*}$, define
$\mathrm{Proj}_{\Gamma}(L)={\\{\mathrm{Proj}_{\Gamma}(w)\mid w\in L\\}}$. The
subword order $\sqsubseteq$ on $\Sigma^{*}$ is defined as $w\sqsubseteq
w^{\prime}$ for $w,w^{\prime}\in\Sigma^{*}$ if $w$ can be obtained from
$w^{\prime}$ by deleting some letters from $w^{\prime}$. For example,
$abba\sqsubseteq bababa$ but $abba\not\sqsubseteq baaba$. The downclosure
$\mathord{\downarrow}{w}$ with respect to the subword order, of a word
$w\in\Sigma^{*}$, is defined as
$\mathord{\downarrow}{w}:=\\{w^{\prime}\in\Sigma^{*}\mid w^{\prime}\sqsubseteq
w\\}$. The downclosure $\mathord{\downarrow}{L}$ of a language
$L\subseteq\Sigma^{*}$ is given by
$\mathord{\downarrow}{L}:=\\{w^{\prime}\in\Sigma^{*}\mid\exists w\in L\colon
w^{\prime}\sqsubseteq w\\}$. Recall that the downclosure
$\mathord{\downarrow}{L}$ of any language $L$ is a regular language [Hai69].
A _multiset_ $\mathbf{m}\colon\Sigma\rightarrow{\mathbb{N}}$ over $\Sigma$
maps each symbol of $\Sigma$ to a natural number. Let ${\mathbb{M}[\Sigma]}$
be the set of all multisets over $\Sigma$. We treat sets as a special case of
multisets where each element is mapped onto $0$ or $1$. For example, we write
$\mathbf{m}={\llbracket\mathtt{a},\mathtt{a},\mathtt{c}\rrbracket}$ for the
multiset
$\mathbf{m}\in{\mathbb{M}[{\\{\mathtt{a},\mathtt{b},\mathtt{c},\mathtt{d}\\}}]}$
with $\mathbf{m}(\mathtt{a})=2$,
$\mathbf{m}(\mathtt{b})=\mathbf{m}(\mathtt{d})=0$, and
$\mathbf{m}(\mathtt{c})=1$. We also use the notation
$\lvert{\mathbf{m}}\rvert=\sum_{\sigma\in\Sigma}\mathbf{m}(\sigma)$. The
Parikh image ${\mathsf{Parikh}}(w)\in{\mathbb{M}[\Sigma]}$ of a word
$w\in\Sigma^{*}$ is the multiset such that ${\mathsf{Parikh}}(w)(\mathtt{a})$
is the number of times $\mathtt{a}$ occurs in $w$.
Given two multisets $\mathbf{m},\mathbf{m}^{\prime}\in{\mathbb{M}[\Sigma]}$ we
define the multiset
$\mathbf{m}\oplus\mathbf{m}^{\prime}\in{\mathbb{M}[\Sigma]}$ for which, for
all $\mathtt{a}\in\Sigma$, we have
$(\mathbf{m}\oplus\mathbf{m}^{\prime})(\mathtt{a})=\mathbf{m}(\mathtt{a})+\mathbf{m}^{\prime}(\mathtt{a})$.
We also define the natural order $\preceq$ on ${\mathbb{M}[\Sigma]}$ as
follows: $\mathbf{m}\preceq\mathbf{m}^{\prime}$ iff there exists
$\mathbf{m}^{\Delta}\in{\mathbb{M}[\Sigma]}$ such that
$\mathbf{m}\oplus\mathbf{m}^{\Delta}=\mathbf{m}^{\prime}$. We also define
$\mathbf{m}^{\prime}\ominus\mathbf{m}$ for
$\mathbf{m}\preceq\mathbf{m}^{\prime}$ analogously: for all
$\mathtt{a}\in\Sigma$, we have
$(\mathbf{m}\ominus\mathbf{m}^{\prime})(\mathtt{a})=\mathbf{m}(\mathtt{a})-\mathbf{m}^{\prime}(\mathtt{a})$.
For $\Gamma\subseteq\Sigma$ we regard $\mathbf{m}\in{\mathbb{M}[\Gamma]}$ as a
multiset of ${\mathbb{M}[\Sigma]}$ where undefined values are sent to $0$.
#### Language Classes and Full Trios
A _language class_ is a collection of languages, together with some finite
representation. Examples are the regular (e.g. represented by finite automata)
or the context-free languages (e.g. represented by pushdown automata
($\mathsf{PDA}$)). A relatively weak and reasonable assumption on a language
class is that it is a _full trio_ , that is, it is closed under each of the
following operations: taking intersection with a regular language, taking
homomorphic images, and taking inverse homomorphic images. Equivalently, a
language class is a full trio iff it is closed under rational transductions
[Ber79] as we explain below.
We assume that all full trios $\mathcal{C}$ considered in this paper are
effective: Given a language $L$ from $\mathcal{C}$, a regular language $R$,
and a homomorphism $h$, we can compute a representation of the languages
$L\cap R$, $h(L)$, and $h^{-1}(L)$ in $\mathcal{C}$.
Many classes of languages studied in formal language theory form effective
full trios. Examples include the regular and the context-free languages
[HMU07], the indexed languages [Aho68, DG86], the languages of higher-order
pushdown automata [Mas74], higher-order recursion schemes ($\mathsf{HORS}$)
[Dam82, HMOS08], Petri nets (both coverability and reachability languages)
[Gre78, Jan79], and lossy channel systems (see Section 4.1). (While
$\mathsf{HORS}$ are usually viewed as representing a tree or collection of
trees, one can also view them as representing a word language, as we explain
in Section 5.)
Informally, a language class defined by non-deterministic devices with a
finite-state control that allows $\varepsilon$-transitions and imposes no
restriction between input letter and performed configuration changes (such as
non-deterministic pushdown automata) is always a full trio: The three
operations above can be realized by simple modifications of the finite-state
control. The deterministic context-free languages are a class that is _not_ a
full trio.
An asynchronous transducer $\mathcal{T}$ is a tuple
$\mathcal{T}=(Q,\Gamma,\Sigma,E,q_{0},F)$ with a finite set of states $Q$,
finite output alphabet $\Gamma$, finite input alphabet $\Sigma$, a finite set
of edges $E\subseteq Q\times\Gamma^{*}\times\Sigma^{*}\times Q$, initial state
$q_{0}\in Q$ and set of final states $F\subseteq Q$. We write
$p\xrightarrow{v|u}q$ if $(p,v,u,q)\in E$ and the machine reads $u$ in state
$p$, outputs $v$ and moves to state $q$. We also write
$p\xrightarrow{w|w^{\prime}}q$ if there are states $q_{0},q_{1},\dots,q_{n}$
and words $u_{1},u_{2},\dots,u_{n},v_{1},v_{2},\dots,v_{n}$ such that
$p=q_{0}$, $q=q_{n}$, $w^{\prime}=u_{1}u_{2}\cdots u_{n}$, $w=v_{1}v_{2}\cdots
v_{n}$ and $q_{i-1}\xrightarrow{v_{i}|u_{i}}q_{i}$ for all $0\leq i\leq n$.
The transduction $T\subseteq\Gamma^{*}\times\Sigma^{*}$ generated by the
transducer $\mathcal{T}$ is the set of tuples
$(v,u)\in\Gamma^{*}\times\Sigma^{*}$ such that $q_{0}\xrightarrow{v|u}q_{f}$
for some $q_{f}\in F$. Given a language $L\subseteq\Sigma^{*}$, we define
$TL:=\\{v\in\Gamma^{*}\;|\;\exists u\in L\colon(v,u)\in T\\}$. A transduction
$T\subseteq\Gamma^{*}\times\Sigma^{*}$ is _rational_ if it is generated by
some asynchronous transducer.
A language class $\mathcal{C}$ is a full trio if and only if it is closed
under each of the following operations:
* •
intersection with a regular language,
* •
taking homomorphic images, and
* •
taking inverse homomorphic images.
It is well known that these two concepts are equivalent:
###### Theorem 1 (Berstel [Ber79]).
A language class is closed under rational transductions if and only if it is a
full trio.
#### Asynchronous Programs: A Language-Theoretic View
We use a language-theoretic model for asynchronous shared-memory programs.
Let $\mathcal{C}$ be an (effective) full trio. An _asynchronous program_
($\mathsf{AP}$) over $\mathcal{C}$ is a tuple
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$,
where $D$ is a finite set of _global states_ , $\Sigma$ is an alphabet of
_handler names_ , $(L_{c})_{c\in\mathfrak{C}}$ is a family of languages over
$\Sigma$ from $\mathcal{C}$, one for each $c\in\mathfrak{C}$ where
$\mathfrak{C}=D\times\Sigma\times D$ is the set of contexts, $d_{0}\in D$ is
the _initial state_ , and $\mathbf{m}_{0}\in{\mathbb{M}[\Sigma]}$ is a
multiset of _initial pending handler instances_.
A configuration $(d,\mathbf{m})\in D\times{\mathbb{M}[\Sigma]}$ of
$\mathfrak{P}$ consists of a global state $d$ and a multiset $\mathbf{m}$ of
pending handler instances. For a configuration $c$, we write $c.d$ and
$c.\mathbf{m}$ for the global state and the multiset in the configuration
respectively. The _initial_ configuration $c_{0}$ of $\mathfrak{P}$ is given
by $c_{0}.d=d_{0}$ and $c_{0}.\mathbf{m}=\mathbf{m}_{0}$. The semantics of
$\mathfrak{P}$ is a labeled transition system over the set of configurations,
with the transition relation
$\xrightarrow{\sigma}\subseteq(D\times{\mathbb{M}[\Sigma]})\times(D\times{\mathbb{M}[\Sigma]})$
given by
$\displaystyle(d,\mathbf{m}\oplus{\llbracket\sigma\rrbracket})\xrightarrow{\sigma}(d^{\prime},\mathbf{m}\oplus\mathbf{m}^{\prime})\quad\text{
if{}f }\quad\exists w\in L_{d\sigma
d^{\prime}}\colon{\mathsf{Parikh}}(w)=\mathbf{m}^{\prime}$
Moreover, we write $(d,\mathbf{m})\rightarrow(d^{\prime},\mathbf{m}^{\prime})$
if there exists a $\sigma\in\Sigma$ with
$(d,\mathbf{m})\xrightarrow{\sigma}(d^{\prime},\mathbf{m}^{\prime})$. We use
$\rightarrow^{*}$ to denote the reflexive transitive closure of the relation
$\rightarrow$. A configuration $c$ is said to be _reachable_ in $\mathfrak{P}$
if $(d_{0},\mathbf{m}_{0})\rightarrow^{*}c$.
Intuitively, the set $\Sigma$ of handler names specifies a finite set of
procedures that can be invoked asynchronously. The shared state takes values
in $D$. When a handler is called asynchronously, it gets added to a bag of
pending handler calls (the multiset $\mathbf{m}$ in a configuration). The
language $L_{d\sigma d^{\prime}}$ captures the effect of executing an instance
of $\sigma$ starting from the global state $d$, such that on termination, the
global state is $d^{\prime}$. Each word $w\in L_{d\sigma d^{\prime}}$ captures
a possible sequence of handlers posted during the execution.
Suppose the current configuration is $(d,\mathbf{m})$. A non-deterministic
scheduler picks one of the outstanding handlers $\sigma\in\mathbf{m}$ and
executes it. Executing $\sigma$ corresponds to picking one of the languages
$L_{d\sigma d^{\prime}}$ and some word $w\in L_{d\sigma d^{\prime}}$. Upon
execution of $\sigma$, the new configuration has global state $d^{\prime}$ and
the new bag of pending calls is obtained by taking $\mathbf{m}$, removing an
instance of $\sigma$ from it, and adding the Parikh image of $w$ to it. This
reflects the current set of pending handler calls—the old ones (minus an
instance of $\sigma$) together with the new ones added by executing $\sigma$.
Note that a handler is executed atomically; thus, we atomically update the
global state and the effect of executing the handler.
Let us see some examples of asynchronous programs. It is convenient to present
these examples in a programming language syntax, and to allow each handler to
have _internal actions_ that perform local tests and updates to the global
state. As we describe informally below, and formally in Appendix A, when
$\mathcal{C}$ is a full trio, internal actions can be “compiled away” by
taking an intersection with a regular language of internal actions and
projecting the internal actions away. Thus, we use our simpler model
throughout.
⬇
1global var turn = ref 0 and x = ref 0;
2let rec s1 () = if * then begin post a; s1(); post b end
3let rec s2 () = if * then begin post a; s2(); post b end else post b
4let a () = if !turn = 0 then begin turn := 1; x := !x + 1 end else post a
5let b () = if !turn = 1 then begin turn := 0; x := !x - 1 end else post b
6
7let s3 () = post s3; post s3
8
9global var t = ref 0;
10let c () = if !t = 0 then t := 1 else post c
11let d () = if !t = 1 then t := 2 else post d
12let f () = if !t = 2 then t := 0 else post f
13
14let cc x = post c; x
15let dd x = post d; x
16let ff x = post f; x
17let id x = x
18let h g y = cc (g (dd y))
19let rec produce g x = if * then produce (h g) (ff x) else g x
20let s4 () = produce id ()
Figure 1. Examples of asynchronous programs
#### Examples
Figure 1 shows some simple examples of asynchronous programs in an OCaml-like
syntax. Consider first the asynchronous program in lines 1–5. The alphabet of
handlers is s1, s2, a, and b. The global states correspond to possible
valuations to the global variables turn and x; assuming turn is a Boolean and
x takes values in $\mathbb{N}$, we have that
$D={\\{0,1\\}}\times{\\{0,1,\omega\\}}$, where $\omega$ abstracts all values
other than ${\\{0,1\\}}$. Since $\mathtt{s1}$ and s2 do not touch any
variables, for $d,d^{\prime}\in D$, we have
$L_{d,\mathtt{s1},d}={\\{\mathtt{a}^{n}\mathtt{b}^{n}\mid n\geq 0\\}}$,
$L_{d,\mathtt{s2},d}={\\{\mathtt{a}^{n}\mathtt{b}^{n+1}\mid n\geq 0\\}}$, and
$L_{d,\mathtt{s1},d^{\prime}}=L_{d,\mathtt{s2},d^{\prime}}=\emptyset$ if
$d^{\prime}\neq d$.
For the languages corresponding to a and b, we use syntactic sugar in the form
of _internal actions_ ; these are local tests and updates to the global state.
In our example, we have $L_{(0,0),\mathtt{a},(1,1)}={\\{\varepsilon\\}}$,
$L_{(1,x),\mathtt{a},(1,x)}={\\{\mathtt{a}\\}}$ for all values of $x$, and
similarly for b. The meaning is that, starting from a global state $(0,0)$,
executing the handler will lead to the global state $(1,1)$ and no handlers
will be posted, whereas starting from a global state in which turn is $1$,
executing the handler will keep the global state unchanged but post an
instance of a. Note that all the languages are context-free.
Consider an execution of the program from the initial configuration
$((0,0),{\llbracket\mathtt{s1}\rrbracket})$. The execution of s1 puts $n$ as
and $n$ bs into the bag, for some $n\geq 0$. The global variable turn is used
to ensure that the handlers a and b alternately update x. When turn is $0$,
the handler for a increments x and sets turn to $1$, otherwise it re-posts
itself for a future execution. Likewise, when turn is $1$, the handler for b
decrements x and sets turn back to $0$, otherwise it re-posts itself for a
future execution. As a result, the variable x never grows beyond $1$. Thus,
the program satisfies the _safety_ property that no execution sets x to
$\omega$.
It is possible that the execution goes on forever: for example, if s1 posts an
a and a b, and thereafter only b is chosen by the scheduler. This is not an
“interesting” infinite execution as it is not fair to the pending a. In the
case of a fair scheduler, which eventually always picks an instance of every
pending task, the program terminates: eventually all the as and bs are
consumed when they are scheduled in alternation. However, if instead we
started with ${\llbracket\mathtt{s2}\rrbracket}$, the program will not
terminate even under a fair scheduler: the last remaining b will not be paired
and will keep executing and re-posting itself forever.
Now consider the execution of s3. It has an infinite fair run, where the
scheduler picks an instance of s3 at each step. However, the number of pending
instances grows without bound. We shall study the _boundedness problem_ ,
which checks if the bag can become unbounded along some run. We also study a
stronger notion of fair termination, called _fair non-starvation_ , which asks
that every _instance_ of a posted handler is executed under any fair
scheduler. The execution of s3 is indeed fair, but there can be a specific
instance of s3 that is never picked: we say s3 can _starve_ an instance.
The program in lines 9–20 is _higher-order_ ($\mathtt{produce}$ and
$\mathtt{h}$ take functions as arguments). The language of $\mathtt{s4}$ is
the set ${\\{\mathtt{c}^{n}\mathtt{d}^{n}\mathtt{f}^{n}\mid n\geq 0\\}}$, that
is, it posts an equal number of $\mathtt{c}$s, $\mathtt{d}$s, and
$\mathtt{f}$s. It is an indexed language; we shall see (Section 5) how this
and other higher-order programs can be represented using higher-order
recursion schemes ($\mathsf{HORS}$). Note that the OCaml types of
$\mathtt{produce}:(\mathtt{o}\rightarrow\mathtt{o})\rightarrow\mathtt{o}\rightarrow\mathtt{o}$
and
$\mathtt{h}:(\mathtt{o}\rightarrow\mathtt{o})\rightarrow\mathtt{o}\rightarrow\mathtt{o}$
are higher-order.
The program is similar to the first: the handlers c, d, and f execute in
“round robin” fashion using the global state t to find their turns. Again, we
use internal actions to update the global state for readability. We ask the
same decision questions as before: does the program ever reach a specific
global state and does the program have an infinite (fair) run? We shall see
later that safety and termination questions remain decidable, whereas fair
termination does not.
## 3\. Decision Problems on Asynchronous Programs
We now describe decision problems on runs of asynchronous programs.
#### Runs, preruns, and downclosures
A prerun of an $\mathsf{AP}$
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ is a
finite or infinite sequence
$\rho=(e_{0},\mathbf{n}_{0}),\sigma_{1},(e_{1},\mathbf{n}_{1}),\sigma_{2},\ldots$
of alternating elements of tuples $(e_{i},\mathbf{n}_{i})\in
D\times{\mathbb{M}[\Sigma]}$ and symbols $\sigma_{i}\in\Sigma$. The set of
preruns of $\mathfrak{P}$ will be denoted $\mathsf{Preruns}(\mathfrak{P})$.
Note that if two asynchronous programs $\mathfrak{P}$ and
$\mathfrak{P}^{\prime}$ have the same $D$ and $\Sigma$, then
$\mathsf{Preruns}(\mathfrak{P})=\mathsf{Preruns}(\mathfrak{P}^{\prime})$. The
_length_ , denoted $|\rho|$, of a finite prerun $\rho$ is the number of
configurations in $\rho$. The $i^{th}$ configuration of a prerun $\rho$ will
be denoted $\rho(i)$.
We define an order $\trianglelefteq$ on preruns as follows: For preruns
$\rho=(e_{0},\mathbf{n}_{0}),\sigma_{1},(e_{1},\mathbf{n}_{1}),\sigma_{2},\ldots$
and
$\rho^{\prime}=(e_{0}^{\prime},\mathbf{n}_{0}^{\prime}),\sigma_{1}^{\prime},(e_{1}^{\prime},\mathbf{n}_{1}^{\prime}),\sigma_{2}^{\prime},\ldots$,
we define $\rho\trianglelefteq\rho^{\prime}$ if $|\rho|=|\rho^{\prime}|$ and
$e_{i}=e_{i}^{\prime},\sigma_{i}=\sigma_{i}^{\prime}$ and
$\mathbf{n}_{i}\preceq\mathbf{n}_{i}^{\prime}$ for each $i\geq 0$. The
_downclosure_ $\mathord{\downarrow}{R}$ of a set $R$ of preruns of
$\mathfrak{P}$ is defined as
$\mathord{\downarrow}{R}=\\{\rho\in\mathsf{Preruns}(\mathfrak{P})\mid\exists\rho^{\prime}\in
R.\;\rho\trianglelefteq\rho^{\prime}\\}$.
A run of an $\mathsf{AP}$
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ is a
prerun
$\rho=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}),\sigma_{2},\ldots$
starting with the initial configuration $(d_{0},\mathbf{m}_{0})$, where for
each $i\geq 0$, we have
$(d_{i},\mathbf{m}_{i})\xrightarrow{\sigma_{i+1}}(d_{i+1},\mathbf{m}_{i+1})$.
The set of runs of $\mathfrak{P}$ is denoted $\mathsf{Runs}(\mathfrak{P})$.
Finally, $\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$ is the downward
closure of $\mathsf{Runs}(\mathfrak{P})$ with respect to $\trianglelefteq$.
An infinite run
$c_{0}\xrightarrow{\sigma_{0}}c_{1}\xrightarrow{\sigma_{1}}\ldots$ is called
_fair_ if for all $i\geq 0$, if $\sigma\in c_{i}.\mathbf{m}$ then there is
some $j\geq i$ such that $c_{j}\xrightarrow{\sigma}c_{j+1}$. That is, whenever
an instance of a handler $\sigma$ is posted, some instance of $\sigma$ is
executed later. Fairness does not guarantee that every specific instance of a
handler is executed eventually. We say that an infinite fair run _starves_ a
handler $\sigma$ if there exists an index $J\geq 0$ such that for each $j\geq
J$, we have (i) $c_{j}.\mathbf{m}(\sigma)\geq 1$ and (ii) whenever
$c_{j}\xrightarrow{\sigma}c_{j+1}$, we have $c_{j}.\mathbf{m}(\sigma)\geq 2$.
In this case, even if the run is fair, a specific instance of $\sigma$ may
never be executed.
Now we give the definitions of the various decision problems.
[Properties of finite runs] The Safety (Global state reachability) problem
asks, given an asynchronous program $\mathfrak{P}$ and a global state
$d_{f}\in D$, is there a reachable configuration $c$ such that $c.d=d_{f}$? If
so, $d_{f}$ is said to be _reachable_ (in $\mathfrak{P}$) and _unreachable_
otherwise. The Boundedness (of the task buffer) problem asks, given an
asynchronous program $\mathfrak{P}$, is there an $N\in\mathbb{N}$ such that
for every reachable configuration $c$, we have $\lvert{c.\mathbf{m}}\rvert\leq
N$? If so, the asynchronous program $\mathfrak{P}$ is _bounded_ ; otherwise it
is _unbounded_. The Configuration reachability problem asks, given an
asynchronous program $\mathfrak{P}$ and a configuration $c$, is $c$ reachable?
[Properties of infinite runs] All the following problems take as input an
asynchronous program $\mathfrak{P}$. The Termination problem asks if all runs
of $\mathfrak{P}$ are finite. The Fair Non-termination problem asks if
$\mathfrak{P}$ has some fair infinite run. The Fair Starvation problem asks if
$\mathfrak{P}$ has some fair run that starves some handler.
Our main result in this section shows that many properties of an asynchronous
program $\mathfrak{P}$ only depend on the downclosure
$\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$ of the set
$\mathsf{Runs}(\mathfrak{P})$ of runs of the program $\mathfrak{P}$. The proof
is by induction on the length of runs. Please see Appendix B for details. For
any $\mathsf{AP}$
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$, we
define the $\mathsf{AP}$
$\mathord{\downarrow}{\mathfrak{P}}=(D,\Sigma,(\mathord{\downarrow}{L_{c}})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$,
where $\mathord{\downarrow}{L_{c}}$ is the downclosure of the language $L_{c}$
under the subword order.
###### Proposition 2.
Let $\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
be an asynchronous program. Then
$\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}=\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$.
In particular, the following holds. (1) For every $d\in D$, $\mathfrak{P}$ can
reach $d$ if and only if $\mathord{\downarrow}{\mathfrak{P}}$ can reach $d$.
(2) $\mathfrak{P}$ is terminating if and only if
$\mathord{\downarrow}{\mathfrak{P}}$ is terminating. (3) $\mathfrak{P}$ is
bounded if and only if $\mathord{\downarrow}{\mathfrak{P}}$ is bounded.
Intuitively, safety, termination, and boundedness is preserved when the
multiset of pending handler instances is “lossy”: posted handlers can get
lost. This corresponds to these handlers never being added to the task buffer.
However, if a run demonstrates reachability of a global state, or non-
termination, or unboundedness, in the lossy version, it corresponds also to a
run in the original problem (and conversely).
In contrast, simple examples show that configuration reachability, fair
termination, and fair non-starvation properties are not preserved under
downclosures.
## 4\. General Decidability Results
In this section, we characterize those full trios $\mathcal{C}$ for which
particular problems for asynchronous programs over $\mathcal{C}$ are
decidable. Our decision procedures will use the following theorem, summarizing
the results from [GM12], as a subprocedure.
[[GM12]] Safety, boundedness, configuration reachability, termination, fair
non-termination, and fair non-starvation are decidable for asynchronous
programs over regular languages.
### 4.1. Safety and termination
Our first main result concerns the problems of safety and termination.
###### Theorem 3.
Let $\mathcal{C}$ be a full trio. The following are equivalent:
1. (1)
Safety is decidable for asynchronous programs over $\mathcal{C}$.
2. (2)
Termination is decidable for asynchronous programs over $\mathcal{C}$.
3. (3)
Emptiness is decidable for $\mathcal{C}$.
###### Proof 4.1.
We begin with “1$\Rightarrow$3”. Let $K\subseteq\Sigma^{*}$ be given. We
construct
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
such that $\mathbf{m}_{0}={\llbracket\sigma\rrbracket}$,
$D=\\{d_{0},d_{1}\\}$, $L_{d_{0},\sigma,d_{1}}=K$ and $L_{c}=\emptyset$ for
$c\neq(d_{0},\sigma,d_{1})$. We see that $\mathfrak{P}$ can reach $d_{1}$ iff
$K$ is non-empty. Next we show “2$\Rightarrow$3”. Consider the alphabet
$\Gamma=(\Sigma\cup\\{\varepsilon\\})\times\\{0,1\\}$ and the homomorphisms
$g\colon\Gamma^{*}\to\Sigma^{*}$ and $h\colon\Gamma^{*}\to\\{\sigma\\}^{*}$,
where for $x\in\Sigma\cup\\{\varepsilon\\}$, we have $g((x,i))=x$ for
$i\in\\{0,1\\}$, $h((x,1))=\sigma$, and $h((x,0))=\varepsilon$. If
$R\subseteq\Gamma^{*}$ is the regular set of words in which exactly one
position belongs to the subalphabet
$(\Sigma\cup\\{\varepsilon\\})\times\\{1\\}$, then the language
$K^{\prime}:=h(g^{-1}(K)\cap R)$ belongs to $\mathcal{C}$. Note that
$K^{\prime}$ is $\emptyset$ or $\\{\sigma\\}$, depending on whether $K$ is
empty or not. We construct
$\mathfrak{P}=(D,\\{\sigma\\},(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
with $D=\\{d_{0}\\}$, $\mathbf{m}_{0}={\llbracket\sigma\rrbracket}$,
$L_{d_{0},\sigma,d_{0}}=K^{\prime}$ and all languages $L_{c}=\emptyset$ for
$c\neq(d_{0},\sigma,d_{0})$. Then $\mathfrak{P}$ is terminating iff $K$ is
empty.
To prove “3$\Rightarrow$1”, we design an algorithm deciding safety assuming
decidability of emptiness. Given asynchronous program $\mathfrak{P}$ and state
$d$ as input, the algorithm consists of two semi-decision procedures: one
which searches for a run of $\mathfrak{P}$ reaching the state $d$, and the
second which enumerates regular overapproximations $\mathfrak{P}^{\prime}$ of
$\mathfrak{P}$ and checks the safety of $\mathfrak{P}^{\prime}$ using Section
4. Each $\mathfrak{P}^{\prime}$ consists of a regular language $A_{c}$
overapproximating $L_{c}$ for each context $c$ of $\mathfrak{P}$. We use
decidability of emptiness to check that $L_{c}\cap(\Sigma^{*}\setminus
A_{c})=\emptyset$ to ensure that $\mathfrak{P}^{\prime}$ is indeed an
overapproximation.
Input: Asynchronous program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ over
$\mathcal{C}$, state $d\in D$
run concurrently
begin /* find a safe overapproximation */
foreach _tuple $(A_{c})_{c\in\mathfrak{C}}$ of regular languages
$A_{c}\subseteq\Sigma^{*}$_ do
if _$L_{c}\cap(\Sigma^{*}\setminus A_{c})=\emptyset$ for each
$c\in\mathfrak{C}$_ then
if
_$\mathfrak{P}^{\prime}=(D,\Sigma,(A_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
does not reach $d$_ then
return _$d$ is not reachable._
begin /* find a run reaching $d$ */
foreach _prerun $\rho$ of $\mathfrak{P}$_ do
if _$\rho$ is a run that reaches $d$_ then
return _$d$ reachable._
Algorithm 1 Checking Safety
Algorithm 1 clearly gives a correct answer if it terminates. Hence, we only
have to argue that it always does terminate. Of course, if $d$ is reachable,
the first semi-decision procedure will terminate. In the other case,
termination is due to the regularity of downclosures: if $d$ is not reachable
in $\mathfrak{P}$, then Proposition 2 tells us that
$\mathord{\downarrow}{\mathfrak{P}}$ cannot reach $d$ either. But
$\mathord{\downarrow}{\mathfrak{P}}$ is an asynchronous program over regular
languages; this means there exists a safe regular overapproximation and the
second semi-decision procedure terminates.
To prove “3$\Rightarrow$2”, we adopt a similar method as for safety. Algorithm
2 for termination consists of two semi-decision procedures. By standard well-
quasi-ordering arguments, an infinite run of an asynchronous program
$\mathfrak{P}$ is witnessed by a finite self-covering run. The first semi-
decision procedure enumerates finite self-covering runs (trying to show non-
termination). The second procedure enumerates regular asynchronous programs
$\mathfrak{P}^{\prime}$ that overapproximate $\mathfrak{P}$. As before, to
check termination of $\mathfrak{P}^{\prime}$, it applies the procedure from
Theorem 4. Clearly, the algorithm’s answer is always correct. Moreover, it
gives an answer for every input. If $\mathfrak{P}$ does not terminate, it will
find a self-covering sequence. If $\mathfrak{P}$ does terminate, then
Proposition 2 tells us that $\mathord{\downarrow}{\mathfrak{P}}$ is a
terminating finite-state overapproximation. This implies that the second
procedure will terminate in that case.
Input: Asynchronous program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ over
$\mathcal{C}$
run concurrently
begin /* find a terminating overapproximation */
foreach _tuple $(A_{c})_{c\in\mathfrak{C}}$ of regular languages
$A_{c}\subseteq\Sigma^{*}$_ do
if _$L_{c}\cap(\Sigma^{*}\setminus A_{c})=\emptyset$ for each
$c\in\mathfrak{C}$_ then
if
_$\mathfrak{P}^{\prime}=(D,\Sigma,(A_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
terminates_ then
return _$\mathfrak{P}$ terminates._
begin /* find a self-covering run */
foreach _prerun $\rho$ of $\mathfrak{P}$_ do
if _$\rho$ is a self-covering run_ then
return _$\mathfrak{P}$ does not terminate._
Algorithm 2 Checking Termination
Let us point out a particular example. The class $\mathcal{L}$ of languages of
lossy channel systems is defined like the class of languages of Well-
Structured Transition Systems (WSTS) with upward-closed sets of accepting
configurations as in [GRB07], except that we only consider lossy channel
systems [ABJ98] instead of arbitrary WSTS. Then $\mathcal{L}$ forms a full
trio with decidable emptiness. Although downclosures of lossy channel
languages are not effectively computable (an easy consequence of [May03]), our
algorithm employs Theorem 3 to decide safety and termination.
### 4.2. Boundedness
###### Theorem 4.
Let $\mathcal{C}$ be a full trio. The following are equivalent:
1. (1)
Boundedness is decidable for asynchronous programs over $\mathcal{C}$.
2. (2)
Finiteness is decidable for $\mathcal{C}$.
###### Proof 4.2.
Clearly, the construction for “1$\Rightarrow$3” of Theorem 3 also works for
“1$\Rightarrow$2”: $\mathfrak{P}$ is unbounded iff $K$ is infinite.
For the converse, we first note that if finiteness is decidable for
$\mathcal{C}$ then so is emptiness. Given $L\subseteq\Sigma^{*}$ from
$\mathcal{C}$, consider the homomorphism
$h\colon(\Sigma\cup{\\{\lambda\\}})^{*}\to\Sigma^{*}$ with
$h(\mathtt{a})=\mathtt{a}$ for every $\mathtt{a}\in\Sigma$ and
$h(\lambda)=\varepsilon$. Then $h^{-1}(L)$ belongs to $\mathcal{C}$ and
$h^{-1}(L)$ is finite if and only if $L$ is empty: in the inverse
homomorphism, $\lambda$ can be arbitrarily inserted in any word. By Theorem 3,
this implies that we can also decide safety. As a consequence of considering
only full trios, it is easy to see that the problem of _context reachability_
reduces to safety: a context $c=(d,\sigma,d^{\prime})\in\mathfrak{C}$ is
_reachable in $\mathfrak{P}$_ if there is a reachable configuration
$(d,\mathbf{m})$ in $\mathfrak{P}$ with $\mathbf{m}(\sigma)\geq 1$. This
latter condition can be checked by moving from $d$ to a special state via a
$\sigma$ transition.
Input: Asynchronous program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ over
$\mathcal{C}$
$F\leftarrow\emptyset$, $I\leftarrow\emptyset$;
foreach _context $c=(d,\sigma,d^{\prime})\in\mathfrak{C}$_ do
if _$L_{c}$ is infinite_ then
if _$c$ is reachable in $\mathfrak{P}$_ then /* using algorithm for safety */
return _$\mathfrak{P}$ is unbounded._
$I\leftarrow I\cup\\{c\\}$
else
$F\leftarrow F\cup\\{c\\}$
foreach _context $c\in F$_ do
foreach _finite set $A\subseteq\Sigma^{*}$_ do /* find a finite $A$ with
$L_{c}=A$ */
if _$L_{c}\cap(\Sigma^{*}\setminus A)=\emptyset$ and
$L_{c}\cap\\{w\\}\neq\emptyset$ for each $w\in A$_ then
$A_{c}\leftarrow A$;
break __;
/* end inner foreach */
foreach _context $c\in I$_ do
$A_{c}\leftarrow\emptyset$
if
_$\mathfrak{P}^{\prime}=(D,\Sigma,(A_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
is bounded_ then
return _$\mathfrak{P}$ is bounded._
else
return _$\mathfrak{P}$ is unbounded._
Algorithm 3 Checking Boundedness
We now explain Algorithm 3 for deciding boundedness of a given aysnchronous
program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$. For
every context $c$, we first check if $L_{c}$ is infinite (feasible by
assumption). This paritions the set of contexts of $\mathfrak{P}$ into sets
$I$ and $F$ which are the contexts for which the corresponding language
$L_{c}$ is infinite and finite respectively. If any context in $I$ is
reachable, then $\mathfrak{P}$ is unbounded. Otherwise, all the reachable
contexts have a finite language. For every finite language $L_{c}$ for some
$c\in F$, we explicitly find all the members of $L_{c}$. This is possible
because any finite set $A$ can be checked with $L_{c}$ for equality.
$L_{c}\subseteq A$ can be checked by testing whether
$L_{c}\cap(\Sigma^{*}\setminus A)=\emptyset$ and
$L_{c}\cap(\Sigma^{*}\setminus A)$ effectively belongs to $\mathcal{C}$. On
the other hand, checking $A\subseteq L_{c}$ just means checking whether
$L_{c}\cap\\{w\\}\neq\emptyset$ for each $w\in A$, which can be done the same
way. We can now construct asynchronous program $\mathfrak{P}^{\prime}$ which
replaces all languages for contexts in $I$ by $\emptyset$ and replaces those
corresponding to $F$ by the explicit description. Clearly
$\mathfrak{P}^{\prime}$ is bounded iff $\mathfrak{P}$ is bounded (since no
contexts from $I$ are reachable) and the former can be decided by Section 4.
We observe that boundedness is strictly harder than safety or termination:
There are full trios for which emptiness is decidable, but finiteness is
undecidable, such as the languages of reset vector addition systems [DFS98]
(see [TZ19] for a definition of the language class) and languages of lossy
channel systems.
### 4.3. Configuration reachability and liveness properties
Theorems 3 and 4 completely characterize for which full trios safety,
termination, and boundedness are decidable. We turn to configuration
reachability, fair termination, and fair starvation. We suspect that it is
unlikely that there is a simple characterization of those language classes for
which the latter problems are decidable. However, we show that they are
decidable only for a limited range of infinite-state systems. To this end, we
prove that decidability of any of these problems implies (a) decidability of
the others as well, and also implies (b) the decidability of a simple
combinatorial problem (called the $Z$-intersection problem) that is known to
be undecidable for many expressive classes of languages.
Let $Z\subseteq\\{\mathtt{a},\mathtt{b}\\}^{*}$ be the language
$Z=\\{w\in\\{\mathtt{a},\mathtt{b}\\}^{*}\mid|w|_{\mathtt{a}}=|w|_{\mathtt{b}}\\}$.
The _$Z$ -intersection problem_ for a language class $\mathcal{C}$ asks, given
a language $K\subseteq\\{\mathtt{a},\mathtt{b}\\}^{*}$ from $\mathcal{C}$,
whether $K\cap Z\neq\emptyset$. Informally, $Z$ is the language of all words
with an equal number of $\mathtt{a}$s and $\mathtt{b}$s and the
$Z$-intersection problem asks if there is a word in $K$ with an equal number
of $\mathtt{a}$s and $\mathtt{b}$s.
###### Theorem 5.
Let $\mathcal{C}$ be a full trio. The following statements are equivalent:
1. (1)
Configuration reachability is decidable for asynchronous programs over
$\mathcal{C}$.
2. (2)
Fair termination is decidable for asynchronous programs over $\mathcal{C}$.
3. (3)
Fair starvation is decidable for asynchronous programs over $\mathcal{C}$.
Moreover, if decidability holds, then $Z$-intersection is decidable for
$\mathcal{C}$.
###### Proof 4.3.
We prove Theorem 5 by providing reductions among the three problems and
showing that $Z$-intersection reduces to configuration reachability. We use
diagrams similar to automata to describe asynchronous programs. Here, circles
represent global states of the program and we draw an edge
$d$$d^{\prime}$$\sigma|L$
in case we have $L_{d,\sigma,d^{\prime}}=L$ in our asynchronous program
$\mathfrak{P}$. Furthermore, we have $L_{d,\sigma,d^{\prime}}=\emptyset$
whenever there is no edge that specifies otherwise. To simplify notation, we
draw an edge $d$$d^{\prime}$$w|L$ in an asynchronous program for a word
$w\in\Sigma^{*}$, $w=\sigma_{1}\ldots\sigma_{n}$ with
$\sigma_{1},\ldots,\sigma_{n}\in\Sigma$, to symbolize a sequence of states
$d$$2$$\cdots$$n$$d^{\prime}$$\sigma_{1}|\\{\varepsilon\\}$$\sigma_{2}|\\{\varepsilon\\}$$\sigma_{n-1}|\\{\varepsilon\\}$$\sigma_{n}|L$
which removes ${\llbracket\sigma_{1},\ldots,\sigma_{n}\rrbracket}$ from the
task buffer and posts a multiset of handlers specified by the language $L$.
Proof of “2$\Rightarrow$1” Given an asynchronous program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ and
a configuration $(d_{f},\mathbf{m}_{f})\in D\times{\mathbb{M}[\Sigma]}$, we
construct asynchronous program $\mathfrak{P}^{\prime}$ as follows. Let
$\mathtt{z}$ be a fresh letter and let
$\mathbf{m}_{f}={\llbracket\sigma_{1},\ldots,\sigma_{n}\rrbracket}$. We obtain
$\mathfrak{P}^{\prime}$ from $\mathfrak{P}$ by adding a new state
$d^{\prime}_{f}$ and including the following edges:
$d_{f}$$d^{\prime}_{f}$$\mathtt{z}\sigma_{1}\cdots\sigma_{n}|\\{\mathtt{z}\\}$$\mathtt{z}|\\{\mathtt{z}\\}$
Starting from $(d_{0},\mathbf{m}_{0}\oplus{\llbracket\mathtt{z}\rrbracket})$,
the program $\mathfrak{P}^{\prime}$ has a fair infinite run iff
$(d_{f},\mathbf{m}_{f})$ is reachable in $\mathfrak{P}$. The ‘if’ direction is
obvious. Conversely, $\mathtt{z}$ has to be executed in any fair run $\rho$ of
$\mathfrak{P}^{\prime}$ which implies that $d^{\prime}_{f}$ is reached by
$\mathfrak{P}^{\prime}$ in $\rho$. Since only $\mathtt{z}$ can be executed at
$d^{\prime}_{f}$ in $\rho$, this means that the multiset is exactly
$\mathbf{m}_{f}$ when $d_{f}$ is reached during $\rho$. Clearly this initial
segment of $\rho$ corresponds to a run of $\mathfrak{P}$ which reaches the
target configuration.
Proof of “3$\Rightarrow$2” We construct
$\mathfrak{P}^{\prime}=(D,\Sigma^{\prime},(L^{\prime}_{c})_{c\in\mathfrak{C}^{\prime}},d_{0},\mathbf{m}^{\prime}_{0})$
given
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
over $\mathcal{C}$ as follows. Let
$\Sigma^{\prime}=\Sigma\cup\\{\mathtt{s}\\}$, where $\mathtt{s}$ is a fresh
handler. Replace each edge
$d$$d^{\prime}$$\sigma|L$ by $d$$d^{\prime}$$\sigma|L\cup
L\mathtt{s}$$\mathtt{s}|\varepsilon$
at every state $d\in D$. Moreover, we set
$\mathbf{m}^{\prime}_{0}=\mathbf{m}_{0}\oplus{\llbracket\mathtt{s}\rrbracket}$.
Then $\mathfrak{P}^{\prime}$ has an infinite fair run that starves some
handler if and only if $\mathfrak{P}$ has an infinite fair run. From an
infinite fair run $\rho$ of $\mathfrak{P}$, we obtain an infinite fair run of
$\mathfrak{P}^{\prime}$ which starves $\mathtt{s}$, by producing $\mathtt{s}$
while simulating $\rho$ and consuming it in the loop. Conversely, from an
infinite fair run $\rho^{\prime}$ of $\mathfrak{P}^{\prime}$ which starves
some $\tau$, we obtain an infinite fair run $\rho$ of $\mathfrak{P}$ by
omitting all productions and consumptions of $\mathtt{s}$ and removing two
extra instances of $\mathtt{s}$ from all configurations.
Proof of “1$\Rightarrow$3” From
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ over
$\mathcal{C}$, for each subset $\Gamma\subseteq\Sigma$ and $\tau\in\Sigma$, we
construct an asynchronous program
$\mathfrak{P}_{\Gamma,\tau}=(D^{\prime},\Sigma^{\prime},(L_{c})_{c\in\mathfrak{C}^{\prime}},d^{\prime}_{0},\mathbf{m}^{\prime}_{0})$
over $\mathcal{C}$ such that a particular configuration is reachable in
$\mathfrak{P}_{\Gamma,\tau}$ if and only if $\mathfrak{P}$ has a fair infinite
run $\rho_{\Gamma,\tau}$, where $\Gamma$ is the set of handlers that is
executed infinitely often in $\rho_{\Gamma,\tau}$ and $\rho_{\Gamma,\tau}$
starves $\tau$. Since there are only finitely many choices for $\Gamma$ and
$\tau$, decidability of configuration reachability implies decidability of
fair starvation. The idea is that a run $\rho_{\Gamma,\tau}$ exists if and
only if there exists a run
$(d_{0},\mathbf{m}_{0})\xrightarrow{\sigma_{1}}\cdots\xrightarrow{\sigma_{n}}(d_{n},\mathbf{m}_{n})=(e_{0},\mathbf{n}_{0})\xrightarrow{\gamma_{1}}(e_{1},\mathbf{n}_{1})\xrightarrow{\gamma_{2}}\cdots\xrightarrow{\gamma_{k}}(e_{k},\mathbf{n}_{k}),$
(1)
where $\bigcup_{i=1}^{k}\\{\gamma_{i}\\}=\Gamma$, for each $1\leq i\leq k$
$\mathbf{n}_{i}\in{\mathbb{M}[\Gamma]}$,
$\mathbf{m}_{n}\preceq\mathbf{n}_{k}$, and for each $i\in\\{1,\ldots,k\\}$
with $\gamma_{i}=\tau$, we have $\mathbf{n}_{i-1}(\tau)\geq 2$. In such a run,
we call
$(d_{0},\mathbf{m}_{0})\xrightarrow{\sigma_{1}}\cdots\xrightarrow{\sigma_{n}}(d_{n},\mathbf{m}_{n})$
its _first phase_ and
$(e_{0},\mathbf{n}_{0})\xrightarrow{\gamma_{1}}\cdots\xrightarrow{\gamma_{k}}(e_{k},\mathbf{n}_{k})$
its _second phase_.
Let us explain how $\mathfrak{P}_{\Gamma,\tau}$ reflects the existence of a
run as in Eq. 1. The set $\Sigma^{\prime}$ of handlers of
$\mathfrak{P}_{\Gamma,\tau}$ includes $\Sigma$, $\bar{\Sigma}$ and
$\hat{\Sigma}$, where $\bar{\Sigma}=\\{\bar{\sigma}\mid\sigma\in\Sigma\\}$ and
$\hat{\Sigma}=\\{\hat{\sigma}\mid\sigma\in\Sigma\\}$ are disjoint copies of
$\Sigma$. This means, a multiset ${\mathbb{M}[\Sigma^{\prime}]}$ contains
multisets
$\mathbf{m}^{\prime}=\mathbf{m}\oplus\bar{\mathbf{m}}\oplus\hat{\mathbf{m}}$
with $\mathbf{m}\in{\mathbb{M}[\Sigma]}$,
$\bar{\mathbf{m}}\in{\mathbb{M}[\bar{\Sigma}]}$, and
$\hat{\mathbf{m}}\in{\mathbb{M}[\hat{\Sigma}]}$. A run of
$\mathfrak{P}_{\Gamma,\tau}$ simulates the two phases of $\rho$. While
simulating the first phase, $\mathfrak{P}_{\Gamma,\tau}$ keeps two copies of
the task buffer, $\mathbf{m}$ and $\bar{\mathbf{m}}$. The copying is easily
accomplished by a homomorphism with $\sigma\mapsto\sigma\bar{\sigma}$ for each
$\sigma\in\Sigma$. At some point, $\mathfrak{P}_{\Gamma,\tau}$ switches into
simulating the second phase. There, $\bar{\mathbf{m}}$ remains unchanged, so
that it stores the value of $\mathbf{m}_{n}$ in Eq. 1 and can be used in the
end to make sure that $\mathbf{m}_{n}\preceq\mathbf{n}_{k}$.
Hence, in the second phase, $\mathfrak{P}_{\Gamma,\tau}$ works, like
$\mathfrak{P}$, only with $\Sigma$. However, whenever a handler
$\sigma\in\Sigma$ is executed, it also produces a task $\hat{\sigma}$. These
handlers are used at the end to make sure that every $\gamma\in\Gamma$ has
been executed at least once in the second phase. Also, whenever $\tau$ is
executed, $\mathfrak{P}_{\Gamma,\tau}$ checks that at least two instances of
$\tau$ are present in the task buffer, thereby ensuring that $\tau$ is
starved.
In the end, a distinguished final state allows $\mathfrak{P}_{\Gamma,\tau}$ to
execute handlers in $\Gamma$ and $\bar{\Gamma}$ simultaneously to make sure
that $\mathbf{m}_{n}\preceq\mathbf{n}_{k}$. In its final state,
$\mathfrak{P}_{\Gamma,\tau}$ can execute handlers
$\hat{\gamma}\in\hat{\Gamma}$ and $\gamma\in\Gamma$ (without creating new
handlers). In the final configuration, there can be no $\hat{\sigma}$ with
$\sigma\in\Sigma\setminus\Gamma$, and there has to be exactly one
$\hat{\gamma}$ for each $\gamma\in\Gamma$. This guarantees that (i) each
handler in $\Gamma$ is executed at least once during the second phase, (ii)
every handler executed in the second phase is from $\Gamma$, and (iii)
$\mathbf{m}_{n}$ contains only handlers from $\Gamma$ (because handlers from
$\bar{\Sigma}$ cannot be executed in the second phase).
Let us now describe $\mathfrak{P}_{\Gamma,\tau}$ in detail. We have
$\Sigma^{\prime}=\Sigma\cup\bar{\Sigma}\cup\hat{\Gamma}\cup\\{\mathtt{z}\\}$,
where $\mathtt{z}$ is a fresh letter. The set of states is
$D^{\prime}=D\cup\tilde{D}\cup\\{d_{f}\\}$, where $\tilde{D}=\\{\tilde{d}\mid
d\in D\\}$ is a disjoint copy of $D$ for simulating the second phase, and
$d_{f}$ is a fresh state. Moreover, we change the languages $L_{c}$ in the
following way:
$\leavevmode\hbox to58.48pt{\vbox
to20.1pt{\pgfpicture\makeatletter\hbox{\hskip 7.9pt\lower-7.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.5pt}{0.0pt}\pgfsys@curveto{7.5pt}{4.14218pt}{4.14218pt}{7.5pt}{0.0pt}{7.5pt}\pgfsys@curveto{-4.14218pt}{7.5pt}{-7.5pt}{4.14218pt}{-7.5pt}{0.0pt}\pgfsys@curveto{-7.5pt}{-4.14218pt}{-4.14218pt}{-7.5pt}{0.0pt}{-7.5pt}\pgfsys@curveto{4.14218pt}{-7.5pt}{7.5pt}{-4.14218pt}{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.60243pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{50.17912pt}{0.0pt}\pgfsys@curveto{50.17912pt}{4.14218pt}{46.8213pt}{7.5pt}{42.67912pt}{7.5pt}\pgfsys@curveto{38.53694pt}{7.5pt}{35.17912pt}{4.14218pt}{35.17912pt}{0.0pt}\pgfsys@curveto{35.17912pt}{-4.14218pt}{38.53694pt}{-7.5pt}{42.67912pt}{-7.5pt}\pgfsys@curveto{46.8213pt}{-7.5pt}{50.17912pt}{-4.14218pt}{50.17912pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67912pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.3067pt}{-4.25pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{7.9pt}{0.0pt}\pgfsys@lineto{32.77916pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.77916pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.46864pt}{3.7pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma|L$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}\leadsto~{}~{}\leavevmode\hbox
to86.93pt{\vbox to21.18pt{\pgfpicture\makeatletter\hbox{\hskip
7.9pt\lower-7.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.5pt}{0.0pt}\pgfsys@curveto{7.5pt}{4.14218pt}{4.14218pt}{7.5pt}{0.0pt}{7.5pt}\pgfsys@curveto{-4.14218pt}{7.5pt}{-7.5pt}{4.14218pt}{-7.5pt}{0.0pt}\pgfsys@curveto{-7.5pt}{-4.14218pt}{-4.14218pt}{-7.5pt}{0.0pt}{-7.5pt}\pgfsys@curveto{4.14218pt}{-7.5pt}{7.5pt}{-4.14218pt}{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.60243pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{78.6319pt}{0.0pt}\pgfsys@curveto{78.6319pt}{4.14218pt}{75.27408pt}{7.5pt}{71.1319pt}{7.5pt}\pgfsys@curveto{66.98972pt}{7.5pt}{63.6319pt}{4.14218pt}{63.6319pt}{0.0pt}\pgfsys@curveto{63.6319pt}{-4.14218pt}{66.98972pt}{-7.5pt}{71.1319pt}{-7.5pt}\pgfsys@curveto{75.27408pt}{-7.5pt}{78.6319pt}{-4.14218pt}{78.6319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{71.1319pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{67.75948pt}{-4.25pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{7.9pt}{0.0pt}\pgfsys@lineto{61.23193pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{61.23193pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.35864pt}{3.7pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\sigma\bar{\sigma}|h^{\pm}(L)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
where $h^{\pm}\colon\Sigma^{*}\to(\Sigma\cup\bar{\Sigma})^{*}$ is the
homomorphism with $\sigma\mapsto\sigma\bar{\sigma}$ for every
$\sigma\in\Sigma$. This means, for every context $c=(d,\sigma,d^{\prime})$, we
have $d\xrightarrow{\sigma\bar{\sigma}|h^{\pm}(L)}d^{\prime}$ in
$\mathfrak{P}_{\Gamma,\tau}$. Note that since $\mathcal{C}$ is a full trio, it
is in particular closed under homomorphisms. Hence, $h^{\pm}(L)$ belongs to
$\mathcal{C}$. Moreover, $\mathfrak{P}_{\Gamma,\tau}$ can spontaneously switch
to simulating the second phase: For each $d\in D$, we have
$d$$\tilde{d}$$\mathtt{z}|\\{\mathtt{z}\\}$
Here, the handler $\mathtt{z}$ merely allows us to move to state $\tilde{d}$
without interfering with the other handlers. In the second phase,
$\mathfrak{P}_{\Gamma,\tau}$ simulates $\mathfrak{P}$ slightly differently. We
perform the following replacement:
$\leavevmode\hbox to58.48pt{\vbox
to20.1pt{\pgfpicture\makeatletter\hbox{\hskip 7.9pt\lower-7.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.5pt}{0.0pt}\pgfsys@curveto{7.5pt}{4.14218pt}{4.14218pt}{7.5pt}{0.0pt}{7.5pt}\pgfsys@curveto{-4.14218pt}{7.5pt}{-7.5pt}{4.14218pt}{-7.5pt}{0.0pt}\pgfsys@curveto{-7.5pt}{-4.14218pt}{-4.14218pt}{-7.5pt}{0.0pt}{-7.5pt}\pgfsys@curveto{4.14218pt}{-7.5pt}{7.5pt}{-4.14218pt}{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.60243pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{50.17912pt}{0.0pt}\pgfsys@curveto{50.17912pt}{4.14218pt}{46.8213pt}{7.5pt}{42.67912pt}{7.5pt}\pgfsys@curveto{38.53694pt}{7.5pt}{35.17912pt}{4.14218pt}{35.17912pt}{0.0pt}\pgfsys@curveto{35.17912pt}{-4.14218pt}{38.53694pt}{-7.5pt}{42.67912pt}{-7.5pt}\pgfsys@curveto{46.8213pt}{-7.5pt}{50.17912pt}{-4.14218pt}{50.17912pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67912pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.3067pt}{-4.25pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$d^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{7.9pt}{0.0pt}\pgfsys@lineto{32.77916pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.77916pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.46864pt}{3.7pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma|L$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}\leadsto~{}~{}\begin{cases}\leavevmode\hbox
to86.93pt{\vbox to20.1pt{\pgfpicture\makeatletter\hbox{\hskip
7.9pt\lower-7.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.5pt}{0.0pt}\pgfsys@curveto{7.5pt}{4.14218pt}{4.14218pt}{7.5pt}{0.0pt}{7.5pt}\pgfsys@curveto{-4.14218pt}{7.5pt}{-7.5pt}{4.14218pt}{-7.5pt}{0.0pt}\pgfsys@curveto{-7.5pt}{-4.14218pt}{-4.14218pt}{-7.5pt}{0.0pt}{-7.5pt}\pgfsys@curveto{4.14218pt}{-7.5pt}{7.5pt}{-4.14218pt}{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.77779pt}{-3.61111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tilde{d}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{78.6319pt}{0.0pt}\pgfsys@curveto{78.6319pt}{4.14218pt}{75.27408pt}{7.5pt}{71.1319pt}{7.5pt}\pgfsys@curveto{66.98972pt}{7.5pt}{63.6319pt}{4.14218pt}{63.6319pt}{0.0pt}\pgfsys@curveto{63.6319pt}{-4.14218pt}{66.98972pt}{-7.5pt}{71.1319pt}{-7.5pt}\pgfsys@curveto{75.27408pt}{-7.5pt}{78.6319pt}{-4.14218pt}{78.6319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{71.1319pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{67.58412pt}{-4.38889pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tilde{d}^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{7.9pt}{0.0pt}\pgfsys@lineto{61.23193pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{61.23193pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.91725pt}{3.7pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\sigma|L\hat{\sigma}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\text{if
$\sigma\neq\tau$}\\\ \leavevmode\hbox to86.93pt{\vbox
to20.1pt{\pgfpicture\makeatletter\hbox{\hskip 7.9pt\lower-7.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.5pt}{0.0pt}\pgfsys@curveto{7.5pt}{4.14218pt}{4.14218pt}{7.5pt}{0.0pt}{7.5pt}\pgfsys@curveto{-4.14218pt}{7.5pt}{-7.5pt}{4.14218pt}{-7.5pt}{0.0pt}\pgfsys@curveto{-7.5pt}{-4.14218pt}{-4.14218pt}{-7.5pt}{0.0pt}{-7.5pt}\pgfsys@curveto{4.14218pt}{-7.5pt}{7.5pt}{-4.14218pt}{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.77779pt}{-3.61111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tilde{d}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@fill{0.9}\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0.3,0.3,0.3}\pgfsys@color@gray@stroke{0.3}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{78.6319pt}{0.0pt}\pgfsys@curveto{78.6319pt}{4.14218pt}{75.27408pt}{7.5pt}{71.1319pt}{7.5pt}\pgfsys@curveto{66.98972pt}{7.5pt}{63.6319pt}{4.14218pt}{63.6319pt}{0.0pt}\pgfsys@curveto{63.6319pt}{-4.14218pt}{66.98972pt}{-7.5pt}{71.1319pt}{-7.5pt}\pgfsys@curveto{75.27408pt}{-7.5pt}{78.6319pt}{-4.14218pt}{78.6319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{71.1319pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{67.58412pt}{-4.38889pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tilde{d}^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{7.9pt}{0.0pt}\pgfsys@lineto{61.23193pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{61.23193pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.21701pt}{3.7pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\tau\tau|L\tau\hat{\tau}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\text{if
$\sigma=\tau$}\end{cases}$ (2)
Note that since $\mathcal{C}$ is a full trio, the languages $Lv=\\{uv\mid u\in
L\\}$ belong to $\mathcal{C}$ for every word $v$. Finally,
$\mathfrak{P}_{\Gamma,\tau}$ can spontaneously switch to its distinguished
state $d_{f}$, so that we have for every $\tilde{d}\in\tilde{D}$ and every
$\gamma\in\Gamma$:
$\tilde{d}$$d_{f}$$\mathtt{z}|\\{\varepsilon\\}$$\gamma\bar{\gamma}|\\{\varepsilon\\}$$\gamma|\\{\varepsilon\\}$$\hat{\gamma}|\\{\varepsilon\\}$
(3)
The initial configuration of $\mathfrak{P}_{\Gamma,\tau}$ is
$(d_{0},\mathbf{m}^{\prime}_{0})$, where
$\mathbf{m}^{\prime}_{0}=\mathbf{m}_{0}\oplus\bar{\mathbf{m}}_{0}\oplus{\llbracket\mathtt{z}\rrbracket}$,
where $\bar{\mathbf{m}}_{0}$ is obtained from $\mathbf{m}_{0}$ by replacing
every occurrence of $\sigma\in\Sigma$ in $\mathbf{m}_{0}$ with $\bar{\sigma}$.
The final configuration is $(d_{f},\mathbf{m}_{f})$, where
$\mathbf{m}_{f}\in{\mathbb{M}[\hat{\Gamma}]}$ is the multiset with
$\mathbf{m}_{f}(\hat{\gamma})=1$ for every $\gamma\in\Gamma$. It is now
straightforward to check that $(d_{f},\mathbf{m}_{f})$ is reachable in
$\mathfrak{P}_{\Gamma,\tau}$ if and only if $\mathfrak{P}$ has an infinite
fair run that starves $\tau$.
Decidability of $Z$-intersection To complete the proof of Theorem 5, we reduce
$Z$-intersection to configuration reachability. Given
$K\subseteq\\{\mathtt{a},\mathtt{b}\\}^{*}$ from $\mathcal{C}$, we construct
the asynchronous program
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ over
$\mathcal{C}$ where $D=\\{d_{0},0,1\\}$,
$\Sigma=\\{\mathtt{a},\mathtt{b},\mathtt{c}\\}$, by including the following
edges:
$d_{0}$$0$$1$$\mathtt{c}|K$$\mathtt{a}|\\{\varepsilon\\}$$\mathtt{b}|\\{\varepsilon\\}$
The initial task buffer is $\mathbf{m}_{0}={\llbracket\mathtt{c}\rrbracket}$.
Then clearly, the configuration $(0,{\llbracket\rrbracket})$ is reachable in
$\mathfrak{P}$ if and only if $K\cap Z\neq\emptyset$.
If the construction seems abstract, recall the example from Section 2: the
procedure s1 plays the role of $K$ and generates strings from its language in
${\\{\mathtt{a},\mathtt{b}\\}}^{*}$. The procedures $\mathtt{a}$ and
$\mathtt{b}$ take turns to ensure there is an equal number of them; the states
$0$ and $1$ are values of turn.
Theorem 5 is useful in the contrapositive to show undecidability. For example,
one can show undecidability of $Z$-intersection for languages of lossy channel
systems (see Section 4.1): One expresses reachability in a non-lossy FIFO
system by making sure that the numbers of enqueue- and dequeue-operations
match. Thus, for asynchronous programs over lossy channel systems, the
problems of Theorem 5 are undecidable. We also use Theorem 5 in Section 5 to
conclude undecidability for higher-order asynchronous programs, already at
order $2$.
## 5\. Higher-Order Asynchronous Programs
We apply our general decidability results to asynchronous programs over
(deterministic) higher-order recursion schemes ($\mathsf{HORS}$). Kobayashi
[Kob09] has shown how higher-order functional programs can be modeled using
$\mathsf{HORS}$. In his setting, a program contains instructions that access
certain resources. The path language of the $\mathsf{HORS}$ produced by
Kobayashi is the set of possible sequences of instructions. For us, the input
program contains post instructions (which post handlers i.e. add tasks to the
task buffer) and we translate higher-order programs with post instructions
into a $\mathsf{HORS}$ whose path language is used as the language of
handlers.
We recall some definitions from [Kob09]. The set of types is defined by the
grammar $A\ :=\ \mathtt{o}\mid A\rightarrow A$. The order
$\operatorname{ord}(A)$ of a type $A$ is inductively defined as
$\operatorname{ord}(\mathtt{o})=0$ and $\operatorname{ord}(A\rightarrow
B):=\max(\operatorname{ord}(A)+1,\operatorname{ord}(B))$. The arity of a type
is inductively defined by $\operatorname{arity}(\mathtt{o})=0$ and
$\operatorname{arity}(A\rightarrow B)=\operatorname{arity}(B)+1$. We assume a
countably infinite set $\mathrm{Var}$ of typed variables $x:A$. For a set
$\Theta$ of typed symbols, the set $\tilde{\Theta}$ of terms generated from
$\Theta$ is the least set which contains $\Theta$ such that whenever $s:A\to
B$ and $t:A$ belong to $\tilde{\Theta}$, then also $s\,t:B$ belongs to
$\tilde{\Theta}$. By convention the type
$\mathtt{o}\rightarrow\ldots(\mathtt{o}\rightarrow(\mathtt{o}\rightarrow\mathtt{o}))$
is written
$\mathtt{o}\rightarrow\ldots\rightarrow\mathtt{o}\rightarrow\mathtt{o}$ and
the term $((t_{1}t_{2})t_{3}\cdots)t_{n}$ is written $t_{1}t_{2}\cdots t_{n}$.
We write $\bar{x}$ for a sequence $(x_{1},x_{2},\ldots,x_{n})$ of variables.
A higher-order recursion scheme ($\mathsf{HORS}$) is a tuple
$\mathscr{S}=(\Sigma,\mathcal{N},\mathcal{R},S)$ where $\Sigma$ is a set of
typed terminal symbols of types of order 0 or 1, $\mathcal{N}$ is a set of
typed non-terminal symbols (disjoint from terminal symbols), $S:\mathtt{o}$ is
the start non-terminal symbol and $\mathcal{R}$ is a set of rewrite rules
$Fx_{1}x_{2}\cdots x_{n}\twoheadrightarrow t$ where
$F:A_{1}\rightarrow\cdots\rightarrow A_{n}\rightarrow\mathtt{o}$ is a non-
terminal in $\mathcal{N}$, $x_{i}:A_{i}$ for all $i$ are variables and
$t:\mathtt{o}$ is a term generated from
$\Sigma\cup\mathcal{N}\cup\\{x_{1},x_{2},\ldots,x_{n}\\}$. The order of a
$\mathsf{HORS}$ is the maximum order of a non-terminal symbol. We define a
rewrite relation $\twoheadrightarrow$ on terms over $\Sigma\cup\mathcal{N}$ as
follows: $F\bar{a}\twoheadrightarrow t[\bar{x}/\bar{a}]$ if
$F\bar{x}\twoheadrightarrow t\in\mathcal{R}$, and if $t\twoheadrightarrow
t^{\prime}$ then $ts\twoheadrightarrow t^{\prime}s$ and $st\twoheadrightarrow
st^{\prime}$. The reflexive, transitive closure of $\twoheadrightarrow$ is
denoted $\twoheadrightarrow^{*}$. A sentential form $t$ of $\mathscr{S}$ is a
term over $\Sigma\cup\mathcal{N}$ such that $S\twoheadrightarrow^{*}t$.
If $N$ is the maximum arity of a symbol in $\Sigma$, then a (possibly
infinite) tree over $\Sigma$ is a partial function $tr$ from
$\\{0,1,\ldots,N-1\\}^{*}$ to $\Sigma$ that fulfills the following conditions:
$\varepsilon\in\operatorname{dom}(tr)$, $\operatorname{dom}(tr)$ is closed
under prefixes, and if $tr(w)=a$ and $\operatorname{arity}(a)=k$ then
$\\{j\mid wj\in\operatorname{dom}(tr)\\}=\\{0,1,\ldots,k-1\\}$.
A _deterministic_ $\mathsf{HORS}$ is one where there is exactly one rule of
the form $Fx_{1}x_{2}\cdots x_{n}\rightarrow t$ for every non-terminal $F$.
Following [Kob09], we show how a deterministic $\mathsf{HORS}$ can be used to
represent a higher-order pushdown language arising from a higher-order
functional program. Sentential forms can be seen as trees over
$\Sigma\cup\mathcal{N}$. A sequence $\Pi$ over $\\{0,1,\ldots,N-1\\}$ is a
path of $tr$ if every finite prefix of $\Pi\in\operatorname{dom}(tr)$ and
$\Pi$ is maximal. The set of finite paths in a tree $tr$ will be denoted
$\mathsf{Paths}(tr)$. Unless otherwise specified, a path is finite in our
context. Associated with any path $\Pi=n_{1},n_{2},\ldots,n_{k}$ is the word
$w_{\Pi}=tr(n_{1})tr(n_{1}n_{2})\cdots tr(n_{1}n_{2}\cdots n_{k}).$
We will also write $w_{\Pi}(\bar{m})$ to denote the letter $tr(\bar{m})$ where
$\bar{m}=n_{1}n_{2}\ldots n_{l}$ for some $l\leq k$. We associate a value tree
$\mathcal{T}_{\mathscr{S}}$ associated with a deterministic $\mathsf{HORS}$
$\mathscr{S}$ in the following way. For a sentential form $t$ define the
finite tree $t^{\bot}$ by case analysis: $t^{\bot}=f$ if $f$ is a terminal
symbol and $t^{\bot}=t_{1}^{\bot}t_{2}^{\bot}$ if $t=t_{1}t_{2}$ and
$t_{1}^{\bot}\neq\bot$, and $t^{\bot}=\bot$ otherwise.
Intuitively, the tree $t^{\bot}$ is obtained by replacing any subterm whose
root is a non-terminal by the nullary symbol $\bot$. We define the partial
order $\leq_{t}$ on $\operatorname{dom}(\Sigma)\cup\\{\bot\\}$ by
$\bot\leq_{t}a$ for all $a\in\operatorname{dom}(\Sigma)$, which is extended to
trees by
$t\leq_{t}s\iff\forall\bar{n}\in\operatorname{dom}(t):\bar{n}\in\operatorname{dom}(s)\land
t(\bar{n})\leq_{t}s(\bar{n})$
The value tree generated by a $\mathsf{HORS}$ $\mathscr{S}$ is denoted
$\mathcal{T}_{\mathscr{S}}$ and is obtained by repeated application of the
rewrite rules to the start symbol $S$. To make this formal, we write
$\operatorname{lub}(T)$ for the least upper bound of a collection $T$ of trees
with respect to the order $\leq_{t}$. Then we set
$\mathcal{T}_{\mathscr{S}}:=\operatorname{lub}(\\{t^{\bot}\mid
S\rightarrow^{*}t\\})$. Any $\mathsf{HORS}$ for which the value tree
$\mathcal{T}_{\mathscr{S}}$ does not contain any $\bot$ symbol is called
_productive_. All $\mathsf{HORS}$ dealt with in this paper are assumed to be
productive. Checking if a given $\mathsf{HORS}$ is productive is decidable,
see, e.g., [Gre16].
Let $\Sigma_{1}:=\\{a\in\Sigma\mid\operatorname{arity}(a)=1\\}$. The path
language $\mathcal{L}_{\mathsf{p}}(\mathscr{S})$ of a deterministic
$\mathsf{HORS}$ $\mathscr{S}$ is defined as
$\\{\mathrm{Proj}_{\Sigma_{1}}(w_{\Pi})\mid\Pi\in\mathsf{Paths}(\mathcal{T}_{\mathscr{S}})\\}$.
The tree language $\mathcal{L}_{\mathsf{t}}(\mathscr{S})$ associated with a
(nondeterministic) $\mathsf{HORS}$ is the set of sentential forms of
$\mathscr{S}$ which contain only labels from $\Sigma$.
The deterministic $\mathsf{HORS}$ corresponding to the higher-order function
$\mathtt{s4}$ from Figure 1 is given by
$\mathscr{S}=(\Sigma,\mathcal{N},\mathcal{R},S)$, where
$\displaystyle\Sigma=$
$\displaystyle\\{\mathtt{br}:\mathtt{o}\rightarrow\mathtt{o}\rightarrow\mathtt{o},\mathtt{c},\mathtt{d},\mathtt{f}:\mathtt{o}\rightarrow\mathtt{o},\mathtt{e}:\mathtt{o}\\}$
$\displaystyle\mathcal{N}=$
$\displaystyle\\{S:\mathtt{o},F:(\mathtt{o}\rightarrow\mathtt{o})\rightarrow\mathtt{o}\rightarrow\mathtt{o},H:(\mathtt{o}\rightarrow\mathtt{o})\rightarrow\mathtt{o}\rightarrow\mathtt{o},I:\mathtt{o}\rightarrow\mathtt{o}\\}$
$\displaystyle\mathcal{R}=$ $\displaystyle\\{S\twoheadrightarrow
F\;I\;\mathtt{e},I\;x\twoheadrightarrow
x,F\;G\;x\twoheadrightarrow\mathtt{br}(F\;(H\;G)\;(\mathtt{f}x))\;(G\;x),H\;G\;x\twoheadrightarrow\mathtt{c}(G(\mathtt{d}x))\\}$
The path language
$\mathcal{L}_{\mathsf{p}}(\mathscr{S})={\\{\mathtt{c}^{n}\mathtt{d}^{n}\mathtt{f}^{n}\mid
n\geq 0\\}}$. To see this, apply the reduction rules to get the value tree
$\mathcal{T}_{\mathscr{S}}$ shown on the right:
$\displaystyle S$ $\displaystyle\twoheadrightarrow
F\;I\;\mathtt{e}\twoheadrightarrow\mathtt{br}\;(F\;(HI)\;(\mathtt{fe}))\;(I\mathtt{e})$
$\displaystyle\twoheadrightarrow\mathtt{br}\;(F\;(HI)\;(\mathtt{fe}))\;\mathtt{e}$
$\displaystyle\twoheadrightarrow\mathtt{br}\;(\mathtt{br}\;(F\;(H(HI))\;(\mathtt{f}(\mathtt{fe})))\;(HI(\mathtt{fe})))\;\mathtt{e}$
$\displaystyle\twoheadrightarrow\mathtt{br}\;(\mathtt{br}\;(F\;(H(HI))\;(\mathtt{f}(\mathtt{fe})))\;(\mathtt{c}(I(\mathtt{d(fe)}))))\;\mathtt{e}$
$\displaystyle\twoheadrightarrow\mathtt{br}\;(\mathtt{br}\;(F\;(H(HI))\;(\mathtt{f}(\mathtt{fe})))\;(\mathtt{c(d(fe))}))\;\mathtt{e}$
$\displaystyle\twoheadrightarrow\cdots$
$\mathtt{br}$$\mathtt{e}$$\mathtt{br}$$\mathtt{c}$$\mathtt{br}$$\mathtt{d}$$\mathtt{f}$$\mathtt{e}$$..$$..$
A $\mathsf{HORS}$ $\mathscr{S}$ is called a _word scheme_ if it has exactly
one nullary terminal symbol $\mathsf{e}$ and all other terminal symbols
$\tilde{\Sigma}$ are of arity one. The word language
$\mathcal{L}_{\mathsf{w}}(\mathscr{S})\subseteq\tilde{\Sigma}^{*}$ defined by
$\mathscr{S}$ is $\mathcal{L}_{\mathsf{w}}(\mathscr{S})=\\{a_{1}a_{2}\cdots
a_{n}\mid(a_{1}(a_{2}\cdots(a_{n}(\mathsf{e}))\cdots))\in\mathcal{L}_{\mathsf{t}}(\mathscr{S})\\}$.
We denote by $\mathcal{H}$ the class of languages
$\mathcal{L}_{\mathsf{w}}(\mathscr{S})$ that occur as the word language of a
higher-order recursion scheme $\mathscr{S}$. Note that path languages and
languages of word schemes are both word languages over the set
$\tilde{\Sigma}$ of unary symbols considered as letters. They are connected by
the following proposition, a proof of which is given in Appendix C.222 The
models of $\mathsf{HORS}$ (used in model checking higher-order programs
[Kob09]) and word schemes (used in language-theoretic exploration of
downclosures [HKO16, CPSW16]) are somewhat different. Thus, we show an
explicit reduction between the two formalisms.
###### Proposition 6.
For every order-$n$ $\mathsf{HORS}$
$\mathscr{S}=(\Sigma,\mathcal{N},S,\mathcal{R})$ there exists an order-$n$
word scheme
$\mathscr{S}^{\prime}=(\Sigma^{\prime},\mathcal{N}^{\prime},S^{\prime},\mathcal{R}^{\prime})$
such that
$\mathcal{L}_{\mathsf{p}}(\mathscr{S})=\mathcal{L}_{\mathsf{w}}(\mathscr{S}^{\prime})$.
A consequence of [Kob09] and Prop. 6 is that the “post” language of higher-
order functional programs (i.e. the language corresponding to sequences of
newly spawned tasks) can be modeled as the language of a word scheme. Hence,
we define an _asynchronous program over $\mathsf{HORS}$_ as an asynchronous
program over the language class $\mathcal{H}$ and we can use the following
results on word schemes.
###### Theorem 7.
$\mathsf{HORS}$ and word schemes form effective full trios [CPSW16]. Emptiness
[KO11] and finiteness [Par17] of order-$n$ word schemes are ${(n-1)}\mathchar
45\relax\mathsf{EXPTIME}$-complete.
Now Theorems 3 and 4, together with Proposition 6 imply the decidability
results in Corollary 8. The undecidability result is a consequence of Theorem
5 and the undecidability of the $Z$-intersection problem for indexed languages
or equivalently, order-2 pushdown automata as shown in [Zet15]. Order-2
pushdown automata can be effectively turned into order-2 OI grammars [DG86],
which in turn can be translated into order-2 word schemes [Dam82]. A direct
(and independent) proof of undecidability of the $Z$-intersection problem for
order-2 word schemes has appeared in [Kob19, Theorem 4].
###### Corollary 8.
For asynchronous programs over $\mathsf{HORS}$: (1) Safety, termination, and
boundedness are decidable. (2) Configuration reachability, fair termination,
and fair starvation are undecidable already at order-2.
## 6\. A Direct Algorithm and Complexity Analysis
We say that _downclosures are computable_ for a language class $\mathcal{C}$
if for a given description of a language $L$ in $\mathcal{C}$, one can compute
an automaton for the regular language $\mathord{\downarrow}{L}$. A consequence
of Proposition 2 and Theorem 4 is that if one can compute downclosures for
some language class, then one can avoid the enumerative approaches of Section
4 and get a “direct algorithm.” The direct algorithm replaces each handler by
its downclosure and then invokes the decision procedure summarized in Theorem
4.
### 6.1. Higher-Order Recursion Schemes
The direct algorithm for asynchronous programs over $\mathsf{HORS}$ relies on
the recent breakthrough results on computing downclosures. [[Zet15, HKO16,
CPSW16]] Downclosures are effectively computable for $\mathcal{H}$.
Unfortunately, current techniques do not yet provide a complexity upper bound
based on the above theorem. To understand why, we describe the current
algorithm for computing downclosures. In [Zet15], it was shown that in a full
trio $\mathcal{C}$, downclosures are computable if and only if the _diagonal
problem_ for $\mathcal{C}$ is decidable. The latter asks, given a language
$L\subseteq\Sigma^{*}$, whether for every $k\in{\mathbb{N}}$, there is a word
$w\in L$ with $|w|_{\sigma}\geq k$ for every $\sigma\in\Sigma$. The diagonal
problem was then shown to be decidable for higher-order pushdown automata
[HKO16] and then for word schemes [CPSW16]. However, the algorithm from
[Zet15] to compute downclosures using an oracle for the diagonal problem
employs enumeration to compute a downclosure automaton. Thus, the enumeration
is hidden inside the downclosure computation.
We conjecture that downclosures can in fact be computed in elementary time
(for word schemes of fixed order). This would imply an elementary time
procedure for asynchronous programs over $\mathsf{HORS}$ as well.
### 6.2. Context Free Languages
For handlers over context-free languages, given e.g., as pushdown automata,
Ganty and Majumdar show an ${\mathsf{EXPSPACE}}$ upper bound. Precisely, the
algorithm of [GM12] constructs for each handler a polynomial-size Petri net
with certain guarantees (forming a so-called _adequate family of Petri nets_)
that accepts a Parikh equivalent language. These Petri nets are then used to
construct a larger Petri net, polynomial in the size of the asynchronous
program and the adequate family of Petri nets, in which the respective
property (safety, boundedness, or termination) can be phrased as a query
decidable in $\mathsf{EXPSPACE}$. A natural question is whether there is a
downclosure-based algorithm with the same asymptotic complexity.
Our goal is to replace the Parikh-equivalent Petri nets with Petri nets
recognizing the downclosure of a language. It is an easy consequence of
Proposition 2 that the resulting Petri nets can be used in place of the
adequate families of Petri nets in the procedures for safety, termination, and
boundedness of [GM12]. Thus, if we can ensure these Petri nets are polynomial
in the size of the handler, we get an ${\mathsf{EXPSPACE}}$ upper bound.
Unfortunately, a finite automaton for $\mathord{\downarrow}{L}$ may require
exponentially many states in the pushdown automaton [BLS15]. Thus a naive
approach gives a $\mathsf{2EXPSPACE}$ upper bound.
We show here that for each context-free language $L$, one can construct in
polynomial time a $1$-bounded Petri net accepting $\mathord{\downarrow}{L}$.
(Recall that a Petri net is $1$-bounded if every reachable marking has at most
one token in each place.) This is a language theoretic result of independent
interest. When used in the construction of [GM12], this matches the
${\mathsf{EXPSPACE}}$ upper bound.
As a byproduct, our translation yields a simple direct construction of a
finite automaton for $\mathord{\downarrow}{L}$ when $L$ is given as a pushdown
automaton. This is of independent interest because earlier constructions of
$\mathord{\downarrow}{L}$ always start from a context-free grammar and produce
(of necessity!) exponentially large NFAs [vL78, Cou91, BLS15].
We begin with some preliminary definitions.
#### Pushdown automata
If $\Gamma$ is an alphabet, we write
$\bar{\Gamma}=\\{\bar{\gamma}\mid\gamma\in\Gamma\\}$. Moreover, if
$x=\bar{\gamma}$, then we define $\bar{x}=\gamma$. For a word
$v\in(\Gamma\cup\bar{\Gamma})^{*}$, $v=v_{1}\cdots v_{n}$,
$v_{1},\ldots,v_{n}\in\Gamma\cup\bar{\Gamma}$, we set
$\bar{v}=\bar{v_{n}}\cdots\bar{v_{1}}$. A _pushdown automaton_ is a tuple
$\mathcal{A}=(Q,\Sigma,\Gamma,E,q_{0},q_{f})$, where $Q$ is a finite set of
_states_ , $\Sigma$ is its _input alphabet_ , $\Gamma$ is its _stack alphabet_
, $E$ is a finite set of _edges_ , $q_{0}\in Q$ is its _initial state_ , and
$F\subseteq Q$ is its set of _final states_. An edge is a four-tuple
$(p,R,v,q)$, where $p,q\in Q$, $R\subseteq\Sigma^{*}$ is a regular language,
and $v\in\Gamma\cup\bar{\Gamma}\cup\\{\varepsilon\\}$. We also write
$p\xrightarrow{R|v}q$
to denote an edge $(p,R,v,q)\in E$. Intuitively, it tells us that from state
$q$, we can read any word in $R$ as input and modify the stack as specified by
$v$. Here, $v\in\Gamma$ means we push $v$ onto the stack. Moreover,
$v\in\Gamma$, $v=\bar{\gamma}$, means we pop $\gamma$ from the stack. Finally,
$v=\varepsilon$ means we do not change the stack. Let us make this formal. A
_configuration_ of $\mathcal{A}$ is a pair $(q,w)$ with $q\in Q$ and
$w\in\Gamma^{*}$. For configurations $(q,w)$ and $(q^{\prime},w^{\prime})$, we
write $(q,w)\xrightarrow{u}(q^{\prime},w^{\prime})$ if there is an edge
$(q,R,v,q^{\prime})$ in $\mathcal{A}$ such that $u\in R$ and (i) if
$v=\varepsilon$, then $w^{\prime}=w$, (ii) if $v\in\Gamma$, then
$w^{\prime}=wv$ and (iii) if $v=\bar{\gamma}$ for $\gamma\in\Gamma$, then
$w=w^{\prime}\gamma$.
A _run_ in $\mathcal{A}$ is a sequence $(q_{0},w_{0}),\ldots,(q_{n},w_{n})$ of
configurations and words $u_{1},\ldots,u_{n}\in\Sigma^{*}$ such that
$(q_{i-1},w_{i-1})\xrightarrow{u_{i}}(q_{i},w_{i})$ for $1\leq i\leq n$. Its
_length_ is $n$ and its _initial_ and _final configuration_ are
$(q_{0},w_{0})$ and $(q_{n},w_{n})$, respectively. The run is said to _read_
the word $u_{1}\cdots u_{n}$. The _stack height_ of the run is defined as
$\max\\{|w_{i}|\mid 0\leq i\leq n\\}$. We call the run _positive_ (resp.
_dually positive_) if $|w_{i}|\geq|w_{0}|$ (resp. $|w_{i}|\geq|w_{n}|$) for
every $1\leq i\leq n$, i.e. if the stack never drops below its initial height
(resp. is always above its final height).
We write $(q,w)\xRightarrow{u}(q^{\prime},w^{\prime})$ for configurations
$(q,w),(q^{\prime},w^{\prime})$ if there is a run with initial configuration
$(q,w)$ and final configuration $(q^{\prime},w^{\prime})$ that reads $u$. If
there is such a run with stack height $\leq h$, then we write
$(q,w)\xRightarrow{u}_{h}(q^{\prime},w^{\prime})$. The _language accepted by
$\mathcal{A}$_ is
$\mathsf{L}(\mathcal{A})=\\{u\in\Sigma^{*}\mid(q_{0},\varepsilon)\xRightarrow{u}(q_{f},\varepsilon)\\}.$
There is also a language accepted with bounded stack height. For
$h\in{\mathbb{N}}$, we define
$\mathsf{L}_{h}(\mathcal{A})=\\{u\in\Sigma^{*}\mid(q_{0},\varepsilon)\xRightarrow{u}_{h}(q_{f},\varepsilon)\\}.$
Thus, $\mathsf{L}_{h}(\mathcal{A})$ is the set of words accepted by
$\mathcal{A}$ using runs of stack height at most $h$.
In order to exploit the symmetry between forward and backward computations in
pushdown automata, we will consider dual pushdown automata. The _dual
automaton_ of $\mathcal{A}$, denoted $\bar{\mathcal{A}}$, is obtained from
$\mathcal{A}$ by changing each edge $p\xrightarrow{R|v}q$ into
$q\xrightarrow{R^{\mathsf{rev}}|\bar{v}}p$. Then
$\mathsf{L}(\bar{\mathcal{A}})=\mathsf{L}(\mathcal{A})^{\mathsf{rev}}$. We
will sometimes argue by _duality_ , which is the principle that every
statement that is true for any $\mathcal{A}$ is also true for any
$\bar{\mathcal{A}}$.
#### Petri nets
A _(labeled) Petri net_ is a tuple
$N=(\Sigma,S,T,\partial_{0},\partial_{1},\lambda,\mathbf{m}_{0},\mathbf{m}_{f})$
where $\Sigma$ is its _input alphabet_ , $S$ is a finite set of _places_ , $T$
is a finite set of _transitions_ , $\partial_{0},\partial_{1}\colon
T\to{\mathbb{M}[S]}$ are maps that specify an _input marking_
$\partial_{0}(t)$ and an _output marking_ $\partial_{1}(t)$ for each
transition $t\in T$, $\lambda\colon T\to\Sigma\cup\\{\varepsilon\\}$ assigns
labels to transitions, and $\mathbf{m}_{0},\mathbf{m}_{f}$ are its _initial_
and _final marking_. More generally, multisets $\mathbf{m}\in{\mathbb{M}[S]}$
are called _markings_ of $N$.
For markings $\mathbf{m}_{1},\mathbf{m}_{2}\in{\mathbb{M}[S]}$ and
$a\in\Sigma\cup\\{\varepsilon\\}$, we write
$\mathbf{m}_{1}\xrightarrow{a}\mathbf{m}_{2}$ if there is a transition $t\in
T$ with $\lambda(t)=a$, $\mathbf{m}_{1}\geq\partial_{0}(t)$, and
$\mathbf{m}_{2}=\mathbf{m}_{1}\ominus\partial_{0}(t)\oplus\partial_{1}(t)$.
Moreover, we write $\mathbf{m}_{1}\xRightarrow{w}\mathbf{m}_{2}$ if there are
$n\in{\mathbb{N}}$, $a_{1},\ldots,a_{n}\in\Sigma\cup\\{\varepsilon\\}$, and
markings $\mathbf{m}^{\prime}_{0},\ldots,\mathbf{m}^{\prime}_{n}$ such that
$w=a_{1}\cdots a_{n}$ and
$\mathbf{m}_{1}=\mathbf{m}^{\prime}_{0}\xrightarrow{a_{1}}\mathbf{m}^{\prime}_{1}\xrightarrow{a_{2}}\cdots\xrightarrow{a_{n}}\mathbf{m}^{\prime}_{n}=\mathbf{m}_{2}$.
Furthermore, we write $\mathbf{m}_{1}\xRightarrow{}\mathbf{m}_{2}$ if there
exists a $w\in\Sigma^{*}$ with $\mathbf{m}_{1}\xRightarrow{w}\mathbf{m}_{2}$.
The _language accepted by $N$_ is
$\mathsf{L}(N)=\\{w\in\Sigma^{*}\mid\mathbf{m}_{0}\xRightarrow{w}\mathbf{m}_{f}\\}$.
For $k\in{\mathbb{N}}$, a Petri net $N$ is _$k$ -bounded_ if for every marking
$\mathbf{m}\in{\mathbb{M}[S]}$ with $\mathbf{m}_{0}\xRightarrow{}\mathbf{m}$,
we have $\lvert{\mathbf{m}}\rvert\leq k$.
Our main results are as follows.
###### Theorem 9 (Succinct Downclosures for PDAs).
Given a pushdown automaton $\mathcal{A}$, one can construct in polynomial time
a pushdown automaton $\hat{\mathcal{A}}$ so that
$\mathord{\downarrow}{\mathsf{L}(\hat{\mathcal{A}})}=\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
Moreover, we have
$\mathsf{L}(\hat{\mathcal{A}})=\mathsf{L}_{h}(\hat{\mathcal{A}})$ for some
bound $h$ that is polynomial in the size of $\mathcal{A}$.
As a pushdown automaton with bounded stack can be simulated by a 1-bounded
Petri net (essentially, by keeping places for each position in the stack), we
get the following corollary and also the promised ${\mathsf{EXPSPACE}}$ upper
bound.
###### Corollary 10.
Given a pushdown automaton $\mathcal{A}$, one can construct in polynomial time
a $1$-bounded Petri net $N$ with
$\mathsf{L}(N)=\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
We now prove Theorem 9. The _augmented automaton_
$\hat{\mathcal{A}}=(Q,\Sigma,\hat{\Gamma},\hat{E},q_{0},q_{f})$ is defined as
follows. We first compute for any $p,q\in Q$ the subalphabet
$\Delta_{p,q}(\mathcal{A})\subseteq\Sigma$ with
$\Delta_{p,q}(\mathcal{A})=\\{a\in\Sigma\mid\exists u\in
M_{p,q}(\mathcal{A}),~{}|u|_{a}\geq 1\\},$
where $M_{p,q}(\mathcal{A})=\\{u\in\Sigma^{*}\mid\exists
v\in\Gamma^{*}\colon(p,\varepsilon)\xRightarrow{u}(p,v),~{}(q,v)\xRightarrow{}(q,\varepsilon)\\}$.
Note that it is easy to construct in polynomial time a pushdown automaton
$\mathcal{A}_{p,q}$ for the language $M_{p,q}(\mathcal{A})$:
$\mathcal{A}_{p,q}$ has a set $Q_{p}\cup Q_{q}$ of states consisting of two
disjoint copies of the states of $\mathcal{A}$. The transitions between two
states in $Q_{p}$ are inherited from $\mathcal{A}$ while for two states in
$Q_{q}$ we replace the input on any transition by $\varepsilon$ but retain the
stack operations from $\mathcal{A}$. The start state is $p\in Q_{p}$ and the
final state is $q\in Q_{q}$. There is an $\varepsilon$-transition from $p\in
Q_{p}$ to $q\in Q_{q}$. This concludes the construction of
$\mathcal{A}_{p,q}$.
Since $a\in\Delta_{p,q}(\mathcal{A})$ iff
$M_{p,q}(\mathcal{A})\cap\Sigma^{*}a\Sigma^{*}\neq\emptyset$, we can decide in
polynomial time whether a given $a\in\Sigma$ belongs to
$\Delta_{p,q}(\mathcal{A})$ by checking the PDA for
$M_{p,q}(\mathcal{A})\cap\Sigma^{*}a\Sigma^{*}\neq\emptyset$ obtained by
product construction for emptiness. Thus, we can compute
$\Delta_{p,q}(\mathcal{A})$ in polynomial time. We construct
$\hat{\mathcal{A}}$ as follows. For any $p,q\in Q$, we introduce a fresh stack
symbol $[p,q]$ and then we add edges
$\displaystyle p\xrightarrow{\Delta_{p,q}(\mathcal{A})^{*}|[p,q]}p,$
$\displaystyle
q\xrightarrow{\Delta_{q,p}(\bar{\mathcal{A}})^{*}|\overline{[p,q]}}q.$ (4)
The following Lemma tells us that $\mathsf{L}(\hat{\mathcal{A}})$ has the same
downward closure as ${\mathsf{\mathcal{A}}}$.
###### Lemma 11.
$\mathsf{L}(\mathcal{A})\subseteq\mathsf{L}(\hat{\mathcal{A}})\subseteq\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
###### Proof 6.1.
Since the inclusion
$\mathsf{L}(\mathcal{A})\subseteq\mathsf{L}(\hat{\mathcal{A}})$ is obvious, we
prove
$\mathsf{L}(\hat{\mathcal{A}})\subseteq\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
We proceed by induction on the number $m$ of executions of new edges (i.e.
those from Eq. 4). More specifically, we show that if
$u\in\mathsf{L}(\hat{\mathcal{A}})$ is accepted using $m$ executions of new
edges, then there is a $u^{\prime}\in\mathsf{L}(\hat{\mathcal{A}})$ such that
$u\sqsubseteq u^{\prime}$ and $u^{\prime}$ is accepted using $<m$ executions
of new edges.
Suppose $u$ is accepted using a run $\rho$ with $m>0$ executions of new edges.
Then $\rho$ must apply one edge
$p\xrightarrow{\Delta_{p,q}(\mathcal{A})|[p,q]}p$ and thus also the edge
$q\xrightarrow{\Delta_{q,p}(\bar{\mathcal{A}})|\overline{[p,q]}}q$ to remove
the letter $[p,q]$ from the stack. Thus, $\rho$ decomposes as
$\rho=\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}$, where $\rho_{2}$ and
$\rho_{4}$ are the executions of the new edges. Let
$u=u_{1}u_{2}u_{3}u_{4}u_{5}$ be the corresponding decomposition of $u$.
The run $\rho_{1}$ must end in state $p$ and with some stack content
$w\in\Gamma^{*}$. Then, $\rho_{3}$ is a run from $(p,w[p,q])$ to $(q,w[p,q])$
and $\rho_{5}$ is a run from $(q,w)$ to $(q_{f},\varepsilon)$ with $q_{f}\in
F$.
Since $u_{2}$ and $u_{4}$ are read while executing the new edges, we have
$u_{2}\in\Delta_{p,q}(\mathcal{A})^{*}$ and
$u_{4}\in\Delta_{q,p}(\bar{\mathcal{A}})^{*}$. We can therefore write
$u_{2}=r_{1}\cdots r_{k}$ and $u_{4}=s_{1}\cdots s_{\ell}$ with
$r_{1},\ldots,r_{k}\in\Delta_{p,q}(\mathcal{A})$ and
$s_{1},\ldots,s_{\ell}\in\Delta_{q,p}(\bar{\mathcal{A}})$. By definition, this
means for each $1\leq i\leq k$, there is a word $\tilde{r}_{i}\in
M_{p,q}(\mathcal{A})$ that contains the letter $r_{i}$. Likewise, for every
$1\leq i\leq\ell$, there is a $\tilde{s}_{i}\in M_{q,p}(\bar{\mathcal{A}})$
that contains $s_{i}$.
Since $\tilde{r}_{i}\in M_{p,q}(\mathcal{A})$ and $\tilde{s}_{j}\in
M_{q,p}(\bar{\mathcal{A}})$ for $1\leq i\leq k$ and $1\leq j\leq\ell$, there
are words $x_{i}$ and $y_{j}$ in $\Gamma^{*}$ so that
$\displaystyle(p,\varepsilon)\xRightarrow{\tilde{r}_{i}}(p,x_{i})$ and
$\displaystyle(q,x_{i})\xRightarrow{}(q,\varepsilon)$
$\displaystyle(p,\varepsilon)\xRightarrow{}(p,y_{j})$ and
$\displaystyle(q,y_{i})\xRightarrow{\tilde{s}_{j}}(q,\varepsilon)$
We can therefore construct a new run
$\rho^{\prime}=\rho_{1}\rho^{\prime}_{2}\rho^{\prime}_{3}\rho^{\prime}_{4}\rho_{5}$,
where
$\displaystyle\rho^{\prime}_{2}:(p,w)\xRightarrow{\tilde{r}_{1}}\cdots\xRightarrow{\tilde{r}_{k}}(p,wx_{1}\cdots
x_{k})\xRightarrow{}\cdots\xRightarrow{}(p,wx_{1}\cdots x_{k}y_{\ell}\cdots
y_{1})$ $\displaystyle\rho^{\prime}_{4}:(q,wx_{1}\cdots x_{k}y_{\ell}\cdots
y_{1})\xRightarrow{\tilde{s}_{1}}\cdots\xRightarrow{\tilde{s}_{\ell}}(q,wx_{1}\cdots
x_{k})\xRightarrow{}\cdots\xRightarrow{}(q,w).$
Moreover, since $\rho_{3}$ is a positive run from $(p,w)$ to $(q,w)$, we
obtain $\rho^{\prime}_{3}$ from $\rho_{3}$ by replacing the prefix $w$ of
every stack by $wx_{1}\cdots x_{k}y_{1}\cdots y_{\ell}$.
Then $\rho^{\prime}$ reads some word
$u_{1}\tilde{r}_{1}\cdots\tilde{r}_{k}fu_{3}\tilde{s}_{1}\cdots\tilde{s}_{\ell}gu_{5}$
for $f,g\in\Sigma^{*}$. Note that since $r_{i}$ occurs in $\tilde{r}_{i}$ and
$s_{i}$ occurs in $\tilde{s}_{j}$, we have
$u=u_{1}u_{2}u_{3}u_{4}u_{5}\sqsubseteq
u_{1}\tilde{r}_{1}\cdots\tilde{r}_{k}fu_{3}\tilde{s}_{1}\cdots\tilde{s}_{\ell}gu_{5}$.
We now show that every word in $\hat{\mathcal{A}}$ is accepted by a run with
bounded stack height.
###### Lemma 12.
$\mathsf{L}(\hat{\mathcal{A}})=\mathsf{L}_{h}(\hat{\mathcal{A}})$, where
$h=2|Q|^{2}$.
Before we prove Lemma 12, we need another observation. Just like Lemma 11, one
can show that for any $p,q\in Q$, we have $M_{p,q}(\mathcal{A})\subseteq
M_{p,q}(\hat{\mathcal{A}})\subseteq\mathord{\downarrow}{M_{p,q}(\mathcal{A})}$
and in particular
$\displaystyle\Delta_{p,q}(\mathcal{A})=\Delta_{p,q}(\hat{\mathcal{A}})$
$\displaystyle\Delta_{q,p}(\bar{\mathcal{A}})=\Delta_{q,p}(\bar{\hat{\mathcal{A}}}),$
(5)
where the second identity follows from the first: Duality yields
$\Delta_{q,p}(\bar{\mathcal{A}})=\Delta_{q,p}(\hat{\bar{\mathcal{A}}})$ and
since $\hat{\bar{\mathcal{A}}}$ and $\bar{\hat{\mathcal{A}}}$ are isomorphic
(i.e. they are the same up to a renaming of stack symbols), we have
$\Delta_{q,p}(\hat{\bar{\mathcal{A}}})=\Delta_{q,p}(\bar{\hat{\mathcal{A}}})$.
We now prove Lemma 12. Let $u\in\mathsf{L}(\hat{\mathcal{A}})$. We show that
any minimal length accepting run $\rho$ reading $u$ must have stack height
$\leq h$ and hence $u\in\mathsf{L}_{h}(\hat{\mathcal{A}})$.
Suppose the stack height of $\rho$ is larger than $h=2|Q|^{2}$.
#### Claim:
$\rho$ decomposes into runs $\rho_{1},\rho_{2},\rho_{3},\rho_{4},\rho_{5}$
reading $u_{1},u_{2},u_{3},u_{4},u_{5}$, respectively, so that there are
$p,q\in Q$ and $w\in\Gamma^{*}$ with
* •
$\rho_{2}$ is a positive run from $(p,w)$ to $(p,wv)$ of length $\geq 2$
* •
$\rho_{3}$ is a positive run from $(p,wv)$ to $(q,wv)$
* •
$\rho_{4}$ is a dually positive run from $(q,wv)$ to $(q,w)$
Let $c_{h}$ be a configuration along $\rho$ with stack height at least
$2|Q|^{2}+1$. Then there exist $|Q|^{2}$ configurations
$c_{2}\xRightarrow{}c_{4}\xRightarrow{}\cdots\xRightarrow{}c_{2|Q|^{2}}\xRightarrow{}c_{h}$
along $\rho$ such that $c_{2i}$ is the last time that the stack height is $2i$
before visiting $c_{h}$. Symmetrically, we have
$c_{h}\xRightarrow{}c^{\prime}_{2|Q|^{2}}\xRightarrow{}\cdots
c^{\prime}_{4}\xRightarrow{}c^{\prime}_{2}$ where $c^{\prime}_{2i}$ is the
first occurrence of stack height $2i$ after visiting $c_{h}$. Clearly by
definition the run between consecutive $c_{2i}$ (resp. $c^{\prime}_{2i}$)
configurations is positive (resp. dually positive). Additionally, the length
of the run between them must be at least two, because the stack heights differ
by two. Considering the pair of states at each $c_{2i},c^{\prime}_{2i}$, there
are $|Q|^{2}$ possibilities. Hence there must exist indices $2i<2j$ such that
the $c_{2i}$ and $c_{2j}$ have the same state $p$ and $c^{\prime}_{2i}$ and
$c^{\prime}_{2j}$ have the same state $q$. It is now clear that
$\rho_{2}\colon c_{2i}\xRightarrow{}c_{2j}$, $\rho_{3}\colon
c_{2j}\xRightarrow{}c^{\prime}_{2j}$ and $\rho_{4}\colon
c^{\prime}_{2j}\xRightarrow{}c^{\prime}_{2i}$ satisfy the conditions of the
claim.
These conditions imply that $u_{2}\in M_{p,q}(\hat{\mathcal{A}})$ and
$u_{4}\in M_{q,p}(\bar{\hat{\mathcal{A}}})$. Therefore, we have
$u_{2}\in\Delta_{p,q}(\hat{\mathcal{A}})^{*}=\Delta_{p,q}(\mathcal{A})^{*}$
and
$u_{4}\in\Delta_{q,p}(\bar{\hat{\mathcal{A}}})^{*}=\Delta_{q,p}(\bar{\mathcal{A}})^{*}$
by Eq. 5.
We obtain the run $\rho^{\prime}$ from $\rho$ as follows. We replace
$\rho_{2}$ by a single execution of the edge
$p\xrightarrow{\Delta_{p,q}(\mathcal{A})^{*}|[p,q]}p$ reading $u_{2}$.
Moreover, we replace $\rho_{4}$ by a single execution of the edge
$q\xrightarrow{\Delta_{q,p}(\bar{\mathcal{A}})^{*}|\overline{[p,q]}}q$. Then
$\rho^{\prime}$ is clearly a run reading $u=u_{1}u_{2}u_{3}u_{4}u_{5}$.
Furthermore, since $\rho_{2}$ has length $\geq 2$, but the single edge used
instead in $\rho^{\prime}$ only incurs length $1$, $\rho^{\prime}$ is strictly
shorter than $\rho$. This is in contradiction to the minimal length of $\rho$.
This completes the proof of Lemma 12 and thus also Theorem 9.
###### Remark 6.2.
The augmented automaton $\hat{\mathcal{A}}$ yields a very simple construction
of a finite automaton (of exponential size) for
$\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$. First, it is easy to
construct a finite automaton for $\mathsf{L}_{h}(\hat{\mathcal{A}})$. Then, by
introducing $\varepsilon$-edges, we get a finite automaton for
$\mathord{\downarrow}{\mathsf{L}_{h}(\hat{\mathcal{A}})}$, which, by Lemmas 11
and 12, equals $\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
It is now straightforward to construct a polynomial size $1$-bounded Petri net
$N=(\Sigma,S,T,\partial_{0},\partial_{0},\mathbf{m}_{0},\mathbf{m}_{f})$ with
$\mathsf{L}(N)=\mathsf{L}_{h}(\hat{\mathcal{A}})$, thus proving Corollary 10.
First, by adding states, we turn $\hat{\mathcal{A}}$ into a pushdown automaton
$\mathcal{A}^{\prime}=(Q^{\prime},\Sigma,\hat{\Gamma},E^{\prime},q_{0},q_{f})$,
where every edge reads at most one letter, i.e. every edge
$p\xrightarrow{R|v}q$ in $\mathcal{A}^{\prime}$ has $R=\\{x\\}$ for some
$x\in\Sigma\cup\\{\varepsilon\\}$ (this is done by ‘pasting’ the automaton for
$R$ in place of the edge). Moreover, we add $\varepsilon$-edges, so that for
every edge $p\xrightarrow{\\{x\\}|v}q$, we also have an edge
$p\xrightarrow{\\{\varepsilon\\}|v}q$. Then clearly
$\mathsf{L}_{h}(\mathcal{A}^{\prime})=\mathord{\downarrow}{\mathsf{L}_{h}(\hat{\mathcal{A}})}=\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
The net $N$ has a place $p$ for each state $p$ of $\mathcal{A}^{\prime}$ and
for each $i\in\\{1,\ldots,h\\}$ and $\gamma\in\hat{\Gamma}$, it has a place
$(i,\gamma)$. Moreover, for each $i\in\\{0,\ldots,h\\}$, it has a place
$s_{i}$. Here, the idea is that a configuration
$c=(p,\gamma_{1}\cdots\gamma_{n})$ of $\mathcal{A}^{\prime}$ with
$\gamma_{1},\ldots,\gamma_{n}\in\hat{\Gamma}$ is represented as a marking
$\mathbf{m}_{c}={\llbracket
p,(1,\gamma_{1}),\ldots,(n,\gamma_{n}),s_{n}\rrbracket}$. We call a marking of
this form a _stack marking_ and will argue that every reachable marking in $N$
is a stack marking. The transitions in $N$ correspond to edges in
$\mathcal{A}^{\prime}$. For each edge $p\xrightarrow{\\{x\\}|v}q$ in
$\hat{\mathcal{A}}$, we add the following transitions:
* •
if $v=\bar{\gamma}$ for some $\gamma\in\hat{\Gamma}$, then we have for every
$1\leq n\leq h$ a transition $t$ with $\partial_{0}(t)={\llbracket
p,(n,\gamma),s_{n}\rrbracket}$, $\partial_{1}(t)={\llbracket
q,s_{n-1}\rrbracket}$, and $\lambda(t)=x$.
* •
if $v\in\hat{\Gamma}$, then for every $0\leq n<h$, we add a transition $t$
with $\partial_{0}(t)={\llbracket p,s_{n}\rrbracket}$,
$\partial_{1}(t)={\llbracket q,(n+1,v),s_{n+1}\rrbracket}$, and
$\lambda(t)=x$.
* •
if $v=\varepsilon$, then we add a transition $t$ with
$\partial_{0}(t)={\llbracket p\rrbracket}$, $\partial_{1}(t)={\llbracket
q\rrbracket}$, and $\lambda(t)=x$.
Then clearly every reachable marking is a stack marking and we have
$c\xrightarrow{x}c^{\prime}$ for configurations $c,c^{\prime}$ of
$\mathcal{A}^{\prime}$ of stack height $\leq h$ if and only if
$\mathbf{m}_{c}\xrightarrow{x}\mathbf{m}_{c^{\prime}}$. Therefore, if we set
$\mathbf{m}_{0}={\llbracket q_{0},s_{0}\rrbracket}$ and
$\mathbf{m}_{f}={\llbracket q_{f},s_{0}\rrbracket}$ as initial and final
marking, we have
$\mathsf{L}(N)=\mathsf{L}_{h}(\mathcal{A}^{\prime})=\mathord{\downarrow}{\mathsf{L}(\mathcal{A})}$.
This completes the proof of Corollary 10.
## References
* [ABJ98] Parosh Aziz Abdulla, Ahmed Bouajjani, and Bengt Jonsson. On-the-fly analysis of systems with unbounded, lossy FIFO channels. In Proceedings of the 10th International Conference on Computer Aided Verification (CAV 1998), pages 305–318, 1998. doi:10.1007/BFb0028754.
* [ABQ09] Mohamed Faouzi Atig, Ahmed Bouajjani, and Shaz Qadeer. Context-bounded analysis for concurrent programs with dynamic creation of threads. In Tools and Algorithms for the Construction and Analysis of Systems, 15th International Conference, TACAS 2009, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009, York, UK, March 22-29, 2009. Proceedings, volume 5505 of Lecture Notes in Computer Science, pages 107–123. Springer, 2009. doi:10.1007/978-3-642-00768-2\\_11.
* [Aho68] Alfred V. Aho. Indexed grammars - an extension of context-free grammars. J. ACM, 15(4):647–671, 1968. doi:10.1145/321479.321488.
* [Ber79] Jean Berstel. Transductions and context-free languages. Teubner-Verlag, 1979.
* [BLS15] Georg Bachmeier, Michael Luttenberger, and Maximilian Schlund. Finite automata for the sub- and superword closure of cfls: Descriptional and computational complexity. In Language and Automata Theory and Applications - 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, volume 8977 of Lecture Notes in Computer Science, pages 473–485. Springer, 2015. doi:10.1007/978-3-319-15579-1\\_37.
* [Cou91] Bruno Courcelle. On constructing obstruction sets of words. Bulletin of the EATCS, 44:178–186, 1991.
* [CPSW16] Lorenzo Clemente, Paweł Parys, Sylvain Salvati, and Igor Walukiewicz. The diagonal problem for higher-order recursion schemes is decidable. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, New York, NY, USA, July 5-8, 2016, pages 96–105. ACM, 2016. doi:10.1145/2933575.2934527.
* [CV07] Rohit Chadha and Mahesh Viswanathan. Decidability results for well-structured transition systems with auxiliary storage. In CONCUR 2007 - Concurrency Theory, 18th International Conference, CONCUR 2007, Lisbon, Portugal, September 3-8, 2007, Proceedings, volume 4703 of Lecture Notes in Computer Science, pages 136–150. Springer, 2007. doi:10.1007/978-3-540-74407-8\\_10.
* [Dam82] Werner Damm. The IO-and OI-hierarchies. Theoretical Computer Science, 20(2):95–207, 1982.
* [DFS98] Catherine Dufourd, Alain Finkel, and Philippe Schnoebelen. Reset nets between decidability and undecidability. In Automata, Languages and Programming, 25th International Colloquium, ICALP’98, Aalborg, Denmark, July 13-17, 1998, Proceedings, volume 1443 of Lecture Notes in Computer Science, pages 103–115. Springer, 1998. doi:10.1007/BFb0055044.
* [DG86] Werner Damm and Andreas Goerdt. An automata-theoretical characterization of the OI-hierarchy. Information and Control, 71(1):1–32, 1986.
* [GM12] Pierre Ganty and Rupak Majumdar. Algorithmic verification of asynchronous programs. ACM Transactions on Programming Languages and Systems (TOPLAS), 34(1):6, 2012. doi:10.1145/2160910.2160915.
* [GRB07] Gilles Geeraerts, Jean-François Raskin, and Laurent Van Begin. Well-structured languages. Acta Informatica, 44(3-4):249–288, 2007. doi:10.1007/s00236-007-0050-3.
* [Gre78] Sheila A. Greibach. Remarks on blind and partially blind one-way multicounter machines. Theoretical Computer Science, 7(3):311 – 324, 1978. doi:10.1016/0304-3975(78)90020-8.
* [Gre16] Charles Grellois. Semantics of linear logic and higher-order model-checking. PhD thesis, Univeristé Denis Diderot Paris 7, 2016.
* [Hai69] Leonard H Haines. On free monoids partially ordered by embedding. Journal of Combinatorial Theory, 6(1):94–98, 1969.
* [HKO16] Matthew Hague, Jonathan Kochems, and C.-H. Luke Ong. Unboundedness and downward closures of higher-order pushdown automata. In Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2016, St. Petersburg, FL, USA, January 20 - 22, 2016, pages 151–163. ACM, 2016. doi:10.1145/2837614.2837627.
* [HMOS08] Matthew Hague, Andrzej S. Murawski, C.-H. Luke Ong, and Olivier Serre. Collapsible pushdown automata and recursion schemes. In Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, 24-27 June 2008, Pittsburgh, PA, USA, pages 452–461, 2008. doi:10.1109/LICS.2008.34.
* [HMU07] John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to automata theory, languages, and computation, 3rd Edition. Pearson international edition. Addison-Wesley, 2007.
* [Jan79] Matthias Jantzen. On the hierarchy of Petri net languages. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 13(1):19–30, 1979. URL: http://www.numdam.org/item?id=ITA_1979__13_1_19_0.
* [JM07] Ranjit Jhala and Rupak Majumdar. Interprocedural analysis of asynchronous programs. In Proceedings of the 34th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2007, Nice, France, January 17-19, 2007, pages 339–350. ACM, 2007. doi:10.1145/1190216.1190266.
* [KO11] Naoki Kobayashi and C.-H. Luke Ong. Complexity of model checking recursion schemes for fragments of the modal mu-calculus. Logical Methods in Computer Science, 7(4), 2011.
* [Kob09] Naoki Kobayashi. Types and higher-order recursion schemes for verification of higher-order programs. In Proceedings of the 36th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2009, Savannah, GA, USA, January 21-23, 2009, pages 416–428, 2009. doi:10.1145/1480881.1480933.
* [Kob19] Naoki Kobayashi. Inclusion between the frontier language of a non-deterministic recursive program scheme and the Dyck language is undecidable. Theoretical Computer Science, 777:409–416, 2019.
* [Mas74] AN Maslov. The hierarchy of indexed languages of an arbitrary level. Doklady Akademii Nauk, 217(5):1013–1016, 1974.
* [May03] Richard Mayr. Undecidable problems in unreliable computations. Theoretical Computer Science, 297(1-3):337–354, 2003.
* [Ong15] Luke Ong. Higher-order model checking: An overview. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 1–15, 2015. doi:10.1109/LICS.2015.9.
* [Par17] Paweł Parys. The complexity of the diagonal problem for recursion schemes. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India, volume 93 of LIPIcs, pages 45:1–45:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. doi:10.4230/LIPIcs.FSTTCS.2017.45.
* [Sip97] Michael Sipser. Introduction to the theory of computation. PWS Publishing Company, 1997.
* [SV06] Koushik Sen and Mahesh Viswanathan. Model checking multithreaded programs with asynchronous atomic methods. In Computer Aided Verification, 18th International Conference, CAV 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings, volume 4144 of Lecture Notes in Computer Science, pages 300–314. Springer, 2006. doi:10.1007/11817963\\_29.
* [TZ19] Ramanathan S. Thinniyam and Georg Zetzsche. Regular separability and intersection emptiness are independent problems. In Proceedings of FSTTCS 2019, volume 150 of Leibniz International Proceedings in Informatics (LIPIcs), pages 51:1–51:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. doi:10.4230/LIPIcs.FSTTCS.2019.51.
* [vL78] Jan van Leeuwen. Effective constructions in well-partially-ordered free monoids. Discrete Mathematics, 21(3):237–252, 1978. doi:10.1016/0012-365X(78)90156-5.
* [Zet15] Georg Zetzsche. An approach to computing downward closures. In ICALP 2015, volume 9135, pages 440–451. Springer, 2015. The undecidability of $Z$ intersection is shown in the full version: http://arxiv.org/abs/1503.01068.
## Appendix A Compiling away internal actions
We have commented in the examples of Section 2 that internal actions and
internal updates of the global state are useful in modeling asynchronous
programs from their programming language syntax. Indeed, we note that the
definition of asynchronous programs in [GM12] additionally uses a separate
alphabet of _internal actions_ , in addition to the alphabet of handler posts.
We show how a model of asynchronous programs with internal actions can be
reduced to our, simpler, model.
Let $\mathcal{C}$ be a language class over an alphabet $\Sigma$ of handler
names. The definition of asynchronous programs with internal actions, as used
by [SV06, JM07, GM12], is as follows.333 The language class $\mathcal{C}$ in
[GM12] is fixed to be the class of context free languages. Their definition
generalizes to any language class. An asynchronous program over $\mathcal{C}$
with _internal actions_ (aka $\mathsf{AP}$ over $\mathcal{C}$ with internal
actions) is a tuple
$\mathfrak{P}=(D,\Sigma,\Sigma_{\mathsf{i}},\mathscr{L},R,d_{0},\mathbf{m}_{0}),$
where $D$, $\Sigma$, $d_{0}$, $\mathbf{m}_{0}$ are as in Definition 2,
$\Sigma_{\mathsf{i}}$ is an alphabet of internal actions disjoint from
$\Sigma$, the set $\mathscr{L}=(L_{\sigma})_{\sigma\in\Sigma}$ consists of
languages from $\mathcal{C}$ (one for each handler $\sigma\in\Sigma)$, and
$R=(D,\Sigma\cup\Sigma_{\mathsf{i}},\delta)$ a deterministic finite state
automaton where $D$ is the set of states, $\Sigma\cup\Sigma_{\mathsf{i}}$ the
alphabet and $\delta$ the transition relation specifying the effect of each
internal action on the global state $D$. We will write
$d\overset{w}{\underset{R}{\Rightarrow}^{*}}d^{\prime}$ to mean that there is
a sequence of transitions with labels $w_{1}w_{2}...w_{n}=w$ in the automaton
$R$ using which we can reach $d^{\prime}$ from $d$.
For alphabets $\Sigma,\Sigma^{\prime}$ with $\Sigma\subseteq\Sigma^{\prime}$,
the Parikh map
${\mathsf{Parikh}}\colon\Sigma^{\prime*}\rightarrow{\mathbb{M}[\Sigma]}$ maps
a word $w\in\Sigma^{\prime*}$ to a multiset ${\mathsf{Parikh}}(w)$ such that
${\mathsf{Parikh}}(w)(a)$ is the number of occurrences of $a$ in $w$ for each
$a\in\Sigma$. For example, ${\mathsf{Parikh}}(abbab)(a)=2$,
${\mathsf{Parikh}}(abbab)(b)=3$ and
${\mathsf{Parikh}}(\varepsilon)={\llbracket\rrbracket}$. For a language $L$,
we define ${\mathsf{Parikh}}(L)={\\{{\mathsf{Parikh}}(w)\mid w\in L\\}}$. If
the alphabet $\Sigma$ is not clear from the context, we write
${\mathsf{Parikh}}_{\Sigma}$ (usually $\Sigma^{\prime}=\Sigma$).
The semantics of a $\mathfrak{P}$ is given as a labeled transition system over
the set of configurations, with a transition relation
$\rightarrow\subseteq(D\times{\mathbb{M}[\Sigma]})\times\Sigma\times(D\times{\mathbb{M}[\Sigma]})$
defined as follows: let
$\mathbf{m},\mathbf{m}^{\prime}\in{\mathbb{M}[\Sigma]}$, $d,d^{\prime}\in D$
and $\sigma\in\Sigma$
$\displaystyle(d,\mathbf{m}\oplus{\llbracket\sigma\rrbracket})\underset{\mathfrak{P}}{\overset{\sigma}{\rightarrow}}(d^{\prime},\mathbf{m}\oplus\mathbf{m}^{\prime})$
iff $\displaystyle\exists w\in(\Sigma\cup\Sigma_{\mathsf{i}})^{*}\colon
d\overset{w}{\underset{R}{\Rightarrow}^{*}}d^{\prime}\land w\in
L_{\sigma}\land\mathbf{m}^{\prime}={\mathsf{Parikh}}_{\Sigma}(w)\enspace.$
We now show that internal actions can be compiled away. Thus, for the
algorithms presented in this paper, we use the more abstract, language-
theoretic version of Definition 2, while we use internal actions as syntactic
sugar in examples.
###### Lemma 13.
Let $\mathcal{C}$ be a full trio. Given an $\mathsf{AP}$
$\mathfrak{P}_{\mathsf{i}}$ with internal actions over $\mathcal{C}$, one can
construct an $\mathsf{AP}$ $\mathfrak{P}$ over $\mathcal{C}$ such that both
have identical sets of runs.
###### Proof A.1.
The proof is along similar lines to that of Lemmas 4.3, 4.5 in [GM12]. Given
$\mathfrak{P}_{\mathsf{i}}=(D,\Sigma,\Sigma_{\mathsf{i}},\mathscr{L},R,d_{0},\mathbf{m}_{0})$
we construct
$\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$ such
that
$\displaystyle(d,\mathbf{m}\oplus{\llbracket\sigma\rrbracket})\underset{\mathfrak{P}}{\overset{\sigma}{\rightarrow}}(d^{\prime},\mathbf{m}\oplus\mathbf{m}^{\prime})\quad\text{
if{}f
}\quad(d,\mathbf{m}\oplus{\llbracket\sigma\rrbracket})\underset{\mathfrak{P}_{\mathsf{i}}}{\overset{\sigma}{\rightarrow}}(d^{\prime},\mathbf{m}\oplus\mathbf{m}^{\prime})$
Let $L(R_{d,d^{\prime}})$ be the language of the automaton $R$ when $d$ is the
initial state and $d^{\prime}$ is the accepting state. We define $L_{d\sigma
d^{\prime}}$ as:
$L_{d\sigma d^{\prime}}:=\mathrm{Proj}_{\Sigma}(L_{\sigma}\cap
L(R_{d,d^{\prime}}))$
First observe that the projection operation is a homomorphism and
$L(R_{d,d^{\prime}}))$ is a regular language; hence by virtue of $\mathcal{C}$
being a full trio $L_{d\sigma d^{\prime}}$ as defined above is in
$\mathcal{C}$. The conditions $\exists
w\in(\Sigma\cup\Sigma_{\mathsf{i}})^{*}\colon
d\overset{w}{\underset{R}{\Rightarrow}^{*}}d^{\prime}\land w\in
L_{\sigma}\land\mathbf{m}^{\prime}={\mathsf{Parikh}}_{\Sigma}(w)$ and $\exists
w\in L_{d\sigma d^{\prime}}\colon{\mathsf{Parikh}}(w)=\mathbf{m}^{\prime}$ are
seen to be equivalent from the definition of $L_{d\sigma d^{\prime}}$,
concluding the proof of the lemma.
## Appendix B Proof of Proposition 2
We prove the following proposition.
Let $\mathfrak{P}=(D,\Sigma,(L_{c})_{c\in\mathfrak{C}},d_{0},\mathbf{m}_{0})$
be an asynchronous program. Then
$\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}=\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$.
In particular,
1. (1)
For every $d\in D$, $\mathfrak{P}$ can reach $d$ if and only if
$\mathord{\downarrow}{\mathfrak{P}}$ can reach $d$.
2. (2)
$\mathfrak{P}$ is terminating if and only if
$\mathord{\downarrow}{\mathfrak{P}}$ is terminating.
3. (3)
$\mathfrak{P}$ is bounded if and only if $\mathord{\downarrow}{\mathfrak{P}}$
is bounded.
###### Proof B.1.
A run of the asynchronous program $\mathfrak{P}$ is defined as a sequence
$c_{0},\sigma_{1},c_{1},\sigma_{2},\ldots$ containing alternating elements of
configurations $c_{i}$ and letters $\sigma_{i}$ beginning with the
configuration $c_{0}=(d_{0},\mathbf{m}_{0})$. First we observe that
$\mathsf{Runs}(\mathfrak{P})\subseteq\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})$
(6)
because every transition enabled in $\mathfrak{P}$ is also enabled in
$\mathord{\downarrow}{\mathfrak{P}}$. Next, we claim:
$\forall\;\rho\in\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})\;\exists\;\rho^{\prime}\in\mathsf{Runs}(\mathfrak{P})\;\rho\trianglelefteq\rho^{\prime}$
(7)
Let
$\rho|_{k}=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}),\sigma_{2},...,\sigma_{k},(d_{k},\mathbf{m}_{k})$
be the $2k+1-$length prefix of $\rho$. We show that for each $\rho|_{k}$ there
exists $\rho^{\prime}_{k}\in\mathsf{Runs}(\mathfrak{P})$ such that
$\rho|_{k}\trianglelefteq\rho^{\prime}_{k}$ and in addition, $\forall
k\;\forall i\leq k\;\rho^{\prime}_{k}(i)=\rho^{\prime}_{k+1}(i)$. We can then
define $\rho^{\prime}(i):=\rho^{\prime}_{i}(i)$ and clearly
$\rho\trianglelefteq\rho^{\prime}$.
We prove by induction on $k$.
Base Case: $\rho|_{0}=\rho^{\prime}|_{0}=(d_{0},\mathbf{m}_{0})$.
Induction Step: Let
$\rho|_{k}=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}),\sigma_{2},...,(d_{k},\mathbf{m}_{k})\in\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})$.
By induction hypothesis there is
$\rho^{\prime}_{k-1}=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}^{\prime}),\sigma_{2},...,(d_{k-1},\mathbf{m}_{k-1}^{\prime})\in\mathsf{Runs}(\mathfrak{P})$
such that $\rho_{k-1}\trianglelefteq\rho^{\prime}_{k-1}$.
$\displaystyle(d_{k-1},\mathbf{m}_{k-1})\overset{\sigma_{k}}{\underset{\mathord{\downarrow}{\mathfrak{P}}}{\rightarrow}}(d_{k},\mathbf{m}_{k})$
$\displaystyle\Rightarrow\exists\mathbf{m}_{k-1}^{\prime\prime}\colon\mathbf{m}_{k-1}=\mathbf{m}_{k-1}^{\prime\prime}\oplus{\llbracket\sigma_{k}\rrbracket}\land(d_{k-1},\mathbf{m}_{k-1}^{\prime\prime}\oplus{\llbracket\sigma_{k}\rrbracket})\overset{\sigma_{k}}{\underset{\mathord{\downarrow}{\mathfrak{P}}}{\rightarrow}}(d_{k},\mathbf{m}_{k})$
$\displaystyle\Rightarrow\exists w\in\Sigma^{*}\colon
w\in\mathord{\downarrow}{L_{d_{k-1}\sigma_{k}d_{k}}}\land{\mathsf{Parikh}}(w)\oplus\mathbf{m}_{k-1}^{\prime\prime}=\mathbf{m}_{k}$
$\displaystyle\Rightarrow\exists w^{\prime}\in\Sigma^{*}\colon w\sqsubseteq
w^{\prime}\land w^{\prime}\in L_{d_{k-1}\sigma_{k}d_{k}}$
$\displaystyle\Rightarrow(d_{k-1},\mathbf{m}_{k-1})\overset{\sigma_{k}}{\underset{\mathfrak{P}}{\rightarrow}}(d_{k},\mathbf{m}_{k}\oplus\mathbf{m}_{\Delta})\text{where
}\mathbf{m}_{\Delta}\oplus{\mathsf{Parikh}}(w)={\mathsf{Parikh}}(w^{\prime})$
$\displaystyle\Rightarrow(d_{k-1},\mathbf{m}_{k-1}^{\prime})\overset{\sigma_{k}}{\underset{\mathfrak{P}}{\rightarrow}}(d_{k},\mathbf{m}_{k}^{\prime})\text{
where
}\mathbf{m}_{k}^{\prime}=\mathbf{m}_{k}\oplus\mathbf{m}_{\Delta}\oplus(\mathbf{m}^{\prime}_{k-1}\ominus\mathbf{m}_{k-1})$
Defining
$\rho^{\prime}_{k}:=\rho^{\prime}_{k-1},\sigma_{k},(d_{k},\mathbf{m}_{k}^{\prime})$
we see that $\rho|_{k}\trianglelefteq\rho^{\prime}_{k}$, completing the proof
of Equation 7. We are now ready to show that
$\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}=\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$.
The direction
$\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}\subseteq\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}$
follows immediately from Equation (6). Conversely, let
$\displaystyle\rho=(s_{0},\mathbf{n}_{0}),\sigma_{1},(s_{1},\mathbf{n}_{1}),\sigma_{2},...\in\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}$
$\displaystyle\Rightarrow$
$\displaystyle\exists\rho^{\prime}\in\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})\;\rho\trianglelefteq\rho^{\prime}$
$\displaystyle\Rightarrow$
$\displaystyle\exists\rho^{\prime\prime}\in\mathsf{Runs}(\mathfrak{P})\;\rho^{\prime}\trianglelefteq\rho^{\prime\prime}\;$
by Equation (7) $\displaystyle\Rightarrow$
$\displaystyle\rho\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$
We have proved that
$\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}=\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$.
We now show that each of the three properties i.e. safety, termination and
boundedness only depend on the downclosure of the runs.
Safety:
$\displaystyle d\text{ is reachable in }\mathfrak{P}$ iff
$\displaystyle\exists\rho=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}),\sigma_{2},...,\sigma_{k},(d_{k},\mathbf{m}_{k})\in\mathsf{Runs}(\mathfrak{P})\colon
d_{k}=d$
By Equation 6, we know $\rho\in\mathsf{Runs}(\mathfrak{P})$ implies
$\rho\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$. Conversely, if
there is
$\rho^{\prime}=(s_{0},\mathbf{n}_{0}),\sigma_{1},(s_{1},\mathbf{n}_{1}),\sigma_{2},...,\sigma_{k},(s_{k},\mathbf{n}_{k})\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$
with $s_{k}=d$, then by Equation 7, there is
$\rho=(d_{0},\mathbf{m}_{0}),\sigma_{1},(d_{1},\mathbf{m}_{1}),\sigma_{1},...,\sigma_{k},(d_{k},\mathbf{m}_{k})\in\mathsf{Runs}(\mathfrak{P})$
with $\rho^{\prime}\trianglelefteq\rho$ which implies $d_{k}=d$. Hence we have
$\displaystyle d\text{ is reachable in }\mathfrak{P}$ iff
$\displaystyle\exists\rho=(s_{0},\mathbf{n}_{0}),\sigma_{1},(s_{1},\mathbf{n}_{1}),\sigma_{2},...,\sigma_{k},(s_{k},\mathbf{n}_{k})\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}\colon
s_{k}=d$
By a similar argument as above we also have:
Termination:
$\displaystyle\mathfrak{P}\text{ does not terminate }$ iff
$\displaystyle\exists\rho\in\mathsf{Runs}(\mathfrak{P})\colon\rho\text{ is
infinite }$ iff
$\displaystyle\exists\rho\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}\colon\rho\text{
is infinite }$
Boundedness:
$\displaystyle\mathfrak{P}\text{ is bounded }$ iff $\displaystyle\exists
N\in{\mathbb{N}}\;\forall\rho\in\mathsf{Runs}(\mathfrak{P})\;\forall
i\;|\rho(2i).m|<N$ iff $\displaystyle\exists
N\in{\mathbb{N}}\;\forall\rho\in\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}\;\forall
i\;|\rho(2i).m|<N$
In each of the three cases, the property only depends on the downclosure and
hence one may equivalently replace $\mathfrak{P}$ by
$\mathord{\downarrow}{\mathfrak{P}}$ since
$\mathord{\downarrow}{\mathsf{Runs}(\mathord{\downarrow}{\mathfrak{P}})}=\mathord{\downarrow}{\mathsf{Runs}(\mathfrak{P})}$.
## Appendix C Proof of Proposition 6
We begin with a simple observation. For every $\mathsf{HORS}$
$\mathscr{S}=(\Sigma,\mathcal{N},S,\mathcal{R})$, there exists another
$\mathsf{HORS}$
$\mathscr{S}^{\prime}=(\Sigma^{\prime},\mathcal{N}^{\prime},S,\mathcal{R}^{\prime})$
where $\Sigma^{\prime}=\\{\mathtt{br},\mathsf{e}\\}\cup\tilde{\Sigma}$ with
$\mathtt{br}$ of arity 2, $\mathsf{e}$ of arity 0 and all symbols in
$\tilde{\Sigma}$ of arity 1; such that
$\mathcal{L}_{\mathsf{p}}(\mathscr{S})=\mathcal{L}_{\mathsf{p}}(\mathscr{S}^{\prime})$.
By rewriting every terminal symbol $A$ of arity $n\geq 2$ using the rule
$Ax_{1}x_{2}\cdots
x_{n}\twoheadrightarrow\mathtt{br}(x_{1}\mathtt{br}(x_{2}\mathtt{br}(\cdots\mathtt{br}(x_{n-1}x_{n}))$
and by replacing the occurrence of every nullary symbol in $\mathscr{S}$ by
$\mathtt{e}$, we get $\mathscr{S}^{\prime}$, which has the same path language.
We assume due to the above observation that
$\Sigma=\\{\mathtt{br},\mathsf{e}\\}\cup\tilde{\Sigma}$ where $\mathtt{br}$ is
binary, $\mathsf{e}$ is nullary and all letters in $\tilde{\Sigma}$ are unary.
Define
$\mathscr{S}^{\prime}=(\Sigma^{\prime},\mathcal{N}^{\prime},S^{\prime},\mathcal{R}^{\prime})$
as
$\Sigma^{\prime}=\tilde{\Sigma}\cup\\{\mathsf{e}\\},\mathcal{N}^{\prime}=\mathcal{N}\cup\\{B:\mathtt{o}\rightarrow(\mathtt{o}\rightarrow\mathtt{o})\\},\mathcal{R}^{\prime}=\\{r[\mathtt{br}/B]\mid
r\in\mathcal{R}\\}\cup\\{Bxy\twoheadrightarrow x,Bxy\twoheadrightarrow y\\}$,
where by $r[\mathtt{br}/B]$ we mean the rule $r$ with $\mathtt{br}$ uniformly
replaced by $B$. Note that the only new non-terminal symbol introduced is $B$,
which is of order 1. Hence the obtained word scheme $\mathscr{S}^{\prime}$ is
of the same order as $\mathscr{S}$.
$\mathcal{L}_{\mathsf{p}}(\mathscr{S})\subseteq\mathcal{L}_{\mathsf{w}}(\mathscr{S}^{\prime})$:
Let $w\in\mathcal{L}_{\mathsf{p}}(\mathscr{S})$. Therefore there exists a
finite path $\Pi$ in a sentential form $t$ such that $\forall\bar{n}\in
dom(\Pi)\;w_{\Pi}(\bar{n})\in\Sigma$. We derive $t^{\prime}:=t[\mathtt{br}/B]$
using the corresponding rules in $\mathcal{R}^{\prime}$. Note that the
corresponding path $\Pi^{\prime}$ in $t^{\prime}$ satisfies
$\forall\bar{n}\in\operatorname{dom}(\Pi^{\prime})w_{\Pi^{\prime}}(\bar{n})\in\Sigma\cup\\{B\\}$.
We then apply either $Bxy\twoheadrightarrow x$ or $Bxy\twoheadrightarrow y$ to
each $B$ in $t^{\prime}$ according to the path $\Pi^{\prime}$ to obtain $w$ in
the word scheme.
$\mathcal{L}_{\mathsf{w}}(\mathscr{S}^{\prime})\subseteq\mathcal{L}_{\mathsf{p}}(\mathscr{S})$:
We define the order $\leq_{\mathsf{pre}}$ on sequences of natural numbers
$\bar{n},\bar{m}$ as $\bar{n}\leq_{\mathsf{pre}}\bar{m}$ if
$\bar{m}=\bar{n}\bar{k}$ for some sequence $\bar{k}$.
Suppose that for given a sentential form $t^{\prime}$ of
$\mathscr{S}^{\prime}$ there exists a sentential form $t$ of $\mathscr{S}$ and
a map $\alpha:\operatorname{dom}(t^{\prime})\rightarrow\operatorname{dom}(t)$
(simply called embedding henceforth) satisfying the following conditions:
* •
$\forall\bar{n}\in\operatorname{dom}(t^{\prime})$
$t^{\prime}(\bar{n})=\begin{cases}B\text{ if
}t(\alpha(\bar{n}))=\mathtt{br}\\\ t(\alpha(\bar{n}))\text{
otherwise}.\end{cases}$ (8)
* •
$\forall\bar{n},\bar{m}\in\operatorname{dom}(t^{\prime}),\bar{n}\leq_{\mathsf{pre}}\bar{m}$
implies
$\alpha(\bar{n})\leq_{\mathsf{pre}}\alpha(\bar{m})\text{, and}$ (9)
$\displaystyle(\forall\bar{l}\;(\bar{n}<_{\mathsf{pre}}\bar{l}<_{\mathsf{pre}}\bar{m})\text{
implies }t^{\prime}(\bar{l})\notin\tilde{\Sigma})\text{ implies }$ (10)
$\displaystyle(\forall\bar{k}\;\alpha(\bar{n})<_{\mathsf{pre}}\bar{k}<_{\mathsf{pre}}\alpha(\bar{m})\text{
implies }t(\bar{k})\notin\tilde{\Sigma})$
Informally, Equations 8, 9 and 10 state the following: $\alpha$ preserves
labels, except for the case of $\mathtt{br}$ where it maps to $B$, $\alpha$
preserves the order $\leq_{\mathsf{pre}}$ and the images of any two nodes with
node labels from $\tilde{\Sigma}$ with no node label from $\tilde{\Sigma}$ in
between are mapped to two such nodes with the same property.
We will show by induction on the length of the derivation that such a pair
$(t,\alpha)$ always exists given some $t^{\prime}$. Let us see how the
existence of such $t$ and $\alpha$ gives us the proposition. Consider a word
$w=w_{1}w_{2}...w_{n}\in\mathcal{L}_{\mathsf{w}}(\mathscr{S}^{\prime})$. In
other words, there is a term
$t^{\prime}=w_{1}(w_{2}...(w_{n}(\mathsf{e}))...)$ such that
$S^{\prime}\underset{\mathscr{S}^{\prime}}{\twoheadrightarrow}^{*}t^{\prime}$.
Corresponding to this, we have a sentential form
$S\underset{\mathscr{S}}{\twoheadrightarrow}^{*}t$ and $\alpha$ satisfying the
given conditions. In particular, there exists a path $\Pi$ with
$\operatorname{dom}(\Pi)\subseteq dom(t)$ which is the path in $t$ connecting
the $\alpha(\varepsilon)$ to $\alpha(0^{n})$. It is immediate that
$\mathrm{Proj}_{\tilde{\Sigma}}(\Pi)=w$. It remains to show the existence of
$t$ and $\alpha$ by induction on the length of derivations.
$t^{\prime}_{0}$$\bar{n}$$u_{0}^{\prime}$$l_{0}^{\prime}$$r_{0}^{\prime}$$Bxy\twoheadrightarrow
x$$\mathscr{S}^{\prime}$$t^{\prime}$$\bar{n}$$u_{0}^{\prime}$$l_{0}^{\prime}$$t_{0}$$\alpha_{0}(\bar{n})$$u_{0}$$l_{0}$$r_{0}$$\alpha_{0}$$t_{0}$$u_{0}$$\alpha(\bar{n})$$l_{0}$$r_{0}$$\alpha$
Figure 2. Construction of embedding $\alpha$ from $\alpha_{0}$ in the case of
the rule $Bxy\twoheadrightarrow x$
$t^{\prime}_{0}$$\bar{n}$$u_{0}^{\prime}$$l_{0}^{\prime}$$r_{0}^{\prime}$$Nxy\twoheadrightarrow
t_{1}$$\mathscr{S}^{\prime}$$t^{\prime}$$\bar{n}$$u_{0}^{\prime}$$\bar{m}$$\bar{k}$$l_{0}^{\prime}$$r_{0}^{\prime}$$t_{1}[x/l^{\prime}_{0},y/r^{\prime}_{0}]$$t_{0}$$\alpha_{0}(\bar{n})$$u_{0}$$l_{0}$$r_{0}$$\alpha_{0}$$Nxy\twoheadrightarrow
t_{1}$$\mathscr{S}$$t$$\alpha(\bar{n})$$u_{0}$$\alpha(\bar{m})$$\alpha(\bar{k})$$l_{0}$$r_{0}$$t_{1}[x/l_{0},y/r_{0}]$$\alpha$
Figure 3. Construction of embedding $\alpha$ from $\alpha_{0}$ when the rule
is $Nxy\twoheadrightarrow t_{1}$
Base case: This is trivial since $t=t^{\prime}=S$.
Induction step: Suppose
$t_{0}^{\prime}\underset{r\in\mathcal{R}^{\prime}}{\twoheadrightarrow}t^{\prime}$
where $t_{0}^{\prime}$ is a sentential form. By induction hypothesis, there is
a sentential form $t_{0}$ of $\mathscr{S}$ and $t_{0}^{\prime}$ embeds into
$t_{0}$ via map $\alpha_{0}$. Assume that the rule $r$ is applied at position
$\bar{n}$ on $t_{0}^{\prime}$. We now have two cases to consider:
Case 1: The rule $r$ is
$Bxy\underset{\mathscr{S}^{\prime}}{\twoheadrightarrow}x$ (the case
$Bxy\underset{\mathscr{S}^{\prime}}{\twoheadrightarrow}y$ being symmetric). By
induction hypothesis, we have $t_{0}$ and and an embedding $\alpha_{0}$ of
$t_{0}^{\prime}$ into $t_{0}$. Refering to Figure 2, we see that $t$ can be
taken to be $t_{0}$ and $\alpha$ maps all nodes in the subtree
$u^{\prime}_{0}$ into $u_{0}$ as before, while the subtree $l^{\prime}_{0}$
rooted at $\bar{n}$ is mapped into $l_{0}$. It is immediate that $\alpha$
preserves the order and since by induction hypothesis $\alpha_{0}(\bar{n})$
has label $\mathtt{br}$, Equation 10 is also satisfied by $\alpha$ since no
new $\tilde{\Sigma}$ labelled nodes have been added.
Case 2: We demonstrate for the case when a non-terminal of arity two is
replaced for an easier reading of the proof: the rule $r$ is
$Nxy\underset{\mathscr{S}^{\prime}}{\twoheadrightarrow}t_{1}$ for some
nonterminal $N\neq B$. Refering to Figure 3, the rule replaces the subtree
rooted at $\bar{n}$ in $t^{\prime}_{0}$ with
$t_{1}[x/l^{\prime}_{0},y/r^{\prime}_{0}]$ where
$l^{\prime}_{0},r^{\prime}_{0}$ are respectively the left and right subtrees
of the subtree rooted at $\bar{n}$ in $u^{\prime}_{0}$. For nodes in the
subtree $u^{\prime}_{0}$ of $t^{\prime}$, $\alpha$ mimics $\alpha_{0}$. By
induction we know that the subtrees $l^{\prime}_{0},r^{\prime}_{0}$ of
$t^{\prime}_{0}$ embed respectively into $l_{0},r_{0}$ of $t_{0}$. Thus
$\alpha$ maps every subtree $l^{\prime}_{0}$ (resp. $r^{\prime}_{0}$) rooted
at $\bar{m}$ (resp. $\bar{k}$) in $t_{1}[x/l^{\prime}_{0},y/r^{\prime}_{0}]$
into the corresponding subtree $l_{0}$ (resp. $r_{0}$) rooted at
$\alpha(\bar{m})$ (resp. $\alpha(\bar{k})$). Note that $\alpha(\bar{m})$ can
be defined as the root of the $i$-th occurrence of $l_{0}$ from left to right
if $\bar{m}$ is the $i$-th occurrence of $l^{\prime}_{0}$ from left to right.
Nodes in $t_{1}[x/l^{\prime}_{0},y/r^{\prime}_{0}]$ which are not in any of
the $l_{0}^{\prime}$ (or $r_{0}^{\prime}$) subtrees (i.e.) between $\bar{n}$
and $\bar{m}$ (or $\bar{k})$) have corresponding nodes in
$t_{1}[x/l_{0},y/r_{0}]$ to which they can be mapped. Label preservation and
order preservation immediately follow by appeal to the induction hypothesis.
In order to see that Equation 10 holds, consider consecutive $\tilde{\Sigma}$
nodes $\bar{n}^{\prime},\bar{m}^{\prime}$ in $t^{\prime}$ (i.e.
$\bar{n}^{\prime}<_{\mathit{pre}}\bar{m}^{\prime}$ and there are no
$\tilde{\Sigma}$ labels in between). If both $\bar{n}^{\prime}$ and
$\bar{m}^{\prime}$ are in $u_{0}^{\prime}$ or in one of the $l^{\prime}_{0}$
(or $r_{0}^{\prime}$) then the induction hypothesis applies. In the case where
$\bar{n}^{\prime}\in u_{0}^{\prime}$ and $\bar{m}^{\prime}\in l_{0}^{\prime}$
(resp. $r_{0}^{\prime}$,) this means that no $\tilde{\Sigma}$ labels are
present in the path from $\bar{n}$ to $\bar{m}$ (resp. $\bar{k}$),
$\bar{n}^{\prime}$ to $\bar{n}$ nor $\bar{m}$ (resp. $\bar{k}$) to
$\bar{m}^{\prime}$. The path from $\alpha(\bar{n})$ to $\alpha(\bar{m})$ is
identical to that from $\bar{n}$ to $\bar{m}$ (resp. $\bar{k}$). The induction
hypothesis also implies that the first $\mathtt{a}$ label for some
$\mathtt{s}\in\tilde{\Sigma}$ in $l_{0}^{\prime}$ must be mapped to first
$\mathtt{a}$ label in $l_{0}$ (or $u_{0}^{\prime}$ does not contain any
$\tilde{\Sigma}$ labels). Hence there are no $\tilde{\Sigma}$ labels between
$\alpha(\bar{m})$ (resp. $\alpha(\bar{k})$) and $\alpha(\bar{m}^{\prime})$ and
similarly between $\alpha(\bar{n}^{\prime})$ and $\alpha(\bar{n})$. The final
case is when either $\bar{n}^{\prime}$ or $\bar{m}^{\prime}$ lies between
$\alpha(\bar{n})$ and $\alpha(\bar{m})$ (resp. $\alpha(\bar{k})$). There are
subcases here to consider when the other point lies in $u_{0}^{\prime}$,
$l_{0}$ or also between $\bar{n}$ and $\bar{m}$. In all of these subcases, it
easily follows that there are no extra $\tilde{\Sigma}$ labels introduced in
between two consecutive nodes which are in the image of $\alpha$.
|
11institutetext: Institut für Theoretische Physik, Goethe Universität, Max-
von-Laue-Str. 1, D-60438 Frankfurt, Germany 22institutetext: Black Hole
Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
33institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany 44institutetext: Tsung-Dao Lee Institute and School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240,
People’s Republic of China 55institutetext: Mullard Space Science Laboratory,
University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK
66institutetext: Department of Astrophysics/IMAPP, Radboud University
Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands 77institutetext:
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park
904, 1098 XH Amsterdam, The Netherlands 88institutetext: Dipartimento di
Fisica “E. Pancini”, Universitá di Napoli “Federico II”, Via Cinthia, I-80126,
Napoli, Italy 99institutetext: INFN Sez. di Napoli, Via Cinthia, I-80126,
Napoli, Italy 1010institutetext: Jodrell Bank Centre for Astrophysics,
University of Manchester, Machester M13 9PL, UK 1111institutetext: Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt,
Germany 1212institutetext: School of Mathematics, Trinity College, Dublin 2,
Ireland
1212email<EMAIL_ADDRESS>
# Using space-VLBI to probe gravity around Sgr A∗
C. M. Fromm 112233 Y. Mizuno 4411 Z. Younsi 5511 H. Olivares 6611 O. Porth
7711 M. De Laurentis 889911 H. Falcke 6633 M. Kramer 331010 L. Rezzolla
1111111212
( Draft 1.0: )
###### Abstract
Context. The Event Horizon Telescope (EHT) will soon provide the first high-
resolution images of the Galactic Centre supermassive black hole (SMBH)
candidate Sagittarius A* (Sgr A∗), enabling us to probe gravity in the strong-
field regime. Besides studying the accretion process in extreme environments,
the obtained data and reconstructed images could be used to investigate the
underlying spacetime structure. In its current configuration, the EHT is able
to distinguish between a rotating Kerr black hole and a horizon-less object
like a boson star. Future developments can increase the ability of the EHT to
tell different spacetimes apart.
Aims. We investigate the capability of an advanced EHT concept, including an
orbiting space antenna, to image and distinguish different spacetimes around
Sgr A∗.
Methods. We use general-relativistic magneto-hydrodynamical (GRMHD)
simulations of accreting compact objects (Kerr and dilaton black holes, as
well as boson stars) and compute their radiative signatures via general-
relativistic radiative transfer (GRRT). To facilitate comparison with upcoming
and future EHT observations we produce realistic synthetic data including the
source variability, diffractive and refractive scattering while incorporating
the observing array, including a space antenna. From the generated synthetic
observations we dynamically reconstructed black hole shadow images using
regularised Maximum Entropy methods. We employ a genetic algorithm to optimise
the orbit of the space antenna with respect to improved imaging capabilities
and u-v-plane coverage of the combined array (ground array and space antenna
and developed a new method to probe the source variability in Fourier space.
Results. The inclusion of an orbiting space antenna improves the capability of
the EHT to distinguish the spin of Kerr black holes and dilaton black holes
based on reconstructed radio images and complex visibilities.
###### Key Words.:
Physical data and processes: Gravitation, Magnetohydrodynamics (MHD),
radiation mechanisms: thermal – Methods: numerical – Galaxies: individual: Sgr
A∗ – Techniques: interferometric
## 1 Introduction
The Event Horizon Telescope presented the first horizon scale images of the
black hole in M87 (Event Horizon Telescope Collaboration et al., 2019a) and
will soon provide the first image of the black hole candidate Sgr A∗ in our
Galaxy. The current configuration of the EHT consists of eight telescope
scattered across Europe, North- and South America and the South Pole (Event
Horizon Telescope Collaboration et al., 2019b). Combining the data recorded
simultaneously by the individual telescopes after the observations, $\mu$as
angular resolution is achieved. These observations enable us, for the first
time, to study accretion processes in M87 and in the centre of our galaxy with
unparalleled resolution and to probe gravity in the strong-field regime. Given
the small number of telescopes participating in the observations, the
intrinsic variability of the source and the interstellar scattering screen,
reconstructing an image and discriminating among different spacetimes is
challenging (Lu et al., 2014, 2016; Mizuno et al., 2018). An interferometer
such as the EHT samples the brightness distribution of an astronomical object
in Fourier-space: the so called _u-v plane_. Due the limited number of
participating telescopes, the u-v plane is sparsely sampled. The sampling of
this u-v plane increases with the number of telescopes and the duration of the
observations. However, an improvement in the resolution of the image can be
only be obtained by increasing the distance between the telescopes, the so-
called baselines, or by increasing the observed frequency. Given the current
configuration of the EHT array, a significant increase in the baselines can
only be achieved by extending baselines to space. Using the ground array while
increasing the observed frequency above 345 GHz is limited by the opacity of
the atmosphere and its water vapour content.
The concept of space-based Very Long Baseline Interferometry (VLBI) including
a ground array has been studied extensively since the early 1970s (see
Schilizzi, 2012, for a historic overview) and successfully launched space
antennas are Highly Advanced Laboratory for Communications and Astronomy
(HALCA) (project VLBI Space Observatory Programme, (VSOP)) (Hirabayashi et
al., 1998) and, more recently, Spektr-R (project RadioAstron) (Kardashev et
al., 2013). Both missions operate at lower frequencies than the EHT (230 GHz)
and in the case of RadioaAstron provide $\rm\mu as$ resolution (Gómez et al.,
2016). Therefore, in this work we consider the increase in the angular
resolution that can be obtained by extending the baselines of the EHT via a
space-based antenna (see also Palumbo et al., 2019). This configuration has
the advantage of using a well-calibrated ground array. For a mission based
entirely on space telescopes, see a recent studies by Roelofs et al. (2019);
Fish et al. (2019). Within this work we address the scientific question as to
whether such a configuration will improve the current ability of the EHT to
distinguish among different theories of gravity using radio observations of
Sgr A∗.
The structure of this paper is as follows: in Sect. 2 we briefly introduce our
GRMHD simulations and GRRT calculations. The procedure for the selection of
the orbit of the space antenna is introduced in Sect. 3 and the results of the
synthetic imaging and data analysis are shown in Sect. 4. We present our
discussion and conclusions in Sect. 5. Sgr A∗ is located at RA:
$17^{h}\,45^{m}\,40.0409^{s}$ and
DEC:$-29^{\circ}\,45^{\prime}\,40.0409^{\prime\prime}$, at a distance of $\rm
8178\pm 13\,stat.\pm 22\,sys.\,pc$ and its central candidate SMBH exhibits a
mass of $M_{\rm bh}=4.14\times 10^{6}M_{\astrosun}$ (Gravity Collaboration et
al., 2019).
## 2 GRMHD and GRRT simulations
We use the state-of-the-art GRMHD code BHAC (Porth et al., 2017; Olivares et
al., 2019) and perform three-dimensional accretion simulations on to Kerr
black holes (${\rm a}_{*}$=0.6 and ${\rm a}_{*}$=0.94), dilaton black holes
and a boson star (Mizuno et al., 2018; Olivares et al., 2020). Here ${\rm
a}_{*}$ is the dimensionless black-hole spin parameter, with $|\rm a_{*}|<1$.
Figure 1: Top: panels A and B show GRRT images for a Kerr black hole with
${\rm a}_{*}$=0.6 including scattering effects ( diffractive and refractive
scattering) for two selected times t=12.6 h and t=23.6 h (dash-dotted line in
the bottom panel) at a frequency of 230 GHz. The average image for 24 hours of
observation is presented in panel C. Bottom: simulated total flux light curve
at 230 GHz for 24 hours of observations. Indicated is a typical EHT
observation with 12 hours duration.
We have chosen a black hole solution in the Einstein-Maxwell-dilaton-axion
(EMDA) theory, which derives from a particular string theory (Okai, 1994;
García et al., 1995). Such black-hole solutions exhibit more diverse physical
properties than others in the literature. For example, they have hair, in
addition to mass, rotation and charge, so as to be an interesting laboratory
for performing black hole experiments and studying possible differences
between Einstein’s gravity and alternative theories of gravity. Moreover,
since both the dilaton and the axion are considered to be candidates for dark
matter, the study of the shadow from a dilaton black hole may provide hints on
the observational properties of the matter in which the black hole is immersed
(Davoudiasl & Denton, 2019).
We also consider a stable boson star, which although classified as a compact
object, is physically completely different from a black hole (Schunck &
Liddle, 1997). It is well known that the gravitational field of a boson star
bends light around itself, creating a region _resembling_ the shadow of a
black hole’s event horizon. Like a black hole, a boson star will accrete
ordinary matter from its surroundings, but its opacity means this matter
(which likely would heat up and emit radiation) would be visible at its
center. There is no significant evidence so far that such stars exist.
However, it may become possible to detect them through VLBI measurements
(Olivares et al., 2020). In theory, a super-massive boson star could exist at
the core of a galaxy, which might explain many of the observed properties of
active galactic nuclei. Boson stars have also been proposed as candidate dark
matter objects, and it has even been hypothesised that the dark matter haloes
surrounding most galaxies might be viewed as enormous boson stars (Levkov et
al., 2018).
The formation of boson stars or other exotic objects is an interesting process
to study. Although we know that at the center of our galaxy there is a highly
compact object, it is important to study different candidates because we might
find that other galaxies could harbour exotic objects composed of “non-
baryonic” matter at their centres (Lee & Koh, 1996).
For spacetimes not described by Einstein’s general theory of relativity, we
use the Rezzolla-Zhidenko metric parameterisation (Rezzolla & Zhidenko, 2014;
Younsi et al., 2016). The initial conditions of all GRMHD simulations for Kerr
and dilaton black holes as well as for the boson star, consist of a torus in
hydrodynamical equilibrium with a weak poloidal magnetic field. In order to
trigger the accretion process the magnetic field in the torus is seeded with a
small perturbation which leads to the formation of the magneto-rotational
instability (MRI). After the saturation of the MRI (${\rm
t}>5000\,\mathrm{GM/c^{2}}$, with gravitational constant, G, mass of the
central object, M, and speed of light c) all simulations show a quasi-
stationary accretion flow (for more details see Mizuno et al., 2018; Olivares
et al., 2020).
In the next step we compute the radiative signature of the accretion process
via GRRT calculations using the BHOSS code (Younsi et al., 2020; Gold et al.,
2020). We use an viewing angle of $\theta=60^{\circ}$, adjust the ion-to-
electron temperature ratio ($T_{i}/T_{e}=3$) and the mass-accretion rate to in
order to obtain the an average flux of 4 Jy at 230 GHz (see supplemental
information in Mizuno et al., 2018, for a detailed description of the GRRT
calculations and fitting procedure). This leads to a flux density variation
between 3 Jy and 5 Jy (see light curve in the bottom panel of Fig. reffluxvar)
which is in agreement with the observed flux density variations 2 Jy – 5.5 Jy
provided by Bower et al. (2015) The GRRT images are computed every 10 M which
corresponds to 200 s for SgrA* for a time span of 4320 M (24 h for SgrA). This
long duration allows for the calculation of several overlapping and two
independent 12 h111typical duration for EHT observations of SgrA∗ which allows
European and American telescopes to participate in the observation.
observational window. In this work we investigate the following spacetimes:
Kerr black holes in general relativity, a dilaton black hole and a boson star.
The variability of the total flux and in the emission structure of the
individual GRRT snapshots, together with the averaged image, is presented in
Fig. 1 and in Table LABEL:GRRTinfo we present an overview of the GRRT images
used.
Table 1: Overview of simulations used Spacetime | ${\rm a}_{*}$ | $\theta$ [deg] | $\Delta t$ [M (s)] | t [M (h)]
---|---|---|---|---
Kerr | 0.60 | 60 | 10 (200) | 4320 (24)
Kerr | 0.94 | 60 | 10 (200) | 4320 (24)
dilaton | 0 | 60 | 10 (200) | 4320 (24)
boson star | 0 | 60 | 10 (200) | 4320 (24)
## 3 Orbit selection and optimisation strategy
In this section of our exploratory paper we present one possible array
optimisation strategy which could lead to improved imaging capabilities for
the EHT and thus may allow us to distinguish among different spacetimes. Thus,
the question arises as to which kind of orbit is required for improving the
imaging of the Galactic Centre. The answer to this question depends strongly
on the observational constraints: the position of the Galactic Centre relative
to the space and ground antennas and the duration of the observations. In the
following we will assume an observation schedule consisting of 12 hours of Sgr
A∗ observations. This long observing time allows European as well as North-
and South American telescopes to participate in the observation.
For the space-EHT concept we consider the EHT 2017 configuration as the ground
array and include the Northern Extended Millimeter Array (NOEMA222http://iram-
institute.org/EN/) and a space antenna. In Table 2 we list each antenna and
its corresponding system equivalent flux density (SEFD) (Event Horizon
Telescope Collaboration et al., 2019b).
Table 2: Effective antenna diameter $d$ (for single telescopes this corresponds to the diameter of the dish) and system equivalent flux density (SEFD) used for the ground and space antennas (see Event Horizon Telescope Collaboration et al., 2019b, for details) Telescope | $d_{\mathrm{eff}}$ [m] | SEFD [Jy]
---|---|---
ALMAa | 73 | 74
APEX | 12 | 4700
JCMT | 15 | 10500
LMT | 32.5 | 4500
NOEMA∗ | 52 | 700
PV | 30 | 1900
SMT | 10 | 17100
SMAb | 14.7 | 6200
SPTc | 6 | 19300
Space Antenna∗ | 8 | 20000
∗ only used for the space-EHT configuration.
a for EHT 2017 ALMA used 37 $\times$ 12 m.
b for EHT 2017 SMA used 6 $\times$ 6 m.
c for EHT 2017 SPT was under-illuminated
with an effective diameter of 6 m.
For the orbiting space antenna we assume a diameter of 8 meters (similar to
the size of Spektr-R), a system temperature of 100 K and an antenna efficiency
of 40%. Given these values, a SEFD of $20\times 10^{3}$ Jy is computed. The
orbit of the space antenna can be described by six orbital elements which can
be divided into orbital shape parameters (semi-major axis, $a$, and
eccentricity, $e$) and orbital orientation parameters which provide the
location and orientation of the orbit in space relative to the equatorial
plane (inclination, $i$, longitude of the ascending node, $\Omega$, argument
of perigee, $\omega$, and the true anomaly, $\vartheta$). The six orbital
elements of the space antenna together with duration of the observations lead
to a seven-dimensional parameter space which has to be searched to obtain an
optimal configuration. This task can be formulated as a constrained non-linear
optimisation problem and written as follows:
$\displaystyle\begin{array}[]{ll@{}ll}\mathrm{minimize:}&\displaystyle
f(\@vec{x})\,,&\\\ \mathrm{subject\ to:}&\displaystyle g_{j}(\@vec{x})\leq
0\,,&&j=1,...,n\,,\\\ &\displaystyle x_{L,i}\leq x_{i}\leq
x_{R,i}\,,&&i=1,...,m\,,\end{array}$ (4)
where $\@vec{x}$ is a seven-dimensional vector containing the model
parameters, i.e.,
$\@vec{x}\equiv\left[t_{\mathrm{obs}},a,e,i,\Omega,\omega,\vartheta\right]^{T}$,
$f(\@vec{x})$ is the objective function (minimisation function),
$g_{j}(\@vec{x})$, are the constraints and $x_{L,i}$ and $x_{R,i}$ are lower
and upper boundaries for the model parameters. In Table LABEL:parabound we
report the boundaries used during the optimisation.
Table 3: Parameter boundaries used for the optimisation $t_{\rm obs}$ [h] | $a$ [$10^{3}$km] | $e$ | $i$ [deg] | $\Omega$ [deg] | $\omega$ [deg] | $\vartheta$ [deg]
---|---|---|---|---|---|---
0 -12 | 0.4-100 | 0-0.9 | 0-180 | 0-360 | 0-360 | 0-360
Improving the imaging capabilities of the EHT can be translated into an
increased angular resolution and a denser sampling of the u-v plane as
compared to the current EHT configuration. The improvement of the imaging
capabilities i.e., better array resolution and u-v plane coverage can be
assessed by computing image metrics between the infinite resolution GRRT image
and the reconstructed image. Therefore, we use for the minimisation process a
combination of the denser sampling in the u-v plane (minimising the distance
between u-v points) and the improved imaging (improved image metrics). For the
image metrics we used the normalised cross correlation coefficient (nCCC) and
the structural dissimilarity measure (DSSIM)333smaller values indicate a
better image agreement (Wang et al., 2004). The minimisation function is given
by:
$f\left(\@vec{x}\right)=\mathrm{DSSIM}\left(\@vec{x}\right),$ (5)
and we use the following constraints444the constraints are fulfilled if the
$g_{i}<0$ to speed up the optimisation procedure:
$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\mathrm{DSSIM}\left(\@vec{x}\right)-0.75\times\mathrm{DSSIM_{EHT2017}},$
(6) $\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle\Delta r_{\rm
uv,\,max}\left(\@vec{x}\right)-\Delta r_{\rm uv,\,max}^{\rm EHT2017}\,.$ (7)
The first constraint ensures that the DSSIM is improved by at least 25% as
compared to image reconstructed using the EHT 2017 configuration. The denser
sampling of the u-v plane is addressed by the second constraint, $g_{2}$,
where we enforce that the projected distance between the u-v points, $\Delta
r$, is minimised with respect to u-v sampling of the EHT 2017 configuration.
During the optimisation we generate for each set of parameters e.g., for each
$\left(\@vec{x}\right)$ synthetic visibilities including both, diffractive and
refractive scattering (Johnson, 2016) and reconstruct the image using the
EHTim package 555https://github.com/achael/eht-imaging (Chael et al., 2016,
2018).
Due to the numerical costs we use during the optimisation a single GRRT image
instead of a series of GRRT images (or GRRT movie) and the array configuration
and additional parameters used during the data generation and image
reconstruction are listed in Table 2 and 4. The procedure of the image
reconstruction follows the approach of Chael et al. (2016): initialisation of
the imaging using as gaussian prior with a FWHM of 70 $\mu$as and the repeated
re-initialisation of the imaging using the previously obtained image convolved
with half the nominal array resolution. During the orbit optimisation we use
two re-initialisation loops and we align the reconstructed image with the GRRT
image prior to the computation of the DSSIM.
Table 4: Parameters used for the data generation and image reconstruction
synthetic data generation
---
$t_{\rm int}$ [s] | $\Delta\nu$ [GHz] | $t_{\rm gap}^{a}$ [s] | $\nu_{\rm obs}$ [GHz] | gain off setb
12 | 4, 8∗ | 200 | 230 | 0.1
image reconstruction
data | weightingc | regularizer | weightingc
visibilities | 100 | simple entropy | 2
bi-spectra | 10 | simple entropy | 1
∗ only used for the space-EHT configuration.
a time difference between GRRT images
b antenna gains are drawn form a normal distribution with
mean=1.0 and standard deviation=0.1 (gain off set).
c weighting factor for data terms and image entropies
used during the image reconstruction.
Figure 2: Result of the orbit optimisation for 12 hours of Sgr A∗
observations. Top row from left to right: GRRT image (panel a), reconstructed
image with interstellar scattering (including both, diffractive and refractive
scattering during the generation of the synthetic visibilities) using the EHT
2017 configuration convolved with 75% (red shading) of the nominal array
resolution (light grey shading, panel b) and reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the space-EHT
configuration convolved with 75% (red shading) of the nominal array resolution
(light grey shading, panel c) for a dilaton black hole. Bottom row from left
to right: satellite orbit as seen from SgrA⋆ with orbital parameters and
orbital period (panel d), satellite ground track (red lines for 12 hours and
grey ones for 24 hours) and ground array antennas (blue points, panel e) and
u-v sampling for the ground array (white points) and the baselines including
the space antenna (red points, panel f).
Given the high dimensionality of this constrained optimisation problem,
gradient-based solvers may become stuck in a local minimum and/or required
large computational resources to map out the gradient of the parameter space
with sufficient resolution to avoid this problem. An elegant method to
circumvent the above mentioned difficulties is based on gradient-free
optimisation algorithms. In this work we apply a genetic algorithm (GA) and in
particular employ the implementation of a Non Sorting Genetic Algorithm II
(NSGA2) to solve the optimisation problem (Deb et al., 2002). For the initial
generation we use 1000 randomly initiated orbits and we evolve them for 100
generations.
The results of our numerical optimisation are presented in Table 5 and Fig. 2.
The improvement of the image reconstruction is clearly visible by comparing
the EHT 2017 reconstructed image (panel b in Fig. 2) with the image obtained
by space-EHT (panel c in Fig. 2). The ground truth GRRT image is presented in
panel a in Fig. 2. The space-EHT image is able to capture and image fine flux
arcs which are smeared out in the EHT 2017 configuration. The visible image
improvements are also reflected in the improved image metrics: the nCCC
increased from 0.92 to 0.96 and the DSSIM decreased by 40% from 0.252 to 0.152
(see also constraint $g_{1}$ Eq. 7). Notice that increased nCCC and decreased
DSSIM corresponds to a better image matching. The satellite orbit as seen from
Sgr A⋆ (note that the Earth is viewed from $-30^{\circ}$) is presented in
panel d in Fig. 2. In panel e in Fig. 2 the satellite ground track (projection
of the satellite orbit onto the surface of the Earth) of the space antenna
(red points for 12 hours and grey ones for 24 hours) and the ground antennas
are indicated by the blue points. The sampling of the u-v plane is presented
in panel f in Fig. 2 where the white points indicate the u-v tracks of the
ground array and the red points the u-v tracks including the space antenna.
The addition of the space antenna is not only extending the u-v sampling up to
25 G$\lambda$ but also adds short and intermediate baselines to the array as
required by the constraint $g_{2}$ (see Eq. 7) of the optimisation process
which also improve the imaging capabilities.
Table 5: Optimised satellite orbit for 12 hours of Sgr A∗ observation. a [km] | i [∘] | e | $\Omega$ [∘] | $\omega$ [∘] | $\vartheta$ [∘] | $t_{\mathrm{obs,start/stop}}$ [UT] | $\mathbf{T_{p}}$ [h]
---|---|---|---|---|---|---|---
14900 | 67 | 0.5 | 46 | 70 | 330 | 04:00 – 16:00 | 5.03
The orbit optimisation procedure suggested an elliptical orbit with a semi-
major axis of $a=14900$ km with an eccentricity of $e=0.5$ at an inclination
of 67∘. The optimal time span of the SgrA⋆ observation is obtained between
4:00 UT and 16:00 UT. The orbital period of the satellite is $\rm T_{p}=5.03$
h. Notice, that elliptical orbits with long orbital periods have also been
used for VSOP ($e$=0.6, $\rm T_{p}=6.3$ h) and the Spekt-R ($e=0.9$, $\rm
T_{p}$=200 h) (Hirabayashi et al., 1998; Kardashev et al., 2013). In Appendix
A we illustrate the impact of different satellite orbits on the image
reconstruction. Our results differ from the results of Palumbo et al. (2019)
which used a circular orbit with semi-major axis $a=6652$ km at an inclination
of 61∘. The similarity in the inclination is due to the declination (DEC) of
SgrA⋆ of -29∘ and a face-on orbit is given by a inclination
$\sim\mathrm{DEC}\pm 90^{\circ}$. The difference in the eccentricity, $e$, and
semi-major axis, $a$, can be explained by the different models used for SgrA⋆.
Palumbo et al. (2019) uses a Schwarzschild black hole (a=0) at an inclination
of 10∘. In this work we use four different physical models: GR black holes
(a=0.6 and a=0.94), a Dilaton black hole and a boson star all at an
inclination of $60^{\circ}$. Due to the larger relativistic effects at higher
inclination (Doppler boosting) the GRRT images show a large left-right
asymmetry e.g., the flux distribution is more compact as compared to black
holes seen at an inclination of $10^{\circ}$. In addition the boson star used
in our work shows the most compact flux distribution around $10\,\mathrm{\mu
as}$ (see panel i in Fig. 4). Since we require in our optimisation to resolve
and recover the compact structure of the black holes and at the same time the
boson star we need to improve the angular resolution (longer base lines) and
the u-v plane coverage (short baselines). Using one satellite this can be
achieved by an elliptical orbit. In our case the perigee is located at a
height of $\sim 1100\,\mathrm{km}$ and the apogee at a height of $\sim
22300\,\mathrm{km}$. The space-ground tracks of this orbit are presented in
panel f of Fig. 2 and the plot shows clearly the improved u-v coverage (short
baselines with $\sqrt{u^{2}+v^{2}}<9\,G\lambda$ and long base lines
$\sqrt{u^{2}+v^{2}}>9\,G\lambda$).
Using this satellite orbit and the observing time span from 4:00-16:00 UT the
reconstructed images for all black holes and the boson star clearly improve
the quality of the reconstructed images (better image metrics as compared to
the EHT 2017 array) and are able to recover the structure seen in their
infinite resolution GRRT counterparts (see Fig. 4).
After the optimisation of the space-EHT e.g., defining the orbit of the space
antenna and the observing time we create synthetic data taking the source
variability during the course of the EHT observation into account. Therefore
we create for each of the space-times under investigation a movie from the
GRRT images which covers 12 h of observations (first 216 GRRT images are used,
see indicated time 12 h time span in Fig. 1). From this movie we create the
synthetic visibilities using the same parameters as used during the orbit
optimisation (see Table 2 and 4). As a consequence of creating the synthetic
data from a variable source the visibility amplitude (VA) of the zero
baselines (ALMA-APEX, JCMT-SMA) is variable and not stationary as in the case
of using an 12 h averaged GRRT image for the data creation. This behaviour is
also true for non-zero baselines and for the closure phases (CP). In Fig. 3 we
compare the VA and CP created from GRRT movie and from its static average
frame for the ALMA-SMT baseline and for the ALMA-LMT-SMT triangle. The
differences in the behaviour of the VA and CP are clearly visible and the most
striking difference can be seen at t=10 UT.
Figure 3: Comparison between synthetic data generation from a dynamical movie
(blue lines) and from its static average frame (black lines) for the
visibility amplitude (top) and closure phase (bottom) for Kerr black hole with
${\rm a}_{*}$=0.6.
## 4 Results
Given the underlying variability in the flux density and also in source size
(see Fig. 1) of the different models, standard VLBI imaging approaches which
assume a static source during the course of the observations are likely to
fail reconstructing an image. Therefore we follow the approach of Johnson et
al. (2017) and use dynamical imaging to reconstruct an average image from the
time variable data. Since we are mainly interested in the average image
obtained during the observation and our GRRT images do not show significant
flaring events666we define a flare as doubling of the flux density from one
frame to another we use the $\mathcal{R}_{\Delta I}$ regularizer with $D_{2}$
distance metric and complex visibilities as data product (see Johnson et al.,
2017, for more details). We reconstructed 24 frames with a duration of 30
minutes each for a total duration of 12 h. For the first initialisation of the
imaging we applied a circular gaussian with FWHM=70 $\mu$as prior image and
for repeated re-initialisation of the imaging we use as prior the previously
obtained average image computed from all 24 frames convolved with the half of
the nominal array resolution. For the dynamical reconstruction we applied 5
re-initialisation loops.
### 4.1 Image Plane comparison
In order to quantify the ability to test different space-times we compute the
structural dissimilarity measure (DSSIM) (Wang et al., 2004) between the GRRT
and reconstructed images. Before computing the DSSIM we perform an image
alignment using the normalised cross correlation coefficient (nCCC) (see,
e.g., Mizuno et al., 2018). Both image metrics are reported in the panels of
Fig. 4 and in Table LABEL:results. In Table LABEL:results we list the DSSIM
and nCCC values computed for the different image combinations and for the
different array configurations. If we could clearly distinguish among
different theories based on the reconstructed images, the smallest DSSIM and
largest nCCC values should be obtained for equal image pairs e.g., boson star
– boson star (the diagonal in Table LABEL:results). However, for the EHT 2017
configuration this is only true for the boson star. Thus based on
reconstructed images obtained from synthetic data which take the source
variability into account it is difficult for the EHT 2017 configuration to
distinguish the spin of Kerr black holes and to differentiate a Kerr black
from a dilaton black hole (see very similar values in
Figure 4: Synthetic black hole images for Sgr A∗ for a Kerr black hole with
${\rm a}_{*}$=0.94 (panels a-c), a Kerr black hole with ${\rm a}_{*}$=0.6
(panels d-f), a non-rotating dilaton black hole (panels g-i) and for a stable,
non-rotating boson star (panels j-l). From left to right and for all rows:
GRRT image (panels a, d, g, j), dynamically reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the EHT 2017
configuration convolved with 75% (red shading) of the nominal beam size (light
grey shading, panels b, e, h, k) and dynamically reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the space-EHT
configuration convolved with 75% (red shading) of the nominal beam size (light
grey shading, panels c, f, i, l)
Table LABEL:results). Including a space antenna improves the ability to
distinguish among the different spacetimes, especially for the Kerr black hole
with high spin. Given the obtained image metrics the space-EHT concept can
distinguish the spin of the Kerr black holes, but it is still difficult to
discriminate between a Kerr black hole with spin ${\rm a}_{*}$=0.6 and a
dilaton black hole (see Table LABEL:results).
By comparing the image metrics obtained for EHT 2017 and the space-EHT concept
two different behaviours can be found: improved image metrics (decreased DSSIM
and increased nCCC) and worsen ones (increased DSSIM and decreased nCCC). For
equal image pairs e.g., dilaton-dilaton, the image metrics for the space-EHT
concept improved as compared to the EHT 2017 (as discussed above). However for
unequal image pairs there are two different behaviours. For example in the
boson star - dilaton case the image metrics improved. In contrast to the
dilaton - boson star pair where the image metrics become worse. This behaviour
could be understood in the following way: The boson star image is very compact
as compared to the dilaton model (see left column in Fig. 4). Due to the
limited resolution of the EHT 2017 array the intrinsically large structure of
the dilaton model is smeared and scattered out to an even larger structure
(compare panel i and j in Fig. 4). However the improved imaging capabilities
of the space-EHT concept leads to a more compact and less smeared out source
structure which is reflected by a better matching between the boson star and
the dilaton black hole. The contrary happens in the dilaton - boson star case:
the EHT 2017 blurs the true boson star structure to a large size which matches
better the dilaton structure as compared to sharper more compact boson star
image provided by the space-EHT concept. As a result the image metrics for
this image pair will become worse as compared to the EHT 2017. A similar
behaviour can be seen for Kerr ${\rm a}_{*}$=0.6 - boson star and to some
extent for dilaton - Kerr ${\rm a}_{*}$=0.94 and Kerr ${\rm a}_{*}$=0.6 - Kerr
${\rm a}_{*}$=0.94. In Appendix B we provide a more detailed study on the
variation of the image metrics with respect to intrinsic source size and array
resolution.
Table 6: Results of the cross-comparison of the synthetic images using the EHT 2017 and the advanced EHT configuration. The values correspond to the DSSIM and the numbers in brackets indicated the normalised cross correlation coefficient (nCCC). Small values for the DSSIM and numbers close to 1 for the nCCC indicate well-matched images (lowest DSSIM values are indicated in bold). | boson star | dilaton | Kerr (${\rm a}_{*}$=0.6) | Kerr (${\rm a}_{*}$=0.94)
---|---|---|---|---
_EHT 2017 configuration_
boson star | 0.03 (0.98) | 0.17 (0.79) | 0.18 (0.81) | 0.10 (0.88)
dilaton | 0.16 (0.83) | 0.13 (0.92) | 0.13 (0.92) | 0.12 (0.93)
Kerr (${\rm a}_{*}$=0.6) | 0.16 (0.85) | 0.14 (0.91) | 0.13 (0.92) | 0.12 (0.94)
Kerr (${\rm a}_{*}$=0.94) | 0.09 (0.89) | 0.18 (0.83) | 0.18 (0.85) | 0.09 (0.93)
_EHT including space antenna_
boson star | 0.03 (0.98) | 0.13 (0.88) | 0.14 (0.85) | 0.09 (0.89)
dilaton | 0.17 (0.78) | 0.10 (0.96) | 0.11 (0.96) | 0.12 (0.88)
Kerr (${\rm a}_{*}$=0.6) | 0.17 (0.80) | 0.11 (0.96) | 0.10 (0.96) | 0.12 (0.91)
Kerr (${\rm a}_{*}$=0.94) | 0.09 (0.89) | 0.13 (0.89) | 0.13 (0.92) | 0.07 (0.96)
### 4.2 Fourier Plane comparison
Given that an interferometer measures the Fourier transform of the brightness
distribution of an astronomical source, a more direct comparison between
images of different space-times can be obtained in Fourier space. The
turbulent nature of the accretion process in the GRMHD simulations manifests
itself in large variations in the total flux density and in its flux density
distribution (see Fig. 1 for Kerr ${\rm a}_{*}$=0.6). In this work we include
the variability of the source in the scoring procedure. Therefore we modify
the scheme used in Event Horizon Telescope Collaboration et al. (2019c) for
the analysis of the recent EHT M87 observations. The main modification is that
the for the comparison with the synthetic data we use a GRRT movie generated
from a series of GRRT images spanning the observing time given by the
synthetic data set. In this work typically 216 frames (for an EHT observation
of 12 h) are used and we slide along our GRRT data series for the different
spacetimes where an increment of 5 frames is used. The increment of 5 frames
is justified by the fact that the typical correlation times of the GRRT images
are around 50 M. This implies that around this time the GRRT images can be
regarded as independent realisations of the accretion flow and thus the
different movies created from the sliding window can be considered as
uncorrelated. The first step in the scoring procedure is to create complex
visibilities from the GRRT movie, taking into account the array configuration,
the observing schedule and interstellar scattering, including both,
diffractive and refractive scattering (see also Table 2 and 4). From the
complex visibilities we compute the visibility amplitude, VA, and from closed
antenna triangles the closure phase, CP. The latter is of great importance in
measuring the structural variation within the source. In the second step we
minimise the $\chi^{2}$ for VA and CP between the snapshots and the individual
frames (images) of the GRRT movie by allowing the total flux, the position
angle and the black hole mass777Actually, we vary the plate scale,
$\mu=m_{bh}/d_{bh}$, during the scoring. Assuming a fixed distance $d_{bh}$ to
SgrA∗, we compute the black hole mass, $m_{bh}$, from $\mu$. to vary. Notice
that once the values for the flux scaling, position angle and black hole mass
are set, they are kept constant for all frames of the GRRT movie in order to
ensure the consistency of the movie. In addition to the image scaling, we
perform antenna gain calibrations. We limit the variation in the individual
antenna gains to be between 50% and 150% in order to avoid the compensation of
structural differences among the models by large gain variations. The scoring
is carried out with the well tested GENA pipeline developed for the image
matching of EHT and VLBA observations (Event Horizon Telescope Collaboration
et al., 2019d, c; Fromm et al., 2019) and we focus our analysis on the
synthetic observations including a space-antenna. The scheme for this kind of
scoring which includes the source variability (hereafter movie scoring) is
illustrated in Fig. 5.
Figure 5: Illustration of the movie scoring scheme. From the GRRT data we
select via a sliding window a set of images from which a movie is created.
From this movie complex visibilities are generated and the $\chi^{2}$ between
the synthetic data is computed while allowing the movie to rotate and vary in
source size and flux density. The $\chi^{2}$ are minimised either using an
evolutionary algorithm (EA) or an MCMC scheme. After the optimisation the
sliding window is advanced until the end of GRRT data set is reached.
In Fig. 6 we show an example for movie scoring for Kerr ${\rm a}_{*}$=0.94
synthetic data to the Kerr ${\rm a}_{*}$=0.94 model. The visibility amplitude
is plotted in the top panel and the closure phase in the second panel. The
blue points correspond to the synthetic data and green ones to the movie. In
the third panel we present the calibrated gains for the antennas involved in
the observations. The bottom panel shows the static average frame from the
Kerr movie (GRRT image left and convolved on the right). The obtained values
for the mass, the flux scaling and position angle are indicated in the top
panel. The recovered values are in good agreement with the injected values. A
more detailed self-test of the movie scoring scheme is presented in the
Appendix C.
Figure 6: Scoring result of the Kerr ${\rm a}_{*}$=0.94 synthetic data to the
Kerr ${\rm a}_{*}$=0.94 model. The panels show from top to bottom the
visibility amplitude, the closure phase, the calibrated gains and the average
image of the Kerr ${\rm a}_{*}$=0.94 movie (GRRT left and convolved right).
Figure 7: Results for the Kerr ${\rm a}_{*}$=0.6 test. The panels show the
distribution of total $\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ (top) and the
position angle, $\phi$ (bottom). In each violin the left hand side corresponds
to ${K^{\rm syn}_{0.6}}-K_{0.6}$ and the numbers above the violins indicate
the results of the two-sided K-S test (see text for further details). The red
line in the bottom panel corresponds to the initial position angle, $\phi=0$
of the GRRT images used to create synthetic data for ${K^{\rm syn}_{0.6}}$.
Figure 8: As in Fig. 7, now for the Kerr ${\rm a}_{*}$=0.94 test.
Figure 9: Results for the Kerr ${\rm a}_{*}$=0.6 -dilaton black hole test with
varying black hole mass. The panels show the distribution of total
$\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ (top left), the position angle,
$\phi$ (top right) and the black hole mass (bottom). In each violin the left
hand side corresponds to ${K^{\rm syn}_{0.6}}-K_{0.6}$ and the right hand side
to ${K^{\rm syn}_{0.6}}-D_{0}$. The numbers above the violins indicate the
results of the two-sided K-S test. The red line in the middle and right panel
corresponds to the initial position angle $\phi=0$ and initial black hole mass
$M_{\rm bh}=4.14\times 10^{6}M_{\astrosun}$ used for the GRRT images.
During our analysis we fit the synthetic data generated from the first 12 h of
our GRRT data set to the entire data by advancing the sliding window with an
increment of 5 frames. This allows us to statistically quantify the difference
between the synthetic data generated from the first 12 h to the entire GRRT
data set including the source variability. In the next step we keep the
synthetic data and score it against the GRRT images of remaining three
different spacetimes. Finally, we obtain the $\chi^{2}$-distribution for VA,
CP, the position angle (measured from north through east) and the mass
distribution for various combinations of synthetic data and the entire GRRT
image sets for different spacetimes. Based on these distributions we perform a
Kolmogorov–Smirnov (K-S) test to determine if the synthetic data under
investigation is in agreement of being drawn from one of the four spacetimes.
Below we provide a short example which explains the generation of the violin
plots and the obtained p-value from the K-S test. Assume two normal
distributions one with mean at 5.5 and standard deviation of 0.25 and another
one with mean at 5.7 and standard deviation of 0.35. Now draw 20 random
numbers for the the distribution and create a violin plot (the left half of
the violin is the histogram for the first distributions and the right half
corresponds to the second distribution). In this case the violin looks nearly
symmetrical with only a slight shift and the computed K-S test provides a
p-value of 0.49. This value implies that we cannot reject the hypothesis that
the values are from the same “overall” distribution or in your case the
underlying model (spacetime) is the same. (see left most violin in the top
panel of Fig. 7). On the contrary if the mean of the second distribution is
located at 12 the two wings of the violin are disjoint and the K-S test
provides a p-value of 5$\times 10^{-10}$. This value indicates that the two
distributions are not draw from the same “overall” distributions. In our case
this would imply that the underlying model (space-time) is not the same (see
right most violin in Fig. 7).
In the following we label the different spacetimes by their first letter and a
subscript indicates their spin, for example $K_{0.6}$ for a Kerr black hole
with spin ${\rm a}_{*}$=0.6. We indicated the synthetic data by a superscript
“syn” for example ${K^{\rm syn}_{0.6}}$ for the synthetic data of the Kerr
black hole with spin ${\rm a}_{*}$=0.6 generated from the first 12 h. Using
this notation, scoring the synthetic data Kerr ${\rm a}_{*}$=0.6 against the
entire data set of Kerr ${\rm a}_{*}$=0.94 is given by ${K^{\rm
syn}_{0.6}}-K_{0.94}$.
#### 4.2.1 Application to Kerr ${\rm a}_{*}$=0.6 (test 1)
In a first application of the movie scoring technique we want to test whether
we can distinguish the synthetic data for a Kerr black hole with spin ${\rm
a}_{*}$=0.6 from a Kerr black hole with spin a∗=0.94, a dilaton black hole and
from a boson star. Therefore, we score the synthetic data for Kerr black with
spina∗=0.6 against the data set for the different space times and perform the
KS test. Given the known black hole mass for the black hole in SgrA⋆ and its
small uncertainty from the GRAVITY experiment (Gravity Collaboration et al.,
2019) we keep for the black hole mass fixed for this first test and perform
the K-S test only on the total $\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ and
the position angle.
Figure 7 shows the results for the Kerr ${\rm a}_{*}$=0.6 test. In the panels
the left half of each violin (blue) corresponds to ${K^{\rm
syn}_{0.6}}-K_{0.6}$ and the right half (from left to right) to ${K^{\rm
syn}_{0.6}}-D_{0}$, ${K^{\rm syn}_{0.6}}-K_{0.94}$ and ${K^{\rm
syn}_{0.6}}-B_{0}$, respectively. The number above the violins indicates the
results of the two-sided K-S test. Within each half-violin the long-dashed
lines indicate the mean and the short dashed line denotes the inter-quartile
range. The top panel displays the $\chi^{2}$ distribution, the bottom panel
the distribution of the position angle, $\phi$. The red line in the bottom
panel indicates initial position angle of images which are used to create the
synthetic data set. For ${K_{0.6}}-D_{0}$ the $\chi^{2}$ show a very similar,
only slightly shifted shape. In contrast to ${K_{0.6}}-K_{0.94}$ and
${K_{0.6}}-B_{0}$ where the distributions are only marginally overlapping. The
distributions for the position angle $\phi$ are in all cases overlapping with
the distribution of the truth model (${K^{\rm syn}_{0.6}}-K_{0.6}$, blue
violins).
This behaviour can be explained by the very similar source size for the Kerr
${\rm a}_{*}=0.6$ and dilaton black hole in contrast to Kerr black hole with
spin ${\rm a}_{*}=0.94$ and the boson star (see left column in Fig 4). In
order to improve the $\chi^{2}$ during the optimisation (or MCMC) step the
image is rotated a bit more in the case of the Kerr ${\rm a}_{*}=0.94$ and the
boson star data set. The results of the K-S test can be found in Table
LABEL:ksresults.
The results of the movie scoring indicate that the space-EHT concept can
clearly distinguish fast from slow spinning black holes as well as from boson
stars.
#### 4.2.2 Application to Kerr ${\rm a}_{*}$=0.94 (test 2)
The second test we use the synthetic data from the Kerr black hole with spin
${\rm a}_{*}$=0.94 and we test if it can be distinguished from either a Kerr
black hole with spin ${\rm a}_{*}$=0.6, a dilaton black hole or a boson star.
The following movie scorings are computed: ${K^{\rm syn}_{0.94}}-K_{0.94}$
(synthetic data and GRRT data set are from the same spacetime), ${K^{\rm
syn}_{0.94}}-D_{0}$, ${K^{\rm syn}_{0.94}}-K_{0.6}$ and ${K^{\rm
syn}_{0.94}}-B_{0}$. The $\chi^{2}$ and position angle distribution of all
test pairs of models are clearly shifted (see Fig. 8). This behaviour is also
reflected in the small numbers for the K-S test (see Table
LABEL:ksresults).The shift in the $\chi^{2}$ distributions can be explained by
the difference in source size for the Kerr ${\rm a}_{*}$=0.6 and dilaton black
hole. In order to match the source size of the Kerr black hole with spin ${\rm
a}_{*}$=0.94 the dilaton and Kerr black hole with spin ${\rm a}_{*}$=0.6 would
require a smaller black hole mass $m_{\rm bh}<4.14\times
10^{6}\,M_{\astrosun}$. Since we do not allow the black hole mass to adjust
the only way to improve the $\chi^{2}$ is to rotate the GRRT movies. An
improved $\chi^{2}$ is obtained by rotating the GRRT movies were the mean of
the $\phi$ distributions is located at -15∘ ( Kerr black hole with spin ${\rm
a}_{*}$=0.6) and -25∘ (dilaton black hole). Similar, the boson star would
require a larger black hole mass to fit the Kerr ${\rm a}_{*}$=0.94 synthetic
data.
Given the distributions and the result of the K-S test the space-EHT concept
can distinguish among all three models if the truth model is a Kerr black hole
with spin ${\rm a}_{*}$=0.94.
#### 4.2.3 Application to Kerr ${\rm a}_{*}$=0.6 with varying black hole mass
(test 3)
In order to test the variations in the black hole mass required to make the
Kerr ${\rm a}_{*}$=0.6 indistinguishable for the space-EHT concept from the
dilaton black hole we performed a third test where we restrict ourselves to
Kerr ${\rm a}_{*}$=0.6 and dilaton black holes and allow the black hole mass
to adjust during the optimisation. The result for this test is presented in
Fig. 9. As expected the adjusted black hole mass for the dilaton black hole
improved the p-values for the K-S test on all quantities ($\chi^{2}_{\rm
tot}$, $\phi$ and $m_{\rm bh}$). However this would require a black hole mass
of $m_{\rm bh}=4.05\times 10^{6}\,M_{\astrosun}$ which is not in agreement
with the current measurements of GRAVITY (Gravity Collaboration et al., 2019).
Table 7: Results of the K-S tests for Kerr ${\rm a}_{*}$=0.6 and Kerr ${\rm
a}_{*}$=0.9 as truth images. The K-S tests are performed on the distribution
of the total $\chi^{2}_{\rm VA+CP}$ (visibility amplitude and closure phases)
and the distribution of the black hole mass, $m_{\rm bh}$. truth model
${K^{\rm syn}_{0.6}}$ (test 1)
---
| ${K^{\rm syn}_{0.6}}-D_{0}$ | ${K^{\rm syn}_{0.6}}-K_{0.94}$ | ${K^{\rm syn}_{0.6}}-B_{0}$
p-value $\left(\chi^{2}\right)$ | $0.46$ | $1.6\times 10^{-9}$ | $1.3\times 10^{-9}$
p-value $\left(\phi\right)$ | $0.55$ | $0.22$ | $0.12$
truth model ${K^{\rm syn}_{0.94}}$ (test 2)
| ${K^{\rm syn}_{0.94}}-D_{0}$ | ${K^{\rm syn}_{0.94}}-K_{0.6}$ | ${K^{\rm syn}_{0.94}}-B_{0}$
p-value $\left(\chi^{2}\right)$ | $7.0\times 10^{-7}$ | $1.0\times 10^{-7}$ | $1.6\times 10^{-9}$
p-value $\left(\phi\right)$ | $1.2\times 10^{-10}$ | $7.5\times 10^{-10}$ | $2.8\times 10^{-4}$
truth model ${K^{\rm syn}_{0.6}}$ (test 3)
| ${K^{\rm syn}_{0.6}}-D_{0}$
p-value $\left(\chi^{2}\right)$ | 0.74
p-value $\left(\phi\right)$ | 0.78
p-value $\left(m_{\rm bh}\right)$ | 0.94
## 5 Discussion and Summary
In this exploratory paper we address the question if an orbiting space antenna
will improve the capabilities of the EHT to distinguish among different space
times around SgrA⋆. Our proposed optimisation procedure suggested an
elliptical orbit with an eccentricity of e=0.5 and a semi-major axis of
a=14900 km. By including the suggested space antenna the baselines of the EHT
array could be extended beyond 10,000 km and thus increase the angular
resolution and the imaging capabilities of the array considerably (see right
column in Fig. 4). As can be seen in Table LABEL:results the space-EHT is able
to distinguish the spin of Kerr black holes (best image metrics can be found
for equal image pairs). For example the DSSIM for the Kerr ${\rm a}_{*}$=0.6 –
Kerr ${\rm a}_{*}$=0.6 dropped from 0.13 to 0.10 while the Kerr ${\rm
a}_{*}$=0.6 – Kerr ${\rm a}_{*}$=0.94 is 0.12. The improvement on the imaging
capabilities of the space-EHT is also noticeable in the change of the computed
nCCC: for Kerr ${\rm a}_{*}$=0.6 from 0.91 to 0.96 while for the Kerr ${\rm
a}_{*}$=0.94 case the nCCC decreased from 0.94 to 0.91. A similar behaviour is
found for the other equal image pairs e.g., Kerr ${\rm a}_{*}$=0.94 – Kerr
${\rm a}_{*}$=0.94 as compared to the non-equal image pairs e.g., Kerr ${\rm
a}_{*}$=0.94 – boson star (see last row in Table LABEL:results). For the ${\rm
a}_{*}$=0.6 Kerr black hole and the dilaton black hole the image metrics
improved on very similar: the DSSIM value: from 0.13 to 0.10 and the nCCC from
0.92 to 0.96. Given these very similar number is currently not possible to
distinguish both spacetimes based solely on their reconstructed images.
To circumvent the limitations of reconstructed average images in
distinguishing different theories of gravity, we additionally performed a
detailed comparison in Fourier space including the source variability.
We created synthetic data from the first 12 h of the GRRT images of the two
different Kerr black holes and scored them against 12 h movies creates from
the GRRT images of all four spacetimes. Fitting the synthetic data for the
Kerr black hole with spin ${\rm a}_{*}$=0.6 against its entire GRRT data set
leads to a $\chi^{2}$ distribution with a mean value of 5.6 and standard
deviation of 0.5 (see blue violins in Fig 7). The position angle distribution
peaks at a value of $\phi=0.7^{\circ}\pm 4^{\circ}$. This value is close the
initial value we used for the creation of the synthetic data and can be
regarded as confirmation of the fitting routine.
We performed two-sided K-S tests to investigate the hypothesis that the
synthetic data is drawn from a spacetime other than Kerr. The result for the
Kerr ${\rm a}_{*}$=0.6 tests reveal very small numbers for the synthetic data
generated from Kerr ${\rm a}_{*}$=0.9 and a boson star, both for the
$\chi^{2}$ and for the position angle (see Fig 7 and Table LABEL:ksresults).
Thus we could reject the null-hypothesis and conclude that Kerr ${\rm
a}_{*}$=0.9 and a boson star can be distinguished from an ${\rm a}_{*}$=0.6
Kerr black hole. The K-S test between the ${\rm a}_{*}$=0.6 Kerr and the
dilaton black holes provides values significantly larger than for the Kerr
${\rm a}_{*}$=0.94 and a Boson star. The mean values of the dilaton
distributions are shifted by roughly one standard deviation compared to the
truth distribution (Kerr ${\rm a}_{*}$=0.6). Given the obtained values, it is
likely (54% level) that we can distinguish an ${\rm a}_{*}$=0.6 Kerr black
hole from a dilaton black hole. In an additional test we allowed the black
hole mass to adjust and in order to probe the mass variation which would lead
to undistinguishable data sets e.g., we can not differentiate between a Kerr
black hole with spin ${\rm a}_{*}$=0.6 and a dilaton black hole. The shift
between the $\chi^{2}$ and $\phi$ distribution decreased which leads to larger
p-values for the K-S test (see Table LABEL:ksresults and Fig. 9). In addition
to the previous test we also obtain the distribution for the black hole mass.
The obtained mean black hole mass for Kerr black hole with spin ${\rm
a}_{*}$=0.6 is $m_{\rm bh}=4.1^{+0.05}_{-0.15}\times 10^{6}\,M_{\astrosun}$
which is in good agreement with used black hole mass during the radiative
transport calculations and a confirmation of the movie scoring method. In
order to make both theories of gravity undistinguishable to 74% the mean black
hole mass should decrease to $m_{\rm bh}=4.02\times 10^{6}\,M_{\astrosun}$.
However, given the mass boundaries from the GRAVITY experiment this black hole
mass is outside the allowed range. Similar to the ${\rm a}_{*}$=0.6 Kerr case,
we perform K-S tests on the fast spinning Kerr black hole with ${\rm
a}_{*}$=0.94. Again, we obtain the distribution for the $\chi^{2}$ and the
position angle $\phi$ and computed the K-S test. All distributions for this
test are clearly offset from the truth distribution (see top panel in Fig. 8),
which is also reflected in the very small numbers obtained for the p-value
(see Table LABEL:ksresults). However, given the small p-values we conclude
that the fast-spinning Kerr black hole can be clearly distinguished from the
slower-spinning Kerr black hole, the dilaton and the boson star.
In summary, we have presented a future EHT concept which includes a space
antenna and tested the capabilities of this new array to investigate the
different possible spacetimes around Sgr A∗ via radio images and complex
visibilities. The orbit of the satellite is computed from a non-linear
optimisation using a genetic algorithm and constraints on the u-v-plane
filling, observation time and improved image metrics computed between the GRRT
image and the reconstructed image. We generated synthetic data for three
different theories of gravity, namely: a Kerr black hole (general relativity),
a dilaton black hole and a boson star. The synthetic data generated was
created from 12 h of GRRT movie in order to properly include the source
variabily and a dynamical image reconstruction was applied following Johnson
et al. (2017) using ehtim. From the dynamical reconstructed images an average
image was computed and compared to the average frame of the GRRT movies using
DSSIM and nCCC metrics. The image plane comparison was accompanied by a more
detailed and robust analysis in the Fourier plane using the newly developed
movie scoring method in GENA. A K-S test was performed on the $\chi^{2}$
position angle and mass distributions, in order to investigate the possibility
of distinguishing among different spacetimes and therefore different theories
of gravity. The space-EHT concept presented in this study has been shown to
both improve the imaging capabilities while including the source variability
of the array and improve our ability to distinguish among (and potentially
exclude) certain solutions and theories of gravity.
Our ability to image time-variable sources and probe the underlying spacetime
will be further improved by the addition of additional ground antennas for
example the African Millimetre Telescope (ATM) (Backes et al., 2016) or
several telescopes placed at dedicated locations across the globe as planed by
the next generation EHT (ngEHT) (Blackburn et al., 2019). Extending the GRRT
images series will allow us to address the source variability on larger non-
overlapping time windows improving the statistics. Increasing the observing
frequency ($\nu>500$ GHz) will reduce the interstellar scattering and also
allow us to study more deeply general relativistic effects on horizon scales
since the obtained image is less effected by the properties of the accretion
flow and the radiation microphysics. However, due to the opacity of the Earths
atmosphere such a concept would require an entirely space-based VLBI concept
(see e.g., Roelofs et al., 2019; van der Gucht et al., 2019).
In addition to horizon scale images of the EHT and future space-EHT concepts
further constraints on the properties of the spacetime around Sgr A⋆ can be
obtained from a pulsar in a tight orbit (orbital period $\sim 1$ yr ) around
the galactic centre (Liu et al., 2012). Combined, both measurements e.g.,
horizon scale images and pulsar timing would allow tightly constraints of the
properties of spacetime around SgrA⋆ (Psaltis et al., 2016).
###### Acknowledgements.
CMF thanks E. Ros, R. Porcas and D. Palumbo for fruitful discussion and
comments on the manuscript. Support comes from the ERC Synergy Grant
“BlackHoleCam - Imaging the Event Horizon of Black Holes” (Grant 610058). CMF
is supported by the black hole Initiative at Harvard University, which is
supported by a grant from the John Templeton Foundation. ZY acknowledges
support from a Leverhulme Trust Early Career Fellowship.
## References
* Backes et al. (2016) Backes, M., Müller, C., Conway, J. E., et al. 2016, in The 4th Annual Conference on High Energy Astrophysics in Southern Africa (HEASA 2016), 29
* Blackburn et al. (2019) Blackburn, L., Doeleman, S., Dexter, J., et al. 2019, arXiv e-prints, arXiv:1909.01411
* Bower et al. (2015) Bower, G. C., Markoff, S., Dexter, J., et al. 2015, ApJ, 802, 69
* Chael et al. (2018) Chael, A. A., Johnson, M. D., Bouman, K. L., et al. 2018, ApJ, 857, 23
* Chael et al. (2016) Chael, A. A., Johnson, M. D., Narayan, R., et al. 2016, ApJ, 829, 11
* Davoudiasl & Denton (2019) Davoudiasl, H. & Denton, P. B. 2019, Phys. Rev. Lett., 123, 021102
* Deb et al. (2002) Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. 2002, Trans. Evol. Comp, 6, 182
* Event Horizon Telescope Collaboration et al. (2019a) Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2019a, The Astrophysical Journal, 875, L1
* Event Horizon Telescope Collaboration et al. (2019b) Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2019b, ApJ, 875, L2
* Event Horizon Telescope Collaboration et al. (2019c) Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2019c, ApJ, 875, L6
* Event Horizon Telescope Collaboration et al. (2019d) Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2019d, ApJ, 875, L5
* Fish et al. (2019) Fish, V. L., Shea, M., & Akiyama, K. 2019, arXiv e-prints, arXiv:1903.09539
* Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24
* Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
* Fromm et al. (2019) Fromm, C. M., Younsi, Z., Baczko, A., et al. 2019, A&A, 629, A4
* García et al. (1995) García, A., Galtsov, D., & Kechkin, O. 1995, Phys. Rev. Lett., 74, 1276
* Gold et al. (2020) Gold, R., Broderick, A. E., Younsi, Z., et al. 2020, ApJ, 897, 148
* Gómez et al. (2016) Gómez, J. L., Lobanov, A. P., Bruni, G., et al. 2016, ApJ, 817, 96
* Gravity Collaboration et al. (2019) Gravity Collaboration, Abuter, R., Amorim, A., et al. 2019, A&A, 625, L10
* Hirabayashi et al. (1998) Hirabayashi, H., Hirosawa, H., Kobayashi, H., et al. 1998, Science, 281, 1825
* Johnson (2016) Johnson, M. D. 2016, ApJ, 833, 74
* Johnson et al. (2017) Johnson, M. D., Bouman, K. L., Blackburn, L., et al. 2017, ApJ, 850, 172
* Kardashev et al. (2013) Kardashev, N. S., Khartov, V. V., Abramov, V. V., et al. 2013, Astronomy Reports, 57, 153
* Lee & Koh (1996) Lee, J.-W. & Koh, I.-G. 1996, Phys. Rev. D, 53, 2236
* Levkov et al. (2018) Levkov, D. G., Panin, A. G., & Tkachev, I. I. 2018, Phys. Rev. Lett., 121, 151301
* Liu et al. (2012) Liu, K., Wex, N., Kramer, M., Cordes, J. M., & Lazio, T. J. W. 2012, ApJ, 747, 1
* Lu et al. (2014) Lu, R.-S., Broderick, A. E., Baron, F., et al. 2014, ApJ, 788, 120
* Lu et al. (2016) Lu, R.-S., Roelofs, F., Fish, V. L., et al. 2016, ApJ, 817, 173
* Mizuno et al. (2018) Mizuno, Y., Younsi, Z., Fromm, C. M., et al. 2018, Nature Astronomy, 2, 585
* Okai (1994) Okai, T. 1994, Progress of Theoretical Physics, 92, 47
* Olivares et al. (2019) Olivares, H., Porth, O., Davelaar, J., et al. 2019, A&A, 629, A61
* Olivares et al. (2020) Olivares, H., Younsi, Z., Fromm, C. M., et al. 2020, MNRAS, 497, 521
* Palumbo et al. (2019) Palumbo, D. C. M., Doeleman, S. S., Johnson, M. D., Bouman, K. L., & Chael, A. A. 2019, ApJ, 881, 62
* Porth et al. (2017) Porth, O., Olivares, H., Mizuno, Y., et al. 2017, Computational Astrophysics and Cosmology, 4, 1
* Psaltis et al. (2016) Psaltis, D., Wex, N., & Kramer, M. 2016, ApJ, 818, 121
* Rezzolla & Zhidenko (2014) Rezzolla, L. & Zhidenko, A. 2014, Phys. Rev. D, 90, 084009
* Roelofs et al. (2019) Roelofs, F., Falcke, H., Brinkerink, C., et al. 2019, A&A, 625, A124
* Schilizzi (2012) Schilizzi, R. 2012, in Resolving The Sky - Radio Interferometry: Past, Present and Future, 10
* Schunck & Liddle (1997) Schunck, F. E. & Liddle, A. R. 1997, Physics Letters B, 404, 25
* van der Gucht et al. (2019) van der Gucht, J., Davelaar, J., Hendriks, L., et al. 2019, arXiv e-prints, arXiv:1910.13236
* Wang et al. (2004) Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. 2004, IEEE Transactions on Image Processing, 13, 600
* Younsi et al. (2020) Younsi, Z., Porth, O., Mizuno, Y., Fromm, C. M., & Olivares, H. 2020, in Perseus in Sicily: From Black Hole to Cluster Outskirts, ed. K. Asada, E. de Gouveia Dal Pino, M. Giroletti, H. Nagai, & R. Nemmen, Vol. 342, 9–12
* Younsi et al. (2016) Younsi, Z., Zhidenko, A., Rezzolla, L., Konoplya, R., & Mizuno, Y. 2016, Phys. Rev. D, 94, 084025
## Appendix A Influence of the orbital parameters on the imaging capabilities
In Fig. 10 we illustrate the impact of the orbital parameters on the imaging
capabilities of the space-EHT concept and the tradeoffs between u-v coverage
and array resolution. We keep the orientation parameters of the orbit fixed to
the results of our orbit optimisation (see Table 5) and change the semi-major
axis, $a$, and eccentricity, $e$.
Figure 10: Reconstruction of a boson star image for SgrA⋆ (ground truth image
in panel a) for three different satellite orbits; small orbit (second column),
intermediate orbit (third column) and large orbit (fourth column). The columns
show from top to bottom the reconstructed image convolved with 75% of the
nominal array resolution, the uv-coverage (white: ground array baselines, red:
space baselines) and the satellite orbit as seen from SgrA⋆.
An increase in the semi-major axis, $a$ (see panel j in Fig. 10), leads to
higher resolution (see beam in panel d in Fig. 10). At the same time the
larger semi-major axis, $a$, which corresponds to a larger orbital period
$T_{p}=2\pi\sqrt{a^{3}/\left(GM_{\rm Earth}\right)}$ reduces the u-v coverage
(see panel g in in Fig. 10) . This leads to a worse image metric as for
example an medium size orbit (compare image metrics in panel c and d in Fig.
10). On the other hand, a small semi-major axis (see panel h in Fig. 10) leads
to a dense u-v coverage (panel e in Fig. 10) but does not provide high angular
resolution (see panel b in Fig. 10). A trade-off between resolution and u-v
coverage is an intermediate size semi-major axis (see third column in in Fig.
10). For each of the three satellite orbits the image metrics are calculated
which prioritise the orbit with the best image metrics here the orbit with a
semi-major axis of $a=14861$ km and an eccentricity, $e=0.5$.
## Appendix B DSSIM and nCCC
In order to explore the variation of the DSSIM and the nCCC we performe a
small parameter space study using averaged GRRT images. The GRRT images are
taken from Kerr BHs with spin ${\rm a}_{*}$=0.6 and ${\rm a}_{*}$=0.94 as well
as from the dilaton black hole. To mimic different observing arrays with
improved imaging capabilities we convolve the average images with different
beam sizes888we applied a circular gaussian ranging from 30 $\mu$as to 1
$\mu$as. The convolved images are compared to the ground truth GRRT images and
the DSSIM and the nCCC values are computed.
Figure 11: DSSIM and nCCC variation study. Top panel: ground truth images for
Kerr ${\rm a}_{*}$=0.6, Kerr ${\rm a}_{*}$=0.94 and dilaton black holes (left
most column) and convolved with decreasing beams (30 $\mu$as to 1 $\mu$as).
Middle panel: DSSIM and nCCC calculations using the average image of Kerr BH
with ${\rm a}_{*}$=0.6 as ground truth image. Bottom panel: DSSIM and nCCC
calculations using the average image of Kerr BH with ${\rm a}_{*}$=0.94 as
ground truth image.
In Fig. 11 we show the results for this study. In the top panel we display the
average GRRT images for the Kerr black holes (first and second row) and for
the dilaton black hole (third row). Note that the images are individually
normalised by their flux density maximum. The first column presents the ground
truth GRRT image followed by a series of convolved images using beams with
decreasing size (beam size is indicated above each image). With decreasing
beam size (equal to the improved imaging capabilities of the array) the images
become sharper and finer image features become visible. In the second panel of
Fig. 11 we compute the image metrics assuming that the ground truth image
corresponds to a Kerr BH with spin ${\rm a}_{*}$=0.6. In the panel the red
colour corresponds to the DSSIM and blue to the nCCC while the line style
indicates the image pairs, e.g., red dashed line stands for the DSSIM computed
between Kerr BH with ${\rm a}_{*}$=0.6 and Kerr BH ${\rm a}_{*}$=0.94. As
expected the DSSIM values decrease with smaller beam sizes while the nCCC
increases for all image pairs. However at a beam size of $\sim 16\,\mu$as the
DSSIM values for the Kerr BH with ${\rm a}_{*}$=0.94 increases and at the same
time the nCCC decreases. This behaviour can be understood by the different
source sizes of the ground truth images (see left most column in the top
panel). The compact source structure of the Kerr BH with ${\rm a}_{*}$=0.94 is
smeared out to larger extents during the convolution with beam with size
$>16\,\mu$as which matches better the ground truth structure of the Kerr BH
with ${\rm a}_{*}$=0.6. Thus, the DSSIM is decreasing while the nCCC is
increasing. However at a beam size of $\sim 16\,\mu$as the behaviour is
inverted. For the models with similar ground truth sizes the DSSIM is
monotonically decreasing while the nCCC is continuously increases.
If the size of the ground truth image is smaller than the intrinsic size of
the models to which it is compared, e.g., Kerr BH with ${\rm a}_{*}$=0.94 and
Kerr BH ${\rm a}_{*}$=0.6 no change in the DSSIM and nCCC behaviour is found:
the DSSIM is always decreasing and the nCCC is continuously increasing with
decreasing beam size (see bottom panel in Fig. 11).
## Appendix C Movie scoring self-test
To valid the developed movie scoring method we perform a self-test. For the
self-test we generate synthetic visibilities from the Kerr black hole with
spin ${\rm a}_{*}$=0.94. The start frame for the synthetic data generation is
shifted by 1.7 hours (30 frames). During the generation of the synthetic data
we take interstellar scattering (including both, diffractive and refractive
scattering), thermal noise and gain variations into account (see also Table
4). For a successful self-test the movie scoring method should recover the
correct starting frame, the black hole mass and the orientation of the images
used during the synthetic data generation. We during the radiative transfer we
use a black hole mass of $4.14\times 10^{6}\,M_{\astrosun}$ and an orientation
angle of $\phi=0^{\circ}$. During the self-test we allow the black hole mass,
$m_{\rm bh}$, the rotation angle, $\phi$ and the flux normalisation, S to
adjust during the scoring. For the optimisation we apply a MCMC method
(Foreman-Mackey et al. 2013; Foreman-Mackey 2016) using 100 MCMC walkers for
500 iterations and perform gain-calibration.
Figure 12: Posterior distributions for the black hole mass, $m_{\rm bh}$, the
position angle, $\phi$, and the flux scaling, $S_{\rm scale}$. The sliding
window is located at $\Delta t=1.7$ h (30 frames) recovering the time range
used for the synthetic data generation (see Text).
The movie scoring method successfully recovered the start frame and in Fig. 12
we present the posterior distribution for the black hole mass, the orientation
angle and the flux scaling for the GRRT movie starting at frame 30. The black
hole mass and the orientation angle are recovered within $1\sigma$ proofing
the applicability of the developed method to recover black hole parameters
from a source with varying flux density and structure. The posterior
distribution of the flux scaling, $S_{\rm scale}$, reflects the influence of
interstellar scattering and antenna gain variations taken into account during
the synthetic data generation. Interstellar scattering smears out the flux
density, thus reducing the measured visibility amplitude as compared to an un-
scattered case. In order to match the visibility amplitude during the scoring
the flux scaling factor is smaller than one. The additional variations due to
the antenna gain variations are compensated by the applied self-calibration.
In addition we show in Fig. 13 the posterior distribution for the scaling
parameters ($m_{\rm bh}$, $\phi$ and $S_{\rm scale}$) obtained from the entire
data set for the Kerr black with ${\rm a}_{*}$=0.94. The posterior
distribution for the black hole mass, $m_{\rm bh}$, and the flux scaling,
$S_{\rm scale}$ show a similar distribution as for the best starting frame
model (see Fig. 12) except for the three times larger uncertainties for the
black hole mass. However the distribution of the position angle, $\phi$, shows
a second local maximum around $\phi=3^{\circ}$. This can be explained by the
variability of the source or more precise the variation of the position angle
of the flux centroid. In Fig. 14 we show the variation of the image centroid
position angle relative to the data set used for the synthetic data. For
sliding windows different that the one used for the data generation (indicated
by the black arrow in Fig. 14) the centroid position angle is lager angle (see
green dashed line in Fig.14). During the optimisation within the movie scoring
method this angle difference is compensated by a rotating the images which
explains the second maximum in the position angle posterior distribution in
Fig. 13.
Figure 13: Posterior distributions for the black hole mass, $m_{\rm bh}$, the
position angle, $\phi$, and the flux scaling, $S_{\rm scale}$ for the entire
Kerr black hole ${\rm a}_{*}$=0.94 data set.
Figure 14: Variation of the image centroid position angle relative to the one
from the sliding window used for the generation of the synthetic data
(indicated by the black arrow). The mean centroid position angle for a second
sliding window with 10 hour offset is indicated in green.
|
# Mindless Attractor: A False-Positive Resistant Intervention for Drawing
Attention Using Auditory Perturbation
Riku Arakawa 0000-0001-7868-4754 The University of TokyoTokyoJapan arakawa-
<EMAIL_ADDRESS>and Hiromu Yakura 0000-0002-2558-735X University
of TsukubaTsukubaJapan<EMAIL_ADDRESS>
(2021; 2021)
###### Abstract.
Explicitly alerting users is not always an optimal intervention, especially
when they are not motivated to obey. For example, in video-based learning,
learners who are distracted from the video would not follow an alert asking
them to pay attention. Inspired by the concept of _Mindless Computing_ , we
propose a novel intervention approach, _Mindless Attractor_ , that leverages
the nature of human speech communication to help learners refocus their
attention without relying on their motivation. Specifically, it perturbs the
voice in the video to direct their attention without consuming their conscious
awareness. Our experiments not only confirmed the validity of the proposed
approach but also emphasized its advantages in combination with a machine
learning-based sensing module. Namely, it would not frustrate users even
though the intervention is activated by false-positive detection of their
attentive state. Our intervention approach can be a reliable way to induce
behavioral change in human–AI symbiosis.
Mindless computing, Human attention, Computational intervention, Machine
learning-based sensing, Video-based learning
††copyright: none††journalyear: 2021††doi:
10.1145/xxxxxxx.xxxxxxxx††journalyear: 2021††copyright:
acmlicensed††conference: CHI Conference on Human Factors in Computing Systems;
May 8–13, 2021; Yokohama, Japan††booktitle: CHI Conference on Human Factors in
Computing Systems (CHI ’21), May 8–13, 2021, Yokohama, Japan††price:
15.00††doi: 10.1145/3411764.3445339††isbn: 978-1-4503-8096-6/21/05††ccs:
Human-centered computing Auditory feedback††ccs: Human-centered computing
Interaction techniques††ccs: Applied computing Interactive learning
environments
## 1\. Introduction
For decades, video-based communication has been expected to take over face-to-
face communication (Fish et al., 1992; Whittaker, 1995). In particular,
schools have leveraged video-based learning to provide educational
opportunities for distanced students, as massive open online courses have done
(Martin, 2012; Guo et al., 2014). Moreover, the recent COVID-19 pandemic has
precipitated the transition to video-based communication for the purpose of
preventing infection (Coombs, 2020; Kodama, 2020), especially in the context
of education (Kerres, 2020; Goldschmidt, 2020). However, it has been noted
that people often have trouble maintaining their attention in video-based
communications (Kuzminykh and Rintel, 2020a, b), as they can concurrently
perform other tasks, like texting or accessing social media using a smartphone
(Oeppen et al., 2020).
Considering the increasing demand for video-based learning, it would be
fruitful if computers can help learners pay attention to a video. Here, recent
advances in machine learning techniques have enabled the automatic estimation
of a user’s attention level from a video of their face (Thomas and Jayagopi,
2017). On the other hand, it is not trivial how to intervene with learners
using computers based on the estimation result. A straightforward approach is
to explicitly alert them when they seem to not be paying attention to the
video, as Xiao and Wang (Xiao and Wang, 2016) did. However, unlike the
critical situations targeted in conventional studies of alert designs (Cobus
et al., 2018; Saint-Lot et al., 2020), users of video-based learning systems
would not hesitate to ignore such alerts, especially when they are focused on
side tasks. For example, Xiao and Wang (Xiao and Wang, 2016) reported that
their intervention approach in their user study was described as unhelpful by
some participants who were less motivated. In other words, the efficacy of the
alerting approach would depend on the user’s motivation to actively take part,
and such interventions would not be an optimal intervention for inducing
behavioral change.
Looking back to the nature of human communications, we often change the tone
of our voices intentionally to draw listeners’ attention (Xu, 2005). Based on
this observation, we anticipate that we can help learners return their
attention to videos by computationally changing the tone of voice during
video-based learning situations. This approach is inspired by the concept of
Mindless Computing—behavior-changing technologies that leverage human biases
or unconscious behaviors—proposed by Adams et al. (Adams et al., 2015). Given
that Mindless Computing does not consume a user’s conscious awareness to be
effective, Adams et al. (Adams et al., 2015) stated that it does not rely on
the user’s motivation, whereas many of the current persuasive technologies
have a strong reliance on user motivation and are likely to fail. In addition,
the independence from the user’s conscious awareness enables such behavior
influencing to work without interfering with the user’s main task, which suits
our situation (i.e., use during video-based learning).
Furthermore, we argue that this mindless intervention approach has a high
affinity with sensing modules based on machine learning techniques. That is,
if we explicitly alert users, they can be distracted and frustrated by
misinformed alerts caused by erroneous false-positive detection, which can
lead them to ignore the result of a machine learning module (Dietvorst et al.,
2015; De-Arteaga et al., 2020). On the other hand, the mindless approach
designed based on human nature does not necessarily consume users’ conscious
awareness, and such negative effects due to false positives can thus be
mitigated.
In this paper, we propose a novel intervention approach, Mindless Attractor,
which computationally leverages the nature of our speech communication, and
examine its effectiveness in the case of helping users in video-based learning
return their attention to the video. For this purpose, we first determined its
requirements and possible designs so as to reduce the time that users are
distracted in a mindless manner. We then conducted an experiment to confirm
that the proposed intervention was effective in helping users refocus their
attention without consuming conscious awareness. We also combined this
mindless intervention with a machine learning-based sensing module and
evaluated its effectiveness in the context of false-positive detection, in
comparison to a conventional alerting approach. The series of experiments
presented the advantages of the proposed approach, especially in combination
with machine learning techniques. Based on the results, we discuss
implications for the HCI community, emphasizing the importance of the mindless
intervention approach in the era of human–AI symbiosis.
## 2\. Related Work
To situate our work, we first examine previous literature on interaction
techniques for video-based learning, particularly those focusing on learners’
attention. We then review conventional alert-based techniques for drawing
human attention and discuss why they would not fit our purposes. We also
explore previous studies regarding the nature of human speech communication,
as this is a foundation of our mindless approach for drawing users’ attention.
### 2.1. Attention-Related Interaction Techniques for Video-Based Learning
As mentioned in Section 1, opportunities for video-based communication are
increasing, and many interaction techniques have thus been proposed to enhance
the experience of such communications. Some prior studies have proposed
interaction techniques centering on the context of participants’ attention
(D’Mello et al., 2012; Sharma et al., 2016; Xiao and Wang, 2016), as it has
been pointed out that people often have difficulty maintaining their attention
during video-based communication (Kuzminykh and Rintel, 2020a, b). These
techniques benefit from the significant effort that has been devoted to
estimating participants’ attentiveness based on visual cues, such as face
movement (Thomas and Jayagopi, 2017), body postures (Zaletelj and Kosir,
2017), and gaze (Bidwell and Fuchs, 2011; Hutt et al., 2017; Veliyath et al.,
2019). They then use the estimation results to enhance learners’ performance,
for instance in the case of video-based learning, as it is widely acknowledged
that learners’ attention and engagement are strongly related to their learning
performance (Baker et al., 2010; D’Mello et al., 2012).
For example, Gaze Tutor is a gaze-reactive intelligent tutoring system for
video-based learning (D’Mello et al., 2012). Using a conventional eye tracker,
it estimates the learner’s attention level based on gaze direction by applying
a simple rule assuming that off-screen gaze patterns imply distraction. When
the system detects that the learner is not focusing on the video, the tutor
agent stops the video and alerts them explicitly (e.g., by saying “Please pay
attention”). Although their experiment showed its effectiveness in reorienting
participants’ attention, the intervention method left room for improvement, as
the authors mentioned in their discussion. Specifically, they found individual
differences in the efficacy of the alert-based intervention, including that
some participants never followed the alerts. Accordingly, the authors noted
that alternate intervention approaches, including indirect feedback, could be
implemented. Another example that computationally utilizes the estimated
attention level during video-based learning was provided by Sharma et al.
(Sharma et al., 2016). Similar to Gaze Tutor, their system provided users with
direct feedback, such as simple red rectangles on the screen, with the purpose
of improving users’ attention.
As can be inferred from these studies, previous research has mainly considered
explicit alerting as an intervention method for video-based learning. However,
the findings from these studies complement our concern, which is discussed in
Section 1 based on the results of Xiao et al. (Xiao and Wang, 2016). That is,
such interventions have a reliance on users’ motivation; they may not work
effectively when we cannot assume that all users are motivated to change their
behavior. In Section 2.2, we will explain why the reliance occurs based on the
discussion by Adams et al. (Adams et al., 2015), which in turn motivated us to
explore a better intervention approach for video-based learning situations.
### 2.2. Alerting Techniques for Drawing Human Attention
Drawing users’ attention is one of the crucial components of human-computer
interaction, not limited to video-based learning. Many researchers have dealt
with a wide range of topics in this area, such as Internet advertisements
(Nettelhorst and Brannon, 2012), smartphone notifications (Stothart et al.,
2015), and alerting systems (Guo et al., 2002). Consequently, previous studies
have developed many methods suitable for individual situations using diverse
perceptual modalities.
One of the most popular strategies is to provide users with visual
stimulation. For example, Red Alert is a visual alerting system which uses a
translucent orange-red flash to mask a screen, designed to warn pilots
potential collisions in air traffic control (Saint-Lot et al., 2020). Audio
stimuli have also been favorably employed as a means to alert users. BBeep is
a collision-avoidance system that can emit a beep sound to alert pedestrians
around a visually impaired user to clear the way (Kayukawa et al., 2019).
Another strategy is the use of the tactile modality. BuzzWear is a wrist-worn
tactile display to notify users on the go by combining different parameters of
the tactile stimulus (Lee and Starner, 2010). As can be observed in these
examples, most systems adopt explicit stimuli to notify users, assuming that
they will take action after their attention is drawn to the target.
However, Adams et al. (Adams et al., 2015) pointed out that such alerting
strategies would not be optimal when used within persuasive technologies
designed to influence user behavior. Unlike critical situations (e.g., air
traffic control) where it can be expected that users will be motivated to
follow an alert from a computer, not all scenarios for inducing behavioral
change can assume that users are motivated to do so. In such cases, an alert
that requires the user’s conscious awareness and effort to work effectively
would likely fail due to lack of motivation or potentially counteract positive
aspects of the intervention by frustrating them. Thus, the authors recommended
the Mindless Computing strategy of leveraging human biases or unconscious
behaviors, which diminishes reliance on users’ conscious awareness. It also
enables a user intervention without interfering with users’ ongoing activity,
whereas alerting users explicitly can interrupt such activity. Furthermore,
they complimented the advantage of the mindless approach by mentioning that
such interventions have long-term effectiveness, even though users are aware
of the biases behind the interventions (Wansink and van Ittersum, 2007).
This point is common to the previous studies for video-based learning in
regards to the reliance on learners’ motivation, which is mentioned in Section
2.1. That is, as conventional alerting approaches are requiring learners’
conscious awareness to be effective, they would have an option not to follow
the intervention. Therefore, for the purpose of helping learners return their
attention, we explore a new computational approach that intervenes without
consuming their conscious awareness. This led us to make use of the nature of
human speech communication.
### 2.3. Speech Communication Techniques for Drawing Human Attention
Speech is one of the most natural modalities of human communication. It
consists not only of linguistic aspects but also of paralinguistic aspects,
such as pitch, volume, and speed, which play an important role in conveying
nuance or emotion (Trager, 1958). Though the use of paralinguistic aspects is
a natural habit that does not necessarily require our conscious processes
(Poyatos, 1993), it is also a common practice to intentionally create changes
in such paralinguistic parameters while speaking so as to draw listeners’
attention (International, 2011). The relationship between speech parameters
and their effects in terms of drawing attention has generated considerable
research interest in understanding human speech communication. For example, Xu
(Xu, 2005) confirmed that an increase in pitch when starting a new topic can
draw listeners’ attention. Moreover, a similar effect of drawing attention has
also been observed in infants hearing the speech of their mothers, who
naturally vary their pitch (Sullivan and Horowitz, 1983). The idea that humans
unconsciously respond to paralinguistic cues is further supported by Zatoree
and Gandour (Zatorre and Gandour, 2007), who verified that human neural
mechanisms are sensitive to such spectral and temporal acoustical properties.
Based on these results, we speculate that leveraging this nature of human
speech communication by computationally varying speech parameters can draw
listeners’ attention in a natural manner. More specifically, if a person
losing their attention to a video hears speech with altered pitch or volume,
they will naturally respond to such a change, regardless of their motivation
to pay attention. Such an intervention approach is in line with the concept of
Mindless Computing (Adams et al., 2015) and thus is expected to work without
depending on users’ motivation. In the following section, we further elaborate
on the rationale for our design of using alterations of human speech to draw
attention in video-based learning situations.
## 3\. Mindless Attractor
In this paper, we propose Mindless Attractor for the purpose of helping users
in video-based learning situations return their attention to the video.
Inspired by the concept of Mindless Computing (Adams et al., 2015), it
leverages the nature of speech communication to intervene with users. In this
section, we present the details of Mindless Attractor, starting by discussing
why the mindless approach should be considered and what requirements should be
fulfilled.
### 3.1. Why Mindless?
As we stated in Section 1, our research aim is to support video-based
learning, given the growing demand for it, by establishing a suitable
computational intervention for users who are not paying attention to the
video. The difficulty is that we cannot assume all users to be highly
motivated to follow such an intervention for maintaining attention, which we
mentioned in Section 2.2 as the reason that conventional alerting approaches
would not be suitable. Thus, we need to consider an intervention approach that
does not rely on users’ motivations. In addition, even when a user is not
focusing on the video, intervention approaches that interrupt the user should
be avoided since such approaches might lead them to miss subsequent content.
These points led us to adopt an approach based on Mindless Computing (Adams et
al., 2015) that leverages human biases or unconscious behaviors to induce
behavioral change. Since such an intervention approach does not consume the
user’s conscious awareness to be effective, it is considered less reliant on
their motivation to pay attention. Moreover, it enables us to design a less
interruptive intervention than explicit alerts, as Adams et al. (Adams et al.,
2015) confirmed that their mindless approach using auditory feedback could
influence people’s behavior when talking without annoying them.
Furthermore, we presume that the mindless approach will reveal a new advantage
when integrated with a sensing module based on machine learning techniques, as
mentioned in Section 1. More specifically, although machine learning systems
enable various sensing scenarios, humans tend to evaluate such systems’
mistakes more severely than human mistakes (Dietvorst et al., 2015). In
addition, the trust that machine learning systems lose as a result of their
failure is usually greater than the trust they gain from their success (Yu et
al., 2017). Consequently, people often become less hesitant to override
outputs from machine learning systems after seeing their failures (De-Arteaga
et al., 2020). Moreover, it has been suggested that people with a high level
of cognitive load will have less trust in interactions with machine learning
systems (Zhou et al., 2017). These discussions imply the risk posed by the
false-positive detection of the sensing module in intervening with users—that
is, mistakenly alerting them in an explicit manner during video-based learning
situations would frustrate them and lead them to disregard the alerts. On the
other hand, since the mindless approach does not consume conscious awareness,
unlike the alerting approach, it might mitigate the negative effects caused by
false positives.
We therefore suppose that the mindless approach would be suitable as an
intervention in the context of video-based learning. In particular, we believe
that this is a plausible solution to the current situation where effective
interventions for video-based learning have not been well investigated, as
discussed in Section 2.1.
### 3.2. Designing Mindless Attractor
To design the mindless approach leveraging human biases or unconscious
behaviors, we exploited the nature of human speech communication. Our design
is based on the following requirements we considered in view of using the
mindless approach in video-based learning situations.
Avoid interruption due to interventions.:
Considering that video-based learning is sometimes delivered in the form of
live streams or in a synchronous manner (Baturay, 2015), interrupting users
due to interventions should be avoided, as it can cause them to miss
information and counteract our aim of helping them pay attention. This
requirement is one reason to eliminate the use of alerting approaches, as we
discussed their interruptive aspect in Section 3.1.
Use a modality that users will not neglect.:
To intervene with users who are not paying attention to the video, it is
important to use a modality that is always reachable for users. In this
regard, though it is possible to leverage human perceptual bias to design the
mindless approach by showing something on a display, this would not be
suitable because the user can take their eyes off the display, especially when
performing other tasks using a smartphone (Oeppen et al., 2020). On the other
hand, it seems more unlikely that the user would not hear the audio due to
muting it while in video-based learning situations.
Function without external devices.:
Though the use of external devices would extend the range of possible
interventions, such as using a tactile stimulus (Lee and Starner, 2010), it
raises an additional cost to utilize the interventions. Therefore, it is
desirable to design an intervention that could be integrated into video-based
learning situations without requiring external devices.
As we reviewed in Section 2.3, it has been suggested that humans unconsciously
respond to paralinguistic cues in speech, such as a change in pitch, volume,
and speed. In our case, we considered perturbing the pitch or volume of the
voice in the video to help users refocus their attention. We did not use speed
because it would be difficult to maintain time-series consistency when video-
based learning is conducted in a synchronous manner (e.g., live lectures
(Baturay, 2015)).
In addition, the perturbation is enabled and disabled repeatedly when the user
is seemingly not paying attention to the video, as Adams et al. (Adams et al.,
2015) emphasized the importance of cues to trigger different perceptions and
sensations in designing mindless approaches. Otherwise, if we activated the
perturbation once when the user became distracted and kept it thereafter, the
user would have less opportunity to refocus their attention as they became
acclimated to the changed pitch or volume.
### 3.3. Implementation
We used Python and PyAudio111https://people.csail.mit.edu/hubert/pyaudio/docs/
to perturb the audio signal in real time. The audio signal was captured in 16
kHz and the perturbation process was activated each 1/16 sec to ensure that
the perturbed signal was delivered without significant delay. The pitch shift
was performed using a library named
rubberband222https://github.com/breakfastquay/rubberband through time-shifting
and resampling the signal via Fourier transform. The volume change was
performed by directly multiplying the waveform double or halve. Our source
code is publicized at a GitHub
repository333https://github.com/hiromu/MindlessAttractor.
In addition, as we mentioned in Section 1 and Section 3.1, our mindless
intervention approach is expected to incorporate a sensing module that
monitors users’ behavior and detects when they are distracted. The detailed
implementation of the sensing module is later explained in Section 6.3.
## 4\. Hypotheses
Up to this point, we have introduced Mindless Attractor, which is designed as
an intervention for users during video-based learning that incorporates a
sensing module based on machine learning techniques. It computationally
perturbs the pitch and volume of the voice in the video in real time to
refocus users’ attention when they seem to be distracted from the video. Our
design rationale for the proposed approach, which we discussed in Section 3.2,
imposes the following hypotheses, which need to be verified to ensure the
validity and effectiveness of the proposed approach.
First, as we discussed in Section 3.1, our proposal is based on the concept of
Mindless Computing (Adams et al., 2015) so as to ensure that the intervention
works without relying on user motivation and without interrupting users. To
satisfy these points, we should examine whether Mindless Attractor can
influence users’ behavior in a mindless manner, i.e., without consuming their
conscious awareness.
> H1: Mindless Attractor is an effective means to refocus the attention of
> users in video-based learning situations without consuming their conscious
> awareness.
If H1 holds, we have two choices for inducing behavioral change in users
(i.e., drawing their attention back to the video): alerting users in an
explicit manner or intervening in a mindless manner. Here, as we discussed in
Section 3.1, we expect that the proposed approach will be favored over
alerting approaches when combined with a machine learning-based sensing module
that detects when users are losing attention. More specifically, the fact that
such a sensing module may produce false positives implies the risk of
mistakenly intervening in users, which can be annoying when we alert them
explicitly. Thus, we posit our second hypothesis:
> H2: Mindless Attractor is not only an effective means to refocus users’
> attention but is also preferred by users when combined with a machine
> learning-based sensing module, while the alerting approach is not accepted.
If these hypotheses are supported, we can pave the way for intervening with
users in real time to support their participation during video-based learning.
With this motivation, we evaluated these hypotheses by conducting a series of
experiments.
## 5\. Experiment I: Evaluation of H1
### 5.1. Design
To evaluate H1, we conducted an experiment that replicated video-based
learning situations. We used a within-participant design comparing a treatment
condition using Mindless Attractor with a control condition that did not
intervene in participants. Then, H1 is supported if the following two points
are confirmed: Mindless Attractor helps participants refocus their attention,
and Mindless Attractor does not consume participants’ conscious awareness.
### 5.2. Measure
We prepared two measures corresponding to the above two points to be
confirmed: recovery time and cognitive workload.
#### 5.2.1. Recovery Time
This metric indicates the time that it took for participants to return their
attention to the video after losing focus. If Mindless Attractor helps
participants refocus their attention, the time that they are distracted should
be shortened in comparison to the case in which no intervention was taken.
To compute this metric, we collected human annotations for each participant
denoting whether the participant was paying attention or not. As we explain in
the detailed procedure description in Section 5.5, an experimenter observing
the state of the participants annotated in real time so that the recovery time
could be calculated later.
#### 5.2.2. Cognitive Workload
This metric was used to evaluate whether Mindless Attractor consumed the
participants’ conscious awareness or not. Measuring cognitive workload is
common in the previous studies proposing alerting approaches (Saint-Lot et
al., 2020; Lee and Starner, 2010). Whereas they aimed to show that their
proposed approaches exhibited lower workload compared to other possible
approaches, we compared the metric between the control and treatment
conditions. If the cognitive workload in the treatment condition is not
significantly different from that in the control condition, it suggests that
Mindless Attractor does not consume participants’ conscious awareness. In our
study, we used the NASA-TLX questionnaire (Hart and Staveland, 1988; Byers et
al., 1989) to measure cognitive workload, in the same manner as the previous
studies (Saint-Lot et al., 2020; Lee and Starner, 2010).
We note that it would be possible to evaluate whether Mindless Attractor
consumes the participants’ conscious awareness by asking them whether they
noticed the perturbation. However, to do so, we would need to conceal from the
participants that they would be subject to an intervention, which would create
an unrealistic situation if we consider the practical applications of the
proposed approach. More specifically, it is unlikely that users in video-based
learning situations would be subject to interventions without opt-in consent;
that is, they would use Mindless Attractor of their own accord to focus on
videos or at least would be notified about the possibility of the
intervention. In addition, as we mentioned in Section 2.2, Adams et al. (Adams
et al., 2015) explained that the mindless approaches work regardless of
whether a user knows their mechanisms or not, as they do not depend on the
user’s conscious awareness. Thus, we used this measure based on NASA-TLX and
also notified the participants beforehand that they would be subject to
interventions.
### 5.3. Material
To replicate a video-based learning situation, we prepared a video recording
of a 30-minute lecture on urban sociology. As this experiment was conducted
remotely, the video was presented to the participants using the screen-sharing
function of Zoom444https://zoom.us/.
By following the implementation we described in Section 3.3, we also prepared
a client software that modifies Zoom’s audio output to perform our
intervention. This software captures and perturbs the audio output in real
time when it receives an activation command from a control server via
WebSocket. Here, we conducted a pilot study in the same manner as Adams et al.
(Adams et al., 2015) to find the best parameters for intervening without
causing distractions. Consequently, we implemented four perturbation patterns:
halving or doubling the volume and lowering or raising the pitch by one tone.
The software then activates one of the four patterns randomly so as to enable
the comparison of their effectiveness for helping the participants refocus
their attention. Since Zoom automatically removes noises and extracts voices,
we confirmed that our naïve implementation of pitch shifting based on fast
Fourier transform would be sufficient for the purposes of this experiment.
We further prepared an experimenter console in the control server to record
annotations concerning whether the participant was paying attention or not.
The console was implemented to enable sending the activation and deactivation
command to the client software when the participant started to divert their
attention from the video and refocused their attention, respectively.
### 5.4. Participants
This experiment involved 10 participants, three of whom were female. They were
recruited via online communication in a local community where over 100
university students gather. As described later in Section 5.5, our
experimental procedure required participants to be observed by a remote
experimenter so that their state of attention could be annotated. Therefore,
we asked them to prepare a PC with a webcam in a quiet room as well as to
enable their faces to be captured.
### 5.5. Procedure
Each participant underwent one session of watching the 30-minute video using a
computer connected over Zoom, as we mentioned in Section 5.3. To replicate the
usual situation of video-based learning, in which learners have some reasons
to watch the video, we told participants in advance that they would be asked
to write a few sentences summarizing the video. At the same time, we asked
them to bring their smartphones and told them that we would not prohibit the
use of smartphones so that they could be distracted as usual (Oeppen et al.,
2020).
As depicted in Figure 1, each session was divided into two parts of 15 minutes
each: one with no intervention and another involving interventions. To
normalize the order effect, we balanced the order of the two parts: five
participants first experienced the part with no intervention, and the others
first experienced the part involving interventions. After each part, the
participant was asked to write a summary and fill out the questionnaire
measuring cognitive workload. Note that these two parts do not correspond to
the control and treatment conditions, as explained in the following
paragraphs.
Figure 1. Example illustration of the procedure for our first experiment. (A)
Half of participants first experienced the part with no intervention and then
experienced the part involving interventions, and (B) the others followed the
reversed order.
Example illustration of the procedure for our first experiment. After each
participant installed the client with opt-in consent at the beginning, they
experienced two parts: one with no intervention and another involving
interventions. In the part involving interventions, an experimenter annotated
whether the participant was paying attention to the video. Half of
participants followed this order, and the others first experienced the part
involving interventions and then experienced the part with no intervention.
In the part involving interventions, an experimenter observed the state of a
participant, including their use of smartphones, and annotated whether they
were paying attention to the video or not. When the experimenter pressed a
button on the experimenter console to record the timestamp at which the
participant diverted their attention from the video, the console assigned
either the control or treatment condition with a 50% probability of each. Note
that the selected condition was concealed from the experimenter in order to
avoid the experimenter bias in the annotations. If the treatment condition was
assigned, the console sent the activation command to the client, and the
client then repeatedly enabled and disabled one of the four perturbation
patterns every 3 seconds, as explained in Section 3.2. This intervention
continued until the client received the deactivation command indicating that
the experimenter pressed another button to record the participant’s recovery
from the distraction. On the other hand, if the control condition was
assigned, no command was sent to the client. Consequently, based on the
assigned conditions and the recorded timestamps, the recovery time could be
calculated and compared.
The other part (with no intervention) was prepared to evaluate the cognitive
workload. We compared its cognitive workload score with that of the part
involving interventions, which were activated on a random basis. If the
intervention did not consume the participant’s conscious awareness, the scores
of the two parts would not be significantly different.
In addition, at the end of the session, we asked the participants for their
comments about their feelings or anything they noticed. In total, the entire
session took about an hour to complete.
### 5.6. Results
Table 1. Comparison of the recovery time and cognitive workload score between the control and treatment conditions. The treatment condition involved the mindless intervention. Measure | Treatment | Control | $p$-value
---|---|---|---
Recovery time | $17.71~{}s$ ($\pm 10.52~{}s$) | $32.25~{}s$ ($\pm 16.92~{}s$) | ¡ 0.0001
Cognitive workload | $26.00$ ($\pm 10.32$) | $27.00$ ($\pm 9.13$) | 0.5212
#### 5.6.1. Recovery Time
Figure 2. Distribution of the recovery time across each participant and the
experimental conditions.
Distribution of the recovery time across each participant and the experimental
conditions. The participants tended to spend more time recovering their
attention in the control condition.
As shown in Table 1, the proposed intervention significantly shortened the
recovery time according to the unpaired $t$-test (Cohen’s $d=1.0044$,
$p<0.0001$). The distribution of the recovery time is shown in Figure 2, which
also confirms this reduction. This result supports that Mindless Attractor
helped participants refocus their attention.
We also investigated which of the four perturbation patterns (i.e., halving or
doubling the volume and lowering or raising the pitch by one tone) effectively
helped participants refocus their attention. We examined the last perturbation
pattern before each time the participant returned their attention and counted
their occurrence, as shown in Table 2. This examination is based on our
assumption that the intervention just before the participant’s attention
returned is the cause of the change in the participant’s state. According to
the $\chi^{2}$-test comparing with the total occurrence, the results were not
significantly different in that each pattern equally helped participants
recover their attention (Cramer’s $V=0.1220$, $p=0.2794$). In other words, we
can conclude that there was no significant difference in the effectiveness of
the four perturbation patterns.
Table 2. Occurrence of the four perturbation patterns that were executed just before participants returned their attention. The comparison with the total occurrence suggests that there was no significant difference in effectiveness ($p=0.2794$). Perturbation | Halve the volume | Double the volume | Lower the pitch | Raise the pitch
---|---|---|---|---
Occurrence just before | 19 | 7 | 14 | 16
participants returned their attention
Total occurrence | 50 | 47 | 50 | 55
#### 5.6.2. Cognitive Workload
We also could not find a significant difference in participants’ cognitive
load scores according to the paired $t$-test (Cohen’s $d=0.2110$, $p=0.5212$),
as presented in Table 1. That is, it is suggested that Mindless Attractor did
not consume participants’ conscious awareness or at least did not negatively
affect participants’ cognitive load by consuming their conscious awareness.
Thus, in combination with the effect on the recovery time, H1 was supported.
#### 5.6.3. Comments
We also examined the comments that the participants wrote at the end of the
experiment. At first, we realized that three participants mentioned that they
did not notice any intervention, although they were informed of the
intervention beforehand. Interestingly, the recovery time for these three
participants also showed a significant difference (Cohen’s $d=0.8105$,
$p=0.0122$) between the treatment ($15.88~{}s$ on average) and control
($28.34~{}s$ on average) conditions. Thus, it is suggested that the mindless
approach worked even when it was not noticed by participants, further
supporting that Mindless Attractor did not consume the participants’ conscious
awareness. This point not only corroborates H1 but also shows consistency with
the discussion by Adams et al. (Adams et al., 2015).
It was also interesting that, although five participants mentioned that they
noticed the changes in volume, no participant recognized the changes in pitch.
That is, although no significant difference was found between the
effectiveness of the four perturbation patterns in Table 2, their
noticeability varied, suggesting further room for investigation.
Nevertheless, no participants regarded the mindless intervention as disruptive
or annoying; rather, two participants made positive comments about it:
> I found it useful because it naturally brought my attention back to the
> video when I thought something might have changed in the speech. (P1)
> It was nice as it made me feel like…the computer was recommending me to
> concentrate, rather than warning me. (P4)
In particular, the latter comment suggested that the mindless approach can
mitigate the negative effect that might be caused by false-positive detection
when combined with a machine learning-based sensing module. These results
motivated us to conduct a second experiment to evaluate this possibility, as
discussed in Section 4 when posing H2.
## 6\. Experiment II: Evaluation of H2
### 6.1. Design
To evaluate H2, we conducted an experiment that replicated a video-based
learning situation in the same manner as Section 5. However, in this case, we
combined a machine learning-based sensing module rather than manually
activating interventions and compared the effects of the mindless approach and
the alerting approach. Here, we used a within-participant design over three
conditions: mindless, alerting, and control (no intervention). We added the
control condition to confirm that the proposed approach was at least effective
in contributing to refocusing users’ attention as an automated system
controlled by a machine learning-based sensing module. H2 is thus supported if
the following two points are confirmed: Mindless Attractor helps participants
refocus their attention, and participants favor Mindless Attractor over the
alerting approach.
### 6.2. Measure
Similar to the first experiment, we measured time with regards to whether
participants were paying attention. However, we introduced a different
approach for evaluating the time factor, i.e., total distracted time instead
of the recovery time. In addition to this, we introduced a measure for
behavioral intention.
#### 6.2.1. Total Distracted Time
Although we have confirmed that Mindless Attractor can help participants
return their attention, it is desirable to investigate whether the total time
that they are distracted during video-based learning is decreased. In other
words, it may be possible that, though the mindless approach shortened the
recovery time, the participants were distracted more frequently, especially
when the mindless approach was combined with a machine learning-based sensing
module having a risk of false positives.
To compute this metric, we collected human annotations for each participant,
as we did in Section 5, and aggregated the duration when the participants were
not paying attention. If the total distracted time in the mindless condition
is significantly shorter than in the control condition, it is suggested that
Mindless Attractor can make users more likely to pay attention, even in
combination with a machine learning-based sensing module.
It should be noted that, due to the false negatives of such a sensing module,
there would be a case when the intervention is not triggered even when the
participant is actually losing their attention and a case when the
intervention is deactivated before the participant refocus. Therefore,
calculating the recovery time as in Section 5.6 is not appropriate in this
second experiment, further rationalizing the introduction of the total
distracted time as a different metric.
#### 6.2.2. Behavioral Intention
This metric was prepared to evaluate whether the mindless approach was favored
over the alerting approach. The concept of behavioral intention is guided by
the Technology Acceptance Model (Davis, 1989), which explains users’ attitudes
towards technologies, and is frequently used to evaluate how likely
individuals are to use the technologies. We used the questionnaire to measure
behavioral intention in the same manner as the previous studies (Venkatesh et
al., 2003). If this score in the mindless condition is significantly better
than that in the alerting condition, we can confirm that Mindless Attractor
can be favored over the alerting approach, especially when it works as an
automated system with a sensing module.
### 6.3. Material
Similar to our first experiment, we prepared a video recording of a 30-minutes
lecture on social sciences. The experiment was conducted remotely and the
video was presented using Zoom’s screen-sharing function, as in the first
experiment. However, in this second experiment, we developed a system that
automatically detected the status of participants’ attention. To implement
this sensing module, we followed previous studies that estimated participants’
attentiveness based on their visual cues, which we reviewed in Section 2.1.
Specifically, we analyzed the video stream of face images of each participant
by leveraging machine learning techniques that can detect their head pose in
real time. If the module detected that the participant was looking off the
screen, the system judged that the participant was failing to pay attention to
the video lecture, and activated an intervention.
Figure 3 illustrates how the system processed the video streams of
participants and intervened in them. Videos were processed in a frame-by-frame
manner. First, a human face was detected and located in each frame using a
deep learning model, RetinaFace (Deng et al., 2020). We used this model
because it achieves state-of-the-art performance and its pretrained model is
publicly released. Face alignment was then performed to obtain facial
keypoints using a deep learning model proposed by Fang et al. (Fang et al.,
2017) that is also known to estimate keypoints with high accuracy. Finally,
based on the estimated facial keypoints, the head pose was calculated by
solving a perspective-n-point problem. These calculations were performed using
a dedicated computation server with an NVIDIA V100 Tensor Core GPU.
Next, the estimated head pose was passed to the experimenter’s PC, a
conventional laptop with a 2.2 GHz Intel Core i7 processor. This PC checked
whether the passed head direction was off-screen or not. The experimenter had
conducted a calibration process beforehand to calculate the threshold for this
judgment, in which participants were asked to track a red circle that appeared
and moved along the edge of the screen. Participants were told to track the
circle by moving their head, i.e., not following it only by moving their gaze.
We then calculated the maximum head rotations for each direction (top–down and
left–right) and regarded them as the range where the head is toward the
screen. In other words, when the estimated head pose was out of this range,
then the system judged that the participant was looking off the screen, and
thus, losing their attention.
While the participants were watching the video, changes in their state–i.e.,
whether they were looking at the screen or not–were shared with another
control server maintaining a WebSocket connection with the client software.
The control server then correspondingly sent activation or deactivation
commands in the same manner as the first experiment. All of the above
processes were performed in real time with a frame rate of 15 FPS.
Figure 3. Architecture of the entire system we implemented for the second
experiment.
Architecture of the entire system we implemented for the second experiment.
From each frame of the video capturing the participant’s face, a GPU server
detects the face position, calculates the face alignment, and estimates the
head pose. Based on the head pose, an experimenter PC activates interventions
via a control server.
In addition to the sensing module, we implemented an intervention to
explicitly alert users in the client software, to be compared with our
proposed approach. In this case, the client software played a short beep for
0.1 seconds, which followed the previous study’s use of a beep alert (Kayukawa
et al., 2019), rather than perturbing the audio output. Once the alert was
activated, it replayed the same beep every 3 seconds until it received the
deactivation command, in the same manner as the mindless condition.
### 6.4. Participants
This experiment involved 20 participants, five of whom were female. They were
recruited in the same manner as we did in the first experiment. Eight of the
participants participated in our first experiment, which had been held at
least two weeks before this experiment. The participants were asked to prepare
a PC in a quiet room and to enable their faces to be captured with a webcam,
as in the first experiment.
### 6.5. Procedure
Similar to the first experiment, each participant experienced a session of
watching the 30-minute video using a computer connected over Zoom. As before,
we told participants in advance that they would be asked to write a few
sentences summarizing the video and also allowed them to bring and use their
smartphones in the session.
As illustrated in Figure 4, each session consisted of three parts lasting 10
minutes each: one with no intervention, another with the mindless approach,
and a third with the alerting approach. The order of these three parts was
automatically randomized among participants, as we will describe later in this
section. After each session, participants were asked to write the summary.
They were also asked to fill out the questionnaire measuring behavioral
intention when they finished a part with either the mindless or alerting
approach. We compared the scores between the two conditions to examine which
approach participants favored.
Before starting the first session, the experimenter performed a calibration
process to determine the threshold for whether the participant’s head pose was
out of the screen, as described in Section 6.3. The experimenter explained
that the participants should not move their PC until the entire process was
complete and advised them to find a comfortable position before the
calibration process started.
In each of the three parts, the experimenter manually annotated whether the
participant was paying attention to the video lecture, similar to the first
experiment. To avoid bias, the experimenter was blind to which of the three
conditions had been applied to the participant. Specifically, the control
server (see Figure 3) decided the order of conditions in each session, and the
experimenter did not have access to this information until the session ended.
The obtained annotations were used to calculate the total distracted time for
each part.
In addition, our developed machine learning-based sensing module triggered
interventions to the participants in either the alerting or mindless
condition, as described in Section 6.3. In the alerting condition,
participants were exposed to the beep sound when the system judged that they
were losing attention, whereas they were exposed to perturbations in the
speech in the mindless condition. In the control condition (i.e., that with no
intervention), the client system did not intervene. In each part, the sequence
of the system’s judgment was recorded along with timestamps, which we later
used to assess the accuracy of the sensing module by comparing it with the
human annotations.
Finally, at the end of the session, we asked the participants for their
comments about their feelings or anything they noticed. In total, the entire
session took about an hour to complete.
Figure 4. Example illustration of the procedure for our second experiment.
Each participant was randomly assigned to one of six possible orders of the
three conditions.
Example illustration of the procedure for our second experiment. After each
participant installed the client and performed the calibration process at the
beginning, they experienced three conditions: mindless, alerting, and control.
Each participant was randomly assigned to one of six possible orders of the
three conditions.
### 6.6. Results
#### 6.6.1. Sensing Accuracy
Table 3. Confusion matrix between the human annotations and the detection results of the machine learning-based module in regard to participants’ attentive state. | | Detection result
---|---|---
| | Attentive | Distracted
Human annotations | Attentive | 435.4 min (68.5 %) | 78.0 min (12.3 %)
Distracted | 51.4 min (8.1 %) | 70.7 min (11.1 %)
We first examined the accuracy of our machine learning-based sensing module in
detecting participants’ attentive state. We compared the human annotations and
the detection results of the module and obtained Table 3. Though our aim is
not to develop a detection system, the accuracy across all the participants
was 79.6 %, which was relatively close to the previous study (Thomas and
Jayagopi, 2017) that achieved the accuracy of 82–85 % using only head pose. We
note that the accuracy varied among participants (64.9–93.0 %), which implies
that some environmental factors (e.g., the distance to camera or lighting
conditions) might largely affect the detection results. At the same time, the
sensing module exhibited a lot of false-positive detection, as its precision
was 47.6 %, which suited our aim to investigate the effect of Mindless
Attracter while having a risk of false positives.
#### 6.6.2. Total Distracted Time
Figure 5. Comparison of participants’ total distracted time. We found
significant differences between the control condition and the other
conditions.
This bar chart with standard errors shows participant’s total distracted time
during each session, comparing the mindless, alerting, and control condition.
The time for the control condition is significantly longer than that for the
mindless and alerting condition (**). The time for the mindless and alerting
condition is not significant.
Figure 6. Comparison of how many times participants got distracted. We found
no significant difference between the three conditions.
This bar chart with standard errors shows how many times participants got
distracted during each session, comparing the mindless, alerting, and control
condition. There is no significant difference between each condition.
Figure 7. Comparison of participants’ scores of the behavioral intention. We
found a significant difference between the mindless and alerting conditions.
This bar chart with standard errors shows the scores of the behavioral
intention, comparing the mindless and alerting condition. The score for the
mindless condition is significantly higher than that for the alerting
condition (**).
Next, based on the human annotations, we calculated the total distracted time
for each participant, as presented in Figure 7. We found a significant
difference among the three conditions according to ANOVA ($F(2,57)=8.5773$,
$\eta^{2}=0.2313$, $p=0.0005$), and thus conducted a post-hoc test. As a
result, the control condition showed significant differences against the
mindless and alerting conditions (Cohen’s $d=1.1795$, $p=0.0013$ and Cohen’s
$d=1.0828$, $p=0.0032$, respectively). On the other hand, we found no
significant difference between the mindless and alerting conditions.
From this result, it was confirmed that Mindless Attractor is an effective
means to refocus users’ attention even when combined with a machine learning-
based sensing module, as the mindless condition significantly reduced the
total distracted time than the control condition. In addition, it is notable
that Mindless Attractor would work effectively as well as the conventional
alerting approaches since the mindless and alerting conditions showed similar
distracted times.
We also examined how many times the participants got distracted because it was
possible that our interventions increased the frequency even though the total
distracted time was reduced. As shown in Figure 7, we did not find significant
differences among the three conditions ($F(2,57)=0.1796$, $\eta^{2}=0.0062$,
$p=0.8360$). It can be explained as follows: the participants were almost
equally likely to lose focus in all the three conditions; but, if there was an
intervention, they often refocused their attention to the video earlier, as
confirmed in the first experiment; as a result, their distraction time in the
mindless and alerting conditions was significantly reduced than the control
conditions. From these results, we conclude that H2 was supported in terms of
the effectiveness of Mindless Attractor.
#### 6.6.3. Behavioral Intention
Lastly, we compared participants’ scores of the behavioral intention between
the mindless and alerting conditions. As presented in Figure 7, we found a
significant difference (Cohen’s $d=0.7025$, $p=0.0054$) according to the
paired $t$-test. That is, compared to the alerting approach, the participants
showed their stronger intentions to use the implemented system when it is
combined with the mindless approach. This result supports that Mindless
Attractor is much preferred by users than the alerting approach, as we
hypothesized as H2.
#### 6.6.4. Comments
The above results coincided with H2; that is, Mindless Attractor helps
participants refocus their attention and it is favored over a conventional
alerting approach. In addition, the comments obtained at the end of the
experiment corroborated H2, especially in regard to the unacceptability of the
alerting approach.
> I felt like the beep sound made me lose focus. It was frustrating,
> especially when I was concentrating. (P9)
> The beep felt like noise because it overlapped the speech though I wanted to
> listen to what was being said. As a result, my concentration was more
> disrupted than the case that I had not used the system. (P12)
> I thought the one with the beep sound might be a good signal until halfway
> through, but then it came to ring repeatedly even though I was
> concentrating. As a result, I stopped caring about the sound. (P2)
These comments confirmed our anticipation; that is, explicitly alerting users
based on false-positive detection makes them distracted and frustrated, which
can lead them to ignore the intervention. In addition, one participant
suggested that such negative effects can be caused even when the intervention
was activated by accurate detection:
> I was disgusted by the alarm, which rang when I was using my smartphone for
> googling a word I never heard. (P8)
In contrast, the mindless condition was totally favored, as follows:
> In the part [of the mindless condition], I felt like I was able to focus on
> the lecture relatively well. (P12)
> I did not notice much of a change in the audio, but when I compare the three
> parts, I seemed to be able to maintain my concentration the most. I think
> having such a system that brings back my attention without making a big deal
> will help me stay focused in usual situations. (P3)
> When the pitch of the speech became higher, I paid attention to the video as
> I felt strange a little. It did not provide a sense of being angry, compared
> to the beep alarm. (P11)
These comments corresponded to the comparison of the scores of behavioral
intention (Figure 7).
Furthermore, 17 of 20 participants agreed they often have trouble maintaining
their attention and computationally solving it would be beneficial, like:
> I find it difficult to maintain my attention in such online situations
> because of the lack of eyes around. (P1)
In addition, they suggested that the proposed approach can be used outside
video-based learning situations.
> I thought it would be nice to be able to introduce a similar system in
> offline situations. I will appreciate it if some device such as a smartwatch
> helps me refocus when I am losing my attention from an important
> conversation. (P4)
The obtained comments not only supported the effectiveness of Mindless
Attractor through supporting H2 but also highlighted the further potential of
the proposed approach.
## 7\. Discussion
So far, by verifying H1 and H2, we have demonstrated that Mindless Attractor
works effectively as a novel intervention approach to support users’
participation during video-based learning. In this section, we contemplate the
findings of our study, envision future application scenarios, and discuss
limitations and directions for future work to further pave the way for
supporting users in video-based communication.
### 7.1. Necessity of Mindless Intervention in Machine Learning-Based Systems
The results of our second experiment supported H2: Participants favored the
proposed mindless approach, while the alerting approach was not accepted.
Specifically, the obtained comments suggested that participants were annoyed
by the alerts when they were triggered by false positives of the sensing
module. In other words, mistakenly intervening in an explicit manner while
users are concentrated on the main task can unnecessarily consume their
conscious awareness and eventually disrupt their experience.
Indeed, such failures in designing automated systems based on machine
learning-based sensing modules have been pointed out in a recent guideline for
human–AI interaction (Amershi et al., 2019). That guideline emphasized the
importance of considering that such AI-infused systems may demonstrate
unpredictable behaviors due to false positives and false negatives.
Consequently, it was suggested that an effective approach in designing AI-
infused systems is to enable users to dismiss the undesired functions
instantly. In light of this, our proposed mindless approach can be a promising
direction that follows this guideline, as it does not consume users’ conscious
awareness, letting them not mind the mistakenly triggered interventions
without much cognitive workload. Therefore, we believe that Mindless Attractor
can support users as a novel intervention method integrated with machine
learning-based systems in various cases, not limited to the presented case
(i.e., video-based learning).
### 7.2. Application Scenarios
As mentioned in Section 1, the importance of helping participants be attentive
during video-based communication has been emphasized in various contexts. In
this regard, we believe that Mindless Attractor can be used effectively not
only in video-based learning but also in other situations using video-based
communication. For example, it can be employed to help participants in video-
based meetings be more attentive in the same manner as shown in this study.
Here, we note that a few studies have aimed to provide real-time feedback to
participants in meetings (Schiavo et al., 2014; Samrose et al., 2017). For
example, CoCo is a system designed to achieve balanced participation through
feedback, such as showing a pie chart representing the participation ratio
that can be estimated from speaking length and frequency (Samrose et al.,
2017). Similar to the discussion we had with regard to video-based learning,
these techniques of providing explicit feedback require participants to be
motivated to change their behavior, i.e., to be more attentive to the meetings
based on the feedback. Therefore, we can expect that Mindless Attractor will
be a promising alternative approach in that it does not consume participants’
conscious awareness during meetings, even when combined with machine learning-
based sensing systems.
Furthermore, we envision a future where Mindless Attractor can be utilized in
everyday interpersonal interactions. If we can assume that wearing earphones
in our daily life become more popular, it is possible to perturb the sound
they hear to utilize Mindless Attractor. For example, once the system detects
that the user is failing to pay attention during a conversation based on their
behavioral or physiological data, the envisioned system can intervene in a
mindless manner by modifying the voice they hear. Note that such demand for
offline use was indeed observed in one participant’s comment (P4) in our
second experiment.
It is noteworthy that we verified the effectiveness of Mindless Attractor in
the experiments in which users used it with prior consent. This lets us
imagine further practical applications utilizing Mindless Attractor as an opt-
in function. More specifically, it would allow users to selectively turn the
system on and off on their own, according to their situations and motivations.
For example, if a user attends an important lecture or meeting and thinks that
they need the assistance, they can actively allow themselves to be exposed to
the mindless intervention by turning on the system. In other words, our
results, which showed that the mindless approach worked with opt-in consent,
will pave the way for the user-centered exploitation of computational
interventions with which users can augment their levels of attention.
### 7.3. Limitations and Future Work
Though our experiments have demonstrated that Mindless Attractor is a
promising approach, there are some limitations. Initially, further
investigations involving a greater number of participants and diverse lecture
content are desirable to generalize our results. For example, if a lecture is
so attractive that learners are not distracted from the video, the proposed
approach would not be necessary, while at worst it would not be harmful, as
its impact on cognitive load was not observed in Section 5.6.
Secondly, our approach and evaluations are based on the discussion of Mindless
Computing proposed by Adams et al. (Adams et al., 2015), considering users
whose motivation for obeying the intervention is not always assumed. In fact,
we have given some consideration to the experimental designs so that the
participants would not become much motivated to the video, like allowing the
use of smartphones. Thus, we skipped the measurement of the participants’
motivation in our studies. However, this means that their results would not
necessarily guarantee the universal effectiveness of the proposed method for
users with any levels of motivation. Thus, evaluating participants’ motivation
and exploring its correlation with the efficacy of Mindless Attractor can be a
promising future work.
In addition, the accuracy of the machine learning-based sensing module in the
second experiment can be improved using the latest techniques (Thomas and
Jayagopi, 2017; Zaletelj and Kosir, 2017; Hutt et al., 2017; Veliyath et al.,
2019). In this study, we used a naïve approach based on head pose to
investigate the effect of the proposed approach with false-positive detection.
Although our sensing approach achieved a certain level of accuracy, as
discussed in Section 6.6, there is room to further sophisticate the algorithm.
It remains to be explored how users would feel if the alerting approach is
combined with a much more accurate sensing module. Nevertheless, we believe
that our mindless approach can be an effective intervention because false
positives will still remain.
In relation to this, it is noteworthy that recent works have proposed methods
for drowsiness detection from human visual cues (Ghoddoosian et al., 2019).
Thus, it can be explored in future work whether Mindless Attractor can help
participants who get sleepy during video-based learning, by integrating such a
detection technique in the sensing module. Examining the boundary of the
effectiveness of the proposed approach in such a situation would inform us of
further possible approaches, such as a hybrid of the mindless and alerting
interventions.
We also acknowledge that refining the design of alerts can mitigate the
negative impact suggested in the second experiment. While we used a simple
beep as an alert, alternative methods to inform users in less annoying manners
are possible. In particular, Weiser and Brown conceptualized “calm technology”
as a more acceptable communication channel from computers (Weiser and Brown,
1995, 1997). For example, alerting users with less explicit sounds (e.g.,
birds chirping) could be preferred to a simple beep sound. In addition, if we
ignore the requirement of using the auditory modality, showing a status lamp
on display is an alternative to inform users that they are losing attention.
However, as Adams et al. pointed out, these techniques require users’
conscious awareness (e.g., interpreting the status based on the lamp) to
induce behavioral change (Adams et al., 2015), while mindless computing does
not. Therefore, Mindless Attractor can be differentiated from alerting
approaches in that it can work without consuming users’ conscious awareness,
as suggested in the first experiment (see Section 5.6). That said, it is
desirable to explore sophisticated alerting approaches to draw further
implications in comparison to our mindless approach.
At the same time, the design of the mindless intervention has also room for
exploration. Currently, as explained in Section 3.2, we decided to perturb the
pitch or volume of the voice based on the nature of human speech
communication. Though we did not statistically examine the results due to the
small number of perturbations activated for each participant, there were
individual differences in terms of their effectiveness, which would imply the
possibility of personalizing the intervention patterns. Moreover, human brains
are known to show a special response to a self-voice (Conde et al., 2015) or a
familiar voice (Beauchemin et al., 2006). Thus, a possible intervention might
involve computationally modifying a voice so as to be similar to a self-voice
or familiar voice when learners are not paying attention. This can be achieved
through recent techniques for high-fidelity real-time voice conversion (Toda
et al., 2012; Arakawa et al., 2019).
Looking toward production deployment, investigating whether the proposed
approach that helps learners pay attention contributes to their learning
performance could be a future study. Considering that previous studies
adopting explicit feedback to help learners pay attention have shown a
positive impact on performance (Baker et al., 2010; Xiao and Wang, 2016), our
mindless approach can be expected to have a positive effect. This is because
the mindless approach exhibited an effect on distracted time comparable to
that of the alerting approach in Section 6.6, while showing no significant
impact on the cognitive load in Section 5.6. Examining the long-term effect of
the proposed approach is also suggested for future work. Though our design is
based on the concept of Mindless Computing, which Adams et al. (Adams et al.,
2015) have described as having long-term effectiveness, it is difficult to
deny, without further investigation, the possibility that users will become
acclimated to the perturbations. However, even in this case, the combination
with voice conversion we mentioned above could be a remedy, as it enables as
many patterns of interventions as the number of conversion targets.
## 8\. Conclusion
We presented a novel intervention approach, Mindless Attractor, which helps
users refocus their attention in a mindless manner. The approach leverages the
nature of human speech communication and perturbs the voice that users hear
when they are losing their attention. Our first experiment confirmed the
effectiveness of Mindless Attractor in a video-based learning context by
showing that it helped users refocus their attention without consuming their
conscious awareness. Moreover, through a comparison with a conventional
alerting approach, our second experiment further supported the efficacy of our
proposed mindless approach when integrated as an automated system with a
machine learning-based sensing module. Based on the results of the
experiments, we discussed implications for utilizing mindless interventions,
especially in tandem with machine learning-based sensing modules, and
envisioned future application scenarios. Our findings and discussion pave the
way for developing novel mindless interventions that can be harnessed in
human–AI symbiosis.
###### Acknowledgements.
This work is partially supported by JST ACT-X, Grant Number JPMJAX200R, Japan.
Several components of the sensing module used in this study were offered by
ACES Inc., Japan.
## References
* (1)
* Adams et al. (2015) Alexander Travis Adams, Jean Marcel dos Reis Costa, Malte F. Jung, and Tanzeem Choudhury. 2015\. Mindless computing: designing technologies to subtly influence behavior. In _Proceedings of the 2015 ACM International Joint Conference on Pervasive and Ubiquitous Computing_. ACM, New York, NY, 719–730. https://doi.org/10.1145/2750858.2805843
* Amershi et al. (2019) Saleema Amershi, Kori Inkpen, Jaime Teevan, Ruth Kikin-Gil, Eric Horvitz, Dan Weld, Mihaela Vorvoreanu, Adam Fourney, Besmira Nushi, Penny Collisson, Jina Suh, Shamsi Iqbal, and Paul N. Bennett. 2019. Guidelines for Human-AI Interaction. In _Proceedings of the 37th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 3. ACM, New York, NY, 1–13. https://doi.org/10.1145/3290605.3300233
* Arakawa et al. (2019) Riku Arakawa, Shinnnosuke Takamichi, and Hiroshi Saruwatari. 2019\. Implementation of DNN-based real-time voice conversion and its improvements by audio data augmentation and mask-shaped device. In _Proceedings of the 10th ISCA Speech Synthesis Workshop_. ISCA, Baixas, France, 93–98. https://doi.org/10.21437/SSW.2019-17
* Baker et al. (2010) Ryan Shaun J. D. Baker, Sidney K. D’Mello, Ma. Mercedes T. Rodrigo, and Arthur C. Graesser. 2010\. Better to be frustrated than bored: The incidence, persistence, and impact of learners’ cognitive–affective states during interactions with three different computer-based learning environments. _International Journal of Human-Computer Studies_ 68, 4 (2010), 223–241. https://doi.org/10.1016/j.ijhcs.2009.12.003
* Baturay (2015) Meltem Huri Baturay. 2015\. An Overview of the World of MOOCs. _Procedia – Social and Behavioral Sciences_ 174 (2015), 427–433. https://doi.org/10.1016/j.sbspro.2015.01.685
* Beauchemin et al. (2006) Maude Beauchemin, Louis De Beaumont, Phetsamone Vannasing, Aline Turcotte, Claudine Arcand, Pascal Belin, and Maryse Lassonde. 2006. Electrophysiological markers of voice familiarity. _The European Journal of Neuroscience_ 23, 11 (2006), 3081–—3086. https://doi.org/10.1111/j.1460-9568.2006.04856.x
* Bidwell and Fuchs (2011) Jonathan Bidwell and Henry Fuchs. 2011. _Classroom analytics: Measuring student engagement with automated gaze tracking_. Technical Report. Department of Computer, University of North Carolina at Chapel Hill, Chapel Hill, NC. 1–17 pages. https://doi.org/10.13140/RG.2.1.4865.6242
* Byers et al. (1989) James C. Byers, Alvah C. Bittner, and Susan G. Hill. 1989\. Traditional and raw task load index (TLX) correlations: Are paired comparisons necessary?. In _Proceedings of the 1989 Annual International Industrial Ergonomics and Safety Conference_. Taylor & Francis, Philadelphia, PA, 481–485.
* Cobus et al. (2018) Vanessa Cobus, Hannah Meyer, Swamy Ananthanarayan, Susanne Boll, and Wilko Heuten. 2018\. Towards reducing alarm fatigue: peripheral light pattern design for critical care alarms. In _Proceedings of the 10th Nordic Conference on Human-Computer Interaction_. ACM, New York, NY, 654–663. https://doi.org/10.1145/3240167.3240218
* Conde et al. (2015) Tatiana Conde, Óscar F. Gonçalves, and Ana P. Pinheiro. 2015\. Paying attention to my voice or yours: An ERP study with words. _Biological Psychology_ 111 (2015), 40–52. https://doi.org/10.1016/j.biopsycho.2015.07.014
* Coombs (2020) Crispin Coombs. 2020\. Will COVID-19 be the tipping point for the Intelligent Automation of work? A review of the debate and implications for research. _International Journal of Information Management_ 102182 (2020), 1–4. https://doi.org/10.1016/j.ijinfomgt.2020.102182
* Davis (1989) Fred D. Davis. 1989\. Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology. _MIS Quarterly_ 13, 3 (1989), 319–340. https://doi.org/10.2307/249008
* De-Arteaga et al. (2020) Maria De-Arteaga, Riccardo Fogliato, and Alexandra Chouldechova. 2020. A Case for Humans-in-the-Loop: Decisions in the Presence of Erroneous Algorithmic Scores. In _Proceedings of the 38th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 509. ACM, New York, NY, 1–12. https://doi.org/10.1145/3313831.3376638
* Deng et al. (2020) Jiankang Deng, Jia Guo, Evangelos Ververas, Irene Kotsia, and Stefanos Zafeiriou. 2020. RetinaFace: Single-Shot Multi-Level Face Localisation in the Wild. In _Proceeedings of the 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition_. IEEE, New York, NY, 5202–5211. https://doi.org/10.1109/CVPR42600.2020.00525
* Dietvorst et al. (2015) Berkeley J. Dietvorst, Joseph P. Simmons, and Cade Massey. 2015\. Algorithm aversion: People erroneously avoid algorithms after seeing them err. _Journal of Experimental Psychology: General_ 144, 1 (2015), 114–126. https://doi.org/10.1037/xge0000033
* D’Mello et al. (2012) Sidney D’Mello, Andrew Olney, Claire Williams, and Patrick Hays. 2012. Gaze tutor: A gaze-reactive intelligent tutoring system. _International Journal of Human-Computer Studies_ 70, 5 (2012), 377–398. https://doi.org/10.1016/j.ijhcs.2012.01.004
* Fang et al. (2017) Haoshu Fang, Shuqin Xie, Yu-Wing Tai, and Cewu Lu. 2017\. RMPE: Regional Multi-person Pose Estimation. In _Proceedings of the 2017 IEEE International Conference on Computer Vision_. IEEE, New York, NY, 2353–2362. https://doi.org/10.1109/ICCV.2017.256
* Fish et al. (1992) Robert S. Fish, Robert E. Kraut, Robert W. Root, and Ronald E. Rice. 1992. Evaluating Video as a Technology for Informal Communication. In _Procedings of the 10th ACM CHI Conference on Human Factors in Computing Systems_. ACM, New York, NY, 37–48. https://doi.org/10.1145/142750.142755
* Ghoddoosian et al. (2019) Reza Ghoddoosian, Marnim Galib, and Vassilis Athitsos. 2019\. A Realistic Dataset and Baseline Temporal Model for Early Drowsiness Detection. In _Proceedings of the 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops_. IEEE, New York, NY, 178–187. https://doi.org/10.1109/cvprw.2019.00027
* Goldschmidt (2020) Karen Goldschmidt. 2020\. The COVID-19 Pandemic: Technology use to Support the Wellbeing of Children. _Journal of pediatric nursing_ 53 (2020), 88–90. https://doi.org/10.1016/j.pedn.2020.04.013
* Guo et al. (2002) Linsong Guo, Qin Zhang, and Shuxia Han. 2002. Agricultural Machinery Safety Alert System Using Ultrasonic Sensors. _Journal of Agricultural Safety and Health_ 8, 4 (2002), 385–396. https://doi.org/10.13031/2013.10219
* Guo et al. (2014) Philip J. Guo, Juho Kim, and Rob Rubin. 2014. How video production affects student engagement: an empirical study of MOOC videos. In _Proceedings of the 1st ACM Conference on Learning @ Scale_. ACM, New York, NY, 41–50. https://doi.org/10.1145/2556325.2566239
* Hart and Staveland (1988) Sandra G. Hart and Lowell E. Staveland. 1988. Development of NASA-TLX (Task Load Index): Results of Empirical and Theoretical Research. _Advances in Psychology_ 52 (1988), 139–183. https://doi.org/10.1016/s0166-4115(08)62386-9
* Hutt et al. (2017) Stephen Hutt, Caitlin Mills, Nigel Bosch, Kristina Krasich, James R. Brockmole, and Sidney K. D’Mello. 2017. “Out of the Fr-Eye-ing Pan”: Towards Gaze-Based Models of Attention during Learning with Technology in the Classroom. In _Proceedings of the 25th ACM International Conference on User Modeling, Adaptation and Personalization_. ACM, New York, NY, 94–103. https://doi.org/10.1145/3079628.3079669
* International (2011) Toastmasters International. 2011\. Your Speaking Voice: Tips for Adding Strength and Authority to Your Speaking Voice. https://www.toastmasters.org/resources/your-speaking-voice. Accessed: August 19, 2019.
* Kayukawa et al. (2019) Seita Kayukawa, Keita Higuchi, João Guerreiro, Shigeo Morishima, Yoichi Sato, Kris Kitani, and Chieko Asakawa. 2019. BBeep: A Sonic Collision Avoidance System for Blind Travellers and Nearby Pedestrians. In _Proceedings of the 37th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 52. ACM, New York, NY, 1–12. https://doi.org/10.1145/3290605.3300282
* Kerres (2020) Michael Kerres. 2020\. Against All Odds: Education in Germany Coping with COVID-19. _Postdigital Science and Education_ 2, 3 (2020), 690–694. https://doi.org/10.1007/s42438-020-00130-7
* Kodama (2020) Mitsuru Kodama. 2020\. Digitally transforming work styles in an era of infectious disease. _International Journal of Information Management_ 102172 (2020), 1–6. https://doi.org/10.1016/j.ijinfomgt.2020.102172
* Kuzminykh and Rintel (2020a) Anastasia Kuzminykh and Sean Rintel. 2020a. Classification of Functional Attention in Video Meetings. In _Proceedings of the 38th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 419. ACM, New York, NY, 1–13. https://doi.org/10.1145/3313831.3376546
* Kuzminykh and Rintel (2020b) Anastasia Kuzminykh and Sean Rintel. 2020b. Low Engagement As a Deliberate Practice of Remote Participants in Video Meetings. In _Extended Abstracts of the 38th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 321. ACM, New York, NY, 1–9. https://doi.org/10.1145/3334480.3383080
* Lee and Starner (2010) Seungyon Claire Lee and Thad Starner. 2010. BuzzWear: alert perception in wearable tactile displays on the wrist. In _Proceedings of the 28th ACM CHI International Conference on Human Factors in Computing Systems_. ACM, New York, NY, 433–442. https://doi.org/10.1145/1753326.1753392
* Martin (2012) Fred G. Martin. 2012\. Will massive open online courses change how we teach? _Commununications of the ACM_ 55, 8 (2012), 26–28. https://doi.org/10.1145/2240236.2240246
* Nettelhorst and Brannon (2012) Stephen C. Nettelhorst and Laura A. Brannon. 2012. The effect of advertisement choice, sex, and need for cognition on attention. _Computers in Human Behavior_ 28, 4 (2012), 1315–1320. https://doi.org/10.1016/j.chb.2012.02.015
* Oeppen et al. (2020) Rachel S. Oeppen, Graham Shaw, and Peter A. Brennan. 2020\. Human factors recognition at virtual meetings and video conferencing: how to get the best performance from yourself and others. _British Journal of Oral and Maxillofacial Surgery_ 58, 6 (2020), 643–646. https://doi.org/10.1016/j.bjoms.2020.04.046
* Poyatos (1993) Fernando Poyatos. 1993\. _Paralanguage: A Linguistic and Interdisciplinary Approach to Interactive Speech and Sounds_. John Benjamins Publishing Company, Amsterdam, Netherlands.
* Saint-Lot et al. (2020) Julie Saint-Lot, Jean-Paul Imbert, and Frédéric Dehais. 2020\. Red Alert: A Cognitive Countermeasure to Mitigate Attentional Tunneling. In _Proceedings of the 38th ACM CHI Conference on Human Factors in Computing Systems_ , Vol. 580. ACM, New York, NY, 1–6. https://doi.org/10.1145/3313831.3376709
* Samrose et al. (2017) Samiha Samrose, Ru Zhao, Jeffery White, Vivian Li, Luis Nova, Yichen Lu, Mohammad Rafayet Ali, and Mohammed E. Hoque. 2017. CoCo: Collaboration Coach for Understanding Team Dynamics during Video Conferencing. _Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies_ 1, 4 (2017), 160:1–160:24. https://doi.org/10.1145/3161186
* Schiavo et al. (2014) Gianluca Schiavo, Alessandro Cappelletti, Eleonora Mencarini, Oliviero Stock, and Massimo Zancanaro. 2014. Overt or subtlefi: supporting group conversations with automatically targeted directives. In _Proceedings of the 19th ACM International Conference on Intelligent User Interfaces_. ACM, New York, NY, 225–234. https://doi.org/10.1145/2557500.2557507
* Sharma et al. (2016) Kshitij Sharma, Hamed S. Alavi, Patrick Jermann, and Pierre Dillenbourg. 2016. A gaze-based learning analytics model: in-video visual feedback to improve learner’s attention in MOOCs. In _Proceedings of the 6th ACM International Conference on Learning Analytics & Knowledge_. ACM, New York, NY, 417–421. https://doi.org/10.1145/2883851.2883902
* Stothart et al. (2015) Cary Stothart, Ainsley Mitchum, and Courtney Yehnert. 2015\. The attentional cost of receiving a cell phone notification. _Journal of Experimental Psychology: Human Perception and Performance_ 41, 4 (2015), 893–897. https://doi.org/10.1037/xhp0000100
* Sullivan and Horowitz (1983) Joseph W. Sullivan and Frances Degen Horowitz. 1983. The effects of intonation on infant attention: the role of the rising intonation contour. _Journal of Child Language_ 10, 3 (1983), 521–534. https://doi.org/10.1017/s0305000900005341
* Thomas and Jayagopi (2017) Chinchu Thomas and Dinesh Babu Jayagopi. 2017. Predicting student engagement in classrooms using facial behavioral cues. In _Proceedings of the 1st ACM SIGCHI International Workshop on Multimodal Interaction for Education_. ACM, New York, NY, 33–40. https://doi.org/10.1145/3139513.3139514
* Toda et al. (2012) Tomoki Toda, Takashi Muramatsu, and Hideki Banno. 2012\. Implementation of Computationally Efficient Real-Time Voice Conversion. In _Proceedings of the 13th Annual Conference of the International Speech Communication Association_. ISCA, Baixas, France, 94–97.
* Trager (1958) George L. Trager. 1958\. Paralanguage: A first approximation. _Studies in Linguistics_ 13 (1958), 1–12.
* Veliyath et al. (2019) Narayanan Veliyath, Pradipta De, Andrew A. Allen, Charles B. Hodges, and Aniruddha Mitra. 2019. Modeling Students’ Attention in the Classroom using Eyetrackers. In _Proceedings of the 2019 ACM Southeast Regional Conference_. ACM, New York, NY, 2–9. https://doi.org/10.1145/3299815.3314424
* Venkatesh et al. (2003) Viswanath Venkatesh, Michael G. Morris, Gordon B. Davis, and Fred D. Davis. 2003. User Acceptance of Information Technology: Toward a Unified View. _MIS Quarterly_ 27, 3 (2003), 425–478. https://doi.org/10.2307/30036540
* Wansink and van Ittersum (2007) Brian Wansink and Koert van Ittersum. 2007. Portion Size Me: Downsizing Our Consumption Norms. _Journal of the American Dietetic Association_ 107, 7 (2007), 1103–1106. https://doi.org/10.1016/j.jada.2007.05.019
* Weiser and Brown (1995) Mark Weiser and John Seely Brown. 1995. _Designing calm technology_. Technical Report. Xerox PARC, Palo Alto, CA. 1–5 pages.
* Weiser and Brown (1997) Mark Weiser and John Seely Brown. 1997. The coming age of calm technology. In _Beyond calculation_. Springer, New York, NY, 75–85. https://doi.org/10.1007/978-1-4612-0685-9_6
* Whittaker (1995) Steve Whittaker. 1995\. Rethinking video as a technology for interpersonal communications: theory and design implications. _International Journal of Human-Computer Studies_ 42, 5 (1995), 501–529. https://doi.org/10.1006/ijhc.1995.1022
* Xiao and Wang (2016) Xiang Xiao and Jingtao Wang. 2016. Context and cognitive state triggered interventions for mobile MOOC learning. In _Proceedings of the 18th ACM International Conference on Multimodal Interaction6_. ACM, New York, NY, 378–385. https://doi.org/10.1145/2993148.2993177
* Xu (2005) Yi Xu. 2005. Speech melody as articulatorily implemented communicative functions. _Speech Communication_ 46, 3-4 (2005), 220–251. https://doi.org/10.1016/j.specom.2005.02.014
* Yu et al. (2017) Kun Yu, Shlomo Berkovsky, Ronnie Taib, Dan Conway, Jianlong Zhou, and Fang Chen. 2017\. User Trust Dynamics: An Investigation Driven by Differences in System Performance. In _Proceedings of the 22nd International Conference on Intelligent User Interfaces_. ACM, New York, NY, 307–317. https://doi.org/10.1145/3025171.3025219
* Zaletelj and Kosir (2017) Janez Zaletelj and Andrej Kosir. 2017. Predicting students’ attention in the classroom from Kinect facial and body features. _EURASIP Journal on Image and Video Processing_ 2017 (2017), 80\. https://doi.org/10.1186/s13640-017-0228-8
* Zatorre and Gandour (2007) Robert J. Zatorre and Jackson T. Gandour. 2007. Neural specializations for speech and pitch: moving beyond the dichotomies. _Philosophical Transactions of the Royal Society B: Biological Sciences_ 363, 1493 (2007), 1087–1104. https://doi.org/10.1098/rstb.2007.2161
* Zhou et al. (2017) Jianlong Zhou, Syed Z. Arshad, Simon Luo, and Fang Chen. 2017\. Effects of Uncertainty and Cognitive Load on User Trust in Predictive Decision Making. In _Proceeding of the 16th IFIP TC 13 International Conference on Human-Computer Interaction_. Springer, Cham, Switzerland, 23–39. https://doi.org/10.1007/978-3-319-68059-0_2
|
11institutetext: Kassel University, Germany
# Commutative Event Sourcing vs. Triple Graph Grammars
Sebastian Copei Albert Zündorf
###### Abstract
This paper proposes Commutative Event Sourcing as a simple and reliable
mechanism for model synchronisation, bidirectional model to model
transformations, incremental updates, and collaborative editing. Commutative
Event Sourcing is a restricted form of a Triple Graph Grammar where the rules
or editing commands are either overwriting or commutative. This restriction
gets rid of a lot of Triple Graph Grammar complexity and it becomes possible
to implement model synchronisation manually. Thus, you are not restricted to
Java as your programming language and you do not need to use a proprietary
library, framework, or tool. You do not even have to dig into graph grammar
theory.
###### Keywords:
Event Sourcing Triple Graph Grammars Bidirectional Model Transformations.
## 1 Introduction
Whenever you have two tools that each deploy their own meta model but
interchange related or overlapping data you face the problem of model
synchronisation: whenever some common data is modified in one tool, you want
to update the corresponding data in the other tool. Note, when both models use
the same meta model, the problem of model synchronisation becomes closely
related to model versioning and to the merging of concurrent model changes and
to collaborative editing.
In the area of bidirectional (BX) transformations there are various approaches
attacking the problem of model synchronisation cf. [5, 1]. Among the various
approaches, we consider Triple Graph Grammars (TGGs) [16] to be the most
mature and most practical solution with a lot of tool support [12]. Recent
development in TGG tools provide support for incremental model synchronisation
[13], i.e. the effort for model synchronisation is proportional to the model
change performed. In [9] Fritsche et al present recent advances in achieving
incremental model synchronisation.
While TGGs have a lot of mature tool support and a very sound theory, it is
quite complex to implement a TGG tool and to apply a TGG approach within your
own application. You will probably fail to implement your own approach and you
will need to use some existing tool that requires you to adopt a lot of tool
specific prerequisites (e.g. the Eclipse Modeling Framework [17]) and to learn
a lot about (triple) graph grammar theory in order to write down appropriate
TGG rules.
This paper proposes Commutative Event Sourcing (CES) as an alternative
approach to model synchronisation. As we will discuss, CES is a restricted
form of a TGG where the order of rule application is commutative. This
facilitates the implementation of CES tremendously, such that you may adopt
our approach without the need of using a proprietary tool and without graph
grammar theory. You just follow some design patterns that we propose and
implement your own incremental model synchronisation manually.
## 2 Triple Graph Grammars (TGGs)
This section revisits the work of Fritsche et al [9]. The running example of
[9] and of this paper is the synchronisation of Java package and Java class
models with JavaDoc folders and files. Figure 1 shows the class diagrams for
the two models as used in our implementation of this example.
Figure 1: Classes for Java and JavaDoc structures plus edit commands
The left of Figure 1 shows the class model for the Java packages tool.
Basically there is the class JavaPackage that may have multiple subPackages.
In addition, a JavaPackage may have many classes of type JavaClass. Our class
model extends the example from [9] with id attributes and with a vTag
attribute. The latter will be used to discuss some editing or merge conflicts
that are not handled by [9]. The id attributes are used for referencing across
tools. The id attributes are the first means that we introduce in order to
facilitate the model synchronisation task.
The left of Figure 1 also shows the ModelCommand classes HaveRoot,
HaveSubUnit, and HaveLeaf. These classes are part of our CES approach and will
be discussed in Section 4. Our approach uses the Command pattern from [10].
Thus, we have command classes that provides methods for command execution and
command objects that protocol each command execution and its actual
parameters. When we serialize these command objects and exchange them with
other applications we frequently call them events. Thus, if we receive an
event and deserialize it, it becomes a command object that then may be
executed. In the following we will use command and event in a mixed way. The
right of Figure 1 shows the classes Folder and DocFile with the associations
subFolders and files that model the JavaDoc structures. Again we have the same
ModelCommand classes as for Java packages.
Figure 2: Triple Graph Grammar (like) rules for Java and JavaDoc structures
Figure 2 shows a slight modification of the Triple Graph Grammar rules used in
[9] to solve the model synchronisation problem for our Java to JavaDoc
example. There are a HaveRoot, a HaveSubUnit, and a HaveLeaf rule. Each rule
shows its name and its parameters in a hexagon in the middle of the rule. Each
rule has a left and a right subrule. Each subrule specifies three possible
operations: run, remove, and parse. We will walk through these operation types
one by one.
The run operation for a subrule tries to create the specified situation. Thus,
all green parts of the subrule are going to be created, all black parts are
required to already exist, and all blue parts must not exist (or may need to
be removed by the run operation). Thus, the left subrule of the HaveRoot rule
describes that on execution a JavaPackage object p shall be created in the
Java model. The id attribute of p is copied from the id parameter of the rule.
The blue parts of the HaveRoot rule require that there must be no JavaPackage
pp attached to p via a pPack link. Thus we shall reset this link. Our manual
implementation of this subrule is shown in Listing 1 See [3] for a complete
reference.
⬇
1package JavaPackages;
2public class HaveRoot extends ModelCommand {
3 @Override
4 public Object run(JavaPackagesEditor editor) {
5 JavaPackage p = (JavaPackage) editor
6 .getOrCreate(JavaPackage.class, this.getId());
7 p.setPPack(null);
8 return p;
9 }
10 …
Listing 1: Manual implementation of HaveRoot rule for Java packages
The run method of our HaveRoot command uses an editor to getOrCreate the
desired JavaPackage object. Our editor maintains a hash table storing id \-
object pairs, cf. Section 6. This hash table is e.g. used in the HaveSubUnit
command to look up the required pp JavaPackage, cf. method getObjectFrame in
Line 8 of Listing 2 and Section 6. Note, in the left subrule of the
HaveSubUnit rule of Figure 2 the upper JavaPackage pp is shown in black color.
This means the execution of the subrule requires that pp exists as context for
the creation of the sub package p. The green pPack link requires that p needs
to be connected to pp via an pPack link (which simultaneously creates the
subPackages link in the reverse direction).
⬇
1package JavaPackages;
2public class HaveSubUnit extends ModelCommand {
3 @Override
4 public Object run(JavaPackagesEditor editor) {
5 JavaPackage p = (JavaPackage) editor
6 .getOrCreate(JavaPackage.class, this.getId());
7 JavaPackage pp = (JavaPackage) editor
8 .getObjectFrame(JavaPackage.class, this.parent);
9 p.setPPack(pp);
10 return p;
11 }
12 …
Listing 2: Manual implementation of HaveSubUnit rule for Java packages
For completeness, Listing 3 shows the HaveLeaf command.
⬇
1package JavaPackages;
2public class HaveLeaf extends ModelCommand {
3 @Override
4 public Object run(JavaPackagesEditor editor)
5 {
6 JavaClass c = (JavaClass) editor
7 .getOrCreate(JavaClass.class, this.getId());
8 JavaPackage p = (JavaPackage) editor
9 .getObjectFrame(JavaPackage.class, this.parent);
10 c.setPack(p);
11 c.setVTag(this.vTag);
12 return c;
13 }
14 …
Listing 3: Manual implementation of HaveLeaf rule for Java packages
In Listing 4 we create a number of command objects and initialize their
parameters, appropriately. Then we ask an appropriate editor to execute the
commands. The editor adds the commands to its command store and then calls
their run method, cf. Section 4. This results in the object structure shown on
the left of Figure 3.
⬇
1package javaPackagesToJavaDoc;
2…
3public class TestPackageToDoc {
4 private void startSituation(JavaPackagesEditor editor) {
5 ModelCommand cmd = new HaveRoot().setId(”org”);
6 editor.execute(cmd);
7 cmd = new HaveSubUnit().setParent(”org”).setId(”fulib”);
8 editor.execute(cmd);
9 cmd = new HaveSubUnit().setParent(”fulib”).setId(”serv”);
10 editor.execute(cmd);
11 cmd = new HaveLeaf().setParent(”serv”)
12 .setVTag(”1.0”).setId(”Editor”);
13 editor.execute(cmd);
14 }
15 …
Listing 4: Invoking triple rules or commands Figure 3: Objects for Java and
JavaDoc structures plus commands (colors cf. Section 4)
Listing 5 shows our manual implementation of the HaveRoot command for JavaDoc
structures. This implements the right subrule of the HaveRoot triple rule of
Figure 2. Lines 5 to 7 of Listing 5 are quite similar to the corresponding
JavaPackages command. They just create a Folder instead of a JavaPackage.
Lines 8 to 13 of Listing 5 remove a potentially existing DocFile d. Such an
object d might have been created by previous command executions. The blue
parts of the right subrule of our HaveRoot triple rule require that after rule
execution such a DocFile must not (no longer) exist. Listing 6 and Listing 7
show the manual implementation of the other two JavaDoc subrules.
⬇
1package JavaDoc;
2public class HaveRoot extends ModelCommand {
3 @Override
4 public Object run(JavaDocEditor editor) {
5 Folder f = (Folder) editor
6 .getOrCreate(Folder.class, this.getId());
7 f.setPFolder(null);
8 String docId = this.getId() + ”.Doc”;
9 DocFile d = f.getFromFiles(docId);
10 if (d != null) {
11 editor.removeModelObject(d.getId());
12 f.withoutFiles(d);
13 }
14 return f;
15 }
16 …
Listing 5: Manual implementation of HaveRoot rule for JavaDoc
⬇
1package JavaDoc;
2public class HaveSubUnit extends ModelCommand {
3 @Override
4 public Object run(JavaDocEditor editor) {
5 Folder f = (Folder) editor
6 .getOrCreate(Folder.class, this.getId());
7 Folder pf = (Folder) editor.
8 getObjectFrame(Folder.class, parent);
9 f.setPFolder(pf);
10 String docId = this.getId() + ”.Doc”;
11 DocFile d = (DocFile) editor
12 .getOrCreate(DocFile.class, docId);
13 d.setContent(this.getId() + ”␣docu”);
14 f.withFiles(d);
15 return f;
16 }
17 …
Listing 6: Manual implementation of HaveSubUnit rule for JavaDoc
⬇
1package JavaDoc;
2public class HaveLeaf extends ModelCommand {
3 @Override
4 public Object run(JavaDocEditor editor) {
5 DocFile d = (DocFile) editor
6 .getOrCreate(DocFile.class, this.getId());
7 Folder f = (Folder) editor
8 .getObjectFrame(Folder.class, parent);
9 d.setFolder(f);
10 d.setVersion(vTag);
11 return d;
12 }
13 …
Listing 7: Manual implementation of HaveLeaf rule for JavaDoc
We might invoke the JavaDoc commands as we have done this for the JavaPackages
in Listing 4. Alternatively, in Line 12 of Listing 8 we lookup the list of
commands that our javaPackagesEditor has collected while building the start
situation. Then Line 13 uses a simple Yaml encoder to serialize these commands
into a string in Yaml format. You may use any JSON based serialization,
either. The serialization task is pretty simple, as our commands use string
based parameters only and have no references to other objects. Finally, Line
14 calls method loadYaml on the javaDocEditor. Method loadYaml turns the
passed string into JavaDoc command objects and executes these. The result is
shown on the right of Figure 3.
⬇
1package javaPackagesToJavaDoc;
2…
3public class TestPackageToDoc {
4 @Test
5 public void testFirstForwardExample()
6 {
7 JavaPackagesEditor javaPackagesEditor =
8 new JavaPackagesEditor();
9 startSituation(javaPackagesEditor);
10 JavaDocEditor javaDocEditor = new JavaDocEditor();
11 Collection commands = javaPackagesEditor
12 .getActiveCommands().values();
13 String yaml = Yaml.encode(commands);
14 javaDocEditor.loadYaml(yaml);
15 …
16 }
17 …
Listing 8: Invoking triple rules or commands
In a Triple Graph Grammar (TGG) tool based approach, you would just provide
the triple rules shown in Figure 2. Then the corresponding commands would be
derived using either a code generator or a TGG interpreter. See [3] for an
example of such an interpreter. Using a TGG tool, you get not only the forward
execution of rules but also remove and parse functionality. Removing the
effects of a TGG subrule basically requires to remove all model parts that
have been created for green rule elements on the forward execution. In a
manual implementation, you have to implement the remove step yourself and it
must be consistent to the run operation. Listings 9, 10, and 11 show our
manual implementation of the remove operations for the JavaPackages commands.
For the JavaDoc commands see [3].
⬇
1package JavaPackages;
2public class HaveRoot extends ModelCommand {
3 …
4 @Override
5 public void remove(JavaPackagesEditor editor)
6 {
7 editor.removeModelObject(this.getId());
8 }
9 …
Listing 9: Manual implementation of HaveRoot.remove() for Java packages
⬇
1package JavaPackages;
2public class HaveSubUnit extends ModelCommand {
3 …
4 @Override
5 public void remove(JavaPackagesEditor editor)
6 {
7 JavaPackage p = (JavaPackage) editor
8 .removeModelObject(this.getId());
9 p.setPPack(null);
10 }
11 …
Listing 10: Manual implementation of HaveSubUnit.remove() for Java packages
⬇
1package JavaPackages;
2public class HaveLeaf extends ModelCommand {
3 …
4 @Override
5 public void remove(JavaPackagesEditor editor)
6 {
7 JavaClass c = (JavaClass) editor
8 .removeModelObject(this.getId());
9 c.setPack(null);
10 }
11 …
Listing 11: Manual implementation of HaveLeaf.remove() for Java packages
Usually, TGG rule applications depend on each other. For example if we call
remove on the JavaPackages HaveRoot command of our example, this would remove
the org JavaPackage from our model, cf. Figure 3. This would leave the fulib
JavaPackage without a parent. Thus the HaveSubUnit TGG rule does no longer
match for the fulib object. This is important as on model synchronisation the
right sub rule of HaveSubUnit creates the fulibDoc DocFile which is no longer
valid. To repair this, a standard TGG approach has to remove dependant rule
applications whenever their context becomes invalid. Thus on remove of the
HaveRoot command, a standard repair step would also remove the HaveSubUnit
command for the fulib JavaPackage and in turn remove the commands for the serv
and for the editor objects. Via model synchronisation all JavaDoc objects will
be removed, either. To keep the lower model parts, one would apply a HaveRoot
command on fulib and rerun the HaveSubUnit and HaveLeaf commands on serv and
editor. Formally, the whole model would be deleted and reconstructed. In [9]
this is called a cascading delete and [9] proposes sophisticated theory and
means to avoid this cascading delete via so called short-cut-repair rules. In
our manual implementation it would suffice to run a HaveRoot command on fulib
in order to repair the situation. This is achieved as Line 7 of Listing 1 and
Lines 7 to 13 of Listing 5 carefully remove all model parts that correspond to
the blue parts of the HaveRoot TGG rule, cf. Figure 2, i.e. all model parts
that might stem from a previous application of a HaveSubUnit rule. In a manual
implementation you have to spot the overlap of the HaveRoot and the
HaveSubUnit rules yourself and you have to design theses rules and their
manual implementation very carefully in order to circumvent cascading deletes.
Commutative Event Sourcing will help you to achieve this as discussed in
Sections 3 and 4. [9] does this for you automatically which is a really great
job.
Whenever you edit a model directly without using the TGG rules or the
corresponding commands and you want to do a new model synchronisation, you
need to parse the changed model in order to identify which TGG rule
applications are now valid. Again TGG tools do this parsing for you. Basically
all green and black parts of a TGG subrule must be matched and the blue parts
must not be there. In general TGG parsing has to deal with rule dependencies,
too. Usually, TGG parsing needs to find all applications of so-called ”root”
rules (that do not depend on other rules), first. Then TGG parsing, inspects
the surroundings of ”root” rule applications and tries to find applications of
rules that use the ”root” rules in their context. In turn, you apply rules
where the context has become available. In our example this results in some
kind of top-down parsing starting with the org JavaPackage and descending to
subPackages and classes, recursively. In general, TGG parsing may be even more
complicated, cf. [16]. Again Commutative Event Sourcing allows us to
facilitate parsing considerably as we will discuss in Section 7.
Thus in our manual implementation we ignore the order of rule applications for
now. For a single rule, parsing is relatively simple. Listing 12 shows the
manual implementation of the parse method for our HaveRoot command for
JavaPackages. Our editor calls the parse methods of our commands when
appropriate, cf. Section 7. On such a call, the editor passes an object that
may have been modified and needs parsing as parameter. Thus Line 6 of the
parse method of Listing 12 first ensures that the current object is a
JavaPackage. Similarly, Line 10 ensures that there is no pPack. If the current
package has no sub packages and contains no JavaClasses, we consider it
garbage. Therefore, Line 15 to 17 return a RemoveCommand with the
corresponding object id. Without such a garbage collection mechanism, the
users would have to invoke RemoveCommands manually in order to get rid of
model objects. Finally Line 20 to 22 create a HaveRoot command and retrieve
its id parameter from the model and return it.
⬇
1package JavaPackages;
2public class HaveRoot extends ModelCommand {
3 …
4 @Override
5 public ModelCommand parse(Object currentObject) {
6 if (! (currentObject instanceof JavaPackage)) {
7 return null;
8 }
9 JavaPackage currentPackage = (JavaPackage) currentObject;
10 if (currentPackage.getPPack() != null) {
11 return null;
12 }
13 if (currentPackage.getClasses().isEmpty()
14 && currentPackage.getSubPackages().isEmpty()) {
15 ModelCommand modelCommand = new RemoveCommand()
16 .setId(currentPackage.getId());
17 return modelCommand;
18 }
19 // yes its me
20 ModelCommand modelCommand = new HaveRoot()
21 .setId(currentPackage.getId());
22 return modelCommand;
23 }
24 …
Listing 12: Manual implementation of the parse step for HaveRoot for Java
packages
Listing 13 shows the parse method of the HaveSubUnit command for JavaPackages.
Note, that this method is slightly simpler as the garbage collection is done
by the HaveRoot command. You find the parse method of the HaveLeaf command for
JavaPackages in [3]. We leave the implementation of the parse methods for the
JavaDoc commands as an exercise for the interested reader.
⬇
1package JavaPackages;
2public class HaveSubUnit extends ModelCommand {
3 …
4 @Override
5 public ModelCommand parse(Object currentObject) {
6 if ( ! (currentObject instanceof JavaPackage)) {
7 return null;
8 }
9 JavaPackage currentPackage = (JavaPackage) currentObject;
10 if (currentPackage.getPPack() == null) {
11 return null;
12 }
13 ModelCommand modelCommand = new HaveSubUnit()
14 .setParent(currentPackage.getPPack().getId())
15 .setId(currentPackage.getId());
16 return modelCommand;
17 }
18 …
Listing 13: Manual implementation of the parse step for HaveSubUnit for Java
packages
## 3 Commutative Event Sourcing (CES) Theory
Event Sourcing has been proposed by [7] and [20] as a means for communication
between multiple domains or (micro) services. Event Sourcing may also be used
as mechanism for model persistence. Basically, a program or service logs
relevant operations as events and these events are then transferred to other
programs or services that react with appropriate operations on their site. In
order to use this idea for model synchronisation, we just log all editing
operations / commands on one model and then send these editing events to some
other model and perform similar changes there. This is clearly related to the
Triple Graph Grammar approach discussed in Section 2. However, Event Sourcing
has some additional requirements that ultimately lead us to Commutative Event
Sourcing.
Figure 4: Collaborative Editing
To connect multiple applications, Event Sourcing is frequently based on some
message broker mechanism. Depending on the quality of service that your
message broker provides, messages may be lost or received in wrong order and
messages may be received multiple times. For example, there may be a (HaveRoot
org) event and a (HaveSubUnit fulib org) event and for some reasons you
receive only the latter, cf. Figure 4. In [15] we deal with this problem by
adding time stamps to all events / commands and by extending each event with
the time stamp of its predecessor. Thus if you receive event (HaveSubUnit
fulib org 13:02 13:01) which has been raised at 13:02 and which has a
predecessor command that has been raised at 13:01 and you have not yet
received the 13:01 command, then you postpone the execution of the 13:02 event
and ask for re-submission of the 13:01 event, cf. Figure 4. Time stamps also
allow to detect duplicated receipt of the same event. In addition, if there
are two independent tools (e.g. the editors of Alice and Bob) that raise event
(HaveLeaf Editor fulib 1.0 13:10 13:02) and (HaveLeaf Editor fulib 1.1 13:11
13:02) you have a merge conflict and the time stamps allow you to resolve such
merge conflicts, deterministically.
While time stamps are a good idea, in [15] postponing command execution until
all predecessor commands have arrived was quite tricky: you have to detect the
missing of a predecessor, you have to ask for resubmission you have to wait
for the predecessor to arrive, there may be a cascade of pre-predecessors you
also have to ask and to wait for, finally you apply the commands in their
correct order. Well, unless there is collaborative editing of multiple
editors: with collaborative editing, when you receive a command with time
stamp 13:30 (and predecessor 13:11) and you already got the 13:11 event, there
still might be e.g. an event with time stamp 13:23 (and also predecessor
13:11) that just did not reach you, yet. Thus, to execute collaborative
commands in a correct timely order opens a new can of worms.
To overcome the problem of rule ordering, we want our commands to be executed
in any order, i.e. to become commutative. Thus, we want to get rid of TGG rule
dependencies. One reason for dependencies between TGG rule applications is
that rule execution requires that some context (black) parts must already
exist in order to connect new elements to their context. For example, a
(HaveLeaf Editor serv) command for JavaPackages needs to connect the new
Editor JavaClass to an (existing) serv JavaPackage via a pack - classes link,
cf. Figure 2 and Listing 3. In our approach we are able to execute the
HaveLeaf Editor serv command before the serv JavaPackage is created. We
achieve this by using ids for model objects and by having a hash table of all
model objects and by using getOrCreate and getObjectFrame operations that do a
hash table lookup for the required object (id) and create the object if it is
missing, cf. Line 9 of Listing 3. The details of the getOrCreate and
getObjectFrame mechanism are explained in Section 7. Thus, our (HaveLeaf
Editor serv) command creates a context object with id serv and creates the
pack link to it. When we execute the (HaveSubUnit serv fulib) command later,
it uses getOrCreate(serv) to retrieve the serv JavaPackage that has already
been created by our HaveLeaf command and to connect the serv JavaPackage to
its pPack fulib. The fulib JavaPackage is again retrieved (or created) via
getObjectFrame.
This getOrCreate mechanism is inspired by QVT Relations [11] and [14], however
QVT Relations allows to combine any attributes that may be used as keys while
we restrict this to just ids, in order to facilitate a manual implementation.
In addition, QVT Relations still deploys a hierarchy of rules which requires
considerable implementation effort for the handling of rule dependencies.
Due to our getOrCreate, mechanism together with some additional rule
restrictions that are discussed in Section 4, in our approach it is possible
to execute commands in any order, i.e. our commands are commutative. Having
commutative commands gets rid of the effort for command ordering. And it
facilitates parsing, cf. Section 7.
There is one additional problem with general event sourcing: usually your
event store grows over time. For example if you have a (HaveLeaf Editor serv
1.1) command that creates an Editor object and assigns 1.1 to its vTag and
than you have a new (HaveLeaf Editor serv 1.2) command that just changes the
vTag to 1.2 and later on you change vTag again and again. Now your event store
may contain a large number of HaveLeaf events that only differ in their vTag
parameter. Actually, it would suffice to keep only the latest HaveLeaf command
in order to recreate the final model or to synchronize with some other model.
In our approach, we want a simple mechanism to identify events that overwrite
each other in order to be able to restrict the size of our event store to be
proportional to the size of our model. Therefore, our implementation uses
command ids and in our implementation two commands that have the same id
parameter must overwrite each other’s effects such that it suffices to keep
one event per command id in our store, cf. Section 4 and Listing 14.
Next, we provide some theory that specifies the requirements for Commutative
Event Sourcing and compressed event histories, formally. Section 4 then shows
simple patterns that achieve a sound implementation of Commutative Event
Sourcing. A previous version of our theory has been published in [2]. However,
this paper restructures our theory in large parts and it even enriches our
formal requirements in order to further facilitate the implementation of a
Commutative Event Sourcing application.
First let us set up some basic notations: like [18] we use capital letters
such as $M$, $N$ for metamodels i.e. for sets of models (that adhere to a
common class diagram). $\emptyset$ denotes an empty model. We denote events
with $e$ and the set of all possible events with $E$. An event
$e=(t,i,x_{1},...,x_{n})$ has an event type $t\in T$, an event identifier
$i\in\mathbb{CHAR^{*}}$ (i.e. some string), and a number of parameters
$x_{i}\in\mathbb{R}\,\cup\,\mathbb{CHAR^{*}}$, i.e. parameters are arbitrary
numbers or strings. We will also use series of events
$\overline{e}=(e_{1},...,e_{n})$ and denote by $\overline{E}$ the set of all
possible event series with events from $E$. Similarly we use sets of events
$\tilde{e}=\\{e_{1},...,e_{n}\\}\in\mathcal{P}(E)$.
Events may be applied to (or synchronized with) models via the function
$\textit{apply}(e,m)$, which generates a possibly new model $m^{\prime}$.
Furthermore, we define the application of an event series
$\overline{e}=(e_{1},\ldots,e_{n})$ to a model $m$ as
$\textit{apply}(\overline{e},m)=\textit{apply}(e_{n},\ldots,\textit{apply}(e_{2},\textit{apply}(e_{1},m))\ldots)$.
Similarly, the application of a set of events $\tilde{e}$ is defined as the
applications of all its elements in some order.
Now, we want a simple mechanism that allows us to restrict the size of our
command histories. Therefore, Definition 2 requires that events with the same
id overwrite each others effects, i.e. one renders the other ineffective. This
allows us to maintain our commands within a hash table and if we run a new
command with an already used id, Definition 2 allows us to overwrite the old
hash table entry in our command store with the new command.
###### Definition 1
(effective events) An event $e_{p}$ is called ineffective within an event
series $\overline{e}$ at position $p$ iff
$\textit{apply}(\overline{e}\setminus
e_{p},\emptyset)=\textit{apply}(\overline{e},\emptyset)$.
We call $e_{p}$ effective, otherwise.
We write $\tilde{f}=\textit{effective}(\tilde{e})$ to denote the series
$\tilde{f}$ that is derived from $\tilde{e}$ by removing all ineffective
events.
###### Definition 2
(overwriting) We call events $e_{1}=(t_{1},i_{1},...)$ and
$e_{2}=(t_{2},i_{2},...)$ with identifier $i_{1}=i_{2}$ overwriting within an
event series $\overline{e}$, if the earlier event is ineffective in
$\overline{e}$.
Next we want to get rid of command dependencies, i.e. we want to be able to
execute a command as soon as we receive it without waiting for other commands
to establish a required context. Similarly, we would like to store our
(unordered) command hash table persistently and to reload and rerun all
commands on tool start up without bothering with command order.
###### Definition 3
(commutativity) We call events commutative, if for all models $m\in M$ and all
events $e_{1},e_{2}\in E$ holds that
$\textit{apply}(e_{2},\textit{apply}(e_{1},m))=\textit{apply}(e_{1},\textit{apply}(e_{2},m))$
Definition 3 just states that command execution must be commutative. Section 4
will show how to achieve this in your implementation.
###### Definition 4
(active event sets $\tilde{e}$)
For any event series $\overline{e}$ where for all $e_{1},e_{2}$ in
$\textit{effective}(\overline{e})$ holds
(1) $e_{1}$ and $e_{2}$ have distinct identifiers, (i.e. all events in
$\overline{e}$ are overwriting) and
(2) $e_{1}$ and $e_{2}$ are commutative
we define the set of active events $\tilde{e}$ = { $e$ in
$\textit{effective}(\overline{e})$ }.
We write $\tilde{e}=\textit{activeSet}(\overline{e})$ to denote the set of
active events $\tilde{e}$ derived from $\textit{effective}(\overline{e})$.
We are now ready to define $M_{\texttt{CES}}$ the set of all models that may
be created via Commutative Event Sourcing:
###### Definition 5
($M_{\textit{CES}}$) A model $m=\textit{apply}(\overline{e},\emptyset)$ is
supporting Commutative Event Sourcing if and only if
for all event series $\overline{e}_{1}$ and $\overline{e}_{2}$ with
(1) $m=\textit{apply}(\overline{e}_{1},\emptyset)$ and
$m=\textit{apply}(\overline{e}_{2},\emptyset)$ and
(2) all events with the same identifier in $\overline{e}_{1}$ and
$\overline{e}_{2}$ are overwriting and
(3) all events in $\textit{effective}(\overline{e}_{1})$ and
$\textit{effective}(\overline{e}_{2})$ are commutative.
holds there exists a unique set of events $\tilde{e}$ with
$\tilde{e}=\textit{activeSet}(\overline{e}_{1})$ and
$\tilde{e}=\textit{activeSet}(\overline{e}_{2})$.
In addition we require a function $\textit{parse}:M\to\mathcal{P}(E)$ such
that $\textit{parse}(m)=\tilde{e}$.
We say $m\in M_{\textit{CES}}$.
This means, for any $m\in M_{\textit{CES}}$ there is a uniquely defined event
set $\tilde{e}$ such that $m=\textit{apply}(\tilde{e},\emptyset)$ and it is
possible to derive $\tilde{e}$ from $m$ via parsing. Thus, we require a
bidirectional mapping between the command store and its model and it must be
possible to reconstruct the command store from the model through parsing.
Now we are able to define whether two models are ”equivalent” or synchronized.
Two models are synchronized if their active event sets are equal. We may also
opt for partial synchronisation e.g. we may restrict ourselves to certain
event types. Then two models are partially synchronized if their active event
sets restricted to the desired event types are equal. This allows e.g. to have
additional HaveContent commands in our JavaDoc tool that add the actual
content to our JavaDoc Files. This additional information is not synchronized
with our JavaPackages tool, cf. [3].
Overall, we restrict ourselves to commands that have to be implemented in a
very specific way. However, this should not restrict the kind of models that
may be generated, too severe. We will discuss this in our conclusions.
## 4 Achieving Commutative Event Sourcing
This section provides some simple design rules to achieve Commutative Event
Sourcing. Our design rules are based on the notion of an increment. An
increment is a set of attributes and links that are edited through a single
command execution. With respect to our getOrCreate mechanism, objects are part
of an increment if their id attribute is a part of the increment. In Figure 3
we have color coded different increments and their commands.
To achieve commutativity for our commands, we first look on commutativity for
TGG rules. Basically, a TGG rule has a context (black) part that shall already
exist and that is not modified and a main (green and blue) part that defines
its increment, i.e. the attributes and links that will be edited on rule
execution. In general, a TGG rule may have an arbitrary complex context
(black) part. You may require multiple objects that shall be connected by
various links and you may have additional attribute constraints and
application conditions. However, such complex context parts potentially create
rule dependencies: for example, if a rule requires some link within its
context part and later this link is removed, the rule application is no longer
valid and it must be removed, too. This may cause cascading deletes. To avoid
such rule dependencies, we restrict the context (black) part of our rules to
simple objects with an id attribute constraint. No links between context
objects and no general attribute constraints. Similarly, we restrict the
negative context (blue) part to simple objects with id constraints. Again no
links and no general attribute conditions.
The main (green and blue) parts of a rule result in active changes of
attribute values and links. This may interfere with another rule, if the other
rule changes the same attributes or links, differently. In that case rule
application order would matter. Thus, we require that the increment edited by
the main rule parts must not overlap with the increment of other rule
applications, i.e. no two rule applications shall edit the same attribute or
link.
Together, simple contexts and not overlapping main parts achieve
commutativity. The prove of this claim is current work. It would also be great
to have a compile time check that validates a set of triple graph grammar
rules for commutativity. In addition, we need criteria and checks for
overwriting TGG rules, i.e. for rules that edit the same increment in
different ways. However, if you implement the commands manually, such TGG
criteria and checks will not apply for you. So far, to validate the
commutativity of your TGG rules, you will have to execute them in different
order and to check whether all orders achieve the same result. Due to our
experience, executing a reasonable sequence of events once in order and once
in reverse order and comparing the result works very well in order to detect
non-commutative rules.
During the manual implementation of our commands, it is straight forward to
restrict ourselves to simple context (black) objects: either do not check for
attribute values or links or if you do check an attribute, consider it as a
part of your increment, i.e. you edit it.
To achieve non-overlapping increments, we first introduce a convention: each
command shall have exactly one core object that has the same id as the command
itself. Based on this convention, to find non-overlapping increments we start
with some reasonably representative object model example. Then we choose some
object as the core object of our first command and we mark the id attribute of
this object with a certain color. In Figure 3 we may e.g. start with the
Editor object in the lower left corner and mark its id attribute with orange.
Next, we look for other attributes of the same object that our command may
initialize in the same step and mark them with the same color. In our example,
we choose the vTag attribute of our Editor object. Next, we look for to-one
links that connect our core object to other objects. In our example we choose
the pack - classes link attached to our Editor object and mark it with orange,
too. Now we look for other core objects of the same type and try to mark a
similar increment induced by this new core object. In Figure 3 there is no
other JavaClass object, thus we go on with a new core object of a different
type, e.g. with the serv object of type JavaPackage. The serv object has no
other attributes but a pPack - subPackages link of cardinality to-one. We
color the id attribute and the link in blue. Now we look at the fulib object
on the left of Figure 3 which is also of type JavaPackage. Here we use green
color to mark a similar increment. If we look at the org object there is no
pPack - subPackages link attached to it. However, there is an empty field
within the org object that could hold a pPack link. To make the org increment
similar to the other JavaPackage increments, we mark the empty space above the
org object with yellow color. Now we have colored all attributes and all links
of our JavaPackages example model and no attribute and no link is marked by
two colors. Thus, we have identified non-overlapping increments that cover all
elements of our example model. So far, this approach only adds to-one
associations to increments. In general, a model may deploy many-to-many
associations. To cover a many-to-many association we introduce dedicated rules
that consist of two context (black) nodes and a green link between theses
nodes (to create the link) or a blue link between these nodes (to delete the
link). These rules shall not contain any other elements. As these rules just
edit a single link, they are either commutative or overwriting by
construction.
We are now ready to define our commands: for each core object we create a
command object with a certain type. For each attribute of an increment, the
command object gets an appropriate parameter attribute. For each link of our
increment the command gets a parameter attribute that holds the id of the
target object. Usually, we use one command type for each increment type or for
each core object type. If there are two core objects of the same type which
handle the attributes and links of their increment in different ways, you
might use two different command types or one command type which distinguishes
the two cases, internally. In our example, we use command type HaveRoot for
the org object and command type HaveSubUnit for the fulib and the serv object.
These two commands differ in the parent parameter and in their handling of the
pPack link: the former deletes the pPack link and the latter creates it.
Generally, increments (with core objects) of the same type must be similar,
i.e. they edit similar sets of attributes and links. Accordingly, commands of
different types that edit increments of the same type must edit similar sets
of attributes and links. Therefore, the HaveRoot command has to assign a
defined value (i.e. null) to the pPack field of its core object. Actually, in
a manual implementation we would probably use only one command type for
JavaPackage objects, as it is easy to handle both cases in one implementation.
However, the example stems from [9] and [9] uses TGG rules and with TGGs you
need different rules for different cases. Thus, we use two command types to
facilitate the comparison.
Note, as all commands that edit the same increment type (or core objects with
the same type) edit similar sets of attributes and links, our commands are
overwriting by construction: if two commands have the same id, they edit the
same core object and the same increment and as discussed all parts of the
increment get a well defined value. Thus, one command will overwrite all
attributes and links that have been edited by the other command and thus it
suffices to keep only the last command in our command store.
Let’s now have a look at the JavaDoc example model at the right side of Figure
3. Given the increments of the left model, we now have to identify the
corresponding increments on the right model. For the orange increment on the
bottom, this is the Editor DocFile with its id attribute, its version
attribute and its folder - files link. In Figure 3 this is easily spotted by
comparing object ids and attribute values and link targets and names. Usually,
you have only one example model as a start and you construct the target model
”incrementally”. Thus, you identify which objects, attributes and links need
to be added to the target model in order to represent the information that is
provided by the source increment or by the corresponding command and its
parameters. And you reuse the ids of the source model within the target model
in order to establish the desired correspondences.
Note, the content attribute of our Editor object on the right is not marked
with orange. Actually, the content attribute of a DocFile has no
correspondence within the JavaPackages model and thus it will not be addressed
by model synchronisation but we will introduce a separate HaveContent command
within the JavaDoc model later on, cf. [3].
For the HaveSubUnit increments or commands the situation is somewhat more
complicated: Within the JavaDoc model, a sub package is represented by a
Folder object and by a special DocFile that describes the sub package. If we
look e.g. at the blue serv increment in Figure 3, within the JavaDoc model
this increment is represented by the serv object (of type Folder) and by the
serv.Doc object (of type DocFile) and by its content attribute and the
attached folder - files and pFolder - subFolders links. Thus, our increment
contains two model objects, the core object with id serv and a dependent
object with id serv.Doc. In general, a Folder object may contain multiple
DocFile objects. In order to identify the DocFile object that describes the
Folder itself, our example uses the convention that the describing DocFile has
an id that is equal to the id of its Folder plus a ".Doc" suffix, cf. Figure 3
and Listing 6. With this convention it is still possible to identify all parts
of an increment by starting at some model object and collecting the attached
parts. If you start e.g. with the serv.Doc object you may identify the
corresponding serv object either using our naming convention or using the
folder - files link. If you start with the serv object you use the naming
convention to identify the serv.Doc object. As in our JavaDoc model a
HaveSubUnit increment covers a DocFile object and as the HaveRoot increment
and command address the same kind of increment, the HaveRoot commands needs to
edit (i.e. remove) a potentially attached DocFile object, too, cf. Listing 5.
Note, the version attribute of our serv.Doc object is not colored as the
corresponding JavaPackage and the corresponding HaveSubUnit command do not
have any version information.
To summarize, you start with core objects and collect attributes, to-one links
(and neighbors) that form an increment, i.e. that are edited together. For
many-to-many associations you add dedicated rules. Core objects of the same
type should have increments of similar structure. For such increments you
introduce a command with parameter attributes that correspond to attribute
values and link targets. If there are alternative cases, you may use different
command types or choose the desired variant from parameter values. This
decision must not rely on context properties. For each increment in some
source model, you identify a corresponding increment within the target model.
Increments shall not overlap and all (relevant) parts of the models shall be
covered. Thereby, you construct commands with simple contexts and non-
overlapping increments, i.e. commutative commands.
You may check the commutativity of your commands by applying a sufficiently
long series of commands once in order and once in reverse order and then you
compare the resulting models. To our experiences, this reveals violations of
the commutativity rule, quite reliably. Note, due to our excessive use of ids,
comparison of two models is quite easy in our case.
## 5 Collaborative Editing and Merge Conflicts
Once you got your commands right, you still have to deal with multiple or
concurrent edits of the same increment. Assume, you have already executed
command (HaveLeaf Editor serv 1.0) and now you want to change the version to
1.1. Thus you run the command (HaveLeaf Editor serv 1.1). In your current
editor the second command would just overwrite the first command and
everything is fine. Unfortunately, model synchronisation may fail, if your
message broker delivers theses two commands in the wrong order. Similarly, you
have a model synchronisation problem, if the two commands are run in two
different editors, concurrently, and you try to synchronize afterwards. As the
two commands are conflicting, i.e. they assign different values to the vTag
attribute of the Editor object, model synchronisation needs some mechanism
that decides which command overwrites the other.
As far as we are aware of, this problem is only recently addressed by Triple
Graph Grammar tools cf. [8]. Actually, [9] explicitly mentions this problem
and puts it in future work. This paper solves this problem with the help of
additional time stamps and with the help of command specific merge strategies.
Listing 14 shows the execute method that we deploy in our editors.
⬇
1package JavaPackages;
2public class JavaPackagesEditor {
3 …
4 public void execute(ModelCommand command) {
5 String id = command.getId();
6 if (id == null) {
7 id = ”obj” + activeCommands.size();
8 command.setId(id);
9 }
10 String time = command.getTime();
11 if (time == null) {
12 time = getTime();
13 command.setTime(time);
14 }
15 ModelCommand oldCommand = activeCommands.get(id);
16 if (oldCommand != null && ! command.overwrites(oldCommand)) {
17 return;
18 }
19 command.run(this);
20 activeCommands.put(id, command);
21 }
22 …
Listing 14: Command execution
When you want to execute a command you actually call editor.execute(cmd) on
the responsible editor. Line 6 of Listing 14 first checks if your command has
an id. If not, we create an id for you. This facilitates testing and iterative
development. Alternatively, you may raise an exception. Next, Line 10 reads
the time stamp of the command. If there is no time stamp yet, we assign the
current time, cf. Line 13111The getTime method of our editor caches the time
it returns. If you call it twice within the same millisecond, it will add an
extra millisecond in order to avoid the same time stamp on multiple commands,
cf. [3]. If you serialize a command later on and send it e.g. to another
editor, the command will already contain the original time stamp.
Now Line 15 does a lookup in our hash table for activeCommands. If we already
have an oldCommand we ask our new command whether it is going to overwrite the
oldCommand (Line 16). Listing 15 shows the default implementation of method
overwrites. Line 4 compares the time stamps and we do not overwrite if the
current command is older than the oldCommand. On equal time stamps (Line 6) it
is probably the same command received twice and there is no need to execute it
again. However, for the unlikely case that two editors concurrently create
different commands for the same id and with the same time stamp, we do a
string compare of the yaml representation of our commands and we do not
execute the current command if the oldCommand is lexically later or equal
(Line 7 to 10).
⬇
1public class ModelCommand {
2 …
3 public boolean overwrites(ModelCommand oldCommand) {
4 if (oldCommand.getTime().compareTo(time) > 0) {
5 return false;
6 } else if (oldCommand.getTime().equals(time)) {
7 String oldYaml = Yaml.encode(oldCommand);
8 String newYaml = Yaml.encode(this);
9 if (oldYaml.compareTo(newYaml) >= 0) {
10 return false;
11 }
12 }
13 return true;
14 }
15 …
Listing 15: Command execution
If there is no oldCommand or if the new command overwrites the oldCommand our
editor executes the new command in Line 19 of Listing 14. Finally, the new
command is added to the hash table of active commands (Line 20).
Thus, if you run (HaveLeaf Editor serv 1.0) in some editor at e.g. 13:36
o’clock and you run (HaveLeaf Editor serv 1.1) at e.g. 13:37 o’clock on the
same editor, the second command will overwrite the first. If you run the two
commands on two different editors, each editor will store its own command. If
you do model synchronisation some time later, on the first editor the 13:37
command will overwrite the 13:36 command while on the other editor the 13:36
command will be ignored as that editor already has the 13:37 command for the
same id. Eventually, both editors have the 13:37 command active and the vTag
of the Editor object will be 1.1. Note, our model synchronisation scheme works
also for two editors with different meta models.
The default implementation of method overwrites shown in Listing 15 resolves
editing or merge conflicts with a ”last edit wins” strategy. While this works
quite frequently, in some cases you may want another strategy for conflict
resolution. For example in case of a seat reservation system you might want a
”first edit wins” strategy. Or in our version number example you may want a
”highest version wins” strategy. Therefore, each command may overwrite the
inherited default implementation of method overwrites and implement its own
strategy. If you do this, ensure that all editors (or models) use the same
strategy during model synchronisation.
## 6 Removing model objects
There is yet another design issue to be discussed. Due to our getOrCreate
mechanism, removing an object from our model is quite tricky. To safely remove
an object from our model we must get rid of the command that created and
initialized it directly (core object and green parts of our TGG rules, cf.
Figure 2). And we must get rid of all usages of this object as context (black
rule part) in all other commands. Generally, we would require that for any
object that is used as a required context (black rule part) by some command
the corresponding set of activeCommands shall contain another command that
explicitly creates and initializes that object (core object and green rule
part). Unfortunately, we deal with unreliable message brokers and some
commands may just not yet have arrived. But we already want to work with our
model. Thus, it would be great to be able to distinguish between fully
initialized model objects and so-called object frames that so far have been
used as context objects, only. To achieve this, our editors deploy one
mapOfModelObjects for explicit model objects and one mapOfFrames for context
objects, cf. Listing 16 and [3]. (The mapOfParsedObjects will be discussed in
Section 7.)
⬇
1package JavaPackages;
2…
3public class JavaPackagesEditor {
4 private Map<String, Object> mapOfModelObjects
5 = new LinkedHashMap<>();
6 private Map<String, Object> mapOfFrames
7 = new LinkedHashMap<>();
8 private Map<String, Object> mapOfParsedObjects
9 = new LinkedHashMap<>();
10 …
11}
Listing 16: Maps for model objects and object frames
When we need an object as context (black rule part) we call method
getObjectFrame on the corresponding editor. Lines 7 to 10 of Listing 17 will
be discussed in Section 7. Line 11 of Listing 17 first tries to retrieve the
desired object from the mapOfModelObjects. If that fails, Line 15 tries to
retrieve the desired object from the mapOfFrames. If this still fails, Line 19
to 24 first use reflection in order to create the desired object and to
initialize its id and then the object is added to the mapOfFrames and
returned.
⬇
1package JavaPackages;
2…
3public class JavaPackagesEditor {
4 …
5 public Object getObjectFrame(Class clazz, String id) {
6 try {
7 Object modelObject = mapOfParsedObjects.get(id);
8 if (modelObject != null) {
9 return modelObject;
10 }
11 modelObject = mapOfModelObjects.get(id);
12 if (modelObject != null) {
13 return modelObject;
14 }
15 modelObject = mapOfFrames.get(id);
16 if (modelObject != null) {
17 return modelObject;
18 }
19 modelObject = clazz.getConstructor().newInstance();
20 Method setIdMethod
21 = clazz.getMethod(”setId”, String.class);
22 setIdMethod.invoke(modelObject, id);
23 mapOfFrames.put(id, modelObject);
24 return modelObject;
25 } catch (Exception e) {
26 throw new RuntimeException(e);
27 }
28 }
29 …
30}
Listing 17: getObjectFrame method
Similarly, when we want to create a model object explicitly (green rule parts)
we call method getOrCreate on the corresponding editor, cf. Listing 18. Again,
Lines 6 to 10 will be discussed in Section 7. Line 11 tries to retrieve the
desired model object from our mapOfModelObjects. If that fails, Line 15 calls
method getObjectFrame which either retrieves the desired object or creates it.
Then Lines 16 and 17 promote the desired object into the mapOfModelObjects and
Line 18 returns it.
⬇
1package JavaPackages;
2…
3public class JavaPackagesEditor {
4 …
5 public Object getOrCreate(Class clazz, String id) {
6 Object modelObject = mapOfParsedObjects.get(id);
7 if (modelObject != null) {
8 mapOfModelObjects.put(id, modelObject);
9 return modelObject;
10 }
11 modelObject = mapOfModelObjects.get(id);
12 if (modelObject != null) {
13 return modelObject;
14 }
15 modelObject = getObjectFrame(clazz, id);
16 mapOfFrames.remove(id);
17 mapOfModelObjects.put(id, modelObject);
18 return modelObject;
19 }
20 …
21}
Listing 18: getOrCreate method
In order to remove an object from our model we call method removeModelObject
on the corresponding editor. Line 6 of Listing 19 tries to remove the object
from our mapOfModelObjects. If that succeeds, Line 8 adds the removed object
to our mapOfFrames (as it may still be used as context by some other command).
⬇
1package JavaPackages;
2…
3public class JavaPackagesEditor {
4 …
5 public Object removeModelObject(String id) {
6 Object oldObject = mapOfModelObjects.remove(id);
7 if (oldObject != null) {
8 mapOfFrames.put(id, oldObject);
9 }
10 return mapOfFrames.get(id);
11 }
12 …
13}
Listing 19: removeModelObject method
Actually, to remove some object from our model, we need to (implement and)
call the remove method of the responsible command in order to remove the
complete increment and to leave our model in a consistent state. In addition,
we have to remove the corresponding command from our activeCommands hash
table. And we have to inform all other tools during subsequent model
synchronisation. In addition, we have to be careful, in order to prevent the
re-execution of the removed command when we receive it again (e.g. from
another tool). Our implementation uses a special RemoveCommand to achieve
this, cf. Listing 20.
⬇
1package JavaPackages;
2…
3public class RemoveCommand extends ModelCommand {
4 public Object run(JavaPackagesEditor editor) {
5 editor.removeModelObject(getId());
6 ModelCommand oldCommand
7 = editor.getActiveCommands().get(getId());
8 if (oldCommand != null) {
9 oldCommand.remove(editor);
10 }
11 return null;
12}
Listing 20: Command execution
We may e.g. call (RemoveCommand c) on our editor and the editor may already
have a (HaveLeaf c sub 1.1) command in its set of activeCommands. Then the
default ”last edit wins” strategy of our editor will find that the
RemoveCommand is later than the HaveLeaf command and it will call run on the
RemoveCommand. Line 5 of Listing 20 assumes per default that the command to be
removed has created at least one model object with the same id as the command
(and the RemoveCommand). Thus, Line 5 calls removeModelObject with that id.
Therefore, simple commands that create and initialize only a single object do
not even have to implement the remove method as the RemoveCommand already does
this job. If you do not want to rely on this default assumption, you may
easily omit this line in your implementation.
Line 7 of Listing 20 retrieves the command to be removed from our
activeCommands and Line 9 calls its remove method. Afterwards, the execute
method of our editor replaces the removed command with the RemoveCommand
within our activeCommands, cf. Line 20 of Listing 14. On model synchronization
the RemoveCommand will be send to the other tool(s) and perform the same
operation there.
Note, we need to keep the RemoveCommand in our activeCommands table until we
are sure that all other tools (and all persistent copies of our activeCommands
table have overwritten their copy of e.g. the HaveLeaf command. If we remove
the RemoveCommand too early and if we receive the overwritten HaveLeaf command
thereafter, we would not notice that the HaveLeaf command has been removed but
we would re-execute it.
## 7 Parsing
Sometimes, you may want to edit your model directly, cf. Line 13 to 24 of
Listing 21. Then you need to parse your (modified) model in order to retrieve
the set of commands that correspond to it and to synchronize your changes with
other models. Parsing basically requires to split your model into increments
where each increment corresponds to a certain command. In our approach, we
follow the convention that each command creates one core model object that
gets the same id as the command. This core model object becomes the nucleus
for each increment. Thus, in our approach the identification of model
increments starts with these core objects and parsing just needs to collect
the remaining parts of the increment.
⬇
1package JavaPackages;
2…
3public class TestPackageToDoc
4 implements PropertyChangeListener {
5…
6private Set changedObjects = new LinkedHashSet();
7@Test
8public void testManualChangesAndParsing() {
9 JavaPackagesEditor javaPackagesEditor
10 = new JavaPackagesEditor();
11 startSituation(javaPackagesEditor);
12 registerModelObjectListener(javaPackagesEditor, this);
13 JavaPackage com = new JavaPackage().setId(”com”);
14 JavaPackage org = (JavaPackage)
15 javaPackagesEditor.getModelObject(”org”);
16 com.withSubPackages(org);
17 JavaPackage fulib = (JavaPackage)
18 javaPackagesEditor.getModelObject(”fulib”);
19 fulib.setPPack(null);
20 JavaClass command = new JavaClass()
21 .setPack(fulib).setVTag(”1.1”).setId(”Command”);
22 JavaClass editorClass = (JavaClass)
23 javaPackagesEditor.getModelObject(”Editor”);
24 editorClass.setVTag(”1.1”);
25 Set allObjects = Yaml.findAllObjects(com, fulib);
26 javaPackagesEditor.parse(changedObjects);
27 …
28}
29…
30}
Listing 21: Command execution
In [9] the parsing of a HaveLeaf rule requires that the JavaPackage that is
reached via a pack link has already been parsed either by a HaveRoot or by an
HaveSubUnit rule. Similarly, the parsing of a HaveSubUnit rule requires that
the JavaPackage attached to the corresponding sub JavaPackage via a pPack -
subPackages link has already been parsed. Thus, in [9] parsing must start with
the HaveRoot rule and then you parse the sub JavaPackages of these root(s) and
then the sub-sub packages until you reach the JavaClass leafs. These parsing
dependencies between general Triple Graph Grammar rules makes the parsing of
Triple Graph Grammars (and especially incremental parsing) very complex.
Compared to general Triple Graph Grammars, parsing of Commutative Event
Sourcing models is a piece of cake. As we have no rule dependencies, even
incremental parsing becomes easy. Therefore, Line 12 of Listing 21 subscribes
a property change listener to our object model. (See [3] for implementation
details.) This property change listener collects all objects affected by the
changes executed in Lines 13 to 24. This allows us to call the parse method of
our editor with just the set of changedObjects in Line 26. Alternatively, Line
25 uses our object serialization mechanism to collect all objects that belong
to the modified model and we could use the set of allObjects within our parse
call.
Line 6 of Listing 22 registers all objects that shall be parsed into our
mapOfParsedObjects (cf. Listing 16, see [3] for details). For each object that
shall be parsed, Line 10 calls method findCommands, which does the actual
parsing. Method findCommands retrieves a set of command prototypes (Line 26 of
Listing 22). Then Line 28 calls the parse method of each command prototype.
These parse methods analyse the current object, whether it is the nucleus of
some increment that matches that command, cf. Listing 12 and Listing 13. If
the command fits, its parse method returns a new copy of the corresponding
command with all command parameters assigned, properly. On success, Line 30 of
Listing 22 collects the parsed command in the set of allCommands. Once we have
identified all commands that correspond to the objects to be parsed, we look
for oldCommands that will be overwritten by the execution of a new command,
cf. Line 14 to 16 of Listing 22. If there is already an old command with the
same parameters, we do not overwrite it in order to keep the old time stamp.
If the new command is actually different (or there is no old command) we
execute it (Line 17) in order to update our set of activeCommands. (Note,
executing a command that has been parsed will edit the corresponding increment
but this edit should just re-assign the values that have been found during
parsing. Thus, if the run and the parse method work consistently, running the
command does no harm (but may also be skipped).)
⬇
1package JavaPackages;
2…
3public class JavaPackagesEditor {
4…
5public void parse(Collection allObjects) {
6 registerParsedObjects(allObjects);
7 ArrayList<ModelCommand> allCommandsFromParsing
8 = new ArrayList<>();
9 for (Object object : allObjects) {
10 findCommands(allCommandsFromParsing, object);
11 }
12 for (ModelCommand command : allCommandsFromParsing) {
13 String id = command.getId();
14 ModelCommand oldCommand = activeCommands.get(id);
15 if (oldCommand == null ||
16 ! equalsButTime(oldCommand, command)) {
17 execute(command);
18 }
19 }
20}
21
22private ModelCommand findCommands(
23 ArrayList<ModelCommand> allCommands,
24 Object object) {
25 ArrayList<ModelCommand> prototypes
26 = haveCommandPrototypes();
27 for (ModelCommand prototype : prototypes) {
28 ModelCommand command = prototype.parse(object);
29 if (command != null) {
30 allCommands.add(command);
31 }
32 }
33 return null;
34}
35…
36}
Listing 22: Command execution
There is still one little design issue to be discussed. In our example, Line
13 of Listing 21 creates the JavaPackage com, directly. Our change listener
will collect the new com object as soon as it is linked to the old model
object org (Line 16). Thus, parsing will create a (HaveRoot com) command. When
we execute this (HaveRoot com) command, its run method calls getOrCreate (Line
6 of Listing 1) to retrieve the desired model object. Now, some parts of our
(testing) program may still hold a reference to the directly created com
object. Thus we want getOrCreate to retrieve this directly created com object
and getOrCreate should not create a new model object. To achieve this, Line 6
of Listing 22 registers all directly changed objects in our mapOfParsedObjects
and method getOrCreate tries to retrieve the required object from there (Lines
6 to 10 of Listing 18). Method getObjectFrame work similarly (Listing 17).
## 8 Conclusions
To revisit the title of this paper, Commutative Event Sourcing may be
considered as just a restricted variant of Triple Graph Grammars where the
commands or rules are either overwriting or commutative. This frees
Commutative Event Sourcing from handling dependencies between commands /
rules. Thereby, model synchronisation, collaborative editing, and even
incremental parsing is facilitated, considerably.
This paper tries to provide sufficient details such that you may implement
Commutative Event Sourcing for your tool(s), manually. You may adapt our
concepts without relying on a special programming language, library,
framework, or tool. You may also copy large parts of our example
implementation from [3]. Actually, our editors are quite generic, only the set
of command prototypes is model specific. [3] also provides a code generator
for editors and the generic parts of commands, if you want to use that. [3]
even provides a simple interpreter for TGG like Commutative Event Sourcing
patterns. However, using this rule interpreter requires a certain learning
curve for writing these patterns and pattern execution is hard to debug. Thus,
beginners may be better of by implementing their commands, manually.
Commutative event sourcing requires that commands / rules are either
overwriting or commutative. This is quite a restriction compared to general
Triple Graph Grammars. But it facilitates implementation. We like to compare
this with the introduction of LALR(k) grammars [6] in compiler construction
that reduced memory consumption within compilers compared to more general
LR(k) grammars. LALR(k) grammars reduce the set of parseable languages,
slightly, but acceptable.
Our approach also makes extensive use of ids for (cross model) object
reference. If you do not want id attributes in your model, your editor may use
an additional objectToId map in order to store and retrieve object ids. We did
so e.g. when using Commutative Event Sourcing for the solution of a model
synchronisation case in the Transformation Tool Contest 2020 [4] and [3]. If
you use ids for cross model referencing, you want to assign these ids directly
and you do not want e.g. a database system generating your ids. Directly
edited ids are usually considered an anti pattern. And yes, this is a design
challenge for Commutative Event Sourcing.
Commutative Event Sourcing considers two models to be equivalent if they
contain the same objects with the same ids and same attribute values and the
same set of links. In case of a to-many association we handle the neighbors as
a set and not as a list, i.e. we ignore the order of the neighbors. If your
commands are commutative and show up in any order it is just hard to achieve a
certain order in a list. This again is a design challenge, e.g. if you want to
show a number of objects in the graphical user interface on two different
tools and you want that both users see the same order. (Well just sort them.)
Neglecting the order of lists is also a challenge for models with a textual
representation: you may not want an arbitrary order of your program statements
(and sorting would not help).
However, we have used Commutative Event Sourcing with great success in [4] and
with some other model synchronisation problems e.g. for BPMN diagrams and a
textual Workflow language [3]. We also used it in our MicroServices course in
Winter Term 2019 / 2020 [19]. According to our experiences, the Commutative
Event Sourcing approach to model synchronisation is quite easy to engineer and
works quite reliable.
## References
* [1] Anjorin, A., Diskin, Z., Jouault, F., Ko, H.S., Leblebici, E., Westfechtel, B.: Benchmarx reloaded: A practical benchmark framework for bidirectional transformations. In: BX@ ETAPS. pp. 15–30 (2017)
* [2] Copei, S., Sälzer, M., Zündorf, A.: Multidirectional transformations for microservices. In: Proc. Dagstuhl Seminar. vol. 18491 (2019)
* [3] Copei, S., Zündorf, A.: Github example implementation. https://github.com/fujaba/fulibServiceGenerator, https://github.com/fujaba/fulibServiceGenerator/blob/master/test/src/test/java/ javaPackagesToJavaDoc/TestPackageToDoc.java, last viewed 31.08.2020
* [4] Copei, S., Zündorf, A.: The fulib solution to the ttc 2020 migration case. arXiv preprint arXiv:2012.05231 (2020)
* [5] Czarnecki, K., Foster, J.N., Hu, Z., Lämmel, R., Schürr, A., Terwilliger, J.F.: Bidirectional transformations: A cross-discipline perspective. In: International Conference on Theory and Practice of Model Transformations. pp. 260–283. Springer (2009)
* [6] DeRemer, F., Pennello, T.: Efficient computation of lalr (1) look-ahead sets. ACM Transactions on Programming Languages and Systems (TOPLAS) 4(4), 615–649 (1982)
* [7] Evans, E.: Domain-driven design: tackling complexity in the heart of software. Addison-Wesley Professional (2004)
* [8] Fritsche, L., Kosiol, J., Möller, A., Schürr, A., Taentzer, G.: A precedence-driven approach for concurrent model synchronization scenarios using triple graph grammars. In: Proceedings of the 13th ACM SIGPLAN International Conference on Software Language Engineering. pp. 39–55 (2020)
* [9] Fritsche, L., Kosiol, J., Schürr, A., Taentzer, G.: Avoiding unnecessary information loss: Correct and efficient model synchronization based on triple graph grammars. arXiv preprint arXiv:2005.14510 (2020)
* [10] Gamma, E., Helm, R., Johnson, R., Vlissides, J.: Design patterns: Elements of reusable software architecture. Reading: Addison-Wesley (1995)
* [11] Group, O.M.: The qvt standard. https://www.omg.org/spec/QVT/, last viewed 31.08.2020
* [12] Hildebrandt, S., Lambers, L., Giese, H., Rieke, J., Greenyer, J., Schäfer, W., Lauder, M., Anjorin, A., Schürr, A.: A survey of triple graph grammar tools. Electronic Communications of the EASST 57 (2013)
* [13] Leblebici, E., Anjorin, A., Schürr, A., Hildebrandt, S., Rieke, J., Greenyer, J.: A comparison of incremental triple graph grammar tools. Electronic Communications of the EASST 67 (2014)
* [14] Macedo, N., Cunha, A.: Implementing qvt-r bidirectional model transformations using alloy. In: International Conference on Fundamental Approaches to Software Engineering. pp. 297–311. Springer (2013)
* [15] Schneider, C., Zündorf, A., Niere, J.: Coobra-a small step for development tools to collaborative environments. In: Workshop on Directions in Software Engineering Environments. Citeseer (2004)
* [16] Schürr, A.: Specification of graph translators with triple graph grammars. In: International Workshop on Graph-Theoretic Concepts in Computer Science. pp. 151–163. Springer (1994)
* [17] Steinberg, D., Budinsky, F., Merks, E., Paternostro, M.: EMF: eclipse modeling framework. Pearson Education (2008)
* [18] Stevens, P.: Bidirectional model transformations in qvt: semantic issues and open questions. Software & Systems Modeling 9(1), 7 (2010)
* [19] University, S.E.G.K.: Youtube playlist labor microservices, winter term 2019 / 2020\. https://www.youtube.com/playlist?list=PLohPa1TMsVqrI0FaMTySbGr02JkQs5MhQ, last viewed 31.08.2020
* [20] Vernon, V.: Implementing domain-driven design. Addison-Wesley (2013)
|
Algebraic curves and foliations
César Camacho111EMAp-FGV, Praia de Botafogo 190, Rio de Janeiro, 22250-900,
RJ, Brazil, cesar.camacho@fgv.br,, Hossein Movasati222IMPA, Estrada Dona
Castorina 110, Rio de Janeiro, 22460-320, RJ, Brazil<EMAIL_ADDRESS>
With an appendix by Claus Hertling333Universität Mannheim, B6, 26, 68159,
Mannheim, Germany<EMAIL_ADDRESS>
###### Abstract
Consider a field ${\mathsf{k}}$ of characteristic $0$, not necessarily
algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame
polynomial $f\in{\mathsf{k}}[x,y]$ with only quasi-homogeneous singularities.
We prove that the space of holomorphic foliations in the plane
$\mathbb{A}^{2}_{\mathsf{k}}$ having $f=0$ as a fixed invariant curve is
generated as ${\mathsf{k}}[x,y]$-module by at most four elements, three of
them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic
and constructs the fourth foliation explicitly. Using Serre’s GAGA and
Quillen-Suslin theorem, we show that for a suitable field extension
${\mathsf{K}}$ of ${\mathsf{k}}$ such a module over ${\mathsf{K}}[x,y]$ is
actually generated by two elements, and therefore, such curves are free
divisors in the sense of K. Saito. After performing Groebner basis for this
module, we observe that in many well-known examples,
${\mathsf{K}}={\mathsf{k}}$.
## 1 Introduction
The geometric analysis of one dimensional foliations in the complex projective
plane $\mathbb{P}^{2}_{\mathbb{C}}$, seen from a global perspective, is a wide
open field of research. In particular, the study of foliations that admit an
algebraic curve as an integral. In [CLS92] foliations whose limit set is an
algebraic curve with hyperbolic holonomy are characterized as rational pull
backs of linear foliations. Curves of high degree can be leaves of foliations
of lower degree, and there is no general statement explaining this phenomenon.
For example A. Lins Neto in [Lin02] considers a curve which is a union of nine
lines supported by two foliations of projective degree $4$. Studying the
pencil generated by these two foliations he answers questions posed by
Poincaré and Painlevé.
In this paper we explore the algebraic aspects of foliations supporting a
fixed algebraic invariant curve. Let ${\mathsf{k}}$ be a subfield of
$\mathbb{C}$ and $\Omega_{\mathbb{A}_{\mathsf{k}}^{2}}^{1}$ the space of
1-forms on $\mathbb{A}_{\mathsf{k}}^{2}$. An element
$\omega\in\Omega_{\mathbb{A}_{\mathsf{k}}^{2}}^{1}$ induces the foliation
$\omega=0$ in the affine variety $\mathbb{A}^{2}_{\mathsf{k}}$. Let
$f\in{\mathsf{k}}[x,y]$ be a polynomial and $C:f(x,y)=0$ the induced curve in
$\mathbb{A}_{\mathsf{k}}^{2}$. We assume that $f$ is a tame polynomial.
###### Definition 1.
([Mov07], [Mov19, §10.6]). A polynomial $f\in{\mathsf{k}}[x,y]$ is called tame
if the following property is satisfied. In the homogeneous decomposition of
$f$
$f=f_{d}+\cdots+f_{2}+f_{1}+f_{0},\ $
into degree $i$ homogeneous pieces $f_{i}$ in the weighted ring
${\mathsf{k}}[x,y],\ \deg(x)=\alpha_{1},\deg(y)=\alpha_{2}$ for some
$\alpha_{1},\alpha_{2}\in\mathbb{N}$, the last homogeneous piece $g:=f_{d}$
has finite dimensional Milnor vector space
$V_{g}:=\frac{{\mathsf{k}}[x,y]}{{\rm jacob}(g)}$.
For $\alpha_{1}=\alpha_{2}=1$ this is equivalent to say that $g$ induces $d$
distinct points in $\mathbb{P}_{\bar{\mathsf{k}}}^{1}$, that is
$g=\prod_{i=1}^{d}(x-a_{i}y)$, where the $a_{i}\in\bar{\mathsf{k}}$ are
pairwise different. In geometric terms this means that the line at infinity
$\mathbb{P}^{1}_{\bar{\mathsf{k}}}:=\mathbb{P}^{2}_{\bar{\mathsf{k}}}\backslash\mathbb{A}_{\bar{\mathsf{k}}}^{2}$
intersects the curve induced by $f=0$ in $\mathbb{P}^{2}_{\mathsf{k}}$
transversely.
Let
(1)
$E_{f}:=\left\\{\omega\in\Omega_{\mathbb{A}_{\mathsf{k}}^{2}}^{1}\Big{|}df\wedge\omega=f\alpha,\
\hbox{ for some }\alpha\in\Omega_{\mathbb{A}_{\mathsf{k}}^{2}}^{2}\right\\}.$
This is the ${\mathsf{k}}[x,y]$-module of differential $1$-forms $\omega$ such
that the foliation induced by $\omega=0$ in $\mathbb{A}^{2}_{\mathsf{k}}$
leaves $f=0$ invariant. By degree of $\omega=Pdx+Qdy$ we mean the affine
degree $\deg(\omega):={\rm max}\\{\deg(P),\deg(Q)\\}$. In this article we
prove that:
###### Theorem 1.
If all the singularities of $f=0$ are quasi-homogeneous then there exists
$\omega_{f}\in E_{f}$ such that $fdx,fdy,df,\omega_{f}$ generate the
${\mathsf{k}}[x,y]$-module $E_{f}$.
The first three foliations $fdx,fdy,df$ are obviously in $E_{f}$ and we call
them trivial foliations. Our proof of Theorem 1 is algorithmic and it computes
$\omega_{f}$. The proof of Theorem 1 only works for curves with quasi-
homogeneous singularities. This follows from an argument due to C. Hertling,
see Appendix A. The curve $x^{5}+y^{5}-x^{2}y^{2}=0$ (due to A’Campo) has a
non quasi-homogeneous singularity at the origin and it turns out that the
conclusion of Theorem 1 is false in this case. For further details see Example
1.
We have implemented the computation of $\omega_{f}$ in a computer. This
$1$-form is usually of high degree and it is not clear how to produce
foliations of lowest possible degree in $E_{f}$. In order to investigate this
problem, we write $E_{f}$ in the standard Groebner basis and take its minimal
resolution. It turns out that in many interesting examples, $E_{f}$ is
actually generated by two foliations $\omega_{0}:=P_{0}dx+Q_{0}dy$ and
$\omega_{\infty}:=P_{\infty}dx+Q_{\infty}dy$. This includes the curves studied
by Lins Neto, Mendes-Pereira, and graphs of polynomial functions. For this we
have prepared the Table 1, for more details regarding this table see §6.
For smooth curves it is easy to see that the foliations given by $fdx,fdy,df$
generate $E_{f}$, see Proposition 3. For curves with only quasi-homogeneous
singularities (including smooth curves) Quillen-Suslin theorem implies that
$E_{f}\otimes_{\mathsf{k}}{\mathsf{K}}$ is actually generated by two
foliations $\omega_{0}$ and $\omega_{\infty}$, for some field extension
${\mathsf{K}}$ of ${\mathsf{k}}$. Computing $\omega_{0}$ and
$\omega_{\infty}$, and in particular the field extension ${\mathsf{K}}$ seems
to be a new problem not treated in the literature. The arguments used in the
proof of Theorem 1 are essentially valid in higher dimensions
$\mathbb{A}^{n}_{\mathsf{k}}$. However, in this paper we are interested in
holomorphic foliations, that is, in those $\omega\in E_{f}$ such that the
integrability condition $\omega\wedge d\omega=0$ holds. This trivially holds
for $n=2$, but for $n>2$ it is a nonlinear identity in $\omega$ and it is not
clear how to interpret results like Theorem 1 in this case.
The article is organized in the following way. In §2 we introduce a basis of
monomials for the Milnor vector space of $f$. In §3 we recall quasi-
homogeneous singularities and apply K. Saito’s theorem to these singularities
in order to get a local freeness statement. We prove Theorem 1 in §4 and
observe that for a curve with A’Campo singularity Theorem 1 does not hold. The
examples presented in Table 1 and even more, are discussed in §6. In §5 we use
Quillen-Suslin theorem and Serre’s GAGA in order to prove that
$E_{f}\otimes_{\mathsf{k}}{\mathsf{K}}$ is actually free for some field
extension ${\mathsf{k}}\subset{\mathsf{K}}$. We discuss this field extension
in the case of a circle. The computer codes of the present paper are written
in Singular, see [GPS01]. For the computations in this paper we have written
the procedures SyzFol, MinFol, BadPrV of foliation.lib which are available on
the second author’s webpage. 444 http://w3.impa.br/$\sim$hossein/foliation-
allversions/foliation.lib
Our sincere thanks go to J. V. Pereira for many useful discussions related to
the topic of the present paper. We thank also C. Hertling for his comments and
corrections to the present article and for writing Appendix A.
Name $f(x,y)=0$ $\left(\begin{array}[]{*{2}{c}}P_{0}&Q_{0}\\\ P_{\infty}&Q_{\infty}\end{array}\right)$ | Figure
---|---
Riccati $f(x)$ $\left(\begin{array}[]{*{2}{c}}1&0\\\ 0&f(x)\end{array}\right)$ |
Quasi-homogeneous $f(x,y)$ $\left(\begin{array}[]{*{2}{c}}f_{x}&f_{y}\\\ -\frac{\alpha_{2}}{d}y&\frac{\alpha_{1}}{d}x\end{array}\right)$ |
Graph $y-f(x)$ $\left(\begin{array}[]{*{2}{c}}ya_{1}(x)+a_{2}(x)&a_{3}(x)\\\ yb_{1}(x)+b_{2}(x)&b_{3}(x)\end{array}\right)$ |
Lins Neto ($a=3$) $(x^{a}-1)(y^{a}-1)(x^{a}-y^{a})$ $\left(\begin{array}[]{*{2}{c}}-y^{a+1}+y&x^{a+1}-x\\\ -x^{a-1}(y^{a}-1)&y^{a-1}(x^{a}-1)\end{array}\right)$ | $(x^{3}-1)(y^{3}-1)(x^{3}-y^{3})=0$
Rose $k=\frac{1}{2}$ $4x^{4}+8x^{2}y^{2}+4y^{4}-4x^{6}-12x^{4}y^{2}-12x^{2}y^{4}-4y^{6}-y^{2}$ $\left(\begin{array}[]{*{2}{c}}8\cdot x^{3}-xy^{2}&11\cdot x^{2}y+2\cdot y^{3}-y\\\ 5\cdot x^{2}y-4\cdot y^{3}+2\cdot y&x^{3}+10\cdot xy^{2}-x\end{array}\right)$ |
Lissajous $2(2x^{2}-a^{2})^{2}-a(2y-a)^{2}(y+a)$ $x=a\cos(3t),y=a\cos(4t)$ $\left(\begin{array}[]{*{2}{c}}-16\cdot xy^{2}-8a\cdot xy+8a^{2}\cdot x&12\cdot x^{2}y+6a\cdot x^{2}-6a^{2}\cdot y-3a^{3}\\\ -16\cdot x^{2}y-8a\cdot y^{2}+4a^{2}\cdot y+4a^{3}&12\cdot x^{3}+6a\cdot xy-9a^{2}\cdot x\end{array}\right)$ | $a=-1$
Deltoid $(x^{2}+y^{2})^{2}+8ax(x^{2}-3y^{2})+18a^{2}(x^{2}+y^{2})-27a^{4}$ $\left(\begin{array}[]{*{2}{c}}x^{2}-3\cdot y^{2}+6a\cdot x+9a^{2}&4\cdot xy-6a\cdot y\\\ -4\cdot xy+6a\cdot y&3\cdot x^{2}-y^{2}+6a\cdot x-9a^{2}\end{array}\right)$ |
Table 1: Curves and foliations
## 2 Milnor vector space
Recall the definition of a tame polynomial in Introduction. In the following
by degree of a polynomial we mean the weighted degree in ${\mathsf{k}}[x,y]$,
$\deg(x)=\alpha_{1},\ \deg(y)=\alpha_{2}$.
###### Proposition 1.
Let $f\in{\mathsf{k}}[x,y]$ be tame polynomial of weighted degree $d$ in
${\mathsf{k}}[x,y]$, $\deg(x)=\alpha_{1},\ \deg(y)=\alpha_{2}$. There is a
basis of monomials $x^{i}y^{j}$ for the Milnor vector space
$V_{f}:=\frac{{\mathsf{k}}[x,y]}{{\rm
jacob}(f)}\cong\frac{\Omega^{2}_{\mathbb{A}_{\mathsf{k}}^{2}}}{df\wedge\Omega^{1}_{\mathbb{A}_{\mathsf{k}}^{2}}}$
with weighted degree $\leq 2d-2\alpha_{1}-2\alpha_{2}$, and among these
monomials, only one monomial is of the highest usual degree
$2d-2\alpha_{1}-2\alpha_{2}$.
###### Proof.
The proposition for $f=g$ a homogeneous polynomial in a weighted ring
${\mathsf{k}}[x,y]$, $\deg(x)=\alpha_{1},\ \deg(y)=\alpha_{2}$ is a classical
fact due to V. Arnold and K. Saito, see [AGV85, Corollary 4, page 200].
Actually in this case, all the monomials of usual degree
$>2d-2\alpha_{1}-2\alpha_{2}$ are zero in $V_{g}$. For an arbitrary $f$ with
the last homogeneous piece $g$ in the same weighted ring, we first find a
monomial basis with the desired property for $V_{g}$ and it follows that the
same set of monomials form a basis of $V_{f}$, see [Mov07, §6] or [Mov19,
Proposition 10.7]. In this context, monomials $x^{i}y^{j}$ with
$i\alpha_{1}+j\alpha_{2}>2d-2\alpha_{1}-2\alpha_{2}$ are not necessarily zero
in $V_{f}$ and we only know that they are equivalent in $V_{f}$ to polynomials
of degree $\leq 2d-2\alpha_{1}-2\alpha_{2}$. ∎
## 3 Quasi-homogeneous singularities
Let $f=0$ be a germ of curve singularity given by a germ of a holomorphic
function $f\in{\cal O}_{\mathbb{C}^{2},0}$.
###### Theorem 2 (K. Saito [Sai80] page 270).
The ${\cal O}_{\mathbb{C}^{2},p}$-module of holomorphic $1$-forms tangent to
$f=0$ is freely generated if and only if it has two elements
$\omega_{0},\omega_{\infty}$ such that
$\omega_{0}\wedge\omega_{\infty}=fdx\wedge dy$.
Quasi-homogeneous singularities are the main example of singularities
satisfying Theorem 2.
###### Definition 2.
A germ of curve singularity $f=0$ is called quasi-homogeneous if there is a
holomorphic change of coordinates $(x,y)$ in $(\mathbb{C}^{2},0)$ such that
$(x(0),y(0))=(0,0)$ and $f(x,y)=a(x,y)\cdot g(x,y),\ \ a\in{\cal
O}_{\mathbb{C}^{2},0},a(0)\not=0$ and $g$ is a quasi-homogeneous polynomial in
$x,y$, that is, there are $\alpha_{1},\alpha_{2}\in\mathbb{Z}$ such that $g$
is homogeneous in the weighted ring $\mathbb{C}[x,y],\ \deg(x)=\alpha_{1},\ \
\deg(y)=\alpha_{2}$.
Let $\eta:=\frac{1}{\deg(g)}(\alpha_{1}xdy-\alpha_{2}ydx)$. We have
$dg\wedge\eta=gdx\wedge dy$, and so by Theorem 2, the module of $1$-forms
tangent to $g=0$ is freely generated by $dg,\eta$.
###### Theorem 3 (K. Saito [Sai71]).
A germ of curve singularity $f=0$ is quasi-homogeneous if and only if $f$
belongs to the Jacobian ideal ${\rm jacob}(f):=\langle\frac{\partial
f}{\partial x},\frac{\partial f}{\partial y}\rangle$.
## 4 Proof of Theorem 1
In $V_{f}$ we consider the linear map $A_{f}$ defined by
$A_{f}:V_{f}\to V_{f},A_{f}([P])=[Pf]$
Let $p(t)$ be the minimal polynomial of $A_{f}$. The critical values of $f$
are exactly the zeros of $p(t)$, see for instance [CLO05, Proposition 2.7 page
150] [Mov19, §10.9]. If $0\in\bar{\mathsf{k}}$ is not a critical value of $f$
then $\\{f=0\\}$ is a smooth curve. Suppose that $f=0$ is singular, and hence,
$A_{f}$ has a non-trivial kernel. We write $p(t)=tq(t)$. By definition of a
minimal polynomial
(2) $fq(f)=0,\ q(f)\not=0\text{ in }V_{f}.$
The polynomial $q(f)$ is of big degree. Proposition 1 implies that we can
simplify $q(f)$ in $V_{f}$ and obtain a polynomial $\Theta_{f}$ of weighted
degree $\leq 2d-2\alpha_{1}-2\alpha_{2}$ which is equal to $q(f)$ in $V_{f}$.
We obtain
(3) $f\Theta_{f}dx\wedge dy=df\wedge\omega_{f},\ $
Note that we do not have any control on the degree of $\omega_{f}$. Note also
that $\Theta_{f}$ and $\omega_{f}$ depend on the monomial basis that we have
chosen in Proposition 1.
The exponent of the affine curve $f=0$ is the number $m$ in $p(t)=t^{m}Q(t),\
Q(0)\not=0$. The exponent of a curve is also the maximum size of the Jordan
blocks of $A_{f}$ associated to the eigenvalue $0$. The exponent of a critical
point $p$ of $f$ with $f(p)=0$ is the minimum number $\tilde{m}$ such that
$f^{\tilde{m}}$ is zero in the local Milnor vector space $V_{p}:={\cal
O}_{\mathbb{A}_{\mathsf{k}}^{2},p}/{\rm jacob}(f)$.
###### Proposition 2.
Let ${\mathsf{k}}$ be an algebraically closed field of characteristic zero. We
have an isomorphism
$V_{f}\cong\oplus_{p\in P_{f}}\frac{{\cal
O}_{\mathbb{A}_{\mathsf{k}}^{2},p}}{{\rm jacob}(f)},$
induced by canonical restriction, where $p$ runs through the set $P_{f}$ of
critical points of $f$. In particular, each piece in the above summand is
invariant under the multiplication by $f$ map $A_{f}$. Let
$\lambda_{1},\lambda_{2},\ldots,\lambda_{k}$ be the critical values of $f$ and
$m_{i},\ i=1,2,\ldots,k$ be the maximum of exponents of the singularities in
$f(x,y)-\lambda_{i}=0$. Then the minimal polynomial of $A_{f}$ is
$p(t):=(t-\lambda_{1})^{m_{1}}(t-\lambda_{2})^{m_{2}}\cdots(t-\lambda_{k})^{m_{k}}$.
###### Proof.
This is an immediate corollary of a well-known fact in commutative algebra,
see [CLO05, Theorem 2.2 ]. It also follows from Max Noether’s theorem, see
[GH94, page 703]. For the surjectivity of the restriction map, we must modify
the proof of Noether’s theorem. Note that $A_{f}-f(p){\rm Id}$ restricted to
$\frac{{\cal O}_{\mathbb{A}_{\mathsf{k}}^{2},p}}{{\rm jacob}(f)}$ is nilpotent
of order which is the exponent of $p$. ∎
Theorem 1 is a consequence of the following:
###### Theorem 4.
We have
1. 1.
If there exists $\Theta\in{\rm Kernel}(A_{f})$ such that $\Theta\cdot
V_{f}=\ker(A_{f})$ then $E_{f}$ is generated by $fdx,fdy,df,\omega_{\Theta}$,
where $f\Theta dx\wedge dy=df\wedge\omega_{\Theta}$.
2. 2.
If the Jordan blocks of $A_{f}$ associated to the eigenvalue $0$ have the same
size then $\Theta_{f}$ satisfies $\Theta_{f}\cdot V_{f}=\ker(A_{f})$.
3. 3.
If all the singularities of $C$ are quasi-homogeneous then the Jordan blocks
of $A_{f}$ associated to the eigenvalue $0$ have the size $1$.
###### Proof.
Proof of 1. For $\omega\in E_{f}$ we have $f\Theta_{1}dx\wedge
dy=df\wedge\omega$ for some polynomial $\Theta_{1}\in{\mathsf{k}}[x,y]$. This
implies that $\Theta_{1}\in\ker(A_{f})$ and by our hypothesis, we have
$\Theta_{1}=P\Theta$ in $V_{f}$ for some $P\in{\mathsf{k}}[x,y]$. We write
this as $\Theta_{1}dx\wedge dy-P\Theta dx\wedge dy=df\wedge\eta$. After
multiplication of this equality by $f$ we get
$(\omega-P\omega_{f}-f\eta)\wedge df=0$ which implies the result. De Rham
lemma, see for instance [Sai76] in the case of homogeneous polynomials and
[Mov19, Proposition 10.3] for tame polynomials, implies that $\omega$ is of
the desired format.
Proof of 2. It is enough to prove that the sequence
(4) $V_{f}\stackrel{{\scriptstyle
q(A_{f})}}{{\rightarrow}}V_{f}\stackrel{{\scriptstyle
A_{f}}}{{\rightarrow}}V_{f}$
is exact. For this we can assume that ${\mathsf{k}}$ is algebraically closed,
as the exactness of a sequence of vector spaces is independent of whether
${\mathsf{k}}$ is algebraically closed or not. It follows from (2) that ${\rm
Image}(q(A_{f}))\subset{\rm Kernel}(A_{f})$. The non-trivial assertion is
${\rm Kernel}(A_{f})\subset{\rm Image}(q(A_{f}))$. Let $V_{f}=V_{1}\oplus
V_{2}$ such that $A_{f}\mid_{V_{1}}$ is nilpotent and $A_{f}\mid_{V_{2}}$ is
invertible. For this we simply write $A_{f}$ in Jordan block format, $V_{1}$
is constructed from the blocks with eigenvalue $0$ and $V_{2}$ from the
remaining blocks. From $p(t)=tq(t)=t^{m}Q(t),\ Q(0)\not=0$, it follows that
$q(A_{f})\mid_{V_{1}}=A_{f}^{m-1}\circ B_{f}$, where
$B_{f}:=Q(A_{f})\mid_{V_{1}}:V_{1}\to V_{1}$ is invertible. Therefore, we can
assume that $p(t)=t^{m},q(t)=t^{m-1}$ and $Q=1$. Now, we use the following
statement in linear algebra. Let $A$ be an $n\times n$ matrix with entries in
$\bar{\mathsf{k}}$ and all eigenvalues equal to zero. Let also $m$ be the
maximal size of its Jordan blocks. Then ${\rm Kernel}(A)={\rm Image}(A^{m-1})$
if and only if all the Jordan blocks of $A$ are of the same size. In order to
see this fact, let
$e_{1}^{1},e_{2}^{1},\ldots,e_{k_{1}}^{1},e_{1}^{2},e_{2}^{2},\ldots,e_{k_{2}}^{2},\ldots\ldots,e_{1}^{s},e_{2}^{s},\ldots,e_{k_{s}}^{s},k_{1}\leq
k_{2}\leq\cdots\leq k_{s}=m.$
be a basis of $\bar{\mathsf{k}}^{n}$ such that $A$ in each block
$e_{1}^{j},\ldots,e_{k_{j}}^{j}$ is a shifting map and
$A_{f}(e_{k_{j}}^{j})=0$. By definition of $m$ we have ${\rm
Image}(A^{m-1})\subset{\rm Kernel}(A)$. Moreover, ${\rm Kernel}(A)\subset{\rm
Image}(A^{m-1})$ if and only if $k_{1}=k_{2}=\cdots=k_{s}$.
Proof of 3. The third part follows from Proposition 2 and the fact that the
exponent of a quasi-homogeneous singularity is $1$. ∎
###### Remark 1.
Let $[\Theta_{i}],\ i=1,2,\ldots,a$ be a basis for the ${\mathsf{k}}$-vector
space $\ker(A_{f})$ and write $df\wedge\omega_{i}=f\Theta_{i}dx\wedge dy$. The
${\mathsf{k}}[x,y]$-module $E_{f}$ is generated by $fdx,fdy,df,\omega_{i},\
i=1,2,\ldots,a$. This is an immediate consequence of definitions.
###### Remark 2.
A careful analysis of the proof of Theorem 1 shows that this theorem is true
for curves $f=0$ such that the multiplication by $f$ map in its Milnor vector
space $V_{f}$ has Jordan blocks of the same size. Jordan blocks of size $1$
correspond to quasi-homogeneous singularities. In Appendix A, C. Hertling
proves that there is no curve singularity such that multiplication by $f$ in
its Milnor vector space has only Jordan blocks of size $2$.
###### Proposition 3.
A foliation which leaves a smooth curve $C\in\mathbb{P}^{2}_{\mathsf{k}}$
invariant is of the form
${\cal F}(Pdf+f\omega),\ \omega\in\Omega_{\mathbb{A}_{k}^{2}}^{1},\
P\in\Omega_{\mathbb{A}_{k}^{2}}^{0}$
in an affine chart
$\mathbb{A}_{\mathsf{k}}^{2}\subset\mathbb{P}^{2}_{\mathsf{k}}$. In other
words the ${\mathsf{k}}[x,y]$-module $E_{f}$ defined in (1) is generated by
$fdx,fdy,df$.
###### Proof.
We take a line in $\mathbb{P}^{2}_{\mathsf{k}}$ which intersects $C$
transversally at $\deg(C)$ points, and hence, in the affine chart which is its
complement, the curve $C$ is given by the tame polynomial $f(x,y)=0$. If
$\omega\wedge df=fPdx\wedge dy$ then $P$ is in the kernel of the map $A_{f}$.
Since $f=0$ is smooth, we conclude that $P$ itself is zero in $V_{f}$, and
hence, $Pdx\wedge dy=df\wedge\alpha$ for some
$\alpha\in\Omega_{\mathbb{A}^{2}_{\mathsf{k}}}^{1}$. Therefore,
$df\wedge(f\alpha-\omega)=0$. De Rham lemma for tame polynomials, see [Mov19,
Proposition 10.3], implies that $\omega$ is of the desired format. ∎
###### Example 1.
The singularity $(0,0)$ of the curve $f:=x^{5}+y^{5}-x^{2}y^{2}=0$ is called
A’Campo singularity and it is not quasi-homogeneous. The polynomial $f$ has
two critical values $s=0,-\frac{16}{3125}$. Over the critical value $s=0$, we
have only the singularity $(0,0)\in\mathbb{A}^{2}_{\mathsf{k}}$ of Milnor
number $11$. In this case $A_{f}$ has $10$ Jordan blocks, one is of size $2$
and $9$ of size $1$. Over the critical value $s=-\frac{16}{3125}$ we have $5$
singularities of Milnor number $1$. In this case, $A_{f}$ has $5$ Jordan
blocks of size $1$. The Macaulay code gives us the following three generators
of $E_{f}$:
loadPackage "VectorFields"
R=QQ[x,y];
f=f=x^5+y^5-x^2*y^2;
derlog(ideal (f));
-->image | -25x2y2+6xy -25x3y+5y3+4x2 -5x4+3xy2 |
--> | -25xy3+5x3+4y2 -25x2y2+6xy -5x3y+2y3 |
The columns $[P,Q]^{\rm tr}$ of the output matrix are vector fields
$P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$. Such three vector
fields generate the $\mathbb{Q}[x,y]$-module of vector fields tangent to
$f=0$. The corresponding $1$-forms must be written as $Pdy-Qdx$. Let
$\omega_{1},\omega_{2},\omega_{3}$ be such $1$-forms. The minimal polynomial
of $A_{f}$ turns out to be $t^{2}(3125t+16)$ and
$\omega_{f}=-\frac{4}{25}x^{2}y\omega_{1}.$
This implies that $\omega_{1}$ is not in the module generated by
$df,fdx,fdy,\omega_{f}$. The real locus of $f=0$ is depicted in Figure 1.
Figure 1: $x^{5}+y^{5}-x^{2}y^{2}=0$
For our computations we have used the following code.
LIB "foliation.lib";
ring r=0, (x,y),dp; poly f=x5+y5-x2*y2;
matrix A=mulmat(f,f); list ll=jordan(A); print(jordanmatrix(ll));
vector v1=[-(-25xy3+5x3+4y2), -25x2y2+6xy]; vector v2=[-(-25x2y2+6xy),-25x3y+5y3+4x2]; vector v3=[-(-5x3y+2y3),-5x4+3xy2];
vector df=[diff(f,x),diff(f,y)]; vector fdy=[0,f]; vector fdx=[f,0]; module m=v1,v2,v3;
division(df,m); division(fdx,m); division(fdy,m);
poly disc=Discrim(f); list l=factorize(disc); disc=x^2*(3125x+16);
ideal I=std(jacob(f)); poly p=reduce(subst(disc/var(1), var(1),f), I);
list l=division(p*f, jacob(f)); l; vector om=[-l[1][2,1], l[1][1,1]]; division(om,m);
module m2=df,fdx,fdy,om; division(v1, m2); |
## 5 A consequence of Quillen-Suslin theorem
In Theorem 1 we have not assumed that ${\mathsf{k}}\subset\mathbb{C}$ is an
algebraically closed field. It turns out that if we use the algebraic closure
of ${\mathsf{k}}$ then $E_{f}$ is free of rank $2$.
###### Proposition 4.
Let $f\in{\mathsf{k}}[x,y]$ be a tame polynomial and assume that $f=0$ is
smooth or at most it has quasi-homogeneous singularities. The
$\bar{\mathsf{k}}[x,y]$-module $E_{f}\otimes_{\mathsf{k}}\bar{\mathsf{k}}$ is
free of rank $2$.
###### Proof.
The ${\mathsf{k}}[x,y]$-module $E_{f}$ is not necessarily projective, however,
$E_{f}\otimes_{\mathsf{k}}\bar{\mathsf{k}}$ turns out to be projective. The
reason is as follows. We compactify $\mathbb{A}^{2}_{\mathsf{k}}$ in
$\mathbb{P}^{2}_{\mathsf{k}}$ and consider the curve $C$ induced by $f=0$ in
$\mathbb{P}^{2}_{\mathsf{k}}$. For $\alpha_{1}=\alpha_{2}=1$ this curve
intersects the line at infinity transversely, and hence, it has no singularity
at infinity. However, for arbitrary $\alpha_{i}$’s it might have singularities
at infinity. We perform a desingularization of singularities of $C$ at
infinity and get a surface $M$ defined over ${\mathsf{k}}$ and with a chart
$\mathbb{A}^{2}_{\mathsf{k}}\subset M$. The curve given by $f=0$ in
$\mathbb{A}^{2}_{\mathsf{k}}$ induces a curve (we call it again $C$) in $M$
and it has now only smooth points in the compactification divisor
$M-\mathbb{A}^{2}_{\mathsf{k}}$. Note that for all these we do not need to
assume that ${\mathsf{k}}$ is algebraically closed. Let $M^{{\rm an}}$ be the
underlying complex manifold of $M/{\mathsf{k}}$ for a fixed embedding
${\mathsf{k}}\subset\mathbb{C}$.
Let ${\cal S}^{\rm alg}$ be the subsheaf of $\Omega^{1}_{M/{\mathsf{k}}}$
containing differential $1$-forms tangent to the curve given by $C$ in $M$. In
a similar way we define ${\cal S}^{\rm an}\subset\Omega^{1}_{M^{\rm an}}$ in
the holomorphic context. By definition ${\cal S}^{\rm an}$ is the
analytification of the algebraic sheaf ${\cal S}^{\rm
alg}\otimes_{\mathsf{k}}\mathbb{C}$. Theorem 2 applied to quasi-homogeneous
singularities implies that ${\cal S}^{\rm an}$ is a locally free sheaf. Note
that here we use the desingularization process as above: the points at
infinity of $f=0$ are smooth. By Serre’s GAGA, ${\cal S}^{\rm
alg}\otimes_{\mathsf{k}}\mathbb{C}$ is also a locally free sheaf. Now we look
at ${\cal S}^{\rm alg}\otimes_{\mathsf{k}}\mathbb{C}$ in the affine chart
$\mathbb{C}^{2}\subset M^{{\rm an}}$ and conclude that ${\cal S}^{\rm
alg}\otimes_{{\mathsf{k}}}\bar{\mathsf{k}}$ is a locally free sheaf in
$\mathbb{A}^{2}_{\bar{\mathsf{k}}}$. For an algebraically closed field
$\bar{\mathsf{k}}$, locally free sheaves over
$\mathbb{A}^{2}_{\bar{\mathsf{k}}}$ are in one to one correspondence with
projective modules: the correspondence is given by taking global sections. Now
by Quillen-Suslin theorem, see for instance [Lan02, Theorem 3.7, page 850]
projective $\bar{\mathsf{k}}[x,y]$-modules are free. Note that for Quillen-
Suslin theorem we do not need that the base field is algebraically closed,
however, for the correspondence with locally free sheaves we need this fact.
∎
###### Remark 3.
In general, we do not know how to write the local generators of ${\cal S}^{\rm
alg}$ around each point, however, for smooth curves this is as follows: In the
chart $U_{0}:f_{x}\not=0$ (resp. $U_{1}:f_{y}\not=0$) it is freely generated
by $df,fdy$ (resp. $df,fdx$). For instance,
$fdx=\frac{f}{f_{x}}df-\frac{f_{y}}{f_{x}}fdy$. If $f=0$ is smooth,
$U_{0},U_{1}$ cover $\mathbb{A}^{2}_{\mathsf{k}}$.
###### Remark 4.
If the module $E_{f}$ has two elements $\omega_{0},\omega_{\infty}$ with
$\omega_{0}\wedge\omega_{\infty}=fdx\wedge dy$ then they generate $E_{f}$ as
${\mathsf{k}}[x,y]$-module. This is as follows. By Theorem 2, the sheaf ${\cal
S}^{{\rm an}}$ is locally free, and hence a similar argument as in Proposition
4 implies that $E_{f}\otimes_{\mathsf{k}}\mathbb{C}$ is generated by two
elements $\check{\omega}_{0},\check{\omega}_{\infty}$. Moreover, these two
elements generate the stalk of ${\cal S}^{\rm an}$ at any point of
$\mathbb{C}^{2}$. It follows that
$\check{\omega}_{0}\wedge\check{\omega}_{\infty}=fdx\wedge dy$. We write
$\omega_{0},\omega_{\infty}$ in terms of
$\check{\omega}_{0},\check{\omega}_{\infty}$ and we get a $2\times 2$ matrix
which has non-zero constant determinant.
Let $\omega_{0},\omega_{\infty}$ be two generators of the
$\bar{\mathsf{k}}[x,y]$-module $E_{f}\otimes_{\mathsf{k}}\bar{\mathsf{k}}$. It
turns out that we have a finite extension ${\mathsf{K}}$ of ${\mathsf{k}}$
such that $\omega_{0}$ and $\omega_{\infty}$ are defined over ${\mathsf{K}}$
and $E_{f}\otimes_{\mathsf{k}}{\mathsf{K}}$ is freely generated by
$\omega_{0},\omega_{\infty}$. It is a natural question to bound
$[{\mathsf{K}}:{\mathsf{k}}]$ in terms of some arithmetic invariants of the
curve $C$. The case of a circle might be enlightening, see Example 2.
## 6 Examples
In this section we explain many examples of curves $f=0$ such that $E_{f}$ is
generated by two elements (including those in Table 1). In all these examples
$E_{f}$ is generated by two foliations $\omega_{0},\omega_{\infty}$ with
$\omega_{0}\wedge\omega_{\infty}=fdx\wedge dy$. The polynomial $1$-form
$\omega_{f}$ is usually big and so we do not reproduce it here. All the
figures in Table 1 are the real locus of $f=0$, except for Lins Neto’s
example, whose figure is an artistic way to depict the arrangement of lines
and it is taken from the original article [Lin02].
###### Example 2.
The module of $1$-forms tangent to $f:=xy-1=0$ is freely generated by
$\omega_{0}:=ydf-fdy=y^{2}dx+dy,\ \omega_{\infty}:=fdx,$
which satisfy $\omega_{0}\wedge\omega_{\infty}=-fdx\wedge dy$. This follows
from Proposition 3 and the identities:
(5) $df=x\omega_{0}+y\omega_{\infty},\ \ \
fdy=f\omega_{0}-y^{2}\omega_{\infty}.$
Over $\mathbb{Q}(i)$ this curve is isomorphic to $f:=x^{2}+y^{2}-1=0$. In
fact, the transformation $(x,y)\mapsto(x+iy,x-iy)$ sends the circle
$x^{2}+y^{2}-1=0$ to $xy-1=0$. The pull-back of the above differential forms
is
$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\Re(\omega_{0})+i\Im(\omega_{0}):=(x^{2}-y^{2}+1)dx+2xydy+i((x^{2}-y^{2}-1)dy-2xydx),$
$\displaystyle\omega_{\infty}$ $\displaystyle=$
$\displaystyle\Re(\omega_{\infty})+i\Im(\omega_{\infty})=fdx+ifdy.$
It would be interesting to prove that the submodule of
$\Omega^{1}_{\mathbb{A}^{2}_{\mathbb{Q}}}$ tangent to $x^{2}+y^{2}-1$ is not
generated by two elements defined over $\mathbb{Q}$. For our computations in
this example we have used
LIB "foliation.lib";
ring r=(0,z), (x,y), dp; minpoly=z^2+1;
poly f=x*y-1; matrix P=transpose(MinFol(f,1)[1]); P;
vector df=[diff(f,x),diff(f,y)]; MinFol(f,1, df); vector fdy=[0,f]; MinFol(f,1, fdy);
poly g=x^2+y^2-1; vector v1=[(x^2-y^2+1),2*x*y]; vector w1=[-2*x*y, (x^2-y^2-1)]; vector v2=[g,0]; vector w2=[0,g];
module m=v1+z*w1,v2+z*w2; division(v1,m); division(w1,m); division(v2,m); division(w2,m);
division(v1-z*w1,m); division(v2-z*w2,m); vector dg=[diff(g,x),diff(g,y)]; division(dg,m); MinFol(g,1, dg);
###### Example 3 (Riccati).
Let us consider the case in which $f$ does not depend on $y$, and hence, $f=0$
is a union of $d$ parallel to $y$ axis lines. In this case $\omega_{0}=dx$ and
$\omega_{\infty}=f(x)dy$. The Riccati foliations given by
$(p_{2}(x)y^{2}+p_{1}(x)y+p_{0}(0))dx+f(x)dy$ are in $E_{f}$.
###### Example 4 (Quasi-homogeneous singularities).
These are curves $C:f=0$ given by a homogeneous tame polynomial $f$ of
weighted degree $d$ in the weighted ring ${\mathsf{k}}[x,y],\
\deg(x)=\alpha_{1},\ \ deg(y)=\alpha_{2}$, see Definition 1. This is also the
algebraic version of Definition 2. In this case $E_{f}$ is generated by
$\omega_{0}=df,\omega_{\infty}:=\frac{\alpha_{1}}{d}xdy-\frac{\alpha_{2}}{d}ydx$.
###### Example 5.
The arrangement of lines given by
$f=(x^{a}-1)(y^{a}-1)(x^{a}-y^{a}).$
for $a=3$ has been studied by Lins Neto in [Lin02]. In this case $E_{f}$ is
generated by
$\omega_{0}:=-(y^{a}-1)x^{a-1})dx+(x^{a}-1)y^{a-1}dy,\ \
\omega_{\infty}:=-(y^{a+1}-y)dx+(x^{a+1}-x)dy,\ $
The birational
$\mathbb{A}_{\mathsf{k}}^{2}\dashrightarrow\mathbb{A}_{\mathsf{k}}^{2}$ given
by $(x,y)\mapsto(\frac{1}{x},\frac{1}{y})$ in the affine chart $(x,y)$ maps
$\omega_{0}$ to $\omega_{\infty}$ and vice versa. For $a=3$ one can prove that
${\cal F}(\omega_{0}+t\omega_{\infty}),\ \ t\in\mathbb{P}^{1}_{\mathsf{k}}$
has a first integral if and only if $t$ is a constant in
$\mathbb{Q}(e^{\frac{2\pi i}{3}})$. The degree of such a first integral has
been computed in [Med13]. Another example due to Lins Neto is
$f=4y^{2}(1-3x)-4x^{3}+(3x^{2}+y^{2})^{2}$ which is up to a linear
transformation is the deltoid in Table 1.
###### Example 6.
The arrangement of lines given by
$f=(x^{2}-1)(y^{2}-1)(x^{2}-(2\tau+1)^{2})(y^{2}-(2\tau+1)^{2})\cdot(x^{2}-y^{2})$
$\cdot(y+1+\frac{1}{\tau}(x+1))(y+1+\tau(x+1))\cdot(y-1+\tau(x-1))(y-1+\frac{1}{\tau}(x-1))$
with $\tau^{2}-\tau-1=0$ is studied in [MP05] and in this case $E_{f}$ is
generated by
$\omega_{0}:=-(y^{2}-(2\tau+1)^{2})(y^{2}-1)(y+(2\tau-1)x)dx+(x^{2}-(2\tau+1)^{2})(x^{2}-1)(x+(2\tau-1)y)dy$
and $\omega_{\infty}$ whose expression can be found in the mentioned
reference. Another example from this reference is
$f=-1728x^{5}+720x^{3}y-80xy^{2}+64(5x^{2}-y)^{2}+y^{3}$. In [KN88] the
authors describe the 1-forms $\omega_{0},\omega_{\infty}$ in another
coordinate system, and it turn out that these are defined over $\mathbb{Q}$.
###### Example 7.
[Graph of a function] In this example we consider a curve of the form $y=f(x)$
which is smooth and of genus zero. Let $d:=\deg(f)$. For $d=2,3,\cdots,11$ we
can verify the following conjecture: For a generic polynomial
$f(x)\in{\mathsf{k}}[x]$ of degree $d$ (in a Zariski open subset of the
parameter space of $f$), the ${\mathsf{k}}[x,y]$-module $E_{f}$ is generated
by
$\omega_{0}:=(ya_{1}(x)+a_{2}(x))dx+a_{3}(x)dy,\ \
\omega_{\infty}:=(yb_{1}(x)+b_{2}(x))dx+b_{3}(x)dy,$
where for $d$ even $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3}\in{\mathsf{k}}[x]$ are
respectively of degree
$\frac{d}{2}-2,\frac{d}{2},\frac{d}{2}-1,\frac{d}{2}-1,\frac{d}{2},\frac{d}{2}$,
and for $d$ odd they are respectively of degree
$\frac{d-3}{2},\frac{d-1}{2},\frac{d-1}{2},\frac{d-5}{2},\frac{d+1}{2},\frac{d-3}{2}$.
This is equivalent to the existence of $a_{i}$ and $b_{i}$’s with
$a_{1}b_{2}-a_{2}b_{1}=-f^{\prime},\ \ \ fa_{1}+a_{2}+f^{\prime}a_{3}=0,\ \
fb_{1}+b_{2}+f^{\prime}b_{3}=0.$
From these three equalities we can derive
$a_{1}b_{3}-a_{3}b_{1}=1,\ \ a_{2}b_{3}-a_{3}b_{2}=-f.$
We can also verify the existence of $\omega_{0}$ and $\omega_{\infty}$ for
random choices of $f\in\mathbb{Q}[x,y]$ of higher degree. The three canonical
foliations can be written in terms of generators:
$\displaystyle(y-f)dx$ $\displaystyle=$ $\displaystyle
b_{3}\omega_{0}-a_{3}\omega_{\infty},$ $\displaystyle(y-f)dy$ $\displaystyle=$
$\displaystyle-(yb_{1}+b_{2})\omega_{0}+(ya_{1}+a_{2})\omega_{\infty}$
$\displaystyle d(y-f)$ $\displaystyle=$ $\displaystyle-
b_{1}\omega_{0}+a_{1}\omega_{\infty}.$
For the computations in this example we have used the following code:
LIB "foliation.lib"; int d=8;
ring r=(0,t(0..d-2)), (x,y), dp; int i=1; int j;
poly f=-y+x^d; for (i=0; i<=d-2; i=i+1){f=f+t(i)*x^i;}
//--Use the next command for a random choice of f.
//--poly f=RandomPoly(list(x,y),d,-19,10); f=subst(f,y,1); f=f-y;
matrix P=transpose(MinFol(f,1)[1]); poly co=det(P)/f; co*f-det(P);
matrix Q1=diff(P,y); intmat degQ1[2][2]; matrix Q2=P-y*Q1; intmat degQ2[2][2];
for (i=1; i<=2; i=i+1)
{for (j=1; j<=2; j=j+1){degQ1[i,j]=deg(Q1[i,j]); degQ2[i,j]=deg(Q2[i,j]);}}
f; P; degQ1; degQ2;
###### Example 8 (Rose with $k=\frac{1}{2}$).
In this case the foliation ${\cal F}(\omega_{0})$ has the first integral
$F:=\frac{(x^{2}+y^{2}-1/3)^{3}}{36x^{2}+9y^{2}-4}$. Its generic fiber is
smooth and has two singular points at infinity. It has three critical values
$t=0,\frac{1}{108},\infty$. The fiber over $t=\frac{1}{108}$ is our initial
curve $f=0$ which has single non-degenerated singularity (Milnor number equal
to one). Since $f=0$ is a rational curve and all the fibers intersects the
line at infinity in the same way, we conclude that the genus of a generic
fiber of $F$ is one. If we set
$\check{\omega}_{0}=(8x-y)dx+(11x+2y-1)dy,\ \
\check{\omega}_{\infty}=(5xy-4y^{2}+2y)dx+(x^{2}+10xy-x)dy,\ \ $
and $\pi:\mathbb{C}^{2}\to\mathbb{C}^{2}:(x,y)\mapsto(x^{2},y^{2})$ then we
have
$\pi^{*}\check{\omega}_{0}=2\omega_{0},\ \
\pi^{*}\check{\omega}_{\infty}=2xy\omega_{\infty},$
This shows that this example is birational to the case of a quasi-homogeneous
singularity. We can use Katz-Grothendieck conjecture for vector fields in
order to investigate whether a foliation by curves has a first integral or
not. For this we have written the code BadPrV which computes the bad and good
primes of a vector field. For instance, before computing the first integral of
${\cal F}(\omega_{0})$ by hand, we used this in order to be sure that such a
foliation has a first integral.
LIB "foliation.lib";
ring r=(0,t), (x,y), dp;
poly f=4*x^4+8*x^2*y^2+4*y^4-4*x^6-12*x^4*y^2-12*x^2*y^4-4*y^6-y^2;
matrix P=transpose(MinFol(f,1)[1]); P;
poly l=(x^2+ y^2-1/3)^3-(36*x^2 + 9*y^2-4)*t; (P[1,1]*diff(l,y)-P[1,2]*diff(l,x))/l;
list vf=P[1,2], -P[1,1]; BadPrV(vf, 40);
The 19th century has produced a lot of curves which are named after many
engineers, astronomer and mathematicians. The website mathcurve.com contains a
rather full list of such curves. Among all these curves $f=0$, those with
$E_{f}$ generated by two elements, seem to be rare. The Lissajous and deltoid
are among them, as shown in Table 1.
## Appendix A Nonexistence of a curve singularity $f$ where all Jordan blocks
of the multiplication by $f$ in its Milnor vector space have size $2$ (By
Claus Hertling)
###### Proposition 5.
Let $f\in\mathbb{C}\\{x,y\\}$ be a holomorphic function germ with $f(0)=0$ and
with an isolated singularity at $0\in\mathbb{C}^{2}$. Let $A_{f}$ be the
multiplication by $f$ in its Milnor vector space
$Q_{f}:=\mathbb{C}\\{x,y\\}/\textup{jacob}(f)$. Then $A_{f}$ is nilpotent and
all Jordan blocks of $A_{f}$ have size 1 or 2. Not all Jordan blocks of
$A_{f}$ have size 2.
Let us call a singularity $f\in\mathbb{C}\\{x,y\\}$ such that all Jordan
blocks of $A_{f}$ have size 2 Jordan block extreme. Proposition 5 says that
such singularities do not exist. If they would exist, one could allow in
Theorem 1 that either all singularities on $C$ are quasi-homogeneous or that
all singularities on $C$ are Jordan block extreme. This follows from Theorem
4.
The nonexistence of Jordan block extreme singularities is not really
surprising. But it is also not trivial. The following proof uses the Gauss-
Manin connection, the spectral numbers and the fact that Hertling’s variance
conjecture on the spectral numbers is true in the case of curve singularities
[Bré04]. The idea is to show that a Jordan block extreme curve singularity
would have spectral numbers which violate the variance conjecture.
Proof of Proposition 5: First we have to review the spectral numbers and their
background. For this, we start with an arbitrary holomorphic function germ
$f\in{\mathcal{O}}_{\mathbb{C}^{n+1},0}=\mathbb{C}\\{x_{0},...,x_{n}\\}$ with
$f(0)=0$ and with an isolated singularity at 0, here $n\geq 1$. Two references
in book form for the following are [AGZV88] and [Her02].
Choose a good representative $\tilde{f}:X\to\Delta$, with
$B_{\varepsilon}\subset\mathbb{C}^{n+1}$ a small ball around 0,
$\Delta=\Delta_{\delta}\subset\mathbb{C}$ a very small disk around 0 (so
$0<\delta\ll\varepsilon\ll 1$) and $X:=B_{\varepsilon}\cap f^{-1}(\Delta)$.
Consider its cohomology bundle
$\bigcup_{t\in\Delta^{*}}H^{n}(\tilde{f}^{-1}(t),\mathbb{C})$, here
$\Delta^{*}:=\Delta-\\{0\\}$. It is a flat vector bundle of rank
$\mu\in\mathbb{N}$. The covariant derivative on it with respect to the
coordinate vector field of the coordinate $t$ on $\Delta$ is called
$\partial_{t}$. We define
$\displaystyle Q_{f}$ $\displaystyle:=$
$\displaystyle{\mathcal{O}}_{\mathbb{C}^{n+1},0}/(\textup{jacob}(f))\quad\textup{Milnor
vector space},$ $\displaystyle\Omega_{f}$ $\displaystyle:=$
$\displaystyle\Omega^{n+1}_{\mathbb{C}^{n+1},0}/df\land\Omega^{n}_{\mathbb{C}^{n+1},0},$
$\displaystyle H_{0}^{\prime\prime}$ $\displaystyle:=$
$\displaystyle\Omega^{n+1}_{\mathbb{C}^{n+1},0}/df\land
d\Omega^{n-1}_{\mathbb{C}^{n+1},0}\quad\textup{Brieskorn lattice},$
$\displaystyle H_{0}^{\prime}$ $\displaystyle:=$ $\displaystyle
df\land\Omega^{n}_{\mathbb{C}^{n+1},0}/df\land
d\Omega^{n-1}_{\mathbb{C}^{n+1},0},$ $\displaystyle V^{>-\infty}$
$\displaystyle:=$ $\displaystyle\\{\textup{germs at 0 of sections in the
cohomology bundle of moderate growth}\\},$ $\displaystyle C^{\alpha}$
$\displaystyle:=$ $\displaystyle\\{\sigma\in
V^{>-\infty}\,|\,(t\partial_{t}-\alpha)^{n+1}(\sigma)=0\\},$ $\displaystyle
V^{\alpha}$ $\displaystyle:=$
$\displaystyle\bigoplus_{\beta\in[\alpha,\alpha+1)}\mathbb{C}\\{t\\}\cdot
C^{\beta},$ $\displaystyle V^{>\alpha}$ $\displaystyle:=$
$\displaystyle\bigoplus_{\beta\in(\alpha,\alpha+1]}\mathbb{C}\\{t\\}\cdot
C^{\beta}.$
$Q_{f}$ and $\Omega_{f}$ are $\mathbb{C}$-vector spaces of dimension $\mu$.
$V^{>-\infty}$ is $\mathbb{C}\\{t\\}[t^{-1}]$-vector space of dimension $\mu$.
It contains the spaces $C^{\alpha}$, $V^{\alpha}$, $V^{>\alpha}$,
$H_{0}^{\prime\prime}$ and $H_{0}^{\prime}$. For any $\alpha\in\mathbb{R}$
$V^{>-\infty}=\bigoplus_{\beta\in[\alpha,\alpha+1)}\mathbb{C}\\{t\\}[t^{-1}]C^{\beta},$
$V^{\alpha}$, $V^{>\alpha}$, $H_{0}^{\prime\prime}$ and $H_{0}^{\prime}$ are
free $\mathbb{C}\\{t\\}$-modules of rank $\mu$. And $V^{\alpha}$ for
$\alpha>-1$, $V^{>\alpha}$ for $\alpha\geq-1$, $H_{0}^{\prime\prime}$ and
$H_{0}^{\prime}$ are free $\mathbb{C}\\{\\{\partial_{t}^{-1}\\}\\}$-modules of
rank $\mu$ ($\mathbb{C}\\{\\{\partial_{t}^{-1}\\}\\}$ is the ring of power
series of Gevrey class 1 in $\partial_{t}^{-1}$.) The Brieskorn lattice
$H_{0}^{\prime\prime}$ satisfies
$V^{>-1}\supset H_{0}^{\prime\prime}\supset V^{n-1}.$
We have the $\mathbb{C}$-vector space isomorphisms
$\displaystyle t:V^{\alpha}$ $\displaystyle\to$ $\displaystyle V^{\alpha+1},$
$\displaystyle t:V^{>\alpha}$ $\displaystyle\to$ $\displaystyle
V^{>\alpha+1},$ $\displaystyle\partial_{t}^{-1}:V^{\alpha}$ $\displaystyle\to$
$\displaystyle V^{\alpha+1}\quad\textup{for }\alpha>-1,$
$\displaystyle\partial_{t}^{-1}:V^{>\alpha}$ $\displaystyle\to$ $\displaystyle
V^{>\alpha+1}\quad\textup{for }\alpha\geq-1,$
$\displaystyle\partial_{t}^{-1}:H_{0}^{\prime\prime}$ $\displaystyle\to$
$\displaystyle H_{0}^{\prime},$ $\displaystyle\Omega_{f}$ $\displaystyle\cong$
$\displaystyle H_{0}^{\prime\prime}/H_{0}^{\prime},$ $\displaystyle
C^{\alpha}$ $\displaystyle\cong$ $\displaystyle
V^{\alpha}/V^{>\alpha}=:\textup{Gr}_{V}^{\alpha}V^{>-\infty}.$
The space $\Omega_{f}$ inherits from $H_{0}^{\prime\prime}\subset V^{>-1}$ a
$V$-filtration, as follows,
$\displaystyle V^{\alpha}\Omega_{f}:=\frac{V^{\alpha}\cap
H_{0}^{\prime\prime}+H_{0}^{\prime}}{H_{0}^{\prime}}\subset\frac{H_{0}^{\prime\prime}}{H_{0}^{\prime}}=\Omega_{f},$
with quotients
$\displaystyle\textup{Gr}_{V}^{\alpha}\Omega_{f}=\frac{V^{\alpha}\Omega_{f}}{V^{>\alpha}\Omega_{f}}\cong\frac{\textup{Gr}_{V}^{\alpha}H_{0}^{\prime\prime}}{\textup{Gr}_{V}^{\alpha}H_{0}^{\prime}}\cong\frac{H_{0}^{\prime\prime}\cap
V^{\alpha}+V^{>\alpha}}{H_{0}^{\prime}\cap V^{\alpha}+V^{>\alpha}}.$
The spectral numbers of the singularity $f$ are the $\mu$ rational numbers
$\alpha_{1},...,\alpha_{\mu}$ with
$\displaystyle\alpha_{1}\leq...\leq\alpha_{\mu}\quad\textup{and}\quad\dim\textup{Gr}_{V}^{\alpha}\Omega_{f}=\sharp\\{j\in\\{1,....,\mu\\}\,|\,\alpha_{j}=\alpha\\}.$
The inclusions $V^{>-1}\supset H_{0}^{\prime\prime}\supset V^{n-1}$ and the
isomorphism
$\Omega_{f}=H_{0}^{\prime\prime}/\partial_{t}^{-1}H_{0}^{\prime\prime}$ show
$-1<\alpha_{1}\leq...\leq\alpha_{\mu}<n.$
Also the symmetry
$\alpha_{j}+\alpha_{\mu+1-j}=n-1,\quad\textup{or,
equivalenty,}\quad\dim\textup{Gr}_{V}^{\alpha}\Omega_{f}=\dim\textup{Gr}_{V}^{n-1-\alpha}\Omega_{f}$
holds (one proof of it uses Varchenko’s description via $H_{0}^{\prime\prime}$
of Steenbrink’s mixed Hodge structure, another one uses K. Saito’s higher
residue pairings). The following variance inequality for the spectral numbers
was conjectured by Hertling [Her02],
$\mu^{-1}\sum_{j=1}^{\mu}(\alpha_{j}-\frac{n-1}{2})^{2}\leq\frac{\alpha_{\mu}-\alpha_{1}}{12}.$
For curve singularities (i.e. $n=1$), it was proved by Brélivet [Bré04].
The action of $t$ on $V^{>-\infty}$ induces an action on $\Omega_{f}$ which is
called $t_{\Omega_{f}}$. It satisfies
$t_{\Omega_{f}}:V^{\alpha}\Omega_{f}\to V^{\alpha+1}\Omega_{f}.$
Therefore $t_{\Omega_{f}}$ is nilpotent. Because of
$V^{\alpha_{1}}\Omega_{f}=\Omega_{f}$ and $V^{>\alpha_{\mu}}\Omega_{f}=0$ and
$\alpha_{1}+(n+1)>\alpha_{\mu}$, $t_{\Omega_{f}}$ has at most Jordan blocks of
size $n+1$.
The $\mu$-dimensional $\mathbb{C}$-vector spaces $Q_{f}$ and $\Omega_{f}$ are
not canonically isomorphic. But the choice of any volume form $u(x){\rm
d}x_{0}...{\rm d}x_{n}=u(x){\rm d}x$ (volume form: $u(0)\neq 0$, i.e.
$u(x)\in{\mathcal{O}}^{*}_{\mathbb{C}^{n+1},0}$) induces an isomorphism
$Q_{f}\to\Omega_{f},\quad g\mapsto g\cdot[u(x){\rm d}x],$
which commutes with the multiplication by $f$ respectively $t$,
$\displaystyle\begin{matrix}Q_{f}&\stackrel{{\scriptstyle\cong}}{{\to}}&\Omega_{f}&g\mapsto
g\cdot[u(x){\rm d}x]\\\ \downarrow A_{f}&&\downarrow t_{\Omega_{f}}&\\\
Q_{f}&\stackrel{{\scriptstyle\cong}}{{\to}}&\Omega_{f}&g\mapsto
g\cdot[u(x){\rm d}x]\end{matrix}$
Therefore $A_{f}$ and $t_{\Omega_{f}}$ have the same Jordan block structure.
Now the review of the spectral numbers and their background is finished.
Now we suppose $n=1$, so $f$ is a curve singularity, and we suppose that $f$
is Jordan block extreme, i.e. all Jordan blocks of $A_{f}$ and
$t_{\Omega_{f}}$ have size 2. We will come to a contradiction. $\mu$ is even.
We have
$V^{>\alpha_{\mu}-1}\Omega_{f}\subset\ker
t_{\Omega_{f}}=\textup{im\,}t_{\Omega_{f}}=t_{\Omega_{f}}(\Omega_{f})=t_{\Omega_{f}}(V^{\alpha_{1}}\Omega_{f})\subset
V^{\alpha_{1}+1}\Omega_{f}.$
Together with $\alpha_{1}+1>0>\alpha_{\mu}-1$, so
$V^{>\alpha_{\mu}-1}\Omega_{f}\supset V^{0}\Omega_{f}\supset
V^{\alpha_{1}+1}\Omega_{f}$, this shows
$V^{>\alpha_{\mu}-1}\Omega_{f}=\ker
t_{\Omega_{f}}=\textup{im\,}t_{\Omega_{f}}=V^{\alpha_{1}+1}\Omega_{f}$
and
$\\{\alpha_{1},...,\alpha_{\mu/2}\\}\subset[\alpha_{1},\alpha_{\mu}-1],\quad\\{\alpha_{\mu/2+1},...,\alpha_{\mu}\\}\subset[\alpha_{1}+1,...,\alpha_{\mu}].$
This distribution of the spectral numbers is already strange. But for a
contradiction to the variance inequality, we need more.
For any element $v\in\Omega_{f}-\\{0\\}$ denote by
$\gamma(v)\in\\{\alpha_{1},...,\alpha_{\mu}\\}$ the unique number $\gamma$
such that $v\in V^{\gamma}\Omega_{f}-V^{>\gamma}\Omega_{f}$, and then denote
by
$\textup{Gr}_{V}^{\gamma(v)}v\in\textup{Gr}_{V}^{\gamma(v)}\Omega_{f}-\\{0\\}$
the class of $v$ in $\textup{Gr}_{V}^{\gamma(v)}\Omega_{f}$.
We choose elements $v_{1},...,v_{\mu/2}\in\Omega_{f}$ such that
$\gamma(v_{j})=\alpha_{j}$ and such that the classes
$\textup{Gr}_{V}^{\alpha_{1}}v_{1},...,\textup{Gr}_{V}^{\alpha_{\mu/2}}v_{\mu/2}$
form a $\mathbb{C}$-vector space basis of the sum of quotients
$\sum_{\alpha\in\\{\alpha_{1},...,\alpha_{\mu/2}\\}}Gr_{V}^{\alpha}\Omega_{f}$.
Then any linear combination $v=\sum_{j=1}^{\mu/2}\lambda_{j}v_{j}$ with
$(\lambda_{1},...,\lambda_{\mu/2})\neq 0$ is nonzero and has
$\gamma(v)\in\\{\alpha_{1},...,\alpha_{\mu/2}\\}$, so it is not in
$V^{>\alpha_{\mu}-1}=\ker t_{\Omega_{f}}$. Therefore the vector space
$\sum_{j=1}^{\mu/2}\mathbb{C}\cdot v_{j}$ has dimension $\mu/2$, and
$\Omega_{f}=\Bigl{(}\bigoplus_{j=1}^{\mu/2}\mathbb{C}\cdot
v_{j}\Bigr{)}\oplus\ker t_{\Omega_{f}}.$
Recall $\ker t_{\Omega_{f}}=\textup{im\,}t_{\Omega_{f}}$ and that this
subspace of $\Omega_{f}$ has dimension $\mu/2$. Therefore
$\ker
t_{\Omega_{f}}=\textup{im\,}t_{\Omega_{f}}=\bigoplus_{j=1}^{\mu/2}\mathbb{C}\cdot
t_{\Omega_{f}}(v_{j}).$
Obviously $\gamma(t_{\Omega_{f}}(v_{j}))\geq\alpha_{j}+1$, but equality does
not necessarily hold, and the classes
$\textup{Gr}_{V}^{\gamma(t_{\Omega_{f}}(v_{j}))}t_{\Omega_{f}}(v_{j})$ for
$j\in\\{1,...,\mu/2\\}$ are not necessarily linearly independent.
The following claim replaces the basis
$t_{\Omega_{f}}(v_{1}),...,t_{\Omega_{f}}(v_{\mu/2})$ of
$\textup{im\,}t_{\Omega_{f}}$ by a basis $w_{1},...,w_{\mu/2}$ of
$\textup{im\,}t_{\Omega_{f}}$ which fits better to the spectral numbers
$\alpha_{\mu/2+1},...\alpha_{\mu}$.
Claim: (a) There is a lower triangular matrix $(a_{ij})\in
M_{\mu/2\times\mu/2}(\mathbb{C})$ with $a_{ii}=1$ such that the basis
$(w_{1},...,w_{\mu/2})=(t_{\Omega_{f}}(v_{1}),...,t_{\Omega_{f}}(v_{\mu/2}))\cdot(a_{ij})$
of $\textup{im\,}t_{\Omega_{f}}$ satisfies the following: Write
$\beta_{j}:=\gamma(w_{j})$. The classes
$\textup{Gr}_{V}^{\beta_{1}}w_{1},...,\textup{Gr}_{V}^{\beta_{\mu/2}}w_{\mu/2}$
are linearly independent.
(b) Therefore they form a basis of
$\bigoplus_{\alpha\in\\{\alpha_{\mu/2+1},...,\alpha_{\mu}\\}}\textup{Gr}_{V}^{\alpha}\Omega_{f}$,
and therefore there is a bijection
$\sigma:\\{1,...,\mu/2\\}\to\\{\mu/2+1,...,\mu\\}$ with
$\alpha_{\sigma(j)}=\beta_{j}$ for $j\in\\{1,...,\mu/2\\}$.
(c) $\beta_{j}\geq\alpha_{j}+1$ for $j\in\\{1,...,\mu/2\\}$.
Proof of the Claim: (a) The vectors $w_{j}$ are constructed inductively in the
order $w_{\mu/2},w_{\mu/2-1},...,w_{1}$. The first step
$w_{\mu/2}=t_{\Omega_{f}}(v_{\mu/2})$ is trivial. Suppose that the vectors
$w_{j+1},...,w_{\mu/2}$ (and the corresponding entries $a_{im}$) have been
constructed. One constructs $w_{j}$ be a sequence of steps which give
$w_{j}^{(0)}:=t_{\Omega_{f}}(v_{j}),w_{j}^{(1)},...,w_{j}^{(k)}=w_{j}$ for
some $k\geq 0$. Each of these vectors is in
$t_{\Omega_{f}}(v_{j})+\bigoplus_{i\geq j+1}\mathbb{C}\cdot w_{i}$. Suppose
that $w_{j}^{(l)}$ has been constructed. If
$Gr_{V}^{\gamma(w_{j}^{(l)})}w_{j}^{(l)}\notin\bigoplus_{i:\,i\geq
j+1,\beta_{i}=\gamma(w_{j}^{(l)})}\mathbb{C}\cdot\textup{Gr}_{V}^{\beta_{i}}w_{i},$
then $l=k$ and $w_{j}^{(l)}=w_{j}$. If
$Gr_{V}^{\gamma(w_{j}^{(l)})}w_{j}^{(l)}\in\bigoplus_{i:\,i\geq
j+1,\beta_{i}=\gamma(w_{j}^{(l)})}\mathbb{C}\cdot\textup{Gr}_{V}^{\beta_{i}}w_{i},$
one adds to $w_{j}^{(l)}$ a suitable linear combination $\sum_{i:\,i\geq
j+1,\beta_{i}=\gamma(w_{j}^{(l)})}a_{ij}\cdot w_{i}$, such that the sum
$w_{j}^{(l+1)}$ satisfies $\gamma(w_{j}^{(l+1)})>\gamma(w_{j}^{(l)})$. This
construction stops at some $w_{j}^{(k)}=w_{j}$.
(b) This follows immediately from part (a).
(c) The construction in the proof of part (a) gives
$\beta_{j}=\gamma(w_{j})=\gamma(w_{j}^{(k)})>..>\gamma(w_{j}^{(0)})=\gamma(t_{\Omega_{f}}(v_{j}))\geq\alpha_{j}+1$
(in the case $k=0$ without the strict inequalities). This finishes the proof
of the Claim. ($\Box$)
Now we can estimate the variance of the spectral numbers. It is
$\displaystyle\mu^{-1}\sum_{j=1}^{\mu}\alpha_{j}^{2}=\mu^{-1}\sum_{j=1}^{\mu/2}(\alpha_{j}^{2}+\beta_{j}^{2})=\mu^{-1}\sum_{j=1}^{\mu/2}\Bigl{(}(\alpha_{j}+\frac{1}{2}-\frac{1}{2})^{2}+(\beta_{j}-\frac{1}{2}+\frac{1}{2})^{2}\Bigr{)}$
$\displaystyle=$
$\displaystyle\mu^{-1}\sum_{j=1}^{\mu/2}\Bigl{(}(\alpha_{j}+\frac{1}{2})^{2}+(\alpha_{j}+\frac{1}{2})(-1)+\frac{1}{4}+(\beta_{j}-\frac{1}{2})^{2}+(\beta_{j}-\frac{1}{2})\cdot
1+\frac{1}{4}\Bigr{)}$ $\displaystyle=$
$\displaystyle\mu^{-1}\sum_{j=1}^{\mu/2}\Bigl{(}\frac{1}{2}+(\alpha_{j}+\frac{1}{2})^{2}+(\beta_{j}-\frac{1}{2})^{2}+(\beta_{j}-\alpha_{j}-1)\Bigr{)}$
$\displaystyle\geq$
$\displaystyle\mu^{-1}\sum_{j=1}^{\mu/2}\frac{1}{2}=\frac{1}{4}\qquad\textup{(here
part (c) of the claim is used).}$
This does not fit to the variance inequality for curve singularities [Bré04],
$\mu^{-1}\sum_{j=1}^{\mu}\alpha_{j}^{2}\leq\frac{\alpha_{\mu}-\alpha_{1}}{12}<\frac{2}{12}=\frac{1}{6}.$
We arrive at a contradiction. A Jordan block extreme curve singularity does
not exist. $\Box$
## References
* [AGV85] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps, Volume I. Classification of critical points, caustics and wave fronts. Boston, MA: Birkhäuser, 1985.
* [AGZV88] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps. Monodromy and asymptotics of integrals Vol. II, volume 83 of Monographs in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1988.
* [Bré04] Thomas Brélivet. The Hertling conjecture in dimension 2. math/0405489, 2004.
* [CLO05] David A. Cox, John Little, and Donal O’Shea. Using algebraic geometry. 2nd ed, volume 185. New York, NY: Springer, 2nd ed. edition, 2005.
* [CLS92] C. Camacho, Alcides Lins Neto, and P. Sad. Foliations with algebraic limit sets. Ann. Math. (2), 136(2):429–446, 1992.
* [GH94] Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.
* [GPS01] G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 2.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001. http://www.singular.uni-kl.de.
* [Her02] Claus Hertling. Frobenius manifolds and moduli spaces for singularities, volume 151 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2002.
* [KN88] Ryoichi Kobayashi and Isao Naruki. Holomorphic conformal structures and uniformization of complex surfaces. Math. Ann., 279(3):485–500, 1988.
* [Lan02] Serge Lang. Algebra. 3rd revised ed, volume 211. New York, NY: Springer, 3rd revised ed. edition, 2002.
* [Lin02] A. Lins Neto. Some examples for the Poincaré and Painlevé problems. Ann. Sci. Éc. Norm. Supér. (4), 35(2):231–266, 2002.
* [Med13] Liliana Puchuri Medina. Degree of the first integral of a pencil in $\mathbb{P}^{2}$ defined by Lins Neto. Publ. Mat., Barc., 57(1):123–137, 2013.
* [Mov07] H. Movasati. Mixed Hodge structure of affine hypersurfaces. Ann. Inst. Fourier (Grenoble), 57(3):775–801, 2007.
* [Mov19] H. Movasati. A Course in Hodge Theory: with Emphasis on Multiple Integrals. Available at author’s webpage. 2019.
* [MP05] L. G. Mendes and J. V. Pereira. Hilbert modular foliations on the projective plane. Comment. Math. Helv., 80(2):243–291, 2005.
* [Sai71] Kyoji Saito. Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math., 14:123–142, 1971.
* [Sai76] Kyoji Saito. On a generalization of de-Rham lemma. Ann. Inst. Fourier (Grenoble), 26(2):vii, 165–170, 1976.
* [Sai80] Kyoji Saito. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci., Univ. Tokyo, Sect. I A, 27:265–291, 1980.
|
# Constructing totally $p$-adic numbers of small height
S. Checcoli S. Checcoli, Institut Fourier, Université Grenoble Alpes, 100 rue
des Mathématiques, 38610 Gières, France<EMAIL_ADDRESS>and A. Fehm A. Fehm, Institut für Algebra, Fakultät Mathematik, Technische
Universität Dresden, 01062 Dresden, Germany<EMAIL_ADDRESS>
###### Abstract.
Bombieri and Zannier gave an effective construction of algebraic numbers of
small height inside the maximal Galois extension of the rationals which is
totally split at a given finite set of prime numbers. They proved, in
particular, an explicit upper bound for the lim inf of the height of elements
in such fields. We generalize their result in an effective way to maximal
Galois extensions of number fields with given local behaviour at finitely many
places.
## 1\. Introduction
Let $h$ denote the absolute logarithmic Weil height on the field
$\overline{\mathbb{Q}}$ of algebraic numbers. We are interested in explicit
height bounds for elements of $\overline{\mathbb{Q}}$ with special local
behaviour at a finite set of primes. The first result in this context is due
to Schinzel [Sch73] who proved a height lower bound for elements in the field
of totally real algebraic numbers $\mathbb{Q}^{\rm tr}$, the maximal Galois
extension of $\mathbb{Q}$ in which the infinite prime splits totally. More
precisely, he showed that every $\alpha\in\mathbb{Q}^{\rm tr}$ has either
$h(\alpha)=0$ or
$h(\alpha)\geq\frac{1}{2}\log\left(\frac{1+\sqrt{5}}{2}\right).$
Explicit upper and lower bounds for the limit infimum of the height of
algebraic integers in $\mathbb{Q}^{\rm tr}$ are given in [Smy80, Smy81,
Fla96].
In [BZ01] Bombieri and Zannier investigate the analogous problem for the
$p$-adic numbers. More precisely, in [BZ01, Theorem 2] they prove the
following:
###### Theorem 1.1 (Bombieri–Zannier).
Let $p_{1},\dots,p_{n}$ be distinct prime numbers, for each $i$ let $E_{i}$ be
a finite Galois extension of $\mathbb{Q}_{p_{i}}$, and $L$ the maximal Galois
extension of $\mathbb{Q}$ contained in all $E_{i}$. Denote by $e_{i}$ and
$f_{i}$ the ramification index and inertia degree of
$E_{i}/\mathbb{Q}_{p_{i}}$. Then
$\liminf_{\alpha\in
L}h(\alpha)\;\geq\;\frac{1}{2}\cdot\sum_{i=1}^{n}\frac{\log(p_{i})}{e_{i}(p_{i}^{f_{i}}+1)}.$
In the special case $E_{i}=\mathbb{Q}_{p_{i}}$, Bombieri and Zannier in [BZ01,
Example 2] show that the lower bound in Theorem 1.1 is almost optimal. More
precisely:
###### Theorem 1.2 (Bombieri–Zannier).
Let $p_{1},\dots,p_{n}$ be prime numbers and let $L$ be the maximal Galois
extension of $\mathbb{Q}$ contained in all $\mathbb{Q}_{p_{i}}$. Then
$\liminf_{\alpha\in
L}h(\alpha)\;\leq\;\sum_{i=1}^{n}\frac{\log(p_{i})}{p_{i}-1}.$
Other proofs, refinements and generalizations were given in [Fil14, Pot15,
FP15, FP19, PS19] See also [Smy07] for a general survey on the height of
algebraic numbers. In Remark 1.4 we will discuss in detail the contribution
[Fil14] and how it compares to our work.
The goal of this note is to generalize in an effective way the upper bound
Theorem 1.2 to general $E_{i}$, and to further replace the base field
$\mathbb{Q}$ by an arbitrary number field. Our main result is the following:
###### Theorem 1.3.
Let $K$ be a number field and let $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ be
distinct primes ideals of the ring of integers $\mathcal{O}_{K}$ of $K$. For
each $i$, let $E_{i}$ be a finite Galois extension of the completion $F_{i}$
of $K$ at $\mathfrak{p}_{i}$. Denote by $e_{i}$ and $f_{i}$ the ramification
index and the relative inertia degree of $E_{i}/F_{i}$ and write
$q_{i}=|\mathcal{O}_{K}/\mathfrak{p}_{i}|=p_{i}^{{f(\mathfrak{p}_{i}|p_{i})}}$.
Then for the maximal Galois extension $L$ of $K$ contained in all $E_{i}$,
$\liminf_{\alpha\in
L}h(\alpha)\;\leq\;\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}.$
More precisely, let
$C=\max\left\\{[K:\mathbb{Q}],|\Delta_{K}|,\max_{i}(e_{i}f_{i}),\max_{i}q_{i}^{f_{i}}\right\\}$
where $\Delta_{K}$ is the absolute discriminant of $K$. Then for every
$0<\epsilon<1$ there exist infinitely many $\alpha\in\mathcal{O}_{L}$ of
height
(1.1)
$h(\alpha)\leq\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+13nC^{2n+2}\frac{\log\left([K(\alpha):K]\right)}{[K(\alpha):K]}+\begin{cases}0,&n=1\\\
n\epsilon,&n>1\end{cases}.$
Namely, for every $\rho\geq 3C^{n}$ there exists such $\alpha$ of degree
(1.2) $\rho\leq[K(\alpha):K]\leq\begin{cases}C\rho,&n=1\\\ \rho^{\frac{(4\log
C)^{n+1}}{\log^{n}(1+\epsilon)}},&n>1\end{cases}.$
Note that in the special case $K=\mathbb{Q}$ and $E_{i}=\mathbb{Q}_{p_{i}}$ we
reobtain Theorem 1.2, except that Theorem 1.3 appears stronger in that the
result is effective and the $\liminf$ can be taken over algebraic integers.
However, an inspection of the proof of Bombieri and Zannier shows that it is
effective as well and does in fact produce algebraic integers.
###### Remark 1.4.
Theorem 1.3 provides an effective version of a result of Fili [Fil14, Theorem
1.2]. The bound in that result seems to differ from ours by the factor
$e(\mathfrak{p}_{i}|p_{i})$, and [Fil14, Theorem 1.1] (and similarly [FP15,
Theorem 9]) also states a variant of Theorem 1.1 which contradicts our Theorem
1.3, but according to Paul Fili (personal communication) this is merely an
error in normalization in [Fil14] and [FP15] that became apparent when
comparing to our result, and the $e_{v}$ in the denominator of Theorems 1.1,
1.2, and Conjecture 1 of [Fil14] (and similarly in the statements of [FP15])
should have been the absolute instead of the relative ramification index. When
this correction is made, the lower bound of [Fil14, Theorem 1.1] agrees with
the one in [FP19, Theorem 13], and the upper bound of [Fil14, Theorem 1.2]
agrees with the one in Theorem 1.3.
In any case, Fili’s proof of [Fil14, Theorem 1.2] uses capacity theory on
analytic Berkovich spaces and does not provide explicit bounds on the degree
and the height of a sequence of integral elements in the $\liminf$. Instead,
our effective proof is more elementary and is inspired by Bombieri and
Zannier’s effective proof of Theorem 1.2. To the best of our knowledge,
Theorem 1.3 is the only result currently available that gives a bound on the
height in terms of the degree of such a sequence of $\alpha$, except for the
case where $K=\mathbb{Q}$ and $E_{i}=\mathbb{Q}_{p_{i}}$ for all $i$, where
such a bound can be deduced from [BZ01].
We also remark that our use of [Fil14, Theorem 1.2] in [CF21] is limited to
the cases where [Fil14, Theorem 1.2] agrees with Theorem 1.3.
The paper is organised as follows. In Section 2 we collect all the preliminary
results needed to prove Theorem 1.3, namely: a consequence of Dirichlet’s
theorem on simultaneous approximation (Proposition 2.1), a bound for the size
of representatives in quotient rings of rings of integers (Proposition 2.2), a
variant of Hensel’s lemma (Proposition 2.3), a bound for the height of a root
of a polynomial defined over a number field in terms of its coefficients
(Proposition 2.4), and a construction of special Galois invariant sets of
representatives of residue rings of local fields (Proposition 2.5).
The proof of Theorem 1.3 is carried out in Section 3. We briefly sketch it
here for clarity. Following Bombieri and Zannier’s strategy, given $\rho\geq
3C^{n}$ we construct a monic irreducible polynomial $g\in\mathcal{O}_{K}[X]$
such that
1. (i)
its degree is upper and lower bounded in terms of $\rho$ as the degree of
$\alpha$ in Theorem 1.3,
2. (ii)
the complex absolute value of all conjugates of its coefficients is
sufficiently small, and
3. (iii)
all its roots are contained in all $E_{i}$.
In Bombieri and Zannier’s proof of Theorem 1.2, (i) and (ii) were achieved by
using the Chinese Remainder Theorem to deform the polynomial
$\prod_{i=1}^{\rho}(X-i)$ into an irreducible polynomial of the same degree
with coefficients small enough to give the desired bound for the height of the
roots. Then a variant of Hensel’s lemma was applied to show that the roots of
the constructed polynomial are still in $\mathbb{Q}_{p_{i}}$ for each $i$.
In our generalisation, the degree of the polynomial is carefully chosen to
obtain (i) in Section 3.1 via Proposition 2.1 (necessary only if $n>1$, which
leads to the better bounds in the case $n=1$). The polynomial $g$ satisfying
(ii) is then constructed in Section 3.2: We start with polynomials
$\prod_{\alpha\in\tilde{A}_{i}}(X-\alpha)$, where now
$\tilde{A}_{i}\subseteq\mathcal{O}_{E_{i}}$ is a set constructed using
Proposition 2.5. These polynomials are then merged into an irreducible
polynomial $g$ by applying the Chinese Remainder Theorem and Proposition 2.2
to bound the size of its coefficients. Property (iii) is verified in Section
3.3, using Proposition 2.3. Finally, Proposition 2.4 is applied to show that
$g$ has a root $\alpha$ of height bounded from above as desired.
## 2\. Notation and preliminaries
We fix some notation. If $K$ is a number field or a non-archimedean local
field we let $\mathcal{O}_{K}$ denote the ring of integers of $K$. For an
ideal $\mathfrak{a}$ of $\mathcal{O}_{K}$ we denote by
$N(\mathfrak{a})=|\mathcal{O}_{K}/\mathfrak{a}|$ its norm. For a nonzero prime
ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$, we denote by $v_{\mathfrak{p}}$ the
discrete valuation on $K$ with valuation ring
$(\mathcal{O}_{K})_{\mathfrak{p}}$ normalized such that
$v_{\mathfrak{p}}(K^{\times})=\mathbb{Z}$. If $L/K$ is an extension of number
fields and $\mathfrak{P}$ is a prime ideal of $\mathcal{O}_{L}$ lying above a
prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$ we denote by
$e(\mathfrak{P}|\mathfrak{p})$ and $f(\mathfrak{P}|\mathfrak{p})$ the
ramification index and the inertia degree. For an extension $E/F$ of non-
archimedean local fields we denote the ramification index and the inertia
degree also by $e(E/F)$ and $f(E/F)$.
### 2.1. Auxiliary results
In this section we collect the preliminary results we need to prove Theorem
1.3. These results are not related to each other and we list them in this
section following their order of appearance in the proof of Theorem 1.3.
###### Proposition 2.1.
Let $x_{1},\dots,x_{n}$ be integers greater than $1$. For every $\rho\geq 3$
and $0<\epsilon<1$ there exist positive integers $r,k_{1},\dots,k_{n}$ such
that $r\geq\rho$ and, for all $i$, $r\leq x_{i}^{k_{i}}\leq(1+\epsilon)r$ and
$k_{i}\leq\frac{2^{2n+1}\log^{n}(\max_{j}x_{j})\log(\rho)}{\log(x_{i})\log^{n}(1+\epsilon)}.$
###### Proof.
Say $x_{1}=\max_{i}x_{i}$. Let $\alpha_{i}=2\log(\rho)/\log(x_{i})$ and
$Q=\lceil 2\log(x_{1})/\log(1+\epsilon)\rceil$. By the simultaneous Dirichlet
approximation theorem [Sch80, Chapter II, Section 1, Theorem 1A] there exist
positive integers $q,k_{1},\dots,k_{n}$ with $1\leq q<Q^{n}$ such that
$|q\alpha_{i}-k_{i}|\leq Q^{-1}$ for all $i$, and thus
$|2\log(\rho)q-\log(x_{i}^{k_{i}})|\leq\log(1+\epsilon)/2$. Letting
$r=\min_{i}x_{i}^{k_{i}}$, one has $\log(r)\geq 2\log(\rho)q-1\geq\log(\rho)$
for $q\geq 1$ and $\rho\geq 3$. In addition, for all $i$,
$0\leq\log(x_{i}^{k_{i}})-\log(r)\leq\log(1+\epsilon)$, hence $r\leq
x_{i}^{k_{i}}\leq(1+\epsilon)r$. Finally $k_{i}\leq q\alpha_{i}+1\leq
2q\alpha_{i}\leq 2\log(\rho)Q^{n}\log(x_{i})^{-1}$ and replacing $Q$ we get
the desired bound. ∎
The next proposition deals with bounds for the absolute value of small
representatives for quotient rings.
###### Proposition 2.2.
Let $K$ be a number field of degree $m=[K:\mathbb{Q}]$. Given a nonzero ideal
$\mathfrak{a}$ of $\mathcal{O}_{K}$, there exists a set of representatives $A$
of $\mathcal{O}_{K}/\mathfrak{a}$ such that, for every $a\in A$ and every
$\sigma\in{\rm Hom}(K,\mathbb{C})$, one has
$|\sigma(a)|\leq\delta_{K}N(\mathfrak{a})^{1/m}$
where $\delta_{K}=m^{\frac{3}{2}}{2}^{\frac{m(m-1)}{2}}\sqrt{|\Delta_{K}|}$.
###### Proof.
This is an immediate consequence of well-known results on lattice reduction.
For instance, [BFH17, Proposition 15] gives for every
$\alpha\in\mathcal{O}_{K}$, an element $a\in\mathcal{O}_{K}$ with
$\alpha-a\in\mathfrak{a}$ such that
$\sqrt{\sum_{\sigma}|\sigma(a)|^{2}}\leq
m^{\frac{3}{2}}{\ell}^{\frac{m(m-1)}{2}}\sqrt{|\Delta_{K}|}N(\mathfrak{a})^{1/m}$
where $\ell$ depends on certain parameters $\eta\in\
(1/2,1),\delta\in(\eta^{2},1)$ and $\theta>0$ coming from applying a variant
of the LLL-reduction algorithm as in [CSV12, Theorem 5.4] and [NSV11, Theorem
7] (see also [BFH17, §4, p.595]) to a $\mathbb{Z}$-basis of $\mathfrak{a}$. In
particular, choosing $\eta=2/3,\delta=7/9$ and $\theta=(\sqrt{19}-4)/3$, we
have $\ell=2$, which gives the claimed upper bound for $|\sigma(a)|$. ∎
The following proposition is a variant of Hensel’s lemma.
###### Proposition 2.3.
Let $E$ be a finite extension of $\mathbb{Q}_{p}$, $\mathfrak{P}$ the maximal
ideal of $\mathcal{O}_{E}$ and $v=v_{\mathfrak{P}}$. Let $f\in E[X]$ and
$x_{0}\in E$. Assume there exist $a,b\in\mathbb{Z}$ such that
1. (i)
$v(f(x_{0}))>a+b$,
2. (ii)
$v(f^{\prime}(x_{0}))\leq a$,
3. (iii)
$v(f^{(\nu)}(x_{0})/\nu!)\geq a-(\nu-1)b$ for every $\nu\geq 2$.
Then there exists $x\in E$ with $f(x)=0$ and $v(x-x_{0})>b$.
###### Proof.
This can be proved precisely as the special case $E=\mathbb{Q}_{p}$ in [BZ01,
Lemma 1]. Alternatively, one can reduce this to one of the standard forms of
Hensel’s lemma as follows. Let $\beta\in E$ with $v(\beta)=b$. Then
$g(X):=(\beta f^{\prime}(x_{0}))^{-1}f(\beta X+x_{0})$ is in
$\mathcal{O}_{E}[X]$ by $(i)$-$(iii)$ and has a simple zero $X=0$ modulo
$\mathfrak{P}$ by $(i)$ and $(ii)$, hence by Hensel’s lemma $g$ has a zero
$x^{\prime}\in\mathfrak{P}$, and $x=\beta x^{\prime}+x_{0}$ is then the
desired zero of $f$. ∎
The final proposition in this subsection gives a bound for the height of the
roots of a polynomial with small algebraic coefficients.
###### Proposition 2.4.
Let $K$ be a number field and let
$f(X)=X^{m}+a_{m-1}X^{m-1}+\ldots+a_{0}\in\mathcal{O}_{K}[X]$. If $B\geq 1$
with $|\sigma(a_{i})|<B$ for every $i$ and every $\sigma\in{\rm
Hom}(K,\mathbb{C})$, then $f$ has a root $\alpha$ with
$h(\alpha)\leq\frac{\log(B\sqrt{m+1})}{m}.$
###### Proof.
Let $M_{K}=M_{K}^{0}\cup M_{K}^{\infty}$ be the set of (finite and infinite)
places of $K$ and let $d=[K:\mathbb{Q}]$. For a place $v\in M_{K}$, denote by
$d_{v}=[K_{v}:\mathbb{Q}_{v}]$ the local degree. Let
$\hat{h}(f)=\log\left(\prod_{v\in M_{K}}M_{v}(f)^{d_{v}/d}\right)$
where if $v$ is non-archimedean $M_{v}(f)=\max_{i}(|a_{i}|_{v})$, while if $v$
is archimedean and corresponds to the embedding $\sigma\in{\rm
Hom}(K,\mathbb{C})$, $M_{v}(f)$ is the Mahler measure $M(\sigma(f))$ of the
polynomial $\sigma(f)$.
By [Zan09, Appendix A, section A.2, pag. 210] we have that, if
$\alpha_{1},\ldots,\alpha_{m}\in\overline{\mathbb{Q}}$ are the roots of $f$
(with multiplicities), then $\hat{h}(f)=\sum_{i=1}^{m}h(\alpha_{i})$ where $h$
denotes the usual logarithmic Weil height. Thus, if $\alpha$ is a root of $f$
of minimal height, then
(2.1) $h(\alpha)\leq\frac{\hat{h}(f)}{m}.$
Let $\sigma_{1},\ldots,\sigma_{r}$ and
$\tau_{1},\overline{\tau_{1}},\ldots,\tau_{s},\overline{\tau_{s}}$ be,
respectively, the real and pairwise conjugate complex embeddings of $K$ in
$\mathbb{C}$, so that $d=r+2s$. As $f$ has coefficients in $\mathcal{O}_{K}$,
$M_{v}(f)\leq 1$ if $v$ is non-archimedean and we have that
$\displaystyle\hat{h}(f)$ $\displaystyle\leq\log\left(\prod_{v\in
M_{K}^{\infty}}M_{v}(f)^{d_{v}/d}\right)=\log\left(\prod_{i=1}^{r}M(\sigma_{i}(f))^{1/d}\cdot\prod_{j=1}^{s}M(\tau_{j}(f))^{2/d}\right)=$
$\displaystyle=\log\left(\prod_{\sigma\in{\rm
Hom}(K,\mathbb{C})}M(\sigma(f))^{1/d}\right).$
By [Zan09, Section 3.2.2, formula (3.7)] and by our hypothesis on $B$, we have
that $M(\sigma(f))\leq B\sqrt{m+1}$ for all $\sigma\in{\rm
Hom}(K,\mathbb{C})$, thus $\hat{h}(f)\leq\log(B\sqrt{m+1})$ and, plugging this
bound into (2.1), we conclude. ∎
### 2.2. Representatives of residue rings of local fields
This subsection contains the technical key result needed to construct the
local polynomials in the proof of Theorem 1.3. Let $E/F$ be a Galois extension
of non-archimedean local fields with Galois group $G$. Let $\mathfrak{p}$ be
the maximal ideal of $\mathcal{O}_{F}$, $\mathfrak{P}$ be the maximal ideal of
$\mathcal{O}_{E}$ and for $k\in\mathbb{N}$ denote by
$\pi_{k}:\mathcal{O}_{E}\rightarrow\mathcal{O}_{E}/\mathfrak{P}^{k}$ the
residue map.
It is known that one can always find a $G$-invariant set of representatives of
the residue field $\mathcal{O}_{E}/\mathfrak{P}$, e.g. the Teichmüller
representatives. As long as the ramification of $E/F$ is tame, one can also
find $G$-invariant sets of representatives of each residue ring
$\mathcal{O}_{E}/\mathfrak{P}^{k}$, but if the ramification is wild, this is
not necessarily so. We will therefore work with the following substitute for
such a $G$-invariant set of representatives:
###### Proposition 2.5.
Let $E/F$ be a Galois extension of non-archimedean local fields and define
$G,\mathfrak{p},\mathfrak{P},\pi_{k}$ as above. Let $d$ be a multiple of
$|G|$. There exists a constant $c$ such that for every $k$ there is
$A\subseteq\mathcal{O}_{E}$ such that
1. (1)
$A$ is $G$-invariant,
2. (2)
all orbits of $A$ have length $|G|$,
3. (3)
$\pi_{k}|_{A}:A\rightarrow\mathcal{O}_{E}/\mathfrak{P}^{k}$ is $d$-to-1 and
onto, and
4. (4)
$\pi_{k+c}|_{A}$ is injective.
Moreover, if $F$ is a $p$-adic field, one can choose
$c\leq
e(\mathfrak{P}|\mathfrak{p})\left(d+|G|+\frac{e(\mathfrak{p}|p)}{p-1}+1\right).$
###### Proof.
Note that $G$ naturally acts on $\mathcal{O}_{E}$ and on
$\mathcal{O}_{E}/\mathfrak{P}^{k}$, and that $\pi_{k}$ is $G$-equivariant. Fix
some primitive element $\alpha\in\mathcal{O}_{E}^{\times}$ of $E/F$ and a
uniformizer $\theta\in\mathcal{O}_{F}$ of $v_{\mathfrak{p}}$, and let
$\displaystyle e$ $\displaystyle=$ $\displaystyle
e(\mathfrak{P}|\mathfrak{p}),$ $\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\max_{1\neq\sigma\in
G}v_{\mathfrak{P}}(\alpha-\sigma\alpha),(\mbox{with }c_{0}=0\mbox{ if }G=1),$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\lceil|G|+c_{0}/e\rceil,\mbox{ and }$ $\displaystyle c$
$\displaystyle=$ $\displaystyle e(d+c_{1}).$
Let $k\in\mathbb{N}$ be given. The desired set $A$ is obtained by applying the
following Claim in the case $X=\mathcal{O}_{E}/\mathfrak{P}^{k}$:
###### Claim.
For every $G$-invariant subset $X\subseteq\mathcal{O}_{E}/\mathfrak{P}^{k}$
there exists a $G$-invariant subset $A\subseteq\mathcal{O}_{E}$ with all
orbits of length $|G|$ such that $\pi_{k+c}|_{A}$ is injective and
$\pi_{k}|_{A}$ is $d$-to-$1$ onto $X$.
We prove the Claim by induction on $|X|$: If $X=\emptyset$, $A=\emptyset$
satisfies the claim. If $X\neq\emptyset$ take $x\in X$ and let
$X^{\prime}=X\setminus Gx$, where $Gx$ denotes the orbit of $x$ under $G$. By
the induction hypothesis there exists $A^{\prime}\subseteq\mathcal{O}_{E}$
satisfying the claim for $X^{\prime}$. Choose $a\in\pi_{k}^{-1}(x)$ and let
$k_{0}=\lceil\frac{k}{e}\rceil$. Then
$n_{0}:=\min\\{n\geq 0:v_{\mathfrak{P}}(a-\sigma a)\neq
e(k_{0}+n)+v_{\mathfrak{P}}(\alpha-\sigma\alpha)\;\forall 1\neq\sigma\in
G\\}<|G|,$
as $v_{\mathfrak{P}}(a-\sigma a)-v_{\mathfrak{P}}(\alpha-\sigma\alpha)$
attains less than $|G|$ many distinct values. Thus for $1\neq\sigma\in G$,
$\displaystyle
v_{\mathfrak{P}}((a+\theta^{n_{0}+k_{0}}\alpha)-\sigma(a+\theta^{n_{0}+k_{0}}\alpha))$
$\displaystyle=$ $\displaystyle\min\\{v_{\mathfrak{P}}(a-\sigma
a),e(n_{0}+k_{0})+v_{\mathfrak{P}}(\alpha-\sigma\alpha)\\}$
$\displaystyle\leq$ $\displaystyle e(n_{0}+k_{0})+c_{0}$ $\displaystyle<$
$\displaystyle k+ec_{1},$
so if we replace $a$ by $a+\theta^{n_{0}+k_{0}}\alpha$, we can assume without
loss of generality that $\pi_{k+ec_{1}}$ is injective on $Ga$ and that
$|Ga|=|G|$. If we now let
$A=A^{\prime}\cup\\{\sigma(a)+\theta^{k_{0}+c_{1}+j}:\sigma\in G,0\leq
j<d/|G_{x}|\\}$
where $G_{x}$ is the stabilizer of $x$, then $\pi_{k}|_{A}$ is $d$-to-$1$ onto
$X=X^{\prime}\cup Gx$ and $A$ is $G$-invariant with all orbits of length
$|G|$. As
$k+ec_{1}\leq e(k_{0}+c_{1}+j)<k+c,$
we have that $\pi_{k+c}|_{A}$ is injective.
Now, if $F$ is a $p$-adic field and if we chose $\alpha\in\mathcal{O}_{E}$ to
be also a generator of $\mathcal{O}_{E}$ as a $\mathcal{O}_{F}$-algebra, by
[Ser80, Chap.IV, Ex. 3(c)], one has the explicit bound $c_{0}\leq
e(\mathfrak{P}/{p})/(p-1)$ which implies the stated bound for $c$. ∎
###### Remark 2.6.
Note that if (4) holds for some $c,k,A$, then also for $c^{\prime},k,A$ for
any $c^{\prime}\geq c$.
## 3\. Proof of Theorem 1.3
Using the notation of Theorem 1.3, for every $1\leq i\leq n$, let
$\mathfrak{P}_{i}$ be the maximal ideal of $\mathcal{O}_{E_{i}}$, $v_{i}$ the
extension of $v_{\mathfrak{P}_{i}}$ to an algebraic closure of $E_{i}$,
$G_{i}={\rm Gal}(E_{i}/F_{i})$ and $d=\prod_{i=1}^{n}|G_{i}|$. Let $C$ be the
constant from Theorem 1.3, note that $C\geq 2$, and let $c=4C^{n+1}$. Fix an
integer $\rho\geq 3C^{n}$ and note that $\rho/d\geq 3$ since $d\leq C^{n}$. If
$n=0$ let $\epsilon=0$, otherwise fix $0<\epsilon<1$.
### 3.1. Choosing the right degree
If $n>1$ we apply Proposition 2.1 to $x_{i}={q_{i}}^{f_{i}}$ to obtain
positive integers $r>\rho/d$ and $k_{1},\ldots,k_{n}$ such that for every $i$,
1. (i)
$r\leq q_{i}^{f_{i}k_{i}}\leq(1+\epsilon)r$ and
2. (ii)
$k_{i}\leq 2^{2(n+1)}(\log C)^{n}\frac{\log(\rho/d)}{\log^{n}(1+\epsilon)}$,
where we used that, for every $i$, $\log
2\leq\log(x_{i})=\log({q_{i}^{f_{i}}})\leq\log C$. It follows that
$\log(\rho/d)\leq\log(r)\leq(4\log
C)^{n+1}\frac{\log(\rho/d)}{\log^{n}(1+\epsilon)}.$
If $n=1$ we instead set $r=q_{1}^{f_{1}k_{1}}$, where
$k_{1}=\lceil\log(\rho/d)/\log(q_{1}^{f_{1}})\rceil$, so that (i) holds with
$\epsilon=0$, and
$\log(\rho/d)\leq\log(r)\leq\log(\rho/d)+\log(q_{1}^{f_{1}}).$
Using that $(4\log C)^{n+1}\geq\log^{n}(1+\epsilon)$, we conclude
(3.1) $\rho\leq dr\leq\begin{cases}C\rho,&n=1\\\ \rho^{\frac{(4\log
C)^{n+1}}{\log^{n}(1+\epsilon)}},&n>1\end{cases}$
### 3.2. Construction of the polynomial $g$
We first want to prove the following:
###### Claim.
For every $i$, there exists a polynomial $g_{i}\in\mathcal{O}_{K}[X]$ of
degree $dr$ whose set of roots ${A_{i}}$ satisfies
1. (a)
$A_{i}\subseteq E_{i}$,
2. (b)
$v_{i}({\alpha}-{\beta})<k_{i}+c$ for all ${\alpha},{\beta}\in{A_{i}}$ with
${\alpha}\neq{\beta}$, and
3. (c)
$v({g_{i}}^{\prime}(\alpha))\leq
d\left(\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+c\right)$ for every
${\alpha}\in{A_{i}}$.
###### Proof of the claim.
As
$e(\mathfrak{P}_{i}|\mathfrak{p}_{i})(d+|G_{i}|+\frac{e(\mathfrak{p}_{i}|p_{i})}{p_{i}-1}+1)\leq
C(C^{n}+C+C+1)\leq 4C^{n+1}=c$, by Proposition 2.5 there is a
$G_{i}$-invariant set $A_{i}^{\prime}\subseteq\mathcal{O}_{E_{i}}$ with all
orbits of length $|G_{i}|$ such that
${A_{i}^{\prime}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}$ is
$d$-to-$1$ and
${A_{i}^{\prime}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}+c}$ is
injective. As $|{A_{i}^{\prime}}|=dq_{i}^{f_{i}k_{i}}$, $|G_{i}|$ divides $d$,
and $r\leq q_{i}^{f_{i}k_{i}}$, there exists a $G_{i}$-invariant subset
${\tilde{A}_{i}}\subseteq{A_{i}^{\prime}}$ with $|\tilde{A_{i}}|=dr$. Let
$\tilde{g_{i}}=\prod_{\alpha\in\tilde{A_{i}}}(X-\alpha)\in\mathcal{O}_{F_{i}}[X].$
We first prove that conditions (a)-(c) hold for $\tilde{g_{i}}$ and the set
$\tilde{A_{i}}$, instead of $g_{i}$ and $A_{i}$. Note that
$\tilde{g_{i}}\in\mathcal{O}_{F_{i}}[X]$ is monic of degree $dr$ and that
condition (a) holds for $\tilde{A_{i}}$ by construction. Moreover, as the map
$\tilde{A_{i}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}+c}$ is
injective, we have that condition (b) is also satisfied for $\tilde{A_{i}}$.
As for condition (c), note that the valuation $v_{\mathfrak{P}_{i}}$ on
$\mathcal{O}_{E_{i}}$ induces a map
$\bar{v}:(\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}})\setminus\\{0\\}\rightarrow\\{0,\dots,k_{i}-1\\}$
such that ${v}_{\mathfrak{P}_{i}}(\gamma)=\bar{v}(\pi_{k_{i}}(\gamma))$ for
all $\gamma\in\mathcal{O}_{E_{i}}\setminus\mathfrak{P}_{i}^{k_{i}}$, where
$\pi_{k_{i}}$ denotes the residue map
$\mathcal{O}_{E_{i}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}$.
Now
$\displaystyle v_{\mathfrak{P}_{i}}(\tilde{g_{i}}^{\prime}(\alpha))$
$\displaystyle=$
$\displaystyle\sum_{\alpha\neq\beta\in\tilde{A_{i}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$
$\displaystyle\leq$ $\displaystyle\sum_{\alpha\neq\beta\in
A_{i}^{\prime}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$ $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\alpha\neq\beta\in
A_{i}^{\prime}}}{{\pi_{k_{i}}(\alpha)=\pi_{k_{i}}(\beta)}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)+\sum_{\stackrel{{\scriptstyle\alpha\neq\beta\in
A_{i}^{\prime}}}{{\pi_{k_{i}}(\alpha)\neq\pi_{k_{i}}(\beta)}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$
$\displaystyle<$ $\displaystyle(d-1)\cdot(k_{i}+c)+d\cdot\sum_{{0\neq
a\in\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}}}\bar{v}(a)$
and
$\displaystyle\sum_{{0\neq
a\in\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}}}\bar{v}(a)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k_{i}-1}|\\{a:\bar{v}(a)=j\\}|\cdot
j\quad=\quad\sum_{j=0}^{k_{i}-1}\sum_{l=1}^{j}|\\{a:\bar{v}(a)=j\\}|$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k_{i}-1}\sum_{j=l}^{k_{i}-1}|\\{a:\bar{v}(a)=j\\}|\quad=\quad\sum_{l=1}^{k_{i}-1}|\\{a:\bar{v}(a)\geq
l\\}|$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{k_{i}-1}(q_{i}^{f_{i}(k_{i}-l)}-1)\quad=\quad\sum_{l=0}^{k_{i}-1}q_{i}^{f_{i}l}-k_{i}\quad=\quad\frac{1-q_{i}^{f_{i}k_{i}}}{1-q_{i}^{f_{i}}}-k_{i}$
and plugging this into the previous inequality gives condition (c) for
$\tilde{g}_{i}$.
As $\mathcal{O}_{K}$ is dense in $\mathcal{O}_{F_{i}}$ with respect to
$v_{i}$, we obtain a monic polynomial ${g_{i}}\in\mathcal{O}_{K}[X]$ of degree
$dr$ arbitrarily close to $\tilde{g_{i}}$. Let ${A_{i}}$ be the set of roots
of ${g_{i}}$. By the continuity of roots [EP05, Theorem 2.4.7] we can achieve
that the roots of ${g_{i}}$ are arbitrarily close to the roots of
$\tilde{g_{i}}$, in particular that conditions (b) and (c) are satisfied by
$g_{i}$ and $A_{i}$. Moreover, by Krasner’s lemma [Lan94, Ch.II, §2,
Proposition 4], we can in addition achieve condition (a), completing the proof
of the claim. ∎
Now, let $p_{0}$ be the smallest prime number not in the set
$\\{p_{1},\ldots,p_{n}\\}$ and let $\mathfrak{p}_{0}$ be a prime ideal of
$\mathcal{O}_{K}$ above $p_{0}$. Fix a monic polynomial
$g_{0}\in\mathcal{O}_{K}[X]$ of degree $dr$ whose reduction modulo
$\mathfrak{p}_{0}$ is irreducible. Let
(3.2)
$m_{i}=\frac{d}{e_{i}}\left(\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+k_{i}+2c\right)$
and
$\mathfrak{a}=\mathfrak{p}_{0}\mathfrak{p}_{1}^{m_{1}}\cdots\mathfrak{p}_{n}^{m_{n}}.$
By the Chinese Remainder Theorem and Proposition 2.2 there exists a monic
polynomial $g\in\mathcal{O}_{K}[X]$ such that
1. (1)
$\deg g=dr$,
2. (2)
$g\equiv g_{0}\mbox{ mod }\mathfrak{p}_{0}[X]$,
3. (3)
$g\equiv{g_{i}}\mbox{ mod }\mathfrak{p}_{i}^{m_{i}}[X]$ for $i=1,\dots,n$, and
4. (4)
$|\sigma(a)|\leq\delta_{K}N(\mathfrak{a})^{1/[K:\mathbb{Q}]}$ for every
coefficient $a$ of $g$ and every $\sigma\in{\rm Hom}(K,\mathbb{C})$,
where
$\delta_{K}=[K:\mathbb{Q}]^{\frac{3}{2}}{2}^{\frac{[K:\mathbb{Q}]([K:\mathbb{Q}]-1)}{2}}\sqrt{|\Delta_{K}|}$.
Note that (2) implies that $g$ is irreducible. In particular, we get from (1)
and (3.1) that every root $\alpha$ of $g$ satisfies the degree bound (1.2) of
Theorem 1.3.
### 3.3. The roots of $g$ are in $E_{i}$ for every $i$.
We claim that the conditions of Proposition 2.3 hold for the field $E_{i}$,
the polynomial $g$ and $x_{0}=\alpha\in{A_{i}}$ (which lies in $E_{i}$ by
condition (a)) by setting $a=v_{i}({g_{i}}^{\prime}(\alpha))$ and
$b=k_{i}+c-1$. Indeed, by (c)
(3.3) $\displaystyle a=v_{i}({g_{i}}^{\prime}(\alpha))\leq
d\cdot\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+dc<e_{i}m_{i}-b,$
and, writing $g-{g_{i}}=t_{i}$ with $t_{i}\in\mathfrak{p}_{i}^{m_{i}}[X]$ by
(3) of Section 3.2, we have $g(\alpha)=t_{i}(\alpha)$ and therefore
$v_{i}(g(\alpha))\geq e_{i}m_{i}>a+b$, so condition (i) holds. Similarly for
condition (ii), we have
$g^{\prime}(\alpha)=t_{i}^{\prime}(\alpha)+{g_{i}}^{\prime}(\alpha)$ and since
$v_{i}(t_{i}^{\prime}(\alpha))\geq
e_{i}m_{i}>v_{i}({g_{i}}^{\prime}(\alpha)),$
we conclude that
$v_{i}(g^{\prime}(\alpha))=v_{i}({g_{i}}^{\prime}(\alpha))=a$. Now for
$\nu\geq 2$ write
${g_{i}}^{(\nu)}(\alpha)=\nu!{g_{i}}^{\prime}(\alpha)\sum_{\begin{subarray}{c}B\subseteq{A_{i}}\\\
|B|=\nu-1\\\ \alpha\notin B\end{subarray}}\prod_{\beta\in
B}(\alpha-\beta)^{-1}.$
Thus
$v_{i}({{g_{i}}^{(\nu)}(\alpha)}/{\nu!})\geq
a-(\nu-1)\max_{\beta\neq\alpha}v_{i}(\alpha-\beta){\geq}a-(\nu-1)b$
where the last inequality holds by (b). Moreover, using (3.3), we get
$v_{i}({t_{i}^{(\nu)}(\alpha)}/{\nu!})\geq e_{i}m_{i}-v_{i}(\nu!)\geq
a+b-\frac{e(\mathfrak{P}_{i}|p_{i})\nu}{p_{i}-1}\geq a-(\nu-1)b$
where the last inequality holds since $b\geq
c\geq\frac{e(\mathfrak{P}_{i}|p_{i})}{p_{i}-1}$. Thus
$v_{i}({g^{(\nu)}(\alpha)}/{\nu!})\geq\min\left\\{v_{i}({{g_{i}}^{(\nu)}(\alpha)}/{\nu!}),v_{i}({t_{i}^{(\nu)}(\alpha)}/{\nu!})\right\\}\geq
a-(\nu-1)b$
fulfilling condition (iii).
So Proposition 2.3 gives ${\alpha}^{\prime}\in E_{i}$ with
$g({\alpha}^{\prime})=0$ and $v_{i}(\alpha^{\prime}-\alpha)>b$. As
$v_{i}(\alpha-\beta)\leq b$ for all $\beta\in{A_{i}}\setminus\\{\alpha\\}$ by
(b), we conclude that ${\alpha}^{\prime}\neq{\beta}^{\prime}$ for all
$\alpha\neq\beta$. Hence $g$ has precisely $|{A_{i}}|=dr$ many roots in
$E_{i}$. As this holds for every $i$ and $g$ totally splits in the maximal
Galois extension $L$ of $K$ that is contained in all $E_{i}$. Moreover, as
$g\in\mathcal{O}_{K}[X]$, all roots of $g$ are actually in $\mathcal{O}_{L}$.
### 3.4. Bounding the height of the roots of $g$
From condition (4) of Section 3.2, for every coefficient $a$ of $g$ and every
$\sigma\in{\rm Hom}(K,\mathbb{C})$, we have
$|\sigma(a)|\leq
B:=\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\cdot\prod_{i=1}^{n}N(\mathfrak{p}_{i})^{m_{i}/[K:\mathbb{Q}]}.$
By Proposition 2.4, $g$ has a root $\alpha\in\mathcal{O}_{L}$ with $h(\alpha)$
bounded by
(3.4) $\frac{\log(B\sqrt{\deg g+1})}{\deg
g}\leq\frac{\log\left(\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\sqrt{\deg
g+1}\right)}{\deg g}+\sum_{i=1}^{n}\frac{m_{i}}{\deg
g}\cdot\frac{\log(q_{i})}{[K:\mathbb{Q}]}.$
By the definition of $m_{i}$ in (3.2) and recalling $\deg g=dr$ from (1), we
have
(3.5) $\displaystyle\frac{m_{i}}{\deg g}$ $\displaystyle=$
$\displaystyle\frac{1}{e_{i}}\cdot\frac{1}{q_{i}^{f_{i}}-1}\cdot\frac{q_{i}^{f_{i}k_{i}}-1}{r}+\frac{k_{i}}{e_{i}r}+\frac{2c}{e_{i}r}.$
Condition (i) of Section 3.2 implies that
$\frac{q_{i}^{f_{i}k_{i}}-1}{r}\leq 1+\epsilon.$
Moreover,
$\frac{k_{i}}{e_{i}r}=\frac{\log(q_{i}^{f_{i}k_{i}})}{e_{i}rf_{i}\log(q_{i})}\leq\frac{2d}{e_{i}f_{i}\log(q_{i})}\cdot\frac{\log(\deg
g)}{\deg g}\leq 3C^{n}\frac{\log(\deg g)}{\deg g}$
Finally,
$\frac{2c}{e_{i}r}\leq\frac{8dC^{n+1}}{e_{i}\deg g}\leq\frac{8C^{2n+1}}{\deg
g}.$
Therefore, substituting in (3.5), and recalling that $C\geq 2$ and $\rho\geq
3$, we have
$\frac{m_{i}}{\deg
g}\leq\frac{1}{e_{i}(q_{i}^{f_{i}}-1)}+\frac{\epsilon}{e_{i}(q_{i}^{f_{i}}-1)}+11C^{2n+1}\frac{\log(\deg
g)}{\deg g}.$
Thus the second summand in (3.4) can be bounded as
$\displaystyle\sum_{i=1}^{n}\frac{m_{i}}{\deg
g}\cdot\frac{\log(q_{i})}{[K:\mathbb{Q}]}\leq\sum_{i=1}^{n}$
$\displaystyle\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+n\epsilon+11nC^{2n+2}\frac{\log(\deg
g)}{\deg g}.$
As for the first summand in (3.4), note that
$N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\leq p_{0}$ where $p_{0}$ is the
smallest prime number not in the set $\\{p_{1},\ldots,p_{n}\\}$, which, by
Bertrand’s postulate, can be bounded by $p_{0}<2\max_{i}p_{i}\leq 2C$.
Moreover,
$\delta_{K}=[K:\mathbb{Q}]^{\frac{3}{2}}{2}^{\frac{[K:\mathbb{Q}]([K:\mathbb{Q}]-1)}{2}}\sqrt{|\Delta_{K}|}\leq
C^{2}\cdot 2^{\frac{C(C-1)}{2}}$
and thus, as $C\geq 2$,
$\displaystyle\frac{\log\left(\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\sqrt{\deg
g+1}\right)}{\deg g}$ $\displaystyle\leq$
$\displaystyle\frac{\log\left(C^{3}2^{\frac{C^{2}-C+2}{2}}\sqrt{\deg
g+1}\right)}{\deg g}\leq 2nC^{2n+1}\frac{\log(\deg g)}{\deg g}.$
Therefore, from (3.4) we get
$h(\alpha)\leq\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+n\epsilon+13nC^{2n+2}\frac{\log(\deg
g)}{\deg g},$
so $\alpha$ satisfies the height bound (1.1) of Theorem 1.3 (recalling that
$\epsilon=0$ if $n=1$).
## Acknowledgments
The authors thank Lukas Pottmeyer for pointing out the results in [Fil14,
FP15, FP19], Paul Fili for the exchange regarding [Fil14], and Philip Dittmann
for helpful discussions on $p$-adic fields as well as for suggesting the short
proof of Proposition 2.3. The first author’s work has been funded by the ANR
project Gardio 14-CE25-0015.
## References
* [BFH17] J. Biasse, C. Fieker and T. Hofmann. On the computation of the HNF of a module over the ring of integers of a number field. J. Symb. Comp. 80(3):581–615, 2017.
* [BZ01] E. Bombieri and U. Zannier. A note on heights in certain infinite extensions of $\mathbb{Q}$. Rend. Mat. Acc. Lincei 12:5–15, 2001.
* [CSV12] X.-W. Chang, D. Stehlé and G. Villard. Perturbation analysis of the QR factor R in the context of LLL lattice basis reduction. Math. Comput. 81(279):1487–1511, 2012.
* [CF21] S. Checcoli and A. Fehm. On the Northcott property and local degrees. To appear in Proc. Amer. Math. Soc., 2021.
* [EP05] A. J. Engler and A. Prestel. Valued Fields. Springer, 2005.
* [Fil14] P. Fili. On the heights of totally $p$-adic numbers. J. Théor. Nombres Bordeaux 26(1):103–109, 2014.
* [FP15] P. Fili and C. Petsche. Energy Integrals Over Local Fields and Global Height Bounds. International Mathematics Research Notices 2015(5):1278–1294, 2015.
* [FP19] P. A. Fili and L. Pottmeyer. Quantitative height bounds under splitting conditions. Trans. Amer. Math. Soc. 372:4605–4626, 2019.
* [Fla96] V. Flammang. Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers. Math. Comp. 65(213):307–311, 1996.
* [Lan94] S. Lang. Algebraic Number Theory. Second Edition. Springer, 1994.
* [NSV11] A. Novocin, D. Stehlé and G. Villard. An LLL-reduction algorithm with quasi-linear time complexity. In: STOC’11–Proceedings of the 43rd ACM Symposium on Theory of Computing. ACM, New York, pp. 403–412, 2011.
* [PS19] C. Petsche and E. Stacy. A dynamical construction of small totally p-adic algebraic numbers. J. Number Theory 202:27–36, 2019.
* [Pot15] L. Pottmeyer. Heights and totally $p$-adic numbers. Acta Arith. 171(3):277–291, 2015.
* [Sch73] A. Schinzel. On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith. 24:385–399, 1973. Addendum, ibidem, 26, 329–361, 1973.
* [Sch80] W. M. Schmidt. Diophantine Approximation. Springer, 1980.
* [Ser80] J.-P. Serre. Local Fields. Springer, 1980.
* [Smy80] C. J. Smyth. On the measure of totally real algebraic integers. J. Austral. Math. Soc. Ser. A 30(2):137–149, 1980.
* [Smy81] C. J. Smyth. On the measure of totally real algebraic integers. II. Math. Comp. 37(155):205–208, 1981.
* [Smy07] C. J. Smyth. The Mahler measure of algebraic numbers: a survey. In: McKee, Smyth (eds.), Number Theory and Polynomials. Cambridge University Press. pp. 322–349, 2007.
* [Zan09] U. Zannier. Lecture Notes on Diophantine Analysis (with an Appendix by F. Amoroso). Edizioni della Normale, 2009.
|
Dedicated to the memory of Uffe V. Haagerup
# C*-dynamical rapid decay
Erik Christensen Erik Christensen, Mathematics Institute, University of
Copenhagen, Copenhagen, Demark<EMAIL_ADDRESS>
###### Abstract.
Some well known results by Haagerup, Jolissaint and de la Harpe may be
extended to the setting of a reduced crossed product of a C*-algebra
${\mathcal{A}}$ by a discrete group $G.$ We show that for many discrete
groups, which include Gromov’s hyperbolic groups and finitely generated
discrete groups of polynomial growth, an inequality of the form
$\|X\|\leq C\sqrt{\sum_{g\in G}(1+|g|)^{4}\|X_{g}\|^{2}}$
holds for any finitely supported operator $X$ in the reduced crossed product.
###### Key words and phrases:
Haagerup, C*-algebra, discrete group, crossed product, rapid decay
###### 2010 Mathematics Subject Classification:
Primary: 46L55, 37A55. Secondary: 22D55, 46L07.
## 1\. Introduction
Any result in classical harmonic analysis naturally raises the question, does
this extend to a non commutative setting ? In the situation of a discrete
dynamical system, you may work with the group of integers ${\mathbb{Z}}$ which
acts on a compact space as the group of homeomorphisms generated by a single
homeomorphism. By Gelfand’s fundamental theorem we know that the set of
compact topological spaces correspond to unital commutative C*-algebras, and
then the classical discrete dynamical system as described above can be made
non commutative in 2 ways, either by studying a general non commutative
C*-algebra equipped with a single *-automorphism or by investigating
properties of an action of a general discrete group of homeomorphisms on a
compact space. The construction named reduced crossed product of a C*-algebra
by a discrete group encodes a set up for the study of the action of a general
discrete group on a C*-algebra.
Here we will extend results, by Haagerup from [8] on free non abelian groups
${\mathbb{F}}_{d},$ by Jolissaint [11] on the concept named rapid decay and by
de la Harpe [10] on hyperbolic groups, from the setting of a reduced group
C*-algebra to the setting of a reduced crossed product C*-algebra. The
C*-algebra ${\mathcal{A}}$ upon which the discrete group $G$ acts by
*-automorphisms $\alpha_{g}$ may be abelian or non abelian, and the case where
the algebra is the trivial ${\mathcal{A}}={\mathbb{C}}I$ is the reduced group
C*-algebra case.
Uffe Haagerup’s article [8] has been very influential in the study of discrete
groups and von Neumann algebras of type II${}_{1},$ and his name and results
are linked to many fundamental mathematical properties such as the Haagerup
approximation property [3], [9], the completely bounded approximation
property [5] and the Haagerup tensor product [16]. We will not go into the
study of any of these aspects but concentrate on an extension of the first
basic result from [8] and lift it from the reduced group case to the reduced
crossed product case for discrete hyperbolic groups acting on general
C*-algebras. In this setting we prove that there exists an $C>0,$ depending on
the group only, such that for any linear combination $X=\sum L_{g}X_{g}$ in
the algebraic reduced crossed product we have the following inequality
(1.1) $\|X\|\leq C\sqrt{\sum_{g\in G}(1+|g|)^{4}\|X_{g}\|^{2}}\,$
which is a direct generalization of Haagerup’s estimate in the group algebra
case, since for these groups $C<2.$ In the article [11] Jolissaint showed
that, in the group algebra case, inequalities like (1.1) may be obtained for
many other discrete groups, which are not free, and he introduced the property
named rapid decay for a discrete group $G$ with a length function $|g|,$ if
there exist positive constants $C,r$ such that for any $x=\sum
x_{g}\lambda_{g}$ in ${\mathbb{C}}[G]$
(1.2) $\|x\|_{op}\leq C\sqrt{\sum(1+|g|)^{2r}|x_{g}|^{2}}.$
Shortly after Jolissaint had obtained his results they were extended to the
setting of hyperbolic groups by de la Harpe [10], and it turns out that it is
possible to extend both Jolissaint’s and de la Harpe’s results to the crossed
product setting. The basic insight which makes this possible was formulated by
Jollisaint in his lemmas 3.2.2 and 3.2.3. Then de la Harpe proved that these
lemmas are true in the setting of discrete finitely generated hyperbolic
groups, so Jolissaint’s findings could be extended. Here we have collected the
statements from Jolissaint’s basic lemmas into a property (J) which a discrete
group with a length function may have, and then we prove that the inequality
(1.1) holds, in any reduced crossed product of a C*-algebra and a group
satisfying property (J). It seems possible to us that property (J) implies
hyperbolicity, but we have very little experience in dealing with such a
question.
We have also considered reduced crossed products of C*-algebras by discrete
finitely generated groups which have polynomial growth. In this case it is
possible to get the following - much better - result. If a discrete finitely
generated group has polynomial growth then there exist $C>0,$ $s>0$ such that
for any finitely supported operator $X=\sum L_{g}X_{g}$ in a reduced crossed
product $C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G)$ we have
(1.3) $\displaystyle\|X\|\leq C$ $\displaystyle\|\sum_{g\in
G}(1+|g|)^{2s}L_{g}X_{g}X_{g}^{*}L_{g}^{*}\|^{\frac{1}{2}}$
$\displaystyle\|X\|\leq C$ $\displaystyle\|\sum_{g\in
G}(1+|g|)^{2s}X_{g}^{*}X_{g}\|^{\frac{1}{2}}$
The advantage of (1.3) over (1.2) is that usually
$\|\sum_{g\in C_{k}}X_{g}^{*}X_{g}\|^{\frac{1}{2}}<\big{(}\sum_{g\in
C_{k}}\|X_{g}^{*}X_{g}\|\big{)}^{\frac{1}{2}}.$
## 2\. Notation and norms
Most of the content of this section is well known in the setting of a reduced
group C*-algebra, but here we deal with a reduced crossed product C*-algebra.
In order to be able to generalize the methods from Haagerup’s paper [8], we
need a characterization of the operator norm in a reduced crossed product of a
C*-algebra by a discrete group, and this result, which we think might be known
but unpublished, is presented in Proposition 2.4 in a self contained frame. We
start with a well known operator theoretical version of Cauchy-Schwarz’
inequality, [16].
###### Definition 2.1.
Let $H,\,K$ be Hilbert spaces, $J$ be an index set and $(a_{\iota})_{(\iota\in
J)}$ a family of bounded operators in $B(H,K).$
* (i)
If the sum $\sum_{\iota}a_{\iota}^{*}a_{\iota}$ is ultrastrongly convergent in
$B(H)$ we say that the family $(a_{\iota})$ is column bounded with column norm
$\|(a_{\iota})\|_{c}:=\|\sum a_{\iota}^{*}a_{\iota}\|^{\frac{1}{2}}.$
* (ii)
If the sum $\sum_{\iota}a_{\iota}a_{\iota}^{*}$ is ultrastrongly convergent in
$B(K)$ we say that the family $(a_{\iota})$ is row bounded with row norm
$\|(a_{\iota})\|_{r}:=\|\sum a_{\iota}a_{\iota}^{*}\|^{\frac{1}{2}}.$
###### Proposition 2.2.
Let $H,\,K$ be Hilbert spaces, $J$ be an index set and $(a_{\iota})_{(\iota\in
J)},$ $(b_{\iota})_{(\iota\in J)}$ be column bounded families of operators in
$B(H,K).$
* (i)
The sum $\sum_{\iota}a_{\iota}^{*}b_{\iota}$ is ultrastrongly convergent in
$B(H)$ and the operator norm of the sum satisfies
$\|\sum_{\iota}a_{\iota}^{*}b_{\iota}\|\leq\|(a_{\iota})\|_{c}\|(b_{\iota})\|_{c}.$
* (ii)
There exists a column bounded family $(e_{\iota})_{(\iota\in J)}$ of column
norm at most $1$ such that for the positive bounded operator $h$ on $H$
defined by $h:=(\sum_{\iota}a_{\iota}^{*}a_{\iota})^{\frac{1}{2}}$ we have for
each $\iota\in J,\,a_{\iota}=e_{\iota}h$ and the sum
$\sum_{\iota}e_{\iota}^{*}e_{\iota}$ equals the range projection of $h.$
* (iii)
Let $k:=(\sum_{\iota}b_{\iota}^{*}b_{\iota})^{\frac{1}{2}},$ then there exists
a contraction $c$ in $B(H)$ such that
$\sum_{\iota}b_{\iota}^{*}a_{\iota}=kch.$
###### Proof.
The families $(a_{\iota})$ and $(b_{\iota})$ represent bounded column
operators in the operator space $M_{(J,1)}\big{(}B(H,K)\big{)}$ and the
statement (i) follows from properties of the operator product.
The statement (ii) follows from the polar decomposition applied to the column
operator $(a_{\iota}).$
The result in statement (ii) may be applied to the column operator
$(b_{\iota})$ such that each $b_{\iota}=f_{\iota}k,$ we can then define a
contraction $c$ in $B(H)$ by $c:=\sum_{\iota}f_{\iota}^{*}e_{\iota}$ and
statement (iii) follows. ∎
The rest of this article takes place in the setting of the reduced crossed
product of a C*-algebra ${\mathcal{A}}$ by a discrete group $G,$ which acts on
${\mathcal{A}}$ by the *-automorphisms $\alpha_{g}.$ We made a study of the
properties of this crossed product in the article [4] and we will use most of
the notation and several of the results of that article below. A basic point
of view in Section 2 of [4] is that there are many facts related to properties
of the coefficients of a Fourier series which generalize to properties of the
coefficients of an element in a reduced discrete C*-crossed product.
We recall that any element $X$ in
${\mathcal{C}}:=C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G)$ has a Fourier
series expansion $X\sim\sum_{g\in G}L_{g}X_{g},$ where the sum is convergent
in the norm $\|.\|_{\pi}$ described in Proposition 2.2 of [4]. A simple
computation shows that for $X\sim\sum L_{g}X_{g}$ we have that the column and
row norms of the family $(L_{g}X_{g})_{(g\in G)}$ may be computed as
(2.1) $\displaystyle\|(L_{g}X_{g})_{(g\in G)}\|_{c}^{2}$
$\displaystyle=\|\pi(X^{*}X)\|=\|(X^{*}X)_{e}\|$
$\displaystyle\|(L_{g}X_{g})_{(g\in G)}\|_{r}^{2}$
$\displaystyle=\|\pi(XX^{*})\|=\|(XX^{*})_{e}\|.$
In particular we notice the following proposition.
###### Proposition 2.3.
Let $X\sim\sum_{g}L_{g}X_{g}$ be an element in ${\mathcal{C}}$ then the sum
converges in the column norm.
We will use the $\pi-$norm or column norm to estimate the operator norm in the
computations to come, so we need the following proposition. It may be known to
several people, but may be in a slightly different setting. We are not aware
of an explicit formulation as the one we present, but the results of [17] have
a similar flavour. We do also think that people who prefer to look at
completely positive mappings as correspondences or operator bimodules may know
the result, but still we are missing a reference.
###### Proposition 2.4.
Let ${\mathcal{B}}$ be a C*-algebra, $H$ a Hilbert space and
$\pi:{\mathcal{B}}\to B(H)$ a completely positive and faithful mapping then
$\forall
b\in{\mathcal{B}}:\|b\|=\sqrt{\sup\\{\|\pi(x^{*}b^{*}bx)\|\,:\,\|\pi(x^{*}x)\|\leq
1\\}}.$
###### Proof.
We may suppose that ${\mathcal{B}}$ is a subalgebra of $B(K)$ for some Hilbert
space $K,$ then since $\pi$ is completely positive and faithful there exists
by Stinespring’s result [18] a faithful representation $\rho$ of
${\mathcal{B}}$ on $H$ and a bounded operator $C$ in $B(K,H)$ such that
$\forall b\in{\mathcal{B}}:\quad\pi(b)=C^{*}\rho(b)C.$
We may define a semi norm, say $n,$ on ${\mathcal{B}}$ by
(2.2) $\forall b\in{\mathcal{B}}:\quad
n(b):=\sup\\{\|\rho(bx)C\|:\|\rho(x)C\|\leq 1\\}.$
Since $\|C^{*}\rho(y^{*}y)C\|=\|\rho(y)C\|^{2}$ for any $y$ in ${\mathcal{B}}$
and $\pi$ is faithful we get that $n$ is a norm and that
$\forall
b\in{\mathcal{B}}:n(b)=\sqrt{\sup\\{\|\pi(x^{*}b^{*}bx)\|\,:\,\|\pi(x^{*}x)\|\leq
1\\}}.$
On the other hand the definition (2.2) implies that for $b,d$ in
${\mathcal{B}}$ we have $n(bd)\leq n(b)n(d)$ so $n$ is an algebra norm and
$n(b)\leq\|b\|.$ For any pair $b,x$ in ${\mathcal{B}}$ with
$\|\rho(x)C\|^{2}=\|\pi(x^{*}x)\|\leq 1$ we have
$\displaystyle n(b^{*}b)$
$\displaystyle\geq\|\rho(b^{*}bx)C\|\geq\|C^{*}\rho(x^{*}b^{*}bx)C\|=\|\rho(bx)C\|^{2},\text{
so }$ $\displaystyle n(b^{*})n(b)$ $\displaystyle\geq n(b^{*}b)\geq n(b)^{2}.$
From here it follows that $n(b)=n(b^{*})$ and then $n(b^{*}b)=n(b)^{2},$ and
the completion say $\hat{\mathcal{B}}$ of ${\mathcal{B}}$ with respect to the
C*-norm $n$ becomes a C*-algebra such that the inclusion of ${\mathcal{B}}$ in
$\hat{\mathcal{B}}$ is a contractive faithful *-homomorphism and hence an
isometry. The proposition follows. ∎
In Haagerup’s and Jolissaint’s articles, [8], [11] they use the symbol $*$ to
denote the operator product in a group algebra, since this is really a
convolution product. In our article on crossed product C*-algebras [4] the
operator product is an invisible dot and the $*$ is used to denote the
Hadamard product, which in the notation from above takes the form
$\big{(}\sum_{g}L_{g}X_{g}\big{)}*\big{(}\sum_{h}L_{h}Y_{h}\big{)}:=\sum_{f}L_{f}X_{f}Y_{f}.$
We will use this convention here, too.
## 3\. The property (J)
The basic ideas in the arguments to come are taken from Haagerup’s article,
and in the setting of a non commutative free group it is clear that for two
group elements $x,y$ with reduced words $x=x_{1}\dots x_{k}$ and $y=y_{1}\dots
y_{l}$ the number of cancellations, say $p,$ needed to spell to $xy$ gives the
spelling of $xy$ directly as $xy=x_{1}\dots x_{(k-p)}y_{(p+1)}\dots y_{l}.$ In
Jolissaint’s article he uses this observation in a very clever original way,
and he shows that this idea may be generalized to work up to a controllable
error in some groups of isometries on a Riemannian manifold with bounded
strictly negative sectional curvature. Then de la Harpe showed that
Jolissaint’s method of dealing with cancellations works in any finitely
generated discrete group which is hyperbolic, as defined by Gromov, [7] and
[6]. Here we will instead take these results as the basis for the definition
of a property we have named (J).
We will now define the setting in which we will use Haagerup’s, Jolissaint’s
and de la Harpe’s ideas. We define the cancellation number in a general group
with a length function as follows.
###### Definition 3.1.
Let $G$ be a group with a length function $g\to|g|.$ For $g,h$ in $G$ the
cancellation number $c(g,h)$ of the pair is defined as the non negative
integer $p(g,h)$ which satisfies
$2p(g,h)\leq|g|+|h|-|gh|<2p(g,h)+2.$
It follows from the properties of a length function that $0\leq
p(g,h)\leq\min\\{|g|,|h|\\},$ and the cancellation number divides the
cartesian product $G\times G$ into a sequence of disjoint subsets
$(P_{p})_{(p\in{\mathbb{N}}_{0})}$ defined by $P_{p}:=\\{(g,h)\in G\times
G\,:\,p(g,h)=p.\\}.$
Following Jolissaint we define certain subsets of a group $G$ with a length
function $|g|$ as follows
###### Definition 3.2.
* * (i)
$\forall r\geq 0:\quad\quad\quad\quad\,\,\,\,\,B_{r}:=\\{g\in G\,:\,|g|\leq
r\\},$
* (ii)
$\forall k\geq 0:\,\quad\quad\quad\,\,\,\,\quad C_{k}:=\\{g\in
G\,:\,k-1<|g|\leq k\\},$
* (iii)
$\forall k\geq 0\,\forall\alpha\geq 0:\quad C_{k,\alpha}:=\\{g\in
G\,:\,k-\alpha\leq|g|\leq k+\alpha\\},$
We can now collect the sufficient conditions, we have have dragged out of
[11], into a property we name (J). We will not focus on which groups that may
satisfy the property (J), but we think that the survey article on rapid decay
[2] by Chatterji will show that many groups do have property (J). On the other
hand we will sketch arguments which show that Jolissaint’s and de la Harpe’s
examples do have property (J). It is easy to see that free non abelian groups
do have property (J) with the extra very nice properties that the constants of
the definition satisfy $\alpha=\beta=\gamma=0$ and $N=1.$
###### Definition 3.3.
Let $G$ be a discrete group with a length function $g\to|g|.$ The pair
$(G,|\cdot|)$ has property, (J) if
(3.1) $\displaystyle\exists\alpha>0,\beta>0,\gamma>0,:$ $\displaystyle(i)\,$
$\displaystyle\forall g\in G,\forall 0\leq s<|g|+1,\,\exists u_{(g,s)}\in
C_{s,\alpha}$ $\displaystyle(ii)\,$ $\displaystyle
v_{(g,s)}:=u_{(g,s)}^{-1}g\in C_{(|g|+1-s),\beta}$ $\displaystyle(iii)\,$
$\displaystyle\forall p\in{\mathbb{N}}_{0},\forall(a,b)\in
P_{p}\,\exists\,c(a,b)\in C_{p,\gamma}\text{ s. t. }$ $\displaystyle
a=u_{(ab,(|a|-p))}c(a,b),\,b=c(a,b)^{-1}v_{(ab,(|a|-p))}.$ (3.2)
$\displaystyle\forall\mu>0,\nu>0\,\exists N\in{\mathbb{N}}:$
$\displaystyle\forall b\in G\forall\,0\leq p\leq|b|\,:\,\,|\\{(c,v)\in
C_{p,\mu}\times C_{(|b|-p),\nu}:c^{-1}v=b\\}|\leq N.$
###### Remark 3.4.
It is important for the following proofs in the next section to notice that
the group element $u_{(g,s)}$ is determined uniquely by $g$ and $s,$ and then
$v_{(g,s)}=u_{(g,s)}^{-1}g$ is also determined by $g$ and $s$ only.
The interesting thing about the factors $c(a,b)$ is that they always
approximately satisfy $|c(a,b)|=p.$
In Jolissaint’s proof the group element $u_{(g,s)}$ is chosen geometrically,
as we sketch now. On the geodesic which connects a point $m$ in the manifold
with its image $g(m),$ one chooses the point $n_{s}$ which has the distance
$s$ to $g(m).$ Then $u_{(g,s)}$ is chosen such that the distance between
$n_{s}$ and $u_{(g,s)}^{-1}\big{(}g(m)\big{)}$ is minimal among the distances
between $n_{s}$ and the set $\\{u(m)\,:\,u\in G\\}.$ We will not continue to
quote Jolissaints proof, but show that a group $G,$ which is hyperbolic with
respect to a given word length, has the property (J). The proof follows that
of de la Harpe [10], but in order to explicitly establish the property (J)
from the definition above, we repeat part of it here. We will use the
following notation. For a real number $s,$ the expression $\lfloor s\rfloor$
means the largest integer dominated by $s.$
###### Lemma 3.5.
Let $G$ be a finitely generated discrete group which is hyperbolic with
respect to word length. Then $G$ has the property (J).
###### Proof.
For a group element $g$ written in reduced form as $g=g_{1}\dots g_{|g|}$ and
a real $s,\,0\leq s<|g|+1$ we define
$u_{(g,s)}:=\begin{cases}e\quad\quad\quad\,\,\,\text{ if }0\leq s<1\\\
g_{1}\dots g_{\lfloor s\rfloor}\text{ if }1\leq s<|g|+1,\end{cases}$
and we find that $u_{(g,s)}\in C_{s,1}$ so $\alpha=1$ may be used. Similarly
we find
$v_{(g,s)}=\begin{cases}g_{(\lfloor s\rfloor+1)}\dots g_{|g|}\text{ if
}\,\,0\leq s<|g|\\\ e\quad\quad\quad\,\,\quad\quad\text{ if }|g|\leq
s<g+1,\end{cases}$
and we get $v_{(g,s)}\in C_{(|g|+1-s),1},$ so $\beta=1$ is possible.
Given a non negative integer $p$ and a pair of group elements $(a,b)\in P_{p}$
with $ab=g,$ then for $k:=|a|,$ $l:=|b|$ there exists $c\in\\{0,1\\}$ such
that $|g|=k+l-2p-c.$ Then for $u_{(g,k-p)}$ we may apply the lemma at the
bottom of page 771 in [10], to see that there exists an $M\geq 0,$ independent
of $k,l,p,c$ such that for $c(a,b):=u_{(g,k-p)}^{-1}a$ the following
inequalities hold
(3.3) $p\leq|c(a,b)|\leq p+M,$
so $c(a,b)\in C_{p,M}.$
In order to establish the property (3.2) we remark, that in our case we have
$v_{(g,(k-p))}\in C_{(|g|+1-k+p),1}=C_{(l-p-c+1),1},$ which means
$c(a,b)\in C_{p,M},\,\,v_{g,(k-p)}\in C_{(|b|-p),2}\text{ and
}c(a,b)v_{(g,(k-p))}=b.$
The result then follows from item (ii) in the lemma of [10].
∎
## 4\. Rapid decay
The most basic example of the phenomenon named rapid decay by Paul Jolissaint
in [11] is presented quite early in many courses on Fourier series. The
example tells, that if $f(t)$ is a differentiable complex
$2\pi-$periodic function on ${\mathbb{R}},$ then its Fourier series is
uniformly convergent. This is proven via the following argument based on the
Cauchy-Schwarz inequality, as follows. Let $f(t)$ have the Fourier series
$f(t)\sim\sum_{\mathbb{Z}}c_{n}e^{int},$ then the derivative $f^{\prime}(t)$
has the Fourier series $f^{\prime}(t)\sim\\\ \sum_{\mathbb{Z}}inc_{n}e^{int},$
and the sequence of complex numbers $(nc_{n})_{(n\in{\mathbb{Z}})}$ is in
$\ell^{2}({\mathbb{Z}}).$ Since for $n\neq 0$ we may write
$c_{n}=\frac{1}{n}(nc_{n}),$ we get that the sequence
$(c_{n})_{(n\in{\mathbb{Z}})}$ is in $\ell^{1}({\mathbb{Z}})$ with
$\|(c_{n})\|_{1}\leq|c_{0}|+\frac{\pi}{\sqrt{3}}\|(nc_{n})\|_{2}.$ If we
translate this to the setting of the discrete group ${\mathbb{Z}}$ equipped
with the natural length function $|n|,$ we find that the group algebra
$C^{*}_{r}({\mathbb{Z}})$ may be identified with the complex continuous
$2\pi-$periodic functions on the real axis and an element $x$ in the group
algebra which correspond to a differentiable function has a presentation as a
uniformly convergent sum $x=\sum_{(n\in{\mathbb{Z}})}x_{n}\lambda_{n}.$ This
example may be generalized to the setting of a discrete group with a length
function when the content of the example is formulated as in the following
proposition.
###### Proposition 4.1.
Let $x\sim\sum_{\mathbb{Z}}x_{n}\lambda_{n}$ be an operator in
$C^{*}_{r}({\mathbb{Z}}).$ If the sequence
$\big{(}(1+|n|)x_{n}\big{)}_{(n\in{\mathbb{Z}})}$ is in
$\ell^{2}({\mathbb{Z}}),$ then the series is uniformly convergent and
$\|x\|\leq\big{(}\frac{\pi^{2}}{3}-1\big{)}^{\frac{1}{2}}\big{(}\sum_{n\in{\mathbb{Z}}}|x_{n}|^{2}(1+|n|)^{2}\big{)}^{\frac{1}{2}}.$
We recall from Chatterji’s survey article [2], Definition 2.9, in a modified
form.
###### Definition 4.2.
Let $G$ be a discrete group with a length function $|g|,$ then $G$ has the
rapid decay property, with respect to $|g|$ if there exists positive constants
$C,s$ such that for any operator $x=\sum_{g}x_{g}\lambda_{g}$ in the reduced
group C*-algebra and with finite support we have
$\|x\|\leq C\sqrt{\sum_{g\in G}|x_{g}|^{2}(1+|g|)^{2s}}.$
It is immediate that this definition may be extended to the setting of a
reduced crossed product of a C*-algebra by a discrete group in several ways.
We have played with 3 possibilities of definition, but only been able to
obtain results for the 2 of them, which we define below. The third possibility
is mentioned after the definition.
###### Definition 4.3.
Let $G$ be a discrete group with a length function $g\to|g|,$ such that $G$
acts on a C*-algebra ${\mathcal{A}}$ via a group of *-automorphisms
$\alpha_{g}.$ The reduced crossed product
${\mathcal{C}}:=C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G)$ has rapid decay of
operator type or scalar type if there exist positive constants $C,s$ such
that for any operator $X=\sum L_{g}X_{g}$ with finite support:
operator type: $\displaystyle\|X\|\leq C\|\sum_{g\in
G}(1+|g|)^{2s}(L_{g}X_{g}X_{g}^{*}L_{g}^{*}+X_{g}^{*}X_{g})\|^{\frac{1}{2}}$
scalar type: $\displaystyle\|X\|\leq C\bigg{(}\sum_{g\in
G}(1+|g|)^{2s}\|X_{g}\|^{2}\bigg{)}^{\frac{1}{2}}$
If one of the properties above holds for any C*-algebra ${\mathcal{A}}$
carrying an action $\alpha_{g}$ of $G$ we say that $G$ possesses complete
rapid decay of respectively operator type and scalar type.
In the very first version of the article we thought that we could prove that
free non abelian groups do have a sort of mixed rapid decay defined as
mixed type: $\displaystyle\|X\|\leq
C\bigg{(}\sum_{k=0}^{\infty}(1+|k|)^{2s}\|\sum_{g\in
C_{k}}\big{(}L_{g}X_{g}X_{g}^{*}L_{g}^{*}+X_{g}^{*}X_{g}\big{)}\|\bigg{)}^{\frac{1}{2}}.$
Unfortunately we were wrong, but it may still be that some discrete groups
with non polynomial growth satisfy such a condition, and that would be very
helpful in the study of multipliers of the form $M_{\varphi}$, as it follows
from the proof of Proposition 7.1.
The complete operator type of rapid decay may be established for finitely
generated groups with polynomial growth by a simple imitation of the proofs
from the group algebra case. For other groups it seems impossible to us to
establish the operator type of rapid decay outside the reduced group algebra
case. We establish the complete scalar type of rapid decay for discrete
groups which possess property (J) with respect to a length function.
## 5\. complete operator rapid decay in a reduced crossed product of a
C*-algebra by a finitely generated group with polynomial growth
Here we modify some of Jolissaint’s results from his section 3.1 of [11] to
cover our situation.
###### Theorem 5.1.
Let ${\mathcal{A}}$ be a C*-algebra, $G$ a finitely generated discrete group
with polynomial growth which has an action $\alpha_{g}$ on ${\mathcal{A}}$ as
a group of *-automorphisms. There exists positive reals $M,s$ such that for
any finitely supported operator $X=\sum L_{g}X_{g}$ in
${\mathcal{C}}:=C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G):$
$\displaystyle\|X\|$ $\displaystyle\leq M\|\sum_{g\in
G}(1+|g|)^{s+2}L_{g}X_{g}X^{*}_{g}L^{*}_{g}\|^{\frac{1}{2}}$
$\displaystyle\|X\|$ $\displaystyle\leq M\|\sum_{g\in
G}(1+|g|)^{s+2}X^{*}_{g}X_{g}\|^{\frac{1}{2}}.$
###### Proof.
The polynomial growth implies that there exists positive reals $C,s$ such that
$|C_{k}|\leq C(1+k)^{s},$ then the following manipulations are standard, and
the proof follows from Proposition 2.2 as follows
$\displaystyle X=$ $\displaystyle\sum_{k=0}^{\infty}\sum_{g\in
C_{k}}\frac{1}{(1+k)|C_{k}|^{\frac{1}{2}}}\big{(}(1+k)|C_{k}|^{\frac{1}{2}}L_{g}X_{g}\big{)}$
$\displaystyle\leq$ $\displaystyle\sqrt{C}\sum_{k=0}^{\infty}\sum_{g\in
C_{k}}\frac{1}{(1+k)|C_{k}|^{\frac{1}{2}}}\big{(}(1+k)^{(1+\frac{s}{2})}{\frac{1}{2}}L_{g}X_{g}\big{)}$
$\displaystyle\|X\|\leq$
$\displaystyle\frac{\pi\sqrt{C}}{\sqrt{6}}\|\sum_{k=0}^{\infty}\sum_{g\in
C_{k}}(1+k)^{(2+s)}X_{g}^{*}X_{g}\|^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle\frac{\pi\sqrt{C}}{\sqrt{6}}\sqrt{2}\|\sum_{g\in
G}(1+|g|)^{(2+s)}X_{g}^{*}X_{g}\|^{\frac{1}{2}}$
so $M:=\frac{\pi\sqrt{2C}}{\sqrt{6}},$ may be used. The inequality involving
$L_{g}X_{g}X_{g}^{*}L_{g}^{*}$ follows in the same way. ∎
## 6\. Complete scalar rapid decay for discrete groups with property (J)
The basic result in this section is the proposition just below, and this is a
combined extension of some of the first results in [8].
###### Proposition 6.1.
Let $(G,|\cdot|)$ be a discrete group with a length function satisfying the
property (J). There exists a positive constant $N$ such that for any action of
$G$ as a group of *-automorphisms $\alpha_{g}$ on a C*-algebra
${\mathcal{A}},$ any non negative integer $k$ and any operator $X=\sum_{g\in
G}L_{g}X_{g}$ in ${\mathcal{C}}:=C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G)$
with finite support in $C_{k}:$
$\|X\|\leq N(1+k)\sqrt{\sum\|X_{g}\|^{2}}.$
###### Proof.
We will let $Y$ denote any finitely supported element in ${\mathcal{C}}$ with
column or $\pi-$norm at most 1. By Proposition 2.4 it is sufficient to bound
the $\pi-$norm of $XY$ in order to bound the operator norm $\|X\|.$ Using the
cancellation numbers, the operator $XY$ may be written as a sum of $k+1$
summands $S_{p}$ defined as follows
$\displaystyle XY$ $\displaystyle=\sum_{a\in C_{k}}\sum_{b\in
G}L_{a}X_{a}L_{b}Y_{b}$ (6.1) $\displaystyle=\sum_{p=0}^{k}\sum_{\\{(a,b)\in
P_{p}\,:a\in C_{k}\\}}L_{a}X_{a}L_{b}Y_{b}$
$\displaystyle=\sum_{p=0}^{k}S_{p}$
Then let us fix a $p$ in the set $\\{0,1,\dots,k\\}.$ By the property (J)
there exists $\alpha>0,\hat{\beta}>0,M>0$ such that to each group element $g,$
each $p$ and each pair of group elements $(a,b)\in P_{p}$ with $ab=g$ and
$a\in C_{k}$ there exist group elements
$u_{(g,(k-p))},\,v_{(g,(k-p))},\,c(a,b)$ such that
(6.2) $\displaystyle u_{(g,(k-p))}\in C_{(k-p),\alpha},\,\quad
v_{(g,(k-p))}\in C_{(|g|+1-k+p),\hat{\beta}},\quad c(a,b)\in C_{p,M},$
$\displaystyle a=u_{(g,(k-p))}c(a,b),\,\quad b=c(a,b)^{-1}v_{(g,(k-p))}.$
We have $|g|=|a|+|b|-2p-c$ with $c\in\\{0,1\\},$ hence since $|a|=k$ we get
$|g|+1-k+p=|b|-p+1-c$ so for $\beta:=\hat{\beta}+1$ we have
(6.3) $v_{(g,(k-p))}\in C_{(|b|-p),\beta}.$
In Haagerup’s case with free groups we get $u_{(g,(k-p))}=a_{1}\dots
a_{(k-p)},\newline \,c(a,b)=a_{(k-p+1)}\dots
a_{k},\,v_{(g,(k-p))}=b_{(p+1)}\dots b_{|b|},$ with
$u_{(g,(k-p))}\in C_{(k-p)},\,c(a,b)\in C_{p},\,v_{(g,(k-p))}\in C_{(|b|-k)}.$
In the case of a free non commutative group the 2 first elements $u,c$ are
determined by $a,p$ and the last 2 elements $c,v$ are determined by $b,p,$ but
in the general case $a,p$ does not determine the pair $u_{(g,(k-p))},c(a,b)$
nor does the pair $b,p$ determines the pair $c(a,b),v_{(g,(k-p))}.$ The
condition (3.2) is designed to deal with this problem.
We can now start the estimation, and we will use the results of Proposition
2.2, so for a given $g$ we define a positive operator $Q_{u_{(g,(k-p))}}$ by
(6.4) $\displaystyle Q_{u_{(g,(k-p))}}^{2}:$ $\displaystyle=\sum_{\\{(a,b)\in
P_{p}:a\in C_{k},\,ab=g\\}}L_{a}X_{a}X_{a}^{*}L_{a}^{*}.$
To each pair $(a,b)$ in $P_{p}$ with $a$ in $C_{k},$ and $ab=g$ there exists a
contraction $q(a,b)$ such that $L_{a}X_{a}=Q_{u_{(g,(k-p))}}q(a,b)$ with
$\sum_{\\{(a,b)\in P_{p}:a\in C_{k},ab=g\\}}q(a,b)q(a,b)^{*}\leq I.$
Analogously we define $R_{v_{(g,(k-p))}}$ as the positive operator which is
given by
(6.5) $\displaystyle R_{v_{(g,(k-p))}}^{2}$ $\displaystyle=\sum_{\\{(a,b)\in
P_{p}:a\in C_{k}\,ab=g\\}}Y_{b}^{*}Y_{b}$
To each group element $g$ and each pair $(a,b)$ in $P_{p}$ with $g=ab$ and $a$
in $C_{k},$ there exists a contraction $r(a,b)$ such that
$L_{b}Y_{b}=r(a,b)R_{v_{(g,(k-p))}}$ with
$\sum_{\\{(a,b)\in P_{p}:a\in C_{k},ab=g\\}}r(a,b)^{*}r(a,b)\leq I,$
and according to Proposition 2.2 we may define a contraction operator $m_{g}$
by
(6.6) $\forall g\in G:\quad m_{g}:=\sum_{\\{(a,b)\in P_{p}:a\in
C_{k}\,ab=g\\}}q(a,b)r(a,b).$
When combining these equations we get
(6.7) $S_{p}(g)=Q_{u_{(g,(k-p))}}m_{g}R_{v_{(g,(k-p))}}$
and then
(6.8) $\displaystyle\sum_{g\in G}S_{p}(g)^{*}S_{p}(g)=\sum_{g\in
G}R_{v_{(g,(k-p))}}m_{g}^{*}Q_{u_{(g,(k-p))}}^{2}m_{g}R_{v_{(g,(k-p))}}\text{by
}\|m_{g}\|\leq 1$ $\displaystyle\leq$ $\displaystyle\sum_{g\in
G}\|Q_{u_{(g,(k-p))}}^{2}\|R_{v_{(g,(k-p))}}^{2}\text{ by }(\ref{Q})\text{ and
}(\ref{GG})$ $\displaystyle\leq$ $\displaystyle\sum_{g\in
G}\big{(}\sum_{(a,b)\in
P_{p}:ab=g}\|X_{u_{(g,(k-p))}c(a,b)}\|^{2}\big{)}\cdot$
$\displaystyle\big{(}\sum_{(e,f)\in
P_{p}:ef=g}Y_{c(e,f)^{-1}v_{(g,(k-p))}}^{*}Y_{c(e,f)^{-1}v_{(g,(k-p))}}\big{)}\text{
split to sum over }h,\,g$ $\displaystyle\leq$ $\displaystyle\big{(}\sum_{g\in
G}\sum_{(a,b)\in P_{p}:ab=g}\|X_{u_{(g,(k-p))}c(a,b)}\|^{2}\big{)}\cdot$
$\displaystyle\big{(}\sum_{h\in G}\sum_{(e,f)\in
P_{p}:ef=h}Y_{c(e,f)^{-1}v_{(g,(k-p))}}^{*}Y_{c(e,f)^{-1}v_{(g,(k-p))}}\big{)}$
In the last 2 sums depending on $g$ and $h$ respectively, one can via the
property (3.2) get an upper bound on the number of times each element of the
form $\|X_{a}\|^{2}$ or $Y_{f}^{*}Y_{f}$ appears in the sum. Let $a$ in
$C_{k}$ be given, then the number of solutions to the equation
$u\in C_{(k-p),\alpha},\,c\in C_{p,M}:\quad uc=a$
is at most $N.$ Similarly for each $f$ the number of solutions to the equation
$v\in C_{(|f|-p),\beta},\,c\in C_{p,M}:\quad c^{-1}v=f$
is at most $N,$ and hence by (6.8)
(6.9) $\displaystyle\|S_{p}\|_{\pi}\leq$ $\displaystyle N\big{(}\sum_{a\in
C_{k}}\|X_{a}\|^{2}\big{)}^{\frac{1}{2}}\|Y\|_{\pi}\text{ so }$
$\displaystyle\|XY\|_{\pi}\leq$ $\displaystyle(k+1)N\big{(}\sum_{a\in
C_{k}}\|X_{a}\|^{2}\big{)}^{\frac{1}{2}}\|Y\|_{\pi}\text{ by Proposition
}\ref{cpnorm}$ $\displaystyle\|X\|\leq$ $\displaystyle(k+1)N\big{(}\sum_{a\in
C_{k}}\|X_{a}\|^{2}\big{)}^{\frac{1}{2}},$
and the proposition follows
∎
In the case of a free non abelian group ${\mathbb{F}}_{d}$ with any set,
finite or infinite, of generators the proposition above holds with $N=1.$ The
reason is that for a pair $(a,b)$ in $P_{p}$ with $a\in C_{k}$ the
decompositions $a=u_{(g,(k-p))}c(a,b)$ and $b=c(a,b)^{-1}v_{(g,(k-p))}$ are
described in an exact form just below the equation (6.3), such that the
constants $\alpha,\,\beta,\,\gamma$ in Definition 3.3 may be used with value
0. The exact decomposition also shows that in this case there will only be one
solution to the equation (3.2) so we get $N=1$ in this case, and we may note
the following corollary.
###### Corollary 6.2.
Let ${\mathcal{A}}$ be a C*-algebra with an action $\alpha_{g}$ of
${\mathbb{F}}_{d},$ the free non commutative group with $d$ generators, then
for any $k\in{\mathbb{N}}_{0}$ and any $X$ in $C^{*}_{r}({\mathcal{A}})$ with
finite support in $C_{k}$:
$\|X\|\leq(k+1)\big{(}\sum_{a\in C_{k}}\|X_{a}\|^{2}\big{)}^{\frac{1}{2}}.$
It is worth to remark that the result in the corollary above in the case of
the trivial C*-algebra ${\mathcal{A}}={\mathbb{C}}I$ gives exactly the content
of Lemma 1.5 in [8]. The content of Lemma 1.3 in [8] is named the Haagerup
property in [15]. We may also here add a corollary describing the form the
Haagerup property takes in the setting of a reduced crossed product of a
C*-algebra by a discrete group with the property (J) This is the natural
extension of Jolissaint’s Proposition 3.2.4 in [11].
###### Corollary 6.3.
For $k,l,m$ non negative integers and any pair of finitely supported elements
$X=\sum_{g}L_{g}X_{g}$ with support in $C_{k}$ and $Y=\sum_{g}L_{g}Y_{g}$ with
support in $C_{l}:$
$\displaystyle\|M_{\chi_{m}}*\big{(}XY\big{)}\|\leq N\big{(}\sum_{a\in
C_{k}}\|X_{a}\|^{2}\big{)}^{\frac{1}{2}}\|\sum_{b\in
C_{l}}Y_{b}^{*}Y_{b}\|^{\frac{1}{2}}$
$\displaystyle\|M_{\chi_{m}}*\big{(}XY\big{)}\|\leq N\|\sum_{a\in
C_{k}}L_{a}X_{a}X_{a}^{*}L_{a}^{*}\|^{\frac{1}{2}}\big{(}\sum_{b\in
C_{l}}\|Y_{b}\|^{2}\big{)}^{\frac{1}{2}}$
When $G$ is a free non abelian group $N=1.$
###### Proof.
Let $a\in C_{k}$ and $b\in C_{l}$ be such that $ab\in C_{m}$ then there exists
uniquely determined $c\in\\{0,1\\}$ and $p$ in ${\mathbb{N}}_{0}$ such that
$k+l-m=2p+c.$
In particular at most one $S_{p}\neq 0,$ and the first inequality of the
corollary follows from the proof of the theorem. The second follows from the
first when applied to $(Y^{*}X^{*}).$ The free group statement follows from
the corollary just above. ∎
The fact that there are 2 inequalities above indicates to us that there might
be a hope for the desired inequality below to be true for a finitely
supported $X$ with support in $C_{k}$ and a finitely supported $Y$ with
support in $C_{l}$ we hope that
desired inequality $\displaystyle\|M_{\chi_{m}}*\big{(}XY\big{)}\|\leq
N\|\sum_{a\in C_{k}}L_{a}X_{a}X_{a}^{*}L_{a}^{*}\|^{\frac{1}{2}}\|\sum_{b\in
C_{l}}Y_{b}^{*}Y_{b}\|^{\frac{1}{2}}$
We may then continue and consider general elemnts $X$ with finite support.
###### Theorem 6.4.
Let $G$ be a discrete group with a length function such that $G$ satisfies the
condition (J). There exists an $M>0$ such that for any action $\alpha_{g}$ of
$G$ on a C*-algebra ${\mathcal{A}}$ and any $X=\sum_{g\in G}L_{g}X_{g}$ of
finite support in $C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G):$
$\|X\|\leq M\sqrt{\sum_{g\in G}(1+|g|)^{4}\|X_{g}\|^{2}}.$
###### Proof.
We may proceed as in the proof of Lemma 1.5 in [8], so
$\displaystyle\|X\|\leq$ $\displaystyle\sum_{k=0}^{\infty}\|\sum_{g\in
C_{k}}X_{g}\|$ $\displaystyle\leq$ $\displaystyle
N\sum_{k=0}^{\infty}(k+1)^{-1}\bigg{(}(k+1)^{2}\big{(}\sum_{g\in
C_{k}}\|X_{g}\|^{2}\big{)}^{\frac{1}{2}}\bigg{)}$ $\displaystyle\leq$
$\displaystyle
N\frac{\pi}{\sqrt{6}}\bigg{(}\sum_{k=0}^{\infty}(k+1)^{4}\sum_{g\in
C_{k}}\|X_{g}\|^{2}\bigg{)}^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle
N\frac{\pi}{\sqrt{6}}\sqrt{2}\bigg{(}\sum_{g\in
G}(1+|g|)^{4}\|X_{g}\|^{2}\bigg{)}^{\frac{1}{2}}$
and the theorem follows, since for $g\in C_{k}$ we have $k-1<|g|\leq k.$ ∎
Again there is a sharper estimate in the case of a free non abelian group.
###### Corollary 6.5.
If $G$ is a free non abelian group the constant $M$ may be chosen as $M=2.$
## 7\. Applications
The theory of rapid decay for group C*-algebras has been applied to various
types of approximation properties for operator algebras [2], [3] [5], [9], and
many research articles are based on, or inspired by these works.
Jolissaint realized from the beginning [12] that the rapid decay property
makes it possible to base some K-theoretical computations on a subalgebra of
rapidly decreasing operators, and this was then used by Lafforgue [14] in his
fundamental work on the Baum-Connes conjecture.
The construction of a spectral triple for a reduced group C*-algebra of a
discrete group, which occurs in Connes’ non commutative geometry, has an
obvious candidate if the group has a length function. It seems natural to use
the property rapid decay to get some information on the properties of this
spectral triple, and such attempts have appeared in [1] and [15]. It is
interesting to see that the so-called Haagerup condition of [15] is the
content of the very basic Lemma 1.3 of [8] and of Proposition 3.2.4 of [11].
Here it is contained in the corollary 6.3.
It is not our intent to pursue possible extensions of the results based on
rapid decay from the group algebra case to the crossed product setting, but we
have made one easy observation, which may be applied to a possible extension
of some of the approximation properties.
In Haagerup’s first article [8] he shows in Lemma 1.7 that for a function
$\varphi$ on a free non abelain group $G$ the multiplier $M_{\varphi}$ is
bounded if $\sup\\{|\varphi(g)|(1+|g|)^{2}:g\in G\\}$ is finite and
$\|M_{\varphi}\|\leq 2\sup\\{|\varphi(g)|(1+|g|)^{2}:g\in G\\}.$ This result
makes it possible for him to cut the completely positive multiplier of norm 1
given by $M_{\varphi_{\lambda}}$ with $\varphi_{\lambda}(g):=e^{-\lambda|g|}$
to the subsets $B_{n},$ and in this way he obtains a bounded approximate
multiplier unit consisting of functions with finite support. We can not obtain
such a nice result here because Haagerup’s estimate is based on the fact that
in the group algebra case we have $\|\lambda(f)\|\geq\|f\|_{2},$ and the
analogous statement for crossed products is not true. We can get a result
which is is similar to Haagerup’s Lemma 1.7 for a group action which has
operator rapid decay.
###### Proposition 7.1.
Let ${\mathcal{A}}$ be a C*-algebra, G a discrete group with a length function
and $\alpha_{g}$ an action of the group on ${\mathcal{A}}$ such that the
reduced crossed product has operator rapid decay with coefficients $C,s.$ If a
complex function $\varphi$ on the group satisfies
$m:=\sup\\{|\varphi(g)|(2+|g|)^{(s+1)}\,:\,g\in G\\}<\infty$ then the
multiplier $M_{\varphi}$ on $C^{*}_{r}({\mathcal{A}}\rtimes_{\alpha}G)$ is
bounded and satisfies $\|M_{\varphi}\|\leq 2Cm.$ If the action of $\alpha_{g}$
has complete operator rapid decay, then $M_{\varphi}$ is completely bounded
with $\|M_{\varphi}\|_{cb}\leq 4Cm.$
###### Proof.
Suppose $\varphi$ is given with $m$ finite then for any $X=\sum L_{g}X_{g}$
with finite support
(7.1) $\displaystyle\|M_{\varphi}*X\|^{2}$ $\displaystyle\leq
C^{2}\|\sum_{g\in
G}(1+|g|)^{2s}|\varphi(g)|^{2}(L_{g}X_{g}X_{g}^{*}L_{g}^{*}+X_{g}^{*}X_{g})\|$
$\displaystyle\leq C^{2}\|\sum_{g\in
G}(1+|g|)^{-2}m^{2}(L_{g}X_{g}X_{g}^{*}L_{g}^{*}+X_{g}^{*}X_{g})\|$
$\displaystyle\leq 4C^{2}m^{2}\sum_{k=0}^{\infty}(1+k)^{-2}\|\sum_{g\in
C_{k}}L_{g}X_{g}X_{g}^{*}L_{g}^{*}+X_{g}^{*}X_{g}\|$ $\displaystyle\leq
8C^{2}m^{2}\frac{\pi^{2}}{6}\|X\|^{2}$ $\displaystyle\leq
16C^{2}m^{2}\|X\|^{2}.$
If $G$ possesses complete operator rapid decay, the action $\alpha_{g}$ on
${\mathcal{A}}$ may be lifted to actions on $M_{n}({\mathcal{A}}),$ which all
have operator rapid decay with coefficients $C,s$ and the result follows. ∎
## References
* [1] C. Antonescu, E. Christensen,Metrics on group C*-algebras and a non-commutative Arzelà-Ascoli theorem, J Funct. Anal. 214 (2004), 247-–259.
* [2] I. Chatterji, Introduction to the rapid decay property. Around Langlands correspondences, Contemp. Math. Amer. Math. Soc. 691 (2017), 53 – 72.
* [3] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette, Groups with the Haagerup property, Progress in Mathematics, 197 (2001), BirkhäuserVerlag, Basel.
* [4] E. Christensen, The block Schur product is a Hadamard product, Math. Scand. 126 (2020), 603-–616.
* [5] M. G. Cowling, U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math.96 (1989), 507-–549.
* [6] E. Ghys, P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, ed., Birkhäuser Boston Inc., Boston, MA, 1990.
* [7] M. Gromov, Hyperbolic groups, Essays in group theory, S. M. Gersten, ed., Springer,1987.
* [8] U. Haagerup, An example of a non nuclear C*-Algebra, which has the metric approximation property, Inv. Math. 50 (1979), 279–293
* [9] U. Haagerup, J. Kraus Approximation properties for group C*-Algebras and group von Neumann algebras, Transactions Amer. Math. Soc. 334 (1994),667–699
* [10] P. de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint, C. R. Acad. Sci. Paris, 307 1988, 771–774,
* [11] P. Jolissaint, Rapidly decreasing functions in reduced C*-algebras of groups, Trans. Amer. Math. Soc. 317 (1990), 167 – 196.
* [12] P. Jolissaint, K-Theory of reduced C*-Algebras and rapidly decreasing functions on groups, K-Theory 2 (1989), 723-–735.
* [13] R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras, Academic Press, 1986.
* [14] V. Lafforgue, KK-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002), 1–95.
* [15] N. Ozawa, M. A. Rieffel, Hyperbolic group C*-algebras and free-product C*-algebras as compact quantum metric spaces, Canad. J. Math. 57 (2005), 1056–1079.
* [16] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Univ. Press, Cambridge, 2002\.
* [17] F. Pop, A. M. Sinclair, R. R. Smith, Norming C*-algebras by C*-subalgebras, J. Funct. Anal. 175 (2000), 168–196.
* [18] W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.
|
# Evolution of a Non-Hermitian Quantum Single-Molecule Junction at Constant
Temperature
Andrea Grimaldi Dipartimento di Scienze Matematiche e Informatiche, Scienze
Fisiche e Scienze della Terra, Università degli Studi di Messina, viale F.
Stagno d’Alcontres 31, 98166 Messina, Italy Alessandro Sergi Dipartimento di
Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra,
Università degli Studi di Messina, viale F. Stagno d’Alcontres 31, 98166
Messina, Italy Istituto Nazionale di Fisica Nucleare, Sez. di Catania, 95123
Catania, Italy Institute of Systems Science, Durban University of Technology,
P. O. Box 1334, Durban 4000, South Africa Antonino Messina Dipartimento di
Matematica ed Informatica dell’Università di Palermo, Via Archirafi 34,
I-90123 Palermo, Italy<EMAIL_ADDRESS>
###### Abstract
This work concerns the theoretical description of the quantum dynamics of
molecular junctions with thermal fluctuations and probability losses To this
end, we propose a theory for describing non-Hermitian quantum systems embedded
in constant-temperature environments. Along the lines discussed in [A. Sergi
_et al_ , _Symmetry_ 10 518 (2018)], we adopt the operator-valued Wigner
formulation of quantum mechanics (wherein the density matrix depends on the
points of the Wigner phase space associated to the system) and derive a non-
linear equation of motion. Moreover, we introduce a model for a non-Hermitian
quantum single-molecule junction (nHQSMJ). In this model the leads are mapped
to a tunneling two-level system, which is in turn coupled to a harmonic mode
(_i.e._ , the molecule). A decay operator acting on the two-level system
describes phenomenologically probability losses. Finally, the temperature of
the molecule is controlled by means of a Nosé-Hoover chain thermostat. A
numerical study of the quantum dynamics of this toy model at different
temperatures is reported. We find that the combined action of probability
losses and thermal fluctuations assists quantum transport through the
molecular junction. The possibility that the formalism here presented can be
extended to treat both more quantum states ($\sim 10$) and many more classical
modes or atomic particles ($\sim 10^{3}-10^{5}$) is highlighted.
Keywords: molecular junction; non-Hermitian quantum mechanics; open quantum
system dynamics; quantum thermodynamics
PACS: 31.15.xg; 02.60.Cb, 05.30.-d; 05.60.Gg, 03.67.Pp
MSC: 80M99; 81-08; 81-10, 81P99
## I Introduction
Molecular junctions are nano-devices composed of metal or semiconductor
electrodes known as leads hystory ; thoss . A single molecule simulates a
conducting bridge between the leads. Such a nanostructure is almost perfectly
suited to study non-equilibrium quantum transport hystory ; thoss .
Various approaches satisfactorily befit the investigation of the phenomenology
of molecular junctions thoss . In this work, we introduce a simple toy model
of a molecular junction thoss in order to perform a qualitative study aimed
at singling out universal features of such systems. To this end, the operator-
valued Wigner formulation of quantum mechanics quasi_lie ; zhang-balescu ;
balescu-zhang is particularly useful to embed quantum toy-models in quantum
phase space baths since it can simultaneously describe dissipative phenomena
and thermal fluctuations. A quantum-classical molecular junction model has
been studied in Ref. hanna2 .
In general, theories of quantum transport offer an acceptable representation
of excitations and state population transfer. They usually assume that what is
transferred does not disintegrate or disappear from the system. Such a
physical property is expressed through the conservation of probability. On the
contrary, by definition, the probability in quantum systems with gain and loss
is not conserved. Such a feature can be incorporated in the theory by
introducing non-Hermitian Hamiltonians gamow_1928 ; carlbenderpt ;
mostafazadeh_2002 ; subotnik ; zelovich ; zelovich2 . Recently, the
development of non-Hermitian quantum mechanics (NHQM) has been propelled by
various experiments on many systems (see rotter_15 ; moiseyev_11 and
references therein). While in this work a non-equilibrium quantum statistical
mechanics of molecular junctions is adopted, linear equations of motion are
used in Refs. subotnik ; zelovich ; zelovich2 . Conceptually, this can be
understood in terms of the different theoretical targets: In Refs. subotnik ;
zelovich ; zelovich2 the target is to study non-equilibrium transport _per
se_. Instead, our work is aimed at showing the possibility of a statistical
mechanical description for non-Hermitian quantum systems in classical bath
as_15 .
Since probability non-conserving quantum systems are open quantum systems
rotter_15 , it is natural to formulate NHQM in terms of the density matrix
as_15 ; sz_13 ; sz_15 ; sz_16 ; entropy2 ; grima_18 ; nh_emwave ; sust .
Typically, a non-Hermitian Hamiltonian is either derived by means of the
Feshbach formalism f-58 ; f-62 or it is postulated as an _ansatz_ (see Refs.
sz_13 ; as_15 ; sz_15 ; sz_16 ; entropy2 ; grima_18 ; nh_emwave ; sust ). It
is remarkable that there is a third possibility according to which a non-
Hermitian Hamiltonian may arise from stroboscopic measurements performed on an
ancilla-subsystem added to a quantum system, a very interesting process
capable of generating entanglement noi . Given a non-Hermitian Hamiltonian and
assuming that the Schrödinger equation is still valid, one derives a
probability non-conserving equation of motion for the density matrix of the
system noi . However, in order to meet the need of establishing a proper
quantum statistical theory, this non-Hermitian equation of motion for the
density matrix is generalized into a non linear-one as_15 ; sz_13 ; sz_15 ;
sz_16 ; entropy2 ; grima_18 ; nh_emwave ; sust . Essentially, this is the
origin of the difference between the linear approach of subotnik ; zelovich ;
zelovich2 and the non-linear approach of as_15 ; sz_13 ; sz_15 ; sz_16 ;
entropy2 ; grima_18 ; nh_emwave ; sust .
Various investigations on thermal effects in molecular junctions are found in
the literature jtherm3 ; jtherm4 ; jtherm5 . However, to the best of our
knowledge, thermal effects have been investigated separately from probability
losses of molecular junctions. In light of the previous considerations, the
purpose of this work is to combine the non-linear equation for the density
matrix of a system with a non-Hermitian Hamiltonian and temperature control in
quantum phase space dynamics ap . Temperature control is technically
implemented by means of the Nosé-Hoover chain nhc thermostat, formulated in
terms of quasi-Lie brackets quasi_lie . The unified formalism here reported is
developed for studying a greater variety of phenomena than those accessible to
theories treating thermal fluctuations and probability losses separately. In
detail, the theory is obtained through the density matrix approach to non-
Hermitian quantum mechanics as_15 ; sz_13 ; sz_15 ; sz_16 ; entropy2 ;
grima_18 ; nh_emwave ; sust and it is expressed in terms of the operator-
valued formulation of quantum mechanics quasi_lie ; zhang-balescu ; balescu-
zhang ; as_15 . For clarity, we highlight the three mathematical ingredients
that must be combined for building our formulation. First, we need an approach
for describing quantum operators in terms of operator-valued Wigner phase
space functions. The conceptual framework of this specific theory assumes an
Eulerian picture where a Hilbert space, or a space of quantum discrete states,
is defined for each point $X=(Q,P)$ of phase space. Accordingly, the operator-
valued Wigner function quasi_lie ; zhang-balescu ; balescu-zhang may also be
called phase-space-dependent density matrix. The consistent coupling of
quantum and phase space degrees of freedom requires that quantum dynamics in
the Hilbert space associated to $X$ also depends non-locally on the quantum
dynamics associated to different points $X^{\prime}$. Likewise, the dynamics
of $X$ does not only depend on its quantum space but it also depends on the
quantum spaces of other points $X^{\prime}$ and _viceversa_. In practice, such
a dependence can be implemented through a variety of algorithms theochem ;
sstp ; algo12 ; algo15 . When the Hamiltonian is quadratic, one can expand the
operator-valued Moyal bracket b3 ; b4 up to linear order in $\hbar$ and
obtain quantum equations of motion mackernan .
The second ingredient entering our theory is the requirement that the degrees
of freedom in phase space evolve in time satisfying the constraint determined
by the constant thermodynamic temperature. The temperature control of the
classical degrees of freedom is achieved _in silico_ by means of the Nosé-
Hoover chain algorithm nhc . We find such an approach advantageous because the
formulation of the Nosé-Hoover chain algorithm can be realized within the
theoretical framework given by the quasi-Lie brackets quasi_lie . Such
brackets allow one to generalize the Nosé-Hoover chain algorithm, originally
formulated for classical systems only, to the more general case of the
operator-valued formulation of quantum mechanics ap ; b3 ; b4 .
The third ingredient, of course, is that the modeling of the leads provides a
non-Hermitian quantum system _per se_.
When all these three theoretical features are simultaneously taken into
account, we obtain an equation that generalizes the one arising from the
quantum-classical quasi-Lie bracket quasi_lie because it can describe
probability gain/loss as_15 too. Moreover, since such an equation keeps the
thermodynamic temperature of the phase space degrees of freedom constant by
means of the Nosé-Hoover chain algorithm quasi_lie ; nhc ; b3 ; b4 , it
generalizes the equations used in Refs. as_15 ; sz_13 ; sz_15 ; sz_16 ;
entropy2 ; grima_18 ; nh_emwave ; sust , which did not implement any
thermodynamic constraint.
In this paper we investigate the time evolution of a single-molecule junction
toy-model represented by a non-Hermitian Hamiltonian with a parametric phase
space dependence. The isolated leads are represented by a two-level system. In
this situation, the transport of the occupation probability of one or the
other state can only occur through quantum tunneling. The molecule connecting
the two leads is modeled by a harmonic oscillator. When we consider only the
coupling between the harmonic mode and the two-level system, the total system
is considered closed. In this case, transport can take place both _via_
tunneling and _via_ the channel given by the oscillator. In order to build a
complete non-Hermitian quantum single-molecule junction model at constant
temperature, the closed spin-boson system is transformed into an open quantum
system by means of two theoretical steps. The first step is to consider a
decay operator acting on the two-level system. This operator introduces
another transport channel that does not conserve probability and then breaks
time-reversibility.
The second step is to put the harmonic mode into contact with a heat bath. Of
course, this condition forces the molecule to have the same temperature of the
bath and to experience fluctuations according to the canonical ensemble
probability distribution. In practice, the action of the constant-temperature
bath on the molecule is implemented by means of the Nosé-Hoover chain
thermostat. For this reason, in the following we will use wording such as
“heat bath”, “constant-temperature environment” or “Nosé-Hoover chain
thermostat” interchangeably.
The final complete non-Hermitian Hamiltonian is designed to represent two
lossy leads interacting with a thermalized molecule.
The organization of the paper is the following: In Sec. II we give a short
summary of the density matrix approach to the dynamics of a system governed by
a non-Hermitian Hamiltonian. In Sec. III we describe briefly how to treat non-
Hermitian quantum systems interacting with classical degrees of freedom. We
present the main theoretical results of this work in Sec. IV, where we
introduce the mathematical formalism describing the quantum dynamics of a non-
Hermitian quantum system interacting with classical degrees of freedom at
constant temperature. In Sec. V we introduce the non-Hermitian quantum single-
molecule junction model. The results of the numerical calculations are
reported in Sec. VI. Finally, our conclusions are given in Sec. VII.
## II Quantum dynamics of non-Hermitian systems
Consider a quantum system described by a non-Hermitian Hamiltonian operator
$\hat{\cal H}=\hat{H}-i\hat{\Gamma}\;,$ (1)
where $\hat{H}$ and $\hat{\Gamma}$ are Hermitian operators ($\hat{\Gamma}$ is
the decay operator). In the presence of a probability sink or source, the
dynamics of the system is defined in terms of the following equations (we set
$\hbar=1$):
$\displaystyle\frac{d}{dt}|\Psi(t)\rangle$ $\displaystyle=$
$\displaystyle-i\hat{\cal H}|\Psi(t)\rangle\;,$ (2)
$\displaystyle\frac{d}{dt}\langle\Psi(t)|$ $\displaystyle=$ $\displaystyle
i\langle\Psi(t)|\hat{\cal H}^{\dagger}\;,$ (3)
where $|\Psi(t)\rangle$ and $\langle\Psi(t)|$ are state vectors of the system.
Equations (2) and (3) reduce to the standard Schrödinger equation when
$\hat{\Gamma}=0$.
The density matrix approach to non-Hermitian quantum systems as_15 ; sz_13 ;
sz_15 ; sz_16 ; entropy2 ; grima_18 ; nh_emwave ; sust is obtained upon
introducing at any t>0 a Hermitian, semipositive defined, but non-normalized
density matrix
$\hat{\Omega}(t)=|\Psi(t)\rangle\langle\Psi(t)|\;.$ (4)
The equation of motion for $\hat{\Omega}$ sz_13 ; grima_18 ; sust is obtained
by taking the time derivative of Eq. (4) and using Eq. (1):
$\frac{d}{dt}\hat{\Omega}(t)=-i\left[\hat{H},\hat{\Omega}(t)\right]-\left[\hat{\Gamma},\hat{\Omega}(t)\right]_{+}\;,$
(5)
where $[...,...]$ is the commutator and $[...,...]_{+}$ is the anticommutator.
Equation (5) is linear but, nevertheless, it describes an open quantum system.
This is possible since it is defined in terms of a decay operator that
physically describes the presence of probability sources/sinks, arising from
the hidden system with which the open quantum system interacts.
Equation (5) is non-Hermitian, breaks time reversibility and does not conserve
the trace of $\hat{\Omega}(t)$. In fact, from Eq. (5) one can easily derive
$\frac{d}{dt}{\rm Tr}\left(\hat{\Omega}(t)\right)=-2{\rm
Tr}\left(\hat{\Gamma}\hat{\Omega}(t)\right)\;.$ (6)
In order to obtain a well-founded statistical mechanics of non-Hermitian
quantum systems, one can introduce the normalized density matrix
$\hat{\rho}(t)=\frac{\hat{\Omega}(t)}{{\rm
Tr}\left(\hat{\Omega}(t)\right)}\;.$ (7)
This operator $\hat{\rho}(t)$ obeys the non-linear equation
$\frac{d}{dt}\hat{\rho}(t)=-i\left[\hat{H},\hat{\rho}(t)\right]-\left[\hat{\Gamma},\hat{\rho}(t)\right]_{+}+2\hat{\rho}(t){\rm
Tr}\left(\hat{\Gamma}\hat{\rho}(t)\right)\;.$ (8)
As a consequence of its very definition and of Eq. (8), ${\rm
Tr}(\hat{\rho}(t))=1$ at all times. Equation (8) breaks time reversal symmetry
and is non-linear. In practice, the formalism trades off the non-conservation
of probability in Eq. (5) with the non-linearity of Eq. (8). As the linear Eq.
(5), the non-linear Eq. (8) describes the dynamics of an open quantum system.
Consistently, averages of arbitrary operators, _e.g._ , $\hat{\chi}$, can be
calculated by means of the normalized density matrix $\hat{\rho}$ as
$\langle\hat{\chi}\rangle_{(t)}={\rm
Tr}\left(\hat{\rho}(t)\hat{\chi}\right)\;.$ (9)
Equation (9) establishes the foundations of the statistical interpretation of
non-Hermitian quantum mechanics.
## III Non-Hermitian quantum systems set in phase space
Non-Hermitian quantum systems can be embedded in phase space as_15 . In the
following, we sketch the derivation given in as_15 . Let us consider a system
described by a hybrid set of coordinates $(\hat{x},X)$, where $\hat{x}$ are
quantum coordinates (operators) and $X=(Q,P)$ are phase space coordinates (an
obvious multidimensional notation is used in order not to clutter formulas
with too many indices). In this specific case, the $X$s describe the phase
space coordinates of the system. As we have already discussed in the
Introduction, the description of the system, using the operator-valued-
formulation of quantum mechanics, can be established _via_ an Eulerian point
of view according to which a space of state vectors is defined for any point
$X$ of phase space, zhang-balescu ; balescu-zhang ; quasi_lie so that
$\hat{x}$ (and operators defined in terms of $\hat{x}$) can act on it. In such
a hybrid space there are operators that depend only on $\hat{x}$, which we
denote with the symbol $\hat{\phantom{x}}$ on top of the operator, _e.g._ ,
$\hat{\Gamma}$. There are pure phase space functions which we denote by their
arguments, _e.g._ , $H_{\rm B}(X)$, or $G(X,t)$ when there is also an explicit
time dependence. Finally, there are operators that depend both on $\hat{x}$
and (parametrically) on $X$. We denote the latters with the symbol
$\tilde{\phantom{x}}$ on top, _e.g._ ,
$\tilde{\Omega}(t)\equiv\hat{\Omega}(X,t)$.
In the non-Hermitian case, we consider a Hamiltonian operator with parametric
dependence on phase space
$\displaystyle\tilde{\cal H}$ $\displaystyle=$ $\displaystyle\hat{H}+H_{\rm
B}(X)+\tilde{H}_{I}-i\hat{\Gamma}$ (10) $\displaystyle=$
$\displaystyle\tilde{H}_{\rm S}-i\hat{\Gamma}\;,$
where $\hat{H}$ is the Hermitian Hamiltonian of the quantum subsystem, $H_{\rm
B}(X)$ is the Hamiltonian of the degrees of freedom in phase space,
$\tilde{H}_{I}$ describes the interaction between the quantum system and the
bath, and $\tilde{H}_{\rm S}=\hat{H}+H_{\rm B}(X)+\tilde{H}_{I}$. Of course,
$\tilde{\cal H}$ is the total non-Hermitian Hamiltonian of the system with
$\hat{\Gamma}$ as the decay operator.
When the operator $\hat{\Gamma}$ acts on the quantum dynamical variables of
the non-Hermitian quantum system, it has been shown as_15 that, as a
generalization of the equations given in Refs. zhang-balescu ; balescu-zhang ;
quasi_lie , the equation of motion becomes
$\frac{\partial}{\partial t}\tilde{\Omega}(t)=-i\left[\tilde{H}_{\rm
S},\tilde{\Omega}(t)\right]-\left[\hat{\Gamma},\tilde{\Omega}(t)\right]_{+}+\frac{1}{2}\left\\{\tilde{H}_{\rm
S},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}-\frac{1}{2}\left\\{\tilde{\Omega}(t),\tilde{H}_{\rm
S}\right\\}_{\mbox{\boldmath$\cal B$}}\;,$ (11)
where $\tilde{\Omega}(t)$ is the phase-space-dependent non-normalized density
matrix of the system and
$\mbox{\boldmath$\cal B$}=\left[\begin{array}[]{cc}{\bf 0}&{\bf 1}\\\ -{\bf
1}&{\bf 0}\end{array}\right]$ (12)
is the symplectic matrix written in block form. Using it, one can write the
Poisson bracket in the form adopted in Eq. (18):
$\left\\{\tilde{H}_{\rm S},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}\equiv\sum_{I,J=1}^{2N}\frac{\partial\tilde{H}_{\rm S}}{\partial
X_{I}}{\cal B}_{IJ}\frac{\partial\tilde{\Omega}(t)}{\partial X_{J}}\;.$ (13)
The dimension of phase space is $2N$.
The trace of the phase-space-dependent density matrix $\tilde{\rm T}{\rm r}$
is defined as
$\tilde{\rm T}{\rm r}\left(\tilde{\Omega}(t)\right)={\rm Tr}^{\prime}\int
dX\tilde{\Omega}(X,t)\;,$ (14)
where ${\rm Tr}^{\prime}$ denotes a partial trace over the quantum dynamical
variables $\hat{x}$ and $\int dX$ is the integral in phase space. The trace
$\tilde{\rm T}{\rm r}\left(\tilde{\Omega}(t)\right)$ obeys the equation of
motion
$\frac{d}{dt}\tilde{\rm T}{\rm r}\left(\tilde{\Omega}(t)\right)=-2\tilde{\rm
T}{\rm r}\left(\hat{\Gamma}\tilde{\Omega}(t)\right)\;.$ (15)
The result given in Eq. (15) is proven in Appendix A.
Let us now introduce a phase-space-dependent normalized density matrix
$\tilde{\rho}(X,t)$:
$\tilde{\rho}(X,t)=\frac{\tilde{\Omega}(X,t)}{\tilde{\rm T}{\rm
r}\left(\tilde{\Omega}(X,t)\right)}\;.$ (16)
Quantum statistical averages of an arbitrary phase-space-dependent operator
$\tilde{\chi}$ can now be calculated as
$\langle\tilde{\chi}\rangle_{(t)}=\tilde{\rm T}{\rm
r}\left(\tilde{\rho}(t)\tilde{\chi}\right)\;.$ (17)
The phase-space-dependent normalized density matrix obeys the non-linear
equation of motion below
$\frac{\partial}{\partial t}\tilde{\rho}(t)=-i\left[\tilde{H}_{\rm
S},\tilde{\rho}(t)\right]-\left[\hat{\Gamma},\tilde{\rho}(t)\right]_{+}+2\hat{\rho}(t)\tilde{\rm
T}{\rm
r}\left(\hat{\Gamma}\tilde{\rho}(t)\right)+\frac{1}{2}\left\\{\tilde{H}_{\rm
S},\tilde{\rho}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}-\frac{1}{2}\left\\{\tilde{\rho}(t),\tilde{H}_{\rm
S}\right\\}_{\mbox{\boldmath$\cal B$}}\;.$ (18)
## IV Non-Hermitian quantum systems set in a constant-temperature phase space
Equations (11) and (18) describe the dynamics of non-Hermitian quantum systems
embedded in phase space as_15 . When considering open quantum systems in more
general settings, it is common to consider the effects of thermal
fluctuations. Phase space coordinates undergo thermal fluctuations when their
microscopic equilibrium state is described by the canonical distribution
function. In this ensemble the thermodynamic temperature is constant. Within
the framework of the operator-valued Wigner formulation of quantum mechanics
zhang-balescu ; balescu-zhang ; quasi_lie , temperature constraints are
efficiently formulated by means of quasi-Lie brackets quasi_lie ; b3 ; b4 . In
the classical case, deterministic thermostats, such as the Nosé-Hoover chain
thermostat nhc , are also implemented by means of quasi-Lie brackets b1 ; b2 ;
aspvg . Below, we briefly explain how this is achieved. In _silico_ (_i.e._ ,
on the computer), temperature control is realized by augmenting the dimensions
of phase space using just a few additional degrees of freedom. The new higher
dimensional phase space is known as extended phase space. We indicate points
in the extended phase space by means of the symbol $X^{\rm e}$. In the case of
the Nosé-Hoover chain thermostat, upon introducing two fictitious coordinates
$\Lambda_{1},\Lambda_{2}$ with their associated momenta $\Pi_{1},\Pi_{2}$, we
have the following point in the extended phase space:
$X^{\rm e}=(Q,\Lambda_{1},\Lambda_{2},P,\Pi_{1},\Pi_{2})\;.$ (19)
To lighten the notation, the phase space coordinates of the Nosé-Hoover chain
thermostat are denoted as ${\cal
Y}=(\Lambda_{1},\Lambda_{2},\Pi_{1},\Pi_{2})$. For the reader, the symbol
$\cal Y$ denotes a calligraphic $Y$. A generic operator acting upon the
extended phase space is identifiable by the apex e. Considering a classical
system with a potential term $V(Q)$ and Hamiltonian $H_{\rm
B}(X)=P^{2}/2+V(Q)$, the extended phase space Hamiltonian (see b1 ; b2 ; aspvg
and references therein) includes the Nosé-Hoover chain energy nhc ; b1 and is
defined as
$\displaystyle H^{\rm e}(X^{\rm e})$ $\displaystyle=$ $\displaystyle H_{\rm
B}(X)+\sum_{K=1}^{2}\left(\frac{\Pi_{K}^{2}}{2\mu_{K}}\right)+gk_{\rm
B}T\Lambda_{1}+k_{\rm B}T\Lambda_{2}$ (20) $\displaystyle=$ $\displaystyle
H_{\rm B}(X)+H_{nhc}({\cal Y})\;,$
where $\mu_{K}$ are fictitious inertial parameters, $T$ is the termodynamic
temperature, $k_{\rm B}$ is the Boltzmann constant, $N$ is the number of
physical coordinates $Q$ that are thermalized. The equation of motion in
extended phase space can be written as:
$\dot{X}_{K}^{\rm e}=\left\\{X_{K}^{\rm e},H^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm
e}}=\sum_{I,J=1}^{2(N+2)}\frac{\partial X_{K}^{\rm e}}{\partial X_{I}^{\rm
e}}{\cal B}_{IJ}^{\rm e}\frac{\partial H^{\rm e}}{\partial X_{J}^{\rm
e}}=\sum_{J=1}^{2(N+2)}{\cal B}_{KJ}^{\rm e}\frac{\partial H^{\rm e}}{\partial
X_{J}^{\rm e}}\;,$ (21)
where the first and second equality define a quasi-Lie bracket. The last
equality enlighten the matrix structure of the equations of motion. This last
form also allows one to compactly define the compressibility of phase space:
$\kappa^{\rm e}\equiv\sum_{I=1}^{2(N+2)}\frac{\partial\dot{X}_{I}^{\rm
e}}{\partial X_{I}^{\rm e}}=\sum_{I,J=1}^{2(N+2)}\frac{\partial{\cal
B}_{IJ}^{\rm e}}{\partial X_{I}^{\rm e}}\frac{\partial H^{\rm e}}{\partial
X_{J}^{\rm
e}}=-N\frac{\dot{\Pi}_{1}}{\mu_{1}}-\frac{\dot{\Pi}_{2}}{\mu_{2}}\;,$ (22)
The antisymmetric matrix $\mbox{\boldmath$\cal B$}^{\rm e}$, entering Eqs.
(21) and (22), defined as
$\mbox{\boldmath$\cal B$}^{\rm e}=\left[\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ -1&0&0&0&-P&0\\\ 0&-1&0&P&0&-\Pi_{1}\\\
0&0&-1&0&\Pi_{1}&0\end{array}\right]\;,$ (23)
is an antisymmetric phase-space-dependent matrix generalizing the symplectic
matrix in Eq. (12). Equation (21) is called a quasi-Hamiltonian equation
because, while conserving $H^{\rm e}(X^{\rm e})$, it cannot be derived from
the Hamiltonian formalism alone: in order to write the equation of motion
(21), together with $H^{\rm e}(X^{\rm e})$, one also needs the antisymmetric
matrix $\mbox{\boldmath$\cal B$}^{\rm e}$ in Eq. (23). The equation of motion
for the distribution function $f(X^{\rm e},t)$ is
$\frac{\partial}{\partial t}f(t)=-\left\\{f(t),H^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}-\kappa^{\rm e}f(t)\;.$ (24)
Given the discussion above, our task becomes that of generalizing Eq. (24) to
a non-Hermitian quantum system. The phase-space-dependent Hamiltonian of the
quantum system must be defined on the extended phase space of the Nosé-Hoover
Chain thermostat:
$\displaystyle\tilde{\cal H}^{\rm e}$ $\displaystyle=$
$\displaystyle\hat{H}+H_{\rm B}(X)+\tilde{H}_{I}+H_{\rm nhc}({\cal
Y})-i\hat{\Gamma}$ (25) $\displaystyle=$ $\displaystyle\tilde{H}_{\rm S}^{\rm
e}-i\hat{\Gamma}\;.$
A phase-space-dependent non-normalized density matrix $\tilde{\Omega}^{\rm
e}(t)$ can be introduced as well. We postulate that $\tilde{\Omega}^{\rm
e}(t)$ obeys the equation of motion
$\frac{\partial}{\partial t}\tilde{\Omega}^{\rm e}(t)=-i\left[\tilde{H}_{\rm
S}^{\rm e},\tilde{\Omega}^{\rm
e}(t)\right]-\left[\hat{\Gamma},\tilde{\Omega}^{\rm
e}(t)\right]_{+}+\frac{1}{2}\left\\{\tilde{H}_{\rm S}^{\rm
e},\tilde{\Omega}^{\rm e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm
e}}-\frac{1}{2}\left\\{\tilde{\Omega}^{\rm e},\tilde{H}_{\rm S}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}-\kappa_{\rm S}^{\rm
e}\tilde{\Omega}^{\rm e}(t)\;.$ (26)
In absence of quantum degrees of freedom, Eq. (26) reduces correctly to the
Eq. (24). Moreover, when $(\Lambda_{I},\Pi_{I})\to 0$, with $I=1,2$, (which
means that the Nosé-Hoover thermostat is not applied), Eq. (26) reduces
unerringly to Eq. (11).
The trace $\tilde{\rm T}{\rm r}^{\rm e}$ of the density matrix
$\tilde{\Omega}^{\rm e}(t)$ involves an integral over the extended phase
space:
$\tilde{\rm T}{\rm r}^{\rm e}\left(\tilde{\Omega}^{\rm e}(t)\right)={\rm
Tr}^{\prime}\int dX^{\rm e}\tilde{\Omega}^{\rm e}(X^{\rm e},t)\;.$ (27)
The equation of motion of the trace defined in Eq. (27) has the same structure
of Eq. (15):
$\frac{d}{dt}\tilde{\rm T}{\rm r}^{\rm e}\left(\tilde{\Omega}^{\rm
e}(t)\right)=-2\tilde{\rm T}{\rm r}^{\rm
e}\left(\hat{\Gamma}\tilde{\Omega}^{\rm e}(t)\right)\;.$ (28)
In Appendix B it is shown how Eq. (28) is obtained.
The phase-space-dependent normalized density matrix in the extended phase
space, $\tilde{\rho}^{\rm e}(X^{\rm e},t)$, is defined in analogy with Eq.
(16) as
$\tilde{\rho}^{\rm e}(X^{\rm e},t)=\frac{\tilde{\Omega}^{\rm e}(X^{\rm
e},t)}{\tilde{\rm T}{\rm r}^{\rm e}\left(\tilde{\Omega}^{\rm e}(X^{\rm
e},t)\right)}\;.$ (29)
In analogy with Eq. (17), quantum statistical averages are calculated as
$\langle\tilde{\chi}^{\rm e}\rangle_{(t)}^{\rm e}=\tilde{\rm T}{\rm r}^{\rm
e}\left(\tilde{\rho}^{\rm e}(t)\tilde{\chi}^{\rm e}\right)\;.$ (30)
The phase-space-dependent normalized density matrix in Eq. (29) obeys the non-
linear equation of motion:
$\displaystyle\frac{\partial}{\partial t}\tilde{\rho}^{\rm e}(t)$
$\displaystyle=$ $\displaystyle-i\left[\tilde{H}_{\rm S}^{\rm
e},\tilde{\rho}^{\rm e}(t)\right]-\left[\hat{\Gamma},\tilde{\rho}^{\rm
e}(t)\right]_{+}+2\tilde{\rho}^{\rm e}(t)\tilde{\rm T}{\rm r}^{\rm
e}\left(\hat{\Gamma}\tilde{\rho}^{\rm e}(t)\right)$ (31) $\displaystyle+$
$\displaystyle\frac{1}{2}\left\\{\tilde{H}_{\rm S}^{\rm e},\tilde{\rho}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm
e}}-\frac{1}{2}\left\\{\tilde{\rho}^{\rm e},\tilde{H}_{\rm S}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}-\kappa_{\rm S}^{\rm
e}\tilde{\rho}^{\rm e}(t)\;.$
We remark that Eqs. (26) and (31) are two important results of this work since
these equations define the dynamics of a non-Hermitian quantum system embedded
in a classical bath at constant temperature.
## V Model of a Non-Hermitian Quantum Single-Molecule Junction at Constant
Temperature
In this section we formulate a molecular junction toy-model described by a
non-Hermitian Hamiltonian denoted by $\tilde{\cal H}^{\rm m}$. The various
energy contributions entering the Hermitian terms in part of the Hamiltonian
model are:
$\displaystyle\hat{H}^{\rm m}$ $\displaystyle=$
$\displaystyle-\Delta\hat{\sigma}_{z}\;,$ (32) $\displaystyle H_{\rm B}^{\rm
m}(X)$ $\displaystyle=$
$\displaystyle\frac{P^{2}}{2}+\frac{\omega^{2}}{2}Q^{2}\;,$ (33)
$\displaystyle\tilde{H}_{\rm I}^{\rm m}(Q)$ $\displaystyle=$ $\displaystyle-
cQ\hat{\sigma}_{x}\;,$ (34) $\displaystyle H_{\rm nhc}^{\rm m}({\cal Y})$
$\displaystyle=$
$\displaystyle\sum_{K=1}^{2}\left(\frac{\Pi_{K}^{2}}{2\mu_{K}}+k_{B}T\Lambda_{K}\right)\;.$
(35)
The two-level system Hamiltonian $\hat{H}^{\rm m}$ describes quantum transport
between the leads in terms of the variation of the population of a ground and
an excited state. The extended phase space point $X^{\rm m}$ is defined as in
Eq. (21) but the model considers a single $Q$ coordinate and its conjugate
momentum $\Pi$. Transport in the isolated two-level system takes place through
tunneling. The inclusion in the model description of the decay operator
$\hat{\Gamma}^{\rm m}$ opens a channel through which the population of the
states can disappear forever from the two-level system. The explicit form of
$\hat{\Gamma}^{\rm m}$ will be given later on when we transform to the
adiabatic basis. This subsystem is also coupled to a harmonic mode, with free
Hamiltonian $H_{\rm B}^{\rm m}(Q)$, by means of an interaction Hamiltonian
$\tilde{H}_{\rm I}^{\rm m}$. The harmonic mode opens another channel for the
transport of the population between the leads. In addition to this, the
harmonic mode is embedded in a thermal bath _via_ the Nosé-Hoover chain
thermostat with energy $H_{\rm nhc}^{\rm m}$. The complete non-Hermitian
Hamiltonian model $\tilde{\cal H}^{\rm m}$ reads
$\displaystyle\tilde{\cal H}^{\rm m}(X^{\rm m})$ $\displaystyle=$
$\displaystyle\hat{H}^{\rm m}+H_{\rm B}^{\rm m}(X)+\tilde{H}_{\rm I}^{\rm
m}(Q)+H_{\rm nhc}^{\rm m}({\cal Y})-i\hat{\Gamma}^{\rm m}$ (36)
$\displaystyle=$ $\displaystyle\tilde{H}_{\rm S}^{\rm m}(X^{\rm
m})-i\hat{\Gamma}^{\rm m}\;,$
where the last equality also defines the Hermitian part of the Hamiltonian
model, _i.e._ ,
$\displaystyle\tilde{H}_{\rm S}^{\rm m}(X^{\rm m})=\hat{H}^{\rm m}+H_{\rm
B}^{\rm m}(X)+\tilde{H}_{\rm I}(Q)+H_{\rm nhc}^{\rm m}({\cal Y})\;.$ (37)
To the authors’ knowledge, the Hamiltonian model in Eq. (36) provides a novel
non-linear approach to the modeling of lossy molecular junctions in thermal
baths. Figure 1 displays a pictorial representation of the non-Hermitian
quantum single-molecule junction (nHQSMJ) model at constant temperature.
The abstract dynamics of the nHQSMJ model is obtained upon using the non-
Hermitian Hamiltonian (36) in Eq. (31). The abstract equations of motion can
be represented using different basis sets. The approach described in Ref.
theochem is based on the representation in the adiabatic basis. We also adopt
such an approach here. The adiabatic Hamiltonian is defined as
$\tilde{h}_{\rm ad}^{\rm m}(Q)=\hat{H}^{\rm m}+H_{\rm B}^{\rm
m}(X)+\tilde{H}_{\rm I}^{\rm m}(Q)-\frac{P^{2}}{2}\;,$ (38)
and the adiabatic basis is introduced through the eigenvalue equation
$\tilde{h}_{\rm ad}^{\rm
m}(Q)|\Phi_{\alpha}(Q)\rangle=E_{\alpha}|\Phi_{\alpha}(Q)\rangle\;.$ (39)
In practice, the adiabatic states are a kind of _dressed_ states of the
quantum system when it interacts with approximately “frozen” classical degrees
of freedom. The operator $\hat{\Gamma}^{\rm m}$ is chosen in a
phenomenological way. In the adiabatic basis it is taken as
$\hat{\Gamma}^{\rm m}=\frac{\gamma}{2}\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]\;.$ (40)
In this way we describe a model where the excited state is in contact with a
sink while the ground state is left unperturbed.
The abstract equation of motion for the non-normalized density matrix of the
model is
$\displaystyle\frac{\partial}{\partial t}\tilde{\Omega}^{\rm m}(t)$
$\displaystyle=$ $\displaystyle-i\left[\tilde{H}_{\rm S}^{\rm
e},\tilde{\Omega}^{\rm m}(t)\right]-\left[\Gamma^{\rm m},\tilde{\Omega}^{\rm
m}(t)\right]_{+}+\frac{1}{2}\left\\{\tilde{H}_{\rm S}^{\rm
m},\tilde{\Omega}^{\rm m}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm m}}$ (41)
$\displaystyle-$ $\displaystyle\frac{1}{2}\left\\{\tilde{\Omega}^{\rm
m},\tilde{H}_{\rm S}^{\rm m}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm
m}}-\kappa_{\rm S}^{\rm m}\tilde{\Omega}^{\rm m}(t)$ $\displaystyle=$
$\displaystyle\left(-i\tilde{\cal L}^{\rm m}-i\hat{\cal L}^{\Gamma^{\rm
m}}\right)\tilde{\Omega}^{\rm m}(t)\;.$
The antisymmetric matrix $\mbox{\boldmath$\cal B$}^{\rm m}$ has the same
structure of that in Eq. (23), but it considers only the physical degrees of
freedom of the harmonic modes. Accordingly, the phase space compressibility of
the model becomes $\kappa_{\rm S}^{\rm
m}=-(\dot{\Pi}_{1}/\mu_{1})-(\dot{\Pi}_{2}/\mu_{2})$.
Equation (41) introduces the two Liouville operators:
$\displaystyle-i\tilde{\cal L}^{\rm m}$ $\displaystyle=$
$\displaystyle-i\left[\tilde{H}_{\rm S}^{\rm
e},...\right]+\frac{1}{2}\left\\{\tilde{H}_{\rm S}^{\rm
m},...\right\\}_{\mbox{\boldmath$\cal B$}^{\rm
m}}-\frac{1}{2}\left\\{...,\tilde{H}_{\rm S}^{\rm
m}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm m}}-\kappa^{\rm
m}\left(...\right)\;,$ (42) $\displaystyle-i\hat{\cal L}^{\Gamma^{\rm m}}$
$\displaystyle=$ $\displaystyle-\left[\Gamma^{\rm m},...\right]_{+}\;.$ (43)
The equation of motion (41) is represented into the adiabatic basis as
$\displaystyle\frac{d}{dt}\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m}(t)$
$\displaystyle=$ $\displaystyle\sum_{\beta\beta^{\prime}}\left(-i\tilde{\cal
L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\rm m}-i\hat{\cal
L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\Gamma^{\rm
m}}\right)\tilde{\Omega}_{\beta\beta^{\prime}}^{\rm m}(t)\;.$ (44)
In Eq. (44) we introduced the two Liouville super-operators $-i\tilde{\cal
L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\rm m}$ and $-i\hat{\cal
L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\Gamma^{\rm m}}$. We first
consider $\tilde{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\rm m}$.
We have
$\displaystyle-i\tilde{\cal L}^{\rm
m}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ $\displaystyle=$
$\displaystyle\left(-i\omega_{\alpha\alpha^{\prime}}-i\tilde{L}_{\alpha\alpha^{\prime}}^{\rm
m}-\kappa^{\rm
m}\right)\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}+\tilde{\cal
T}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\;,$ (45)
where $\omega_{\alpha\alpha^{\prime}}=E_{\alpha}(Q)-E_{\alpha^{\prime}}(Q)$ is
the Bohr frequency. The operator
$\displaystyle i\tilde{L}_{\alpha\alpha^{\prime}}^{\rm m}$ $\displaystyle=$
$\displaystyle P\frac{\partial}{\partial
Q}+\frac{1}{2}\left(F_{\alpha}(Q)+F_{\alpha^{\prime}}(Q)\right)\frac{\partial}{\partial
P}+P\frac{\Pi_{1}}{\mu_{1}}\frac{\partial}{\partial P}+$ (46)
$\displaystyle-\frac{\Pi_{1}}{\mu_{1}}\frac{\partial}{\partial\Lambda_{1}}-\left(P^{2}-k_{B}T\right)\frac{\partial}{\partial\Pi_{1}}+\Pi_{1}\frac{\Pi_{2}}{\mu_{2}}\frac{\partial}{\partial\Pi_{1}}+$
$\displaystyle-\frac{\Pi_{2}}{\mu_{2}}\frac{\partial}{\partial\Lambda_{2}}-\left(\frac{\Pi_{1}^{2}}{\mu_{1}}-k_{B}T\right)\frac{\partial}{\partial\Pi_{2}}$
is a classical-like Liouville operator generating the dynamics of the physical
coordinates $X$ under the feedback of the fictitious coordinates $\cal Y$. In
Eq. (46) we have also introduced the Hellmann-Feynman force:
$F_{\alpha}(Q)=-\frac{\partial E_{\alpha}(Q)}{\partial Q}\;.$ (47)
This is the force acting on the harmonic oscillator when state $\alpha$ is
occupied.
The adiabatic surfaces transition operator $\tilde{\cal
T}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ is purely off-diagonal. In the
adiabatic basis it is expressed as
$\displaystyle{\cal T}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$
$\displaystyle=$
$\displaystyle\delta_{\alpha^{\prime}\beta^{\prime}}P\cdot{\cal
C}_{\alpha\beta}(Q)\left(1+\frac{1}{2}\tilde{\cal
S}_{\alpha\beta}\cdot\frac{\partial}{\partial P}\right)$ $\displaystyle+$
$\displaystyle\delta_{\alpha\beta}P\cdot{\cal
C}_{\alpha^{\prime}\beta^{\prime}}^{*}(Q)\left(1+\frac{1}{2}\tilde{\cal
S}_{\alpha^{\prime}\beta^{\prime}}^{*}\cdot\frac{\partial}{\partial
P}\right)\;,$
where
$\displaystyle{\cal C}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\langle\Phi_{\alpha}(Q)|\frac{\partial}{\partial
Q}|\Phi_{\beta}(Q)\rangle\;,$ (48) $\displaystyle\tilde{\cal S}_{\alpha\beta}$
$\displaystyle=$
$\displaystyle\frac{\left(E_{\alpha}-E_{\beta}\right)}{P\cdot{\cal
C}_{\alpha\beta}(Q)}{\cal C}_{\alpha\beta}(Q)\;.$ (49)
Equation (48) defines the adiabatic coupling vector ${\cal C}_{\alpha\beta}$.
This vector quantifies the superposition of the adiabatic states when $Q$
changes. Equation (49) introduces the vector $\tilde{\cal S}_{\alpha\beta}$.
If dimensional coordinates were introduced, $\tilde{\cal S}_{\alpha\beta}$
would have the dimension of momentum. Roughly speaking, $\tilde{\cal
S}_{\alpha\beta}$ provides the momentum variation when jumping from one
adiabatic surface to the other.
We now consider $\hat{\cal L}^{\Gamma^{\rm m}}$. In general, a decay operator
$\tilde{\Gamma}_{\alpha,\beta}$ can always be decomposed in a diagonal part
$\tilde{\Gamma}_{\alpha\beta}^{\rm
d}=\tilde{\Gamma}_{\alpha\beta}\delta_{\alpha\beta}\;,$ and an off-diagonal
part $\tilde{\Gamma}_{\alpha\beta}^{\rm
o}=\tilde{\Gamma}_{\alpha\beta}\left(1-\delta_{\alpha\beta}\right)\;,$ so that
$\tilde{\Gamma}_{\alpha,\beta}\equiv\tilde{\Gamma}_{\alpha\beta}^{\rm
d}+\tilde{\Gamma}_{\alpha\beta}^{\rm o}\;.$ However, in the case of the decay
operator of the nHQSMJ model adopted here, and defined in Eq. (40), we have
$\left(\tilde{\Gamma}^{\rm m}\right)_{\alpha\beta}^{\rm o}=0$. Hence, we have
$i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{\Gamma^{\rm
m}}=\gamma_{\alpha\alpha^{\prime}}^{\rm
m}\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}\;,$ (50)
where $\gamma_{\alpha\alpha^{\prime}}^{\rm m}=\Gamma_{\alpha\alpha}^{\rm
m}+\Gamma_{\alpha^{\prime}\alpha^{\prime}}^{\rm m}$.
Using Eqs. (45) and (50), Eq. (44) becomes
$\displaystyle\frac{d}{dt}\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m}(t)$
$\displaystyle=$
$\displaystyle-\sum_{\beta\beta^{\prime}}\left[\left(i\omega_{\alpha\alpha^{\prime}}+\gamma_{\alpha\alpha^{\prime}}^{\rm
m}+i\tilde{L}_{\alpha\alpha^{\prime}}^{\rm m}+\kappa_{\rm S}^{\rm
m}\right)\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}+\tilde{\cal
T}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\right]\tilde{\Omega}_{\beta\beta^{\prime}}^{\rm
m}(t)\;.$ (51)
Equation (51) provides the equation of motion for the density matrix of the
nHQSMJ model at constant temperature. This equation can be implemented through
a variety of algorithms theochem ; sstp ; algo12 ; algo15 .
## VI Calculations and Results
We consider a situation where the molecule is weakly coupled to the leads.
This implies that neglecting the action of the transition operator
$\tilde{\cal T}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ in Eq. (51) the
phase-space-dependent density matrix is propagated by means of the sequential
short-time propagation (SSTP) algorithm theochem ; sstp as
$\displaystyle\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m}(t)$
$\displaystyle=$ $\displaystyle\prod_{j=1}^{n_{\rm
step}}\sum_{\alpha\alpha^{\prime}}e^{-i\left(\omega_{\alpha\alpha^{\prime}}-i\gamma_{\alpha\alpha^{\prime}}^{\rm
m}-i\kappa^{\rm m}+\tilde{L}_{\alpha\alpha^{\prime}}^{\rm
m}\right)\tau}\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m},(0)$ (52)
$\displaystyle=$ $\displaystyle\prod_{j=1}^{n_{\rm
step}}\sum_{\alpha\alpha^{\prime}}e^{-i\tilde{\cal
L}_{\alpha\alpha^{\prime}}^{\rm
m,(0)}\tau}\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m}(0)\;,$
where $\tau=t/n_{\rm step}$ and the non-Hermitian adiabatic Liouville super-
operator is $-i\tilde{\cal L}_{\alpha\alpha^{\prime}}^{\rm
m(0)}=-i\omega_{\alpha\alpha^{\prime}}-\gamma_{\alpha\alpha^{\prime}}^{\rm
m}-\kappa_{\rm S}^{\rm m}-i\tilde{L}_{\alpha\alpha^{\prime}}^{\rm m}$. Quantum
statistical averages can be calculated as
$\displaystyle\tilde{\rm T}{\rm r}^{\rm m}\left(\tilde{\rho}^{\rm
m}(t)\tilde{\chi}^{\rm m}\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{\alpha\alpha^{\prime}}\int dX^{\rm
m}\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm m}(X^{\rm
m},t)\tilde{\chi}_{\alpha^{\prime}\alpha}(X^{\rm m})}{\sum_{\sigma}\int
dX^{\rm m}\tilde{\Omega}_{\sigma\sigma}^{\rm m}(X^{\rm m},t)}$ (53)
$\displaystyle=$
$\displaystyle\frac{\sum_{\alpha\alpha^{\prime}}\langle\tilde{\Omega}_{\alpha\alpha^{\prime}}^{\rm
m}(t)\tilde{\chi}_{\alpha^{\prime}\alpha}\rangle^{\rm
m}}{\sum_{\sigma}\langle\tilde{\Omega}_{\sigma\sigma}^{\rm m}(t)\rangle^{\rm
m}}\;,$
where the bracket $\langle...\rangle^{\rm m}$ stands for the average in the
extended phase space of the model. We implement Eq. (53) in the following way
theochem ; sstp . Once the quantum initial state is assigned for every point
of the extended phase space, the calculation of quantum averages can be
performed by sampling the initial $X^{\rm e}$ with a Monte Carlo algorithm.
The point $X^{\rm m}$ is then propagated over the assigned quantum state for a
time length $t$. The time step $\tau=t/n_{\rm step}$ in Eq. (53) must be
chosen small enough to minimize the numerical error. In the calculation we
take the following uncorrelated form of the initial phase-space-dependent
density matrix
$\tilde{\Omega}^{\rm e}(X^{\rm m},0)=\hat{\Omega}^{\rm m}(0)\Omega_{\rm
B}^{\rm m}(X,0)\Omega_{\rm nhc}^{\rm m}({\cal Y},0)\;,$ (54)
where
$\displaystyle\hat{\Omega}^{\rm m}(0)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right]\;,$ (57)
$\displaystyle\Omega_{\rm B}^{\rm m}(X,0)$ $\displaystyle=$
$\displaystyle\frac{\tanh(\beta\omega/2)}{\pi}\exp\left(-\frac{\tanh(\beta\omega/2)}{\omega}H_{\rm
B}^{\rm m}(X)\right)\;,$ (58) $\displaystyle\Omega_{\rm nhc}^{\rm m}({\cal
Y},0)$ $\displaystyle=$
$\displaystyle\frac{\prod_{J=1}^{2}\delta(\Lambda_{J})\exp(-\Pi_{J}^{2}/2)}{Z_{\cal
Y}^{\rm m}}$ (59)
and
$\int d{\cal
Y}\prod_{J=1}^{2}\delta(\Lambda_{J})\exp(-\Pi_{J}^{2}/2\sigma^{2})$ (60)
Calculations were performed with fixed values of $\gamma=0.1$, $\omega=1/3$,
$\Delta=1$, $c=0.007$, time step $\tau=0.005$, number of time steps $n_{\rm
step}=10^{4}$, and number of Monte Carlo steps $n_{\rm mcs}=25\cdot 10^{3}$.
All the values of the parameters are given in adimensional units. Twenty
different values of the inverse temperature were considered upon chosing
$\beta_{l+1}=1/T_{l+1}=\beta_{0}(1+l)$, with $l=0,...,19$ and
$\beta_{0}=0.0005$. For each $\beta_{l}$ we have investigated four different
types of dynamics: i) Unitary dynamics. ii) Constant-temperature dynamics.
iii) Non-unitary dynamics. iv) Non-unitary dynamics at constant temperature.
Initial conditions are chosen with the two-level system in its lowest
eigenvector and the bath in a thermal state. It is useful to introduce
$\hat{\Xi}^{m}$ and $\hat{\cal X}^{\rm m}$, the non-normalized and normalized
reduced density matrices, respectively. We write their matrix elements in the
adiabatic basis as
$\displaystyle\Xi_{\alpha\alpha^{\prime}}^{m}$ $\displaystyle=$
$\displaystyle\int dX^{\rm m}\tilde{\Omega}_{\alpha\alpha^{\prime}}\;,$ (61)
$\displaystyle{\cal X}_{\alpha\alpha^{\prime}}^{m}$ $\displaystyle=$
$\displaystyle\int dX^{\rm m}\tilde{\rho}_{\alpha\alpha^{\prime}}\;.$ (62)
Both reduced density matrices can be easily calculated by means of our
numerical approach.
In order to check the numerical scheme, we calculated the evolution in time of
$\tilde{\rm T}{\rm r}(\tilde{\Omega}^{\rm m}(t))$. The results are shown in
Fig. 2. When the decay operator is set to zero, Panels (a) and (b), the trace
of the density matrix is conserved with extremely good numerical precision
both when the dynamics is unitary and when the temperature is controlled. This
provides a convincing indication that the our algorithm conserves important
dynamical invariants. Panels (c) and (d) in Fig. 2 show the decrease in time
of $\tilde{\rm T}{\rm r}(\tilde{\Omega}^{\rm m}(t))$ non-unitary
dynamics.Moreover, there is no appreciable difference between unitary and
constant-temperature dynamics.
In Fig. (3) we show the time-evolution of $\Xi_{\alpha\alpha}^{\rm m}(t)$ and
${\cal X}_{\alpha\alpha}^{\rm m}(t)\rangle^{\rm m}$, with $\alpha=1,2$,
defined in Eq. (61) and (62), respectively. Panels (a) and (b) display
$\Xi_{11}^{\rm m}(t)$ and $\Xi_{22}^{\rm m}(t)$, respectively. Initially, both
adiabatic states are equally occupied. However, according to Eq. (40), only
the occupation of the excited state experiences depletion, see in Panel (a).
Panels (c) and (d) of Fig. (3) show ${\cal X}_{11}^{\rm m}(t)$ and ${\cal
X}_{22}^{\rm m}(t)$, respectively. The behaviour of ${\cal X}_{11}^{\rm m}(t)$
is hardly distinguishable from that of $\Xi_{11}^{\rm m}(t)$. However, the
situation is different for ${\cal X}_{22}^{\rm m}(t)$ in Panel (d). As a
matter of fact, while $\Xi_{22}^{\rm m}(t)$ is constant, ${\cal X}_{22}^{\rm
m}(t)$ increases so that the $\tilde{\rm T}{\rm r}^{\rm m}(\tilde{\rho}^{\rm
m}(t))={\cal X}_{11}^{\rm m}(t)+{\cal X}_{22}^{\rm m}(t)=1$.
In Fig. (4) we show the real part of the reduced off-diagonal element ${\rm
Re}({\cal X}_{12}^{\rm m}(t))$. For the chosen values of the parameters,
dephasing occurs on the same time scale of the population’s depletion of the
excited state.
Figure (5) displays the time evolution of the population difference for
$\beta=0.0075$. Below each plot showing the time evolution of the population
difference, the Fourier transform is also displayed. The Fourier transform of
Panel (a) displays two peaks: one at small frequency and one at higher
frequency. The Rabi-like oscillations of the population difference at distant
time are signs of the absence of stable quantum transport in the nHQSMJ model.
Nosé-Hoover chain dynamics, shown in Panel (b), suppresses the peak at small
frequency. The Rabi-like oscillations of the population difference at distant
time are less pronounced: they take place at relatively high frequency, as the
Fourier transform shows, around zero. This indicates that in the presence of
thermal noise the population difference oscillates around zero so that
transport is stable. This can be seen as a particular instance of environment-
assisted quantum transport: remarkably, the noise provided by an environment
can defeat Anderson localization anderson and assist quantum transport. The
idea that the noise of the environment could enhance quantum transport,
instead of suppressing it, has been postulated to explain the high efficiency
of energy transport in photosynthetic systems plenio . Obviously, the noise-
enhancement of quantum transport can take place only below a certain
threshold. Above it, transport is suppressed by the quantum Zeno effect zeno .
In Panel (c) Hamiltonian dynamics of the bath is combined with a non-zero
decay operator acting on the two-level system. As it can be seen, the decay
operator suppresses the high frequencies in the Rabi-like oscillations of the
population difference at distant time. Such oscillations show that, even in
this case, quantum transport is not very stable. Comparison of the dynamics in
Panel (b) and in Panel (c) shows that there is a difference between thermal
and probability sink dissipation: the first suppresses low frequency
oscillation while the second damps high frequency oscillations. As one can
expect the combination of the two dynamics, shown in Panel (d), leads to the
suppression of both frequencies, increasing the efficiency and the stability
of the population transport. Calculations for $\beta=0.0025$ are reported in
Fig. (7). The results are indistinguishable to the human eye from those
obtained at $\beta=0.0075$. This reinforces the conclusion that quantum
transport in the nHQSMJ model is enhanced by coupling the two-level system
with a probability sink and controlling the temperature of the molecule.
## VII Conclusions
In this work, we construct a theory for studying non-Hermitian phase-space-
dependent quantum systems at constant temperature. This theory is based on an
operator-valued Wigner formulation of quantum mechanics or, in other words, on
a phase-space-dependent density matrix. The condition of constant temperature
in phase space is achieved by means of the Nosé-Hoover chain thermostat. A
mathematical result of the formalism is the derivation of the non-linear
equation of motion for the normalized phase-space-dependent density matrix of
the system. We remark that this result, considering a constant temperature
bath, improves the theory developed in Ref. as_15 where the temperature
fluctuations are not constrained. These latter situation is somewhat
unrealistic in common experiments.
This theory is applied to a model of a non-Hermitian quantum single-molecule
junction. We emphasize that this model treats probability loss and thermal
fluctuations on the same level. In detail, the model comprises a two-level
system with a probability sink coupled to a thermalized harmonic mode. The
structure of our model was conjectured upon drawing an analogy with the
process of noise-assisted transport noise ; noise2 ; noise3 . In our case, we
expect that the assisted quantum transport arises from the combined action of
the probability sink and the constant-temperature fluctuating molecule.
Indeed, we observe a transport enhancement _in silico_ upon simulating
numerically the non-Hermitian quantum single-molecule junction model for a
range of temperatures, while keeping the values of the other parameters
constant. The Fourier transformed signal in frequency space shows that the
Nosé-Hoover chain thermostat suppresses the slow frequencies while the
probability sink damps the high frequency oscillations of the population
difference of the non-Hermitian quantum single-molecule junction model at
constant temperature.
The non-Hermitian quantum single-molecule junction model at constant
temperature introduced in this paper can more accurately be classified as a
particular instance of a class of models. As a matter of fact, it can be
easily generalized by considering a greater number of quantum states ($\sim
10$) and an even greater number of classical modes ($\sim 10^{3}-10^{5}$). Of
course, more sophisticated algorithms would have to be used for calculating
the evolution of the density matrix. We also note that spin models that are
similar but simpler than the non-Hermitian quantum single-molecule junction
model introduced in this paper can be solved analytically when obeying certain
symmetry properties hn_16 ; gv_17 ; gm_17 ; gm_18 ; gm_19 ; gv_19 ; gv2_19 ;
hn_18 ; hn2_18 . It would be interesting to investigate whether the extension
of these models to non-Hermitian quantum mechanics would still be analytically
treatable. We defer such generalizations to future work.
Figure 1: Pictorial representation of the nHQSMJ model. The leads are
represented by two gray triangles. The spring portrays the molecule connecting
the leads. The arrows with the cyan cusps depict the state of the TLS,
depending on its position along the junction. When the TLS is in the left
lead, the arrow points downward. As the TLS moves toward the right lead, the
arrow rotates until it reaches the upward position, which implies a complete
transfer to the right lead. The sink below the right lead absorbs the TLS
probability in an irreversible way. In the formalism, the sink is represented
by a Hermitian decay operator. The decay operator acts on quantum states that
in turn determine probabilities. Hence, the action of the sink unfolds upon an
ensemble of trajectories of the TLS system. The picture also shows a box
separating the molecular junction from the environment. The drawing of the box
resembles on purpose that of an oven with a thermostat on top. The thermostat
temperature is set at the same value of the $T_{\rm R}$ temperature of the
environment. The thermostat controls the temperature inside the oven so that
it is equal to that of the environment. The yellow color inside the oven and
around it, in the environment, conveys the idea of thermal equilibrium.
Figure 2: Plot of $\tilde{\rm T}{\rm r}(\tilde{\Omega}(t))$. Number of Monte
Carlo sampled phase space initial conditions $X^{\rm e}$ points $=2500$, time
step $\tau=0.005$, $\gamma=0.1$, $\omega=1/3$, $\Delta=1$, coupling constant
$c=0.007$, inverse temperature $\beta=0.0050$. The red line shows the average
trend while the cyan area around it, which is hardly visible, displays the
statistical error of the calculation. Panels (a) and (b) show the results of
the calculations when the dynamics takes place for the isolated single-
molecule junction (SMJ) and for the SMJ at constant temperature (SMJ+NHC),
respectively. Panel (c) shows the result of the calculation when a decay
operator acts onto the SMJ (SMJ+nH). Finally, in Panel (d) all types of
dynamics are considered together (SMJ+nH+NHC). It is easy to see that the
trace is conserved when the Hamiltonian is Hermitian. Otherwise, the
population drops to about one half as the probability is subtracted only from
the excited state. Figure 3: The plots show the diagonal elements of the
density matrices resulting from nH+NHC dynamics. Panels (a) and (b) show the
behavior of the non-normalized density matrix elements $\Xi_{11}(t)$ and
$\Xi_{22}(t)$, respectively. While Panels (c) and (d) display the behavior of
the normalized density matrix elements ${\cal X}_{11}^{\rm m}(t)$ and ${\cal
X}_{22}^{\rm m}(t)$ . Number of iterations $=10000$, number of Monte Carlo
points $=2500$, time step $\tau=0.005$, $\gamma=0.1$, $\omega=1/3$,
$\Delta=1$, $c=0.007$, $\beta=0.0050$. The red line is the average trend, the
cyan area (almost invisible to the eye) is the error. In (a) the effects of
the decay operator $\hat{\Gamma}^{\rm m}$ can be observed. The trend in (b)
clarifies the cause for the trend of the trace in Fig.(2). In (c) the
difference with (a) is not evident, as the trace decreases directly because of
the decrease of $\Omega_{11}$. The comparison between (d) and (b) further
clarifies that the cause for the population loss of the excited state is the
decay operator and not energy transfer. Figure 4: The plot shows the average
of the real part of ${\cal X}_{12}^{\rm m}(t)$, the off-diagonal element of
the reduced normalized density matrix (NHC+NH system). Number of iterations
$=10000$, number of Monte Carlo points $=2500$, time step $\tau=0.005$,
$\gamma=0.1$, $\omega=1/3$, $\Delta=1$, $c=0.007$, $\beta=0.0050$. The red
line is the average trend, the cyan area is the error. The trend clearly shows
decoherence. Figure 5: The plot shows the average population transfer vs.
time for different types of dynamics. The values of the parameters of the
calculation are (NDS=10000, NMCP=2500, $\tau=0.005$, $\gamma=0.1$,
$\omega=1/3$, $\Delta=1$, $g=0.007$, $\beta=0.0075$). The red line indicates
the average trend, the cyan area represents the error. The blue line denotes
the Fourier transform of the time evolution of the average value (the data are
considered after $t=20$). Panel (a) displays the results for the isolated SMJ.
One can observe that the values oscillate with low and high frequency. Panel
(b) displays the dynamics of the SMJ with temperature control (SMJ+NHC). The
thermostat facilitates the population transfer and stabilizes its average. The
low frequency terms are highly damped while the high frequency ones are
unaltered. Panel (c) shows that the effect of the decay operator (SMJ+nH) is
to cancel the high frequency term. Finally, in (d), it is shown that the union
of the NHC thermostat and decay operator (SMJ+nH+NHC) determines a stable
tranfer. Both the low and the high frequency terms are strongly damped.
Figure 6: The plot shows the average population transfer vs. time for
different types of dynamics. The values of the parameters of the calculation
are (NMDS= 10000, NMCP = 2500, $\tau=0.005$, $\gamma=0.1$, $\omega$ = 1/3,
$\Delta$ = 1, g = 0.007, $\beta$ = 0.0050). The red line indicates the average
trend, the cyan area represents the error. The blue line denotes the Fourier
transform of the time evolution of the average value (the data are considered
after $t=20$). Panel (a) displays an average population transfer for the SMJ
model where low frequency oscillations are strongly damped. In Panel (b) it is
shown that a higher temperature damps high frequency oscillations (SMJ+NHC).
Statistical error appears to be somewhat greater. Panel (c) shows that the
effect of the decay operator is to damp high frequency oscillations (SMJ+nH).
Panel (d) shows that the combined action of the NHC thermostat and the decay
operator (SMJ+nH+NMJ) determines a stable transfer process. Both low and high
frequency oscillations are strongly damped, except for a very weak oscillation
at a very low frequency. Figure 7: The plot shows the average population
transfer vs. time for different types of dynamics. NMDS=10000, NMCP=2500,
$\tau=0.005$, $\gamma=0.1$, $\omega=1/3$, $\Delta=1$, $g=0.007$,
$\beta=0.0025$. The red line is the average trend, the cyan area is the error.
The blue line is the Fourier transform, carried out after the stabilization of
the trend (from $t=20$ onward). In Panel (a) the population difference
oscillates with low and high frequency contributions (SMJ). Panel (b) shows
that the thermostat (SMJ+NHC) facilitates the population transfer and
stabilizes it more than in the case of lower temperatures. The low frequency
terms are highly damped while the high frequency ones are unaltered. Panel (c)
shows that the effect of the decay operator (SMJ+nH) is to cancel the high
frequency term. Finally, Panel (d) shows that SMJ+nH+NHC displays a stable
transfer: both the low and the high frequency terms are strongly damped.
Funding: This research received no external funding.
Acknowledgments: AM and AS express their thanks to Hiromichi Nakazato for
carefully reading the manuscript and for numerous stimulating suggestions. All
the authors are grateful to Dr. Marika Matalone for drawing Figure 1.
## Appendix A Time evolution of the trace of $\tilde{\Omega}(t)$
The trace of Eq. (11):
$\displaystyle\frac{d}{dt}\tilde{\rm T}{\rm r}\left(\tilde{\Omega}(t)\right)$
$\displaystyle=$ $\displaystyle-2\tilde{\rm T}{\rm
r}\left(\hat{\Gamma}\tilde{\Omega}(X,t)\right)+\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{H}^{\rm
e},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal B$}}\right)$ (63)
$\displaystyle-$ $\displaystyle\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{\Omega}(t),\tilde{H}\right\\}_{\mbox{\boldmath$\cal
B$}}\right)\;.$
The last term in the right hand side of Eq. (63) can be transformed as follows
$\displaystyle-\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{\Omega}(t),\tilde{H}\right\\}_{\mbox{\boldmath$\cal
B$}}\right)$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\sum_{I,J=1}^{2N}\frac{\partial\tilde{\Omega}(t)}{\partial X_{I}}{\cal
B}_{IJ}\frac{\partial\tilde{H}}{\partial X_{J}}\right)=-\frac{1}{2}\tilde{\rm
T}{\rm r}\left(\sum_{I,J=1}^{2N}\frac{\partial\tilde{\Omega}(t)}{\partial
X_{J}}{\cal B}_{JI}\frac{\partial\tilde{H}}{\partial X_{I}}\right)$ (64)
$\displaystyle=$ $\displaystyle\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\sum_{I,J=1}^{2N}\frac{\partial\tilde{H}}{\partial X_{I}}{\cal
B}_{IJ}\frac{\partial\tilde{\Omega}(t)}{\partial
X_{J}}\right)=\frac{1}{2}\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{H},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}\right)$
Substituting Eq. (64) in the right hand side of Eq. (63), one obtains
$\displaystyle\frac{d}{dt}\tilde{\rm T}{\rm r}\left(\tilde{\Omega}(t)\right)$
$\displaystyle=$ $\displaystyle-2\tilde{\rm T}{\rm
r}\left(\hat{\Gamma}\tilde{\Omega}(X,t)\right)+\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{H},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}\right)\;.$ (65)
However, the last term in the right hand side of Eq. (65) is zero:
$\displaystyle\tilde{\rm T}{\rm
r}\left(\left\\{\tilde{H},\tilde{\Omega}(t)\right\\}_{\mbox{\boldmath$\cal
B$}}\right)$ $\displaystyle=$ $\displaystyle\tilde{\rm T}{\rm
r}\left(\sum_{I,J=1}^{2N}\frac{\partial\tilde{H}}{\partial X_{I}}{\cal
B}_{IJ}\frac{\partial\tilde{\Omega}(t)}{\partial X_{J}}\right)=-\tilde{\rm
T}{\rm r}\left(\sum_{I,J=1}^{2N}\frac{\partial^{2}\tilde{H}}{\partial
X_{I}\partial X_{J}}{\cal B}_{IJ}\tilde{\Omega}(t)\right)$ (66)
$\displaystyle=$ $\displaystyle 0$
because of the antisymmetry of $\cal B$. Equation (66) allows one to obtain
Eq. (15).
## Appendix B Time evolution of the trace of $\tilde{\Omega}^{\rm e}(t)$
The trace of Eq. (26) is
$\frac{d}{dt}\tilde{\rm T}{\rm r}^{\rm e}\left(\tilde{\Omega}^{\rm
e}(t)\right)=-2\tilde{\rm T}{\rm r}^{\rm
e}\left(\hat{\Gamma}\tilde{\Omega}^{\rm e}(t)\right)+\frac{1}{2}\tilde{\rm
T}{\rm r}^{\rm e}\left(\left\\{\tilde{H}^{\rm e},\tilde{\Omega}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)-\frac{1}{2}\tilde{\rm
T}{\rm r}^{\rm e}\left(\left\\{\tilde{\Omega}^{\rm e},\tilde{H}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)-\tilde{\rm T}{\rm
r}^{\rm e}\left(\kappa^{\rm e}\tilde{\Omega}^{\rm e}(t)\right)\;.$ (67)
One can find the following identity
$\displaystyle\tilde{\rm T}{\rm r}^{\rm e}\left(\left\\{\tilde{\Omega}^{\rm
e},\tilde{H}^{\rm e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)$
$\displaystyle=$ $\displaystyle\tilde{\rm T}{\rm r}^{\rm
e}\left(\sum_{I,J=1}^{2(N+2)}\frac{\partial\tilde{\Omega}^{\rm e}}{\partial
X_{I}^{\rm e}}{\cal B}_{IJ}^{\rm e}\frac{\partial\tilde{H}^{\rm e}}{\partial
X_{J}^{\rm e}}\right)=\tilde{\rm T}{\rm r}^{\rm
e}\left(\sum_{I,J=1}^{2(N+2)}\frac{\partial\tilde{\Omega}^{\rm e}}{\partial
X_{J}^{\rm e}}{\cal B}_{JI}^{\rm e}\frac{\partial\tilde{H}^{\rm e}}{\partial
X_{I}^{\rm e}}\right)$ (68) $\displaystyle=$ $\displaystyle\tilde{\rm T}{\rm
r}^{\rm e}\left(\sum_{I,J=1}^{2(N+2)}\frac{\partial\tilde{H}^{\rm e}}{\partial
X_{I}^{\rm e}}{\cal B}_{JI}^{\rm e}\frac{\partial\tilde{\Omega}^{\rm
e}}{\partial X_{J}^{\rm e}}\right)=-\tilde{\rm T}{\rm r}^{\rm
e}\left(\sum_{I,J=1}^{2(N+2)}\frac{\partial\tilde{H}^{\rm e}}{\partial
X_{I}^{\rm e}}{\cal B}_{IJ}^{\rm e}\frac{\partial\tilde{\Omega}^{\rm
e}}{\partial X_{J}^{\rm e}}\right)$ $\displaystyle=$ $\displaystyle-\tilde{\rm
T}{\rm r}^{\rm e}\left(\left\\{\tilde{H}^{\rm e},\tilde{\Omega}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)$
Inserting Eq. (68) in Eq. (67) one obtains
$\frac{d}{dt}\tilde{\rm T}{\rm r}^{\rm e}\left(\tilde{\Omega}^{\rm
e}(t)\right)=-2\tilde{\rm T}{\rm r}^{\rm
e}\left(\hat{\Gamma}\tilde{\Omega}^{\rm e}(t)\right)+\tilde{\rm T}{\rm r}^{\rm
e}\left(\left\\{\tilde{H}^{\rm e},\tilde{\Omega}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)-\tilde{\rm T}{\rm
r}^{\rm e}\left(\kappa^{\rm e}\tilde{\Omega}^{\rm e}(t)\right)\;.$ (69)
One can now find the following identity:
$\displaystyle\tilde{\rm T}{\rm r}^{\rm e}\left(\left\\{\tilde{H}^{\rm
e},\tilde{\Omega}^{\rm e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)$
$\displaystyle=$ $\displaystyle-\tilde{\rm T}{\rm r}^{\rm
e}\left(\left\\{\tilde{H}^{\rm e},\tilde{\Omega}^{\rm
e}\right\\}_{\mbox{\boldmath$\cal B$}^{\rm e}}\right)=-\tilde{\rm T}{\rm
r}^{\rm e}\left(\sum_{I,J=1}^{2(N+1)}\frac{\partial\tilde{H}^{\rm e}}{\partial
X_{I}^{\rm e}}{\cal B}_{IJ}^{\rm e}\frac{\partial\tilde{\Omega}^{\rm
e}}{\partial X_{J}^{\rm e}}\right)$ (70) $\displaystyle=$
$\displaystyle\tilde{\rm T}{\rm r}^{\rm
e}\left(\sum_{I,J=1}^{2(N+1)}\frac{\partial{\cal B}_{IJ}^{\rm e}}{\partial
X_{J}^{\rm e}}\frac{\partial\tilde{H}^{\rm e}}{\partial X_{I}^{\rm
e}}\tilde{\Omega}^{\rm e}\right)=\tilde{\rm T}{\rm r}^{\rm e}\left(\kappa^{\rm
e}\tilde{\Omega}^{\rm e}\right)\;.$
Equation (70) is obtained exploiting the identity
$\sum_{I,J=1}^{2(N+1)}{\cal B}_{IJ}^{\rm e}\frac{\partial^{2}\tilde{H}^{\rm
e}}{\partial X_{I}^{\rm e}\partial X_{J}^{\rm e}}\equiv 0\;,$ (71)
which derives from the fact that the trace of a symmetric matrix times an
antisymmetric matrix is identically zero. Finally, substituting Eq. (70) in
Eq. (69), one obtains Eq. (28).
## References
* (1) Ratner, M. A brief history of molecular electronics. Nature Nanotechnology 2013, 8, 378.
* (2) Thoss, M.; Evers, F. Perspective: Theory of quantum transport in molecular junctions. J. Chem. Phys. 2018, 148, 030901.
* (3) Sergi, A.; Hanna, G.; Grimaudo, R.; Messina, A. Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths. Symmetry 2018, 10, 518.
* (4) Zhang, W. Y.; Balescu, R. Statistical Mechanics of a spin-polarized plasma. J. Plasma Phys. 1988, 40, 199.
* (5) Balescu, R.; Zhang, W. Y. Kinetic equation, spin hydrodynamics and collisional depolarization rate in a spin polarized plasma. J. Plasma Phys. 1988, 40, 215.
* (6) Carpio-Martínez, P.; Hanna, G. Nonequilibrium heat transport in a molecular junction: A mixed quantum-classical approach. J. Chem. Phys. 2019, 151, 074112.
* (7) Gamow, G. Zur Quantentheorie des Atomkernes. Zeitschrift fur Physik 1928, 51, 204–212.
* (8) Bender, C. M. PT Symmetry: In Quantum And Classical Physics, World Scientific: London, 2019.
* (9) Mostafazadeh, A. Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 2002, 43, 205–214.
* (10) Subotnik, J. E.; Hansen, T.; Hansen, M. A.; Nitzan, A. Nonequilibrium steady state transport via the reduced density matrix operator. J. Chem. Phys. 2009, 130, 144105.
* (11) Zelovich, T.; Kronik, L.; Hod, O. State Representation Approach for Atomistic Time-Dependent Transport Calculations in Molecular Junctions. J. Chem. Theory and Computation 2014, 10, 2927.
* (12) Zelovich, T.; Hansen, T.; Liu, Z.-F.; Neaton, J. B.; Kronik, L.; Hod, O. Parameter-free driven-von Neumann approach electronic transport simulations in open quantum systems. J. Chem. Phys. 2017, 146, 092331.
* (13) Moiseyev, N. Non-Hermitian quantum mechanics. Cambridge University Press: Cambridge, Great Britain, 2011.
* (14) Rotter, I.; Bird, J. P. A review of progress in the physics of open quantum systems: theory and experiment. Reports on Progress in Physics 2015, 78, 114001.
* (15) Sergi, A. Embedding quantum systems with a non-conserved probability in classical environments, Theor. Chem. Acc. 2015, 134, 79.
* (16) Sergi, A.; Zloshchastiev, K. G. Non-Hermitian Quantum Dynamics of a Two-Level System and Models Of Dissipative Environments. International Journal of Modern Physics B 2013, 27, 1350163.
* (17) Sergi, A.; Zloshchastiev, K. G. Time correlation functions for non-Hermitian quantum systems. Phys. Rev. A 2015, 91, 062108.
* (18) Sergi, A.; Zloshchastiev, K. G. Quantum entropy of systems described by non-Hermitian Hamiltonians. JSTAT 2016, 3, 033102.
* (19) Sergi A.; Giaquinta, P. V. Linear Quantum Entropy and Non-Hermitian Hamiltonians. Entropy 2016, 18, 451(11pp).
* (20) Grimaudo, R.; de Castro, A. S. M.; Kuś, M.; Messina, A. Exactly solvable time-dependent pseudo-Hermitian su(1,1) Hamiltonian models. Phys. Rev. A 2018, 98, 033835.
* (21) Zloshchastiev, Konstantin G. Quantum-statistical approach to electromagnetic wave propagation and dissipation inside dielectric media and nanophotonic and plasmonic waveguides. Phys. Rev. B 2016, 94, 115136.
* (22) Zloshchastiev, K. G. Sustainability of Environment-Assisted Energy Transfer in Quantum Photobiological Complexes. Ann. Phys. 2017, 529, 1600185.
* (23) Feshbach, H. Ann. Phys. 1958, 5, 357.
* (24) Feshbach, H. Ann. Phys. 1962, 19, 287.
* (25) Grimaudo, R.; Messina, A.; Sergi, A.; Vitanov, N.; Filippov, S. Two-qubit entanglement generation through non-Hermitian Hamiltonians induced by repeated measurements on an ancilla. Entropy 2020, 22, 1184(1-18).
* (26) Berfield, J. P.; Ratner, M. A. Forty years of molecular electronics: Non-equilibrium heat and charge transport at the nanoscale. Phys. Status Solidi B 2013, 250, 2249.
* (27) Kamenetska, M.; Widawsky, J. R.; Dell’Angela, M.; Frei, M. Temperature dependent tunneling conductance of single molecule junctions. J. Chem. Phys. 2017, 146, 092311.
* (28) G. T. Craven and A. Nitzan, Electron transfer at thermally heterogeneous molecule-metal interfaces. J. Chem. Phys. 146 092305 (2017).
* (29) Sergi A.; Petruccione, F. Nosé-Hoover dynamics in quantum phase space. Journal of Physics A 2008, 41, 355304(14pp).
* (30) Martyna, G. J.; Klein, M. L.; Tuckerman, M. Nosé-Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys. 1992, 97, 2635.
* (31) Sergi, A.; Mac Kernan, D.; Ciccotti, G.; Kapral, R. Simulating Quantum Dynamics in Classical Environments. Theor. Chem. Acc. 2003, 110, 49-58.
* (32) Mac Kernan, D.; Kapral, R.; Ciccotti, G. Sequential short-time propagation of quantum–classical dynamics. J. Phys. Condens. Matter 2002, 14, 9069–9076.
* (33) Sergi A.; Petruccione, F. Sampling Quantum Dynamics at Long Time. Phys. Rev. E 2010, 81, 032101(4pp).
* (34) Uken, D. A.; Sergi, A.; Petruccione, F. Filtering Schemes in the Quantum-Classical Liouville Approach to Non-adiabatic Dynamics. Phys. Rev. E 2013, 88, 033301(7pp).
* (35) Sergi, A. Non-Hamiltonian Commutators in Quantum Mechanics, Phys. Rev. E 2005, 72, 066125.
* (36) A. Sergi, Deterministic constant-temperature dynamics for dissipative quantum systems. J. Phys. A 2007, 40, F347.
* (37) Mac Kernan, D.; Ciccotti, G.; Kapral, R. Surface-hopping dynamics of a spin-boson system. J. Chem. Phys. 2002, 116, 2346.
* (38) Sergi, A.; Ferrario, M. Non-Hamiltonian Equations of Motion with a Conserved Energy. Phys. Rev. E 2001, 64, 056125.
* (39) Sergi, A. Non-Hamiltonian Equilibrium Statistical Mechanics. Phys. Rev. E 2003, 67, 021101.
* (40) Sergi, A.; Giaquinta, P. V. On the geometry and entropy of non-Hamiltonian phase space. JSTAT 2007, 02, P02013(20pp).
* (41) Anderson, P. W. Phys. Rev. 1958, 109, 1492.
* (42) Huelga, S.; Plenio, M. Contemporary Physics 2013, 54, 181.
* (43) Sudarshan, E. C. G.; Misra, B. The Zeno’s paradox in quantum theory. J. Math. Phys. 1977, 18, 756.
* (44) Caruso, F.; Chin, A. W.; Datta, A.; Huelga, S. F.; Plenio, M. B. Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport. J. Chem. Phys. 2009, 131, 105106.
* (45) Viciani, S.; Lima, M.; Bellini, M.; Caruso, F. Observation of Noise-Assisted Transport in an All-Optical Cavity-Based Network. Phys. Rev. Lett. 2015, 115, 083601.
* (46) Chin, A. W.; Datta, A.; Caruso, F.; Huelga, S. F.; Plenio, M. B. New Journal of Physics 2010, 12, 065002(16pp).
* (47) Grimaudo, R.; Messina, A.; Nakazato, H. Exactly solvable time-dependent models of two interacting two-level systems. Phys. Rev. A 2016, 94, 022108.
* (48) Grimaudo, R.; Messina, Ivanov, P. A.; Vitanov, N. V. Spin-1/2 sub-dynamics nested in the quantum dynamics of two coupled qutrits, J. Phys. A 2017, 50, 175301.
* (49) Grimaudo, R.; Belousov, Y.; Nakazato, H.; Messina, A. Time evolution of a pair of distinguishable interacting spins subjected to controllable and noisy magnetic fields. Ann. Phys. 2017, 392, 242-259.
* (50) Grimaudo, R.; Lamata, L.; Solano, E.; Messina, A. Cooling of many-body systems via selective interactions. Phys. Rev. A 2018, 98, 042330.
* (51) Grimaudo, R.; Isar, A.; Mihaescu, T.; Ghiu, I.; Messina, A. Dynamics of quantum discord of two coupled spin-1/2’s subjected to time-dependent magnetic fields. Results in Physics 2019, 13, 102147.
* (52) Grimaudo, R.; Vitanov, N. V.; Messina, A. Coupling-assisted Landau-Majorana-Stückelberg-Zener transition in a system of two interacting spin qubits. Phys. Rev. B 2019, 99, 174416.
* (53) R. Grimaudo, N. V. Vitanov, and A. Messina, Landau-Majorana-Stc̈kelberg-Zener dynamics driven by coupling for two interacting qutrit systems, Phys. Rev. B 99, 214406 (2019).
* (54) Suzuki, T.; Nakazato, H.; Grimaudo, R.; Messina, A. Analytic estimation of transition between instantaneous eigenstates of quantum two-level system. Scientific reports 2018, 8 1-7.
* (55) Grimaudo, R.; Magalh$\tilde{\rm a}$es de Castro, A. S.; Nakazato, H.; Messina, A. Classes of Exactly Solvable Generalized Semi-Classical Rabi Systems. Annalen der Physik 2018, 530, 1800198.
|
∎
11institutetext: P. O. Hess 22institutetext: Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, Circuito Exterior, C.U., A.P. 70-543,
04510 México D.F., Mexico
and
Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universität,
Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany 22email:
<EMAIL_ADDRESS>33institutetext: E. López-Moreno 44institutetext:
Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico-City,
Mexico
# Axial and polar modes for the ring down of a Schwarzschild black hole with
an $r$ dependent mass-function ††thanks: DGAPA-PAPIIT IN100418
Peter O. Hess Enrique López-Moreno
(Received: date / Accepted: date)
###### Abstract
The axial and polar modes for the ring down of a Schwarzschild black hole are
calculated, by first deriving the Regge-Wheeler and Zerilli equations,
respectively, and finally applying the Asymptotic Iteration Method (AIM). We
were able to reach up to 500 iterations, obtaining for the first time
convergence for a wide range of large damping modes. The General Relativity
(GR) and a particular version of an extended model with an $r$-dependent mass-
function are compared. This mass-function allows an analytical solution for
the Tortoise coordinate. The example of the mass-function corresponds to the
leading correction for extended theories and serves as a starting point to
treat other $r$-dependent parameter mass-functions.
###### Keywords:
General Relativity axial modes polar modes
††journal: General Relativity and Gravitation
## 1 Introduction
General Relativity (GR) is one of the best tested theories, which accounts for
the observations in the solar system will and also its prediction of
gravitational waves maggiore was confirmed recently abbot1 ; abbot2 . The
first indirect proof of these waves stems from the 1970’s hulse , through the
observation of changes in the orbital frequency of a neutron star binary. The
black hole merger consists of two phases: the inspiral and the ring-down
phase. For the description of the inspiral phase the result depends very much
on the approximation used maggiore ; hess-2016 or on a non-linear hydro-
dynamic approach rezzolla-book . For the ring-down phase the situation is
”simpler”, because only the stability of the black hole under metric
perturbations has to be studied. The investigation of the ring-down phase can
be traced back to S. Chandrasekhar’s book on the mathematics of black holes
chandra and chandra1975a ; chandra1975b (though, not the first). The
equation for the calculation of the ring-down frequencies where treated in
zerilli for the polar modes and in RW for the axial modes. These equations
permit to calculate the frequencies, solving the eigenvalue problem of the
equation, which is much simpler than to implement a dynamical theory. There
are several methods to solve these equations, as for example the so-called
Asymptotic Iteration Method (AIM) ciftci2003 ; ciftci2005 , with an improved
approach published in cho2012 .
For this reason, we restrict our analysis to the study of the ring-down modes
only and in addition to a non-rotating star, i.e., to the Schwarzschild
metric. We follow closely the method described in the book of S. Chandrasekhar
chandra , Chapter 4 on the perturbations of a Schwarzschild black hole.
An interesting questions is: What changes, when a $r$-dependent mass-function
$m(r)$ is used, instead of a constant mass parameter $m_{0}$? There is a
$1/r^{4}$ leading order, due to the following arguments: a) $1/r^{2}$
corrections are excluded due to observations in the solar system will . b)
Also $1/r^{3}$ corrections are excluded due to adjusting to the inspiral phase
nielsen2018 ; nielsen2019 in the first observed gravitational event. Thus,
the leading corrections are of the order $1/r^{4}$. Other corrections, as
appearing in cosmological models (de Sitter), are not included but the path
explained can be extended to it. It is probable that the constant mass $m_{0}$
is substituted by a function in $r$ as soon as GR is extended and, therefore,
it is interesting to ask what kind of changes one can expect? Does the
$1/r^{4}$ correction to the metric still lead to stable modes? Is
isospectrality between axial and polar modes maintained? These are some of the
motivations of this contribution. To make life easier, we will use a
particular mass-function which still implies an event horizon, enabling us to
extend directly, with few modifications, the calculations reported in chandra
.
Another example is given in hess2009 ; book where the pseudo-complex General
Relativity (pc-GR) is proposed, which adds in the vicinity of a black hole a
distribution of dark energy, which is repulsive and from a certain value of
the coupling constnat halts the collapse of a star, before forming an event
horizon and a singularity. In all applications up to now, for practical
reasons, the coupling constant is chosen such that there is still an event
horizon at $r=\frac{3}{2}m_{0}$. First observable predictions are published in
MNRAS2013 ; MNRAS2014 . More recent descriptions of this model can be found in
highenergy ; universe ; PPNP . One reason for using this theory becomes
obvious in the main body of the text: The mass-function in pc-GR stars with a
$1/r^{4}$ correction and we show that it permits an analytical solution for
the Tortoise coordinate, thus, it provides us with a controlled handling of
the asymptotic limit of the solutions. This property helps to understand the
changes in the spectrum of the Quasinormal Modes (QNM) in the ring-down phase
of a black hole and its stability.
Because the mass-function is such that there is still an event-horizon, one
can proceed in an analog way as in GR. The only question is how to treat the
accumulation of dark energy around a black hole. The distribution defines es
new vacuum and it will be shown that it only depends on the total central mass
through a coupling constant, which is not changed when perturbations are
included.
The paper is organized as follows: In Section 2 the Schwarzschild limit will
be discussed. In Section 3 the easier to treat case of axial modes are
determined and in Section 4 the polar modes. In Section 5 the Regge-Wheeler
and Zerilli equations are derived and the numerical method to solve the
differential equation is shortly explained. In Section 6 the asymptotic limit
of the corresponding solution is calculated, using an analytical solution for
the Tortoise coordinate. The spectrum of axial and polar modes are determined,
within GR and its extension. We will show that the isospectral symmetry
observed in GR, namely that the frequencies of the axial and polar modes are
the same, is not maintained for an $r$-dependent mass-function, though, some
similar structures are present. In Section 7 the Conclusions are drawn.
## 2 The Schwarzschild solution
Following the notation of chandra ; chandra1975a ; chandra1975b , the length
element is given by
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
e^{2\nu}(dt)^{2}-e^{2\psi}\left(d\phi-\omega dt-
q_{2}dx_{2}-q_{3}dx_{3}\right)^{2}$ (1)
$\displaystyle-e^{2\mu_{2}}(dx_{2})^{2}-e^{2\mu_{3}}(dx_{3})^{2}~{}~{}~{},$
where $dx_{2}=r$ and $dx_{3}=\theta$, the azimuth angle. The functions $\nu$,
$\phi$, $\omega$, $q_{2}$, $q_{3}$, $\mu_{2}$ and $\mu_{3}$ depend, in
general, on $t$, $x_{2}=r$ and $x_{3}=\theta$, though, in this contribution we
restrict to a spherical symmetry. The components of the Ricci and the Einstein
tensor can be retrieved from the book by S. Chandrasekhar chandra , in terms
of the above functions. Care has to be taken in comparing notations: With the
convention of chandra , the tensor components of thr Riemann and Einstein
tensor are all at the lower position, but by directly calculating these
components, one can show that they are equivalent to $R^{\mu}_{~{}\nu}$ and
$G^{\mu}_{~{}\nu}$, which is also explained in chandra . This is important, as
will be seen further below.
The metric perturbations are introduced around the generalized Schwarzschild
metric
$\displaystyle e^{2\nu}$ $\displaystyle=$ $\displaystyle
e^{-2\mu_{2}}~{}=~{}\left(1-\frac{2m(r)}{r}\right)~{}=~{}\frac{\Delta}{r^{2}}$
$\displaystyle e^{\mu_{3}}$ $\displaystyle=$ $\displaystyle
r~{},~{}e^{\psi}~{}=~{}r{\rm sin}\theta$ $\displaystyle{\rm and}$
$\displaystyle\omega$ $\displaystyle=$ $\displaystyle
q_{2}~{}=~{}q_{3}~{}=~{}0~{},~{}\Delta~{}=~{}r^{2}-2m(r)r~{}~{}~{}.$ (2)
The $m(r)$ is the parameter mass-function, which is $m_{0}$ in the GR case but
will depend on $r$ for a generalized form, as will be used later. It is
preferable to define the dimensionless coordinate $y=\frac{r}{m_{0}}$ and the
dimensionless mass-function $m(y)=\frac{m(r)}{m_{0}}$ (for simplicity, we use
the same letter $m$ for the mass-function).
Later, we will propose a particular mass-function, namely
$m(y)=\left(1-\frac{b}{6y^{3}}\right)$. for $b=\frac{81}{8}$ the $g_{00}$
component of the metric is zero at $y=\frac{3}{2}$, i.e., it exhibits an event
horizon. In conclusion, the object under study is a black hole with an event
horizon at $\frac{3}{2}m_{0}$. Thus, the advantage in using $b=\frac{81}{8}$
is that there is still an event horizon and considerations can be limited to
the outside region of the black hole.
Using an arbitrary $b$ the Ricci and Einstein tensor components satisfy the
equations
$\displaystyle R^{\mu}_{~{}\nu}$ $\displaystyle=$ $\displaystyle
M^{\mu}_{\nu}~{},~{}G^{\mu}_{~{}\nu}~{}=~{}8\pi T^{m}u_{~{}\nu}~{}~{}~{},$ (3)
with
$\displaystyle\left(M^{\mu}_{~{}\nu}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-\frac{b}{y^{6}}&0&0&0\\\
0&-\frac{b}{y^{6}}&0&0\\\ 0&0&\frac{b}{2y^{6}}&0\\\ 0&0&0&\frac{b}{2y^{6}}\\\
\end{array}\right)$ (8) $\displaystyle 8\pi\left(T^{\mu}_{~{}\nu}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-\frac{b}{2y^{6}}&0&0&0\\\
0&-\frac{2b}{y^{6}}&0&0\\\ 0&0&\frac{b}{y^{6}}&0\\\ 0&0&0&\frac{b}{y^{6}}\\\
\end{array}\right)~{}=~{}8\pi\left(\begin{array}[]{cccc}-\varrho^{\Lambda}&0&0&0\\\
0&p^{\Lambda}_{r}&0&0\\\ 0&0&p^{\Lambda}_{\vartheta}&0\\\
0&0&0&p^{\Lambda}_{\vartheta}\\\ \end{array}\right)~{}~{}~{},$ (17)
where $\varrho^{\Lambda}$ is a dark energy density, $p^{\Lambda}_{r}$ the
radial pressure and $p^{\Lambda}_{\vartheta}$ a tangential pressure. Note that
$b$ is an interaction constant which couples the amount of dark energy
accumulating near the black hole to the central mass of the black hole. Thus,
as the gravitational constant, the $b$ is fixed, not subject to variation. The
$r$-dependence of the dark energy determines the vacuum structure, different
to the one in GR, but still invariable.
Perturbations are introduced in first order contributions for $\delta\omega$,
$\delta q_{2}$, $\delta q_{3}$, $\delta\mu_{2}$, $\delta\mu_{3}$ and
$\delta\psi$. Axial waves (negative parity) are related to the perturbations
of $\delta\omega$, $\delta q_{2}$ and $\delta q_{3}$, while polar waves
(positive parity) are related by the perturbations $\delta\nu$,
$\delta\mu_{2}$, $\delta\mu_{3}$ and $\delta\psi$ chandra . When the variation
is applied to the components of the Ricci and Einstein tensor, considering
that the interaction constant can not be varied because it only depends on the
total central mass which is not changed. As a consequence, we have $\delta
R^{\mu}_{~{}\nu}=0$ and $\delta G^{\mu}_{~{}\nu}=0$, as in the standard GR
chandra .
## 3 Axial modes: Regge-Wheeler equation
To follow this section, please consult the book of S. Chandrasekhar chandra .
The function $\nu(r)$, $\psi(r)$, $\mu_{2}(r)$, $\mu_{3}(r)$, $\omega(r)$,
$q_{2}(r)$ and $q_{3}(r)$ are maintained as general functions in $r$. In
chandra the Riemann and Ricci tensor components are written in terms of these
functions, as are the Einstein tensor components $G^{\mu}_{~{}\nu}$. Only at
the very last the explicit functions are substituted by their expressions in
the Schwarzschild metric, which in this contribution is of the form listed in
(2).
Demanding the invariance of these components under variation, leads to $\delta
R^{1}_{~{}2}=0$ and $\delta R^{1}_{~{}3}=0$ chandra (see also discussion in
Section 2, which in turn results in the equations (using the same notation as
in chandra )
$\displaystyle\left(e^{3\psi+\nu-\mu_{2}-\mu_{3}}Q_{23}\right)_{\mid 3}$
$\displaystyle=$ $\displaystyle-e^{3\psi-\nu+\mu_{3}-\mu_{2}}Q_{02\mid 0}$
$\displaystyle\left(e^{3\psi+\nu-\mu_{2}-\mu_{3}}Q_{23}\right)_{\mid 2}$
$\displaystyle=$ $\displaystyle+e^{3\psi-\nu+\mu_{2}-\mu_{3}}Q_{03\mid
0}~{}~{}~{},$ (18)
where ”$\mid k$” denotes the usual derivative with respect to the variable
$x_{k}$ and the $Q_{ab}$ are defined as
$\displaystyle Q_{ab}$ $\displaystyle=$ $\displaystyle q_{a\mid b}-q_{b\mid
a}~{},~{}Q_{a0}~{}=~{}a_{a\mid 0}-\omega_{\mid a}~{}~{}~{}.$ (19)
With the ansatz
$\displaystyle Q(t,r,\theta)$ $\displaystyle=$ $\displaystyle\Delta Q_{23}{\rm
sin}^{3}\theta~{}=~{}\Delta(q_{2\mid 3}-q_{3\mid 2}){\rm
sin}^{3}\theta~{}~{}~{},$ (20)
using (2), we arrive at the equations
$\displaystyle\frac{1}{r^{4}{\rm sin}~{}3\theta}\frac{\partial
Q}{\partial\theta}$ $\displaystyle=$ $\displaystyle-\left(\omega_{\mid
2}-q_{2\mid 0}\right)_{\mid 0}$ $\displaystyle\frac{\Delta}{r^{4}{\rm
sin}~{}3\theta}\frac{\partial Q}{\partial r}$ $\displaystyle=$
$\displaystyle+\left(\omega_{\mid 3}-q_{3\mid 0}\right)_{\mid 0}~{}~{}~{}.$
(21)
Eliminating $\omega$ and assuming a time dependence of $e^{i\omega t}$, we
arrive at
$\displaystyle r^{4}\frac{\partial}{\partial
r}\left(\frac{\Delta}{r^{4}}\frac{\partial Q}{\partial r}\right)+{\rm
sin}^{3}\theta\frac{\partial}{\partial\theta}\left(\frac{1}{{\rm
sin}^{3}\theta}\frac{\partial
Q}{\partial\theta}\right)+\omega^{2}\frac{r^{4}}{\Delta}Q$ $\displaystyle=$
$\displaystyle 0~{}~{}~{}.$ (22)
With the ansatz
$\displaystyle Q(t,\theta)$ $\displaystyle=$ $\displaystyle
Q(r)C_{l+2}^{-\frac{3}{2}}(\theta)~{}~{}~{},$ (23)
where $C_{l+2}^{-\frac{3}{2}}$ is a Gegenbauer function, we obtain for the
final equation
$\displaystyle\Delta\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\frac{dQ}{dr}\right)-\mu^{2}\frac{\Delta}{r^{4}}Q+\omega^{2}Q$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle{\rm with}$
$\displaystyle\mu^{2}~{}=~{}2n~{}=~{}(l-1)(l+2)$ $\displaystyle~{}~{}~{}.$
(24)
Further, setting
$\displaystyle Q(r)$ $\displaystyle=$ $\displaystyle rZ^{(-)}$ (25)
and defining ($r_{*}$ is the Tortoise coordinate)
$\displaystyle\frac{d}{dr_{*}}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{r^{2}}\frac{d}{dr}~{}~{}~{},$ (26)
we arrive finally at the Regge-Wheeler equation RW
$\displaystyle\left(\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}\right)Z^{(-)}$
$\displaystyle=$ $\displaystyle V^{(-)}Z^{(-)}~{}~{}~{}.$ (27)
The Eq. (26) is the definition of the Tortoise coordinate, which for an $r$
dependent $m$ has not the simple form as the one exposed in the book of
Chandrasekhar chandra and in chandra1975a . Later it will be shown that in
certain cases also analytic solutions may exist.
The potential $V^{(-)}$ is derived in the Appendix A and is given by
$\displaystyle V^{(-)}(r)$ $\displaystyle=$
$\displaystyle\mu^{2}\frac{\Delta}{r^{4}}-\frac{\Delta}{r}\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\right)~{}~{}~{}.$
(28)
Using the definition of $\Delta$ and applying the derivatives leads to
$\displaystyle V^{(-)}(r)$ $\displaystyle=$
$\displaystyle\frac{\Delta}{r^{5}}\left[(\mu^{2}+2)r-6m(r)+2m^{\prime}(r)r\right]~{}~{}~{}.$
(29)
The prime indicates a derivation in $r$ and (29) reduces to the result by
Chandrasekhar chandra ; chandra1975a when the derivative of $m(r)$ is zero,
i.e., when it is constant. The upper index $(-)$ refers to axial (negative
parity) modes.
Thus, the derivation of the equation for the axial modes is in complete
analogy to the derivation presented in chandra . The changes in the formulas
are minimal.
## 4 Polar modes: Zerilli equation
In contrast to the axial modes, for obtaining the differential equation of the
polar modes the procedure is more involved. The final equation will have a
similar form as in (27), however, with a quite complex potential. S.
Chandrasekhar proved chandra that the frequencies of the polar oscillations
are the same as for axial modes, which is one of the reasons in most cases
only the axial modes are calculated. Using the extended mass-function, It will
be shown that axial and polar modes are not equal anymore, though, some
similarities can be conjectured. For that reason, the axial and polar modes
have to be treated separately. Again, we will closely follow the path exposed
in chandra and mainly mention key points, deviations and approximations.
To obtain the Zerilli equation zerilli for the polar modes, we proceed in the
same manner as in chandra , section 24b and of chandra1975a . The variations
$\delta R^{0}_{~{}2}$, $\delta R^{0}_{~{}3}$, $\delta R^{2}_{~{}3}$, $\delta
G^{2}_{~{}2}$ and $\delta R^{1}_{~{}1}$ lead to the identical equations as
given in chandra ($\delta R^{\mu}_{~{}\nu}=0$, $\delta G^{\mu}_{~{}\nu}=0$
and see discussion in Section 2). New functions are introduced, varying $\nu$,
$\mu_{2}$, $\mu_{3}$ and $\psi$ (equations (36)-(39) in chapter 24 of chandra
), restricting to the quadrupole mode ($l=2$) of the multipole expansion:
$\displaystyle\delta\nu$ $\displaystyle=$ $\displaystyle N(r)P_{l}({\rm
cos}\theta)$ $\displaystyle\delta\mu_{2}$ $\displaystyle=$ $\displaystyle
L(r)P_{l}({\rm cos}\theta)$ $\displaystyle\delta\mu_{3}$ $\displaystyle=$
$\displaystyle\left[T(r)P_{l}+V(r)P_{l\mid\theta\mid\theta}\right]$
$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\left[T(r)P_{l}+V(r)P_{l\mid\theta}{\rm
cot}\theta\right]~{}~{}~{}.$ (30)
We are also lead to the relation (equation (43) in chandra )
$\displaystyle T-V+L$ $\displaystyle=$ $\displaystyle 0~{}~{}~{},$ (31)
which reduces the number of linear independent functions by one.
The following steps in chandra will remain the same. One reason is that the
general structure is not affected by the modified metric term $e^{2\nu}$,
e.g., only the function $\nu$ and its derivative $\nu_{\mid r}=\nu^{\prime}$
appear and not their explicit dependence on $m(r)$, it is implicit. In Eq.
(48) of chandra a new function is defined in substitution of $V$, namely
$\displaystyle X$ $\displaystyle=$ $\displaystyle
nV~{}=~{}\frac{1}{2}(l-1)(l+2)V~{}~{}~{}.$ (32)
A relation of the derivatives of these defined functions is given in (52)-(54)
of chandra , which we will repeat here, because the coefficients in these
equations do depend on $m(r)$ and, which is new, its radial derivative
$m^{\prime}(r)$:
$\displaystyle N_{\mid r}$ $\displaystyle=$ $\displaystyle aN+bL+cX$
$\displaystyle L_{\mid r}$ $\displaystyle=$
$\displaystyle\left(a-\frac{1}{r}+\nu_{\mid
r}\right)N+\left(b-\frac{1}{r}-\nu_{\mid r}\right)L+cX$ $\displaystyle X_{\mid
r}$ $\displaystyle=$ $\displaystyle-\left(a-\frac{1}{r}+\nu_{\mid
r}\right)N-\left(b+\frac{1}{r}-2\nu_{\mid
r}\right)L-\left(c+\frac{1}{r}-\nu_{\mid r}\right)X~{}~{}~{}.$
The coefficients and $\nu_{\mid r}$, have now new contributions due to the
mass-function, and are given by
$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{n+1}{r-2m(r)}$
$\displaystyle b$ $\displaystyle=$
$\displaystyle-\frac{1}{r}-\frac{n}{r-2m(r)}+\frac{m(r)}{r(r-2m(r))}+\frac{m^{2}(r)}{r(r-2m(r))^{2}}+\omega^{2}\frac{r^{3}}{(r-2m(r))^{2}}$
$\displaystyle-\frac{m^{\prime}(r)}{r-2m(r)}-\frac{2m(r)m^{\prime}(r)}{(r-2m(r))^{2}}+\frac{r(m^{\prime}(r))^{2}}{(r-2m(r))^{2}}$
$\displaystyle c$ $\displaystyle=$
$\displaystyle-\frac{1}{r}+\frac{1}{r-2m(r)}+\frac{m^{2}(r)}{r(r-2m(r))^{2}}+\omega^{2}\frac{r^{3}}{(r-2m(r))^{2}}$
$\displaystyle+\frac{m^{\prime}(r)\left[-2m(r)+m^{\prime}(r)r\right]}{(r-2m(r))^{2}}$
$\displaystyle\nu_{\mid r}$ $\displaystyle=$
$\displaystyle\frac{m(r)}{r(r-2m(r))}$ (34)
$\displaystyle-\frac{m^{\prime}(r)}{r-2m(r)}~{}~{}~{}.$
In the second row of each factor the new contributions appear, if any,
depending on the derivatives of $m(r)$. The results reduce to the one in
chandra when $m^{\prime}(r)$ is set to zero.
In what follows, we write some of the expressions needed to derive the Zerilli
equations in explicit form, because of new contributions due to the dependence
of $m(r)$ on $r$:
$\displaystyle\left(L+X\right)_{\mid r}$ $\displaystyle=$
$\displaystyle-\left(\frac{2}{r}-\frac{m(r)}{r(r-2m(r))}+\frac{m^{\prime}(r)}{r-2m(r)}\right)L$
$\displaystyle-\left(\frac{1}{r}-\frac{m(r)}{r(r-2m(r))}+\frac{m^{\prime}(r)}{r-2m(r)}\right)X$
$\displaystyle=$
$\displaystyle-\left(\frac{2}{r}-\frac{m(r)}{r(r-2m(r))}+\frac{m^{\prime}(r)}{r-2m(r)}\right)\left(L+X\right)+\frac{n}{r}V$
$\displaystyle=$
$\displaystyle-\frac{1}{r(r-2m(r))}\left\\{\left[2r-5m(r)+rm^{\prime}(r)\right]L\right.$
$\displaystyle\left.+\left[r-3m(r)+rm^{\prime}(r)\right]X\right\\}$
$\displaystyle X_{\mid r}$ $\displaystyle=$
$\displaystyle-\frac{\left[nr+3m(r)-rm^{\prime}(r)\right]}{r(r-2m(r))}N-\frac{(n+1)}{r-2m(r)}X$
$\displaystyle-\left[-\frac{m(r)}{r(r-2m(r))}+\frac{\left(m^{2}(r)+\omega^{2}r^{4}\right)}{r(r-2m(r))^{2}}+\frac{m^{\prime}(r)}{r-2m(r)}\right.$
$\displaystyle\left.-\frac{2m(r)m^{\prime}(r)}{(r-2m(r))^{2}}+\frac{r(m^{\prime}(r))^{2}}{(r-2m(r))^{2}}-\frac{n}{r-2m(r)}\right]\left(L+X\right)$
In order to obtain the Zerilli equation in GR, one defines a function
$Z^{(+)}$ as a particular combination of $N$, $V$, $X$ and $L$ (see (58) and
(59) in chandra ). After that one calculates the first and second derivative
with respect to $r_{*}$, using the above equations of derivatives for $N$, $L$
and $X$.
In chandra ; chandra1975a only the ansatz for $Z^{(+)}$ is presented, without
a derivation. Here, we provide the foundation for this ansatz of the general
Zerilli equation, which includes in the limit of a constant mass-function the
GR.
The combination of $Z^{(+)}$ in terms of the functions $V$, $L$ and $X$ is
chosen such that, when the second order derivative in $r_{*}$ is applied, the
only contributions left is solely proportional to $Z^{(+)}$. Due to the new
contributions in the derivatives of $m(r)$, the ansatz for the linear
combination changes to
$\displaystyle Z^{(+)}$ $\displaystyle=$
$\displaystyle\alpha(r)N+\beta(r)V+\gamma(r)(L+X)~{}~{}~{}.$ (36)
On how the functions $\alpha(r)$, $\beta(r)$ and $\gamma(r)$ are determined is
explained in the Appendix B. Also, the ansatz proposed by S. Chanbrasekhar
chandra will be derived, in the limit of $m(r)=m_{0}$.
In a first step, the first derivative ($\frac{dZ^{(+)}}{dr_{*}}$) and second
derivative ($\frac{d^{2}Z^{(+)}}{dr_{*}^{2}}$) have to be calculated, using
(26). The factors proportional to $\omega^{2}$ are determined and a solution
of $\alpha(r)$ and $\beta(r)$ is found. In a second step a differential
equation for $\gamma(r)$ is set up, where the solution will depend on the
mass-function used. For a constant mass the GR solution of chandra ;
chandra1975a is recovered. This is a rather lengthy, but straightforward,
calculation done in the Appendix B and C. but better done with MATHEMATICA
mat11 ; matdetails .
In the first step, for $\alpha(r)$ and $\beta(r)$ we obtain (see Appendix B)
$\displaystyle\alpha(r)$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\beta(r)$ $\displaystyle=$ $\displaystyle r~{}~{}~{}.$ (37)
In the second step, we take the $\omega^{2}$ independent term, obtained after
having applied $\left[\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}\right]$ to
$Z^{(+)}$, which leads to the
$\displaystyle
V_{N}^{(+)}N(r)+V^{(+)}_{V}rV(r)+V^{(+)}_{LX}\gamma(r)(L(r)+X(r))~{}~{}~{},$
(38)
where $V_{N}^{(+)}$ depends on $\alpha(r)=0$, $\beta(r)=r$ and $\gamma(r)$.
Because $\alpha(r)=0$, the $Z^{(+)}$ is only a combination in $V(r)$ and
$(L(r)+X(r))$, the factor of $N(r)$ has to vanish. This condition leads to
$\displaystyle\gamma(r)$ $\displaystyle=$
$\displaystyle-\frac{r^{2}}{\left(nr+3m(r)-rm^{\prime}(r)\right)}\left(1+\frac{2m^{\prime}(r)-rm^{\prime\prime}(r)}{n}\right)~{}~{}~{}.$
(39)
For the linear combination in $V(r)$ and $(L(r)+X(r))$ to be written as
$V^{(+)}Z^{(+)}$, the two potential factors in (38) have to be equal, which
leads to the condition
$\displaystyle G(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2r^{5}\gamma(r)}\left\\{(-12\gamma(r)^{2}(r-2m(r))(-3m(r)+r(2+m^{\prime}(r)))\right.$
(40) $\displaystyle\left.+2r\gamma(r)(r-2m(r))(m(r)(23-8m^{\prime}(r))\right.$
$\displaystyle\left.+r(-6+4\gamma^{\prime}(r)-11m^{\prime}(r)+rm^{\prime\prime}(r))+r^{2}(8m(r)^{2}(6\gamma^{\prime}(r)-r\gamma^{\prime\prime}(r))\right.$
$\displaystyle\left.+4rm(r)(-1+2m^{\prime}(r)-2\gamma^{\prime}(r)(5+2m^{\prime}(r))\right.$
$\displaystyle\left.+2r\gamma^{\prime\prime}(r)-rm^{\prime\prime}(r))+r^{2}(2(1+4\gamma^{\prime}(r)-2m^{\prime}(r))(1+m^{\prime}(r))\right.$
$\displaystyle\left.-2r\gamma^{\prime\prime}(r)+r(1+2m^{\prime}(r))m^{\prime\prime}(r)))\right\\}=0~{}~{}~{}.$
As noted above, for a constant mass $m(r)=m_{0}$, this equation is identically
fulfilled for the expression given in chandra ; chandra1975a . In Appendix C
we will see that this solution satisfies the condition (40) for a wide range
of $r$, save near $r=\frac{3}{2}m_{0}$, with a small error, though. This shows
that the Zerilli equation can be constructed approximately.
Finally, the Zerilli equation acquires the form
$\displaystyle\left[\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}\right]Z^{(+)}$
$\displaystyle=$ $\displaystyle V^{(+)}Z^{(+)}~{}~{}~{},$ (41)
which describes the polar modes with an $r$ dependent mass-function $m(r)$. As
we will see further below, the axial and polar modes, though different, still
share similar structures.
In terms of the mass-function $m(r)$, using (39) the potential for the polar
modes is given by
$\displaystyle V^{(+)}(r)=$
$\displaystyle((r-2m(r))(18(m(r))^{3}+3r(m(r))^{2}(6n-12m^{\prime}(r)+5rm^{\prime\prime}(r)-$
$\displaystyle
4r^{2}m^{\prime\prime\prime}(r))+r^{3}(2n^{2}(1+n)-8(m^{\prime}(r))^{3}+(m^{\prime}(r))^{2}(8+6n+3rm^{\prime\prime}(r))+$
$\displaystyle
y(m^{\prime\prime}(r)(-n(6+n)+2rm^{\prime\prime}(r))+2nrm^{\prime\prime\prime}(r))$
$\displaystyle-2m^{\prime}(r)(-4n+r((3+n)m^{\prime\prime}(r)+rM^{\prime\prime\prime}(r))))+$
$\displaystyle
2r^{2}m(r)(3n^{2}+9(m^{\prime}(r))^{2}+r(m^{\prime\prime}(r)(-3+5n-2rm^{\prime\prime}(r))$
$\displaystyle+(3-2n)rm^{\prime\prime\prime}(r))+m^{\prime}(r)(-12n+r(m^{\prime\prime}(r)+2rm^{\prime\prime\prime}(r))))))/$
$\displaystyle(r^{4}(3m(r)+r(n-m^{\prime}(r)))^{2})~{}~{}~{}.$ (42)
In what follows, the dimensionless coordinate
$\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{r}{m_{0}}~{}=~{}\frac{y_{eh}}{1-\xi}$ (43)
is used, where $y_{eh}$ is the position of the event horizon and the variable
$\xi$ has the range $[0,1]$. When $\xi=0$, then $y=y_{eh}$ and when $\xi$
tends to 1, the coordinate $y$ tends to $+\infty$. The particular mass-
function used, corresponds to an event horizon at $y_{\rm eh}=\frac{3}{2}$.
The reason for using the coordinate $\xi$ with a compact support lies in the
use of the AIM method, explained further below. It guarantees a better
convergence of an iterative equation.
## 5 Constructing the Regge-Wheeler and Zerilli equations
In this section the final form of the differential equations, used for the
axial and polar modes, will be derived.
In subsection 5.2 the Asymptotic Iteration Method (AIM) is resumed, which
solves a differential equation of second order. In what follows, the Regge-
Wheeler/Zerilli equation is rewritten in a form, which is practical for the
AIM.
For the function $m(r)$ the particular form, defining $y=\frac{r}{m_{0}}$,
$\displaystyle m(r)$ $\displaystyle=$ $\displaystyle m_{0}m(y)$ $\displaystyle
m(y)$ $\displaystyle=$ $\displaystyle 1-\frac{27}{32y^{3}}~{}~{}~{},$ (44)
is used, which exhibits an event horizon at $y=\frac{3}{2}$.
### 5.1 Rewriting the differential equation
First, the explicit form of the differential equation is derived, noting that
the Regge-Wheeler and Zerilli equation can be, in general, written as
$\displaystyle\left[\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}-V^{(\pm)}(r)\right]Z^{(\pm)}$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}.$ (45)
Let us use, for a moment, the more general mass-function
$m(y)=\left(1-\frac{b}{12y^{3}}\right)$, which for $b=\frac{81}{8}$ acquires
the one of (44). It was used in universe for the study of phase transitions
from GR to pc-GR. The relation of $y=\frac{r}{m_{0}}$ to a variable $\xi$ with
a compact support for the range of integration is defined as in (43), where
$y_{eh}(b)$ is the position of the event horizon as a function of the
parameter $b$. It is the solution of the condition for the event horizon in
the Schwarzschild case, i.e.,
$\displaystyle y^{4}-2y^{3}+\frac{b}{6}$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}.$ (46)
Using the Wolfram MATHEMATICA code mat11 ; matdetails , the solution is
$\displaystyle y_{eh}(b)$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{2\sqrt{3}}\left[\left(3+\frac{2b}{\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}}+\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}\right)\right]^{\frac{1}{2}}$
(47)
$\displaystyle+\frac{1}{2}\left[\left(2-\frac{2b}{3\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}}-\frac{1}{3}\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}\right.\right.$
$\displaystyle\left.\left.+\frac{2\sqrt{3}}{\left(\left[\left(3+\frac{2b}{\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}}+\left(9b+\sqrt{81b^{2}-8b^{3}}\right)^{\frac{1}{3}}\right)\right]^{\frac{1}{2}}\right)}\right)\right]^{\frac{1}{2}}~{}~{}~{}.$
The second order derivative with respect to the Tortoise coordinate is
$\displaystyle m_{0}^{2}\frac{d^{2}}{dr_{*}^{2}}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{r^{2}}\frac{d}{dr}\frac{\Delta}{r^{2}}\frac{d}{dr}$
$\displaystyle=$
$\displaystyle\left(1-\frac{2}{y}m(y)\right)^{2}\frac{d^{2}}{dy^{2}}+\left(1-\frac{2}{y}m(y)\right)\frac{2}{y^{2}}\left(m(y)-ym^{\prime}(y)\right)\frac{d}{dy}~{}~{}~{}.$
The prime refers to a derivative in $y$.
Defining the dimensionless expressions
$\displaystyle{\tilde{\omega}}$ $\displaystyle=$ $\displaystyle
m_{0}\omega~{},~{}{\tilde{V}}(y)~{}=~{}m_{0}^{2}V(r)~{}~{}~{},$ (49)
the differential equation acquires the form
$\displaystyle\left[\left(1-\frac{2}{y}m(y)\right)^{2}\frac{d^{2}}{dy^{2}}+\right.$
$\displaystyle\left.\left(1-\frac{2}{y}m(y)\right)\frac{2}{y^{2}}\left(m(y)-ym^{\prime}(y)\right)\frac{d}{dy}+{\tilde{\omega}}^{2}-{\tilde{V}}^{(\pm)}(y)\right]Z(y)$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}.$ (50)
The next step is to substitute $y$ by $\xi$. We also define $m(\xi)$ as
$m(y(\xi))$ and $m^{\prime}(\xi)$ as the same as $m^{\prime}(y)$ (in order to
avoid more definitions of functions, we use the same letter $m$). Doing so,
leads after some manipulations to
$\displaystyle\left[\frac{d^{2}}{d\xi^{2}}-\frac{2}{(1-\xi)}\left(1-\frac{(1-\xi)}{y_{eh}(b)}\frac{\left(m(\xi)-(y_{eh}/(1-\xi))m^{\prime}(\xi)\right)}{\left(1-\frac{2(1-\xi)}{y_{eh}(b)}m(\xi)\right)}\right)\frac{d}{d\xi}\right.$
$\displaystyle\left.\frac{(y_{eh}(b))^{2}}{(1-\xi)^{4}}\frac{\left({\tilde{\omega}}^{2}-{\tilde{V}}^{(\pm)}(\xi)\right)}{\left(1-\frac{2(1-\xi)}{y_{eh}(b)}m(\xi)\right)^{2}}\right]Z(\xi)$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}.$
Remember that the prime refers to the derivative with respect to $y$ and notto
$\xi$.) Note also, that $V^{(\pm)}(r)=V^{(\pm)}(y(\xi))/m_{0}^{2}$, where
$V^{(\pm)}(y(\xi))$ is dimensionless.
In practical calculations the function (44) is used, which results in the
function $m(\xi)$
$\displaystyle m(\xi)$ $\displaystyle=$ $\displaystyle
1-\frac{(1-\xi)^{3}}{4}~{}~{}~{}.$ (52)
For the potentials $V^{(\pm)}$ in terms of $\xi$, we obtain, the expressions
enlisted in matdetails .
### 5.2 AIM
The AIM was introduced by H. Ciftci, R. L. Hall, and N. Saad ciftci2003 ;
ciftci2005 , for solving second order differential equations of the form
cho2012
$\displaystyle f^{\prime\prime}(\xi)$ $\displaystyle=$
$\displaystyle\lambda_{0}(\xi)f^{\prime}(\xi)+s_{0}(\xi)f(\xi)~{}~{}~{},$ (53)
with $\xi$ as the variable.
Deriving both sides $p$ times, $p$ being an integer, leads to an equivalent
differential equation
$\displaystyle f^{(p+1)}(\xi)$ $\displaystyle=$
$\displaystyle\lambda_{p-1}(\xi)f^{\prime}(\xi)+s_{p-1}(\xi)f(\xi)~{}~{}~{},$
(54)
with
$\displaystyle\lambda_{p}(\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\prime}_{p-1}(\xi)+s_{p-1}(\xi)+\lambda_{0}(\xi)\lambda_{p-1}(\xi)$
$\displaystyle s_{p}(\xi)$ $\displaystyle=$ $\displaystyle
s^{\prime}_{p-1}(\xi)+s_{0}(\xi)\lambda_{p-1}(\xi)~{}~{}~{}.$ (55)
Convergence is achieved, when the ratio of $s_{p}(x)$ and $\lambda_{p}(x)$
does not change, with $p$ the iteration number. Once achieved, the
Quantization Condition reads
$\displaystyle s_{p}(\xi)\lambda_{p-1}(\xi)-s_{p-1}(\xi)\lambda_{p}(\xi)$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}.$ (56)
This expression depends on $\xi$, which can be chosen arbitrarily, and has to
be resolved for the frequencies $\omega$. Because of its $\xi$-dependence, it
is a very subtle task to obtain convergence rapidly. The problem was resolved
partially in cho2012 , expanding the $\lambda_{p}$ and $s_{p}$ in a Taylor
series around a point $\xi$, defining
$\displaystyle\lambda_{p}(\rho)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}c_{p}^{i}(\xi-\rho)^{i}$ $\displaystyle
s_{p}(\rho)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}d_{p}^{i}(\xi-\rho)^{i}~{}~{}~{}.$ (57)
Substituting this into (55) leads to a new recursion relation for the
coefficients $c_{p}^{i}$ and $d_{p}^{i}$ and a new quantization condition
$\displaystyle d_{p}^{0}c_{p-1}^{0}-d_{p-1}^{0}c_{p}^{0}$ $\displaystyle=$
$\displaystyle 0~{}~{}~{}.$ (58)
The clear advantage of (58) lies in the fact that (58) depends only on the
frequencies and is a polynomial in the frequencies. Thus, the determination of
the $\omega$-spectrum is restricted to solve (58). However, the result still
depends on the point of expansion $\xi$ in the Taylor series (57).
In order to obtain a ”quick” convergence, the following rules should be
observed, which are the results of the experience of others cho2012 and ours:
* •
A compact support for the range of the coordinate should be used, i.e. the
coordinate $\xi$, which is zero at the event horizon and approaches 1 for
$r\rightarrow\infty$. In a Taylor expansion, this prevents too large
deviations to the real potential function at the limits $\xi=0$ and $\xi=1$.
I.e., it is important to describe the potential well near the limits.
* •
The asymptotic behavior for $r_{*}\rightarrow\pm\infty$ of the wave function
should be extracted as exactly as possible. Analytic solutions are of course
the best.
* •
For the expansion around a point $\xi$, the maximum or minimum of the
potential is recommended as a starting point. However, shifting it to its
vicinity at larger values may give better convergence. The following criterion
helps, namely that with an increasing number of iterations the lower
frequencies are not changing any more for low values of $-\omega_{I}$.
* •
Using MATHEMATICA, only rational numbers are allowed. In case of irrational
numbers, it is recommended to approximate them by rational ones, otherwise
MATHEMATICA develops numerical instabilities.
## 6 Spectrum of the Regge-Wheeler equation (axial modes) and the Zerilli
equations (polar modes)
The Regge-Wheeler equation for the axial modes and Zerilli equations for the
polar modes are solved, with the help of the AIM. In a first step the
asymptotic limit is discussed and taken into account in the definition of the
$Z^{(\pm)}$-functions, which leads to the final form of the differential
equation to solve.
We will present calculations with different iteration numbers, which allows to
judge the convergence of the iteration method and see trends for large
iteration numbers.
### 6.1 The asymptotic limit
The wave solution must satisfy the condition
$\displaystyle\Psi$ $\displaystyle\rightarrow$
$\displaystyle\left\\{\begin{array}[]{c}e^{+i\omega r_{*}}~{},~{}{\rm
for}~{}r_{*}\rightarrow+\infty~{}(r\rightarrow\infty)\\\ e^{-i\omega
r_{*}}~{},~{}{\rm
for}~{}r_{*}\rightarrow-\infty~{}(r\rightarrow\frac{3}{2}m_{0})\end{array}\right.~{}~{}~{}.$
(61)
The time dependence for both limits is $e^{-i\omega t}$. This implies that for
a complex $\omega=\omega_{R}+i\omega_{I}$, the time dependence has the form
$\displaystyle e^{-i\omega t}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{R}t}e^{+\omega_{I}t}~{}~{}~{},$ (62)
i.e., for an exponential decreasing function the imaginary part of the
frequency ($\omega_{I}$) has to be negative ($\omega_{I}<0$), otherwise, there
is no damping, i.e., no stable mode. The Schwarzschild solution is stable
under the perturbations when no positive $\omega_{I}$ solution appears.
The integrated relation of the Tortoise coordinate, defined in (26),
$y_{*}=\frac{r_{*}}{m_{0}}$ to the variable $y=\frac{r}{m_{0}}$ is given by
$\displaystyle y_{*}$ $\displaystyle=$
$\displaystyle\int\frac{dy}{\left(1-\frac{2m(y)}{y}\right)}~{}~{}~{}.$ (63)
For $m(y)=1$, i.e. GR, the solution is well known, namely
$\displaystyle y_{*}$ $\displaystyle=$ $\displaystyle y+2{\rm
ln}\left(\frac{y}{2}-1\right)~{}~{}~{},$ (64)
For $m(y)=\left(1-\frac{27}{32y^{3}}\right)$, there is surprisingly also a
solution
$\displaystyle y_{*}$ $\displaystyle=$ $\displaystyle y+2{\rm
ln}\left(\frac{y}{\frac{3}{2}}-1\right)$ (65) $\displaystyle+2{\rm
ln}\left(3\right)-\frac{3}{2}+\frac{9}{(12-8y)}-\frac{{\rm
arctan}\left[\frac{1+2y}{\sqrt{2}}\right]}{4\sqrt{2}}~{}~{}~{},$
which was obtained, using MATHEMATICA mat11 . When some of the parameters in
$m(y)$ are changed, no solution can be found. That only this particular ansatz
of $m(r)$ provides an analytic solution for the Tortoise coordinate and not
any other with a different parametrization, is quite a surprise which we would
like to understand.
Changing to the variable $\xi=1-\frac{3}{2y}$, the relation of the Tortoise
coordinate to $\xi$ is
$\displaystyle y_{*}$ $\displaystyle=$ $\displaystyle\frac{3}{2(1-\xi)}+2{\rm
ln}(\frac{\xi}{1-\xi})$ (66) $\displaystyle+2{\rm
ln}(3)-\frac{3}{2}-\frac{3(1-\xi)}{4\xi}-\frac{\rm{arctan}\left[\frac{4-\xi}{\sqrt{2}(1-\xi)}\right]}{4\sqrt{2}}~{}~{}~{}.$
#### 6.1.1 Limit $r_{*}\rightarrow+\infty$
We again define $y=\frac{r}{m_{0}}$ ($y_{*}=\frac{r_{*}}{m_{0}}$) and
substitute $y_{*}$ by (66). We also define
$\displaystyle{\tilde{\omega}}$ $\displaystyle=$ $\displaystyle
m_{0}\omega~{}~{}~{}.$ (67)
For $y_{*}\rightarrow+\infty$, the $\xi$ tends to 1, i.e, terms proportional
to $\frac{1}{(1-\xi)}$ and ${\rm ln}(1-\xi)$ dominate. This leads to the
asymptotic form ($e^{i\omega r}=e^{i\widetilde{\omega}y_{*}}$)
$\displaystyle e^{+i\widetilde{\omega}y_{*}}$ $\displaystyle\rightarrow$
$\displaystyle
e^{i\frac{3{\tilde{\omega}}}{2(1-\xi)}}(1-\xi)^{-2i{\tilde{\omega}}}~{}~{}~{}.$
(68)
The other terms in (66) can be neglected, because they are either constant or
approach a constant value in this limit.
#### 6.1.2 Limit $r_{*}\rightarrow-\infty$
In this case, the $y$ approaches the event horizon.
Using the above obtained expression for $y_{*}$, we obtain the additional
terms, taking into account that terms proportional to $\frac{1}{\xi}$ and
${\rm ln}(\xi)$ dominate, namely
$\displaystyle
e^{i\frac{3\widetilde{\omega}(1-\xi)}{4\xi}}\xi^{-2i\widetilde{\omega}}~{}~{}~{}.$
(69)
The $e^{i\frac{3}{2}}$ and $e^{-i\widetilde{\omega}\frac{{\rm
arctan}\left[\frac{(2+\xi)}{\sqrt{2}(1-\xi)}\right]}{4\sqrt{2}}}$
contributions are skipped because the first is a constant and the second has
the limit $e^{-i\widetilde{\omega}{\rm arctan}\sqrt{2}}$ for $\xi=0$, thus, it
does not effect the asymptotic limit.
#### 6.1.3 Final ansatz of the asymptotic behavior
Thus, extracting the complete asymptotic limits, the new ansatz for the wave
function is
$\displaystyle Z^{(\pm)}(\xi)$ $\displaystyle=$ $\displaystyle
e^{i\frac{3{\tilde{\omega}}}{2(1-\xi)}}(1-\xi)^{-2i{\tilde{\omega}}}e^{i\frac{3\widetilde{\omega}(1-\xi)}{4\xi}}\xi^{-2i\widetilde{\omega}}P^{(\pm)}(\xi)~{}~{}~{},$
(70)
with the new wave-function $P^{(\pm)}(\xi)$, whose differential equations are
set up and solved with the help of the AIM matdetails .
### 6.2 Axial and polar potential
In Fig. 1 the potentials obtained for the axial and polar modes are depicted.
In the first row the potential for the axial modes in the GR-case
($m(r)=m_{0}$) is shown. Because in GR, the axial and polar modes are equal,
it is sufficient to plot only this potential. In the lower row, left panel,
the potential $V^{(-)}$ in pc-GR is depicted and in the right panel the
potential $V^{(+)}$ for the polar modes in pc-GR. As noted, all these
potentials have similar characteristics, leading to the conjecture that the
frequency spectrum of axial and polar modes still may share some common
structure. However, as we will see, the axial and polar modes are not
isospectral, which is a surprise, considering that in GR they are, What the
deep origin of this difference is cannot be eaxplained at this moment.
Isopectrality is related to supersymmetric transformations of a Schrödinger
type of potential cooper1995 but with non-bound states. Isospectrality
between the axial and polar modes for a more general type of metrics (de
Sitter and Anti-de Sitter) was proven in moulin2020 . A similar procedure we
plan to apply for the present study, in order to see which potentials give
rise to isospectrality.
Figure 1: First row: Potential for the axial modes in GR. Second row: The
potential for the axial modes ($V^{(-)}$, left panel) and the polar modes
($V^{(+)}$, right panel).
### 6.3 Spectrum of the Regge-Wheeler equation: axial modes
Using the AIM, we obtained the spectrum for 198 (red dots) and 200 iterations
(blue dots), plotted in Fig. 2, which compares GR (upper row) with pc-GR
(lower row). While convergence is observed for the low damping modes
($\mid\widetilde{\omega}_{I}\mid$ small), there is still no convergence
obtained for the high damping modes. In Figure 3 the axial modes are depicted
for 400 iterations. Above, the GR (left panel) is compared to pc-GR (right
panel) for a large range of $-\widetilde{\omega}_{I}$. In the lower row the
same is plotted but for a restricted range of $-\widetilde{\omega}_{I}$. The
left panel (GR) reproduces the Figure 2 in kokkotash and of Figure 5 in
konoplya2011 , where also distinct methods to resolve the differential
equation are resumed.
Comparing GR with pc-GR, the structure shares still some common features,
namely that the figure reminds at a fish with its head to the right and its
tail to the left. However, the head is moving further to the left when the
number of iterations is increased, thus, it is not a physical property and has
to be rejected. The left panel in Fig. 3 shows the result for GR and the right
one for pc-GR. Very interesting is, that the structure of a raising branch for
pc-GR from low to large damping modes is also stable. This branch can be
approximated by continuum and represents a definite feature of pc-GR, which is
not present in GR.
At low damping, convergence is obtained and the frequencies in pc-GR are
comparable in size to GR. This changes for the large damping modes, where the
frequencies are significantly larger in pc-GR than in GR.
Figure 2: Spectrum of the axial modes for 200 iterations. The horizontal axis
corresponds to minus the imaginary part of the frequency, which has to be
positive in order to represent a damped, stable oscillation. No negative
values are present, rendering the system stable. The vertical axis depicts the
real part of the frequency. The first row shows the axial modes in GR, while
the second row depicts the frequencies in pc-GR. The left panel shows the
frequency distribution for a larger range of $-\widetilde{\omega}_{I}$ while
the right panel is restricted to small damping modes.
Figure 3: Spectrum of the axial modes for 400 iterations. The horizontal axis
corresponds to minus the imaginary part of the frequency and the vertical axis
to the real part of the complex frequency. In each row, the left panel is for
GR and the right one for pc-GR. Note, that compared to 200 iterations, the
onset of the ”head” is shifted far to the right in $-\widetilde{\omega}_{I}$,
showing that this part is of no physical significance. For more details, in
the lower row a zoom to a restricted range is shown.
### 6.4 Spectrum of the Zerilli equation: polar modes
In Figure 4 the polar frequency modes in pc-GR are depicted. The structure is
similar to the one for the axial modes (see (3)), as expected when comparing
the two potentials. In Figure 4 the red dots correspond to 200 iterations, the
green dots to 300 and the blue dots to 400 iterations. Note, that convergence
is clearly obtained for low values of $-\widetilde{\omega}_{I}$ and up to 30
the convergence is also acceptable. Thus the feature of a raising curve for
large damping is confirmed. The ”fish head”, however, has moved further to the
right, showing its unphysical nature.
Figure 4: Spectrum of the polar modes for 200 (red dots), 300 (green dots)
and 400 (blue dots) iterations. The horizontal axis corresponds to minus the
imaginary part of the frequency, which has to be positive for a damped
oscillation. No negative values are present, rendering the system stable. The
vertical axis depicts the real part of the frequency. The left panel shows the
frequency distribution for a larger range of $-\widetilde{\omega}_{I}$ while
the right panel is restricted to small damping modes. Note, how the ”fish-
head” moves further to the right when the iteration number is increased.
Figure 5: Comparison of axial to polar modes in pc-GR (left and right panel,
respectively), depicting 300 (red dots), 400 (green dots) and 500 (blue dots)
iterations. The upper row shows a large range in $-\widetilde{\omega}_{I}$ and
the lower row is a zoom to the lowest region. Note the structural similarity
of the axial to polar modes, suggesting an equivalence, though not perfect, as
in GR.
In Fig. 5 a comparison of axial to polar modes within pc-GR is shown. In the
upper row, with a wide range of $-\widetilde{\omega}_{I}$, the structure seems
to be similar up to large values of $-\widetilde{\omega}_{I}$. In the lower
row a Zoom to small values of $-\widetilde{\omega}_{I}$ is depicted. The
$\widetilde{\omega}_{R}$ polar modes are in general larger in pc-GR than in
GR.
That there is a branch in the frequency spectrum which has larger frequencies
$\widetilde{\omega}_{R}$ in pc-GR than in GR is of importance: The real part
of the frequency is given by $\widetilde{\omega}_{R}$ = $m_{0}\omega_{R}$.
Using the frequency $\nu=250$Hz and transforming it to units in km-1, one
obtains an $\omega_{R}=5.24~{}10^{-3}$km-1. In the first gravitational event
observed a mass of the united system of about $m_{0}=60$ solar masses was
reported. This gives a value of $\widetilde{\omega}_{R}=0.47$. This is the
kind of order we also obtain. However, he mass $m_{0}$ was obtained recurring
to the GR, i.e., it is theory based. When in a distinct theory a larger real
frequency $\widetilde{\omega}_{R}$ is obtained, keeping $\omega_{R}$ the same,
a larger mass is deduced. This also implies a larger release in energy and a
larger deduced luminous distance, as suggested in hess-2016 .
Where the observed distribution of frequencies lies, is a matter of the
dynamics of a black hole merger. A usual assumption is that all modes can in
principle be excited, but only the low damping modes survive. In this
scenario, no difference between axial and polar modes are expected and the
results will be very similar to GR, However, when the dynamics permits to
excite principally large damping modes, there might be some hope to
distinguish the theories and clarification can come from explicit numerical
studies. In case, large damping modes are excited only, one way to detect a
difference is to search, for case of large damping modes, for simultaneous
light events in the same region of the sky where the merger is observed. If
consistently this light event is at larger distances than the deduced event,
using GR, then it will be in favor of the existence of additional terms in the
metric. This depends also on the requirement that a light event is produced,
requiring some mass distribution near to the event, as an accretions disc.
## 7 Conclusions
Axial and polar modes where calculated over a wide range of damping, within
the General Relativity (GR) and possible extensions, involving a parametric
mass-function $m(r)$, whose leading term correction is proportional to
$1/r^{4}$. In particular the pseudo-complex General Relativity (pc-GR), leads
to such a particular extension of the parameter mass-function. This mass-
function includes a coupling constant of the central mass to the dark energy
and it was chosen such that still an event-horizon exists, resulting in an
easier treatment of the QNM, analog to the one in standard GR. The Regge-
Wheeler equation for the axial modes and the Zerilli equation for the polar
modes were derived, with their corresponding potentials. After having
constructed $\gamma(r)$, it was shown that axial and polar modes, though
different, still share some common features. Isospectrality is not maintained,
a feature we still would like to understand and we refer to future work in
progress. The modes were found to be stable, implying that the corrections to
the metric lead to consistent results.
Adding a further $r$-dependence to the mass-function leads to a branch of
frequencies at high damping, resulting in larger deduced masses than in GR,
while for low damping no large differences are observed.
Assuming that all frequencies are excited, only the low damping modes survive,
resulting in no detectable differences between pcGR and GR. However, when the
frequencies distribution in a merger lies in the large damping region, the
deduced masses in the extended version are larger, implying also a larger
distance to a gravitational wave event. If one detects at the same time of
this event a light emission, the two observations result in a different
distance using GR.
The present results serve as a starting point to understand the changes
involved in the frequency distribution of the ring down modes in extending the
theory of GR, which results in a parametric mass-function $m(r)$ in the metric
components.
In the Appendices explicit derivation of the ansatz for $Z^{(+)}$ is given,
which includes the one proposed by S. Chandrasekhar in chandra ; chandra1975a
.
In a future publication, we will address the pc-Kerr metric, with however more
involved equations. This will pose a problem to the numerical method used.
## Acknowledgments
Financial support from DGAPA-PAPIIT (IN100421 and IN114821) is acknowledged.
## Appendix A: Axial potential
Using $Q=rZ^{(-)}$ gives for the first term in (24)
$\displaystyle\Delta\frac{d}{dr}\frac{\Delta}{r^{4}}\frac{dQ}{dr}~{}=~{}\Delta\frac{d}{dr}\frac{\Delta}{r^{4}}\left[r\frac{d}{dr}Z^{(-)}+Z^{(-)}\right]$
$\displaystyle~{}=~{}\Delta\frac{d}{dr}\left[\frac{\Delta}{r^{3}}\frac{dZ^{(-)}}{dr}+\frac{\Delta}{r^{4}}Z^{(-)}\right]$
$\displaystyle~{}=~{}\Delta\left[\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\right)Z^{(-)}+\frac{\Delta}{r^{4}}\frac{dZ^{(-)}}{dr}\right]+\Delta\left[\frac{1}{r}\frac{\Delta}{r^{2}}\frac{dZ^{(-1)}}{dr}\right]$
$\displaystyle~{}=~{}\Delta\left[\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\right)Z^{(-)}+\frac{\Delta}{r^{4}}\frac{dZ^{(-)}}{dr}\right]-\left(\frac{\Delta}{r^{2}}\right)^{2}\frac{dZ^{(-)}}{dr}+\frac{\Delta}{r}\frac{d}{dr}\frac{\Delta}{r^{2}}\frac{dZ^{(-)}}{dr}~{}~{}~{}.$
(71)
Using that $\frac{\Delta}{r^{2}}\frac{d}{dr}=\frac{d}{dr_{*}}$, we arrive
finally at
$\displaystyle
r\frac{d^{2}Z^{(-)}}{dr_{*}^{2}}+\Delta\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\right)Z^{(-)}~{}~{}~{}.$
(72)
This has to be substituted into the differential equation (24), leading to
$\displaystyle
r\frac{d^{2}}{dr_{*}^{2}}Z^{(-)}+\Delta\frac{d}{dr}\left(\frac{\Delta}{r^{4}}\right)Z^{(-)}-\mu^{2}\frac{\Delta}{r^{3}}Z^{(-)}+r\omega^{2}Z^{(-)}$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}.$ (73)
Dividing by $r$ and reordering the terms in this differential equation, leads
to (27), with the potential given in (29).
## Appendix B: Ansatz for the polar mode wave function
The MATHEMATICA code, used to derive the equations in this section, can be
retrieved from matdetails .
The general ansatz (36) is used, also valid for a constant mass-function, for
which the expression in chandra ; chandra1975b is recovered.
Using (36), namely
$\displaystyle Z^{(+)}(r)$ $\displaystyle=$
$\displaystyle\alpha(r)N(r)+\beta(r)V(r)+\gamma(r)LX(r)$ (74)
and applying the operator $\frac{d^{2}}{dr_{*}^{2}}$ to $Z^{(+)}(r)$, we found
matdetails that the factor of $N(r)$ does not depend on the frequency squared
$\omega^{2}$, only the factors of $V(r)$ and $LX(r)$ do. We use the
definitions
$\displaystyle LX(r)$ $\displaystyle=$ $\displaystyle L(r)+X(r)$
$\displaystyle X(r)$ $\displaystyle=$ $\displaystyle nV(r)~{}~{}~{}.$ (75)
Concentrating only on the component proportional to $\omega^{2}$, one should
obtain for the factor of $V(r)$ and $LX(r)$ the result $-\omega^{2}$
$\left(\beta(r)V(r)+\gamma(r)LX(r)\right)$. The factor obtained after the
application of $\frac{d^{2}}{dr_{*}^{2}}$ onto $V(r)$ is
$\left(n\alpha(r)-\beta(r)\right)$, which must be equated to $-\beta(r)$. This
demands
$\displaystyle\alpha(r)=0~{}~{}~{}.$ (76)
This automatically reduces (74) to $Z^{(+)}(r)=\beta(r)V(r)+\gamma(r)LX(r)$.
For the factor of $LX(r)$, restricting to the one proportional to
$\omega^{2}$, and using (76) we obtain
$-2\beta(r)+n\gamma(r)+2r\beta^{\prime}(r)$ = $n\gamma(r)$ (the prime refers
to the derivative in $r$), which leads to the differential equation
$\displaystyle\beta^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\beta~{}~{}~{},$ (77)
with the solution
$\displaystyle\beta(r)$ $\displaystyle=$ $\displaystyle r~{}~{}~{}.$ (78)
The $\gamma(r)$ function was determined earlier and is given in (39) As seen
in Appendix C, this solution satisfies the equation $G(r)=0$ (40) in a wide
range of $r$, save near the point $r=\frac{3}{2}m_{0}$, implying that the
Zerilli equation can be approximately constructed, with a quite small error.
## Appendix C: Comparison of $V^{(+)}_{V}$ with $V^{(+)}_{LX}$, using (39) for
$\gamma(r)$
Figure 6: The left hand side depicts the two functions $V(r)$ (green online)
and $VLX(r)$ (red online) in a wide range of $r$. The right panel shows the
difference $\left(V(r)-VLX(r)\right)$ in a limited region in r.
In this appendix we analyze the functions $V^{(+)}(r)$, $V^{(+)}(r)$ and their
difference, which is proportional to the condition (40).
On the left panel of Fig. 6 the two potentials $V_{V}^{(+)}$ and
$V_{LX}^{(+)}$ are compared for a wide range of $r$. The potentials were
extended to $r<\frac{3}{2}$ in order to appreciate the agreement of both
potentials, using the $\gamma(r)$ as given in (39). The two functions agree
very well in a wide range of $r$. The only difference appears near
$r=\frac{3}{2}$, which is shown in a reduced scale on the right hand side of
the figure, where, the difference $G(r)$ =
$\left(V_{V}^{(+)}-V_{LX}^{(+)}\right)$ is depicted. All in all, the
expression for $\gamma(r)$ works very well, however with the grain of salt
that the Zerilli equation is not identically satisfied.
## References
* (1) C. M. Will, The confrontation between general relativity and experiment, Living Rev. Relativ. 9, 3 (2006).
* (2) M. Maggiore, Gravitational Waves , Oxford University Press, Oxford, (2008).
* (3) Abbott B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).
* (4) Abbott B. P. et al. (LIGO Scientific Collaboration and The Virgo Collaboration), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116, 241103 (2016).
* (5) R. A. Hulse, J. H. Taylor, A high sensitivity pulsar survey, ApJ 191, L59 (1974).
* (6) R. Adler, M. Bazin and M. Schiffer, Introduction to General Relativity, McGraw-Hill, New York, (1975).
* (7) C. W. Misner, K. S, Thorne and J. A. Wheeler. Gravitation, H. W. Freeman and Company, San Francisco, (1973).
* (8) P. O. Hess, The black hole merger event GW150914 within a modified theory of general relativity, MNRAS 462, 3026 (2016).
* (9) L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, Oxford, UK, (2013)
* (10) S. Chandrasekhar, The mathematical Theory of Black Holes, International Series of Monographs on Physics 69, Clarendon Press, Oxford, (1983).
* (11) S. Chandraskhar, On the Equations Governing the Perturbations of the Schwarzschild Black Hole, Proc. Roy. Soc. London, Series A, Math. and Phys. Sciences 343, 289 (1975).
* (12) S. Chandrasekhar and S. Detweiler, The Quasi-Normal Modes of the Schwarzschild Black Hole, Proc. Roy. Soc. London, Series A, Math. and Phys. Sciences 344, 441 (1975).
* (13) F. J. Zerilli, Effective potential for even-parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24, 737 (1970).
* (14) T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
* (15) H. Ciftci, R. L. Hall and N. Saad, N., Asymptotic iteration method for eigenvalue problems, J. Phys. A 36, 11807 (2003).
* (16) H. Ciftci, R. L. Hall and N. Saad, Perturbation theory from an iteration method, Phys. Lett. A 340, 388 (2005).
* (17) H. Cho, A. S. Cornell, J. Doukas, T.-R. Huang and W. Naylor, A New Approach to Black Hole Quasinormal Modes: A Review of the Asymptotic Iteration Method, Adv. in Math. Phys. 2012, 281705 (2012).
* (18) P. O.Hess, W. Greiner, Pseudo-complex General Relativity, Int. J. Mod. Phys. E 18, 51 (2009).
* (19) Hess P. O., Schäfer M., Greiner W., Pseudo-Complex General Relativity, Springer, Heidelberg, Germany, (2015).
* (20) T. Schönenbach, G. Caspar, P.O. Hess, T. Boller, A. Müller, W. Greiner, MNRAS 430, 2999 (2013)
* (21) T. Schönenbach, G. Caspar, P.O. Hess, T. Boller, A. Müller, W. Greiner, MNRAS 442, 121–130 (2014).
* (22) P. O. Hess, Adv. in High Energy Phys., Review on the pseudo-complex General Relativity and dark energy, 2019, 1840360 (2019).
* (23) P. O. Hess, E. López-Moreno, Kerr Black Holes within a Modified Theory of Gravity, Universe 5, 191 (2019).
* (24) P. O. Hess, Progr. Part. and Nucl. Phys., Alternatives to Einstein’s General Relativity Theory, 114, 103809 (2020).
* (25) MATHEMATICA 11.3.0.0; Wolfram Research Foundation: Champaign, IL, USA, 2018.
* (26) https://github.com/peterottohess/gravitationalwavesSchwarzschild/
* (27) K. D. Kokkotash and B. G. Schmidt, Quasi-normal modes of stars and black holes, Living Reviews in Relativity 2, article no. 2 (1999).
* (28) R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83, 793 (2011).
* (29) A. Nielsen, O. Birnholz, Testing pseudo-complex General Relativity with gravitational waves, AN 339, 298 (2018).
* (30) A. Nielsen, O. Birnholz, Gravitational wave bounds on dirty black holes, AN 340, 116 (2019).
* (31) F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and Quantum Mechanics, Phys. Rep. 251, 267 (1995).
* (32) F. Moulin and A. Barrau, Analytical proof of the isospectrality of quasimnormal modes for Schwarzschild-de Sitter-Anti-de Sitter specetimes, Gen. Rel. and Grav. 52, 82 (2020).
|
# Statistics for $S_{n}$ acting on $k$-sets
Nick Gill Department of Mathematics, University of South Wales, Treforest,
CF37 1DL, U.K<EMAIL_ADDRESS>and Bianca Lodà Department of
Mathematics, University of South Wales, Treforest, CF37 1DL, U.K.
<EMAIL_ADDRESS>
###### Abstract.
We study the natural action of $S_{n}$ on the set of $k$-subsets of the set
$\\{1,\dots,n\\}$ when $1\leq k\leq\frac{n}{2}$. For this action we calculate
the maximum size of a minimal base, the height and the maximum length of an
irredundant base.
Here a _base_ is a set with trivial pointwise stabilizer, the _height_ is the
maximum size of a subset with the property that its pointwise stabilizer is
not equal to the pointwise stabilizer of any proper subset, and an
_irredundant base_ can be thought of as a chain of (pointwise) set-stabilizers
for which all containments are proper.
###### Key words and phrases:
permutation group; height of a permutation group; relational complexity; base
size
## 1\. Introduction
In this note we study three statistics pertaining to primitive permutation
groups. Our main theorem gives the value of these three statistics for the
permutation groups $S_{n}$ acting (in the natural way) on the set of
$k$-subsets of the set $\\{1,\dots,n\\}$.
Before we state our main result, let us briefly define the three statistics in
question (more complete definitions, as well as some background information,
are given in §1.1): suppose that $G$ is a finite permutation group on a set
$\Omega$. We define, first, $\mathrm{B}(G,\Omega)$ to be the maximum size of a
minimal base for the action of $G$; we define, second, $\mathrm{H}(G,\Omega)$
to be the maximum size of a subset $\Lambda\subseteq\Omega$ that has the
property that its pointwise stabilizer is not equal to the pointwise
stabilizer of any proper subset of $\Lambda$; we define, third,
$\mathrm{I}(G,\Omega)$ to be the maximum length of an irredundant base for the
action of $G$.
Our main result is the following.
###### Theorem 1.1.
Let $k$ and $n$ be positive integers with $1\leq k\leq\frac{n}{2}$. Consider
$S_{n}$ acting in the natural way on $\Omega_{k}$, the set of $k$-subsets of
$\\{1,\dots,n\\}$.
1. (1)
$\mathrm{I}(S_{n},\Omega_{k})=\begin{cases}n-1,&\textrm{if $\gcd(n,k)=1$};\\\
n-2,&\textrm{otherwise}.\end{cases}$
2. (2)
${\rm
B}(S_{n},\Omega_{k})=\mathrm{H}(S_{n},\Omega_{k})=\begin{cases}n-1,&\textrm{if
}k=1;\\\ n-2,&\begin{array}[]{l}\textrm{if }k=2\textrm{ or }\\\ \textrm{if
}k\geq 3\textrm{ and }n=2k+2;\end{array}\\\
n-3,&\textrm{otherwise}.\end{cases}$
### 1.1. Definition of statistics
Throughout, consider a finite permutation group $G$ on a set $\Omega$. Let
$\Lambda=\\{\omega_{1},\dots,\omega_{k}\\}\subseteq\Omega$; we write
$G_{(\Lambda)}$ or $G_{\omega_{1},\omega_{2},\dots,\omega_{k}}$ for the
pointwise stabilizer.
If $G_{(\Lambda)}=\\{1\\}$, then we say that $\Lambda$ is a _base_. We say
that a base is a _minimal base_ if no proper subset of it is a base. We denote
the minimum size of a minimal base $\mathrm{b}(G,\Omega)$, and the maximum
size of a minimal base $\mathrm{B}(G,\Omega)$.
We say that $\Lambda$ is an _independent set_ if its pointwise stabilizer is
not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We
define the _height_ of $G$ to be the maximum size of an independent set, and
we denote this quantity $\mathrm{H}(G,\Omega)$.
Given an ordered sequence of elements of $\Omega$,
$[\omega_{1},\omega_{2},\dots,\omega_{\ell}]$, we can study the associated
_stabilizer chain_ :
$G\geq G_{\omega_{1}}\geq G_{\omega_{1},\omega_{2}}\geq
G_{\omega_{1},\omega_{2},\omega_{3}}\geq\dots\geq
G_{\omega_{1},\omega_{2},\dots,\omega_{\ell}}.$
If all the inclusions given above are strict, then the stabilizer chain is
called _irredundant_. If, furthermore, the group
$G_{\omega_{1},\omega_{2},\dots,\omega_{\ell}}$ is trivial, then the sequence
$[\omega_{1},\omega_{2},\dots,\omega_{\ell}]$ is called an _irredundant base_.
The length of the longest possible irredundant base is denoted
$\mathrm{I}(G,\Omega)$. Note that, defined in this way, an irredundant base is
not a base (because it is an ordered sequence, not a set).
Let us make some basic observations. First, it is easy to verify the following
inequalities:
(1.1)
$\mathrm{b}(G,\Omega)\leq\mathrm{B}(G,\Omega)\leq\mathrm{H}(G,\Omega)\leq\mathrm{I}(G,\Omega).$
Second, it is easy to see that
$\Lambda=\\{\omega_{1},\omega_{2},\dots,\omega_{k}\\}$ is independent if and
only if the pointwise stabilizer of $\Lambda$ is not equal to the pointwise
stabilizer of $\Lambda\setminus\\{\omega_{i}\\}$ for all $i=1,\dots,k$. Third,
any subset of an independent set is independent.
### 1.2. Some context
Our interest in the statistics considered here was stimulated by our study of
yet another statistic, the _relational complexity_ of the permutation group
$G$, denoted $\mathrm{RC}(G,\Omega)$. This statistic was introduced in
[CMS96]; it can be defined as the least $k$ for which $G$ can be viewed as the
automorphism group of a homogeneous relational structure whose relations are
$k$-ary [Che16].
It is an exercise to confirm that
$\mathrm{RC}(G,\Omega)\leq\mathrm{H}(G,\Omega)+1$ for any permutation group
$G$ on a set $\Omega$ [GLS21]. Anecdotally it would seem that
$\mathrm{RC}(G,\Omega)$ tends to track $\mathrm{H}(G,\Omega)+1$ rather
closely: it often seems to equal this value or to be rather close to it. In
this respect Theorem 1.1 tells us that the action of $S_{n}$ on the set of
$k$-sets is an aberration: in [Che00], Cherlin calculates that
$\mathrm{RC}(S_{n},\Omega_{k})=\lfloor\log_{2}k\rfloor+2$; asymptotically this
is very far from the value for the height that is given in Theorem 1.1.
In a different direction, an earlier result with Spiga, along with work of
Kelsey and Roney-Dougal, asserts that the statistics $\mathrm{H}(G,\Omega)$
and $\mathrm{I}(G,\Omega)$ satisfy a particular upper bound whenever $G$ is
primitive and not in a certain explicit family of permutation groups [GLS21,
KRD]. Ultimately we would like to calculate the value of
$\mathrm{H}(G,\Omega)$ and $\mathrm{I}(G,\Omega)$ for all of the permutation
groups in this explicit family; our calculation of
$\mathrm{H}(S_{n},\Omega_{k})$ and $\mathrm{I}(S_{n},\Omega_{k})$ is the first
step in this process.
The one statistic that we have neglected in our study is
$\mathrm{b}(G,\Omega)$. The value of this statistic for the actions under
consideration has not been completely worked out, although significant
progress has been made (see [CGG+13, Hal12] as well as [BC11] and the
references therein). On the other hand, for those primitive actions of $S_{n}$
for which a point-stabilizer acts primitively in the natural action on
$\\{1,\dots,n\\}$, the value of $\mathrm{b}(G,\Omega)$ is known [BGS11].
Finally it is worth mentioning that, in general, $\ell(G)$, the maximum length
of a chain of subgroups in a group $G$, is an upper bound for
$\mathrm{I}(G,\Omega)$ for any faithful action of the group $G$ on a set
$\Omega$. It is known that $\ell(S_{n})=\lfloor\frac{3n-1}{2}\rfloor-b_{n}$,
where $b_{n}$ is the number of $1$’s in the binary expansion of $n$ [CST89].
### 1.3. Acknowledgments
The work of the first author was supported by EPSRC grant EP/R028702/1.
## 2\. The proof
In this section we prove Theorem 1.1. Throughout the proof we will write $G$
for $S_{n}$. We need some terminology.
Suppose that $\Delta=\\{\delta_{1},\dots,\delta_{\ell}\\}$ is a set of non-
empty subsets of $\\{1,\dots,n\\}$. We define $\mathcal{P}_{\Delta}$, the
_partition associated with $\Delta$ on $\\{1,\dots,n\\}$_, to be the partition
of $\\{1,\dots,n\\}$ associated with the equivalence relation $\sim$ given as
follows: for $x,y\in\\{1,\dots,n\\}$, we have $x\sim y$ if and only if for all
$i=1,\dots,\ell$, $x\in\delta_{i}\Longleftrightarrow y\in\delta_{i}$. If
$\Delta$ is empty, then we define $\mathcal{P}_{\Delta}$ to be the partition
with a single part of size $n$.
It is an easy exercise to check that, first, $\mathcal{P}_{\Delta}$ can be
obtained by taking intersections of all elements of $\Delta$; second, the
pointwise stabilizer of $\Delta$ in $S_{n}$ is simply the stabilizer of all
parts of $\mathcal{P}_{\Delta}$.
If $i,j\in\\{1,\dots,n\\}$ and $\omega\in\Omega_{k}$, then we will say that
$\omega$ _splits_ $i$ and $j$ if $|\\{i,j\\}\cap\omega|=1$. In particular, if
$\omega$ splits $i$ and $j$ then for any set $\Delta$ such that
$\omega\in\Delta\subseteq\Omega_{k}$, we have $i\not\sim j$ where $\sim$ is
the equivalence relation associated with $\Delta$.
### 2.1. The result for $\mathrm{I}(G,\Omega_{k})$
We will use the terminology above and begin with a couple of lemmas.
###### Lemma 2.1.
Let $d,e\in\mathbb{Z}^{+}\cup\\{0\\}$ with $e>d$, let $H$ be a permutation
group on the set $\\{1,\dots,n\\}$, let
$\Delta=\\{\delta_{1},\dots,\delta_{d}\\}$ be a set of non-empty subsets of
$\\{1,\dots,n\\}$, let $\Lambda$ be a set of non-empty subsets of
$\\{1,\dots,n\\}$ that contains $\Delta$ and let
$\Lambda\setminus\Delta=\\{\lambda_{d+1},\dots,\lambda_{e}\\}.$ Suppose that
$H\gneq H_{\delta_{1}}\gneq H_{\delta_{1},\delta_{2}}\gneq\cdots\gneq
H_{\delta_{1},\dots,\delta_{d}}\gneq
H_{\delta_{1},\dots,\delta_{d},\lambda_{d+1}}\gneq
H_{\delta_{1},\dots,\delta_{d},\lambda_{d+1},\lambda_{d+2}}\gneq\cdots\gneq
H_{(\Lambda)}.$
If $\mathcal{P}_{\Delta}$ has $r$ parts and $\mathcal{P}_{\Lambda}$ has $s$
parts, then $|\Lambda|=e\leq d+s-r$.
Note that if $\Delta$ is empty, then the lemma applies with $d=0$ and $r=1$,
and we obtain that $|\Lambda|\leq s-1$.
###### Proof.
For $i=1,\dots,d$, let $\mathcal{P}_{i}$ be the partition associated with the
set $\\{\delta_{1},\dots,\delta_{i}\\}$ and, for $i=d+1,\dots,e$, let
$\mathcal{P}_{i}$ be the partition associated with the set
$\Delta\cup\\{\lambda_{d+1},\lambda_{d+2},\dots,\lambda_{i}\\}$. Since all the
containments are proper, $\mathcal{P}_{i+1}$ has at least one more part than
$\mathcal{P}_{i}$ for all $i=1,\dots,e-1$. There are $|\Delta|$ containments
up to $H_{(\Delta)}=H_{\delta_{1},\dots,\delta_{d}}$ and then the number of
containments after that is at most $s-r$. The result follows. ∎
###### Lemma 2.2.
Let $\ell\in\mathbb{Z}^{+}$, let $g=\gcd(n,k)$, let
$\omega_{1},\dots,\omega_{\ell}$ be $k$-subsets of $\Omega$ and let
$\mathcal{P}_{i}$ be the partition associated with
$\\{\omega_{1},\dots,\omega_{i}\\}$. If $\mathcal{P}_{i+1}$ has exactly one
more part than $\mathcal{P}_{i}$ for all $i=1,\dots,k-1$, then all parts of
$\mathcal{P}_{\ell}$ have size divisible by $g$.
###### Proof.
We proceed by induction. Observe that $\mathcal{P}_{1}$ has two parts, one of
size $k$ and the other of size $n-k$. Both $k$ and $n-k$ are divisible by $g$
and so the result is true for $i=1$.
Let $i\in\\{1,\dots,k-1\\}$ and assume that all parts of $\mathcal{P}_{i}$
have size divisible by $g$. The property that $\mathcal{P}_{i+1}$ has exactly
one more part than $\mathcal{P}_{i}$ implies that, with precisely one
exception, if $P$ is a part of $\mathcal{P}_{i+1}$, then $|\omega_{i+1}\cap
P|\in\\{0,|P|\\}$. In other words
$\omega_{i+1}=P_{1}\cup\dots\cup P_{m}\cup X,$
where $m$ is some integer, $P_{1}\dots,P_{m+1}$ are parts of $\mathcal{P}_{i}$
and $X$ is a proper subset of part $P_{m+1}$. Note that, by assumption,
$|\omega_{i+1}|=k$ is divisible by $g$. What is more the inductive hypothesis
implies that $|P_{1}|,\dots,|P_{m}|$ are divisible by $g$, hence the same is
true of $|X|$. But now $\mathcal{P}_{i+1}$ has the same parts as
$\mathcal{P}_{i}$ except that part $P_{m+1}$ has been replaced by two parts,
$X$ and $P_{m+1}\setminus X$, both of which have size divisible by $g$. The
result follows. ∎
We are ready to prove item (1) of Theorem 1.1. First we let $\Lambda$ be an
independent set and observe that Lemma 2.1 implies that $|\Lambda|\leq n-1$;
thus $\mathrm{I}(G,\Omega_{k})\leq n-1$.
Next we suppose that $\gcd(n,k)=1$ and we must show that there exists a
stabilizer chain
$G\gneq G_{\omega_{1}}\gneq G_{\omega_{1},\omega_{2}}\gneq
G_{\omega_{1},\omega_{2},\omega_{3}}\gneq\dots\gneq
G_{\omega_{1},\omega_{2},\dots,\omega_{n-1}},$
with $\omega_{1},\dots,\omega_{n-1}\in\Omega_{k}$. Observe that if such a
chain exists, then, writing $\mathcal{P}_{i}$ for the partition associated
with $\\{\omega_{1},\dots,\omega_{i}\\}$, it is clear that, for
$i=1,\dots,n-2,$ the partition $\mathcal{P}_{i+1}$ has exactly one more part
than $\mathcal{P}_{i}$.
We show the existence of such a chain by induction on $k$: If $k=1$, then the
result is obvious. Write $d=\lfloor n/k\rfloor$ and write $n=dk+r$. For
$i=1,\dots,d$, we set
$\omega_{i}=\\{(i-1)k+1,\dots,ik\\}.$
The stabilizer of these $d$ sets is associated with a partition,
$\mathcal{P}_{d}$, of $d$ parts of size $k$ and one of size $r$. Now we will
choose the next sets, $\omega_{d+1},\dots,\omega_{d+k-1}$, so that they all
contain the part of size $r$ and so that the remaining $k-r$ points in each
are elements of $\omega_{1}$. Observe that, since $(n,k)=1$, we know that
$(k,k-r)=1$. Now the inductive hypothesis asserts that we can choose $k-1$
subsets of $\omega_{1}$, all of size $k-r$, so that the corresponding
stabilizer chain in $\mathop{\mathrm{Sym}}(\\{1,\dots,k\\})$ is of length
$k-1$, i.e. so that the corresponding chain of partitions of $\\{1,\dots,k\\}$
has the property that each partition has exactly one more part than the
previous.
We can repeat this process for $\omega_{2},\dots,\omega_{d}$, at the end of
which we have constructed a stabilizer chain of length $dk$ for which the
associated partition, $\mathcal{P}_{dk}$, has $dk$ parts of size $1$ and $1$
part of size $r$. A further $r-1$ subgroups can be added to the stabilizer
chain by stabilizing sets of form
$\\{1,\dots,k-1,dk+i\\}$
for $i=1,\dots,r-1$. We conclude that $\mathrm{I}(G,\Omega_{k})=n-1$ if
$\gcd(n,k)=1$.
On the other hand, let us see that $\mathrm{I}(G,\Omega_{k})\geq n-2$ in
general. Define
$\omega_{i}=\begin{cases}\\{1,\dots,k-1,k+i-1\\},&\textrm{ if
}i=1,\dots,n-k;\\\ \\{i-(n-k),k+1,\dots,2k-1\\}&\textrm{ if
}i=(n-k)+1,\dots,n-2.\end{cases}$
It is easy to check that the corresponding stabilizer chain
$G\geq G_{\omega_{1}}\geq G_{\omega_{1},\omega_{2}}\geq
G_{\omega_{1},\omega_{2},\omega_{3}}\geq\dots\geq
G_{\omega_{1},\omega_{2},\dots,\omega_{n-2}}$
is irredundant for $G=S_{n}$.
Finally we assume that $\gcd(n,k)=g>1$ and we show that
$\mathrm{I}(G,\Omega_{k})\leq n-2$. We must show that it is not possible to
construct a stabilizer chain of length $n-1$; as we saw above such a chain
would have the property that at every stage the corresponding partition
$\mathcal{P}_{i+1}$ has exactly one more part than $\mathcal{P}_{i}$.
Suppose that we have a stabilizer chain with the property that, for all $i$,
$\mathcal{P}_{i+1}$ has exactly one more part than $\mathcal{P}_{i}$. Now
Lemma 2.2 implies that all parts of $\mathcal{P}_{i}$ have size divisible by
$g$, for all $i$. We see immediately that such a stabilizer chain is of length
at most $n/g-1<n-1$ and we are done.
### 2.2. Preliminaries for $\mathrm{B}(G,\Omega_{k})$ and
$\mathrm{H}(G,\Omega_{k})$
Note, first, that for the remaining statistics the result for $k=1$ is
immediate; thus we assume from here on that $k>1$. Note, second, that to prove
what remains we need to show that there exists a lower bound for ${\rm
B}(G,\Omega_{k})$ which equals an upper bound for $\mathrm{H}(G,\Omega_{k})$.
### 2.3. The case $k=2$
First assume that $k=2$, and observe that
$\Big{\\{}\\{1,2\\},\\{1,3\\},\dots,\\{1,n-1\\}\Big{\\}}$
is a minimal base for $S_{n}$ acting on $\Omega_{2}$, and we obtain the
required lower bound.
For the upper bound on $\mathrm{H}(G,\Omega_{2})$ we let $\Lambda$ be an
independent set. We construct a graph, $\Gamma_{\Lambda},$ on the vertices
$\\{1,\dots,n\\}$ as follows: there is an edge between $i$ and $j$ if and only
if $\\{i,j\\}\in\Lambda$.
###### Lemma 2.3.
Suppose that $H$ is a permutation group on $\Omega$ and consider the natural
action of $H$ on $\Omega_{2}$. If $\Lambda$ is an independent subset of
$\Omega_{2}$ with respect to the action of $H$, then $\Gamma_{\Lambda}$
contains no loops.
###### Proof.
Suppose that $[i_{1},\dots,i_{\ell}]$ is a loop in the graph, i.e.
$E_{j}=\\{i_{j},i_{j+1}\\}$ is in $\Lambda$ for $j=1,\dots,\ell-1$, along with
$E_{\ell}=\\{i_{1},i_{\ell}\\}$. Now observe that if $E_{j}$ is removed from
$\Lambda$ for some $j$, then the stabilizer of the resulting set,
$\Lambda\setminus\\{E_{j}\\}$, fixes the two vertices contained in $E_{j}$.
But this implies that $\Lambda$ is not independent, a contradiction. ∎
We apply Lemma 2.3 to the action of $G=S_{n}$ on $\Omega_{2}$ and conclude
that the graph $\Gamma_{\Lambda}$ is a forest. If $\Gamma_{\Lambda}$ is
disconnected, then the result follows immediately. Assume, then, that
$\Gamma_{\Lambda}$ is connected, i.e. it is a tree on $n$ vertices. In this
case there are $n-1$ edges and we calculate directly that the point-wise
stabilizer of $\Lambda$ is trivial. But now, observe that if we remove any set
from $\Lambda$, then the point-wise stabilizer remains trivial. This is a
contradiction and the result follows.
### 2.4. A lower bound for $\mathrm{B}(G,\Omega_{k})$ when $k>2$
Assume for the remainder that $k>2$. We prove the lower bound first: first
observe that the following set, of size $n-3$, is a minimal base with respect
to $S_{n}$ (note that there are $n-k-1$ sets listed on the first row, and
$k-2$ listed altogether on the second and third):
(2.1)
$\left\\{\begin{array}[]{c}\Big{\\{}1,2,\dots,k-1,k\Big{\\}},\Big{\\{}1,2,\dots,k-1,k+1\Big{\\}},\dots,\Big{\\{}1,2,\dots,k-1,n-2\Big{\\}},\\\
\Big{\\{}1,n-(k-1),n-(k-2),n-(k-3),\dots,n-1\Big{\\}},\Big{\\{}2,n-(k-1),n-(k-2),n-(k-3),\dots,n-1\Big{\\}},\\\
\dots,\Big{\\{}k-2,n-(k-1),n-(k-2),n-(k-3),\dots,n-1\Big{\\}}\end{array}\right\\}.$
To complete the proof of the lower bound, we must deal with the case $n=2k+2$.
For this we observe that the following set, which is of size $n-2=2k$, is a
minimal base:
(2.2) $\Big{\\{}\\{1,\dots,k+1\\}\setminus\\{i\\}\mid
i=1,\dots,k\Big{\\}}\bigcup\Big{\\{}\\{k+2,\dots,2k+2\\}\setminus\\{i\\}\mid
i=k+2,\dots,2k+1\Big{\\}}$
### 2.5. An upper bound for $\mathrm{H}(G,\Omega_{k})$ when $k>2$
We must prove that, if $n\geq 2k$, then an independent set in $\Omega_{k}$ has
size at most $n-3$, except when $n=2k+2$, in which case it has size at most
$n-2$. It turns out that it is easy to get close to this bound in a much more
general setting, as follows.
###### Lemma 2.4.
Let $n\geq 2$, let $\Delta$ be a set of subsets of $\Omega=\\{1,\dots,n\\}$
and suppose that $\Delta$ is independent with respect to the action of
$G=S_{n}$ on the power set of $\Omega$. Then one of the following holds:
1. (1)
There exists $\delta\in\Delta$ with $|\delta|\in\\{1,n-1\\}$.
2. (2)
$|\Delta|\leq n-2$.
###### Proof.
Let us suppose that (1) does not hold; we will prove that (2) follows. Note
that if $\delta\in\Delta$ with $|\delta|>\frac{n}{2}$, then we can replace
$\delta$ with $\Omega\setminus\delta$ and the resulting set will still be
independent. Note too that, since $\Delta$ is independent, all sets in
$\Delta$ are non-empty. Thus we can assume that $1<|\delta|\leq\frac{n}{2}$
for all $\delta\in\Delta$.
Suppose that there exist $\delta_{1},\delta_{2}\in\Delta$ such that
$\delta_{1}\cap\delta_{2}\neq\emptyset$. Since
$|\delta_{1}|,|\delta_{2}|\leq\frac{n}{2}$ this means that
$\Omega\setminus(\delta_{1}\cup\delta_{2})\neq\emptyset$. We conclude that
$\mathcal{P}_{\\{\delta_{1},\delta_{2}\\}}$ contains 4 parts. Now the result
follows from Lemma 2.1.
Suppose, instead, that $\delta_{1}\cap\delta_{2}=\emptyset$ for all distinct
$\delta_{1},\delta_{2}\in\Delta$. If $|\Delta|\geq 1$, then $\mathcal{P}$ has
at most $n-1$ parts and the result follows from Lemma 2.1. If $|\Delta|=0$,
then the result is true since we assume that $n\geq 2$. ∎
To improve the upper bound in Lemma 2.4 (2) from $n-2$ to $n-3$ we will need
to do quite a bit of work (and we will need to deal with some exceptions). In
what follows we set $\Lambda$ to be an independent set in $\Omega_{k}$ and, to
start with at least, we drop the requirement that $n\geq 2k$.
As when $k=2$, it is convenient to think of $\Lambda$ as being the set of
hyperedges in a $k$-hypergraph, $\Gamma_{\Lambda}$, with vertex set
$\Omega=\\{1,\dots,n\\}$. From here on we will write “edge” in place of
“hyperedge”. We think of two edges as being _incident_ in $\Gamma_{\Lambda}$
if they intersect non-trivially. If $\Delta$ is a set of edges in this graph
(i.e. $\Delta\subseteq\Lambda$), then the _span_ of $\Delta$ is the set of
vertices equalling the union of all edges in $\Delta$.
Write $\Gamma_{C_{1}},\dots,\Gamma_{C_{\ell}}$ for the connected components of
$\Gamma_{\Lambda}$; in particular $\ell$ is the number of connected components
in $\Gamma_{\Lambda}$. For $\Gamma_{C_{i}}$, we write $C_{i}$ for the vertex
set and $\Lambda_{C_{i}}$ for the edge set.
In what follows we repeatedly use the fact that if
$\lambda_{1},\dots,\lambda_{j}$ are elements of the independent set $\Lambda$,
then we must have
$G\gneq G_{\lambda_{1}}\gneq G_{\lambda_{1},\lambda_{2}}\gneq\cdots\gneq
G_{\lambda_{1},\lambda_{2},\dots,\lambda_{j}}.$
This in turn means that, for all $i=1,\dots,j-1$, the partition
$\mathcal{P}_{\lambda_{1},\dots,\lambda_{i+1}}$ has more parts than
$\mathcal{P}_{\lambda_{1},\dots,\lambda_{i}}$.
###### Lemma 2.5.
1. (1)
If $\Gamma_{\Lambda}$ has a connected component with exactly one edge, then
$|\Lambda|\leq n-3$.
2. (2)
If $\ell\geq 3$, then $|\Lambda|\leq n-3$.
3. (3)
Suppose that $\ell=2$, that $|\Gamma_{C_{i}}|\geq 2$ for $i=1,2$, and that
there exist incident edges $E_{1},E_{2}$ in $\Lambda_{C_{1}}$ such that the
span of $\\{E_{1},E_{2}\\}$ is not equal to $C_{1}$. Then $|\Lambda|\leq n-3$.
###### Proof.
For (1), observe that $\mathcal{P}_{\Lambda}$ has at most $n-k+1$ parts. Then
Lemma 2.1 yields the result.
We may assume, then, that any connected component of $\Lambda$ either contains
at least 2 edges or none. To prove (2) we go through the possibilities:
* •
If there are at least 3 components with no edges, then,
$\mathcal{P}_{\Lambda}$ has at most $n-2$ parts and Lemma 2.1 yields the
result.
* •
Suppose there are 2 components with no edges and at least 1 component,
$C_{1}$, containing 2 edges. Let $E_{1},E_{2}$ be incident edges in
$\Lambda_{C_{1}}$ and observe that $\mathcal{P}_{\\{E_{1},E_{2}\\}}$ contains
4 parts while $\mathcal{P}_{\Lambda}$ contains at most $n-1$ parts; Lemma 2.1
yields the result.
* •
Suppose there are at least 3 components in total and at least 2 components,
$C_{1}$ and $C_{2}$, containing 2 edges. Let $E_{1},E_{2}$ be incident edges
in $\Lambda_{C_{1}}$, let $F_{1},F_{2}$ be incident edges in $\Lambda_{C_{2}}$
and observe that $\mathcal{P}_{\\{E_{1},E_{2},F_{1},F_{2}\\}}$ contains 7
parts while $\mathcal{P}_{\Lambda}$ contains at most $n$ parts; Lemma 2.1
yields the result.
We have proved (2). For (3) let $E_{1},E_{2}$ be incident edges in
$\Lambda_{C_{1}}$ for which the span of $\\{E_{1},E_{2}\\}$ is not equal to
$C_{1}$, let $F_{1},F_{2}$ be incident edges in $\Lambda_{C_{2}}$. We can see
that $\Lambda\setminus(E_{1}\cup E_{2}\cup F_{1}\cup F_{2})$ is non-empty;
thus $\mathcal{P}_{\\{E_{1},E_{2},F_{1},F_{2}\\}}$ contains 7 parts and the
result again follows from Lemma 2.1. ∎
The next result deals with a particular case when $\Lambda$ is connected.
###### Lemma 2.6.
If $|\Lambda|\geq 2$ and $\Omega$ is spanned by 2 incident edges, then
$|\Lambda|\leq\begin{cases}n-1,&\textrm{if }n=k+1;\\\ n-2,&\textrm{if
}n>k+1.\end{cases}$
###### Proof.
Since $\Omega$ is spanned by 2 incident edges, we have $k>|\Omega|/2$. Observe
that, for any $\lambda\in\Lambda$, the set $\Omega\setminus\lambda$ is a
subset of size $|\Omega|-k$. Since the pointwise stabilizers in
$\mathop{\mathrm{Sym}}(\Omega)$ of the subsets $\lambda$ and
$\Omega\setminus\lambda$ are equal, we obtain that
$\overline{\Lambda}=\\{\Omega\setminus\lambda\mid\lambda\in\Lambda\\}$
is an independent set with respect to the action of
$\mathop{\mathrm{Sym}}(\Omega)$ on $\Omega_{j}$ where $j=|\Omega|-k$. Since
$j<|\Omega|/2<k$ it is clear that $\Omega$ is not spanned by 2 edges in
$\overline{\Lambda}$. If $j\geq 1$, then Lemma 2.4 implies that
$|\overline{\Lambda}|=|\Lambda|=|\Lambda|\leq n-2$, as required. If $j=1$,
then the result is obvious. ∎
From here on we impose the condition that $n\geq 2k$.
###### Lemma 2.7.
Suppose that $n\geq 2k$, that $\ell=2$ and that $|\Gamma_{C_{i}}|\geq 2$ for
$i=1,2$. Then
$|\Lambda|\leq\begin{cases}n-2,&\textrm{if $n=2k+2$};\\\
n-3,&\textrm{otherwise}.\end{cases}$
###### Proof.
Item (3) of Lemma 2.5 yields this result in the case where there exist
$i\in\\{1,2\\}$ and incident edges $E_{1},E_{2}$ in $\Lambda_{C_{i}}$ such
that the span of $\\{E_{1},E_{2}\\}$ is not equal to $C_{i}$. Thus we may
assume that, for $i=1,2$ and for distinct $E_{1},E_{2}\in\Gamma_{C_{i}}$, the
span of $\\{E_{1},E_{2}\\}$ is equal to $C_{i}$.
We claim that for each $i=1,2$, the set $\Lambda_{C_{i}}$ is independent with
respect to the action of $\mathop{\mathrm{Sym}}(C_{i})$ on $C_{i}$. To see
this observe that $\Lambda=\Lambda_{C_{1}}\cup\Lambda_{C_{2}}$ and that, by
definition, $\Lambda_{C_{2}}$ must be an independent set for
$H:=G_{(\Lambda_{C_{1}})}$. But $H=H_{0}\times\mathop{\mathrm{Sym}}(C_{2})$
where $H_{0}<\mathop{\mathrm{Sym}}(C_{1})$. Now if
$\Delta\subseteq\Lambda_{C_{2}}$, then
$H_{(\Delta)}=H_{0}\times\mathop{\mathrm{Sym}}(C_{1})_{(\Delta)}$. In
particular, if $\Delta_{1},\Delta_{2}\subseteq\Lambda_{C_{2}}$, then
$H_{(\Delta_{1})}=H_{(\Delta_{2})}$ if and only if
$\mathop{\mathrm{Sym}}(C_{2})_{(\Delta_{1})}=\mathop{\mathrm{Sym}}(C_{2})_{(\Delta_{2})}$.
This implies immediately that $\Lambda_{C_{2}}$ is independent with respect to
the action of $\mathop{\mathrm{Sym}}(C_{2})$ on $C_{2}$, and the same argument
works for $C_{1}$.
Now we apply Lemma 2.6 to these two actions. We conclude that, for each
$i=1,2$, either $|C_{i}|=k+1$ and $|\Lambda_{C_{i}}|\leq|C_{i}|-1$, or else
$|\Lambda_{C_{i}}|\leq|C_{i}|-2$. The result now follows from the fact that
$|\Lambda|=|\Lambda_{C_{1}}|+|\Lambda_{C_{2}}|$ and $n=|C_{1}|+|C_{2}|$. ∎
Notice that Lemma 2.7 attends to the strange appearance of “$n=2k+2$” in the
statement of item (2) of Theorem 1.1. Before we prove Theorem 1.1 we need one
more lemma.
###### Lemma 2.8.
Suppose that $n\geq 2k$. Suppose that either $\Lambda$ is connected, or else
it has two connected components, exactly one of which is a single isolated
point. Then $|\Lambda|\leq n-3$.
###### Proof.
Note that the supposition, along with the fact that $n\geq 2k$, implies that
$\Omega$ contains 2 incident edges and $\Omega$ is not spanned by 2 edges.
Consider $E_{1},E_{2}$, a pair of incident edges in $\Lambda$. Let
$\Pi=\\{E_{1},E_{2}\\}$ and observe that the parts of $\mathcal{P}_{\Pi}$ are
(2.3) $R:=E_{1}\cap E_{2},\,\,S:=E_{1}\setminus(E_{1}\cap
E_{2}),\,\,T:=E_{2}\setminus(E_{1}\cap E_{2})\,\,\textrm{ and
}U:=\Omega\setminus(E_{1}\cup E_{2});$
in particular $|\mathcal{P}_{\pi}|=4$.
If $i$ and $j$ are distinct elements of $\Omega$ that are unsplit by any
element of $\Lambda$, then $\mathcal{P}_{\Lambda}$ has at most $n-1$ parts.
Then Lemma 2.1 implies that $|\Lambda|\leq n-3$ as required. Thus we assume
that all distinct elements of $\Omega$ are split by an element of $\Lambda$.
These observations imply that we can write
(2.4)
$\Lambda=\\{E_{1},E_{2}\\}\cup\Lambda_{R}\cup\Lambda_{S}\cup\Lambda_{T}\cup\Lambda_{U},$
where, for $X\in\\{R,S,T,U\\}$, $\Lambda_{X}$ is the set of elements in
$\Lambda$ that split pairs of distinct elements in $X$.
If there exists $E_{3}\in\Lambda$ such that
$\mathcal{P}_{\\{E_{1},E_{2},E_{3}\\}}$ contains $6$ parts, then Lemma 2.1
implies that $|\Lambda|=n-3$ and we are done. Thus we assume that
$\mathcal{P}_{\\{E_{1},E_{2},E_{3}\\}}$ contains 5 parts for all choices of
$E_{3}\in\Lambda\setminus\\{E_{1},E_{2}\\}$. In particular if
$E_{3}\in\Lambda_{X}$, then $E_{3}$ does not split any pairs of elements in
$\Lambda_{Y}$ for $Y\in\\{R,S,T,U\\}\setminus X$. This means, first, that if
$E_{3}\cap Y\neq\emptyset$ for some $Y\in\\{R,S,T,U\\}\setminus X$, then
$E_{3}\supset Y$; it means, second, that the sets
$\Lambda_{R},\Lambda_{S},\Lambda_{T}$ and $\Lambda_{U}$ are pairwise disjoint.
Set $x:=|R|$, so $|S|=|T|=k-x$. Observe that, since $n\geq 2k$ we must have
$|U|\geq x$. We split into two cases and we will show that our assumptions to
this point lead to a contradiction.
1\. Suppose that we can choose $E_{1},E_{2}$ so that $1<x$. This means, in
particular that both $R$ and $U$ have cardinality at least $2$; hence
$\Lambda_{R}$ and $\Lambda_{U}$ are all non-empty.
Let $E_{3}\in\Lambda_{R}$. By counting we must have
$E_{3}=(E_{3}\cap R)\cup U\,\,\textrm{ or }\,\,(E_{3}\cap R)\cup S\cup T.$
Let $E_{4}\in\Lambda_{U}$. By counting we must have
$E_{4}=(E_{4}\cap U)\cup R\ \,\,\textrm{ or }\,\,(E_{4}\cap U)\cup
S\,\,\textrm{ or }\,\,(E_{4}\cap U)\cup T\,\,\textrm{ or }\,\,(E_{4}\cap
U)\cup S\cup T.$
We will go through the various combinations and show that, in every case, the
set $\\{E_{1},E_{2},E_{3},E_{4}\\}$ is not independent, thereby giving our
contradiction. In what follows $g\in G_{E_{1},E_{3},E_{4}}$.
Consider, first, the possibilities for $E_{3}$.
1. (E3A)
Suppose that $E_{3}=(E_{3}\cap R)\cup U$. Then $g$ stabilizes $(E_{1}\cup
E_{3})^{C}=T$ and $E_{3}\setminus(E_{1}\cap E_{3})=U$.
2. (E3B)
Suppose that $E_{3}=(E_{3}\cap R)\cup S\cup T$. Then $g$ stabilizes
$E_{3}\setminus(E_{1}\cap E_{3})=T$ and $(E_{1}\cup E_{3})^{C}=U$.
Thus, in all cases, $g$ stabilizes both $T$ and $U$. Now consider the
possibilities for $E_{4}$.
1. (E4A)
Suppose that $E_{4}=(E_{4}\cap U)\cup R$. Then $g$ stabilizes $E_{1}\cap
E_{4}=R$ and hence also $R\cup T=E_{2}$. This contradicts independence (the
pointwise stabilizer of $\\{E_{1},E_{3},E_{4}\\}$ is equal to the pointwise
stabilizer of $\\{E_{1},E_{2},E_{3},E_{4}\\}$).
2. (E4B)
Suppose that $E_{4}=(E_{4}\cap U)\cup S$. Then $g$ stabilizes $E_{1}\cap
E_{4}=S$, hence also $\Omega\setminus(S\cup T\cup U)=R$, hence also $R\cup
T=E_{2}$. We have the same contradiction.
3. (E4C)
Suppose that $E_{4}=(E_{4}\cap U)\cup S\cup T$. Then $g$ stabilizes $E_{1}\cap
E_{4}=S$, hence also $\Omega\setminus(S\cup T\cup U)=R$, hence also $R\cup
T=E_{2}$. We have the same contradiction.
4. (E4D)
Suppose that $E_{4}=(E_{4}\cap U)\cup T$. For this final case we swap $g$ with
an element $h\in G_{E_{2},E_{3},E_{4}}$, we swap $S$ with $T$ and we swap
$E_{1}$ with $E_{2}$. Now, with these changes, the arguments for (E3A) and
(E3B) tell us that $h$ stabilizes both $S$ and $U$. Next the argument for
(E4B) tells us that $h$ stabilizes $R$, hence also $R\cup S=E_{1}$. Now we
again have a contradiction (the pointwise stabilizer of
$\\{E_{2},E_{3},E_{4}\\}$ is equal to the pointwise stabilizer of
$\\{E_{1},E_{2},E_{3},E_{4}\\}$).
2\. Suppose that $|E_{1}\cap E_{2}|\in\\{0,1\\}$ for all distinct
$E_{1},E_{2}\in\Lambda$. This is the remaining case. We fix an incident pair
$E_{1}$ and $E_{2}$ and observe that $\Lambda_{S}$ and $\Lambda_{T}$ are non-
empty. Let $E_{3}\in\Lambda_{S},E_{4}\in\Lambda_{T}$; observe that
$E_{3}=(E_{3}\cap S)\cup U$ and $E_{4}=(E_{4}\cap T)\cup U$. But then
$|U|\leq|E_{3}\cap E_{4}|\leq 1$, hence $|U|=1$. This implies that
$|E_{3}|=|E_{3}\cap S|+|U|=|E_{3}\cap E_{1}|+|U|=1+1=2.$
Thus $k=2$, a contradiction. ∎
###### Proof of Theorem 1.1 (2) for $k\geq 3$.
The work in §2.4 implies that we need only prove an upper bound for
$\mathrm{H}(S_{n},\Omega_{k})$. Lemma 2.5 (2) yields the result if $\ell\geq
3$. Lemma 2.8 yields the result if $\ell=1$. Assume, then, that $\ell=2$.
Lemma 2.7 yields the result if each component contains at least 2 edges. Lemma
2.5 (1) yields the result if there is a component with 1 edge. Lemma 2.8
yields the result if exactly one of the components has 0 edges. Finally if
both components have 0 edges, then the fact that $n\geq 2k$ implies the
result. ∎
## 3\. The alternating group
One naturally wonders to what extent the results given here extend to the
action of $A_{n}$ on $k$-sets. Throughout this section $k$ and $n$ will be
positive integers with $k\leq\frac{n}{2}$.
For irredundant bases we can adjust the proof given in §2.1, making use of the
following easy fact: suppose that $\mathcal{P}_{i}$ and $\mathcal{P}_{j}$ are
partitions corresponding to a set of $k$-subsets in $\\{1,\dots,n\\}$ as
described at the start of §2. Let $H_{i}$ (resp. $H_{j}$) be the stabilizer in
$A_{n}$ of all parts of $\mathcal{P}_{i}$ (resp. $\mathcal{P}_{j}$). If
$H_{i}=H_{j}$, then either $\mathcal{P}_{i}=\mathcal{P}_{j}$ or else the two
partitions are of type $1^{n}$ or $1^{n-2}2^{1}$.
Let us show how this observation yields the required result.
###### Proposition 3.1.
$\mathrm{I}(A_{n},\Omega_{k})=\begin{cases}n-2,&\textrm{if $\gcd(n,k)=1$};\\\
\max(2,n-3),&\textrm{otherwise}.\end{cases}$
###### Proof.
Let $G=S_{n}$ and suppose that
$G\gneq G_{\omega_{1}}\gneq G_{\omega_{1},\omega_{2}}\gneq
G_{\omega_{1},\omega_{2},\omega_{3}}\gneq\dots\gneq
G_{\omega_{1},\omega_{2},\dots,\omega_{e}}$
is a stabilizer chain corresponding to an irredundant base
$[\omega_{1},\dots,\omega_{e}]$. The observation implies that we have
$A_{n}\gneq G_{\omega_{1}}\cap A_{n}\gneq G_{\omega_{1},\omega_{2}}\cap
A_{n}\gneq G_{\omega_{1},\omega_{2},\omega_{3}}\cap A_{n}\gneq\dots\gneq
G_{\omega_{1},\omega_{2},\dots,\omega_{e-1}}\cap A_{n}\geq
G_{\omega_{1},\omega_{2},\dots,\omega_{e}}\cap A_{n}$
and hence either $[\omega_{1},\dots,\omega_{e}]$ or
$[\omega_{1},\dots,\omega_{e-1}]$ is an irredundant base for $A_{n}$. This
implies that $\mathrm{I}(A_{n},\Omega_{k})=\mathrm{I}(S_{n},\Omega_{k})-1$ and
Theorem 1.1 implies that
$\mathrm{I}(A_{n},\Omega_{k})\geq\begin{cases}n-2,&\textrm{if
$\gcd(n,k)=1$};\\\ n-3,&\textrm{otherwise}.\end{cases}$
Now we will give an upper bound for $\mathrm{I}(A_{n},\Omega_{k})$. Let
$[\omega_{1},\dots,\omega_{e}]$ be an irredundant base for the action of
$A_{n}$ on $\Omega_{k}$. Then the observation above implies that
$\mathcal{P}_{\\{\omega_{1},\dots,\omega_{e-1}\\}}$ contains at most $n-2$
parts. Applying Lemma 2.1 with $\Lambda=\\{\omega_{1},\dots,\omega_{e-1}\\}$
and $\Delta=\emptyset$ implies that $e-1=|\Lambda|\leq n-3$ and so $e\leq
n-2$. This yields the result when $\gcd(n,k)=1$.
Suppose now that $\gcd(n,k)=g>1$ and that $[\omega_{1},\dots,\omega_{n-2}]$ is
an irredundant base for the action of $A_{n}$ on $\Omega_{k}$; we must show
that then $n=4$. The observation above implies that
$\mathcal{P}_{\\{\omega_{1},\dots,\omega_{i+1}\\}}$ has exactly one more part
than $\mathcal{P}_{\\{\omega_{1},\dots,\omega_{i}\\}}$ for $i=1,\dots,n-4$ and
$\mathcal{P}_{\\{\omega_{1},\dots,\omega_{n-2}\\}}$ has exactly two more parts
than $\mathcal{P}_{\\{\omega_{1},\dots,\omega_{n-3}\\}}$. Lemma 2.2 implies
that all parts of $\mathcal{P}_{\\{\omega_{1},\dots,\omega_{n-3}\\}}$ are
divisible by $g$. But the type of
$\mathcal{P}_{\\{\omega_{1},\dots,\omega_{n-3}\\}}$ is either $2^{2}1^{n-4}$
or $3^{1}1^{n-3}$ and we conclude that $(n,k)=(4,2)$ as required. The proof is
completed by observing that $\mathrm{I}(A_{4},\Omega_{2})=2$. ∎
For the other statistics in question it is easy to pin the value down within
an error of 1; the next result does this.
###### Proposition 3.2.
Suppose that $k$ and $n$ are positive integers with $k\leq\frac{n}{2}$. Then
(3.1)
$\mathrm{H}(S_{n},\Omega_{k})-1\leq\mathrm{B}(A_{n},\Omega_{k})\leq\mathrm{H}(A_{n},\Omega_{k})\leq\mathrm{H}(S_{n},\Omega_{k}).$
###### Proof.
The first inequality is obtained by observing that if we excise the final set
from (2.1) and (2.2), then we obtain a minimal base for $A_{n}$.
The second inequality is elementary; it was given in (1.1). The third
inequality, likewise, is an easy consequence of the definition of height. ∎
All that remains, then, is to establish which of the two possible values holds
for each value of $k$ and $n$. Let us consider the situation for small values
of $k$:
1. ($k=1$)
It is immediate that
$\mathrm{B}(A_{n},\Omega_{1})=\mathrm{H}(A_{n},\Omega_{1})=n-2$, the smaller
of the two possible values.
2. ($k=2$)
We claim that in this case
$\mathrm{B}(A_{n},\Omega_{2})=\mathrm{H}(A_{n},\Omega_{2})=\begin{cases}n-3,&\textrm{if
$n\neq 4$;}\\\ 2,&\textrm{if $n=4$.}\end{cases}$
Thus, provided $n\neq 4$, we again obtain the smaller of the two possible
values. To justify our claim we note first that the value for $n=4$ is easy to
obtain. When $n>4$ it is sufficient to prove that
$\mathrm{H}(A_{n},\Omega_{2})\leq n-3$. To see this we let $\Lambda$ be an
independent set and we form the graph $\Gamma_{\Lambda}$ as in §2.3. Lemma 2.3
implies that, since $\Lambda$ is independent, the graph $\Gamma_{\Lambda}$ is
a forest. If this forest has 3 or more connected components, then the result
follows immediately, so we suppose that there are at most 2 components. If one
of these components consists of a single edge, then deleting this edge results
in a set of $2$-sets whose pointwise stabilizer in $A_{n}$ is trivial; this is
a contradiction of the fact that $\Lambda$ is independent. If all components
contain $0$ edges or at least $2$ edges, then it is easy to check that either
$n=4$ or else deleting a leaf edge results in a set of $2$-sets whose
pointwise stabilizer in $A_{n}$ is trivial. Again this is a contradiction and
we are done.
3. ($k=3$)
We claim that in this case
$\mathrm{B}(A_{n},\Omega_{3})=\mathrm{H}(A_{n},\Omega_{3})=n-3$ which is the
larger of the two possible values, except when $n=8$. To justify our claim we
note first that the value for $n=8$ is easy to obtain. When $n\neq 8$ it is
sufficient to prove that $\mathrm{B}(A_{n},\Omega_{3})\geq n-3$. This follows
simply by observing that the following set is a minimum base of size $n-3$:
$\Big{\\{}\,\\{1,2,3\\},\,\\{1,2,4\\},\,\dots,\,\\{1,2,n-1\\}\,\Big{\\}}.$
We have not investigated the case $k\geq 4$.
Finally, referring to §1.2, we remark that in [Che16], Cherlin calculated
$\mathrm{RC}(A_{n},\Omega_{k})$ precisely (correcting an earlier calculation
in [Che00]). The comments above imply that for all $k$ and $n$ with
$k\leq\frac{n}{2}$ we have
$\mathrm{H}(A_{n},\Omega_{k})\leq\mathrm{RC}(A_{n},\Omega_{k})\leq\mathrm{H}(A_{n},\Omega_{k})+1.$
Thus, unlike $S_{n}$, the relational complexity of the action of $A_{n}$ on
$k$-sets does indeed track height.
## References
* [BC11] Robert F. Bailey and Peter J. Cameron. Base size, metric dimension and other invariants of groups and graphs. Bulletin of the London Mathematical Society, 43(2):209–242, 2011\.
* [BGS11] Timothy C. Burness, Robert M. Guralnick, and Jan Saxl. On base sizes for symmetric groups. Bull. Lond. Math. Soc., 43(2):386–391, 2011.
* [CGG+13] José Cáceres, Delia Garijo, Antonio González, Alberto Márquez, and María Luz Puertas. The determining number of Kneser graphs. Discrete Math. Theor. Comput. Sci., 15(1):1–14, 2013.
* [Che00] Gregory Cherlin. Sporadic homogeneous structures. In The Gelfand Mathematical Seminars, 1996–1999. Dedicated to the memory of Chih-Han Sah, pages 15–48. Boston, MA: Birkhäuser, 2000.
* [Che16] Gregory Cherlin. On the relational complexity of a finite permutation group. J. Algebraic Combin., 43(2):339–374, 2016.
* [CMS96] Gregory L. Cherlin, Gary A. Martin, and Daniel H. Saracino. Arities of permutation groups: Wreath products and $k$-sets. J. Comb. Theory, Ser. A, 74(2):249–286, art. no. 0050, 1996.
* [CST89] Peter J. Cameron, Ron Solomon, and Alexandre Turull. Chains of subgroups in symmetric groups. J. Algebra, 127(2):340–352, 1989.
* [GLS21] N. Gill, B. Lodà, and P. Spiga. On the height and relational complexity of a finite permutation group. Nagoya Mathematical Journal, pages 1–40, 2021.
* [Hal12] Zoltán Halasi. On the base size for the symmetric group acting on subsets. Stud. Sci. Math. Hung., 49(4):492–500, 2012.
* [KRD] Veronica Kelsey and Colva M. Roney-Dougal. On the height and relational complexity of finite primitive permutation groups. Preprint: https://arxiv.org/abs/2107.14208.
|
1School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam
686560, India.
2Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India.
3Physical Research Laboratory, Ahmedabad, Gujarat 380009, India.
4Indian Institute of Technology, Bombay 400076, India.
5Tata Institute of Fundamental Research, Mumbai 400005, India
# AstroSat-CZTI as a hard X-ray Pulsar Monitor
Anusree K.G.1,* Bhattacharya D.2 Rao A.R.2,5 Vadawale S.3 Bhalerao V.4
Vibhute A.2
###### Abstract
The Cadmium Zinc Telluride Imager (CZTI) is an imaging instrument onboard
AstroSat. This instrument operates as a nearly open all-sky detector above 60
keV, making possible long integrations irrespective of the spacecraft
pointing. We present a technique based on the AstroSat-CZTI data to explore
the hard X-ray characteristics of the $\gamma$-ray pulsar population. We
report highly significant ($\sim 30\sigma$) detection of hard X-ray (60–380
keV) pulse profile of the Crab pulsar using $\sim$5000 ks of CZTI observations
within 5 to 70 degrees of Crab position in the sky, using a custom algorithm
developed by us. Using Crab as our test source, we estimate the off-axis
sensitivity of the instrument and establish AstroSat-CZTI as a prospective
tool in investigating hard X-ray characteristics of $\gamma$-ray pulsars as
faint as 10 mCrab.
###### keywords:
Pulsars: individual (Crab, PSR J0534+2200)—Calibration—hard X-ray—piggyback
data—LAT pulsars
<EMAIL_ADDRESS>
30 October 202018 January 2021
12.3456/s78910-011-012-3 #### 000 0000 1– 1
## 1 Introduction
Pulse profiles of rotation powered pulsars are shaped by the geometry of the
emission region as well as the radiation processes at work in pulsar
magnetosphere (Watters and Romani, 2011, Pierbattista et al., 2014). The
emission is usually broadband, covering radio through $\gamma$-rays. However,
emission at different wavebands typically arises in different regions of the
magnetosphere, giving rise to pulse profiles that are wavelength dependent.
There exist a variety of magnetospheric models (e.g. Cheng, Ho & Ruderman
1986, Muslimov & Harding 2004, Pétri 2011, Cerutti & Beloborodov 2017) which
differ in their prediction of the shape and distribution of acceleration and
radiation zones. Broadband Spectral Energy Distribution (SED) and the energy
dependence of the pulse shape and arrival time provide clues to the
distribution of these zones and have been used to constrain the magnetospheric
geometry and to discriminate between theoretical models (e.g. Romani &
Yadigaroglu 1995, Cheng et al., 2000, Abdo et al., 2009c, Abdo et al. 2010c,
Bai & Spitkovsky 2010, YuanJie et al., 2012, Pierbattista et al., 2012, 2014).
The number of known $\gamma$-ray pulsars stood at just seven until the launch
of NASA’s Fermi Large Area Telescope (LAT) in 2008. Since then, the continuous
accumulation of data, together with highly efficient searching algorithm, has
resulted in the number of pulsars detected by LAT in the $\gamma$-ray band to
mount to 253 (Ray et al., 2020), of which 71 have no radio counterpart. The
modelling of $\gamma$-ray and radio emission together can provide important
constraints on the global magnetospheric properties (e.g. see Pétri & Mitra
2020). For radio-quiet $\gamma$-ray pulsars, the magnetospheric X-ray emission
provides the only additional clue to the emission process. The radiation
energy output of a rotation powered pulsar typically peaks in the
X-ray/$\gamma$-ray region. However, only 18 pulsars have been detected in the
X-ray band till date (Caraveo 2014, Kuiper and Hermsen 2015) and the SED of
many of them are sparsely sampled. Using ephemerides determined from LAT data,
four of the LAT radio-quiet pulsars have been observed at photon energies
$<$20 keV (Lin et al., 2010; Caraveo et al., 2010; Marelli et al., 2014; Lin
et al., 2013 and Lin et al., 2014). These fall in the category of “Geminga-
like” pulsars, for which the profile and spectra are known at soft X-rays
(i.e., $<$20keV) and $\gamma$-rays alone (PSR B0633+17, Halpern and Holt 1992;
Bertsch et al., 1992). Unfortunately, at energies above 20 keV, the major
X-ray observatories have had relatively low sensitivity. Additional deep
observations in the hard X-ray band would therefore augment the existing
information, providing a better estimate of spectral shape and bolometric
luminosity of these and other pulsars hitherto undetected in X-rays. Three
major emission mechanisms operate in pulsar magnetospheres, namely synchrotron
radiation, curvature radiation and inverse-Compton scattering. A well-measured
SED can distinguish the relative contributions of these components, leading to
a model of the particle energy distribution in the emission zones.
To increase the sample of pulsar detections in the hard X-ray/Soft
$\gamma$-ray bands, and to investigate how their properties fit in the general
picture emerging from the theoretical studies of the Fermi’s young gamma-ray
pulsars, we need “open all-sky X-ray detectors”. The pulsar spectra are steep
at high energies. In general, photon flux of the young/middle-aged LAT
$\gamma$-ray pulsars can be represented by a power-law with a simple
exponential cutoff, i.e.
$F_{\gamma}=k\cdot(E_{\gamma}/E_{0})^{\Gamma}\cdot\exp(-(E_{\gamma}/E_{c})^{\beta})$
where $\beta\approx 1$ and $E_{\gamma}$ is the photon energy, $E_{0}$ a
normalisation energy, $E_{c}$ the cutoff energy and $k$ is the normalization.
The photon index $\Gamma$ has been found to lie in the range $-0.4$ to $-1.7$
(Kuiper and Hermsen, 2015). Even with a highly sensitive detector with sub-
second timing resolution, it takes extremely long exposures to detect a
significant number of photons from a typical $\gamma$-ray pulsar. Such long
integrations are not affordable by any of the missions at present. Open all-
sky detectors, on the other hand, can collect photons during other
observations, making it possible to search for these pulsars.
The first Indian multi-wavelength Satellite AstroSat was launched in 2015 with
five instruments onboard (Singh et al., 2014). One of them, the Cadmium Zinc
Telluride Imager (CZTI; Bhalerao et al., 2017), can detect photons in 20–380
keV energy range. Its housing and collimators are made of Aluminium alloy and
thin Tantalum shields that define its low-energy field of view but allow
sufficient uncollimated penetration above $\sim 60$ keV to make CZTI an
excellent wide-angle monitor at higher energies, covering roughly one-third of
the sky at all times. This monitoring capability has been leveraged to detect
many transients, including over 300 Gamma-Ray Bursts (Sharma et al., this
volume). A more detailed description of CZTI can be found in Bhalerao et al.,
2017. The aim of this paper is to assess the suitability of CZTI for the
detection and study of pulsars in the hard X-ray band, using its off-axis
detection capability.
During CZTI pointing observations, photons from candidate hard X-ray sources
that shine in through the walls will come to piggyback on any ongoing
observation, with varying sensitivity depending on the pointing. Arrival times
of the photons from a pulsar carry the signature of its spin period, enabling
us to search the data for hard X-ray pulsars with known ephemeris. This
presents the detection of Crab Pulsar in the energy range 60-380 keV from off-
axis CZTI observations, using a custom algorithm developed by us. The Crab
pulsar (PSRJ0534+2200) is well-studied at all wavelengths. Using Crab as our
calibration source, we determine the off-axis sensitivity of the instrument.
This paper is structured as follows. Section 2 describes the instruments used
in this work and the analysis of data, followed by a discussion of results in
section 3 which includes the comparison of hard X-ray pulse profiles of the
Crab pulsar obtained from CZTI with $\gamma$-ray profiles from LAT data.
Finally, we assess the potential of AstroSat-CZTI in the investigation of hard
X-ray counterparts of $\gamma$-ray pulsars.
## 2 Data and analysis
This work is based on data from AstroSat-CZTI pointing observations that were
released for public use on or before 30th April 2019. We have also used
publicly available archival data from NASA’s Fermi-LAT mission. All material
informations about the instruments and data, along with the characteristics of
analysis methods, are described in this section. Moreover, we also discuss
some of our checks against the vulnerabilities anticipated during long
integration.
Table 1:
Telescope | Energy range | Start MJD | Stop MJD | Exposure(ks)
---|---|---|---|---
AstroSat-CZTI( Crab at nominal pointing) | 30-60 keV | 57290 | 58232 | 519
AstroSat-CZTI(Crab within 5-70 degree of nominal pointing) | 60-380 keV | 57290 | 58232 | 4752
Fermi-LAT( within 1 degree of $\gamma$-ray position of Crab pulsar) | 0.1-300 GeV | 57290 | 58500 | 5962
Brief summary of the observations. Participating telescopes and their
instruments, the energy range chosen for the work, range of MJDs for which the
data have been collected for this work and the total exposure achieved are
listed
### 2.1 AstroSat-CZTI
CZTI is the AstroSat instrument primarily designed for simultaneous hard X-ray
imaging and spectroscopy of celestial X-ray sources in the energy band 20-150
keV. Its functioning employs the technique of coded mask imaging. The CZTI
instrument bears a two-dimensional coded aperture mask (CAM) above its
pixellated, 5-mm thick solid-state CZT detector modules spread over four
quadrants. Passive collimators are placed in each quadrant of CZTI that
support the coded mask. The set up defines a 4.6 deg $\times$ 4.6 deg field of
view in $\sim$20-100 keV range with an angular resolution of $\sim$ 8 arcmins.
A critical aspect of the collimators and mask is that these are designed to be
effective up to $\sim 100$ keV above which they become transparent. The
transparency being a function of energy and angle of incidence, appreciable
sensitivity for off-axis sources extends down to $\sim 60$ keV (see
Bhattacharya et al., 2018).
The $976cm^{2}$ of the detector’s total geometric area is distributed over
16384 pixels, with 4096 pixels in each independent quadrant. After the launch,
about 15 percent of the pixels were disabled for having shown excessive
electronic noise. Nearly 25 percent of the remaining pixels seemed to have an
inadequate spectroscopic response. Considering that the coded mask has a
$\sim$ 50% open fraction, the total effective area at normal incidence is
$\sim 420cm^{2}$ in all active pixels at energies below 100 keV. The detected
events are recorded with a time resolution of 20 $\mu$s. The absolute
timestamp assigned to CZTI events is estimated to have a jitter of $\sim
3\,\mu$s RMS (Bhattacharya, 2017) and a fixed offset with respect to Fermi of
$650\pm 70\,\mu$s (Basu et al., 2018).
#### 2.1.1 Analysis of CZTI data:
All available CZTI pointing observations during MJD 57366-58362, with the Crab
pulsar within 5-80 degrees from normal incidence, were selected for this work,
resulting in a total exposure of 5096ks. The merged Level-1 data of all the
selected CZTI observations were reduced to Level-2 using standard CZTI
analysis software. Details and the sequence of analysis modules can be found
in the latest version of the CZTI user-guide available at the ASSC
website111http://astrosat-ssc.iucaa.in/. There are intervals during pointing
observation where data is absent due to SAA passage and data transmission
loss. There are also intervals when the earth occults the target source. To
generate science products from such observations, identifying such intervals
and removing the data for that duration by adequately accounting for the gaps
is essential. This task is performed by the module cztgtigen. It generates
Good Time Interval (GTI) files based on various threshold parameters. We
generated custom GTI files to take into consideration the earth occultation of
the off-axis source and filtered the original event file accordingly. These
filtered, clean event files were used in the subsequent steps of analysis.
In this work, we have used AstroSat CZTI observations with on-axis pointing of
the Crab pulsar as well as others when the pulsar is off-axis at angles
between 5 to 70 deg (see below for the choice of this angle range). We have
also used Fermi-LAT observations of the Crab pulsar spanning a similar time
range for comparison. A brief description of the data sets used is provided in
Table 1.
The arrival times of the CZTI events at the spacecraft were converted to those
at the Solar-system Barycenter using the JPL DE200 Solar-system ephemeris.
This was done using the tool as1bary, a version of NASA/HEASOFT AXBARY
package, customised for AstroSat, using the well-known astrometric position of
the Crab pulsar.
We developed a custom code to fold the barycentered event data spanning many
months but including frequent gaps. This code assigns an absolute phase to
each recorded event using the polynomial model timing solutions known as SSB
polycos generated using tempo2 (Hobbs et al.,2006). Polycos predict a pulsar’s
parameters at a particular epoch. A polyco file contains pulsar ephemerides
over a short period, typically hours, in simple polynomial expansion. All the
polycos were generated at an epoch in the centre of each 6-hour interval in
this work. These 6-hour ephemerides were then used for folding the LAT data as
well as the CZTI data. As an initial test of the custom code, Fermi-LAT
$\gamma$-ray events for a few pulsars were folded and compared with their
published timing models. The SSB polycos were generated from LAT timing models
available publicly at Fermi’s website. 222
https://confluence.slac.stanford.edu/display/GLAMCOG/LAT+Gamma-
ray+Pulsar+Timing+Models. Finally, the code was tested on AstroSat-CZTI data.
For this, publicly available AstroSat-CZTI data from six pointing observations
of Crab pulsar (Table 1, first row) were selected and reduced using the
default CZTI data analysis pipeline. The 30-60 keV pulse profile, thus
obtained, is shown in Figure 2(a). The Crab pulsar ephemeris derived from
Fermi observations and used in this work are given in Table 2 along with the
reference epochs.
To compare the pulse profiles between hard X-rays and $\gamma$-rays, we folded
all the events extracted from AstroSat-CZTI observations and those obtained
from Fermi observation within MJD 57290–58500 using the $\gamma$ ray ephemeris
mentioned above. For all CZTI data analysis, we have directly combined the
data of all four quadrants, as they run on a synchronised time reference (see
Figure 1). We initially folded the off-axis data separately for every
10-degree angle interval and found that when the source is located beyond 70
degrees from the pointing axis, the signal-to-noise ratio in a given
integration time drops significantly below those for smaller off-axis angles.
We, therefore, decided to restrict the accumulation of off-axis data to the
angle range of 5 to 70 deg (below 5 deg the source would appear in the main
FoV).
With the phase reference as presented in Table 3, we consider the “off-pulse”
region of the profile, phase 0.5 through 0.8, as the background. The signal to
noise ratio is calculated as the ratio of the peak count to the standard
deviation of the counts in the off-pulse region. We also divided the obtained
X-ray events into several different energy bands to check consistency, as
shown in Figure 2(b).
Figure 1: Phase histograms of PSR J0534+2200 in 60-380 keV band obtained from
5-70 degree off-angle data (this work) from the four different quadrants (Q0,
Q1, Q2 and Q3) of CZTI operating as independent detectors. All the quadrants
are found to be time aligned with no measurable relative delay
Table 2: Fermi-LAT $\gamma$-ray timing solution of PSR J0534+2200 by Fermi
Timing Observers. The numbers in parentheses denote $1\sigma$ errors in the
parameters
. Parameters Right ascension, $\alpha$ 05:34:31.94 Declination, $\delta$
+22:00:52.1 Valid MJD range 54686-58767 Pulse frequency, $f~{}(s^{-1}$
29.7169027333 (0.0000148379) First derivative , $\dot{f}~{}(10^{-10}s^{-2})$
-3.71184342371 (4.6591493867e-17) Second derivative
,$\ddot{f}~{}(10^{-20}s^{-3})$ 3.3226958153 (1.1931513864e-24) Third
derivative , $\dddot{f}~{}(10^{-28}s^{-4})$ 1.2860657170 (9.2842295751e-32)
PEPOCH 55555 POSEPOCH 50739 DMEPOCH 55107.807158553 TZRMJD 56730.15526586924
Solar system ephemeris model DE405 Time system TDB
### 2.2 Fermi-LAT
The LAT instrument is described by Atwood et al., (2008). We have used the
already publicly available data from LAT. A unique value of the LAT data is
that a pulsar’s discovery in $\gamma$-rays often enables the immediate
measurement of the pulsar parameters over the ten-year span in which the LAT
has been operating. LAT data have been used to find precise timing solutions
for many pulsars, including radio-quiet and radio-faint pulsars (Ray et al.,
2011; Kerr et al., 2015; Clark et al., 2017).
#### 2.2.1 Analysis of Fermi-LAT data
In order to get a significant $\gamma$-ray profile to compare with the CZTI
results, we took all available photon data for the LAT source PSRJ0534+2200
from MJD 57290 through 58500, bracketing the CZTI data used in this work. It
leads to a total effective exposure time of 5962-kilo seconds, as the
observatory scans the entire sky once every three hours. We used the HEADAS-
FTOOLS 333http://heasarc.gsfc.nasa.gov/ftools on HEAsoft-ver 6.27. (Blackburn,
1995) to perform the data reduction. We obtained the Fermi-LAT data in the
energy range of 0.1-300 GeV within a circular region of interest (ROI) with a
1-degree radius from the decided $\gamma$-ray position of PSR J0534+2200. We
used Pass 8 data and selected events in the ”Source” class (i.e. event class
2). We also excluded the events with zenith angles larger than 105 degrees to
reduce the contamination by the Earth albedo $\gamma$-rays. We used $gtbary$
tool in fermi science tools to apply barycentric corrections to photon arrival
times in LAT event files using corresponding Fermi orbit files. After
barycentering the events using fermi science tool
444https://fermi.gsfc.nasa.gov/ssc/data/analysistools/overview.html $gtbary$,
the absolute phase of each event in 0.1-300 GeV was determined by the custom
code developed by us, with 6 hour SSB polycos generated using the timing
parameters in Table 2
## 3 Discussion of Results
Table 3: Phase component definitions for the Crab pulsar (Abdo et al., 2010)
adopted in this study
Component | Abbreviation | Phase Interbval | Width
---|---|---|---
Peak 1 | P1 | 0.87 - 1.07 | 0.20
Peak 2 | P2 | 0.27 - 0.47 | 0.20
Bridge | Bridge | 0.098 - 0.26 | 0.162
Off Pulse | OP | 0.52 - 0.87 | 0.35
(a)
(b)
Figure 2: The pulse profile of the Crab pulsar observed with Fermi-LAT (black)
and the AstroSat-CZTI (purple). The integration time in the 0.1–300 GeV LAT
profile is $\sim$6000 ks, while that in the 60–-380 keV CZTI profiles is
$\sim$5000 ks. The bottom panel of Figure 2(a) shows the 30-60 keV profile
obtained by folding $\sim$600 ks of on-axis pointing observations of the Crab
pulsar with AstroSat-CZTI. In CZTI profiles, data from multiple observations
during MJD 57290-58232 with PSR J0534+2200 within 5–70 degree from pointing
axis have been accumulated. Two cycles in 512 phase bins are plotted for
clarity. The energy dependence of the profile is apparent here: below 60 keV
the first peak P1 (near phase 1.0) dominates, and emission in the “bridge”
phase interval is moderate, while above 60 keV the second peak dominates with
significant “bridge” emission. At MeV energies, P1 starts to dominate again
with strongly reduced bridge emission. A detailed reference to the energy
dependence of Crab pulse profile is available in Kuiper et al., (2001).
Figure 2 shows the folded pulse profile of the Crab pulsar (Period: $\sim$ 33
ms) using the LAT and the CZTI instruments. The CZTI profiles (purple in
colour) accumulate data from multiple observations spread over several months,
with Crab position within 5 to 70 degrees away from the nominal pointing
direction. The total integration time in these CZTI profiles is about 5000 ks,
giving a signal to noise ratio of $\sim 30$ in the energy integrated profile
and over $\sim 15$ in the energy-resolved ones. The long-known energy
dependence of the Crab pulse profile can be seen when compared with the softer
bands (e.g. 30-60 keV in CZTI) in Figure 2(a). The left peak is taller at
lower energies while the right peak dominates at high energies. The bridge
emission connecting the two peaks is also seen relatively stronger at higher
X-ray energies. The normalised light curves in three separate hard X-ray bands
are shown in Figure 2(b). The right peak(P2) grows, but the left peak (P1)
again starts to dominate at very high energies, beyond ($\sim$10 MeV) in the
$\gamma$-regime, as seen in Figure 2(a).
For a more quantitative comment on the morphology change observed, we
determined the intensity ratios P2/P1 and Bridge/P1 in three separate hard
X-ray bands shown in Figure 2(b), adopting the phase interval definitions of
(Abdo et al., 2010) shown in Table 3. The values obtained for P2/P1 ratios are
1.301$\pm$0.002, 1.340$\pm$0.002, 1.385$\pm$0.002 and that for Bridge/P1 are
0.536$\pm$0.002, 0.574$\pm$0.002, 0.609$\pm$0.003 in 60-100 keV, 100-150 keV
and 150-380 keV bands respectively. The gradual increase of both the ratios is
consistent with that reported by Kuiper et al., (2001). This validates the
accuracy and stability of AstroSat-CZTI clocks and the robustness of our
custom algorithm for long-term phase-connected analysis with available
accurate $\gamma$-ray/radio timing models.
Integrating the well established broken power-law model spectrum for Crab
(Ulmer et al., 1995), we calculated the hard X-ray flux in 60-380 keV band to
be 0.011 ph cm-2s-1. The observed CZTI detection count rates of $0.876\pm
0.01$ cps in 60-380 keV band translates to an average CZTI off-axis effective
area of $\sim$80 cm2, averaged over 5-70 degree off-axis. This is about 20
percent of the on-axis effective area at lower energies.
Assuming a Crab-like spectrum and based on the observed count rates mentioned
above, we estimate the possible integration time required for a 5$\sigma$
detection of a 10 mCrab off-axis pulsed source with CZTI to be $\sim 14$ Mega
seconds. With the continuous accumulation of CZTI data since launch, one can
foresee the detection of more than 90 percent of the $\gamma$-ray pulsars from
the second Fermi/LAT pulsar catalog in the CZTI hard X-ray (60-380 keV) band,
as shown in Figure 3.
## 4 Conclusion
Figure 3: Detectability of Fermi-LAT pulsars with the CZTI. The green
horizontal line marks the estimated flux of the faintest detectable hard X-ray
pulsar using five years of archival AstroSat-CZTI data. The red diamonds
represent the hard X-ray pulsar candidates discovered/observed in the soft
X-ray band ($<$20 keV). The brightest $\gamma$-ray pulsars Vela
(PSRJ0835-4510) and Geminga(PSRJ0633+1746), with fluxes of 7 $\&$ 3 Crab
respectively, have been omitted from the figure to clearly depict the rest of
the population.
We have presented a successful attempt to detect the Crab pulsar by the
AstroSat CZT Imager from pointings where the pulsar was off-axis, at angles
ranging from 5 to 70 degrees. This is possible because the Collimator and the
housing of the CZTI become gradually transparent at energies above 60 keV,
turning it into an all-sky detector at higher energies. Accumulating an off-
axis exposure of $\sim 5$ Ms, we obtain a $\sim 30\sigma$ pulse profile of the
Crab pulsar in the 60–380 keV energy band. Energy-resolved pulse profiles
constructed at multiple sub-bands reproduce the known energy dependence of the
profile shape. This demonstrates the capability of AstroSat-CZTI to act as a
hard X-ray pulsar monitor, in 60-380 keV band, with an average off-axis
effective area of $\sim$80 cm2. Our results establish that the CZTI time
stamps possess sufficient long-term stability to carry out phase connected
timing spanning many years. We estimate that with continued accumulation of
data, it will become possible for the CZTI to detect pulsars with hard X-ray
fluxes down to $\sim 10$ mCrab, thus making a large majority of Fermi/LAT
pulsars accessible for study in the 60–380 keV energy band. Such a survey is
currently ongoing with several successful detections already made. These
results will be reported elsewhere.
## Acknowledgements
This publication makes use of data from the CZTI onboard Indian astronomy
mission AstroSat, archived at the Indian Space Science Data Centre (ISSDC).
The CZT Imager instrument was built by a TIFR-led consortium of institutes
across India, including VSSC, ISAC, IUCAA, SAC, and PRL. The Indian Space
Research Organisation funded, managed and facilitated the project. We extend
our gratitude to CZTI POC team members at IUCAA for helping with the
augmentation of data. We thank Fermi Timing Observers Paul Ray and Kerr Mattew
for their timely and favourable response in providing LAT ephemeris for Crab
pulsar and helping with queries related to SSB polyco generation using tempo2.
We thank the anonymous referee for his/her valuable suggestions to improve the
paper. We would like to thank Avishek Basu, Karthik Rajeev and Atul Mohan for
useful discussions. We thank IUCAA HPC facility where we carried out all the
analysis. Anusree K. G. acknowledges support for this work from DST-INSPIRE
Fellowship grant, IF170239, under Ministry of Science and Technology, India.
## References
* [1] Abdo, A. A., Ackermann, M., Atwood, W. B., Bagagli, R. et al., 2009, The Astrophysical Journal, 696, 1084.
* [2] Abdo, A. A., et al., 2010, The Astrophysical Journal Supplement Series, 208, 17.
* [3] Abdo, A. A., Ackermann, M., Ajello, M. et al., 2010, The Astrophysical Journal, 708, 1254.
* [4] Atwood, W. B., Abdo, A. A., Ackermann, M. et al., 2009, The Astrophysical Journal, 697, 1071.
* [5] Bai, X. N. & Spitkovsky, A., 2010, The Astrophysical Journal, 715, 1282
* [6] Basu, A., Joshi, B.C., Bhattacharya, D. et al., 2018, Astronomy and Astrophysics, 617, A22.
* [7] Bertsch, D. L., Brazier, K. T. S., Fichtel, C. E. et al., 1992, Nature, 357, 306.
* [8] Bhalerao, V., Bhattacharya, D., Vibhute, A. et al., 2017, Journal of Astrophysics and Astronomy, 38, 31.
* [9] Bhattacharya, D. (2017). Journal of Astrophysics and Astronomy, 38, 51.
* [10] Bhattacharya, D., Dewangan, G.C., Antia H.M. et al., 2018, AstroSat Handbook, http://www.iucaa.in/`~`astrosat/AstroSat_handbook.pdf
* [11] Blackburn, J. K. (1995). Astronomical Data Analysis Software and Systems IV, 77, 367.
* [12] Caraveo, P. A. (2014). Gamma-ray pulsar revolution. Annual Review of Astronomy and Astrophysics, 52.
* [13] Caraveo, P. A., De Luca, A., Marelli, M. et al., 2010, The Astrophysical Journal Letters, 725, L6.
* [14] Cerutti, B., & Beloborodov, A. M., 2017, Space Science Reviews, 207, 111.
* [15] Cheng, K.S., Ho, C. & Ruderman, M., 1986, The Astrophysical Journal, 300, 522.
* [16] Cheng, K.S., Ruderman, M. & Zhang, L., 2000, The Astrophysical Journal, 537, 964.
* [17] Clark, C. J., Wu, J., Pletsch, H. J. et al., 2017, The Astrophysical Journal, 834, 106.
* [18] Du, Y., Qiao, G., & Wang, W., 2012, The Astrophysical Journal, 748, 84.
* [19] Halpern, J. P., & Holt, S. S., 1992, Nature, 357, 222.
* [20] Harding, A., & Muslimov, A., 2004, cosp, 35, 562.
* [21] Hewish, A., Bell, S. J., Pilkington, J. D. et al., 1968. Nature, 217, 709.
* [22] Hobbs, G. B., Edwards, R. T., & Manchester, R. N., 2006, Monthly Notices of the Royal Astronomical Society, 369, 655.
* [23] Kerr, M., Ray, P. S., Johnston, S., 2015, The Astrophysical Journal, 814, 128.
* [24] Kuiper, L., Hermsen, W., Cusumano, G. et al., 2001, Astronomy & Astrophysics, 378, 918.
* [25] Kuiper, L., & Hermsen, W., 2015, Monthly Notices of the Royal Astronomical Society, 449, 3827.
* [26] Lin, L. C., Huang, R. H., Takata, J. et al., 2010, The Astrophysical Journal Letters, 725, L1.
* [27] Lin, L. C. C., Hui, C. Y., Hu, C. P. et al., 2013, The Astrophysical Journal Letters, 770, L9.
* [28] Lin, L. C. C., Hui, C. Y., Li, K. T. et al., 2014, The Astrophysical Journal Letters, 793, L8.
* [29] Marelli, M., Harding, A., Pizzocaro, D. et al., 2014, The Astrophysical Journal, 795, 168.
* [30] Muslimov, A.G. & Harding, A.K., 2004, The Astrophysical Journal, 606, 1143
* [31] Pétri, J., 2011, Monthly Notices of the Royal Astronomical Society, 412, 1870
* [32] Pétri, J. & Mitra, D., 2020, Monthly Notices of the Royal Astronomical Society, 491, 80.
* [33] Pierbattista, M., Grenier, I. A., Harding, A. K., & Gonthier, P. L., 2012, Astronomy & Astrophysics, 545, A42.
* [34] Pierbattista, M., Grenier, I., Harding, A., & Gonthier, P., 2014, radio and $\gamma$-ray light-curve morphology of young pulsars: comparing Fermi observations and simulated populations. Saint-Petersburg, Russia July 28–August 1, 2014, 99.
* [35] Ray, P. S., Kerr, M., Parent, D. et al., 2011, The Astrophysical Journal Supplement Series, 194, 17.
* [36] Ray, P.S., Smith D.A. et al., 2020, https://confluence.slac.stanford.edu/display/GLAMCOG/ Public+List+of+LAT-Detected+Gamma-Ray+Pulsars/
* [37] Romani, R. & Yadigaroglu, I.-A., 1995, The Astrophysical Journal, 438, 314.
* [38] Singh, K.P., Tandon, S.N., Agrawal, P.C. et al., 2014, SPIE, 9144E, 1S.
* [39] Ulmer, M. P., Matz, S. M., Grabelsky, D. A. et al., 1995, The Astrophysical Journal, 448, 356.
* [40] Watters, K. P., & Romani, R. W., 2011.The Astrophysical Journal, 727, 123.
|
# Generic power series on subsets of the unit disk
Balázs Maga and Péter Maga Eötvös Loránd University, Department of
Analysis, Pázmány Péter sétány 1/C, Budapest, H–1117 Hungary<EMAIL_ADDRESS>MTA Alfréd Rényi Institute of Mathematics, POB 127, Budapest H-1364, Hungary
<EMAIL_ADDRESS>
###### Abstract.
In this paper, we examine the boundary behaviour of the generic power series
$f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense
of Baire category. Notably, we prove that for any open subset $U$ of the unit
disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense
set of $\mathbb{C}$. As it is demonstrated, this conclusion does not
necessarily hold for arbitrary open sets accumulating to the unit circle. To
complement these results, a characterization of coefficient sets having this
property is given.
###### Key words and phrases:
complex power series, boundary behaviour, Baire category
###### 2010 Mathematics Subject Classification:
Primary 30B30; Secondary 28A05, 54H05
The first author was supported by the ÚNKP-20-3 New National Excellence
Program of the Ministry for Innovation and Technology from the source of the
National Research, Development and Innovation Fund, and by the Hungarian
National Research, Development and Innovation Office–NKFIH, Grant 124003. The
second author was supported by the Premium Postdoctoral Fellowship of the
Hungarian Academy of Sciences, the MTA Rényi Intézet Automorphic Research
Group, and by the Hungarian National Research, Development and Innovation
Office–NKFIH, Grants FK 135218, K 119528.
## 1\. Introduction
Let $\Lambda\subseteq\mathbb{C}$ be a bounded subset with at least two
elements, endowed with its usual subspace topology. Moreover, assume that the
product space
$\Omega:=\Omega_{\Lambda}:=\bigtimes_{n=0}^{\infty}\Lambda$
is a Baire space. Due to Alexandrov’s theorem (e.g. [5, p. 408]), this holds,
for example, if $\Lambda$ is $G_{\delta}$. These general conditions on
$\Lambda$ will be assumed throughout the paper.
For any $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$ we can define the
power series
$f(z):=f_{\bm{\lambda}}(z):=\sum_{n=0}^{\infty}\lambda_{n}z^{n}.$
The resulting function is clearly holomorphic in the open unit disk
$D:=\\{|z|<1\\}$. Roughly speaking, we are interested in the generic behaviour
of $f$ in terms of Baire category near the boundary $\partial D=S$. The
genericity is understood as follows: if a property holds for a set of power
series corresponding to a residual set of configurations in $\Omega$, we say
it is generic.
This work is a direct continuation of our previous paper [6], in which we
investigated the typical boundary behaviour of real power series with
coefficients chosen from a finite set $\Lambda$. (Actually, the set of
coefficients was denoted by $D$ in that paper, we opted to introduce this
notational modification as the symbol $D$ is customarily preserved for the
unit disk in this setup.) While the probabilistic aspects of the problem was
our main focus (considering the uniform distribution over $\Lambda$), we
proved straightforward results in terms of Baire category as well ([6, Theorem
3]). Notably, if $\Lambda$ has both positive and negative elements, for the
generic power series $f$ we have $\limsup_{1-}f=+\infty$ and
$\liminf_{1-}f=-\infty$, while if $\Lambda\subseteq[0,+\infty)$ (resp.
$\Lambda\subseteq(-\infty,0]$), we have $\lim_{1-}f=+\infty$ (resp.
$\lim_{1-}f=-\infty$). It was natural to consider the same problem in the
complex setup as well, which is a more natural habitat of power series. (We
note that while those results were stated for finite $\Lambda$ exclusively,
the proofs can be generalized in a straightforward manner for any $\Lambda$
for which $\bigtimes_{n=0}^{\infty}\Lambda$ is a Baire space.)
A direct prelude to our results is given in [1]. Even though the main focus of
that paper is the theory of random power series, where it presents spectacular
results about natural boundaries without assuming independence, it contains
the following result relevant to our setting:
###### Proposition 1 ([1, Theorem 1.9]).
For generic $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the power
series $f_{\bm{\lambda}}$ has a natural boundary on $\partial D$. That is it
cannot be analytically continued through any of the boundary points of the
convergence domain.
Another predecessor of this line of research is [2], in which results are
stated and proved about generic complex power series. For a more detailed
historical summary of the topic we also refer to that paper. We recall that
while the problem of the “general behaviour” dates back to Borel, the more
thorougly examined probabilistic question was worked out by several authors,
as presented in [3]. In terms of Baire category, the first results were
provided in [4]. The setup of [4] slightly differs from ours, as the
topological vector space $H(D)$ in which Baire category is investigated is the
space of all functions which are holomorphic in $D$ with the topology of
locally uniform convergence. (The $\Omega$ we consider corresponds to a
subspace of it.) Their main results stated that generically, $S$ is a natural
boundary and $f(D)=\mathbb{C}$. These results were generalized by [2], in
particular:
###### Proposition 2 ([2, Proposition 3.2]).
Assume that $U\subseteq D$ is open and $\overline{U}\cap S\neq\emptyset$. Then
for generic $f\in H(D)$ we have $f(U)=\mathbb{C}$ generically. (For a set
$B\subseteq\mathbb{C}$, its closure is denoted by $\overline{B}$ throughout
the paper.)
Our main goal is strengthening Proposition 1 in a similar manner. The proof of
Proposition 2 relies on Runge’s theorem on polynomial approximation, which is
out of reach in our setup that poses restrictions on the permissible
holomorphic functions. Consequently, when one would like to verify similar
results for $\Omega$, different techniques are required. As we will see, this
leads to somewhat weaker theorems: roughly, we could only verify that $f(U)$
is an open, dense set instead of being equal to $\mathbb{C}$. Our first main
result is the following.
###### Theorem 1.
Assume that $U\subseteq D$ is open with an accumulation point $\zeta\in S$
with non-vanishing imaginary part. Then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f(U)\subseteq\mathbb{C}$ is dense and open.
As we will see below, the conclusion of Theorem 1 does not hold for all open
sets accumulating to $S$. In order to allow 1 or $-1$ to be the accumulation
point on the boundary, certain conditions on $\Lambda$ should be introduced.
The necessary and sufficient conditions are summarised by our other theorems.
By a real line, we mean a one-dimensional real affine subspace of
$\mathbb{C}$, while by a real half-plane we mean one of the two components of
the complement of a real line. We refer to its closure as a closed real half-
plane.
###### Theorem 2.
If $\Lambda$ is not contained by a real line, and $U\subseteq D$ is open with
$-1\in\overline{U}$, then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f_{\bm{\lambda}}(U)\subseteq\mathbb{C}$ is dense and open. If $\Lambda$ is
contained by a real line, there exists an open $U\subseteq D$ with
$-1\in\overline{U}$ for which $f_{\bm{\lambda}}(U)$ evades a closed real half-
plane, regardless of the choice of $\bm{\lambda}$.
###### Theorem 3.
If $\Lambda$ is not contained by a closed real half-plane of the form
$\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$, and $U\subseteq D$ is open
with $1\in\overline{U}$, then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f_{\bm{\lambda}}(U)\subseteq\mathbb{C}$ is dense and open. If $\Lambda$ is
contained by a closed real half-plane of the form $\\{z:\text{ }\alpha\leq\arg
z\leq\alpha+\pi\\}$, then there exists an open $U\subseteq D$ with
$1\in\overline{U}$ for which $f_{\bm{\lambda}}(U)$ evades a closed real half-
plane, regardless of the choice of $\bm{\lambda}$.
We note that the openness of $f(U)$ for generic $f$ is a trivial consequence
of the open mapping theorem for analytic functions in each of the cases, as
such $f$ are non-constant. This implies that the real task in these questions
is proving the density of the images.
We fix some notation for the rest of the paper. Throughout,
$D_{r}=\\{|z|<r\\}$ for general disks centered at the origin, while its
boundary is denoted by $S_{r}$ (that is, $D=D_{1}$, $S=S_{1}$.) We use the
notation $\text{proj}_{n}(G)$ for the projection of
$G\subseteq(\lambda_{n})_{n=0}^{\infty}\in\Omega$ to the $n$th coordinate of
the product.
For any $\zeta\in S$ and any $N\in\mathbb{N}$, define
$\\{0,1\\}_{N}[\zeta]:=\left\\{\sum_{j=N}^{\infty}a_{j}\zeta^{j}:a_{j}\in\\{0,1\\},\
a_{j}=0\text{ with finitely many exceptions}\right\\}.$
A terminology we are going to use frequently is the following. For sets
$A,B\subseteq\mathbb{C}$ and some $\varepsilon>0$, we say that $A$ is an
$\varepsilon$-net of $B$, if for any $w\in B$, there exists some $z\in A$ such
that $|z-w|<\varepsilon$.
Below we will also use the notation $e(x)=e^{2\pi ix}$ for $x\in\mathbb{C}$.
Acknowledgement. We are grateful to the anonymous referee for their careful
reading of the manuscript and their suggestions to improve the exposition.
## 2\. Preliminary statements
We fix the following notation: for any $c\in\mathbb{C}$, set
$\Lambda+c:=\\{\lambda+c:\lambda\in\Lambda\\},\qquad
c\Lambda:=\\{c\lambda:\lambda\in\Lambda\\},$
and
$\bm{\lambda}+c=(\lambda_{n})_{n=0}^{\infty}+c:=(\lambda_{n}+c)_{n=0}^{\infty},\qquad
c\bm{\lambda}=c(\lambda_{n})_{n=0}^{\infty}c:=(c\lambda_{n})_{n=0}^{\infty}.$
Set also
$f_{\Lambda}(U):=\bigcup_{\bm{\lambda}\in\Omega}f_{\bm{\lambda}}(U).$
Our first lemma concerns the effect on $f_{\bm{\lambda}}(U)$ of certain
modifications on $\Lambda$, enabling us to circumvent some of the technical
burden in the proof of our theorems through replacing arbitrary $\Lambda$’s by
simpler ones.
###### Lemma 1.
1. (a)
Let $U\subseteq D$ be open. Assume that $f_{\bm{\lambda}}(U)$ is dense in
$\mathbb{C}$. Then for any $0\neq c\in\mathbb{C}$, $f_{c\bm{\lambda}}(U)$ is
also dense in $\mathbb{C}$.
2. (b)
Let $(U_{k})_{k=1}^{\infty}$ be a shrinking sequence of open subsets of $D$
such that $\text{diam}(U_{k})\to 0$ and none of them accumulates to $1$.
Moreover, assume that all of the sets $f_{\bm{\lambda}}(U_{k})$ are dense in
$\mathbb{C}$. Then for any $c\in\mathbb{C}$, all of the sets
$f_{\bm{\lambda}+c}(U_{k})$ are dense in $\mathbb{C}$.
3. (c)
Assume that $f_{\Lambda}(U)$ evades a closed real half-plane for some open
$U\subseteq D$ with $1\notin\overline{U}$. Then for any $c\in\mathbb{C}$, the
same holds for $f_{\Lambda+c}(U)$.
4. (d)
If the assumption of a (resp. b) is a generic property in $\Omega_{\Lambda}$,
then its implication is a generic property in $\Omega_{c\Lambda}$ (resp.
$\Omega_{\Lambda+c}$).
###### Remark 1.
The assumption of Lemma 1 b looks a bit complicated and one may wonder if it
can be formulated in a much simpler way, akin to Lemma 1 a. Notably, one can
intuitively believe that assuming $\overline{U}\cap
S\setminus\\{1\\}\neq\emptyset$, the density of $f_{\bm{\lambda}}(U)$ in
$\mathbb{C}$ implies that $f_{\bm{\lambda}+c}(U)$ is also dense in
$\mathbb{C}$ for any $c\in\mathbb{C}$. (This formulation would be directly
usable in the proof of Theorem 1 and 2 to translate $\Lambda$.) However, this
claim is false: a simple counterexample is given by
$\Lambda=\\{0,1\\},\qquad U=D$
and
$f_{\bm{\lambda}}(z)=\sum_{k=0}^{\infty}z^{2k+1}=\frac{z}{1-z^{2}}.$
Indeed, $f_{\bm{\lambda}}(z)=\alpha$ leads to a quadratic equation such that
its roots has product -1. Consequently, it has a root in $\overline{D}$, which
quickly yields the density of $f_{\bm{\lambda}}(D)$. However,
$f_{\bm{\lambda}-\frac{1}{2}}(z)=f_{\bm{\lambda}}(z)-\frac{1}{2}\cdot\frac{1}{1-z}=-\frac{1}{2(1+z)},$
for which $f_{\bm{\lambda}-\frac{1}{2}}(U)$ is clearly not dense in
$\mathbb{C}$.
###### Proof of Lemma 1.
Statement a follows trivially from the relation
$f_{c\lambda}(U)=cf_{\lambda}(U)$.
To prove b and c, consider the mapping
(1) $f_{\bm{\lambda}}\mapsto f_{\bm{\lambda}+c},\qquad
f_{\bm{\lambda}+c}(z)=f_{\bm{\lambda}}(z)+\frac{c}{1-z}=f_{\bm{\lambda}}(z)+g(z).$
Now if $(U_{k})_{k=1}^{\infty}$ is a sequence satisfying the conditions of b,
we clearly have $\text{diam}(g(U_{k}))\to 0$, which easily implies the
statement due to (1).
As for c, observe that in (1), $g(U)$ is bounded under the assumptions.
Therefore if $f_{\Lambda}(U)$ is a subset of a closed real half-plane, then so
is $f_{\Lambda+c}(U)$.
Finally, let us observe that d is obvious. ∎
The following three lemmata serve as a preparation to the proof of Theorem 1.
###### Lemma 2.
Assume that $(H_{j})_{j=1}^{\infty}$ is a sequence of dense subsets of $S$.
Then $\bigcup_{k=1}^{\infty}\sum\limits_{j=1}^{k}H_{j}$ is dense in
$\mathbb{C}$, where $\sum\limits_{j=1}^{k}H_{j}$ denotes the Minkowski sum of
$H_{1},...,H_{k}$.
###### Proof.
The proof follows from three simple observations:
* •
for any $k\geq{1}$,
$\overline{\left(\sum_{j=1}^{k}H_{j}\right)}=\sum_{j=1}^{k}\overline{H_{j}};$
* •
$S_{1}+S_{1}=\overline{D_{2}}$;
* •
$D_{r}+S_{r^{\prime}}=D_{r+r^{\prime}}$ for $0<r^{\prime}<r$.
Putting together these claims yields that
$\overline{\left(\sum_{j=1}^{k}H_{j}\right)}=\overline{D_{k}}$
for $k\geq 2$. Consequently,
$\overline{\left(\bigcup_{k=1}^{\infty}\sum\limits_{j=1}^{k}H_{j}\right)}=\mathbb{C}$
clearly holds. ∎
###### Lemma 3.
Let $\zeta\in S$ be different from $\pm 1,\pm i,\pm\omega,\pm\omega^{2}$,
where $i=e(1/4)$, $\omega=e(1/3)$. Then for any $N\in\mathbb{N}$,
$\\{0,1\\}_{N}[\zeta]$ is dense in $\mathbb{C}$, i.e. $\\{0,1\\}_{N}[\zeta]$
is an $\varepsilon$-net of $\mathbb{C}$ for any $\varepsilon>0$.
###### Proof.
If $\zeta=e(x)$ with $x\in\mathbb{R}\setminus\mathbb{Q}$, then the statement
follows simply from Lemma 2. Indeed, if $w\in\mathbb{C}$ and $\varepsilon>0$
be given, our goal is to approximate $w$ with error smaller than $\varepsilon$
with a finite sum of the form given in the statement. Setting
$H_{1}:=H_{2}:=\ldots:=\\{\zeta^{N},\zeta^{N+1},\zeta^{N+2},\ldots\\}$
in Lemma 2, we obtain a certain $z:=\zeta^{n_{1}}+\ldots+\zeta^{n_{k}}$ such
that $|z-w|<\varepsilon$. Possibly there are repetitions among the $n_{j}$’s,
but any $\zeta^{n_{j}}$ can be replaced with another $\zeta^{n_{j^{\prime}}}$
on the cost of an arbitrarily small error, which altogether verifies the
claim.
If $\zeta=e(x)$ with $x\in\mathbb{Q}$, a root of unity different from $\pm
1,\pm i,\pm\omega,\pm\omega^{2}$, then its degree over $\mathbb{Q}$ is greater
than $2$. In particular, $\zeta+\zeta^{-1}\in\mathbb{R}\setminus\mathbb{Q}$,
and since $\zeta,\zeta^{-1}$ can be obtained as arbitrarily large powers of
$\zeta$, we see that $\\{0,1\\}_{N}[\zeta]$ is dense in $\mathbb{R}$ for any
$N\in\mathbb{N}$. Then obviously $\\{0,1\\}_{N}[\zeta]$ is dense also in
$\zeta\mathbb{R}$, hence in $\mathbb{C}$, too. ∎
###### Lemma 4.
Let $\zeta\in S$ be different from $\pm 1$. Then $\\{0,1\\}_{N}[\zeta]$ is a
$1$-net of $\mathbb{C}$ for any $N\in\mathbb{N}$.
###### Proof.
If $\zeta\neq\pm i,\pm\omega,\pm\omega^{2}$, then the statement is obvious
from Lemma 3. Otherwise, we may assume $N=0$, and if $\zeta=\pm i$ (resp.
$\zeta=\pm\omega$ or $\zeta=\pm\omega^{2}$), then
$\mathbb{Z}[\zeta]\subset\mathbb{C}$ is the lattice of Gaussian (resp.
Eulerian) integers, which are known from elementary geometry to satisfy that
for any $w\in\mathbb{C}$, there exist $a_{0},a_{1}\in\mathbb{Z}$ such that
$\left|w-(a_{0}+a_{1}\zeta)\right|<1.$
Again, the potientially negative coefficients can be switched to a sum of
positive ones by recording
$-1=\sum_{j=1}^{11}\zeta^{j},\qquad-\zeta=\sum_{j=2}^{12}\zeta^{j},$
and repetitions can be treated via $\zeta^{k}=\zeta^{k+12}$. ∎
The following three lemmata serve as a preparation to the proof of Theorem 3.
###### Lemma 5.
If $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real half-plane
of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$, then we can find
$\Delta_{0}=\Delta_{4},\Delta_{1},\Delta_{2},\Delta_{3}$ elements of $\Lambda$
such that
(2) $0\leq\arg{\frac{\Delta_{j+1}}{\Delta_{j}}}<\pi$
for any $0\leq j\leq 3$.
###### Proof.
Multiplying $\Lambda$ by a nonzero scalar does not change the assumption, nor
the implication. Consequently, we can assume $1\in\Lambda$. Let
$\Delta_{0}=1$. By the condition on $\Lambda$, $\\{\lambda:\Im\lambda>0\\}$ is
nonempty. Consequently, we can define
$\beta:=\sup_{\lambda\in\Lambda,\text{ }\Im\lambda>0}\arg{\lambda}.$
Now by the same argument, $\\{\lambda\in\Lambda:\text{
}\beta<\arg\lambda<\beta+\pi\\}$ is nonempty. However, due to the definition
of $\beta$, we can deduce that $\\{\lambda\in\Lambda:\text{
}\pi\leq\arg\lambda<\beta+\pi\\}$ is nonempty. Let $\Delta_{2}$ be an element
of it. Due to the definition of $\beta$, we can find $\Delta_{1}$ such that
(2) is satisfied for $j=0,1$. Now if $\arg{\Delta_{2}}>\pi$, the choice
$\Delta_{3}=\Delta_{2}$ guarantees that it is also satisfied for $j=2,3$ and
we are done. Otherwise we can choose $\Delta_{3}$ to be any element of the
necessarily nonempty $\\{\lambda\in\Lambda,\text{ }\Im\lambda<0\\}$, which
yields (2) for $j=2,3$. ∎
###### Lemma 6.
Assume that $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Then
there exists an appropriate $R>0$ with the property that for any
$z\in\mathbb{C}$ satisfying $|z|>R$, we can find $\lambda\in\Lambda$ such that
$|z+\lambda|<|z|$.
###### Proof.
Fix $\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3}$ as guaranteed by Lemma 5.
Let
$\alpha_{0}:=\max_{0\leq j\leq 3}\arg{\frac{\Delta_{j+1}}{\Delta_{j}}}<\pi.$
Now, if $z\neq 0$ is arbitrary, we can find $0\leq j\leq 3$ such that for
$\Delta_{j}$ we have
$\frac{\pi+\alpha_{0}}{2}\leq\arg\frac{\Delta_{j}}{z}<\frac{3\pi-\alpha_{0}}{2}$.
Consequently, if we consider the triangle determined by $0,z,z+\Delta_{j}$, we
have that the angle at $z$ is smaller than the right angle, and its size is
bounded away from $\frac{\pi}{2}$ by some positive quantity. As the set of all
$\Delta_{j}$’s is bounded, this implies that if $|z|\geq R$ for large enough
$R$, then the side $[0,z]$ of the triangle is larger than the side
$[0,z+\Delta_{j}]$. Defining $R$ accordingly proves the lemma. ∎
Note that if $R$ is sufficient for some $\Lambda$ in the setup of Lemma 6,
then $cR$ is sufficient for $c\Lambda$. In particular, if $c<1$, the same $R$
can be used.
###### Lemma 7.
Assume that $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Then
there exists $R^{*}>0$ such that for any $z$, $|z|<1$, and $0\leq
n_{0}<n_{1}<\ldots$, there exists $(\lambda_{n_{j}})_{j=0}^{\infty}$,
$\lambda_{n_{j}}\in\Lambda$ such that
$\left|\sum_{j=0}^{\infty}\lambda_{n_{j}}z^{n_{j}}\right|\leq R^{*}$.
###### Proof.
We prove that $R^{*}=R+\sup_{\lambda\in\Lambda}|\lambda|$ is sufficient, where
$R$ is the one guaranteed by Lemma 6. Due to the note following its proof, the
same $R$ can be used for any coefficient set of the form $z^{n}\Lambda$.
For $z=0$, the claim is trivial, regardless of the choice of
$(\lambda_{n_{j}})_{j=0}^{\infty}$. Hence fix $z\neq 0$, $|z|<1$. The proof
depends on a recursive construction of the sequence $(\lambda_{n_{j}})$.
Notably, let $\lambda_{n_{0}}\in\Lambda$ be arbitrary, and assume
$\lambda_{n_{0}},\ldots,\lambda_{n_{k}}$ are already defined. If
$\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|\leq R,$
then $\lambda_{n_{k+1}}\in\Lambda$ can be chosen arbitrarily as well.
Otherwise, we apply Lemma 6 to $z^{n_{k+1}}\Lambda$ to define
$\lambda_{n_{k+1}}$ such that
$\left|\sum_{j=0}^{k+1}\lambda_{n_{j}}z^{n_{j}}\right|<\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|.$
These choices obviously guarantee that
$\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|\leq
R+\max_{\lambda\in\Lambda}|\lambda|=R^{*},$
regardless of the value of $k$. Consequently, the same bound holds for the sum
of the series as well. ∎
## 3\. Proof of Theorem 1
Assume first that $0,1\in\Lambda$. For any fixed $w\in\mathbb{C}$ and
$\varepsilon>0$, we introduce
$A_{w,\varepsilon}:=\\{\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}:\text{there
exists some $\tau\in U$ such that
$|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$}\\}.$
Fixing $w\in\mathbb{C}$ and $\varepsilon>0$, we introduce the abbreviation
$A:=A_{w,\varepsilon}$, and prove below that it is open and dense in $\Omega$.
To see that $A$ is open in $\Omega$, let
$\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}\in A$, i.e. for some $\tau\in U$,
$\varepsilon_{0}:=|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$. Let $N$ be large
enough to satisfy that
$\sum_{n>N}\sup\\{|\lambda|:\lambda\in\Lambda\\}\tau^{n}<\frac{\varepsilon-\varepsilon_{0}}{2}.$
Also, choose $\delta>0$ in such a way that if
$|\lambda_{n}^{\prime}-\lambda_{n}|<\delta$ for all $n\leq N$, then
$\left|\sum_{n\leq N}\lambda_{n}\tau^{n}-\sum_{n\leq
N}\lambda_{n}^{\prime}\tau^{n}\right|<\frac{\varepsilon-\varepsilon_{0}}{2}.$
Clearly, if
$\bm{\lambda}^{\prime}=(\lambda_{n}^{\prime})_{n\in\mathbb{N}}\in\bigtimes_{n\leq
N}\\{\lambda_{n}^{\prime}:|\lambda_{n}^{\prime}-\lambda_{n}|<\delta\\}\times\bigtimes_{n>N}\Lambda\subseteq\Omega,$
then
$|f_{\bm{\lambda}^{\prime}}(\tau)-w|<\varepsilon,$
hence $\bm{\lambda}^{\prime}\in A$, which shows that $A$ is open.
Now we prove that $A$ is dense in $\Omega$. It suffices to show that $A$
intersects any set of the form
$G:=\\{\lambda_{0}\\}\times\ldots\times\\{\lambda_{N}\\}\times\bigtimes_{n>N}\Lambda\subseteq\Omega.$
Let us fix some $\pm 1\neq\zeta\in\overline{U}\cap S$ throughout the proof.
Our goal is to find an element $\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}\in
G$ and some $\tau\in U$ such that $|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$. We
immediately prescribe $|\tau-\zeta|<\delta$ with $0<\delta<2/5$ chosen in such
a way that
$\left|\sum_{n\leq N}\lambda_{n}\tau^{n}-\sum_{n\leq
N}\lambda_{n}\zeta^{n}\right|<\frac{\varepsilon}{2}$
is guaranteed by $|\tau-\zeta|<\delta$.
Relabeling our original $w$ (shifting it by $-\sum_{n\leq
N}\lambda_{n}\zeta^{n}$), and rescaling $\varepsilon$, we have to find a
sequence $(\lambda_{n})_{n>N}\in\\{0,1\\}^{n>N}$ and some $\tau\in U$ in such
a way that
(3)
$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-\zeta|<\delta.$
Now we go by cases according to Lemmata 3–4 about the nature of $\zeta$. To
avoid notational difficulties, assume that $w\neq 0$ (which is not a real
restriction, since if anything but zero can be arbitrarily approximated, then
so can zero).
If $\zeta\neq\pm i,\pm\omega,\pm\omega^{2}$, then by Lemma 3, we may find and
fix a _finite_ sequence $(\lambda_{n})_{N<n<K}$ satisfying
$\left|\sum_{N<n<K}\lambda_{n}\zeta^{n}-w\right|<\frac{\varepsilon}{2},\qquad\lambda_{n}\in\\{0,1\\},$
and then if $|\tau-\zeta|$ is small enough (consistently with the earlier
prescribed $|\tau-\zeta|<\delta$),
$\left|\sum_{N<n<K}\lambda_{n}\tau^{n}-\sum_{N<n<K}\lambda_{n}\zeta^{n}\right|<\frac{\varepsilon}{2},\qquad|\tau-\zeta|<\delta,$
which, setting $\lambda_{K}:=\lambda_{K+1}:=\ldots:=0$, together clearly imply
(3).
Now assume that $\zeta\in\\{\pm i,\pm\omega,\pm\omega^{2}\\}$. Fix a finite
set $W$ which is a $1$-net of $D_{10|w|/\varepsilon}$. By Lemma 4, for any
$w^{\prime}\in W$, we may find and fix a _finite_ sequence
$(\lambda_{n})_{N<n<K}(w^{\prime})$ satisfying
$\left|\sum_{N<n<K}\lambda_{n}(w^{\prime})\zeta^{n}-w^{\prime}\right|<1,\qquad\lambda_{n}(w^{\prime})\in\\{0,1\\},$
where the upper bound $K$ on the coefficient indices is uniform over
$w^{\prime}\in W$ (this can be achieved, since $W$ is finite). Now if
$|\tau-\zeta|$ is small enough (consistently with the earlier prescribed
$|\tau-\zeta|<\delta$), then
$\left|\sum_{N<n<K}\lambda_{n}(w^{\prime})\tau^{n}-\sum_{N<n<K}\lambda_{n}(w^{\prime})\zeta^{n}\right|<1\text{
for all $w^{\prime}\in W$},\qquad|\tau-\zeta|<\delta.$
Fix now $M$ in such a way that $\varepsilon/5<|\tau^{M}|<\varepsilon/3$ (this
is possible, since $\delta<2/5$ implies $|\tau|>3/5$, and that $\tau$ has a
power in the indicated annulus). Then
$\left\\{\sum_{N<n<K}\lambda_{n}(w^{\prime})\tau^{n+M}:w^{\prime}\in
W\right\\}$
gives rise to an $\varepsilon$-net of $D_{2|w|}$. In particular, choosing the
appropriate $w^{\prime}$ (for which the sum in the last display is closest to
$w$), and setting
$\lambda_{n}:=\begin{cases}\lambda_{n},\qquad&\text{if $n\leq N$,}\\\
\lambda_{n-M}(w^{\prime}),&\text{if $N+M<n<K+M$,}\\\
0&\text{otherwise,}\end{cases}$
(3) is achieved.
To sum up, any $A_{w,\varepsilon}$ is open and dense, which in turn implies
that the set
$\bigcap_{w\in\mathbb{Q}+\mathbb{Q}i}\bigcap_{k=1}^{\infty}A_{w,1/k}$
is residual, hence the proof of Theorem 1 is complete, at least, when
$0,1\in\Lambda$. This, however, immediately gives rise to the general case by
applying Lemma 1 a–b. Indeed, the proof of the density of $A$ presented above
guarantees $\tau$’s arbitrarily close to $\zeta$, which means that the
condition of Lemma 1 b is satisfied.
A fairly straightforward consequence of Theorem 1 is the following:
###### Corollary 1.
For the generic $(\lambda_{n})_{n=0}^{\infty}\in\Omega$, for any $\zeta\in S$
and $w\in\mathbb{C}$ there exists $(\zeta_{k})_{k=1}^{\infty}\subseteq D$ with
$\zeta_{k}\to\zeta$ such that $f(\zeta_{k})\to w$.
The proof is left as a simple exercise to the reader, it suffices to rely on
the case of irrational arguments. With a slightly different formulation,
proving this statement was Problem 9 at the prestigious Miklós Schweitzer
Memorial Competition for Hungarian university students in 2020, proposed by
the authors. Complete solutions were given by Márton Borbényi and Attila
Gáspár, who were awarded the two first prizes of the contest, and to whom we
congratulate hereby. A direct solution is available at [7] in Hungarian.
## 4\. Proof of Theorem 2
Due to the open mapping theorem, it suffices to prove that the density of
$f(U)$ holds generically.
Due to Lemma 1 a–b, we can assume $0,1\in\Lambda$. First we will consider the
case when $\Lambda$ is contained by a real line. Following our assumption,
this means $\Lambda\subseteq\mathbb{R}$.
We will define $U=\bigcup_{k=1}^{\infty}U_{k}$ where
$U_{k}=\left\\{z:-\frac{k-1}{k}<\Re z<0,\text{
}\pi-\alpha_{k}<\arg{z}<\pi+\alpha_{k}\right\\}$
for $\alpha_{k}>0$ to be fixed later. (The lower bound on $\Re z$ is somewhat
arbitrary, separation from $-1$ is relevant only.) Right now we specify only
that $\alpha_{k}$ is small enough to guarantee that $\overline{U_{k}}\subseteq
D$.
As each $U_{k}$ is open, the same holds for $U$. Now consider any $z\in
U_{k}$. As $\overline{U_{k}}\subseteq S$, we can choose $N$ large enough to
have
(4) $\left|\sum_{n=2N}^{\infty}z^{n}\right|<1.$
We choose $\alpha_{k}$ based on the choice of $N$ such that
$2N\alpha_{k}<\arcsin\left(\frac{1}{N}\right).$
This clearly implies that
$-\arcsin\left(\frac{1}{N}\right)<\arg{\left(\sum_{n=0}^{N-1}z^{2n}\right)}<\arcsin\left(\frac{1}{N}\right)$
and
$\pi-\arcsin\left(\frac{1}{N}\right)<\arg{\left(\sum_{n=0}^{N-1}z^{2n+1}\right)}<\pi+\arcsin\left(\frac{1}{N}\right).$
However, the absolute value of each of these partial sums is at most $N$,
which yields that their imaginary part is at most $1$. Consequently,
$\Im\left(\sum_{n=0}^{2N-1}z^{n}\right)\leq 2.$
By the same argument for any $(\lambda_{n})_{n=0}^{\infty}$ we have
$\Im\left(\sum_{n=0}^{2N-1}\lambda_{n}z^{n}\right)\leq
2\sup_{\lambda\in\Lambda}|\lambda|.$
Taking (4) into consideration implies
$\Im\left(\sum_{n=0}^{\infty}\lambda_{n}z^{n}\right)\leq
3\sup_{\lambda\in\Lambda}|\lambda|.$
As it holds for any $k$ and $z\in U_{k}$, we have it for any $z\in U$, which
concludes the proof of the first part.
In the other direction, our argument will be similar to the one we have given
in the proof of Theorem 1. Indeed, defining the set $A$ as above, and
following the argument verbatim, it suffices to find a sequence
$(\lambda_{n})_{n>N}\in\\{0,1\\}^{n>N}$ and some $\tau\in U$ in such a way
that
$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-(-1)|<\delta.$
Fix an element $\lambda\in\Lambda$ with non-vanishing imaginary part; its
existence is guaranteed by the assumption that $0,1\in\Lambda$ and
$\Lambda\nsubseteq\mathbb{R}$. Consider the lattice
$\\{a+b\lambda:\text{ }a,b\in\mathbb{Z}\\}.$
It obviously gives a $\delta$-net of $\mathbb{C}$ for some $\delta>0$.
Consequently,
$\\{\xi(a+b\lambda):\text{ }a,b\in\mathbb{Z}\\}$
gives an $\varepsilon/2$-net of $\mathbb{C}$ for $|\xi|=\varepsilon_{0}$ if
$\varepsilon_{0}>0$ is small enough. Fix $\varepsilon_{0}$ accordingly, and
fix $R$ such that $|w|<\varepsilon_{0}R$. Now it is clear that one can find
$m_{R}$ such that for
$C^{m_{R}}=\\{a+b\lambda:\text{ }a,b\in\mathbb{Z},\text{ }|a|,|b|<m_{R}\\},$
$\xi C^{m_{R}}$ gives an $\varepsilon/2$-net of $D_{R}$ for
$|\xi|=\varepsilon_{0}$.
Now let us notice that for large enough $M_{R}$, any element of $C^{m_{R}}$
can be written in the form
$\lambda\sum_{j=1}^{k}(-1)^{n_{j}}+\sum_{j^{\prime}=1}^{l}(-1)^{n^{\prime}_{j^{\prime}}},$
where the exponents used are pairwise distinct, and
$N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}$ for $j=1,\ldots,k$ and
$j^{\prime}=1,\ldots,l$. Denote the set of such sums by $C(-1)$, and motivated
by this, let
$C(z)=\left\\{\lambda\sum_{j=1}^{k}z^{n_{j}}+\sum_{j^{\prime}=1}^{l}z^{n^{\prime}_{j^{\prime}}}:\text{
}N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}\text{ are distinct}\right\\}.$
As $C(z)$ is determined by finitely many continuous functions of $z$, if
$|(-1)-\tau|$ is small enough, $\xi C(\tau)$ gives a $\varepsilon$-net of
$D_{\varepsilon_{0}R}$ for any $\xi,\text{ }|\xi|=\varepsilon_{0}$. On the
other hand, we can choose $\tau$ in any neighborhood of $-1$ so that
$|\tau|^{M}=\varepsilon_{0}$ for some $M>0$. Consequently, we have that
$\tau^{M}C(\tau)$ forms a $\varepsilon$-net of $D_{\varepsilon_{0}R}$.
By definition, this implies that there exist pairwise distinct numbers
$N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}$ for $j=1,...,k$, $j^{\prime}=1,...,l$
such that
$\left|\left(\lambda\sum_{j=1}^{k}\tau^{n_{j}+M}+\sum_{j^{\prime}=1}^{l}\tau^{n^{\prime}_{j^{\prime}}+M}\right)-w\right|<\varepsilon.$
Now if we define $(\lambda_{n})_{n=N+1}^{\infty}$ such that
$\lambda_{n}=\lambda$ if and only if $n=n_{j}+M$ for some $1\leq j\leq k$,
moreover, $\lambda_{n}=1$ if and only if $n=n^{\prime}_{j^{\prime}}+M$ for
some $1\leq j^{\prime}\leq l$, and otherwise $\lambda_{n}=0$, then we
immediately obtain $|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$, which concludes
the proof.
## 5\. Proof of Theorem 3
Due to the open mapping theorem, it suffices to prove that the density of
$f(U)$ holds generically.
First we will consider the case when $\Lambda$ is contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Due
to Lemma 1 a, we can assume that this half-plane is $\\{\Re z\geq 0\\}$.
We will define $U=\bigcup_{k=1}^{\infty}U_{k}$ where
$U_{k}=\left\\{z:0<\Re z<\frac{k-1}{k},\text{
}-\alpha_{k}<\arg{z}<\alpha_{k}\right\\}$
for $\alpha_{k}>0$ to be fixed later. (The upper bound on $\Re z$ is somewhat
arbitrary, separation from $1$ is relevant only.) Right now we specify only
that $\alpha_{k}$ is small enough to guarantee that $\overline{U_{k}}\subseteq
S$.
From this point, the proof of this part is basically a simplified version of
the proof of the same part of the proof of Theorem 2. Notably, for $z\in
U_{k}$ we can find a threshold index such that the tail sum is very small due
to $|z|$ being bounded away from 1, and then by choosing $\alpha_{k}$ to be
small enough, we can control the argument of the preceding terms. (The
relative simplicity in this case is due to the fact these arguments are all
near 0, instead of being near to 0 and $\pi$ alternatingly.) Based on these
estimates, the real part of $f_{\bm{\lambda}}(z)$ can be bounded from below,
regardless of $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}$ and $z\in U_{k}$,
which concludes the proof of the first part.
We now prove the second part of the statement of the theorem. Defining the set
$A$ as in the proof of Theorem 1 and following the argument verbatim, it
suffices to find a sequence $(\lambda_{n})_{n>N}\in\Lambda^{n>N}$ and some
$\tau\in U$ in such a way that
$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-1|<\delta.$
Define $\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3}\in\Lambda$ as guaranteed
by Lemma 5. Denote their set by $V$, and define the convex polygon
$P=\text{conv}(V)$. Due to the choice of $V$, $0\in\operatorname{int}(P)$,
that is $D_{r}\subseteq V$ for small enough $r$.
Notice that if $P$ has diameter $\delta$, then $V$ is clearly a $\delta$-net
of $P$. Moreover, for any $m$ we have that the Minkowski sum $\sum_{i=1}^{m}V$
is a $\delta$-net of $\sum_{i=1}^{m}P$: the proof proceeds by induction,
capitalizing on the simple observation that
$\sum_{i=1}^{m}P=V+\sum_{i=1}^{m-1}P$. It clearly yields that
$\sum_{i=1}^{m}V$ is a $\delta$-net of $D_{mr}$ as well for any $m$.
Consequently, $\xi\cdot\sum_{i=1}^{m}V$ gives an $\varepsilon/2$-net of
$D_{\varepsilon_{0}mr}$ for any $m$ and for $|\xi|=\varepsilon_{0}$, if
$\varepsilon_{0}>0$ is small enough. Fix $\varepsilon_{0}$ accordingly, noting
that it does not depend on $m$. Now fix $R^{*}$ as guaranteed by Lemma 7, and
based on the choice of $R^{*}$ and $\varepsilon_{0}$, fix $m$ such that
$|w|+R^{*}<\varepsilon_{0}mr$. Let us remark that any element of
$\sum_{i=1}^{m}V$ is expressible in the seemingly complicated form
$\sum_{j=0}^{3}\Delta_{j}k_{j}=\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}1^{n^{(j)}_{l}},$
where each $0\leq k_{j}\leq m$, and the exponents $n^{(j)}_{l}$ are pairwise
distinct and their union equals $\\{N+1,N+2,...,N+m\\}$. Let us denote the set
of such combinations by $C(1)$, and motivated by this, let
$C(z)=\left\\{\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}z^{n^{(j)}_{l}}:\text{
}0\leq k_{j}\leq m,\text{ }N<n^{(j)}_{l}\leq N+m\text{ are all distinct and
their union is }\\{N+1,N+2,...,N+m\\}\right\\}.$
As $C(z)$ is determined by finitely many continuous functions of $z$, if
$|1-\tau|$ is small enough, $\xi C(\tau)$ gives a $\varepsilon$-net of
$D_{\varepsilon_{0}mr}$ for any $\xi,\text{ }|\xi|=\varepsilon_{0}$. On the
other hand, we can choose $\tau$ in any neighborhood of $1$ so that
$|\tau|^{M}=\varepsilon_{0}$ for some $M>0$. Consequently, we have that
$\tau^{M}C(\tau)$ forms an $\varepsilon$-net of $D_{\varepsilon_{0}mr}$.
So far the coefficients of the power series we would like to define are fixed
for the indices $(i)_{i=0}^{N}$. Motivated by the previous paragraph, we would
like to set aside the indices $(N+i+M)_{i=1}^{m}$. Notably, these are the
indices which are intimately connected to the lastly defined
$\tau^{M}C(\tau)$. Consequently, we apply Lemma 7 at this point for $\tau$ and
the complementary sequence $(N+1,N+2,...,N+M,N+m+M+1,N+m+M+2,...)$: we can
find elements of $\Lambda$ corresponding to these indices,
$(\lambda_{n})_{n=N+1}^{N+M}$, $(\lambda_{n})_{n=N+m+M+1}^{\infty}$ such that
$|w_{1}|\leq R^{*},\qquad\text{where}\qquad
w_{1}:=\sum_{n=N+1}^{N+M}\lambda_{n}z^{n}+\sum_{n=N+m+M+1}^{\infty}\lambda_{n}z^{n}.$
Consequently, $|w-w_{1}|\leq|w|+R^{*}<\varepsilon_{0}R$. This implies that
there exist numbers
$N<n^{(j)}_{l}\leq N+m$
for $j=0,1,2,3$ and $l=1,...,k_{j}$, such that their union fills
$\\{N+1,N+2,...,N+m\\}$ without any repetitions (that is the numbers
$n^{(j)}_{l}$ are pairwise distinct for all the possible choices of $j,l$),
and
(5)
$\left|\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}\tau^{n^{(j)}_{l}+M}-(w-w_{1})\right|<\frac{\varepsilon}{2}.$
What remains from the definition of $(\lambda_{n})$ is fixing
$(\lambda_{n})_{n=N+1+M}^{N+m+M}$, which we carry out now based on (5).
Notably let $\lambda_{n}=\Delta_{j}$ if and only if $n=n^{(j)}_{l}+M$ for some
$1\leq l\leq k_{j}$. Then by the definition of $w_{1}$ we obtain
$|\sum_{n=N+1}^{\infty}\lambda_{n}\tau^{\prime n}-w|<\varepsilon$. This
concludes the proof.
## 6\. Concluding remarks
Even though with a careful separation of cases we managed to generalize the
most natural result given by Theorem 1, our results are far from being
complete. In our view, the most interesting open problem related to them is
whether $f(U)=\mathbb{C}$ holds generically in the setup of our theorems,
similarly to what is proved in [2]. As $f$ is uniformly locally bounded in
$D$, we clearly cannot rely on techniques similar to the ones seen there,
hence answering this question requires additional ideas.
Another interesting aspect partially inspired by this paper is whether we can
rearrange the quantifiers in our statements to some extent. More explicitly,
each of our theorems addresses the question of what the generic image is of a
fixed open set. It would be desirable to find extensions of this result, for
example a nontrivial family of open sets such that generically, the image of
each of them is dense.
## References
* [1] J. Breuer , B. Simon, Natural boundaries and spectral theory, Adv. Math. 226, (2011), 4902–4920.
* [2] J.-P. Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math. 80, (2000), 143–182.
* [3] J.-P. Kahane, Some Random Series of Functions, Heath, Mass., 1968; 2nd edn. Cambridge Univ. Press, 1985, (1993).
* [4] S. Kierst, E. Szpilrajn, Sur certaines singularités desfonctions analytiques uniformes, Fund. Math. 21, (1933), 267-294.
* [5] K. Kuratowski, Topologie, Vol. 1, 4th ed., PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, (1966).
* [6] B. Maga, P. Maga, Random power series near the endpoint of the convergence interval, Publ. Math. Debrecen, 93 (3-4). (2018), 413-424. ISSN 0033-3883
* [7] https://www.bolyai.hu/Schweitzer2020_elozetes_megoldasok.pdf
|
SEPAR – Covid-19]Studying the course of Covid-19 by a recursive delay approach
M. Kreck, E. Scholz]Matthias Kreck$^{\dag}$, Erhard Scholz $^{\ddag}$
In an earlier paper we proposed a recursive model for epidemics; in the present paper we generalize this model to include the asymptomatic or unrecorded symptomatic people, which we call dark people (dark sector). We call this the SEPAR$_d$-model. A delay differential equation version of the model is added; it allows a better comparison to other models. We carry this out by a comparison with the classical SIR model and indicate why we believe that the SEPAR$_d$ model may work better for Covid-19 than other approaches.
In the second part of the paper we explain how to deal with the data provided by the JHU, in particular we explain how to derive central model parameters from the data. Other parameters, like the size of the dark sector, are less accessible and have to be estimated more roughly, at best by results of representative serological studies which are accessible, however, only for a few countries.
We start our country studies with Switzerland where such data are available.
Then we apply the model to a collection of other countries, three European ones (Germany, France, Sweden), the three most stricken countries from three other continents (USA, Brazil, India). Finally we show that even the aggregated world data can be well represented by our approach.
At the end of the paper we discuss the use of the model. Perhaps the most striking application is that it allows a quantitative analysis of the influence of the time until people are sent to quarantine or hospital. This suggests that imposing means to shorten this time is a powerful tool to flatten the curves.
[2]Mathematisches Institut der Universität Bonn and Mathematisches Institut der Universität Frankfurt, Germany<EMAIL_ADDRESS>[3]University of Wuppertal, Faculty of Math./Natural Sciences, and Interdisciplinary Centre for History and Philosophy of Science<EMAIL_ADDRESS>
§ INTRODUCTION
There is a flood of papers using the standard S(E)IR models for describing the outspread of Covid-19 and for forecasts. Part of them is discussed in <cit.>. We propose alternative delay models and explain the differences.
In <cit.> we have proposed a discrete delay model for an epidemic which we call SEPAR-model (in our paper we called it $SEPIR$ model). In this paper we explained why and under which conditions the model is adequate for an epidemic. In the present note we add two new compartments reflecting asymptomatic or symptomatic, but not counted, infected which we call the dark sector. We call this model the generalized SEPAR-model, abbreviated SEPAR$_d$, where $d$ stands for dark. This is our main new contribution. We will discuss the role of the dark sector in a theoretical comparison of the SEPAR$_d$-model with the SEPAR model. We will see, that – as expected – as long as the number of susceptibles is nearly constant, the difference of the two models is small, but in the long run it matters.
A second topic in this paper is a comparison with the standard SIR model. This comparison has two aspects, a purely theoretical one by comparing the different fundaments on which the models are based, and a numerical one. For comparing two models it is helpful to derive them from similar inputs. For this we pass from the discrete model leading to difference equations to a continuous model, replacing difference equations by differential equations. These differential equations fit into the general approach developed by Kermack/McKendrick in <cit.>, as we learnt from O. Diekmann. The analytic model resulting from our discrete model has been introduced independently by J. Mohring and coauthors <cit.> and, more recently, by B. Shayit and M. Sharma <cit.>. Also F. Balabdaoui and D. Mohr work with a discrete delay approach with additional compartments and a stratification into different age layers adapted to the Swiss context <cit.>.
Recently y R. Feßler has written a paper <cit.> in which different differential equation models are discussed and compared, including the classical S(E)IR-model and the analytic version of our model.
Some hints to earlier papers on the analytic delay approach can be found there.
In the second part of the present paper we apply the SEPAR model to selected countries and to the aggregated data of the world. To do so we first lay open how to pass from the data provided by the Humdata project of the JHU to the model parameters. The data themselves are obviously not reflecting the actual outspread correctly, which is most visible by the lower numbers of reported cases during weekends. But in addition there are aspects of the data which need to be corrected like for example a delay of reporting of recovered cases. All this is discussed carefully. Reliable data about the size of the dark sector are only available in certain countries where such studies were carried out. We found such studies for Germany and Switzerland, for other countries we estimate these numbers as good as we can. The case of Switzerland is particular interesting since the effect of the dark sector which started to play a non-negligible role for the overall dynamics of the epidemic in the later part of 2020 for the majority of the countries discussed here (India, USA, Brazil, France) can be studied there particularly well. For that reason we begin with this country and discuss the role of the dark sector in detail.
The paper closes with a discussion about what one can learn from the applications of the SEPAR model. We address three topics: The role of the constancy intervals, the role of the dark sector and the the influence of the time between infection and quarantine. The latter is perhaps the most striking application of our model offering a door for flattening the curves by sending people faster into quarantine, a restriction which imposes much less harm to the society than other means.
PART I: Theoretical framework
tocchapterPart I: Theoretical framework
§ THE SEPAR MODEL AND ITS COMPARISON WITH OTHER MODELS
§.§ The SEPAR$_d$ model
We begin by pointing out that we have changed our notation from <cit.>. The compartment consisting of those who are isolated after sent to quarantine or hospital, which there was called $I$, is now being denoted by $A$ like actually infected, in some places also described – although a bit misleading – as “active” cases (e.g. in Worldometer). This is why we speak now of the SEPAR model rather than of SEPIR.
Let us first recall the compartments introduced in <cit.>. We observe 5 compartments which we call $S$, $E$, $P$, $A$, $R$, which people pass through in this order: Susceptibles in compartment $S$ moving after infection to compartment $E$, where they are exposed but not infectious, after they are infected by people from compartment $P$ which comprises the actively infectious people, those which propagate the virus. After $e$ days they move from $E$ to the compartment $P$, where they stay for $p$ days. After diagnosis they are sent into quarantine or hospital and become members of the compartment $A$, where they no longer contribute to the spread of the virus although they are then often counted as the actual cases of the statistics. In order not to overload the model with too many details, we pass over the recording delay between diagnosis and the day of being recorded in the statistics. After another $q$ days the recorded infected move from compartment $A$ to the compartment $R$ of removed (recovered or dead).
We add two more compartments reflecting the role of the dark sector. There are two types of infected people, those who will at some moment be tested and counted, and those who are never tested, which we call people in the dark. This suggests to decompose compartment $P$ into two disjoint sub-compartments: $P_c$ of people who after $p_c$ days will be tested and counted and move to compartment $A$, and the collection $P_d$ of people who after a longer period of $p_d$ days get immune and so move into a new compartment $R_d$ of removed people in the dark. To distinguish these removed people in the dark from those who come from compartment $P$ after recovery or death we introduce another new compartment $R_c$ of those removed people who occur in the statistics. Of course $R = R_c \cup R_d$.
The introduction of the dark sector in addition to the sector of counted people leads to the picture that for the infected persons leaving compartment $E$ there is a branching process: a certain fraction $\alpha(k)$ of people from $E$ moves to compartment $P_c$ at day $k$, whereas the fraction $ 1 - \alpha(k)$ of people moves to compartment $P_d$.
The existence of these compartments is a fundamental assumption which distinguishes the SEPAR$_d$ model from many other models including standard SIR. The existence of these compartments is closely related to our picture of an epidemic like Covid-19. Of course this is a simplification. If one assumes that the passage from compartment $S$ to compartment $E$ takes place at a certain moment, the duration of the stay in the next compartments varies from case to case. But it looks natural to take the average of these durations leading for the different lengths $e$, $p_c$, $p_d$, and $q$. All these have to be estimated from available information.
Once one has agreed to this there is another fundamental assumption. This concerns the dynamics of the epidemic. Each person in compartment $P$ has a certain average number $\kappa (k)$ of contacts at day $k$. Depending on the strength of the infectious power of an individual the contacts will lead to newly exposed people. It is natural to model this development of the strength of infectiousness by a function $A(\tau)$, which measures the strength $\tau$ days after entry into compartment $P$. Again we simplify this very much, by replacing $A(\tau)$ by a constant $\gamma$, the average value of this assumed function. We will discuss this assumption later on in the light of information available for Covid-19. Given the parameters $\kappa (t)$ and $\gamma$ our next assumption is that, if we ignore the dark sector and set $\eta (k): = \gamma \kappa(k)$, the dynamics of the infection can be described by the following formula:
E_new (k) = η(k-1) S(k-1)/NP(k-1).
Here $E_{new}(k)$ is the number of additional members of compartment $E$ at day $k$ infected at day $(k-1)$ by people from compartment $P$ and $N$ the total number of the population. This is a very plausible formula. We call $\eta(k) $ the daily strength of infection. It is an integrated expression for the averaged contact behaviour of the population and the aggressiveness of the virus.
This is the dynamics if we ignore the dark sector. But members of compartment $P_d$ also infect. We assume that the contacts are equal to those in compartment $P_c$. But the average of the strength of infection of people from $P_d$ may be smaller than for those in compartment $P_c$, since in general they can be expected to stay longer in their compartment until they are immune and the strength of infection goes further down. Thus we introduce a separate measure $\gamma_c$ for those in compartment $P_c$ and $\gamma_d = \xi \gamma_c$, with $0\leq \xi \leq 1$, for those in compartment $P_d$.
Using this the equation (1) has to be replaced by:
E_new (k) = s(k-1)η(k-1) [P_c(k-1) + ξP_d(k-1)].
Given this infection equation the rest of the model just describes the time shifting passage of infected from one compartment into the next and counts their cardinality at day $k$. As usual we denote the latter by $S(k)$, $E(k)$, $P_c(k)$, $P_d(k)$, $A(k)$, $R_c(k)$ and $R_d(k)$. How such a translation is justified is explained in <cit.>. So we can just write down the self explaining formulas here:
Introduction (definition) of the discrete SEPAR$_{d}$ model:
Let $e$, $p_c$, $p_d$, $q$ be integers standing for the duration of staying in the corresponding compartments, $0 \leq \alpha (k) \leq 1$ be branching ratios at day $k$ between later registered infected and those which are never counted, $\eta (k)$ be positive real numbers describing the daily strength of infection for $k\geq 0$, while $\eta(k)= 0 $ for $k<0$, and $ \xi \le 1 $ a non-negative real number. Using $s(k)=\frac{S(k)}{N}$ the quantities $S(k),\, E(k)$, $E_{new}(k), \, P_c(k),\,P_d(k) , \, A(k),\, R_c(k),\,R_d(k) $ of the SEPAR$_{d}$ model are given by
a) the start condition:
since the model is recursive we need an input for the first $e+p_d$ days (which we shift to negative values of $k$), i.e. start data
$E_{start}(k)$ for
$1-(e+p_d) \leq k \leq 0$, while $E_{start}(k)=0$ for all $k>0$,
$P_c(k)=P_d(k)=A(k) = R_c(k)= R_d(k)=0$ (or some other well defined start values, cf. sec. <ref>) for $ k < 1-(e+p_d) $;
b) the recursion scheme for $k \geq 1-(e+p_d)$:
E_new(k) = s(k-1)η(k-1) [P_c(k-1) +ξP_d(k-1)] + E_start(k)
E(k) = E(k-1) + E_new(k) - E_new(k-e)
P_c(k) =
P_c(k-1) + α(k) E_new(k-e) - α(k-p_c) E_new(k-e-p_c)
P_d(k) =
P_d(k-1) +(1- α(k) ) E_new(k-e) - (1- α(k-p_d) ) E_new(k-e-p_d)
A(k) = A(k-1) + α(k-p_c) E_new(k-e-p_c) - α(k-p_c-q) E_new(k-e-p_c-q)]
R_c(k) = R_c(k-1) + α(k-p_c-q) E_new(k-e-p_c-q)
R_d(k) = R_d(k-1) + (1-α(k-p_d)) E_new(k-e-p_d)
S(k) = N - E(k) - P(k) - A(k) - R(k)
P(k)=P_c(k)+P_d(k) , R(k)=R_c(k)+R_d(k)
The reason for this definition is easy to see. Additional people in, for example, compartment $P_c$ at the day $k$ are $(P_c)_{new}(k) = \alpha(k) E_{new} (k-e)$, while $\alpha(k) E_{new}(k-(e+p))$ move to the next compartment. Similar formulas hold for the compartments $P_d$, $A$, $R_c$ and $R_d$.
An important parameter in an epidemic is the reproduction number $\rho$, the number of people infected by a single infectious person during its life time. If we assume that $\kappa (k) $ and $s(k)$ may be considered as constant during $p_d$ days about $k$, we can derive this number from the equations. It is
\[\rho (k) =(1+ \delta)^{-1} \kappa(k)\, s(k) \big( \gamma_c p_c + \delta \gamma _d p_d \big) \,,
\]
where $\delta = \frac{1-\alpha}{\alpha}$.
For $s(k)=1$ it is usually called the basic reproduction number, in order to distinguish it from the effective reproduction number with $s(k) < 1$. In part I of this paper we usually mean the basic reproduction number when we speak of reproduction number, while in the part II the decreasing $s(k)$ hast to be taken into account and we usually speak of the effective reproduction number, also without use of the attribute “effective”.
If we set $\alpha(k) = 1$, $P_d = 0$ and $R_d =0$, we obtain the SEPAR model without dark sector as a special case of the SEPAR$_{d}$ model. Effects of vaccination can easily be implemented by sending the according number of persons directly from $S$ to $R$.
For later use the following observation is useful. The number of people in a given compartment at day $k$ is the sum of additional entries at previous days, for example
E(k) = \sum_{j=0}^{e-1} E_{new}(k-j),
P_c(k) = \alpha(k) \sum_{j=0}^{p_c-1} E_{new}(k-e-j),
and so on for $P_d(k)$ and $A(k)$.
We abbreviate
H(k) = E(k) + P(k) + A(k) + R(k) ,
the number of herd immunized (without vaccination). Then the recursion scheme implies:
H(k) - H(k-1) = E_{new}(k)
Putting this into the formula above: $
E(k) = \sum_{j=0}^{e-1} E_{new}(k-j)
$, we obtain:
E(k) = H(k) - H(k-e)
and similarly
P_c(k) = α(k) H(k-e) - α(k-p_c) H(k-e-p_c)),
P_d(k) = (1 - α(k)) H(k-e) - (1- α(k-p_d)) H(k-e-p_d)),
A(k) = \alpha(k-p_c)\, (H(k-e-p_c) - \alpha(k-p_c-q)\, H(k-e-p_c-q)).
Using $R_c(k) - R_c(k-1) = \alpha(k-p_c-q) E_{new} (k-e-p_c-q)= \alpha(k-p_c-q) H(k-e-p_c-q) - \alpha(k-p_c-q-1)\,H(k-e-p_c-q-1)$ we conclude:
R_c(k) = \alpha(k-p_c-q) H(k-e-p_c-q)
and similarly
R_d(k) =(1-\alpha(k-p_d)) H(k-e-p_d).
This gives a very simple structure of the model in terms of a single recursion equation.
SEPAR$_d$-model: The recursion scheme of the SEPAR$_d$ model is given by a single recursion equation:
H(k)- H(k-1) =
s(k-1)η(k-1) ( α(k-1) H(k-1-e) - α(k-1-p_c) H(k-1-e-p_c)
ξ [ (1 - α(k-1)) (H(k-1-e) -(1- α(k-1-p_d)) H(k-1-e-p_d)) ]
the functions $S(k)$, $E(k)$, $P_c(k) $, $P_d(k) $, $A(k) $, $R_c(k)$ and $R_d(k)$ are given in terms of $H(k)$ by the equations above.
If we pass from a daily recursion to a infinitesimal recursion, replacing the difference equation by a differential equation, we obtain the continuous recursion scheme, where now all functions are differentiable functions of the time $t$:
H'(t) = s(t)η(t) ( α(t) H(t-e) - α(t-p_c) H(t-e-p_c))
ξ [ (1 - α(t)) H(t-e) -(1- α(t-p_d)) H(t-e-p_d) ]
In both cases, discrete and continuous, consistent start conditions in an interval of length $e+p$ have to be added. For the discrete case see sec. <ref>.
If we remove the dark sector, the continuous model was independently obtained in <cit.>.
The branching ration $\alpha$ and with it the number $\delta \approx \frac{1-\alpha}{\alpha}$ of unrecorded infected for each newly recorded one varies drastically in space and time, roughly in the range $1 \leq \delta \leq 50$. For Switzerland and Germany serological studies in late 2020 conclude $\delta \approx 2$, for the USA a recent study finds $\delta \approx 8$ and in part of India (Punjab) a serological study found values indicating $\delta \approx 50$.[For Germany see <cit.>, for Switzerland <cit.> announcing a forthcoming study of Corona-Immunitas, for USA <cit.> and for India a report in ANI <https://www.aninews.in/news/national/general-news/second-sero-survey-finds-2419-pc-of-punjab-population-infected-by-covid-1920201211181032/> retrieved 12/21 2020. ]
For our choice of the model parameter see below, section <ref>.
Besides the determination of $\alpha$ one needs to know the difference between $p_c$ and $p_d$ and between $\eta_c (k)$ and $ \eta _d(k)$, if one wants to apply the $SEPAR_d$-model. As explained above we estimate $p_c = 7$.
The mean time of active infectivity of people who are not quarantined seems to be not much longer, although in some cases it is. According to the study <cit.> “no isolates were obtained from samples taken after day 8 (after occurrence of symptoms) in spite of ongoing high viral loads”.
This allows to work with an estimate $p_d = 10$, and so it is not much larger than $p_c$. A comparison of the $SEPAR_d$ model with a simplified version, where we assume $p_c = p_d=: p$ and $\eta_c (k) = \eta _d(k) = :\eta(k)$ shows that with these values the difference is very small (see figure <ref>). In the following we therefore work with the simplified $SEPAR_:d$ model setting $p_c = p_d=: p$.
Recent studies indicate that the number of asymptomatic infected is often as low as about 1 in 5 symptomatic unrecorded and is thus much smaller than originally expected <cit.>. Although asymptomatic infected are there reported to be considerably less infective than the symptomatic ones, their relatively small number among all unreported cases justifies to work in the simplified dark model with the assumption $ \eta _d(k) \approx \eta_c (k) = :\eta(k)$. If we set $P(k) = P_c(k) + P_d(k)$ as above, we see that $P_c(k) = \alpha(k) P(k)$ and $P_d(k) = (1-\alpha(k)) P(k)$.
Comparison of SEPAR$_d$ model for $A(k)$ between dark sector with $p_c=7,\, p_d=10, \, \xi =0.9$ (solid blue) and simplified dark sector $p_c= p_d =p=7, \, \xi=1$ (dashed blue), assuming constant $\eta$.
In part II we discuss how the time dependent parameter $\eta(k)$ can be derived from the data and a rough estimate of the dark factor $\delta$ can be arrived at, although it lies in the nature of the dark sector that information is difficult to obtain.
A comparison of the SEPAR model ($\delta=0$) with the simplified SEPAR$_d$ model is given in fig. <ref> for a constant parameter $\alpha = 0.2$, respectively dark factor $\delta=4$ and constant reproduction coefficient $\rho=3$. This illustrates the influence of the dark factor from the theoretical viewpoint.
Comparison of the course of an epidemic with constant reproduction number $\rho=3$ without dark sector (dashed), and with dark sector $\delta=4$ (solid lines): Left counted number of actual infected $A_c(k)$ (blue). Right: total number of confirmed infected $A_{tot \, c}(k)=A_c(k)+R_c(k)$ (brown).
§.§ The S(E)IR models and their assumptions
Whereas the derivation of the SEPAR$_d$-model is based on the idea of disjoint compartments, infected people pass through in time, there is a different approach with goes back to the seminal paper
<cit.>. A special case is the standard SIR-model or SEIR model. It seems that most people use this as a black box without observing the assumptions on which it is built. One should keep these assumptions in mind whenever one applies a model. There is a modern and easy to understand paper by Breda, Diekmann, and de Graaf with the title: On the formulation of epidemic models (an appraisal of Kermack and McKendrick) <cit.>, which explains the general derivation. In the introduction the authors state that the Kermack/McKendrick paper was cited innumerable times and continue: "But how often is it actually read? Judging from an incessant misconception of its content one is inclined to conclude: hardly ever! If one observes the principles from which the S(E)IR models are derived one should be hesitant to apply it to Covid-19.
Following <cit.> we shortly repeat the assumptions on which the general Kermack/McKendrick approach is based . The general model considers a function
S(t) := \text {density (number per unit area) of susceptibles at time t}
and related to this a function
the force of infection at time t.
By definition, the force of infection is the probability per unit of time that a susceptible becomes infected. So, if numbers are large enough to warrant a deterministic description, we have
I_{new}(t) = F(t)S(t),
where $I_{new}(t)$ is defined as the number of new cases per unit of time and area. The functions $S$ and $I$ are related by the equation:
S'(t) = -F(t) S(t)
Then the central modelling ingredient is introduced:
\[\begin{split} A(\tau ) := \text{expected contribution to the force of infection} \, \tau \, \text{units of time ago.}\end{split}
\]
Alone from this ingredient an integral differential equation is derived, which gives the model equations. For more details we also refer to a recent paper by Robert Feßler who derived the integral equation independently <cit.>.
Already here we see a different view of an epidemic. No compartments and their cardinality are mentioned; in their place the authors mention only certain functions.
If the function $A$ is assumed to decay exponentially,
A(\tau) = \alpha e^{-\beta \tau}
with constants $\alpha, \, \beta$.
The model derived from this input is called the standard SIR-model. It leads to two ordinary differential equations in the variable $t$:
I' = \alpha s I - \beta I
R' = \beta I
$$S(t) = N - I(t) - R(t),$$
where $N$ as before is the number of the population and $s(t)=\frac{S(t)}{N}$.
For the standard SEIR-model there is an additional function $E(t)$ measuring the
exposed and the input function is now
A(\tau) = \alpha \frac {\beta} {\beta - \gamma} (e^{- \gamma \tau } - e^{-\beta \tau} )
This leads to 3 ordinary differential equations in $t$
E' = \alpha s I - \beta E
I' = \beta E - \gamma I
R' = \gamma I
$$S(t) = N - E(t) - I(t) - R(t),$$
where $N$ as before is the number of the population. The infection function $A(\tau)$ considered here determines the convolution part of an integral kernel in Feßler's approach mentioned above.
If Breda et al. are right, readers should be critical to papers applying the S(E)IR models without explaining why the models, given their fundaments, are applicable. As far as we can see, the assumption of an exponential decay $A(t)$ is often not mentioned by authors applying it in situations where it would be necessary to discuss whether this assumption can be reasonably made. In a situation like Covid-19 where infectious people are isolated as soon as possible, it seems questionable whether this assumption holds. We are surprised that in most of the papers we have seen, which apply the S(E)IR model to an analysis of Covid 19, this problem is not even mentioned. This includes the papers of the group around Viola Priesemann which play an important part in the discussion about how to deal with Covid 19 in Germany <cit.>,
§.§ Comparing SIR with SEPAR
When we want to compare the SIR models with the delay SEPAR model we have to lower, in a first step, the number of compartments by removing $E$, $P$ and $A$ and to replace them by a single compartment, called $I$, of infected people which are at the same time infectious. For this model we assume that infected susceptibles move right away to compartment $I$, where they stay for $p$ days. In contrast to the SEPAR model it is assumed that these people are counted as actual infected people at the moment they are infected. After $p$ days they are counted as recovered or dead. So it is a strong simplification of the SEPAR model, but it follows the same pattern as the SEPAR model since it is a delay model. We call it d-SIR-model (“d-”for delay) to distinguish it from the standard SIR-model. The equations for this model are based on the same principles as the SEPAR model:
The continuous delay d-SIR model: Let $p$ be a positive real number standing for the duration of staying in the compartment $I$ of infected and infectious people. Let $\eta(t)$ be a differentiable function measuring the strength of infection (including the effects of social constraints). The quantities $S(t),\, I(t),\, R(t) $ of the delay SIR model are given by
a) the start condition:
A differentiable function $ I(t)$ for $0\le t< p$ ,
b) and the delay differential equations:
I'(t) = η(t) s(t) I(t)
- η(t-p) s(t-p) I(t-p)
R'(t) = η(t-p)s(t-p) I(t-p)
S(t) = N - I(t) -R(t)
To compare this model with the standard SIR-model above we note that also the d-SIR model (like the continuous SEPAR$_d$ model) can be derived from the principles of Kermack/McKendrick, as explained in <cit.>.
One only has to take the product of the characteristic function of the interval $[0,p]$ with $\eta = \gamma \kappa$ as
the function $A(\tau)$. To compare the two models one has to relate the input parameters. In the case of the d-SIR model they are $\eta $ (for the comparison we assume that the contact rate is constant) and $p$, whereas for the SIR-model they are $\alpha$ and $\beta$. The role of $\eta$ is that of $\alpha$ in the SIR-model, so we set $\alpha = \eta$. There are several ways to relate the paramter $\beta$ of the SIR model with $p$ occurring in the d-SIR model. One is to assume that the total force of infection has to be the same if they describe the same developments, i.e. with the function $A(\tau)$ which is the product of the characteristic function of the interval $[0,p]$ with $\eta$ one has the condition:
\int _0^ \infty \alpha e^{-\beta t} dt = \int _0^ \infty A(t) dt
Then the second relation:
\frac \alpha \beta = p\, \eta = p \, \alpha$ and thus
\beta = \frac 1 p.
In both cases the reproduction number is $\frac \alpha \beta = p\, \alpha = p\, \eta$.
If one applies this then there is a problem to find parameters so that at least at the beginning the two models are approximatively equal. Thus one can relate the tow models in a second way by choosing the parameters so that this is the case. For this we fix values for $\alpha $ and $\beta $ and chose the start conditions of the d-SIR model so that they agree with the SIR-model during the first days. By construction of the SIR-model the function $I$ is nearly an exponential function as long as the function $S$ is nearly constant. Thus we chose the same exponential function as start values for the d-SIR model.
Then the question is whether there are differences of the model curves in the long run and how large the differences are.
One should expect that the assumptions of an exponential decay regulating the strength of infection of an infectious person in the case of the SIR-model versus a period of $p$ days, where the strength of infection is constant and after that goes immediately down to $0$ in the case of the delay d-SIR, should result in higher values for the functions $I(t)$ and $I_{tot}(t)= I(t) +R(t)$, the total number of infected until time $t$ of the SIR model. The following graphics in which we assume a constant reproduction rate slightly above $1$ show, in fact, a dramatic difference supporting the expectation.
A similar observation can be found in <cit.>
Comparison of SIR (black) and dSIR (blue) for $s(t) \approx 1$ and constant coefficients with identical exponential increase and population size $N= 80$ M. Left: Number of infected $I(t)$. Right: $H(t)/N$ with $H(t)$ total number of infected up to time $t$. Parameter values: $N=8$ M, $I_0=1 $ k, $\alpha= \eta = 0.15, \, \beta= 0.1 $; $p= 8.11$ for dSIR.
This has an interesting consequence for a situation in which a high rate of immunity is achieved either by “herd” effects or by vaccination. According to a simple SIR model with constant reproduction number $\rho=1.5$ and a population of 80 million people (like in Germany) a little bit more than $0.6 \cdot 80 = 24$ million people would have to be infected or vaccinated to achieve herd immunity, whereas according to the delay model “only” about $0.3 \cdot 80=12$ million have to be infected (see fig. <ref>).
The difference corresponds to the fact that equal initial exponential growth is related to different reproduction rates in the two models of the example given: $\rho_{SIR}= \frac{\alpha}{\beta}=1.5$ and $\rho_{d-SIR}=p \, \eta \approx 1.2$. Such a difference matters because, according to the plausibility arguments given above, the delay model may very well be more realistic than the SIR model for Covid-19.
Next we discuss the differences between the SIR model and the delay SIR model during a time when $s$ is still approximately equal to $1$, both have constant reproduction numbers, and $\alpha = \eta$ like above.
Moreover we assume that the initial growths functions of
both models are approximately identical to the same exponential function (because both are designed to modelling the same growth process).
In reality one observes longer periods in the data where the reproduction number is approximately constant until it changes in a short transition period to a new approximately
constant value. Such changes may be due to containment measures (non-pharmaceutical interventions) imposed by governments, which influence the contact rate $\kappa (t)$. In the next graphics we show the effect of such a change for both models.
Comparison of $I(t)$ for SIR (top left) and dSIR (top right) for $s(t) \approx 1$ and constant coefficients for $t\leq 29$ and $t\geq 31$ with identical exponential increase in the initial upswing. Reduction of reproduction rate by 40 % in both cases. Bottom: SIR black, dSIR blue. Parameter values: $N=8$ M, $I_0=1 $ k, $\alpha_1= \eta_1 = 0.15, \, \beta_1 = \beta_2 =0.1, \; \alpha_2=\eta_2 =0.09, \, \beta_2 = \beta_1$; dSIR $p= 8.11$.
In figure <ref> we let the dSIR and the SIR curves start with identical exponential functions based on constant reproduction numbers. Then we lower the reproduction rate by 40 percent within three days. As expected the SIR curves have a cusp since one exponential function jumps into another, whereas the dSIR equation due to the delay character shows a slightly smoother transition. The second and more dramatic effect is that a similar phenomenon like in the long term comparison can be observed: the SIR solution is far above the dSIR solution. The reason seems to be the same, the different assumptions made by the two approaches about the decay of the strength of infection.
These considerations show that the choice of the model may result in important differences for the medium and long range development of the epidemic. We have given arguments why we consider the delay SIR model more realistic for Covid-19 than the standard SIR approach.
But also the delay SIR model has defects when comparing it to the data. The reason is that in reality it is not the case that an infected person gets infectious the same day, and also it takes some time until an infectious person shows symptoms. This
speaks in favour of the delay SEPAR approach.
In part II we apply the SEPAR$_d$ model to data of selected countries. Here the model shows its high quality. Since data about the dark sector are insecure we check how much the dark sector influences the overall dynamics of the epidemic in the discussed countries up to the present (until the end of 2020) and choose the dark factor of the model on the basis of the analysis and given estimates for the respective countries.
PART II: Applications of the SEPAR model
tocchapterPart II: Application
§ DETERMINING EMPIRICAL PARAMETERS FOR THE MODEL
§.§ JHU data
§.§.§ The basic data sets (JHU)
The worldwide data
provided by the Humdata project (Humanitarian Data Exchange) of the Johns Hopkins University provides data on the development of the Covid-19 pandemic for more than 200 countries and territories.[<https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases>] The data are compressed into 3 basic data sets for each country/territory
\[\mathit{Conf}(k), \; Rec(k), \; D(k) \,
\]
where $\mathit{Conf}(k)$ denotes the total number of confirmed cases until the day $k$ (starting from January 22, 2020), $Rec (k)$ the number of reported recovered cases and $D(k)$ the number of reported deaths until the day $k$.
The last two entries can be combined to the number of redrawn persons of the epidemic, captured by the statistic,
\[ \hat{R}(k) = Rec(k)+ D(k)\, .
\]
Empirical quantities derived from the JHU data set will be endowed with a hat, like $\hat{R}$, to distinguish them from the corresponding model quantity, here $R$.
The (first) differences of $\mathit{Conf}(k)$ encode the daily numbers of newly reported and acknowledged cases:
Â_new(k) = 𝐶𝑜𝑛𝑓(k)- 𝐶𝑜𝑛𝑓(k-1)
The other way round, the number of confirmed cases is the complete sum of newly reported ones, and may be considered as the total number of acknowledged cases
Â_tot(k) = ∑_j=1^k Â_new(j) = 𝐶𝑜𝑛𝑓(k) ,
while the difference
Â(k)= 𝐶𝑜𝑛𝑓(k)-R̂(k) ,
is the empirical number of acknowledged, not yet redrawn, actual cases. Some authors call it the number of “active cases”;[E.g. in <cit.> …A similar identification underlies the numbers for the active case in the Worldometer
but this is misleading because
the phase of effective infectivity is usually over as soon as an infection is diagnosed and the person is quarantined.
The number $Rec(k)$ of recovered people is often reported with much less care than the daily new cases and the deaths.
By this reason the recorded number of redrawn, $\hat{R}(k)$, may be heavily distorted, with the result that neither itself nor the derived numbers $\hat{A}(k)$ can be be taken at face value. The most reliable basic data remain therefore
\[ \hat{A}_{new}(k) \;, \quad \mathit{Conf}(k) \qquad \mbox{and} \quad D(k) \; .
\]
Even $\hat{A}_{new}(k)$ has its peculiarities due to the weekly cycle of reporting activities.
In this paper we abstain from discussing mortality rates and consider the first two data sets of the mentioned three only. $\hat{R}$ and $\hat{A}$ play an important role for a complete image of an epidemic, but they are reliable only for a few countries; for the majority of countries they have to be substituted or complemented by more adequate quantities derived from the basic data (see eq. <ref>).
§.§.§ Smoothing the weekly oscillations of $\hat{A}_{new}$
For all countries the reported number of daily new infections shows a characteristic 7-day oscillation resulting from the reduction of tests over weekends and the related delay of transmission of data. A 3-day sliding average suppresses fluctuations on a day-to-day scale and shows the weekly oscillations even more clearly.
Acknowledged cases $\hat{A}_{new}(k)$ for Germany (left), sliding 3-day and 7-day averages $\hat{A}_{new,3}(k)$, $\hat{A}_{new,3}(k)$ (middle) and $\hat{A}_{new,7}(k)$ (right)
In some countries these oscillations are corrected for transmission delay by central institutions,[This is done by the Robert Koch Institut for the German case.]
but such corrections are not implemented in the JHU data. A simple method for smoothing the weekly oscillations consists in using sliding centred 7-day averages:
Â_new,7(k) = 1/7 ∑_j=-3^3 Â_new(k+j)
and similarly for the centred 3-day average $\hat{A}_{new,3}(k)$.
Note that in order to avoid a time shift effect which would arise from using a purely backward sliding average, we use a sliding average over 3 days forward and 3 days back. For most countries this suffices for carving out the central tendency of the new infections quite clearly (fig. <ref>).
For some countries already the daily fluctuations of $\hat{A}_{new}$ are extreme.
The French data even indicate negative values for $\hat{A}_{new}$ for certain days, although this ought to be excluded by principle. Such effects indicate a highly unreliable system of data recording and transmission; they may be due to ex post corrections of earlier exaggeration of transmitted numbers.
But even under such extreme conditions
the sliding 7-day average leads to reasonable information on the mean motion of the new infections, so that we don't have to exclude such countries from further consideration (sec. <ref>).
§.§ Data evaluation
§.§.§ The “actual” cases in the statistical sense
The difference $\hat{A}(k)=\hat{A}_{tot}(k)-\hat{R}(k)$ (<ref>) can in principle be considered as an expression for the number of actual cases; but
it is corrupted by the fact that the number of daily recovering $Rec(k)$ is irregularly reported.
For a critical investigation of this number we start from the truism that any actually infected person recorded at day $k$
has been counted among the $\hat{A}_{new}(l)$ at some earlier day, $l\leq k$. The smallest number $\hat{q}(k)$ of days preceding $k$ (including the latter) necessary for supplying sufficient large numbers of infected $\hat{A}(k)$,
q̂(k)= min_l [ ∑_j=0^l Â_new(k-j) ≥Â(k) ] ,
is a good indicator for the mean time of sojourn in the collective of infected which are recorded as “actual cases”. As long as the number of severely ill among all infected persons is relatively small and the time of severe illness well constrained, we may expect that the mean time of actual illness does not deviate much from the time of prescribed minimal time of isolation $q_{min}$ for infected persons. In the case of Covid-19 $q(k)$ surpasses $q_{min}\approx 14$ only moderately for India, Germany, Switzerland etc. (fig. <ref>). For many other countries $\hat{q}(k)$ behaves differently. It starts near the time of quarantine or isolation but increases for a long time monotonically with the development of the epidemic, before often – although not always – it starts to decrease again after the (local) peak of a wave has been surpassed. This is the case, e.g., for Italy and the US; in the last case the deviation is extreme, $\hat{q}(k)$ surpasses 100 and shoots up a little later (see fig. <ref>)
Daily values $\hat{q}(k)$ for the mean time of being statistically counted as an actual case for India (left) and Germany (right)
Daily values of the mean time $\hat{q}(k)$ of being statistically counted as actual infected for Italy (left) and USA (right)
This effect cannot be attributed to medical reasons; the major contribution rather results from
the unreliability of the statistical book keeping: With the growing overload of the health system, the time of recovery of registered infected persons is being reported with an increasing time delay, sometimes not at all (e.g., Sweden, UK). The difference
$ \hat{A}(k) = \hat{A}_{tot}(k) -\hat{R}(k) $
gets increasingly confounded by the lack of correctness in the numbers $Rec(k)$. In these countries it is an expression of the number of “statistically actual” cases only with, at best, an indirect relation to the real numbers of people in quarantine or hospital.
The information gathered for Covid-19 proposes the existence of a stable mean time $q$ of isolation of infected persons (including hospital) for long periods in each country. It is usually a few days longer than the official duration of quarantine prescribed by the health authorities.
Given $q$, the sum
Â_q(k) = ∑_j=0^q-1 Â_new(k-j)
can be used as an estimate of the number of infected recorded persons who are in isolation or hospital at the day $k$.
Here we do not use 7- or 3- day averages, because the summation compensates the daily oscillations anyhow. The accordingly corrected number of redrawn $\hat{R}_q$ is of course given by
R̂_q = Â+ R̂-Â_q .
Figure <ref> shows $\hat{A}(k), \, \hat{A}_q(k)$
for the USA (with $q=15$). It demonstrates the difference between $\hat{A}(k)$ (dark blue) and $\hat{A}_q(k)$ (bright blue) and shows that $\hat{A}_q(k)$ is a more reliable estimate of actually infected than the numbers $\hat{A}(k)$ (the “active cases” of the Worldometer).
$\hat{A}(k)$ (dark blue), $\hat{A}_q(k)$ (bright blue) for the USA ($q=15$)
For countries with reliable statistical recording of the recovered we find $\hat{q}(k) \approx const$. In this case we choose this constant as the value for the model $q$. For other countries one may use a default value, inferred from comparable countries with a better status of recording the $Rec(k)$ data (i.e. $\hat{A}(k) \approx \hat{A}_q(k)$).
§.§.§ Simplifying assumptions on the duration $e$ of exposition and the duration $p$ of effective infectivity
For Covid-19 it is known that there is a period of duration say $e$ between the exposition to the virus, marking the beginning of an infection, and the onset of active infectivity. Then a period of propagation, i.e. effective infectivity, with duration $p$ follows, before the infection is diagnosed, the person is isolated in quarantine or hospital and can no longer contribute to the further spread of the virus.
Although one might want to represent the mentioned durations by stochastic variables with their respective distributions and mean values, we use here the mean values only and make the simplifying assumption of constant $e$ and $p$ approximated by the nearest natural numbers.
The Robert Koch Institute estimates the mean time from infection to occurrence of symptoms to about 4 days <cit.>, (5.). This is divided into $e$ plus the time from getting infectious to the occurrence of symptoms. According to studies already mentioned above the latter is estimated as $2$ days, so as a consequence we estimate $e=2$. In section <ref> we generically use $p=7$.
We have checked the stability of the model under a change of the conventions of parameter choice inside the mentioned intervals.
§.§.§ Estimate of the daily strength of infection
As announced in part I we work with the simplified SEPAR$_d$ model. This means that the duration in compartments $P_c$ and $P_d$ is equal, here denoted by $p$, and also the strength of infection is assumed to be equal: $\eta_c = \eta_d = \eta$. Furthermore, if $P(k) = P_c(k) + P_d(k)$ there is a branching ratio $\alpha$, which has to be estimated for each country, such that $P_c (k)= \alpha P(k)$ and $P_d(k) = (1-\alpha)P(k)$. For every counted infected there are then
\[ \delta= \frac{1-\alpha}{\alpha}
\]
uncounted ones. We call $\delta$ the dark factor.
Once $e$ and $p$ are given (or fixed by convention inside their intervals) one can determine the empirical strength of infection $\eta(k)$ using the model equations. Namely
\eta(k) = \frac {E_{new}(k+1) }{s(k) P(k)}.
In terms of the total number of infected $H(k)$ (see (<ref>) and with constant $\alpha$ this is
η(k) =
H(k+1)-H(k)/s(k) (α(H(k-e) -H(k-(e+p_c))) + ξ (1-α)(H(k-e) -H(k-(e+p_d))) ) )
For the simplified SEPAR$_d$ model we have $\alpha E_{new}(k) = A_{new}(k+e+p)$ and $\alpha P(k) = \sum_{j=1}^p\hat{A}_{new}(k+j)$. Thus $\alpha $ cancels and we obtain:
η(k) = A_new(k+e+p) /s(k) ∑_j=1^p A_new(k+j)
Denoting, as before, the values we obtain from the data by $\hat \eta (k)$ etc. we obtain
η̂(k) = Â_new(k+e+p) /ŝ(k) ∑_j=1^pÂ_new(k+j) η̂_7(k) = Â_new,7(k+e+p) /ŝ(k) ∑_j=1^pÂ_new(k+j),
where in the second equation we work with the weekly averaged data.
An estimation of the total number of new infections induced by an infected person during the effective propagation time (and thus the whole time of illness) is then
ρ̂(k) = ∑_j=0^p-1 η̂ (k+j) ŝ(k+j) ;
and similarly for $\hat{\rho}_7(k)$. In the following we generally use the latter but write just $\hat{\rho}(k)$. Note also that the determination of the strength of infection $\hat{\eta}(k)$ by (<ref>) needs an estimation of the dark factor $\delta$ because the latter enters into the ratio of susceptibles $\hat{s}(k)$,while it cancels in the calculation of the empirical reproduction rate $\hat{\rho}(k)$ (<ref>).
This is an empirical estimate for the reproduction number $\rho(k)$.
In periods of nearly constant daily strength of infection one may use the approximation
ρ̂_̂7̂(k) ≈p η̂_̂7̂(k) ŝ(k) = p A_new,7(k+e+p)/∑_j=1^p A_new(k+j)
Inspection of transition periods between constancy intervals for Covid 19 shows that this approximation is also feasible in such phases of change. For $p=7$
this variant of the reproduction number stands in close relation to the reproduction numbers used by the Robert Koch Institut, see appendix <ref>), which gives additional support to this choice of the parameter.
§.§ SEPAR$_d$ parameters
For modelling Covid-19 in the simplified dark approach we use the parameter choice $e=2,\; p \,(=p_c=p_d) =7$ as explained in sec. <ref>. Where we differentiate between $p_c$ and $p_d$ we usually use $p_c=7$ and $p_d=10$.
The value for $q$ depends on the reported mean duration of reported infected being counted as “actual (active)” case for each country (sec. <ref>); in the following reports it usually lies between 10 and 17. For each country we let the recursion start at the first day $k_0$ at which the reported new infections become “non-sporadic” in the sense that no zero entries appear at least in the next $e+p_d$ days ($\hat{A}_{new}(k)\neq 0$ for $k_0 \leq k \leq k_0+(e+p_d)$). For $k\geq k_0-1$ the values $\hat{\eta}(k)$ are calculated in the simplified case, $p=p_c=p_d$ according to (<ref>). Otherwise the formula above (<ref>) has to be used.
§.§.§ Start values
For the numerical calculations we use the recursion (<ref>), with start values given by time shifted numbers of the statistically reported confirmed cases, expanded by the dark factor, for $k$ in the interval $J_{-1}= [k_{-1}, k_0 -1]$ where $k_{-1}= k_0-1-(e+p_d)$:
\[ \hat{H}(k)=(1+\delta)\mathit{Conf}(k+e+p_c)= (1+\delta)\hat{A}_{tot}(k+e+p_c)
\]
Note that the time step parametrization in the introduction/definition of the SEPAR model in sec. <ref> works with $k_0= 1$.
If we set the model parameters $\eta(k)=0$ for $k< k_0-1$ and $\eta(k)=\hat{\eta}(k)$ for $k\geq k_0-1$, the recursion reproduces the data exactly, due to the definition of the coefficients. Then and only then it becomes tautological.
Already if we use coefficients $\hat{\eta}_7(k)$ for $k\geq k_0$ from time averaged numbers of daily newly reported according to eq. (<ref>), the model ceases to be tautological. In this case the parameter $\eta_0=\eta (k_0-1)$ may be used for optimizing (root mean square error) the result for the total number of reported infected $A_{tot}$ in comparison with the empirical data (<ref>).
The model acquires conditional predictive ability, if longer intervals of constant coefficients are chosen.
One may prefer to replace $A_{new}$ by the 7-day averages $\hat{A}_{new,7}$ by introducing
\[\mathit{Conf}_7(k)=\sum_{j=k_{-1}}^k \hat{A}_{new,7}(j) \; \; (+const)
\]
and $\hat{H}_7(k)= (1+\delta)\,Conf_7(k+e+ p_c)$. For constant $\alpha(k)=\alpha$ the replacement of the $\hat{A}_{new}$ in the denominator of (<ref>) then boils down to defining
\[ \hat{\eta}_7(k)= \frac{\hat{H}_7(k+1)- \hat{H}_7(k)}{\hat{s}(k)
\Big(\alpha \big( \hat{H}_7(k-e) -\hat{H}_7(k-(e+p_c))\big) + \xi (1-\alpha) \, \big(\hat{H}_7(k-e) -\hat{H}_7(k-(e+p_d)) \big) \Big) }
\]
Then the model becomes tautological also for the use of daily varying $\hat{\eta}_7(k)$ and ceases to be so only after introducing constancy intervals as described in the next subsection.
§.§.§ Main intervals
The number of empirically determined parameters can be drastically reduced by approximating the daily changing empirical infection coefficients $\hat{\eta}_7(k)$ ($k= k_0, k_0+1, \ldots$) by constant model values $\eta_j$ ($1 \leq j \leq l$) in appropriately chosen intervals $J_1, \ldots J_l$. We call them the
constancy, or main, intervals of the model. Their choice is crucial for arriving at a full-fledged non-tautological model of the epidemic.
We thus choose time markers $k_j$ (“change points”) for the beginning of such intervals and durations $\Delta_j$ for the transition between two successive constancy intervals, such that:
\[ J_j =[ k_j, \, k_{j+1}-\Delta_{j+1}]\; , \qquad j= 1, \ldots , l
\]
In the main interval $J_j$ the model strength of infection $\eta_j$ (this is the constant daily strength of infection during this time) are generically chosen as the arithmetical mean of the empirical values $\mathit{ mean} \{ \hat{\eta}(k)\; | \; k \in J_j\}$. Small deviations of the mean, inside the 1 $\sigma$ range of the $\hat{\eta}$-fluctuations in the interval, are admitted if in this way the mean square error of the model $A_{tot}$ can be reduced noticeably. The dates $t_j$ of change points $k_j$ can be read off heuristically from the graph of the $\hat{\eta}_7$ and may be improved by an optimization procedure.
$k_1$ is chosen as the first day of a period in which the daily strength of infection can reasonably be approximated by a constant. In the initial interval $J_0=[k_0, \, k_1 -1]$ the model uses the empirical daily strength of infection: $\eta(k)=\hat{\eta}_7(k)$ for $k\in J_0$. In the transition intervals $[ k_j - \Delta_j, \, k_j ]$ the model strength of infection is gradually, e.g. linearly, lowered from $\eta_{j-1}$ to $\eta_j$.[Here we use the smoothing function described in <cit.>. Also an elementary optimization procedure for determining the main intervals is described in this paper.]
For the labelling of the days $k$ there are two natural choices; the JHU day count starting with $k=1$ at 01/22/2020, or a country adapted count such that $k_0=1$, where $k_0$ labels the first day for which the reported new infections become non-sporadic (see above). Both choices have their pros and contras; in the following we make use of both in different contexts, declaring of course which one is being used.
§.§.§ Influence of the dark sector
Proceeding in this way involves an indirect observance of the dark sector's contribution to the infection dynamics of the visible sector. An unknown part of the counted new infections $\hat{E}_{new}(k)$ is causally due to contacts with infectious persons in $P_d$ of the dark sector, eqs. (<ref>) or (<ref>)).
According to estimates of epidemiologists there is a wide spectrum of possibilities worldwide, $0 \leq \delta \leq 100$, for Covid-19, while we have only rough guesses for the different countries. In the following country reports we work with the simplified dark approach, $p_d=p_c=p=7$ and $\xi=1$ ($\gamma_c =\gamma_d$) and use estimates for the dark factor $\delta$ explained in the respective country section.
§ SELECTED COUNTRIES/TERRITORIES
In our collection we include four small or medium sized European countries (Switzerland, Germany, France, Sweden), three large countries from three different continents (USA, Brazil, India), and a model for the aggregated data of all world countries and territories. For the first country analysed here, Switzerland, relatively reliable data on the dark sector are accessible. We take it as an example for a discussion of the effects of different assumptions on the branching ration $\alpha$, respectively the factor $\delta$, of the dark sector. For the other countries we lay open on which considerations our choice of the model the dark factor is based.
§.§ Four European countries; Switzerland Germany, France, Sweden
The four European countries discussed here show different features with regard to the epidemic: Switzerland and Germany have a relatively well organized health and data reporting system; the overall course of the epidemic with wave peaks in early April and in early November 2020 and a moderately controlled phase in between is is typical for most other European countries. France, in contrast, shows surprising features in the documentation of statistically recorded new infections (negative entries in the first half of the year); and Sweden has been chosen because of a containment strategy of its own. In the case of Switzerland and Germany first results of representative serological studies are available. They allow a more reliable estimate of the size of the dark sector than in most other cases. We therefore start our discussion with these countries.
§.§.§ Switzerland
The numbers of reported new infected ceased to be sporadic in Switzerland at February 29, 2020; we take this as our day $t_0= 1$. For the reported daily new infections (3-day and 7-day sliding averages) see figure <ref>.
At Feb 28 recommendations for hygiene etc. were issued by the Swiss government and large events prohibited, including Basel Fasnacht (carnival). These regulations were already active at our day $t_0=1$ (Feb 29) and explain the rapid fall of the empirical strength of infection (7-day averages) $\hat{\eta}_7(k)$ at the very beginning of our period. During the next 15 days a series of additional general regulations were taken: Mar 13, ban of assemblies of more than 100 persons, lockdown of schools; Mar 16 ($t=12$) lockdown of shops, restaurants, cinemas; Mar 20 ($t=16$), gatherings of more than 5 persons prohibited. This sufficed for lowering the growth rate quite effectively.
The SEPAR reproduction number started from peak values close to $5$ and fell down to below the critical value at March 23, the day $t_1=24$ in the country count (fig. <ref>, left). Here it remained with small oscillations until mid May, after which it rose (we choose $t_2=85$, May 23, as the next time mark), with strong oscillations until late June, before it was brought down to close to 1 in late June ($t_3= 158$, June 23). A long phase of slow growth ($\rho \approx 1.1$) followed until mid September. An extremely swift rise of the reproduction number to values above 2 in late September ($t_4=242$, Sep. 26) brought the number of new infections to heights formerly unseen in Switzerland. In spite of great differences among the differently affected regions (Kantone) the epidemic was brought under control at the turn to November ($t_5=272$, Oct. 27).
Daily new reported cases for Switzerland, 3-day sliding averages $\hat{A}_{new,3}(k)$ and 7-day averages $\hat{A}_{new,7}(k)$.
Left: Empirical reproduction rates $\hat{\rho}(k)$ for Switzerland. Right: Daily strength of infection $\hat{\eta}_7(k)$ (yellow) with model parameters $\eta_j$ (black dashed) in the main intervals $J_j$ for $p=7$ and $\delta=2$.
The empirical values of the infection strengths $\hat{\eta}_7(k)$ determined on the basis of the 7-day sliding averages $\hat{A}_{new,7}$ depend on estimates of the dark factor $\delta$ (cf. eq. <ref>). Recent serological studies summarized in <cit.>
conclude $\delta \approx 2$. We choose this value as generic for the simplified SEPAR$_d$ model. The values of
$\hat{\eta}_7(k)$, assuming a dark factor $\delta=2$, are shown in fig. <ref>, right. How its values are affected by different assumptions for the dark sector can be inspected by comparing with the results for $\delta =0$ and $\delta=4$ (fig. <ref>). The influence of the dark sector on $\hat{\eta}_7$ becomes visible only late in the year 2020.
Comparison of empirical values $\hat{\eta}_7(k)$ assuming dark factor $\delta=0$ (left) and $\delta=4$ (right); for comparison with generic choice $\delta=2$ see last figure.
With the time markers between different growth phases of the epidemic indicated above we choose the following main intervals for our model:
$J_0= [1, 22], \; J_1=[24, 78],\; J_2= [85, 117], \; J_3=[119, 204], J_4= [211, 230 ], \; J_5=[242, 264],\; J_6= [270, 321] $, end of data ($t_{eod}=321$) at Jan 14, 2021. The model values $\eta_j$ in the Intervals $J_j$ are essentially the mean values of $\hat{\eta}_7(k)$ in the respective interval, where small deviations inside the 1 sigma domain are admitted if the (root mean square) fit to the empirical data $\hat{A}_{tot}$ can be improved. They are given in the table below and indicated in fig. <ref> (black dashed lines). (Note that $\eta_0$ has no realistic meaning; it is a free parameter of the start condition for modelling on the basis of the 7-day averages $\hat{\eta}_7(k)$, see sec. <ref>.) The reproduction rates $\rho_j$ in the table refer to the beginning of the intervals; in later times the decrease of $s(k)$ can lower the reproduction rates until the end of the intervals considerably. In the case of Switzerland the latter crosses the critical threshold 1 inside the last constancy interval (see below).
8|c|Model $\eta_j$ and $\rho_j$ in intervals $J_j$ for Switzerland
$J_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$
$\eta_j$ -0.127 0.095 0.203 0.156 0.265 0.132 0.165
$\rho_j$ — 0.66 1.41 1.08 1.82 0.86 1.01
The course of the new infections is well modelled with these values (fig. <ref>); the same holds for the total number of counted infected (fig. <ref> below).
Model reconstruction for the 7-day averages of the number of new infected $A_{new}(k)$ (black dashed) in comparison with the empirical data, 3-day averages, $\hat{A}_{new,3}(k)$ (solid red) for Switzerland.
The count of days $\hat{q}(k)$, necessary for filling up the numbers of reported actual infected by sums of newly infected during the directly preceding days shows a relatively stable value, $\hat{q}(k) \approx 15$, until early November. Since then the mean time of reported sojourn in the department $A$ increases steeply (fig. <ref>, left). Initially this may have been due to a rapid increase of severely ill people during the second wave of the epidemic; but the number does not fall again with the stabilization in December 2021. A comparison of the statistically recorded actual infected $\hat{A}(k)$ with the $q$-corrected one $\hat{A}_q(k)$ (eq. <ref>) shows an ongoing increase of the first while the second one falls again after a sharp peak in early November (same figure, right). We conclude from this that from December 2020 onward the numbers of recovered are no longer reliably reported even in Switzerland.
Left: Daily values of the mean time of statistically actual infection $\hat{q}(k)$ for Switzerland. Right: Comparison of reported infected $\hat{A}$ (dark blue) and $q$-corrected number ($q=15$) of recorded actual infected $\hat{A}_q$ (bright blue) from the JHU data in Switzerland.
Both quantities cam be well modelled in our approach (fig. <ref>), although we consider $\hat{A}_q(k)$ as a more reliable estimate for actually ill persons.
Left: Number of actual infected $\hat{A}(k)$ (blue) for Switzerland, recorded by the JHU data, and model values calculated with time dependent $q(k)=\hat{q}(k)$ (black dashed).
Right: Empirical values, $q$-corrected with $q=15$, for statistically actual cases $\hat{A}_q(k)$ and the corresponding model values $A(k)$ (black dashed).
On this basis, the SEPAR$_d$ model expresses the development of the epidemic in Switzerland on the basis of only 6 constancy intervals for the parameters $\eta$ for the whole year 2020. The three main curves of the total number of recorded infected, $A_{tot}(k)$, the number of redrawn $R(k)$ and the number of diseased counted as statistical “actual” cases $A(k)$ are shown in figure <ref>.
Empirical data (solid coloured lines) and model values (black dashed) in Switzerland for numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}$ (bright green), and actual numbers $\hat{A}$, according to the statistic (blue).
This may encourage to look at a conditional 30-day prediction for Switzerland given by the SEPAR$_d$ model, the condition being the hypothesis of no considerable change in the contact behaviour of the population and no increasing influence of new virus mutations, i.e., a continuation of the recursion with $\eta_6$, the strength of infection in the last constancy interval (fig. <ref>).
30-day prediction for $A_{new}, \, A_q$ (top) and $A_{tot}$ (bottom) for Switzerland, assuming dark factor $\delta=2$; empirical values coloured solid lines, model black dashed (boundaries of 1-sigma region prediction dotted).
The figures show clearly that, under the generic assumption $\delta=2$ for Switzerland, the ratio of infected $s(k)$ starts to suppress the rise of new and actual infections already in January 2021 even for the upper bound of the 1-sigma estimate for the parameter $\eta_7$ (fig. <ref>, top). Of course the question arises what would be changed assuming different values for the dark factor. Figure <ref> shows how strongly the ratio of susceptibles is influenced by the choice of $\delta$ already at the end of 2020.
Development of ratio of susceptibles $s(k)$ for Switzerland (model values), assuming dark factor $\delta=0$ (dotted), $\delta=2$ (solid line) and $\delta=4$ (dashed).
The results for $\delta=0$ and $\delta=4$ of the model values of reported new infected, $A_{new}(k)$ (black dotted or dashed as above) in comparison with the 3-day averages of the JHU data, $\hat{A}_{new,3}(k)$, is shown in figure <ref>. Remember that the model expresses the dynamics of 7-day averages of the newly infected.
30-day prediction for $A_{new}$ for Switzerland, assuming dark factor $\delta=0$ (left), respectively $\delta=4$ (right); empirical values $\hat{A}_{new,3}$ coloured solid lines, model values black dashed (boundaries of 1-sigma region dotted).
Assuming a negligible dark sector ($\delta=0$) and the upper boundary of the 1-sigma interval of $\eta_7$, the number of new infected would continue to rise deeply into the first quarter of 2021. In all other cases the effective reproduction number is suppressed below the critical value by the ratio of susceptibles $s(k)$ already at the turn to the new year. We conclude that the role of the dark sector starts to have qualitative impact on the development of the epidemic in Switzerland already at this time.
§.§.§ Germany
The epidemic entered Germany (population 83 M) in the second half of February 2020; the recorded new infections seized to be sporadic at $t_0=$ Feb. 25, the day $k_0=1$ in our country count. With the health institutions being set in a first alarm state and public advertising of protective behaviour, the initial reproduction rate (as determined in our model approach) fell swiftly from roughly $\rho \approx 4$ to about 1.5 until mid March.
At May 25 ($t_1=30$) it dropped below 1 and stayed there, with an exceptional week (dominated by a huge infection cluster in the meet factory Tönnies) until late June 2020 (fig. <ref> left). The peak of the first wave was reached by the 7-day averages of new infections $\hat{A}_{new,7}$ at March 30; 6 days later, i.e. April 6, the local maximum for the actual numbers of reported infected $\hat{A}$ followed.
Serological studies in the region Munich indicate that during the
first half of 2020
the ratio of counted people was about $\alpha =0.25 $ in Germany, i.e. for each counted person there were $\delta \approx \frac{1-\alpha}{\alpha}=3$ persons entering the dark sector <cit.>. Of course this ratio varies in space and time, for example if the number of available tests increases or, the other way round, it is too small for a rapidly increasing number of infected people. The rapid expansion of testing,in Germany during early summer seems to have increased the branching ratio to about $\alpha \approx 0.5$. In follow up investigations the authors of the Munich study come to the conclusion that during the next months the ratio of unreported infected has decreased considerably; this brought the factor $\delta$ down to $\approx 1$ in early November 2020.[Short report in <cit.>. This is consistent with the result in <cit.>.]
Since the dark segment influences our model mainly through its contribution to lowering of the ratio of susceptibles $s(k)$, it is nearly negligible in the first six months of the epidemic. We therefore simplify the empirical findings by setting $\delta=1$ for the SEPAR$_d$ model of Germany.
Contrary to a widely pronounced assessment (including by experts) according to which the epidemic was well under control until September, the reproduction rate rose to $\rho \approx 1.3$ already in July and the first half of August. Only the low level of daily new infections, reached in late May, covered up the expanding tendency and gave the impression of a negligible increase. After a short lowering interlude about mid August
the rise came back in late August with $\rho \approx 1.2$, before it accelerated in late September, brought the reproduction rate to about 1.5, and led straight into the second wave. Here the weekly oscillations of recorded new infections rose to an amplitude not conceived before (fig.
<ref>). The levels of the mean daily strength of infection used in the model (proportional to the corresponding reproduction rates) are well discernible in the next figure <ref>, right.
Left: Empirical reproduction rates $\hat{\rho}(k)$ for Germany (yellow). Right: Daily strength of infection $\hat{\eta}(k)$ for Germany (yellow) with model parameters $\eta_j$ in the main intervals $J_j$ (black dashed); critical value $1/p$ of $\eta$ dotted.
Daily new reported cases for Germany, 3-day sliding averages $\hat{A}_{new}(k)$ and 7-day averages $\hat{A}_{new,7}(k)$.
The dates $t_j$ of the time markers $k_j$ used for our model in the German case are $t_0 =$ 02/25, 2020, $t_1= $ 03/24, $t_2= $ 04/26, $t_3= $ 06/06, $t_4= $ 06/16, $t_5= $ 07/05, $t_6= $ 08/17, $t_7= $ 08/28, $t_8= $ 09/27, $t_9= $ 10/31, $t_{10}= $ 11/28, $t_{11}= $ 12/14, end of data here $t_{eod}= $ 01/15, 2021. In the country day count, where $t_0=1$ ($\sim$ 35 in JHU day count) the main intervals are $J_0= [1, 27], \; J_1=[29, 59],\; J_2= [62, 100], \; J_3=[103, 108]; J_4= [113, 128], \; J_5=[132, 165],\; J_6= [175, 179], \; J_7=[186, 207]; J_8= [216, 241], \; J_9=[250, 270],\; J_{10}= [278, 285], \, J_{11}= [294,326] $.
The strength of infection and reproduction numbers in the main intervals are given in the following table.
13|c|$\eta_0$ and model $\eta_j$, $\rho_j$ in intervals $J_j$ for Germany
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_{8}$ $J_9$ $J_{10}$ $J_{11}$
$\eta_j$ 0 0.108 0.119 0.281 0.109 0.174 0.131 0.158 0.215 0.141 0.170 0.139
$\rho_j$ — 0.75 0.83 1.96 0.76 1.21 0.91 1.10 1.49 0.97 1.15 0.93
With these parameters the SEPAR model reproduces (pre- or better “post”-dicts) the averaged daily new infections and the total number of reported infected well (fig. <ref>), while one has to be more careful for treating the reported number of actual infected $\hat{A}$.
Left: Daily new reported infected (3-day averages) for Germany; empirical $\hat{A}_{new}$ solid red, model $A_{new}$ black dashed.Right: Total number of reported infected; empirical $\hat{A}_{tot}$ solid brown, model $A_{tot}$ black dashed.
If one checks the mean duration of being recorded as actual case in the JHU statistics for Germany by eq. (<ref>) one finds a good approximation $q \approx 15$ after the early phase; but the result also indicates that in May/June, and again in the second half of November, the reported duration of the infected state surpassed this value (fig. <ref> left). Accordingly the empirical data $\hat{A}$ and the corrected ones $\hat{A}_q$ ($q=15$) drop apart in late November (same figure, right).
Left: Daily values of the mean time of statistically actual infection $\hat{q}(k)$ for Germany. Right: Comparison of reported infected $\hat{A}$ (dark blue) and $q$-corrected number ($q=15$) of recorded actual infected $\hat{A}_q$ (bright blue) from the JHU data in Germany.
In consequence the model value for the actual infected $A(k)$ agree with the JHU data $\hat{A}(k)$ only if the model uses time varying values $\hat{q}(k)$ (fig. <ref> left), while the $q$-corrected numbers $\hat{A}_q(k)$ are well reproduced by the model with constant $q=15$ (same figure, right).
Left: Statistically actual cases $\hat{A}(k)$ for Germany and the corresponding model values $A(k)$ (black dashed) calculated with time dependent model values for $q$ (see text).
Right: Empirical values, $q$-corrected, for actual cases $\hat{A}_q(k)$ and the corresponding model values $A(k)$ (black dashed), $q=15$.
As a result, the 3 model curves representing the total number of (reported) infected $A_{tot}$, the redrawn $R$ and the actual infected fit the German data well, if the last two are compared with the $q$-corrected empirical numbers $\hat{A}_q$ (fig. <ref>).
Empirical data (solid coloured lines) and model values (black dashed) for Germany: numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{Rq}$ (bright green), and $q$-corrected actual numbers $\hat{A}_q$ (bright blue) .
Conditional predictions for $A_{new}, \, A_q$ and $A_{tot}$, assuming no essential change of the behaviour, contact rates and the reproduction number from the last main interval $J_{10}$ are given in fig. <ref>. The dotted lines indicate the boundaries of the prediction for the 1-$\sigma$ domain for the variations of the values of $\hat{\eta}$ in the last main interval $J_{10}$.
30-day prediction for $A_{new}, \, A$ (top) and $A_{tot}$ (bottom) for Germany; empirical values coloured solid lines, model black dashed (boundaries of 1-sigma region prediction dotted).
§.§.§ France
The overall picture of the epidemic in France (population 66 M) is similar to other European countries. But the
French JHU data show anomalies which are not found elsewhere: The differences of two consecutive values of the confirmed cases, which ought to represent the number of newly reported, is sometime negative!
This happens in particular in the early phase of the epidemic (until June 2020) where, e.g., $\hat{A}_{new}(58)=-2206$, and there are other days with negative entries. Presumably this is due to ex-post data corrections necessary in the first few months of the epidemic. Later on
the control over the documented data seems to have been improved; negative values are avoided, although null entries in $\hat{A}_{new}$ still appear.
So the French data are a particular challenge to any modelling approach.
Even in this extreme case the smoothing by 7-day sliding averages works well, as shown in fig. <ref>.
Left: Number of daily newly reported in France $\hat{A}_{new}(k)$. Right: 7-day sliding averages of new infections $\hat{A}_{new,7}(k)$ for France.
Another surprising feature of the French (JHU) statistics is an amazing increase in the number of days which infected persons are being counted as “actual cases”. It starts close to 15, but shows a monotonous increase until late October where a few downward outliers appear, before the curve turns moderately down in early November 2020 (fig. <ref>, left).
Left: Daily values of the mean time of statistically actual infection $\hat{q}(k)$ for France. Right: Comparison of reported infected $\hat{A}$ (dark blue) and $q$-corrected number ($q=19$) of recorded actual infected $\hat{A}_q$ (bright blue) from the JHU data in France.
As a consequence the peak of the first wave in early April (clearly visible in the number of newly reported) is suppressed in the curve of the actual infected; $\hat{A}(k)$ has no local maximum in the whole period of our report. It even continues to rise, although with a reduced slope, after the second peak of the daily newly reported, $\hat{A}_{new}$, in early November.
The decrease of the slope of $\hat{A}$ starts shortly after this peak, accompanied by a local maximum of the
$q$-corrected values for the actual infected reaches $\hat{
A}_q$ (fig. <ref>). Both effects seem to be due to a downturn of $\hat{q}(k)$ . This extreme behaviour of the data cannot be ascribed to medical reasons; quite obviously it results from a high degree of uncertainty in data taking and recording in the French health system.
Left: Empirical reproduction rates $\hat{\rho}(k)$ for France. Right: Daily strength of infection are shown in figure <ref> for France (yellow) with model parameters $\eta_j$ in the main intervals $J_j$ (black dashed).
The reproduction numbers are shown in figure <ref>, left. For the determination of the daily strength of infection we have to fix a value for the dark factor. Lacking data from representative serological studies in France we assume that it is larger than in Switzerland and
choose as a reference value for the model $\delta=4$. With this value the determination of the $\hat{\eta}_7(k)$ are given in figure <ref>, right, here again with dashed black markers for the periods modelled by constancy intervals in our approach.
The markers of change times are here
$t_0 =$ 02/25, 2020, $t_1= $ 05/17, $t_2= $ 06/15, $t_3= $ 07/21, $t_4= $ 08/22, $t_5= $ 09/29, $t_6= $11/03, $t_7= $11/27, end of data $t_{eod}= $ 12/30, 2020. In the country day count, $t_0 =1$ ($\sim 35$ in JHU day count), the main intervals are $J_0= [1, 82], \; J_1=[83, 110],\; J_2= [112, 146], \; J_3=[148, 173]; J_4= [180, 213], \; J_5=[218, 234],\; J_6= [253, 267],\; J_7= [277, 310] $.
The model strength of infection $\eta_j$ and corresponding reproduction numbers $\rho_j$ for the main intervals are given in the following table.
9|c|$\eta_0$ and model $\eta_j$, $\rho_j$ in intervals $J_j$ for France
$a_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$
$\eta_j$ 2.008 0.149 0.158 0.203 0.170 0.211 0.118 0.186
$\rho_j$ — 1.03 1.09 1.39 1.16 1.40 0.71 1.07
Left: Daily new reported number of infected for France (7-day averages); empirical $\hat{A}_{new}$ solid red, model $A_{new}$ black dashed. Right: Total number of reported infected (brown); empirical $\hat{A}_{tot}$ solid, model $A_{tot}$ black dashed.
Empirical values for statistically actual $q$-corrected cases $\hat{A}_q(k)$ and the corresponding model values $A(k)$ (black dashed) for France.
An overall picture of the French development of new infections and total number of recorded cases is given in fig. <ref>. The so-called “actual” cases are well modelled in our approach (fig.<ref>), if the reference are the $q$-corrected numbers $\hat{A}_q(k)$ of actual infected (or if time dependent durations $q(k)$ (read off from the JHU data, $q(k)=\hat{q}(k)$) are used). For a combined graph of the 3 curves see fig. <ref>.
Empirical data (solid coloured lines) and model values (black dashed) in France for numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}$ (bright green), and $q$-corrected actual numbers $\hat{A}_q$ (bright blue).
§.§.§ Sweden
Sweden (population 10 M) has chosen a path of its own for containing Covid-10, significantly different from most other European countries. In the first half year of the epidemic no general lockdown measures were taken; the general strategy consisted in advising the population to reduce personal contacts and to go into self-quarantine, if somebody showed symptoms which indicate an infection with the SARS-CoV-2 virus. One might assume that the number of undetected infected, the dark sector, could be larger than in other European countries. As we see below such a hypothesis is not supported by the analysis of the data in the framework of our model.
Under the conditions of the country (in particular the relative low population density in Sweden) the first wave of the epidemic was fairly well kept under control, if we abstain from discussing death rates like in the rest of this paper. Once the initial phase was over (with reproduction numbers already lower than in comparable countries, but still up to about $\rho \approx 2$), the reproduction rate was close to 1 for about two weeks in late June, and even lower in early July (fig. <ref>, right). In the second half of October 2020, however, Sweden was hit by a second wave like all other European countries. After a sharp rise of the daily new infections in early October (fig. <ref>, left), the Swedish government decided to decree a (partial) lockdown. By this the reproduction number, which already in late August had risen to above 1.2 and went up to 1.45 in mid October, was brought down to the former range ($\rho \approx 1$), although now on a much higher level of actual infected than during the first wave.
Left: Daily new reported cases $\hat{A}_{new}(k)$ (JHU data) for Sweden and 7-day sliding averages $\hat{A}_{new,7}(k)$. Right: empirical reproduction rates $\hat{\rho}(k)$ for Sweden.
The higher the assumptions for the dark sector, the larger the calculated values for $\hat{\eta}(k)$ on the basis of the same data, and vice versa. Figure <ref> shows the differences of $\hat{\eta}_7(k)$ in the case of Sweden under the hypotheses $\delta = 0, \; 4, \; 15, \; 25$. Until July/August 2020 the four curves show minor differences and indicate relative stable values for the infection strengths leading to reproduction rates close to 1. In September/October the values rose considerably; they went down after the October lockdown only under the assumption of a small dark sector, $\delta=0$ or $4$, while for the larger dark factors $\delta=15, \; 25$ the values of the daily strength of infection continues to increase. This seems implausible.[Such a hypothesis for the dark sector could be explained only by a drastic and irresponsible change of contact behaviour of the Swedish population or an increased infectivity of the virus. Neither of these explanations is supported by available empirical evidence.]
We therefore choose $\delta =4$ also for Sweden.
Daily strength of infection (sliding 7-day averages) $\hat{\eta}_7(k)$ for Sweden, assuming different dark factors $\delta$.
Top: $\delta=0$ (left) and $\delta=4$ (right). Bottom: $\delta=15$ (left) and $\delta=25$ (right).
In our definition the new infections in Sweden ceased to be sporadic at $t_0 =$ 03/02, 2020, the day 41 in the JHU day count. After a month of strong ups and downs of the strength of infection, the approach of constancy intervals gains traction; with time markers of the main intervals $t_1= $ 03/31, $t_2= $ 05/17, $t_3= $ 06/18, $t_4= $ 07/17, $t_5= $ 08/25, $t_6= $ 10/10, $t_7= $ 11/05, $t_8= $ 12/12,
end of data $t_{eod}= $ 12/30, 2020. In the country count the main intervals are $J_0= [1, 29], \; J_1=[30, 75],\; J_2= [77, 99], \; J_3=[109, 126]; J_4= [138, 169], \; J_5=[177, 218],\; J_6= [223, 214],\; J_7= [223, 284],\; J_8= [286, 303] $.
10|c|Model $\eta_0$ and $\eta_j$, $\rho_j$ in intervals $J_j$ for Sweden
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$
$\eta_j$ 0.659 0.146 0.174 0.095 0.149 0.177 0.232 0.178 0.150
$\rho_j$ — 1.01 1.20 0.65 1.00 1.19 1.54 1.13 0.84
Although one might want to refine the constancy intervals, already these intervals lead to a fairly good model reconstruction of the mean motion of new infections and the total number of infected (fig. <ref>). Note that since early September the reported numbers of new infections show strong weekly oscillations between null at the weekends and high peaks in the middle of the week.
Left: Daily new reported infected for the Sweden, empirical 3-day averages $\hat{A}_{new,3}$ solid red; model $A_{new}$ black dashed. Right: Total number of reported infected (brown); empirical $\hat{A}_{tot}$ solid, model $A_{tot}$ dashed.
The JHU statistics does not register recovered people for Sweden at all; only deaths are reported. In consequence the usual interpretation of (<ref>) as characterizing the “actual” infected breaks down for Sweden[The same holds for the UK.] and the estimation (<ref>) for the mean duration of illness becomes meaningless (fig. <ref>, left). An indication of the extent of reported actual diseased is given by the $q$-corrected number $\hat{A}_q$ (same figure, right).
In this sense, the synopsis with a collection of the “3 curves” can be given for Sweden like for any other country (fig. <ref>).
Of course the model reproduces the empirical values $\hat{A}(k)$ even in such an extreme case if the time dependent empirical values of (<ref>) are used for the the model calculation, $q(k)=\hat{q}(k)$, while it reconstructs the $q$-corrected numbers for the estimate of actually infected $\hat{A}_q(k)$ if the respective constant is used, here $q=15$ (fig. <ref>).
Left: Daily values of the mean time of statistically actual infection $\hat{q}(k)$ for Sweden. Right: Comparison of reported infected $\hat{A}$ (dark blue) and $q$-corrected number ($q=15$) of recorded actual infected $\hat{A}_q$ (bright blue) from the JHU data in Sweden.
Left: Empirical data $\hat{A}(k)$ for Sweden (blue) and model values $A(k)$ determined with time varying $q(k)=\hat{q}(k)$ (black dashed).
Right: Empirical values, $q$-corrected, for statistically actual cases $\hat{A}_q(k)$ and the corresponding model values $A_q(k)$ (black dashed).
Left: Numbers of totally infected $\hat{A}_{tot}$ (brown), reported redrawn $\hat{R}$ (bright green) – here deaths only – and the difference $\hat{A}$ (blue) for Sweden; model values black dashed. Right: the same for $\hat{A}_{tot}$ (brown), $q$-corrected $\hat{A}_q$ (bright blue) and the redrawn $\hat{R}_q$ (green) as the difference (model values black dashed).
§.§ The three most stricken regions: USA, Brazil, India
In this section we give a short analysis of the course of the pandemic during 2020 for the three countries which have to bemoan the largest numbers of deceased and huge numbers of infected (USA, Brazil, India). We expected higher dark factors $\delta$ than for the European countries discussed above and checked this expectation by the same heuristic approach as used for Sweden, i.e. by a comparative judgement of the changes of the empirically determined strength of infection $\hat{\eta}_7(k)$, which result from different assumptions of the values for $\delta$. To our surprise we found no clear evidence for an overall larger dark factor for the USA than for the European countries and work here with $\delta =4$, while for India there are strong indications of a large dark factor which we estimate as $\delta\approx 35$ (see below). For Brazil we consider $\delta \approx 8$ a reasonable choice for the overall development of the epidemic. Don't forget, however, that all these are plausibility considerations which are not based on representative serological studies.
§.§.§ USA
At the beginning of the pandemics the United States of America (population
333 M) suffered a rapid rise of infections with an initial reproduction rate well above 5. In early April 2020 this dynamics was broken and a slow decrease started for about 2 months. In mid June a second wave with an upswing for about a month and a reproduction rate shortly below $1.3$ followed. In late August and early September the subsequent downswing faded out. After a short phase of indecision the beginnings of a third wave became clearly visible; it lasted until (at least) mid December. Figure <ref> shows the daily new infections, the reproduction rate $\hat{\rho}(k)$ determined from the JHU data and the strength of infection $\hat{\eta}_7(k)$ assuming $\delta=4$ .
Top: 3-day and 7-day sliding averages of daily new reported cases, $\hat{A}_{new, 3}(k)$, $\hat{A}_{new, 7}(k)$, for the USA. Bottom, left: Empirical reproduction rates $\hat{\rho}(k)$ for the USA. Right: Daily strength of infection $\hat{\eta}_7(k)$ for the USA (yellow), with $\delta=4$; model parameters $\eta_j$ in the main intervals $J_j$ black dashed.
Daily strength of infection (sliding 7-day averages) $\hat{\eta}_7(k)$ for USA, assuming different dark factors:
$\delta=0$ (left), $\delta=4$ (middle), and $\delta=8$ (right).
Figure <ref> displays three variants of $\hat{\eta}_7(k)$ for $\delta= 0, \; 4, \; 8$.
The third one shows an implausible increase for the strength of infection at the end of the year, which would seem reasonable only if one of the new, more aggressive mutants of the virus had started to spread in the USA in September 2020 already. Without further evidence we do not assume such a strong case. As $\delta=0$ contradicts all evidences collected on unreported infected, we choose $\delta=4$ for the SEPAR$_d$ model of the USA. Also here we find a moderate increase of the mean level of the $\hat{\eta}_7$.
To judge whether this may be due to the inconsiderate behaviour of part of the US population (supporters of the outgoing president) or the first influences of a virus mutation and/or still other factors is beyond the scope if this paper and our competence.
In section <ref> it was already noted that the estimation of the time of sojourn in the “actual” state of infectivity, suggested by the statistics for the USA, leads to surprising effects. It rises from about 15 in March 2020 to above 100 in early November, with a moderate platform in between; then it starts falling,before it makes an abrupt jump (fig <ref>, left). The jump of $\hat{q}(k)$ is an artefact of a change in the record keeping: from December 14, 2020 onward the reporting of data of recovered people was given up ($Rec(k)=0$ for date of $k$ after 2020/12/14). Of course this jump is also reflected in the numbers of recorded actual infected $\hat{A}(k)$ (same figure, right).
Left: Daily values of the mean time of statistically actual (“active”) infection $\hat{q}(k)$ for USA. Right: Empirical data (JHU) for statistically actual cases $\hat{A}(k)$ (dark blue) versus $q$-corrected ones, $q=15$, $\hat{A}_q(k)$ (bright blue) for the USA.
The main (constancy) intervals of the
model are visible in the graph of the daily strength of infection $\hat{\eta}_7(k)$ in fig. <ref>, bottom right.
The dates of the time marker between the intervals are:
$t_0$= 02/29 2020; $t_1=$ 04/02, $t_2=$ 06/10,
$t_3=$ 07/19, $t_4=$ 09/04, $t_5=$ 10/21 $t_6=$ 11/11, end of data $t_{eod}=$ 12/30.
Expressed in terms of the country day count with $k_0=1$ ($\sim 39$
in the JHU day count)
the main intervals for the USA are:
$ J_0=[1 , 33], \, J_1=[34, 92], \, J_2 = [103, 118], \,J_3 = [142, 185], \,
J_4 = [189, 222] ,$ $\, J_5 = [236, 249], \, J_6 = [257, 322]
The start parameter for the strength of infection $\eta_0$ and a slightly adapted choice of parameter values $\eta_j$ inside the 1-sigma domain of the respective interval $J_j$ are given by the following table.
8|c|Model $\eta_0$ and $\eta_j$, $\rho_j$ in intervals $J_j$ for the USA
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$
$\eta_j$ -1.011 0.140 0.189 0.143 0.172 0.215 0.194
$\rho_j$ — 0.97 1.28 0.94 1.08 1.30 1.12
Ratio of susceptibles $s(k)$ (model values) for the USA, $\delta=4$.
Here, as for the other countries, the $\rho_j$ denote the model reproduction rates at the beginning of the respective intervals. Assuming the dark factor $\delta=4$, the effective reproduction rate $\rho(k)$ changes considerably from the beginning of $J_6$ until the end of the year, $s(11/11\, 2020)=0.82$ to $s(12/31)=0.67$, (cf.fig. <ref>).
In consequence, the reproduction rate at the end of the year is down to $\rho(12/31)=0.86$; and the newly reported infected are expected to reach a peak in late December (fig. <ref>, right). Apparently this does not agree with the data, while the total number of infections is well reproduced by the SEPAR$_d$ model (same figure, left). The difference for the new infected may be an indication that our estimate of the dark factor $\delta$ is wrong or the strength of infection at the beginning of the new year 2021 is drastically boosted, e.g., by a rapid spread of a new, more aggressive mutant of the virus.
Left: Total number of infected (brown); empirical $\hat{A}_{tot}$ solid, model $A_{tot}$ dashed. Right: Daily newly reported for the USA, 7-day averages (red); empirical $\hat{A}_{new}$ solid red, model $_{new}$ black dashed ($\delta=4$).
As noted above, the recovered people are
documented with increasingly large time delays in the records for the USA. Therefore it seems preferable to compare the model value of actually infected, $A(k)$, with $\hat{A}_q(k)$ rather than with $\hat{A}(k)$ (fig. <ref>). The empirical determined values of $\hat{A}_q(k)$ are shown in bright blue in the figure. They are marked by three local maxima indicating the peak values of three waves of the epidemics in the USA. These peaks are blurred in the graph of $\hat{A}$ because too many of the effectively redrawn are dragged along as acute cases in the statistics. Due to under-reporting at the end of the year, the local extremum in December may be fuzzier than it appears here. But keep in mind that the SEPAR$_d$ model with $\delta=4$ predicts a local maximum inside the interval $J_6$, if the contact behaviour and the resulting strength of infectivity $\eta_6$ do not change considerably.
Number of $q$-corrected actual infected for the USA; empirical $\hat{A}_q$ (bright blue) and model values $A_q$ (black dashed).
Figure <ref>, left,
shows the 3 curves for the model values (black dashed) of the total number of infected $A_{tot}$, the actually infected $A$ in terms of estimates with constant $q=15$, and the redrawn $R$, all of them compared with the corresponding values $\hat{A}_{tot}(k),\; \hat{A}_q(k), \;\hat{R}_q(k)$ determined from the JHU data (coloured solid lines).
By using time dependent values $q(k)$, like, e.g., in the case of Germany, SEPAR$_d$ is able to
model the statistically “actual” cases also here (fig. <ref>, right).
Because of the growing fictitiousness of the numbers $\hat{A}(k)$ in the case of the USA we prefer, however, to look at the corrected values $\hat{A}_q(k)$, as stated already.
Left: numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}_q$ (green), and $q$-corrected actual numbers $\hat{A}_q$ (bright blue). Right: Numbers of totally infected $\hat{A}_{tot}$ (brown), reported redrawn $\hat{R}$ (brigth green), and actual numbers $\hat{A}$ of the statistic (blue) for the USA. Empirical data (solid coloured lines) and model values (black dashed).
§.§.§ Brazil
The documentation of newly reported became non-sporadic in Brazil (population 212 M) at $t_0=$ March 15, 2020.
A first peak for the officially recorded number of actual infected $\hat{A}(k)$
was surpassed in early August 2020 with a decreasing phase until late October, after which a second wave started (fig. <ref> left). In contrast to the USA we find here a comparatively stable estimate for the mean time $\hat{q}(k) \approx 14$ (fig. <ref> right).
Left: Acute infected $\hat{A}(k)$ recorded by the statistics (dark blue) in comparison with $q$-equalized number $\hat{A}_q(k)$ (bright blue) for Brazil ($q=14$). Right: Empricial estimate $\hat{q}(k)$ of mean duration of active infective according to the statistics for Brazil.
The outlier peak of $\hat{A}(k)$ about October 25 appears also as an exceptional peak in the $\hat{q}(k)$. Apparently it is due to an interruption of writing-off actual infected to the redrawn (compare fig. <ref>).
Up to this exceptional phase there is a close incidence between the $\hat{A}(k)$ and the $q$-corrected number $\hat{A}_q(k)$. The minimum of the mean square difference is acquired for $q=14$. The numbers of newly reported $\hat{A}_{new}(k)$ show strong daily fluctuations which are smoothed by the 7-day sliding average $\hat{A}_{new,7}(k)$ (fig. <ref>).
Daily varying numbers $\hat{A}_{new}(k)$ (left)versus 7-day sliding averages $\hat{A}_{new,7}(k)$ (right) of new infections for Brazil.
Daily strength of infection (sliding 7-day averages) $\hat{\eta}_7(k)$ for Brazil, assuming different dark factors.
Top: $\delta=0$ (left) and $\delta=4$ (right). Bottom: $\delta=8$ (left) and $\delta=15$ (right).
A comparison of different strengths of infection $\hat{\eta}_7(k)$ indicates a value between $\delta=0$ and $\delta=8$ as a plausible choice (fig. <ref>). For higher values an unnatural increase of the infection strength would appear close to the end of the year (if not due to a new mutant.
As we assume that in Brazil the dark factor is higher than in European countries, we use $\delta=8$ as a plausible model hypothesis.
Left: Empirical reproduction rates $\hat{\rho}(k)$ for Brazil (orange).
Right: Daily strength of infection, empirical $\hat{\eta}_7(k)$ (yellow), for Brazil assuming $\delta=8$, and model parameters $\eta_j$ in the main intervals $J_j$ (black dashed).
The daily reproduction numbers $\hat{\rho}(k)$ (independent of $\delta$) start from a lower level than for many other countries, slightly above 2. They show a relatively stable downward trend, falling below 1 for a few days in early June and for longer periods after June 22, 2020 (fig. <ref>, left).
But already at the end of March, when the reproduction rate was still considerably above 1 ($t_0 = $ March 30), its downward trend was already slow enough to allow for approximation by constancy intervals.
In the case of Brazil the initial interval starts at $t_0=$ 03/15, 2020. The following time can be subdivided into main intervals $J_j$ in which the averaged daily strength of infection $\hat{\eta}(k)$ can be replaced by their mean values $\eta_j$, starting with $t_1=$ 03/30. The main intervals are separated by the days
$t_2=$ 05/14, $t_3=$ 06/22, $t_4=$ 07/11, $t_5=$ 07/19 $t_6=$ 08/27, $t_7=$ 08/10, $t_8=$ 10/29, $t_9=$ 12/13, 2020. They
can well be discerned in fig. <ref>, right, showing the daily strength of infection $\hat{\eta}_7(k)$ (yellow) and their mean values (black dashed) in these intervals.
In the country day count $k_0=1$ ($\sim 70$ in the JHU count) the main intervals are
$J_0=[1, 15], \,J_1=[16, 59], \; \; J_2 = [61, 97], \, J_3 = [100, 115], \, J_4 = [119, 122], \,
J_5 =[127, 164],
J_6= [166, 204],\, J_7= [206, 220], \, J_8= [222, 271], \, J_9= [274, t_{eod}] $,
here with the end of data $t_{eod}=292$.
The start parameter $\eta_0$ and the model reproduction numbers $\rho_j$ in the respective interval $J_j$ are
11|c|Model $\eta_0$ and $\eta_j$ and $\rho_j$ in $J_j$ for Brazil
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_6$ $J_8$ $J_9$
$\eta_j$ 0.031 0.202 0.164 0.142 0.191 0.143 0.141 0.148 0.180 0.172
$\rho_j$ — 1.42 1.14 0.98 1.30 0.96 0.94 0.97 1.17 1.00
Also here the $\rho_j$ designate reproduction numbers at the beginning of the $j$-the interval and the fall of $s(k)$ makes the reproduction number cross the critical value in the last main interval (cf. <ref>). This does not mean that it will stay there.
Model values of the ratio of susceptibles $s(k)$ for Brazil ($\delta=8$).
The resulting model curves and their relationship to the empirical data for new infections and acual infections are shown in fig. <ref>.
A panel of the three curves $A_{tot}, \, A, R$ is shown in fig. <ref>. Here one sees clearly that the outlier bump of $\hat{A}(k)$ is accompanied by an inverse outlier in $\hat{R}(k)$.
Left: Empirical values (3-day average) for daily newly reported for Brazil $\hat{A}_{new,3}$ (solid red line)) and model values black dashed). Right: Actual cases $\hat{A}$ (blue), model values (black dashed).
Empirical data (coloured solid lines) for the numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}$ (bright green), actual infected $\hat{A}$ (blue) and the respective model values (black dashed) for Brazil ($\delta=8$).
§.§.§ India
The recorded data on $\hat{A}_{new}(k)$ for India (population 1387 M) start to be non-sporadic at $t_0=$ March 4, 2020. From this time on we find a steady growth of the number of reported actual infected $\hat{A}(k)$ until early September. Because the size of the country and the life conditions in large parts of it a comparatively high number of unrecorded infected may be assumed, with a dark factor at the order of magnitude $\delta \sim 10$ probably $20 \leq \delta \leq 50$.[A serological investigation of over 4000 inhabitants found 24 % infected (from which over 90 % were asymptomatic). With about 140 $k$ reported infected in a population of roughly 31 $M$ this amounts to a dark factor $\delta \approx 50$ (source ANI retrieved 12/21 2020 <https://www.aninews.in/news/national/general-news/second-sero-survey-finds-2419-pc-of-punjab-population-infected-by-covid-1920201211181032/>).]
In early September the tide changed and a nearly monotonous decline of actual infected started. With the exception of a short intermediate dodge the decline continues at the end of 2020 (fig. <ref>, left). Although in late December 2020 there were only about 0.7 % recorded infected in India, the high quota of unreported infected poses the question whether the downturn in late summer may already be due to a the decrease of the fraction of susceptibles $s(k)$.
Before we discuss this point let us remark that the time of being statistically recorded as actual case is relatively stable in the Indian data, with a good approximative constant value $q\approx 11$ (fig. <ref>, right). In consequence $\hat{A}(k)$ does not differ much from $\hat{A}_q(k)$ (same fig. left). This allows to use the recorded data $\hat{A}$ in the following without the proviso to be made in the case of the USA.
Left: Actual infected $\hat{A}(k)$ recorded by the statistics (dark blue) in comparison with $q$-equalized number $\hat{A}_q(k)$ (bright blue) for India ($q=11$). Right: Empiricial estimate $\hat{q}(k)$ of mean duration of actual infectived according to the statistics, i.e. in $\hat{A}$, for India.
The reproduction numbers and the corresponding daily strength of infection of the model are derived from the 7-day sliding averages of newly reported. Figure <ref> shows both the daily varying $\hat{A}_{new}$ and the averaged $\hat{A}_{new,7}$.
Top: Empirical number of daily new infections $\hat{A}_{new}$ and 7-day sliding average $\hat{A}_{new,7}$ for India.
The reproduction number
fell rapidly from roughly 3.5 at the beginning to below 1.5 in early April, and 1.2 in late May, after which it continued to decrease with minor fluctuations. In early September it dropped below the critical value 1, where it stayed with few exceptional fluctuations until December (fig. <ref>). It runs, of course, parallel to the daily strength of infection $\hat{\eta}(k)$ calculated from the 7-day averages if abstraction is made from the dark sector, $\delta=0$ (fig. <ref>, left). Here we confront it with the more realistic graph of $\hat{\eta}_7$ calculated under the assumption of a dark sector.
Left: Empirically determined reproduction rates $\hat{\rho}(k)$ for India (iorange). Right: Empirical infections strength $\hat{\eta}_7$ (yellow) assuming a dark sector with $\delta=35$; model values for $\eta$ black dotted.
Left: Empirical daily strength of infection $\hat{\eta}_7(k)$ for India (yellow) with no dark sector, i.e. assuming $\delta =0$.
Right: Empirical strength of infection $\hat{\eta}_7(k)$ for India (yellow), assuming a dark sector with factor $\delta = 35$
Such an idealized scenario with $\delta =0$ is shown in fig <ref>, left.
If, on the other hand, the empirical daily strength of infection
are determined under the more realistic assumption of a non-negligible dark sector, e.g. $\delta = 35$, the picture is different (same figure, right). Here one finds a daily strength of infection moderately fluctuating in a narrow band between 10 and 20 % above the critical value $\eta_{crit} = \frac{1}{7} \approx 0.147 $, rather than dropping below it in early September like in the first case. We do not know of any indications for a changing contact behaviour of the population in India; neither can we assume a decreasing aggressiveness of the virus. Therefore the first scenario ($\delta \approx 0$) looks highly unrealistic. In both cases the empirically determined reproduction rates $\hat{\rho}(k)$ are the same (fig. <ref>). In the second case the fall of $\hat{\rho}(k)$ below 1 in early September is due to the lowering of $\hat{s}(k)$, i.e. as an effect of an incipient herd immunization.
But how can that be with a herd immunization quota of $(1+\delta) \frac{\hat{A}_{tot}(k)}{N} \approx 12\, \%$, even with $\delta = 35$, in September 2020?[In 12/2020 it was already twice as much.]
The reason lies in the comparatively low overall daily strength of infection. Between May and December 2020 it fluctuated between 10 and 20 % above the critical level (fig. <ref>, right). Even if part of the low level had to be ascribed to an intentional under-reporting of the $\hat{A}_{new}(k)$, this would mean an increasing size of the dark sector; the overall effect would be the same.[Only a permanently increasing amount of under-reporting could emulate a fake picture of a non-existing downswing of the epidemic for several months. We exclude such a hypothesis.]
For modelling the epidemic in India we can do with 7 constancy intervals.
After the initial interval $ J_0=[1 , 30]$ in the day count of the country ($k_0 \sim 43$ in the JHU count) the main intervals are
$J_1=[31, 70], \, J_2 = [74, 137], \, J_3 =[142, 161], \, J_4= [169, 179], \, J_5= [191, 228], \, J_6= [234, 256], \, J_7= [260, k_{eod}]$,
with end of data $k_{eod}=297$. The date of the interval separators are $t_0=$ 03/04, $t_1= $ 04/03, $t_2=$ 05/16, $t_3= $ 07/23, $t_4=$ 08/19, $t_5=$ 09/10, $t_6=$ 10/23, $t_7=$ 11/18, end of data $t_{eod}= $ 12/26 2020.
Similar to the Brazilian case, the reproduction rate surpasses 1 in the first four main intervals; only in mid September a downswing of the epidemic started, interrupted by an intermediate dodge at the beginning of November. In mid September the total number of acknowledged infected was roughly $A_{tot}(240) \approx 5 \; M$, about 3.8 per mill of the total population; but with a dark quota of $\delta=35$ the total number of infected had probably already risen above the 10 % margin (see above).
The start parameter of the model $\eta_0$ is chosen according to the best adaptation to the 7-day averaged data (without a claim for a directly realistic interpretation) and the parameter values $\eta_j$ essentially as the mean values of the $\hat{\eta}(k)$ in the respective interval $J_j$. They are given in the table.
9|c|Model $\eta_0$ and $\eta_j$, $\rho_j$ in $J_j$ for India
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$
$\eta_j$ -0.138 0.192 0.175 0.164 0.177 0.157 0.182 0.166
$\rho_j$ — 1.34 1.2 1.07 1.12 0.90 0.99 0.87
With these parameters the SEPAR model leads to a convincing reconstruction of the epidemic in India. This is shown by the graph showing the
three curves $A_{tot}, \, A, R$ (fig. <ref>).
Empirical data (colored solid lines) for the numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}$ (bright green), actual infected $\hat{A}$ (blue), and the respective model values (black dashed) for India. All with dark sector, $\delta=35$ and constancy intevals (see main text).
The numbers of newly reported $\hat{A}_{new}(k)$ and the number of actual cases $\hat{A}(k)$, including a conditional prediction for the next 30 days on the basis of the last infection strength $\eta_6$ (fig. <ref>). Black dotted the boundaries of the 1 $\sigma$ domain for the $\hat{\eta}$ -variation in $J_6$. In the case of India the data show exceptional low variability inside the constancy intervals. Thus the width of the 1 $\sigma$ domain is smaller than in any of the other countries considered.
Left: Daily newly reported for India, empirical $\hat{A}_{new}$ (solid red line) and model $A_{new}$ (black dashed). Right: Reported actual cases for India, empirical $\hat{A}$ (blue) versus model $A$ (black dashed). Both with dark sector, $\delta=35$ and 30-day conditional prediction assuming no large vchange of the infection strenght in $J_6$, the last main interval.
With a dark sector roughly as large as assumed in the model ($\delta = 35$) the ratio of susceptibles went down in late 2020 to $s(k) \approx 0.7$ (fig <ref>). This is the background for the reassuring prognosis for the development in India at the beginning of 2021 (fig. <ref>).
Ratio of susceptibles $s(k)$ for India.
§.§ Aggregated data of the World
Let us now see how the aggregated data of all countries and territories documented in the JHU data resource can be analysed in our framework, and how they are reproduced by the SEPAR$_d$ model. For the sake of simplicity we speak simply of the World.
The number of daily new infections $\hat{A}_{new}$ shows clearly three or four steps, expressed by phases of accelerated growth of $\hat{A}_{new}$ between February and December 2020 (fig. <ref>): March (European countries), May to July (two waves in the US, bridged by rising numbers in Brazil and India), October (second wave in Europe and Brazil, third wave in US), and less visible the January/February wave in China and South-Korea.
Daily new reported cases $\hat{A}_{new}(k)$ for the World and 7-day sliding averages $\hat{A}_{new,7}(k)$.
These steps of steeper increase of the daily new infections correspond to local peaks or elevated levels of the mean strength of infection and reproduction numbers. The first two peaks of the mean reproduction numbers with $\rho_{peak-1} \approx 2$ in January (China) and $\rho_{peak-2} \approx 2.5$ in March (Europe) are followed by much lower phases of elevated levels from early May to early July, respectively in October 2020 (fig. <ref>).
Bottom, left: Empirical reproduction rates $\hat{\rho}(k)$ for the World (orange). Right: Daily strength of infection $\hat{\eta}(k)$ for the World (yellow) with model parameters $\eta_j$ in the main intervals $J_j$ (black dashed).
We let the model start at $t_0 =$ 01/25, 2020, the fourth day of the JHU day count, and use the following time separators for the main (constancy) intervals
$t_1= $ 03/29, $t_2= $ 05/06, $t_3= $ 07/19, $t_4= $ 10/02, $t_5= $ 11/10, $t_6=11/26 $, end of data $t_{eod}= $ 12/11, 2020. In the count adapted to the $t_0$ chosen here the main intervals are $J_0= [1, 64], \; J_1=[65, 99],\; J_2= [103, 170], \; J_3=[177, 240]; J_4= [252, 276], \; J_5=[291, 303],\; J_6= [307, 322], ] $. The parameters $\eta_j$ and the corresponding mean reproduction numbers in these intervals are give by the following table.
8|c|Model $\eta_0$ and $\eta_j$, $\rho_j$ in intervals $J_j$ for the World
$\eta_0$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$
$\eta_j$ 0.325 0.145 0.156 0.145 0.160 0.144 0.158
$\rho_j$ — 1.01 1.09 1.02 1.12 1.01 1.11
Left: 3-day averages of daily new reported infected for the the World (red); empirical $\hat{A}_{new}$ solid red, model $A_{new}$ black dashed. Right: Total number of reported infected (brown); empirical $\hat{A}_{tot}$ solid, model $A_{tot}$ dashed.
Because of the lack of reliable reporting for recovering dates in several countries, among them some large ones like the USA, we cannot expect a balanced value for the sojourn in the state of actual disease, documented in the statistics. Fig. <ref>, left shows that the estimated values $\hat{q}$ keeps close to 15 or even 20 until late March. Later on the weight of the countries with reliable documentation of recovering dates is large enough to keep the mean number of $\hat{q}(k)$ between 30 and 40, even with the rise of the pandemic after May 2020 (fig. <ref>, left). Accordingly the $q$-corrected number of actually infected $\hat{A}_q$ separate from the the ones given directly by the statistics $\hat{A}(k)$ only in mid April. Since October 2020 they difference between the two is rising progressively (same figure, right).
Left: Daily values of the mean time of statistically actual infection $\hat{q}(k)$ for the World. Right: Comparison of reported infected $\hat{A}$ (dark blue) and $q$-corrected number ($q=15$) of recorded actual infected $\hat{A}_q$ (bright blue) from the JHU data in the World.
Like in the case of those countries which have an unreliable documentation of the actual state of infected (e.g. US, Sweden, …) we can here reconstruct the statistically given number $\hat{A}(k)$ by the model value $A(k)$ by using time variable durations $q(k)=\hat{q}(k)$. This is being displayed in the graph of the 3 curves of the World (fig. <ref>).
Empirical data (solid coloured lines) and model values (black dashed) for the World: numbers of totally infected $\hat{A}_{tot}$ (brown), redrawn $\hat{R}$ (bright green), and numbers of those which are statistically displayed as actually infected $\hat{A}$ (blue).
§ DISCUSSION
The data evaluation in sec. <ref> shows clearly that the SEPAR$_d$ model works well for countries or territories with widely differing conditions and courses of the epidemic. For the “tautological” application of the model with daily changing coefficients of infection $\eta(k)$ this is self-evident, while it is not so for the use of a restricted number of constancy intervals. The examples studied in this paper show that in this mode of application the model is well-behaved, able to characterize the mean motion of an epidemic and to analyse its central dynamic. In the country studies we have shown that this is the case not only for the number of acknowledged daily new reported, our $A_{new}(k)$ but also for data which, in the standard SIR approach, are not easily interpretable like the number of actual infected persons, $A(k)$or the $q$-normalized number $A_q(k)$.
What is the $SEPAR_d$ model good for? It is clear that it cannot predict the future. The main reason for this is that nobody knows how the contact rates are changing in the future. It allows – though – a prediction under assumptions. In the different countries we carried this out with different scenarios.
The main value of the model is as a tool for analysing the development, and to learn from such an analysis. We will discuss three such topics:
– the role of constancy intervals
– the role of the dark sector
– the influence of the time between infection and quarantine
§.§.§ The role of constancy intervals
The empirical values of the infection strength $\hat{\eta}(k)$ are calculated from data on reported new infected and are therefore subject to irregularities in data taking and reporting. The most drastic consequences of this are the obvious weekly fluctuations. Different methods can be applied to smooth these weekly fluctuation, sliding 7-day averages (used here), stochastic estimate used by the RKI (see appendix), band filter etc. Independent of the applied method there remain effects (e.g. non- weekly reporting delays) which distort the calculated numbers away from being correct empirical values for the intended quantities (e.g. $\eta(k)=\gamma \kappa(k)$). Even if they were, one would encounter day to day fluctuations resulting from the variation of intensities of contacts and of the strengths of infectiousness involved, which one is not really interested in if one wants to gain insight into the dynamics of the epidemic. For this one needs to distil a cross-sectional picture of the process. In our approach this is achieved by constructing constancy intervals (main intervals) $J_j$ and model strengths of infection $\eta_j$, read off from the data, and to apply the infection recursion (<ref>).
§.§.§ The role of the dark sector
With increasing numbers of herd immunized, the influence of the dark sector on the ratio $s(k)$ of susceptibles in the total population gains increasing weight, in particular for countries in which a high dark ratio $\delta$ may be expected. In most of the European countries studied here we find the ratio of recorded infected at the order of magnitude of 1 % all over the year 2020. With the non-reported ones added it can easily rise to the order of magnitude 10 % and start to have visible effects. If our estimated values of the dark factor $\delta$ are not utterly wrong, our model calculation shows that in nearly all countries of the study, Germany being the only exception, the development of the epidemic is already noticeably influenced by the dark sector at the beginning of the year 2021. The latter contributes essentially to turning the tide of the reported new infected, if one assumes constant contact ratios $\kappa(k)$ and mean infection strength $\gamma$ of the virus. Of course the appearance of new mutants may change $\gamma$, and counteract the decrease of the numbers of infected predicted by the model. This seems to be the main problem for the early months of 2021.
Ratio of susceptibles $s(k)=S(k)/N$ for Germany (dashed) and Switzerland (solid line) at the end of the year 2020, assuming a dark ratio $\delta=2$ for Switzerland and $\delta=1$ for Germany.
This becomes particularly succinct by a comparing the Swiss situation with Germany at the end of the year 2020 (fig. <ref>). In both countries containment measures were taken after a rise of the reproduction rate to 1.4 to 1.5 in late September /early October, although with different degrees of resoluteness and results (figs. <ref>, <ref>).
The weight of the dark sector is much stronger for the non-European countries of our study. In the case of the USA and Brazil it has started to suppress the effective reproduction number below the critical value 1, according to our model assumptions on the dark factor. But even if one would set it dowm to $\delta=1$ or 2 the effect would already occur, although a bit later and weaker. That this is not yet reflected in the numbers of newly infected may have different reasons; one of it would, of course be, that the model can no longer be trusted in this region. Others have been mentioned in the country section. And finally it could be that persons infected some months ago need not necessarily be immune against a second attack. If virologists come to this conclusion, the whole model structure would need a revision. At the moment it is too early to envisage such a drastic step.
§.§.§ The influence of the time between infectivity and quarantine
A central input into the $SEPAR_d$ model is the assumption that there is a rather short period of length $p_c$, where people, who later are positively tested, are infectious. This is closely related to the fact that people with positive test results are sent to quarantine or hospital. One can wonder what would happen, if the time between infectivity and quarantine or hospital is changed.
It is a bit confusing, but there are two answers to this question. To explain the difference we recall the role of $p_c$ in our model. We usually derive the $\eta$ parameters from the data (eq. <ref>). For the reproduction number (<ref>) in the simplified $SEPAR_d$-model with $p_c=p_d=p$ and a constant coefficient $\eta$ this means:
\[\rho(k) = p \, \eta \, s(k) = (p_c\, \frac{\eta}{2} +p_d \frac{\eta}{2}) \, s(k)
\]
Here we assume that $p_c$ is given. This number is only a rough estimate and may be chosen slightly differently. So, for each choice of the estimated number $p_c$ one gets model curves and one might ask, how much these model curves differ, in particular how much the reproduction rates would differ. The answer is: not very much. The reason is that $p_c$ enters implicitly also in the formula (<ref>) for $\eta$, since the denominator is a sum over $p_c$ values of the daily newly infected. If we assume that this number is constant (which often is approximately the case) then in the denominator we would have the factor $p_c$ and in the formula for $\rho$ it cancels out. Thus in this situation the reconstruction of $\rho$ from $p_c$ and $\eta$ is independent of the choice of $p_c$. If the values of the newly infected changes more drastically this is not the case and one has to use the general formula (<ref>), but the difference is not dramatic. So the first answer to the question is: A different estimation for $p_c$ does not have a noticeable influence for the model curves.
For understanding the second very different answer we have to recall that $\eta$ may be interpreted as the product of the contact rate $\kappa$ (as measured in the model) and the strength of the infection $\gamma$. If we assume that $\gamma $ is constant, the change of $p_c$ discussed above amounts to a change of the model-$\kappa$, which does not express a changing contact behaviour. This means that our measure for the contact rate is related to our choice of $p_c$.
Now we come to the second answer. Here we assume that the contact rate remains the same, the contact behaviour of the society is not changed. But suppose that by some new regulations the value of $p_c$ is changed. Then, as expected, if the contact behaviour is unchanged the reproduction number changes proportionally and so the curves are different. This second answer is what we are interested in here. Let us assume that one finds means by which the time until the people go to quarantine or hospital is reduced. Then less contacts take place and so the curves are flattened. This fact is well known, e.g., <cit.>.[We thank S. Anderl for the hint.]
But how much?
For answering this question we have taken the model description for Germany, lowering the value of $p_c$ from $7$ to $6$ days from a certain moment on. Here we have to discuss an important point. One can only influence the time until quarantine or hospital for those who are registered, while the infected people who end up in the dark sector behave as before. At this moment we have to give up our assumption that $p_c$ = $p_d$. So, from a certain moment on we assume that $p_d$ is still 7 but $p_c$ is 6.
We have carried this out in two different scenarios for the expected numbers of daily new recorded infected $A_{new}(k)$ and the numbers of actual infected $A(k)$. In the first one we compare the past development in Germany during the year 2020 with a fictitious reduction of $p_c$ from $7$ to $6$ during May 2020, keeping $p_d = 7$ fixed (fig. <ref>). In a second one we take a look into the future, perpetuating the contact rate of the last constancy interval, i.e., assuming that the contact behaviour of the population is unchanged for a while and assume the same fictitious reduction as above in the second half of January 2021 (fig. <ref>). This doesn't mean that we make a prediction of the future, our only aim here is to demonstrate what would happen if we could lower $p_c$ from 7 to 6. The lowering of the numbers of infected, newly recorded and actual ones, are very impressive.
Model calculations for reported new infected $A_{new}(k)$ (left) and reported actual infected $A(k)$ (right) for Germany. Solid lines with parameter values given in sec. <ref>($p_c=7$ all over the year 2020). Dashed $p_c=7$ from March to May, $p_c=6$ from August onward, smooth transition in June.
Model calculations for reported new infected $A_{new}(k)$ (left) and reported actual infected $A(k)$ (right) for Germany (30 days prediction on the basis of data available 14 Jan 2021). Solid lines with parameter values given in the sec. <ref>, in particuilar $p_c=7$. Dashed $p_c=7$ from March 2020 to 15 Jan 2021, $p_c=6$ from February 2021 onward, smooth transition in between.
In the past none of the regulations imposed by the German federal authorities made an attempt to reduce the time until people got to quarantine or hospital aside from raising the number of tests. Our considerations suggest to make a serious attempt in this direction. It has the big advantage that it does not require additional restrictions of the majority of the population and can be expected to be very effective at the same time.
§ APPENDIX
§.§.§ Comparison with RKI reproduction numbers
The estimates of the reproduction numbers for Germany by the Robert Koch Institut (RKI), Berlin, are based on an approach using the generation time as crucial delay time.
The generation time $\tau_g$ of an epidemic is defined as the mean time interval between a primary infection and the secondary infections induced by the first one; similarly the length $\tau_s$ of the serial interval as the mean time between the onset of symptoms of a primary infected and the symptom onset of secondary cases.
There are various methods to determine time dependent effective reproduction numbers on the basis of stochastic models for infections using both intervals. In our simplified approach with constant $e$ and $p$ these numbers correspond to $\tau_g=\tau_s=e+ \frac{p-1}{2}$.
The RKI calculation uses a method of its own for a stochastic estimation of the numbers of newly infected, called $E(t)$, from the raw data of newly reported cases, described in <cit.>.
The calculation of the reproduction numbers works with these $E(t)$ and assumes constant generation time and serial intervals of equal lengths $\tau_g=\tau_s=4$
<cit.>.[For $e=2$ this would correspond to $p=5$, while for $\tau_g=\tau_s=5$ we arrive at our $p=7$.]
Two versions of reproduction numbers are being used, a day-sharp and therefore “sensitive” one
$\rho_{rki,\, 1} (t) = \frac{E(t)}{E(t-4)}$,
and a weekly averaged one,
\[ \rho_{rki,\, 7} (t) = \frac{\sum_{j=0}^6 E(t-j)}{\sum_{j=0}^6 E(t-4-j)} \, ,
\]
which we refer to in the following simply as $\rho_{rki}(t)$.
The paper remarks that the RKI reproduction numbers (“$R$-values”) $\rho_{rki}(u)$ indexed by the date $u$ of calculation refer to a period of infection which, after taking the incubation period $\iota$ between 4 and 6 days into account, lies between $u-16, \ldots, u-8$ (with central day $u-12$ in the interval). We reformulate this redating by setting
ρ̂_rki (t-12) = ∑_j=0^6 E(t-j)/∑_j=0^6 E(t-4-j) , ,
For a comparison with the SEPAR reproduction numbers we write (<ref>) as
\[\hat{ \rho} (k-(e+p+3)))= \frac{\frac{p}{7}\, \sum_{j=0}^{6}\hat{A}_{new}(k-j) }{\sum_{j=0}^{p-1} \hat{A}_{new}(k-(e+2)-j) } \, ,
\]
which for $e=2,\, p=7$ boils down to
\[ \hat{\rho} (k-12))= \frac{ \sum_{j=0}^{6}\hat{A}_{new}(k-j) }{\sum_{j=0}^6 \hat{A}_{new}(k-4-j) } \; .
\]
This is very close to (<ref>). The main differences lie in the usage of different raw data bases (RKI versus JHU) and the adjustment of the raw data (stochastic redistribution $E(t)$ versus sliding 7-day averages $\hat{A}_{new,7}$). This may explain the differences in the level of low or high plateaus shown in fig. <ref> (with 1 day additional time shift).
Empirical reproduction numbers $\hat{\rho}_7(k)$ of the SEPAR$_q$ model for Germany (orange) and reproduction numbers $\rho_{rki}(k -13)$ (7-day averages) of the RKI (blue).
In this sense, our model supports the claim of the RKI that their reproduction numbers can be used as indicators of “a trend analysis of the epidemic curve” <cit.>.
We thank Odo Diekmann for discussing our thoughts as non-experts at an early stage of this work; he helped us to understand compartment models better. Moroever, we appreciate the exchange with Stephan Luckhaus, and thank Robert Schaback, Robert Feßler, Jan Mohring, and Matthias Ehrhardt for hints and discussions.
Calculations and graphics were made with Mathematica 12.
an der Heiden Hamouda2020anderHeiden_ea:2020
an der Heiden, Matthias Osamah Hamouda. 2020.
“Schätzung der aktuellen Entwicklung der SARS-CoV-2-Epidemie in
Deutschland – Nowcasting 23.” RKI Bulletin pp. 10–16.
[Breda, Diekmann de Graaf]Breda, Diekmann
de Graaf2012Breda/Diekmann:2012
Breda, Dimitri, Odo Diekmann W.F. de Graaf. 2012.
“On the formulation of epidemic models (an appraisal of Kermack and
McKendrick).” Journal of Biological Dynamics 6, Suppl 2:103–117.
[Contreras et al.]Contreras, Dehning, Loidolt Priesemann et al.2020Contreras-ea:2020challenges
Contreras, Sebastian, Jonas Dehning, Matthias Loidolt Viola
Priesemann et al. 2020.
“The challenges of containing SARS-CoV-2 via
[Dehning et al.]Dehning, Zierenberg, Spitzner, Pinhero Neto,
Wilczek Priesemann et al.2020Dehning-ea:2020
Dehning, Jonas, Johannes Zierenberg, Paul Spitzner, Joao Pinhero Neto, Michael
Wilczek Viola Priesemann et al. 2020.
“Inferring change points in the spread of COVID-19 reveals the
effectiveness of interventions.” Science p. 10.1126/science.abb9789.
Fessler, Robert. 2020.
“A general integral equation model for epidemics.”.
Kaiserslautern: Fraunhofer Institut für Techno- und
Wirtschaftsmathematik ITWM
HelmholtzZentrum, München. 2020.
“Die Dynamik der COVID-19 Pandemie im Blick. Zwischenergebnisse zur
zweiten Runde der KoCo19-Antikörperstudie. Pressemitteilung (23. 12. 2020):
Prospektive COVID-19 Kohorte München (KoCo19).”.
Kermack McKendrick1927Kermack/McKendrick:1927
Kermack, William O. Anderson G. McKendrick. 1927.
“A contribution to the mathematical theory of epidemics.” Proceedings of the Royal Society of London 115(772):700–721.
Kreck Scholz2020K-S
Kreck, Matthias Erhard Scholz. 2020.
“Proposal of a recursive compartment model of epidemics and
applications to the Covid-19 pandemic.”.
Kuster, Sabine. 2021.
“Rund 1,7 Millionen Schweizer haben sich bereits mit dem Coronavirus
infiziert – bis zur Herdenimmunität dauert es noch.” Tagblatt St
Gallen .
retrieved Jan 12, 2021.
[Mohring et al.]Mohring, Wegener, Gramsch Schöbel2020Mohring_ea:2020
Mohring, Jan, Raimund Wegener, Simone Gramsch Anita Schöbel.
“Prognosemodelle für die Corona-Pamdemie.”.
Kaiserslautern: Fraunhofer Institut für Techno- und
Wirtschaftsmathematik ITWM
<https://www.itwm.fraunhofer.de/de/suche.html?_charset_=UTF-8 numberResults=10 page=1 scope=ITWM lang=de queryString=Mohring+prognosemodelle>.
Nogrady, Bianca. 2020.
“What the data sy about asymptomatic Covid infections.” Nature (587):534–535.
[Radon, Saathoff Pritschl]Radon, Saathoff
Radon, Katja, Elmar Saathoff Michael Pritschl. 2020.
“Protocol of a population-based prospective COVID-19 cohort study
Munich, Germany.” BMC Public Health .
[Reese et al.]Reese, Iuliano, Patei Garg et
Reese, Heather, Danielle Iuliano, Neha Patei Shika Garg et al.
“Estimated incidence of COVID-19 illness and hospitalization –
United States, February– September, 2020.” Clinical Infectious
Diseases .
Robert Koch Institut2020aRKI:Steckbrief
Robert Koch Institut, RKI. 2020a.
“Epidemiologischer Steckbrief zu SARS-CoV-2 und COVID-19.”
Robert Koch Institut2020bRKI:2020Erlaeuterung
Robert Koch Institut, RKI. 2020b.
“Erläuterung der Schätzung der zeitlich variierenden
Reproduktionszahl $R$.”.
Robert Koch Institut2020cRKI:2020SerologischeStudie
Robert Koch Institut, RKI. 2020c.
“Serologische Untersuchungen von Blutspenden auf Antikörper gegen
SARS-CoV-2 (SeBluCo-Studie). Zwischenauswertung Datenstand 05.11.2020.”.
Schaback, Robert. 2020.
“On COVID-19 modeling.” Jahresbericht der Deutschen
Mathematiker-Vereinigung (122):167––205.
[Wölfel, Corman Guggemos et al]Wölfel, Corman
Guggemos et al2020Woelfel_ea:2020
Wölfel, R., V.M. Corman W. Guggemos et al. 2020.
“Virological assessment of hospitalized patients with COVID-2019.”
Nature (581):465–469.
Example discussion: USA? – Phases of rapid increase followed by long phases of slow decrease
India? Long phase of increase with turn of the tide in the same constancy interval
Xmas ff. in Germany – fake decrease 10 days before X-mas, increase between Xmas and new year, difficult to disentangle effect of Xmas contacts and two more stable effects: less discipline in contact reduction than in March + possibly new virus mutant – tendency of early January only to be read off from constancy interval for $\hat{\eta}$ fully in January ?
|
# Lattice and Electronic properties of VO2 with the SCAN(+$U$) approach
Sooran Kim<EMAIL_ADDRESS>Department of Physics Education, Kyungpook
National University, Daegu 41566, Korea
###### Abstract
Appropriate consideration of the electron correlation is essential to
reproduce the intriguing metal-insulator transition accompanying the Peierls-
type structural transition in VO2. In the density functional theory-based
approach, this depends on the choice of the exchange-correlation functional.
Here, using a newly developed strongly constrained and appropriately norm
(SCAN) functional, we investigate the lattice and electronic properties of the
metallic rutile phase of VO2 ($R$-VO2) from the first-principles calculations.
We also explored the role of the Coulomb correlation $U$. By adding $U$, we
found that the phonon instability properly describes the Peierls-type
distortions. The orbital-decomposed density of states presents the orbital
selective behavior with the SCAN+$U$, which is susceptible to the one-
dimensional Peierls distortion. Our results suggest that even with the SCAN
functional, the explicit inclusion of the Coulomb interaction is necessary to
describe the structural transition of VO2.
VO2, Density functional theory, SCAN functional, Phonon, Structural transition
## I INTRODUCTION
Vanadium dioxide, VO2 is one of the most intensively studied transition metal
oxides due to its interesting metal-insulator transition accompanied by a
structural transition at 340 K Mortin59 ; Goodenough60 . VO2 is crystallized
in the rutile structure ($R$-VO2) with metallic behavior at high temperature
McWhan74 . At a low temperature below 340 K, it exhibits an insulating
monoclinic phase ($M_{1}$-VO2) Longo70 . Figure 1 illustrates the two crystal
structures of VO2. The key distortion in the structural transition is related
to V atoms having dimerization along the $c$-axis of $R$-VO2 and zigzag
distortion, which are typical Peierls distortion Eyert02 ; Pintchovski78 .
There has been an extensive discussion on the strong interplay between
electron correlations and structural transition in VO2 Biermann05 ;
Lazarovits10 ; Kim13 ; Haverkort05 . To explain the phase transition and
insulating behavior of $M_{1}$-VO2, theoretical approaches beyond the density
functional theory (DFT) level such as DFT+$U$ Liebsch05 ; Kim13 (the on-site
Coulomb interaction $U$), $GW$ Continenza99 ; Gatti07 ; Sakuma08 , hybrid
functionalEyert11 ; Budai14 , and the dynamical mean-field theory Biermann05 ;
Tomczak08 ; Lazarovits10 ; Belozerov12 have been employed. Not only the
electronic structures but also harmonic Kim13 ; Budai14 ; Mellan19 and
anharmonic Budai14 ; Lee17 phonon properties were reported within the
generalized-gradient approximation (GGA) and GGA+$U$ frameworks. Especially,
Kim et al. suggested that the Coulomb correlation $U$ plays an essential role
in producing the Peierls-type structural transition Kim13 .
Previous theoretical works demonstrate that appropriate consideration of the
electron correlation is essential for the sound description of the electronic
and structural transition of VO2. Recently, a new strongly constrained and
appropriately normed (SCAN) functional was proposed, which is a non-empirical
meta-GGA introducing the kinetic energy density and satisfying all known exact
constraints Sun15 ; Sun16 . This new functional shows the improved performance
on energetics and structural properties of binary oxides Hinuma17 and
perovskite ferroelectrics Zhang17 as well as the band gaps and absolute
voltages of cathode materials Chakraborty18 over GGA(+$U$) functional. Since
SCAN does not require a tunable parameter $U$ to explain key properties of
typical transition metal oxides, employing the SCAN functional increases the
predictive power while keeping the first-principles character from the view of
the ab initio community Varignon19 . SCAN, however, also has an intrinsic
self-interaction error as in the GGA functional Zhang18 , particularly from
the local Coulomb interaction. The Coulomb interaction $U$ is often introduced
in the SCAN to reduce the error Isaacs20 ; Gautam18 ; Olivia20 . Therefore,
the SCAN functional needs to be carefully tested for each property and phase
of different materials.
For VO2, there have been previous studies of the energetics and electronic
structures using the SCAN Ilkka17 ; Olivia20 ; Stahl20 ; Ganesh20 ; mondal20 .
Notably, Mondal et al. reported vibrational properties of VO2 with SCAN
functional mondal20 . Their phonon band of $R$-VO2 with the SCAN is almost
same as using GGA functional and does not exhibit an evidence for the Peierls-
like structural distortion mondal20 . Therefore, considering an ongoing
discussion on the SCAN, it would be worth exploring the lattice properties of
the high-temperature phase, $R$-VO2 with the SCAN(+$U$) approach.
In this paper, we investigate the lattice dynamics and electronic structure of
$R$-VO2 to demonstrate the performance of the newly proposed SCAN functional
on VO2. We also considered various $U$ values to examine the effect of Coulomb
interaction with the SCAN. From the phonon dispersion curves with the SCAN and
the SCAN+$U$, we find that phonon soft modes, which implies the structural
instability, are significantly changed by adding the Coulomb interaction.
Specifically, the phonon soft mode with the SCAN+$U$ shows the lattice
displacements that are related to the Peierls-type structural transition
whereas the soft mode with the SCAN does not. Furthermore, we investigate the
orbital-decomposed density of states and present the orbital redistribution
with the inclusion of $U$, resulting in the Peierls-like distortion.
## II COMPUTATIONAL METHOD
All density functional theory (DFT) calculations were performed by the Vienna
ab initio simulation package (VASP) implementing pseudo potential band method
Kresse96 ; Kresse962 . We employed SCAN Sun15 ; Sun16 as an exchange-
correlation functional and further included $+U$ to account for the correlated
$d$ orbitals of V atom within the Dudarev method Dudarev98 . The various
$U_{eff}=U-J$ (0.0, 1.0, 2.0, 2.1, 2.2, 2.3, 2.5, 3.1 eV) are chosen for
detailed investigation of the $U$ effect with the SCAN functional. The
$U_{eff}$ of 2.5 eV is used for Figures unless otherwise specified. The energy
cut for the plane waves and the k-point sampling are 520 eV and $8\times
8\times 12$ in the Monkhorst-Pack grid, respectively. The structures are fully
relaxed including lattice parameters and atomic positions from the initial
experimental $R$-VO2 structure McWhan74 .
Phonopy was employed for phonon study where the dynamic matrix and the force
constants are obtained with the finite displacements method based on the
Hellmann-Feynman theorem Togo08 . The $2\times 2\times 2$ supercell of the
$R$-VO2 and the $4\times 4\times 6$ k-point sampling were used for the phonon
calculations.
## III RESULTS AND DISCUSSION
Figure 2 shows the phonon dispersion curves of $R$-VO2 with the SCAN and
SCAN+$U$ approaches. Phonon bands for both cases present phonon soft modes
with imaginary frequencies, indicating the structural instability. These
instabilities are in agreement with the experimentally unstable $R$-VO2 phase
at the low temperature. The phonon soft modes are mostly related to the V
atoms as shown in the phonon density of states (DOS). The detailed
displacements by the phonon soft modes will be discussed later. The phonon
bands, however, with the SCAN and SCAN+$U$ clearly show the discrepancy at the
$q$-points where the instability occurs. The phonon soft modes are obtained at
$\Gamma$, $M$, and $X$ without $U$ while the soft modes exist in the whole
Brillouin zone except for the $\Gamma$ point with the SCAN+$U$. Such behaviors
are similar to the previous reports with GGA and GGA+$U$ approachesKim13 . The
phonon dispersion curves with the GGA and GGA+$U$ exhibit the phonon soft
modes at ($\Gamma$, $M$, $X$) and ($A$, $Z$, $R$), respectively Kim13 .
Since the phonon softening at the $R$ point was reported to explain the
structural transition from $R$-VO2 to $M_{1}$-VO2 Brews70 ; Hearn72 ;
Terauchi78 ; Gervais85 ; Kim13 , our phonon results suggest that, even with
the SCAN functional, the inclusion of the Hubbard parameter $U$ is necessary
to describe the structural transition of VO2. Namely, SCAN without adding
explicit $U$ could not fully include the electron correlation effect in VO2.
These results raise the question of when SCAN requires additional correction
such as $U$ and van der Waals interaction, in the ongoing discussion about
SCAN functional Olivia20 ; Gautam18 ; Isaacs20 ; Peng16 ; Hermann18 ; kim20 .
To investigate the $U$ effect on the phonon bands in more detail, we have
plotted the lowest phonon frequencies at $\Gamma$ and $R$ with varying the
effective $U$, $U_{eff}$ as shown in Fig. 3. As increasing $U_{eff}$, the
phonon soft modes disappear at $\Gamma$ while the soft modes appear at the $R$
point. Specifically, the phonon soft mode at $R$ starts to occur with
$U_{eff}\geq$ 2.1 eV. When $U_{eff}\geq$ 2.3 eV, the soft optical modes at
$\Gamma$ disappear and the lowest phonon mode at $\Gamma$ is the acoustic mode
with a frequency of 0 meV. The lowest phonon modes at $A$ and $Z$ exhibit a
similar trend with the soft modes at $R$. These results suggest that at least
$U_{eff}$ of $\sim$2.5 eV is required to reproduce the structural Peierls
transition in VO2 within SCAN+$U$ approach.
The displacements by the lowest phonon soft modes at $\Gamma$ with the SCAN
and $R$ with the SCAN+$U$ are illustrated in Fig. 4(a) and (b), respectively.
The distortions at both $\Gamma$ and $R$ points are related to V atoms as
shown in the phonon DOS of Fig. 2. However, the detailed displacements are
different from each other. The phonon mode at $\Gamma$ exhibits collinear
displacements of atoms along the $c$-axis while the corresponding displacement
at $R$ shows the simultaneous dimerization and zigzag distortion of V atoms.
The ordering of the dimerization by the phonon soft mode at $R$ also agrees
well with the experimental distortion. In detail, there are two degenerated
phonon soft modes at $R$ with the SCAN+$U$ as shown in Fig. 4(c) and (d). One
can see the dimerization in half of V atoms and zigzag distortion in another
half of V atoms in each mode. By the linear combination of two modes, the
dimerization and zigzag distortion in V ions are obtained at the same time as
in Fig. 4(b). Again, it suggests that the SCAN+$U$ framework is required to
describe the experimental structural distortion. The phonon soft mode with the
lowest frequency at $A$ and $Z$ has the displacement of V-V dimerization but
not zigzag distortion. The lattice displacements by the lowest phonon soft
modes at $M$ and $X$ are related to neither dimerization nor zigzag
distortion.
We further investigate the electronic DOS of $R$-VO2 within the SCAN(+$U$)
functional. Figure 5 shows the total and partial DOS of $R$-VO2. Both DOS with
the SCAN and SCAN+$U$ exhibit the metallic states in agreement with the
experiments. The inclusion of $U$ does not notably change the hybridization
between V and O atoms as in Fig.5 (a) and (b). However, the orbital-decomposed
DOS in Fig. 5(c) and (d) show a clear difference between SCAN and SCAN+$U$.
The occupation of $d_{x^{2}-y^{2}}$ whose lobe is along with the V-V dimer
significantly increases with the inclusion of $U$. This pronounced one-
dimensional orbital distribution can result in the Peierls-type distortion.
Furthermore, the orbital selective character with adding $U$ is consistent
with the phonon soft modes at $R$, $A$, and $Z$ points and was also reported
in the previous GGA+$U$ calculationsKim13 .
## IV CONCLUSIONS
In conclusion, we investigate the lattice and electronic properties of the
metallic rutile phase of VO2 and demonstrate that the direct inclusion of the
Coulomb interaction $U$ in the SCAN functional is essential to reproduce the
experimental structural transition in VO2. The phonon bands with the SCAN+$U$
exhibit the phonon soft mode at $R$ whose lattice displacement is the
dimerization and zigzag distortion of V atoms in agreement with the
experiments. On the other hand, the phonon soft modes with the SCAN does not
produce such displacements. Furthermore, the SCAN+$U$ calculation provides the
orbital-selective distribution of V $d$ orbitals, which is suitable for the
Peierls-type distortion and consistent with the phonon results. We hope that
this work can stimulate further study on the performance of the SCAN
functional for transition metal oxides.
###### Acknowledgements.
We thank Bongjae Kim and Kyoo Kim for helpful discussion. This research was
supported by Kyungpook National University Research Fund, 2018.
## References
* (1) F. J. Morin, Phys. Rev. Lett.3, 34 (1959).
* (2) J. B. Goodenough, Phys. Rev.117, 1442 (1960).
* (3) D. B. McWhan, M. Marezio, J. P. Remeika, and P. D. Dernier, Phys. Rev. B 10, 490 (1974).
* (4) J. Longo and P. Kierkegaard, Acta Chem. Scand. 24, 420 (1970).
* (5) V. Eyert, Annalen der Physik 11, 650 (2002).
* (6) F. Pintchovski, W. Glaunsinger, and A. Navrotsky, J. Phys. Chem. Solids 39, 941 (1978).
* (7) S. Biermann, A. Poteryaev, A. I. Lichtenstein, and A. Georges, Phys. Rev. Lett. 94, 026404 (2005).
* (8) B. Lazarovits, K. Kim, K. Haule, and G. Kotliar, Phys. Rev. B 81, 115117 (2010).
* (9) S. Kim, K. Kim, C.-J. Kang, and B. I. Min, Phys. Rev. B 87, 195106 (2013).
* (10) M. W. Haverkort et al., Phys. Rev. Lett. 95, 196404 (2005).
* (11) A. Liebsch, H. Ishida, and G. Bihlmayer, Phys. Rev. B 71, 085109 (2005).
* (12) A. Continenza, S. Massidda, and M. Posternak, Phys. Rev. B 60, 15699 (1999).
* (13) M. Gatti, F. Bruneval, V. Olevano, and L. Reining, Phys. Rev. Lett. 99, 266402 (2007).
* (14) R. Sakuma, T. Miyake, and F. Aryasetiawan, Phys. Rev. B 78, 075106 (2008).
* (15) V. Eyert, Phys. Rev. Lett. 107, 016401 (2011).
* (16) J. D. Budai et al., Nature 515, 535 (2014).
* (17) J. M. Tomczak, F. Aryasetiawan, and S. Biermann, Phys. Rev. B 78, 115103 (2008)
* (18) A. S. Belozerov, M. A. Korotin, V. I. Anisimov, and A. I. Poteryaev, Phys. Rev. B 85, 045109 (2012).
* (19) T. A. Mellan, H. Wang, U. Schwingenschlögl, and R. Grau-Crespo, Phys. Rev. B 99, 064113 (2019).
* (20) S. Lee et al., Science 355, 371 (2017).
* (21) J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
* (22) J. Sun et al., Nature Chemistry 8, 831 (2016).
* (23) Y. Hinuma et al., Phys. Rev. B 96, 094102 (2017).
* (24) Y. Zhang, J. Sun, J. P. Perdew, and X. Wu, Phys. Rev. B 96, 035143 (2017).
* (25) A. Chakraborty, M. Dixit, D. Aurbach, and D. T. Major, npj Computational Materials 4, 60 (2018).
* (26) J. Varignon, M. Bibes, and A. Zunger, Nature Communications 10, 1658 (2019).
* (27) Y. Zhang et al., npj Computational Materials 4, 9 (2018).
* (28) E. B. Isaacs, S. Patel, and C. Wolverton, Phys. Rev. Materials 4, 065405 (2020).
* (29) G. Sai Gautam and E. A. Carter, Phys. Rev. Materials 2, 095401 (2018).
* (30) O. Y. Long, G. Sai Gautam, and E. A. Carter, Phys. Rev. Materials 4, 045401 (2020).
* (31) I. Kylänpää et al., Phys. Rev. Materials 1, 065408 (2017).
* (32) B. Stahl and T. Bredow, J. Comput. Chem. 41, 258 (2020).
* (33) P. Ganesh et al., Phys. Rev. B 101, 155129 (2020).
* (34) W. R. Mondal et al., arXiv:2008.08725.
* (35) G. Kresse and J. Furthmüller Phys. Rev. B 54, 11169 (1996).
* (36) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
* (37) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
* (38) A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008).
* (39) J. R. Brews, Phys. Rev. B 1, 2557 (1970).
* (40) C. J. Hearn, J. Phys. C 5, 1317 (1972).
* (41) H. Terauchi and J. B. Cohen, Phys. Rev. B 17, 2494 (1978).
* (42) F. Gervais and W. Kress, Phys. Rev. B 31, 4809 (1985).
* (43) H. Peng, Z.-H. Yang, J. P. Perdew, and J. Sun, Phys. Rev. X 6, 041005 (2016).
* (44) J. Hermann and A. Tkatchenko, Journal of Chemical Theory and Computation 14, 1361 (2018).
* (45) B. Kim, K. Kim, and S. Kim, arXiv:2007.04652.
Figure 1 Crystal structures of VO2. (a) The rutile phase in the P42/mnm space
group ($R$-VO2) McWhan74 . The dotted arrow indicates the local axis following
Ref. Eyert02 (b) The monoclinic phase in the P21/c space group ($M_{1}$-VO2)
Longo70 . The V-V dimerization and zigzag distortion in the monoclinic phase
are indicated by the bond between V atoms. The blue and red balls represent V
and O atoms, respectively.
Figure 2 Phonon dispersion curves and phonon density of states of $R$-VO2 with
(a) the SCAN and (b) the SCAN+$U$. The imaginary phonon frequencies in both
figures imply the structural instability.
Figure 3 The lowest phonon frequencies at $q=\Gamma$ and $R$ as a function of
$U_{eff}$.
Figure 4 (a) Lattice displacements by the phonon soft mode with the lowest
frequency at $\Gamma$ with the SCAN. (b) A linearly superposed displacement by
the lowest two degenerated phonon soft modes at $R$ as shown in (c) and (d)
with the SCAN+$U$.
Figure 5 The total and atomic decomposed DOS of $R$-VO2 with (a) the SCAN and
(b) the SCAN+$U$. The $d$ orbital-decomposed DOS of $R$-VO2 with (c) the SCAN
and (d) the SCAN+$U$.
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
|
∎
11institutetext: Arzu Kurt
11email<EMAIL_ADDRESS>
1 Department of Physics, Bolu Abant İzzet Baysal University, Bolu 14030,
Turkey.
# Two-time correlation functions beyond quantum regression theorem: Effect of
external noise
Arzu Kurt1
(Received: date / Accepted: date)
###### Abstract
We present the results of a study of the two-time correlation functions of a
dichotomously driven two-level system in contact with a thermal bath by using
corrections beyond quantum regression theorem. In the strong system-
environment coupling regime, it is found that the noise parameters could be
tuned to control the magnitude of corrections at low environmental
temperatures. Motional averaging and narrowing effect of the external noise
were observed on the absorption and emission spectra of the two-state system.
Furthermore, effects similar to the destruction of tunneling and noise
enhancement of transport are observed in the dynamics of the two-time
correlations.
###### Keywords:
Quantum regression theorem telegraph noise non-Markovian dynamics
## 1 Introduction
Developments in spectroscopy techniques have increased the need to develop
effective methods to compute the multi-time correlation functions which encode
large amounts of crucial information on the measured spectra Scully97 (1, 2,
3, 4, 5, 6), that might not be obtained from the expectation values of the
system operators. Use of two-time correlation functions to model measured
spectra range from the optical spectrum of the semiconductor quantum dot
Bertelot2006 (7) and optical emission in quantum-dot-cavity systems Winger2009
(8) to Mollow triplet spectra of a resonantly driven quantum dot in a
microcavity Ulrich2011 (9). Jorgensen et. al.’s work on the phonon emission
spectrum of the spin-boson modelJorgensen2020 (10) is a recent example of
modeling the spectral response of open quantum systems by using two-time
correlation functions (TTCFs) which has a long history. TTCFs have, also, been
used in inferring direct quantum dynamics in experimental investigation of the
long-lived electronic quantum coherence for a photosynthetic pigment-protein
structure known as Fenna-Matthews-Olsen (FMO) complex Brixner2005 (11).
Besides, the authors Li2013 (4) reported a study of motional averaging and
narrowing in the situation where the energy levels of qubit system are
modulated by a random telegraph noise in experimental set-up.
Many different approaches to calculate the TTCFs have been developed over the
yearsOnsager31a (12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23). Quantum
regression theorem (QRT) provides a direct relation between the dynamics of
single-time expectation values of system operators and their two-time
correlation functions in Markovian systems and is used extensively in the
earlier modelling attempts Onsager31a (12, 13). Validity of QRT in formulation
of TTCFs for non-Markovian systems was questioned by various groups Ford1996
(24, 25, 26, 27) which lead to various proposals to amend it to account for
the memory effects Alonso2005 (14, 15, 16, 17, 18, 19, 20, 21, 22, 23). de
Vega and Alonso deVega2008 (22) used the stochastic Schrödinger equation
approach to derive time evolution equations for single and two-time
correlation functions of arbitrary system operators and used it to calculate
the spectral properties of a two level atom weakly interacting with a highly
non-Markovian electromagnetic environment. This model was criticized as being
valid only for a limited class of system-environment coupling operators and
the zero temperature environment by Goan et. al. goan11 (18) who used a second
order master equation approach to systematically improve the QRT evolution
equations by including terms that account for the memory effects. They have
found considerable differences in time-evolution of the TTCFs obtained in QRT
and the improved QRT models even for the weak system-bath coupling regime. Jin
et. al. derived time-local equations of motion for the two-time non-Markovian
correlation function of arbitrary system operators in open quantum systems and
used them to demonstrate that they could account for the typical non-Markovian
effects in current fluctuation spectra of single and double quantum dots.
Jin2016 (23). Cosacchi et. al. Cosacchi2018 (28) have improved the iterative
path integral algorithm for the numerically exact computation of multi-time
correlation functions introduced by Shao and Makri Shao2002 (29) and used it
to study spectra of a two-level system simultaneously in contact with
Markovian and non-Markovian environments. Their method was found to reproduce
the non-Markovian features in the emission of a quantum dot in contact with
acoustic phonons. Recent attempts to include multi-time measurement scenario
without applying QRT, also exist Pollock2018 (30, 31, 10). Jorgensen and
Pollock suggest, for example, an alternative formulation based on time-
evolving matrix product operator(TEMPO) algorithm to both effectively
calculate multi-time correlation functions for more general systems and
indicate to enhance the efficiency of the method by directly computing the the
emission spectra in non-Markovian case for the spin-boson model beyond QRT
Jorgensen2019 (31). In order to incorporate multi-time correlation functions
Jorgensen and Pollock developed a powerful protocol based on the transfer-
tensor framework Pollock2018 (30) by examining full of steady-state
correlations for the spin-boson model without computing inhomogeneous terms.
Also, it is studied the dynamics of system-bath correlations in the spirit of
the non-Markovianity relative entropy measurement Jorgensen2020 (10).
Moreover, various authors suggested the possibility of using the violation of
QRT as an indication of non-Markovianity of the system dynamics Guarnieri2014
(32, 33, 34, 28). Guarnieri et. al. Guarnieri2014 (32) compared the validity
of QRT with CP-divisibility and distinguishability non-Markovianity measures
for the exactly solvable pure dephasing spin boson model and showed that QRT
might be violated even when the measures indicate Markovian dynamics. Chen and
Manirul Ali have used the non-Markovian TTCFs developed by Goan et. al. goan11
(18) to study the violation of Leggett-Garg inequality for a two-level system
in a non-Markovian dephasing environment and reported that the violation
decreases with increasing environmental coupling Chen2014 (33). Also, Kurt
Kurt2020 (35) has considered the violation of QRT for the spin-boson model in
the strong coupling regime by using the method developed in Ref. goan11 (18)
and found that beyond QRT corrections were small for the range of studied
parameters.
Most of the studies on the non-Markovian two-time correlation functions deals
with systems weakly interacting with a non-Markovian thermal environment or
Markovian plus another non-Markovian environment. However, some realistic
systems, such as FMO light harvesting complexes, are in contact with a thermal
environment which might be Markovian or non-Markovian as well as driven by
external disturbances which can not be described as a collection of harmonic
oscillators Jang2018 (36). Aim of the present work is to investigate the
behaviour of two-time correlation functions for a two-state system (TSS) in a
thermal bath when the TSS energy gap is modulated by an external random
telegraph signal in the strong coupling regime. Noise averaged time-
convolutionless master equation with QRT plus corrections introduced by Goan
et. al. goan11 (18) is used to examine the effect of the external noise on the
QRT violations as a function of system, environment and noise parameters.
The paper is organized as the following. We present the model in the polaron
frame and the time evolution of a single time as well as two-time correlations
functions in non-Markovian case in Sec. 2. In Sec. 3, we display the result
including the averaged over the noise realization for the TTCFs for the non-
Markovian system quantities. We briefly summarize the paper in Sec. 4.
## 2 Model
We consider a two-state system (TSS) that is driven by a random telegraph
noise (RTN), which in turn is in contact with a structured bath. The
Hamiltonian describing the closed system (TSS + bath) is:
$H=H_{S}(t)+H_{B}+H_{I},$ (1)
where $H_{S}(t)$ is time dependent system Hamiltonian, $H_{B}$ is the
Hamiltonian of the thermal bath which consists of a set of independent
harmonic oscillators, and interaction Hamiltonian between these two is
described by $H_{I}$. Using the units with $\hbar=1$, the total system
Hamiltonian is:
$H=\frac{\epsilon(t)}{2}\sigma_{z}+\frac{V}{2}\sigma_{x}+\sum_{k}\omega_{k}\,b^{\dagger}_{k}\,b_{k}+\sigma_{z}\sum_{k}g_{k}\left(b^{\dagger}_{k}+b_{k}\right),$
(2)
where $V$ is the tunneling matrix element between the two states, $\sigma_{i}$
are the Pauli spin matrices, $b^{\dagger}_{k}(b_{k})$ are creation
(annihilation) operators of the $k^{th}$ bath mode, $g_{k}$ is the interaction
strength between the system and the $k$th mode of the environmental
oscillators. Here, $\epsilon(t)=\epsilon_{0}+\Omega\,\alpha(t)$ where
$\epsilon_{0}$ is a bare energy difference between the states of the TSS and
the time-dependent term accounts for the random telegraph noise modulation of
the state energies. RTN has two states with noise amplitude $\pm\Omega$. Its
character is described by zero average ($\langle\alpha(t)\rangle=0$) and
exponentially decaying auto-correlation function
($\langle\alpha(t)\alpha(t^{\prime})\rangle=e^{-\nu|t-t^{\prime}|}$) with
noise frequency $\nu$ that is a measure of number of flippings of the noise
signal in unit time interval.
In the present study, we consider strong interaction between the TSS and the
bath and use polaron transformation to express the interaction Hamiltonian in
a form that lets one use it in perturbative master equation. In the polaron
frame, the total time-dependent Hamiltonian of Eq. (1) can be expressed as:
$\displaystyle H^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
H_{S}^{{}^{\prime}}(t)+H_{B}^{{}^{\prime}}+H_{I}^{{}^{\prime}},$ (3)
$\displaystyle=$
$\displaystyle\frac{\epsilon(t)}{2}\sigma_{z}+V_{r}\,\sigma_{x}+\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\sigma_{+}\,B_{-}+\sigma_{-}\,B_{+},$
where the superscript $"\,^{\prime}\,"$ means that the operator
$O^{{}^{\prime}}$ is in the polaron frame. After this point, we will drop the
superscript notation, for simplicity. $\sigma_{\pm}$ are the flip operators of
the TSS. $B_{\pm}$ are the bath correlation operators and $V_{r}$ is the
reduced tunneling matrix element, and given as a function of system-bath
coupling strength $g_{k}$ with the $k^{th}$ oscillator mode respectively:
$\displaystyle B_{\pm}$ $\displaystyle=$ $\displaystyle\langle
e^{\mp\sum_{k}\frac{2\,g_{k}}{\omega_{k}}\left(b_{k}^{\dagger}-b_{k}\right)}\rangle_{R},$
(4) $\displaystyle V_{r}$ $\displaystyle=$ $\displaystyle
V\exp\left(-\frac{1}{4\pi}\int_{0}^{\infty}\frac{J(\omega)}{\omega^{2}}\coth\left(\beta\omega\right)\right)$
(5)
where $\langle\dots\rangle_{R}$ implies averaging over the environmental
degrees of freedom. It is significant to note that the reduced tunneling rate
$V_{r}$ is equal to zero for the bath spectral densities of which frequency
exponent is less than two which is the case in the current study.
The evolution equation of the reduced density matrix of the TSS in the
interaction picture can be written as
$\frac{d}{d\,t}\rho_{S}(t)=-\int_{0}^{t}dt_{1}\,\mathrm{Tr}_{R}{\left[H_{I}(t),\left[H_{I}(t_{1}),\rho_{S}\otimes\rho_{R}\right]\right]}.$
(6)
where $\mathrm{Tr}_{R}$ indicates partial trace over the environmental modes.
In the Schrödinger picture, the evolution equations of any system operator $A$
in the second order can be given by
$\displaystyle\frac{d}{d\,t_{1}}\langle A(t_{1})\rangle$ $\displaystyle=$
$\displaystyle i\,\mathrm{Tr}_{S\otimes
R}\left(\\{\left[H_{S},A\right]\\}(t_{1})\,\rho_{T}(0)\right)$ (7)
$\displaystyle+$ $\displaystyle\int_{0}^{t_{1}}\,d\tau\,\mathrm{Tr}_{S\otimes
R}\left(\\{\tilde{H}_{I}(\tau-
t_{1})\left[A,H_{I}\right]\\}(t_{1})\rho_{T}(0)\right.$
$\displaystyle\left.+\\{\left[H_{I},A\right]\tilde{H}_{I}(\tau-
t_{1})\\}(t_{1})\rho_{T}(0)\right).$
Here,
$\tilde{H}_{I}(t)=V\left(\sigma_{-}(t)\,B_{+}(t)+\sigma_{+}(t)\,B_{-}(t)\right)$
which describes the time evolution in the interaction picture in the polaron
frame. Tr with the symbol $S\otimes R$ refers to a trace over the Hilbert
space of the total system. Curly brackets in Eqs. (7) and (8) indicate that
the expression should be evaluated in the Heisenberg picture and its time
should be taken as given in the post bracket. It is straightforward to express
the TTCFs in non-Markovian regime by using the result of Refs.goan11 (18, 21):
$\displaystyle\frac{d}{d\,t_{1}}\langle A(t_{1})B(t_{2})\rangle$
$\displaystyle=$ $\displaystyle i\,\mathrm{Tr}_{S\otimes
R}\left(\\{\left[H_{S},A\right]\\}(t_{1})B(t_{2})\,\rho_{T}(0)\right)$ (8)
$\displaystyle+$ $\displaystyle\int_{0}^{t_{1}}\,d\tau\,\mathrm{Tr}_{S\otimes
R}\left(\\{\tilde{H}_{I}(\tau-
t_{1})\left[A,H_{I}\right]\\}(t_{1})B(t_{2})\rho_{T}(0)\right.$
$\displaystyle\left.+\\{\left[H_{I},A\right]\tilde{H}_{I}(\tau-
t_{1})\\}(t_{1})B(t_{2})\rho_{T}(0)\right)$ $\displaystyle+$
$\displaystyle\int_{0}^{t_{2}}d\tau\,\mathrm{Tr}_{S\otimes
R}\left(\\{\left[H_{I},A\right]\\}(t_{1})\\{\left[B,\tilde{H}_{I}(\tau-
t_{2})\right]\\}(t_{2})\,\rho_{T}(0)\right).$
Here, the first two terms on the right hand side of Eq. (8) belong to the QRT
terms while the last term accounts for the corrections for the non-Markovian
effects. The last integral term in Eq. (8) is the source of violation of the
quantum regression theorem. By using the set of TTCFs in non-Markovian
approximation in Eq.(8) we obtain the time evolution of TTCFs of system
operators of dichotomously driven spin-boson model in non-Markovian case and
average those equations over the large number of noise trajectories in the
following section.
## 3 Results
In the present study, we will analyze the dynamics of
$\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ for a TSS which is in
contact with a thermal bath at temperature $T$. We will consider the strong
system-bath coupling regime and use a structured spectral density for
describing the interaction. The energy levels of the TSS is assumed to be
driven stochastically with an RTN signal. By using the Hamiltonian Eq. (3) in
the master equation Eq. (8) goan11 (18) with $A=\sigma_{z}$ and
$B=\sigma_{z}$, one can obtain following set of four coupled differential
equations for various TTCFs and $\langle\sigma_{z}(t)\rangle$ which include
beyond the quantum regression theorem contributions:
$\displaystyle\frac{d}{d\,t_{1}}\langle\sigma_{z}(t_{1})\,\sigma_{z}(t_{2})\rangle$
$\displaystyle=$
$\displaystyle-\Gamma_{1}(t_{1})\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle-\Gamma_{2}(t_{1})\langle\sigma_{z}(t_{2})\rangle$
(9)
$\displaystyle+4\,\Gamma_{3}(t_{1},t_{2})\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle+4\,\Gamma_{4}(t_{1},t_{2})\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle$
$\displaystyle\frac{d}{d\,t_{1}}\langle\sigma_{z}(t_{1})\rangle$
$\displaystyle=$
$\displaystyle-\Gamma_{1}(t_{1})\langle\sigma_{z}(t_{1})\rangle-\Gamma_{2}(t_{1}),$
(10)
$\displaystyle\frac{d}{d\,t_{1}}\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle$
$\displaystyle=$
$\displaystyle\left[i\,\left(\epsilon_{0}+\Omega\,\alpha(t_{1})\right)-\Gamma_{5}(t_{1})\right]\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle$
(11)
$\displaystyle+\Gamma_{3}(t_{1},t_{2})\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle,$
$\displaystyle\frac{d}{d\,t_{1}}\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$
$\displaystyle=$
$\displaystyle-\left[i\,\left(\epsilon_{0}+\Omega\,\alpha(t_{1})\right)+\Gamma_{5}(t_{1})^{*}\right]\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$
(12)
$\displaystyle+\Gamma_{4}(t_{1},t_{2})\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle,$
We should note that derivation of Eqs. (9)-(12) is simplified by the fact that
the reduced tunneling rate $V_{r}$ in Eq. (3) equals to zero for the chosen
spectral density. All the terms in Eqs. (9)-(12) that contain $\Gamma_{i}$
coefficients with $i=3$ and 4 are due to the corrections to the QRT (the last
line of Eq. (8)) while those with $i=1,2,5$ and $6$ account for the quantum
regression theorem. The environmental correlation functions $\Gamma_{i}(t)$
are derived from thermal averages of various system and bath operators as:
$\displaystyle\Gamma_{1}(t_{1})$ $\displaystyle=$ $\displaystyle
4\,V^{2}\int_{0}^{t_{1}}d\tau\,e^{-Q_{2}\left(t_{1}-\tau\right)}\cos{\left[Q_{1}(t_{1}-\tau)\right]}\,\cos{\left[f(t_{1},\tau)\right]},$
$\displaystyle\Gamma_{2}(t_{1})$ $\displaystyle=$ $\displaystyle
4\,V^{2}\int_{0}^{t_{1}}d\tau\,e^{-Q_{2}\left(t_{1}-\tau\right)}\,\sin{\left[Q_{1}(t_{1}-\tau)\right]}\,\sin{\left[f(t_{1},\tau)\right]},$
$\displaystyle\Gamma_{3}(t_{1},t_{2})$ $\displaystyle=$ $\displaystyle
V^{2}\int_{0}^{t_{2}}d\tau\,e^{-Q_{2}\left(t_{1}-\tau\right)+i\,Q_{1}\left(t_{1}-\tau\right)}\,e^{i\,f(t_{2},\tau)},$
$\displaystyle\Gamma_{4}(t_{1},t_{2})$ $\displaystyle=$ $\displaystyle
V^{2}\int_{0}^{t_{2}}d\tau\,e^{-Q_{2}\left(t_{1}-\tau\right)+i\,Q_{1}\left(t_{1}-\tau\right)}\,e^{-i\,f(t_{2},\tau)},$
$\displaystyle\Gamma_{5}(t_{1})$ $\displaystyle=$ $\displaystyle
2\,V^{2}\int_{0}^{t_{1}}d\tau\,e^{-Q_{2}\left(t_{1}-\tau\right)}\,\cos{\left[Q_{1}(t_{1}-\tau)\right]}\,e^{i\,f(t_{1},\tau)},$
where
$\displaystyle Q_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega^{2}}\sin(\omega
t),$ (13) $\displaystyle Q_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega^{2}}\coth\left(\frac{\beta\,\omega}{2}\right)(1-\cos(\omega
t)),$ (14) $\displaystyle f(t,\tau)$ $\displaystyle=$
$\displaystyle\epsilon_{0}(t-\tau)+\Omega\,\int_{t}^{\tau}d\zeta\,\alpha(\zeta),$
(15)
where $Q_{1,2}(t)$ are the imaginary and the real parts of the bath
correlation function. $f(t,\tau)$ accounts for the static as well as
stochastically fluctuating system energy gap. Spectral density $J(\omega)$
characterizes the interaction between the TSS and the environment as a
function of environmental modes. We choose $J(\omega)$ as the structured
spectral density. In this model, the environment is described by single
harmonic oscillator (with frequency $\omega_{0}$) which is in contact with an
Ohmic thermal bath that broadens its energy levels Garg1985 (37):
$J(\omega)=8\,\kappa^{2}\frac{\gamma\,\omega_{0}\,\omega}{(\omega^{2}-\omega_{0}^{2})^{2}+4\gamma^{2}\,\omega^{2}},$
(16)
where $\kappa$ is the magnitude of system-bath coupling, $\omega_{0}$ is the
frequency of the central harmonic oscillator, $\gamma$ is the broadening of
the oscillator levels due to interaction with the environment. A rough measure
of the strength of the interaction between the TSS and its environment is the
reorganization energy of the bath
($E_{r}=\int_{0}^{\infty}d\omega\,J(\omega)/\omega$). The reorganization
energy for the chosen $J(\omega)$ is equal to $\kappa^{2}/\omega_{0}$.
### 3.1 Stochastic Averaging
The RTN signal $\alpha(t)$ appears in the coupled differential equations that
describe the dynamics of the two-time correlation functions in Eqs.(9)-(12) as
bare coefficients as well as its integral in definition of $\Gamma_{i}(t)$
coefficients. One can either use an ensemble averaging approach to find the
noise averaged TTCFs by solving Eqs. (9)-(12) for an appropriately controlled
number of different realizations of RTN noise and average the solutions or use
probability density of RTN signal to average the coupled differential
equations and solve those averaged equations. Thanks to theorems due to
Bourret-Frisch Bourret73 (38) which provide a factorization for the multi-time
correlation functions of dichotomous process and Loginov-Shapiro Shapiro78
(39) which relates the average of time evolution operator of the noise with
that of the system operators, Eqs. (9)-(12) can be averaged exactly over the
RTN Magazzu2017 (40). The averaging process effectively doubles the number of
correlation functions by coupling the correlations of the noise signal
$\alpha(t)$ with the system operator correlations. We will adopt the single
average notation to describe both the correlation and noise averaging, i. e.,
$\langle\sigma_{z}(t)\rangle$ denotes noise averaged correlation of
$\sigma_{z}$ at times $t_{1}$ and $t_{2}$ in the rest of the paper. The set of
RTN averaged coupled differential equations for
$\langle\sigma_{z}(t_{1})\rangle$ and
$\langle\alpha(t_{1})\sigma_{z}(t_{2})\rangle$ are:
$\displaystyle\frac{d}{d\,t_{1}}\langle\sigma_{z}(t_{1})\rangle$
$\displaystyle=$
$\displaystyle-\Gamma_{1}(t_{1})\langle\sigma_{z}(t_{1})\rangle+\Gamma_{2}(t_{1})\langle\alpha(t_{1})\,\sigma_{z}(t_{1})\rangle-\Gamma_{3}(t_{1}),$
$\displaystyle\frac{d}{d\,t_{1}}\langle\alpha(t_{1})\sigma_{z}(t_{1})\rangle$
$\displaystyle=$
$\displaystyle-(\nu+\Gamma_{1}(t_{1}))\,\langle\alpha(t_{1})\sigma_{z}(t_{1})\rangle+\Gamma_{2}(t_{1})\langle\sigma_{z}(t_{1})\rangle-\Gamma_{4}(t_{1}).$
(17)
The noise averaged coupled differential equations for the six two-time
correlation functions can be expressed as:
$\frac{d}{dt_{1}}Y=A(t_{1},t_{2})\cdot Y+b(t_{1},t_{2})$ (18)
where $Y$ is a column vector whose elements are the different correlations:
$\displaystyle
Y=\left(\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle,\langle\alpha(t_{1})\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle,\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle,\langle\alpha(t_{1})\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle,\right.$
$\displaystyle\left.\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle,\langle\alpha(t_{1})\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle\right)^{T}$
and $A$ is the time-dependent coefficient matrix of the noise-averaged coupled
differential equations:
$\displaystyle
A=\left(\begin{array}[]{cccccc}-\Gamma_{1,1}&\Gamma_{1,2}&-4\Gamma_{3,1}&-4\Gamma_{3,2}&4\Gamma_{4,1}&4\Gamma_{4,2}\\\
\Gamma_{1,2}&-\left(\nu+\Gamma_{1,1}\right)&4\Gamma_{3,2}&4\Gamma_{3,1}&-4\Gamma_{4,2}&4\Gamma_{4,1}\\\
\Gamma_{4,1}&-\Gamma_{4,2}&(i\epsilon_{0}-\Gamma_{5,1})&(i\Omega+\Gamma_{5,2})&0&0\\\
-\Gamma_{4,2}&\Gamma_{4,1}&(i\Omega+\Gamma_{5,2})&-(\nu-i\epsilon_{0}+\Gamma_{5,1})&0&0\\\
\Gamma_{3,1}&\Gamma_{3,2}&0&0&-(i\epsilon_{0}+\Gamma_{6,1})&-(i\Omega+\Gamma_{6,2})\\\
\Gamma_{3,2}&\Gamma_{3,1}&0&0&-(i\Omega+\Gamma_{6,2})&-(\nu+i\epsilon_{0}+\Gamma_{6,1})\end{array}\right)$
where all $\Gamma$s in $A$ depend on time $t_{1}$ except $\Gamma_{3,}$ and
$\Gamma_{4,}$ which have double time arguments $(t_{1},t_{2})$. The
$b(t_{1},t_{2})$ term in Eq. (18) contributes to the change in
$\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ and
$\langle\alpha(t_{1})\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ terms and is
found to be:
$\displaystyle
b=\left(-\Gamma_{2,1}(t)g_{1}(t_{2})-e^{-\nu|t-t_{2}|}\Gamma_{2,2}(t)g_{2}(t_{2}),-\Gamma_{2,2}(t)g_{1}(t_{2})-e^{-\nu|t-t_{2}|}\Gamma_{2,1}(t)g_{2}(t_{2}),0,0,0,0\right)^{T}$
where $g_{1}(t)=\langle\sigma_{z}(t)\rangle$ and
$g_{2}(t)=\langle\alpha(t)\sigma_{z}(t)\rangle$. $\Gamma_{i,j}$s of $A$
involve the time evolution operator
$S(t,t^{\prime})=\exp\left[-i\,\Omega\int_{t^{\prime}}^{t}d\tau\alpha(\tau)\right]$
of the Kubo oscillator and are given as:
$\displaystyle\Gamma_{1,1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{cc}(t-\tau)S_{0}(t-\tau)$
$\displaystyle\Gamma_{1,2}(t)$ $\displaystyle=$ $\displaystyle
i\int_{0}^{t}d\tau\,\mathcal{E}_{cs}(t-\tau)S_{1}(t-\tau)$
$\displaystyle\Gamma_{2,1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{ss}(t-\tau)S_{0}(t-\tau)$
$\displaystyle\Gamma_{2,2}(t)$ $\displaystyle=$ $\displaystyle
i\int_{0}^{t}d\tau\,\mathcal{E}_{sc}(t-\tau)S_{1}(t-\tau)$
$\displaystyle\Gamma_{3,1}(t,t_{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{2}}d\tau\,\mathcal{E}_{f+}(t-\tau)e^{i\epsilon(t_{2}-\tau)}S_{0}(t_{2}-\tau)$
$\displaystyle\Gamma_{3,2}(t,t_{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{2}}d\tau\,\mathcal{E}_{f+}(t-\tau)e^{i\epsilon(t_{2}-\tau)}S_{1}(t_{2}-\tau)$
(20) $\displaystyle\Gamma_{4,1}(t,t_{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{2}}d\tau\,\mathcal{E}_{f-}(t-\tau)e^{-i\epsilon(t_{2}-\tau)}S_{0}(t_{2}-\tau)$
$\displaystyle\Gamma_{4,2}(t,t_{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{2}}d\tau\,\mathcal{E}_{f-}(t-\tau)e^{-i\epsilon(t_{2}-\tau)}S_{1}(t_{2}-\tau)$
$\displaystyle\Gamma_{5,1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{c-}(t-\tau)S_{0}(t-\tau)$
$\displaystyle\Gamma_{5,2}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{c-}(t-\tau)S_{1}(t-\tau)$
$\displaystyle\Gamma_{6,1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{c+}(t-\tau)S_{0}(t-\tau)$
$\displaystyle\Gamma_{6,2}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\,\mathcal{E}_{c+}(t-\tau)S_{1}(t-\tau)$
where
$\mathcal{E}_{cs}(t)=e^{-Q_{2}(t)}\mathrm{(c)os}\left(Q_{1}(t)\right)\,\mathrm{(s)in}\left(\epsilon_{0}\,t\right)$,
$\mathcal{E}_{c\pm}(t)=e^{-Q_{2}(t)}\mathrm{cos}\left(Q_{1}(t)\right)\,e^{\pm\,i\,\epsilon_{0}\,t}$,
and $\mathcal{E}_{f\pm}(t)=e^{-Q_{2}(t)}e^{i(Q_{1}(t)\pm\epsilon_{0}\,t)}$.
$S_{0}(t)$ and $S_{1}(t)$ in kernel definitions are, also, dichotomous noise
propagators Goychuk95 (41) which can be expressed as:
$\displaystyle S_{0}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\eta}\left(\nu_{+}e^{-t\,\nu_{-}/2}-\nu_{-}e^{-t\,\nu_{+}/2}\right),$
$\displaystyle S_{1}(t)$ $\displaystyle=$ $\displaystyle
i\frac{\Omega}{\nu}\left(e^{-t\,\nu_{+}/2}-e^{-t\,\nu_{-}/2}\right)$ (21)
where $\eta=\sqrt{\nu^{2}-4\Omega^{2}}$, $\nu_{+}=\nu+\eta$, and
$\nu_{-}=\nu-\eta$.
One should note that the $\Gamma_{i,j}$ that depends on a single time argument
stem from the non-Markovian quantum regression theorem, while the ones with
double time argument arise as corrections to the QRT. They will be referred to
as QRT$+$ in the rest of the current paper. Also, when there is no external
noise, $S_{0}(t)=1$ and $S_{1}(t)=0$, so all the $\Gamma_{i,2}$ are zero.
---
(a) $\epsilon_{0}=1,\,\beta=0.02,\,\nu=0.01$
---
(b) $\epsilon_{0}=0,\,\beta=50,\,\nu=0.01$
---
(c) $\epsilon_{0}=1,\,\beta=0.02,\,\nu=1$
---
(d) $\epsilon_{0}=0,\,\beta=50,\,\nu=1$
Figure 1: Dynamics of the real part of two-time correlations
$\langle\sigma_{z}(t)\sigma_{z}(t_{2})\rangle$,
$\langle\sigma_{+}(t)\sigma_{-}(t_{2})\rangle$ and
$\langle\sigma_{-}(t)\sigma_{+}(t_{2})\rangle$ for the QRT (dashed line) and
QRT+ (solid line) approaches for various combinations of the TSS bias, the
noise frequency and temperature of the environment. Inset shows the power
spectra of $\langle\sigma_{+}(t)\sigma_{-}(t_{2})\rangle$ (solid line) and
$\langle\sigma_{+}(-)\sigma_{+}(t_{2})\rangle$ (dashed line) correlations
calculated with the QRT+ corrections. The noise amplitude $\Omega=0.75$ for
all the figures.
For the present study, we consider the strong system-environment coupling
regime with $\omega_{0}=1$, $\kappa=2$ along with a relatively large
dispersion $\gamma=0.5$ and investigate the low ($\beta=50$) and the high
temperature ($\beta=0.02$) as well as the biased ($\epsilon_{0}=1$) and the
non-biased ($\epsilon_{0}=1$) cases. With these coupling parameters, the short
time approximationGarg1985 (37) can be used to write the bath correlation
functions in a simpler form as $Q_{1}(t)=E_{r}t$ and $Q_{2}(t)=-\xi t^{2}$
where
$\xi=E_{r}/\beta+\kappa^{2}/\left(\pi\sqrt{\omega_{0}^{2}-\gamma^{2}}\right)\omega_{0}\,\mathrm{Im}\left[\Psi\left(0,1+\beta\left(\gamma+i\sqrt{\omega_{0}^{2}-\gamma^{2}}\right)/(2\pi)\right)\right].$
For this choice of the bath correlation functions, the $\Gamma_{i,j}$
coefficients in Eq. (3.1) can be expressed as sum of error functions of
various sum and difference combinations of the TSS bias, noise frequency and
the bath reorganization energy. But, the expressions are too cumbersome and
will not be displayed here.
We start the discussion of the results with presentation of the various
qualitatively different forms of time-dependence of the two-time correlation
functions calculated in non-Markovian QRT and QRT+ approximations in Fig.
1a-d. While $\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ is found to
decay exponentially with a rate that depends on the temperature, the bias and
the properties of the noise for the strong system-bath coupling,
$\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle$ and
$\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$ display richer dynamics. It
can be seen from the figures that the difference between the QRT and QRT+
approximations for all three correlation functions at low temperature is
higher than the difference at high temperature. The difference between the two
will be discussed in more detail below.
---
(a) $\beta=50$
---
(b) $\beta=0.02$
Figure 2: Noise amplitude and color dependence of absorption spectra at low
and high temperature for the non-biased TSS.
The dynamics of a two state system driven by RTN can be deduced intuitively at
the asymptotic limits with the help of the concept of the noise color or the
Kubo number $K=\Omega/\nu$. As the noise jumps of magnitude $\Delta
E=\hbar\Omega$ occur with average time spacing $\Delta t=1/\nu$, the energy-
time uncertainty relation ($\Delta E\Delta t=\hbar\Omega/\nu\geq\hbar/2$)
would make resolving those fluctuations impossible at the fast jumping limit.
Resulting motional averaging leads to emission or absorption at dynamically
averaged $\epsilon_{0}$ frequency although the system is at one of the states
$\epsilon_{0}\pm\Omega$ at all times. At the slow noise limit, the mean time
between the jump events is long and the dynamics of the system can be
considered as the static average at frequencies $\epsilon_{0}\pm\Omega$.
Fourier transform of $\langle\sigma_{-}(t+\tau)\sigma_{+}(t)\rangle$ and
$\langle\sigma_{+}(t+\tau)\sigma_{-}(t)\rangle$ are measures of absorption and
emission spectrum, respectively. The insets in Fig. 1a-d display the power
spectrum of the correlations
$\langle\sigma_{+}(t_{1})\sigma_{-}(t_{2})\rangle$ and
$\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$. The first row where $K=75$
is the slow noise (strongly colored noise) limit and one can observe emission
and absorption phenomena at frequencies $\epsilon_{0}\pm\Omega$ clearly. The
plots in the second row display the dynamics of the correlation at
intermediate noise color with $K=0.75$ for which the motional averaging starts
to dominate the power spectrum as the displayed spectra tend to broaden and
become a single peak at frequency $\epsilon_{0}$. We display the $\Omega$, $K$
and frequency dependence of the absorption spectrum at high and low
temperatures in Fig.2. As can be seen from the plots, the single peak to
double peak behaviour start emerging around $K\approx 0.7$ independent of the
bath temperature.
---
(a) $\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$
---
(b) $\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$
Figure 3: Noise frequency $\nu$ and amplitude $\Omega$ dependence of the
relative difference between the QRT and QRT+ ($\epsilon_{0}=0,\,\beta=50$).
Thick line indicates $\nu=2\Omega$.
There has been some discussion on the relevance of QRT$+$ terms on the non-
Markovianity of the dynamics of the system, which suggests that the difference
can be used as a measure of non-Markovianity Ali2015 (34, 28, 32, 33). As a
rough measure of the difference between the two dynamics, we adopt a heuristic
approach and define
$\Delta_{c}=100\left|\frac{I_{\textrm{QRT}}(c)-I_{\textrm{QRT+}}(c)}{I_{\textrm{QRT}}(c)}\right|,$
(22)
where $I_{QRT}(c)$ is the integral of the absolute value of the correlation
function $c$ obtained as solution of the QRT formulation. $\Delta_{c}$ is
found to be highly sensitive to the environmental temperature for the system
and the other environment parameters considered in the present study. It is
found to be close to zero ($<0.25$) at high temperature ($\beta=0.02$) for all
three correlation functions. Figure 3 displays $\Delta_{zz}$ and $\Delta_{-+}$
as function of the noise frequency and amplitude at low temperature
($\beta=50$) for the non-biased system. $\Delta_{zz}$ and $\Delta_{-+}$ are
found to show similar qualitative and quantitative dependence on noise
parameters. Interestingly, for both correlations, the difference display a
death and revival cycle as function of the noise amplitude. The cycle has a
weak dependence on the noise frequency which is peculiar because one would
expect effect of noise to be dependent on its color or Kubo number.
---
(a) $\epsilon_{0}=0,\,\beta=50$
---
(b) $\epsilon_{0}=1,\,\beta=50$
---
(c) $\epsilon_{0}=0,\,\beta=0.02$
---
(d) $\epsilon_{0}=1,\,\beta=0.02$
Figure 4: Noise frequency $\nu$ and amplitude $\Omega$ dependence of the
inverse life time of the $\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$.
The two-time correlation function
$\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ decays exponentially for
the system and environment parameters chosen for the present study as can be
seen from Figs. 1a-d. We fit the time dependence of
$\langle\sigma_{z}(t_{1})\sigma_{z}(t_{2})\rangle$ to a function of the form
$p+(1-p)e^{-kt}$ where $p$ is the limit of
$\langle\sigma_{z}(t_{1})\sigma_{z}(1)\rangle$ as $t_{1}\to\infty$ and $k$ is
the decay parameter of the correlation. We display the noise frequency and
amplitude dependence of this exponential decay parameter $k$ at different
temperature and TSS bias parameters in Fig. 4 a-d. It can be seen from the
sub-figures that $k$ decreases towards zero at high noise amplitude
irrespective of the bath temperature, TSS bias and the noise frequency which
is reminiscent of the noise induced destruction of tunneling Berry95 (42).
Effect of $\nu$ and $\Omega$ on $k$ is found to be observably different at low
and high temperatures. At high temperature, $k$ is almost independent of the
bias and the noise frequency and it decreases with increasing noise amplitude
monotonously while one can observe an enhancement of the transport especially
for the low-frequency noise for both the biased and non-biased TSS at low
temperature. This finding is another example of so called ENAQT Rebentrost2009
(43) which describes noise assisted enhancement of quantum transport.
---
(a) $\epsilon_{0}=0,\,\beta=50$
---
(b) $\epsilon_{0}=1,\,\beta=50$
---
(c) $\epsilon_{0}=0,\,\beta=0.02$
---
(d) $\epsilon_{0}=1,\,\beta=0.02$
Figure 5: Noise frequency $\nu$ and amplitude $\Omega$ dependence of the
decoherence rate for $\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$.
We found that the real part of the correlation
$\langle\sigma_{-}(t_{1})\sigma_{+}(t_{2})\rangle$ could be fitted to a
function of the form $e^{-\Lambda
t}\left(a_{1}\cos{\left(\omega_{1}t\right)}+a_{2}\cos{\left(\omega_{2}t\right)}\right)$
which could in return be used to extract the exponential decay rate $\Lambda$
which is similar to the decoherence rate of the $\langle\sigma_{x}\rangle$.
Figure 5 indicates the noise amplitude and frequency dependence of $\Lambda$
at high and low temperatures for the biased and non-biased TSS. The $\Lambda$
value at low $\Omega$ might be considered as the $1/T_{2}$ Wang2005 (44) of
the system. As expected, its value at high temperature is higher than at low
temperature. Decoherence rate $\Lambda$ is found to be high at large $\nu$ and
$\Omega$ values irrespective of the environmental temperature and the bias of
the TSS which is expected because dynamics is dominated by the noise. An
interesting decrease of decoherence rate can be observed at low temperatures
for the slow noise for both the biased and non-biased TSS (Fig. 5a and b).
Also, for the very weak noise strength, the decoherence rate displays a
resonance structure as function of the noise frequency as can be seen in all
four sub-plots in Fig. 5.
## 4 Conclusions
We have investigated the effect of random telegraph noise on the two-time
correlation functions of a strongly coupled spin-boson model with structured
spectral density. Both the quantum regression theorem and corrections to the
QRT terms were included in the formulation of the master equations for the
correlations. One of the important findings of the study is that the QRT+
corrections are found to be qualitatively insignificant and quantitatively
small. As the temperature of the bath or the strength of the noise is
increased the difference tends to zero with some structure in the noise
dependence at low temperature which is an indication of noise induced non-
Markovianity. The motional averaging and narrowing caused by the random
telegraph noise driving of the TSS was observed in the emission and absorption
spectra of the system as function of the noise color with a transition region
around $K=0.7$. The decoherence rate of the spectra were found to strongly
depend on the noise parameters as they are expected to be directly
proportional to the noise frequency $\nu$ and the noise strength
$\Omega^{2}/\nu$ in the slow and fast jumping limits. We have, also, studied
the decay rate of the two-time correlation of $\sigma_{z}$ which can be an
indicator of the transport and found that slow noise can enhance transport at
low temperature. The reported findings of this study might be beneficial in
the open quantum systems when the physical properties corresponding to the
two-time correlation functions are of interests and the system-environment
interaction might not be described by using a single thermal bath.
## 5 Acknowledgments
The author would like to acknowledge many useful comments and discussions with
Prof. Dr. Resul Eryiğit.
## References
* (1) M. O. Scully and M. S. Zubairy, _Quantum Optics_ (Cambridge University Press, 1997).
* (2) H. J. Carmichael, _Statistical Methods in Quantum Optics_ (Springer Press, 1999).
* (3) C. W. Gardiner and P. Zoller, _Quantum Noise_ (Springer-Verlag, 2000).
* (4) J. Li, M.P. Silveri, K.S. Kumar, J.-M Pirkkalainen, A. Vepsäläinen, W.C. Chien, J. Tuorila, M.A. Sillanpää, P.J. Hakonen, E.V. Thuneberg, and G.S. Paraoanu, “Motional averaging in a superconducting qubit,” Nat. commun. 4, 1420 (2013).
* (5) L. Li, M. J.W. Hall, and H. M. Wisemanv, “Concepts of quantum non-markovianity: A hierarchy,” Phy. Rep. 759, 1–51 (2018).
* (6) I. de Vega and D. Alonso, “Dynamics of non-markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
* (7) A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, and J. M. Gérard, “Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot,” Nat. Phys. 2, 759–764 (2006).
* (8) M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot–cavity system,” Phys. Rev. Lett. 103, 207403 (2009).
* (9) S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Dephasing of triplet-sideband optical emission of a resonantly driven $\mathrm{InAs}/\mathrm{GaAs}$ quantum dot inside a microcavity,” Phys. Rev. Lett. 106, 247402 (2011).
* (10) Mathias R. Jørgensen and Felix A. Pollock, “A discrete memory-kernel for multi-time correlations in non-markovian quantum processes,” Phys. Rev. A 102, 052206 (2020).
* (11) T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434, 625–628 (2005).
* (12) L. Onsager, “Reciprocal relations in irreversible processes. i.” Phys. Rev. 37, 405 (1931).
* (13) M. Lax, “Formal theory of quantum fluctuations from a driven state,” Phys. Rev. 129, 2342–2348 (1963).
* (14) D. Alonso and I. de Vega, “Multiple-time correlation functions for non-markovian interaction: Beyond the quantum regression theorem,” Phys. Rev. Lett. 94, 200403 (2005).
* (15) I. de Vega and D. Alonso, “Non-markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit,” Phys. Rev. A 73, 022102 (2006).
* (16) D. Alonso and I. de Vega, “Hierarchy of equations of multiple-time correlation functions,” Phys. Rev. A 75, 052108 (2007).
* (17) A. A. Budini, “Operator correlations and quantum regression theorem in non-markovian lindblad rate equations,” J. Stat. Phys. 131, 51–78 (2008).
* (18) H.-S. Goan, P.-W. Chen, and C.-C. Jian, “Non-markovian finite-temperature two-time correlation functions of system operators: Beyond the quantum regression theorem,” J. Chem. Phys. 134, 124112 (2011).
* (19) P.-W. Chen, C.-C. Jian, and H.-S. Goan, “Non-markovian dynamics of a nanomechanical resonator measured by a quantum point contact,” Phys. Rev. B 83, 115439 (2011).
* (20) H.-S. Goan, C.-C. Jian, and P.-W. Chen, “Non-markovian finite-temperature two-time correlation functions of system operators of a pure-dephasing model,” Phys. Rev. A 82, 012111 (2010).
* (21) D. P. S. McCutcheon, “Optical signatures of non-markovian behavior in open quantum systems,” Phys. Rev. A 93, 022119 (2016).
* (22) I. de Vega and D. Alonso, “Emission spectra of atoms with non-markovian interaction: Fluorescence in a photonic crystal,” Phys. Rev. A 77, 043836 (2008).
* (23) J. Jin, C. Karlewski, and M. Marthaler, “Non-markovian correlation functions for open quantum systems,” New J. Phys. 18, 083038 (2016).
* (24) G. W. Ford and R. F. O’Connell, “There is no quantum regression theorem,” Phys. Rev. Lett. 77, 798–801 (1996).
* (25) G. W. Ford and R. F. O’Connell, “Calculation of correlation functions in the weak coupling approximation,” Ann. Phys. 276, 144–151 (1999).
* (26) G. W. Ford and R. F. O’Connell, “Driven systems and the lax formula,” Opt. Commun. 179, 451–461 (2000).
* (27) Ya.M. Blanter and M. Büttiker, “Shot noise in mesoscopic conductors,” Phys. Rep. 336, 1–166 (2000).
* (28) M. Cosacchi, M. Cygorek, F. Ungar, A. M. Barth, A. Vagov, and V. M. Axt, “Path-integral approach for nonequilibrium multitime correlation functions of open quantum systems coupled to markovian and non-markovian environments,” Phys. Rev. B 98, 125302 (2018).
* (29) J. Shao and N. Makri, “Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems,” J. Chem. Phys. 116, 507 (2002).
* (30) F. Pollock, C. A. Rodriguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Non-markovian quantum processes: Complete framework and efficient characterization,” Phys. Rev. A 97, 012127 (2018).
* (31) M. R. Jørgensen and F. A. Pollock, “Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-markovian path integrals,” Phys. Rev. Lett. 123, 240602 (2019).
* (32) G. Guarnieri, A. Smirne, and B. Vacchini, “Quantum regression theorem and non-markovianity of quantum dynamics,” Phys. Rev. A 90, 022110 (2014).
* (33) P.-W. Chen and Md. M. Ali, “Investigating leggett-garg inequality for a two level system under decoherence in a non-markovian dephasing environment,” Sci. Rep. 4, 6165 (2014).
* (34) Md. M. Ali, P.-Y. L., M. W.-Y. Tu, and W.-M. Zhang, “Non-markovianity measure using two-time correlation functions,” Phys. Rev. A 92, 062306 (2015).
* (35) A. Kurt, “Non-Markovian Corrections to Quantum Regression Theorem for the Strong Coupling Spin-Boson Model,” SAUJS 24, 596–604 (2020).
* (36) S. J. Jang, M. Benedetta, “Delocalized excitons in natural light-harvesting complexes,” Rev. Mod. Phys. 90, 035003 (2018).
* (37) A. Garg, J. N. Onuchic, and V. J. Ambegaokar, “Effect of friction on electron transfer in biomolecules,” J. Chem. Phys. 83, 4491–4503 (1985).
* (38) R. C. Bourret, U. Frisch, and A. Pouquet, “Browian motion of harmonic oscillator with stochastic frequency,” Physica 65, 303–320 (1973).
* (39) V. E. Shapiro and V. M. Loginov, “”formulae of differentiation” and their use for solving stochastic equations,” Physica A 91, 563 (1978).
* (40) L. Magazzù, P. Hänggi, B. Spagnolo, and D. Valenti, “Quantum resonant activation,” Phys. Rev. E 95, 042104 (2017).
* (41) I. A. Goychuk, E. G. Petrov, and V. May, “Dynamics of the dissipative two-level system driven by external telegraph noise,” Phys. Rev. E 52, 2392–2400 (1995).
* (42) M. Berry, “Two-state quantum asymptotics,” Ann. N. Y. Acad. Sci. 755, 303 (1995).
* (43) P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum transport,” New J. Phys. 11, 033003 (2009).
* (44) X. R. Wang, Y. S. Zheng, and Sun Yin, “Spin relaxation and decoherence of two-level systems,” Phys. Rev. B 72, 121303 (2005).
|
# Impact of Dark Photon Emission on Massive Star Evolution and Pre-Supernova
Neutrino Signal
A. Sieverding A. Sieverding<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA E. Rrapaj Department of
Physics, University of California, Berkeley, CA 94720, USA School of Physics
and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA G. Guo
Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan Y.-Z. Qian
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA
###### Abstract
We study the effects of additional cooling due to the emission of a dark
matter candidate particle, the dark photon, on the final phases of the
evolution of a $15\,M_{\odot}$ star and resulting modifications of the pre-
supernova neutrino signal. For a substantial portion of the dark photon
parameter space the extra cooling speeds up Si burning, which results in a
reduced number of neutrinos emitted during the last day before core collapse.
This reduction can be described by a systematic acceleration of the relevant
timescales and the results can be estimated semi-analytically in good
agreement with the numerical simulations. Outside the semi-analytic regime we
find more complicated effects. In a narrow parameter range, low-mass dark
photons lead to an increase of the number of emitted neutrinos because of
additional shell burning episodes that delay core collapse. Furthermore,
relatively strong couplings produce a thermonuclear runaway during O burning,
which could result in a complete disruption of the star but requires more
detailed simulations to determine the outcome. Our results show that pre-
supernova neutrino signals are a potential probe of the dark photon parameter
space.
## 1 Introduction
Dark matter, which represents more than 80% (Tanabashi et al., 2018) of the
matter density of the universe and whose nature remains one of the biggest
mysteries in physics, could be part of a dark sector which weakly interacts
with Standard Model (SM) particles. Such scenarios of dark sectors naturally
appear in many extensions of the SM where dark matter particles only interact
with the SM via a mediator. There is a rich experimental program searching for
signatures of such mediators (Essig et al., 2013; Alexander et al., 2016;
Aaboud et al., 2019). However, if the coupling to SM particles is too weak,
these particles could evade the searches and remain hidden. Astrophysical
probes can greatly extend the reach of the search for dark matter candidates,
trading the precision associated with the controlled environment of a
laboratory for the vast range of densities and temperatures of stars (Raffelt,
1996). In one of the simplest extensions of the SM, the dark sector interacts
with ordinary matter through the exchange of light vector bosons that couple
to SM conserved currents (Holdom, 1986; Rajpoot, 1989; Nelson & Tetradis,
1989; Batell et al., 2014a). Dark matter is charged under a local U(1)
symmetry in which the mediator couples to the SM electric charge $Q$, and is
described by the spin-one field $A^{D}_{\mu}$, called the dark photon, which
mixes kinetically with the standard photon $A_{\mu}$. Dark photons are
characterized by two independent parameters, their mass $m_{A}$ and the
reduced coupling strength with normal matter $\varepsilon$. Several general
dark matter searches have established experimental bounds on this parameter
space (Batell et al., 2014b; Essig et al., 2013).
Complementary constraints are provided by core collapse supernovae (SNe). For
instance, parameters that would result in a noticeable reduction of the
observed neutrino burst duration from SN 1987A can be excluded (Sung et al.,
2019; Chang et al., 2017; Rrapaj & Reddy, 2016; Raffelt, 1990; Choplin et al.,
2017). Additional constraints have been derived from observational signatures
of our sun and other low-mass stars (An et al., 2013). If beyond SM particles
can be produced at relatively low temperatures, they would affect stellar
evolution during He-burning and onward. This phase lasts millions of years and
is sampled broadly by observations that can be used to derive stringent
constraints (Raffelt & Dearborn, 1987). Dark matter particles that are more
massive, however, can only be produced at much higher temperatures and
therefore only affect the short, advanced stages of the evolution of more
massive stars.
This work is a continuation of Rrapaj et al. (2019) to study the potential
impact of the dark photon on the final stages of a $15\,M_{\odot}$ star.
During these stages temperatures in the star become high enough for dark
photons to be emitted from electron-positron pair annihilation while the
densities are still far below the regime where other processes, e.g.,
bremsstrahlung, become important. We therefore only include dark photons from
pair annihilation in this work and consider values of $m_{A}$ ranging from
$2m_{\mathrm{e}}$ to $10\,\mathrm{MeV}$, where $m_{\mathrm{e}}$ is the
electron mass. Due to energy-momentum conservation, dark photons with masses
below $2m_{\mathrm{e}}$ cannot be produced by electron-positron pair
annihilation and the temperatures needed for the emission of particles with
masses more than $10\,\mathrm{MeV}$ are not reached before core collapse. We
explore a wide range of coupling strengths $\varepsilon$, between $10^{-13}$
and $10^{-6}$, thereby probing a part of the parameter space unconstrained by
neutrino observations of SN 1987A (Chang et al., 2017; Hardy & Lasenby, 2017).
If the dark photon is lighter than all other particles in the dark sector it
decays back into electron-positron pairs, either still inside the star or
later in the interstellar medium. In the latter case, the $\gamma$-ray
signature of the subsequent annihilation of positrons can be used to derive
constraints (DeRocco et al., 2019). If the dark photons decay inside the star,
this may act as an additional heating mechanism leading to further constraints
from the observed explosion energies of SNe (Sung et al., 2019). In contrast
to these scenarios, we assume that the dark sector contains lighter particles
that the dark photon ultimately decays into, as also discussed in Rrapaj et
al. (2019). In this case, any energy carried away by the dark photon is leaked
into the dark sector, and the emission of the dark matter particles always
acts as a cooling mechanism without additional signatures. Under this
assumption the constraints discussed by DeRocco et al. (2019) and Sung et al.
(2019) do not apply.
As a messenger of the stellar interior, we look at the emission of pre-SN
neutrinos and find that, in a large part of our selected dark photon parameter
space, the extra cooling leads to a speed-up of the final burning stages and
systematically reduces the number of neutrinos emitted during the last day
before core collapse. We also show that this reduction can be estimated with
good accuracy just based on the baseline stellar model by adjusting the time-
integration of the neutrino luminosity. Our results suggest that this effect
may be used to constrain the dark photon parameter space if pre-SN neutrinos
are detected in the future. In a very small region of the parameter space we
also find that the effect of the extra cooling results in a slight increase of
the neutrino emission. In addition, for strong couplings unstable and
explosive O burning occurs. While the latter case may potentially also provide
constraints on the dark photon parameters, improved simulations are required
to better determine the final outcome and observational signatures.
Our paper is organized as follows. In §2 we describe the setup for our
calculations. In §3 we discuss the details of our fiducial model of the
evolution of a $15\,M_{\odot}$ star calculated without extra cooling. In §4 we
provide an overview of the relevant parameter space and discuss details of the
three types of effects that we find. In §5 we briefly outline the possibility
of deriving constraints from future observations of pre-SN neutrinos and
summarize our results.
## 2 Calculations
We implement a tabulation of the dark photon emission rates from Rrapaj et al.
(2019) in the stellar evolution and hydrodynamics code KEPLER (Weaver et al.,
1978; Woosley & Heger, 2007) and calculate the evolution of a $15\,M_{\odot}$
star with an initial composition of solar metallicity (Lodders & Palme, 2009).
The mass loss prescription is based on Nieuwenhuijzen & de Jager (1990) and
the mixing length for convection is equal to the pressure scale height. Semi-
convection is treated as described in Woosley & Weaver (1988), limiting the
convective diffusion coefficient to $10\%$ of the thermal value (see also
Woosley et al. 2002). Overshoot and thermohaline mixing (Kippenhahn et al.,
1980) are also included. Neutrino energy loss and the resulting total neutrino
luminosity are based on Itoh et al. (1996).
We evolve the models until the onset of core collapse, which we define as the
point when the infall velocity exceeds $5000\,\mathrm{km}/\mathrm{s}$. This
limiting value is higher than the value of $1000\,\mathrm{km}/\mathrm{s}$ used
in previous studies (Woosley & Heger, 2007; Woosley et al., 2002) because the
additional dark matter cooling tends to accelerate contraction.
Figure 1: Track of core temperature vs. density for our fiducial stellar model
without extra cooling. The time until core collapse is indicated at several
points. In the region above the dotted contours for two sets of $m_{A}$ and
$\varepsilon$, the corresponding dark photon emission dominates neutrino loss.
In order to explore the dark photon parameter space we have calculated more
than 470 stellar models and to limit the computational cost we only use an
approximate 19-isotope nuclear reaction network to calculate the energy
generation rates. At a temperature of 3.5 GK the code switches to solving for
the composition in quasi-statistical equilibrium (QSE) (Hix et al., 1998). For
even higher temperatures and when O is depleted full nuclear statistical
equilibrium (NSE) is assumed. Furthermore, we limit the number of zones in our
models to 2000.
## 3 Fiducial model
Figure 1 shows the track of the fiducial stellar model without extra cooling
in terms of central temperature $T_{\mathrm{C}}$ and density
$\rho_{\mathrm{C}}$. This track is in good agreement with the results from
Woosley & Heger (2007) and also with those from different stellar evolution
codes, such as MESA (Paxton et al., 2015). For the parameters that we are
studying, dark photon production is not relevant for temperatures below $\sim
1$ GK. The dotted contours in Figure 1 indicate where energy loss due to dark
photon emission equals the neutrino loss, marking the boundary of the
temperature and density regime where dark photon loss would dominate. This
boundary depends on the values of $m_{A}$ and $\varepsilon$. For instance,
lighter dark photons with stronger coupling strengths start to dominate at
lower temperatures. The details of these contours were discussed by Rrapaj et
al. (2019). As illustrated for $m_{A}=2\,\mathrm{MeV}$ and
$\varepsilon=10^{-11}$ in Figure 1, dark photons may already become important
right after central C depletion, which occurs around 64 years before core
collapse. Therefore, for context, we provide a short description of the
evolution after C depletion.
Figure 2: Kippenhahn diagram of the fiducial stellar model for the last 60
years before collapse. Hatched areas denote convective and semi-convective
regions that are indicative of nuclear burning. Blue (red) background color
indicates regions where cooling (nuclear energy generation) dominates.
The convective burning phases over the last 60 years are indicated in Figure
2, where hatched areas indicate convection and the color code shows the net
energy generation or loss. After C core and shell burning, Ne burning ignites
at the center at $T_{\mathrm{C}}=1.35$ GK, around 1500 days before collapse
under slightly degenerate conditions with a value of the degeneracy parameter
$\eta=\mu_{\mathrm{e}}/k_{\mathrm{B}}T\approx 4$, where $\mu_{\mathrm{e}}$ is
the chemical potential of electrons, $T$ the temperature and $k_{\mathrm{B}}$
the Boltzmann constant. The partial electron degeneracy leads to an initially
rapid rise of the temperature, followed by core expansion at almost constant
temperature. Once Ne is depleted, neutrino cooling leads to a decrease in
temperature before the core continues to contract and eventually heats up
again. This leads to the loop in the $T_{\mathrm{C}}$-$\rho_{\mathrm{C}}$
diagram shown in Figure 1. Ne burning is visible as the small hatched peak in
Figure 2, which also shows that this episode is very short.
About 650 days before collapse, O burning ignites centrally at
$T_{\mathrm{C}}=1.7$ GK, which leads to a rapid rise of the central
temperature and subsequent expansion. In contrast to the Ne-burning loop, at
the end of O burning in the core the temperature does not decrease and the
core is stabilized by shell burning. Once the shell burning ceases, neutrino
loss reduces $T_{\mathrm{C}}$ while the core contracts. At higher densities,
neutrino loss is suppressed and the track resumes its upward climb, while a
second O-burning shell is ignited. Between core O depletion and Si ignition,
He and C shell burning are still active.
Figure 3: Stellar profiles of our fiducial 15 $M_{\odot}$ model at core
collapse. The top panel shows temperature and density and the bottom panel
mass fractions for the most important isotopes.
Convective Si burning ignites at the center around 3.5 days before collapse,
after the second O-burning shell is extinguished. Around 5.6 hours before
collapse, central Si burning finishes and leaves a hot $1.1M_{\odot}$ Fe core
behind. This initial core grows further by shell burning that ignites about
one hour before collapse and continues until the core mass exceeds its
effective Chandrasekhar limit and collapses. Figure 3 shows the temperature,
density, and mass fraction profiles of the most important isotopes of our
fiducial model at core collapse. The He core encompasses 4.26 $M_{\odot}$, the
C/O core 3.02 $M_{\odot}$, the O/Ne core up to 2.70 $M_{\odot}$, and the Fe
core encompasses 1.60 $M_{\odot}$ at a radius of $1336$ km.
Multi-dimensional hydrodynamics simulations of the last minutes before core
collapse (Yadav et al., 2020; Yoshida et al., 2019; Müller et al., 2016) have
shown deviations from spherically symmetric models using mixing length theory,
but we do not expect our results to be qualitatively affected by these
differences.
Because the evolution of the stellar core is mostly decoupled from the surface
during the final stages, neutrinos are unique messengers that may provide
detailed information about the processes and conditions in the core shortly
before collapse (e.g., Guo & Qian, 2016; Kato et al., 2020). Current and near-
term neutrino detectors are expected to be able to detect the neutrinos from a
nearby massive star only within the last day before collapse (e.g., Guo et
al., 2019; Kato et al., 2020). The top panel of Figure 4 shows the neutrino
luminosity during the last 10 days before core collapse for our fiducial
model. In general, the luminosity increases as the star contracts and heats
up, following the track in Figure 1. The two peaks visible in Figure 4 are
caused by the ignition of nuclear burning that leads to expansion and cooling,
temporarily delaying collapse. The peak at around 3.5 days before collapse
corresponds to Si ignition at the center and the peak at about one hour before
collapse corresponds to the ignition of Si shell burning.
Here we focus on the $\bar{\nu}_{\mathrm{e}}$ for two reasons. Firstly, future
scintillation detectors are likely to observe the pre-SN
$\bar{\nu}_{\mathrm{e}}$, mainly by inverse beta decay (IBD,
$\bar{\nu}_{\mathrm{e}}+\mathrm{p}\rightarrow\mathrm{e}^{+}+\mathrm{n}$).
Secondly, towards the end of the life of a massive star
$\bar{\nu}_{\mathrm{e}}$ are mostly produced by electron-positron pair
annihilation. In contrast to the emission of $\nu_{\mathrm{e}}$, this thermal
process is relatively insensitive to the details of the stellar composition
and does not depend on the uncertainties related to electron capture on
nuclei. We calculate the spectral neutrino flux from pair annihilation as in
Guo & Qian (2016), using temperature, density, and electron fraction profiles
as functions of time from our stellar models. The $\bar{\nu}_{\mathrm{e}}$
luminosity due to pair annihilation alone is shown in the top panel of Figure
4 in comparison to the total neutrino luminosity. At one day before collapse,
$\bar{\nu}_{\mathrm{e}}$ constitute almost 30% of the total luminosity. This
fraction decreases towards collapse, as $\nu_{\mathrm{e}}$ from electron
captures become increasingly important. At $10\,\mathrm{s}$ before collapse,
however, $\bar{\nu}_{\mathrm{e}}$ from pair annihilation still account for
$10\,\%$ of the total neutrino luminosity.
Figure 4: Top panel: Neutrino luminosities during the last 10 days before
collapse from our fiducial model, without considering flavor transformations,
for all neutrino species and processes as well as for only
$\bar{\nu}_{\mathrm{e}}$ from pair annihilation. Bottom panel: Expected IBD
event rates at JUNO for $\bar{\nu}_{\mathrm{e}}$ from pair annihilation
assuming a distance of $500\,\mathrm{pc}$ to the source. The expected event
rate depends on the neutrino mass ordering (normal or inverted hierarchy). The
backgrounds from geo-neutrinos and nearby reactors are indicated by the
horizontal gray lines.
Based on the $\bar{\nu}_{\mathrm{e}}$ fluxes from the stellar evolution
models, we calculate the expected pre-SN neutrino signal at the Jiangmen
Underground Neutrino Observatory (JUNO) following Guo et al. (2019). With the
spectral number luminosity $\phi_{\bar{\nu}_{\mathrm{e}}}(E_{\nu},t)$ of the
star, the expected energy-differential event rate at a distance $d$ is
$\frac{d^{2}N}{dE_{\nu}dt}=\frac{1}{4\pi
d^{2}}\epsilon_{\mathrm{eff}}N_{p}\sigma_{\rm
IBD}(E_{\nu})\phi_{\bar{\nu}_{\mathrm{e}}}(E_{\nu},t),$ (1)
where $\epsilon_{\mathrm{eff}}=0.73$ is the detector efficiency and
$N_{p}=1.45\times 10^{33}$ is the number of protons based on 20 kt of active
detector material with a $12\,\%$ proton fraction (An et al., 2016). The IBD
cross-section $\sigma_{\rm IBD}(E_{\nu})$ is calculated as in Guo & Qian
(2016). For the event rate we integrate Equation (1) over
$E_{\nu}=1.8$–$4\,\mathrm{MeV}$, which is the optimal energy window for
detection. Due to flavor transformations the detection rate depends on the
neutrino mass ordering and the expected number of events for the normal
hierarchy is about 3.4 times higher than for the inverted hierarchy (e.g., Guo
et al., 2019). Figure 4 shows the expected event rates from our fiducial model
assuming a distance of $500\,\mathrm{pc}$. We also show the main sources of
background, geo-neutrinos and $\bar{\nu}_{\mathrm{e}}$ from nearby reactors.
For the inverted hierarchy about 15 events are expected during the last day,
which are approximately the same as the background. For the normal hierarchy,
however, about 50 events are expected during the last day, which are a factor
of $\sim 3$ more than the background. Therefore, a pre-SN neutrino signal may
be detected above the background during the final day before core collapse,
especially in the case of the normal mass hierarchy.
## 4 Results
We have calculated stellar models for a grid of the dark photon parameters
$m_{A}$ and $\varepsilon$ laid out in Figure 5. For
$3\,\mathrm{MeV}<m_{A}<10\,\mathrm{MeV}$ we have looked at increments of
$0.5\,\mathrm{MeV}$ and for $2\,m_{\mathrm{e}}<m_{A}\leq 3\,\mathrm{MeV}$ we
have taken smaller steps of $0.1\,\mathrm{MeV}$. For $\varepsilon$ we looked
at logarithmic intervals of $0.5\,\mathrm{dex}$ spanning values from
$10^{-13}$ to $10^{-6}$. For reference, the region of the parameter space that
is already excluded by the detection of the neutrinos from SN 1987A (Chang et
al., 2017) is indicated as the top gray shaded area in Figure 5. Note also
that the region corresponding to $m_{A}<2\,m_{\mathrm{e}}$ is not studied
here. The different symbols in Figure 5 indicate the outcomes of the models
that we have calculated.
Figure 5: Overview of the results for a grid of the dark photon parameters
$m_{A}$ and $\varepsilon$. The gray shaded area at the top indicates the
parameter space already excluded by the detection of the neutrinos from SN
1987A (Chang et al., 2017). Note also that we do not consider values of
$m_{A}<2\,m_{\mathrm{e}}$. The green squares mark models that are not
noticeably affected by the dark photon emission. The blue circles mark models
that exhibit a significant reduction of pre-SN neutrino emission (see §4.1),
whereas the red crosses mark the three cases with noticeably increased
neutrino emission (see §4.2). For the red shaded area in the top left corner,
runaway nuclear burning can be expected from simple arguments (see §4.3) but
full exploration is beyond the scope of this paper. The red stars indicate
models that feature such explosive behavior.
We find three qualitatively different ways in which the extra cooling changes
the late phases of stellar evolution and the last-day pre-SN neutrino signal.
The green squares in Figure 5 mark models that exhibit only negligible
deviations from the fiducial model. In contrast, the blue circles mark cases
in which the pre-SN neutrino emission is reduced by more than $30\,\%$ due to
accelerated Si burning. Part of the parameter space with the blue circles is
currently unconstrained, suggesting that future observations of pre-SN
neutrinos may impose new constraints on the dark photon properties.
Only three models with small dark photon masses, indicated by the red crosses
in Figure 5, exhibit a slight increase in the pre-SN neutrino emission. As we
will show in §4.2, this result is caused by an additional shell burning
episode that delays core collapse. For sufficiently small dark photon masses
and intermediate coupling strengths, the dark photon emission predominantly
originates not from the stellar center, but from the region of the final the
O/Ne shell.
For the parameters in the red shaded region in the upper left corner of Figure
5, the nuclear timescale exceeds the hydrodynamic timescale during O burning
and thermonuclear runaway is to be expected, as we will explain in §4.3. We
did not perform full calculations to cover this whole region due to the
complications associated with this explosive scenario. While the final fate of
the corresponding models requires further investigation beyond the scope of
this paper, we can speculate that all these cases may even lead to
thermonuclear SNe and the disruption of the whole star.
In general, energy loss determines the timescales of the advanced burning
stages in massive star evolution (Woosley & Heger, 2007). The balance between
energy generation due to nuclear burning and energy loss determines the
temperature of hydrostatic burning and thus the rates at which nuclear
reactions occur. Furthermore, energy loss determines how fast the stellar core
can contract because part of the resulting gain in gravitational binding
energy needs to be radiated away. Thus, emission of dark matter particles that
increases the energy loss rate generally leads to a speed-up of stellar
processes. The consequences of this acceleration, however, depend on where the
speed-up occurs and on whether the star can adjust sufficiently fast. Below we
give more detailed explanations for the various effects that we observe.
### 4.1 Reduced neutrino emission
As long as the stellar path to core collapse is guided by energy loss, an
additional cooling agent, such as dark photon emission, leads to a speed-up of
the evolution. Once the core becomes hot enough, the bulk of the energy loss
is taken over by the dark matter particles instead of the neutrinos, and the
accelerated evolution leaves less time for neutrino emission.
Figure 6: Kippenhahn diagram for the model with the parameters
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$. The onset of convective Si
burning in the core, indicated by the hatched region starting about 0.9 day
before collapse, is much closer to core collapse than in the fiducial model
without extra cooling shown in Figure 2.
As an example, Figure 6 shows the convective burning phases for a calculation
with the parameters $m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$.
Comparison with Figure 2 shows that the stellar structure remains largely the
same as in the fiducial model, but the evolution timescales change greatly.
The period between the onset of convective Si burning and collapse is only
around 0.9 day in Figure 6, compared to 3.5 days in the fiducial model (see
Figure 2). That between the end of convective Si burning at the center and
collapse is less than 3 hours, compared to 8 hours in the fiducial model. The
radial extent of the convective regions, on the other hand, remains almost
identical. The slight reduction of the extent of the convective zone due to
the more efficient energy loss, along with the shorter time for shell burning
due to the faster evolution of the core, results in a slightly smaller final
Fe core mass of $1.58\,M_{\odot}$, compared with $1.60\,M_{\odot}$ for the
fiducial model. With the remnant mass estimated by the point where the entropy
decreases below $4\,k_{\mathrm{B}}/$baryon (Woosley et al., 2002), dark photon
emission slightly reduces the potential neutron star mass from
$1.78\,M_{\odot}$ to $1.76\,M_{\odot}$.
For the above values of $m_{A}$ and $\varepsilon$, the overall changes to the
stellar structure relative to the fiducial model are minor. In contrast, the
integrated number of neutrinos emitted during the last day is already
decreased by a factor of two. This result shows that the main effect of the
additional cooling is a speed-up of Si burning and core contraction that
scales with the energy loss rate. This speed-up of the evolutionary clock
reduces the integrated neutrino emission with dark photons also contributing
to the energy loss. The effects on the $\bar{\nu}_{\mathrm{e}}$ luminosity
from pair annihilation and the associated signal are illustrated in Figure 7
for models with $m_{A}=2\,\mathrm{MeV}$ and a range of $\varepsilon$ values.
In addition to a suppression of the luminosity, some features of the signal
are shifted closer to the time of collapse. This shift is clearly visible for
the peak associated with Si shell burning, which appears at about one hour
before collapse for the fiducial model (see §3), but at only
$30\,\mathrm{minutes}$ and $12\,\mathrm{minutes}$ before collapse for
$\varepsilon=10^{-12}$ and $10^{-11.5}$, respectively. Note that for
$\varepsilon=10^{-11}$, the evolution of the $\bar{\nu}_{\mathrm{e}}$
luminosity becomes less smooth due to complications discussed in §4.3.
Figure 7: Evolution of $\bar{\nu}_{\mathrm{e}}$ luminosity from pair
annihilation and associated event rate as in Figure 4, but for models
including dark photons with $m_{A}=2\,\mathrm{MeV}$. The event rate is
calculated assuming the normal neutrino mass hierarchy and a distance of
$500\,\mathrm{pc}$. With increasing values of $\varepsilon$ the luminosity is
suppressed more and the peak associated with Si shell burning moves closer to
the time of collapse, indicating the acceleration of the evolution.
For all dark photon masses covered in our study, we find the same type of
reduction of the neutrino emission, and the necessary coupling strength to
achieve the same reduction increases with mass. In the following we show that
this reduction can be estimated semi-analytically, just based on the fiducial
stellar model. Without dark photon cooling, the timescales during the final
stages of evolution are determined mostly by the neutrino loss. We expect
$dt\propto L_{\nu}(t)^{-1}$, where $L_{\nu}(t)$ is the total neutrino
luminosity from the whole star at time $t$. With the additional dark photon
cooling, we expect
$dt^{\prime}=\frac{L_{\nu}(t)}{L_{\nu}(t)+L_{A}(t)}dt,$ (2)
where $L_{A}(t)$ is the dark photon luminosity from the whole star. This
change affects the results of any time-integrated quantity in two ways.
Firstly, the time measure itself is changed. Secondly, the effective
integration limits are changed. The time of core collapse, $t_{\mathrm{CC}}$,
marks the final time with a well-defined physical condition. In order to
accumulate a fixed time interval $\Delta t$, e.g., one day, up to
$t_{\mathrm{CC}}$, the starting point needs to be modified such that
$\Delta
t=\int\limits_{t_{0}}^{t_{\mathrm{CC}}}dt=\int\limits_{t_{0}^{\prime}}^{t_{\mathrm{CC}}}dt^{\prime}.$
(3)
With $dt^{\prime}<dt$ due to dark photon cooling, we have
$t_{0}^{\prime}<t_{0}$, so $\Delta t$ samples earlier stages of the evolution.
The above prescription allows us to estimate the energy emitted in
$\bar{\nu}_{\mathrm{e}}$ from pair annihilation during the last day before
collapse as
$E_{\mathrm{1day}}^{\prime}=\int\limits_{t_{0}^{\prime}}^{t_{\mathrm{CC}}}L_{\bar{\nu}_{\mathrm{e}},\mathrm{pair}}(t)\left(\frac{L_{\nu}(t)}{L_{\nu}(t)+L_{A}(t)}\right)dt,$
(4)
where $L_{\bar{\nu}_{\mathrm{e}},\mathrm{pair}}(t)$ is the
$\bar{\nu}_{\mathrm{e}}$ luminosity due to pair annihilation from the whole
star. Note that $L_{\nu}(t)$ and $L_{A}(t)$ are based on the temperature and
density profiles of the fiducial model without dark photon emission. This
approximation is valid as long as the effects of extra cooling on stellar
structure are negligible. In this case, the change of the
$\bar{\nu}_{\mathrm{e}}$ emission can be estimated semi-analytically without
computing new stellar models with the additional energy loss.
Figure 8: Energy carried away by $\bar{\nu}_{\mathrm{e}}$ over the last day
before collapse relative to the fiducial model. Semi-analytic estimates based
on Eq. (4) are displayed as solid curves and compared to the results from full
simulations (symbols connected with dotted line segments). The gray horizontal
bar indicates no change from the fiducial model.
Figure 8 displays the energy emitted in $\bar{\nu}_{\mathrm{e}}$ from pair
annihilation over the last day before core collapse, normalized to the
fiducial model, as a function of the dark photon coupling strength for a range
of $m_{A}$. The symbols show the results from full stellar evolution
calculations and demonstrate that increasing dark photon emission leads to a
reduction of the neutrino emission. The solid curves show the estimates based
on Eq. (4), which successfully describe the onset of the effects of dark
photons as well as the overall trend for many of the calculated models,
especially when the reduction of the neutrino emission is $\lesssim 50\%$.
Deviations from the actual calculations are expected when the dark photon
cooling starts to change the temperature and density profiles for sufficiently
large values of $\varepsilon$. For a large part of the dark photon parameter
space, however, the main effect is the relatively straightforward reduction of
the neutrino emission without significant changes to the stellar structure as
discussed above. Those cases are indicated by the blue circles in Figure 5 and
cover an unconstrained region of the parameter space.
Figure 8 also shows large deviations of the estimates from the actual
calculations for $m_{A}=1.2$ MeV. As we will explain in §4.2, these deviations
are due to changes of the shell-burning evolution.
### 4.2 Increased neutrino emission
Figure 9: Kippenhahn diagram for the model with the parameters
$m_{A}=1.2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$. At 0.25 days before
collapse, an additional convective O burning shell ignites between the
enclosed mass coordinates of $1.6\,M_{\odot}$ and $2\,M_{\odot}$, which delays
core collapse and allows more time for neutrino emission.
As discussed in §4.1, the reduction of the pre-SN neutrino emission results
almost entirely from the modification of the timescales of core burning and
contraction. Below we show that the deviations from this explanation for low
$m_{A}$ (see Figure 8) are due to the appearance of shell burning phases with
a stabilizing effect, which leads to the cases of the increased neutrino
emission marked by the red crosses in Figure 5.
For the dark photon parameters under our consideration, only three models with
$m_{A}=2\,m_{\mathrm{e}}$, $1.1\,\mathrm{MeV}$, and $1.2\,\mathrm{MeV}$, all
with $\varepsilon=10^{-12}$, have an increase of $\lesssim 50\%$ in the pre-SN
neutrino emission relative to the fiducial model. The comparison of the
neutrino emission with the fiducial model is shown for the case of
$m_{A}=1.2\,\mathrm{MeV}$ in Figure 8. The Kippenhahn diagram for this case is
shown in Figure 9, which reveals an additional episode of convective O shell
burning between an enclosed mass of $1.6\,M_{\odot}$ and $2\,M_{\odot}$. This
episode starts at $0.25\,\mathrm{days}$ before collapse, following the end of
core Si burning, but before the onset of Si shell burning. Such an episode
occurs neither in the fiducial model nor in the model with the reduced
neutrino emission shown in Figure 6. Heating by this additional O-shell
burning phase relieves the pressure on the Si-core and delays the ignition of
Si-shell burning and hence core collapse. This delay gives the star more time,
thereby increasing the total number of pre-SN neutrinos emitted.
For $m_{A}=1.2\,\mathrm{MeV}$, the additional O-shell burning phase also
occurs for larger values of the dark photon coupling, but it cannot compensate
for the effects of the accelerated burning and core contraction. Consequently,
the integrated neutrino emission for such cases is still reduced (see Figure
8), although not by as much as estimated from Eq. (4), which does not take
into account the additional O-shell burning. We can only find a net increase
of the pre-SN neutrino emission for a very narrow range of $\varepsilon$,
which is strong enough to cause an additional burning episode, but weak enough
to limit the acceleration of the evolution.
In order to understand the apparent association of the additional burning
episodes with the lightest dark photons in our study, we look at the time-
integrated energy loss as a function of the enclosed mass $M_{r}$ inside
radius $r$,
$E_{\mathrm{loss},x}(M_{r})=\int\limits_{t_{0}}^{t_{CC}}\dot{q}_{x}(M_{r},t)dt,$
(5)
where $x$ is either $\nu$ for the neutrino with the energy loss rate
$\dot{q}_{\nu}(M_{r},t)$ or $A$ for the dark photon with the energy loss rate
$\dot{q}_{A}(M_{r},t)$, and $t_{0}$ is the time of C ignition when the energy
loss due to neutrinos and dark photons becomes relevant. Figure 10 shows the
ratio
$R_{A/\nu}(M_{r})=E_{\mathrm{loss,A}}(M_{r})/E_{\mathrm{loss},\nu}(M_{r})$
relative to the value for the central zone $R_{A/\nu}(M_{r,0})$ based on the
fiducial model. Due to the scaling of $R_{A/\nu}(M_{r})$ with $\varepsilon$,
the quantity $R_{A/\nu}(M_{r})/R_{A/\nu}(M_{r,0})$ is independent of
$\varepsilon$. It is, however, quite sensitive to $m_{A}$.
Figure 10: Ratio of dark photon to neutrino energy loss rates
$R_{A/\nu}(M_{r})$ (normalized to the central zone) as a function of the
enclosed mass for the fiducial stellar model. The background colors indicate
the shells with different compositions shown in Figure 3. For
$m_{A}<2\,\mathrm{MeV}$, the strongest dark photon emission originates not
from the Fe core, but from the Si and O/Ne shells due to the suppression at
high densities.
For masses above $2\,\mathrm{MeV}$ the emission of dark photons dominates that
of neutrinos mostly in the final Fe core because the temperature is generally
higher at smaller radii. For masses below $2\,\mathrm{MeV}$, however, the dark
photon emission dominates further outside the core. This different behavior
for the lower values of $m_{A}$ can be understood as follows. Because the
number of $e^{\pm}$ pairs is reduced in degenerate conditions, dark photon
production by pair annihilation is suppressed at high densities in the core
(see Rrapaj et al. 2019 for details). This suppression does not occur in the
region at lower densities outside the core, but the temperature of this region
can only facilitate significant emission of the lower-mass dark photons.
Therefore, shell burning phases are accelerated and additional burning
episodes with stabilizing effects as described above tend to occur for
$m_{A}<2\,\mathrm{MeV}$. For heavier dark photons, their production requires
energetic $e^{\pm}$ pairs that can only be provided by the final core burning
stages. Therefore, the assumption that the effect of dark photon cooling is
limited to the core burning regions is justified for $m_{A}>2\,\mathrm{MeV}$,
but the evolution of the outer regions is noticeably affected for smaller dark
photon masses.
The stabilizing effect of additional shell burning phases due to the
accelerated evolution outside the core cannot be easily captured by a semi-
analytical prescription like the one presented in §4.1. Detailed stellar
evolution calculations are essential to the discovery and understanding of
this effect.
### 4.3 Runaway O Burning
Figure 11: Tracks of central temperature (upper panels) and degeneracy
parameter $\eta$ (bottom panels) vs. central density for models with
$m_{A}=2\,\mathrm{MeV}$ and different values of $\varepsilon$. For reference,
the fiducial model is shown in gray in all the panels.
The stellar models with those dark photon parameters indicated by the red
stars in Figure 5 exhibit a thermonuclear runaway during O burning. This
result may be understood as follows. With the dark photon cooling, the
accelerated burning timescales get shorter than the convective and eventually
also the hydrodynamic timescales. In addition, the increased electron
degeneracy further prevents the star from adjusting its structure in response
to the nuclear energy release. Consequently, the central temperature rises
rapidly to $\sim 5$ GK and O is exhausted in a convective region covering less
than the innermost $0.1M_{\odot}$ of material. Afterwards, O burning
propagates outward in a thin shell as a deflagration front. Figure 11 shows
the evolutionary tracks of $T_{\mathrm{C}}$ and $\rho_{\mathrm{C}}$ for
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$, $10^{-11}$, $10^{-10.5}$,
and $10^{-10}$, respectively, in comparison with the fiducial model. The
degeneracy parameter $\eta$ is also shown in the bottom panels of this figure.
The onset of the runaway can be seen clearly for the largest two values of
$\varepsilon$: $T_{\mathrm{C}}$ almost vertically increases to $\sim
5\,\mathrm{GK}$ when we stopped the calculations due to the required very
small time steps.
Compared to the fiducial model, the tracks for $\varepsilon=10^{-12}$ in
Figure 11 do not show much deviation, but the density tends to be higher for
the same temperature, thereby increasing the degeneracy. While the track of
$T_{\mathrm{C}}$ vs. $\rho_{\mathrm{C}}$ for the fiducial model already
exhibits a loop, which arises from burning under degenerate conditions (see
also Figure 1), the track for $\varepsilon=10^{-11}$ shows two such loops and
several backward kinks resulting from shell flashes, signifying ignition of
shell burning under degenerate conditions. For $\varepsilon=10^{-10.5}$, O
burning becomes unstable and the central temperature explosively increases up
to $\sim 5$ GK. It can be seen from Figure 11 that in this case the core
expands somewhat initially, but the decrease of density fails to stop the
runaway heating. As $T_{\mathrm{C}}$ increases rapidly, convection also
quickly becomes inefficient to distribute the energy released from the nuclear
reactions. The detailed evolution in this case is discussed in Appendix A. We
find the same type of behavior as in this case for even larger values of
$\varepsilon$. For example, for $\varepsilon=10^{-10}$, the density response
to the rise in temperature due to O burning is barely visible in Figure 11,
indicating a further acceleration of the runaway process.
While the runaway burning described above results from the interplay of
several factors, including the competition of burning timescales with
convective and hydrodynamic timescales, as well as partial electron
degeneracy, below we present a simple argument for the association of a
thermonuclear runaway with large values of $\varepsilon$ and provide a
conservative estimate for the dark photon luminosity (per unit mass) at which
hydrostatic burning is no longer possible.
In configurations of stable nuclear burning, the energy release from nuclear
reactions is compensated by energy losses, leading to a stable consumption of
the nuclear fuel. For advanced burning phases without dark photons, the
temperature for hydrostatic burning can be estimated by equating the neutrino
energy loss and nuclear energy release (Woosley et al., 2002). Assuming
$\rho=10^{6}\left(T/\rm{GK}\right)^{3}\ \rm{g/cm}^{3}$ (6)
for O burning, the intersection of the O-burning energy generation and the
neutrino loss rate gives the typical O-burning temperature of $\approx 1.8$ GK
(see blue curves in Figure 12).
The additional energy loss due to dark photons changes the temperature for
hydrostatic O burning. Due to the very steep temperature dependence of the
nuclear reaction rate, a small change of temperature is sufficient to
compensate for a large increase of the energy loss. As shown in Figure 12, a
factor of $\approx 100$ increase of the energy loss for
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-11}$ only slightly increases the
nominal O-burning temperature to $\approx 2\,\mathrm{GK}$. Nevertheless, this
temperature increase is responsible for the speed-up of the evolution
discussed in §4.1. Of course, the burning temperature must increase
substantially for strong dark photon couplings. For example, it reaches
$\approx 3\,\mathrm{GK}$ for $\varepsilon=10^{-9}$ (see Figure 12).
Complications arise if the timescale $\tau_{\mathrm{n}}$ of nuclear burning
becomes shorter than the hydrodynamic timescale on which stellar structure can
adjust to the temperature change. We take the latter timescale to be the free-
fall timescale
$\tau_{\mathrm{ff}}=\frac{1}{\sqrt{G\rho}},$ (7)
where the density $\rho$ can be estimated from the temperature assuming Eq.
(6). The timescale of nuclear burning can be estimated as
$\tau_{\mathrm{n}}=XQ/\dot{q}_{\mathrm{nuc}},$ (8)
where $Q=5\times 10^{17}$ erg/g (Woosley et al., 2002) is the effective energy
release, $X=0.7$ is the mass fraction of the 16O fuel, and
$\dot{q}_{\mathrm{nuc}}$ is the specific energy generation rate per unit mass.
For O burning, $\tau_{\rm{n}}=\tau_{\rm{ff}}$ occurs at a critical specific
energy generation rate of
$\dot{q}_{\mathrm{nuc,crit}}\approx 9\times 10^{16}(T/{\rm GK})^{3/2}\
\rm{erg/g/s}.$ (9)
The contour for $\tau_{\rm{n}}=\tau_{\rm{ff}}$ is shown in Figure 12. It
intersects the curve for the energy release due to O burning at $T\approx 3$
GK and $\dot{q}_{\rm{nuc,crit}}\approx 4.6\times 10^{17}\,\rm{erg/g/s}$. At
this temperature and the corresponding density from Eq. (6), if the specific
dark photon energy loss rate exceeds $\dot{q}_{\rm{nuc,crit}}$, i.e.,
$\dot{q}_{A}(T=3\,\rm{GK})>4.6\times 10^{17}\ \rm{erg/g/s},$ (10)
a thermonuclear runaway is expected. As shown in Figure 12, this result occurs
for $m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-9}$.
Figure 12: Comparison of the specific energy generation rate for O burning
with the loss rates for neutrinos and for dark photons with
$m_{A}=2\,\mathrm{MeV}$. Equating the energy generation and loss rates gives
the nominal nuclear burning temperatures. For the red shaded area at the top,
the nuclear burning timescale $\tau_{\rm n}$ is shorter than the free-fall
timescale $\tau_{\mathrm{ff}}$, and a thermonuclear runaway is expected. See
text for detail.
The red shaded region in Figure 5 indicates the general parameter space of
$m_{A}$ and $\varepsilon$ for which the condition in Eq. (10) is fulfilled.
While the boundary of the region roughly reflects the appearance of the red
stars indicating runaway O burning, the stellar models already exhibit such
burning for much weaker couplings for $m_{A}\lesssim 2$ MeV. This result may
be explained by the complications due to electron degeneracy, which tends to
prevent the star from reacting to nuclear energy release, and due to the
competition of nuclear burning with convection. Both these complications are
ignored in the simple picture assumed for deriving Eq. (10). At O ignition,
our fiducial model already exhibits some degree of degeneracy with $\eta\sim
2$ (see Figure 11). When dark photon cooling is included, the temperature of
stable burning increases, which requires the stellar core to contract to a
higher density. Because degeneracy rises more steeply with density than the
temperature, values up to $\eta\sim 8$ is reached in the core at O-burning
temperatures when O burning becomes unstable. As the adjustment of the core is
slowed down, the O burning luminosity exceeds the maximum luminosity that can
be transported by convection (see Appendix A), which also favors a local
runaway.
The boundary of the red shaded region in Figure 5 reflects the dependence of
the specific dark photon energy loss rate on $m_{A}$ and $\varepsilon$. It
also tends to predict too small threshold values of $\varepsilon$ for runaway
burning for higher values of $m_{A}$. Specifically, for $(m_{A}/{\rm
MeV},\varepsilon)=(4,10^{-7})$, $(4.5,10^{-6.5})$, $(4.5,10^{-6})$, and
$(5,10^{-6})$ that are above the boundary, although O burning leads to a rapid
rise in temperature, the dark photon energy loss increases fast enough to
avoid a thermonuclear runaway.
## 5 Discussion and conclusions
We have studied the impact of extra cooling due to beyond SM particles on the
evolution of a $15\,M_{\odot}$ star. Specifically, we have implemented the
dark photon emission from $e^{\pm}$ pair annihilation in the stellar evolution
code KEPLER, assuming that these particles decay into other dark sector
components, thereby representing an additional mechanism of energy loss for
the star. We have considered dark photon masses of $m_{A}=2\,m_{\mathrm{e}}$
to 10 MeV and couplings of $\varepsilon=10^{-13}$ to $10^{-6}$, and found that
the dark photon can affect the O and Si burning phases. There are three types
of potentially observable effects, which are summarized in Figure 5.
Figure 13: Constraints on dark photon parameters. The light gray region would
be excluded if more than 17 pre-SN neutrinos were detected from a $15\
M_{\odot}$ star at a distance of $500\,\mathrm{pc}$ (the case of
$m_{A}<2\,m_{\mathrm{e}}$ is not considered). The dark gray regions have been
excluded by big bang nucleosynthesis (BBN, Fradette et al. 2014) and the
detection of the neutrinos from SN 1987A (Sung et al., 2019).
For broad ranges of $m_{A}$ and $\varepsilon$, the extra cooling allows the
star to contract faster and increases the temperature at which nuclear burning
proceeds in equilibrium with energy losses. These effects speed up the burning
processes and reduce the number of neutrinos emitted during the last day
before core collapse. We have developed a semi-analytical approach that
describes the reduction of the pre-SN neutrino emission for the relevant
ranges of $m_{A}$ and $\varepsilon$ (see §4.1).
We further find that, due to the density dependence of the emission process,
dark photons with $m_{A}<2\,\mathrm{MeV}$ are also produced outside the core,
which causes additional shell burning episodes, thereby increasing the pre-SN
neutrino emission for a very narrow range of parameters (see §4.2).
Because extra cooling increases the nominal nuclear burning temperatures,
sufficiently strong dark photon couplings are expected to produce a
thermonuclear runaway (see §4.3). We have found many cases of runaway O
burning in our models and speculate that thermonuclear SNe may be the outcome
in these cases (see Appendix A). Finding the actual outcome and possible
observables in these cases, however, requires more refined simulations that
can properly follow the propagation of narrow nuclear burning fronts. Were
dark photon cooling to cause complete disruption of a star, the relevant
parameter space may be constrained by the observed inventory of neutron stars
and stellar black holes, which requires most massive stars to undergo core-
collapse SNe.
The first type of effect, i.e., the suppression of the pre-SN neutrino
emission by dark photon cooling, may be constrained by the detection of such
neutrinos. In the most conservative approach, this suppression means that part
of the parameter space is only consistent with a non-detection of pre-SN
neutrinos. Therefore, any positive detection of a pre-SN neutrino signal from
a well-understood progenitor excludes this part of the dark photon parameter
space. For illustration, we take our $15\,M_{\odot}$ model to explode as a
core-collapse SN at a distance of $500\,\mathrm{pc}$. Assuming the normal
neutrino mass hierarchy, we can find the values of ($m_{A},\varepsilon$) that
reduce the pre-SN neutrino emission sufficiently to give an expected number of
events less than the background count of about $17$ events, in contrast to
about 50 events without dark photon cooling. The corresponding parameter space
is bounded by the red curve in Figure 13. The detection a number of events
significantly above the background level would exclude this part of the
parameter space, if the progenitor is similar to the model we have studied
here. Note that a small fraction of this parameter space may lead to a
thermonuclear runaway, and therefore, could be constrained by the observation
of a core-collapse rather than thermonuclear SN.
Pre-SN neutrinos have not been detected yet. Our results, however, indicate
that their observation in the future may offer a unique probe of not only the
dark photon but also other dark matter particles that are efficiently produced
in the temperature regime of $\sim 1$–$10\,\mathrm{GK}$ during stellar
evolution.
This work was supported in part by the US Department of Energy [DE-
FG02-87ER40328 (UM)]. Calculations were carried out at the Minnesota
Supercomputing Institute. ER was supported by the NSF (PHY-1630782) and the
Heising-Simons Foundation (2017-228). GG acknowledges support from the
Academia Sinica by Grant No. AS-CDA-109-M11. We thank Alexander Heger for
providing access to the KEPLER code.
## Appendix A A model with Runaway O Burning
Figure 14: Snapshot profiles of the stellar core for the model with
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-10.5}$. Time progresses from
left to right and convective regions are shown as red shaded areas. The top
row shows the temperature and density profiles, the second row shows the
nuclear energy release and combined neutrino and dark photon loss, the third
row shows the mass fractions of the key isotopes, and the bottom row shows the
velocity profile, where positive (outward) velocities are displayed as solid
curves and negative (infall) velocities as dashed curves. Panel (a) shows the
beginning of O burning when the energy release from nuclear reactions just
exceeds the combined neutrino and dark photon loss. Panel (b) shows a snapshot
275 s after panel (a), when the energy generation at the center starts to
decrease as O is exhausted. The high luminosity cannot be distributed
efficiently by convection and only the region of the inner 300 km finishes O
burning. Panel (c) is 82 s after panel (b) and shows that the burning is
compressed into a narrow burning front, where the temperature jumps from
$3\,\mathrm{GK}$ to $5\,\mathrm{GK}$.
In order to understand the phenomenon of runaway O burning, we investigate the
model for $m_{A}=2$ MeV and $\varepsilon=10^{-10.5}$ in more detail. Figure 14
shows a sequence of snapshots from the evolution of this model. The top row
shows the temperature and density profiles, the second row from the top shows
the nuclear energy release and combined energy loss due to neutrinos and dark
photons, the third row shows the mass fractions of the major isotopes to
illustrate the progress of nuclear burning, and the bottom row shows the
velocity profile, where positive (outward) velocities are shown as solid
curves and negative (infall) velocities are represented by dashed curves. The
red shaded regions are convective zones. Time runs from panel (a) on the left
to panel (c) on the right, spanning $357\,\mathrm{s}$. Panel (a) shows the
onset of O burning, which is confined to a convective region reaching up to
1000 km and encompassing about 0.1 $M_{\odot}$ of material. At $T\approx
2\,\mathrm{GK}$ and $\rho\approx 8\times 10^{7}\,\mathrm{g/cm}^{3}$, the
energy release from nuclear reactions exceeds the combined neutrino and dark
photon loss. In response to the excess heat, positive velocities develop,
indicating the onset of core expansion. Panel (b), however, shows that O
burning proceeds much faster than the adjustment of stellar structure. The
fuel in the center is almost depleted and the temperature has risen to
$4.7\,\mathrm{GK}$ while the density has only been reduced by about $40\,\%$
compared to panel (a).
At a temperature above $4\,\mathrm{GK}$, the luminosity due to O burning
exceeds the maximum value that can be transported by convection (Woosley &
Heger, 2015)
$L_{\mathrm{max}}\approx 4\pi r^{2}\rho v_{\mathrm{conv}}fC_{\mathrm{P}}T,$
(A1)
where $v_{\mathrm{conv}}$ is the convective velocity, $C_{\mathrm{P}}T$ is the
thermal energy content, and $f\ll 1$ indicates the efficiency of convection to
remove the internal energy. With $f=0.1$ we find
$L_{\mathrm{max}}<10^{50}\mathrm{erg/s}$, corresponding to an energy
generation rate of $\dot{q}_{\mathrm{max}}\approx 5\times
10^{17}\,\mathrm{erg/g/s}$ for the conditions of O burning assuming an initial
core mass of $0.1\,M_{\odot}$. The nuclear energy release in panel (b) exceeds
this luminosity already by four orders of magnitude and convection cannot
distribute the heat and the fuel throughout the convective region on the
burning timescale. Therefore, O is only depleted in the innermost
$300\,\mathrm{km}$ despite that the convective zone reaches out to
$2000\,\mathrm{km}$.
Without efficient convection, the very rapid burning gets confined into a
narrow burning front at the bottom of the nominally convective layer, as shown
in panel (c) of Figure 14. The change of composition and continuing energy
generation also show that Si burning immediately follows because of the high
temperature of the central region. In our model, the burning region consists
only of a few zones and the fuel in one zone is consumed before the zone above
it ignites. The propagation of the burning depends critically on the heat
transport and would need to be described by a model for flame propagation. For
a very narrow burning front, multi-dimensional effects are important for the
propagation of such a deflagration Fryxell & Woosley (1982) and the outcome is
highly sensitive to turbulence and mixing at the boundary of the burning front
(Jones et al., 2013, 2016). The bottom row of Figure 14 also shows that
significant outward velocities develop but remain subsonic. The above
situation cannot be adequately simulated by our model due to the lack of
resolution and the employed small reaction network and mixing-length treatment
of convection. Therefore, our results on the final outcome are speculative.
We have followed the calculation of our model further and find that the shell
burning steepens into a supersonic shock once it reaches the steep density
gradient at the upper edge of the convective zone around the radius of 2000
km, where Ne burning also provides additional energy. The outward velocity
reaches several 1000 km/s, which takes the pressure off the core, eventually
facilitating a rapid expansion and cooling. The $T_{\mathrm{C}}$ drops below
0.1 GK and $\rho_{\mathrm{C}}$ below $10^{5}$ g/cm3. When the shock reaches
the H envelope we find that the kinetic energy of $3.6\times 10^{50}$ erg
exceeds the gravitational binding energy of $1.4\times 10^{50}$ erg for the
star. The shock may thus disrupt the whole star or at least unbind a
significant fraction of the H envelope and lead in either case to an optical
transient. Even if the core eventually collapses, the runaway O burning would
delay the collapse by many hours up to days. Here we caution again that our
model is probably inadequate to describe this situation accurately and a
better treatment is needed to predict potentially observable signatures.
## References
* Aaboud et al. (2019) Aaboud, M., et al. 2019, JHEP, 05, 142, doi: 10.1007/JHEP05(2019)142
* Alexander et al. (2016) Alexander, J., et al. 2016, in Dark Sectors 2016 Workshop: Community Report
* An et al. (2016) An, F., An, G., An, Q., et al. 2016, Journal of Physics G Nuclear Physics, 43, 030401, doi: 10.1088/0954-3899/43/3/030401
* An et al. (2013) An, H., Pospelov, M., & Pradler, J. 2013, Physics Letters B, 725, 190, doi: 10.1016/j.physletb.2013.07.008
* Batell et al. (2014a) Batell, B., deNiverville, P., McKeen, D., Pospelov, M., & Ritz, A. 2014a, Phys. Rev., D90, 115014, doi: 10.1103/PhysRevD.90.115014
* Batell et al. (2014b) Batell, B., Essig, R., & Surujon, Z. 2014b, Phys. Rev. Lett., 113, 171802, doi: 10.1103/PhysRevLett.113.171802
* Chang et al. (2017) Chang, J. H., Essig, R., & McDermott, S. D. 2017, JHEP, 01, 107, doi: 10.1007/JHEP01(2017)107
* Choplin et al. (2017) Choplin, A., Coc, A., Meynet, G., et al. 2017, A&A, 605, A106, doi: 10.1051/0004-6361/201731040
* DeRocco et al. (2019) DeRocco, W., Graham, P. W., Kasen, D., Marques-Tavares, G., & Rajendran, S. 2019, Journal of High Energy Physics, 2019, 171, doi: 10.1007/JHEP02(2019)171
* Essig et al. (2013) Essig, R., Mardon, J., Papucci, M., Volansky, T., & Zhong, Y.-M. 2013, Journal of High Energy Physics, 2013, 167, doi: 10.1007/JHEP11(2013)167
* Fradette et al. (2014) Fradette, A., Pospelov, M., Pradler, J., & Ritz, A. 2014, Phys. Rev. D, 90, 035022, doi: 10.1103/PhysRevD.90.035022
* Fryxell & Woosley (1982) Fryxell, B. A., & Woosley, S. E. 1982, ApJ, 261, 332, doi: 10.1086/160344
* Guo & Qian (2016) Guo, G., & Qian, Y.-Z. 2016, Phys. Rev. D, 94, 043005, doi: 10.1103/PhysRevD.94.043005
* Guo et al. (2019) Guo, G., Qian, Y.-Z., & Heger, A. 2019, Physics Letters B, 796, 126, doi: 10.1016/j.physletb.2019.07.030
* Hardy & Lasenby (2017) Hardy, E., & Lasenby, R. 2017, JHEP, 02, 033, doi: 10.1007/JHEP02(2017)033
* Hix et al. (1998) Hix, W. R., Khokhlov, A. M., Wheeler, J. C., & Thielemann, F.-K. 1998, ApJ, 503, 332, doi: 10.1086/305968
* Holdom (1986) Holdom, B. 1986, Physics Letters B, 166, 196 , doi: http://dx.doi.org/10.1016/0370-2693(86)91377-8
* Itoh et al. (1996) Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996, ApJ, 102, 411, doi: 10.1086/192264
* Jones et al. (2016) Jones, S., Röpke, F. K., Pakmor, R., et al. 2016, A&A, 593, A72, doi: 10.1051/0004-6361/201628321
* Jones et al. (2013) Jones, S., Hirschi, R., Nomoto, K., et al. 2013, ApJ, 772, 150, doi: 10.1088/0004-637X/772/2/150
* Kato et al. (2020) Kato, C., Ishidoshiro, K., & Yoshida, T. 2020, Annual Review of Nuclear and Particle Science, 70, annurev, doi: 10.1146/annurev-nucl-040620-021320
* Kippenhahn et al. (1980) Kippenhahn, R., Ruschenplatt, G., & Thomas, H. C. 1980, A&A, 91, 175
* Lodders & Palme (2009) Lodders, K., & Palme, H. 2009, Meteoritics and Planetary Science Supplement, 72, 5154
* Müller et al. (2016) Müller, B., Viallet, M., Heger, A., & Janka, H.-T. 2016, ApJ, 833, 124, doi: 10.3847/1538-4357/833/1/124
* Nelson & Tetradis (1989) Nelson, A. E., & Tetradis, N. 1989, Phys. Lett., B221, 80, doi: 10.1016/0370-2693(89)90196-2
* Nieuwenhuijzen & de Jager (1990) Nieuwenhuijzen, H., & de Jager, C. 1990, A&A, 231, 134
* Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, The Astrophysical Journal Supplement Series, 220, 15, doi: 10.1088/0067-0049/220/1/15
* Raffelt (1996) Raffelt, G. 1996, Stars as laboratories for fundamental physics: The astrophysics of neutrinos, axions, and other weakly interacting particles (University of Chicago Press), 1–664
* Raffelt (1990) Raffelt, G. G. 1990, Phys. Rept., 198, 1, doi: 10.1016/0370-1573(90)90054-6
* Raffelt & Dearborn (1987) Raffelt, G. G., & Dearborn, D. S. P. 1987, Phys. Rev. D, 36, 2211, doi: 10.1103/PhysRevD.36.2211
* Rajpoot (1989) Rajpoot, S. 1989, Phys. Rev. D, 40, 2421, doi: 10.1103/PhysRevD.40.2421
* Rrapaj & Reddy (2016) Rrapaj, E., & Reddy, S. 2016, Phys. Rev., C94, 045805, doi: 10.1103/PhysRevC.94.045805
* Rrapaj et al. (2019) Rrapaj, E., Sieverding, A., & Qian, Y.-Z. 2019, Phys. Rev. D, 100, 023009, doi: 10.1103/PhysRevD.100.023009
* Sung et al. (2019) Sung, A., Tu, H., & Wu, M.-R. 2019, Phys. Rev. D, 99, 121305, doi: 10.1103/PhysRevD.99.121305
* Tanabashi et al. (2018) Tanabashi, M., Hagiwara, K., Hikasa, K., et al. 2018, Phys. Rev. D, 98, 030001, doi: 10.1103/PhysRevD.98.030001
* Weaver et al. (1978) Weaver, T. A., Zimmerman, G. B., & Woosley, S. E. 1978, ApJ, 225, 1021
* Woosley & Heger (2007) Woosley, S., & Heger, A. 2007, Physics Reports, 442, 269 , doi: https://doi.org/10.1016/j.physrep.2007.02.009
* Woosley & Heger (2015) Woosley, S. E., & Heger, A. 2015, ApJ, 810, 34, doi: 10.1088/0004-637X/810/1/34
* Woosley et al. (2002) Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Reviews of Modern Physics, 74, 1015, doi: 10.1103/RevModPhys.74.1015
* Woosley & Weaver (1988) Woosley, S. E., & Weaver, T. A. 1988, Phys. Rep., 163, 79, doi: 10.1016/0370-1573(88)90037-3
* Yadav et al. (2020) Yadav, N., Müller, B., Janka, H. T., Melson, T., & Heger, A. 2020, ApJ, 890, 94, doi: 10.3847/1538-4357/ab66bb
* Yoshida et al. (2019) Yoshida, T., Takiwaki, T., Kotake, K., et al. 2019, ApJ, 881, 16, doi: 10.3847/1538-4357/ab2b9d
|
# Conceptualization and cases of study on cyber operations against the
sustainability of the tactical edge
Marco Antonio Sotelo Monge<EMAIL_ADDRESS>University of MurciaFaculty of
Computer ScienceCampus de Espinardo s/nMurcia, Spain30100 and Jorge Maestre
Vidal<EMAIL_ADDRESS>IndraDigital LabsAvenida de Bruselas, 35Alcobendas,
Madrid, Spain28108
###### Abstract.
The last decade consolidated the cyberspace as fifth domain of operations,
which extends its preliminarily intelligence and information exchange purposes
towards enabling complex offensive and defensive operations
supported/supportively of parallel kinetic domain actuations. Although there
is a plethora of well documented cases on strategic and operational
interventions of cyber commands, the cyber tactical military edge is still a
challenge, where cyber fires barely integrate to the traditional joint
targeting cycle due among others to long planning/development times,
asymmetric effects, strict target reachability requirements, or the fast
propagation of collateral damage; the latter rapidly deriving on hybrid
impacts (political, economic, social, etc.) and evidencing significant socio-
technical gaps. In this context, it is expected that tactical clouds
disruptively facilitate cyber operations at the edge while exposing the rest
of the digital assets of the operation to them. On these grounds, the main
purpose of the conducted research is to review and in depth analyze the risks
and opportunities of jeopardizing the sustainability of the military tactical
clouds at the edge by cyber operations. Along with a 1) comprehensively
formulation of the researched problematic, the study 2) formalizes the
Tactical Denial of Sustainability (TDoS) concept; 3) introduces the phasing,
potential attack surfaces, terrains and impact of TDoS attacks; 4) emphasizes
the related human and socio-technical aspects; 5) analyzes the
threats/opportunities inherent to their impact on the cloud energy efficiency;
6) reviews their implications at the military cyber thinking for tactical
operations; 7) illustrates five extensive CONOPS that facilitate the
understanding of the TDoS concept; and given the high novelty of the discussed
topics, it 8) paves the way for further research and development actions.
Cyber Defence, Economical Denial of Sustainability, Military operations,
Situational Awareness, Tactical Denial of Sustainability
††copyright: none††ccs: Security and privacy Network security††ccs: Networks
Network architectures
## 1\. Introduction
The increasingly digitalization of the military sector is settling the socio-
technical foundations towards translating an information advantage, enabled in
part by network-centric and fusion warfare paradigms, into a competitive
advantage on the joint theatres of operations, secondly supporting the
provisioning of intelligence, ISTAR and C2 capabilities thorough the military
edge (Hoeben, 2017). Characterized as NATO C3 (NATO, 2019) and beyond
services, the cyber tactical capabilities may range from logistic,
surveillance and reconnaissance, to distributed and federated light
algorithmic able to bring automatism as support for decision making to kinetic
tactical effectors. But the inventory, provisioning, operation, upkeep and
removal of these services is highly technological and data dependant,
typically requiring costly data management and distribution procedures (Hui et
al., 2017) hardly affordable on Beyond Line of Sight (BLOS) operational
scenarios. In this context, growing concepts like Cloud Computing (CC),
Software-Defined Networks (SDN), Network Function Virtualization (NFV) or
Self-Organizing Networking (SON) (Sotelo Monge et al., 2018) became key
enablers for digital tactical capability integration and orchestration at the
tactical edge via Tactical Clouds; from which vulnerabilities and cyber attack
surfaces are inherited (Maestre Vidal and Sotelo Monge, 2018) and exposed to
cyber adversarial (Mandt, 2017)(Lenders et al., 2015).
With the motivation of contributing to understand the intersection between the
cyber domain and the rest of the operational terrains, the conducted research
covered by this paper explores the escalation of a specific family of cyber
threats to the Tactical Cloud concept: the Economical Denial of Sustainability
(EDoS) attacks; and their evolution on the military context as Tactical Denial
of Sustainability (TDoS) actions. The conducted research extends the work
preliminary presented to the research community and cyber defence
practitioners in (Maestre Vidal and Sotelo Monge, 2020b), compiling the widely
received feedback and increasing its original scope from raw technological
aspects up to human and environmental aspects. Given the high interest it
aroused, the paper additionally includes a section dedicated to the cyber
military thinking concerning offensive/defensive denial of sustainability at
the edge; and novel illustrative concepts of operation. The following
enumerates the main contributions of the conducted research.
* •
The state-of-the-art on Tactical Clouds and the impact situations on the
cyberspace regarding the rest of kinetic operational domains is widely
reviewed and discussed.
* •
The term Tactical Denial of Sustainability (TDoS) is formally introduced as
evolution of the EDoS concepts to serve tactical objectives.
* •
The impact dimensions of TDoS related actuation at technological, tactical,
operational and strategic level are studied in-depth.
* •
The concepts Digital Tactical Capabilities, Tactical Cloud Provisioning
Similarity, Maintenance-based Tactical Denial of Sustainability, and
Deployment-based Tactical Denial of Sustainability are formulated and
discussed.
* •
The implications of TDoS against the human sustainability of cyber military
tactical operations are analyzed, concluding in the formulation of the
paradigms Enterprise-based Tactical Denial of Sustainability, Organizational-
based Tactical Denial of Sustainability and Individual-based Tactical Denial
of Sustainability.
* •
The close relationship between TDoS and the energy efficiency and
environmental sustainability of tactical clouds is studied
* •
Guidelines for TDoS related offensive and defensive cyber military thinking at
the edge are proposed.
* •
Five illustrative Concepts of Operations (CONOPs) are detailed (hybrid, proxy
and symmetrical war scenarios) as cases of study, which highlight the
vertical/horizontal propagation and effect of TDoS at different operational
contexts, including digital tactical capability supplying and their potential
in exposing the Infrared (IR) signature of stealthy actors .
The paper is organized into nine sections, being the first of them the present
introduction. Section II describes the emerging paradigms on tactical cloud
computing, as well as the basis of the EDoS attacks. Section III formalizes
the TDoS concept, related attack surfaces and impact dimensions. Section IV
discusses the potential of TDoS against jeopardizing the human sustainability
of tactical clouds. Section V explores the energy efficiency and environmental
impacts of TDoS. Section VI addresses the implication of TDoS in terms of
cyber military thinking at the tactical Edge. Section VII illustrates five
complementary CONOPs where TDoS became a key tactical military enabler.
Section VIII widely discusses the CONOPs presented at Section VII. Finally,
Section IX summarizes the reached conclusions and foreseen research actions.
Before starting reading the rest of this paper, the authors want to let the
reader know that the vision about military thinking, military operations on
the cyberspace and tactical cloud operation does not correspond in any way to
the transposition of the doctrine of any particular army nor coalition force,
being a vision of their own, resulting from the experience acquired after
several years of experience in related sectors. The concepts of operation that
are introduced are fictitious scenarios with a mere didactic purpose aiming on
facilitating the understanding of the TDoS concept and its potential on
military situations. Any deductible Rules Of Engagement (ROE), jus ad bellum,
or jus in bello premise do not intentionally align with any particular
international agreement.
Figure 1. Example of W-EDoS attack on a tactical cloud Figure 2. Example of
I-EDoS attack on a tactical cloud
## 2\. Background
This section describes the Tactical Cloud Computing paradigms, including its
essential concepts, technological dependencies, and foreseen challenges/gaps.
On the other hand, the Economic Denial of Sustainability (EDoS) threats are
reviewed, emphasizing their key element and features in the scope of being
adapted to BLOS tactical environments.
### 2.1. Tactical Clouds
The Tactical Cloud concepts born in the need to meet the critical
communication needs at the tactical edge of military BLOS operations (Smith et
al., [n. d.]), preliminarily bringing capabilities that among others shall
support mobility, heterogeneous waveform, routing policy, or the scalability
of the computational resources thorough the edge. Similarly to MANETs/FANETs,
military BLOS communications are defined under the condition of no base
station, for which edge computing and decision-making were typically combined
(Oubbati et al., 2017). Because of their high mobility, their nodes can be
deployed quickly, temporarily organize and have strong ability to resist
destruction and self-healing, being characterized by enabling distributed
peer-to-peer tactical networks, allowing cognitive/dynamical changes in
topology and information multi-hop transmission; but also inherited some of
the key drawbacks of BLOS deployments, like energy or bandwidth limitations
(Wang et al., [n. d.]). Because of this, they embrace emerging paradigms like
Ground Centric Networking (GCC), Cloudlets and Self-Orgaizing Networking
(SON). The first provides resilient, bandwidth-efficient communication to
users participating in interest groups which have numerous applications to
military tactical edge networks due that army units are organized in a
hierarchical manner (Chen et al., 2016). Cloudlets can be viewed as a “data
centers in a box” that “brings the cloud closer” to the battle (Chi et al.,
2018). The proximity of a cloudlet to its associated mobile devices is the key
to its value in hostile environments (Satyanarayanan et al., 2013). Their
original motivation was to reduce end-to-end latency of cloud offload from
mobile devices for applications that are both resource-intensive and latency
sensitive, additionally posing less cyber-physical attack surfaces and
sources, enabling self-organization and facilitating the ICT heterogeneity. On
the other hand, Self-Organizing Networks (SON) arose with the goal of moving
forward from traditional manual management processes towards an automatic and
dynamic perspective (Maestre Vidal et al., 2018a), for which Software Defined
Networking (SDN) reduced the network management complexity by decoupling the
control channels, and Network Function Virtualization (NFV) allowed decoupling
the software implementation of Network Functions (NF) from the underlying
hardware, providing flexibility in the management of the network resources
(Satyanarayanan et al., 2013). When deployed at the tactical edge, these
capabilities facilitate the deployment of cloud infrastructure to enable
distributed mission command leveraging common software/hardware solutions, and
provisioning C3 services thorough conventional and BLOS operational
circumstances (Hess, J. and Kiser, A. and Bouhafa, E.M. and Williams, S.,
2019). As highlighted in (Layton, 2018), modern conflict will increasingly
rely on advanced information technologies, highlighting four essential
pillars: networks, combat cloud, multi-domain battle and fusion warfare. In
this context, the Tactical Cloud improves the capabilities for acquiring
situational awareness, makes long-range engagements more practical, moves
advanced surveillance, data collection and AI to the tactical edge, and
supports the development of multi-domain Common Operational Pictures (COPs).
However, and as it will be discussed thorough this paper, tactical clouds
inherit part of the attack surface of the technological ecosystem that
integrates, as is the case of cybersecurity issues and several threats against
the cloud sustainability, as is the case of EDoS.
### 2.2. Economic Denial of Sustainability
The Economical Denial of Sustainability (EDoS) attacks are novel situations
preliminarily hypothesized by Hoff (Hoff, C., 2008)(Hoff, C., 2009), which at
the dawn of the cloud computing ecosystem, echoed of potential hostile
situations where the attacker may attempt to impact on the cloud computing
sustainability by maliciously enforcing economic costs derived from increasing
the economic costs derived from both maintenance and provision of the services
offered (Hui et al., 2017). Despite their novelty, EDoS caught the attention
of the research community, which framed their modus operandi as part of the
Reduction of Quality (RoQ) (Bremler-Barr et al., [n. d.]) and Fraudulent
Resource Consumption (FRC) threats (Somani et al., 2016)(Singh and Rehman, [n.
d.])(Somani et al., 2017). In (Sotelo Monge et al., 2019)(Maestre Vidal et
al., 2018b) the problem of EDoS was reviewed in the context of the Self-
Organizing Network (SON) paradigm, which presented one of the studies closer
to the TDoS paradigms. Accordingly, SON deployments may be jeopardized for
causing Workload-based EDoS (W-EDoS) and Instantiation-based (I-EDoS), W-EDoS
being caused by maliciously forcing huge workloads (e.g. execution of high
complexity algorithms, exploitation of software vulnerabilities, etc.) that
require the vertical-escalation of the SON capabilities (see Fig. 1); while
I-EDoS is caused by the massive deployment of unnecessary (referred to as
lazy) VNFs, which was demonstrated by poisoning the telemetric services of the
SON Network Function Virtualization Infrastructure (NFVI) (see Fig. 2). As
highlighted in (Somani et al., 2016), beyond the economic impact, EDoS attacks
entail several cross-cutting situations, which among others concern the
computational capabilities of the cloud, performance, latency, connectivity,
availability; and from the socio-technical aspect negatively affect the trust
between customers and Digital Service Providers (DSP) (in both direction).
Although in (Somani et al., 2016) was demonstrated that the implementation of
cybersecurity measures based on predicting the behavior of the protected
system, constructing adaptive thresholds, and clustering of VNFs instances
based on productivity, were effective enough to reveal EDoS threats (Sotelo
Monge et al., 2017), their prevention, detection, mitigation and attribution
still entail important research challenges, to which is added that the
bibliography does not include a large collection of publications focused on
the defense against EDoS threats. The studies that address this problem
usually assume metrics at network-level, usually confusing features for EDoS
identification with those that typically detect flooding-based DDoS behaviors
(Bhingarkar and Shah, [n. d.])(Bawa and Manickam, 2015)(Baig et al., 2016).
## 3\. Tactical Denial of Sustainability
Bearing in mind the semantical similarities between Flash Crowds, DoS/DDoS and
EDoS threats, in (Sotelo Monge et al., 2019) a discriminative criterion was
introduced: the CRoWN indicators. Accordingly, the key differentiation between
those situations rely on the four parameter directly linked to the client-
server model and the distributed provisioning of resources: clients ($C$),
requests ($R$), workload that entails their resolution ($W$) and the network
functions necessary for their processing ($NF$). Accordingly, W-EDoS are
characterized by unexpected workloads ($W$), while I-EDoS are characterized by
the suspicious deployment of new functionalities on the distributed
environment. As highlighted in Section 2.2., these threats operate on the
cyberspace, attempting to achieve technological goals. According to the
military theory, they may entail potential threats at the Technological level
of war. Beyond the original scope of the EDoS attacks, the TDoS situations
target to impact on the Tactical level of war (Zhou et al., 2019) by
jeopardizing the sustainability of the capabilities provisioned by the
Tactical Clouds, so they embrace tactical-level actuations focused on
jeopardizing the decisions and actions that shall originally create advantages
when in contact with or in proximity to the enemy. Although this research
hypostatizes about the vertical propagation of EDoS to TDoS, other tactical
level situations may derive on TDoS.
An illustrative example of EDoS to TDoS is illustrated in Fig, 3. Accordingly,
a TDoS threat aimed on reducing the ISR, C2 and Situational Awareness
capabilities supplied by tactical nodes, having the potential effect of
vertically impacting on the operational capabilities and strategical
objectives that drive a military mission. With this purpose, the attacker
propagates a vertical threat from the network and datacenter business
processes to the tactical plane. The impact on the tactical level is inferred
by jeopardized Virtual Network Functions (VNFs) deployed at the Tactical edge,
which implement the tactical services (3GPP, 2008)(ETSI, 2014). These services
feed operational intelligence, C2 and join planning capabilities, from which
the fulfillment of the strategic objectives depends. In this context, by
achieving a conventional I-EDoS situation (Maestre Vidal et al., 2018b), the
attacker forced the instantiation of redundant VNFs, with among others,
heavily impact on the energy efficiency of the mobile tactical infrastructure
that enables network and datacenter operations. They support the IST, SA, C2
and Decision-making services deployed at the Tactical Edge, which usability
will be reduced as the redundant VNFs pointless consume energy. Consequently,
and as depicted in the example, the motivation of EDoS and TDoS is different.
The first of them attempts to infer a computational overhead in the VNFs,
their supportive infrastructure and their related business processes; and the
second attempts to reduce the usability of the tactical services deployed at
the edge, which is achieved by depleting the energy supply of the Tactical
Cloud enablers. The connection between EDoS and TDoS is the energy depletion,
where EDOs became the “cause” and TDoS the effect. As commented before, it is
expected that non EDoS related attack vectors lead to TDoS situations, so they
entail different but related families of threats. So TDoS is a Tactical-level
threat potentially unchained by from technological-level situations up to
horizontally propagated tactical level threats, being significantly beyond the
scope of EDoS.
Figure 3. Example of vertical propagation of a TDoS threat
### 3.1. Digital Tactical Capabilities
The digital tactical capabilities entail the primarily attack surface fo the
TDoS threats. The conducted research assumed that these capabilities when
deployed on tactical nodes resemble a lifecycle similar to that of the VNFs
(Cappanera et al., 2019), which among others shall enable rapid deployment of
new functionalities, the scalability of the existing services, shall be
interoperable and increase the tactical flexibility (at all C3 Taxonomy
levels); from the back-end capabilities to the Operational context. However, a
digital tactical capability may be habilitated by various different VNFs,
services, or other technological/tactical assets, which lifecycle at least
shall embrace some of the following phases:
* •
Onboarding. End-to-end provisioning of tactical capabilities from dual-use
inventories and/or repositories beyond the operational environment, to the
tactical edge. This requires a full technological interoperability, where the
onboarded capabilities that support the feature are properly harmonized
regarding the preliminarily deployed CIS services and assets.
* •
Deployment. The onboarded capabilities require the installation, configuration
and coordination of their related CIS enablers. On the other hand, they shall
coordinate and cooperate with the rest of the C3 capabilities covered by the
Tactical Cloud, all of this being covered at the deployment stage.
* •
Surveillance. In order to identify potential risks and malfunctions, the
deployed tactical capabilities should be real-time monitored and verified.
This is also important to acquire tactical awareness and facilitating the
tactical onboarding and deployment of further C3 capabilities.
* •
Adaptation. The Tactical Cloud shall enable scalability and flexibility enough
to guarantee the proper provisioning and operation of the tactical
capabilities. At the adaptation stage, correction, optimizations, and
scalability/flexibility issues are continuously conducted, which occur
reactive/proactively and driven by automatisms or human intervention (Maestre
Vidal and Sotelo Monge, [n. d.]).
* •
Undeployment. The deployed tactical services may require to be removed after a
certain time period expires, when the accomplished their purport (are they are
not needed for the rest of the supported missions phases), or when operational
circumstances require it (e.g. as result of enforcing tactical/technological
CoAs.
### 3.2. Conceptualization
The rapid provisioning of ondemand digital tactical capabilities thorough the
tactical edge is one of the key features of a Tactical Cloud, which processes
usually are close to cyber expeditionary warfare actuations (in particular,
when operating at BLOS). Under the tactical expectations caused by a Rapid
Deployment Force (Deng et al., 2019), digital tactical capabilities shall be
onboarded and deployed at the distributed tactical nodes. In these grounds,
the holistic Tactical Denial of Sustainability concept is formally defined as
highlighted in Lemma 2, which depends of the Tactical Communications
Similarity relationship formalized in Lemma 1.
###### Lemma 1.
Tactical Cloud Provisioning Similarity
In analogy with EDoS situations, it is expected that during a TDoS attack the
digital tactical capabilities pose great resemblance regarding the normal and
legitimate circumstances (Ahmed et al., 2016) If this is not the case, the
tactical disruptions may be caused by alternative operational conditions
(denial of service attempts, massive demand by legitimate circumstances, flash
crowds, etc.). So let the state $A$ that depicts the normal operability of the
tactical communication networks, the State $B$ only can resemble a TDoS
situation if: $nA_{A}\sim nA_{B}$, $nA_{A}(t)\sim nA_{b}(t)$, $nT_{A}\sim
nT_{b}$ and $nT_{A}(t)\sim nT_{b}(t)$, where:
* •
$nA$: total number of tactical actions that require certain digital tactical
capability.
* •
$nA(t)$: distribution of the number of tactical actions that require certain
digital tactical capability as a mission progress.
* •
$nT$: total number of tactical capabilities that requests certain digital
tactical capability (horizontal dependencies).
* •
$nT(t)$: distribution of the number of tactical capabilities that requests
certain digital tactical capability as a mission progress (horizontal
dependencies).
Hereinafter this relationship is referred to as Tactical Cloud Provisioning
Similarity (abbreviated as TPS), so if all the aforementioned conditions are
satisfied, it is possible to state that A and B are tactical-based similar
(TPS(A,B)).
###### Lemma 2.
Tactical Denial of Sustainability
Let the state A that depicts the normal and expected behavior of a capability
delivered by services enabled by a Tactical Cloud, B depicts a Tactical Denial
of Sustainability situation (abbreviated as TDoS) if TCS(A,B), and if all the
following conditions are satisfied: $C_{A}\ll C_{B}$ and $C_{A}(t)\nsim
C_{B}(t)$; or $tD_{A}\ll tD_{B}$ and $tD_{A}(t)\nsim tD_{B}(t)$; where:
* •
$nC$: total cost of upkeeping a particular digital tactical capability.
* •
$nC(t)$: distribution of. the cost of upkeeping a particular digital tactical
capability as a mission progress.
* •
$tD$: total number of tactical nodes that supply certain digital tactical
capability.
* •
$tD(t)$: distribution of the number of tactical nodes that supply certain
digital tactical capability as a mission progress.
In these grounds, the conducted research hypothesized about at least a pair of
modus operandi able to trigger a potential TDoS situations, which rely on
threatening the hybrid upkeep costs (economical, energy, materials, etc.)
and/or and the distribution of the digital tactical capabilities thorough the
tactical edge. The are defined as Maintenance-based Tactical Denial of
Sustainability (U-TDoS) and Deployment-based Tactical Denial of Sustainability
(D-TDoS).
###### Lemma 3.
Maintenance-based Tactical Denial of Sustainability Let the state A that
depicts the normal and expected behavior of a capability delivered by services
enabled by a Tactical Cloud, B depicts a Maintenance-based Tactical Denial of
Sustainability situation (abbreviated as U-TDoS) if TCS(A,B), and if the
following conditions are satisfied: $C_{A}\ll C_{B}$ and $C_{A}(t)\nsim
C_{B}(t)$. This can be read as that under conventional provisioning conditions
(i.e. there are not outlying observations in terms of the ongoing/planned
actions nor the number of tactical capabilities that depends on digital
tactical capabilities), the upkeep of a digital tactical service became
significatively ($C_{A}\ll C_{B}$) and unexpectively ($C_{A}(t)\nsim
C_{B}(t)$) costly at some point of the supported missions.
###### Lemma 4.
Deployment-based Tactical Denial of Sustainability Let the state A that
depicts the normal and expected behavior of a capability delivered by services
enabled by a Tactical Cloud, B depicts a Deployment-based Tactical Denial of
Sustainability situation (abbreviated as D-TDoS) if TCS(A,B), and if the
following conditions are satisfied: $tD_{A}\ll tD_{B}andtD_{A}(t)\nsim
tD_{B}(t)$. This can be read as that under conventional provisioning
conditions (i.e. there are not outlying observations in terms of the
ongoing/planned actions nor the number of tactical capabilities that depends
on digital tactical capabilities), the tactical nodes that supply certain
digital tactical capability somehow increase significatively ($tD_{A}\ll
tD_{B}$) and unexpectedly ($tD_{A}(t)\nsim tD_{B}(t)$) at some point of the
supported missions.
### 3.3. Causality and attack vectors
Both U-TDoS and D-TDoS may by triggered by multiple and heterogeneous
potential threat vectors, which can be linked with vertical (technical,
operational) and/or horizontal (tactical) decisions or hostile actions. The
technical reasons rely on the vertical propagation from cyber-physical treats
to the tactical plane, which should be discoverable by mission-centric risk
identification and assessment capabilities (Javonik et al., [n. d.]), and
against responding presents a high dependency of the acquisition of a mission-
centric cyber situational awareness (Silva and Jacob, [n. d.]). In this
context, it is expected that the conventional threat situations on Cloud
Computing environments deployed at tactical edge may bring TDoS scenarios,
among them Denial of Service (DoS), Advanced Persistent Threats (APTs),
Insiders , etc. (Maestre Vidal and Sotelo Monge, 2020a)(Maestre Vidal et al.,
2016) The EDoS attacks have the potential of became the key enabler of U-TDoS
and D-TDoS, which as demonstrated in (Sotelo Monge et al., 2019), may be
enforced by combining some of the aforementioned malicious actions. The
propagations vertical from the operational plane are directly related with the
decision-made as the mission evolves, provisioning (e.g. sources of digital
tactical capabilities to be onboarded, and the interference of operational-to-
tactical C2 communication (Lopes et al., [n. d.]). For example, the
capabilities that enable operational assessment determines the progress of a
joint force toward mission accomplishment. An adversarial counterintelligence
action may lead to redefine and develop operational plans that place the
Tactical Cloud infrastructure in bad terrains, where the communication cost
between tactical nodes rapidly drain the energy on which the ICT assets
depend. For example, in terms of U-TDoS this may be directly caused by the
need for greater energy consumption in order to enable longer-distance
communications. On the other hand, D-TDoS may be the consequence of deploying
redundant capabilities. The latter may be the cause of Operational-to-Tactical
vertical propagations, as well as the consequence of tactical situations. For
example, in order to trigger TDoS the adversarial effectors may tactically
force to dynamically redistribute the mobile ally tactical nodes, thus leading
to a situational similar to that enforced at operational level. In either
cases (vertical and horizontal propagations), the denial of the sustainability
of the Tactical Cloud have the potential of become a critical tactical
problems, on which digital tactical capabilities like C2, ISR or tools for
acquiring situational awarenessdepends. It is important to highlight that the
TDoS causality and attack vectors may sinificantly vary depending on the stage
of the digital tactical capabilities lifecycle. For example, the attack
vectors targeting the onboarding stage are mostly related with the
provisioning of the capability dependencies, from which mitigation to protect
the end-to-end digital supply chain is critical. In the opposite, attempting
to weaponize the adaptation or undeployment stages are mostly related with
contextual operational, tactical and technological decision, where the
adversarial may attempt to force that the rival enforce particular CoAs in
order to gain tactical terrains o made the rival weak them.
### 3.4. Impact Dimensions
Unlike most tactical threats, a TDoS situations have the potential of
impacting on multiple and hybrid dimensions, highlighting among them the
mission economics, safety, security, energy efficiency or socio-technical
features. The economic impacts are related with the software licenses, the
scalability/expandability of the digital tactical capabilities, personnel
costs, materials, etc. which are expected to pursue deterring purposes
(Brantly, [n. d.]) as the missions supported by the jeopardized Tactical Cloud
extends over time. Alternatively, economic and material expenditures may lead
to the prioritization of tactical services, thus exposing new attack surfaces
and tactical blind spots (Downes, 2018). They can derive on new safety and
security situations that the adversarial may weaponized towards gain tactical
terrains. The environmental and energy related consequences of TDoS situations
on Tactical Clouds can be extrapolated with the economic/material aspects of
the mission, which extends towards side-channel related consequences that may
facilitate adversarial ISR and Electronic Warfare actions (Samaras et al.,
2019a). They may also enforce tactical CoAs like in-flight refuelling, battery
recharges, etc. on the ITC infrastructure and its carrying units/facilities,
causing delays, exposing effectors and disrupting the original planning of the
military missions. Finally, the socio-technical impact is related with the
Tactical Cloud ergonomics and the human understanding of the Common
Operational Picture (COP) (Ulrik, [n. d.]). For example, by forcing that the
rival deploys a huge number of heterogeneous capabilities at different levels
of the tactical environment may be possible to make understanding and
prioritization difficult when making decisions, which may be perceived as a
socio-technological jamming action. Another more complicated TDoS action
against socio-technical dimensions is to enforce that the adversarial deploys
digital tactical services that require higher-levels of human capacitation.
This would cause that more specific, and hard-to-find profiles lose the focus
on other tactical situations.
## 4\. TDoS against the human sustainability of tactical clouds
As pointed out by the European Union Institute for Security Studies (European
Union Institute for Security Studies, 2020), ”the digitalization of the
military sector should not be seen as silver bullet for every problem facing
Europe’s militaries and a human dimension will be required for politico-
strategic guidance and maintaining the morale of troops”. It is inherent to
cross-dimensional issues and challenges where novel contributions are not
exempt of potentially disruptive changes at all the levels of the Joint
Capabilities Integration and Development System (JCIDS): doctrine,
organization, training, materiel, leadership and education, personnel,
facilities, etc. These changes are directly linked with implications on the
human barriers on the acceptance and adoption of the military tactical clouds,
all of them needed to be analyzed and considered at design, planning and
enforcement of joint operations on the cyberspace. Some of them are enumerated
below:
Lack of trust to share capabilities and information between military and
civilian actors. The capability of acquiring military sovereign in cyberspace
is controversial, since the exclusivity of classical state sovereignty runs
contrary to the spirit of the internet, which rests on the concept of
unrestricted interconnectivity and net neutrality. On the other hand, most of
the required infrastructure and services is managed and/or belongs to the
private sector, which places a variety of legal, regulatory, and accepted
self-limiting obstacles, hindering public/private cooperation in cyber defense
and counterattack (Center for Strategic Leadership (CSL), US Army War College,
2016). Therefore, the development of collaborative platforms to facilitate the
information exchange between stakeholders plays a key role in the pursuit of
an aligned, connected and prepared cyber defence environment. However, the
lack of trust between different public and private actors in each country is
seen as the primary inhibitor to cross-sectorial and cross-border
collaboration, and intense competition and distrust from business rivals often
prevent information exchange and cooperation between different private sector
stakeholders (Brogaard, 2018). Companies and organizations are hesitant to
share capabilities with defence stakeholders because of their law enforcement
and supervisory functions, as well as fear of sanctions under national laws or
competition. In contrast, fear of adverse media coverage is another reason why
organizations are reluctant to expose operational information to the public
sector and the general public. A clear example of lack of trust and civil
acceptation on this cooperation is illustrated in the repudiation of a set of
employees, customers and investors of MIT Media Lab with the United States
Department of Defence on Project Maven, where AI capabilities were attempted
to be developed for supporting combat drone operations (Bretl et al., 2019).
Country-specific & cultural differences. Notions of privacy, security and
safety depend mainly on context and legislation and differ between cultures
and countries, even within Europe The last decade has also seen the emergence
of novel procedures and concepts of operations with a clear country-specific
vision, as is the case of the regulation and enforcement of ”active
cyberdefence” tactics. Lacking a commonly accepted definition, a number of
states (e.g., the USA, the UK), international organisations, such as the
Cooperative Cyber Defence Centre of Excellence, and scholars have provided
explanations that help qualify (Wanner and Ghernaouti, 2020), but they are
still far from having a common perception. On the other hand, questions such
as infrastructure, security, policy on the transmission of data and sensitive
information, awareness and education are not only technical but also, in no
small extent, socio-economic and political; linked to JCDP interpretation and
enforcement. Finally, it is worth to highlight that many of the national
regulations which concern cybersecurity are designed in a similar fashion to
regulations relating to public safety and security. Kharlamov and Pogrebna
(Kharlamov, 2019) hypothesized that a human values-based framework for
cybersecurity governance (also applied to Civil-Military Cooperation (CMC))
shall at least ask the following issues: the problem of coexistence between an
individual and a group (society), coexistence between an individual and social
fabric responsibilities (responsibility) and coexistence between human beings
and nature (nature); these answers inherently linked to cultural/regional
singularities and heritages.
Social perception of digitalization and AI as threats. Although the
digitalization of the defence sector is bringing proven advantages in terms of
sustainability, efficiency, quality and performance, it can be perceived by
society as a threat, which is directly transposed to the tactical capabilities
relying on it. From the workforce standpoint, the main concern is
technological barriers and the risk of job loss, but citizens’ point of view,
one of the main problems will be the lack of privacy and the risk of
continuous surveillance. There is a cross-sectorial resistance to change.
Other barriers to social acceptance stem from the emergence of security
problems attributed to AI-related processes, which can range from the handling
of sensitive information to the presence of cognitive biases in decision
stages (European Commission, 2019) This is the case of (Center of Naval
Analysis, U.S. Navy, Department of Defense (DOD), 2017) and (Ekelhof, 2018),
which analysed revealed operational failures, partly due to the incorrect use
of automatic response systems. On the other hand, in (Broeders, D. and Sergei,
B.and Georgieva, I., 2019) the risk and (un)intended of intelligence agencies
behaviours in cyberspace are reviewed in detail, among others concluding that
there is a need of exploring possibilities to define objects and organisations
that should be off limits for cyber operations, in line with the ‘Geneva’
style of the regulation of responsible state behaviour; also suggesting to
resist the temptation of replying in kind to influence operations that target
the integrity of public information, since integrity of data and information –
much more than confidentiality and availability – touches on core values of
democratic societies and resembles a very negative social perception.
Misuse and other human factors. Morality and ethics have occasionally been
identified as precursors to or inhibitors of important phenomena in the
behavioral security domain, which in absence may derive in insider threat
triggering situations categorized as traitors and masqueraders, the first
(traitors) resembling legitimate workforce trying to gain privileges for
accessing restricted assets/capabilities with malicious purpose, and the
second (masqueraders) entails external actors that are somehow able to
impersonate authorized users and disrupt planed/ongoing tactical operations
(Maestre Vidal and Sotelo Monge, 2020a; Maestre Vidal et al., 2016). Both
scenarios are catalyzed by the lack of training, motivation, understanding and
awareness of the personnel; which are triggered by assuming the gaps and
challenges described above. On the other hand, human factors may mislead
operations relying on digitalization and tactical clouds inherent in the human
cognition and capability to react to a massive digest of information. In this
regard, Research on Human Cybersecurity (HCS) behavior suggests that for
example, time pressure is one of the important driving factors behind non-
secure decisions (Chowdhury et al., 2020). The research on facilitating the
acquisition of cybersituational awareness also revealed the need for
capabilities able to adapt the presentation of cross-domain operational
pictures to the decision-making contexts, so human beings behind decisions can
efficiently consider the most relevant information (Daton Medenou et al.,
2020).
### 4.1. Enterprise, Organizational and Individual TDoS threats
In 2015, Gen. Odierno commissioned the U.S. Army Human Dimension Strategy
(AHDS) that outlined the way ahead for development of human dimension assuming
a time horizon of ten years (United States Army Combined Arms Center, 2015).
The paper defined the human dimension as the cognitive, physical, and social
components of the Army’s trusted professionals and teams, emphasizing the need
to “optimize the human performance of every soldier” and “build cohesive teams
of trusted professionals who thrive in ambiguity and chaos”. As response, the
strategy defined the US Cognitive Dominance Line of Effort (LOE) concerning
objectives and tasks that equip Army personnel with the intellectual aptitude,
cultural understanding, physical toughness, and resilience to adapt and thrive
in ambiguity and chaos. The conceptualization of the TDoS on the cognitive
dominance can be abstracted according to the model describe by C. Song (Song,
2018), which by exploring the transformation of armies for attract retain and
develop both civilian and military personnel, perceived model armies as
“systems,” being larger military forces as “System of Systems” (SoS); i.e.
“assemblages of components which individually may be regarded as systems, and
which possess two additional properties - operational independence of the
components and managerial independence of the components”. Accordingly, the
human impact derived from adversarial activities on the tactical edge can be
framed as enterprise, organizational and individual levels, which are
enunciated at the following lemmas.
###### Lemma 1.
Enterprise-based Tactical Denial of Sustainability
Let the state $A$ that depicts the normal and expected behavior of a
capability delivered by services enabled by a Tactical Cloud, $B$ depicts a
Enterprise-based Tactical Denial of Sustainability situation (abbreviated as
E-TDoS) if jeopardizes the army human capital lifecycle management system
concerning recruiting, trusting, training, educating, developing, promoting
and/or retaining both military and civilian talent (U.S. Defense Acquisition
University Huntsville United States, 2020).
###### Lemma 2.
Organizational-based Tactical Denial of Sustainability
Let the state $A$ that depicts the normal and expected behavior of a
capability delivered by services enabled by a Tactical Cloud, $B$ depicts a
Organizational-based Tactical Denial of Sustainability situation (abbreviated
as O-TDoS) if jeopardizes the effectiveness, efficiency and upkeep of
governance and business practices aimed on accelerating communication,
decision-making, and DOTMLPF-P integration for military human resource
management.
###### Lemma 3.
Individual-based Tactical Denial of Sustainability
Let the state $A$ that depicts the normal and expected behavior of a
capability delivered by services enabled by a Tactical Cloud, $B$ depicts a
Individual-based Tactical Denial of Sustainability situation (abbreviated as
I-TDoS) if jeopardizes the independent actions of each individual within the
army. The AHDS organizeds the individual situations into three core groups,
which may overlap: social, cognitive and physical situations (United States
Army Combined Arms Center, 2015).
### 4.2. Socio-technical dimensions and attack vectors of TDoS
The adversarial actions against the military enterprise (E-TDoS) and
organizational (O-TDoS) by targeting the tactical cloud inherently entail
vertical propagation from tactical to operational and strategical goals
thorough exploiting cross-frontier vulnerabilities across the Political,
Military, Economic, Social, Informational and Infrastructure (PMESII) spectrum
(Multinational Capability Development Campaign project, 2017). Consequently,
they are connected with hybrid warfare situations where the cyberspace
operates as a highway interface that links the “grey zone “ with cybernetic
assets deployed at the tactical environments on which the tactical cloud relay
on, or which are enabled by this interface; and operational/strategical
capabilities vertically connected with their related technical actions.
Therefore, tactical clouds enable network-centric warfare tactics able to
trigger hybrid consequences (Otaiku, 2018). Some of the most probable
nonviolent hybrid threat instruments inferable from digital tactical
capabilities are reviewed in (Monaghan, 2019), as is the case of services
linked to media/propaganda, military intelligence, communication and
networking, or economical sustainability. By making their upkeep more
expensive, forcing the fraudulent instantion of poisoned related VNFs or
sinkholing targeted groups of tactical services, the conventional U-TDoS and
D-TDoS effects derived from cyber attacks like I-EDoS and W-EDoS have the
potential of impact on the military enterprise and organization from their
bases (comprimising due to cognitive and physical factors their recruitment
effectiveness, trustworthiness, poisoning of the aquired institutional picture
of the theaters of operations, etc.). The U.S. Institute for the Study of War
highlighted a subset of behavioral patterns for grey zone operations that can
be directly transposed to the enforcement on E-TDoS and O-TDoS: 1) creation of
maximum uncertainty at the tactical edge: covert and clandestine actions
preferred, 2) maintenance of deniability of E-TDoS and O-TDoS as long as
possible, 3) gradually escalating pressure, indirectly triggering “domestic”
violence as necessary, 4) staying below the threshold of war or triggers of
intervention (at least while E-TDoS and O-TDoS aim to proxy/hybrid goals), 5)
when aiming to trigger direct human reactions, mixing “carrots” and “sticks ,
and 6) last resort or when “invited” conventional military force. Bearing all
this in mind, the close dependencies between cyberspace and information
warfare makes the tactical clouds perfect platforms for deploying related
defensive effectors, in the same time that expose a complex potential attack
surface (Barret and Mansfield, 2019).
The hybrid warfare and the use of cyber assets as part of it is one of the
most important factors for understanding the future arc of conflict (Danyk et
al., 2017), which beyond jeopardizing the socio-technical dimensions of the
military enterprise and organization layer, may directly impact on the
individual at any level (including technological and tactical) resulting in
I-TDoS situations. A good example of the adoption of cyber effectors against
socio-cognitive aspects was observed during the Russo-Ukrainian conflict at
2014, where combat actions in Illovaysk and Debalcevo were preceded by a
significant burst of activity in information space. These theaters of
operations identified negative information on key authorities of Armed Forces
of Ukraine and government representatives as a cyber aggression, coupled with
disinformation from proxies and false fronts on the internet. In related
contexts, a tactical cloud resemble a perfect vector for deploying
disinformation actions while jeopardizing the required ICT capabilities
towards maximizing their dissemination effect. These enablers were discussed
in deep in (Danyk et al., 2017), highlighting anonymous claims to authority,
news items manipulated with half-truths, repetition of messages, information
overload, cyber-pseudo operations (government posing as insurgents), sock-
puppeting (government agents playing the role of online commentators), and
astro-turfing (creating of false grassroots movements).
Finally, I-TDoS attacks triggered from tactical clouds or against them may
take advantage of the data processing capabilities bring by the digitalization
of the battlefield for disrupting the human ability of understanding the
operational picture; especially if this is facilitated by AI-driven
automatism. For example, from (Stevens, 2020) can be inferred that data
dependent AI enablers may be manipulated or poisoned for affecting the human
decision-making activities that are feed by them; TDoS actions may lead to
similar results by overload of targeted edge services and leading the victim
to develop a biased perception of the operational situation. Related risks
inherent in data forgery and adversarial AI tactics when applying federated
machine learning at the tactical edge are described in (Cirincione and Verma,
2019), where it is emphasized that tactical cloud services may face rapidly
changing, never-before-seen situations, where pre-existing training data will
quickly become ineffective (e.g. tactical training data is noisy, incomplete,
erroneous, etc.); and exposes adversarial opportunities for enforcing
deception tactics on the collected information required for it update/upgrade
(Yan, 2020). From the TDoS perspective, this may occur among others by forcing
the fraudulent instantiation of VNFs from poisoned VF inventories (e.g. third-
party digital marketplaces) thus jeopardizing the data processing or
information exchange processes; or by injecting via I-EDoS a false perception
of activity able to create and/or exploit blind spots in the automatism. As
more extended examples, the cases of study presented as CONOPs in Section 7
hypothesize on additional vectors against the human sustainability framed in
the concrete mission narratives that they describe.
## 5\. Jeopardizing the energy efficiency of tactical clouds
As indicated by Samaras et al, “Energy has played a role in every facet of war
from troops in garrison and defensive planning to mobilization and attack. The
need to deliver adequate and timely energy supplies to military
forces—particularly to those in the most forward-deployed locations—has long
existed as a strategic vulnerability to the success of military campaigns”
(Samaras et al., 2019b). Energy resembles a critical aspect to consider from
classical and modern defense planning; and at all warfare levels (strategical,
operational, tactical and technological). The digitalization of the sector and
management of tactical clouds is not exempt of these considerations; in fact,
the instantiation, upkeep and removal of digital tactical capabilities
typically demands a movement of high volume of data across different
geographically separated nodes which create a huge burden on the network
infrastructure with strong energy and environmental implications (Aujl and
Kumar, 2018). Since to extended the tactical cloud lifetime with complete
connectivity and coverage entails a critical edge terrain (Thomas et al.,
2019), ongoing and ambitions defense projects like the DARPA’s Near Zero Power
RF and Sensor Operations (N-ZERO) (U.S. Defense Advanced Research Projects
Agency (DARPA), [n. d.]) address the challenges of extending the energy
efficiency, coverage and connectivity of related enablers (cyber-physical
sensors, wireless communication or distribute data storage capabilities,
etc.). In this context, Wang et al. [80] pointed out three core actions
towards minimizing the energetic and environmental impacts derived from
tactical digital information processing: 1) to enforce effective workload
classification so that “latency-hungry” and realistic end user requests can be
provisioned to edge devices and “resource-hungry” requests are relayed towards
central computing capabilities (e.g. data centers); 2) to optimal scheduling
the classified workload in edge–cloud environment in order to achieve trade-
off between energy efficiency and effectiveness; and 3) to migrate tactical
requests to centralized computing capabilities when the digital tactical
capabilities are enabled by resource-limited edge devices. But despite the
challenges inherent in the deployment, supply and maintenance of digital
tactical services, adversaries may aggravate related situations by enforcing
hostile actions against the tactical cloud lifecycle and energetic
dependencies. Threats like the depletion-of-battery attacks (Shakhov and Koo,
2018), wormhole attacks (Gupta and Pathak, 2016), clone attack (nodes
replication attack) (Zeng et al., 2010), vampire attacks (Nisha et al., 2016),
or denial-of-sleep attack for keeping the sensor nodes awake to consume more
energy (Capossele et al., 2016); demonstrated effectiveness when causing
energy depletion at dual use data processing services.
As proven in (Sotelo Monge et al., 2019), the EDoS threats that target the
instantiation (I-EDoS) and workload (W-EDoS) of virtualized serviced
deployable at the edge can effectively cause cost overrun linked to fraudulent
CPU, memory, connectivity, bandwidth, etc. usage; all of them transversely
derived in malicious energy expenditures. Based on them, specific hostile
actions against digital tactical services can be driven by the M-TDoS and
D-TDoS actions; thus compromising any phase of the tactical services
provisioning and upkeep (onboarding, deployment, surveillance, adaptation,
etc.). It is important to consider that the Allied Joint Doctrine for the
Planning of Operations explicitly refers to the critically of the sustainment
concept as: “No COA is complete without a plan to sustain it properly. The
sustainment concept is more than just gathering information on various
logistic and personnel services.”. In this context, the Allied Joint Doctrine
for the Conduct of Operations (NATO, [n. d.]b) points out success conditions
dependant on the energetic sustainability of the digital tactical services,
among them the Ally unity of effort, concentration of force, flexibility,
surprise, agility, or synchronization. Any of the aforementioned technical-
level threats against the energetic sustainability of the cloud can be
vertically propagated to tactical and operational situations and dimensions,
some examples of this being illustrated in the CONOPs presented in Section 7.
## 6\. TDoS in Military Thinking at the cyber Tactical Edge
Kallberg and Cook (Kallberg and Cook, 2017) pointed out four core premises to
be assumed when conducting cyber military operations: 1) lack of object
permanence, which undermines the concept of maneuver; 2) limited or absent
measurement of effectiveness in offensive cyber; 3) conflicts that are
executed at computational speed, removing the time window that would allow for
meaningful strategic leadership; 4) anonymity that makes the parties to the
conflict unsure who is the other party. These assumptions make the
conventional C2 thinking directly linked to OODA (Observe, Orient, Decide,
Act) loop weaken, since they compromise their four tenants, and thus project
the need for significant changes at doctrine-level as the defene sectors is
digitalized. Complementarily, and as remarked by Schulze (Schulze, 2020), the
“tactical cyber operations are difficult to integrate into the traditional
target cycle of conventional forces due to their long planning and development
time. Traditional weapons only need to be targeted once; tactical cyber
operations must provide permanent covert access to a hacked system”. Because
they need significant preparatory and enforcement time, nowadays cyber actions
are mostly perceived as strategic decisions. From the tactical level, the fact
that they require a constant spatial proximity with the adversarial CIS
infrastructure make their execution under traditional circumstances very
challenging; and this is exactly the point where the tactical cloud and
digital tactical capabilities entails a promising enabler. Since the research
presented in this paper focused in the tactical edge and its digital tactical
capabilities, it is possible to assume that these conditions will also impact
on the tactical operations at the cyber edge. In this context, TDoS should be
considered as potential effectors operable at offensive and defensive
doctrine-level. In order to contribute to their adoption, as well as for the
better understanding of their potential, the following introduces some key
considerations on their application at offensive and defensive thinking
processes.
### 6.1. TDoS in Offensive Thinking
The historic of military thinking has been mostly approached by different ways
and variations of offensive thinking, most of them inherited from the widely
adoption of OODA loop based C2 processes; as OODA is driven by the idea of
getting inside the enemy’s decision cycle (Weissmann, 2019). This loops has
successfully applied defensively, but requiring to observe the attacker
actions and with the requirement of disposing the capability of fast orient,
decide and taking (counter) actions, in this way presenting an effective
prevention model. But as discussed in (Kallberg and Cook, 2017), TDoS
operations and the nature of the cyberspace rely on assumptions that suggest
the need for extending the conventional offensive models, which shall
accommodate to the doctrine on Offensive Cyberspace Operations (OCO) (NATO,
[n. d.]a). Accordingly, TDoS attack capabilities shall create fires in and
thorough cyberspace, occasionally leading to desired physical alterations, the
latter potentially deriving in cascading effects (including collateral
effects) in the rest of kinetic warfare domains. These actions shall encompass
a number of task, action and processes (e.g. targeting, coordination,
deconfliction) according to the enforced fire doctrine (Armed Forces of the
United States, [n. d.]), typically aiming on jeopardizing the sustainability
of adversary digital tactical capabilities, or triggering first-order
sustainability effects to initiate carefully controlled cascading situations
at strategical, operational or tactical levels.
TDoS fires may deny (degradate, disrupt, destroy) or manipulate controls or
information towards cause deception, decoying, conditioning, spoofing,
falsification, and other similar effects. Layton (Layton, 2018) discussed four
different illustrative approaches for offensive thinking applications from
tactical clouds at 5th air warfare operations, which can easily benefit from
TDoS effectors: 1) Attack Adversary Sensing Grids: offensive TDoS can
contribute to reduce their autonomy, degrade their capability of collecting
and managing information, manipulate their observations, and from the latter,
trigger cascade hybrid impacts (loss of confidence, propaganda, jeopardizing
supply chains, injection of noise for diffusing the information from which the
operator constructs the operational image, etc.). 2) Attack an Adversary’s
Long-Range Strike Systems: Layton conceptualized the strike systems as an
adversary grid of short-range nodes but considerably fewer long-range strike
nodes. Accordingly, long-rage nodes became a practical target set for friendly
forces to engage and achieve meaningful tactical results, which infrastructure
and CIS level assets can be targeted by each of the offensive actions
described above.
Layton also remarked that long-range strike are more expensive and difficult
to replace, typically depending of more complex monitoring and analytical
capabilities. 3) Rapidly Attrite Adversary Forces: This is one of the most
challenging scenario discussed by Layton , where higher loses may be affected
in order to gain a fast victory, typically by making simultaneous attacks
converge as parallel air warfare. Significant intelligence on the adversarial
tactical cloud is needed, and cyber effectors shall be able to appropriately
scale towards absorb combat losses to continue with the tempo. This scenario
will mostly require TDoS capabilities able to deplete the resources of the
attacked cloud and thus forcing tactical decisions (U-TDoS, I-TDoS) while
feeding information able to create confusion and disguise the rest of the
convergence actions (I-TDoS, O-TDoS, etc.). Finally, Layton discussed the 4)
Force Horizontal Escalation offensive scenario, which is the closer to the
TDoS concept. In this context, TDoS may force adversarial capabilities do
distribute over the theater of operations and/or move from specific terrains,
so allies can exploit the triggered situation for reorient their efforts and
address high-value regions. Socio-technical TDoS actions and attacks reducing
the autonomy, energy or digital resources of the tactical cloud and/or its
sensors grids entail promising support for these actions. More detailed
examples of TDoS offensive thinking against tactical clouds can be found in
the Concepts of Operation presented at the next section.
### 6.2. TDoS in Defensive Thinking
The adoption of a defensive thinking posture may be subject to different
conditions, including military superiority, tactical restrictions or strict
Rules of Engagement (ROE). As occurs with conventional cyber situations,
attackers have the advantage of surprise, and plenty of time to gather the
cyber threat intelligence needed to success on the TDoS actions (enumeration
of attack surfaces, discovery of vulnerabilities, customization of exploits,
etc.) (De Santo et al., 2021). On the other hands, defenders tend to decide
the cyber operational context (hardening, decoys, passive security procedures,
etc.) and can recall intelligent on the attacker modus operandi (Layton,
2018). As defined at the UK AJP-5 (NATO, [n. d.]c) “in defensive operations,
the defending force reaches its culminating point when it no longer has the
capability to mount a counter offensive or defend successfully and needs to be
reinforced, disengaged or withdrawn to avoid defeat”. Transposed to the
tactical cloud concept, the goal is to defeat the threat of a specific
adversary and/or return the cloud or the affected digital tactical services to
a secure and functional state.
Similarly to the NATO JP3-20 (NATO, [n. d.]a), defensive actions on tactical
clouds may entail Internal Defensive Measures (IDM), Defensive Response
Actions (DRA), or the combination of both of them at owned or ally cyberspace.
The first occur within the defended edge infrastructure and capabilities, are
focalized into dynamically reconfirm or reestablish the security of degraded,
compromised or threatened assets by rerouting, restoring, isolation, decoying,
etc. In (Sotelo Monge et al., 2019) passive defensive tactics able to discover
and thwart denial of economical sustainability attacks against commercial
cloud environment were introduced. Similarly to other state of the art
solutions, they focused on prevent the fraudulent scaling of the computational
resources expenditures by identifying “lazy and unproductive” virtual machines
while isolating those that may contribute to the horizontal propagation of the
threats. Although it is out of the scope of this paper to propose and discuss
potential anti TDOs procedures and algorithmic, it has sense to hypothesize
that related solutions can apply against the U-TDoS and D-TDoS modus operandi
introduced at Section 3. In particular, U-TDoS defense shall focalize on
preventing and detecting fraudulent and unexpected maintenance issues on the
deployed digital tactical capabilities; while D-TDoS mitigation may address
the optimization of the digital tactical service instantiation and the dynamic
removal/recovery of those fraudulent instantiated.
On the other hand, DRAs actions are characterized by taking place out of the
defended portion of the edge, and without authorization of the owner or
responsible of the external assets. These defensive procedures tends to
require very precise objectives, strict ROEs and a proper justification, since
they may be confused with cyber offensive actions or settled out of just ad
bellum. Active defence against TDoS shall focalize on the sources of the
hostile activities (which may be a network, hostile digital tactical
capabilities, etc.) and block, trace, neutralize, etc. the enemy physical or
digital assets linked to the TDoS attack affection. General purpose active
response procedures may serve for this aim, while more specific actions may
explore SON enablers towards orchestrate quarantine regions, sinkholes, etc.
guided from isolate the source regions within the external CIS edge. Active
TDoS may attempt to scale to socio-technical aspect at tactical level (I-TDoS)
or vertically propagating up to organizational or strategic levels (O-TDoS,
E-TDoS)
Figure 4. Hybrid TDoS fires agaisnt adversarial ISR operations
## 7\. Cases of study
In the grounds of the COPD doctrine (NATO, 2010), this sections embraces the
purpose of defining illustrative Concepts of Operation (CONOPs) for
hypothesized tactical offensive/defensive capabilities able to develop TDoS
situations on adversarial Combat Clouds. The concepts explore to jeopardize
key elements of fusion warfare and the Joint Aerial Layer Network (JALN)
(Arbiv et al., [n. d.]) linked to C3 services critical to the proper
achievement of tactical goals. In order to facilitate the understanding to the
Tactical Cloud applications and their direct dependency with the
sustainability of their digital tactical capabilities, different scenarios
have been developed assuming hybrid warfare (Wither, 2016), proxy warfare
(Fox, 2019) and asymmetrical warfare (Bruneau, 2017) situations.
As remarked in Section 1, it is important to highlight that the introduced
cases of study are fictitious scenarios with a mere didactic purpose, where
most of the assets and processes inherent in related military operations have
been omitted for simplicity and in the shake of learning, only describing the
technological and procedural aspects directly linked to the TDoS situations.
As in the previous sections of the project, any deductible Rules Of Engagement
(ROE), jus ad bellum, or jus in bello premise do not intentionally align with
any particular international agreement. Factions, terrains and actors
described do not correspond in any way to real entities.
### 7.1. CONOP#1: Jeopardizing the adversarial energy efficiency at ISR
operations
Background: A peace agreement between two confronted nations has resulted in a
fragile ceasefire. The ally faction strategically decided to take the
opportunity for secretly rearming reinforce their anti-aircraft artillery
systems with a novel target development system. The ally intelligence reported
that the adversarial faction is conducted ISR actions in the surrounding of
the anti-craft effectors supported by a Tactical Cloud.
Current situation: The confronted factions have a common military doctrine. In
both cases the humanitarian situation is approaching crisis proportions.
* •
Own Forces. The ally forces are less technological advanced. The adversarial
reached air, space, and sea superiority, but ally forces have land
superiority.
* •
Co-operating and neutral actors. Civilian supporters may cooperate in easy
surveillance and logistic operations.
* •
Assumptions. 1) Conventional adversarial air ISR operations are not capable of
discovering the new anti-craft effectors. However, certain digital tactical
capabilities may reveal these assets if properly deployed at the edge by a
Tactical Cloud. 2) The hybrid computing services that enable the Tactical
Cloud are vulnerable against certain W-EDoS attacks, which can be enforced
from conventional ITC infrastructure. 3) The mobile tactical nodes are feed by
short/mid duration electronic batteries, which energetic consumption varies on
the workload of the C3 services they enable.
* •
Limitations and constraints. The ally forces have not a Tactical Cloud nor
physical way to disrupt the operations of the adversarial digital tactical
nodes.
* •
Tactical end-state and objectives. The end-state is achieved when the anti-
aircraft effectors are properly rearmed. The tactical objective of the
operation is to prevent that de digital tactical services discover the
rearming operations.
Mission: The digital tactical capabilities able to discover the rearming anti-
aircraft artillery shall be denied or delayed enough time for properly
completing the rearming actions without being notified by the adversarial ISR
actions (see Fig. 4).
Execution: The commander purpose is to delay the activities of the digital
tactical capabilities as long as needed for rearming of the anti-aircraft
artillery systems without being discovered by the adversarial. With this
purpose, the commander will conduct U-TDoS by firing W-EDoS attacks to the
technological infrastructure that supports the adversarial digital tactical
capabilities. Their overload shall increase their energetic dependence, thus
forcing the need for more frequent recharge and battery provisioning, which
will delay the tactical activities the enable.
Tasks: 1) Enumeration and exploitation of the cyber attack surfaces of the CIS
infrastructure that enables the adversarial tactical nodes. 2) Exploitation of
the identified vulnerabilities and fire of W-EDoS cyber attacks against them.
3) Coordination with the command of the anti-aircraft artillery rearming
operation towards ensure that the cyber fires are prolonged until the end-
state is reached. 4) Remove any evidence about this cooperation, and the
offensive modus operandi.
Coordination instructions: Cyber Threat Intelligence is dynamically gathered
and distributed between the participant cyber commands. Information about the
status of the rearming actions is shared between their commanders. A
collaborative cyber situational awareness is developed, including a join COP.
The liaison, engagement rules, communication procedures and reporting systems
are those comprised by the allied doctrines.
Integrated support systems: Cyber fires are commanded from the ally cyber
command HQ.
### 7.2. CONOP#2: TDoS against massive disinformation proxy tactics
Background: Two factions are in long-term confrontation despite the bad
societal perspective of the situation. Consequently, they use to rely on
proxy-war tactics like funding, military training, arming, etc. triggering the
hostile actions by co-operating states and related stakeholders. In this
context, local adversarial militias are called to arms through broadcasted
cyber propaganda, which among others is issued by digital tactical
capabilities serving a Tactical Cloud deployed on the conflict zone. The
propaganda is delivered by spam services able to discover and exploit common
vulnerabilities on the Data Link and Network Layers (Ethernet, ZigBee, WLAN,
etc.) (Manesh and Kaabouch, 2019), injecting them on the civilian
technological devices.
Current situation: The direct confronted factions have a common military
doctrine. Their proxy cooperators are in a direct hostile situation that
involves both military and civilian actions.
* •
Own Forces. The proxy cooperators are in a similar technological capacitation.
The superiority of the cyberspace is in dispute, as is the case of that on the
rest of the operational domains.
* •
Co-operating and neutral actors. The ally proxy state facilitates its ICT
communication infrastructure, which among others includes 5G capabilities
connecting large metropolitan areas (Barona López et al., 2017).
* •
Assumptions. 1) The enumeration and intelligence services reported that the
adversarial propaganda systems dynamically detect vulnerable end-to-end
metropolitan communications, and inject the propaganda as man-in-the-middle
actions. 2) The propaganda reached the civilians via unwanted ads, pop-ups and
other classical spam actions. 3) The targeted metropolitan areas have large
Cloud-Ran (C-RAN) infrastructure with real time network virtualization
capabilities (Tang et al., 2019). 3) The C-RAN virtualizations horizontal-
scale, so during the TDoS actions they shall not suffer computational nor
energy restrictions. 4) The adversarial Tactical Cloud nodes have strong
computational limitations, which are overcomed by instantiation of on demand
digital tactical services thorough the edge.
* •
Limitations and constraints. The ally forces have not a Tactical Cloud nor
DDoS/jamming capabilities able to fight back or prevent the emission of cyber
propaganda.
* •
Tactical end-state and objectives. The end-state is achieved when the cyber
propaganda stops to reach the civilian population. The mission objectives is
to prevent that the emitted cyber propaganda onboard on civilian devices.
Mission: The ally cyber commands shall cooperated with the proxy ally state
towards weaponing its available 5G infrastructure towards TDoS-based mitigate
the propaganda emitted by the adversarial Combat Cloud (see Fig. 5).
Execution: The commander purpose is to dissipate the adversarial cyber
propaganda before it reaches the civilian devices by exploiting D-TDoS. In
particular, the existing C-RAN capabilities of the proxy state shall be able
to procedural generate decoy networks and end-point behaviors, so the Tactical
Cloud direct their activities against them rather than the real civilians
communications. If the number of decoys is large enough 1) most of the
propaganda will be redirected to cyber sinkholes, and 2) the Tactical Cloud
will need to expand enough for covering most of the faked civilian assets.
Since its scalability is limited, this will lead to a D-TDoS situation.
Tasks: 1) Synchronization with the proxy state cyber commands. 2) Enumeration
of the adversarial propaganda actions and the behavior of its enabling
Tactical Cloud. 3) Development and fire of decoy networks, end-users and ICT
virtual infrastructure attempting to cause a larger deployment on digital
tactical nodes. 4) Sinkhole the propaganda that reach the faked assets. 5)
Remove any evidence about this cooperation, and the offensive modus operandi.
Coordination instructions: Cyber Threat Intelligence is dynamically
distributed between the participant cyber commands and the proxy effectors. A
collaborative cyber situational awareness is developed, including a join COP.
Integrated support systems: Cyber fires are supported by the ally cyber
command HQ. However they are directly commanded by the proxy state Minister of
Defence (MoD).
Figure 5. TDoS against massive disinformation proxy tactics
### 7.3. CONOP#3: TDoS fires supported by air combat operations
Background: Two rival factions are engaged in a short term conflict, which
included direct and proxy hostile actions. Their military capabilities are
symmetrical, and both are able to combine conventional kinetic actions with
operations on the cyberspace. Both actions have symmetrical Tactical Cloud
capabilities. The ally intelligence services reported that the adversarial is
preparing a big drone offensive against critical terrains thorough the
conflict area
Current situation: The direct confronted factions have similar military
doctrine.
* •
Own Forces. The direct confronted factions have similar technological
capacitation and effectors. The ally faction is closed to reach the
superiority of the cyberspace. On the opposite, the adversarial faction is
closer to reach air superiority. The rest of the operational domains are under
balanced dispute.
* •
Co-operating and neutral actors. There are not potential cooperators nor
neutral actors. Military engagements are far from civilian actors, typically
under BLOS conditions.
* •
Assumptions. 1) The foreseen adversarial offensive will be enforced by a swarm
of Remotely Piloted Aircraft Systems (RPAS) (Perez-Castan et al., 2019). 2)
The RPAS will rely on digital tactical services provided by a Tactical Cloud,
that among others will enable their air-to-air communications and C2 services.
3) Under close combat engagements, the digital tactical services will
prioritize enabling real-time communications and C2 actions above cyber
defence and/or other cross-cutting services (Heinl, [n. d.]). 4) Due to the
BLOS condition, the adversarial Tactical Cloud vulnerabilities are only
exploitable by close dynamic tactical nodes, this disabling any potential
counteraction from fixed conventional ITC infrastructure. 5) Due to the
adversarial air superiority, the ally air forces are only able to delay the
adversarial of reaching its goals, which suggests the need for a joint action.
* •
Limitations and constraints. The enforcement of jamming and denial of service
related CoAs is not recommended by the analysts, since it may jeopardize the
ally defence communications, and would not be able to cover the entire space
of operations.
* •
Tactical end-state and objectives. The end-state is achieved when the
adversarial air actions are neutralized or no longer risks the targeted key
terrains. The mission objective is to disrupt the foreseen adversarial
offensive actions in order to protect their targeted terrains.
Mission: A joint operation between the air forces and the cyber commands shall
defence the key terrains targeted by the adversarial air operations (Fig. 6).
Execution: A joint defence between ally air forces and cyber commands shall
contain the RPAS threat until a close combat situation exposes exploitable
attack surfaces on the adversarial digital tactical capabilities. When this
occurs the ally Tactical Cloud will onboard and deploy cyber weapons able to
trigger W-EDoS and I-EDoS on the adversarial VNFs orchestration services that
enable C2 and air-to-air communications between the RPAS that integrate the
hostile swarm. The denial of the adversarial tactical sustainability shall
force it to retire. If the tactical deployment of cyber weapons on the ally
Tactical Cloud is fast and effective enough, when the enemy retires the ally
air forces will have contained the kinetic impact on the protected terrains.
Tasks: 1) Synchronization between air and cyber effectors. 2) Detection of the
RPAS-driven large-scale offensive against key ally terrains. 3) enforcement of
air close combat situations in order to press the adversarial Tactical Cloud
towards prioritizing Real-time and C2 communication digital tactical services
above those related with cyber defense. 4) Enumeration of the cyber
vulnerabilities of the adversarial Tactical Cloud. 5) Exploitation of the
identified cyber attack surfaces towards causing W-EDoS and I-EDoS. 6) Extend
their propagated TDoS situations as much as possible in order to force the
adversarial retirement. 7) Removal of evidences of the conducted cyber
operations.
Coordination instructions: A collaborative cyber situational awareness is
developed, including a join COP. The liaison, engagement rules, communication
procedures and reporting systems are those comprised by the allied doctrines.
Integrated support systems: Cyber fires are commanded from the ally cyber
command HQ, and air actions are commanded from the air command HQ. They are
coordinated from a join operaton HQ.
Figure 6. TDoS fires supported by air combat operations
### 7.4. CONOP#4: Jeopardizing the Digital Supply Chain
Background: At a mid-term conflict between an ally and adversarial faction,
human intelligence sources reported an incoming enemy infiltration, which will
be preceded by advanced ISR actions driven by never-seen-before hostile
digital tactical services enabled by the sensing grid, from an adversarial
tactical cloud. The disruptive digital tactical capabilities are no deployable
from the current adversarial tactical repository of Virtual Functions (VFs),
so they require to be requested and provisioned from a digital supply chain
directly connected to a global “distribution market” (Bondan et al., 2019).
Current situation: The confronted factions have a similar military doctrine
concerning tactical cyberspace operations..
* •
Own Forces. The confronted factions have similar technological capabilities
and effectors. There is not a clear cyberspace superiority. Ally forces have
fixed ICT infrastructure able to reach the adversarial VNF orchestration and
management capabilities. Adversarial digital tactical services coordinated
from them are able to reach an external artefact repository (VF marketplaces),
from which new digital tactical capabilities can be provisioned to the
tactical edge.
* •
Co-operating and neutral actors. . Civilian neutral actor and ICT
infrastructure.
* •
Assumptions. 1) The adversarial disruptive ISR functions may change the curse
of the cyberspace dominance. Once deployed, these capabilities can be
reconfigured and adapted to the operational context. 2) Adversarial digital
tactical capabilities as VNFs link a centralized digital artefact repository
(VF marketplace) with the tactical repository of VFs. 3) The tactical
adversarial tactical cloud is susceptible to I-EDoS attacks, which may allow
to fraudulent instantiate tactical VNFs on the edge. 4) The main adversarial
repository is distributed into a pair of physical clusters. 5) One of the
clusters contained the original VFs that supply the ISR capabilities to be
requested, while the other contains Data Aggregation VNFs (Markakis et al.,
2017) that include a backdoor operable by the ally cyber commands that allow
manipulate the information collected and processed at the edge.
* •
Limitations and constraints. Any physical actions (e.g. electronic warfare)
against the adversarial tactical cloud, the digital supply infrastructure or
their communication processes shall be conducted carefully enough so that the
supply of the fraudulent VF is not affected.
* •
Tactical end-state and objectives. The end-state is achieved when the digital
new ISR effectors deployed at the adversarial tactical cloud can be
manipulated by the ally cyber command. The mission objective is to disrupt the
forthcoming ISR action reported by human intelligence actors, thus
preventing/deterring their subsequent kinetic infiltrations.
Mission: An operation between cyber commands unit operating fixed ICT
infrastructure and the ally tactical cloud shall intercede in the adversarial
ISR operations enabled by digital tactical capabilities, and from a hostile
tactical cloud (see Fig. 7).
Execution: In order to deter the adversarial incursion, the Ally cyber command
with enforce an offensive response action against the digital supply chain of
the adversarial faction, for which the manipulation of the data provided by
its new ISR capabilities is essential. In this context, the Ally cyber command
will weaponize its fixed ICT capabilities towards firing an I-EDoS attack
against the adversarial tactical cloud, which shall increase the computational
cost of processing the multi-sourced ISR data. This shall lead to the
adversarial VM Manager to request Data Aggregation VNFs (Markakis et al.,
2017) from a secondary VFs market, which stores jeopardized images for Data
Aggregation. When the adversarial Orchestrator and VM Manager of the enemy
tactical cloud instantiated them, the Ally cyber commands shall be able to
exploit a backdoor on the corrupted Data Aggregators that allows forging the
perceived information at the edge (Singh et al., 2020). Consequently, the
enemy will acquire feeds leading to a fraudulent operational picture
manipulated towards persuading the infiltration.
Tasks: 1) Synchronization between the cyber command effectors available in
range to the adversarial edge. 2) I-EDoS attack against the enemy VFN
Orchestrator. 3) Wait for the enemy deployment of jeopardized data aggregation
VNFs on their edge. 4) Exploitation of the backdoors on the jeopardized data
aggregation VNFs. 5) Cyber sensed data poisoned aiming on infer a wrong
operational picture. 6) Once the enemy infiltration is canceled, removal of
evidences of the conducted cyber operation.
Figure 7. TDoS and the Digital Supply Chain
Coordination instructions: Information exchange on cyber and human
intelligence, and the Ally cyber ISR actions for confirming both, the
instantiation of jeopardized VNFs and the cancelation of the enemy
infiltration, are distributed between the participants and effectors. The
liaison, engagement rules, communication procedures and reporting systems are
those comprised by the allied doctrines.
Integrated support systems: Cyber fires and cyber ISR are commanded from the
ally cyber command HQ. Human intelligence is distributed thorough Federated
Mission Networking (FMN).
Figure 8. TDoS exposing the IR signature of adversarial drones
### 7.5. CONOP#5: Intensifying the IR signature of small hostile effectors
Background: An ally faction will attempt to maneuver around a metropolitan
area dominated by hostile insurgents. Under the assumption of benefit from the
surprise factor, it is expected that the targeted region will be mainly
defenced by light infantry and few field artillery effectors. The insurgents
are escorted by polyvalent small UAVs which digital payload adaptable and
reconfigurable via NFV technologies; which allow customized recognition VFs
onboarding as support to the deployed artillery effectors (Nogales et al.,
2018). The VNF provisioning is enabled by a low intensity tactical cloud
deployed at the insurgent edge on fixed ICT infrastructure.
Current situation: There is high asymmetry between the involved factions,
where insurgents mostly focus on wear out and slow down the opponent with the
expectation of taking advantage of side hybrid warfare actions.
* •
Own Forces. Allies have superiority at all battle dimensions. Insurgents
mostly rely on guerrilla tactics supported by proxy actor technology.
* •
Co-operating and neutral actors. Insurgents are technologically supported by
proxy actors. Since the maneuvers will be conducted on a metropolitan area,
there is a high likelihood of encountering civilian neutral actors on a
complex urban terrain.
* •
Assumptions. 1) If the insurgents discover the ally maneuvers, they will rely
on their smalls UAVs for supporting their field artillery support fires. 2)
UAVs operate on digital tactical services deployed as VNFs, and supplied by
the adversarial tactical cloud. 3) Adversarial digital tactical services are
susceptible to W-EDoS attacks. 4) The adversarial small UAVs reveal a very
weak infrared (IR) signature, so they can barely been deactivated by the ally
anti-drone effectors. 5) The higher the intensity of the VNFs workload, the
higher IR signature visibility of the small UAV that carries them (Lykou et
al., 2020). 6) The beast suitable way for the ally cyber command to reach the
insurgent fixed ICT infrastructure is to deploy a tactical cloud on the
operational metropolitan area. This cloud shall allow to instantiate and
trigger the offensive digital tactical capabilities able to cause TDoS by
W-EDoS on the adversarial services deployed at their edge.
* •
Limitations and constraints. The enforcement of jamming and denial of service
related CoAs is not recommended by the analysts, since it may jeopardize the
ally maneuver communications and impede the own cyber command operations. This
also may derive on cyber EW collateral damage.
* •
Tactical end-state and objectives. The end-state is achieved when the kinetic
maneuvers conclude. The mission objective is to disrupt the insurgent field
artillery operations by exposing their support to targeting drones to the ally
anti-drone systems.
Mission: Ally cyber commands shall support the urban kinetic maneuvers by
facilitating the interception of the adversarial small UAVs. This will be
conducted by triggering TDoS situations on the VFs they enable so their IR
signature became more perceptible by the ally anti-drone effectors (see Fig.
8).
Execution: In order to support the ally maneuver, offensive digital tactical
capabilities will be deployed on an urban tactical cloud. These capabilities
shall allow cyber commands to poison the information exchanges between the
fixed insurgent ICT infrastructure and the small UAV digital payload. The
poisoning shall derived on W-EDoS (Sotelo Monge et al., 2019), so the CPU and
memory usage of the general purpose processors of the drones intensify
(Nogales et al., 2018), thus making their IR signature more detectable
(temperature, emissivity, etc.) (Lykou et al., 2020). Then it is up to the
ally anti-drone fires to intercept the small UAVs, the insurgent field
artillery losing their support to targeting..
Tasks: 1) Synchronization between the cyber command and the kinetic
deployment. 2) Identification of the insurgent fixed ICT infrastructure that
enables their tactical cloud. 3) Deployment of offensive digital tactical
capabilities able to reach the discovered infrastructure. 3) The cyber
offensive effectors poison the telemetric and orchestration communications
between the adversarial VF Manager and orchestrator and the VFs on the drones,
leading to cause W-EDoS. 4) The ally IR sensors detect the small drones, since
their workloads fraudulent intensified intensifying their signature. 5) The
Ally anti-drone effectors neutralize the small UAVs via High Power Microwave
(HPM) able to fire Electromagnetic Pulses (EMP) or guided ammunition (Kang et
al., 2020). 6) The cyber commands remove any evidence of the cyber offensive
action.
Coordination instructions: Information exchange between cyber commands and the
kinetic maneuvers for confirming the presence of hostile small UAVs and the
effectiveness of the cyber actions on them. Anti-drone system shall confirm
the effect of their fires at both cyber command and the kinetic maneuvers. The
liaison, engagement rules, communication procedures and reporting systems are
those comprised by the allied doctrines.
Integrated support systems: Cyber fires and maneuvers are commanded from the
ally cyber command HQ. Cyber Threat Intelligence is distributed thorough
Federated Mission Network (FMN).
## 8\. Discussion
The following discussed the key aspects, main opportunities and challenges
reasoned from the CONOPs involving TDoS illustrated at Section 7.
### 8.1. Rationale about CONOP #1: jeopardizing the adversarial energy
efficiency at ISR operations
The scenario depicts a TDoS encounter aiming on disrupting the adversarial ISR
actions enabled by hostile digital tactical capabilities deployed on an
adversarial tactical cloud; in particular, VFs instantiable from a remote
cyber asset inventory. The narrative of the CONOP introduces a hybrid
situation, where a temporal peace agreement suggests that the opposite
factions operations shall not be attributable (if possible not discoverable),
in this way preventing an escalation of the conflict. The response of the ally
faction against the enemy ISR actions resembles an Offensive Cyber defence
Operation (OCO), where the attacker attempts to degrade the surveillance and
recognition power of the adversarial faction. From a technical perspective,
the allies may rely on both W-EDoS and I-EDoS attacks against the adversarial
cloud, the first of them causing overload of CPU and Memory usage resulting in
additional energy depletion; and the second causing overload in the VF Manager
instantiation capabilities, thus resulting in energy depletion and the
adversarial Cloudlets and data aggregation clusterheads. It is not expected
direct socio-technical TDoS, although the presented mission will disrupt the
adversarial capabilities for acquiring an accurate operational picture.
From a tactical dimension, the narrative is mainly focused on justifying the
opportunity of firing an M-TDoS situation on the adversarial tactical ISR
capabilities, which is expected to lead the enemy decision-makers to enforce
responsive CoAs to manage an unexpected energy depletion. The mission planner
hypothesises that the enemy contingency plan will include to call back the
effectors that carry the ISR sensor, so under a perfectly synchronization
action, the ally artillery can replace its effectors. In general terms, the
mission success depends on a wide set of premises (W-EDoS/I-EDoS able to
jeopardize the enemy VFs, correct predictions on the adversary response,
proper synchronization between ally cyber commands and artillery units, etc.),
which makes it success difficult.
As is usual in cyberspace operations, the mission will require a great
preparation effort (simulation, vulnerability analysis, study of adversarial
doctrine, etc.), but it will depend on fast execution times. Side conditions
that have not been considered on the narrative despite entailing critical
implementation challenges shall response questions like: how to assess the
effectiveness of the involved cyber fires?, how ally forces will notice when
the adversarial ISR effectors will be temporally out of the artillery units
rage? Is there some potential cyber collateral damage? Or how to impede the
attribution of the ally cyber command actions?. These questions can only be
answered when digging into the particular operational conditions, which are
out of the scope of the CONOP#1 description. According to the mission
narrative, it is assumed that joint commanders find feasible way of solving
them.
### 8.2. Rationale about CONOP#2: TDoS against massive disinformation proxy
tactics
The CONOP introduces mission that depicts a Civil-Military Co-operation
(CIMIC) situations, where framed within a proxy warfare conflict, a faction
supported by ally cyber commands shall take advantage of TDoS tactics in order
to counter a massive disinformation campaign orchestrated by and adversarial
tactical cloud. Given the proxy nature of the conflict, the ally cyber command
shall operate undercovered, so it is essential that its implication on the
TDoS fires must not be attributable. The CONOP describes a planned mission
that define Internal Defensive Measures (IDM) able to deplete the adversarial
capabilities for reaching the citizens on the targeted urban areas. The
operational environment is a metropolitan area with fully functional 5G
infrastructure and an operational civilian C-RAN layer, from which allies are
able to effectively deploy defensive digital tactical capabilities. Based on
this, defenders have a greater and more escapable CIS infrastructure, which
shall be able to manage the hostile situation in properly weaponized.
From the technical perspective, W-EDoS attacks are expected to be able to
jeopardize the workload of the adversarial digital tactical capabilities while
I-EDoS may reduce their potential of instantiate and smart distribute
offensive VFs as the operational circumstances demand. However, they may
potentially reveal the interference of the allies on a proxy faction, so the
commander decided to rely on passive defensive actions: i.e. decoys, honeypots
and sinkholes. From a tactical perspective, M-TDoS may contribute to reduce
their workload. But the commander assumes that the most effective
countermeasures will be based on deception triggering D-TDoS situations: it is
assumed that the C-RAN infrastructure can better scale (in terms of
connectivity, bandwidth, CPU usage, etc.) than the adversarial tactical
infrastructure, so in an ideal case, to reach a one-to-one (hostile service –
to – decoy situation) shall guarantee the mission success.
From the socio-technical perspective, the adversarial propaganda attempt to
directly impact on the citizen perception of the conflict, but it will be very
challenging to assess and evaluate the impact of the related enforced actions.
The deception tactics operated by the defenders may misguide the attacker
understanding of the social perceptions of the conflict. for example, they may
lead them to think that the propaganda is being delivered, but that it is not
causing any effect on the population, which can be discouraging and even
negatively impacting on the morale of the attacker and its commitment to the
cause. This may result in directly I-TDoS at both civilian and military
dimensions while causing information chaos tentatively derived by misinformed
decisions and failed communications (O-TDoS). In a higher level, this may
vertically propagate as a rebound effect on the human capital life-cycle and
HUMINT operations (E-TDoS).
Overall, the CONOP#2 is significantly easier to implement that the previous
one, since its success does not heavily depends on strong hypothesis on the
adversarial modus operandi and technological conditions. It does not require
such a great synchronization between ally cyber commands and related actions.
However, it suffers the challenges inherent in CIMIC operations: most of the
necessary infrastructure has mainly civilian use and industrial property,
there is a large likelihood of causing cyber collateral damage, Rules Of
Engagement (ROE) must consider strong civilian conditions, there is an
increased risk of information leaks, some of the civil actors may not be
sufficiently prepared to provide support in this type of scenario, etc..
### 8.3. Rationale about CONOP #3: TDoS fires supported by air combat
operations
The CONOP describes the mission plans for a join operation that makes converge
an ally cyber offensive operation with combat air interception against and
aerial air attack against an ally critical terrain. The presented mission
outline a very challenging situation that demands a perfect synchronization
and coordination of the involved effectors, otherwise potentially resulting in
the loss of significant air assets (in the worse scenario, the loss of a
critical terrain). In this context, it is expected that cyber commands take
advantage of the exposure of attack surfaces on the adversarial digital
tactical capabilities inherent to their resource reallocation for serving the
drone swarm C2 and communications while dogfighting. The mission planner idea
was to trigger a TDoS scenario that force the enemy to call back its effector
due to battery loss earlier than expected, which requires that the ally air
effectors maintain a dogfighting situation the time needed by the cyber
commands to complete their action.
Although the mission is very risky, given the air superiority of the attacker,
the allies are left with few options for keeping their terrain. For example,
the CONOP description assumes limitations concerning the application of EW
Courses of Actions due to the full-spectrum criticality of the electromagnetic
spectrum for operation-level communications. On the other hand, there are not
indications about the availability of anti-drone systems, so it can be assumed
that defenders lack these capabilities. The air interception operations cannot
be sustained for long, so for a technical perspective, W-EDoS and I-EDoS fires
shall convergently and strong impact on the exposed attack surfaces leading to
M-TDoS and D-TDoS in all possible ways. From a socio-technical perspective,
I-TDoS and O-TDoS impacts are expected, since it can be hypothesized that the
unexpected depletion of the drone swam batteries may weaken the individual and
operational trust on these enablers and the offensive operation that depends
on them. This may affect decision-making, mislead the enemy lessons learned,
and derive into E-TDoS cross-cutting impacts at all enterprise levels: order
technical revisions/maintenance of the jeopardized assets, lack of trust on
related industrial providers, questioning the preparedness of the operators
and the awareness of the offensive decision-makers, etc.
### 8.4. Rationale about CONOP #4: Jeopardizing the Digital Supply Chain
The proposed mission illustrates an alternative scenario to the CONOPs above
that prompts the tactical advantages of TDoS when exploiting its synergies
with offensive cyber actions against the cyber digital supply chain, the
latter focalizing on the enemy capabilities of on demand provisioning their
tactical services deployable at the edge as VFs. On these grounds, the
scenario describes a symmetric conflict scenario where the allies attempt to
plan and enforce fire TDoS fires able to reduce the adversarial ISR power,
which is enabled by a distributed sensor grids orchestrated by a tactical
cloud. The mission has a medium difficulty level, and its most challenging
aspect depends on previous actions (preconditions); among them the need for
preliminarily jeopardizing the VF images on the digital cornerstone that
provisions the tactical operation, effectively weaponization of the poisoned
VFs, cover operation of the cyber command that weaken the detection and
attribution of the remote ally cyber operations, trust in the attacker not
discovering the data/processing forgeries and exploiting them in terms of
counterintelligence, or certainly on the VF Manager/orchestrator instantiation
Data Aggregation services against ISR sustainability issues. The CONOP
indicates limitations concerning the use of EW tactics against the adversarial
tactical cloud, so all the cyber actions will consider from data link layer
(OSI model Layer 2) upwards. Collateral damage may be mainly due to the
exposure of sensitive information, and the instantiation of the jeopardized
VFs in contexts beyond the scope of the described operation (which may include
civilian operations, especially if the cyber asset inventory is linked to
dual-use repositories).
From the technical perspective, the mission describes a principal TDoS action
to be enforced by the ally cyber command: to fire undercovered W-EDoS and
I-EDoS against the adversarial ISR VFs until the VF Manager and Orchestrator
decide to call supportive Data aggregation services. The expected effect of
W-EDoS is to increase (not denying) the consumption of computation resources
per VNF, while I-EDoS shall fraudulently increase the horizontal escalation of
the services making grow the ISR expenditures via upkeep of “lazy” services.
From a technical perspective, the first will be linked to M-TDoS on the
adversarial digital tactical capabilities while the second will vertically
propagate a D-TDoS situation.
Since the counterintelligence operation attempts to disrupt the capability of
the enemy of perceiving the operational picture, the presence of I-TDoS
implications in inherent, which among others may include jeopardizing the
individual capabilities for acquiring situational awareness, reducing the
personnel trust on the supportive tactical cloud and its sensor grid, or
triggering cognitive/social disruptions by forgery and injection of sensitive
information and propaganda. The related O-TDoS impacts are linked to the
quality of the operational information exchanges and decisions, and the E-TDoS
consequences will derive on the claim of additional enterprise-level needs
(additional preparedness, replacement of effectors, difficulties in further
recruiting campaigns, etc.) and diverse hybrid threats (politics economics,
social, etc.)
### 8.5. Rationale about CONOP #5: Intensifying the IR signature of small
hostile effectors
The presented scenario depicts an asymmetric conflict where allies have
superiority at all battle dimensions, which is mostly negated by an urban
warfare confrontation against the adversarial insurgent faction. The mission
plan reveals that insurgents are supported by field artillery, which is able
to weaponize recognition small UAVs for guiding its fire life-cycle. The ally
commander relies on a heavy assumption about the side-impact of
deploying/maintaining digital tactical services at the edge: that a
significant increase or oscillation on the VFs CPU/Energy efficiency will make
more visible their IR signature. This hypothesis is supported by state-of-the-
art publications and simulation, so it has sense to infer that TDoS fires may
offensively trigger these variations, resulting in a tactical advantage for
the ally anti-drone effectors. The mission describes a related cyber offensive
operation as support to a kinetic deployment via reducing the adversarial
field artillery power.
The CONOP does not provides evidence about how UAVs may support artillery
(EO/IR target identification, damage assessment, etc.) nor if side-channels
attacks derive from the inferred electromagnetic emanations may reveal
additional information beyond the sole IR signature, the later tentatively
enabling interesting related courses of action. Although direct EW actions
against them are not supported because of the assumption of potentially
damaging the maneuver tactical communications, collateral damage may impact on
civilian infrastructure (cyber), and ”false positives” at IR signature
recognition may derive in fires against neutral targets, resulting in
physical, logical or even personal damage and their social implications. The
CONOP does not state that the cyber offensive shall be enforced undercover,
but this is recommended in order to thwart the lessons learned by the enemy,
which for example may have repercussions on the surprise factor at future
related actions.
The mission entails medium difficulty mostly due to large preparatory
dependencies (simulation and modelling of IR signatures of VFs, enumeration of
attack surfaces and their exploitation for TDoS purpose, etc.), the difficulty
inherent on operating an ally tactical cloud on a metropolitan region, and the
reachability of the cyber targets. From a technical perspective, W-EDoS
attacks will hardly focus on disrupting the VFs workload sustainability while
I-EDoS attacks may only target communication cluster heads, since the
instantiation of “lazy” nodes may create confusion and difficult the IR
signature recognition. In analogy, M-TDoS shall be vertically propagated from
the technical plane due to W-EDoS impacts, while D-TDoS will be linked to
D-TDOs. The mission does not describe direct socio-technical goals, but
significant side effects can be expected. For example, I-TDoS may be featured
by individual lack of trust on the supportive small UAVs, confusion by not
knowing why their assets are being detected or demotivation as the enemy
infantry can’t be properly supported by their field artillery. The
organizational side impacts (O-TDoS) will be mainly linked to inaccurate
decision making and a wrong understanding of the opposite faction
capabilities. At enterprise-level, most E-TDoS consequences will be related to
wrongly involving preparedness and capacitation actions.
## 9\. Conclusions
This paper presented the evolution of the conventional denial of
sustainability attacks towards the emerging Tactical Clouds, highlighting
their typology, potential attack surfaces and impacts when released on the
military edge. In this context, the concept Tactical Denial of Sustainability
(TDoS) have been developed, including two raw technical variations:
Maintenance-based Tactical Denial of Sustainability (U-TDoS) and Deployment-
based Tactical Denial of Sustainability (D-TDoS); and their
horizontal/vertical socio-technical propagations as Enterprise (E-TDoS),
Organizational (O-TDoS) and Individual (I-TDoS) levels. The paper also
revealed the close synergy between TDoS and energy efficiency situations, as
well as how the later may jeopardize the tactical availability of tactical
clouds.
Their potential causality and impact on tactical military operations have been
highlighted by five illustrative Concepts of Operation (CONOPS), covering
their weaponization on hybrid, proxy and symmetrical operations. Given the
important gap in the bibliography on related topics, the high level of secrecy
surrounding the Tactical Cloud paradigm, and the novelty of the technological
ecosystem in which it relies on; the scope of this publication has been to
introduce the aforementioned topics to both military and general public with
the clear purpose of incenting further research and further exploring the
potential of the Tactical Cloud concept in the modern conflict scenarios.
Because of this, many aspects of particular interest to the authors have not
been treated as thoroughly as they would have liked, including: to describe in
more detail the related ICT enablers and their inter dependencies, to explain
related vertical/horizontal TDoS propagation assessment capabilities, or to
in-depth review the potential tactical and CIS Courses of Action (CoAs)
against adversarial actions trying to jeopardize the sustainability of the
digital tactical assets. They will be developed as future research, and
relaying on particular military capacitation.
## Disclaimer
The contents reported in the paper reflect the opinions of the authors and do
not necessarily reflect the opinions of the respective agencies, institutions
or companies.
## References
* (1)
* 3GPP (2008) 3GPP. 2008. 3GPP TS 32.500 Self-Organising Networks (SON): Concepts and requirements. www.3gpp.org/DynaReport/32500.htm.
* Ahmed et al. (2016) M. Ahmed, A. Naser, and J. Hu. 2016. A survey of network anomaly detection techniques. _Journal of Network and Computer Applications_ 60 (2016), 19–31.
* Arbiv et al. ([n. d.]) S. Arbiv, T. Amin, T. Goff, D. Street, I. Pedan, L. Bresser, T. Gibbons, B.N. Cheng, and C. Timmerman. [n. d.]. Data Collection and Analysis Framework for Mobile Ad Hoc Network Research. In _Proceedings of the IEEE Military Communications Conference (MILCOM)_.
* Armed Forces of the United States ([n. d.]) Armed Forces of the United States. [n. d.]. Joint Publication 3-09, J̈oint Fire Support.̈
* Aujl and Kumar (2018) G.S. Aujl and N. Kumar. 2018. MEnSuS: An efficient scheme for energy management with sustainability of cloud data centers in edge-cloud environment. _Future Generation Computer Systems_ 86 (2018), 1279–1300.
* Baig et al. (2016) Z.A. Baig, S.M. Sait, and F. Binbeshr. 2016. Controlled access to cloud resources for mitigating Economic Denial of Sustainability (EDoS) attacks. _Computer Networks_ 97 (2016), 31–47.
* Barona López et al. (2017) L.I. Barona López, A.L. Valdivieso Caraguay, J. Maestre Vidal, and M.A. Sotelo Monge. 2017. Towards Incidence Management in 5G Based on Situational Awareness. _Future Internet_ 9(1), 3 (2017).
* Barret and Mansfield (2019) D. Barret and A. Mansfield. 2019. The Challenge and Opportunities of Standing on Cloud – Finding our Warfighting Advantage. _The Cyber Defense Review_ 4(1) (2019), 15–22.
* Bawa and Manickam (2015) P.S. Bawa and S. Manickam. 2015. Critical Review of Economical Denial of Sustainability (EDoS) Mitigation Techniques. _Journal of Computer Science_ 11(7) (2015), 855–862.
* Bhingarkar and Shah ([n. d.]) A.S. Bhingarkar and B.D. Shah. [n. d.]. A survey: Securing cloud infrastructure against edos attack. In _Proceedings of the International Conference on Grid Computing and Applications (GCA)_.
* Bondan et al. (2019) L. Bondan, M.F. Franco, L. Marcuzzo, G. Venancio, R.L. Santos, R.J. Pfitscher, E.J. Scheid, B. Stiller, F.D. Turk, E.P. Duarte, A.E. Schaeffer-Filho, and L.Z. dos Santos, C.R.P.and Granville. 2019. FENDE: Marketplace-Based Distribution, Execution, and Life Cycle Management of VNFs. _IEEE Communications Magazine_ 57(1) (2019).
* Brantly ([n. d.]) A.F. Brantly. [n. d.]. The cyber deterrence problem. In _Proceedings of the 10th International Conference on Cyber Conflict (CyCon)_.
* Bremler-Barr et al. ([n. d.]) A. Bremler-Barr, E. Brosh, and M. Sides. [n. d.]. DDoS attack on cloud auto-scaling mechanisms. In _Proceedings of the 17 IEEE Conference on Computer Communications (INFOCOM 2017)_.
* Bretl et al. (2019) T. Bretl, l. Righetti, and R. Madhavan. 2019. Epstein, Project Maven, and Some Reasons to Think About Where We Get Our Funding [Ethical, Legal, and Societal Issues]. _IEEE Robotics & Automation Magazine_ 26(4) (2019), 8–13.
* Broeders, D. and Sergei, B.and Georgieva, I. (2019) Broeders, D. and Sergei, B.and Georgieva, I. 2019. Foreign Intelligence in the Digital Age. Navigating a State of ‘Unpeace’. https://ssrn.com/abstract=3493612. The Hague Program For Cyber Norms Policy Brief.
* Brogaard (2018) L. Brogaard. 2018\. Business Value in Public-Private Partnerships: The Positive Impact of Trust and Task-Relevant Competencies on Business Outcomes in PPPs. _International Public Management Journal_ (2018), 1–26. https://doi.org/10.1080/10967494.2018.1457107
* Bruneau (2017) E. Bruneau. 2017\. The enemy as animal: Symmetric dehumanization during asymmetric warfare. _PLoS One_ 12(7) (2017). https://doi.org/dx.doi.org/10.1371%2Fjournal.pone.0181422
* Capossele et al. (2016) A. Capossele, V. Cervo, C. Petrioli, and D. Spenza. 2016\. Counteracting Denial-of-Sleep Attacks in Wake-Up-Radio-Based Sensing Systems. In _Proceedings of the IEEE International Conference on Sensing, Communication, and Networking_. London, UK, 1–9.
* Cappanera et al. (2019) P. Cappanera, F. Paganelli, and F. Paradiso. 2019\. VNF placement for service chaining in a distributed cloud environment with multiple stakeholders. _Computer Communications_ 133 (2019), 24–40.
* Center for Strategic Leadership (CSL), US Army War College (2016) Center for Strategic Leadership (CSL), US Army War College. 2016. rethinking sovereignity in the context of cyberspace. https://www.hsdl.org/?abstract&did=802916.
* Center of Naval Analysis, U.S. Navy, Department of Defense (DOD) (2017) Center of Naval Analysis, U.S. Navy, Department of Defense (DOD). 2017. Insights for the Third Offset: Addressing Challenges of Autonomy and Artificial Intelligence in Military Operations. https://apps.dtic.mil/sti/citations/AD1041043.
* Chen et al. (2016) B.N. Chen, G. Kuperman, P. Deutsch, L. Mercer, and A. Narula-Tam. 2016. Group-centric networking: addressing information sharing requirements at the tactical edge. _IEEE Communications Magazine_ 54(10), 145–151 (2016).
* Chi et al. (2018) F. Chi, X. Wang, W. Cai, and V.C.M. Leung. 2018\. Ad-Hoc Cloudlet Based Cooperative Cloud Gaming. _IEEE Transactions on Cloud Computing_ 6(3), 625–639 (2018).
* Chowdhury et al. (2020) N.H. Chowdhury, M.T.P. Adam, and T. Teubner. 2020. Time pressure in human cybersecurity behavior: Theoretical framework and countermeasures. _Computers & Security_ 97 (2020). https://doi.org/10.1016/j.cose.2020.101963
* Cirincione and Verma (2019) G. Cirincione and D. Verma. 2019. Federated machine learning for multi-domain operations at the tactical edge. In _Proceedings of SPIE 11006_. Baltimore, MA, US. https://doi.org/10.1117/12.2526661
* Danyk et al. (2017) Y. Danyk, T. Maliarchuk, and C. Briggs. 2017. Hybrid War: High-tech, Information and Cyber Conflicts. _Connections_ 16(2) (2017), 5–24.
* Daton Medenou et al. (2020) Calzado Mayo V.M. Garcia Balufo M. Daton Medenou, R., M. Paramo Castrillo, F.J. Gonzalez Garrido, A. Luis Martinez, D. Nevado Catalan, A. Hu, D.S. Rodriguez-Bermejo, J. Maestre Vidal, G.R. Pasqual De Riquelme, A. Berardi, P. De Santis, F. Torelli, and S. Llopis Sanchez. 2020. CYSAS-S3: a novel dataset for validating cyber situational awareness related tools for supporting military operations. In _Proceedings of the 15th International Conference on Availability, Reliability and Security (ARES)_. Dublin, Ireland, 1–9.
* De Santo et al. (2021) D. De Santo, C.S. Malavenda, S.P. Romano, and C. Vecchio. 2021. Exploiting the MIL-STD-1553 avionic data bus with an active cyber device. _Computers & Security_ 100 (2021). https://doi.org/10.1016/j.cose.2020.102097
* Deng et al. (2019) X. Deng, Z. Yu, R. Tang, X. Qian, K Yuan, and S. Liu. 2019. An Optimized Node Deployment Solution Based on a Virtual Spring Force Algorithm for Wireless Sensor Network Applications. _Sensors_ 19(8), 1817 (2019).
* Downes (2018) C. Downes. 2018\. Strategic Blind–Spots on Cyber Threats, Vectors and Campaigns. _The Cyber Defense Review_ 3 (2018), 79–104.
* Ekelhof (2018) M.A.C. Ekelhof. 2018\. Lifting the Fog of Targeting Autonomous Weapons and Human Control through the Lens of Military Targeting. _Naval War College Review_ 71(3) (2018), 61–94.
* ETSI (2014) ETSI. 2014. GS NFV-MAN 001 V1.1.1. Network Functions Virtualisation (NFV): Management and Orchestration. https://www.etsi.org/deliver/etsi_gs/NFV-MAN/001_099/001/01.01.01_60/gs_NFV-MAN001v010101p.pdf.
* European Commission (2019) European Commission. 2019\. Ethics Guidelines for Trustworthy AI. https://ec.europa.eu/digital-single-market/en/news/ethics-guidelines-trustworthy-ai.
* European Union Institute for Security Studies (2020) European Union Institute for Security Studies. 2020. Digitalising Defence: Protecting Europe in the age of quantum computing and the cloud. https://www.iss.europa.eu/content/digitalising-defence.
* Fox (2019) A.C. Fox. 2019. Conflict and the Need for a Theory of Proxy Warfare. _Journal of Strategic Security_ 12(1) (2019), 44–71.
* Gupta and Pathak (2016) C. Gupta and P. Pathak. 2016. Movement based or neighbor based technique for preventing wormhole attack in MANET. In _Proceedings of IEEE Symposium on Colossal Data Analysis and Networking_. Indore, India.
* Heinl ([n. d.]) C.H. Heinl. [n. d.]. Artificial (intelligent) agents and active cyber defence: Policy implications. In _Proceedings of the 6th International Conference On Cyber Conflict (CyCon 2014)_.
* Hess, J. and Kiser, A. and Bouhafa, E.M. and Williams, S. (2019) Hess, J. and Kiser, A. and Bouhafa, E.M. and Williams, S. 2019. The Combat Cloud: Enabling Multidomain Command and Control acreoss the Range of Military Operations. https://apps.dtic.mil/dtic/tr/fulltext/u2/1042210.pdf.
* Hoeben (2017) B.A. Hoeben. 2017\. 5th Generation Air C2 and ISR. _Royal Australian Air Force_ (2017).
* Hoff, C. (2008) Hoff, C. 2008\. Cloud Computing Security: From DDoS (Distributed Denial Of Service) to EDoS (Economic Denial of Sustainability). http://rationalsecurity.typepad.com/blog/2008/.
* Hoff, C. (2009) Hoff, C. 2009\. A Couple of Follow-Ups On The EDoS (Economic Denial Of Sustainability) Concept… http://rationalsecurity.typepad.com/blog/edos/.
* Hui et al. (2017) Kin-Ping Hui, Damien Phillips, and Asanka Kekirigoda. 2017\. Beyond line-of-sight range extension with OPAL using autonomous unmanned aerial vehicles. In _MILCOM 2017-2017 IEEE Military Communications Conference (MILCOM)_. IEEE, 279–284.
* Javonik et al. ([n. d.]) M. Javonik, J. Komarkova, and M. Husak. [n. d.]. Decision Support for Mission-Centric Cyber Defence. In _Proceedings of the 15rd International Conference on Availability, Reliability and Security (ARES)_.
* Kallberg and Cook (2017) J. Kallberg and T.S. Cook. 2017. The Unfitness of Traditional Military Thinking in Cyber. _IEEE Access_ 5 (2017), 8126–8130.
* Kang et al. (2020) H. Kang, J. Joung, J. Kim, J. Kang, and Y. Soo Choo. 2020. Protect Your Sky: A Survey of Counter Unmanned Aerial Vehicle Systems. _IEEE Access_ 8 (2020), 168671–168710.
* Kharlamov (2019) G. Kharlamov, A.and Pogrebna. 2019\. Using human values‐based approach to understand cross‐cultural commitment toward regulation and governance of cybersecurity. _Regulation & Governance_ (2019).
* Layton (2018) P. Layton. 2018\. Fifth-generation air warfare. _Australian Defence Force Journal_ 204 (2018), 23–32.
* Lenders et al. (2015) Vincent Lenders, Axel Tanner, and Albert Blarer. 2015\. Gaining an edge in cyberspace with advanced situational awareness. _IEEE Security & Privacy_ 13, 2 (2015), 65–74.
* Lopes et al. ([n. d.]) R.R.F. Lopes, P.H. Balaraju, and P. Sevenich. [n. d.]. Creating and Handling Ever-changing Communication Scenarios in Tactical Networks. In _Proceedings of the 15th International Conference on the Design of Reliable Communication Networks (DRCN)_.
* Lykou et al. (2020) G. Lykou, D. Moustakas, and D. Gritzalis. 2020. Defending Airports from UAS: A Survey on Cyber-Attacks and Counter-Drone Sensing Technologies. _Sensors_ 20(12), 3537 (2020).
* Maestre Vidal et al. (2018a) J. Maestre Vidal, A.L.S. Orozco, and L.J.G. Villalba. 2018a. Adaptive artificial immune networks for mitigating DoS flooding attacks. _Swarm and Evolutionary Computation_ 38 (2018), 94–108.
* Maestre Vidal et al. (2016) J. Maestre Vidal, A.L. Sandoval Orozco, and L.J. Garcia Villalba. 2016\. Online masquerade detection resistant to mimicry. _Expert Systems with Applications_ 61 (2016), 162–180.
* Maestre Vidal and Sotelo Monge (2018) J. Maestre Vidal and L.J.G. Sotelo Monge, M.A.and Villalba. 2018\. A Novel Pattern Recognition System for Detecting Android Malware by Analyzing Suspicious Boot Sequences. _Knowledge-Based Systems_ 150 (2018), 198–217.
* Maestre Vidal and Sotelo Monge ([n. d.]) J. Maestre Vidal and M.A. Sotelo Monge. [n. d.]. Framework for Anticipatory Self-Protective 5G Environments. In _Proceedings of the 14th nternational Conference on Availability, Reliability and Security (ARES)_.
* Maestre Vidal and Sotelo Monge (2020a) J. Maestre Vidal and M.A. Sotelo Monge. 2020a. Obfuscation of Malicious Behaviors for Thwarting Masquerade Detection Systems Based on Locality Features. _Sensors_ 20(7), 2084 (2020).
* Maestre Vidal and Sotelo Monge (2020b) J. Maestre Vidal and M.A. Sotelo Monge. 2020b. Denial of sustainability on military tactical clouds. In _Proceedings of the 15th International Conference on Availability, Reliability and Security (ARES)_. Dublin, Ireland, 1–9.
* Maestre Vidal et al. (2018b) J. Maestre Vidal, M. A. Sotelo Monge, and L. J. García Villalba. 2018b. Detecting Workload-based and Instantiation-based Economic Denial of Sustainability on 5G environments. In _Proceedings of the 13th International Conference on Availability, Reliability and Security (ARES)_. Hamburg, Germany, Article 50, 50:1–50:8 pages.
* Mandt (2017) Erick J Mandt. 2017\. Integrating cyber-intelligence analysis and active cyber-defence operations. _Journal of Information Warfare_ 16, 1 (2017), 31–48.
* Manesh and Kaabouch (2019) M.R. Manesh and N. Kaabouch. 2019. Cyber-attacks on unmanned aerial system networks: Detection, countermeasure, and future research directions. _Computers & Security_ 85 (2019), 386–401.
* Markakis et al. (2017) E.K. Markakis, K. Karras, N. Zotos, A. Sideris, T. Moysiadis, A. Corsaro, G. Alexiou, C. Skianis, G. Mastorakis, C.X. Mavromoustakis, and E. Pallis. 2017\. FENDE: Marketplace-Based Distribution, Execution, and Life Cycle Management of VNFs. _IEEE Communications Magazine_ 55(7) (2017), 173–179.
* Monaghan (2019) S. Monaghan. 2019\. Countering Hybrid Warfare: So What for the Future Joint Force? _National Defense University Press, PRISM_ 8(2) (2019), 82–98.
* Multinational Capability Development Campaign project (2017) Multinational Capability Development Campaign project. 2017. Countering Hybrid Warfare Project:Understanding Hybrid Warfare. https://www.gov.uk/government/publications/countering-hybrid-warfare-project-understanding-hybrid-warfare.
* NATO ([n. d.]a) NATO. [n. d.]a. Allied Joint Doctrine for cyberspace operation (AJP-3.20).
* NATO ([n. d.]b) NATO. [n. d.]b. Allied Joint Doctrine for the Conduct of Operations (AJP-3).
* NATO ([n. d.]c) NATO. [n. d.]c. Allied Joint Publication for the the Planning of Operations (AJP-5).
* NATO (2010) NATO. 2010. Allied Command Operations: Comprehensive Operations Planning Directive COPD. https://info.publicintelligence.net/NATO-COPD.pdf.
* NATO (2019) NATO. 2019. C3 Taxonomy Baseline 3.1”. AC/322-D(2019)0034. https://www.nato.int/cps/en/natohq/topics_157573.htm.
* Nisha et al. (2016) A. Nisha, V. Vaishali, T. Shivaranjani, and P. Subathra. 2016\. The effect of vampire attacks on distance vector routing protocols for wireless ad hoc sensor networks. In _Proceedings of the IEEE International Conference on Science Technology Engineering and Management_. Chennai, India, 587–594.
* Nogales et al. (2018) B. Nogales, V. Sanchez-Aguero, I. Vidal, and F. Valera. 2018. Adaptable and Automated Small UAV Deployments via Virtualization. _Sensors_ 18(12), 5116 (2018).
* Otaiku (2018) A.A. Otaiku. 2018\. A Framework for Hybrid Warfare: Threats, Challenges and Solutions. _Journal of Defense Management_ 8(3) (2018), 1–13.
* Oubbati et al. (2017) O.S. Oubbati, A. Lakas, F. Zhou, M. Gunes, and M.B. Yagoubi. 2017. A survey on position-based routing protocols for Flying Ad hoc Networks (FANETs). _Vehicular Communications_ 10, 29–56 (2017).
* Perez-Castan et al. (2019) J.A. Perez-Castan, F.G. Comendador, A. Rodriguez-Sand, I.A. Cabrera, and J. Torrecilla. 2019\. RPAS conflict-risk assessment in non-segregated airspace. _Safety Science_ 111, 7–16 (2019).
* Samaras et al. (2019a) C. Samaras, W.J. Nuttall, and M. Bazilian. 2019a. Energy and the military: Convergence of security, economic, and environmental decision-making. _Energy Strategy Reviews_ (2019). https://doi.org/doi.org/10.1016/j.esr.2019.100409
* Samaras et al. (2019b) C. Samaras, W.J. Nuttall, and M. Bazilian. 2019b. Energy and the military: Convergence of security, economic, and environmental decision-making. _Energy Strategy Reviews_ 25, 100409 (2019).
* Satyanarayanan et al. (2013) M. Satyanarayanan, G. Lewis, E. Morris, S. Simanta, J. Boleng, and K. Ha. 2013\. The Role of Cloudlets in Hostile Environments. _IEEE Pervasive Computing_ 12(4), 40–49 (2013).
* Schulze (2020) M. Schulze. 2020\. Cyber in War: Assessing the Strategic, Tactical, and Operational Utility of Military Cyber Operations. In _Proceedings of the 12th International Conference on Cyber Conflict (CyCon)_. Estonia, Estonia.
* Shakhov and Koo (2018) V. Shakhov and I. Koo. 2018\. Depletion-of-Battery Attack: Specificity, Modelling and Analysis. _Sensors_ 18(6), 1849 (2018).
* Silva and Jacob ([n. d.]) F.R.L. Silva and P. Jacob. [n. d.]. Mission-Centric Risk Assessment to Improve Cyber Situational Awareness. In _Proceedings of the 13rd International Conference on Availability, Reliability and Security (ARES)_.
* Singh et al. (2020) J. Singh, A. Refaey, and A. Shami. 2020. Multilevel Security Framework for NFV Based on Software Defined Perimeter. _IEEE Network_ 34(5) (2020), 114–119.
* Singh and Rehman ([n. d.]) S. Singh, P.and Manickam and S.U. Rehman. [n. d.]. A survey of mitigation techniques against Economic Denial of Sustainability (EDoS) ttack on cloud computing architecture. In _Proceedings of the 3rd International Conference on Reliability, Infocom Technologies and Optimization (ICRITO)_.
* Smith et al. ([n. d.]) W. Smith, G. Kuperman, M. Chan, E. Morgan, N. Nguyen, H.and Schear, B. Vu, A. Weinert, and D. Whisman. [n. d.]. Cloud computing in tactical environments. In _Proceedings of the IEEE Military Communications Conference (MILCOM)_.
* Somani et al. (2016) G. Somani, M.S. Gaur, D. Sanghi, and M. Conti. 2016\. DDoS attacks in cloud computing: Collateral damage to non-targets. _Computer Networks_ 109 (2016), 157–171.
* Somani et al. (2017) G. Somani, M.S. Gaur, D. Sanghi, M. Conti, and R. Buyya. 2017. DDoS attacks in cloud computing: Issues, taxonomy, and future directions. _Computer Communications_ 107 (2017), 30–48.
* Song (2018) C. Song. 2018. A System Analysis of the US Army’s human dimension Strategy. In _Proceedings of the International Annual Conference of the American Society for Engineering Management (ASEM)_. Coeur d’Alene, Idaho, USA.
* Sotelo Monge et al. (2017) M.A. Sotelo Monge, J. Maestre Vidal, and L.J.G. Villalba. 2017\. Entropy-Based Economic Denial of Sustainability Detection. _Entropy_ 19(2), 12 (2017).
* Sotelo Monge et al. (2018) M. A. Sotelo Monge, J. Maestre Vidal, and L. J. García Villalba. 2018\. A novel Self-Organizing Network solution towards Crypto-ransomware Mitigation. In _Proceedings of the 13th International Conference on Availability, Reliability and Security (ARES)_. Hamburg, Germany, Article 48, 48:1–48:10 pages.
* Sotelo Monge et al. (2019) M. A. Sotelo Monge, J. Maestre Vidal, and G. Martínez Pérez. 2019\. Detection of economic denial of sustainability (EDoS) threats in self-organizing networks. _Computer Communications_ 145 (2019), 284–308.
* Stevens (2020) T. Stevens. 2020\. Knowledge in the grey zone: AI and cybersecurity. _Digital War_ (2020). https://doi.org/10.1057/s42984-020-00007-w
* Tang et al. (2019) J. Tang, T.Q.S. Quek, T.H. Chang, and B. Shim. 2019\. Systematic Resource Allocation in Cloud RAN With Caching as a Service Under Two Timescales. _IEEE Transactions on Communications_ 67(11), 7755–7770 (2019).
* Thomas et al. (2019) D. Thomas, R. Shankaran, M. Orgun, and W. Hitchens, M.and Ni. 2019\. Energy-Efficient Military Surveillance: Coverage Meets Connectivity. _IEEE Sensors Journal_ 10(10) (2019), 3902–3911.
* Ulrik ([n. d.]) S. Ulrik. [n. d.]. The cyber deterrence problem. In _Proceedings of the 22nd International Command and Control Research and Technology Symposium (ICCRTS): Frontiers of C2_.
* United States Army Combined Arms Center (2015) United States Army Combined Arms Center. 2015. The Human Dimension Strategy. https://caccapl.blob.core.usgovcloudapi.net/web/character-development-project/repository/human-dimension-strategy-2015.pdf.
* U.S. Defense Acquisition University Huntsville United States (2020) U.S. Defense Acquisition University Huntsville United States. 2020. Education, Experience, and Organizational Enhancements Necessary to Develop the Army Acquisition Workforce. https://apps.dtic.mil/sti/citations/AD1106554.
* U.S. Defense Advanced Research Projects Agency (DARPA) ([n. d.]) U.S. Defense Advanced Research Projects Agency (DARPA). [n. d.]. Near Zero Power RF and Sensor Operations (N-ZERO).
* Wang et al. ([n. d.]) J.N. Wang, A. Narula-Tam, and R. Byan. [n. d.]. Interconnecting Heterogeneous MANET Networks at the Tactical Edge. In _Proceedings of the IEEE Military Communications Conference (MILCON)_.
* Wanner and Ghernaouti (2020) B. Wanner and S. Ghernaouti. 2020. Active Defensive Measures in Cyberspace– a Swiss Case Study. _European Cybersecurity Journal_ 6(2) (2020).
* Weissmann (2019) M. Weissmann. 2019\. Mirror, mirror on the wall, who is the most offensive of them all? – Explaining the offensive bias in military tactical thinking. _Defence Studies_ (2019), 170–188. https://doi.org/10.1080/14702436.2019.1599287
* Wither (2016) J.K. Wither. 2016\. Making Sense of Hybrid Warfare. _Connections_ 15(2) (2016), 73–87.
* Yan (2020) G. Yan. 2020. The impact of Artificial Intelligence on hybrid warfare. _Small Wars & Insurgencies_ 31(4) (2020), 898–917.
* Zeng et al. (2010) Y. Zeng, J. Cao, S. Zhang, S. Guo, and L. Xie. 2010. Random-Walk Based Approach to Detect Clone Attacks in Wireless Sensor Networks. _IEEE Journal on Selected Areas in Communications_ 28 (2010), 677–691.
* Zhou et al. (2019) Zejian Zhou, Lijun Qian, and Hao Xu. 2019. Intelligent Decentralized Dynamic Power Allocation in MANET at Tactical Edge based on Mean-Field Game Theory. In _MILCOM 2019-2019 IEEE Military Communications Conference (MILCOM)_. IEEE, 604–609.
|
††thanks: Currently at Computational Physics Inc.
# A Multi-Center Quadrature Scheme for the Molecular Continuum
H. Gharibnejad University of Maryland, Department of Chemistry and
Biochemistry, College Park, MD, USA N. Douguet Kennesaw State University,
Department of Physics, Marietta, GA, USA University of Central Florida,
Department of Physics & CREOL, Orlando, FL, U.SA B. I. Schneider National
Institute of Standards and Technology, Applied and Computational Mathematics
Division, Gaithersburg, MD, USA J. Olsen Aarhus University, Department of
Chemistry, Aarhus, Denmark L. Argenti University of Central Florida,
Department of Physics & CREOL, Orlando, FL, U.SA
###### Abstract
A common way to evaluate electronic integrals for polyatomic molecules is to
use Becke’s partitioning scheme [J. Chem. Phys. 88, 2547 (1988)] in
conjunction with overlapping grids centered at each atomic site. The Becke
scheme was designed for integrands that fall off rapidly at large distances,
such as those approximating bound electronic states. When applied to states in
the electronic continuum, however, Becke scheme exhibits slow convergence and
it is highly redundant. Here, we present a modified version of Becke scheme
that is applicable to functions of the electronic continuum, such as those
involved in molecular photoionization and electron-molecule scattering, and
which ensures convergence and efficiency comparable to those realized in the
calculation of bound states. In this modified scheme, the atomic weights
already present in Becke’s partition are smoothly switched off within a range
of few bond lengths from their respective nuclei, and complemented by an
asymptotically unitary weight. The atomic integrals are evaluated on small
spherical grids, centered on each atom, with size commensurate to the support
of the corresponding atomic weight. The residual integral of the interstitial
and long-range region is evaluated with a central master grid. The accuracy of
the method is demonstrated by evaluating integrals involving integrands
containing Gaussian Type Orbitals and Yukawa potentials, on the atomic sites,
as well as spherical Bessel functions centered on the master grid. These
functions are representative of those encountered in realistic electron-
scattering and photoionization calculations in polyatomic molecules.
Numerical approximation and analysis, Numerical Simulation, Scattering theory,
Photoionization and excitation, Electronic Excitation and Ionization of
Molecules
###### pacs:
02.60.-x, 02.60.Cb, 03.65.Nk, 32.80.-t, 34.50.Gb
## I INTRODUCTION
Numerical integration is required for many electronic structure calculations
where orbitals other than Gaussians are needed Saad _et al._ (2010). Most
modern quantum chemistry packages employ Gaussian-type orbitals (GTOs) to
describe molecular bound states, a computationally efficient choice, since
electronic integrals involving only GTOs may be evaluated analytically Alder
_et al._ (1963); Helgaker _et al._ (2012). Although optimum GTO basis sets
have already been designed to reproduce highly excited Rydberg states Kaufmann
_et al._ (1989), nodeless GTOs are ill-suited to describe continuum functions.
Indeed, extended sets of GTOs can reproduce at most a handful of the
characteristic radial oscillations of continuum functions before becoming
linearly dependent Marante _et al._ (2014).
Modern approaches to represent continuum wave functions in molecular systems
rely on a hybrid basis set that comprises both GTOs, centered on each atom and
on the molecular center of gravity, as well as numerical functions such as
Bessel functions, Coulomb functions Rescigno _et al._ (1995), B-splines
Marante _et al._ (2014, 2017); Mašín _et al._ (2020) or finite-element basis
functions Rescigno _et al._ (2005), to properly describe both the short and
long-range behavior of continuum orbitals. Such hybrid approaches allow one to
take advantage of the capabilities of modern quantum chemistry packages to
compute neutral and ionic bound states but require the electronic integrals
between hybrid functions to be evaluated numerically. Numerical quadrature
algorithms are often easier to code and parallelize than analytic techniques
but can converge slowly due to the sharp variations of molecular orbitals and
interaction potentials in the proximity of the atomic nuclei. These
limitations are particularly severe in single-center expansion techniques
Bishop (1967). To achieve uniform numerical convergence, therefore, it is
essential to employ quadrature grids that selectively cluster at all nuclei
where the wave function does not vanish.
Several techniques have been developed to evaluate hybrid integrals Collins
and Schneider (1983); Gianturco _et al._ (1994); Yip _et al._ (2010, 2014).
The simplest approach consists in expanding any polycentric GTO function in
terms of a monocentric auxiliary basis Mašín _et al._ (2020). As already
noted, this approach converges slowly, since representing the sharp variation
of a molecular orbital in the neighborhood of an off-center nucleus with
acceptable accuracy requires a large set of angular momenta or angular
integration points as well as high radial resolution near each nucleus. As a
consequence, the auxiliary basis or grid must be chosen to cover the whole
molecular region, both close to and far away from the atoms, thus drastically
increasing the number of integration points. An alternative approach is to
confine the numerical basis to a radial range outside a spherical region that
encompasses all the nuclei Marante _et al._ (2017). In this approach, the
hybrid integrals are easy to evaluate, but the Gaussian basis must be able to
reproduce the continuum orbitals in the whole internal region, thus limiting
the size of the molecular system. Neither of these approaches, therefore,
offers a fully satisfactory solution to the construction of scattering
functions for general molecules. In particular, they are poorly suited to
describe molecular dissociation.
To account for the sharp variation of poly-centric molecular wave functions
close to the nuclei, it is possible to separately integrate the region in
close proximity of each nucleus. Along this direction, pioneered by Boys and
Rajagopal Boys and Rajagopal (1965), are partitioning schemes that divide the
molecular space into Voronoi polyhedra, each containing a single atom
Boerrigter _et al._ (1988); Averill and Painter (1989); Pederson and Jackson
(1990); te Velde and Baerends (1992). The integral within the sphere inscribed
in the polyhedron centered on an atom is evaluated using single-center
techniques on the atomic site. While the Voronoi method effectively tackles
the nuclear region, the evaluation of the residual polyhedron integral outside
the sphere is still a challenge, due to the complex shape of the Voronoi
boundary. Furthermore, the use of Voronoi polyhedra is still not suited to the
electronic continuum.
In 1988, Becke has proposed an integration scheme that divides space into
overlapping “fuzzy” Voronoi polyhedra Becke (1988); Becke and Dickson (1988),
i.e., a set of smooth positive weight functions $w_{\alpha}(\vec{r})$, one per
nucleus, defining a partition of unity, $1=\sum_{\alpha}w_{\alpha}(\vec{r})$,
with the additional requirement that each weight function vanishes at all but
one of the nuclear positions $R_{\beta}$,
$w_{\alpha}(\vec{R}_{\beta})=\delta_{\alpha\beta}$. Using Becke’s
partitioning, it is possible to split any electronic integral
$I=\int\mathrm{d}^{3}\vec{r}f(\vec{r})$ into separate atomic components,
$I=\sum_{\alpha}I_{\alpha}$,
$I_{\alpha}=\int\mathrm{d}^{3}rw_{\alpha}(\vec{r})f(\vec{r})$, whose argument
is regular everywhere except at a single nucleus, and is accurately
discretized with a spherical quadrature grid centered on that nucleus. Becke
scheme, which is easier to implement than methods based on non-overlapping
Voronoi polyhedra, has been demonstrated to be at least as accurate and
efficient when applied to density functional calculations Franchini _et al._
(2013), and is now prevalent in the density-functional methods of common
quantum chemistry packages Karlström _et al._ (2003); Shao _et al._ (2015);
te Velde _et al._ (2001); Werner _et al._ (2019); Barca _et al._ (2020);
Balasubramani _et al._ (2020). Becke scheme is also a promising starting
point to evaluate more general multi-center integrals, such as the bound-free
and free-free integrals that appear in scattering and photoionization problems
Rescigno _et al._ (1995).
In the original formulation of Becke scheme, the atomic weights do not vanish
asymptotically, and hence each atomic grid must cover a region that may be as
large as the whole molecular electronic density in the integral. This
circumstance limits the applicability of the method in ionization problems,
where the electronic density is finite even at a large distance from the
molecule and each atomic grid would consequently need to be as large as the
whole quantization volume for the continuum. Furthermore, the integral suffers
from slow convergence due to the mismatch between the boundary of the
quantization volume and that of the atomic spherical grids.
In this work, we propose an extension of Becke scheme that circumvents this
difficulty and which is tailored to treat photoionization of polyatomic
systems. This new scheme defines a partition of unity in which the atomic
weights $w^{\prime}_{\alpha}(\vec{R}_{\beta})=\delta_{\alpha\beta}$ are
confined to the molecular region, $\sum w_{\alpha}^{\prime}(\vec{r})\simeq 0$,
$\forall r>R_{\textsc{Mol}}$, and complemented by a positive external weight,
$w_{0}^{\prime}(\vec{r})=1-\sum_{\alpha}w_{\alpha}^{\prime}(\vec{r})$, with
$w_{0}^{\prime}(\vec{R}_{\alpha})=0$, $\forall\alpha$. In this new approach,
the atomic grids can be confined within spherical shells of the size of the
first one or two nearest-neighbor atoms, whereas the complementary
interstitial and asymptotic portion of the integrand are regular everywhere
and hence can be discretized with a single master spherical grid, which is
typically placed at the center of gravity of the molecule. The method has all
the advantages of the Becke scheme near the nuclei and is capable of treating
integrals which have appreciable contributions far from the atomic centers by
employing a single-center master grid at larger distance. As a result, the
method is more efficient than the original scheme, and it is flexibile in the
choice of the shell’s radii. In the mid- to long-range region, where only a
limited number of multipolar terms contribute, the central grid can, by
itself, describe the numerical integration with a limited number of angular
points. The use of Becke weights in the present approach is essential to allow
distinct atomic shells to overlap.
In the next section we illustrate the modified Becke schemes, present two
variants of the methods, and discuss their accuracy and efficiency for a set
of integrals of representative functions. The functions are chosen to
represent both typical bound state integrals as well as those involving
molecular continuum functions. In Sec. III we present test results for a
multi-center model that mimics the NO2 molecule. In Sec. IV we offer our
conclusions.
## II A modified Becke integration scheme
Let us summarize how the weights in the original Becke scheme are defined
Becke (1988) for a polyatomic system with $N$ nuclei, centered at
$\vec{R}_{a}$, $\vec{R}_{b}$, etc. The elliptic coordinate
$\mu_{ab}(\vec{r})=\frac{r_{a}-r_{b}}{R_{ab}},$ (1)
where $r_{a}=|\vec{r}-\vec{R}_{a}|$ and $R_{ab}=|\vec{R}_{a}-\vec{R}_{b}|$,
quantifies how close a position vector $\vec{r}$ is to one or the other of two
nuclei, $\vec{R}_{a}$ and $\vec{R}_{b}$. The quantity $\mu_{ab}(\vec{r})$ is
bounded by $+1$, along the semi-axis
$\ell_{ab}=\\{(1+x)\vec{R}_{a}-\vec{R}_{b},x>0\\}$, on the side of atom $a$,
and $-1$, along the semi-axis $\ell_{ba}$. The level surfaces of
$\mu_{ab}(\vec{r})=c$ are hyperboloids. The plane $\mu_{ab}(\vec{r})=0$
orthogonally bisects the segment connecting the two nuclei. Consider now a
non-increasing positive step function $s(\mu):[-1,1]\to[0,1]$ that transitions
from $s(-1)=1$ to $s(1)=0$ and is flat at the two ends of the interval,
$\mathrm{d}s/\mathrm{d}\mu|_{\pm 1}=0$. The function $s[\mu_{ab}(\vec{r})]$ is
then unity for $\vec{r}=\vec{R}_{a}$ and zero for $\vec{r}=\vec{R}_{b}$. To
define the domain of any atom $a$, therefore, one considers the product of
such switch-off functions for all the other atoms in the molecule,
$P_{a}(\vec{r})=\prod_{b\neq a}s[\mu_{ab}(\vec{r})],\qquad
P_{a}(\vec{R}_{b})=\delta_{ab}.$ (2)
Finally, from the $P_{a}(\vec{r})$, a partition of unity
$\\{w_{a}(\vec{r})\\}$ is readily obtained as
$w_{a}(\vec{r})=P_{a}(\vec{r})/\sum_{b}P_{b}(\vec{r}),\quad\sum_{a}w_{a}(\vec{r})=1.$
(3)
Becke suggested a hierarchy of increasingly sharper polynomial step functions,
$s_{k}(\mu)=[1-f_{k}(\mu)]/2$, $k\in\mathbb{N}$ Becke (1988), with
$f_{1}(\mu)=(3\mu-\mu^{3})/2,\quad f_{k+1}(\mu)=f_{k}[f_{1}(\mu)],$ (4)
which we will also use in the modified Becke scheme.
Let’s now consider a hybrid basis set composed of GTOs and numerical
functions. The matrix element of a local one-body operator $\hat{o}$ between a
molecular orbital $\phi(\vec{r})$ and a numerical function $\chi(\vec{r})$ can
be partitioned as
$\begin{split}\langle\phi|\hat{o}|\chi\rangle&=\sum_{a}\langle\phi|w_{a}\hat{o}|\chi\rangle=\\\
&=\sum_{a}\int
w_{a}(\vec{r})\,\mathrm{d}^{3}r\,\phi^{*}(\vec{r})\hat{o}\,\chi(\vec{r})\,.\end{split}$
(5)
Due to the presence of the weighting factor $w_{a}(\vec{r})$, the argument of
this integral smoothly approaches zero at every nucleus other than $a$. It is
then natural to discretize each atomic integral
$\langle\phi|w_{a}\hat{o}|\chi\rangle$ in terms of a sum over the points of a
spherical grid centered on $a$. The atomic integrals are expected to converge
faster with respect to the number of angular and radial points of the
corresponding atomic grid, which clusters at $\vec{R}_{a}$, than if a single
spherical grid was used for all the atoms in the molecule. As mentioned in the
introduction, this considerable improvement in convergence comes at the cost
of summing over the points of $N$ grids, instead of a single one. Furthermore,
the weight functions introduce an additional modulation of the integrand,
which can also affect convergence.
Let’s examine how these ideas impact the calculation of the two-body integral
$[\phi_{1}\chi_{1}|\phi_{2}\chi_{2}]=\int\mathrm{d}^{3}r\int\mathrm{d}^{3}r^{\prime}\frac{\rho_{1}(\vec{r}^{\prime})\rho_{2}(\vec{r})}{|\vec{r}-\vec{r}^{\prime}|},\quad\rho_{i}=\phi_{i}\chi_{i},$
(6)
where we assume for simplicity that all the functions, and with them the
charge densities, are real. This problem is normally divided into two parts:
i) the solution of the Poisson equation for one distribution of charge, say
$\rho_{1}$,
$\Phi_{1}(\vec{r})=\int\mathrm{d}^{3}r^{\prime}\frac{\rho_{1}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|},$
(7)
or, equivalently,
$\nabla^{2}\Phi_{1}=-4\pi\rho_{1}\quad{\rm
with}\quad\lim_{r\to\infty}\Phi_{1}(\vec{r})=0,$ (8)
and ii) the subsequent evaluation of the electrostatic energy of the second
charge in the field of the first,
$[\rho_{1}|\rho_{2}]=\int\mathrm{d}^{3}r\,\,\Phi_{1}(\vec{r})\rho_{2}(\vec{r}).$
(9)
In this case, the single-center expansion is dramatically inadequate. Imagine
computing the electrostatic repulsion between two distributions of charge with
an appreciable concentration near the same off-center nucleus. The integral
between the potential and the charge density has the same poor convergence as
in the one-body case. Worse still, the solution of the Poisson equation would
be drastically inaccurate in proximity of the off-center nucleus, where the
potential is known to exhibit large and rapidly changing values.
Figure 1: Switching cutoff function, Eq.(13), shown for a range of powers $n$
where $R_{At}$ is the limit of the atomic grid.
This problem is solved using Becke scheme, since both the source term of the
Poisson equation and the integrand of the electrostatic potential can be
partitioned in atomic components,
$\begin{split}\nabla^{2}\Phi_{1,a}&=-4\pi\,w^{B}_{a}\rho_{1},\\\
[\rho_{1}|\rho_{2}]&=\sum_{a}\int\mathrm{d}^{3}r\,\,\Phi_{1,a}\,w^{B}_{a}\,\rho_{2}+\\\
&+\sum_{a,b\neq
a}\int\mathrm{d}^{3}r\,\,\Phi_{1,a}\,w^{B}_{b}\,\rho_{2}.\end{split}$ (10)
Here, $w^{B}_{a}$ are the Becke weights defined for atom $a$. The charge
distribution $w^{B}_{a}\,\rho_{1}$ and the corresponding potential
$\Phi_{1,a}$ vary rapidly only near $\vec{R}_{a}$. The solution of the Poisson
equation, therefore, can be accurately sought after in a reference frame
centered on $\vec{R}_{a}$, e.g., by expanding it in a basis of auxiliary
spherical functions. The second integration concerns the energy of a charge
distribution close to a different center than the one from which the
electrostatic potential originates. The potential is smooth at all points
where the charge density is finite. This last component, therefore, is
integrated to high precision using the atomic grid centered on $b$.
Since the weight functions $w^{B}_{a}(\vec{r})$ do not normally vanish
asymptotically, the sphere within which the Poisson equation must be solved,
or the integral evaluated, is as large as that of a single-center expansion.
Once again, while this approach does eliminate the largest source of
inaccuracy of the single-center expansion, it also multiplies the number of
integration grids and with it the number of grid points needed to reach
convergence. More importantly, this partitioning scheme fails to take
advantage of the multipolar character of the interaction between well
separated charges, requiring the solution of the Poisson equation over the
full configuration space for each atom in the molecule. These two aspects are
severe limitations in scattering calculations, where the wavefunction must be
evaluated at radial distances significantly larger than the molecule itself.
As a consequence, this approach can be predicted to scale poorly with the
number of atoms in the molecule, as well as with the diameter of the molecular
region, which can easily reach large dimensions even for few atoms, in the
case of molecular dissociation Laqua _et al._ (2018) or loosely bound van-
der-Waals complexes.
To treat integrals between basis functions that do not decay asymptotically,
such as those involving continuum functions, it is essential for the atomic
weights to decay rapidly, and be complemented by a single master weight and
grid at large distances. The focus of the present study is to modify the Becke
scheme to address the issues that arise when continuum functions are present,
namely: i) its slow convergence, due to the mismatch between grids’ and
integration domain; ii) its redundancy, due to most of the points in the
atomic grids that encompass the whole integration domain having negligible
weight; iii) its inability to give rise to a natural partitioning of two-body
interactions in short-range terms and long-range multipolar terms. This is
accomplished by switching off the atomic weights $w_{a}(\vec{r})$ with an
attenuation function that decreases with the distance from the molecular
region. In addition, we introduce a single-center master grid, which accounts
for those parts of configuration space outside the atomic centers and whose
spherical boundary is ideally suited to enforce the asymptotics of scattering
functions.
We begin by modifying the Becke weight function, $w^{B}_{a}$ as,
$w^{At}_{a}(\vec{r})=w^{B}_{a}(\vec{r})\,f^{At}(|\vec{r}-\vec{R}_{a}|),$ (11)
where $f^{At}$ is a positive bell-shaped, monotonically decreasing radial
function that starts from $1$ at the origin, with several zero derivatives,
and becomes negligible beyond a given radius $R_{At}$:
$\begin{split}&f^{At}(x)\geq 0,\quad f(0)=1,\\\ &(f^{At})^{(n)}(0)=0,\quad
n<n_{max},\\\ &f^{At}(x)\simeq 0\quad\forall\,\,x>R_{At}.\end{split}$ (12)
One such possible function is
$\begin{split}&f^{At}(x;\alpha,n)=1-(1-e^{-\alpha x^{2}})^{n},\\\
&\mathrm{for}\,\,x\ll\alpha^{-1/2},\,\,f^{At}(x;\alpha,n)\sim
1-\alpha^{n}x^{2n}+o(x^{4n}),\\\
&\mathrm{for}\,\,x\gg\alpha^{-1/2},\,\,f^{At}(x;\alpha,n)\sim n\,e^{-\alpha
x^{2}}.\end{split}$ (13)
Here $x=|r-R_{a}|/R_{At}$, where $R_{At}$ is the radius of the atomic grid
associated with atomic nucleus at $R_{a}$. The value of $\alpha$ such that
$w^{At}_{a}(\vec{r})$ is smaller than a certain threshold $\delta$ beyond a
given radius $R_{At}$ may be computed using,
$f^{At}(R_{At})=\delta\quad\implies\quad\alpha=R_{At}^{-2}\ln(n/\delta).$ (14)
Examples of such functions for different order $n$ are shown in Fig. 1. The
introduction of the complementary weight function,
$w^{At}_{0}(\vec{r})=1-\sum_{a}w^{At}_{a}(\vec{r}),$ (15)
restores the partition of unity with the desired properties of the grid at
long range,
$\sum_{a=0,N}w^{At}_{a}(\vec{r})=1,\quad\lim_{r\to\infty}w^{At}_{a>0}(\vec{r})=0,\quad\lim_{r\to\infty}w^{At}_{0}(\vec{r})=1.$
(16)
The partitioning scheme that uses this partition of unity,
$\\{w^{At}_{a}\\}_{a\in\\{0,1,2,\ldots,N\\}}$, will be denoted scheme 1.
Alternatively (scheme 2), one can use a single switch-off function
$f^{Mol}(\vec{r})$ for the whole molecule,
$w^{Mol}_{a}(\vec{r})=w^{B}_{a}(\vec{r})f^{Mol}(\vec{r}).$ (17)
The switch-off function $f^{Mol}(\vec{r})$ can be easily defined from any
scalar field $V(\vec{r})$ that diverges at all nuclei and vanishes at large
distances, such as
$f^{Mol}(\vec{r})=f^{Mol}(\vec{r};\alpha,n)=f^{At}\left(N/V(\vec{r});\alpha,n\right),$
(18)
with the electrostatic potential due to singly-charged nuclei being a possible
choice,
$V(\vec{r})=\sum_{a=1}^{N}|\vec{r}-\vec{R}_{a}|^{-1}.$ (19)
In this case, $w_{0}$ coincides with $1-f^{Mol}(\vec{r};\alpha,n)$. In Sec.
III, we examine the performance of the partition of unity based on both the
atomic (1) and the molecular (2) schemes.
Both approaches have manifest advantages over the original Becke scheme.
First, all integrals over a spherical volume are much more accurate, since the
quadrature grid matches the integration domain, whereas in the original Becke
scheme the polycentric atomic grids contain but do not coincide with the
integration domain, resulting in a first-order integration scheme, where the
error is inversely proportional to the number of integration points, for any
integrand that does not vanish smoothly before reaching the boundary of the
integration volume. Second, in the Poisson equation for any $a>0$,
$\nabla^{2}\Phi_{1,a}(\vec{r})=-4\pi\rho_{1}(\vec{r})w^{At}_{a}(\vec{r}),$
(20)
the term on the RHS is essentially zero beyond the finite radius $R_{At}$ of
any atom. As a consequence, the solution for $r>R_{At}$ is conveniently
expressed in terms of a multipolar expansion, which is analytic and easy to
evaluate,
$\Phi_{1,a}(\vec{r})=\sum_{\ell m}\frac{M_{\ell m;1,a}}{r^{\ell+1}}Y_{\ell
m}(\hat{r})$ (21a) $M_{\ell m;1,a}=\frac{4\pi}{2\ell+1}\int r^{\ell}Y_{\ell
m}^{*}(\hat{r})w_{a}(\vec{r})\rho_{1}(\vec{r})\,\mathrm{d}^{3}r.$ (21b)
This approach significantly reduces the size of the sphere around each atom,
and with it the region of space where the Poisson equation has to be solved
numerically and is sensitive to the distribution of the grid points.
## III Numerical results
This section is devoted to compare the performance of the modified Becke
schemes 1 and 2 with the original Becke approach and with a single center
spherical grid for a set of integrals representative of those encountered in a
scattering or photoionization calculation. We consider integrands that
exponentially decrease at large distances, such as nuclear centered Gaussians,
that oscillate to large radial distances, such as spherical Bessel functions
of varying energy located on the central grid, as well as functions exhibiting
singularities at the atomic nuclei. In the latter case, we consider integrals
involving multicenter Yukawa potentials, where the result is known
analytically for very large integration volumes, and which are analogous to
nuclear-attraction integrals. The application of the modified Becke methods to
bielectronic integrals and to the calculation of physical observables such as
molecular photoionization cross sections is beyond the scope of the present
work and will be examined elsewhere.
We carry out the numerical tests using as a reference the nuclei of the NO2
molecule in its equilibrium configuration in the $yz$ plane, as specified in
Table 1.
Table 1: Atom Coordinates (a.u.) Atom | $y$ | $z$
---|---|---
N | -0 | -0.6909
O1 | -2.1403 | -0.3022
O2 | -2.1403 | -0.3022
Every atomic integration region is subdivided into a set of spherical shells.
The 3D grid points within a given shell are a product of a Gauss-Legendre
radial and Lebedev angular Lebedev (1975, 1976, 1977); Lebedev and Skorokhodov
(1992); Lebedev (1995); Lebedev and Laikov (1999) set of points. Different
shells have have the same number of angular points, but can differ in the
number of radial points.
Figure 2: Original and modified Becke’s weight distributions for NO2 molecule
placed at coordinate points of Table 1. Atomic grid sizes are $R=10$ a.u. The
original Becke scheme defines atomic weights that in general do not vanish at
infinity (a-c). The atomic based modified weights (scheme 1) and the molecular
based weights (scheme 2) are given in (d-g) and (h-k), respectively for the
same atomic parameters. Panes (g) and (k) give the complementary weights for
the central grid in the two modified schemes. The solid circles represent the
location of atomic nuclei.
Fig. 2 shows the range of the original and modified weight functions in NO2.
Whereas the weights in the modified schemes 1 and 2 (middle and lower panels,
respectively) rapidly drop to zero as the distance from the their atomic
centers of reference increases, the original Becke weights
$w_{a}^{B}(\vec{r})$ (upper panels) do not vanish asymptotically at all. For
integrands that do not decrease rapidly outside the molecular region, such as
those involving Rydberg and scattering orbitals, therefore, the original Becke
scheme requires disproportionately large integration grids for each and every
atom. In the modified Becke schemes, the atomic weights smoothly approach zero
at the atomic shell boundaries, set at a fixed distance $R$ from each nucleus.
As a consequence, the new atomic weights maintain their ability to describe
compact functions near the nuclei, while the central grid captures integrands
of long-range components. In scheme 1, the complementary weight $w_{0}^{At}$
(panel g, Fig. 2) exhibits comparatively sharp features at the intersection
between the boundary support of atomic weights, thus affecting the convergence
of the integral as a function of the angular points in the central grid. This
problem is less prominent for larger atomic spheres but it is exacerbated if
the atomic weights are strongly confined. The bottom panels in Fig. 2 show the
three atomic (h-j) and the complementary weight (k) for scheme 2. Even if the
parameters are chosen to yield more localized atomic weights than in scheme 1,
the hole profile in the complementary weight is considerably smoother, which
suggests a smaller number of angular points is needed to reach convergence, if
strongly localized atomic weights are required.
Let us begin by considering $s$-type GTOs using the same grid on each atom,
defined with $N_{r}$ number of radial and $N_{\Omega}$ number of angular grid
points, for a total number of points $N_{G}=N_{R}\cdot N_{\Omega}$. The number
of angular points $N_{\Omega}$ is dictated by the maximum angular momentum
$\ell_{\rm max}$ for which the Lebedev quadrature is exact. The density of
radial points of the atomic shell is larger near the nucleus, to achieve
higher accuracy for the compact functions. The GTOs are placed at each atomic
center with different exponential factors $\alpha$ to cover the cases from
compact to diffuse orbitals. For each scheme, we set a convergence tolerance
for all GTOs of at least 10${}^{-10}\%$.
In the single-center expansion, the radial grids extends to $R_{c}=80$ a.u. to
cover the most diffuse GTOs, and it has a smaller radial spacing in the
molecular region. A Lebedev quadrature of $\ell_{\rm max}=131$ is necessary to
maximize the integration accuracy for the compact GTOs. The results, in Table
2, show that while the single-center expansion can accurately describe the
GTOs on the nitrogen atom, it fails to represent GTOs with exponent
$\alpha\geq 200$ on the oxigen atoms, which are furher away from the expansion
center. Increasing the number of radial points does not improve the accuracy
of the most compact GTOs. Additional tests using products of $\theta$ and
$\varphi$ grids, with up to ten times more points than the Lebedev grids used
in Table 2, still have errors larger than 10-5 for the compact Gaussians.
In the original Becke scheme, with atomic shells extending to $R=80$ a.u.,
Table 2 shows that quite accurate results can be achieved with an $\ell_{\rm
max}=83$. The contrast with the single-center expansion is particularly
dramatic for the GTOs centered on the oxygen atoms, since the integration
accuracy improves by eight orders of magnitude. In general, all of the Becke
schemes perform significantly better than the single-center expansion, and the
modified Becke scheme reach high accuracy with fewer points than the original
Becke scheme. In addition, the modified Becke scheme does not require a high
density of radial points near the origin of the central grid.
Table 2: Relative error between numerical and analytical results for integration of Gaussian functions over a range of GTO exponents, $\alpha$, using a single-center grid (SC), the original Becke scheme and the modified Beckes’s scheme with the associated total number of grid points, NG, (see text for details). The notation $[n]$ stands for $\times 10^{n}$. Atom | $\alpha$ | SC | Becke | Mod. Becke
---|---|---|---|---
N | 1000 | 1.76 [-13] | 4.97 [-11] | 5.38 [-11]
500 | 2.52 [-14] | 2.55 [-11] | 9.82 [-11]
200 | 4.61 [-14] | 4.71 [-11] | 7.92 [-11]
100 | 5.83 [-14] | 3.03 [-13] | 4.09 [-11]
10 | 1.72 [-13] | 1.96 [-13] | 1.33 [-13]
1 | 1.41 [-13] | 1.72 [-13] | 7.29 [-14]
0.1 | 2.08 [-14] | 1.79 [-13] | 4.43 [-14]
0.01 | 4.97 [-13] | 2.76 [-13] | 2.59 [-13]
O | 1000 | 3.72 [-2] | 4.91 [-11] | 4.22 [-11]
500 | 8.05 [-4] | 1.80 [-11] | 4.52 [-11]
200 | 1.26 [-6] | 5.39 [-11] | 2.78 [-11]
100 | 2.80 [-12] | 3.32 [-11] | 4.14 [-11]
10 | 7.88 [-14] | 2.27 [-13] | 7.28 [-13]
1 | 1.23 [-13] | 7.97 [-14] | 4.07 [-14]
0.1 | 7.38 [-14] | 1.90 [-13] | 6.49 [-14]
0.01 | 4.82 [-13] | 2.55 [-13] | 2.67 [-13]
$N_{G}$ | | 1,801,100 | 2,602,500 | 1,790,440
$\ell_{\rm max}$ | | 131 | 83 | 83
Fig. 3 compares the two modified Becke schemes for three different Gaussians
centered on the O atom. Since the modified Becke schemes contain a central
grid, one can considerably shrink the sizes of the atomic radii, here chosen
as $R=10$a.u. For the more compact Gaussians, the modified Becke scheme I does
better than scheme II, whereas the opposite is true for the diffuse Gaussians.
Overall, however, scheme I shows better control of the error over a broader
range of exponential parameters.
Figure 3: Modified-Becke’s accuracy convergence for scheme 1 vs. scheme 2 as
a function of angular momentum number $\ell_{\rm max}$ of the central grid.
The test is for an O atom in the NO2 molecule placed at coordinate points of
Table 1. Atomic grid sizes are $R=10~{}$a.u. set in a central grid of of size
$R=80~{}$a.u.
In the rest of the article, therefore, we will focus on Scheme I.
It is instructive to analyze the dependence of the modified Becke scheme on
the size of the atomic shells, where the transition from the near-nuclei Becke
scheme to a single-center grid method occurs. Using different shells with
radii $R=2.5$, $5.0$, $10.0$ and $20.0$ a.u., the requested accuracy for all
GTOs was always reached with the modified scheme with less than 2 million
points. As the radius $R$ decreases, the density of grid points at small
distances must be increased for both the atomic and the complementary grid, to
efficiently (see panel (d) of Fig. 2) represent the variation of the switch-
off function $f^{At}$. In fact, using a smaller radius $R$ allowed us to reach
higher accuracy ($\sim 10^{-11}$ instead of $\sim 10^{-9}$) for the most
compact GTOs when compared with the case $R=10$ a.u. shown in Table 2. In
general, however, the accuracy is excellent for all shell radii.
Let us now consider the accuracy of the different weighting schemes for
evaluating integrals of the Yukawa potential,
$V_{b}(\vec{r})=\frac{e^{-|\vec{r}-\vec{R}_{b}|}}{|\vec{r}-\vec{R}_{b}|}.$
(22)
This test is more demanding than Gaussian integrals due to the Coulomb
singularity at the atomic centers. The results are shown in Table 3, where we
used the same parameters as previously for the Gaussian tests. Both the
original and the modified Becke schemes are superior in accuracy by many
orders of magnitude than the single-center grid method, and, as expected for a
fast-decreasing integrands such as the Yukawa potential, the modified Becke
scheme is comparable in accuracy to the original.
Table 3: Relative error between numerical and analytic results for the integration of Yukawa integrals on N and O atoms for the three different schemes. Chem. Symb. | Single-Center | Becke | Mod. Becke
---|---|---|---
N | 1.65 [-5] | 2.90 [-13] | 1.99 [-14]
O | 2.01 [-4] | 3.25 [-13] | 1.97 [-14]
So far, we have shown the equivalence of the original and the modified Becke
scheme using functions that decay at large radial distances. The goal,
however, is to develop a weighting scheme that is effective not only for the
short-range functions examined in previous sections, but also for the
continuum functions found in scattering and photoionization calculations, such
as spherical Bessel functions, B-splines or finite element DVR functions.
Photoelectrons can easily have de Broglie wavelengths of the order of one
atomic unit, i.e., comparable to the width of the switch-off function used in
the modified Becke schemes, and they rapidly oscillate over large (indeed,
infinite) radial distances. A few examples illustrate the superior performance
of the modified Becke scheme when continuum functions are involved. In the
first test, we compute the overlap of the spherical Bessel function
$j_{0}(kr)=\sin(kr)/kr$ with various Gaussian functions, for Gaussian
exponents between $.002$ and $.005$ and radial photoelectron momentum $k$
varying from $1$eV to $400$eV (1a.u.$\simeq$27.21 eV). The atomic spheres were
chosen to have a radius of $R=5$ a.u.. While the original Becke scheme can
compute these integrals, the modified scheme could do so with a reduction of
$60$ to $90$% in the number of points for all energies. These results
illustrate the efficiency of the master grid to accurately capture the
behavior of these integrands at large radial distances.
For the second test, shown in Table 4, we compute the integral over a
spherical region of the two functions $j_{0}(r)$, and $rj_{1}(r)$ where
$j_{\ell}(r)$ are spherical Bessel functions. Our reason for testing the
integral of $rj_{1}(r)$ rather than $j_{1}(r)$ itself, is that $j_{1}(r)$ has
a discontinuous radial derivative at the origin, something that would never
occur in practice using a set of full 3D basis functions. The dramatic
improvement in accuracy of the modified vs the original Becke method, is
clear. The poor performance of the original Becke scheme is due to the
mismatch between the boundary of the integration region, which is dictated by
the physics of the problem, and the boundary of off-center spherical
quadrature grids. Apart from the larger number of points required by the Becke
scheme, the method simply will not be usable for the computation of the
integrals needed in scattering and photoionization processes with general
numerical bases.
Table 4: Relative error between numerical and analytic results for the integration of spherical Bessel integrals for the Becke and modified Becke schemes. function | Method | | Total
---
radial points
| angular points
---
per radial shell
| Total num.
---
grid points
| Total points inside
---
integration region
| %diff with
---
analytical
$j_{0}$ | Becke | 1000 | 974 | 974000 | 769744 | 4.11[-2]
Mod. Becke | 900 | 974 | 876600 | 876600 | 3.52[-12]
$r\cdot j_{1}$ | Becke | 2000 | 974 | 1948000 | 1370186 | 1.16[-2]
Mod. Becke | 1800 | 974 | 1753200 | 1753200 | 1.92[-11]
## IV Conclusions and Outlook
In this paper, we have developed a new variant of the popular Becke molecular
integration method which is able to efficiently integrate functions containing
integrands that do not vanish asymptotically, such as those appearing in
typical scattering and photoionization calculations. This required a
modification of the Becke weighting to confine the grid points on the atomic
sites to molecular dimensions and to augment the atomic grid with a single,
master grid to capture the interstitial and long range behavior of the
integrand. The new approach has a number of appealing properties that should
be stressed. In addition to its high accuracy for integrands which do not
decay rapidly away from the atomic centers, it provides a very efficient
approach to solving the Poisson equation and ultimately to the calculation of
the hybrid one-and-two electron integrals encountered in scattering
calculations.
The numerical study confirmed that single-center grids are incapable of
providing very accurate integrations with integrands having strongly localized
integrands close to the atomic centers. Integration approaches which partition
space into atomic sub-regions are far superior, as already shown by Becke. By
weighting each atomic region, the molecular integral may be decomposed into a
sum of atomic-like contributions, which can be easily discretized to produce
an accurate answer. For integrands which vanish sufficiently rapidly at large
distances, the original weighting method of Becke which partitions the atomic
regions into “fuzzy”, overlapping regions, works remarkably well. For
integrals that extend over large regions of space, such as those that appear
in scattering calculations, however, the Becke scheme becomes unnecessarily
expensive, as a consequence of the atomic Becke weights having a non-vanishing
amplitude at large distances, and inaccurate, since the boundary of the
integration grids do not match the boundary of a single spherical integration
domain. We have presented two extensions of the Becke method that confine the
atomic weights to the molecular region, completes them with a complementary
central grid and weight and accurately and economically captures the large-
radius region, where the charge densities are regular. The new weighting
schemes produce quite accurate results for test functions such as compact
Gaussian functions, Yukawa potentials, and spherical Bessel functions, which
are representative of the functions encountered in realistic molecular
scattering problems. The method scales favorably with the number of atoms in
the molecule and with the size of the integration volume, when compared with
the original Becke scheme, as shown in the Appendix. Furthermore, it is
expected to yield uniformly accurate results even for dissociating molecules.
The efficiency of the modified schemes compared to the original Becke scheme
increases rapidly with the size of the molecule as a consequence of more
localized nature of the weighted charge distributions, allowing simple
multipole methods to be used away from the atomic sites. For one-body
integrals, the efficiency increases linearly with the number of atoms. In
Appendix A, we provide informed estimates of the computational cost for both
the Becke and Modified Becke methods in calculating the two-electron
integrals. The estimates show that the modified Becke method can lead to a
several order of magnitude speedup over the Becke method when applied to these
integrals.
## V Acknowledgments
The work of N.D. and L.A. was supported by the United States National Science
Foundation under NSF grant No. PHY-1607588, by the DOE CAREER grant No. DE-
SC0020311 and the work of B.I.S and H.G. was supported by the Department of
Commerce, National Institute of Standards and Technology.
## Appendix A Computational Cost
This appendix estimates and compares the computational costs of evaluating
one-electron and two-electron integrals in the Becke and Modified Becke
schemes. Table 5 defines some parameters employed in this analysis.
Table 5: Parameters relevant to the cost of the Becke ($b$) and Modified Becke ($mb$) integrations. Variable | Definition | Value
---|---|---
$N_{at}$ | Number of atoms |
$R_{cg}$ | Radius of central grid | $25-50$ a.u.
$R_{at}$ | Radius of atom | $5-10$ a.u.
$r_{g}$ | Ratio of radii | $R_{at}/R_{cg}\lesssim 0.2$
$N_{cg}$ | Number of points in central grid |
$N_{ag}^{b}$ | Number of atomic points in |
| Becke scheme | $\approx N_{cg}$
$N_{ag}^{mb}$ | Number of atomic points in |
| modified Becke scheme | $N_{cg}r_{g}^{3}\approx N_{cg}/125$
$N^{b}$ | Total number of Becke points | $\approx N_{cg}N_{at}$
$N^{mb}$ | Total number of modified | $N_{cg}+N^{mb}_{ag}N_{at}$
| Becke points | = $N_{cg}[1+N_{at}\,r_{g}{{}^{3}}]$
$N_{op}$ | flops to evaluate auxiliary functions |
| in the Poisson Equation solution |
$N_{nn}$ | Number of nearest neighbor atoms |
In the definition of $N^{b}$, we have made the reasonable assumption that, in
the Becke scheme, we will need the same number of points per atom as is needed
for the central grid in the modified Becke scheme. Indeed, each atomic grid in
the Becke scheme needs to cover the entire computational volume. In the
modified Becke scheme, on the other hand, only the central grid covers the
entire computational volume, while the individual atomic grids are restricted
to an atomic radius. We also assume that the number of grid points scales
proportionally to the volume of the enclosed computational region.
The size of the central grid is dictated by the asymptotics of a scattering
problem and needs to be sufficiently large that one may neglect interchannel
coupling. Here, we assume $r_{g}=0.2$ as a reasonable upper bound to the ratio
of the atomic and central grid radii. As a consequence, in the evaluation of
single-electron integrals, the computational cost of the Modified Becke
approach is dominated by the sum over the points of the master grid, which is
equivalent to the sum over a single atomic grid in the original Becke
approach. For one-electron integrals, therefore, the Modified Becke scheme is
more efficient than the original Becke scheme by a factor comparable to the
number of atoms in the system.
If the reduction of the computational cost of one-electron integrals is
already appreciable, the gain for two-electron integrals is dramatic. In the
Modified Becke scheme, a two-electron integral $[\rho|\rho^{\prime}]$ is
evaluated by partitioning the densities as follows
$[\rho|\rho^{\prime}]=[\rho_{0}|\rho^{\prime}_{0}]+\sum_{a}\left([\rho_{a}|\rho^{\prime}_{0}]+[\rho_{a}^{\prime}|\rho_{0}]\right)+\sum_{a,b\in
I^{NN}_{a}}[\rho_{a}|\rho^{\prime}_{b}]+\sum_{a,b\notin
I^{NN}_{a}}[\rho_{a}|\rho^{\prime}_{b}],$ (23)
where $\rho_{0}=w_{0}\rho$, $\rho_{a}=w_{a}\rho$, etc. [compare with (16)] and
$I^{NN}_{a}$ is the set of “nearest-neighbor” atomic indices whose integration
spheres intersect that of atom $a$, including $a$ itself. The evaluation of
each term $[\rho_{a}|\rho^{\prime}_{b}]$ requires two steps: i) the
determination of the potential $V_{a}$, originating from $\rho_{a}$, and ii)
the integration of $V_{a}\rho_{b}$. Evaluating the potential $V_{a}(\vec{r})$
at points outside the support of $\rho_{a}$ is inexpensive, since the
potential can be expanded in multipoles. To evaluate the potential within the
support of $\rho_{a}$, on the other hand, it is necessary to solve the Poisson
equation (PE) by the inverting a discretized radial Laplace operator and
applying it to every spherical density component, $\rho_{a,\ell m}(r)$, where
$\rho_{a}(\vec{r})=\sum_{\ell m}\rho_{a,\ell m}(r)Y_{\ell m}$. Finally, step
(2) requires evaluating $V_{a}$ at the points in sphere $b$ and then carrying
out the final integration by summing over all points in sphere $b$. In the
original Becke scheme, all atomic grids overlap and all have the same large
size. In modified Becke, only one grid is large, and all the other
interactions are either between few overlapping spheres, one of which is
small, or between many disjoint small spheres. Most of the interaction,
therefore, is multipolar. The computation cost for each step for both schemes
is listed in Table 6, where $N_{nn}=N_{at}^{-1}\sum_{a}o(I^{NN}_{a})$ is the
average number of nearest neighbors. Here, $o(I)$ stands for the number of
elements of set of indexes $I$. Using the value of
$N^{mb}_{ag}=N_{cg}\,r_{g}^{3}$ given in Table 5, the number of operations in
the modified Becke method, $N_{\mathrm{flop}}^{mb}$, compared to Becke, is
$\frac{N_{\mathrm{flop}}^{mb}}{N_{\mathrm{flop}}^{b}}\approx\frac{N_{nn}}{N_{at}}\Big{[}r_{g}{{}^{6}}+\frac{r_{g}{{}^{3}}}{N_{nn}(N^{mb}_{ag})^{1/3}}.\Big{]}$
(24)
Since $r_{g}\lesssim 0.2$, the Modified Becke scheme requires three to five
orders of magnitude less operations than the original Becke scheme.
Table 6: Comparison of Computational Cost of Becke and Modified Becke Method. The third row shows the individual contributions of the overlapping, non-overlapping and central grids. The number of radial and angular points in a spherical grid with $N$ points is estimated as approximately $N^{1/3}$ and $N^{2/3}$, respectively. | Becke | Mod. Becke
---|---|---
Tabulation of $\rho$ | $N^{b}=N_{at}N_{cg}$ | $N^{mb}=N_{cg}+N_{at}N^{mb}_{ag}$
Solution of $V_{a}$ | $N_{at}[N^{b}_{ag}]{{}^{4/3}}\approx N_{at}[N_{cg}]{{}^{4/3}}$ | $N_{at}[N^{mb}_{ag}]^{4/3}+[N_{cg}]{{}^{4/3}}\approx[N_{cg}]{{}^{4/3}}$
Evaluation of $V_{a}$ | $N_{at}[N_{at}-1]/2[N^{b}_{ag}]^{2}N_{op}$ | $N_{nn}N_{at}[N^{mb}_{ag}]^{2}N_{op}$
| $\approx N_{at}[N_{at}-1]/2[N_{cg}]^{2}N_{op}$ | \+ $N_{at}[N_{at}-N_{nn}-1][N^{mb}_{ag}]^{5/3}$
| | $+N_{at}N_{cg}[N^{mb}_{ag}]^{2/3}$
Final Integration | $\sim N^{b}$ | $\sim N^{mb}$
Total | $\approx(N_{at}^{2}N_{cg}^{2}N_{op})$ | $\approx N_{nn}N_{at}[N^{mb}_{ag}]^{2}N_{op}+N_{at}N_{cg}[N^{mb}_{ag}]^{2/3}$
## References
* Saad _et al._ (2010) Y. Saad, J. Chelikowsky, and S. Shontz, SIAM Review 52, 3 (2010).
* Alder _et al._ (1963) B. Alder, S. Fernbach, and M. Rotenberg, _Methods in Computational Physics._ (Academic Press; New York, 1963, 1963).
* Helgaker _et al._ (2012) T. Helgaker, P. Jorgensen, and J. Olsen, _Molecular Electronic-Structure Theory_ (John Wiley and Sons Ltd, the Altrium, Southern Gate, Chichester, West Sussex, England, 2012).
* Kaufmann _et al._ (1989) K. Kaufmann, W. Baumeister, and M. Jungen, Journal of Physics B: Atomic, Molecular and Optical Physics 22, 2223 (1989).
* Marante _et al._ (2014) C. Marante, L. Argenti, and F. Martín, Phys. Rev. A 90, 012506 (2014).
* Rescigno _et al._ (1995) T. N. Rescigno, B. H. L. III, and C. W. McCurdy, _Modern Electronic Structure Theory 1_ , 1st ed. (World Scientific, Singapore, 1995).
* Marante _et al._ (2017) C. Marante, M. Klinker, T. Kjellsson, E. Lindroth, J. González-Vázquez, L. Argenti, and F. Martín, Phys. Rev. A 96, 022507 (2017).
* Mašín _et al._ (2020) Z. Mašín, J. Benda, J. D. Gorfinkiel, A. G. Harvey, and J. Tennyson, Computer Physics Communications 249, 107092 (2020).
* Rescigno _et al._ (2005) T. N. Rescigno, D. A. Horner, F. L. Yip, and C. W. McCurdy, Phys. Rev. A 72, 052709 (2005).
* Bishop (1967) D. M. Bishop (Academic Press, 1967) pp. 25 – 59.
* Collins and Schneider (1983) L. A. Collins and B. I. Schneider, Phys. Rev. A 27, 101 (1983).
* Gianturco _et al._ (1994) F. A. Gianturco, R. R. Lucchese, and N. Sanna, The Journal of Chemical Physics 100, 6464 (1994).
* Yip _et al._ (2010) F. L. Yip, C. W. McCurdy, and T. N. Rescigno, Phys. Rev. A 81, 053407 (2010).
* Yip _et al._ (2014) F. L. Yip, C. W. McCurdy, and T. N. Rescigno, Phys. Rev. A 90, 063421 (2014).
* Boys and Rajagopal (1965) S. Boys and P. Rajagopal, Advances in Quantum Chemistry 2, 1 (1965).
* Boerrigter _et al._ (1988) P. M. Boerrigter, G. Te Velde, and J. E. Baerends, International Journal of Quantum Chemistry 33, 87 (1988).
* Averill and Painter (1989) F. W. Averill and G. S. Painter, Physical review B 39, 8115 (1989).
* Pederson and Jackson (1990) M. R. Pederson and K. A. Jackson, Phys. Rev. B 41, 7453 (1990).
* te Velde and Baerends (1992) G. te Velde and E. J. Baerends, Journal of Computational Physics 99, 84 (1992).
* Becke (1988) A. D. Becke, J. Chem. Phys. 88, 2547 (1988).
* Becke and Dickson (1988) A. D. Becke and R. M. Dickson, The Journal of Chemical Physics 89, 2993 (1988).
* Franchini _et al._ (2013) M. Franchini, P. H. T. Philipsen, and L. Visscher, Journal of Computational Chemistry 34, 1819 (2013), https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.23323 .
* Karlström _et al._ (2003) G. Karlström, R. Lindh, P.-Å. Malmqvist, B. O. Roos, U. Ryde, V. Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, and L. Seijo, Computational Materials Science 28, 222 (2003).
* Shao _et al._ (2015) Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D. Jacobson, I. Kaliman, R. Z. Khaliullin, T. Kuś, A. Landau, J. Liu, E. I. Proynov, Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. W. III, P. M. Zimmerman, D. Zuev, B. Albrecht, E. Alguire, B. Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist, K. Brandhorst, K. B. Bravaya, S. T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H. Chien, K. D. Closser, D. L. Crittenden, M. Diedenhofen, R. A. D. Jr., H. Do, A. D. Dutoi, R. G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M. W. D. Hanson-Heine, P. H. P. Harbach, A. W. Hauser, E. G. Hohenstein, Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U. Lao, A. D. Laurent, K. V. Lawler, S. V. Levchenko, C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan, A. Luenser, P. Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian, A. V. Marenich, S. A. Maurer, N. J. Mayhall, E. Neuscamman, C. M. Oana, R. Olivares-Amaya, D. P. O’Neill, J. A. Parkhill, T. M. Perrine, R. Peverati, A. Prociuk, D. R. Rehn, E. Rosta, N. J. Russ, S. M. Sharada, S. Sharma, D. W. Small, A. Sodt, T. Stein, D. Stück, Y.-C. Su, A. J. W. Thom, T. Tsuchimochi, V. Vanovschi, L. Vogt, O. Vydrov, T. Wang, M. A. Watson, J. Wenzel, A. White, C. F. Williams, J. Yang, S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y. Zhang, X. Zhang, Y. Zhao, B. R. Brooks, G. K. L. Chan, D. M. Chipman, C. J. Cramer, W. A. G. III, M. S. Gordon, W. J. Hehre, A. Klamt, H. F. S. III, M. W. Schmidt, C. D. Sherrill, D. G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A. T. Bell, N. A. Besley, J.-D. Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P. Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Liang, C. Ochsenfeld, V. A. Rassolov, L. V. Slipchenko, J. E. Subotnik, T. V. Voorhis, J. M. Herbert, A. I. Krylov, P. M. W. Gill, and M. Head-Gordon, Molecular Physics 113, 184 (2015).
* te Velde _et al._ (2001) G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. Fonseca Guerra, S. J. A. van Gisbergen, J. G. Snijders, and T. Ziegler, Journal of Computational Chemistry 22, 931 (2001).
* Werner _et al._ (2019) H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, _et al._ , “Molpro, version 2019.2, a package of ab initio programs,” (2019).
* Barca _et al._ (2020) G. M. J. Barca, C. Bertoni, L. Carrington, D. Datta, N. De Silva, J. E. Deustua, D. G. Fedorov, J. R. Gour, A. O. Gunina, E. Guidez, T. Harville, S. Irle, J. Ivanic, K. Kowalski, S. S. Leang, H. Li, W. Li, J. J. Lutz, I. Magoulas, J. Mato, V. Mironov, H. Nakata, B. Q. Pham, P. Piecuch, D. Poole, S. R. Pruitt, A. P. Rendell, L. B. Roskop, K. Ruedenberg, T. Sattasathuchana, M. W. Schmidt, J. Shen, L. Slipchenko, M. Sosonkina, V. Sundriyal, A. Tiwari, J. L. Galvez Vallejo, B. Westheimer, M. Wloch, P. Xu, F. Zahariev, and M. S. Gordon, The Journal of Chemical Physics 152, 154102 (2020).
* Balasubramani _et al._ (2020) S. G. Balasubramani, G. P. Chen, S. Coriani, M. Diedenhofen, M. S. Frank, Y. J. Franzke, F. Furche, R. Grotjahn, M. E. Harding, C. Hättig, A. Hellweg, B. Helmich-Paris, C. Holzer, U. Huniar, M. Kaupp, A. Marefat Khah, S. Karbalaei Khani, T. Müller, F. Mack, B. D. Nguyen, S. M. Parker, E. Perlt, D. Rappoport, K. Reiter, S. Roy, M. Rückert, G. Schmitz, M. Sierka, E. Tapavicza, D. P. Tew, C. van Wüllen, V. K. Voora, F. Weigend, A. Wodyński, and J. M. Yu, The Journal of Chemical Physics 152, 184107 (2020).
* Laqua _et al._ (2018) H. Laqua, J. Kussmann, and C. Ochsenfeld, Journal of Chemical Physics 149, 204111 (2018).
* Lebedev (1975) V. Lebedev, USSR Computational Mathematics and Mathematical Physics 15, 44 (1975).
* Lebedev (1976) V. Lebedev, USSR Computational Mathematics and Mathematical Physics 16, 10 (1976).
* Lebedev (1977) V. I. Lebedev, Siberian Math. J. 18, 99 (1977).
* Lebedev and Skorokhodov (1992) V. Lebedev and A. Skorokhodov, Doklady Mathematics 45, 587 (1992).
* Lebedev (1995) V. I. . Lebedev, Doklady Mathematics 50, 283 (1995).
* Lebedev and Laikov (1999) V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).
|
∎
e1 _Email address:_<EMAIL_ADDRESS>
11institutetext: National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA 22institutetext: Department of
Theoretical Physics, IFIN-HH, Reactorului 30, 077125 Mǎgurele-Bucharest,
Romania
# In-medium $\Delta(1232)$ potential, pion production in heavy-ion collisions
and the symmetry energy
M.D. Cozmaaddr1,addr2,e1 and M.B. Tsang addr1
(Received: date / Accepted: date)
###### Abstract
Using the dcQMD transport model, the isoscalar and isovector in-medium
potentials of the $\Delta$(1232) baryon are studied and information regarding
their effective strength is obtained from a comparison to experimental pion
production data in heavy-ion collisions below 800 MeV/nucleon impact energy.
The best description is achieved for an isoscalar potential moderately more
attractive than the nucleon optical potential and a rather small isoscalar
relative effective mass m${}^{*}_{\Delta}\approx$ 0.45. For the isovector
component only a constraint between the potential’s strength at saturation and
the isovector effective mass difference can be extracted, which depends on
quantities such as the slope of the symmetry energy and the neutron-proton
effective mass difference. These results are incompatible with the usual
assumption, in transport models, that the $\Delta$(1232) and nucleon
potentials are equal. The density dependence of symmetry energy can be studied
using the high transverse momentum tail of pion multiplicity ratio spectra.
Results are however correlated with the value of neutron-proton effective mass
difference. This region of spectra is shown to be affected by uncertain model
ingredients such as the pion potential or in-medium correction to inelastic
scattering cross-sections at levels smaller than 10$\%$. Extraction of precise
constraints for the density dependence of symmetry energy above saturation
will require experimental data for pion production in heavy-ion collisions
below 800 MeV/nucleon impact energy and experimental values for the high
transverse momentum tail of pion multiplicity ratio spectra accurate to better
than 5$\%$.
###### pacs:
21.65.Mn Nuclear matter equations of state 21.65.Cd Nuclear matter asymmetric
matter 25.70.-z Heavy-ion nuclear reactions, low and intermediate energy
††journal: Eur. Phys. J. A
## 1 Introduction
The isospin dependent part of the equation of state of nuclear matter (asy-
EoS), commonly known as the symmetry energy (SE) remains among the most
debated topics in nuclear physics. Its relevance for the structure of rare
isotopes, dynamics of heavy-ion collisions and properties of neutron stars and
associated phenomena has been long recognized and has prompted numerous
experimental and theoretical studies Li:2008gp ; Lattimer:2006xb ;
Baldo:2016jhp ; Lattimer:2015nhk . By combining results for various
experimental observables with phenomenological models Chen:2005ti ;
Trippa:2008gr ; Tsang:2012se ; Brown:2013mga ; Zhang:2013wna ;
Danielewicz:2013upa ; Morfouace:2019jky and theoretical many-body simulations
of nuclear matter Kruger:2013kua ; Drischler:2016djf ; Drischler:2015eba a
consistent description of SE at sub-saturation densities has been achieved.
The recent observation of a binary neutron star merger by the LIGO-VIRGO
collaboration TheLIGOScientific:2017qsa ; Abbott:2018exr has opened up the
possibility of studying the asy-EoS in the vicinity of twice saturation
density (2$\rho_{0}$) by means of correlations between tidal polarizability of
neutron stars ($\Lambda$), their radii and ultimately symmetry energy
Lattimer:2015nhk ; Fattoyev:2017jql . However, a unique correspondence between
$\Lambda$ and the SE does not exist, due to a degeneracy of the sensitivity to
the slope ($L$) and curvature ($K_{sym}$) parameters of the asy-EoS around
2$\rho_{0}$ Zhang:2018vbw . Nuclear physics laboratory experiments,
astrophysical observations and theoretical studies are thus needed to provide
lacking complementary information. More recently, developments of theoretical
many-body calculations based on chiral effective interactions have made
predictions of the asy-EoS up to 2$\rho_{0}$ with unprecedented accuracy
possible Drischler:2020hwi , calling for independent confirmation of these
results.
Heavy-ion collisions (HIC) provide an unique opportunity to study nuclear
matter at densities exceeding $\rho_{0}$ in the laboratory. To this end
several promising observables have been identified: the ratio of neutron-to-
proton yields of squeezed out nucleons Yong:2007tx , charged pion multiplicity
ratio (PMR) and its spectral ratio Li:2004cq ; Hong:2013yva , elliptic flow
related observables Li:2002qx and others. Using neutron-to-proton and
neutron-to-charged particles elliptic flow ratios compatible constraints for
the value of $L$ have been extracted using different transport models
Russotto:2011hq ; Wang:2014rva ; Russotto:2016ucm ; Cozma:2017bre .
Extrapolations to 2$\rho_{0}$ are still uncertain due to limited experimental
accuracy and suboptimal average density probed by these observables in AuAu
collisions at 400 MeV/nucleon impact energy.
The charged pion multiplicity ratio has attracted considerable attention from
the community. Reaching at a consistent picture for the density dependence of
SE has been however elusive up to this moment Xiao:2009zza ; Feng:2009am ;
Xie:2013np ; Hong:2013yva ; Song:2015hua ; Cozma:2016qej . Numerous studies
have attempted to remedy the problem, but have only succeeded in unvealing the
sensitivity of PMR to additional model ingredients Song:2015hua ;
Cozma:2016qej ; Cozma:2014yna ; Zhang:2017mps ; Zhang:2017nck ; Zhang:2018ool
; Ikeno:2016xpr ; Ikeno:2019mne ; Cui:2019dmk . In recent years, the Transport
Model Evaluation Project (TMEP) has aimed towards understanding differences
between existing models and formulating benchmark calculations that every
realistic model should reproduce Xu:2016lue ; Zhang:2017esm ; Ono:2019ndq .
The model used in this study is part of that effort.
In Refs. Cozma:2016qej ; Cozma:2014yna a Quantum Molecular Dynamics (QMD)
model has been employed in an attempt to explain the FOPI experimental pion
production data Reisdorf:2010aa by inclusion of threshold effects
Ferrini:2005jw ; Ferini:2006je that arise as a consequence of imposing total
energy conservation of the system. This requirement is often not properly
treated in semi-classical transport models, in spite of its relevance for the
existence of thermodynamic equilibrium Zhang:2017nck . The crucial ingredients
for the computation of threshold effects are the in-medium potential energy of
nucleons, resonances (only $\Delta$(1232) close to the vacuum production
threshold) and pions.
The knowledge of the isoscalar $\Delta$(1232) potential (ISDP) is uncertain,
with empirical information contradicting microscopical calculations
O'Connell:1990zg ; Bodek:2020wbk ; Hirata:1977hg ; Horikawa:1980cv ;
Oset:1987re ; GarciaRecio:1989xa ; deJong:1992wm ; Baldo:1994fk .
Discrepancies among results of microscopical models have also been noted and
are often related to details of how pion-nucleon and pion-nucleon-delta
couplings have been extracted from few-body experimental data. In particular,
including (or omitting) processes such as $N\Delta\rightarrow N\Delta$,
$N\Delta\rightarrow\Delta\Delta$, $NN\rightarrow\Delta\Delta$ and
$\Delta\Delta\rightarrow\Delta\Delta$ in models used to describe nucleon-
nucleon scattering data was proven to have an impact on the determined
strength of the $\Delta$ potential Baldo:1994fk . No information is avaiblable
about the isovector component of the $\Delta$(1232) potential (IVDP). These
quantities are also relevant for determining the threshold density above which
$\Delta$(1232) occurs in neutron stars, with impact on the maximum mass of
such objects Drago:2014oja ; Cai:2015xga ; Zhu:2016mtc ; Kolomeitsev:2016ptu ;
Li:2019tjx and in the analysis of neutrino physics experimental data
Bodek:2020wbk .
In view of the above, it is customary to set, in transport models, the
$\Delta$(1232) potential (DPOT) in terms of that of nucleons using a simple
Ansatz based on the decay channels of this resonance into nucleon-pion pairs
Li:2002yda . The significance of this assumption was recognized and a large
sensitivity of PMR to the magnitude of these potentials was evidenced in Ref.
Cozma:2014yna . Subsequently, it was shown that the density dependence of the
SE can be studied by using PMR supplemented by the ratio of average transverse
momenta of charged pions Cozma:2016qej . The latter observable is needed in
order to constrain the strength of IVDP, which was varied using a scaling
parameter. In that study the ISDP was kept fixed, equal to that of the
nucleon, in spite of previously proven dependence of PMR on its strength
Cozma:2014yna .
Extracting the asy-EoS from low and intermediate energy regime experiments is
further complicated by uncertainties stemming from the rather poorly
constrained momentum/energy dependence of nuclear interactions, usually
quantified in terms of effective masses Morfouace:2019jky ; Li:2013ola ;
Li:2014qta ; Zhang:2015qdp ; Zhang:2017hvh ; Li:2018lpy and the degeneracy of
effects induced by the isoscalar mass, the neutron-proton effective mass
difference ($\delta m^{*}_{np}$) and the density dependence of SE on
observables Li:2018lpy ; Kong:2017nil ; Malik:2018juj .
The present study builds on the results of Refs. Cozma:2016qej ; Cozma:2014yna
. The goal is to describe all pionic observables, not just ratios of
multiplicities or average transverse momenta, in an attempt to reduce residual
model dependence originating from the isoscalar part of the interaction. To
achieve this goal the DPOT is treated as an independent quantity. For both
isoscalar and isovector components freedom is built into parametrizations as
to allow independent assigning of potential depths at saturation and effective
masses. Details of the transport model, parametrizations used for DPOT and
benchmarking calculations for nucleonic observables are presented in Section
2. The observables relevant for constraining of DPOT parameters and their
extraction from experimental data are described in Section 3. In Section 4 the
feasibility of constraining the density dependence of SE from pionic
observables is reassessed, together with a study of the impact of other
relevant model parameters, such as $\delta m^{*}_{np}$. A section devoted to
summary and conclusions follows.
## 2 The model
### 2.1 Transport model
Quantum molecular dynamics transport models provide a semi-classical framework
for theoretical description of heavy ion reactions by accounting for relevant
quantum aspects such as stochastic scattering and Pauli blocking of nucleons.
They deliver a solution for the time dependence of the density matrix of the
system by the method of the Weyl transformation applied to the many-body
Schrödinger equation. Generally, the expectation values for the position and
momentum operators can be shown to satisfy the classical Hamiltonian equations
of motion deGroot:1972aa ; Hartnack:1997ez . These can be factorized to each
particle by approximating the total wave-function of the system as the product
of individual nucleon wave functions, represented by Gaussian wave packets of
finite spread in phase space,
$\displaystyle\frac{d\vec{r}_{i}}{dt}=\frac{\partial\langle
U_{i}\rangle}{\partial\vec{p}_{i}}+\frac{\vec{p}_{i}}{m},\qquad\frac{d\vec{p}_{i}}{dt}=-\frac{\partial\langle
U_{i}\rangle}{\partial\vec{r}_{i}}\,.$ (1)
The average of the potential operator is understood to be taken over the
entire phase-space and weighted by the Wigner distribution of particle $i$.
The potential operator $U_{i}$ is in this case the sum of the Coulomb and
strong interaction potential operators.
In the present study a variant developed over the last couple of years, dubbed
dcQMD, is used Cozma:2017bre ; Cozma:2016qej ; Cozma:2014yna . It traces its
origin to the Tübingen QMD model transport model developed in the 90’s and
early 2000’s Khoa:1992zz ; UmaMaheswari:1997ig ; Fuchs:2000kp ;
Shekhter:2003xd .
In the present model, the relativistic relation between mass, energy and
momentum is used in all kinematic equations. Consequently the kinetic term in
Eq. (1) is replaced by its relativistic counterpart. To be complete, the
effective classical Hamiltonian reads
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\sqrt{p_{i}^{2}+m_{i}^{2}}+\sum_{i,j,j>i}\,\bigg{[}\frac{A_{u}+A_{l}}{2}+\tilde{\tau}_{i}\,\tilde{\tau}_{j}\,\frac{A_{l}-A_{u}}{2}\bigg{]}\,u_{ij}$
$\displaystyle+$
$\displaystyle\sum_{i,j,j>i}\bigg{[}(C_{l}+C_{u})+\tilde{\tau}_{i}\,\tilde{\tau}_{j}\,(C_{l}-C_{u})\bigg{]}\frac{u_{ij}}{1+(\vec{p_{i}}-\vec{p_{j}})^{2}/\Lambda^{2}}$
$\displaystyle+$
$\displaystyle\sum_{i}\,\frac{B}{\sigma+1}\,[1-x\tilde{\tau}_{i}\,\beta_{i}\,]\,u_{i}^{\sigma}+\frac{D}{3}\,[1-y\tilde{\tau}_{i}\,\beta_{i}]\,u_{i}^{2}+\sum_{i,j,j>i}U_{ij}^{Coul}$
where $\tilde{\tau}_{i}$=-$\tau_{i}/T_{i}$, $u_{ij}=\rho_{ij}/\rho_{0}$ is the
partial relative interaction density of particles $i$ and $j$ with
$u_{i}=\sum_{j\neq i}u_{ij}$ and $\beta_{i}$ is the isospin asymmetry at the
location of particle $i$. Here $T_{i}$ and $\tau_{i}$ denote the isospin and
isospin projection of particle $i$ respectively. It is straightforward to show
that the momentum independent part of the interaction leads to the expression
of the energy per particle presented in Eq. (2.2) up to symmetry potentials of
second and higher order. The momentum dependent term above represents a finite
particle number approximation to the corresponding expression in Eq. (2.2).
The scattering term includes elastic and inelastic two-baryon collisions
($N+N\rightarrow N+N$, $N+N\rightarrow N+R$, $N+R\rightarrow N+R^{\prime}$,
etc.), resonance decays into a pion-nucleon or pion-resonance pairs
($R\rightarrow N+\pi$ and $R\rightarrow R^{\prime}+\pi$) and single pion
absorption reactions ($\pi+N\rightarrow R$). Collision processes that consist
of 3-particle initial or final states (as for example non-resonant background
pion production $N+N\rightarrow N+N+\pi$) have not been considered. Non-
resonant pion production contributions are needed at invariant masses close to
the production threshold to describe experimental data Engel:1996ic ;
Shyam:1996id ; Effenberger:1996im . Their inclusion in the scattering term is,
in the context of using the geometrical Bertsch prescription for collision
validation Bertsch:1988ik and requirement of conservation of total energy of
the system Cozma:2014yna , technically challenging, leading to a significant
slow down of computations, and has thus not been attempted.
The vacuum Li-Machleidt Li:1993rwa ; Li:1993ef and Cugnon ${\it et\;al.}$
Cugnon:1980rb parametrizations of elastic nucleon-nucleon cross-sections are
used below and above pion production threshold respectively. They are modified
in nuclear matter using an empirical factor depending on density and relative
momentum, but not isospin asymmetry. Such a modification has been found
necessary to describe stopping and flow observables at low and intermediate
energy heavy-ion collision Barker:2016hqv ; Basrak:2016cbo ; Wang:2013wca ;
Li:2011zzp . The FU3FP4 parametrization in Ref. Li:2011zzp has been found to
lead to the best description of stopping and flow, see Section 2.3. For this
choice, elastic cross-sections are multiplied by a factor depending on the
local density $\rho$ and relative momentum $p$ of the scattering nucleons
$\displaystyle F(\rho,p)=\left\\{\begin{array}[]{ll}1&\textrm{if {\it p} $>$
1.0 GeV/c}\\\ \frac{F_{\rho}-1}{1+(p/p_{0})^{\kappa}}+1&\textrm{if {\it p}
$\leq$ 1.0 GeV/c}\end{array}\right.$ (5) $\displaystyle\mathrm{with}\qquad
F_{\rho}=\lambda+(1-\lambda)\,Exp[-\frac{\rho}{\zeta\,\rho_{0}}]\,.$
The parameters in the above expression take the following values: $p_{0}$=0.30
GeV/c, $\kappa$=8, $\lambda$=1/6 and $\zeta$=1/3. Theoretically computed
medium-modified cross-sections Li:1993rwa ; Li:1993ef fail to lead to a good
description of stopping at low impact energies.
The Huber ${\it et\,\,al.}$ parametrizations for vacuum inelastic nucleon-
nucleon cross sections Huber:1994ee are used. They lead to charged pion
production cross-section that underpredict experimental values for $nn$/$pp$
and $np$ reactions by 20$\%$ and 40$\%$ respectively, at an impact energy of
400 MeV/nucleon. The discrepancy can be alleviated by including non-resonant
background contributions. Charged pions emitted in HIC originate predominantly
from $nn$/$pp$ collisions since for these channels production cross-sections
are an order of magnitude larger than in $np$ reactions. Consequently,
explicit non-resonant terms to pion production multiplicities can be neglected
at the impact energies of interest for this study, as their omission can, as a
first approximation, be compensated by modifying the strength of the
$\Delta$(1232) potentials (see Section 3.1). This approximation becomes better
as the invariant mass of colliding baryons increases. High energy pions may
thus be a probe of the equation of state less impacted by this type of model
uncertainties.
Inelastic nucleon-nucleon cross-sections are modified in-medium by using a
scaling factor that depends on the effective masses of the scattering baryons,
in agreement with the results of the one-pion exchange microscopical model of
Ref. Larionov:2003av . Within this model in-medium modified inelastic
$NN\rightarrow N\Delta$ cross-sections have been determined by including
effects such as in-medium corrections to the pion propagator, vertex
corrections and in-medium effective masses. The dominant effect could be
described by a correction factor depending on effective masses of initial and
final state-baryons and of the medium-modified invariant mass obtained by
replacing canonical with kinetic momenta.
The dynamics of the present model is non-relativistic and consequently
modifications of the invariant mass using a relativistic mean field approach
is not possible. Instead we follow the approach in Ref. Cozma:2014yna
developed to ensure total energy conservation of the system, which naturally
leads to threshold effects and in-medium modifications of cross-sections. The
central assumption of the approach is that no true two-body scattering
processes exist, but rather they are modified by interaction with the rest of
the system. Due to energy exchange with the fireball the initial- and final-
state invariant masses of the two scattering particles ($s_{ini}$ and
$s_{fin}$), determined using vacuum masses and momenta, differ. Considering
the fact that vacuum inelastic cross-section for resonance excitation
increases with the invariant mass, the contribution involving two-particles
scattering with the higher invariant mass dominates the total scattering
amplitude. This approximation is best close to threshold and was estimated to
be valid up to impact energies of about 800 MeV/nucleon. Therefore the medium
modified invariant mass used to determine cross-sections reads
$s^{*}=Max(s_{ini},s_{fin})$. In Ref. Cozma:2014yna it was shown that
$s_{fin}-s_{ini}>0$ for EoS’es that are not too soft ($L>$0 MeV). The above
Ansatz thus translates into contributions that involve energy exchanges with
the fireball in the initial state, followed by inelastic scattering of the two
baryons, dominating the total scattering amplitudes of resonance excitation.
The expression for the in-medium inelastic cross-sections thus reads
$\displaystyle\sigma_{NN\to
N\Delta}^{(med)}(s^{*})=\frac{\mu^{(ini)*}}{\mu^{(ini)}}\,\frac{\mu^{(fin)*}}{\mu^{(fin)}}\,\sigma_{NN\to
N\Delta}^{(vac)}(s^{*})$ (6)
with starred and regular variables corresponding to in-medium and vacuum
quantities and $\mu$ denoting the reduced mass of the system. A similar
expression for the modification factor was obtained in Refs. Schulze:1997zz ;
Persram:2001dg ; Li:2005jy on qualitative grounds for elastic nucleon-nucleon
cross-sections. For effective masses the non-relativistic formula is used e.g.
$m^{*}={m}/{(1.0+\frac{m}{p}\frac{dU}{dp})}$. The density dependence of
effective masses has only a rather small impact on pion multiplicities, in
spite of modification factors that amount to values in the range of 0.5-0.7 at
saturation. Such substantial decreases of cross-sections are partially
compensated by having also smaller absorption $N\Delta\to NN$ rates. The
impact of in-medium modifications of inelastic cross-sections on pion
observables due to isospin asymmetry dependence of effective masses were found
to be small during tests and have been therefore neglected in the present
study.
The cross-section for the resonance absorption reaction $NR\rightarrow NN$ is
determined using a detailed balance formula Danielewicz:1991dh ,
$\displaystyle\frac{d\sigma^{(NR\rightarrow
NN)}}{d\Omega}(s^{*})=\frac{1}{4}\frac{m_{R}\,p_{NN}^{2}}{p_{NR}}\frac{d\sigma^{(NN\rightarrow
NR)}}{d\Omega}(s^{*})\times$ (7)
$\displaystyle\qquad\quad\bigg{(}\frac{1}{2\pi}\int_{m_{N}+m_{\pi}}^{\sqrt{s_{ini}}-m_{N}}dMM\,p^{\prime}_{NR}\,A_{R}(M)\bigg{)}^{-1}.$
Due to the difference between $s_{ini}$ and $s_{fin}$ momenta $p_{NN}$ and
$p_{NR}$ have to be evaluated using the invariant masses of the $NN$ (final)
and $NR$(initial) states respectively. Such a prescription can be understood
since, in the expression for the cross-section of a 2-body reaction
$NR\rightarrow NN$, $p_{NR}$ originates from the evaluation of the incoming
flux, while $p_{NN}$ arises from the final-state phase space.
The pion decay width of resonances is determined using the expression
Weil:2016zrk
$\displaystyle\Gamma_{R\rightarrow N\pi}(\sqrt{s})=\Gamma_{R\rightarrow
N\pi}(\sqrt{s_{0}})\,\frac{\sqrt{s_{0}}}{\sqrt{s}}\,\frac{p^{3}}{p_{0}^{3}}\,\frac{p_{0}^{2}+\Lambda^{2}}{p^{2}+\Lambda^{2}},$
(8)
depending on the invariant mass $\sqrt{s}$ and its pole mass value
$\sqrt{s_{0}}$; $p$ and $p_{0}$ are the corresponding pion momenta in the rest
frame of the resonance. The above formula is a particular case of a more
general expression Manley:1992yb for a value of the orbital angular momentum
of the pion-nucleon system equal to 1. The quantity $\Lambda$ is computed
using
$\displaystyle\Lambda=\sqrt{(m_{R}-m_{N}-m_{\pi})^{2}+\Gamma^{2}/4.0},$ (9)
where $m_{R}=1.232$ GeV and $\Gamma=0.115$ GeV (pole mass properties of the
resonance, $\Delta(1232)$ in this case); similarly for other resonances
(N(1440), etc). In the parent TuQMD model, as well a in previous publications
Cozma:2016qej ; Cozma:2014yna , a formula for the width that is close to the
Huber parametrization Huber:1994ee had been used. It leads to pion absorption
cross-sections close to threshold that are too large, by a factor close to 2,
as compared to the experimental data. At invariant masses in the vicinity of
the resonance’s mass pole realistic values are obtained. The above
parametrization for the resonance decay width solves the mentioned problem. It
is worth noting that a modification of the resonance decay width does not
require a refit of the Huber OBE model as long as double $\Delta$ production
is negligible, since the difference can be absorbed in the $\pi N\Delta$
vertex form-factor.
The above expression for the decay width employs a generic variable $s$. For
the resonance decay $R\rightarrow N\pi$ and pion absorption $\pi N\rightarrow
R$ terms in the transport model the expression is evaluated using a modified
invariant mass $s^{*}=Max(s_{ini},s_{fin})$ supplemented by the same
argumentation as for baryon-baryon scattering.
Contributions of pion optical potentials have been included by using the
Ericson-Ericson parametrization to describe their density, isospin asymmetry
and momentum dependence, see Ref. Cozma:2016qej for all relevant details. The
set of parameter values for the optical potential commonly known as Batty-1
Batty:1978aa has been used extensively in this work, with one exception. In
Section 4 the effective S-wave model set of parameters (denoted S’)
Cozma:2016qej has been used to study the residual model dependence on $p_{T}$
spectra of PMR. Mean field propagation of pions is treated similarly to that
of nucleons, by associating a Gaussian wave function to them, whose width has
been set such that the ratio of pion-to-proton charge radii is close to its
experimental value Cozma:2016qej .
Threshold effects have been accounted for within the global energy
conservation (GEC) scenario introduced in Ref. Cozma:2014yna and which has
been briefly presented above. It has been checked that such a scenario is
compatible with a system of nucleons, $\Delta$(1232)s and pions reaching
chemical equilibrium. Specifically, this has been achieved by performing
numerical checks of detailed balance. To this end, nuclear matter in a box at
temperature T=60 MeV has been simulated using the full model. The initial
state of the system consisted of nucleons and pions with relative multiplicity
abundances of 90$\%$ and 10$\%$ respectively. Detailed balanced for the
reactions $N+N\leftrightarrow N+\Delta$ and $\Delta\leftrightarrow\pi N$ was
shown to be fulfilled at a few percent level after a time lapse of about 100
fm/c which signals that chemical equilibrium has been reached. With
appropriate settings the model reproduces the benchmark results of the TMEP
Collaboration Xu:2016lue ; Zhang:2017esm ; Ono:2019ndq .
### 2.2 Baryon in-medium interactions
The same parametrization for the equation of state of nuclear matter as in
Cozma:2017bre is used. The potential part reads
$\displaystyle\frac{E}{N}(\rho,\beta)$ $\displaystyle=$ $\displaystyle
A_{u}\frac{\rho(1-\beta^{2})}{4\rho_{0}}+A_{l}\frac{\rho(1+\beta^{2})}{4\rho_{0}}$
$\displaystyle+\frac{B}{\sigma+1}\frac{\rho^{\sigma}}{\rho_{0}^{\sigma}}\,(1-x\beta^{2})+\frac{D}{3}\frac{\rho^{2}}{\rho_{0}^{2}}\,(1-y\beta^{2})$
$\displaystyle+\frac{1}{\rho\rho_{0}}\sum_{\tau,\tau^{\prime}}C_{\tau\tau^{\prime}}\\!\\!\int\\!\\!\int
d^{\\!\>3}\vec{p}\,d^{\\!\>3}\vec{p}\\!\;^{\prime}\frac{f_{\tau}(\vec{r},\vec{p})f_{\tau^{\prime}}(\vec{r},\vec{p}\\!\;^{\prime})}{1+(\vec{p}-\vec{p}\\!\;^{\prime})^{2}/\Lambda^{2}}.$
Its analytic form is similar to MDI Gogny-inspired parametrizations Das:2002fr
; Xu:2014cwa , but differs from these by an extra density-dependent but
momentum-independent term, proportional to the $D$ parameter, that has been
introduced in order to allow independent variations of the slope $L$ and
curvature $K_{sym}$ parameters of the symmetry energy, while keeping the
neutron-proton isovector effective mass difference fixed.
Figure 1: Momentum (left panel) and density dependence (right panel) of ISDP
for several choices of depth and effective isoscalar mass, as discussed in the
text, compared to the nucleon isoscalar potential with the compressibility
modulus set to $K_{0}$=245 MeV. Each explanatory key applies to both plots. In
the left panel, the nucleon potential in symmetric matter $U_{0}$ and the
standard choice $U_{0}^{\Delta}$ almost coincide. In the right, panel the
ISDPs $U_{0}^{\Delta}$ corresponding to different effective masses show
similar density dependence.
The corresponding single-particle nucleon potential is given by
$\displaystyle U_{\tau}(\rho,\beta,p)$ $\displaystyle=$ $\displaystyle
A_{u}\frac{\rho_{\tau^{\prime}}}{\rho_{0}}+A_{l}\frac{\rho_{\tau}}{\rho_{0}}$
$\displaystyle+B\,\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{\sigma}(1-x\beta^{2})+8\tau
x\frac{B}{\sigma+1}\frac{\rho^{\sigma-1}}{\rho_{0}^{\sigma}}\beta\rho_{\tau^{\prime}}$
$\displaystyle+D\,\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{2}(1-y\beta^{2})+8\tau
y\frac{D}{3}\frac{\rho}{\rho_{0}^{2}}\beta\rho_{\tau^{\prime}}$
$\displaystyle+\frac{2C_{\tau\tau}}{\rho_{0}}\int
d^{\\!\>3}\vec{p}\\!\;^{\prime}\,\frac{f_{\tau}(\vec{r},\vec{p}\\!\;^{\prime})}{1+(\vec{p}-\vec{p}\\!\;^{\prime})^{2}/\Lambda^{2}}$
$\displaystyle+\frac{2C_{\tau\tau^{\prime}}}{\rho_{0}}\int
d^{\\!\>3}\vec{p}\\!\;^{\prime}\,\frac{f_{\tau^{\prime}}(\vec{r},\vec{p}\\!\;^{\prime})}{1+(\vec{p}-\vec{p}\\!\;^{\prime})^{2}/\Lambda^{2}}.$
In the above expressions $\rho$, $\beta$ and $p$ denote the density, isospin
asymmetry and momentum variables respectively. The label $\tau$ designates the
isospin component of the nucleon and takes the value $\tau$=-1/2 (1/2) for
neutrons (protons). For cold nuclear matter it holds
$f_{\tau}(\vec{r},\vec{p})=(2/h^{3})\Theta(p_{F}^{\tau}-p)$, with
$p_{F}^{\tau}$ the Fermi momentum of nucleons with isospin $\tau$.
Figure 2: Momentum (left panel) and density dependence (right panel) of the
leading order symmetry potential $U^{\Delta}_{sym,1}$. The leading order
nucleon symmetry potential corresponding to $L$=60.5 MeV and $K_{sym}$=-81.0
MeV is also shown for comparison. Each explanatory key applies to both plots.
In the left panel, the leading order nucleon symmetry potential $U_{sym,1}$
and the standard choice IVDP $U^{\Delta}_{sym,1}$ almost coincide. In the
right, panel the IVDPs $U^{\Delta}_{sym,1}$ corresponding to different
effective mass differences show a very similar density dependence.
It is common practice, within the framework of transport models, to set the
resonance potentials in terms of the nucleonic one. This choice is guided by
the decay channels of the resonance in question into a final state comprising
a nucleon and a pion Li:2002yda . This approach is particularly well suited
for the $\Delta$(1232) baryon which has a branching ratio close to 1 for the
$\Delta\rightarrow N\pi$ decay channel. It is nevertheless applied to the
entire list of resonances included in the given transport model. To be
specific,
$\displaystyle U^{R}_{\tau}(\rho,\beta,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-\tau/T)\;U_{-\frac{1}{2}}(\rho,\beta,p)$
$\displaystyle+$
$\displaystyle\frac{1}{2}(1+\tau/T)\;U_{\frac{1}{2}}(\rho,\beta,p),$
where $T$ and $\tau$ are the isospin and its desired projection for the
resonance in question; $U_{-\frac{1}{2}}$ and $U_{\frac{1}{2}}$ represent the
neutron and proton potentials respectively, whose expressions can be read from
Eq. (2.2). For an isospin $T$=3/2 resonance it leads to
$\displaystyle\begin{array}[]{lcrcrcrcrl}U_{\Delta^{-}}&=&U_{-\frac{1}{2}}&&&=&U_{is}&+&U_{iv}&\\\
U_{\Delta^{0}}&=&\frac{2}{3}\,U_{-\frac{1}{2}}&+&\frac{1}{3}\,U_{\frac{1}{2}}&=&U_{is}&+&\frac{1}{3}U_{iv}&\\\
U_{\Delta^{+}}&=&\frac{1}{3}\,U_{-\frac{1}{2}}&+&\frac{2}{3}\,U_{\frac{1}{2}}&=&U_{is}&-&\frac{1}{3}U_{iv}&\\\
U_{\Delta^{++}}&=&&&U_{\frac{1}{2}}&=&U_{is}&-&U_{iv}&,\end{array}$ (17)
which can be split into iso-scalar and iso-vector contributions, denoted above
by $U_{is}$ and $U_{iv}$. Their expression can be readily found out to be
$\displaystyle U_{is}(\rho,\beta,p)$ $\displaystyle=$
$\displaystyle\frac{A_{u}+A_{l}}{2}\,\frac{\rho}{\rho_{0}}+B\,\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{\sigma}\,(1-x\beta^{2})$
$\displaystyle+D\,\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{2}\,(1-y\beta^{2})$
$\displaystyle+\frac{C_{l}+C_{u}}{\rho_{0}}\,[\;I(p,p_{F}^{n})+I(p,p_{F}^{p})\,],$
$\displaystyle U_{iv}(\rho,\beta,p)$ $\displaystyle=$
$\displaystyle\frac{A_{l}-A_{u}}{2}\,\frac{\rho}{\rho_{0}}\,\beta-2x\frac{B}{\sigma+1}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{\sigma}\beta$
$\displaystyle-2y\frac{D}{3}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{2}\beta$
$\displaystyle+\frac{C_{l}-C_{u}}{\rho_{0}}\,[\;I(p,p_{F}^{n})-I(p,p_{F}^{p})\,],$
with the following notations: $C_{l}=C_{1/2,1/2}=C_{-1/2,-1/2}$,
$C_{u}=C_{1/2,-1/2}=C_{-1/2,1/2}$, $p_{F}^{n}$ and $p_{F}^{p}$ represent the
Fermi momenta of neutrons and protons respectively; $I(p,p_{F}^{\tau})$ stands
for the integrals appearing in Eq. (2.2), for which an analytic expression can
be derived for the case of zero-temperature nuclear matter
$\displaystyle I(p,p_{F}^{\tau})=\int
d^{\\!\>3}\vec{p}\\!\;^{\prime}\,\frac{f_{\tau}(\vec{r},\vec{p}\\!\;^{\prime})}{1+(\vec{p}-\vec{p}\\!\;^{\prime})^{2}/\Lambda^{2}}$
(20)
$\displaystyle=\frac{2\pi}{h^{3}}\Lambda^{3}\Bigg{[}\frac{\Lambda^{2}+p_{F}^{2}(\tau)-p^{2}}{2\Lambda
p}\,\mathrm{ln}\,\frac{\Lambda^{2}+[p+p_{F}(\tau)]^{2}}{\Lambda^{2}+[p-p_{F}(\tau)]^{2}}$
$\displaystyle+\frac{2p_{F}(\tau)}{\Lambda}+2\,\Bigg{(}\mathrm{arctan}\frac{p-p_{F}(\tau)}{\Lambda}-\mathrm{arctan}\frac{p+p_{F}(\tau)}{\Lambda}\Bigg{)}\Bigg{]}.$
It can be easily seen that the expression above is an odd function of
$p_{F}^{\tau}$. As a result the isoscalar and isovector potentials above are
even and odd functions in the isospin asymmetry variable $\beta$ respectively,
as required by charge symmetry.
It is worth stressing that in the above equations $U_{is}$ and $U_{iv}$ are
identical to the corresponding nucleonic potentials, as a direct consequence
of the Ansatz in Eq. (2.2), and the parameters appearing in their expressions
are therefore determined by reproducing nuclear matter properties.
The nucleonic potential in Eq. (2.2) can be expanded in a Taylor series in
terms of the isospin asymmetry parameter around the point $\beta$=0
$\displaystyle U_{\tau}(\rho,\beta,p)$ $\displaystyle=$ $\displaystyle
U_{0}(\rho,p)+\sum_{i=1,\infty}\,U_{sym,i}(\rho,p)\,(2\tau\,\beta)^{i}\,.$
(21)
The first two terms, $U_{0}(\rho,p)$ and $U_{sym,1}(\rho,p)$, represent the
nucleon potential in isospin symmetric nuclear matter and the first-order
symmetry potential respectively. Their expressions can be derived from those
for the isoscalar and isovector nucleon potentials in Eq. (2.2) and Eq. (2.2)
using the relations
$\displaystyle U_{0}(\rho,p)$ $\displaystyle=$ $\displaystyle
U_{is}(\rho,\beta=0,p)\,,$ (22) $\displaystyle U_{sym,1}(\rho,p)$
$\displaystyle=$ $\displaystyle\lim_{\beta\to
0}\frac{U_{iv}(\rho,\beta,p)}{\beta}\,.$
Naturally, for the case of the Ansatz used in Eq. (2.2) we have
$U_{0}^{R}(\rho,p)=U_{0}(\rho,p)$ while
$U_{sym,1}^{R}(\rho,p)$=$U_{sym,1}(\rho,p)$ once the replacement
$2\tau\rightarrow\tau/T$ is made in Eq. (21).
In this study we depart from the usually made assumption in transport models,
briefly presented above, that $U_{is}$ and $U_{iv}$ entering Eq. (17) are the
corresponding nucleon potentials. We do however assume that their expressions
in terms of density, isospin asymmetry and momentum are the same but different
values for the coupling parameters. In order to make this distinction clear we
add a superscript “$\Delta$” to relevant quantities, in particular
$U_{is}^{\Delta}$, $U_{iv}^{\Delta}$, $U_{0}^{\Delta}$ and
$U_{sym,1}^{\Delta}$. We allow the freedom that the density and momentum
dependence of resonance potentials be different at intermediate and long
ranges as well as at densities below twice saturation density. We do however
require, for a standard case labeled accordingly where distinction is
relevant, that their high density part is similar to that of nucleons, in view
of their similar quark structure. This approach is different from the one
pursued in Refs. Cozma:2016qej ; Cozma:2014yna where both the isoscalar and
isovector components of the DPOT were modified by a scaling factor.
In the following we present details of how the values of parameters entering
in Eq. (2.2) and Eq. (2.2) are fixed in this study. There are six free
parameters entering the expression of the ISDP $U_{is}^{\Delta}$:
$(A_{l}+A_{u})/2$, $B$, $\sigma$, $D$, $C_{u}+C_{l}$ and $\Lambda$ (a
superscript “$\Delta$” is in order for each of these parameters, but is
omitted). For simplicity we set $D$=0.0 MeV and $\sigma$=1.465. The remaining
four are determined by requiring that certain values for the isoscalar
effective mass of the resonance $m_{\Delta}^{*}$ and the potential in
symmetric matter at suitable values for density and momentum,
$\tilde{U}_{0}^{\Delta}\equiv U_{0}^{\Delta}(\rho_{0},p=0)$,
$U_{0}^{\Delta}(2\rho_{0},p=0)$ and $U_{0}^{\Delta}(\rho_{0},p=\infty)$, are
described. The quoted value for $\sigma$ ensures that the density dependence
of the resulting ISDP is close to that of the nucleon once the values at the
three above mentioned points fulfill this requirement too.
The expression of the IVDP $U_{iv}^{\Delta}$ contains four additionally free
parameters: $(A_{l}-A_{u})/2$, $C_{l}-C_{u}$, $x$ and $y$ (again, the
“$\Delta$” superscript is omitted). The value of the last one is irrelevant in
the context of setting $D$=0 MeV. The remaining three are determined by
requiring definite values for $\tilde{U}_{sym,1}^{\Delta}\equiv
U_{sym,1}^{\Delta}(\rho_{0},p=0)$, $U_{sym,1}^{\Delta}(2\rho_{0},p=0)$ and the
isovector mass-splitting $\delta
m^{*}_{\Delta}=(m^{*}_{\Delta^{-}}-m^{*}_{\Delta^{++}})/m_{\Delta}$, the last
quantity being evaluated at saturation density and $\beta$=0.5. The second
order symmetry potential $U_{sym,2}^{\Delta}$ impacts the value of the
isovector mass-splitting at a few percent level since its contribution to the
symmetry potential is smaller than 10$\%$ irrespective of the value of
$\beta$.
The values for the ten model parameters for the case when the DPOT is similar
to the nucleon’s up to twice saturation density and for kinetic energies up to
1.0 GeV are presented in Table (1). For the isovector part, the quoted
parameter values lead to nucleon in-medium interactions that correspond to a
density dependence of SE with a slope $L$=60.5 MeV and curvature parameter
$K_{sym}$=-81.0 MeV.
Table 1: Input quantities and their values (first and second columns) used to set the DPOT together with the model parameters appearing in Eq. (2.2) and Eq. (2.2) and their determined values (third and fourth columns). This set of parameters leads to ISDP and IVDP that resemble the nucleonic potentials closely. Quantities denoted by capital letters are expressed in units of MeV, while the rest are dimensionless. The effective mass $m_{\Delta}^{*}$ is expressed in units relative to the vacuum value of the mass of the $\Delta$(1232) isobar. Input | Parameters
---|---
$m_{\Delta}^{*}$ | 0. | 65 | $\Lambda$ | 700. | 98
$U_{0}^{\Delta}(\rho_{0},p=0)$ | -67. | 0 | $C_{l}+C_{u}$ | -153. | 82
$U_{0}^{\Delta}(2\rho_{0},p=0)$ | -55. | 0 | $A_{l}+A_{u}$ | -26. | 15
$U_{0}^{\Delta}(\rho_{0},p=\infty)$ | +75. | 0 | $B$ | 88. | 08
| . | | $D$ (fixed) | 0. | 0
| . | | $\sigma$ (fixed) | 1. | 465
$\delta m_{\Delta}^{*}$ | 0. | 175 | $C_{l}-C_{u}$ | 125. | 50
$U_{1,sym}^{\Delta}(\rho_{0},p=0)$ | +45. | 0 | $A_{l}-A_{u}$ | -109. | 97
$U_{1,sym}^{\Delta}(2\rho_{0},p=0)$ | +67. | 5 | $x$ | 0. | 140
| . | | $y$ (fixed) | 0. | 0
In Fig. (1) the momentum and density dependence of ISDP in symmetric nuclear
matter $U_{0}^{\Delta}$ for several cases is presented. The corresponding
nucleon potential, $U_{0}$, is also shown for reference. A standard
$U_{0}^{\Delta}$ that corresponds to a potential depth at saturation and zero
momentum $\tilde{U}_{0}^{\Delta}$=-67.0 MeV and an isoscalar effective mass
$m^{*}_{\Delta}$ = 0.65 has been defined. It mirrors both the momentum and
density dependence of the nucleon $U_{0}$ potential, as can be seen from the
left and right panels of Fig. (1) respectively. Sensitivity of pionic
observables to $U_{0}^{\Delta}$ will be studied by varying its depth at
saturation $\tilde{U}_{0}^{\Delta}$ in the interval [-100.0,0.0] MeV and the
isoscalar effective mass in the range [0.45,0.85]. The potentials
corresponding to the limits of these intervals are shown in Fig. (1).
Modification of the ISDP depth at saturation induces also a drastic change of
the density dependence. Additionally, two potentials denoted as “stiff density
dependence” and “soft density dependence” are also shown. They have been
constructed by modifying the value of the potential at twice saturation
density $U^{\Delta}_{0}(2\rho_{0},p=0)$ to -5 MeV for stiff and -105 MeV for
soft and allowing for a non-zero value of the $D$ parameter while keeping
parameters $\sigma$ and $y$ fixed to the values quoted in Table (1). This
procedure ensures that the IVDP remains unchanged. The model parameters have
been adjusted such as to modify only the density dependence above saturation,
while keeping the potential depth at saturation and half-saturation (both at
zero momentum) and the isoscalar effective mass fixed. These two potentials
will be used to study the impact of stiff and soft supranormal density
dependence of the $U_{0}^{\Delta}$ potential on pionic observables.
Similarly, in Fig. (2) the momentum and density dependence of the leading
order symmetry potential of $\Delta$(1232) $U^{\Delta}_{sym,1}$ is shown. A
standard choice $U^{\Delta}_{sym,1}$ potential is defined by requiring that
its strength at saturation and zero momentum is
$\tilde{U}^{\Delta}_{sym,1}$=45.0 MeV and the isovector mass splitting amounts
to $\delta m^{*}_{\Delta}$=0.175. The corresponding nucleon potential, that
leads to a density dependence of symmetry energy with a slope $L$=60.5 MeV and
curvature parameter $K_{sym}$=-81.0 MeV, is shown for comparison. Sensitivity
of pionic observables to $U_{sym,1}^{\Delta}$ will be studied by varying its
strength at saturation $\tilde{U}^{\Delta}_{sym,1}$ in the interval
[-15.0,90.0] MeV and the isovector mass splitting $\delta m^{*}_{\Delta}$ in
the range [-0.10,0.30]. Also in this case, two potentials labeled “stiff” and
“soft” density dependence have been constructed by modifying the potential
strength at twice saturation density, while keeping the values at saturation
and half-saturation fixed, all this at p=0. The choices of
$U^{\Delta}_{sym,1}(2\rho_{0},p=0)$ equal to 117.5 MeV and 17.5 MeV have been
made for the stiff and soft cases respectively. Technically this was achieved
by modifying the value of quantity $D\,y$ (redefined as a variable independent
of $D$) while keeping $D$ equal to zero.
Figure 3: (Left Panel) Model dependence for proton stopping in central AuAu
collisions as a function of the impact energy per nucleon. (Right Panel)
System size dependence of stopping for protons in central collision for
systems of different masses. The FOPI experimental data Reisdorf:2010aa have
been plotted for comparison.
Figure 4: (Left Panel) Model dependence for elliptic flow of protons.
Theoretical curves have the same meaning as those in the left panel of Fig.
(3). (Right Panel) System size dependence of elliptic flow of protons.
Experimental data are taken from Ref. FOPI:2011aa .
### 2.3 Benchmarking the nucleonic sector
The time evolution of heavy-ion collisions at impact energies of a few hundred
MeV/nucleon is governed by nucleonic degrees of freedom. In order to
realistically describe pion production at these energies it is crucial that
nucleonic multiplicity spectra are accurately reproduced in order to have the
correct invariant mass spectra of two-body collisions. To this end, before
embarking on a study of pion production, a theoretical transport model would
have to pass the test of comparing predictions for nucleonic observables to
experimental data. In particular a proper description of stopping and flow
observables is mandatory.
In a previous publication theoretical predictions for transverse and elliptic
flows for 197Au+197Au at an impact energy of 400 MeV/nucleon were compared to
experimental FOPI data in the context of extracting constraints for the
density dependence of the symmetry energy Cozma:2017bre . In this Section, the
theory-experiment comparison is extended by investigating stopping and system
size dependence of observables in the 150 to 1000 MeV/nucleon impact energy
range.
In the left panel of Fig. (3) theoretical predictions for the stopping
observable $varxz$ of protons in central 197Au+197Au are presented and
compared to experimental data Reisdorf:2010aa . The impact of relevant models
ingredients is shown in order to assess model uncertainties. The full model
predictions (full curve) describe low impact energy data very well. At the
higher end of the incident energy interval a slight underprediction is however
noticeable. For comparison, full model predictions employing a different Pauli
blocking algorithm that estimates occupancy fractions making use of the
Gaussian wave function associated to each nucleon rather than the standard
TuQMD algorithm Cozma:2017bre are presented (dashed curve). The difference is
small at all incident energies. The importance of threshold effects and the
multi-nucleon correlations they induce is underlined by comparing the
predictions of the model with these effects switched off (dash-dotted curve)
to the full model (full curve). Their impact is larger at lower incident
energies, the difference between the two calculations amounting to about
10$\%$. The magnitude of the effect is surprising in view of the fact that
shifts of the invariant mass of the colliding nucleons amounts to a few MeV
Cozma:2014yna . At a basic level the effect is a consequence of stronger
energy dependence of elastic collision and the nucleon optical potential at
lower incident energies.
The impact of in-medium modifications of cross-sections is demonstrated by
switching off these effects to inelastic channels and then additionally also
to the elastic ones. As expected, in-medium modifications of inelastic cross-
section affect stopping observables only above 500 MeV/nucleon impact energy.
The Ansatz of relating these medium corrections to effective masses induces an
energy dependence of $varxz$ that deviates visibly from the experimental one
even though the absolute magnitudes are still reproduced. At low impact
energies, modifications of elastic nucleon-nucleon cross-sections are crucial
to describe experimental data and a momentum dependence of these effects
appears to be mandatory. Similar conclusions have been reached in other
studies Barker:2016hqv ; Wang:2013wca ; Li:2011zzp .
The same analysis has also been performed for deuteron and triton stopping in
197Au+197Au collisions for which experimental measurement are available
Reisdorf:2010aa . The relevance of the above discussed model ingredients
remains similar, however the overall description of the experimental data is
poorer. Deuteron stopping is under-predicted by approximately 15$\%$, while
for tritons the deviation increases to 35$\%$. This is not surprising in the
context of triton multiplicities being under-estimated by a factor of about 2
Cozma:2017bre by the model. Switching off in-medium effects on cross-sections
reduces the discrepancy considerably but the induced energy dependence of the
observable at the lower limit for the impact energy is not realistic.
The right panel of Fig. (3) presents predictions of the full model for proton
stopping in central collisions for three different systems: 197Au+197Au,
58Ni+58Ni and 40Ca+40Ca. Experimental results at impact energies for which
data are available Reisdorf:2010aa are also shown. A generally good agreement
between theory and experiment is observed.
Turning to elliptic flow, in the left panel of Fig. (4) predictions for
transverse momentum dependent elliptic flow of protons in 197Au+197Au
collision at an impact energy of 250 MeV/nucleon are presented. Similarly as
for stopping, the impact of certain model ingredients is shown. Only in-medium
modifications of elastic cross-sections lead to a significant departure from
the full model predictions, while threshold effects and different approaches
of computing the nucleon occupancy have a negligible impact. The full model is
in almost in perfect agreement to the corresponding experimental data
FOPI:2011aa . By comparing the left panels of Fig. (3) and Fig. (4) it is
evident that a simultaneous description of both stopping and elliptic flow is
not possible by solely introducing in-medium modifications of elastic cross-
sections. The inclusion of threshold effects appears almost indispensable. As
the incident energy is increased the impact of in-medium modifications of
elastic cross-sections on elliptic flow decreases, a good description of the
experimental data is still achieved Cozma:2017bre . Investigation of elliptic
flow of deuterons and tritons has lead to the same conclusions.
The right panel of Fig. (4) presents predictions for rapidity dependent
elliptic flow at an impact energy of 400 MeV/nucleon for three systems:
197Au+197Au, 96Ru+96Ru and 40Ca+40Ca. Experimental data are available only for
the first and third systems FOPI:2011aa . An excellent description of
197Au+197Au data is observed, the strength of the predicted elliptic flow of
protons for 40Ca+40Ca collisions is slightly weaker than the experimental one.
A similar picture is valid for the elliptic flow of deuterons for the same
reactions.
A similar study has been performed for transverse flow. None of the model
ingredients studied above have a significant impact for this observable and
consequently the quality of the description of the experimental data is
similar to that of Ref. Cozma:2017bre for all impact energies in the range of
interest and for all light cluster species for which experimental data have
been reported in Ref. FOPI:2011aa .
Figure 5: Dependence of the total charged pion multiplicity in CaCa collisions
(top panels), ratio of charged pion multiplicities of AuAu to CaCa (middle
panels) and double charged pion multiplicity ratio of AuAu to CaCa (bottom
panels) on $\Delta$ (1232) potential depths at saturation
$\tilde{U}_{is}^{\Delta}$, $\tilde{U}_{iv}^{\Delta}$, effective mass
parameters $m^{*}_{\Delta}$ and $\delta m^{*}_{\Delta}$, compressibility
modulus of symmetric nuclear matter $K_{0}$ and slope of symmetry energy at
saturation $L$ for central ($b_{0}<$0.15) collisions at 400 MeV/nucleon impact
energy. The corresponding experimental values Reisdorf:2010aa are depicted by
horizontal bands.
## 3 Impact of the $\Delta$ (1232) potential on pionic observables
The magnitudes of ISDP and IVDP are poorly known at best, as already
emphasized in previous sections. It is thus mandatory to identify a sufficient
number of observables to extract both the values for the parameters used to
fix these potentials and those describing the density dependence of the EoS.
Originally, the charged pion multiplicity ratio was proposed as an observable
for extracting the value of the slope parameter $L$ of SE Li:2004cq . In a
previous publication Cozma:2016qej the average transverse momentum of charged
pions was used to constrain the strength of the IVDP relative to the nucleon
symmetry potential. This observable has however been proven very sensitive to
the pion optical potential. To avoid additional model dependence we will
restrict the present study to multiplicity related observables only. An
obvious choice is the total charged pion multiplicity, for which experimental
data exist for several systems at various impact energies Reisdorf:2010aa .
Owing to the limited applicability of the approximations used in taking into
account threshold effects Cozma:2014yna the upper limit of the impact energy
will be restricted to 800 MeV/nucleon. Consequently the following available
experimental data for given systems and impact energies in MeV/nucleon can be
used: 40Ca40Ca (400, 600, 800), 96Ru96Ru (400), 96Zr96Zr (400) and 197Au197Au
(400, 600, 800). These systems have the following average isospin asymmetry:
0.0 (CaCa), 0.08 (RuRu), 0.17 (ZrZr) and 0.20 (AuAu) allowing the study of
both ISDP and IVDP. The rather broad range of impact energies will also
facilitate the study of their momentum dependence. In the near future
experimental data with significantly better accuracy for 108Sn112Sn,
112Sn124Sn and 132Sn124Sn at an impact energy of 270 MeV/nucleon, slightly
below threshold, will become available Shane:2014tsa and potentially provide
tighter constraints.
### 3.1 Relevant observables
For each system two independent observables can be constructed from charged
pion multiplicities: total charged pion multiplicity (PM) and charged pion
multiplicity ratio (PMR). For two systems at the same impact energy, one
neutron rich and one neutron deficient, two observables, dependent on the two
PMs and two PMRs can be defined: the ratio of total charged pion
multiplicities and double charged pion multiplicity ratio. Each of them are
useful in studying the impact of DPOT on pionic observables.
In the top panels of Fig. (5) the sensitivity of PM to DPOT parameters,
compressibility modulus of symmetric nuclear matter and slope $L$ of the
symmetry energy is presented. The standard choice for the six mentioned
parameters is (see also Table (1)): $\tilde{U}_{0}^{\Delta}$=-67.0 MeV,
$\tilde{U}_{sym,1}^{\Delta}$=45.0 MeV, $m^{*}_{\Delta}$=0.65, $\delta
m^{*}_{\Delta}$=0.175, $K_{0}$=245 MeV and $L$=60 MeV. The values of the first
four parameters lead to ISDP and IVDP that resemble closely the corresponding
ones of nucleon. The calculations presented in the figure were performed by
varying the indicated parameter within a reasonable interval, as represented
by the abscissa of the corresponding plot, while the values of the other five
parameters are kept unchanged to their standard ones. The PM displays
considerable sensitivity to the isoscalar potential depth at saturation
$\tilde{U}_{0}^{\Delta}$ and the value of the isoscalar effective mass
$m^{*}_{\Delta}$. There is a comparably much smaller sensitivity, of the order
of 10-20$\%$, to the strength of IVDP at saturation
$\tilde{U}_{sym,1}^{\Delta}$ and to the value of the compressibility modulus.
The sensitivities to the remaining two parameters, isovector mass difference
$\delta m^{*}_{\Delta}$ and slope parameter $L$ are negligibly small. The
sensitivity to $\tilde{U}_{sym,1}^{\Delta}$ is small, but not negligible, for
the 40Ca40Ca system. In fact the smallest sensitivity to this parameter was
found for 96Ru96Ru. The slope of the dependence of PM on
$\tilde{U}_{sym,1}^{\Delta}$ for 197Au197Au has an opposite sign to that
derived from the top panel of Fig. (5) for 40Ca40Ca. This suggests that the
net effect is the result of two opposite trends related to the average isospin
asymmetry and fluctuations. It is concluded that PM is suitable to fix the
parameters of ISDP. Secondary order corrections due to IVDP are not negligible
and will have to be accounted for.
In the middle panels of Fig. (5) a similar analysis is presented for the ratio
of total charged pion multiplicities of 197Au197Au to 40Ca40Ca at impact
energy of 400 MeV/A. The important feature of this observable is that the huge
dependence on ISDP evidenced for total pion multiplicities of individual
systems almost cancel out, with a remaining residual sensitivity of about
10$\%$. The dominant variations are related to the isovector potential depth
at saturation $\tilde{U}_{sym,1}^{\Delta}$ and isovector effective mass
difference $\delta m^{*}_{\Delta}$. Sensitivity to the equation of state
parameters $K_{0}$ and $L$ is also in this case limited to about 10$\%$.
Constraining the isovector potential parameters using this observable appears
feasible but model dependence is not negligible due to relatively important
sensitivity to other parameters. Using the standard values for model
parameters the calculation overestimates the experiment by 30$\%$. Varying
IVDP parameters within a conservative interval does not allow a mitigation of
the discrepancy. An investigation of this issue suggests that a possible
resolution may involve modifications of in-medium $\Delta$ production cross-
sections, a stiff density dependence of IVDP above saturation (see below) or a
larger positive value for the neutron-proton effective mass difference (the
standard choice being 0.33$\beta$).
The third sensitivity study was performed for the double charged pion
multiplicity ratio of 197Au197Au to 40Ca40Ca at 400 MeV/nucleon impact energy.
The results are presented in bottom panels Fig. (5) for the same model
parameters as above. For this observable the sensitivity to each of the chosen
model parameters is sizable. This is the only observable of the three that
shows sizable dependence to asy-EoS. However, the extraction of the value of
$L$ is impeded by the unknown values of DPOT parameters. Setting this quantity
equal to that of the nucleon has been in the past a choice of convenience that
generally does not lead to a good description of all available experimental
data. The alternative approach of modifying in-medium $\Delta$ production
cross-section is restricted by existing microscopical models Larionov:2003av
that suggest that such effects are largely governed by scaling laws involving
in-medium effective masses. Such effective modifications of inelastic cross-
sections have been included in the present model.
Figure 6: Dependence of the total charged pion multiplicity in CaCa collisions
(top panel), ratio of charged pion multiplicities of AuAu to CaCa (middle
panel) and double charged pion multiplicity ratio of AuAu to CaCa (bottom
panel) on the density dependence of $U_{0}^{\Delta}$ and $U_{sym,1}^{\Delta}$
potentials above saturation for central ($b_{0}<$0.15) collisions at 400
MeV/nucleon impact energy. Results using the standard choice for the
supranormal density dependence of these potentials are presented in the
leftmost column for comparison. The corresponding experimental values
Reisdorf:2010aa are depicted by horizontal bands.
In Section 2.2 soft and stiff density dependent ISDPs and IVDPs have been
constructed and displayed in Fig. (1) and Fig. (2) respectively. In Fig. (6)
their impact on the three observables discussed above is presented and
compared with predictions for the “standard” case potential whose parameters
values are listed in Table (1). Modifying the density dependence of ISDP above
saturation has an impact on total charged pion multiplicities (PM) of at most
10$\%$, a softer density dependence of the potential leading to an increase of
total pion multiplicities. The ratio of total charged pion multiplicities and
the double charged pion multiplicities ratio of AuAu to CaCa at 400
MeV/nucleon are only marginally affected by a soft/stiff density dependence of
ISDP above saturation. Modification of the density dependence of the IVDP has
a visible impact on all three observables. PM is affected at 5$\%$ level, the
impact on PM(Au)/PM(Ca) is close to $10\%$ and the effect on the double ratio
PMR(Au)/PMR(Ca) is the largest at 20$\%$. Comparison with Fig. (5) reveals
that the sensitivity to the density dependence of IVDP above saturation is
several times smaller than to the magnitude of the symmetry potential at
saturation $\tilde{U}_{sym,1}^{\Delta}$. The same observation is true also for
ISDP by an even larger margin. This provides an a posteriori justification for
the choice of parameters used in this study to fix the density dependence of
DPOT: its strength at saturation and at twice saturation density.
Sensitivity of each of the three observables has also been studied with
respect to the following ingredients of the transport model: pion potential,
neutron-proton effective mass difference and in-medium modification of both
elastic and inelastic cross-sections. The impact is found to lie in the
interval 5-10$\%$ in each case, with the exception of medium modifications of
cross-sections that impact total pion multiplicities at the level of 20$\%$
for light systems such as CaCa. It is concluded that the three observables can
be used to extract information on the strength of the ISDP and IVDP at
saturation $\tilde{U}_{0}^{\Delta}$ and $\tilde{U}_{sym,1}^{\Delta}$ and the
isoscalar effective mass of $\Delta$(1232) in nuclear matter $m^{*}_{\Delta}$.
Extracting constraints for $\delta m_{\Delta}^{*}$ is feasible but dependence
of results on other model parameters (such as the slope parameter of asy-EoS
and the density dependence of IVDP above saturation) cannot be neglected.
Constraints for the density dependence of DPOT above saturation are necessary
for a complete knowledge of this quantity. The results in this Section do
however prove that this is not feasible at present given the small impact on
the studied observables, which is comparable or even smaller than the
uncertainties of other not precisely known model ingredients such as pion
potentials or in-medium cross-sections. Consequently, in the following we will
only present the impact of the high density dependence of DPOT on constraints
for $\tilde{U}_{0}^{\Delta}$, $\tilde{U}_{sym,1}^{\Delta}$, $m_{\Delta}^{*}$
and $\delta m_{\Delta}^{*}$ rather than attempting to extract values for
$U_{0}^{\Delta}$ and $U_{sym,1}^{\Delta}$ at 2$\rho_{0}$.
A similar study has been performed for momentum related observables. Also in
this case the impact of the DPOT is sizable but of comparable relative
magnitude to that of the pion optical potential. Using such observables would
induce important model dependence of results, as already evidenced in Ref.
Cozma:2016qej , and will not be pursued here.
### 3.2 Constraining $\Delta$ (1232) potential parameters
Using the insights of the previous Section we proceed to extract constraints
for the values of DPOT parameters. Results for the isoscalar component are
presented in Fig. (7) as correlations between the isoscalar potential depth at
saturation $\tilde{U}_{0}^{\Delta}$ and isoscalar effective mass
$m_{\Delta}^{*}$. To fix this quantity the available experimental data
comprise those of the following (nearly) isospin symmetric systems: CaCa at
400, 600 and 800 MeV/nucleon and RuRu at 400 MeV/nucleon central collisions
Reisdorf:2010aa . In the left panel constraints for ISDP parameters extracted
from collisions of CaCa at 400 MeV/nucleon are presented in the form of
1-$\sigma$ confidence level contour plots. Besides a calculation employing the
full model, certain model ingredients have been modified or switched off to
test model dependence. Three additional simulations correspond to the full
model making use of a Pauli blocking algorithm based on computation of the
occupancy fraction using the Gaussian wave function associated to each nucleon
Cozma:2017bre , full model without the pion potential and full model with
vacuum inelastic cross-sections. The first two lead to results compatible with
the full model, while for the third the deviation is more important as a
consequence of total pion multiplicities being impacted at 20$\%$ level by in-
medium modifications of inelastic cross-sections. For heavier systems (such as
AuAu) or light systems at higher impact energies the effect of medium
modification of inelastic cross-sections on multiplicities is smaller, in the
10-15 $\%$ range. Additionally, constraints extracted using the soft and stiff
density dependent $U_{0}^{\Delta}$ potentials above saturation, introduced in
Section 2.2, are also presented. The impact of modifying the strength of
$U_{0}^{\Delta}$ at 2$\rho_{0}$ is small, similar in magnitude to the effect
due to the pion potential. Calculations using a soft/stiff density dependence
of $U_{sym,1}^{\Delta}$ are not shown, however results in Fig. (6) allow the
inference that their impact is of similar small magnitude as for
$U_{0}^{\Delta}$. Consequently the PM observable can only be used to study the
$U_{0}^{\Delta}$ potential close or below saturation by determining values for
$\tilde{U}_{0}^{\Delta}$ and $m_{\Delta}^{*}$.
Figure 7: (Left Panel) Model dependence of constraining ISDP depth at
saturation density and zero momentum $\tilde{U}_{0}^{\Delta}$ and isoscalar
effective mass $m_{\Delta}^{*}$ for central CaCa collisions at 400
MeV/nucleon. Results for six different cases are shown: full model, different
Pauli blocking algorithm, no pion potential, no in-medium effects on inelastic
channels cross-sections and soft/stiff density dependence of $U_{0}^{\Delta}$
above saturation. The star represents the choice of potential parameters that
would render the ISDP equal to nucleon’s. (Right Panel) Constraints for the
same parameters by making use of the FOPI experimental data Reisdorf:2010aa
for central collisions of RuRu at 400 MeV/nucleon and CaCa at 400, 600 and 800
MeV/nucleon. The combined result for the four reactions is represented by the
contour plot labeled “combined fit”. For the contour plot labeled “corr comb
fit” corrections due to sensitivity to IVDP have been applied, as described in
the text. All contour plots correspond to 1-$\sigma$ confidence level of
fitting total charged pion multiplicity.
In the attempt to pin down the momentum dependence of ISDP, simulations of
collisions at different impact energies have been performed and compared to
experimental data. Constraints for $\tilde{U}_{0}^{\Delta}$ and
$m_{\Delta}^{*}$ are presented in the right panel of Fig. (7). Simulations for
RuRu at 400 MeV/nucleon have also been performed and since total multiplicity
for this system has displayed the smallest sensitivity to the isovector
component of the potential, as previously mentioned, it has been added to the
comparison. It is evident that the 1-$\sigma$ CL contour plots for CaCa at
different impact energies have slightly different slopes in the
($\tilde{U}_{0}^{\Delta}$,$\,m_{\Delta}^{*}$) plane, converging at smaller
values for $m_{\Delta}^{*}$. A combined fit of the four reactions is sub-
optimal with a minimum value of $\chi^{2}$/point=2.55. As has been evidenced
in the upper panels of Fig. (5) there is still non-negligible residual
dependence on the IVDP strength. It was found that it affects total pion
multiplicities of CaCa by 15, 10 and 5 $\%$ at 400, 600 and 800 MeV/nucleon
impact energies respectively. Once this is taken into account a corrected
combined fit with a minimum value of $\chi^{2}$/point=0.80 is obtained. The
combined fit of the four reactions has somewhat restricted the possible values
for $\tilde{U}_{0}^{\Delta}$ and $m_{\Delta}^{*}$, definite values could
however not be extracted in part because the experimental data carry rather
large uncertainties but also because at higher impact energies the sensitivity
decreases. Near-future availability of experimental data slightly below pion
production threshold for the nearly isospin symmetric 108Sn112Sn system by the
SPIRIT collaboration may improve the present situation significantly. In Fig.
(7) the values of parameters leading to an ISDP equal to that of the nucleon
are depicted by a star symbol. It departs from the favored parameter values of
the combined fit for the four systems at more than 5-$\sigma$ CL. However,
within this approach it is not clear whether this is a model independent
conclusion as the favored values for the potential parameters may be the
result of the fit compensating for some drastic approximations.
In Ref. Bodek:2020wbk the strength of the DPOT was extracted from the study
of experimental data of quasi-elastic scattering of electrons on bound
nucleons in nuclei of different masses: 6Li, 12C, 27Al, 40Ca/Ar and 56Fe. It
was found that the DPOT is more attractive than the empirical nucleon optical
potential and the attraction is stronger for heavier nuclei reflecting higher
probed densities. The attraction is stronger by about 20 MeV at momenta close
to $p$=0.5 GeV/c for the heaviest nuclei for which the analyses was performed.
Additionally, an arguably stronger energy dependence was evidenced for momenta
significantly above the Fermi sea, which may suggest a lower isoscalar
effective mass of $\Delta$(1232). A previous similar study O'Connell:1990zg
has reached similar conclusions. These qualitative results are in full
agreement with findings of the present study for the ISDP, as shown in the
right panel of Fig. (7) due to a similar similar approach.
Comparison with microscopic calculations reveals important differences. Many-
body calculations of pion-nucleus scattering or absorption performed in the
framework of the Delta-hole model arrived at the conclusion of an ISDP less
attractive than that of the nucleon at saturation density:
$\tilde{U}_{0}^{\Delta}\approx$-30 MeV Hirata:1977hg ; Horikawa:1980cv ;
Oset:1987re ; GarciaRecio:1989xa . Contributions such as non-resonant
background pion production, the spin-orbit component of the $\Delta$ potential
and short-range corrections to interaction vertices are crucial for the quoted
result. From the upper panels of Fig. (5) it is evident that by inclusion of
non-resonant background contributions to pion production in the scattering
term of the transport model a less attractive ISDP will be favored. Ab-initio
calculations, that have used well established microscopical nucleon-nucleon
potentials (such as Argonne $v_{28}$) as input, performed within the framework
of the Bethe-Brueckner-Goldstone method Baldo:1994fk or one-boson exchange
nucleon-nucleon potentials in the relativistic Dirac-Brueckner model allowing
a good reproduction of the elastic pion-nucleon $P_{33}$ phase-shift
deJong:1992wm , have arrived at a mildly repulsive ISDP. This is in part due
to dominant repulsive contributions of total isospin I=2, a channel which
cannot be sufficiently constrained by elastic nucleon-nucleon scattering data.
To proceed to extraction of constraints for IVDP parameters
$\tilde{U}_{sym,1}^{\Delta}$ and $\delta m_{\Delta}^{*}$ specific choices need
to be made for ISDP parameters. The choice $\tilde{U}_{0}^{\Delta}$=-78 MeV
and $m_{\Delta}^{*}$=0.45 allows, as evidenced in the right panel of Fig. (7),
a good description of pion multiplicity for isospin symmetric (or nearly so)
systems. In Fig. (8) the favored values for $\tilde{U}_{sym,1}^{\Delta}$ and
$\delta m_{\Delta}^{*}$, resulting from comparing theoretical and experimental
values of PMR for central AuAu collisions at 400 MeV/nucleon incident energy,
are shown as 1-$\sigma$ CL contour plots. Results for three different values
of the slope parameter $L$ of SE are shown, evidencing an important dependence
on this parameter. Additionally, the soft and stiff density dependent
$U_{sym,1}^{\Delta}$ introduced in Section 2.2 lead to differences in the
extracted constraints of similar magnitude as those induced by $L$. It has
been verified that the extracted constraints are sensitive also to the value
of the neutron-proton effective mass difference. Results obtained by fitting
the experimental value of PMR for ZrZr central collisions at 400 MeV/nucleon
are also shown for $L$=60 MeV and the standard density dependence above
saturation. They are compatible with the corresponding ones for AuAu. Adding
the total multiplicity of charged pions for these isospin asymmetric systems
in the fit, slightly restricts the allowed ranges, by disfavoring regions of
higher values for $\tilde{U}_{sym,1}^{\Delta}$. The star symbol represents the
choice for the potential parameters that would lead to an identical isovector
potential for nucleons and $\Delta$(1232) isobars. Constraints extracted for a
widely used value of the slope parameter, $L$=60 MeV, depart from this
commonly made choice, but by a smaller margin compared to the isoscalar case.
Fitting available experimental data of PMR for AuAu at higher impact energy
does not bring additional information mainly due to the larger experimental
error for this observable. The second observable of interest for constraining
the isovector $\Delta$(1232) potential, the ratio of total charged-pion
multiplicities, has proven ineffective, nearly half of the probed parameter
space in Fig. (8) leading to theoretical predictions in accord to experiment.
It becomes clear that a unique extraction of DPOT using multiplicity
observables alone is not possible at present. In principle, the analysis can
be extended to include existing information related to momenta of pions.
Published results for the ratio of average $p_{T}$ of charged pions exists in
the literature Reisdorf:2010aa and have been used for this purpose in the
past Cozma:2016qej . The additional induced model dependence from the
isoscalar channel, as shown in the left panel of Fig. (7), is not negligible
and the extraction of the stiffness of SE will carry an even larger model
dependence. Determining the slope of the SE is in principle possible by
performing a five parameter fit of multiplicity and momentum related
observables. This avenue has been explored. The resulting value for $L$ does
however carry large uncertainties.
Figure 8: Constraints for the IVDP parameters, potential depth at saturation
density and zero momentum $\tilde{U}_{sym,1}^{\Delta}$ and isovector mass
difference $\delta m_{\Delta}^{*}$, extracted from FOPI experimental data
Reisdorf:2010aa for PMR in central ZrZr and AuAu collisions at 400
MeV/nucleon. The star represents the choice of potential parameters that would
render IVDP equal to nucleon’s. All contour plots correspond to 1-$\sigma$
confidence level. Figure 9: Numerical proof that any choice for DPOT
parameters, corresponding to both the isoscalar and isovector potentials, that
lie on the 2-dimensional subspace of the 4-dimensional parameter space that
results from fitting experimental total charged pion multiplicities and pion
multiplicity ratios lead to simulated spectra are the same up to uncertainties
induced by experimental data uncertainties. Parameter values for five such
choices together with the theoretical multiplicity and PMR $p_{T}$ spectra are
shown for mid-central (0.25$\leq b_{0}\leq$0.45) AuAu collisions at 400
MeV/nucleon.The standard values for the parameters determining the density
dependence the EoS of nuclear matter are used ($K_{0}$=245 MeV and $L$=60
MeV).
Close to the pion production threshold it is possible to partially avoid the
issue of the not uniquely extracted DPOT. By fitting experimental
multiplicities the 4-dimensional parameter space fixing DPOT at saturation is
projected onto a 2-dimensional subspace. For any choice of the remaining two
unconstrained parameters pion spectra are almost identical. This is a
consequence of the fact that close to threshold $\Delta$ degrees of freedom
have no impact on the time evolution of the reaction in view of their
scarcity. Consequently, nucleon spectra, which determine the distribution of
invariant masses at which inelastic collisions take place, remain for all
practical purposes unaffected by the depth or momentum dependence of DPOT. At
these energies the DPOT plays the role of normalization constants (zeroth
order moments) for the spectra, allowing for a reduction of model dependence
of higher order moments. Fitting pion multiplicities will thus preserve any
asy-EoS dependence of these quantities. The situation at higher impact
energies, close to 1 GeV/nucleon and above, where the fraction of nucleons
excited to resonances in the high density fireball is non-negligible
Ehehalt:1993cx , is different.
Results of numerical calculations, that prove invariance of spectra to
arbitrary choices of model parameters in the 2-dimensional subspace left
unconstrained after experimental multiplicities have been fitted, are
presented in Fig. (9). A four dimensional fit for PM and PMR for mid-central
AuAu collisions at 400 MeV/nucleon has been performed. Five combinations of
parameter values for $\tilde{U}_{0}^{\Delta}$, $m_{\Delta}^{*}$,
$\tilde{U}_{sym,1}^{\Delta}$ and $\delta m_{\Delta}^{*}$ for which the fit is
perfect ($\chi^{2}$/point=0.0) have been chosen such that the sets of values
are diverse. These choices are listed in the legend of Fig. (9). The resulting
total multiplicity and pion multiplicity ratio spectra as a function of the
transverse momentum $p_{T}$ are shown. They are definitely close to each other
though not identical. Differences are due to experimental accuracy of these
observables that were used to compute the value of $\chi^{2}$/point and to the
interpolation in a 4-dimensional parameter space using a very limited number
of points (3 for each dimension) spanning a rather large parameter space.
Nevertheless, the spectra are for the majority of cases within a few percent
of each other. Means of improving this numerical proof are obvious and with
predictable results. The standard density dependence of $U_{0}^{\Delta}$ and
$U_{sym,1}^{\Delta}$ above saturation has been used. Extending the fit to a
6-dimensional one, thus including two additional parameters that can be used
to change $U_{0}^{\Delta}(2\rho_{0},p=0)$ and
$U_{sym,1}^{\Delta}(2\rho_{0},p=0)$ will lead quantitatively to the same
multiplicity and single ratio spectra.
In practice the following approach will be used. Two parameters of the DPOT,
$m_{\Delta}^{*}$ and $\delta m_{\Delta}^{*}$, will be set to well chosen
values. The remaining two, $\tilde{U}_{0}^{\Delta}$ and
$\tilde{U}_{sym,1}^{\Delta}$ will be uniquely determined from a fit to
experimental data for PM and PMR. It should be stressed that such a procedure
destroys the predictive power of the model. The determined set of parameters
can only be used for the particular combination of systems and impact energies
used in the fit. No firm conclusions can be drawn from a possible description
(or failure to do so) of a different system. In the next Section the choice
$m_{\Delta}^{*}$=0.45 and $\delta m_{\Delta}^{*}$=0.0 will be used.
Figure 10: (Left Panel) Pion multiplicity and (Right Panel) pion multiplicity
ratio $p_{T}$ dependent spectra obtained after fitting DPOT parameters to
multiplicity observables for three values of the SE slope parameter $L$ and
$\delta m_{np}^{*}$=0.33$\,\beta$. For each value of $L$ the impact on spectra
of adding to the fitted observables also transverse momentum related ones
(average combined transverse momentum and average $p_{T}$ ratio) is also
shown. The simulations correspond to mid-central AuAu collisions at 400
MeV/nucleon impact energy. Unpublished FOPI experimental data Reisdorf:2015aa
are represented by full circle symbols (left panel) and band (right panel),
the shown uncertainties being statistical.
## 4 Feasibility of constraining the symmetry energy
### 4.1 Sensitivity to asy-EoS and model dependence
In this Section the sensitivity of transverse momentum PMR spectra to the
density dependence of SE and other relevant model parameters is studied. For
this purpose simulations for mid-central ($0.25<b_{0}<0.45$) AuAu collisions
at an incident energy of 400 MeV/nucleon have been performed. The sole
motivation for the choice of this system has been the availability of
experimental data. Nevertheless, this data set is preliminary and does not
account for systematical uncertainties, only statistical uncertainties being
depicted for experimental data in figures of this Section. They have been
employed in previous similar studies Song:2015hua ; Cozma:2016qej and can be
useful in estimating the feasibility of studying the symmetry energy using
this observable and the accuracy of the transport model.
In Fig. (10) a comparison of model prediction with experimental $p_{T}$
dependent individual pion multiplicity (left panel) and PMR (right panel)
spectra is presented. One set of calculations (full curves) correspond to DPOT
parameters extracted from a fit to PM and PMR observables, as described in
Section 3.2. Simulations for which DPOT parameters have been determined from a
fit of both multiplicity and average transverse momentum observables are also
displayed (dashed curves). The fitted momentum observables are: average
transverse momentum of charged pions $\langle
p_{T}^{c}\rangle=\frac{M_{\pi^{-}}\,\langle
p_{T}^{\pi^{-}}\rangle+M_{\pi^{+}}\,\langle
p_{T}^{\pi^{+}}\rangle}{M_{\pi^{-}}+M_{\pi^{+}}}$ and average transverse
momentum ratio $\frac{\langle p_{T}^{\pi^{+}}\rangle}{\langle
p_{T}^{\pi^{-}}\rangle}$. Each observable contributes to the total $\chi^{2}$
with the same weight. For each set, calculations for three values of the slope
parameter of SE are provided: $L$=15, 60 and 106 MeV. The value for the
neutron-proton effective mass difference has been set to its default value
$\delta m_{np}^{*}$=0.33$\beta$.
The left panel of Fig. (10) presents calculations for $\pi^{-}$ (top plot) and
$\pi^{+}$ (bottom plot) multiplicity spectra. Differences between the two sets
of calculations are largest for $\pi^{+}$ spectra at low and intermediate
$p_{T}$. Theoretical predictions for $\pi^{+}$ spectra are seen to deviate by
important margins from experimental data at large values of $p_{T}$.
Uncertainties in other model parameters, such as the neutron-proton effective
mass difference $\delta m^{*}_{np}$, may explain this discrepancy (see below).
Finer tuning of the symmetric part of EoS, in particular the compressibility
modulus $K_{0}$ and nucleon isoscalar effective mass, preserving a consistent
description of nucleonic observables, may also improve the description at high
$p_{T}$ spectra of both $\pi^{-}$ and $\pi^{+}$ mesons.
In the right panel of Fig. (10) the PMR $p_{T}$ dependent spectrum is
presented. Theoretical predictions display sensitivity to $L$ for all values
of $p_{T}$, in relative terms they are largest at higher transverse momenta
where predictions for the stiffest and softest choices of asy-EoS differ by a
factor of almost 2. At higher $p_{T}$ values the two sets of predictions are
nearly identical suggesting that this range of transverse momenta is free of
model dependence originating in left-over uncertainties of the DPOT. The two
sets of calculations become similar to each other for values of $p_{T}$ for
which the strength in the multiplicity spectrum is below 10$\%$ of its peak
value.
The inclusion of momentum observables in the fit does not allow for a perfect
fit, $Min(\chi^{2})=$0, to be obtained anymore, but the the minimum of the
merit function depends on other model parameters such as $L$ or the strength
of the optical pion potential. The quality of the fit when momentum
observables are included can in principle be improved by also varying
$m_{\Delta}^{*}$ and $\delta m_{\Delta}^{*}$, rather than using the values
mentioned at the end of Section 3.2. In practice the discrepancy between model
and experiment at low/moderate $p_{T}$ is reduced only modestly at the expense
of performing a 4-dimensional fit (explicit calculations have been performed
for the $L$=60 MeV case). The reason lies in the fact that to describe the
spectra, moments of $p_{T}$ multiplicity distributions of order larger or
equal to 1 need to be described by the model. The performed 4-dimensional fit
only ensures that the 0th order moments are reproduced. In principle this
approach can be used to constrain the asy-EoS parameters, but given the strong
dependence of momentum observables on pion optical potentials, constraints
extracted in this manner are rather imprecise Cozma:2016qej , as already
argued in the previous Section. The high $p_{T}$ region appears thus better
suited for studies of the SE. This will become clearer after other sources of
model dependence of predictions in this region will be addressed below.
The sensitivity of PMR spectra to other two parameters of the EoS, $\delta
m^{*}_{np}$and value of SE at $\rho$=0.10 fm-3, is presented in Fig. (11). The
former quantity is varied within a range that includes most constraints for
its value available in the literature: -0.33$\,\beta<\delta
m_{np}^{*}<0.66\,\beta$ Li:2018lpy . Theoretical calculations reveal that the
sensitivity to this parameter is almost as large as to the slope parameter of
SE. This is a hardly surprising result since a different momentum dependence
of the interaction results in different magnitudes of threshold shifts which
in turn have been previously shown to have a large impact on PMR Song:2015hua
; Cozma:2014yna . To our best knowledge the impact of $\delta m^{*}_{np}$ on
PMR has not been previously addressed, which may have contributed to a certain
extent to the conflicting results for the density dependence of SE obtained
using this observable.
Figure 11: Sensitivity of the $p_{T}$ dependent PMR spectra to the magnitude
of neutron-proton effective mass difference $\delta m_{np}^{*}$ and value of
the symmetry energy at $\rho$=0.10 fm-3. The same details regarding the
reaction as for Fig. (10) are in order. Figure 12: Model dependence PMR
spectra to the pion potential and in-medium effects on inelastic cross-
sections. The same details regarding the reaction as for Fig. (10) are in
order.
Figure 13: (Left Panel) Density dependence of the symmetry energy for the
three values of $L$ for which results were shown in Fig. (10). For the $L$=60
MeV case two additional EoS’es that have the same density dependence below
saturation as the standard one but have different values for the slope
parameter $L$ above that point are shown. (Right Panel) Transverse momentum
dependent PMR spectra for the three EoS’es with $L$=60 MeV but with different
values of the slope above saturation for mid-central AuAu collisions at
various impact energies. As the impact energy is increased the transverse
momentum above which spectra are insensitive to residual DPOT dependence (see
Fig. (10)) also increases, requiring computation of spectra up to higher
values of this variable. For AuAu collisions at 300 MeV/nucleon impact energy,
the total charged pion multiplicity and ratio are determined by extrapolating
existing experimental data for central AuAu reactions Reisdorf:2010aa which
leads to the approximate values of 1.0 and 4.25 for the two observables
respectively. To obtain the corresponding result for mid-central collisions
the experimentally observed fact that pion multiplicity divided by the number
of participants is constant as a function of impact parameter is used.
The latter parameter represents a substitute to fixing the symmetry energy at
saturation, a point where it is not accurately known at present. It has been
however possible to extract precise values at sub-saturation densities from
experimental data of static properties of nuclei Trippa:2008gr ; Brown:2013mga
; Zhang:2013wna . Such empirical findings are in good agreement with many body
calculations of the neutron matter EoS that use as input microscopical N3LO
chiral perturbation theory effective potentials Kruger:2013kua ;
Drischler:2016djf ; Drischler:2015eba . The empirical value
S($\tilde{\rho}$)=25.5 MeV, with $\tilde{\rho}$=0.1 fm-3, extracted in Ref.
Brown:2013mga has been used as part of the standard input to the model. The
sensitivity to this parameter has been studied by varying it in the extremely
conservative interval 22.5 MeV $<$ S($\tilde{\rho}$) $<$ 28.5 MeV. Results
plotted in Fig. (11) prove that its magnitude is limited to less than 10$\%$.
If the uncertainty of 1.0 MeV quoted in Ref. Brown:2013mga for
S($\tilde{\rho}$) is taken into account as a more realistic interval of
variation, the sensitivity drops to a few percent.
The sensitivity of results to a few extra model ingredients has additionally
been studied. In Fig. (12) the impact of modifying the pion optical potential,
either by choosing a different S-wave optical potential or discarding it
completely, and switching off in-medium effects on inelastic cross-sections in
PMR spectra is shown. In relative terms the impact of these model ingredients
is largest at low $p_{T}$ values. Nevertheless, in the high $p_{T}$ region, of
interest for SE studies, a 10$\%$ effect is still observed. Additionally it
has been investigated what the impact on PMR spectra of setting the DPOT equal
to nucleon’s or to a rather arbitrary strength ($\tilde{U}_{0}^{\Delta}$=-25
MeV, $m_{\Delta}^{*}$=0.85, $\tilde{U}_{sym,1}^{\Delta}$= 0 MeV and $\delta
m_{\Delta}^{*}$=-0.15) would be (not shown). In either case the deviation from
the standard full model calculation in Fig. (12) amounts to 20$\%$ in the high
$p_{T}$ region. Fitting multiplicity observables to extract DPOT parameters is
thus a minimum requirement to keep model dependence at reasonable levels.
The results presented above lead to the conclusion that studies of PMR cannot
provide a constraint for the density dependence of SE but rather a correlation
of the parameter used to adjust its stiffness (here the slope $L$) with the
value of $\delta m^{*}_{np}$. To lift this degeneracy an independent
constraint or information for the latter quantity needs to be provided from
other sources. Elliptic flow ratios of neutrons-to-charged particles, double
ratio of $n/p$ multiplicity spectra and dipole polarizability of nuclei have
been identified as promising such sources Cozma:2017bre ; Morfouace:2019jky ;
Malik:2018juj . To minimize model dependence, a third observable providing a
constraint for the nucleon isoscalar effective mass may be required.
### 4.2 Probed density and impact energy dependence
PMR ratio has been proposed as a probe of the density dependence of SE above
saturation. A few studies that address this question are available Liu:2014psa
; Yong:2017hak , but neither of these models include threshold effects. A
proof that pion production probes densities significantly above saturation is
provided in the following. In the left panel of Fig. (13) the density
dependence of SE for the three choices of $L$ employed in this Section is
presented. Two additional EoS’es that lead to different density dependence
above saturation for the $L$=60 MeV case have been constructed by modifying
the slope parameter above saturation to $L$=100 MeV and $L$=20 MeV to
reproduce a stiff and a soft density dependence above this point respectively.
The three $L$=60 MeV EoS’es have identical density dependence below saturation
enforced by using in each case also a common value for the curvature parameter
$K_{sym}$.
To avoid numerical problems generated by discontinuous derivatives of the SE
with respect to density, model parameters that govern the density dependence
of SE become $C^{1}$ functions of this variable in a narrow interval around
$\rho_{0}$, its width being set to 0.05$\rho_{0}$. As a consequence,
additional contributions to forces proportional to the derivatives with
respect to density of coupling constants will need to be included to obey
energy conservation. For the two-body term in Eq. (2.1) these corrections lead
to computational requirements that scale with the third power of the number of
nucleons, instead of the second power for the ordinary case. To avoid this
issue, the coupling constant of the two-body term has been kept the same below
and above saturation. Only the three-body contributions, proportional to the
coupling constants $x$ and $y$ in Eq. (2.1), have been modified with the
consequence that above saturation the values of $L$ and $K_{sym}$ cannot be
chosen independently anymore. The advantage of this approach resides in the
fact that energy conservation violation is small, of the order of a few
hundred KeV per event, even without including contributions to forces due to
the density dependence of the two coupling constants in the vicinity of
saturation.
In the right panel of Fig. (13) theoretical values for PMR spectra are
presented for the three asyEoS’es that are identical below saturation but
differ above this point. Results for mid-central AuAu collisions at four
impact energies in the 300-800 MeV/nucleon range are shown. Noteworthy
differences between the stiffest and softest choices for the SE that amount to
a factor close to 2 in the region $p_{T}>$ 0.25 GeV/c are observed. For the
impact energy of 400 MeV/nucleon it is almost as large as the one evidenced in
Fig. (10) for the case when the asy-EoS’es also differ below saturation. This
proves that the information about the high density phase where most
$\Delta$(1232) are first excited is preserved to a large extent in spite of
the fact that pions that survive up to the final state of the reaction
undergo, on average, a few absorption/decay processes. To determine the
density at which PMR is most sensitive to, calculations with different
combinations of values for the slope parameter below and above saturation have
to be performed. The average probed density can then be extracted using the
approach described in Ref. Morfouace:2019jky . The sensitivity to the asy-EoS
above saturation is approximately independent on impact energy advocating
experimental measurements at higher impact energy in view of less required
beam-time for similar statistical accuracy.
## 5 Summary and Conclusions
The dcQMD model, an offspring of the Tübingen QMD transport model, has been
further developed by implementing in-medium nucleonic resonance potentials
that can be set independently of the nucleon optical potential and are
described in terms of intuitive quantities such as potential depths, at
saturation and zero momentum, and effective masses. This effort has been
prompted by results of phenomenological studies and ab-initio calculations
that suggest a $\Delta$ (1232) potential that is different from that of the
nucleon. The two approaches have led however to different results which has
contributed in the past to adopting the Ansatz of equal resonance and nucleon
potentials in semi-classical transport models for heavy-ion collisions at
intermediate impact energies. This model extension has been deemed necessary
as the accurate description of observables carrying information about the
isospin dependent part of the equation of state of nuclear matter requires a
proper understanding of residual effects induced by uncertainties of our
knowledge of the equation of state of symmetric nuclear matter or other
quantities leading to isoscalar contributions to observables.
The upgraded model has been employed in the study of pion production from
slightly above threshold to impact energies of 800 MeV/nucleon. One of the
objectives has been the extraction of ${\it effective}$ isoscalar and
isovector $\Delta$(1232) potential strengths and masses from a comparison to
available experimental data for 40Ca40Ca, 96Ru96Ru, 96Zr96Zr and 197Au197Au
provided by the FOPI Collaboration. The analysis has been performed separately
for the isoscalar and isovector components of the $\Delta$(1232) potential
following the identification of observables that are predominantly sensitive
to one of the two potentials: total charged pion multiplicity for the former
and ratio of total charged multiplicity for systems with different isospin
asymmetry for the latter. The charged pion multiplicity, an observable
proposed in the past for the study of the density dependence of symmetry
energy, has been shown to be equally sensitive to both the isoscalar and
isovector $\Delta$(1232) potentials. It has been shown that available
experimental data for nucleonic observables such as stopping, transverse and
elliptic flow for systems of different masses and at different impact energies
can be accurately described by the model, a pre-requisite for studying pion
emission close to threshold.
The extraction of the isoscalar $\Delta$(1232) potential (ISDP) parameters has
been attempted using the experimental data for total charged pion
multiplicities for 40Ca40Ca and also 96Ru96Ru systems at impact energies of
400, 600 and 800 MeV/nucleon (only the first impact energy for the latter
system). A precise extraction of the potential depth and isoscalar effective
mass was not possible due to sub-optimal accuracy of experimental data and a
decrease of sensitivity at higher impact energies. However, an effective
isoscalar potential that is more attractive and a smaller isoscalar effective
mass are favored for $\Delta$(1232) as compared to those corresponding to the
nucleon. The result is in agreement with similar information extracted from
quasi-elastic electron-nucleus scattering but is incompatible with
microscopical model calculation. A possible reason for the latter is the
omission of non-resonant pion production contributions, which would lead to a
less attractive potential and may also impact its required momentum
dependence.
For the isovector $\Delta$(1232) potential (IVDP) the study has proven more
challenging. Comparing model predictions for the ratio of total charged
multiplicity for systems with different isospin asymmetry to experiment has
only led to extremely loose constraints for the IVDP parameters, spanning half
of the probed parameter space. Using the pion multiplicity ratio for isospin
asymmetric systems instead, more precise constraints, in the form of
correlations between potential depth and isovector effective mass difference,
could be obtained. These have however proven to be rather sensitive to values
of the slope parameter of symmetry energy at saturation, the value of the
neutron-proton effective mass difference and the assumed stiffness for the
density dependence of IVDP above saturation. Adding the total charged pion
multiplicity to the fit was shown to exclude the more repulsive IVDP
scenarios.
Without an accurate knowledge of the $\Delta$(1232) potential a study of the
symmetry energy using multiplicity observables alone is not possible. An
alternative, previously studied in Ref. Cozma:2016qej , is to include average
transverse momentum observables among the fitted quantities. The additional
uncertainties induced by the ISDP results however in even more uncertain
constraints than before.
Studying pion multiplicity ratio spectra has proven more fruitful. It has been
shown that by including average transverse momenta in the fit of DPOT
parameters, a value of $p_{T}$ above which spectra are insensitive to
uncertainties in the $\Delta$(1232) potential can be determined. Residual
model dependence due to pion optical potential and in-medium modifications of
cross-sections uncertainties are below 10$\%$ in this high $p_{T}$ region.
Extraction of information regarding the symmetry energy and related quantities
is thus feasible from a comparison theory-experiment of the high $p_{T}$ tail
of pion multiplicity ratio spectra. It has been shown that due to inclusion of
threshold effects the sensitivity to the value of the neutron-proton effective
mass difference has to be taken into consideration. Without input from other
sources only a correlation between the values of the slope of symmetry energy
and neutron-proton effective mass can be extracted from pion production close
to threshold. The sensitivity to the magnitude of symmetry energy at
$\rho$=0.10 fm-3, the density for which it is kept fixed in the present model,
was found to be small, of the order of a few percent. The sensitivity on the
density dependence of the symmetry energy above saturation was however found
appreciable in spite of the fact that surviving pions undergo, on average, a
few absorption/decay cycles and was proved to be approximately independent of
impact energy.
It is concluded that more accurate experimental data for pion production in
heavy-ion collisions from threshold to 800 MeV/nucleon incident energy, that
provide sufficient statistical accuracy but are below the point where the
fraction of excited nucleons into resonances becomes non negligible, will be
one of the pre-requisites for the extraction of constraints for the symmetry
energy at supranormal densities from terrestrial laboratory experiments.
However, precise information from other sources regarding the momentum
dependence of the isovector component of the nucleon potential will be needed
for providing a precise answer regarding the value of the symmetry energy
around 2$\rho_{0}$.
## 6 Acknowledgments
The authors acknowledge financial support from the U.S. Department of Energy,
USA under Grant Nos. DE-SC00145 30, US National Science Foundation, United
States Grant No. PHY- 1565546\. The research of M.D.C. has been financially
supported in part by the Romanian Ministry of Education and Research through
Contract No. PN 19 06 01 01/2019-2022. M.D.C. acknowledges the hospitality of
NSCL / MSU where part of this study was performed. The authors express their
gratitude to Maria Colonna, Pawel Danielewicz, Justin Estee, Che-Ming Ko,
William Lynch, Hermann Wolter and TMEP Collaboration for stimulating
discussions. The computing resources have been partly provided by the
Institute for Cyber-Enabled Research (ICER) at Michigan State University.
## References
* (1) B.A. Li, L.W. Chen, C.M. Ko, Phys. Rept. 464, 113 (2008). DOI 10.1016/j.physrep.2008.04.005
* (2) J.M. Lattimer, M. Prakash, Phys. Rept. 442, 109 (2007). DOI 10.1016/j.physrep.2007.02.003
* (3) M. Baldo, G.F. Burgio, Prog. Part. Nucl. Phys. 91, 203 (2016). DOI 10.1016/j.ppnp.2016.06.006
* (4) J.M. Lattimer, M. Prakash, Phys. Rept. 621, 127 (2016). DOI 10.1016/j.physrep.2015.12.005
* (5) L.W. Chen, C.M. Ko, B.A. Li, Phys. Rev. C72, 064309 (2005). DOI 10.1103/PhysRevC.72.064309
* (6) L. Trippa, G. Colo, E. Vigezzi, Phys. Rev. C77, 061304 (2008). DOI 10.1103/PhysRevC.77.061304
* (7) M.B. Tsang, et al., Phys. Rev. C86, 015803 (2012). DOI 10.1103/PhysRevC.86.015803
* (8) B.A. Brown, Phys. Rev. Lett. 111(23), 232502 (2013). DOI 10.1103/PhysRevLett.111.232502
* (9) Z. Zhang, L.W. Chen, Phys. Lett. B726, 234 (2013). DOI 10.1016/j.physletb.2013.08.002
* (10) P. Danielewicz, J. Lee, Nucl. Phys. A922, 1 (2014). DOI 10.1016/j.nuclphysa.2013.11.005
* (11) P. Morfouace, et al., Phys. Lett. B 799, 135045 (2019). DOI 10.1016/j.physletb.2019.135045
* (12) T. Krüger, I. Tews, K. Hebeler, A. Schwenk, Phys. Rev. C88, 025802 (2013). DOI 10.1103/PhysRevC.88.025802
* (13) C. Drischler, A. Carbone, K. Hebeler, A. Schwenk, Phys. Rev. C94(5), 054307 (2016). DOI 10.1103/PhysRevC.94.054307
* (14) C. Drischler, K. Hebeler, A. Schwenk, Phys. Rev. C93(5), 054314 (2016). DOI 10.1103/PhysRevC.93.054314
* (15) B. Abbott, et al., Phys. Rev. Lett. 119(16), 161101 (2017). DOI 10.1103/PhysRevLett.119.161101
* (16) B. Abbott, et al., Phys. Rev. Lett. 121(16), 161101 (2018). DOI 10.1103/PhysRevLett.121.161101
* (17) F. Fattoyev, J. Piekarewicz, C. Horowitz, Phys. Rev. Lett. 120(17), 172702 (2018). DOI 10.1103/PhysRevLett.120.172702
* (18) N.B. Zhang, B.A. Li, J. Phys. G 46(1), 014002 (2019). DOI 10.1088/1361-6471/aaef54
* (19) C. Drischler, R. Furnstahl, J. Melendez, D. Phillips, Phys. Rev. Lett. 125(20), 202702 (2020). DOI 10.1103/PhysRevLett.125.202702
* (20) G.C. Yong, B.A. Li, L.W. Chen, Phys.Lett. B650, 344 (2007). DOI 10.1016/j.physletb.2007.05.050
* (21) B.A. Li, G.C. Yong, W. Zuo, Phys. Rev. C71, 014608 (2005). DOI 10.1103/PhysRevC.71.014608
* (22) J. Hong, P. Danielewicz, Phys. Rev. C90(2), 024605 (2014). DOI 10.1103/PhysRevC.90.024605
* (23) B.A. Li, Phys.Rev.Lett. 88, 192701 (2002). DOI 10.1103/PhysRevLett.88.192701
* (24) P. Russotto, et al., Phys. Lett. B697, 471 (2011). DOI 10.1016/j.physletb.2011.02.033
* (25) Y. Wang, C. Guo, et al., Phys. Rev. C89(4), 044603 (2014). DOI 10.1103/PhysRevC.89.044603
* (26) P. Russotto, et al., Phys. Rev. C94(3), 034608 (2016). DOI 10.1103/PhysRevC.94.034608
* (27) M.D. Cozma, Eur. Phys. J. A54(3), 40 (2018). DOI 10.1140/epja/i2018-12470-1
* (28) Z. Xiao, B.A. Li, et al., Phys.Rev.Lett. 102, 062502 (2009). DOI 10.1103/PhysRevLett.102.062502
* (29) Z.Q. Feng, G.M. Jin, Phys.Lett. B683, 140 (2010). DOI 10.1016/j.physletb.2009.12.006
* (30) W.J. Xie, J. Su, L. Zhu, et al., Phys.Lett. B718, 1510 (2013). DOI 10.1016/j.physletb.2012.12.021
* (31) T. Song, C.M. Ko, Phys. Rev. C91(1), 014901 (2015). DOI 10.1103/PhysRevC.91.014901
* (32) M.D. Cozma, Phys. Rev. C95(1), 014601 (2017). DOI 10.1103/PhysRevC.95.014601
* (33) M.D. Cozma, Phys. Lett. B753, 166 (2016). DOI 10.1016/j.physletb.2015.12.015
* (34) Z. Zhang, C.M. Ko, Phys. Rev. C95(6), 064604 (2017). DOI 10.1103/PhysRevC.95.064604
* (35) Z. Zhang, C.M. Ko, Phys. Rev. C 97(1), 014610 (2018). DOI 10.1103/PhysRevC.97.014610
* (36) Z. Zhang, C.M. Ko, Phys. Rev. C 98(5), 054614 (2018). DOI 10.1103/PhysRevC.98.054614
* (37) N. Ikeno, A. Ono, Y. Nara, A. Ohnishi, Phys. Rev. C 93(4), 044612 (2016). DOI 10.1103/PhysRevC.93.044612. [Erratum: Phys.Rev.C 97, 069902 (2018)]
* (38) N. Ikeno, A. Ono, Y. Nara, A. Ohnishi, Phys. Rev. C 101(3), 034607 (2020). DOI 10.1103/PhysRevC.101.034607
* (39) Y. Cui, Y. Zhang, Z. Li, Chin. Phys. C 44(2), 024106 (2020). DOI 10.1088/1674-1137/44/2/024106
* (40) J. Xu, et al., Phys. Rev. C 93(4), 044609 (2016). DOI 10.1103/PhysRevC.93.044609
* (41) Y.X. Zhang, et al., Phys. Rev. C 97(3), 034625 (2018). DOI 10.1103/PhysRevC.97.034625
* (42) A. Ono, et al., Phys. Rev. C 100(4), 044617 (2019). DOI 10.1103/PhysRevC.100.044617
* (43) W. Reisdorf, et al., Nucl. Phys. A 848, 366 (2010). DOI 10.1016/j.nuclphysa.2010.09.008
* (44) G. Ferini, M. Colonna, T. Gaitanos, M. Di Toro, Nucl. Phys. A 762, 147 (2005). DOI 10.1016/j.nuclphysa.2005.08.007
* (45) G. Ferini, T. Gaitanos, M. Colonna, M. Di Toro, H. Wolter, Phys. Rev. Lett. 97, 202301 (2006). DOI 10.1103/PhysRevLett.97.202301
* (46) J. O’Connell, R. Sealock, Phys.Rev. C42, 2290 (1990). DOI 10.1103/PhysRevC.42.2290
* (47) A. Bodek, T. Cai, Eur. Phys. J. C 80(7), 655 (2020). DOI 10.1140/epjc/s10052-020-8236-8
* (48) M. Hirata, F. Lenz, K. Yazaki, Annals Phys. 108, 116 (1977). DOI 10.1016/0003-4916(77)90354-2
* (49) Y. Horikawa, M. Thies, F. Lenz, Nucl. Phys. A 345, 386 (1980). DOI 10.1016/0375-9474(80)90346-2
* (50) E. Oset, L. Salcedo, Nucl. Phys. A 468, 631 (1987). DOI 10.1016/0375-9474(87)90185-0
* (51) C. Garcia-Recio, E. Oset, L. Salcedo, D. Strottman, M. Lopez, Nucl. Phys. A 526, 685 (1991). DOI 10.1016/0375-9474(91)90438-C
* (52) F. de Jong, R. Malfliet, Phys.Rev. C46, 2567 (1992). DOI 10.1103/PhysRevC.46.2567
* (53) M. Baldo, L. Ferreira, Nucl.Phys. A569, 645 (1994)
* (54) A. Drago, A. Lavagno, G. Pagliara, D. Pigato, Phys. Rev. C 90(6), 065809 (2014). DOI 10.1103/PhysRevC.90.065809
* (55) B.J. Cai, B.A. Li, Phys. Rev. C 93(1), 014619 (2016). DOI 10.1103/PhysRevC.93.014619
* (56) Z.Y. Zhu, A. Li, J.N. Hu, H. Sagawa, Phys. Rev. C 94(4), 045803 (2016). DOI 10.1103/PhysRevC.94.045803
* (57) E. Kolomeitsev, K. Maslov, D. Voskresensky, Nucl. Phys. A 961, 106 (2017). DOI 10.1016/j.nuclphysa.2017.02.004
* (58) J.J. Li, A. Sedrakian, Astrophys. J. Lett. 874(2), L22 (2019). DOI 10.3847/2041-8213/ab1090
* (59) B.A. Li, Nucl. Phys. A708, 365 (2002). DOI 10.1016/S0375-9474(02)01018-7
* (60) B.A. Li, X. Han, Phys. Lett. B 727, 276 (2013). DOI 10.1016/j.physletb.2013.10.006
* (61) X.H. Li, W.J. Guo, B.A. Li, L.W. Chen, F.J. Fattoyev, W.G. Newton, Phys. Lett. B 743, 408 (2015). DOI 10.1016/j.physletb.2015.03.005
* (62) Z. Zhang, L.W. Chen, Phys. Rev. C 93(3), 034335 (2016). DOI 10.1103/PhysRevC.93.034335
* (63) Z. Zhang, Y. Lim, J.W. Holt, C.M. Ko, Phys. Lett. B 777, 73 (2018). DOI 10.1016/j.physletb.2017.12.012
* (64) B.A. Li, B.J. Cai, L.W. Chen, J. Xu, Prog. Part. Nucl. Phys. 99, 29 (2018). DOI 10.1016/j.ppnp.2018.01.001
* (65) H.Y. Kong, J. Xu, L.W. Chen, B.A. Li, Y.G. Ma, Phys. Rev. C 95(3), 034324 (2017). DOI 10.1103/PhysRevC.95.034324
* (66) T. Malik, C. Mondal, B. Agrawal, J. De, S. Samaddar, Phys. Rev. C 98(6), 064316 (2018). DOI 10.1103/PhysRevC.98.064316
* (67) S.R. de Groot, L.G. Suttorp, _Foundations of Electrodynamics_ (North-Holland, Amsterdam, 1972)
* (68) C. Hartnack, R.K. Puri, J. Aichelin, J. Konopka, S.A. Bass, H. Stöcker, W. Greiner, Eur. Phys. J. A1, 151 (1998). DOI 10.1007/s100500050045
* (69) D.T. Khoa, N. Ohtsuka, M.A. Matin, A. Faessler, S.W. Huang, E. Lehmann, R.K. Puri, Nucl. Phys. A548, 102 (1992). DOI 10.1016/0375-9474(92)90079-Y,10.1016/0375.9474(92)90079.Y
* (70) V.S. Uma Maheswari, C. Fuchs, A. Faessler, L. Sehn, D.S. Kosov, Z. Wang, Nucl. Phys. A628, 669 (1998). DOI 10.1016/S0375-9474(97)00646-5
* (71) C. Fuchs, A. Faessler, E. Zabrodin, Y.M. Zheng, Phys. Rev. Lett. 86, 1974 (2001). DOI 10.1103/PhysRevLett.86.1974
* (72) K. Shekhter, C. Fuchs, A. Faessler, M. Krivoruchenko, B. Martemyanov, Phys. Rev. C 68, 014904 (2003). DOI 10.1103/PhysRevC.68.014904
* (73) A. Engel, A. Dutt-Mazumder, R. Shyam, U. Mosel, Nucl. Phys. A 603, 387 (1996). DOI 10.1016/0375-9474(96)80008-F
* (74) R. Shyam, U. Mosel, Phys. Lett. B 426, 1 (1998). DOI 10.1016/S0370-2693(98)00297-4
* (75) M. Effenberger, A. Hombach, S. Teis, U. Mosel, Nucl. Phys. A 613, 353 (1997). DOI 10.1016/S0375-9474(96)00408-3
* (76) G. Bertsch, S. Das Gupta, Phys. Rept. 160, 189 (1988). DOI 10.1016/0370-1573(88)90170-6
* (77) G.Q. Li, R. Machleidt, Phys. Rev. C 48, 1702 (1993). DOI 10.1103/PhysRevC.48.1702
* (78) G.Q. Li, R. Machleidt, Phys. Rev. C 49, 566 (1994). DOI 10.1103/PhysRevC.49.566
* (79) J. Cugnon, T. Mizutani, J. Vandermeulen, Nucl. Phys. A 352, 505 (1981). DOI 10.1016/0375-9474(81)90427-9
* (80) B. Barker, P. Danielewicz, Phys. Rev. C 99(3), 034607 (2019). DOI 10.1103/PhysRevC.99.034607
* (81) Z. Basrak, P. Eudes, V. de la Mota, Phys. Rev. C 93(5), 054609 (2016). DOI 10.1103/PhysRevC.93.054609
* (82) Y. Wang, C. Guo, Q. Li, H. Zhang, Z. Li, W. Trautmann, Phys. Rev. C89(3), 034606 (2014). DOI 10.1103/PhysRevC.89.034606
* (83) Q. Li, C. Shen, C. Guo, Y. Wang, Z. Li, J. Łukasik, W. Trautmann, Phys. Rev. C83, 044617 (2011). DOI 10.1103/PhysRevC.83.044617
* (84) S. Huber, J. Aichelin, Nucl. Phys. A 573, 587 (1994). DOI 10.1016/0375-9474(94)90232-1
* (85) A. Larionov, U. Mosel, Nucl. Phys. A 728, 135 (2003). DOI 10.1016/j.nuclphysa.2003.08.005
* (86) H.J. Schulze, A. Schnell, et al., Phys.Rev. C55, 3006 (1997). DOI 10.1103/PhysRevC.55.3006
* (87) D. Persram, C. Gale, Phys.Rev. C65, 064611 (2002). DOI 10.1103/PhysRevC.65.064611
* (88) B.A. Li, L.W. Chen, Phys.Rev. C72, 064611 (2005). DOI 10.1103/PhysRevC.72.064611
* (89) P. Danielewicz, G. Bertsch, Nucl.Phys. A533, 712 (1991). DOI 10.1016/0375-9474(91)90541-D
* (90) J. Weil, et al., Phys. Rev. C 94(5), 054905 (2016). DOI 10.1103/PhysRevC.94.054905
* (91) D. Manley, E. Saleski, Phys. Rev. D 45, 4002 (1992). DOI 10.1103/PhysRevD.45.4002
* (92) C.J. Batty, S.F. Biagi, E. Friedman, S. Hoath, J.D. Davies, G.J. Pyle, G.T.A. Squier, Phys. Rev. Lett. 40, 931 (1978). DOI http://dx.doi.org/10.1103/PhysRevLett.40.931
* (93) C.B. Das, S.D. Gupta, C. Gale, B.A. Li, Phys. Rev. C67, 034611 (2003). DOI 10.1103/PhysRevC.67.034611
* (94) J. Xu, L.W. Chen, B.A. Li, Phys. Rev. C91(1), 014611 (2015). DOI 10.1103/PhysRevC.91.014611
* (95) W. Reisdorf, et al., Nucl. Phys. A 876, 1 (2012). DOI 10.1016/j.nuclphysa.2011.12.006
* (96) R. Shane, et al., Nucl.Instrum.Meth. A784, 513 (2015). DOI 10.1016/j.nima.2015.01.026
* (97) W. Ehehalt, W. Cassing, A. Engel, U. Mosel, G. Wolf, Phys. Rev. C 47, 2467 (1993). DOI 10.1103/PhysRevC.47.R2467
* (98) W. Reisdorf, private communication (2015)
* (99) H.l. Liu, G.C. Yong, D.H. Wen, Phys. Rev. C91(4), 044609 (2015). DOI 10.1103/PhysRevC.91.044609
* (100) G.C. Yong, Y. Gao, G.F. Wei, Y.F. Guo, W. Zuo, J. Phys. G 46(10), 105105 (2019). DOI 10.1088/1361-6471/ab3772
|
# OPUS: an interoperable job control system based on VO standards
Mathieu Servillat1 and Stéphane Aicardi2 and Baptiste Cecconi3 and Marco
Mancini4
###### Abstract
OPUS (Observatoire de Paris UWS System) is a job control system that aims at
facilitating the access to analysis and simulation codes through an
interoperable interface. The Universal Worker System pattern v1.1 (UWS) as
defined by the International Virtual Observatory Alliance (IVOA) is
implemented as a REST service to control the asynchronous execution of a job
on a work cluster. OPUS also follows the recent IVOA Provenance Data Model
recommendation to capture and expose the provenance information of jobs and
results. By following well defined standards, the tool is interoperable and
jobs can be run either through a web interface, or a script, and can be
integrated to existing web platforms. Current instances are used in production
by several projects at the Observatoire de Paris (CTA/H.E.S.S, MASER,
CompOSE).
1LUTH, Observatoire de Paris, CNRS, PSL, 92190 Meudon, France;
<EMAIL_ADDRESS>
2DIO, Observatoire de Paris, PSL, 92190 Meudon, France
2LESIA, Observatoire de Paris, CNRS, PSL, 92190 Meudon, France
2Institut Denis Poisson, CNRS, 37200 Tours, France
## 1 Objective and use cases
Research teams at the Observatoire de Paris often develop customized analysis
and simulation codes for their projects. They also have access to locally
managed computing resources. However, the visibility of the codes sometimes
remains low, with a sub-optimal use of the available resources. The objective
of OPUS is to provide a simple and interoperable access to those codes, from
either a small team or a larger public, on either a local machine or a work
cluster. OPUS executes jobs asynchronously to enable the management of jobs
with long execution duration.
OPUS is developed in the context of the CTA and H.E.S.S projects at the
LUTH111https://voparis-cta-confluence.obspm.fr/display/CD/Analysis+tools. It
is tested with public H.E.S.S. data processed and analysed with tools such as
gammapy, ctools or ctapipe that run on the local cluster. The MASER project
provides a tool222https://maser.lesia.obspm.fr/task-2-modeling-tools/expres
that allows to simulate auroral radio emission dynamic spectra (Louis et al.
2019). The code is available for run-on-demand operations through an OPUS
instance. The online service CompOSE333https://compose.obspm.fr (Typel et al.
2013) provides data tables for different state of the art equations of state
for dense matter. Those data tables can be generated on demand through an OPUS
instance.
## 2 Modularity
The application is composed of several modules connected by different
Application Programming Interfaces (see Figure 1) : i/ a Web Server that
receives job control commands, stores job information and manages jobs on ii/
the Work Cluster that executes the jobs; and iii/ a Web Client that shows job
lists, job details with result preview (see snapshots in Figure 2), and which
includes a job definition editor and an admin interface.
Those modules may be executed on different machines or on the same laptop. The
documentation444https://opus-job-manager.readthedocs.io indicates how to
install and configure the system. The source code is available on
GitHub555https://github.com/ParisAstronomicalDataCentre/OPUS.
Figure 1.: Communications between OPUS modules. Figure 2.: Web client pages to
select a job, view it and visualize a result.
## 3 Interfaces and standards
The IVOA Universal Worker System pattern666https://www.ivoa.net/documents/UWS
(UWS, Harrison & Rixon 2016) defines how to manage asynchronous execution of
jobs on a service. The Web Server implements UWS as a REST API and returns
standard XML outputs. The UWS commands may be sent by the Web Client or using
scripts and command lines.
The IVOA Provenance Data Model777https://www.ivoa.net/documents/ProvenanceDM
(ProvDM, Servillat et al. 2020) is used for job definitions and provenance
tracing. The job definition editor implements the ProvDM description classes
(Usage, Generation, Parameters…). A ProvSAP (Simple Access Protocol) interface
returns graphs built with the voprov Python package. An example of the output
is presented in Figure 3.
In order to manipulate the job definitions, the IVOA service descriptor
serialization is extended to include descriptions of usage and generation of
entities by the job. Such files are VOTable composed of a RESOURCE block (type
= meta, utype = voprov:ActivityDescription), that contains attributes for the
activity description as PARAM blocks, and then GROUP blocks for InputParams,
Usage and Generation. This transposes to a JSON/YAML file exchanged between
web services, a job activity being described by a dictionary of attributes,
that includes sub-dictionaries for parameters, usage and generation.
The Server interfaces with a Work Cluster and its batch queue system (SLURM,
or local jobs are currently supported). User accounts on the Server are
managed from the Client using the System for Cross-domain Identity Management
(SCIM) standards888http://www.simplecloud.info.
## 4 Next developments
OPUS is also a test-bed to develop new functionalities related to the
management of provenance information. In particular, the granularity of the
provenance can be refined by providing detailed internal provenance at the end
of the job, e.g. generated with the logprov Python
package999https://github.com/mservillat/logprov that uses a similar YAML
serialization for job or activity definitions.
In order to better integrate OPUS with other services, we plan to include
federation authentication support (e.g. SAML or OpenID Connect) to the
authentication system.
It is also foreseen to provide a Docker container to deploy the system more
easily, with an automatic setup of the environment.
Figure 3.: Provenance graph returned by OPUS for a sequence of test jobs. Jobs
(Activities) are in blue square boxes, files (Entities) in yellow round shapes
and users (Agents) in light orange boxes. Description classes are shown in
dark orange, and configuration parameters in green.
## References
* Harrison & Rixon (2016) Harrison, P. A., & Rixon, G. 2016, Universal Worker Service Pattern Version 1.1, IVOA Recommendation 24 October 2016
* Louis et al. (2019) Louis, C. K., Hess, S. L. G., Cecconi, B., Zarka, P., Lamy, L., Aicardi, S., & Loh, A. 2019, A&A, 627, A30. 1901.11523
* Servillat et al. (2020) Servillat, M., Riebe, K., Boisson, C., Bonnarel, F., Galkin, A., Louys, M., Nullmeier, M., Renault-Tinacci, N., Sanguillon, M., & Streicher, O. 2020, IVOA Provenance Data Model Version 1.0, IVOA Recommendation 11 April 2020
* Typel et al. (2013) Typel, S., Oertel, M., & Klaehn, T. 2013, arXiv e-prints, arXiv:1307.5715. 1307.5715
|
# A Two-Stage Data Association Approach for 3D Multi-Object Tracking
Minh-Quan Dao, Vincent Frémont
Laboratoire des Sciences du Numérique de Nantes (LS2N)
École Centrale de Nantes
Nantes, 44300
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Multi-object tracking (MOT) is an integral part of any autonomous driving
pipelines because it produces trajectories which has been taken by other
moving objects in the scene and helps predict their future motion. Thanks to
the recent advances in 3D object detection enabled by deep learning, track-by-
detection has become the dominant paradigm in 3D MOT. In this paradigm, a MOT
system is essentially made of an object detector and a data association
algorithm which establishes track-to-detection correspondence. While 3D object
detection has been actively researched, association algorithms for 3D MOT seem
to settle at a bipartie matching formulated as a linear assignment problem
(LAP) and solved by the Hungarian algorithm. In this paper, we adapt a two-
stage data association method which was successful in image-based tracking to
the 3D setting, thus providing an alternative for data association for 3D MOT.
Our method outperforms the baseline using one-stage bipartie matching for data
association by achieving 0.587 AMOTA in NuScenes validation set.
## 1 Introduction
Multi-object tracking have been a long standing problem in computer vision and
robotics community since it is a crucial part of autonomous systems. From the
early work of tracking with hand-craft features, the revolution of deep
learning which results in highly accurate object detection models [1, 2, 3]
has shifted the focus of the field to the track-by-detection paradigm [4, 5].
In the framework of this paradigm, tracking algorithms receive a set of object
detection, usually in the form of bounding boxes, at each time step and they
aim to link detection of the same object across time to form trajectories.
While image-based methods of this paradigm have reached a certain maturity, 3D
tracking is still in its early phase where most of the published approaches
are originated from successful 2D exemplars. The most popular attempt at 3D
tracking with established 2D tracking method is [6] which is an extension of
[4] into 3D space. In these works, the tracking algorithm is made of the
Hungarian algorithm [7] and Kalman filter. While the former finds track-to-
measurement correspondences by solving a linear assignment problem, the later
performs prediction and correction of tracks’ state. An improvement of [6] is
proposed by [8] which replaces the 3D IoU [9] with the Mahalanobis distance as
the cost function for the assignment problem. The idea of handling tracking as
a matching problem is also used in the context of end-to-end learning [10, 11,
12]. [10] solves the tracking task in same fashion as [6]; however, this work
trains a sub network for calculating the cost function of the assignment
problem and the correction step is carried out by another sub network instead
of the Kalman filter.[11, 12] train deep models to predict tracks position in
the following frames along with generating detection and the track-to-
detection correspondences are found by greedy matching.
Even though 3D tracking has been progressed rapidly thanks to the availability
of standardized large scale benchmarks such as KITTI [13], NuScenes [14],
Waymo Open Dataset [15], the focus of the field is placed on developing better
object detection models rather than developing better tracking algorithm as
evidenced in the Table.1. There are two trends can be observed in this table.
First, tracking performance experiences significant boost when a better object
detection model is introduced. Second, the method of AB3DMOT [6] is favored by
most recent 3D tracking systems.
Table 1: Summary of tracking methods which details are published in the leader board of NuScenes and Waymo Open Dataset Dataset | Method Name | Tracking Method | AMOTA | Object Detector | mAP
---|---|---|---|---|---
NuScenes | CenterPoint [11] | Greedy closest-point matching | 0.650 | CenterPoint | 0.603
PMBM* | Poisson Multi-Bernoulli Mixture filter [16] | 0.626 | CenterPoint | 0.603
StanfordIPRL-TRI [8] | Hungarian algorithm with Mahalanobis distance as cost function and Kalman Filter | 0.550 | MEGVII [17] | 0.519
AB3DMOT [6] | Hungarian algorithm with 3D IoU as cost function and Kalman Filter | 0.151 | MEGVII | 0.519
CenterTrack | Greedy closest-point mathcing | 0.108 | CenterNet [18] | 0.388
Waymo | HorizonMOT [19] | 3-stage data associate, each stage is an assignment problem solved by Hungarian algorithm | 0.6345 | AFDet [20] | 0.7711
CenterPoint | Greedy closest-point matching | 0.5867 | CenterPoint | 0.7193
PV-RCNN-KF | Hungarian algorithm and Kalman Filter | 0.5553 | PV-RCNN [21] | 0.7152
PPBA AB3DMOT | Hungarian algorithm with 3D IoU as cost function and Kalman Filter | 0.2914 | PointPillars and PPBA[22] | 0.3530
The reason of AB3DMOT’s popularity is that despite its simplicity, it achieves
competitive result in challenging datasets at significantly high frame rate
(more than 200 FPS). However, such simplicity comes at the cost of the MOT
system being vulnerable to false associations due to occlusion or imperfect
detections which is case for objects in a clutter or far away from the ego
vehicle.
Aware of the shortage of a generic 3D tracking algorithm which can better
handle occlusion and imperfect detections so that to limit the false track-to-
detection correspondence, yet remains relatively simple, we adapt the image-
based tracking method proposed by [23] to the 3D setting. Specifically, this
method is a two-stage data association scheme. In this scheme, each tracked
trajectory is called a tracklet and is assigned a confidence score computed
based on how well associated detection matches with tracklet. The first
association stage aims to establish the correspondence between high confident
tracklets and detection. The second stage matches the left over detection with
the low confident tracklets as well as link low confident tracklets to high
confident ones if they meet a certain criterion.
In this paper, we make two contributions
* •
Our main contribution is the adaptation of an image-based tracking method to
the 3D setting. In details, we exploit a kinematically feasible motion model,
which is unavailable in 2D, to facilitate objects pose prediction. This model
in turn defines the minimal state vector needed to be tracked.
* •
Extensive experiment carried out in various datasets proves the effectiveness
of our approach. In fact, our better performance, compared to AB3DMOT-style
models, show that adding a certain degree of re-identification can improve the
tracking performance while keeping the added complexity to the minimum.
## 2 Related work
A multi-object tracking system in the track-by-detection paradigm consists of
an object detection model, a data association algorithm and a filtering
method. While the last two components are domain agnostic, object detection
models, especially learning-based methods, are tailored to their operation
domain (e.g images or point clouds). This paper targets autonomous driving
where objects pose are required thus interest in 3D object detection models.
However, developing such a model is not in the scope of this paper, instead we
use the detection result provided by baseline models of benchmarks (e.g.
PointPillars of NuScenes) to focus on the data association algorithm and to
have a fair comparison. Interested readers are referred to [24] for a review
of 3D object detection.
Data association via the Hungarian algorithm was early explored in [25] where
a 2-stage tracking scheme was proposed for offline 2D tracking. Firstly,
detections are linked frame-by-frame to form tracklets. The affinity matrix of
the Hungarian algorithm is established by geometric and appearance cue. While
the geometric cue is the 2D Intersection over Union (IoU), the appearance cue
is the correlation between two bounding boxes. Secondly, tracklets are
associated to each other to compensate trajectory fragments and ID switch due
to occlusion. This association is also carried out the Hungarian algorithm.
Due to its batch-processing nature [25] cannot be applied to online tracking,
[4] overcomes this by eliminating the second stage and let objects which
temporarily left the sensor’s field of view reenter with new IDs. Despite its
simplicity, SORT - the method proposed by [4] achieve competitive result in
MOT15[26] with lightning-fast inference speed (260 Hz). The success of SORT
inspired [6] to adapt it to 3D setting by using 3D IoU as the affinity
function. The performance of SORT in 3D setting is later improved in [8] which
shows the use of Mahalanobis distance is superior to 3D IoU. [27] integrates
the 3D version of SORT into a complete perception pipeline for autonomous
vehicles.
The two-stage association scheme is adapted to online tracking in [23] which
proposes a confidence score to quantify tracklets quality. Based on this
score, tracklets are associated with detections or another tracklets, or
terminated. The appearance model learned by ILDA in [23] is improved by deep
learning in the follow-up work [28]. Recently, this association scheme is
revisited in the context of image-based pedestrian tracking by [29] which
proposed to use the rank of the Hankel matrix as tracklets motion affinity.
Differ from [23] and its related works, this paper applies the two-stage
association scheme to online 3D tracking. In addition, we can provide
competitive result despite relying solely on geometric cue to compute tracklet
affinity thanks to the Constant Turning Rate and Velocity (CTRV) motion model
which can accurately predict objects position in 3D space by exploiting their
kinematic.
## 3 Method
### 3.1 Problem Formulation
Online multi-object tracking (MOT) in the sense of track-by-detection aims to
gradually grow the set of tracklets
$\mathbb{T}_{0:t}=\\{\mathcal{T}^{i}\\}_{i=1}^{|\mathbb{T}_{0:t}|}$ by
establishing correspondences with the set of detections received at every time
step $\mathbb{D}_{t}=\\{d_{t}^{j}\\}_{j=1}^{|\mathbb{D}|}$ and updating
tracklets state accordingly. A tracklet is a collection of state vectors
corresponding to the same object $\mathcal{T}^{i}=\\{\mathrm{x}_{k}^{i}|0\leq
t_{s}^{i}\leq k\leq t_{e}^{i}\leq t\\}$, here $t_{s}^{i},t_{e}^{i}$ are
respectively the starting- and ending-time of the tracklet. A detection
$d_{t}^{j}$ at time step $t$ encapsulates information of a 3D bounding box
including the position of its center in a common reference frame $[x,y,z]$,
heading $\theta$, and size $[w,l,h]$. It is worth to notice that in the
context of autonomous driving, objects are assumed to remain in contact with
the ground; therefore, their detections are up-right bounding boxes which
orientation is described by a single number - the heading angle.
The correspondence between $\mathbb{T}_{0:t}$ and $\mathbb{D}_{t}$ can be
formally defined as finding the set $\mathbb{T}_{0:t}^{*}$ that maximizes its
likelihood given $\mathbb{D}_{t}$.
$\mathbb{T}_{0:t}^{*}=\underset{\mathbb{T}_{0:t}}{\mathsf{argmax}}\text{
}\mathtt{p}\left(\mathbb{T}_{0:t}|\mathbb{D}_{t}\right)$ (1)
Due the exponential growth of possible associations between $\mathbb{T}_{0:t}$
and $\mathbb{D}_{t}$, Equation.(1) is computationally intractable after a few
time steps. In this paper, such a correspondence is approximated by the two-
stage data association proposed by [23] as shown in the following.
### 3.2 Two-stage Data Association
#### 3.2.1 Tracklet Confidence Score
The reliability of a tracklet is quantified by a confidence score which is
calculated based on how well associated detections match with its states
across its life span and how long its corresponding object was undetected.
$\mathsf{conf}\left(\mathcal{T}^{i}\right)=\left(\frac{1}{L^{i}}\sum_{k\in[t_{s}^{i},t_{e}^{i}]|v^{i}(k)=1}\Lambda\left(\mathcal{T}^{i},d_{k}^{j}\right)\right)\times\exp\left(-\beta\frac{W}{L_{i}}\right)$
(2)
where $v^{i}(k)$ is a binary indicator which takes 1 if the tracklet has a
detection $d_{k}^{j}$ associated with it at time step $k$, and 0 otherwise.
$L^{i}$ is the number of time step that the traklet gets associated with a
detection. $\Lambda(\cdot)$ is the affinity function which detail will be
presented later. $\beta$ is a tuning parameter which takes high value if the
object detection model is accurate. $W=t-t_{s}^{i}-L^{i}+1$ is the number of
time step that tracklet was undetected (i.e. did not have associated
detection) calculated from its birth to the current time step $t$.
Applying a threshold $\tau^{c}$ this confidence score divides the set
$\mathbb{T}_{0:t}$ into a subset of high confidence tracklets
$\mathbb{T}_{0:t}^{h}=\\{\mathcal{T}^{i,h}|\mathsf{conf}\left(\mathcal{T}^{i}\right)>\tau^{c}\\}$
and a subset of low confidence tracklets
$\mathbb{T}_{0:t}^{l}=\\{\mathcal{T}^{i,l}|\mathsf{conf}\left(\mathcal{T}^{i}\right)\leq\tau^{c}\\}$.
These two subsets are the fundamental elements of the two-stage association
pipeline showed in Figure.1
Figure 1: The pipeline of two-stage data association. The first stage - local
association establish the correspondences between detections at this time step
$\mathbb{D}_{t}$ and high confidence tracklets $\mathbb{T}_{0:t}^{h}$. Then,
global association stage matches each low confidence tracklets
$\mathcal{T}^{i,l}$ with either a high confidence tracklet or a left-over
detection, or terminates it.
#### 3.2.2 Local Association
In this association stage, high confident tracklets are extended by their
correspondence in the set of detections $\mathbb{D}_{t}$. This tracklet-to-
detection is found by solving the a linear assignment problem (LAP)
characterized by the cost matrix $\mathbf{L}$ as following
$\mathbf{L}=\begin{bmatrix}l_{i,j}\end{bmatrix}\in\mathbb{R}^{h\times
d},\text{ with }l_{i,j}=-\Lambda\left(\mathcal{T}^{i},d_{t}^{j}\right)$ (3)
where $h,d$ are respectively the number of high confidence tracklets and the
number of detections. The intuition of this association stage is that because
tracklets with high confidence have been tracked accurately for a number of
time steps, the affinity function can identify if a detection is belong to the
same object as the tracklet with high accuracy, thus limiting the possibility
of false correspondences. In addition, low confidence tracklets are usually
resulted from fragment trajectories or noisy detections, excluding them from
this association stage help reduces the ambiguity.
#### 3.2.3 Global Association
As shown in Figure.1, the global assocaition stage carries out the following
tasks
* •
Matching low confidence tracklets with high confidence ones
* •
Matching low confidence tracklets with detections left over by the local
association stage
* •
Deciding to terminate low confidence tracklets
These tasks are simultaneously solved as a LAP formulated by the following
cost matrix
$\mathbf{G}_{(l+d^{\prime})\times(h+l)}=\begin{bmatrix}\mathbf{A}_{l\times
h}&B_{l\times l}\\\ \infty_{d^{\prime}\times h}&C_{d^{\prime}\times
l}\end{bmatrix}$ (4)
here, $l,d$ are respectively the number low confidence tracklets and
detections left over by the local association stage. Recall $h$ is the number
of high confidence tracklets. Submatrix $\mathbf{A}$ is the cost matrix of the
event where low confidence tracklets are matched with high confidence ones
$\mathbf{A}=[a_{i,j}]\in\mathbb{R}^{l\times h},\text{ with
}a_{i,j}=-\Lambda\left(T^{i,l},T^{j,h}\right)$ (5)
Submatrix $\mathbf{B}$ represents the event where low confidence tracklets are
terminated.
$\mathbf{B}=[b_{i,j}]\in\mathbb{R}^{l\times l},\text{ with
}b_{i,j}=\begin{cases}-\log\left(1-\mathsf{conf}\left(\mathcal{T}^{i}\right)\right),\text{
if }i=j\\\ \infty,\text{ otherwise}\end{cases}$ (6)
Finally, submatrix $\mathbf{C}$ is the cost of the associating low confidence
tracklets with detections left over by local association stage.
$\mathbf{C}=[c_{i,j}]\in\mathbb{R}^{d^{\prime}\times l},\text{ with
}c_{i,j}=-\Lambda\left(\mathcal{T}^{j},d^{i}_{t}\right)$ (7)
The solution to the LAP in this stage and in the Local Association stage is
the association that minimize the cost and can be either found by the
Hungarian algorithm for the optimal solution or by a greedy algorithm which
interatively pick and remove correspondence pair with the smallest cost until
there is no pair has cost less than a threshold (the detail of this greedy
algorithm can be found in [8]).
Once a detection is assocatied with a tracklet, its position and heading is
used to update the tracklet’s state according to the equation of the Kalman
Filter, while its sizes is averaged with tracklet’s sizes in past few time
steps to result in updated sizes. Detections do not get associated in the
global association stage are used to initialize new tracklets.
#### 3.2.4 Affinity Function
Affinity function $\Lambda(\cdot)$ is to compute how similar a detection to a
tracklet or a tracklet to another. As mentioned earlier, due to the lack of
colorful texture in point cloud, the affinity function used in this work is
just comprised of geometric cue. Specifically, it is the sum of position
affinity and size affinity.
$\Lambda(\mathcal{T}^{i},d_{t}^{j})=\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}+\Lambda(\mathcal{T}^{i},d_{t}^{j})^{s}$
(8)
The scheme for computing position affinity between a tracklet and a detection
or between two tracklets are shown in Figure.2.
Figure 2: The computational scheme of position affinity. The filled triangles
(or rectangles) are subsequent states of a tracklet. The colored arrow
represents the time order: the closer to the tip, the more recent the state.
The triangle (or rectangle) in dash line is the state propagated forward (or
backward) in time. The covariance of these propagated states are denoted by
ellipses with the same color. The two-headed arrows indicate the Mahalanobis
distance. In the subfigure (a), the blue circle denotes a detection.
As shown in Figure.2.a, the position affinity
$\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}$ between a tracklet $\mathcal{T}^{i}$
and a detection $d_{t}^{j}$ is defined as the Mahalanobis distance between
tracklet’s last state propagated to the current time step $t$ and the
measurement vector $\mathrm{z}_{t}^{j}$ extracted from $d_{t}^{j}$
$\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}=\left(\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)-\mathrm{z}_{t}^{j}\right)^{T}\cdot\mathrm{S}^{-1}\cdot\left(\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)-\mathrm{z}_{t}^{j}\right)$
(9)
where $\bar{\mathrm{x}}^{i}_{e}$ is last state of tracklet $\mathcal{T}^{i}$
propagated to the current time step using the motion model which will be
presented below. $\mathsf{h}(\cdot)$ is the measurement model computing the
expected measurement using the inputted state and the measurement vector
$\mathrm{z}_{t}^{j}=[x,y,z,\theta]^{T}$. The matrix $\mathcal{S}$ is the
covariance matrix of the innovation (i.e. the difference between expected
measurement $\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)$ and its true
value $\mathrm{z}_{t}^{j}$)
$\mathbf{S}=\mathbf{H}\cdot\mathbf{P}\cdot\mathbf{H}^{T}+\mathbf{R}$ (10)
here, $\mathbf{H}=\delta\mathsf{h}/\delta\mathrm{x}$ is the Jacobian of the
measurement model, $\mathbf{P},\mathbf{R}$ are covariance matrix of
$\bar{\mathrm{x}}^{i}_{e}$ and $\mathrm{z}_{t}^{j}$, respectively.
In the case of two tracklets $\mathcal{T}^{i}$ and $\mathcal{T}^{j}$, assuming
$\mathcal{T}^{j}$ starts after $\mathcal{T}^{i}$ ended, their motion affinity
is, according to Figure.2.b, is the sum of
* •
Mahalanobis distance between the last state of $\mathcal{T}^{i}$ propagated
forward in time and the first state of $\mathcal{T}^{j}$
* •
Mahalanobis distance between the first state of $\mathcal{T}^{j}$ propagated
backward in time and the last state of $\mathcal{T}^{i}$
$\Lambda(\mathcal{T}^{i},\mathcal{T}^{j})^{p}=\Lambda(\mathcal{T}^{j},\bar{\mathrm{x}}^{i}_{e})^{p}+\Lambda(\mathcal{T}^{i},\bar{\mathrm{x}}^{j}_{s})^{p}$
(11)
here, $\bar{\mathrm{x}}^{i}_{e}$ is the last state of tracklet
$\mathcal{T}^{i}$ propagated forward in time to the first time step of
tracklet $\mathcal{T}^{j}$, while $\bar{\mathrm{x}}^{j}_{s}$ is the first
state of tracklet $\mathcal{T}^{j}$ propagated backward in time to the last
time step of tracklet $\mathcal{T}^{i}$.
The size affinity $\Lambda(\mathcal{T}^{i},d_{t}^{j})^{s}$ is computed as
following
$\Lambda\left(\mathcal{T}^{i},d_{t}^{j}\right)^{s}=-\frac{|w_{e}^{i}-w_{t}^{j}|}{w_{e}^{i}+w_{t}^{j}}\cdot\frac{|l_{e}^{i}-l_{t}^{j}|}{l_{e}^{i}+l_{t}^{j}}\cdot\frac{|h_{e}^{i}-h_{t}^{j}|}{h_{e}^{i}+h_{t}^{j}}$
(12)
here, $[w_{e}^{i},l_{e}^{i},h_{e}^{i}]$ are the size of the last state of
tracklet $\mathcal{T}^{i}$, while $[w_{t}^{j},l_{t}^{j},h_{t}^{j}]$ are the
size of the detection $d_{t}^{j}$. In the case of two tracklets
$\mathcal{T}^{i}$ and $\mathcal{T}^{j}$, assuming $\mathcal{T}^{j}$ starts
after $\mathcal{T}^{i}$ ended, there size affinity is
$\Lambda\left(\mathcal{T}^{i},\mathcal{T}^{j}\right)^{s}=-\frac{|w_{e}^{i}-w_{s}^{j}|}{w_{e}^{i}+w_{s}^{j}}\cdot\frac{|l_{e}^{i}-l_{s}^{j}|}{l_{e}^{i}+l_{s}^{j}}\cdot\frac{|h_{e}^{i}-h_{s}^{j}|}{h_{e}^{i}+h_{s}^{j}}$
(13)
The subscript $e,s$ in Equation.(13) respectively denote the ending and
starting state of a tracklet.
### 3.3 Motion Model and State Vector
Exploiting the fact that objects are tracked in 3D space of a common static
reference frame which can be referred to as the world frame, motion of objects
can be described by more kinematically accurate models, compared to the
commonly used Constant Velocity (CV) model. In this work, we use the Constant
Turning Rate and Velocity (CTRV) model to predict motion of car-like vehicles
(e.g. cars, buses, trucks), while keep the CV model for pedestrians.
For a car-like vehicles, its state can be described by
$\mathrm{x}=[x,y,z,\theta,\mathtt{v},\dot{\theta},\dot{z}]^{T}$ (14)
here, $[x,y,z]$ is the location in the world frame of the center of the
bounding box represented by the state vector, $\theta$ is the heading angle,
$\mathtt{v}$ is longitudal velocity (i.e. velocity along the heading
direction), $\dot{\theta},\dot{z}$ are respectively velocity of $\theta$ and
$z$.
The motion on x-y plane of car-like vehicles can be predicted using CTRV as
following
$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\frac{\mathtt{v}}{\dot{\theta}}\left(\sin(\theta+\dot{\theta}\Delta
t)-\sin(\theta)\right)\\\
\frac{\mathtt{v}}{\dot{\theta}}\left(-\cos(\theta+\dot{\theta}\Delta
t)+\cos(\theta)\right)\\\ \dot{z}\Delta t\\\ \dot{\theta}\Delta t\\\ 0\\\ 0\\\
0\\\ \end{bmatrix}$ (15)
where, $\Delta t$ is the sampling time. Note that in Equation.(15), $z$ is
assumed to evolve with constant velocity. In the case of zero turning rate
(i.e. $\dot{\theta}=0$),
$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\mathtt{v}\cos(\theta)&\mathtt{v}\sin(\theta)&\dot{z}\Delta
t&\dot{\theta}\Delta t&0&0&0&\end{bmatrix}^{T}$ (16)
The state vector of a pedestrian is
$\mathrm{x}=\begin{bmatrix}x&y&z&\theta&\dot{x}&\dot{y}&\dot{z}&\dot{\theta}\end{bmatrix}^{T}$
(17)
The motion of pedestrians is predicted according to CV model
$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\dot{x}&\dot{y}&\dot{z}&\dot{\theta}&0&0&0&0\end{bmatrix}^{T}\cdot\Delta
t$ (18)
## 4 Experiments
The effectiveness of our method is demonstrated by benchmarking against the
SORT-style baseline model on 3 large scale datasets: KITTI, NuScenes, and
Waymo. In addition, we perform ablation study using NuScenes dataset to better
understand the impact of each component on our system’s general performance.
### 4.1 Tracking Results
Evaluation Metrics: Classically, MOT systems are evaluated by the CLEAR MOT
metrics [30]. As pointed out by [31] and later by [6], there is a linear
relation between MOTA and object detectors’ recall rate, as a result, MOTA
does not provide a well-rounded evaluation performance of trackers. To remedy
this, [6] proposes to average MOTA and MOTP over a range of recall rate,
resulting in two integral metrics AMOTA and AMOTP which become the norm in
recent benchmarks.
Datasets To verify the effectiveness of our method, we benchmark it on 3
popular autonomous driving datasets which offer 3D MOT benchmark: KITTI,
NuScenes, and Waymo. These datasets are collections of driving sequences
collected in various environment using a multi-modal sensor suite including
LiDAR. KITTI tracking benchmark interests in two classes of object which are
cars and pedestrians. Initially, KITTI tracking was designed for MOT in 2D
images and recently [6] adapts it to 3D MOT. NuScenes concerns a larger set of
objects which comprises of cars, bicycles, buses, trucks, pedestrians,
motorcycles, trailers. Waymo shares the same interest as NuScenes but groups
car-like vehicles into meta class.
Public Detection: As can be seen in Table.1, AMOTA highly depends on the
precision of object detectors. Therefore, to have a fair comparison, the
baseline detection results made publicly available by the benchmarks are used
as the input to our tracking system. Specifically, we use MEGVII detection and
PointPillars with PPBA detection for NuScenes and Waymo, respectively.
The performance of our model compared to the SORT-style baseline model in 3
popular benchmarks are shown in Table.2. As can be seen, our model
consistently outperforms the baseline model in term of the primary metric
AMOTA. The main reason of this is the lower ID switches and trajectory
fragments of ours which shows the better ability of establishing track-to-
detection correspondence compared to SORT-style algorithm.
Table 2: Quantitative performance of our model on KITTI, NuScenes, and Waymo validation set. Dataset | Method | AMOTA$\uparrow$ | AMOTP$\downarrow$ | MT$\uparrow$ | ML$\downarrow$ | FP$\downarrow$ | FN$\downarrow$ | IDS$\downarrow$ | FRAG$\downarrow$
---|---|---|---|---|---|---|---|---|---
KITTI | Ours | 0.415 | 0.691 | N/A | N/A | 766 | 3721 | 10 | 259
AB3DMOT[6] | 0.377 | 0.648 | N/A | N/A | 696 | 3713 | 1 | 93
NuScenes | Ours | 0.583 | 0.748 | 3617 | 1885 | 13439 | 28119 | 512 | 511
StanfordIPRL-TRI[8] | 0.561 | 0.800 | 3432 | 1857 | 12140 | 28387 | 679 | 606
### 4.2 Ablation Study
In this ablation study, default method is the method presented in Section.3
which has
* •
Two stages of data association (local and global). Each stage is formulated as
a LAP and solved by a greedy matching algorithm [8].
* •
The affinity function the sum of position affinity and size affinity (as in
Equation.(8)).
* •
The motion model is Constant Turning Rate and Velocity (CTRV) for car-like
objects (cars, buses, trucks, trailers, bicycles) and Constant Veloctiy (CV)
for pedestrians.
To understand the effect of each component on the system’s general
performance, we modify or remove each of them and carry out experiment with
the rest of the system being kept the same as the default method and the same
hyperparameters. The changes and the resulted performance are shown in
Table.3.
Table 3: Ablation study using NuScenes dataset. Method | AMOTA$\uparrow$ | AMOTP$\downarrow$ | MT$\uparrow$ | ML$\downarrow$ | FP$\downarrow$ | FN$\downarrow$ | IDS$\downarrow$ | FRAG$\downarrow$
---|---|---|---|---|---|---|---|---
Default | 0.583 | 0.748 | 3617 | 1885 | 13439 | 28119 | 512 | 511
Hungarian for LAP | 0.587 | 0.743 | 3609 | 1880 | 13667 | 28070 | 596 | 573
No ReID | 0.583 | 0.748 | 3616 | 1882 | 13429 | 28100 | 504 | 510
Global assoc only | 0.327 | 0.924 | 2575 | 2244 | 26244 | 38315 | 4215 | 3038
Const Velocity only | 0.567 | 0.781 | 3483 | 1966 | 12649 | 29427 | 718 | 606
No size affinity | 0.581 | 0.748 | 3595 | 1904 | 13423 | 28448 | 512 | 508
## References
* [1] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. IEEE transactions on pattern analysis and machine intelligence, 39(6):1137–1149, 2016.
* [2] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg. Ssd: Single shot multibox detector. In European conference on computer vision, pages 21–37. Springer, 2016.
* [3] Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi. You only look once: Unified, real-time object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 779–788, 2016.
* [4] Alex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, and Ben Upcroft. Simple online and realtime tracking. In 2016 IEEE International Conference on Image Processing (ICIP), pages 3464–3468. IEEE, 2016.
* [5] Samuel Scheidegger, Joachim Benjaminsson, Emil Rosenberg, Amrit Krishnan, and Karl Granström. Mono-camera 3d multi-object tracking using deep learning detections and pmbm filtering. In 2018 IEEE Intelligent Vehicles Symposium (IV), pages 433–440. IEEE, 2018.
* [6] Xinshuo Weng, Jianren Wang, David Held, and Kris Kitani. Ab3dmot: A baseline for 3d multi-object tracking and new evaluation metrics. arXiv preprint arXiv:2008.08063, 2020.
* [7] Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83–97, 1955.
* [8] Hsu kuang Chiu, Antonio Prioletti, Jie Li, and Jeannette Bohg. Probabilistic 3d multi-object tracking for autonomous driving, 2020.
* [9] Dingfu Zhou, Jin Fang, Xibin Song, Chenye Guan, Junbo Yin, Yuchao Dai, and Ruigang Yang. Iou loss for 2d/3d object detection. In 2019 International Conference on 3D Vision (3DV), pages 85–94. IEEE, 2019.
* [10] Ming Liang, Bin Yang, Wenyuan Zeng, Yun Chen, Rui Hu, Sergio Casas, and Raquel Urtasun. Pnpnet: End-to-end perception and prediction with tracking in the loop. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11553–11562, 2020.
* [11] Tianwei Yin, Xingyi Zhou, and Philipp Krähenbühl. Center-based 3d object detection and tracking. arXiv preprint arXiv:2006.11275, 2020.
* [12] Wenjie Luo, Bin Yang, and Raquel Urtasun. Fast and furious: Real time end-to-end 3d detection, tracking and motion forecasting with a single convolutional net. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 3569–3577, 2018.
* [13] Andreas Geiger, Philip Lenz, Christoph Stiller, and Raquel Urtasun. Vision meets robotics: The kitti dataset. The International Journal of Robotics Research, 32(11):1231–1237, 2013.
* [14] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11621–11631, 2020.
* [15] Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for autonomous driving: Waymo open dataset. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2446–2454, 2020.
* [16] Angel F. Garcia-Fernandez, Jason L. Williams, Karl Granstrom, and Lennart Svensson. Poisson multi-bernoulli mixture filter: Direct derivation and implementation. IEEE Transactions on Aerospace and Electronic Systems, 54(4):1883–1901, Aug 2018.
* [17] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and Gang Yu. Class-balanced grouping and sampling for point cloud 3d object detection. arXiv preprint arXiv:1908.09492, 2019.
* [18] Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Objects as points, 2019.
* [19] Zhuangzhuang Ding, Yihan Hu, Runzhou Ge, Li Huang, Sijia Chen, Yu Wang, and Jie Liao. 1st place solution for waymo open dataset challenge – 3d detection and domain adaptation, 2020.
* [20] Runzhou Ge, Zhuangzhuang Ding, Yihan Hu, Yu Wang, Sijia Chen, Li Huang, and Yuan Li. Afdet: Anchor free one stage 3d object detection, 2020.
* [21] Shaoshuai Shi, Chaoxu Guo, Li Jiang, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. Pv-rcnn: Point-voxel feature set abstraction for 3d object detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10529–10538, 2020.
* [22] Shuyang Cheng, Zhaoqi Leng, Ekin Dogus Cubuk, Barret Zoph, Chunyan Bai, Jiquan Ngiam, Yang Song, Benjamin Caine, Vijay Vasudevan, Congcong Li, et al. Improving 3d object detection through progressive population based augmentation. arXiv preprint arXiv:2004.00831, 2020.
* [23] Seung-Hwan Bae and Kuk-Jin Yoon. Robust online multi-object tracking based on tracklet confidence and online discriminative appearance learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1218–1225, 2014.
* [24] Eduardo Arnold, Omar Y Al-Jarrah, Mehrdad Dianati, Saber Fallah, David Oxtoby, and Alex Mouzakitis. A survey on 3d object detection methods for autonomous driving applications. IEEE Transactions on Intelligent Transportation Systems, 20(10):3782–3795, 2019.
* [25] Andreas Geiger, Martin Lauer, Christian Wojek, Christoph Stiller, and Raquel Urtasun. 3d traffic scene understanding from movable platforms. IEEE transactions on pattern analysis and machine intelligence, 36(5):1012–1025, 2013.
* [26] Laura Leal-Taixé, Anton Milan, Ian Reid, Stefan Roth, and Konrad Schindler. Motchallenge 2015: Towards a benchmark for multi-target tracking. arXiv preprint arXiv:1504.01942, 2015.
* [27] Antoine Mauri, Redouane Khemmar, Benoit Decoux, Nicolas Ragot, Romain Rossi, Rim Trabelsi, Rémi Boutteau, Jean-Yves Ertaud, and Xavier Savatier. Deep learning for real-time 3d multi-object detection, localisation, and tracking: Application to smart mobility. Sensors, 20(2):532, 2020.
* [28] Seung-Hwan Bae and Kuk-Jin Yoon. Confidence-based data association and discriminative deep appearance learning for robust online multi-object tracking. IEEE transactions on pattern analysis and machine intelligence, 40(3):595–610, 2017.
* [29] Honghong Yang, Jinming Wen, Xiaojun Wu, Li He, and Shahid Mumtaz. An efficient edge artificial intelligence multipedestrian tracking method with rank constraint. IEEE Transactions on Industrial Informatics, 15(7):4178–4188, 2019\.
* [30] Keni Bernardin and Rainer Stiefelhagen. Evaluating multiple object tracking performance: the clear mot metrics. EURASIP Journal on Image and Video Processing, 2008:1–10, 2008\.
* [31] Laura Leal-Taixé, Anton Milan, Konrad Schindler, Daniel Cremers, Ian Reid, and Stefan Roth. Tracking the trackers: an analysis of the state of the art in multiple object tracking. arXiv preprint arXiv:1704.02781, 2017.
|
# Overfitting for Fun and Profit:
Instance-Adaptive Data Compression
Ties van Rozendaal
Qualcomm AI Research
<EMAIL_ADDRESS>&Iris A.M. Huijben11footnotemark: 1
Qualcomm AI Research 22footnotemark: 2
Department of Electrical Engineering
Eindhoven University of Technology
<EMAIL_ADDRESS>&
Taco S. Cohen
Qualcomm AI Research 22footnotemark: 2
<EMAIL_ADDRESS>Equal contributionQualcomm AI Research is an initiative
of Qualcomm Technologies, Inc.Work done during internship at Qualcomm AI
Research
###### Abstract
Neural data compression has been shown to outperform classical methods in
terms of rate-distortion ($RD$) performance, with results still improving
rapidly. At a high level, neural compression is based on an autoencoder that
tries to reconstruct the input instance from a (quantized) latent
representation, coupled with a prior that is used to losslessly compress these
latents. Due to limitations on model capacity and imperfect optimization and
generalization, such models will suboptimally compress test data in general.
However, one of the great strengths of learned compression is that if the
test-time data distribution is known and relatively low-entropy (e.g. a camera
watching a static scene, a dash cam in an autonomous car, etc.), the model can
easily be finetuned or _adapted_ to this distribution, leading to improved
$RD$ performance. In this paper we take this concept to the extreme, adapting
the _full_ model to a _single_ video, and sending model updates (quantized and
compressed using a parameter-space prior) along with the latent
representation. Unlike previous work, we finetune not only the encoder/latents
but the entire model, and - during finetuning - take into account both the
effect of model quantization and the additional costs incurred by sending the
model updates. We evaluate an image compression model on I-frames (sampled at
2 fps) from videos of the Xiph dataset, and demonstrate that full-model
adaptation improves $RD$ performance by $\sim 1$ dB, with respect to encoder-
only finetuning.
## 1 Introduction
The most common approach to neural lossy compression is to train a variational
autoencoder (VAE)-like model on a training dataset to minimize the _expected_
$RD$ cost $D+\beta R$ (Theis et al., 2017; Kingma & Welling, 2013). Although
this approach has proven to be very successful (Ballé et al., 2018), a model
trained to minimize expected $RD$ cost over a full dataset is unlikely to be
optimal for every test instance because the model has limited capacity, and
both optimization and generalization will be imperfect. The problem of
generalization will be especially significant when the testing distribution is
different from the training distribution, as is likely to be the case in
practice.
Suboptimality of the encoder has been studied extensively under the term
_inference suboptimality_ (Cremer et al., 2018), and it has been shown that
finetuning the encoder or latents for a particular instance can lead to
improved compression performance (Lu et al., 2020; Campos et al., 2019; Yang
et al., 2020b; Guo et al., 2020). This approach is appealing as no additional
information needs to be added to the bitstream, and nothing changes on the
receiver side. Performance gains however are limited, because the prior and
decoder can not be adapted.
In this paper we present a method for _full-model instance-adaptive
compression_ , i.e. adapting the _entire_ model to a _single_ data instance.
Unlike previous work, our method takes into account the costs for sending not
only the latent prior, but also the decoder model updates, as well as
quantization of these updates. This is achieved by extending the typical $RD$
loss with an additional model rate term $M$ that measures the number of bits
required to send the model updates under a newly introduced _model prior_ ,
resulting in a combined $RDM$ loss.
As an initial proof of concept, we show that this approach can lead to very
substantial gains in $RD$ performance ($\sim 1$ dB PSNR gain at the same
bitrate) on the problem of I-frame video coding, where a set of key frames,
sampled from a video at 2 fps, are independently coded using an I-frame (image
compression) model. Additionally, we show how the model rate bits are
distributed across the model, and (by means of an ablation study) quantify the
individual gains achieved by including a model-rate loss and using
quantization-aware finetuning.
The rest of this paper is structured as follows. Section 2 discusses the
basics of neural compression and related work on adaptive compression. Section
3 presents our method, including details on the $RDM$ loss, the choice of the
model prior, its quantization, and the (de)coding procedure. In Sections 4 and
5 we present our experiments and results, followed by a discussion in Section
6.
## 2 Preliminaries and Related Work
### 2.1 Neural data compression
The standard approach to neural compression can be understood as a particular
kind of VAE (Kingma & Welling, 2013). In the compression literature the
encoder ${q_{\phi}}({\bm{z}}|{\bm{x}})$ is typically defined by a neural
network parameterized by $\phi$, with either deterministic output (so
${q_{\phi}}({\bm{z}}|{\bm{x}})$ is one-hot) (Habibian et al., 2019) or with
fixed uniform $[0,1]$ noise on the outputs (Ballé et al., 2018). In both
cases, sampling $z\sim{q_{\phi}}({\bm{z}}|{\bm{x}})$ is used during training
while quantization is used at test time.
The latent ${\bm{z}}$ is encoded to the bitstream using entropy coding in
conjunction with a latent prior ${p_{\theta}}({\bm{z}})$, so that coding
${\bm{z}}$ takes about $-\log{p_{\theta}}({\bm{z}})$ bits (up to
discretization). On the receiving side, the entropy decoder is used with the
same prior ${p_{\theta}}({\bm{z}})$ to decode ${\bm{z}}$ and then reconstruct
$\hat{{\bm{x}}}$ using the decoder network ${p_{\theta}}({\bm{x}}|{\bm{z}})$
(note that we use the same symbol $\theta$ to denote the parameters of the
prior and decoder jointly, as in our method both will have to be coded and
added to the bitstream).
From these considerations it is clear that the rate $R$ and distortion $D$ can
be measured by the two terms in the following loss:
$\displaystyle\mathcal{L}_{\textup{RD}}(\phi,\theta)$
$\displaystyle=\beta\underbrace{\mathbb{E}_{{q_{\phi}}({\bm{z}}|{\bm{x}})}\left[-\log{p_{\theta}}({\bm{z}})\right]}_{R}+\underbrace{\mathbb{E}_{{q_{\phi}}({\bm{z}}|{\bm{x}})}\left[-\log{p_{\theta}}({\bm{x}}|{\bm{z}})\right]}_{D}.$
(1)
This loss is equal (up to the tradeoff parameter $\beta$ and an additive
constant) to the standard negative evidence lower bound (ELBO) used in VAE
training. The rate term of ELBO is written as a KL divergence between encoder
and prior, but since $D_{\mathrm{KL}}(q,p)=R-H[q]$, and the encoder entropy
$H[q]$ is constant in our case, minimizing the KL loss is equivalent to
minimizing the rate loss.
Neural video compression is typically decomposed into the problem of
independently compressing a set of key frames (i.e. I-frames) and
conditionally compressing the remaining frames (Lu et al., 2019; Liu et al.,
2020; Wu et al., 2018; Djelouah et al., 2019; Yang et al., 2020a). In this
work, we specifically focus on improving I-frame compression.
### 2.2 Adaptive Compression
A compression model is trained on a dataset ${\mathcal{D}}$ with the aim of
achieving optimal $RD$ performance on test data. However, because of limited
model capacity, optimization difficulties, or insufficient data (resulting in
poor generalization), the model will in general not achieve this goal. When
the test data distribution differs from that of the training data,
generalization will not be guaranteed even in the limit of infinite data and
model capacity, and perfect optimization.
A convenient feature of neural compression however is that a model can easily
be finetuned on new data or data from a specific domain. A model can for
instance (further) be trained after deployment, and Habibian et al. (2019)
showed improved $RD$ gains after finetuning a video compression model on
footage from a dash cam, an approach dubbed _adaptive compression_ (Habibian
et al., 2019).
In adaptive compression, decoding requires access to the adapted prior and
decoder models. These models (or their delta relative to a pretrained shared
model) thus need to be signaled. When the amount of data coded with the
adapted model is large, the cost of signaling the model update will be
negligible as it is amortized. However, a tradeoff exists, the more restricted
the domain of adaptation, the more we can expect to gain from adaptation (e.g.
an image compared to a video or collection of videos). In this paper we
consider the case where the domain of adaptation is a set of I-frames from a
single video, resulting in costs for sending model updates which become very
relevant.
### 2.3 Closing the amortization gap
Coding model updates can easily become prohibitively expensive when the model
is adapted for every instance. However, if we only adapt the encoder or
latents, no model update needs to be added to the bitstream, since the encoder
is not needed for decoding as the latents are sent anyway. We can thus close,
or at least reduce, the _amortization gap_ (the difference between
${q_{\phi}}({\bm{z}}|{\bm{x}})$ and the optimal encoder; Cremer et al. (2018))
without paying any bits for model updates. Various authors have investigated
this approach: Aytekin et al. (2018); Lu et al. (2020) adapt the encoder,
while Campos et al. (2019); Yang et al. (2020b); Guo et al. (2020) adapt the
latents directly. This simple approach was shown to provide a modest boost in
$RD$ performance.
### 2.4 Encoding model updates
As mentioned, when adapting (parts of) the decoder or prior to an instance,
model updates have to be added to the bitstream in order to enable decoding.
Recent works have proposed ways to finetune parts of the model, while keeping
the resulting bitrate overhead small. For instance Klopp et al. (2020) train a
reconstruction error predicting network at encoding time, quantize its
parameters, and add them to the bitstream. Similarly (Lam et al., 2019; 2020)
propose to finetune all parameters or only the convolutional biases,
respectively, of an artifact removal filter that operates after decoding. A
sparsity-enforcing and magnitude-suppressing penalty is leveraged, and
additional thresholding is applied to even more strictly enforce sparsity. The
update vector is thereafter quantized using k-means clustering. Finally, Zou
et al. (2020) finetune the latents in a meta-learning framework, in addition
to updating the decoder convolutional biases, which are quantized by k-means
and thereafter transmitted. All these methods perform quantization post-
training, leading to a potentially unbounded reduction in performance. Also,
albeit the use of regularizing loss terms, no valid proxy for the actual cost
of sending model updates is adopted. Finally, none of these methods performs
adaptation of the full model.
The field of model compression is related to our work as the main question to
be answered is how to most efficiently compress a neural network without
compromising on downstream task performance (Han et al., 2016; Kuzmin et al.,
2019). Bayesian compression is closely related, where the model weights are
sent under a model prior (Louizos et al., 2017; Havasi et al., 2018) as is the
case in our method. Instead of modeling uncertainty in parameters, we however
assume a deterministic posterior (i.e. point estimate). Another key difference
with these works is that we send the model parameter updates relative to an
existing baseline model, which enables extreme compression rates (0.03-0.3
bits/param). This concept of compressing updates has been used before in the
context of federated learning (McMahan et al., 2017; Alistarh et al., 2017) as
well. We distinguish ourselves from that context, as there the model has to be
compressed during every iteration, allowing for error corrections in later
iterations. We only transmit the model updates once for every data instance
that we finetune on.
## 3 Full-Model Instance-Adaptive Compression
In this section we present full-model finetuning on one instance, while
(during finetuning) taking into account both model quantization and the costs
for sending model updates. The main idea is described in Section 3.1, after
which Section 3.2 and 3.3 respectively provide details regarding the model
prior and its quantization. The algorithm is described in Section 3.4.
### 3.1 Finetuning at inference time
Full-model instance-adaptive compression entails finetuning of a set of
_global model_ parameters {$\phi_{\mathcal{D}},\theta_{\mathcal{D}}$}
(obtained by training on dataset ${\mathcal{D}}$) on a single instance
${\bm{x}}$. This results in updated parameters $\phi,\theta$, of which only
$\theta$ has to be signaled in the bitstream. In practice we only learn the
changes with respect to the global model, and encode the _model updates_
$\delta=\theta-\theta_{\mathcal{D}}$ of the decoding model. In order to encode
$\delta$, we introduce a continuous model prior $p(\delta)$ to regularize
these updates, and use the quantized counterpart ${p[\bar{\delta}]}$ for
entropy (de)coding them (more on quantization in Section 3.3).
The overhead for sending quantized model update $\bar{\delta}$ is given by
model rate $\widebar{M}=-\log{p[\bar{\delta}]}$, and can be approximated by
its continuous counterpart $M=-\log{p(\delta)}$ (see Appendix A.1 for
justification). Adding this term to the standard $RD$ loss using the same
tradeoff parameter $\beta$, we obtain the instance-adaptive compression
objective:
$\mathcal{L}_{\textup{RDM}}(\phi,\delta)=\mathcal{L}_{\textup{RD}}(\phi,\theta_{\mathcal{D}}+\bar{\delta})+\beta\underbrace{\left(-\log{p(\delta)}\right)}_{M}.$
(2)
At inference time, this objective can be minimized directly to find the
optimal model parameters for transmitting datapoint ${\bm{x}}$. It takes into
account the additional costs for encoding model updates ($M$ term), and
incorporates model quantization during finetuning ($\mathcal{L}_{\textup{RD}}$
evaluated at $\bar{\theta}=\theta_{\mathcal{D}}+\bar{\delta}$).
### 3.2 Model prior design
A plethora of options exist for designing model prior ${p(\delta)}$ as any
probability density function (PDF) could be chosen, but a natural choice for
modeling parameter update $\delta=\theta-\theta_{\mathcal{D}}$ is to leverage
a Gaussian distribution, centered around the zero-update. Specifically, we can
define the model prior on the updates as a multivariate zero-centered Gaussian
with zero covariance, and a single shared (hyperparameter) $\sigma$, denoting
the standard deviation:
$p({\delta})=\mathcal{N}(\delta\,|\,\mathbf{0},\sigma{\bm{I}})$.
Note that this is equivalent to modeling $\theta$ by
$p(\theta)=\mathcal{N}(\theta\,|\,\theta_{\mathcal{D}},\sigma{\bm{I}})$.
When entropy (de)coding the quantized updates $\bar{\delta}$ under
${p[\bar{\delta}]}$, we must realize that even the zero-update, i.e.
$\bar{\delta}=\mathbf{0}$, is not for free. We define these initial static
costs as ${\widebar{M}_{0}}=-\log p[\bar{\delta}=\mathbf{0}]$. Because the
mode of the defined model prior is zero, these initial costs
${\widebar{M}_{0}}$ equal the minimum costs. Minimization of eq. 2 thus
ensures that – after overcoming these static costs – any extra bit spent on
model updates will be accompanied by a commensurate improvement in $RD$
performance.
Since our method works best when signaling the zero-update is cheap, we want
to increase the probability mass $p[\bar{\delta}=\mathbf{0}]$. We propose to
generalize our earlier proposed model prior to a so-called spike-and-slab
prior (Ročková & George, 2018), which drastically reduces the costs for this
zero-update. More specifically, we redefine the PDF as a (weighted) sum of two
Gaussians – a wide (slab) and a more narrow (spike) distribution:
$\displaystyle
p(\delta)=\frac{p_{\textup{slab}}(\delta)+\alpha~{}p_{\textup{spike}}(\delta)}{1+\alpha},\hskip
14.22636pt\mathrm{with}$ $\displaystyle
p_{\textup{slab}}(\delta)=\mathcal{N}(\delta|\mathbf{0},\sigma{\bm{I}})\hskip
8.5359pt\textrm{and}\hskip
8.5359ptp_{\textup{spike}}(\delta)=\mathcal{N}(\delta|\mathbf{0},\frac{t}{6}{\bm{I}}),$
(3)
where $\alpha\in\mathbb{R}_{\geq 0}$ is a hyperparameter determining the
height of the spiky Gaussian with respect to the wider slab, $t$ is the the
bin width used for quantization (more details in Section 3.3), and
$\sigma>>\frac{t}{6}$. By choosing the standard deviation of the spike to be
$\frac{t}{6}$, the mass within six standard deviations (i.e. $99.7\%$ of the
total mass) is included in the central quantization bin after quantization.
Note that the (slab-only) Gaussian prior is a special case of the spike-and-
slab Gaussian prior in section 3.2, where $\alpha=0$. As such, ${p(\delta)}$
refers to the spike-and-slab prior in the rest of this work. Appendix A.2
compares the continuous and discrete spike-and-slab prior and its gradients.
Adding the spike distribution, not only decreases ${\widebar{M}_{0}}$, it also
more heavily enforces sparsity on the updates via regularizing term $M$ in eq.
2. In fact, a high spike (i.e. large $\alpha$) can make the bits for signaling
a zero-update so small (i.e. almost negligible), that the model effectively
learns to make a binary choice; a parameter is either worth updating at the
cost of some additional rate, or its not updated and the ‘spent’ bits are
negligible.
### 3.3 Quantization
In order to quantize a scalar $\delta_{i}$ (denoted by $\delta$ in this
section to avoid clutter), we use $N$ equispaced bins of width $t$, and we
define the following quantization function:
$\bar{\delta}=Q_{t}(\delta)=\operatorname{clip}\left(\left\lfloor\frac{\delta}{t}\right\rceil\cdot
t,~{}\operatorname{min}=-\frac{(N-1)t}{2},~{}\operatorname{max}=\frac{(N-1)t}{2}\right).$
(4)
As both rounding and clipping are non-differentiable, the gradient of $Q_{t}$
is approximated using the Straight-Through estimator (STE), proposed by Bengio
et al. (2013). That is, we assume $\partial Q_{t}(\delta)/\partial\delta=1$.
The bins are intervals
$Q_{t}^{-1}(\bar{\delta})=[\bar{\delta}-t/2,\bar{\delta}+t/2]$. We view $t$ as
a hyperparameter, and define $N$ to be the smallest integer such that the
region covered by bins (i.e. the interval $[-(N-1)t/2,(N-1)t/2]$), covers at
least $1-2^{-8}$ of the probability mass of $p(\delta)$. Indeed the number of
bins $N$ is proportional to the ratio of the width of $p(\delta)$ and the
width of the bins: $N\propto\sigma/t$. The number of bins presents a tradeoff
between finetuning flexibility and model rate costs $\widebar{M}$, so the
$\sigma/t$-ratio is an important hyperparameter. The higher $N$, the higher
these costs due to finer quantization, but simultaneously the lower the
quantization gap, enabling more flexible finetuning.
Since $\bar{\delta}=Q_{t}(\delta)$, the discrete model prior
${p[\bar{\delta}]}$ is the pushforward of $p(\delta)$ through $Q_{t}$:
${p[\bar{\delta}]}=\int_{Q^{-1}(\bar{\delta})}p(\delta)d\delta=\int_{\bar{\delta}-t/2}^{\bar{\delta}+t/2}p(\delta)d\delta=P(\delta<\bar{\delta}+t/2)-P(\delta<\bar{\delta}-t/2).$
(5)
That is, ${p[\bar{\delta}]}$ equals the mass of $p(\delta)$ in the bin of
$\bar{\delta}$, which can be computed as the difference of the cumulative
density function (CDF) of $p(\delta)$ evaluated at the edges of that bin.
### 3.4 Entropy coding and decoding
After finetuning of the compression model on instance ${\bm{x}}$ by minimizing
the loss in eq. 2, both the latents and the model updates are entropy coded
(under their respective priors $p_{\bar{\theta}}({\bm{z}})$ and
${p[\bar{\delta}]}$) into a bitstream ${\bm{b}}$. Decoding starts by decoding
$\bar{\delta}$ using ${p[\bar{\delta}]}$, followed by decoding ${\bm{z}}$
using $p_{\bar{\theta}}({\bm{z}})$ (where
$\bar{\theta}=\bar{\delta}+\theta_{\mathcal{D}}$), and finally reconstructing
$\hat{{\bm{x}}}$ using $p_{\bar{\theta}}({\bm{x}}\,|\,{\bm{z}})$. The whole
process is shown in Figure 1 and defined formally in Algorithms 1 and 2.
## 4 Experimental setup
### 4.1 Datasets
The experiments in this paper use images and videos from the following two
datasets:
CLIC19 111https://www.compression.cc/2019/challenge/ The CLIC19 dataset
contains a collection of natural high resolution images. It is conventionally
divided into a professional set, and a set of mobile phone images. Here we
merge the existing training folds of both sets and use the resulting dataset
${\mathcal{D}}$ to train our I-frame model. The corresponding validation folds
are used to validate the global model performance.
Xiph-5N 2fps 222https://media.xiph.org/video/derf/ The Xiph dataset contains a
variety of videos of different formats. We select a representative sample of
five videos (_5N_) from the set of 1080p videos (see Appendix B for details
regarding selection of these samples). Each video is temporally subsampled to
2 fps to create a dataset of I-frames, referred to as Xiph-5N 2fps. Frames in
all videos contain $1920\times 1080$ pixels, and the set of I-frames after
subsampling to 2 fps contain between 20 and 42 frames. The five selected
videos are single-scene but multi-shot, and come from a variety of sources.
Xiph-5N 2fps is used to validate our instance-adaptive data compression
framework.
### 4.2 Global model architecture and training
Model rate can be restricted by (among others) choosing a low-complexity
neural compression model, and amortizing the additional model update costs
over a large number of pixels.
The most natural case for full-model adaptation is therefore to finetune a
low-complexity model on a video instance. Typical video compression setups
combine an image compression (I-frame) model to compress key frames, and a
predict- or between-frame model to reconstruct the remaining frames. Without
loss of generality for video-adaptive finetuning, we showcase our full-model
finetuning framework for the I-frame compression problem, being a subproblem
of video compression.
Specifically, we use the (relatively low-complexity) hyperprior-model proposed
by Ballé et al. (2018), including the mean-scale prior (without context) from
Minnen et al. (2018). Before finetuning, this model is trained on the training
fold of the CLIC19 dataset, using the $RD$ objective given in eq. 1. Appendix
C provides more details on both its architecture and the adopted training
procedure.
Figure 1: Visualization of encoding (Algorithm 1) and decoding (Algorithm 2)
of our full-model instance-adaptive method. Each step is denoted with a code,
e.g. E9, which refers to line 9 of the encoding algorithm. EE and ED denote
entropy encoding and decoding, respectively. Both the latent representation
${\bm{z}}$ and the parameter updates $\bar{\delta}$ are encoded in their
respective bitstreams ${\bm{b}}_{\bm{z}}$ and ${\bm{b}}_{\bar{\delta}}$. Model
prior ${p[\bar{\delta}]}$ entropy decodes ${\bm{b}}_{\bar{\delta}}$, after
which the latent prior can decode ${\bm{b}}_{\bm{z}}$.
Algorithm 1 Encoding of ${\bm{x}}$
0: Global model parameters $\\{\phi_{\mathcal{D}},\theta_{\mathcal{D}}\\}$
trained on training set ${\mathcal{D}}$, model parameter quantizer $Q_{t}$,
model prior ${p[\bar{\delta}]}$, datapoint to be compressed ${\bm{x}}$.
0: Compressed bitstream ${\bm{b}}=({\bm{b}}_{\bar{\delta}},{\bm{b}}_{\bm{z}})$
1: Initialize model parameters: $\phi=\phi_{\mathcal{D}}$, and
$\theta=\theta_{\mathcal{D}}$
2: for step in MAX STEPS do
3: Sample single I-frame: ${\bm{x}}_{f}\sim{\bm{x}}$
4: Quantize transmittable parameters: $\bar{\theta}\leftarrow
Q_{t}(\delta)+\theta_{\mathcal{D}}$, with $\delta=\theta-\theta_{\mathcal{D}}$
5: Forward pass: ${\bm{z}}\sim q_{\phi}({\bm{z}}\,|\,{\bm{x}}_{f})$ and
evaluate $p_{\bar{\theta}}({\bm{x}}_{f}\,|\,{\bm{z}})$ and
$p_{\bar{\theta}}({\bm{z}})$.
6: Compute loss $\mathcal{L}_{RDM}(\phi,\delta)$ on ${\bm{x}}_{f}$ according
to eq. 2.
7: Backpropagate using STE for $Q_{t}$, then update $\phi,\theta$ using
gradients $\dfrac{\partial\mathcal{L}_{RDM}}{\partial\phi}$ and
$\dfrac{\partial\mathcal{L}_{RDM}}{\partial\theta}$.
8: end for
9: Compress ${\bm{x}}$ to ${\bm{z}}\sim{q_{\phi}}({\bm{x}})$.
10: Compute quantized model parameters:
$\bar{\theta}=\theta_{\mathcal{D}}+\bar{\delta}$, with
$\bar{\delta}=Q_{t}(\theta-\theta_{\mathcal{D}})$.
11: Entropy encode:
${\bm{b}}_{\bar{\delta}}={\operatorname{enc}}(\bar{\delta};{p[\bar{\delta}]})$
and
${\bm{b}}_{\bm{z}}={\operatorname{enc}}({\bm{z}};p_{\bar{\theta}}({\bm{z}}))$.
Algorithm 2 Decoding of ${\bm{x}}$
0: Global model parameters $\theta_{\mathcal{D}}$ trained on training set
${\mathcal{D}}$, model prior ${p[\bar{\delta}]}$, bitstream
${\bm{b}}=({\bm{b}}_{\bar{\delta}},{\bm{b}}_{\bm{z}})$.
0: Decoded datapoint $\hat{{\bm{x}}}$
1: Entropy decode:
$\bar{\delta}={\operatorname{dec}}({\bm{b}}_{\bar{\delta}};{p[\bar{\delta}]})$.
2: Compute updated parameters:
$\bar{\theta}=\theta_{\mathcal{D}}+\bar{\delta}$.
3: Entropy decode latent under finetuned prior:
${\bm{z}}={\operatorname{dec}}({\bm{b}}_{\bm{z}};p_{\bar{\theta}}({\bm{z}}))$.
4: Decode instance as the mean of the finetuned decoder:
$p_{\bar{\theta}}({\bm{x}}|{\bm{z}})$
### 4.3 Instance-adaptive finetuning
Each instance in the Xiph-5N 2fps dataset (i.e. a set of I-frames belonging to
one video) is (separately) used for full-model adaptation. The resulting model
rate costs are amortized over the set of I-frames, rather than the full video.
As a benchmark, we only finetune the encoding procedure, as it does not induce
additional bits in the bitstream. Encoder-only finetuning can be implemented
by either finetuning the encoding model parameters $\phi$, or directly
optimizing the latents as in Campos et al. (2019). We implement both
benchmarks, as the former is an ablated version of our proposed full-model
finetuning, whereas the latter is expected to perform better due to the
amortization gap Cremer et al. (2018).
Global models trained with $RD$ tradeoff parameter
$\beta\in\\{3e{-3},1e{-3},2.5e{-4},1e{-4}\\}$ are used to initialize the
models that are then being finetuned with the corresonding value for $\beta$.
Both encoder-only and direct-latent finetuning minimize the $RD$ loss as given
in eq. 1. For encoder-only tuning we use a constant learning rate of $1e{-6}$,
whereas for latent optimization a learning rate of $5e{-4}$ is used for the
low bitrate region (i.e. two highest $\beta$ values), and $1e{-3}$ for the
high rate region. In case of direct latent optimization, the pre-quantized
latents are finetuned, which are initialized using a forward pass of the
encoder.
Our instance-adaptive _full-model_ finetuning framework extends encoder-only
finetuning by jointly updating $\phi$ and $\theta$ using the $RDM$ loss from
eq. 2. In this case we finetune the global model that is trained with
$\beta=0.001$, independent of the value of $\beta$ during finetuning.
Empirically, this resulted in negligible difference in performance compared to
using the global model of the corresponding finetuning $\beta$, while it
alleviated memory constraints thanks to the smaller size of this low bitrate
model architecture (see Appendix C). The training objective in eq. 2 is
expressed in bits per pixel and optimized using a fixed learning rate of
$1e{-4}$. The parameters for the model prior were chosen as follows:
quantization bin width $t=0.005$, standard deviation $\sigma=0.05$, and the
multiplicative factor of the spike $\alpha=1000$. We empirically found that
sensitivity to changing $\alpha$ in the range 50-5000 was low. Realize that,
instead of empirically setting $\alpha$, its value could also be solved for a
target initial cost ${\widebar{M}_{0}}$, given the number of decoding
parameters and pixels in the finetuning instance.
All finetuning experiments (both encoding-only and _full-model_) ran for
$100$k steps, each containing one mini-batch of a single, full resolution
I-frame333All frames in the videos in Xiph-5N 2fps are of spatial resolution
$1920\times 1080$. In order to make this shape compatible with the strided
convolutions in our model, we pad each frame to $1920\times 1088$ before
encoding. After reconstructing $\hat{{\bm{x}}}$, it is cropped back to its
original resolution for evaluation.. We used the Adam optimizer (default
settings) (Kingma & Ba, 2014), and best model selection was based on the
$RD{\widebar{M}}$ loss over the set of I-frames.
## 5 Results
### 5.1 Rate-distortion gains
Figure 2a shows the compression performance for encoder-only finetuning,
direct latent optimization, and full-model finetuning, averaged over all
videos in the Xiph-5N 2fps dataset for different rate-distortion tradeoffs.
Finetuning of the entire parameter-space results in much higher $RD$
$\widebar{M}$ gains (on average approximately 1 dB for the same bitrate)
compared to only finetuning the encoding parameters $\phi$ or the latents
directly. Figure 8 in Appendix E shows this plot for each video separately.
Note that encoder-only finetuning performance is on par with direct latent
optimization, implying that the amortization gap (Cremer et al., 2018) is
close to zero when finetuning the encoder model on a moderate number of
I-frames.
Table 1 provides insight in the distribution of bits over latent rate $R$ and
model rate $\widebar{M}$, which both increase for lower values of $\beta$.
However, the relative contribution of the model rate increases for the higher
bitrate regime, which could be explained by the fact that in this regime the
latent rate can more heavily be reduced by finetuning. Figure 2a indeed
confirms that the the total rate reduction is higher for the high bitrate
regime, which thus fully originates from the reduction in latent rate after
finetuning. Table 1 also shows that the static intial costs
${\widebar{M}_{0}}$ only marginally contribute to the total model rate. Figure
2b shows for one video how finetuning progressed over training steps,
confirming that the compression gains already at the beginning of finetuning
cover these initial costs. This effect was visible for all tested videos (see
Appendix E).
(a) (b) (c)
Figure 2: (a) Averaged $RD$ $\widebar{M}$ performance over all videos of the
Xiph-5N 2fps datase for four different rate-distortion tradeoffs with
$\beta=3e{-3}$, $1e{-3}$, $2.5e{-4}$, and $1e{-4}$ (from left to right). Our
_full-model_ finetuning outperforms _encoder-only_ and _direct latent_
optimization with approximately 1 dB gain for the same rate. (b) Finetuning
progression of the sunflower video over time. Between each dot, 500 training
steps are taken, showing that already at the start of finetuning large $RD$
$\widebar{M}$ gains are achieved and the $RD$ $\widebar{M}$ performance
continues to improve during finetuning. (c) Ablation where we show the effect
of both quantization- ($Q_{t}(\delta)$) and model rate aware ($M$ Loss)
finetuning. Case VI shows the upper bound on achievable finetuning performance
when (naively) not taking into account quantization and model update rate.
### 5.2 Ablations
Figure 2c shows several ablation results for one video. Case I is our proposed
full-model finetuning, optimizing the $RDM$ loss, while simultaneously
quantizing the updates to compute distortion (denoted with $Q_{t}(\delta)$).
One can see that not doing quantization-aware finetuning (case II)
deteriorates the distortion gains during evaluation. Removing the (continuous)
model rate penalty from the finetuning procedure (case III) imposes an extreme
increase in rate during evaluation, caused by unbounded growing of the model
rate during finetuning. As such, the models result in a rate even much higher
than the baseline model’s rate, showing that finetuning without model rate
awareness provides extremely poor results. Case IV shows performance
deterioration in the situation of both quantization- and model rate unaware
finetuning. Analyzing these runs while (naively) not taking into account the
additional model update costs (cases V and VI) provides upper bounds on
compression improvement. Case VI shows the most naive bound; $RD$ finetuning
without quantization, whereas case V is a tighter bound that does include
quantization. The gap between V and VI is small, suggesting that the used
quantization strategy does only mildly harm performance.
An ablation study on the effect of the number of finetuning frames is provided
in Appendix D. It reveals that, under the spike-and-slab model prior, full-
model finetuning works well for a large range of instance lengths. Only in the
worst case scenario, when finetuning only one frame in the low bitrate regime,
full-model finetuning was found to be too costly.
### 5.3 Distribution of bits
Figure 3 shows for different (mutually exclusive) parameter groups the
distribution of the model updates $\delta$ (top) and their corresponding bits
(bottom). Interestingly one can see how for (almost) all groups, the updates
are being clipped by our earlier defined quantizer $Q_{t}$. This suggests the
need for large, expensive updates in these parameter groups, for which the
additional $RD$ gain thus appears to outweigh the extra costs. At the same
time, all groups show an elevated center bin, thanks to training with the
spike-and-slab prior (see Appendix F). By design of this prior, the bits paid
for this zero-update are extremely low, which can best be seen in the bits
histogram (Fig. 3-bottom) of the Codec Decoder Weight and Biases. The
parameter updates of the Codec Decoder IGND group are the only ones that are
non-symmetrically distributed across the zero-updated, which can be explained
by the fact that IGDN (Ballé et al., 2016) is an (inverse) normalization
layer. The Codec Decoder Weights were found to contribute most to the total
model rate ${\widebar{M}}$.
Figure 3: Empirical distribution of parameter updates for the model finetuned on the sunflower video with $\beta=2.5e{-4}$. Columns denote parameter groups. Top: Histograms of the model updates $\bar{\delta}$. Bottom: Histogram of bit allocation for $\bar{\delta}$. Subtitles indicate the total number of bits for each parameter group, both expressed in bits per pixel (b/px) and bits per parameter (b/param). $\bm{\beta}$ | PSNR | $R+\widebar{M}$ | $R$ | $\widebar{M}$ | ${\widebar{M}_{0}}$ | $\widebar{M}$ | $\widebar{M}$
---|---|---|---|---|---|---|---
| (dB) | (bits/pixel) | (bits/parameter) | (kB/frame)
3.0e-03 | 34.0 | 0.175 | 0.174 | 0.001 | 0.00033 | 0.026 | 0.39
1.0e-03 | 36.2 | 0.304 | 0.301 | 0.003 | 0.00033 | 0.044 | 0.67
2.5e-04 | 38.9 | 0.601 | 0.593 | 0.008 | 0.00033 | 0.129 | 1.99
1.0e-04 | 40.7 | 0.921 | 0.905 | 0.017 | 0.00033 | 0.282 | 4.35
Table 1: Distribution of bitrate for different rate-distortion tradeoffs
$\beta$, averaged over the videos in the Xiph-5N 2fps dataset. The number of
bits are distributed over the latent rate $R$ and the model rate
$\widebar{M}$, which is computed using the quantized model prior
${p[\bar{\delta}]}$.
## 6 Discussion
This work presented instance-adaptive neural compression, the first method
that enables finetuning of a full compression model on a set of I-frames from
a single video, while restricting the additional bits for encoding the
(quantized) model updates. To this end, the typical rate-distortion loss was
extended by incorporating both model quantization and the additional model
rate costs during finetuning. This loss guarantees pure $RD$ $\widebar{M}$
performance gains after overcoming a small initial cost for encoding the zero-
update. We showed improved $RD$ $\widebar{M}$ performance on all five tested
videos from the Xiph dataset, with an average distortion improvement of
approximately 1 dB for the same bitrate.
Among videos, we found a difference in achieved finetuning $RD$ $\widebar{M}$
gain (see Appendix E). Possible causes can be three-fold. First, the
performance of the global model differs per video, therewith influencing the
maximum gains to be achieved by finetuning. Second, video characteristics such
as (non-)stationarity greatly influence the diversity of the set of I-frames,
thereby affecting the ease of model-adaption. Third, the number of I-frames
differs per video and thus trades off model update costs (which are amortized
over the set of I-frames), with ease of finetuning.
The results of the ablation in Fig. 2c show that the quantization gap (V vs
VI) is considerably smaller than the performance deterioration due to
(additionally) regularizing the finetuning using the model prior (I vs V).
Most improvement in future work is thus expected to be gained by leveraging
more flexible model priors, e.g. by learning its parameters and/or modeling
dependencies among updates.
We showed how instance-adaptive full-model finetuning greatly improves $RD$
$\widebar{M}$ performance for I-frame compression, a subproblem in video
compression. Equivalently, one can exploit full-model finetuning to enhance
compression of the remaining (non-key) frames of a video, compressing the
total video even further. Also, neural video compression models that exploit
temporal redundancy, could be finetuned, as long as the model’s complexity is
low enough to restrict model rate. Leveraging such a low-complexity video
model moves computational complexity of data compression from the receiver to
the sender by heavily finetuning this small model on each video. We foresee
convenience of this computational shift in applications where bandwidth and
receiver compute power is scarce, but encoding compute time is less important.
This use case in practice happens e.g. for (non-live) video streaming to low-
power edge devices. Finding such low-complexity video compression models is
however non-trivial as it’s capacity must still be enough to compress an
entire video. We foresee great opportunities here, and will therefore
investigate this in future work.
## References
* Alistarh et al. (2017) Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Qsgd: Communication-efficient sgd via gradient quantization and encoding. In _Advances in Neural Information Processing Systems_ , pp. 1709–1720, 2017.
* Aytekin et al. (2018) Caglar Aytekin, Xingyang Ni, Francesco Cricri, Jani Lainema, Emre Aksu, and Miska M Hannuksela. Block-optimized variable bit rate neural image compression. In _CVPR Workshops_ , pp. 2551–2554, 2018.
* Ballé et al. (2016) Johannes Ballé, Valero Laparra, and Eero Simoncelli. Density modeling of images using a generalized normalization transformation. In _4th International Conference on Learning Representations, ICLR 2016_ , 2016.
* Ballé et al. (2018) Johannes Ballé, David Minnen, Saurabh Singh, Sung Jin Hwang, and Nick Johnston. Variational image compression with a scale hyperprior. In _International Conference on Learning Representations_ , 2018.
* Bengio et al. (2013) Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. _arXiv preprint arXiv:1308.3432_ , 2013.
* Campos et al. (2019) Joaquim Campos, Simon Meierhans, Abdelaziz Djelouah, and Christopher Schroers. Content adaptive optimization for neural image compression. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ , pp. 0–0, 2019.
* Cremer et al. (2018) Chris Cremer, Xuechen Li, and David Duvenaud. Inference suboptimality in variational autoencoders. In _International Conference on Machine Learning_ , pp. 1078–1086, 2018.
* Djelouah et al. (2019) Abdelaziz Djelouah, Joaquim Campos, Simone Schaub-Meyer, and Christopher Schroers. Neural inter-frame compression for video coding. In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 6421–6429, 2019.
* Guo et al. (2020) Tiansheng Guo, Jing Wang, Ze Cui, Yihui Feng, Yunying Ge, and Bo Bai. Variable rate image compression with content adaptive optimization. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops_ , pp. 122–123, 2020.
* Habibian et al. (2019) Amirhossein Habibian, Ties van Rozendaal, Jakub M Tomczak, and Taco S Cohen. Video compression with rate-distortion autoencoders. In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 7033–7042. openaccess.thecvf.com, 2019.
* Han et al. (2016) Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. _arXiv preprint arXiv:1510.00149_ , 2016.
* Havasi et al. (2018) Marton Havasi, Robert Peharz, and José Miguel Hernández-Lobato. Minimal random code learning: Getting bits back from compressed model parameters. In _International Conference on Learning Representations_ , 2018.
* Kingma & Ba (2014) Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. _ICLR_ , 2014.
* Kingma & Welling (2013) Diederik P Kingma and Max Welling. Auto-Encoding variational bayes. _arXiv preprint arXiv:1312.6114_ , 2013.
* Klopp et al. (2020) Jan P Klopp, Liang-Gee Chen, and Shao-Yi Chien. Utilising low complexity cnns to lift non-local redundancies in video coding. _IEEE Transactions on Image Processing_ , 2020.
* Kuzmin et al. (2019) Andrey Kuzmin, Markus Nagel, Saurabh Pitre, Sandeep Pendyam, Tijmen Blankevoort, and Max Welling. Taxonomy and evaluation of structured compression of convolutional neural networks. _arXiv preprint arXiv:1912.09802_ , 2019.
* Lam et al. (2019) Yat-Hong Lam, Alireza Zare, Caglar Aytekin, Francesco Cricri, Jani Lainema, Emre Aksu, and Miska Hannuksela. Compressing weight-updates for image artifacts removal neural networks. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ , pp. 0–0, 2019.
* Lam et al. (2020) Yat-Hong Lam, Alireza Zare, Francesco Cricri, Jani Lainema, and Miska M Hannuksela. Efficient adaptation of neural network filter for video compression. In _Proceedings of the 28th ACM International Conference on Multimedia_ , pp. 358–366, 2020.
* Liu et al. (2020) Haojie Liu, Han Shen, Lichao Huang, Ming Lu, Tong Chen, and Zhan Ma. Learned video compression via joint spatial-temporal correlation exploration. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pp. 11580–11587, 2020.
* Louizos et al. (2017) Christos Louizos, Karen Ullrich, and Max Welling. Bayesian compression for deep learning. In _Advances in neural information processing systems_ , pp. 3288–3298, 2017.
* Lu et al. (2019) Guo Lu, Wanli Ouyang, Dong Xu, Xiaoyun Zhang, Chunlei Cai, and Zhiyong Gao. Dvc: An end-to-end deep video compression framework. In _The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , June 2019.
* Lu et al. (2020) Guo Lu, Chunlei Cai, Xiaoyun Zhang, Li Chen, Wanli Ouyang, Dong Xu, and Zhiyong Gao. Content adaptive and error propagation aware deep video compression. _arXiv preprint arXiv:2003.11282_ , 2020.
* McMahan et al. (2017) Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In _Artificial Intelligence and Statistics_ , pp. 1273–1282. PMLR, 2017.
* Minnen et al. (2018) David Minnen, Johannes Ballé, and George D Toderici. Joint autoregressive and hierarchical priors for learned image compression. In _Advances in Neural Information Processing Systems_ , pp. 10771–10780, 2018.
* Rippel et al. (2019) Oren Rippel, Sanjay Nair, Carissa Lew, Steve Branson, Alexander G Anderson, and Lubomir Bourdev. Learned video compression. In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 3454–3463, 2019.
* Ročková & George (2018) Veronika Ročková and Edward I George. The spike-and-slab lasso. _Journal of the American Statistical Association_ , 113(521):431–444, 2018.
* Theis et al. (2017) L. Theis, W. Shi, A. Cunningham, and F. Huszár. Lossy image compression with compressive autoencoders. In _International Conference on Learning Representations_ , 2017. URL https://openreview.net/pdf?id=rJiNwv9gg.
* Wu et al. (2018) Chao-Yuan Wu, Nayan Singhal, and Philipp Krahenbuhl. Video compression through image interpolation. In _Proceedings of the European Conference on Computer Vision (ECCV)_ , pp. 416–431, 2018.
* Yang et al. (2020a) Yang Yang, Guillaume Sautière, J Jon Ryu, and Taco S Cohen. Feedback recurrent autoencoder. In _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 3347–3351. IEEE, 2020a.
* Yang et al. (2020b) Yibo Yang, Robert Bamler, and Stephan Mandt. Improving inference for neural image compression. _Advances in Neural Information Processing Systems_ , 33, 2020b.
* Zou et al. (2020) Nannan Zou, Honglei Zhang, Francesco Cricri, Hamed R Tavakoli, Jani Lainema, Miska Hannuksela, Emre Aksu, and Esa Rahtu. L2C–learning to learn to compress. In _2020 IEEE 22nd International Workshop on Multimedia Signal Processing (MMSP)_ , pp. 1–6. IEEE, 2020.
## Appendix A Model rate loss
### A.1 Gradient definition
The proof below shows that $\nabla_{\delta}{\widebar{M}}$ is an unbiased
first-order approximation of $\nabla_{\delta}M$, validating its use during
finetuning.
The gradient of the continuous model rate loss $M$ towards $\delta$ is defined
as:
$\displaystyle\nabla_{\delta}M=-\nabla_{\delta}\log{p(\delta)}=-\frac{\nabla_{\delta}{p(\delta)}}{{p(\delta)}}$
(6)
The gradient of the non-differentiable discrete model rate loss $\bar{M}$
towards $\delta$ can be defined by exploiting the Straight-Through gradient
estimator (Bengio et al., 2013), i.e. $\nabla_{\delta}\bar{\delta}=1$.
As such444Due to cutting (a maximum) of $2^{-8}$ mass from the tails of
${p(\delta)}$ to enable quantization, the difference in cumulative masses in
section A.1 should be renormalized by $1-2^{-8}$. As this is a constant
division inside a $\log$, it results in a subtraction of $\nabla_{\delta}\log
1-2^{-8}=0$. Since this normalization does not influence the gradient, we
omitted it for the sake of clarity.:
$\displaystyle\nabla_{\delta}{\widebar{M}}=-\nabla_{\delta}\log{p[\bar{\delta}]}$
$\displaystyle=-\nabla_{\delta}\log[P(\delta<\bar{\delta}+\frac{t}{2})-P(\delta<\bar{\delta}-\frac{t}{2})]$
$\displaystyle=-\frac{p(\bar{\delta}+\frac{t}{2})-p(\bar{\delta}-\frac{t}{2})}{P(\delta<\bar{\delta}+\frac{t}{2})-P(\delta<\bar{\delta}-\frac{t}{2})}.$
(7)
By first-order approximation we can write:
$\nabla_{\delta}p(\bar{\delta})\approx\frac{p(\bar{\delta}+\frac{t}{2})-p(\bar{\delta}-\frac{t}{2})}{t}$
(8)
and
$P(\delta<\bar{\delta}+\frac{t}{2})-P(\delta<\bar{\delta}-\frac{t}{2})\approx
p(\bar{\delta})t.$ (9)
Using eq. 8 and eq. 9, we can express the gradient of the discrete model rate
costs $\nabla_{\delta}\bar{M}$ as:
$\displaystyle\nabla_{\delta}{\widebar{M}}$
$\displaystyle\approx-\frac{\nabla_{\delta}p(\bar{\delta})t}{p(\bar{\delta})t}=-\frac{\nabla_{\delta}p(\bar{\delta})}{p(\bar{\delta})},$
(10)
which is thus a first-order approximation of $\nabla_{\delta}M$.
### A.2 Continuous vs discrete model rate penalty
The proof provided in the previous section is not restricted to specific
designs for ${p(\delta)}$, and thus holds both for the spike-and-slab prior
including a spike ($\alpha>0$), or the special case where no spike is used
($\alpha=0$).
Figure 4 shows quantization of the model updates (top-left) and its
corresponding gradient (left-bottom), exploiting the Straight-Through gradient
estimator Bengio et al. (2013). Also the discrete (true) model rate
${\widebar{M}}$ and its continuous analogy $M$ (middle-top) with their
corresponding gradients (middle-bottom) are shown for the special case of not
using a spike, i.e. $\alpha=0$. One can see that the continuous model rate
proxy is a shifted version of the discrete costs. Since such a translation
does not influence the gradient (and neither gradient-based optimization), the
continuous model rate loss can be used during finetuning, preventing instable
training thanks to its smooth gradient.
Figure 4: (Left) Illustrative example of the quantization effect and the
corresponding gradient (using the Straight-Through estimator) for parameter
update $\delta$. (Middle) The true bitrate overhead (blue) and its continuous
proxy (orange) of a (slab-only) Gaussian prior ($\alpha=0$) and their
gradients with respect to unquantized $\delta$. (Right) The true bitrate
overhead (blue) and its continuous proxy (orange) of a spike-and-slab prior
($\alpha=4$) and their gradients with respect to unquantized $\delta$. One can
see how the effect of the spike (almost) fully disappears in the gradient of
the quantized bitrate overhead, as the largest amount of mass of the spike
distribution is part of the center quantization bin.
Figure 4-right shows the same figure but for a model prior that includes a
spike. Comparing true (discrete) model rate costs of the slab-only prior
(middle-top), to these costs using the spike-and-slab prior (right-top), shows
how the introduced spike reduces the number of bits to encode the zero-update
(i.e. the center bin), at the cost of making larger updates more expensive in
number of bits. Another interesting phenomena is visible when comparing the
gradients of the discrete and continuous model rates for slab-only (middle-
bottom) versus spike-and-slab prior (right-bottom). The effect of the spike
almost fully disappears in the gradient of the discrete model rate. This is
caused by the fact that most of the spike’s mass is (by design) positioned
inside the center quantization bin after quantization. Mathematically, this
can be seen from filling in section A.1 for the spike-and-slab prior:
$\displaystyle\nabla_{\delta}{\widebar{M}}=$
$\displaystyle-\frac{p_{\textup{slab}}(\bar{\delta}+\frac{t}{2})-p_{\textup{slab}}(\bar{\delta}-\frac{t}{2})+\alpha~{}p_{\textup{spike}}(\bar{\delta}+\frac{t}{2})-\alpha~{}p_{\textup{spike}}(\bar{\delta}-\frac{t}{2})}{P_{\textup{slab}}(\delta<\bar{\delta}+\frac{t}{2})-P_{\textup{slab}}(\delta<\bar{\delta}-\frac{t}{2})+\alpha~{}P_{\textup{spike}}(\delta<\bar{\delta}+\frac{t}{2})-\alpha~{}P_{\textup{spike}}(\delta<\bar{\delta}-\frac{t}{2})}.$
(11)
We can distinguish different behavior of eq. 11 for two distinct ranges of
$\bar{\delta}$:
* •
$\bar{\delta}=0$
Symmetry around zero-update:
$p_{\textup{slab}}(\bar{\delta}+\frac{t}{2})=p_{\textup{slab}}(\bar{\delta}-\frac{t}{2})$
and
$p_{\textup{spike}}(\bar{\delta}+\frac{t}{2})=p_{\textup{spike}}(\bar{\delta}-\frac{t}{2})$.
$\implies\nabla_{\delta}{\widebar{M}}=0$
* •
$\bar{\delta}\neq 0$
Since $t=6\sigma_{\textup{spike}}$:
$p_{\textup{spike}}(\bar{\delta}+\frac{t}{2})\approx 0$,
$p_{\textup{spike}}(\bar{\delta}-\frac{t}{2})\approx 0$,
$P_{\textup{spike}}(\delta<\bar{\delta}+\frac{t}{2})\approx 0$,
$P_{\textup{spike}}(\delta<\bar{\delta}-\frac{t}{2})\approx 0$.
$\implies\nabla_{\delta}{\widebar{M}}=-\frac{p_{\textup{slab}}(\bar{\delta}+\frac{t}{2})-p_{\textup{slab}}(\bar{\delta}-\frac{t}{2})}{P_{\textup{slab}}(\delta<\bar{\delta}+\frac{t}{2})-P_{\textup{slab}}(\delta<\bar{\delta}-\frac{t}{2})}$.
Note that this approximation becomes less tight for large $\alpha$, as the
small probabilities and cumulative densities of the spike distribution are
multiplied by $\alpha$.
From the $\bar{\delta}=0$ case, one can see that inclusion of the spike leaves
the gradient unbiased. The $\bar{\delta}\neq 0$ case shows that the spike does
not influence the gradient in the limit when the spike’s mass is entirely
positioned within the center quantization bin. As the standard deviation of
the spiky Gaussian was chosen to be $\frac{t}{6}$, a total of $0.3\%$ of it’s
mass is in practice being quantized in an off-center quantization bin. This
explains the slight increase of the off-center bins in the gradient of the
discrete model rate costs in Fig. 4 (right-bottom). Comparing the gradient of
the discrete versus continuous model rate costs for the spike-and-slab prior
in Fig. 4 (right-bottom), we can see that the first order approximation
between the two introduces a larger error than in the slab-only case (Fig.
4(middle-bottom)). This can be explained by the fact that the introduced spiky
Gaussian has a higher tangent due to its support being more narrow than that
of the Gaussian slab. Nevertheless, the use of the continuous model rate loss
is preferred for finetuning as it (much more) strictly enforces zero-updates
(thanks to the present gradient peaks around the center bin) than its discrete
counterpart.
## Appendix B Sample selection from Xiph dataset
### B.1 Xiph dataset
The Xiph test videos can be found at https://media.xiph.org/video/derf/. Like
Rippel et al. (2019) we select all 1080p videos, and exclude computer-
generated videos and videos with inappropriate licences, which leaves us with
the following videos:
aspen_1080p | ducks_take_off_1080p50 | red_kayak_1080p | speed_bag_1080p
---|---|---|---
blue_sky_1080p25 | in_to_tree_1080p50 | riverbed_1080p25 | station2_1080p25
controlled_burn_1080p | old_town_cross_1080p50 | rush_field_cuts_1080p | sunflower_1080p25
crowd_run_1080p50 | park_joy_1080p50 | rush_hour_1080p25 | tractor_1080p25
dinner_1080p30 | pedestrian_area_1080p25 | snow_mnt_1080p | west_wind_easy_1080p
### B.2 Xiph-5N 2 fps dataset
Due to computational limits, we draw a sample of five videos, which we refer
to as the Xiph-5N dataset. When drawing such a small sample randomly, a high
probability arises of drawing an unrepresentative sample, including for
example too many videos with either low or high finetuning potential. To
alleviate this problem, we use the following heuristic to select $N(=5)$
videos:
1. 1.
Evaluate the global model’s $RD$ performance on all videos for all values of
$\beta$.
2. 2.
Per $\beta$ value, rank all videos based on their respective
$\mathcal{L}_{RD}$ loss.
3. 3.
For each video, average the $\mathcal{L}_{RD}$ rank over all values of
$\beta$.
4. 4.
Order all videos according to their average rank, and select $N$ videos based
on $N+1$ evenly spaced percentiles.
The global model’s $RD$ performance for all videos from the Xiph dataset is
shown in Figure 5. The five videos part of Xiph-5N are indicated with colors,
and Table 2 provides more details about these five videos. The column titled
_RD-tank percentile_ shows the actual percentile at which the selected videos
are ranked. The computed (target) percentiles for $N=5$ are 1/6, 2/6, $\dots$,
5/6. For each percentile we selected the video closest to these target
percentiles. The last column denotes the number of I-frames after subsampling
to 2 fps. Videos of which the original sampling frequency was not integer
divisible by a factor 2, were subsampled with a factor resulting in the
I-frame sampling frequency closest to 2 fps.
Figure 5: $RD$ performance of global baseline for all datapoints in Xiph and Xiph-5N. Video | RD-rank | Target | width $\times$ height | Original | Duration | Nr. of I-frames
---|---|---|---|---|---|---
| percentile | percentile | $\times$ frames | fps | (s) | at 2 fps
in_to_tree | 0.22 | 0.167 | 1920 $\times$ 1080 $\times$ 500 | 50 | 10.0 | 20
aspen | 0.37 | 0.333 | 1920 $\times$ 1080 $\times$ 570 | 30 | 19.0 | 38
controlled | 0.50 | 0.500 | 1920 $\times$ 1080 $\times$ 570 | 30 | 19.0 | 38
sunflower | 0.69 | 0.667 | 1920 $\times$ 1080 $\times$ 500 | 25 | 20.0 | 42
pedestrian_area | 0.82 | 0.833 | 1920 $\times$ 1080 $\times$ 375 | 25 | 15.0 | 32
AVERAGE | 0.52 | 0.500 | 1920 $\times$ 1080 $\times$ 503 | 32 | 16.6 | 34
Table 2: Characteristics of the five selected videos in Xiph-5N 2fps.
## Appendix C Global model architecture and training
For our neural compression model, we adopt the architecture proposed by Ballé
et al. (2018), including the mean-scale prior from Minnen et al. (2018). We
use a shared hyperdecoder to predict the mean and scale parameters. Like
(Ballé et al., 2018) we use a model architecture with fewer parameters for the
low bitrate regime ($\beta\geq 1e{-3}$). Table 3 indicates both the model
architecture and the number of parameters, grouped per sub-model. The upper
row in this table links the terminology proposed by Ballé et al. (2018) to
conventional VAE terminology which we follow in this work.
Figure 6 provides a visual overview of the model architecture, where we use
${\bm{z}}_{2}$ and ${\bm{z}}_{1}$ to indicate the latent and hyper-latent
space respectively (referred to as ${\bm{y}}$ and ${\bm{z}}$ in the original
paper of Ballé et al. (2018)). Note that even though we adopt a hierarchical
latent variable model, we simplify the notation by defining a single latent
space ${\bm{z}}=\\{{\bm{z}}_{1},{\bm{z}}_{2}\\}$ throughout this work.
During training we adopt a mixed quantization strategy, where the quantized
latent ${\bm{z}}_{2}$ are used to calculate the distortion loss (with their
gradients estimated using the Straight-Trough estimator from Bengio et al.
(2013)) , while we use noisy samples for ${\bm{z}}_{1},{\bm{z}}_{2}$ when
computing the rate loss.
Optimization of eq. 1 on the training fold of the CLIC19 dataset, was done
using the Adam optimizer with default settings (Kingma & Ba, 2014), and took
1.5M steps for the low bitrate models, and 2.0M steps for the high bitrate
models. Each step contained $8$ random crops of $256\times 256$ pixels, and
the initial learning rate was set to $1e{-4}$ and lowered to $1{e-5}$ at
$90\%$ of training.
Figure 6: The mean-scale hyperprior model architecture visualized using the VAE framework. Table 3: Model architecture and parameter count for the adopted hyperprior model. As suggested by Ballé et al. (2018), a distinct architecture is used for the low and high bitrate regime. We report the number of output channels per layer, for the exact model architecture we refer to Ballé et al. (2018). Note that the last column refers to the total number of parameters that needs to be known at the receiver side. | Transmitter | Receiver
---|---|---
| Encoder | Hyper Encoder | Hyperprior | Hyper Decoder | Decoder | Nr. of receiver
| $q_{\phi}({\bm{z}}_{2}|{\bm{x}})$ | $q_{\phi}({\bm{z}}_{1}|{\bm{z}}_{2})$ | $p_{\theta}({\bm{z}}_{1})$ | $p_{\theta}({\bm{z}}_{2}|{\bm{z}}_{1})$ | $p_{\theta}({\bm{x}}|{\bm{z}}_{2})$ | parameters $\theta$
Low bitrate model | | | | | |
Layers x Output channels | 4x192 | 3x128 | 3x3 | 2x128 + 1x256 | 3x192 + 1x3 | -
| Parameter count
---
2.89M | 1.04M | 5.50k | 1.26M | 2.89M | 4.16M
High bitrate model | | | | | |
Layers x Output channels | 4x320 | 3x192 | 3x3 | 2x192 + 1x384 | 3x320 + 1x3 | -
| Parameter count
---
8.01M | 2.40M | 8.26k | 2.95M | 8.01M | 10.97M
## Appendix D Temporal Ablation
In this experiment we investigate the tradeoff between the number of frames
that the model is finetuned on, and the final $RD{\widebar{M}}$ performance.
The higher this number of frames, the higher (potentially) the diversity
(making finetuning more difficult), but the lower the bitrate overhead (in
bits/pixels) due to model updates. This ablation repeats our main experiment
on the sunflower video (Fig. 8) for a varying number of I-frames.
We sample $f$ number of frames (equispaced) from the full video, starting at
the zero’th index. The experiment is run for
$f\in\\{1,2,5,10,25,50,100,250,500\\}$, and the two outmost rate-distortion
tradeoffs: $\beta\in\\{0.003,0.0001\\}$. Note that the original experiment was
done with frames sampled at 2 fps, resulting in $f=42$ for the sunflower
video.
Figure 7: Compression performance as a function of number of finetuning frames
for the sunflower video. The dashed red line indicates sampling at 2 fps (used
in our main experiments).
Figure 7 shows (for the low and high bitrate region) the total
$RD{\widebar{M}}$ loss, and its subdivision in distortion and the different
rate terms, as a function of numbers of finetuning frames. Full-model
finetuning outperforms encoder-only finetuning in all cases, except for $f=1$
in the low bit-rate regime. In this case, the model rate ${\widebar{M}}$ is
causing the total rate $R+{\widebar{M}}$ to become too high to be competitive
with the baselines. This in turn is mainly caused by the initial cost
${\widebar{M}}_{0}$, which can only be amortized over a single frame. In
general, for other values of $f$, these initial costs were found to contribute
only little to the total rate (with even a negligible contribution in the low
and high bitrate regions for respectively $f\geq 25$ and $f\geq 10$).
Note that the global model’s performance varies noticeably as a function of
$f$. Apparently, the first frame of this video is easy to compress, therewith
lowering the total loss for small sets of I-frames. To make fair comparisons,
one should thus only consider relative performance of encoder-only and full-
model finetuning with respect to the $RD$ trained baseline. In line with our
main findings in Fig. 2c, full-model finetuning shows the biggest improvements
for the high bitrate setting.
Interestingly, when comparing the low and high bitrate regimes, the total
relative $RD{\widebar{M}}$ gain of full-model finetuning follows a similar
pattern for varying values of $f$ (higher gain for higher $f$). However, the
subdivision of this gain in rate and distortion gain differs due to leveraging
another tradeoff setting $\beta$. For the high bitrate, mainly distortion is
diminished (row 2), whereas for the low bitrate, rate is predominantly reduced
(row 3). These rate and distortion reduction plots clearly show how the
flexibility of full-model (compared to encoder-only) finetuning can improve
results in various conditions.
This experiment has shown that the potential for full-model finetuning (under
the current model architecture and prior) seems highest for video compression
purposes, as gains are negative (due to relative high static initial costs in
the low bitrate regime) or only marginal (in the high bitrate regime) when
overfitting on a single frame. Yet, we hypothesize that full-model finetuning
could still be useful for (single) image compression as well, given other
choices for the model architecture and/or model prior. Also, the provided
ablation is run on one video only, so further research is needed to
investigate full-model finetuning in an image compression setup.
## Appendix E Rate-distortion finetuning performance per video
Figure 8 shows the $RD$ $\widebar{M}$ plots for the different videos after
finetuning for 100k steps. Full-model finetuning outperforms the global model,
_encoder-only_ and direct latent optimization for all videos. The blue lines,
indicating global model performance, differ per video, which might influence
the finetuning gains, which also differ per video, e.g. controlled_burn versus
sunflower. True entropy-coded results are used to create these graphs, rather
than the computed $RD$ $\widebar{M}$ values. Deviations between entropy-coded
and computed rates were found to be negligible (mean deviation was 1.94e-04
bits/pixel for $R$ and 1.06e-03 bits/pixel for $\bar{M}$). Throughout this
paper, all training graphs and ablations are therefore provided using the
computed values, rather than the entropy coded results.
Figure 9 shows for each of these videos the finetuning progression over
training steps. Also here, differences in performance are visible among
videos. The videos that result in highest finetuning gains, e.g. sunflower,
show quicker performance improvement after the start of finetuning, and also
continue to improve more over time.
Figure 8: $RD$ $\widebar{M}$ performance for instance-adaptive encoder-only
and full-model finetuning, compared with the performance of the global model,
split per video. The instance-adaptive models used to create each graph are
finetuned on the corresponding single video. Figure 9: Progression of
finetuning over time for all videos in Xiph-5N 2fps.
## Appendix F Model Updates Distributions
Figure 10 shows how the (quantized) model updates become much sparser (top
row) when finetuning includes the spike-and-slab model rate loss $M$, compared
to unregularized finetuning (bottom row).
Figure 10: Histograms showing the distribution of quantized model updates
$\bar{\delta}$ when finetuning with (top row) and without (bottom row) the
model rate regularizer $M$.
|
# Practical Provenance in Astronomy
Mathieu Servillat,1 François Bonnarel,2 Mireille Louys,2,3 and Michèle
Sanguillon4
###### Abstract
Recently the International Virtual Observatory Alliance (IVOA) released a
standard to structure provenance metadata, and several implementations are in
development in order to capture, store, access and visualize the provenance of
astronomy data products. This BoF will be focused on practical needs for
provenance in astronomy. A growing number of projects express the requirement
to propose FAIR data (Findable, Accessible, Interoperable and Reusable) and
thus manage provenance information to ensure the quality, reliability and
trustworthiness of this data. The concepts are in place, but now, applied
specifications and practical tools are needed to answer concrete use cases.
During this session we discussed which strategies are considered by projects
(observatories or data providers) to capture provenance in their context and
how a end-user might query the provenance information to enhance her/his data
selection and retrieval. The objective was to identify the development of
tools and formats now needed to make provenance more practical needed to
increase provenance take-up in the astronomical domain.
1Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, CNRS,
Université de Paris, 92190 Meudon, France<EMAIL_ADDRESS>
2Centre de Données astronomiques de Strasbourg, Observatoire Astronomique de
Strasbourg, Université de Strasbourg, CNRS-UMR 7550, Strasbourg, France
3ICube Laboratory, Université de Strasbourg, CNRS-UMR 7357, Strasbourg, France
4Laboratoire Univers et Particules de Montpellier, Université de Montpellier,
CNRS/IN2P3, France
## 1 The path to Open Science
Open science is the movement to make scientific research (including
publications, data, physical samples, and software) and its dissemination
accessible to all levels of an inquiring society, amateur or professional.
In this context, the European Open Science Cloud (EOSC) is an environment for
hosting and processing research data to support science. The aim is to make
research data interoperable and machine actionable following the FAIR
principles111https://www.go-fair.org/fair-principles (Wilkinson et al. 2016),
by federating existing research data infrastructures in Europe and realising a
web of FAIR data and related services for science, .
Within EOSC, the ESCAPE project222https://projectescape.eu (European Science
Cluster of Astronomy & Particle physics ESFRI research infrastructures) brings
together the astronomy, astroparticle and particle physics communities and
puts together a cluster with ESFRI projects with aligned challenges of data-
driven research. Keeping and exposing provenance information is one of those
challenges.
A workshop333https://indico.in2p3.fr/event/21913/page/2641-summary dedicated
to the management of provenance took place in September 2020 within ESCAPE, in
line with this ADASS session on practical provenance. The discussions led to a
clarification of the requirement for structured provenance (§3), a more
precise terminology (§4), and the identification of key concepts associated
with the management of provenance information (§5).
## 2 The IVOA Provenance Data Model
The IVOA has released a recommendation for a Provenance Data Model (Servillat
et al. 2020b). The core model and definition of provenance is as defined by
the W3C (Belhajjame et al. 2013): information about entities, activities, and
people (agents) involved in producing a piece of data or thing, which can be
used to form assessments about its quality, reliability or trustworthiness.
Provenance is related by definition to the origin of a product (where does it
come from?), but also the path followed to generate this product (what has
been done?). Provenance is thus seen as a chain of activities and entities,
used and generated. With the core data model, the basic objectives are
achieved: use of unique identifiers, traceability of the operations,
connection with contacts for further information, citation or acknowledgement.
By following the full IVOA data model, more advanced questions are answered:
What happened during each activity? How was the activity tuned to be executed
properly? What kind of content is in the entities?
The data model is a base for the development of tools and services, see e.g.:
Servillat et al. (2021); Sanguillon et al. (2021); Landais et al. (2021);
Servillat et al. (2020a); Sanguillon et al. (2020).
## 3 Requirement for structured provenance
We often realize too late that there are missing elements or links in the
provenance. The capture of the provenance should thus be structured, as
detailed as possible, and as naive as possible (simply record what happens).
There are clear advantages to retain this information as structured, machine-
readable data, in particular in the context of Open Science:
* •
Explicitly required through the FAIR principles that indicate to include rich
metadata, following standard data model, protocols and formats, with detailed
provenance.
* •
Quality / Reliability / Trustworthiness of the products: the simple fact of
being able to show its provenance is sufficient to give more value to a
product, and if the provenance information is detailed, the value will be
higher.
* •
Reproducibility requirement in many projects: provenance details are essential
to be able to rerun each activity (maybe testing and improving each step);
Having this information, it may not be necessary to keep every intermediate
file that is easily reproducible (hence a possible gain on storage space and
costs).
* •
Debugging: with detailed provenance, it is not necessary to restart from
scratch, as one can locate in the provenance graph the faulty parts or the
products to be discarded, and reprocess only from the identified failing
steps.
## 4 Some terminology
The word "provenance" is used to refer to different aspects depending on the
persons or the goals involved. Provenance may be used for internal data
management, or to improve the scientific exploitation of a data product. It
may be stored inside the data file or separately (external file or database).
We propose here a base for the definition of provenance categories.
* •
full provenance: graph/tree/chain of activities and entities up to the raw
data. This information is not hosted by the entities themselves, but stored on
an external server, or as separate files.
* •
minimum provenance: information attached to an entity as a list of keywords
that gives some context and info on last activity (general process/workflow,
software versions, contacts…), maybe including the list of used entities, so
that a full provenance may be reconstructed from minimum provenance, but such
information is not always preserved.
* •
end-user/specific "provenance": information attached to an entity, as a list
of keywords or data that provides key information to use/analyse the entity
(e.g. for CTA: event class, event type, telescope configuration, sky
conditions, reconstruction method,…). This information may be extracted from
the full provenance, or inversely, used to enrich a reconstructed provenance.
## 5 Applying the model
Different concepts were identified concerning provenance in various use cases.
We distinguish on-top provenance handling, where data products/collections
already exist and is used to reconstruct a provenance graph, and inside
provenance processing, where the provenance information is saved during the
execution of the processing activities.
The use of unique identifiers is recommended and implies to think about
provenance capture at the conception phase of a project. One also has to
evaluate in advance the necessary and sufficient granularity (what steps? what
objects?) and level of details (inclusion of descriptions and configuration
details?).
The challenges in provenance management encompass the capture, storage,
access, and visualization of the provenance information.
## 6 Discussion topics
The discussion was organised around several topics, following the priorities
of the participants as reported in a preliminary questionnaire. Those topics
will be discussed during forthcoming ESCAPE events, in collaboration with IVOA
working groups:
1. Defining the content of a minimum provenance, with a list of keywords related to the last activity and context.
2. Serializing provenance, both in a human- and machine-readable way.
3. Provenance and workflows, with workflow information simply attached to provenance (as used entities), or mapped to the model.
4. Evolving from provenance "on-top" to provenance "inside", e.g. with generic tools to capture provenance at the execution.
5. Provenance storage in a database and related interfaces.
6. Provenance exploration and visualization, based on access protocols (ProvTAP, see Bonnarel et al. 2019, or ProvSAP, see Servillat et al. 2021.), and tools such as the voprov Python package.
### Acknowledgments
We acknowledges support from the ESCAPE project funded by the EU Horizon 2020
research and innovation program (Grant Agreement n.824064). Additional funding
was provided by the INSU (Action Spécifique Observatoire Virtuel, ASOV), the
Action Fédératrice CTA at the Observatoire de Paris and the Paris Astronomical
Data Centre (PADC).
## References
* Belhajjame et al. (2013) Belhajjame, K., B’Far, R., Cheney, J., Coppens, S., Cresswell, S., Gil, Y., Groth, P., Klyne, G., Lebo, T., McCusker, J., Miles, S., Myers, J., Sahoo, S., & Tilmes, C. 2013, PROV-DM: The prov data model, W3C Recommendation. URL http://www.w3.org/TR/prov-dm/
* Bonnarel et al. (2019) Bonnarel, F., Louys, M., Mantelet, G., Nullmeier, M., Servillat, M., Riebe, K., & Sanguillon, M. 2019, in ADASS XXVIII, edited by P. J. Teuben, M. W. Pound, B. A. Thomas, & E. M. Warner, vol. 523 of ASP Conf. Ser., 313
* Landais et al. (2021) Landais, G., Servillat, M., Bonnarel, F., Louys, M., Sanguillon, M., & Michel, L. 2021, in ADASS XXX, edited by J.-E. Ruiz, & F. Pierfederici, vol. TBD of ASP Conf. Ser., 999 TBD
* Sanguillon et al. (2021) Sanguillon, M., Arrabito, L., Boisson, C., Bregeon, J., Kosack, K., & Servillat, M. 2021, in ADASS XXX, edited by J.-E. Ruiz, & F. Pierfederici, vol. TBD of ASP Conf. Ser., 999 TBD
* Sanguillon et al. (2020) Sanguillon, M., Bonnarel, F., Louys, M., Nullmeier, M., Riebe, K., & Servillat, M. 2020, in ADASS XXVII, edited by P. Ballester, J. Ibsen, M. Solar, & K. Shortridge, vol. 522 of ASP Conf. Ser., 545
* Servillat et al. (2021) Servillat, M., Aicardi, S., Cecconi, B., & Mancini, M. 2021, in ADASS XXX, edited by J.-E. Ruiz, & F. Pierfederici, vol. TBD of ASP Conf. Ser., 999 TBD
* Servillat et al. (2020a) Servillat, M., Boisson, C., Lefaucheur, J., Kosack, K., Sanguillon, M., Louys, M., & Bonnarel, F. 2020a, in ADASS XXVII, edited by P. Ballester, J. Ibsen, M. Solar, & K. Shortridge, vol. 522 of ASP Conf. Ser., 199
* Servillat et al. (2020b) Servillat, M., Riebe, K., Boisson, C., Bonnarel, F., Galkin, A., Louys, M., Nullmeier, M., Renault-Tinacci, N., Sanguillon, M., & Streicher, O. 2020b, IVOA Provenance Data Model Version 1.0, IVOA Recommendation 11 April 2020
* Wilkinson et al. (2016) Wilkinson, M. D., Dumontier, M., Aalbersberg, I. J., Appleton, G., Axton, M., Baak, A., Blomberg, N., Boiten, J.-W., da Silva Santos, L. B., Bourne, P. E., et al. 2016, Scientific Data, 3
|
Somerville College Doctor of Philosophy Revised Version, 2020
The absolute/relative debate on the nature of space and time is ongoing for
thousands of years. Here we attempt to investigate space and time from the
information theoretic point of view to understand spatial and temporal
correlations under the relative assumption. Correlations, as a measure of
relationship between two quantities, do not distinguish space and time in
classical probability theory; quantum correlations in space are well-studied
but temporal correlations are not well understood. The thesis investigates
quantum correlations in space-time, by treating temporal correlations equally
in form as spatial correlations and unifying quantum correlations in space and
time. In particular, we follow the pseudo-density matrix formalism in which
quantum states in spacetime are properly defined by correlations from
measurements.
We first review classical correlations, quantum correlations in space and
time, to motivate the pseudo-density matrix formalism in finite dimensions.
Next we generalise the pseudo-density matrix formulation to continuous
variables and general measurements. Specifically, we define Gaussian spacetime
states by the first two statistical moments, and for general continuous
variables spacetime states are defined via the Wigner function representation.
We also define spacetime quantum states in position measurements and weak
measurements for general measurement processes. Then we compare the pseudo-
density matrix formalism with other spacetime formulations: indefinite causal
structures, consistent histories, generalised non-local games, out-of-time-
order correlation functions, and path integrals. We argue that in non-
relativistic quantum mechanics, different spacetime formulations are closely
related and almost equivalent via quantum correlations, except path integrals.
Finally, we apply the pseudo-density matrix formulation to time crystals. By
defining time crystals as long-range order in time, we analyse continuous and
discrete time translation symmetry as well as discuss the existence of time
crystals from an algebraic point of view. Finally, we summarise our work and
provide the outlook for future directions.
# Quantum Correlations in Space-Time: Foundations and Applications
Tian Zhang
###### Abstract
The absolute/relative debate on the nature of space and time is ongoing for
thousands of years. Here we attempt to investigate space and time from the
information theoretic point of view to understand spatial and temporal
correlations under the relative assumption. Correlations, as a measure of
relationship between two quantities, do not distinguish space and time in
classical probability theory; quantum correlations in space are well-studied
but temporal correlations are not well understood. The thesis investigates
quantum correlations in space-time, by treating temporal correlations equally
in form as spatial correlations and unifying quantum correlations in space and
time. In particular, we follow the pseudo-density matrix formalism in which
quantum states in spacetime are properly defined by correlations from
measurements.
We first review classical correlations, quantum correlations in space and
time, to motivate the pseudo-density matrix formalism in finite dimensions.
Next we generalise the pseudo-density matrix formulation to continuous
variables and general measurements. Specifically, we define Gaussian spacetime
states by the first two statistical moments, and for general continuous
variables spacetime states are defined via the Wigner function representation.
We also define spacetime quantum states in position measurements and weak
measurements for general measurement processes. Then we compare the pseudo-
density matrix formalism with other spacetime formulations: indefinite causal
structures, consistent histories, generalised non-local games, out-of-time-
order correlation functions, and path integrals. We argue that in non-
relativistic quantum mechanics, different spacetime formulations are closely
related and almost equivalent via quantum correlations, except path integrals.
Finally, we apply the pseudo-density matrix formulation to time crystals. By
defining time crystals as long-range order in time, we analyse continuous and
discrete time translation symmetry as well as discuss the existence of time
crystals from an algebraic point of view. Finally, we summarise our work and
provide the outlook for future directions.
“Space and time are the pure forms thereof; sensation the matter.”
— Immanuel Kant, Critique of Pure Reason
###### Acknowledgements.
I find myself very lucky to have the opportunity to study at Oxford and around
the world with so many talented mentors and friends, from whom I have learnt
so much. I am very grateful for all the help and support they offer and would
express my sincere gratitude to all who have made my DPhil days special.First
of all, I would like to thank my supervisor, Prof. Vlatko Vedral, for inviting
me to his group, for insightful discussions during these years, and for his
enthusiasm for physics which always inspires me. I would also like to thank
Dr. Tristan Farrow, who is always so kind and thoughtful, like a big elder
brother, not only teaches me how to cope with different academic situations,
but also cares about my personal development. I am grateful to Prof. Oscar
Dahlsten, who has invited me twice to SUSTech in Shenzhen, China, collaborated
with me, especially for his patience on teaching me how to write my first
paper, as well as continually giving me lots of guidance and feedback. I also
thank Dr. Felix Tennie, who has been so patient and helpful to revise my
first-year transfer report and all the suggestions on academic writing. I
would like to thank Dr. Chiara Marletto, who collaborated with me on my first
project on time crystals, for being so helpful and considerate to give me
suggestions on research and writing. I would love to thank the rest of the
group, including Anupama Unnikrishnan, Nana Liu, Christian Schilling, Davide
Girolami, Reevu Maity, Pieter Bogaert, Thomas Elliott, Benjamin Yadin, Aditya
Iyer, Jinzhao Sun, Sam Kuypers, David Felce, and Anicet Tibau Vidal. I would
also like to thank the quantum group in the department of computer science,
especially Prof. Giulio Chiribella and Prof. Jonathan Barrett for insightful
discussion, and Prof. Bob Coecke and lots of others for organising interesting
talks, conferences and Wolfson foundation discussions, from which I benefited
a lot. Then I want to thank Perimeter Institute for Theoretical Physics, in
particular for the visiting graduate fellow program. I would love to thank my
local host and advisor, Prof. Lucien Hardy, for inviting me to Perimeter, for
regular meetings with me to answer my questions and check my progress, for
always being so encouraging and telling me to think of big problems. It is a
great pleasure to meet Prof. Lee Smolin at Perimeter and hold weekly meetings
with him. I learned so many fascinating ideas and deep thoughts from Lee and
gradually started to build on my own research taste. Thanks to Prof. Rafael
Sorkin as well for being so patient and helpful and explaining to me on
quantum measure theory, irreversibility, nonunitary, and lots of sightful
conversation. And thanks to Beni Yoshida, for introducing the wonderful world
of black hole information paradox to me and collaborating with me on the black
hole final state projection proposal. I would also thank Guifre Vidal, for his
guidance on organising the session of black hole information paradox in the
quantum information workshop in Benasque, Spain. I would also love to thank
the whole quantum foundation and quantum information group, Daniel Gottesman,
Robert Spekkens, David Schmid, Tobias Fritz, Denis Rosset, Thomas Galley,
Nitica Sakharwade, Zi-Wen Liu, Nick Hunter-Jones, and lots of others that I am
sorry I cannot list for all, for having lunches and discussions together and
always being so friendly and helpful. I had a brilliant time at Perimeter and
I am so grateful for everyone there. I thank Prof. Xi Dong, for hosting my
visit at Santa Barbara and discussing possible projects on holographic min-
and max-entropy. I also want to thank the organisers and participants of
Boulder Summer School 2018, for organising such a great quantum information
summer school, where I learned so much and broadened my understanding of
quantum information science. Thanks to Felix Leditzky, Graeme Smith, Mark M.
Wilde for the opportunity to present my work at Rocky Mountain Summit on
Quantum Information. Thanks to Hilary Carteret and John Donohue for inviting
me to present my work at Institute of Quantum Computing, University of
Waterloo. And thanks to Prof. Otfried Guehne for inviting me to visit and work
with his group at University of Siegen. I am very grateful to Prof. Renato
Renner at ETH Zurich and Prof. Simon Benjamin at Oxford Materials for being my
examiners and all the discussion and suggestions for my thesis. I would thank
my valuable friends at Oxford, at Perimeter Institute, at Boulder Summer
School, at different conferences, from my undergraduate, Peking University and
even long time before. I wish I could list all their names here but I am
afraid there are too many of them and I am more afraid I may miss some of
their names. Thank you so much, my dearest mum and dad. Thanks to all my
family members for their love and understanding. I have had very difficult
time during my DPhil. I am so grateful that I have received so much love and
support to get through all the hard time. I would express my gratitude, again
to so many people along the way for their companion. And thank you, my reader,
for taking your time to have a look at this thesis.
###### Contents
1. 1 Introduction
2. 2 Quantum correlations in space-time
1. 2.1 Classical correlations
1. 2.1.1 Correlations in probability theory
2. 2.1.2 Correlations in statistical mechanics
2. 2.2 Quantum correlations in space
1. 2.2.1 Basics for quantum mechanics
2. 2.2.2 Bipartite quantum correlations
3. 2.2.3 Multipartite quantum correlations
3. 2.3 Quantum correlations in time
1. 2.3.1 Correlations in quantum field theory
2. 2.3.2 Further possibility for temporal correlations
3. 2.3.3 Towards a unified approach for quantum correlations in space and time
4. 2.4 Pseudo-density matrix formalism
1. 2.4.1 Definition and properties
2. 2.4.2 Characterisation of bipartite correlations in space-time
3. 3 Generalisation of pseudo-density matrix formulation
1. 3.1 Introduction
2. 3.2 Gaussian generalisation of pseudo-density matrix
1. 3.2.1 Preliminaries
2. 3.2.2 Spacetime Gaussian states
3. 3.2.3 Example: vacuum state at two times
4. 3.2.4 Spatial vs temporal Gaussian states
3. 3.3 Pseudo-density matrix formulation for general continuous variables
1. 3.3.1 Preliminaries
2. 3.3.2 Spacetime Wigner function
3. 3.3.3 Spacetime density matrix in continuous variables
4. 3.3.4 Properties
4. 3.4 Generalised measurements for pseudo-density matrix
1. 3.4.1 Position measurements
2. 3.4.2 Weak measurements
5. 3.5 Experimental proposal for tomography
6. 3.6 Comparison and comments
4. 4 Correlations from other spacetime formulations: relation and lesson
1. 4.1 Introduction
2. 4.2 Indefinite causal structures
1. 4.2.1 Preliminaries for process matrix formalism
2. 4.2.2 Correlation analysis and causality inequalities
3. 4.2.3 Postselection and closed timelike curves
4. 4.2.4 Summary of the relation between pseudo-density matrix and indefinite causal structures
3. 4.3 Consistent histories
1. 4.3.1 Preliminaries for consistent histories
2. 4.3.2 Temporal correlations in terms of decoherence functional
4. 4.4 Generalised non-local games
1. 4.4.1 Introduction to non-local games
2. 4.4.2 Quantum-classical non-local & signalling games
3. 4.4.3 Temporal correlations from signalling games
5. 4.5 Out-of-time-order correlations (OTOCs)
1. 4.5.1 Brief introduction to OTOCs
2. 4.5.2 Calculating OTOCs via pseudo-density matrices
3. 4.5.3 Black hole final state proposal
6. 4.6 Path integrals
1. 4.6.1 Introduction to path integrals
2. 4.6.2 Temporal correlations in path integrals are different
7. 4.7 Conclusion and discussion
5. 5 Time crystals as long-range order in time
1. 5.1 Literature review for time crystals
1. 5.1.1 Spontaneous symmetry breaking
2. 5.1.2 Time translation symmetry breaking
3. 5.1.3 Mathematical definitions of time crystals
2. 5.2 Definition: time crystals as long-range order in time
1. 5.2.1 Long-range order
2. 5.2.2 Time crystals in terms of temporal correlations
3. 5.3 Continuous time translation symmetry
1. 5.3.1 General decoherent process
2. 5.3.2 Generalised Mermin-Wagner theorem
4. 5.4 Discrete time translation symmetry
1. 5.4.1 Stabilisation of quantum computation
2. 5.4.2 Quantum error correction of phase flip codes
3. 5.4.3 Floquet many-body localisation
4. 5.4.4 Possible sufficient conditions for general open systems
5. 5.5 An algebraic point of view
1. 5.5.1 Preliminaries
2. 5.5.2 Existence of time crystals
3. 5.5.3 Temporal correlations
6. 6 Conclusion and outlook
7. 7 Choi-Jamiołkowski isomorphism
8. 8 Proofs for the properties for spacetime Wigner functions
1. 8.1 Wigner Representation in Liouville Space
2. 8.2 Proofs for the properties
9. 9 Proof for continuous time translation symmetry in 1+1 dimensions
## Chapter 1 Introduction
What is time?
The intrinsic motivation for all the work in the thesis is to get a little bit
closer to this question.
In general, there are three schools that hold different views on time. As Page
and Wootters argue in their famous “evolution without evolution” paper [1],
all the observables which commute with the Hamiltonian are stationary and the
dynamics of a system we observe can be fully described by stationary
observable dependent upon internal clock readings. Barbour [2] also believes
in the timeless universe where time does not exist and is merely an illusion.
They claim that in general relativity, especially in the equivalent Arnowitt-
Deser-Misner (ADM) formalism [3], the dynamics is embedded in three-
dimensional Riemannian spaces rather than the four-dimensional spacetime since
one dimension can be arbitrarily chosen. Not to mention quantum cosmology [4],
where quantum mechanics is applied to the whole universe, the Wheeler-Dewitt
equation [5] serves as a stationary Schrödinger equation for the wave function
of the universe.
However, Smolin and his colleagues [6, 7] hold the opposite point of view;
that is, time is fundamental in nature. They claim that in the Newtonian
paradigm [8], questions such as why the laws and why these initial conditions
remain unanswered. They believe that the reality of time is important in
selecting the fundamental laws of physics and construct an ultimate theory of
the whole universe instead of part of the universe.
Nevertheless, we have no evidence to judge the above two views on time,
whether time does not exist or time is fundamental so far. Instead in this
thesis, we would take a practical point of view from the lesson of relativity:
time may be treated as an equal footing as space. Both special relativity and
general relativity treat time as part of spacetime and gain beautiful results
which have already been verified. What’s more, treating time operationally
equal as space, also provides one possible method to study time following the
methods for investigating space.
More specifically, we investigate time from the quantum information
perspective in terms of temporal correlations, as an analogue of spatial
correlations. Spatial correlations like entanglement, nonlocality, steering,
and discord, are well-studied in quantum information. We know that physicists
have been working for decades in search for a way to quantise space-time and
trying to build a theory for quantum gravity. That is not the goal for this
thesis. Here, we focus on the quantum information side of spacetime; more
precisely, our topic is restricted to quantum correlations in non-relativistic
space-time.
We start from a particular kind of space-time formulation called pseudo-
density matrix formalism which treats temporal correlations as spatial
correlations, further generalise this formulation to continuous variables and
general measurement processes, compare it with other space-time formulations
via quantum correlations and argue that these non-relativistic space-time
formulations are very much related, and apply the pseudo-density matrix
formalism to time crystals to show its practical power.
The thesis proceeds as follows. In Chapter 2, we introduce quantum
correlations in space-time. We first introduce classical correlations in
probability theory and statistical mechanics. After introducing the basics for
quantum mechanics, we review quantum correlations in space. In bipartite
quantum correlations, we discuss correlation and entanglement, the difference
among Bell nonlocality, steering and entanglement, other quantum correlations
as quantum discord, and formulate the hierarchy of quantum correlations in
space based on operator algebra. We briefly mention multipartite quantum
correlations. Then we move on to quantum correlations in time. From the
correlations in field theory, we explore a further possibility for temporal
correlations and propose a unified approach for quantum correlations in space
and time to motivate pseudo-density matrices. Finally we formally introduce
the pseudo-density matrix formalism.
In Chapter 3, we fully generalise the pseudo-density matrix formalism to
continuous variables and general measurement processes. Pseudo-density matrix
formalism is based on building measurement correlations; the key for
generalisation is to choose the right measurement operators. For the Gaussian
case, we simply extend the correlations to the temporal domain by quadratures
measurements and compare the spatial vs temporal Gaussian states. For general
continuous variables, we use the Wigner function representation and its one-
to-one correspondence with the density matrix formalism to define spacetime
states, and compare the properties of spacetime Wigner functions with the
uniquely determined properties of normal Wigner functions. We further
generalise the formalism for general measurement processes like position
measurements and weak measurements. We also give an experimental proposal for
tomography in the Gaussian case. Before coming to the end, we compare
spacetime states in the generalised pseudo-density matrix formalism and make
further comments.
In Chapter 4, we compare spatial-temporal correlations in pseudo-density
matrix formalism with correlations in other spacetime formulations. In
particular, we analyse indefinite causal structures, consistent histories,
generalised non-local games, out-of-time-order correlation functions, and path
integrals. We aim to argue, in non-relativistic quantum mechanics, spacetime
formulations are closely related via quantum correlations. We also take
lessons from these spacetime formulations and further develop the pseudo-
density matrix formalism. In the section of out-of-time-order correlation
functions, we discuss their possible application in the black hole final state
projection proposal, as one of possible explanations for black hole
information paradox. The path integral approach gives a different
representation of quantum correlations and suggests interesting properties for
quantum measure and relativistic quantum information.
In Chapter 5, we use time crystals as an illustration of temporal
correlations, or more specifically, long-range order in time. We first review
spontaneous symmetry breaking, time translation symmetry breaking and
different mathematical definitions for time crystals. After formally
introducing long-range order, we define time crystals as long-range order in
time in the pseudo-density matrix formulation. To illustrate what time
crystals are, we consider continuous time translation symmetry in terms of
general decoherent processes and a generalised version of Mermin-Wagner
theorem, and discuss discrete time translation symmetry via a stabilisation
protocol of quantum computation, phase flip codes of quantum error correction
and Floquet many-body localisation. We also use an algebraic point of view to
analyse the existence of time crystals.
Chapter 6 is for the conclusion and outlook.
## Chapter 2 Quantum correlations in space-time
### 2.1 Classical correlations
In this section we introduce classical correlations in probability theory and
statistical mechanics. In the classical case, it is not necessary to
distinguish spatial or temporal correlations; that is, classical correlations
are defined whatever the spatio-temporal structures are.
#### 2.1.1 Correlations in probability theory
Now we introduce correlations defined in probability theory based on Ref. [9].
For a discrete random variable $X$ with the probability mass function
$p(x)=P\\{X=x\\}$, the expectation value of $X$ is defined as
$E[X]=\sum_{x:p(x)>0}xp(x)$. For a continuous random variable $X$ with the
probability density function $f(x)$ such that $P\\{a\leq X\leq
b\\}=\int_{a}^{b}f(x)\textrm{d}x$, the expectation value of $X$ is defined as
$E[X]=\int_{-\infty}^{\infty}xf(x)\textrm{d}x$. The variance of $X$ is defined
as $\text{Var}(X)=E[(X-E[X])^{2}]$. This definition is equivalent to
$\text{Var}(X)=E[X^{2}]-(E[X])^{2}$.
For two random variables $X$ and $Y$, the covariance is defined as
$\text{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]$. It is easy to see that
$\text{Cov}(X,Y)=E[XY]-E[X]E[Y]$. Then we define the correlation of $X$ and
$Y$ as
$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}}$
(2.1)
It is also referred to the Pearson product-moment correlation coefficient or
the bivariate correlation, as a measure for the linear correlation between $X$
and $Y$.
#### 2.1.2 Correlations in statistical mechanics
In statistical mechanics [10], the equilibrium correlation function for two
random variables $S_{1}$ at position $\mathbf{x}$ and time $t$ and $S_{2}$ at
position $\mathbf{x}+\mathbf{r}$ and time $t+\tau$ is defined as
$C(\mathbf{r},\tau)=\langle
S_{1}(\mathbf{x},t)S_{2}(\mathbf{x}+\mathbf{r},t+\tau)\rangle-\langle
S_{1}(\mathbf{x},t)\rangle\langle S_{2}(\mathbf{x}+\mathbf{r},t+\tau)\rangle,$
(2.2)
where $\langle O\rangle$ is the thermal average of the random variable $O$; it
is usually averaged over the whole phase space of the system. That is,
$\langle O\rangle=\frac{\int Oe^{-\beta
H(q_{1},\dots,q_{m},p_{1},\dots,p_{n})}\textrm{d}\tau}{\int e^{-\beta
H(q_{1},\dots,q_{m},p_{1},\dots,p_{n})}\textrm{d}\tau},$ (2.3)
where $\beta=1/k_{B}T$, $k_{B}$ is Boltzmann constant and $T$ is the
temperature, $H$ is the Hamiltonian of the classical system in terms of
coordinates $q_{i}$ and their conjugate generalised momenta $p_{i}$, and
$\textrm{d}\tau$ is the volume element of the classical phase space. In
particular, the equal-time spin-spin correlation function for two Ising spins
is given as
$C_{t}(\mathbf{r})=\langle
S(\mathbf{x},t)S(\mathbf{x}+\mathbf{r},t)\rangle-\langle
S(\mathbf{x},t)\rangle\langle S(\mathbf{x}+\mathbf{r},t)\rangle.$ (2.4)
It is used as a measure for spatial coherence for how much information a spin
can influence its distant neighbours. Taking the limit of $\mathbf{r}$ to
infinity, we obtain the long-range order for which correlations remain non-
zero even in the long distance.
### 2.2 Quantum correlations in space
In this section, we introduce quantum correlations in space. First we review
briefly on basics of quantum mechanics. Then we introduce bipartite quantum
correlations, in terms of entanglement, steering, nonlocality and discord. We
also list the hierarchy of spatial quantum correlations in terms of operator
algebra. Finally we mention multipartite quantum correlations in brief.
#### 2.2.1 Basics for quantum mechanics
In this subsection we briefly review the axioms of quantum mechanics and
introduce the concept of quantum states.
##### Axioms of quantum mechanics
We introduce the five axioms of quantum mechanics [11, 12, 13].
(1) The state in an isolated physical system is represented by a vector, for
example, $\ket{\psi}$, in the Hilbert space $\mathcal{H}$ which is a complex
vector space with an inner product. A system is completely described by
normalised state vectors in the Hilbert space.
(2) An observable is represented by an Hermitian operator with
$A^{{\dagger}}=A$.
(3) Suppose the system is measured by a collection of measurement operators
$\\{M_{m}\\}$ with measurement outcomes $\\{m\\}$. With the initial state
$\ket{\psi}$, after measurements the result $m$ comes with the probability
$p(m)=\bra{\psi}M_{m}^{{\dagger}}M_{m}\ket{\psi}$ (2.5)
and the state becomes
$\frac{M_{m}\ket{\psi}}{\sqrt{\bra{\psi}M_{m}^{{\dagger}}M_{m}\ket{\psi}}}$
(2.6)
The measurement operators satisfy
$\sum_{m}M_{m}^{{\dagger}}M_{m}=\mathbbm{1}$, then the probabilities sum to 1.
According to Wigner’s theorem, for any transformation
$\ket{\psi}\rightarrow\ket{\psi^{\prime}}$ in which the probabilities for a
complete set of states collapsing into another complete set
$|\innerproduct{\psi}{\psi_{n}}|^{2}=|\innerproduct{\psi^{\prime}}{\psi^{\prime}_{n}}|^{2}$
hold the same, we may define an operator $U$ such that
$\ket{\psi^{\prime}}=U\ket{\psi}$. Then $U$ is either unitary and linear or
else anti-unitary and anti-linear. Thus, we have
(4) A closed quantum system evolves under unitary transformation. That is, the
state of the system at two times $t_{1}$ and $t_{2}$ are related by a unitary
operator $U$ defined by $U^{{\dagger}}U=UU^{{\dagger}}=\mathbbm{1}$ such that
$\ket{\psi(t_{2})}=U\ket{\psi(t_{1})}.$ (2.7)
It is equivalent to
(4’) A closed quantum system evolves under Schrödinger equation:
$i\hbar\frac{d\ket{\psi}}{dt}=H\ket{\psi}$ (2.8)
$H$ is the Hamiltonian of the quantum system.
In addition, we have another postulate for composite quantum systems.
(5) The Hilbert space of the composite system $AB$ is the tensor product
$\mathcal{H_{A}}\otimes\mathcal{H_{B}}$ of the Hilbert spaces
$\mathcal{H_{A}}$ and $\mathcal{H_{B}}$ for systems $A$ and $B$. That is, if
the system $A$ is in the state $\ket{\psi}_{A}$ and the system $B$ is in the
state $\ket{\phi}_{B}$, then the composite system $AB$ is in the state
$\ket{\psi}_{A}\otimes\ket{\phi}_{B}$.
##### Quantum states
Here we define quantum states for discrete finite systems, and leave
continuous variables to next chapter.
The state vector is defined as before in terms of a normalised vector in the
Hilbert space: $\ket{\psi}\in\mathcal{H}$. A pure state is then given by
$\pi=\ket{\psi}\bra{\psi}\in\mathcal{P}$. If a quantum system is in the state
$\ket{\psi_{i}}$ with the probability $p_{i}$, we call the set
$\\{p_{i},\ket{\psi_{i}}\\}$ as an ensemble of pure states. An arbitrary
quantum state is represented by a density matrix defined as
$\rho=\sum_{i}p_{i}\ket{\psi_{i}}\bra{\psi_{i}}\in\mathcal{D}$ [12]. On the
one hand, the set of all possible states $\mathcal{D}$ is a convex set, that
is, $\mathcal{D}=\text{Conv}\mathcal{P}$; on the other hand, the extreme
points in the state space are pure states, i.e.,
$\mathcal{P}=\text{Extr}\mathcal{D}$ [14]. Note that the convex hull
$\text{Conv}\mathcal{P}$ of the set $\mathcal{P}$ in the complex state space
is defined to be the intersection of all convex sets in the state space that
contain $\mathcal{P}$. An extreme point $x$ of a convex set $\mathcal{D}$ is a
point such that for $y,z\in\mathcal{D}$, $0<\lambda<1$, $x=\lambda
y+(1-\lambda)z$ implies that $x=y=z$. [15]
A simple criterion to check whether the state $\rho$ is pure or mixed is that
$\Tr\rho^{2}=1$ for pure states and $\Tr\rho^{2}<1$ for mixed states [12].
Another measure of mixedness for quantum states is given by the von Neumann
entropy $S(\rho)=-\Tr\rho\log\rho$ [16]. It is non-negative and vanishes if
and only if $\rho$ is a pure state. The von Neumann entropy is concave,
subadditive and strongly subadditive. According to Schumacher’s quantum
noiseless channel coding theorem [17], it is the amount of quantum information
as the minimum compression scheme of rate.
The distinguishability of states [12] is measured by the quantum relative
entropy $D(\rho||\sigma)=\Tr\rho(\log\rho-\log\sigma)$ based on quantum
Stein’s lemma. The quantum relative entropy is jointly convex, non-negative,
and vanishes if and only if $\rho=\sigma$. Other distance measures for quantum
states include the trace distance $D(\rho,\sigma)=\frac{1}{2}\Tr|\rho-\sigma|$
and the fidelity $F(\rho,\sigma)=\Tr\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or
$F(\rho,\sigma)=(\Tr\sqrt{\rho^{1/2}\sigma\rho^{1/2}})^{2}$.
#### 2.2.2 Bipartite quantum correlations
In this subsection, we focus on bipartite quantum correlations. First we
introduce quantum correlation measures based on the distance of the states and
compare correlation with entanglement. Then we compare three types of quantum
correlations: entanglement, steering and Bell nonlocality. After introducing
other measures of quantum correlations such as discord, we use the operator
algebraic language to present the hierarchy of quantum correlations in space.
##### Correlation and entanglement
As an analog of classical correlations in probability theory, the correlation
for the quantum state itself is defined for
$\Gamma=\rho-\rho_{1}\otimes\rho_{2}$ where $\rho_{i}$ is the reduced state
for the subsystem $i(i=1,2)$. The covariance for two observables $A$ and $B$
on the two subsystems separately is then given by
$\text{Cov}(A,B)=\Tr\Gamma A\otimes B$ (2.9)
Recall that Eqn. (2.1)
$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}}$.
Then we say that the state is uncorrelated, if and only if
$\text{Corr}(A,B)=0$ for all observables $A,B$ for two subsystems. This
condition is equivalent to $\langle AB\rangle=\langle A\rangle\langle
B\rangle$, as well as $\rho=\rho_{1}\otimes\rho_{2}$.
For a pure state, if the state is correlated, we call it entanglement. For a
mixed state, the state is uncorrelated if and only if
$\rho=\rho_{1}\otimes\rho_{2}\in\mathcal{D}_{unc}$, otherwise we call it
correlated. At the same time, the state is separable for a possible
decomposition
$\rho=\sum_{i}p_{i}\pi_{1,i}\otimes\pi_{2,i}\in\mathcal{D}_{sep}$; otherwise
it is entangled. Note that $\mathcal{D}_{sep}=\text{Conv}\mathcal{D}_{unc}$.
The correlation measure can be given by the distinguishability as the relative
entropy
$C(\rho)=\min_{\sigma\in\mathcal{D}_{unc}}D(\rho||\sigma)=D(\rho||\rho_{1}\otimes\rho_{2})=S(\rho_{1})+S(\rho_{2})-S(\rho)=I(\rho);$
(2.10)
it is equal to the mutual information of the state. Here we only discuss
whether the state is correlated or not; for general quantum correlations, we
will introduce entanglement, steering, Bell nonlocality and discord later.
##### Entanglement, steering and Bell nonlocality
Here we compare three types of quantum correlations: entanglement, steering
and Bell nonlocality [18].
Bell nonlocality is characterised by the violation of Bell inequalities [19,
20]. In a typical Bell experiment, two spatially separated systems are
measured by two distant observers, say Alice and Bob, respectively. Alice may
select her measurement from several possible ones and denote her choice of
measurement by $x$, and gain the outcome $a$ after the measurement. Bob makes
the measurement denoted by $y$ and gains the outcome $b$. If there exists a
local hidden variable model, then the probability to obtain the results $a$
and $b$ under the measurements $x$ and $y$ can be written as
$p(a,b|x,y)=\int\textrm{d}\lambda p(\lambda)p(a|x,\lambda)p(b|y,\lambda),$
(2.11)
where the hidden variable $\lambda$ gives the probability function
$p(\lambda)$, Alice and Bob yield the outcome under their local probability
distributions with the parameter $\lambda$. Given the measurements $x,y$ and
the outcomes $a,b$, the probabilities $p(a,b|x,y)$ in Eqn. (2.11) satisfy
certain linear inequalities which are referred to Bell inequalities. For some
experiments, for example with a pair of entangled qubits, the local hidden
variable model cannot exist and Bell inequalities are violated.
Entanglement is defined as before when a bipartite state cannot be written in
terms of a convex combination of the tensor product of pure states
$\rho_{AB}=\sum_{i}p_{i}\rho_{i}^{A}\otimes\rho_{i}^{B},$ (2.12)
otherwise the state is separable. General measurements are represented by
positive operator-valued measures (POVMs). A set of POVMs $\\{E_{a|x}\\}$
satisfying $E_{a|x}>0$, $E_{a|x}^{{\dagger}}=E_{a|x}$, and
$\sum_{a}E_{a|x}=\mathbbm{1}$ give the probability of gaining the result $a$
in the state $\rho$ as $p(a)=\Tr(\rho E_{a|x})$. For a separable state, the
probability for the measurements $E_{a|x}$ and $E_{b|y}$ is given as
$p(a,b|x,y)=\sum_{i}p_{i}\Tr(E_{a|x}\rho_{i}^{A})\Tr(E_{b|y}\rho_{i}^{B}).$
(2.13)
It is easy to see that it belongs to the local hidden variable models and
separable states are a convex subset of the local hidden variable states.
Quantum steering in a sense lies in-between of entanglement and Bell
nonlocality, where Alice is described by a classical hidden variable and Bob
makes a quantum mechanical measurement. That is, the probability is given by
$p(a,b|x,y)=\int\textrm{d}\lambda
p(\lambda)p(a|x,\lambda)\Tr(E_{b|y}\sigma_{\lambda}^{B}).$ (2.14)
In the steering scenario, Alice and Bob share a bipartite quantum state
$\rho_{AB}$. For each measurement $x$ and the corresponding outcome $a$ in
Alice’s lab, Bob has the conditional state $\rho_{a|x}$ such that
$\rho_{B}=\sum_{a}\rho_{a|x}$ is independent of Alice’s choice for the
measurement $x$. The state $\rho_{AB}$ is said to be unsteerable or have a
local hidden state model if there is a representation from some parameter
$\lambda$ that
$\rho_{a|x}=\int\textrm{d}\lambda p(\lambda)p(a|x,\lambda)\sigma_{\lambda};$
(2.15)
otherwise the state is steerable. In Eqn. (2.14), the probability can be
rewritten as
$p(a,b|x,y)=\Tr(E_{b|y}\rho_{a|x}),\quad\rho_{a|x}=\int\textrm{d}\lambda
p(\lambda)p(a|x,\lambda)\sigma_{\lambda}^{B};$ (2.16)
thus, the local hidden state model exists.
We can summarise that, the states that have a local hidden variable model and
do not violate Bell inequalities form the convex set of LHV states; the states
that have a local hidden state model and are unsteerable form the convex
subset of LHV states, denoted by LHS states; the separable states form the
convex subset of LHS states.
##### Discord and related measures
Entanglement is crucial in distinguishing quantum correlations from classical
ones; however, it cannot represent for all non-classical correlations, and
even separable states contain correlations which are not fully classical [21].
One of these non-classical correlation measures is quantum discord [22, 23].
Suppose a POVM measurement $E_{a}$ is made on the subsystem $A$ of the initial
state $\rho_{AB}$. For the outcome $a$, Alice observes it with the probability
$p(a)=\Tr(E_{a}\rho_{AB})$ and Bob gains the conditional state
$\rho_{B|a}=\Tr_{A}(E_{a}\rho_{AB})/p(a)$. The conditional entropy then has a
classical-quantum version of definition as
$S(B|\\{E_{a}\\})=\sum_{a}p_{a}S(\rho_{B|a})$. The quantum discord is defined
as
$J(B|A)=\max_{\\{E_{a}\\}}S(B)-S(B|\\{E_{a}\\}).$ (2.17)
It is non-symmetric, non-negative, invariant under local unitary
transformations, and vanishes if and only if the state is classical quantum.
Other measures of quantum correlations include quantum deficit [24],
distillable common randomness [25], measurement-induced disturbance [26],
symmetric discord [27], relative entropy of discord and dissonance [28], and
so on.
##### Hierarchy of spatial correlations
Now we introduce the hierarchy of quantum correlations based on Ref. [29]:
$C_{c}\subseteq C_{q}\subseteq C_{qs}\subseteq C_{qa}\subseteq C_{qc}.$ (2.18)
Here all the sets are convex, and $C_{c}$, $C_{qa}$ are closed. Consider a
two-player non-local game $\mathcal{G}$ with finite input sets
$\mathcal{I}_{A}$, $\mathcal{I}_{B}$, finite output sets $\mathcal{O}_{A}$,
$\mathcal{O}_{B}$ and a function
$V:\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}\rightarrow\\{0,1\\}$.
Suppose the two players, Alice and Bob, after given the inputs
$x\in\mathcal{I}_{A}$ and $y\in\mathcal{I}_{B}$ respectively, cannot
communicate with each other, and return outputs $a\in\mathcal{O}_{A}$ and
$b\in\mathcal{O}_{B}$ respectively. The players win if $V(a,b|x,y)=1$, or lose
if $V(a,b|x,y)=0$. The probabilities $p(a,b|x,y)$ that Alice and Bob return
output $a\in\mathcal{O}_{A}$ and $b\in\mathcal{O}_{B}$ given inputs
$x\in\mathcal{I}_{A}$ and $y\in\mathcal{I}_{B}$ form a collection
$\\{p(a,b|x,y)\\}\subset\mathbb{R}^{\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}}$
called a correlation matrix.
A correlation matrix $\\{p(a,b|x,y)\\}$ is said to be classical under
classical strategies with classical shared randomness. Specifically,
$p(a,b|x,y)=\sum_{i=1}^{k}\lambda_{i}p_{i}(a|x)q_{i}(b|y)\ \text{for
all}(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B},$
(2.19)
for a probability distribution $\\{\lambda_{i}\\}$ on $\\{1,\dots,k\\}$,
probability distributions $\\{p_{i}(a|x)\\}$ on $\mathcal{O}_{A}$ for each
$1\leq i\leq k$ and $x\in\mathcal{I}_{A}$, and probability distributions
$\\{q_{i}(b|y)\\}$ on $\mathcal{O}_{B}$ for each $1\leq i\leq k$ and
$y\in\mathcal{I}_{B}$. Then the set of classical correlation matrices is
denoted by
$C_{c}(\mathcal{O}_{A},\mathcal{O}_{B},\mathcal{I}_{A},\mathcal{I}_{B})$ or
$C_{c}$.
A quantum correlation matrix is constructed under
$p(a,b|x,y)=\bra{\psi}M_{a}^{x}\otimes N_{b}^{y}\ket{\psi}\text{for
all}(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}$
(2.20)
for a quantum state $\ket{\psi}$ on the finite-dimensional Hilbert spaces
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, projective measurements
$\\{M_{a}^{x}\\}_{a\in\mathcal{O}_{A}}$ on $\mathcal{H}_{A}$ for every
$x\in\mathcal{I}_{A}$, and projective measurements
$\\{N_{b}^{y}\\}_{b\in\mathcal{O}_{B}}$ on $\mathcal{H}_{B}$ for every
$y\in\mathcal{I}_{B}$. Then the set of quantum correlation matrices is denoted
by $C_{q}(\mathcal{O}_{A},\mathcal{O}_{B},\mathcal{I}_{A},\mathcal{I}_{B})$ or
$C_{q}$.
If we allow Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ to be
infinite-dimensional, we have another set of correlation matrices denoted by
$C_{qs}$. If we take finite-dimensional correlations to the limit, then the
closure of $C_{q}$ constitutes a new set of correlation matrices denoted by
$C_{qa}$. It is known that $C_{qs}\subseteq C_{qa}$ and $C_{qa}$ is also the
closure of $C_{qs}$ [30].
It is easy to see that $C_{c}\subseteq C_{q}\subseteq C_{qs}\subseteq C_{qa}$.
Bell’s theorem [19] states that $C_{c}\neq C_{q}$. Slofstra [29] suggests that
$C_{q}$ and $C_{qs}$ are not closed; that is, $C_{q}\neq C_{qa}$ and
$C_{qs}\neq C_{qa}$.
We can even drop the restriction on tensor product structures and define
correlation matrices in terms of commuting operators. Then
$p(a,b|x,y)=\bra{\psi}M_{a}^{x}N_{b}^{y}\ket{\psi}\text{for
all}(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}$
(2.21)
for $M_{a}^{x}N_{b}^{y}=N_{b}^{y}M_{a}^{x}$ with
$\\{M_{a}^{x}\\}_{a\in\mathcal{O}_{A}}$ on $\mathcal{H}$ for every
$x\in\mathcal{I}_{A}$, and projective measurements
$\\{N_{b}^{y}\\}_{b\in\mathcal{O}_{B}}$ on $\mathcal{H}$ for every
$y\in\mathcal{I}_{B}$. This set of correlation matrices is denoted by
$C_{qc}$. To determine whether $C_{qc}$ is equal to $C_{q}$, $C_{qs}$ or
$C_{qa}$ is known as Tsirelson’s problem [31, 32]. It is proven that
$C_{qs}\neq C_{qc}$ [33]. A recent result further solves the problem and
concludes that $C_{qa}\neq C_{qc}$ [34].
#### 2.2.3 Multipartite quantum correlations
As a direct generalisation of bipartite separability, full separability [35]
is defined as $n$-separability of $n$ systems $A_{1}\dots A_{n}$:
$\rho_{A_{1}\dots
A_{n}}=\sum_{i=1}^{k}p_{i}\rho_{A_{1}}^{i}\otimes\cdots\otimes\rho_{A_{n}}^{i}$
where $k\leq\text{dim}\mathcal{H}_{A_{1}\dots A_{n}}^{2}$ is known as the
Caratheodory bound. Multipartite quantum correlations are also defined in
terms of subsystems and partitions. Consider a quantum state $\ket{\psi}$ of
$n$ subsystems. If it can be written as the tensor product of $m$ disjoint
subsets $\ket{\psi}=\bigotimes_{i=1}^{m}\ket{\psi_{i}}$, then it is said to be
$m$-separable $(2\leq m\leq n)$. $\ket{\psi}$ is said to be $k$-producible if
the largest subset for $\ket{\psi_{i}}$ has at most $k$ subsystems. For a
mixed state $\rho$, it is $m$-separable or $k$-producible if it has a
decomposition of $m$-separable or $k$-producible pure states [36]. In
particular, the state $\rho_{A_{1}\dots A_{m}}$ is semiseparable if and only
if it is separable under all $1-(m-1)$ partitions:
$\\{I_{1}=\\{k\\},I_{2}=\\{1,\dots,k-1,k+1,\dots,m\\}\\},1\leq k\leq m$.
Multipartite quantum correlations have much more rich structures and a full
characterisation for multipartite quantum correlations remains as an important
open problem.
### 2.3 Quantum correlations in time
In this section we introduce quantum correlations defined in quantum field
theory and explore further possibilities for correlations which are defined in
an even-handed manner for space and time. We also aim towards a unified
approach for quantum correlations in space and time.
#### 2.3.1 Correlations in quantum field theory
In quantum field theory [37], the $n$-point correlation function for the field
operator $\phi(x)$ is usually defined in the ground state $\ket{\Omega}$ as
$C_{n}(x_{1},x_{2},\dots,x_{n})=\bra{\Omega}\mathcal{T}\phi(x_{1})\phi(x_{2})\cdots\phi(x_{n})\ket{\Omega}$
(2.22)
where
$\mathcal{T}O_{1}(t_{1})O_{2}(t_{2})=\theta(t_{1}-t_{2})O_{1}(t_{1})O_{2}(t_{2})+\theta(t_{2}-t_{1})O_{2}(t_{2})O_{1}(t_{1})$
is the time-ordering operator. For example, consider a perturbation for
interacting fields with the Hamiltonian divided by $H=H_{0}+H_{int}$. With the
unitary operator $U(t,t_{0})=e^{iH_{0}(t-t_{0})}e^{-iH(t-t_{0})}$, the
Schrödinger equation is written equivalently as $i\frac{\partial}{\partial
t}U(t,t_{0})=H_{I}(t)U(t,t_{0})$ where
$H_{I}(t)=e^{iH_{0}(t-t_{0})}H_{int}e^{-iH_{0}(t-t_{0})}$. Then the two-point
correlation function is given as
$\bra{\Omega}\mathcal{T}\\{\phi(x)\phi(y)\\}\ket{\Omega}=\lim_{T\rightarrow\infty(1-i\epsilon)}\frac{\bra{0}\mathcal{T}\\{\phi_{I}(x)\phi_{I}(y)\exp[-i\int_{-T}^{T}\mathrm{d}tH_{I}(t)]\\}\ket{0}}{\bra{0}\mathcal{T}\\{\exp[-i\int_{-T}^{T}\mathrm{d}tH_{I}(t)]\\}\ket{0}},$
(2.23)
where $\phi_{I}(x)$ is defined through
$\phi(x)=U^{{\dagger}}(t,t_{0})\phi_{I}(x)U(t,t_{0})$.
#### 2.3.2 Further possibility for temporal correlations
As we can see from statistical mechanics and field theory, correlations are
defined in terms of thermal states or ground states for the background on
spacetime. Here we are thinking of possibilities of generalising temporal
correlations beyond thermal states or ground states.
One possibility comes from autocorrelation functions. In statistics, the
autocorrelation of a real or complex random process $\\{X(t)\\}$ is defined as
the expectation value of the product of the values at two different times
[38]:
$r_{XX}(t_{1},t_{2})=E[X(t_{1})X^{*}(t_{2})].$ (2.24)
Here the complex conjugate guarantees the product to be the square of the
magnitude of the second momentum for $X(t)$ when $t_{1}=t_{2}$. It is possible
to take the expectation values of the product of measurement results for
observables at different times to gain the temporal correlations.
Another choice may be to define quantum states in time. Quantum states are
defined across the whole of space but at one instant of time. We associate a
Hilbert space for each spatially separated system and assign the tensor
product structure; it is possible to associate a Hilbert space for each time
and define quantum states in time. Then we may adopt the usual rule for
calculating spatial correlations to analyse temporal correlations.
#### 2.3.3 Towards a unified approach for quantum correlations in space and
time
In the previous subsection, we have discussed further possibilities of
temporal correlations; here we are looking for a unified approach for quantum
correlations in space and time. Following the discussion on quantum states in
time, we may define quantum states across spacetime. We assume that the tensor
product structure should work for Hilbert spaces at different times. Based on
the Hilbert spaces across spacetime, we may define spacetime quantum states
and unify temporal correlations and spatial correlations in the spacetime
framework. This proposal has already been achieved in the pseudo-density
matrix formalism as we are about to introduce in the next section.
### 2.4 Pseudo-density matrix formalism
In this section, we introduce the pseudo-density matrix formalism [39, 40, 41,
42, 43, 44] as a unified approach for quantum correlations in space and time.
We review the definition of pseudo-density matrices for finite dimensions,
present their properties, take bipartite correlations as an example to
illustrate how the formalism unifies correlations in space-time.
#### 2.4.1 Definition and properties
The pseudo-density matrix formulation is a finite-dimensional quantum-
mechanical formalism which aims to treat space and time on an equal footing.
In general, this formulation defines an event via making a measurement in
space-time and is built upon correlations from measurement results; thus, it
treats temporal correlations just as spatial correlations and unifies spatio-
temporal correlations. As a price to pay, pseudo-density matrices may not be
positive semi-definite.
An $n$-qubit density matrix can be expanded by Pauli operators in terms of
Pauli correlations which are the expectation values of these Pauli operators.
In spacetime, instead of considering $n$ qubits, let us pick up $n$ events,
where a single-qubit Pauli operator is measured for each. Then, the pseudo-
density matrix is defined as
$\hat{R}\equiv\frac{1}{2^{n}}\sum_{i_{1}=0}^{3}...\sum_{i_{n}=0}^{3}\langle\\{\sigma_{i_{j}}\\}_{j=1}^{n}\rangle\bigotimes_{j=1}^{n}\sigma_{i_{j}},$
(2.25)
where $\langle\\{\sigma_{i_{j}}\\}_{j=1}^{n}\rangle$ is the expectation value
of the product of these measurement results for a particular choice of events
with operators $\\{\sigma_{i_{j}}\\}_{j=1}^{n}$.
Similar to a density matrix, it is Hermitian and unit-trace, but not positive
semi-definite as we mentioned before. If the measurements are space-like
separated or local systems evolve independently, the pseudo-density matrix
will reduce to a standard density matrix. Otherwise, for example if
measurements are made in time, the pseudo-density matrix may have a negative
eigenvalue. For example, we take a single qubit in the state $\ket{0}$ at the
initial time and assume the identity evolution between two times. The
correlations are 1 for $\langle\\{I,I\\}\rangle$, $\langle\\{X,X\\}\rangle$,
$\langle\\{Y,Y\\}\rangle$, $\langle\\{Z,Z\\}\rangle$,
$\langle\\{Z,I\\}\rangle$, and $\langle\\{I,Z\\}\rangle$ while all others are
given as 0. Then we construct the pseudo-density matrix for two times as
$R=\begin{bmatrix}1&0&0&0\\\ 0&0&\frac{1}{2}&0\\\ 0&\frac{1}{2}&0&0\\\
0&0&0&0\end{bmatrix},$ (2.26)
with eigenvalues $\\{-\frac{1}{2},0,\frac{1}{2},1\\}$. Thus it is not positive
semi-definite and encodes temporal correlations as a spacetime density matrix.
Furthermore, the single-time marginal of the pseudo-density matrix is given as
the density matrix at that particular time under the partial trace. For any
set of operators $O_{i}$ with eigenvalues $\pm 1$, the expectation values of
the measurement outcomes is given as
$\langle\\{O_{i}\\}_{i=1}^{m}\rangle=\Tr\left[\left(\bigotimes_{i=1}^{m}O_{i}\right)R\right].$
(2.27)
Here $O_{i}$ may be an operator measured on several qubits at the same time.
This suggests that any complete basis of operators with eigenvalues $\pm 1$
has the proper operational meaning for correlations of operators, and thus
serves as a good alternative basis for pseudo-density matrices. Note that
pseudo-density matrices are defined in an operational manner via the
measurements of correlations; a strict mathematical characterisation does not
exist yet. A full investigation on the all possible basis choices remains an
open problem. In the following chapters, we will present several
generalisations of pseudo-density matrices with different measurement basis.
To understand causal relationships, a measure for causal correlations called
causality monotone is proposed, similar to the entanglement monotone. This
causality monotone $f(R)$ is defined when it satisfies the following criteria:
(1) $f(R)\geq 0$. In particular, $f(R)=0$ if $R$ is positive semi-definite;
$f(R)=1$ for a single-qubit closed system at two times.
(2) $f(R)$ is invariant under local unitary operations (thus under a local
change of basis).
(3) $f(R)$ is non-increasing under local operations.
(4) $\sum_{i}p_{i}f(R_{i})\geq f(\sum_{i}p_{i}R_{i})$.
#### 2.4.2 Characterisation of bipartite correlations in space-time
In this subsection we introduce the work on the characterisation for bipartite
correlations in space-time [40]. Here the spatial correlations are given by
all possible two-qubit density matrices, and compared with temporal
correlations in a single-qubit pseudo-density matrix at two times under the
unitary evolution. There is a reflection between spatial correlations and
temporal correlations in the $\langle XX\rangle-\langle ZZ\rangle$ plane of
the correlation space $\\{\langle\sigma_{i}\sigma_{i}\rangle\\}_{i=1}^{3}$.
The spatial correlations given in terms of Pauli measurements are
characterised in Ref. [45]. The two-point correlations
$t_{mn}=\Tr(\rho\sigma_{m}\otimes\sigma_{n})$ form a real matrix $T$. Up to a
unitary rotation, $t_{mn}$ is full characterised by its diagonal terms
$t_{11}$, $t_{22}$, $t_{33}$. For any two-qubit density matrix $\rho$, the $T$
matrix belongs to the tetrahedron $\mathcal{T}_{s}$ with vertices
$\mathbf{t}=(t_{11},t_{22},t_{33})$ given as $(-1,-1,-1)$, $(-1,1,1)$,
$(1,-1,1)$, $(1,1,-1)$. These four vertices correspond to four Bell states.
Now we consider its temporal analog, a pseudo-density matrix for a single
qubit at two times under the unitary evolution. This $T$ matrix is represented
in another tetrahedron $\mathcal{T}_{t}$ with vertices
$\mathbf{t}=(t_{11},t_{22},t_{33})$ given as $(1,1,1)$, $(-1,-1,1)$,
$(-1,1,-1)$, $(1,-1,-1)$. Fig. 2.1 illustrates these relations. On the left,
blue and red tetrahedrons $\mathcal{T}_{s}$ and $\mathcal{T}_{t}$ show all
possible bipartite spatial and temporal correlations. The right figure view
these correlations from the $(-1,-1,-1)-(1,1,1)$ direction. It is easy to see
that the intersection of the spatial and temporal correlations is given by the
purple octahedron representing separable states.
Figure 2.1: Geometrical representation for bipartite correlations in space and
time. The left figure represents the spatial and temporal correlations in the
blue and red tetrahedrons, respectively, in 3D modelling of the correlation
space $\\{\langle\sigma_{i}\sigma_{i}\rangle\\}_{i=1}^{3}$. The right figure
views the correlation from the $(-1,-1,-1)-(1,1,1)$ direction. The
intersection of the spatial and temporal correlations is given by the purple
octahedron representing separable states. Thanks to Zhikuan Zhao for providing
his original figure in Ref. [40].
Similarly, temporal correlations of the single-qubit initial state
$\frac{I}{2}$ under arbitrary CPTP maps can also be mapped back to bipartite
spatial correlations under the partial transpose and given as Fig. 2.1. We
will see the importance of partial transposition in the continuous-variable
generalisation as well. In general, for all possible quantum channel
evolution, the set of temporal correlations strictly contains
$\mathcal{T}_{s}$ and is convex on each edge; that is, in the bipartite case
the set of all possible temporal correlations is larger than the set of all
possible spatial correlations (entanglement).
## Chapter 3 Generalisation of pseudo-density matrix formulation
### 3.1 Introduction
In this chapter we follow the paradigm of the pseudo-density matrix [39],
which is understood as a particular spacetime state. The pseudo-density matrix
uses only a single Hilbert space for each spacetime event defined in terms of
making measurements in spacetime; as a price to pay, it may not be positive
semi-definite. We take the view from Wigner that “the function of quantum
mechanics is to give statistical correlations between the outcomes of
successive observations [46],” and then construct the spacetime states in
continuous variables from the observation of measurements of modes and
generalise the pseudo-density matrix formulation. We give six possible
definitions for spacetime density matrices in continuous variables or
spacetime Wigner functions built upon measurement correlations. The choice of
measurements to make is a major issue here. They should form a complete basis
to extract full information of states in spacetime. One natural choice is the
quadratures, which turn out to be efficient in analysing Gaussian states.
Analogous to the Pauli operators as the basis for a multi-qubit system,
another option in continuous variables would be the displacement operators;
however, they are anti-Hermitian. Instead, we apply their Fourier transform
$T(\alpha)$, twice of displaced parity operators, to the representation of
general Wigner functions. We also initialise the discussion of defining
spacetime states from position measurements and weak measurements based on
previous work on successive measurements [47, 48, 49, 50], motivated by
linking the pseudo-density matrix formalism to the path integral formalism. We
further show that these definitions for continuous variables satisfy natural
desiderata, such as those listed in Ref. [51] for quantum joint states over
time, as well as additional criteria for spacetime states. An experimental
proposal for tomography is presented as well to show how these definitions are
operationally meaningful.
This chapter is based on Ref. [42]. It proceeds as follows. First we define
spacetime Gaussian states via the characterisation of the first two
statistical moments and show that the temporal statistics are different but
related to the spatial statistics. Next we define the spacetime Wigner
function representation and the corresponding spacetime density matrix, and
desirable properties are satisfied analogous to the spatial case. We further
discuss the possibility of defining spacetime states via position measurements
and weak measurements. A tomographical scheme is suggested for experiments.
Then we comment on the pseudo-density matrix paradigm in terms of its
properties and basic assumptions, and show its relation with the Choi-
Jamiołkowski isomorphism and the path integral formalism. We also set up
desirable properties for spacetime quantum states and check whether all the
above definitions satisfy them or not.
### 3.2 Gaussian generalisation of pseudo-density matrix
In this section we review Gaussian representation in continuous variables,
define spacetime Gaussian states motivated from the pseudo-density matrix
formalism, analyse simple examples and the differences and similarities of
spatial and temporal Gaussian states.
#### 3.2.1 Preliminaries
Gaussian states are continuous-variable states with a representation in terms
of Gaussian functions [52, 53, 54]. The first two statistical moments of the
quantum states, the mean value and the covariance matrix, fully characterise
Gaussian states, just as normal Gaussian functions in statistics. The mean
value $\bm{d}$, is defined as the expectation value of the $N$-mode quadrature
field operators $\\{\hat{q}_{k},\hat{p}_{k}\\}_{k=1}^{N}$ arranged in
$\bm{\hat{x}}=(\hat{q}_{1},\hat{p}_{1},\cdots,\hat{q}_{N},\hat{p}_{N})^{T}$,
that is,
$d_{j}=\langle\hat{x}_{j}\rangle_{\rho}\equiv\Tr(\hat{x}_{j}\hat{\rho}),$
(3.1)
for the Gaussian state $\hat{\rho}$. The elements in the covariance matrix
$\bm{\sigma}$ are defined as
$\sigma_{ij}=\langle\hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i}\rangle_{\rho}-2\langle\hat{x}_{i}\rangle_{\rho}\langle\hat{x}_{j}\rangle_{\rho}.$
(3.2)
The covariance matrix $\bm{\sigma}$ is real and symmetric, and satisfies the
uncertainty principle [55] as (note that in this thesis we set $\hbar=1$)
$\bm{\sigma}+i\bm{\Omega}\geq 0,$ (3.3)
in which the elements of $\bm{\Omega}$ is given by commutation relations as
$[\hat{x}_{i},\hat{x}_{j}]=i\hbar\Omega_{ij},$ (3.4)
thus $\bm{\Omega}$ is the $2N\times 2N$ matrix
$\bm{\Omega}\equiv\bigoplus_{k=1}^{N}\bm{\omega}=\begin{bmatrix}\bm{\omega}&\
&\ \\\ \ &\ddots&\ \\\ \ &\ &\bm{\omega}\end{bmatrix}\ \ \text{and}\ \
\bm{\omega}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}.$ (3.5)
This condition also implies the positive definiteness of $\bm{\sigma}$, i.e.,
$\bm{\sigma}>0$. Then we introduce the Wigner representation for Gaussian
states. The Wigner function originally introduced in Ref. [56] is a quasi-
probability distribution in the phase space and the characteristic function
can be given via the Fourier transform of the Wigner function. By definition,
the Wigner representation of a Gaussian state is Gaussian, that is, the
characteristic function and the Wigner function [54] are given by
$\displaystyle\chi(\bm{\xi})$
$\displaystyle=\exp[-\frac{1}{4}\bm{\xi}^{T}(\bm{\Omega}\bm{\sigma}\bm{\Omega}^{T})\bm{\xi}-i(\bm{\Omega}\bm{d})^{T}\bm{\xi}],$
(3.6) $\displaystyle W(\bm{x})$
$\displaystyle=\frac{\exp[-(\bm{x}-\bm{d})^{T}\bm{\sigma}^{-1}(\bm{x}-\bm{d})]}{\pi^{N}\sqrt{\det\bm{\sigma}}},$
(3.7)
where $\bm{\xi},\bm{x}\in\mathbb{R}^{2N}$.
Typical examples of Gaussian states include vacuum states, thermal states and
two-mode squeezed states. A one-mode vacuum state $\ket{0}$ has zero mean
values and the covariance matrix as the $2\times 2$ identity matrix $I$. A
one-mode thermal state with the mean number of photons $\bar{n}$ [52] or the
inverse temperature $\beta$ [53] is defined equivalently as
$\hat{\rho}^{th}(\bar{n})=\sum_{n=0}^{+\infty}\frac{\bar{n}^{n}}{(\bar{n}+1)^{n+1}}\ket{n}\bra{n},$
(3.8)
or
$\hat{\rho}^{th}(\beta)=(1-e^{-\beta})\exp(-\beta\hat{a}^{{\dagger}}\hat{a}),$
(3.9)
where $\hat{a},\hat{a}^{{\dagger}}$ are annihilation and creation operators.
Note that $\beta=-\ln\frac{\bar{n}}{1+\bar{n}}$. The thermal state has zero
mean values and the covariance matrix proportional to the identity as
$(2\bar{n}+1)I$ or $\frac{1+e^{-\beta}}{1-e^{-\beta}}I$, respectively to the
above two definitions. A two-mode squeezed state [53] is generated from the
vacuum state $\ket{0}$ by acting with a two-mode squeezing operator which is
defined as
$\hat{S}_{2}(\xi)=\exp[\xi\hat{a}^{{\dagger}}\hat{b}^{{\dagger}}-\xi^{*}\hat{a}\hat{b}],$
(3.10)
where $\hat{a}^{{\dagger}}$ and $\hat{b}^{{\dagger}}$ ($\hat{a}$ and
$\hat{b}$) are creation (annihilation) operators of the two modes, $\xi$ is a
complex number where $r=|\xi|$ and $\xi=re^{i\psi}$. Then the two-mode
squeezed vacuum state is given as $\hat{S}_{2}(\xi)\ket{00}$. From here we
omit the phase $\psi$ for simplicity. A two-mode squeezed state with a real
squeezed parameter $r$, known as the Einstein-Podolsky-Rosen (EPR) state
$\hat{\rho}^{epr}(r)=\hat{S}_{2}(r)\ket{00}\bra{00}\hat{S}_{2}^{{\dagger}}(r)$,
has zero mean values and the covariance matrix as
$\bm{\sigma}_{tmss}=\begin{bmatrix}\cosh 2r&0&\sinh 2r&0\\\ 0&\cosh
2r&0&-\sinh 2r\\\ \sinh 2r&0&\cosh 2r&0\\\ 0&-\sinh 2r&0&\cosh
2r\end{bmatrix}.$ (3.11)
Taking the partial trace of the two-mode squeezed state, we get the one-mode
thermal state:
$\Tr_{b}[\hat{\rho}^{epr}(r)]=\hat{\rho}_{a}^{th}(\bar{n})=\hat{\rho}_{a}^{th}(\beta),$
where $\bar{n}=\sinh^{2}r$ or $\beta=-\ln\tanh^{2}r$ [53].
#### 3.2.2 Spacetime Gaussian states
Instead of Gaussian states at a specific time as given before, now we define
Gaussian states in spacetime. Suppose that we are given data associated with
single-mode measurements labelled by some index $k=1,\dots,N$. We will use the
same recipe, given the data, to create the spacetime state, whether these
measurements are made on the same mode at different times or whether they are
made on separate modes, or more generally on both different modes and
different times. This follows the pseudo-density matrix paradigm, in which one
wishes to use the same quantum density matrix formalism for all the cases.
Assume that we are given enough data to characterise a Gaussian state fully,
i.e., the mean value and the covariance matrix. The expectation values of all
quadratures are defined as before. The correlation
$\langle\\{\hat{x}_{i},\hat{x}_{j}\\}\rangle$ of two quadratures $\hat{x}_{i}$
and $\hat{x}_{j}$ for two events is defined to be the expectation value for
the product of measurement results on these quadratures. Particularly for
measurements or events at the same time, this correlation is defined via a
symmetric ordering of two quadrature operators. Then the covariance is defined
to be related to this correlation and corresponding mean values as the spatial
covariance.
###### Definition 1.
We define the Gaussian spacetime state in terms of measurement statistics as
being (i) a vector $\bm{d}$ of 2N mean values, with j-th entry
$d_{j}=\langle\hat{x}_{j}\rangle_{\rho}=\Tr(\hat{x}_{j}\rho).$ (3.12)
and (ii) a covariance matrix $\bm{\sigma}$ with entries as
$\sigma_{ij}=2\langle\\{\hat{x}_{i},\hat{x}_{j}\\}\rangle_{\rho}-2\langle\hat{x}_{i}\rangle_{\rho}\langle\hat{x}_{j}\rangle_{\rho}$
(3.13)
where $\langle\\{\hat{x}_{i},\hat{x}_{j}\\}\rangle_{\rho}$ is the expectation
value for the product of measurement results; specifically
$\\{\hat{x}_{i},\hat{x}_{j}\\}=\frac{1}{2}(\hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i})$
for measurements at the same time. To get the reduced state associated with
the mode $k$ one picks out the entries in the $\bm{d}$ and $\bm{\sigma}$
associated with the mode $k$ to create the corresponding Gaussian state of
that mode.
According to the above definition of reduced states, it is easy to see that
the single time marginal is identical to the spatial Gaussian state at that
particular time. This is because the mean values and covariances at one time
in the spacetime case are defined as the same as them in the spatial case.
#### 3.2.3 Example: vacuum state at two times
For a simple example, we take a vacuum state at two times with the identity
evolution in between. A vacuum state is $\ket{0}$ at the initial time $t_{1}$
and under the identity evolution it remains $\ket{0}$ at a later time $t_{2}$.
Remember that a one-mode vacuum state $\ket{0}$ is a Gaussian state with zero
means and the covariance matrix as the identity as stated before. That is, at
a single time $t_{1}$ or $t_{2}$,
$\displaystyle\langle\hat{q}_{1}\rangle=\langle\hat{p}_{1}\rangle=\langle\hat{q}_{2}\rangle=\langle\hat{p}_{2}\rangle=0;$
(3.14)
$\displaystyle\langle\hat{q}_{1}\hat{q}_{1}\rangle=\langle\hat{p}_{1}\hat{p}_{1}\rangle=\langle\hat{q}_{2}\hat{q}_{2}\rangle=\langle\hat{p}_{2}\hat{p}_{2}\rangle=\frac{1}{2},$
$\displaystyle\langle\hat{q}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{q}_{1}\rangle=\langle\hat{q}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{q}_{2}\rangle=0.$
(3.15)
For measurements at both time $t_{1}$ and time $t_{2}$,
$\displaystyle\langle\\{\hat{q}_{1},\hat{q}_{2}\\}\rangle=\langle\\{\hat{q}_{2},\hat{q}_{1}\\}\rangle=\iint\textrm{d}q_{1}\textrm{d}q_{2}q_{1}q_{2}\Tr(\ket{q_{1}}\bra{q_{1}}\ket{0}\bra{0})\Tr(\ket{q_{2}}\bra{q_{2}}\ket{q_{1}}\bra{q_{1}})=\langle\hat{q}_{1}\hat{q}_{1}\rangle=\frac{1}{2},$
$\displaystyle\langle\\{\hat{q}_{1},\hat{p}_{2}\\}\rangle=\langle\\{\hat{p}_{2},\hat{q}_{1}\\}\rangle=\iint\textrm{d}q_{1}\textrm{d}p_{2}q_{1}p_{2}\Tr(\ket{q_{1}}\bra{q_{1}}\ket{0}\bra{0})\Tr(\ket{p_{2}}\bra{p_{2}}\ket{q_{1}}\bra{q_{1}})=0,$
$\displaystyle\langle\\{\hat{p}_{1},\hat{p}_{2}\\}\rangle=\langle\\{\hat{p}_{2},\hat{p}_{1}\\}\rangle=\iint\textrm{d}p_{1}\textrm{d}p_{2}p_{1}p_{2}\Tr(\ket{p_{1}}\bra{p_{1}}\ket{0}\bra{0})\Tr(\ket{p_{2}}\bra{p_{2}}\ket{p_{1}}\bra{p_{1}})=\langle\hat{p}_{1}\hat{p}_{1}\rangle=\frac{1}{2},$
$\displaystyle\langle\\{\hat{p}_{1},\hat{q}_{2}\\}\rangle=\langle\\{\hat{q}_{2},\hat{p}_{1}\\}\rangle=\iint\textrm{d}p_{1}\textrm{d}q_{2}p_{1}q_{2}\Tr(\ket{p_{1}}\bra{p_{1}}\ket{0}\bra{0})\Tr(\ket{q_{2}}\bra{q_{2}}\ket{p_{1}}\bra{p_{1}})=0.$
(3.16)
According to the definition given in Eqn. (3.12, 3.13), the mean values are 0
and the covariance matrix in time is
$\bm{\sigma}_{vs}=\begin{bmatrix}1&0&1&0\\\ 0&1&0&1\\\ 1&0&1&0\\\
0&1&0&1\end{bmatrix}.$ (3.17)
Note that $\bm{\sigma}_{vs}$ is not positive definite and violates the
uncertainty principle of Eqn. (3.3). Thus it is an invalid _spatial_
covariance matrix. This illustrates how the covariance statistics for spatial
and temporal matrices are different, just as bipartite Pauli correlations in
spatial and temporal case are different [45, 40], which makes the study of
temporal statistics particularly interesting.
Since the determinant of the covariance matrix is 0, it is impossible to get
the inverse of the covariance matrix directly to obtain the temporal Wigner
function from Eqn. (3.7). From the mean values and the covariance matrix, we
gain the temporal characteristic function from Eqn. (3.6) as
$\chi(q_{1},p_{1},q_{2},p_{2})=\exp(-p_{1}^{2}-2p_{1}p_{2}-p_{2}^{2}-q_{1}^{2}-2q_{1}q_{2}-q_{2}^{2}),$
(3.18)
Via the Fourier transform, the temporal Wigner function is given as
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\frac{1}{4\pi}\exp(-p_{1}^{2}/4-q_{1}^{2}/4)\delta(-p_{1}+p_{2})\delta(-q_{1}+q_{2}),$
(3.19)
It is easy to check that the temporal Wigner function is normalised to 1:
$\iiiint\mathcal{W}(q_{1},p_{1},q_{2},p_{2})\mathrm{d}q_{1}\mathrm{d}p_{1}\mathrm{d}q_{2}\mathrm{d}p_{2}=1.$
(3.20)
However, if we consider the condition that the Wigner function of a pure state
is bounded by $\pm\frac{2}{h}$, then this temporal Wigner function is invalid.
This may be taken as the temporal signature of the Wigner function.
#### 3.2.4 Spatial vs temporal Gaussian states
Now compare spatial Gaussian states and temporal Gaussian states via a simple
two-mode example. In general, there is not much meaning to comparing an
arbitrary spatial state with an arbitrary temporal state. We need to pick up
the spatial state carefully and figure out its temporal analog. Remember in
the preliminaries we mentioned that taking the partial transpose of a two-mode
squeezed state (or to say, the EPR state), we gain a one-mode thermal state.
Hence, the temporal analog of the two-mode squeezed state will be the one-mode
thermal state at two times. Take the one-mode thermal state as the initial
state at $t_{A}$ and further assume that the evolution between $t_{A}$ and
$t_{B}$ corresponds to the identity operator. The mean values are zero. The
covariance matrix in time becomes
$\bm{\sigma}_{omts}=\begin{bmatrix}\cosh 2r&0&\cosh 2r&0\\\ 0&\cosh 2r&0&\cosh
2r\\\ \cosh 2r&0&\cosh 2r&0\\\ 0&\cosh 2r&0&\cosh 2r\end{bmatrix}.$ (3.21)
Note that again $\bm{\sigma}_{omts}$ is not positive definite and violates the
uncertainty principle.
Compare $\bm{\sigma}_{omts}$ with its spatial analog, the covariance matrix of
the two-mode squeezed state $\bm{\sigma}_{tmss}$. Under the high temperature
approximation as $\beta\rightarrow 0$, $\tanh r\approx 1$ and $\sinh
2r\approx\cosh 2r$. Since
$\hat{q}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{{\dagger}})$ and
$\hat{p}=\frac{i}{\sqrt{2}}(\hat{a}^{{\dagger}}-\hat{a})$, it follows that
$\hat{q}^{T}=\hat{q}$ and $\hat{p}^{T}=-\hat{p}$. If we take the partial
transpose on the first mode, only $\sigma_{24}=\sigma_{42}$ related to
measurements $\hat{p}_{1}$, $\hat{p}_{2}$ change the sign. Note that
$\sigma_{23}=\sigma_{32}$ related to measurements $\hat{p}_{1}$, $\hat{q}_{2}$
remain 0. Then the temporal covariance matrix is equal to the spatial
covariance matrix under the partial transpose and the high temperature
approximation. This can be understood as a continuous-variable analogue on
temporal and spatial correlations of bipartite pseudo-density matrices for the
qubit case [40]. Note that taking the partial trace of a two-qubit maximally
entangled state $\frac{1}{2}\sum_{i,j=0,1}\ket{ii}\bra{jj}$ we get a one-qubit
maximally mixed state $I$; the temporal analog of a two-qubit maximally
entangled state $\frac{1}{2}\sum_{i,j=0,1}\ket{ii}\bra{jj}$ is the one-qubit
maximally mixed state $I$ at two times under the identity evolution, that is
represented by $\frac{1}{2}\sum_{i,j=0,1}\ket{ij}\bra{ji}$. They are invariant
under the partial transpose as well. In the continuous variable context, the
one-mode thermal state under the high temperature approximation is close to
the maximally mixed state $I$. We will come back to this partial transpose
again later via Choi-Jamiołkowski isomorphism.
### 3.3 Pseudo-density matrix formulation for general continuous variables
Now we move on to define spacetime states for general continuous variables. We
first define the spacetime Wigner function by generalising correlations to the
spacetime domain, following the paradigm of pseudo-density matrices. Then
demanding the one-to-one correspondence between a spacetime Wigner function
and a spacetime density matrix, we gain the spacetime density matrix in
continuous variables from the spacetime Wigner function. This spacetime
density matrix in continuous variables can be regarded as the extension of the
pseudo-density matrix to continuous variables. We further analyse the
properties of this spacetime Wigner function based on the corresponding
spacetime density matrix in continuous variables and rediscover the five
properties of a uniquely-determined Wigner function.
#### 3.3.1 Preliminaries
The Wigner function is a convenient representation of non-relativistic quantum
mechanics in continuous variables and is fully equivalent to the density
matrix formalism. The one-to-one correspondence between the Wigner function
and the density matrix [57, 58] states that,
$\displaystyle\hat{\rho}=\int W(\alpha)T(\alpha)\pi^{-1}\textrm{d}^{2}\alpha,$
(3.22) $\displaystyle W(\alpha)=\Tr[\hat{\rho}T(\alpha)].$ (3.23)
Here $T(\alpha)$ is defined as
$T(\alpha)=\int
D(\xi)\exp(\alpha\xi^{*}-\alpha^{*}\xi)\pi^{-1}\textrm{d}^{2}\xi,$ (3.24)
where $D(\xi)$ is the displacement operator defined as
$D(\xi)=\exp(\xi\hat{a}^{{\dagger}}-\xi^{*}\hat{a})$. It can be seen that
$T(\alpha)$ is the complex Fourier transform of $D(\xi)$. Besides, $T(\alpha)$
can be reformulated as $T(\alpha)=2U(\alpha)$ where
$U(\alpha)=D(\alpha)(-1)^{\hat{a}^{{\dagger}}\hat{a}}D^{{\dagger}}(\alpha)$ is
the displaced parity operator. $T(\alpha)$ is Hermitian, unitary, unit-trace,
and an observable with eigenvalues $\pm 2$.
We can also see from Eqn. (3.23) that the Wigner function is the expectation
value of $T(\alpha)$ [59]. For an $n$-mode Wigner function, a straightforward
generalisation is
$W(\alpha_{1},...,\alpha_{n})=\langle\bigotimes_{i=1}^{n}T(\alpha_{i})\rangle,$
(3.25)
as Ref. [60] gives the two-mode version.
#### 3.3.2 Spacetime Wigner function
Let us start to construct the Wigner function in spacetime. It seems a bit
ambitious to merge position and momentum with time in a quasi-probability
distribution at first sight, but we will see that it is possible to treat
instances of time just as how we treat modes. Again we borrow the concept of
events from the pseudo-density matrix in finite dimensions and consider $n$
events instead of $n$ modes. Notice that the only difference between a pseudo-
density matrix and a standard density matrix in construction is the
correlation measure. Here we change the correlation measures of an $n$-mode
Wigner function given in Eqn. (3.25) in a similar way.
###### Definition 2.
Consider a set of events $\\{E_{1},E_{2},...,E_{N}\\}$. At each event $E_{i}$,
a measurement of $T(\alpha_{i})$ operator on a single mode is made. Then for a
particular choice of events with operators $\\{T(\alpha_{i})\\}_{i=1}^{n}$,
the spacetime Wigner function is defined to be
$\mathcal{W}(\alpha_{1},...,\alpha_{n})=\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle,$
(3.26)
where $\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle$ is the expectation value
of the product of the results of the measurements on these operators.
For spatially separated events, the spacetime Wigner function reduces to the
ordinary $n$-mode Wigner function, for the order of product and measurement
does not matter and it remains the same after making a flip (remember that
$n$-mode Wigner function is the expectation value of the measurement results
of the tensor product of these operators). If the measurements are taken in
time, then a temporal Wigner function is constructed under temporal
correlations. Thus, it is a generalisation for the Wigner function to the
spacetime domain.
It is easy to check that the spacetime Wigner function is real and normalised
to 1. Since the measurement results of $T(\alpha_{i})=2U(\alpha_{i})$ is $\pm
2$ (remember that $U(\alpha_{i})$ is the displaced parity operator), the
expectation value of the product of the measurement results is to make
products of $\pm 2$ with certain probability distribution. Thus,
$\mathcal{W}(\alpha_{1},...,\alpha_{n})$ is real.
For the normalisation, we give a proof for the bipartite case, i.e.,
$\int W(\alpha,\beta)\pi^{-2}\textrm{d}^{2}\alpha\textrm{d}^{2}\beta=1;$
(3.27)
for $n$ events, it can be proven directly following the same logic.
As mentioned before, a bipartite spacetime Wigner function reduces to two-mode
Wigner function for two spatially separated events. The normalisation
obviously holds in this case.
For a spacetime Wigner function between two times $t_{1}$ and $t_{2}$, we
assume the initial state $\hat{\rho}$ is arbitrary and the evolution between
$t_{1}$ and $t_{2}$ is an arbitrary CPTP map from $\hat{\rho}$ to
$\mathcal{E}(\hat{\rho})$. At the time $t_{1}$, we measure $T(\alpha)$. Note
that $T(\alpha)=2[\Pi_{2}(\alpha)-\Pi_{1}(\alpha)]$ where
$\Pi_{2}(\alpha)=\sum_{n=0}^{\infty}\ket{2n,\alpha}\bra{2n,\alpha}$ and
$\Pi_{1}(\alpha)=\sum_{n=0}^{\infty}\ket{2n+1,\alpha}\bra{2n+1,\alpha}$. That
is, we make projections $\Pi_{1}(\alpha)$ and $\Pi_{2}(\alpha)$ to the odd and
even subspaces for the eigenvalues $-2$ and $+2$. According to the measurement
postulation, we get the state
$\hat{\rho}_{1}=\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)/\Tr[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]$
with the probability $\Tr[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]$ after
making the measurement of $\Pi_{i}(\alpha)$ $(i=1,2)$. Note that projection
operators $\Pi_{i}(\alpha)=\Pi_{i}^{{\dagger}}(\alpha)$ and
$\Pi_{i}^{2}(\alpha)=\Pi_{i}(\alpha)$. Then from $t_{1}$ to $t_{2}$,
$\hat{\rho}_{1}$ evolves to $\mathcal{E}(\hat{\rho}_{1})$. At the time
$t_{2}$, we measure $T(\beta)$. We make projections $\Pi_{1}(\beta)$ and
$\Pi_{2}(\beta)$ for the eigenvalues $-2$ and $+2$ again. So the temporal
Wigner function, or $\\{T(\alpha),T(\beta)\\}$ correlation, is given by
$\displaystyle\mathcal{W}(\alpha,\beta)=\langle\\{T(\alpha),T(\beta)\\}\rangle$
$\displaystyle=$ $\displaystyle
4\sum_{i,j=1,2}(-1)^{i+j}\Tr[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\Tr\left\\{\Pi_{j}(\beta)\mathcal{E}\left[\frac{\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)}{\Tr[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]}\right]\Pi_{j}(\beta)\right\\}$
$\displaystyle=$ $\displaystyle
4\sum_{i,j=1,2}(-1)^{i+j}\Tr\\{\Pi_{j}(\beta)\mathcal{E}[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\\}$
$\displaystyle=$ $\displaystyle
2\sum_{i=1,2}(-1)^{i}\Tr\\{T(\beta)\mathcal{E}[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\\}$
(3.28)
Now let us check the normalisation property. Note that $\int
T(\beta)\pi^{-1}\textrm{d}^{2}\beta=\int
T(\alpha)\pi^{-1}\textrm{d}^{2}\alpha=I$ and $\mathcal{E}$ is trace-
preserving. Then we have
$\displaystyle\iint\mathcal{W}(\alpha,\beta)\pi^{-2}\textrm{d}^{2}\alpha\textrm{d}^{2}\beta$
$\displaystyle=$ $\displaystyle
2\iint\sum_{i=1,2}(-1)^{i}\Tr\\{T(\beta)\mathcal{E}[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\\}\pi^{-2}\textrm{d}^{2}\alpha\textrm{d}^{2}\beta$
$\displaystyle=$ $\displaystyle
2\int\sum_{i=1,2}(-1)^{i}\Tr\\{\mathcal{E}[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\\}\pi^{-1}\textrm{d}^{2}\alpha$
$\displaystyle=$ $\displaystyle
2\int\sum_{i=1,2}(-1)^{i}\Tr[\Pi_{i}(\alpha)\hat{\rho}\Pi_{i}(\alpha)]\pi^{-1}\textrm{d}^{2}\alpha$
$\displaystyle=$
$\displaystyle\int\Tr[T(\alpha)\hat{\rho}]\pi^{-1}\textrm{d}^{2}\alpha$
$\displaystyle=$ $\displaystyle 1.$ (3.29)
Thus, the normalisation property holds.
#### 3.3.3 Spacetime density matrix in continuous variables
Though it is not always convenient to use the density matrix formalism in
continuous variables, we are still interested in the possible form of
spacetime density matrices as it is the basic construction for states.
Remember that there is a one-to-one correspondence between the Wigner function
and the density matrix. Here we demand that a similar one-to-one
correspondence holds for the spatio-temporal version. Then we can define a
spacetime density matrix in continuous variables from the above spacetime
Wigner function.
###### Definition 3.
A spacetime density matrix in continuous variables is defined as
$\hat{R}=\idotsint\mathcal{W}(\alpha_{1},...,\alpha_{n})\bigotimes_{i=1}^{n}T(\alpha_{i})\pi^{-n}\textrm{d}^{2}\alpha_{1}\cdots\textrm{d}^{2}\alpha_{n}.\\\
$ (3.30)
This follows the direction from a spacetime Wigner function to a spacetime
density matrix in continuous variables just as Eqn. (3.22). Analogous to Eqn.
(3.23), the opposite direction from a spacetime density matrix in continuous
variables to a spacetime Wigner function automatically holds:
$\mathcal{W}(\alpha_{1},...,\alpha_{n})=\Tr\\{[\bigotimes_{i=1}^{n}T(\alpha_{i})]\hat{R}\\}=\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle.$
(3.31)
Now we prove Eqn. (3.31) as a transform from the spacetime density matrix in
continuous variables to the spacetime Wigner function. Applying the definition
of the spacetime density matrix in continuous variables to the middle hand
side of Eqn. (3.31), we get
$\Tr\left\\{\left[\bigotimes_{i=1}^{n}T(\alpha_{i})\right]\hat{R}\right\\}=\Tr[\idotsint\mathcal{W}(\beta_{1},...,\beta_{n})\bigotimes_{i=1}^{n}T(\alpha_{i})T(\beta_{i})\pi^{-n}\textrm{d}^{2}\beta_{1}\cdots\textrm{d}^{2}\beta_{n}\bigg{]}.$
(3.32)
Note that
$T(\alpha)T(\beta)=4\exp[2(\alpha^{*}\beta-\alpha\beta^{*})]D(2\alpha-2\beta),$
(3.33) $\Tr D(\xi)=\pi\delta(\xi_{I})\delta(\xi_{R})=\pi\delta^{(2)}(\xi),$
(3.34)
and $\delta^{(2)}(2\xi)=\frac{1}{4}\delta^{(2)}(\xi)$.
$\displaystyle\Tr\left\\{\left[\bigotimes_{i=1}^{n}T(\alpha_{i})\right]\hat{R}\right\\}$
$\displaystyle=$
$\displaystyle\Tr\\{\idotsint\mathcal{W}(\beta_{1},...,\beta_{n})\bigotimes_{i=1}^{n}4\exp[2(\alpha_{i}^{*}\beta_{i}-\alpha_{i}\beta_{i}^{*})]D(2\alpha_{i}-2\beta_{i})\pi^{-n}\textrm{d}^{2}\beta_{1}\cdots\textrm{d}^{2}\beta_{n}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\idotsint\mathcal{W}(\beta_{1},...,\beta_{n})\prod_{i=1}^{n}4\exp[2(\alpha_{i}^{*}\beta_{i}-\alpha_{i}\beta_{i}^{*})]\delta^{(2)}(2\alpha_{i}-2\beta_{i})\textrm{d}^{2}\beta_{1}\cdots\textrm{d}^{2}\beta_{n}$
$\displaystyle=$ $\displaystyle\mathcal{W}(\alpha_{1},...,\alpha_{n})$
$\displaystyle=$ $\displaystyle\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle.$
(3.35)
Thus, Eqn. (3.31) holds as
$\Tr\\{[\bigotimes_{i=1}^{n}T(\alpha_{i})]\hat{R}\\}=\mathcal{W}(\alpha_{1},...,\alpha_{n})=\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle.$
It is also convenient to define the spacetime density matrix in continuous
variables directly from $T(\alpha)$ operators, without the introduction of a
spacetime Wigner function.
###### Definition 4.
An equivalent definition of a spacetime density matrix in continuous variables
is
$\hat{R}=\idotsint\langle\\{T(\alpha_{i})\\}_{i=1}^{n}\rangle\bigotimes_{i=1}^{n}T(\alpha_{i})\pi^{-n}\textrm{d}^{2}\alpha_{1}\cdots\textrm{d}^{2}\alpha_{n}.$
(3.36)
If we compare this definition with the definition of the pseudo-density matrix
in finite dimensions given as Eqn. (2.25) element by element, we will find a
perfect analogue. This may suggest the possibility for a generalised
continuous-variable version of pseudo-density matrices.
#### 3.3.4 Properties
Now we investigate the properties of the spacetime Wigner function and the
spacetime density matrix for continuous variables.
It is easy to check the spacetime density matrix $\hat{R}$ is Hermitian and
unit-trace. Since $T(\alpha_{i})$ is Hermitian and
$\mathcal{W}(\alpha_{1},...,\alpha_{n})$ is real, $\hat{R}$ is Hermitian. From
the normalisation property of the spacetime Wigner function and the fact that
$T(\alpha_{i})$ has unit trace, we conclude that $\Tr\hat{R}=1$.
Analogous to the normal spatial Wigner function, we analyse the properties for
the spacetime Wigner function. For example, the spacetime Wigner function can
be used as a quasi-probability distribution in calculating the expectation
value of an operator from the spacetime density matrix. For an operator
$\hat{A}$ in the Hilbert space $\mathcal{H}^{\otimes n}$,
$\langle\hat{A}\rangle_{R}=\Tr[\hat{R}\hat{A}]=\iint\mathcal{W}(\alpha_{1},...,\alpha_{n})A(\alpha_{1},...,\alpha_{n})\pi^{-n}\textrm{d}^{2}\alpha_{1}\cdots\textrm{d}^{2}\alpha_{n},$
(3.37)
where
$A(\alpha_{1},...,\alpha_{n})=\Tr\\{[\bigotimes_{i=1}^{n}T(\alpha_{i})]\hat{A}\\}.$
(3.38)
It is obvious that a spacetime Wigner function for a single event does not
discriminate between space and time; that is, for a single event the spacetime
Wigner function is the same as an ordinary one-mode Wigner function in space.
From the following we consider a bipartite spacetime Wigner function and
generalisation to arbitrary events is straightforward.
The five properties to uniquely determine a two-mode Wigner function in Ref.
[61, 62] are: (1) that it is given by a Hermitian form of the density matrix;
(2) that the marginal distributions hold for $q$ and $p$ and it is normalised;
(3) that it is Galilei covariant; (4) that it has corresponding
transformations under space and time reflections; (5) that for two Wigner
functions, their co-distribution is related to the corresponding density
matrices. They all hold in a similar way for a bipartite spacetime Wigner
function and the corresponding spacetime density matrix in continuous
variables. For a bipartite spacetime Wigner function, the five properties are
stated as follows:
###### Property 1.
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ is given by a Hermitian form of the
corresponding spacetime density matrix as
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\Tr[\hat{M}(q_{1},p_{1},q_{2},p_{2})\hat{R}]$
(3.39)
for
$\hat{M}(q_{1},p_{1},q_{2},p_{2})=\hat{M}^{{\dagger}}(q_{1},p_{1},q_{2},p_{2}).$
(3.40)
Therefore, it is real.
###### Property 2.
The marginal distributions of $q$ and $p$ as well as the normalisation
property hold.
$\displaystyle\iint\textrm{d}p_{1}\textrm{d}p_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\bra{q_{1},q_{2}}\hat{R}\ket{q_{1},q_{2}},$
$\displaystyle\iint\textrm{d}q_{1}\textrm{d}q_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\bra{p_{1},p_{2}}\hat{R}\ket{p_{1},p_{2}},$
$\displaystyle\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\Tr\hat{R}=1.$
(3.41)
###### Property 3.
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ is Galilei covariant 111The original
paper [61] uses the word “Galilei invariant”. , that is, if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{q_{1}+a,q_{2}+b}\hat{R}\ket{q^{\prime}_{1}+a,q^{\prime}_{2}+b}$
then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})\rightarrow\mathcal{W}(q_{1}+a,p_{1},q_{2}+b,p_{2})$
and if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\exp\\{[ip^{\prime}_{1}(-q_{1}+q^{\prime}_{1})+ip^{\prime}_{2}(-q_{2}+q^{\prime}_{2})]/\hbar\\}\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}},$
then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})\rightarrow\mathcal{W}(q_{1},p_{1}-p^{\prime}_{1},q_{2},p_{2}-p^{\prime}_{2}).$
###### Property 4.
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ has the following property under space
and time reflections 222Again the original paper [61] uses the word “invariant
under space and time reflections”. : if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{-q_{1},-q_{2}}\hat{R}\ket{-q^{\prime}_{1},-q^{\prime}_{2}}$
then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})\rightarrow\mathcal{W}(-q_{1},-p_{1},-q_{2},-p_{2})$
and if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{q^{\prime}_{1},q^{\prime}_{2}}\hat{R}\ket{q_{1},q_{2}}$
then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})\rightarrow\mathcal{W}(q_{1},-p_{1},q_{2},-p_{2}).$
###### Property 5.
Two spacetime Wigner functions are related to the two corresponding spacetime
density matrices as
$\Tr(R_{1}R_{2})=(2\pi\hbar)\iint\textrm{d}q\textrm{d}p\mathcal{W}_{R_{1}}(q,p)\mathcal{W}_{R_{2}}(q,p),$
(3.42)
for $\mathcal{W}_{R_{1}}(q,p)$ and $\mathcal{W}_{R_{2}}(q,p)$ are spacetime
Wigner functions for spacetime density matrices in continuous variables
$\hat{R}_{1}$ and $\hat{R}_{2}$ respectively.
All these six properties (five plus the previous one for the expectation value
of an operator in this subsection) are proven in Appendix B.
### 3.4 Generalised measurements for pseudo-density matrix
Here we go beyond the pseudo-density matrix formulation, in the sense that we
generalise spatial correlations to the spacetime domain. Nevertheless, we
still follow the idea to build spacetime states upon measurements. We consider
position measurements for a special diagonal case. To reduce the additional
effects caused by measurement processes, we discuss weak measurements and
construct spacetime states from them. Here the connection with path integral
is more obvious.
#### 3.4.1 Position measurements
Besides quadratures and $T(\alpha)$ operators, it is also possible to expand a
continuous-variable density matrix in the position basis since it is an
orthogonal and complete basis. Here we consider a special case which is the
diagonal matrix for convenience.
In principle, a density matrix in the continuous variables can be diagonalised
in the position basis as
$\hat{\rho}=\int_{-\infty}^{\infty}\textrm{d}x\ p(x)\ket{x}\bra{x},$ (3.43)
where
$p(x)=\Tr[\ket{x}\bra{x}\hat{\rho}].$ (3.44)
In the standard theory of quantum mechanics, we assume that the measurement
results are arbitrarily precise to get the probability density $p(x)$ with the
state updated to $\ket{x}\bra{x}$ after the measurement of $\hat{x}$. It is
hard to achieve in the actual setting and imprecise measurements will be
employed in the following discussion.
Then we define the spacetime density matrix in exactly the same way with the
probability density now in the spatio-temporal domain.
###### Definition 5.
Consider a set of $N$ events labelled $\\{E_{1},\cdots,E_{N}\\}$. At each
event $E_{i}$, a measurement of the position operator $\hat{x}_{i}$ is made.
For a particular choice of the event, for example, $\\{E_{i}\\}_{i=1}^{n}$, we
can define the spacetime density matrix from the joint probability of all
these measurements as
$\rho=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\cdots\textrm{d}x_{n}p(x_{1},\cdots,x_{n})\ket{x_{1}}\bra{x_{1}}\otimes\cdots\otimes\ket{x_{n}}\bra{x_{n}}.$
(3.45)
The remaining problem is how to calculate the joint probability
$p(x_{1},\cdots,x_{n})$. For spatially separated events, the problem reduces
to results given by states in ordinary quantum mechanics. So we only need to
consider how to formulate states in time. Successive position measurements
have been discussed properly in the path integral formalism, effect and
operation formalism and multi-time formalism [47, 48].
Based on the discussion in Ref. [48], we consider $n$ events of instantaneous
measurements of $x(t)$ at times $t_{1},\cdots,t_{n}$ ($t_{1}<\cdots<t_{n}$).
In reality, such a measurement cannot be arbitrarily precise; a conditional
probability amplitude called resolution amplitude $\Upsilon(\bar{x}-x)$ is
introduced for $\bar{x}$ as the measurement result with the initial position
of the system at $x$. Denote the state of the system as $\ket{\psi(t)}$ with
the wave function $\psi(x,t)=\bra{x}\psi(t)\rangle$. For a meter prepared in
the state $\ket{\Upsilon}$ with the wave function
$\Upsilon(\bar{x})=\bra{\bar{x}}\Upsilon\rangle$, the total system before the
measurement will be $\ket{\Psi_{i}}=\ket{\Upsilon}\otimes\ket{\psi(t)}$ with
the wave function $\bra{\bar{x},x}\Psi_{i}\rangle=\Upsilon(\bar{x})\psi(x,t)$.
Consider the interaction for the measurement process as $\hat{x}\hat{\bar{p}}$
at some particular time. The total system after the measurement will be
$\ket{\Psi_{f}}=e^{-(i/\hbar)\hat{x}\hat{\bar{p}}}\ket{\Psi_{i}}=\int\textrm{d}xe^{-(i/\hbar)x\hat{\bar{p}}}\ket{\Upsilon}\otimes\ket{x}\psi(x,t)$,
with the wave function
$\bra{\bar{x},x}\Psi_{f}\rangle=\Upsilon(\bar{x}-x)\psi(x,t)=\bra{x}\Upsilon(\bar{x}-\hat{x})\ket{\psi(t)}$.
Following the calculation in Ref. [48], for the wave function of the system
$\psi(x(t_{1}),t_{1})$ at some initial time $t_{1}$, the joint probability for
measurement results $(\bar{x}_{1},\cdots,\bar{x}_{n})$ is given by a path
integral as
$p(\bar{x}_{1},\cdots,\bar{x}_{n})=\int_{t_{1}}^{t_{n}}\mathcal{D}x(t)\left[\prod_{\nu=1}^{n}\Upsilon(\bar{x}_{\nu}-x(t_{\nu}))\right]e^{(i/\hbar)S[x(t)]}\psi(x(t_{1}),t_{1}),$
(3.46)
where
$\int_{t_{1}}^{t_{n}}\mathcal{D}x(t)=\lim_{N\rightarrow\infty}\left[\prod_{k=1}^{N}\int_{-\infty}^{\infty}\textrm{d}x_{k}\right],$
(3.47)
with the insertion of $N-2$ times between the initial time $t_{1}$ and the
final time $t_{n}=t_{N}$; and note that all the measurement times are included
in the insertion. This integral sums over all path $x(t)$ from $x(t_{1})$ to
$x(t_{n})$ with arbitrary initial values $x(t_{1})$ and arbitrary final
positions $x(t_{n})$. Here
$S[x(t)]=\int_{t_{1}}^{t_{n}}\textrm{d}tL(x,\dot{x},t)$ (3.48)
is the action for the path $x(t)$ with the Lagrangian of the system as
$L(x,\dot{x},t)$.
Note that $p(\bar{x}_{1},\cdots,\bar{x}_{n})$ is normalised, i.e.,
$\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}\bar{x}_{1}\cdots\textrm{d}\bar{x}_{n}p(\bar{x}_{1},\cdots,\bar{x}_{n})=1;$
(3.49)
thus, the spacetime density matrix defined above has unit trace.
Here the diagonalised spacetime density matrix in the position basis is fully
equivalent to the path integral formalism. Or we can take this definition as
the transition from the path integral. Thus, this definition suggests a
possible link between the pseudo-density matrix formulation and the path
integral formalism.
#### 3.4.2 Weak measurements
Weak measurements are the measurements that only slightly disturb the state,
with POVM elements close to the identity. They are often continuous. It is
particularly interesting here as weak measurements minimise the influence of
measurements and maximally preserve the information of the original states.
Via weak measurements, we do not need to worry about the change of marginal
states at each time. There are several slightly different mathematical
definitions for weak measurements. Here we follow the convention in the
formulation of effects and operations [63].
Recall that an effect $\hat{F}$ is defined as an operator which satisfies
$\hat{F}^{{\dagger}}=\hat{F}$ and $0<\hat{F}<\mathbbm{1}$. Similar to a
projection, the probability of obtaining the result in the interval
$I=(a,a+\Delta a)$ at time $t$ is writen as
$P(\rho|I,t)=\Tr\\{\hat{F}^{1/2}_{H}(I,t)\hat{\rho}\hat{F}^{1/2}_{H}(I,t)\\},$
(3.50)
And the state evolves to
$\rho^{\prime}=\frac{\hat{F}^{1/2}_{H}(I,t)\hat{\rho}\hat{F}^{1/2}_{H}(I,t)}{\Tr\left\\{\hat{F}^{1/2}_{H}(I,t)\hat{\rho}\hat{F}^{1/2}_{H}(I,t)\right\\}}$
(3.51)
Assume that the disturbance at time $t$ does not affect the discrimination for
$I$ out of the whole range and the reduction postulate holds. At a later time
$t^{\prime}$, we have
$P(\rho^{\prime}|I^{\prime},t^{\prime})=\Tr\left\\{\hat{F}^{1/2}_{H}(I^{\prime},t^{\prime})\hat{\rho^{\prime}}\hat{F}^{1/2}_{H}(I^{\prime},t^{\prime})\right\\}.$
(3.52)
We have the joint probability as
$P(\rho|I,t;I^{\prime},t^{\prime})=\Tr\left\\{\hat{F}^{1/2}_{H}(I^{\prime},t^{\prime})\hat{F}^{1/2}_{H}(I,t)\hat{\rho}\hat{F}^{1/2}_{H}(I,t)\hat{F}^{1/2}_{H}(I^{\prime},t^{\prime})\right\\}.$
(3.53)
Consider the densities of effects
$\textrm{d}\hat{F}(a)=\hat{f}(a)\textrm{d}\mu(a),\qquad\int_{-\infty}^{+\infty}\textrm{d}\mu(a)\hat{f}(a)=\mathbbm{1},$
(3.54)
where $\textrm{d}\mu(a)$ is a measure for the function $\hat{f}(a)$. Then we
have
$\hat{F}(\textrm{d}I,t;\textrm{d}I^{\prime},t^{\prime})=\hat{f}(a,t;a^{\prime},t^{\prime})\textrm{d}\mu(a)\textrm{d}\mu(a^{\prime})=\hat{f}^{1/2}_{H}(a,t)\hat{f}_{H}(a^{\prime},t^{\prime})\hat{f}^{1/2}_{H}(a,t)\textrm{d}\mu(a)\textrm{d}\mu(a^{\prime}),$
(3.55)
$P(\rho|\textrm{d}I,t;\textrm{d}I^{\prime},t^{\prime})=p(\rho|a,t;a^{\prime},t^{\prime})\textrm{d}\mu(a)\textrm{d}\mu(a^{\prime})=\Tr{\hat{f}(a,t;a^{\prime},t^{\prime})\hat{\rho}}\textrm{d}\mu(a)\textrm{d}\mu(a^{\prime}).$
(3.56)
In general,
$P(\rho|\textrm{d}I_{1},t_{1};\cdots;\textrm{d}I_{n},t_{n})=\Tr{\hat{f}(a_{1},t_{1};\cdots;a_{n},t_{n})\hat{\rho}}\textrm{d}\mu(a_{1})\cdots\textrm{d}\mu(a_{n}),$
(3.57)
where
$\displaystyle\hat{f}(a_{1},t_{1};\cdots;a_{n-1},t_{n-1};a_{n},t_{n})=\hat{f}^{1/2}_{H}$
$\displaystyle(a_{1},t_{1})\cdots\hat{f}^{1/2}_{H}(a_{n-1},t_{n-1})\hat{f}_{H}(a_{n},t_{n})$
$\displaystyle\times\hat{f}^{1/2}_{H}(a_{n-1},t_{n-1})\cdots\hat{f}^{1/2}_{H}(a_{1},t_{1}).$
(3.58)
Now following the calculation in Ref. [50], we can define a generalised
observable corresponding to a simultaneous inaccurate measurement of position
and momentum for a density matrix $\hat{\rho}$:
$\hat{F}(T)=\int_{T}\frac{\textrm{d}x\textrm{d}p}{2\pi\hbar}\exp\left[\frac{i}{\hbar}(p\hat{q}-x\hat{p})\right]\hat{\rho}\exp\left[-\frac{i}{\hbar}(p\hat{q}-x\hat{p})\right].$
(3.59)
Take
$\hat{\rho}=C\exp[-\alpha(\hat{q}^{2}+\lambda\hat{p}^{2})],\qquad\alpha,\lambda>0,$
(3.60)
where $C$ is some normalisation factor. We get the density of this generalised
effect-valued measure as
$\hat{f}(q,p)=C\exp\left[-\alpha[(\hat{q}-q)^{2}+\lambda(\hat{p}-p)^{2}]\right],$
(3.61)
where $\textrm{d}\hat{F}(q,p)=\hat{f}(q,p)\textrm{d}\mu(q,p)$. We set
$\alpha=\gamma\tau,$ (3.62)
where $\tau$ is the time interval between two subsequent measurements. When
$\alpha\rightarrow 0$, the measurement is continuous and we call it weak. For
an initial density matrix $\hat{\rho}$ at time $t=0$, we make continuous
measurements in time and find the probability density of obtaining measurement
results $q,p$ at time $t=\tau$ is given by
$p(q,p,\tau|\hat{\rho})=\Tr\mathcal{F}(q,p;\tau)\hat{\rho},$ (3.63)
where
$\displaystyle\mathcal{F}(q,p;\tau)\hat{\rho}=$
$\displaystyle\int\textrm{d}\mu_{G}[q(t),p(t)]\delta\left(q-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tq(t)\right)\delta\left(p-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tp(t)\right)\exp[-\frac{i}{\hbar}\hat{H}\tau]$
$\displaystyle\mathcal{T}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q}_{H}(t)-q(t))^{2}+\lambda(\hat{p}_{H}(t)-p(t))^{2}]\right]\hat{\rho}$
$\displaystyle\mathcal{T}^{*}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q}_{H}(t)-q(t))^{2}+\lambda(\hat{p}_{H}(t)-p(t))^{2}]\right]\exp[\frac{i}{\hbar}\hat{H}\tau],$
(3.64)
here
$\textrm{d}\mu_{G}[q(t),p(t)]=\lim_{N\rightarrow\infty}\left(\frac{\gamma\tau\sqrt{\lambda}}{\pi
N}\prod_{s=1}^{N}\textrm{d}q(t_{s})\textrm{d}p(t_{s})\right),$ (3.65)
and
$\displaystyle\hat{q}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{q}\exp\left[-\frac{i}{\hbar}\hat{H}t\right],$
$\displaystyle\hat{p}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{p}\exp\left[-\frac{i}{\hbar}\hat{H}t\right].$
(3.66)
###### Definition 6.
A possible form for the temporal Wigner function
$W(\bar{x}_{1},\bar{p}_{1},\bar{t}_{1};\dots;\bar{x}_{\nu},\bar{p}_{\nu},\bar{t}_{\nu})$
is given by the probability density of simultaneous measurement results
$\bar{x}_{i},\bar{p}_{i}$ at the time $\bar{t}_{i}$ for $i=1,\dots,\nu$ with
$\hat{\rho}$ as the initial density matrix at the initial time $\bar{t}_{1}$
in Ref. [50]:
$\displaystyle
W(\bar{x}_{1},\bar{p}_{1},\bar{t}_{1};\cdots;\bar{x}_{\nu},\bar{p}_{\nu},\bar{t}_{\nu})$
$\displaystyle=$
$\displaystyle\Tr\mathcal{F}(\bar{x}_{\nu},\bar{p}_{\nu};\bar{t}_{\nu}-\bar{t}_{\nu-1})\mathcal{F}(\bar{x}_{\nu-1},\bar{p}_{\nu-1};\bar{t}_{\nu-1}-\bar{t}_{\nu-2})\cdots\mathcal{F}(\bar{x}_{2},\bar{p}_{2};\bar{t}_{2}-\bar{t}_{1})\mathcal{F}(\bar{x}_{1},\bar{p}_{1};0)\hat{\rho}.$
(3.67)
Here we employ the probability density in weak measurements to define a
temporal Wigner function. This generalises the form of measurements to take.
As shown in the next section, this temporal Wigner function turns out to be a
desirable spacetime quantum state and expand the possibility for relating
generalised measurement theory with spacetime. In general, a unified spacetime
Wigner function defined from weak measurements is possible as well. For n-mode
spatial Wigner function from weak measurements, it is defined as
$W(q_{1},p_{1},\cdots,q_{n},p_{n})=\Tr\mathcal{F}(q_{1},p_{1};0)\otimes\cdots\otimes\mathcal{F}(q_{n},p_{n};0)\hat{\rho}.$
(3.68)
Thus spacetime Wigner function is a mixture of product and tensor product of
$\mathcal{F}$. We obtain the spacetime states from weak measurements. It
follows the paradigm of pseudo-density matrix formalism that spacetime Wigner
function is defined via measurement correlations. Specifically, we make
simultaneous measurements of position and momentum; as a price to pay, we
fixed the average positions and the average momentums for certain time
periods. It is not the usual Wigner function but a generalised version in the
average sense.
### 3.5 Experimental proposal for tomography
Here we propose an experimental tomography for spacetime Gaussian states in
quantum optics. Especially, we construct the temporal Gaussian states, in
terms of measuring mean values and the temporal covariance matrix for two
events in time. The covariance of quadratures are defined in terms of the
correlation of quadratures and mean values. Thus, all we need to measure are
mean values and correlations of quadratures.
With the balanced homodyne detection, we can measure the mean values of single
quadratures $d_{i}=\langle x_{i}\rangle$, the correlation of the same
quadrature $\langle x_{i}x_{i}\rangle$ (the diagonal terms of the covariance
matrix), and the correlation of both position operators or both momentum
operators at two times $\langle q_{j}q_{k}\rangle$ or $\langle
p_{j}p_{k}\rangle$ ($j\neq k$ for this section). Mean values of single
quadratures are measured by the balanced homodyne detection as usual. For
$\langle x_{i}x_{i}\rangle$, we can measure by almost the same method, only do
an additional square for each measurement outcome of $\hat{x}_{i}$. For
$\langle q_{j}q_{k}\rangle$ or $\langle p_{j}p_{k}\rangle$, we record the
homodyne results for a long time with small time steps and calculate the
expectation values of the product the measurement results at two times to get
the correlation.
It is a bit difficult to measure the correlation for a mixture of position and
momentum operators. For such correlations at the same time $t_{j}$, the
measurement of $q_{j}$ and $p_{j}$ cannot be precise due to the uncertainty
principle. An eight-port homodyne detector may be a suggestion; that is, we
split the light into half and half by a 50/50 beam splitter, and measure each
quadrature separately with a local oscillator which is split into two as well
for homodyne detection. However, we cannot avoid the vacuum noise when we
split the light and the local oscillator. A better method for measuring
$q_{j}$ and $p_{j}$ at time $t_{j}$ will be resort to quantum-dense metrology
in Ref. [64]. For the correlation $\langle q_{j}p_{k}\rangle$, we use the same
protocol as before. As the two-time correlation for the same quadrature, we
record the homodyne results for a long time with small time steps and
calculate the expectation values of the product of the measurement results at
two times with a fixed time interval in between to get the correlation.
Then we gain all the correlations to construct the temporal covariance matrix.
The corresponding temporal density matrix or temporal Wigner function is
easily built with mean values and the temporal covariance matrix; thus, we
achieve the experimental tomography.
### 3.6 Comparison and comments
The pseudo-density matrix for $n$ qubits is neatly defined and satisfies the
properties listed in Ref. [51]. These properties are: (1) that it is
Hermitian; (2) that it represents probabilistic mixing; (3) that it has the
right classical limit; (4) that it has the right single-time marginals; (5)
for a single qubit evolving in time, composing different time steps is
associative. For Gaussian spacetime states, the first four properties easily
hold; for the fifth one, it remains true for the Gaussian evolution. For
general continuous variables, except the one for single-time marginals, all
the others hold. This property for single-time marginals is non-trivial. The
correlation of a single Pauli operator for each single-time marginal is
preserved after making the measurement of that Pauli operator. As each single-
time marginal is just the spatial state at that time, the total correlation
for all Pauli operators is independent of the measurement collapse. It is a
perfect coincide.
The relation with the Choi-Jamiołkowski isomorphism is important in deriving
the above properties. Consider a single qubit or mode evolving under a channel
$\mathcal{E}_{B|A}$ from $t_{A}$ to $t_{B}$. Then define an operator $E_{B|A}$
as the Jamiołkowski isomorphism of $\mathcal{E}_{B|A}$:
$E_{B|A}=(\mathcal{E}_{B|A}\otimes\mathcal{I})(\ket{\Phi^{+}}\bra{\Phi^{+}}^{\Gamma})$
(3.69)
where $\ket{\Phi^{+}}$ is the unnormalised maximally entangled state on the
double Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{A}$ at $t_{A}$ and
$\Gamma$ denotes partial transpose.
$\ket{\Phi^{+}}=\sum_{i=0,1}\ket{i}\otimes\ket{i}$ for the qubit case.
$\ket{\Phi^{+}}=\sum_{n=0}^{\infty}\ket{n,\alpha}\otimes\ket{n,\alpha}$ for
continuous variables; in which $\ket{n,\alpha}=D(\alpha)\ket{n}$ with the
displacement operator $D(\alpha)$ and the number eigenstates $\ket{n}$. Then
the spacetime state in terms of pseudo-density matrix formulation is given as
the Jordan product
$R_{AB}=\frac{1}{2}\left[E_{B|A}(\rho_{A}\otimes I_{B})+(\rho_{A}\otimes
I_{B})E_{B|A}\right].$ (3.70)
The qubit version is proved in Ref. [51] and we can follow its argument for
the continuous-variable version we defined above. It is particularly
interesting when we consider temporal correlations for two times. The orders
between $E_{B|A}$ and $\rho_{A}\otimes I_{B}$ automatically suggest a
symmetrised order of operators in two-time correlations. For a special case
that $\rho_{A}$ is maximally mixed as proportional to the identity $I$,
$R_{AB}=E_{B|A}$. Consider the identity evolution $\mathcal{E}_{B|A}$ as
$\mathcal{I}$, then $E_{B|A}=\ket{\Phi^{+}}\bra{\Phi^{+}}^{\Gamma}$. The
spatial and temporal analogue discussed in the Gaussian section is recovered
by partial transpose again.
One thing of particular interest to look at in continuous variables is the
relation between with the pseudo-density matrix formulation and the path
integral formulation. In Ref. [43], we establish the connection between
pseudo-density matrix and decoherence functional in consistent histories. The
only thing left unrelated in different spacetime approaches listed in the
introduction is the path integral formulation. Here consider the propagator
$\bra{y_{2},t_{2}}\hat{U}\ket{y_{1},t_{1}}$, or more specifically, the
absolute square of this propagator as the probability for transforming
$\ket{y_{1}}$ at $t_{1}$ to $\ket{y_{2}}$ at $t_{2}$. The initial state
evolves under the unitary
$\hat{U}=\exp(-\mathcal{T}\int_{t_{1}}^{t_{2}}i\hat{H}\textrm{d}t/\hbar)$. For
the Gaussian case, $\ket{y_{1}}$ at the time $t_{1}$ and $\ket{y_{2}}$ at
$t_{2}$ may be two eigenstates of $\hat{x}$ or $\hat{p}$ or a mixture of them
over a period. For general continuous variables, they should be two
eigenstates of $T(\alpha)$ and $T(\beta)$, that is, a mixture of
$\ket{n,\alpha}$ and $\ket{m,\beta}$.Via this propagator, we can calculate the
two-time correlation. It gives the same results as the pseudo-density matrix
does, which suggests the two formulations may be equivalent.
Ref. [51] suggests five criteria for a quantum state over time to satisfy as
the analog of a quantum state over spatial separated systems. Here we also set
up desirable properties of quantum states in the whole spacetime. The basic
principle is that the statistics calculated using the spacetime state should
be identical to those calculated using standard quantum theory. Note that
Criterion 1, 2, 3, and 6 are adapted from Ref. [51].
###### Criterion 1.
A spacetime quantum state has a Hermitian form, that is, the spacetime density
matrix is self-adjoint and the spacetime Wigner function is given by the
expectation value of a Hermitian operator.
###### Criterion 2.
The probability related to all the measurements at different spacetime events
is normalised to one, that is, the spacetime density matrix is unit-trace and
the spacetime Wigner function is normalised to one.
###### Criterion 3.
A spacetime quantum state represents probabilistic mixing appropriately, that
is, a spacetime state of different systems with a mixture of initial states is
the corresponding mixture of spacetime states for each system, as well as the
mixture of channel evolutions.
###### Criterion 4.
A spacetime quantum state provides the right expectation values of operators.
In particular, it gives the same expectation values of time-evolving operators
as the Heisenberg picture does.
###### Criterion 5.
A spacetime quantum state provides the right propagator/kernel which is the
probability amplitude evolving from one time to another.
###### Criterion 6.
A spacetime quantum state has the appropriate classical limit.
It is easy to check that the Gaussian characterisation satisfies Criterion 1,
2, 3, 5, 6 and the second half of Criterion 3; the first half of Criterion 3
does not hold since the mixture of Gaussian states is not necessarily
Gaussian.
For the Wigner function and corresponding density matrix representation,
Criterion 1, 2, 3, 4, 6 hold. Criterion 5 remains to be further analysed.
All of the Criteria 1-6 hold for position measurements and weak measurements,
though the spacetime density matrix for position measurements assumes
diagonalisation. It seems that the spacetime Wigner function from weak
measurements is best-defined under these criteria.
Note that we have considered whether the single time marginals of a spacetime
quantum state reduce to the spatial state at that particular time. It
unfortunately fails for Definition 2- 6 in general due to a property in the
measurement theory which suggests the irreversibility of the time evolution in
the repeated observations [50]; only the initial time marginal is reduced to
the initial state. Thus, we prefer not to list it as one of the criteria.
## Chapter 4 Correlations from other spacetime formulations: relation and
lesson
### 4.1 Introduction
Now we have already generalised the pseudo-density matrix formalism to
continuous variables and general measurement processes. There are several
other approaches which also tends to treat space and time more equally but
different from the pseudo-density matrix formalism. In this chapter, we
identify the relationship among these spacetime approaches via quantum
correlation in time [65].
The problem of time [66] is especially notorious in quantum theory as time
cannot be treated as an operator in contrast with space. Several attempts have
been proposed to incorporate time into the quantum world in a more even-handed
way to space, including: indefinite causal structures [67, 68, 69, 70, 71,
72], consistent histories [73, 74, 75, 76, 77], generalised quantum games [78,
79], spatio-temporal correlation approches [80, 81], path integrals [82, 83],
and pseudo-density matrices [39, 40, 84, 85]. Different approaches have their
own advantages. Of particular interest here is the pseudo-density matrix
approach for which one advantage is that quantum correlations in space and
time are treated on an equal footing. The present work is motivated by the
need to understand how the different approaches connect via temporal
correlations, so that ideas and results can be transferred more readily.
We accordingly aim to identify mappings between these approaches and pseudo-
density matrices. We ask what kind of relationship these space-time approaches
hold in terms of temporal correlations. Are the allowed temporal correlations
the same or different from each other? If the same, are they equal, or do they
map with each other and what kind of mapping? If different, how different are
they? More specifically, we take temporal correlations represented in
different approaches and find that they are consistent with each other expect
in the path integral formalism. Quantum correlations in time in these
approaches are either exactly equal or operationally equivalent expect those
used in the path integral formalism. By operational equivalence of two
formalisms, we mean the correlations or the probabilities of possible
measurement outcomes with given inputs in these two formalisms are equal. We
find several mappings and relations between these approaches, including (i) we
map process matrices with indefinite causal order directly to pseudo-density
matrices in three different ways; (ii) we show the diagonal terms of
decoherence functionals in consistent histories are exactly the probabilities
in temporal correlations of corresponding pseudo-density matrices; (iii) we
show quantum-classical signalling games give the same probabilities as
temporal correlations measured in pseudo-density matrices; (iv) the
calculation of OTOCs reduces half numbers of steps by pseudo-density matrices;
and (v) correlations in path integrals are defined as expectation values in
terms of the amplitude measure rather than the probability measure as in
pseudo-density matrices and are different from correlations in all the other
approaches. A particular example via a tripartite pseudo-density matrix is
presented to illustrate the unified picture of different approaches except
path integrals. This applies to more complicated cases and provides a unified
picture of these approaches. It also supports the further development of
space-time formalisms in non-relativistic quantum theory. Difference in
correlations between path integrals and other approaches also suggests the
importance of measure choice in quantum theory.
This chapter is based on Ref. [65] and proceeds as follows. We introduce
indefinite causal structures and compare the process matrix formalism with the
pseudo-density matrix formalism in terms of correlation analysis, causality
violation, and postselection in Section 4.2. In Section 4.3, we establish the
relation between pseudo-density matrix and decoherence functional in
consistent histories. We further explore generalised non-local games and build
pseudo-density matrices from generalised signalling games in Section 4.4. In
Section 4.5, we simplify the calculation of out-of-time-order correlations via
pseudo-density matrices. We further argue that the path integral formalism
defines correlations in a different way. Finally we provide a unified picture
under a tripartite pseudo-density matrix except the path integral formalism
and summarise our work and provide an outlook in Section 4.7.
### 4.2 Indefinite causal structures
The concept of indefinite causal structures was proposed as probabilistic
theories with non-fixed causal structures as a possible approach to quantum
gravity [86, 87]. There are different indefinite causal order approaches:
quantum combs [67, 68], operator tensors [69, 88], process matrices [70, 89],
process tensors [90, 71], and super-density operators [72, 91]. Also, Several
of the approaches are closely related [92], for example, quantum channels with
memories [93], general quantum strategies [94], multiple-time states [95, 96,
97], general boundary formalism [98], and quantum causal models [99, 100].
General quantum strategies can be taken as a game theory representation;
multiple-time states are a particular subclass of process matrices; quantum
causal models just use the process matrix formalism. Since there are clear
maps among quantum combs, operator tensors, process tensors, and process
matrices, we just take the process matrix formalism in order to learn from
causality inequalities and postselection. We will investigate its relation
with the pseudo-density matrix and show what lessons we shall learn for
pseudo-density matrices.
#### 4.2.1 Preliminaries for process matrix formalism
The process matrix formalism was originally proposed in Ref. [70] as one of
the indefinite causal structures assuming local quantum mechanics and well-
defined probabilities. The process matrix was defined to take completely
positive(CP) maps to linear probabilities. It is redefined in Ref. [101] in a
more general way as high order transformations, where the definition is
extended to take CP maps to other CP maps. Here we follow as Ref. [101]. We
define bipartite processes first; the multipartite case is obtained directly
or from Ref. [89].
For the bipartite case, consider a global past $P$ and a global future $F$.
Quantum states in the past are transformed to quantum states in the future
through a causally indefinite structure. A process is defined as a linear
transformation take two CPTP maps $\mathcal{A}:A_{I}\otimes
A_{I}^{\prime}\rightarrow A_{O}\otimes A_{O}^{\prime}$ and
$\mathcal{B}:B_{I}\otimes B_{I}^{\prime}\rightarrow B_{O}\otimes
B_{O}^{\prime}$ to a CPTP map
$\mathcal{G}_{\mathcal{A},\mathcal{B}}:A_{I}^{\prime}\otimes
B_{I}^{\prime}\otimes P\rightarrow A^{\prime}_{O}\otimes B_{O}^{\prime}\otimes
F$ without acting on the systems $A_{I}^{\prime}$, $A_{O}^{\prime}$,
$B_{I}^{\prime}$, $B_{O}^{\prime}$. Specifically, it is a transformation that
act on $P\otimes A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes F$.
We introduce the Choi-Jamiołkowski isomorphism [102, 103] to represent the
process in the matrix formalism. Recall that for a CP map
$\mathcal{M}^{A}:A_{I}\rightarrow A_{O}$, its corresponding Choi-Jamiołkowski
matrix is given as
$\mathfrak{C}(\mathcal{M})\equiv[\mathcal{I}\otimes\mathcal{M}^{A}(|\mathbbm{1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\mathbbm{1}|)]\in
A_{I}\otimes A_{O}$ with $\mathcal{I}$ as the identity map and
$|\mathbbm{1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=|\mathbbm{1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{A_{I}A_{I}}\equiv\sum_{j}\ket{j}^{A_{I}}\otimes\ket{j}^{A_{I}}\in\mathcal{H}^{A_{I}}\otimes\mathcal{H}^{A_{I}}$
is the non-normalised maximally entangled state. The inverse is given as
$\mathcal{M}(\rho^{A_{I}})=\Tr[(\rho^{A_{I}}\otimes\mathbbm{1}^{A_{O}})M^{A_{I}A_{O}}]$
where $\mathbbm{1}^{A_{O}}$ is the identity matrix on $\mathcal{H}^{A_{O}}$.
Then $A=\mathfrak{C}(\mathcal{A})$, $B=\mathfrak{C}(\mathcal{B})$, and
$G_{A,B}=\mathfrak{C}(\mathcal{G_{A,B}})$ are the corresponding CJ
representations. We have
$G_{A,B}=\Tr_{A_{I}A_{O}B_{I}B_{O}}[W^{T_{A_{I}A_{O}B_{I}B_{O}}}(A\otimes B)]$
(4.1)
where the process matrix is defined as $W\in P\otimes A_{I}\otimes
A_{O}\otimes B_{I}\otimes B_{O}\otimes F$, $T_{A_{I}A_{O}B_{I}B_{O}}$ is the
partial transposition on the subsystems $A_{I}$, $A_{O}$, $B_{I}$, $B_{O}$,
and we leave identity matrices on the rest subsystems implicit. Note that we
require that $G_{A,B}$ is a CPTP map for any CPTP maps $A$, $B$. This
condition is equivalent to the followings:
$\displaystyle W\geq 0,$ (4.2) $\displaystyle\Tr W=d_{A_{O}}d_{B_{O}}d_{P},$
(4.3) $\displaystyle W=L_{V}(W),$ (4.4)
where $L_{V}$ is defined as a projector
$\displaystyle L_{V}(W)=W-_{F}W+$
${}_{A_{O}F}W+_{B_{O}F}W-_{A_{O}B_{O}F}W-_{A_{I}A_{O}F}W+_{A_{I}A_{O}B_{O}F}W$
$\displaystyle-_{B_{I}B_{O}F}W+_{A_{I}A_{O}B_{O}F}W-_{A_{I}A_{O}B_{I}B_{O}F}W+_{PA_{I}A_{O}B_{I}B_{O}F}W.$
(4.5)
Terms that can exist in a process matrix include states, channels, channels
with memory; nevertheless, local loops, channels with local loops and global
loops are not allowed [70]. A bipartite process matrix can be fully
characterised in the Hilbert-Schmidt basis [70]. Define the signalling
directions $\preceq$ and $\npreceq$ as follows: $A\preceq B$ means $A$ is in
the causal past of $B$, $A\npreceq B$ means it is not; similarly for $\succeq$
and $\nsucceq$. Any valid bipartite process matrix $W^{A_{I}A_{O}B_{I}B_{O}}$
can be given in the Hilbert-Schmidt basis as
$W^{A_{I}A_{O}B_{I}B_{O}}=\frac{1}{d_{A_{I}}d_{B_{I}}}(\mathbbm{1}+\sigma_{A\preceq
B}+\sigma_{A\succeq B}+\sigma_{A\npreceq\nsucceq B})$ (4.6)
where the matrices $\sigma_{A\preceq B}$, $\sigma_{A\succeq B}$, and
$\sigma_{A\npreceq\nsucceq B}$ are defined by
$\displaystyle\sigma_{A\preceq B}$
$\displaystyle\equiv\sum_{ij>0}c_{ij}\sigma_{i}^{A_{O}}\sigma_{j}^{B_{I}}+\sum_{ijk>0}d_{ijk}\sigma_{i}^{A_{I}}\sigma_{j}^{A_{O}}\sigma_{k}^{B_{I}}$
(4.7) $\displaystyle\sigma_{A\succeq B}$
$\displaystyle\equiv\sum_{ij>0}e_{ij}\sigma_{i}^{A_{I}}\sigma_{j}^{B_{O}}+\sum_{ijk>0}f_{ijk}\sigma_{i}^{A_{I}}\sigma_{j}^{B_{I}}\sigma_{k}^{B_{O}}$
(4.8) $\displaystyle\sigma_{A\npreceq\nsucceq B}$
$\displaystyle\equiv\sum_{i>0}g_{i}\sigma_{i}^{A_{I}}+\sum_{i>0}h_{i}\sigma_{i}^{B_{I}}+\sum_{ij>0}l_{ij}\sigma_{i}^{A_{I}}\sigma_{j}^{B_{I}}$
(4.9)
Here $c_{ij},d_{ijk},e_{ij},f_{ijk},g_{i},h_{i},l_{ij}\in\mathbb{R}$. That is,
a bipartite process matrix of the system $AB$ is a combination of an identity
matrix, the matrices where $A$ signals to $B$, where $B$ signals to $A$, and
where $A$ and $B$ are causally separated. It is thus a linear combination of
three possible causal structures.
#### 4.2.2 Correlation analysis and causality inequalities
In this subsection, we analyse correlations in both the process matrix
formalism and the pseudo-density matrix formalism. We first take a special
case with causal order and map correlations in two formalisms to each other.
Then we consider the set of all possible causal correlations forms a causal
polytope. The facets of the causal polytope are defined as causal inequalities
and they are violated in the two formalisms with indefinite causal structures.
##### Correlation analysis
Now we analyse the relation between a process matrix and a pseudo-density
matrix in the causal order. The basic elements in a process matrix are
different laboratories, and the basic elements in a pseudo-density matrix are
different events. We map a process matrix to a pseudo-density matrix in a way
that each lab corresponds to each event.
A process matrix with a single-qubit Pauli measurement taken at each
laboratory is mapped to a finite-dimensional pseudo-density matrix. Compare
them in the bipartite case as an illustration. In the simplest temporal case,
a maximally mixed qubit evolves under the identity evolution between two
times. The process matrix for this scenario is given as
$W=\frac{\mathbbm{1}^{A_{I}}}{2}\otimes[[\mathbbm{1}]]^{A_{O}B_{I}},$ (4.11)
where
$[[\mathbbm{1}]]^{XY}=\sum_{ij}\ket{i}\bra{j}^{X}\otimes\ket{i}\bra{j}^{Y}=\frac{1}{2}(\mathbbm{1}\otimes\mathbbm{1}+X\otimes
X-Y\otimes Y+Z\otimes Z)$. At the same time, the corresponding pseudo-density
matrix is
$R=\frac{1}{4}(I\otimes I+X\otimes X+Y\otimes Y+Z\otimes
Z)=\frac{1}{2}[[\mathbbm{1}]]^{PT}=\frac{1}{2}S,$ (4.12)
where the swap operator $S=\frac{1}{2}(\mathbbm{1}\otimes\mathbbm{1}+X\otimes
X+Y\otimes Y+Z\otimes Z)=[[\mathbbm{1}]]^{PT}$, here $PT$ is the partial
transpose. For an arbitrary state $\rho$ evolving under the unitary evolution
$U$, the process matrix is given as
$W=\rho^{A_{I}}\otimes[[U]]^{A_{O}B_{I}},$ (4.13)
where $[[U]]=(\mathbbm{1}\otimes U)[[\mathbbm{1}]](\mathbbm{1}\otimes
U^{{\dagger}})$. The pseudo-density matrix is given from Ref. [40] as
$R=\frac{1}{2}(\mathbbm{1}\otimes
U)(\rho^{A}\otimes\frac{\mathbbm{1}^{B}}{2}S+S\rho^{A}\otimes\frac{\mathbbm{1}^{B}}{2})(\mathbbm{1}\otimes
U^{{\dagger}})=\frac{1}{2}(\rho^{A}\otimes\frac{\mathbbm{1}^{B}}{2}[[U]]^{PT}+[[U]]^{PT}\rho^{A}\otimes\frac{\mathbbm{1}^{B}}{2}),$
(4.14)
where the partial transpose is taken on the subsystem $A$. Now we compare the
correlations in the two formalisms and check whether they hold the same
information.
The single-qubit Pauli measurement $\sigma_{i}$ for each event in the pseudo-
density matrix has the Choi-Jamiołkowski representation as
$\Sigma^{A_{I}A_{O}}_{i}=P^{+A_{I}}_{i}\otimes
P^{+A_{O}}_{i}-P^{-A_{I}}_{i}\otimes P^{-A_{O}}_{i}$ (4.15)
where $P^{\pm}_{i}=\frac{1}{2}(\mathbbm{1}\pm\sigma_{i})$; that is, to make a
measurement $P^{\alpha}_{i}(\alpha=\pm 1)$ to the input state and project the
corresponding eigenstate to the output system. It is equivalent to
$\Sigma^{A_{I}A_{O}}_{i}=\frac{1}{2}(\mathbbm{1}^{A_{I}}\otimes\sigma_{i}^{A_{O}}+\sigma_{i}^{A_{I}}\otimes\mathbbm{1}^{A_{O}}).$
(4.16)
In the example of a single qubit $\rho$ evolving under $U$, the correlations
from the process matrix are given by
$p(\Sigma^{A_{I}A_{O}}_{i},\Sigma^{B_{I}B_{O}}_{j})=\Tr[(\Sigma^{A_{I}A_{O}}_{i}\otimes\Sigma^{B_{I}B_{O}}_{j})W]=\frac{1}{2}\Tr[\sigma_{j}U\sigma_{i}U^{{\dagger}}];$
(4.17)
while the correlations from the pseudo-density matrix are given as
$\langle\\{\sigma_{i},\sigma_{j}\\}\rangle=\frac{1}{2}\left(\Tr[\sigma_{j}U\sigma_{i}\rho
U^{{\dagger}}]+\Tr[\sigma_{j}U\rho\sigma_{i}U^{{\dagger}}]\right)=\frac{1}{2}\Tr[\sigma_{j}U\sigma_{i}U^{{\dagger}}].$
(4.18)
The last equality holds as a single-qubit $\rho$ is decomposed into
$\rho=\frac{\mathbbm{1}}{2}+\sum_{k=1,2,3}c_{k}\sigma_{k}$. The allowed
spatio-temporal correlations given by the two formalisms are the same; thus,
pseudo-density matrices and process matrices are equivalent in terms of
encoded correlations. In a general case of bipartite systems on $AB$, this
equivalence holds for each case with causal order as $A\preceq B$, $A\succeq
B$, $A\npreceq\nsucceq B$. In principle, their superpositions for arbitrary
process matrices will satisfy the correlation equivalence as well. The only
condition here is that $A$ and $B$ make Pauli measurements in their local
laboratories. Therefore, a process matrix where a single-qubit Pauli
measurement is made at each laboratory corresponds to a finite-dimensional
pseudo-density matrix since the correlations are equal.
For generalised measurements, for example, arbitrary POVMs, a process matrix
is fully mapped to the corresponding generalised pseudo-density matrix; thus,
a process matrix can be always mapped to a generalised pseudo-density matrix
in principle. The process matrix and the corresponding generalised pseudo-
density matrix just take the same measurement process in each laboratory or at
each event. The analysis for correlations is similar.
For a given set of measurements, a process matrix where the measurement is
made in each laboratory hold the same correlations as a generalised pseudo-
density matrix with the measurement made at each event. Thus, a universal
mapping from a process matrix to a pseudo-density matrix for general
measurements is established. However, a pseudo-density matrix in finite
dimensions is not necessarily mapped back to a valid process matrix. As
mentioned before, a valid process matrix excludes the possibilities for post-
selection, local loops, channels with local loops and global loops. Pseudo-
density matrices are defined operationally in terms of measurement
correlations and may allow these possibilities. We will come back to this
point in the discussion for postselection and out-of-time-order correlation
functions.
##### Causal inequalities
In the subsubsection, we introduce the causal polytope formed by the set of
correlations with a definite causal order. Its facets are defined as causal
inequalities [104]. We show that the characterisation of bipartite
correlations is consistent with the previous analysis in the pseudo-density
matrix formalism. We show that causal inequalities can be violated in both of
the process matrix formalism and the pseudo-density matrix formalism.
We follow as Ref. [104]. Recall that we denote Alice in the causal past of Bob
as $A\preceq B$. Now for simplicity, we do not consider relativistic causality
but normal Newton causality. We denote $A\prec B$ for events in Alice’s system
precedes those in Bob’s system. Then Bob cannot signal to Alice, and the
correlations satisfy that
$\forall x,y,y^{\prime},a,\quad p^{A\prec B}(a|x,y)=p^{A\prec
B}(a|x,y^{\prime}),$ (4.19)
where $p^{A\prec B}(a|x,y^{(^{\prime})})=\sum_{b}p^{A\prec
B}(a,b|x,y^{(^{\prime})})$. Similarly, for $B\prec A$, Alice cannot signal to
Bob that
$\forall x,x^{\prime},y,b,\quad p^{A\prec B}(b|x,y)=p^{A\prec
B}(b|x^{\prime},y),$ (4.20)
where $p^{A\prec B}(b|x^{(^{\prime})},y)=\sum_{a}p^{A\prec
B}(a,b|x^{(^{\prime})},y)$.
Correlations of the order $A\prec B$ satisfy the properties of non-negativity
and normalisation, and the no-signaling-to-Alice condition:
$\displaystyle p^{A\prec B}(a,b|x,y)\geq$ $\displaystyle 0,\quad\forall
x,y,a,b;$ (4.21) $\displaystyle\sum_{a,b}p^{A\prec B}(a,b|x,y)=$
$\displaystyle 1,\quad\quad\forall x,y;$ (4.22) $\displaystyle p^{A\prec
B}(a|x,y)=p^{A\prec B}(a|$ $\displaystyle x,y^{\prime}),\quad\forall
x,y,y^{\prime},a.$ (4.23)
Via these linear conditions, the set of correlations $p^{A\prec B}$ forms a
convex polytope. Similarly for the set of correlations $p^{B\prec A}$. The
correlations are defined as causal if it is compatible with $A\prec B$ with
probability $q$ and $B\prec A$ with probability $1-q$, that is, for
$q\in[0,1]$,
$p(a,b|x,y)=qp^{A\prec B}(a,b|x,y)+(1-q)p^{B\prec A}(a,b|x,y),$ (4.24)
where $p^{A\prec B}$ and $p^{B\prec A}$ are non-negative and normalised to 1.
Then the set of causal correlations is the convex hull of the sets of
correlations $p^{A\prec B}$ and $p^{B\prec A}$ and constitutes a causal
polytope.
Suppose that Alice and Bob’s inputs have $m_{A}$ and $m_{B}$ possible values,
their outputs have $k_{A}$ and $k_{B}$ values respectively. The polytope of
$p^{A\prec B}$ has $k_{A}^{m_{A}}k_{B}^{m_{A}m_{B}}$ vertices, of dimension
$m_{A}m_{B}(k_{A}k_{B}-1)-m_{A}(m_{B}-1)(k_{A}-1)$. The polytope of $p^{B\prec
A}$ has $k_{A}^{m_{A}m_{B}}k_{B}^{m_{B}}$ vertices, of dimension
$m_{A}m_{B}(k_{A}k_{B}-1)-(m_{A}-1)m_{B}(k_{A}-1)$. The causal polytope has
$k_{A}^{m_{A}}k_{B}^{m_{A}m_{B}}+k_{A}^{m_{A}m_{B}}k_{B}^{m_{B}}-k_{A}^{m_{A}}k_{B}^{m_{B}}$
vertices, of dimension $m_{A}m_{B}(k_{A}k_{B}-1)$. Consider the bipartite
correlations where a qubit evolves between two times $t_{A}$ and $t_{B}$. We
make a Pauli measurement at each time to record correlations. Given an initial
state of the qubit, we have $m_{A}=m_{B}=1$, $k_{A}=k_{B}=2$. The polytope of
$p^{A\prec B}$ has 4 vertices in 3 dimensions. The same as $p^{B\prec A}$ and
the causal polytope. This result is consistent with the characterisation by
the pseudo-density matrix formalism in Ref. [40].
Now we characterise the causal polytope with $m_{A}=m_{B}=k_{A}=k_{B}=2$. It
has 112 vertices and 48 facets. 16 of the facets are trivial, which imply the
non-negativity of the correlations $p(a,b|x,y)\geq 0$. If we relabel the
inputs and outputs of the systems, the rest of facets are divided into two
groups, each with 16 facets:
$\frac{1}{4}\sum_{x,y,a,b}\delta_{a,y}\delta_{b,x}p(a,b|x,y)\leq\frac{1}{2},$
(4.25)
and
$\frac{1}{4}\sum_{x,y,a,b}\delta_{x(a\oplus y),0}\delta_{y(b\oplus
x),0}p(a,b|x,y)\leq\frac{3}{4},$ (4.26)
where $\delta_{i,j}$ is the Kronecker delta function and $\oplus$ is the
addition modulo 2. They are interpreted into the bipartite "guess your
neighbour’s input" (GYNI) games and "lazy GYNI" (LGYNI) games [104].
Then we show the violation of causal inequalities via process matrix formalism
and pseudo-density matrix formalism. In the process matrix formalism, we take
the global past $P$, the global future $F$, Alice’s ancilla systems
$A_{I}^{\prime}$, $A_{O}^{\prime}$ and Bob’s ancilla systems $B_{I}^{\prime}$,
$B_{O}^{\prime}$ trivial. Then the process matrix correlations are given as
$p(a,b|x,y)=\Tr[W^{T_{A_{I}A_{O}B_{I}B_{O}}}A_{a|x}\otimes B_{b|y}].$ (4.27)
Consider the process matrix
$W=\frac{1}{4}\left[\mathbbm{1}^{\otimes
4}+\frac{Z^{A_{I}}Z^{A_{O}}Z^{B_{I}}\mathbbm{1}^{B_{O}}+Z^{A_{I}}\mathbbm{1}^{A_{O}}X^{B_{I}}X^{B_{O}}}{\sqrt{2}}\right].$
(4.28)
We choose the operations as (here slightly different from Ref. [104]):
$\displaystyle A_{0|0}$ $\displaystyle=B_{0|0}=0,$ (4.29) $\displaystyle
A_{1|0}$ $\displaystyle=B_{1|0}=(\ket{00}+\ket{11})(\bra{00}+\bra{11}),$
(4.30) $\displaystyle A_{0|1}$
$\displaystyle=B_{0|1}=\frac{1}{2}\ket{0}\bra{0}\otimes\ket{0}\bra{0}+\frac{1}{2}\ket{0}\bra{0}\otimes\ket{1}\bra{1},$
(4.31) $\displaystyle A_{1|1}$
$\displaystyle=B_{1|1}=\frac{1}{2}\ket{1}\bra{1}\otimes\ket{0}\bra{0}+\frac{1}{2}\ket{1}\bra{1}\otimes\ket{1}\bra{1}.$
(4.32)
Then
$\displaystyle p_{GYNI}$
$\displaystyle=\frac{5}{16}(1+\frac{1}{\sqrt{2}})\approx 0.5335>\frac{1}{2},$
(4.33) $\displaystyle p_{LGYNI}$
$\displaystyle=\frac{5}{16}(1+\frac{1}{\sqrt{2}})+\frac{1}{4}\approx
0.7835>\frac{3}{4}.$ (4.34)
For a pseudo-density matrix, we consider a similar strategy. Alice has two
systems $X$ and $A$, where $X$ is the ancillary system prepare with
$\ket{x}\bra{x}$. Bob has two systems $Y$ and $B$, where $Y$ is the ancillary
system prepare with $\ket{y}\bra{y}$. Given a pseudo-density matrix
$R=\frac{1}{4}\left[\ket{x}\bra{x}^{X}\otimes\mathbbm{1}^{A}\otimes\ket{y}\bra{y}^{Y}\otimes\mathbbm{1}^{B}+\frac{Z^{X}Z^{A}Z^{Y}\mathbbm{1}^{B}+Z^{X}\mathbbm{1}^{A}X^{Y}X^{B}}{\sqrt{2}}\right],$
(4.35)
we choose the operations as before and gain the success probabilities as
$\displaystyle p_{GYNI}$
$\displaystyle=\frac{5}{16}(1+\frac{1}{\sqrt{2}})\approx 0.5335>\frac{1}{2},$
(4.36) $\displaystyle p_{LGYNI}$
$\displaystyle=\frac{5}{16}(1+\frac{1}{\sqrt{2}})+\frac{1}{4}\approx
0.7835>\frac{3}{4}.$ (4.37)
Again the causal inequalities are violated. This example also highlights
another relationship for the mapping between a process matrix and a pseudo-
density matrix. Instead of an input system and an output system in a process
matrix, the corresponding pseudo-density matrix has an additional ancillary
system for each event. Thus, a process matrix which makes a measurement and
reprepares the state in one laboratory describes the same probabilities as a
pseudo-density matrix with ancillary systems which makes a measurement and
reprepares the state at each event. Another mapping from a process matrix to a
pseudo-density matrix is established by introducing ancillary systems.
#### 4.2.3 Postselection and closed timelike curves
Postselection is conditioning on the occurrence of certain event in
probability theory, or conditioning upon certain measurement outcome in
quantum mechanics. It allows a quantum computer to choose the outcomes of
certain measurements and increases its computational power significantly. In
this subsection, we take the view from postselection and show that a
particular subset of postselected two-time states correspond to process
matrices in indefinite causal order. Postselected closed timelike curves are
presented as a special case.
##### Two-time quantum states
In this subsubsection, we review the two-time quantum states approach [97]
which fixes initial states and final states independent at two times. The two-
time quantum state takes its operational meaning from postselection. Consider
that Alice prepares a state $\ket{\psi}$ at the initial time $t_{1}$. Between
the initial time $t_{1}$ and the final time $t_{2}$, she performs arbitrary
operations in her lab. Then she measures an observable $O$ at the final time
$t_{2}$. The observable $O$ has a non-degenerate eigenstate $\ket{\phi}$.
Taking $\ket{\phi}$ as the final state, Alice discards the experiment if the
measurement of $O$ does not give the eigenvalue corresponding to the
eigenstate $\ket{\phi}$.
Consider that Alice makes a measurement by the set of Kraus operators
$\\{\hat{E}_{a}=\sum_{k,l}\beta_{a,kl}\ket{k}\bra{l}\\}$ between $t_{1}$ and
$t_{2}$. Note that $\\{\hat{E}_{a}\\}$ are normalised as
$\sum_{a}\hat{E}_{a}^{{\dagger}}\hat{E}_{a}=\mathbbm{1}$. The probability for
Alice to gain the outcome $a$ under the pre- and post-selection is given as
$p(a)=\frac{|\bra{\phi}\hat{E}_{a}\ket{\psi}|^{2}}{\sum_{a^{\prime}}|\bra{\phi}\hat{E}_{a^{\prime}}\ket{\psi}|^{2}}.$
(4.38)
Now define the two-time state and the two-time version of Kraus operator as
$\displaystyle\Phi=_{\mathcal{A}_{2}}\bra{\phi}\otimes\ket{\psi}^{\mathcal{A}_{1}}$
$\displaystyle\in\mathcal{H}_{\mathcal{A}_{2}}\otimes\mathcal{H}^{\mathcal{A}_{1}},$
$\displaystyle
E_{a}=\sum_{kl}\beta_{a,kl}\ket{k}^{\mathcal{A}_{2}}\otimes_{\mathcal{A}_{1}}\bra{l}$
$\displaystyle\in\mathcal{H}^{\mathcal{A}_{2}}\otimes\mathcal{H}_{\mathcal{A}_{1}},$
(4.39)
where the two-time version of Kraus operator is denoted by $E_{a}$ without the
hat. An arbitrary pure two-time state takes the form
$\Phi=\sum\alpha_{ij}\
{}_{\mathcal{A}_{2}}\bra{i}\otimes\ket{j}^{\mathcal{A}_{1}}\in\mathcal{H}_{\mathcal{A}_{2}}\otimes\mathcal{H}^{\mathcal{A}_{1}}.$
(4.40)
Then the probability to obtain $a$ as the outcome is given as
$p(a)=\frac{|\Phi\cdot E_{a}|^{2}}{\sum_{a^{\prime}}|\Phi\cdot
E_{a^{\prime}}|^{2}}.$ (4.41)
A two-time density operator $\eta$ is given as
$\eta=\sum_{r}p_{r}\Phi_{r}\otimes\Phi_{r}^{{\dagger}}\in\mathcal{H}_{\mathcal{A}_{2}}\otimes\mathcal{H}^{\mathcal{A}_{1}}\otimes\mathcal{H}_{\mathcal{A}_{1}^{{\dagger}}}\otimes\mathcal{H}^{\mathcal{A}_{2}^{{\dagger}}}.$
(4.42)
Consider a coarse-grained measurement
$J_{a}=\sum_{\mu}E_{a}^{\mu}\otimes
E_{a}^{\mu{\dagger}}\in\mathcal{H}^{\mathcal{A}_{2}}\otimes\mathcal{H}_{\mathcal{A}_{1}}\otimes\mathcal{H}^{\mathcal{A}_{1}^{{\dagger}}}\otimes\mathcal{H}_{\mathcal{A}_{2}^{{\dagger}}}$
(4.43)
where the outcome $a$ corresponds to a set of Kraus operators
$\\{\hat{E}_{a}^{\mu}\\}$. Then the probability to obtain $a$ as the outcome
is given as
$p(a)=\frac{\eta\cdot J_{a}}{\sum_{a^{\prime}}\eta\cdot J_{a^{\prime}}}.$
(4.44)
##### Connection between process matrix and pseudo-density matrix under post-
selection
Now consider postselection applied to ordinary quantum theory. It is known
that a particular subset of postselected two-time states in quantum mechanics
give the form of process matrices within indefinite causal structures [97].
Here we first give a simple explanation for this fact and further analyse the
relation between a process matrix and a pseudo-density matrix from the view of
postselection.
For an arbitrary bipartite process matrix
$W\in\mathcal{H}^{A_{I}}\otimes\mathcal{H}^{A_{O}}\otimes\mathcal{H}^{B_{I}}\otimes\mathcal{H}^{B_{O}}$,
we can expand it in some basis:
$W^{A_{I}A_{O}B_{I}B_{O}}=\sum_{ijkl,pqrs}w_{ijkl,pqrs}\ket{ijkl}\bra{pqrs}.$
(4.45)
For the elements in each Hilbert space, we map them to the corresponding parts
in a bipartite two-time state. For example, we map the input Hilbert space of
Alice to the bra and ket space of Alice at time $t_{1}$, and similarly for the
output Hilbert space for $t_{2}$. That is,
$\displaystyle\ket{i}\bra{p}$
$\displaystyle\in\mathcal{L}(\mathcal{H}^{A_{I}})\rightarrow\bra{p}\otimes\ket{i}\in\mathcal{H}_{A_{1}^{{\dagger}}}\otimes\mathcal{H}^{A_{1}}$
(4.46) $\displaystyle\ket{j}\bra{q}$
$\displaystyle\in\mathcal{L}(\mathcal{H}^{A_{O}})\rightarrow\bra{q}\otimes\ket{j}\in\mathcal{H}_{A_{2}}\otimes\mathcal{H}^{A_{2}^{{\dagger}}}$
(4.47)
Thus, a two-time state
$\eta_{W^{A_{1}A_{2}}}\in\mathcal{H}_{A_{2}}\otimes\mathcal{H}^{A_{1}}\otimes\mathcal{H}^{A_{2}^{{\dagger}}}\otimes\mathcal{H}_{A_{1}^{{\dagger}}}$
is equivalent to a process matrix for a single laboratory $W^{A_{I}A_{O}}$.
The connection with pre- and post-selection suggests one more interesting
relationship between a process matrix and a pseudo-density matrix. For a
process matrix, if we consider the input and output Hilbert spaces at two
times, we can map it to a two-time state. That is, we connect a process matrix
with single laboratory to a two-time state. A pseudo-density matrix needs two
Hilbert spaces to represent two times. For a two-time state $\eta_{12}$, the
corresponding pseudo-density matrix $R_{12}$ has the same marginal single-time
states, i.e., $\Tr_{1}\eta_{12}=\Tr_{1}R_{12}$ and
$\Tr_{2}\eta_{12}=\Tr_{2}R_{12}$. Then we find a map between a process matrix
for a single event and a pseudo-density matrix for two events. Note that in
the previous subsections, we have mapped a process matrix for two events to a
pseudo-density matrix with half Hilbert space for two events, and mapped a
process matrix for two events to a pseudo-density matrix with two Hilbert
spaces at each of two events. This suggests that the relationship between a
process matrix and a pseudo-density matrix is non-trivial with a few possible
mappings.
One question arising naturally here concerns the pseudo-density matrices with
postselection. The definitions for finite-dimensional and Gaussian pseudo-
density matrices guarantee that under the partial trace, the marginal states
at any single time will give the state at that time. In particular, tracing
out all other times in a pseudo-density matrix, we get the final state at the
final time. On the one hand, we may think that pseudo-density matrix
formulation is kind of time-symmetric. On the other hand, the final state is
fixed by evolution; that implies that we cannot assign an arbitrary final
state, making it difficult for the pseudo-density matrix to be fully time-
symmetric. For other generalisation of pseudo-density matrices like position
measurements and weak measurements, the property for fixed final states does
not hold. Nevertheless, we may define a new type of pseudo-density matrices
with postselection. We assign the final measurement to be the projection to
the final state and renormalise the probability. For example, a qubit in the
initial state $\rho$ evolves under a CPTP map
$\mathcal{E}:\rho\rightarrow\mathcal{E}(\rho)$ and then is projected on the
state $\eta$. We may construct the correlations
$\langle\\{\sigma_{i},\sigma_{j},\eta\\}\rangle$ as
$\langle\\{\sigma_{i},\sigma_{j},\eta\\}\rangle=\sum_{\alpha,\beta=\pm
1}\alpha\beta\Tr[\eta P^{\beta}_{j}\mathcal{E}(P^{\alpha}_{i}\rho
P^{\alpha}_{i})P^{\beta}_{j}]/p_{ij}(\eta),$ (4.48)
where $P^{\alpha}_{i}=\frac{1}{2}(\mathbbm{1}+\alpha\sigma_{i})$ and
$p_{ij}(\eta)=\sum_{\alpha,\beta=\pm 1}\Tr[\eta
P^{\beta}_{j}\mathcal{E}(P^{\alpha}_{i}\rho P^{\alpha}_{i})P^{\beta}_{j}]$.
Then the pseudo-density matrix with postselection is given as
$R=\frac{1}{4}\sum_{i,j=0}^{3}\langle\\{\sigma_{i},\sigma_{j},\eta\\}\rangle\sigma_{i}\otimes\sigma_{j}\otimes\eta.$
(4.49)
We further conclude the relation between a process matrix and a pseudo-density
matrices with postselection. A process matrix with postselection for a
laboratory is operationally equivalent to a tripartite postselected pseudo-
density matrix.
##### Post-selected closed timelike curves
We briefly discuss postselected closed timelike curves before we move on to a
summary. Closed timelike curves (CTCs), after being pointed out by Gödel to be
allowed in general relativity [105], have always been arising great interests.
Deutsch [106] proposed a circuit method to study them and started an
information theoretic point of view. Deustch’s CTCs are shown to have many
abnormal properties violated by ordinary quantum mechanics. For example, they
are nonunitary, nonlinear, and allow quantum cloning [107, 108]. Several
authors [109, 110, 111, 112] later proposed a model for closed timelike curves
based on postselected teleportation. It is studied that process matrices
correspond to a particular linear version of postselected closed timelike
curves [113]. In pseudo-density matrices we can consider a system evolves in
time and back; that is the case for calculating out-of-time-order correlation
functions we will introduce later, and different from closed timelike curves
as there is no loop. However, the black hole final state proposal in later
section is very much related. Now we briefly introduce postselected closed
timelike curves and its representation in pseudo-density matrices.
Postselected closed timelike curves can be seen as a “chronology-respecting”
system $S$ and a CTC system $A$ evolving under a unitary $U_{SA}$. Consider
the CTC system $A$ is part of the maximally entangled state
$\ket{\Phi}_{AB}=\sum_{i=0}^{d-1}\frac{1}{\sqrt{d}}\ket{i}\ket{i}$. More
specifically, the system $S$ and $A$ evolve under the unitary $U$ and then we
project the two systems $AB$ onto the state $\ket{\Phi}$ and renormalise the
probability. One assumes that this projection is certain with probability $1$.
Then for the system $S$, $\rho_{S}$ goes in to the state
$\frac{C\rho_{S}C^{{\dagger}}}{\Tr[C\rho_{S}C^{{\dagger}}]}$ where
$C=\Tr_{A}U_{SA}$. In this way, we create a quantum channel from the future to
the past and the CTC qubit goes back in time.
Here we illustrate post-selected closed timelike curves by the pseudo-density
matrices with postselection. It is a two-time process with a postselection. We
assume that at the initial time systems $S$ and $AB$ are prepared. We make a
measurement $P_{i}$. After the unitary evolution $U_{SA}$, we make another
measurement $P_{j}$. Then we project the state to $\ket{\Phi}_{AB}$. The
correlations are represented by
$\langle\\{P_{i},P_{j},\ket{\Phi}\bra{\Phi}_{AB}\\}\rangle=\frac{\sum_{\alpha,\beta}\alpha\beta
p_{ij}^{\alpha\beta}}{\sum_{\alpha,\beta}p_{ij}^{\alpha\beta}},$ (4.50)
where
$p_{ij}^{\alpha\beta}=\Tr[\mathbbm{1}_{S}\otimes\ket{\Phi}\bra{\Phi}_{AB}P_{j}^{\beta}(U_{SA}\otimes\mathbbm{1}_{B})P_{i}^{\alpha}(\rho_{S}\otimes\ket{\Phi}\bra{\Phi}_{AB})P_{i}^{\alpha{\dagger}}(U^{{\dagger}}_{SA}\otimes\mathbbm{1}_{B})P_{j}^{\beta{\dagger}}].$
(4.51)
Here $P_{i}^{\alpha}$ is denoted for the measurement $P_{i}$ with the outcome
$\alpha$ and $P_{j}^{\beta}$ for the measurement $P_{j}$ with the outcome
$\beta$. For simplicity, we consider $P_{i}=P_{j}=\mathbbm{1}$. Then
$\displaystyle p$
$\displaystyle=\Tr[\mathbbm{1}_{S}\otimes\ket{\Phi}\bra{\Phi}_{AB}(U_{SA}\otimes\mathbbm{1}_{B})(\rho_{S}\otimes\ket{\Phi}\bra{\Phi}_{AB})(U^{{\dagger}}_{SA}\otimes\mathbbm{1}_{B})]$
$\displaystyle=\frac{1}{d^{2}}\Tr[C\rho_{S}C^{{\dagger}}]$ (4.52)
where $C=\Tr_{A}U_{SA}$. The result is consistent with Ref. [108]. However,
the role of quantum correlations plays in the closed timelike curves is still
an open problem.
#### 4.2.4 Summary of the relation between pseudo-density matrix and
indefinite causal structures
In this subsection, we have introduced the relation between pseudo-density
matrices and indefinite causal structures. We argue that the pseudo-density
matrix formalism belongs to indefinite causal structures. So far, all other
indefinite causal structures to our knowledge use a tensor product of both
input and output Hilbert spaces, while a pseudo-density matrix only assumes a
single Hilbert space. For a simple example of a qudit at two times, the
dimension used in other indefinite causal structures is $d^{4}$ but for
pseudo-density matrix it is $2d^{2}$. Though other indefinite causal
structures assume a much larger Hilbert space, pseudo-density matrix should
not be taken as a subclass of any indefinite causal structures which already
exist. There are certain non-trivial relation between pseudo-density matrices
and other indefinite causal structures. As we can see from the previous
subsections, it is possible to map a process matrix to a corresponding pseudo-
density matrix in three different ways: one-lab to one-event direct map, one-
lab to one-event with double Hilbert spaces map, and one-lab to two-event map.
###### Claim 1.
A process matrix and the corresponding pseudo-density matrix allow the same
correlations or probabilities in three different mappings.
One obvious difference between a process matrix and a pseudo-density matrix is
that, for each laboratory, a process matrix measures and reprepares a state
while a pseudo-density matrix usually only makes a measurement and the state
evolves into its eigenstate for each eigenvalue with the corresponding
probability. The correlations given by process matrices and pseudo-density
matrices are also the same. Examples in postselection and closed time curves
suggest further similarities. In general, we can understand that the pseudo-
density matrix is defined in an operational way which does not specify the
causal order, thus belongs to indefinite causal structures. We borrow the
lessons from process matrices here to investigate pseudo-density matrices
further. Maybe it will be interesting to derive a unified indefinite causal
structure which takes the advantage of all existing ones.
Nevertheless, the ultimate goal of indefinite causal order towards quantum
gravity is still far reaching. So far, all indefinite causal structures are
linear superpositions of causal structures; will that be enough for quantising
gravity? It is generally believed among indefinite causal structure community
that what is lacking in quantum gravity is the quantum uncertainty for
dynamical causal structures suggested by general relativity. The usual causal
order may be changed under this quantum uncertainty and there is certain
possibility for a superposition of causal orders and even beyond.
Generalisation to relativistic quantum field theory and quantum gravity
remains to be a very exciting open problem.
### 4.3 Consistent histories
In this section we first review consistent histories and then explore the
relation between pseudo-density matrices and consistent histories.
#### 4.3.1 Preliminaries for consistent histories
Consistent histories, or decoherent histories, is an interpretation for
quantum theory, proposed by Griffiths [73, 74], Gell-Mann and Hartle [75, 76],
and Omnes [77]. The main idea is that a history, understood as a sequence of
events at successive times, has a consistent probability with other histories
in a closed system. The probabilities assigned to histories satisfy the
consistency condition to avoid the interference between different histories
and that set of histories are called consistent histories [114, 115].
Consider a set of projection operators $\\{P_{\alpha}\\}$ which are exhaustive
and mutually exclusive:
$\sum_{\alpha}P_{\alpha}=\mathbbm{1},\qquad
P_{\alpha}P_{\beta}=\delta_{\alpha\beta}P_{\beta},$ (4.53)
where the range of $\alpha$ may be finite, infinite or even continuous. For
each $P_{\alpha}$ and a system in the state $\rho$, the event $\alpha$ is said
to occur if $P_{\alpha}\rho P_{\alpha}=\rho$ and not to occur if
$P_{\alpha}\rho P_{\alpha}=0$. The probability of the occurrence of the event
$\alpha$ is given by
$p(\alpha)=\Tr[P_{\alpha}\rho P_{\alpha}].$ (4.54)
A projection of the form $P_{\alpha}=\ket{\alpha}\bra{\alpha}$
($\\{\ket{\alpha}\\}$ is complete) is called completely fine-grained, which
corresponds to the precise measurement of a complete set of commuting
observables. Otherwise, for imprecise measurements or incomplete sets, the
projection operator is called coarse-grained. Generally it takes the form
$\bar{P}_{\bar{\alpha}}=\sum_{\alpha\in\bar{\alpha}}P_{\alpha}$.
In the Heisenberg picture, the operators for the same observables $P$ at
different times are related by
$P(t)=\exp(iHt/\hbar)P(0)\exp(-iHt/\hbar),$ (4.55)
with $H$ as the Hamiltonian of the system. Then the probability of the
occurrence of the event $\alpha$ at time $t$ is
$p(\alpha)=\Tr[P_{\alpha}(t)\rho P_{\alpha}(t)].$ (4.56)
Now we consider how to assign probabilities to histories, that is, to a
sequence of events at successive times. Suppose that the system is in the
state $\rho$ at the initial time $t_{0}$. Consider a set of histories
$[\alpha]=[\alpha_{1},\alpha_{2},\cdots,\alpha_{n}]$ consisting of $n$
projections $\\{P^{k}_{\alpha_{k}}(t_{k})\\}_{k=1}^{n}$ at times
$t_{1}<t_{2}<\cdots<t_{n}$. Here the subscript $\alpha_{k}$ allows for
different types of projections, for example, a position projection at $t_{1}$
and a momentum projection at $t_{2}$. Then the decoherence functional is
defined as
$D([\alpha],[\alpha^{\prime}])=\Tr[P^{n}_{\alpha_{n}}(t_{n})\cdots
P^{1}_{\alpha_{1}}(t_{1})\rho P^{1}_{\alpha^{\prime}_{1}}(t_{1})\cdots
P^{n}_{\alpha^{\prime}_{n}}(t_{n})],$ (4.57)
where
$P^{k}_{\alpha_{k}}(t_{k})=e^{i(t_{k}-t_{0})H}P^{k}_{\alpha_{k}}e^{-i(t_{k}-t_{0})H}.$
(4.58)
It is important in consistent histories because probabilities can be assigned
to histories when the decoherence functional is diagonal. It is easy to check
that
$\displaystyle
D([\alpha],[\alpha^{\prime}])=D([\alpha^{\prime}],[\alpha])^{*},$ (4.59)
$\displaystyle\sum_{[\alpha]}\sum_{[\alpha^{\prime}]}D([\alpha],[\alpha^{\prime}])=\Tr\rho=1.$
(4.60)
The diagonal elements are the probabilities for the histories
$(\rho,t_{0})\rightarrow(\alpha_{1},t_{1})\rightarrow\cdots\rightarrow(\alpha_{n},t_{n})$:
$p(\alpha_{1},\alpha_{2},\dots,\alpha_{n})=D(\alpha_{1},\alpha_{2},\dots,\alpha_{n}|\alpha_{1},\alpha_{2},\dots,\alpha_{n})=D([\alpha],[\alpha])$
(4.61)
Until now, we considered fine-grained projections $P^{k}_{\alpha_{k}}$ for
fine-grained histories. The coarse-grained histories are characterised by the
coarse-grained projections $\bar{P}^{k}_{\bar{\alpha}_{k}}$. To satisfy the
probability sum rules, the probability for a coarse-grained history is the sum
of the probabilities for its fine-grained histories. That is,
$p(\bar{\alpha}_{1},\bar{\alpha}_{2},\dots,\bar{\alpha}_{n})=\sum_{[\alpha]\in[\bar{\alpha}]}p(\alpha_{1},\alpha_{2},\dots,\alpha_{n}),$
(4.62)
where
$\sum_{[\alpha]\in[\bar{\alpha}]}=\sum_{\alpha_{1}\in\bar{\alpha}_{1}}\sum_{\alpha_{2}\in\bar{\alpha}_{2}}\cdots\sum_{\alpha_{n}\in\bar{\alpha}_{n}}.$
(4.63)
On the other hand, we gain the decoherence functional for coarse-grained
histories by directly summing over the fine-grained projections as
$D([\bar{\alpha}],[\bar{\alpha}^{\prime}])=\sum_{[\alpha]\in[\bar{\alpha}]}\sum_{[\alpha^{\prime}]\in[\bar{\alpha^{\prime}}]}D([\alpha],[\alpha^{\prime}]).$
(4.64)
For the diagonal terms,
$D([\bar{\alpha}],[\bar{\alpha}])=\sum_{[\alpha]\in[\bar{\alpha}]}D([\alpha],[\alpha])+\sum_{[\alpha]\neq[\alpha^{\prime}],[\alpha]\in[\bar{\alpha}]}\sum_{[\alpha^{\prime}]\in[\bar{\alpha^{\prime}}]}D([\alpha],[\alpha^{\prime}]),$
(4.65)
where $[\alpha]\neq[\alpha^{\prime}]$ means
$\alpha_{k}\neq\alpha^{\prime}_{k}$ for at least one $k$.
To obey the probability sum rules that all probabilities are non-negative and
summed to $1$, the sufficient and necessary condition is
$\real[D(\alpha_{1},\alpha_{2},\dots,\alpha_{n}|\alpha^{\prime}_{1},\alpha^{\prime}_{2},\dots,\alpha^{\prime}_{n})]=p(\alpha_{1},\alpha_{2},\dots,\alpha_{n})\delta_{\alpha_{1}\alpha^{\prime}_{1}}\cdots\delta_{\alpha_{n}\alpha^{\prime}_{n}}.$
(4.66)
Eqn. (4.66) is called the consistency condition or decoherence condition. Sets
of histories obeying the condition are referred to consistent histories or
decoherent histories. A stronger version of consistency condition is
$D(\alpha_{1},\alpha_{2},\dots,\alpha_{n}|\alpha^{\prime}_{1},\alpha^{\prime}_{2},\dots,\alpha^{\prime}_{n})=p(\alpha_{1},\alpha_{2},\dots,\alpha_{n})\delta_{\alpha_{1}\alpha^{\prime}_{1}}\cdots\delta_{\alpha_{n}\alpha^{\prime}_{n}}.$
(4.67)
The decoherence functional has a path integral representation. With
configuration space variables $q^{i}(t)$ and the action $S[q^{i}]$,
$D([\alpha],[\alpha^{\prime}])=\int_{[\alpha]}\mathcal{D}q^{i}\int_{[\alpha^{\prime}]}\mathcal{D}q^{i^{\prime}}\exp(iS[q^{i}]-iS[q^{i^{\prime}}])\delta(q_{f}^{i}-q_{f}^{i^{\prime}})\rho(q_{0}^{i},q_{0}^{i^{\prime}}),$
(4.68)
where the two paths $q^{i}(t)$, $q^{i^{\prime}}(t)$ begin at $q^{i}_{0}$,
$q^{i^{\prime}}_{0}$ respectively at $t_{0}$ and end at
$q^{i}_{f}=q^{i^{\prime}}_{f}$ at $t_{f}$, and correspond to the projections
$P^{k}_{\alpha_{k}}$, $P^{k}_{\alpha^{\prime}_{k}}$ made at time $t_{k}$
($k=1,2,\dots n$).
#### 4.3.2 Temporal correlations in terms of decoherence functional
The relation with the $n$-qubit pseudo-density matrix is arguably obvious. For
example, consider an $n$-qubit pseudo-density matrix as a single qubit
evolving at $n$ times. For each event, we make a single-qubit Pauli
measurement $\sigma_{i_{k}}$ at the time $t_{k}$. We can separate the
measurement $\sigma_{i_{k}}$ into two projection operators
$P_{i_{k}}^{+1}=\frac{1}{2}(I+\sigma_{i_{k}})$ and
$P_{i_{k}}^{-1}=\frac{1}{2}(I-\sigma_{i_{k}})$ with its outcomes $\pm 1$.
Corresponding to the history picture, each pseudo-density event with the
measurement $\sigma_{i_{k}}$ corresponds to two history events with
projections $P_{i_{k}}^{\alpha_{k}}(\alpha_{k}=\pm 1)$. A pseudo-density
matrix is built upon measurement correlations
$\langle\\{\sigma_{i_{k}}\\}_{k=1}^{n}\rangle$. Theses correlations can be
given in terms of decoherence functionals as
$\displaystyle\langle\\{\sigma_{i_{k}}\\}_{k=1}^{n}\rangle$
$\displaystyle=\sum_{\alpha_{1},\dots,\alpha_{n}}\alpha_{1}\cdots\alpha_{n}\Tr[P_{i_{n}}^{\alpha_{n}}U_{n-1}\cdots
U_{1}P_{i_{1}}^{\alpha_{1}}\rho P_{i_{1}}^{\alpha_{1}}U_{1}^{{\dagger}}\cdots
U_{n-1}^{{\dagger}}P_{i_{n}}^{\alpha_{n}}]$
$\displaystyle=\sum_{\alpha_{1},\dots,\alpha_{n}}\alpha_{1}\cdots\alpha_{n}p(\alpha_{1},\dots,\alpha_{n})$
$\displaystyle=\sum_{\alpha_{1},\dots,\alpha_{n}}\alpha_{1}\cdots\alpha_{n}D([\alpha],[\alpha]),$
(4.69)
where $D([\alpha],[\alpha])$ is the diagonal terms of decoherence functional
with $[\alpha]=[\alpha_{1},\dots,\alpha_{n}]$. Note that here only diagonal
decoherence functionals are taken into account, which coincides with the
consistency condition.
Similar relations hold for the Gaussian spacetime states. For each event, we
make a single-mode quadrature measurement $\hat{q}_{k}$ or $\hat{p}_{k}$ at
time $t_{k}$. We can separate the measurement $\hat{x}_{k}=\int
x_{k}\ket{x_{k}}\bra{x_{k}}\textrm{d}x_{k}$ into projection operators
$\ket{x_{k}}\bra{x_{k}}$ with outcomes $x_{k}$. Then each Gaussian event with
the measurement $\hat{x}_{k}$ corresponds to infinite and continuous history
events with projections $\ket{x_{k}}\bra{x_{k}}$.
$\displaystyle\langle\\{x_{k}\\}_{k=1}^{n}\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\cdots\textrm{d}x_{n}x_{1}\cdots
x_{n}$ $\displaystyle\qquad\qquad\Tr[\ket{x_{n}}\bra{x_{n}}U_{n-1}\cdots
U_{1}\ket{x_{1}}\bra{x_{1}}\rho\ket{x_{1}}\bra{x_{1}}U_{1}^{{\dagger}}\cdots
U_{n-1}^{{\dagger}}\ket{x_{n}}\bra{x_{n}}]$
$\displaystyle=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\cdots\textrm{d}x_{n}x_{1}\cdots
x_{n}p(x_{1},\dots,x_{n})$
$\displaystyle=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\cdots\textrm{d}x_{n}x_{1}\cdots
x_{n}D([x],[x]),$ (4.70)
where $D([x],[x])$ is the diagonal terms of decoherence functional with
$[x]=[x_{1},\dots,x_{n}]$.
For general spacetime states for continuous variables, we make a single-mode
measurement $T(\alpha_{k})$ at time $t_{k}$ for each event. It separates into
two projection operators $P^{+1}(\alpha_{k})$ and $P^{-1}(\alpha_{k})$, then
it follows as the $n$-qubit case.
The interesting part is to apply the lessons from consistent histories to the
generalised pseudo-density matrix formulation with general measurements. We
have argued that the spacetime density matrix can be expanded diagonally in
terms of position measurements as
$\rho=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\cdots\textrm{d}x_{n}p(x_{1},\cdots,x_{n})\ket{x_{1}}\bra{x_{1}}\otimes\cdots\otimes\ket{x_{n}}\bra{x_{n}}.$
(4.71)
It reminds us of the diagonal terms of the decoherence functional. It is
possible to build a spacetime density matrix from all possible decoherence
functionals as
$\rho=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\textrm{d}x_{1}\textrm{d}x^{\prime}_{1}\cdots\textrm{d}x_{n}\textrm{d}x^{\prime}_{n}D(x_{1},\dots,x_{n}|x^{\prime}_{1},\dots
x^{\prime}_{n})\ket{x_{1}}\bra{x^{\prime}_{1}}\otimes\cdots\otimes\ket{x_{n}}\bra{x^{\prime}_{n}}.$
(4.72)
Applying the strong consistency condition to the above equation, we gain Eqn.
(4.71) again. This argues why it is effective to only consider diagonal terms
in position measurements. which is originally taken for convenience.
Similarly, the spacetime Wigner function from weak measurements is easily
taken as a generalisation for the diagonal terms of the decoherence functional
allowing for general measurements. Recall that a generalised effect-valued
measure is represented by
$\hat{f}(q,p)=C\exp\left[-\alpha[(\hat{q}-q)^{2}+\lambda(\hat{p}-p)^{2}]\right].$
(4.73)
The generalised decoherence functional for weak measurements is then given by
$D(q,p,q^{\prime},p^{\prime},\tau|\hat{\rho})=\Tr\left[\mathcal{F}(q,p,q^{\prime},p^{\prime};\tau)\hat{\rho}\right],$
(4.74)
where
$\displaystyle\mathcal{F}(q,p,q^{\prime},p^{\prime};\tau)\hat{\rho}=$
$\displaystyle\int\textrm{d}\mu_{G}[q(t),p(t)]\int\textrm{d}\mu_{G}[q^{\prime}(t),p^{\prime}(t)]\delta\left(q-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tq(t)\right)$
$\displaystyle\delta\left(p-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tp(t)\right)\delta\left(q^{\prime}-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tq^{\prime}(t)\right)\delta\left(p^{\prime}-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tp^{\prime}(t)\right)$
$\displaystyle\exp[-\frac{i}{\hbar}\hat{H}\tau]\mathcal{T}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q}_{H}(t)-q(t))^{2}+\lambda(\hat{p}_{H}(t)-p(t))^{2}]\right]\hat{\rho}$
$\displaystyle\mathcal{T}^{*}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q^{\prime}}_{H}(t)-q^{\prime}(t))^{2}+\lambda(\hat{p^{\prime}}_{H}(t)-p^{\prime}(t))^{2}]\right]\exp[\frac{i}{\hbar}\hat{H}\tau],$
(4.75)
here
$\displaystyle\textrm{d}\mu_{G}[q(t),p(t)]=\lim_{N\rightarrow\infty}\left(\frac{\gamma\tau\sqrt{\lambda}}{\pi
N}\prod_{s=1}^{N}\textrm{d}q(t_{s})\textrm{d}p(t_{s})\right),$ (4.76)
$\displaystyle\textrm{d}\mu_{G}[q^{\prime}(t),p^{\prime}(t)]=\lim_{N\rightarrow\infty}\left(\frac{\gamma\tau\sqrt{\lambda}}{\pi
N}\prod_{s=1}^{N}\textrm{d}q^{\prime}(t_{s})\textrm{d}p^{\prime}(t_{s})\right),$
(4.77)
and
$\displaystyle\hat{q}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{q}\exp\left[-\frac{i}{\hbar}\hat{H}t\right],\qquad\hat{q^{\prime}}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{q^{\prime}}\exp\left[-\frac{i}{\hbar}\hat{H}t\right],$
$\displaystyle\hat{p}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{p}\exp\left[-\frac{i}{\hbar}\hat{H}t\right],\qquad\hat{p^{\prime}}_{H}(t)=\exp\left[\frac{i}{\hbar}\hat{H}t\right]\hat{p^{\prime}}\exp\left[-\frac{i}{\hbar}\hat{H}t\right].$
(4.78)
The diagonal terms under the strong consistency condition reduce to the form
in the previous chapter:
$p(q,p,\tau|\hat{\rho})=\Tr\mathcal{F}(q,p;\tau)\hat{\rho},$ (4.79)
where
$\displaystyle\mathcal{F}(q,p;\tau)\hat{\rho}=$
$\displaystyle\int\textrm{d}\mu_{G}[q(t),p(t)]\delta\left(q-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tq(t)\right)\delta\left(p-\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}tp(t)\right)\exp[-\frac{i}{\hbar}\hat{H}\tau]$
$\displaystyle\mathcal{T}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q}_{H}(t)-q(t))^{2}+\lambda(\hat{p}_{H}(t)-p(t))^{2}]\right]\hat{\rho}$
$\displaystyle\mathcal{T}^{*}\exp\left[-\frac{\gamma}{2}\int_{0}^{\tau}\textrm{d}t[(\hat{q}_{H}(t)-q(t))^{2}+\lambda(\hat{p}_{H}(t)-p(t))^{2}]\right]\exp[\frac{i}{\hbar}\hat{H}\tau].$
(4.80)
Now we conclude the relation between decoherence functionals in consistent
histories and temporal correlations in pseudo-density matrices.
###### Claim 2.
The decoherence functional in consistent histories is the probabilities in
temporal correlations of pseudo-density matrices.
Thus, we establish the relationship between consistent histories and all
possible forms of pseudo-density matrix. From the consistency condition, we
also have a better argument for why spacetime states for general measurements
are defined in the diagonal form. It is not a coincide.
### 4.4 Generalised non-local games
Game theory studies mathematical models of competition and cooperation under
strategies among rational decision-makers [116]. Here we give an introduction
to nonlocal games, quantum-classical nonlocal games, and quantum-classical
signalling games. Then we show the relation between quantum-classical
signalling games and pseudo-density matrices, and comment on the relation
between general quantum games and indefinite causal order.
#### 4.4.1 Introduction to non-local games
The interests for investigating non-local games start from interactive proof
systems with two parties, the provers and the verifiers. They exchange
information to verify a mathematical statement. A nonlocal game is a special
kind of interactive proof system with only one round and at least two provers
who play in cooperation against the verifier. In nonlocal games, we refer to
the provers as Alice, Bob, $\dots$, and the verifier as the referee. In Ref.
[117], nonlocal games were formally introduced with shared entanglement and
used to formulate the CHSH inequality [118]. Here we introduce the CHSH game
as an example and then give the general form of a non-local game.
The CHSH game has two cooperating players, Alice and Bob, and a referee who
asks questions and collects answers from the players. The basic rules of the
CHSH game are as the following:
1) There are two possible questions $x\in\\{0,1\\}$ for Alice and two possible
questions $y\in\\{0,1\\}$ for Bob. Each question has an equal probability as
$p(x,y)=\frac{1}{4},\forall x,\forall y$.
2) Alice answers $a\in\\{0,1\\}$ and Bob $b\in\\{0,1\\}$.
3) Alice and Bob cannot communicate with each other after the game begins.
4) If $a\oplus b=x\cdot y$, then they win the game, otherwise they lose.
For a classical strategy, that is, Alice and Bob use classical resources, they
win with the probability at most $\frac{3}{4}$. Alice and Bob can also adopt a
quantum strategy. If they prepare and share a joint quantum state
$\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ and make local
measurements based on the questions they receive separately, then they can
achieve a higher winning probability $\cos^{2}(\pi/8)\approx 0.854$.
In general, a non-local game $G$ is formulated by $(\pi,l)$ on
$\overrightarrow{nl}=\langle\mathcal{X},\mathcal{Y};\mathcal{A},\mathcal{B};l\rangle,$
(4.81)
where $\mathcal{X}$, $\mathcal{Y}$ are question spaces of Alice and Bob and
$\mathcal{A}$, $\mathcal{B}$ are answer spaces of Alice and Bob. Here
$\pi(x,y)$ is a probability distribution of the question spaces for Alice and
Bob in the form $\pi:\mathcal{X}\times\mathcal{Y}\rightarrow[0,1]$.
$l(a,b|x,y)$ is a function of question and answer spaces for Alice and Bob to
decide whether they win or lose in the form
$l:\mathcal{X}\times\mathcal{Y}\times\mathcal{A}\times\mathcal{B}\rightarrow[0,1];$
for example, if they win, $l=1$; otherwise lose with $l=0$. For any strategy,
the probability distribution for answers $a,b$ of Alice and Bob given
questions $x,y$, respectively, is referred to as the correlation function
$p(a,b|x,y)$ of the form
$p:\mathcal{X}\times\mathcal{Y}\times\mathcal{A}\times\mathcal{B}\rightarrow[0,1].$
(4.82)
with the condition $\sum_{a,b}p(a,b|x,y)=1.$ With a classical source,
$p_{c}(a,b|x,y)=\sum_{\lambda}\pi(\lambda)d_{A}(a|x,\lambda)d_{B}(b|y,\lambda),$
(4.83)
where $d_{A}(a|x,\lambda)$ is the probability of answering $a$ given the
parameter $\lambda$ and the question $b$ and similar for $d_{B}(b|y,\lambda)$;
with a quantum source,
$p_{q}(a,b|x,y)=\Tr[\rho_{AB}(P_{A}^{a|x}\otimes Q_{B}^{b|y})],$ (4.84)
where $\rho_{AB}$ is the quantum state shared by Alice and Bob, $P_{A}^{a|x}$
is the measurement made by Alice with the outcome $a$ given $x$, $Q_{B}^{b|y}$
is the measurement made by Bob with the outcome $b$ given $y$. Then the
optimal winning probability is given by
$\mathbb{E}_{\overrightarrow{nl}}[*]\equiv\max\sum_{x,y}\pi(x,y)\sum_{a,b}l(a,b|x,y)p_{c/q}(a,b|x,y).$
(4.85)
#### 4.4.2 Quantum-classical non-local & signalling games
First we introduce a generalised version of non-local games where the referee
asks quantum questions instead of classical questions (therefore this type of
non-local games are refereed to quantum-classical) [78]. Then we give the
temporal version of these quantum-classical non-local games as quantum-
classical signalling games [79].
##### Quantum-classical non-local games
We now recap the model of quantum-classical non-local games [78], in which the
questions are quantum rather than classical. More specifically, the referee
sends quantum registers to Alice and Bob instead of classical information.
For a non-local game, with the question spaces $\mathcal{X}=\\{x\\}$ and
$\mathcal{Y}=\\{y\\}$, the referee associates two quantum ancillary systems
$X$ and $Y$ such that $\dim\mathcal{H}_{X}\geq|\mathcal{X}|$,
$\dim\mathcal{H}_{Y}\geq|\mathcal{Y}|$, the systems are in the states
$\tau^{x}_{X}=\ket{x}\bra{x}$ and $\tau^{y}_{Y}=\ket{y}\bra{y}$ with the
questions $x\in\mathcal{X}$ and $y\in\mathcal{Y}$. Assume that Alice and Bob
share a quantum state $\rho_{AB}$. Given the answer sets $\mathcal{A}=\\{a\\}$
and $\mathcal{B}=\\{b\\}$ and quantum systems $XA$ and $YB$, Alice and Bob can
make the corresponding POVMs $P^{a}_{XA}$ and $Q^{b}_{YB}$ in the linear
operators on the Hilbert space $\mathcal{H}_{XA}$ and $\mathcal{H}_{YB}$, such
that $\sum_{a}P^{a}_{XA}=\mathbbm{1}_{XA}$ and
$\sum_{b}Q^{b}_{YB}=\mathbbm{1}_{YB}$. Then the probability distribution for
the questions and answers of Alice and Bob, that is, the correlation function
$P(a,b|x,y)$, is given by
$P(a,b|x,y)=\Tr[(P^{a}_{XA}\otimes
Q^{b}_{YB})(\tau^{x}_{X}\otimes\rho_{AB}\otimes\tau^{y}_{Y})].$ (4.86)
Quantum-classical non-local games replace classical inputs with quantum ones,
formulated by $(\pi(x,y),l(a,b|x,y))$ on
$\overrightarrow{qcnl}=\langle\\{\tau^{x}\\},\\{\omega^{y}\\};\mathcal{A},\mathcal{B};l\rangle.$
(4.87)
The referee picks $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ with the probability
distribution $\pi(x,y)$ as the classical-classical non-local game. With a
classical source,
$p_{c}(a,b|x,y)=\sum_{\lambda}\pi(\lambda)\Tr[(\tau^{x}_{X}\otimes\omega^{y}_{Y})(P_{X}^{a|\lambda}\otimes
Q_{Y}^{b|\lambda})];$ (4.88)
with a quantum source,
$p_{q}(a,b|x,y)=\Tr[(\tau^{x}_{X}\otimes\rho_{AB}\otimes\omega^{y}_{Y})(P_{XA}^{a}\otimes
Q_{BY}^{b})].$ (4.89)
The optimal winning probability is, again, given by
$\mathbb{E}_{\overrightarrow{qcnl}}[*]\equiv\max\sum_{x,y}\pi(x,y)\sum_{a,b}l(a,b|x,y)p_{c/q}(a,b|x,y).$
(4.90)
##### Quantum-classical signalling games
In quantum-classical signalling games [79], instead of two players Alice and
Bob, we consider only one player Abby at two successive instants in time. Then
quantum-classical signalling games change the Alice-Bob duo to a timelike
structures of single player Abby with
$\overrightarrow{qcsg}=\langle\\{\tau^{x}\\},\\{\omega^{y}\\};\mathcal{A},\mathcal{B};l\rangle.$
(4.91)
With unlimited classical memory,
$p_{c}(a,b|x,y)=\sum_{\lambda}\pi(\lambda)\Tr[\tau^{x}_{X}P_{X}^{a|\lambda}]\Tr[\omega^{y}_{Y}Q_{Y}^{b|a,\lambda}].$
(4.92)
For admissible quantum strategies, suppose Abby at $t_{1}$ receives
$\tau^{x}_{X}$ and makes a measurement of instruments $\\{\Phi_{X\rightarrow
A}^{a|\lambda}\\}$, and gains the outcome $a$. Then the quantum output goes
through the quantum memory $\mathcal{N}:A\rightarrow B$. The output of the
memory and $\omega^{y}_{Y}$ received by Abby at $t_{2}$ are fed into a
measurement $\\{\Psi_{BY}^{b|a,\lambda}\\}$, with outcome b. Then
$p_{q}(a,b|x,y)=\sum_{\lambda}\pi(\lambda)\Tr[(\\{(\mathcal{N}_{A\rightarrow
B}\circ\Phi_{X\rightarrow
A}^{a|\lambda})(\tau^{x}_{X})\\}\otimes\omega^{y}_{Y})\Psi_{BY}^{b|a,\lambda}].$
(4.93)
The optimal payoff function is, again, given by
$\mathbb{E}_{\overrightarrow{qcsg}}[*]\equiv\max\sum_{x,y}\pi(x,y)\sum_{a,b}l(a,b|x,y)p_{c/q}(a,b|x,y).$
(4.94)
#### 4.4.3 Temporal correlations from signalling games
To compare quantum-classical signalling games with pseudo-density matrices,
first we generalise the finite-dimensional pseudo-density matrices from Pauli
measurements to general positive-operator valued measures(POVMs). Recall that
a POVM is a set of Hermitian positive semi-definite operator $\\{E_{i}\\}$ on
a Hilbert space $\mathcal{H}$ which sum up to the identity
$\sum_{i}E_{i}=\mathbbm{1}_{\mathcal{H}}$. Instead of making a single-qubit
Pauli measurement at each event, we make a measurement
$E_{i}=M_{i}^{a{\dagger}}M_{i}^{a}$ with the outcome $a$. For each event,
there is a measurement
$\mathcal{M}_{i}:\mathcal{L}(\mathcal{H}^{X})\rightarrow\mathcal{L}(\mathcal{H}^{A}),\tau^{x}_{X}\mapsto\sum_{i}M_{i}^{a}\tau^{x}_{X}M_{i}^{a{\dagger}}$
with $\sum M_{i}^{a{\dagger}}M_{i}^{a}=\mathbbm{1}_{\mathcal{H}^{X}}$.
Now we map the generalised pseudo-density matrices to quantum-classical
signalling games. Assume $\omega_{Y}^{y}$ to be trivial. For Abby at the
initial time and the later time, we consider $\Phi_{X\rightarrow
A}^{a}:\tau^{x}_{X}\rightarrow\sum_{i}M_{i}^{a}\tau^{x}_{X}M_{i}^{a{\dagger}}$,
where $\sum M_{i}^{a{\dagger}}M_{i}^{a}=\mathbbm{1}_{\mathcal{H}^{A}}$.
Between two times, the transformation from $A$ to $B$ is given by
$\mathcal{N}:\rho_{A}\rightarrow\sum_{j}N_{j}\rho_{A}N_{j}^{{\dagger}}$ with
$\sum_{j}N_{j}^{{\dagger}}N_{j}=\mathbbm{1}_{\mathcal{H}^{A}}$. Then
$\displaystyle p_{q}(a,b|x,y)$
$\displaystyle=\Tr[\\{(\mathcal{N}_{A\rightarrow B}\circ\Phi_{X\rightarrow
A}^{a})(\tau^{x}_{X})\\}\Psi_{B}^{b|a}]$
$\displaystyle=\sum_{ik}\Tr[\mathcal{N}\\{M_{i}^{a}\tau^{x}_{X}M_{i}^{a{\dagger}}\\}\Psi_{B}^{b|a}]$
$\displaystyle=\sum_{ijk}\Tr[N_{j}M_{i}^{a}\tau^{x}_{X}M_{i}^{a{\dagger}}N_{j}^{{\dagger}}\Psi_{B}^{b|a}]$
(4.95) $\displaystyle\langle\\{\Phi,\Psi\\}\rangle$
$\displaystyle=\sum_{a,b}abp_{q}(a,b|x,y)$ (4.96)
It is the temporal correlation given by pseudo-density matrices. That is, a
quantum-classical signalling game with a trivial input at later time
corresponds to a pseudo-density matrix with quantum channels replacing
measurements for events.
###### Claim 3.
The probability in a quantum-classical signalling game with a trivial input at
later time corresponds to the probability in a pseudo-density matrix where the
state goes through quantum channels instead of measurements.
It is also convenient to establish the relation between generalised games in
time and indefinite causal structures with double Hilbert spaces for each
event. For completeness, we also mention that Gutoski and Watrous [94]
proposed a general theory of quantum games in terms of the Choi-Jamiołkowski
representation, which is an equivalent formulation of indefinite causal order.
### 4.5 Out-of-time-order correlations (OTOCs)
In this section we introduce out-of-time-order correlation functions, find a
simple method to calculation these temporal correlations via the pseudo-
density matrix formalism, and apply the out-of-time-order correlation
functions into the black hole final state projection proposal as one of the
proposals for black hole information paradox.
#### 4.5.1 Brief introduction to OTOCs
Consider local operators $W$ and $V$. With a Hamiltonian $H$ of the system,
the Heisenberg representation of the operator $W$ is given as
$W(t)=e^{iHt}We^{-iHt}$. Out-of-time-order correlation functions (OTOCs) [80,
81] are usually defined as
$\langle VW(t)V^{{\dagger}}W^{{\dagger}}(t)\rangle=\langle
VU(t)^{{\dagger}}WU(t)V^{{\dagger}}U^{{\dagger}}(t)W^{{\dagger}}U(t)\rangle,$
(4.97)
where $U(t)=e^{-iHt}$ is the unitary evolution operator and the correlation is
evaluated on the thermal state $\langle\cdot\rangle=\Tr[e^{-\beta
H}\cdot]/\Tr[e^{-\beta H}]$. Note that OTOC is usually defined for the
maximally mixed state $\rho=\frac{\mathbbm{1}}{d}$. Consider a correlated
qubit chain. Measure $V$ at the first qubit and $W$ at the last qubit. Since
the chain is correlated in the beginning, we have OTOC as $1$ at the early
time. As time evolves and the operator growth happens, OTOC will approximate
to $0$ at the later time.
#### 4.5.2 Calculating OTOCs via pseudo-density matrices
In this subsection we make a connection between OTOCs and the pseudo-density
matrix formalism. Consider a qubit evolving in time and backward, we can get a
tripartite pseudo-density matrix. In particular, we consider measuring $A$ at
$t_{1}$, $B$ at $t_{2}$ and $A$ again at $t_{3}$ and assume the evolution
forwards is described by $U$ and backwards $U^{{\dagger}}$. Then the
probability is given by
$\Tr[AU^{{\dagger}}BUA\rho
A^{{\dagger}}U^{{\dagger}}B^{{\dagger}}UA^{{\dagger}}]=\Tr[AB(t)A\rho
A^{{\dagger}}B^{{\dagger}}(t)A^{{\dagger}}]$ (4.98)
If we assume that $AA^{{\dagger}}=A$, $\rho=\frac{\mathbbm{1}}{d}$, then Eqn.
(4.98) will reduce to the OTOC $\langle AB(t)AB(t)\rangle$.
###### Claim 4.
OTOCs can be represented as temporal correlations in pseudo-density matrices
with half numbers of steps for calculation; for example, a four-point OTOC,
usually calculated by evolving forwards and backwards twice, is represented by
a tripartite pseudo-density matrix with only once evolving forwards and
backwards.
#### 4.5.3 Black hole final state proposal
In this subsection, we briefly review black hole information paradox and final
state projection proposal, and use the relation between OTOCs and pseudo-
density matrices to analyse OTOC in the final state proposal.
##### Review of black hole information paradox
Hawking showed that black holes emit exactly thermal radiations [119].
Consider that a black hole initially in a pure state evolves unitarily. The
fact that the radiation emitted outside the black hole is in a mixed state is
not surprising when we take the black hole interior and the outside radiation
as the whole system. However, the problem appears when the black hole fully
evaporates and only thermal radiation is left. The final state is a mixed
state. We find that a pure state evolves into a mixed state in the black hole
evaporation; that is, in a closed system the unitarity is violated. This is
the black hole information paradox [120, 121].
A few possible solutions have been proposed for the information paradox. For
example, some people believe that there is fundamental non-unitarity in the
universe and the information is just lost. Another possibility might be that
information is stored in a Planck-sized remnant [122] and we need to apply an
unknown quantum gravity theory to solve it. Also, information might be stored
in a baby universe [123, 124] which carries away the collapsing matter as well
as the information. Or, information is encoded in the correlations between the
early and late radiation [125, 126].
##### Final state projection proposal
To solve the black hole information paradox, one possible proposal is the
black hole final state projection proposal [127, 128, 129, 130]. The matter
that forms the black hole lives in the Hilbert space $\mathcal{H}_{M}$ with
the dimension $N=e^{S_{BH}}$ where $S_{BH}$ is the black hole entropy. The
evaporation of the black hole, usually formulated by a semiclassical
approximation to field fluctuations, divides the fluctuation fields into
$\mathcal{H}_{in}$ and $\mathcal{H}_{out}$, inside and outside the event
horizon respectively. Each of them has dimension $N=e^{S_{BH}}$ as well. The
(Unruh) state $\ket{\Phi}_{in\otimes out}$ on
$\mathcal{H}_{in}\otimes\mathcal{H}_{out}$ is the maximally entangled state
$\ket{\Phi}_{in\otimes
out}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\ket{i}_{in}\ket{i}_{out}$ (4.99)
where $\ket{i}_{in}$, $\ket{i}_{out}$ are orthonormal bases in
$\mathcal{H}_{in}$ and $\mathcal{H}_{out}$. In the final state projection
proposal, Horowitz and Maldacena attempt to construct the unitary evaporation
$\ket{m}_{M}\rightarrow S_{jm}\ket{j}_{out},$ (4.100)
to solve the problem of information paradox. In particular, they impose a
final state boundary condition at the singularity and project the state in
$\mathcal{H}_{M}\otimes\mathcal{H}_{in}$ to a super-normalised maximally
entangled state
$\bra{BH}=N^{1/2}\sum_{m,i}S_{im}\bra{m}_{M}\ket{i}_{in}=N\bra{\Phi}_{M\otimes
in}(S\otimes\mathbbm{1}).$ (4.101)
The whole process is formulated as
$\displaystyle\ket{m}$
$\displaystyle\rightarrow\ket{m}_{M}\ket{\Phi}_{in\otimes out}$
$\displaystyle\rightarrow\bra{BH}_{M,in}\left(\ket{m}_{M}\ket{\Phi}_{in\otimes
out}\right)$ $\displaystyle=S_{jm}\ket{j}.$ (4.102)
Thus, a unitary process for evaporation is achieved with this final state
projection.
##### OTOC analysis for final state proposal
Now we apply the above OTOC analysis to the final state proposal. First, we
write the initial state as
$\rho=\ket{\psi}\bra{\psi}_{M}\otimes\ket{\Phi}\bra{\Phi}_{in\otimes out},$
(4.103)
and the final state as
$\sigma=\ket{\Phi}\bra{\Phi}_{M\otimes in}\otimes
S\ket{\psi}\bra{\psi}S^{{\dagger}}_{out}.$ (4.104)
From the initial state to the final state, there is an evolution described by
$U=S_{M}\otimes\mathbbm{1}_{in}\otimes\mathbbm{1}_{out}$, and a projection
$P=\ket{BH}\bra{BH}_{M,in}\otimes\mathbbm{1}_{out}$. Suppose $S$ is a Haar
random unitary, we have
$\Tr[PU\rho U^{{\dagger}}P^{{\dagger}}]=1.$ (4.105)
Here we assume $S_{M}=S^{{\dagger}}$. Now we consider the OTOC between the
initial time and the final time. It can be computed in the pseudo-density
matrix formulation by assuming the evolution forwards and backwards. Thus we
measure $A$ at $t_{1}$, let the state evolve under $U$, after that we make the
final state projection $P$ at $t_{2}$, then the state evolves under
$U^{{\dagger}}$, and we measure $A$ at $t_{3}$. That is,
$OTOC=\langle
AP(t)A^{{\dagger}}P^{{\dagger}}(t)\rangle=\Tr[AU^{{\dagger}}PUA\rho
A^{{\dagger}}U^{{\dagger}}P^{{\dagger}}U].$ (4.106)
For simplicity, we take $A$ as the identity operator. Then again we have
$OTOC=\Tr[PU\rho U^{{\dagger}}P^{{\dagger}}]=1.$ (4.107)
Consider the measurement $A$ is acted on the outside radiation part as
$A=\mathbbm{1}_{M\otimes in}\otimes\ket{\psi}\bra{\psi}_{out}$, then
$OTOC=\Tr[APUA\rho A^{{\dagger}}U^{{\dagger}}P^{{\dagger}}]=1.$ (4.108)
Note that $[A,P]=0$. $P$ is acted on the matter and inside radiation while $A$
is acted on the outside radiation. The out-of-time-order correlation remains
unchanged. This suggests there is no operator growth from the interior of the
black hole to outside. This is consistent with the preservation of the
information and unitarity. However, we notice that the projection is onto a
supernormalised state; it remains doubt whether this is physical enough to be
achieved.
### 4.6 Path integrals
The path integral approach [82] is a representation of quantum theory, not
only useful in quantum mechanics but also quantum statistical mechanics and
quantum field theory. It generalises the action principle of classical
mechanics and one computes a quantum amplitude by replacing a single classical
trajectory with a functional integral of infinite numbers of possible quantum
trajectories. Here we argue that the path integral approach of quantum
mechanics use amplitude as the measure in correlation functions rather than
probability measure in the above formalisms.
#### 4.6.1 Introduction to path integrals
Now we briefly introduce path integrals and correlation functions in this
formalism [83]. Consider a bound operator in a Hilbert space
$U(t_{2},t_{1})(t_{2}\geq t_{1})$ as the evolution from time $t_{1}$ to
$t_{2}$, which satisfies the Markov property in time as
$U(t_{3},t_{2})U(t_{2},t_{1})=U(t_{3},t_{1}),\forall\ t_{3}\geq t_{2}\geq
t_{1}\qquad U(t,t)=\mathbbm{1}.$ (4.109)
We further assume that $U(t,t^{\prime})$ is differentiable and the derivative
is continuous:
$\left.\frac{\partial U(t,t^{\prime})}{\partial
t}\right|_{t=t^{\prime}}=-H(t)/\hbar$ (4.110)
where $\hbar$ is a real parameter, and later identified with Planck’s
constant; $H=i\tilde{H}$ where $\tilde{H}$ is the quantum Hamiltonian. Then
$U(t^{\prime\prime},t^{\prime})=\prod_{m=1}^{n}U[t^{\prime}+m\epsilon,t^{\prime}+(m-1)\epsilon],\qquad
n\epsilon=t"-t^{\prime}.$ (4.111)
The position basis for $\hat{q}\ket{q}=q\ket{q}$ is orthogonal $\langle
q^{\prime}\ket{q}=\delta(q-q^{\prime})$, and complete
$\int\textrm{d}q\ket{q}\bra{q}=\mathbbm{1}$. We have
$\bra{q^{\prime\prime}}U(t^{\prime\prime},t^{\prime})\ket{q^{\prime}}=\int\prod_{k=1}^{n-1}\textrm{d}q_{k}\prod_{k=1}^{n}\bra{q_{k}}U(t_{k},t_{k-1})\ket{q_{k-1}}$
(4.112)
with $t_{k}=t^{\prime}+k\epsilon,q_{0}=q^{\prime},q_{n}=q^{\prime\prime}$.
Suppose that the operator $H$ is identified with a quantum Hamiltonian of the
form
$H=\hat{\bm{p}}^{2}/2m+V(\hat{\bm{q}},t)$ (4.113)
where $\bm{p},\bm{q}\in\mathbb{R}^{d}$. We have
$\bra{\bm{q}}U(t,t^{\prime})\ket{\bm{q^{\prime}}}=\left(\frac{m}{2\pi\hbar(t-t^{\prime})}\right)^{d/2}\exp[-\mathcal{S}(\bm{q})/\hbar]$
(4.114)
where
$\mathcal{S}(\bm{q})=\int_{t^{\prime}}^{t}\textrm{d}\tau[\frac{1}{2}m\dot{\bm{q}}^{2}(\tau)+V(\bm{q}(\tau),\tau)]+O((t-t^{\prime})^{2}),$
(4.115)
and
$\bm{q}(\tau)=\bm{q}^{\prime}+\frac{\tau-t^{\prime}}{t-t^{\prime}}(\bm{q}-\bm{q}^{\prime}).$
(4.116)
We consider short time slices, then
$\bra{\bm{q^{\prime\prime}}}U(t^{\prime\prime},t^{\prime})\ket{\bm{q^{\prime}}}=\lim_{n\rightarrow\infty}\left(\frac{m}{2\pi\hbar\epsilon}\right)^{dn/2}\int\prod_{k=1}^{n-1}\textrm{d}^{d}q_{k}\exp[-\mathcal{S}(\bm{q},\epsilon)/\hbar],$
(4.117)
with
$\mathcal{S}(\bm{q},\epsilon)=\sum_{k=0}^{n-1}\int_{t_{k}}^{t_{k+1}}\textrm{d}t[\frac{1}{2}m\dot{\bm{q}}^{2}(t)+V(\bm{q}(t),t)]+O(\epsilon^{2}).$
(4.118)
Introducing a linear and continuous trajectory
$\bm{q}(t)=\bm{q}_{k}+\frac{t-t_{k}}{t_{k+1}-t_{k}}(\bm{q}_{k+1}-\bm{q}_{k})\
\ \text{for}\ \ t_{k}\leq t\leq t_{k+1},$ (4.119)
we can rewrite Eqn. (4.118) as
$\mathcal{S}(\bm{q},\epsilon)=\int_{t^{\prime}}^{t^{\prime\prime}}\textrm{d}t[\frac{1}{2}m\dot{\bm{q}}^{2}(t)+V(\bm{q}(t),t)]+O(n\epsilon^{2}).$
(4.120)
Taking $n\rightarrow\infty$ and $\epsilon\rightarrow 0$ with
$n\epsilon=t^{\prime\prime}-t^{\prime}$ fixed, we have
$\mathcal{S}(\bm{q})=\int_{t^{\prime}}^{t^{\prime\prime}}\textrm{d}t[\frac{1}{2}m\dot{\bm{q}}^{2}(t)+V(\bm{q}(t),t)]$
(4.121)
as the Euclidean action. The path integral is thus defined as
$\bra{\bm{q}^{\prime\prime}}U(t^{\prime\prime},t^{\prime})\ket{\bm{q}^{\prime}}=\int_{\bm{q}(t^{\prime})=\bm{q}^{\prime}}^{\bm{q}(t^{\prime\prime})=\bm{q}^{\prime\prime}}[\textrm{d}\bm{q}(t)]\exp(-\mathcal{S}(\bm{q})/\hbar),$
(4.122)
where a normalisation of $\mathcal{N}=(\frac{m}{2\pi\hbar\epsilon})^{dn/2}$ is
hidden in $[\textrm{d}\bm{q}(t)]$.
The quantum partition function $\mathcal{Z}(\beta)=\Tr e^{-\beta H}$ ($\beta$
is the inverse temperature) can be written in terms of path integrals as
$\displaystyle\mathcal{Z}(\beta)$ $\displaystyle=\Tr e^{-\beta H}=\Tr
U(\hbar\beta,0)=\int\textrm{d}q^{\prime\prime}\textrm{d}q^{\prime}\delta(\bm{q}^{\prime\prime}-\bm{q}^{\prime})\bra{\bm{q}^{\prime\prime}}U(\hbar\beta,0)\ket{\bm{q}^{\prime}}$
$\displaystyle=\int_{\bm{q}(0)=\bm{q}(\hbar\beta)}[\textrm{d}q(t)]\exp[-\mathcal{S}(\bm{q})/\hbar].$
(4.123)
The integrand $e^{-\mathcal{S}(\bm{q})/\hbar}$ is a positive measure and
defines the corresponding expectation value as
$\langle\mathcal{F}(q)\rangle=\mathcal{N}\int[\textrm{d}q(t)]\mathcal{F}(q)\exp[-\mathcal{S}(\bm{q})/\hbar],$
(4.124)
where $\mathcal{N}$ is chosen for $\langle 1\rangle=1$. Moments of the measure
in the form as
$\langle q(t_{1})q(t_{2})\cdots
q(t_{n})\rangle=\mathcal{N}\int[\textrm{d}q(t)]q(t_{1})q(t_{2})\cdots
q(t_{n})\exp[-\mathcal{S}(\bm{q})/\hbar]$ (4.125)
are the $n$-point correlation function. Suppose for the finite time interval
$\beta$ periodic boundary conditions hold as $q(\beta/2)=q(-\beta/2)$. The
normalisation is given as $\mathcal{N}=\mathcal{Z}^{-1}(\beta)$. Then we
define
$Z^{(n)}(t_{1},\cdots,t_{n})=\langle q(t_{1})\cdots q(t_{n})\rangle.$ (4.126)
The generating functional of correlation functions is
$\displaystyle\mathcal{Z}(f)$
$\displaystyle=\sum_{n=0}\frac{1}{n!}\int\textrm{d}t_{1}\cdots\textrm{d}t_{n}Z^{(n)}(t_{1},\cdots,t_{n})f(t_{1})\cdots
f(t_{n})$
$\displaystyle=\sum_{n=0}\frac{1}{n!}\int\textrm{d}t_{1}\cdots\textrm{d}t_{n}\langle
q(t_{1})\cdots q(t_{n})\rangle f(t_{1})\cdots f(t_{n})$
$\displaystyle=\left\langle\exp\left[\int\textrm{d}tq(t)f(t)\right]\right\rangle$
(4.127)
Note that the $n$-point quantum correlation functions in time also appear as
continuum limits of the correlation functions of $1D$ lattice in classical
statistical models. The path integral formalism represents a mathematical
relation between classical statistical physics on a line and quantum
statistical physics of a point-like particle at thermal equilibrium. This is
the first example of the quantum-classical correspondence which maps between
quantum statistical physics in $D$ dimensions and classical statistical
physics in $D+1$ dimensions [83].
#### 4.6.2 Temporal correlations in path integrals are different
Here we take two-point correlations functions:
$\langle
q(t_{1})q(t_{2})\rangle=\frac{\int[\textrm{d}q(t)]q(t_{1})q(t_{2})\exp[-\mathcal{S}(\bm{q})/\hbar]}{\int[\textrm{d}q(t)]\exp[-\mathcal{S}(\bm{q})/\hbar]}$
(4.128)
In the Gaussian representation of pseudo-density matrices, temporal
correlation for $q_{1}$ at $t_{1}$ and $q_{2}$ at $t_{2}$ with the evolution
$U$ and the initial state $\ket{q_{1}}$ is given as
$\displaystyle\langle\\{q_{1},q_{2}\\}\rangle$
$\displaystyle=\int\textrm{d}q_{1}\textrm{d}q_{2}q_{1}q_{2}|\bra{q_{2}}U\ket{q_{1}}|^{2}$
$\displaystyle=\frac{\int\textrm{d}q_{1}\textrm{d}q_{2}q_{1}q_{2}\left|\int_{q(t_{1})=q_{1}}^{q(t_{2})=q_{2}}[\textrm{d}q(t)]\exp[-\mathcal{S}(\bm{q})/\hbar]\right|^{2}}{\left|\int[\textrm{d}q(t)]\exp[-\mathcal{S}(\bm{q})/\hbar]\right|^{2}}$
(4.129)
Correlations are defined as the expectation values of measurement outcomes.
However, path integrals and pseudo-density matrices use different positive
measure to calculate the expectation values. The correlations in path
integrals use the amplitude as the measure, while in pseudo-density matrices
the measure is the absolute square of the integrated amplitudes, or we say the
probability.
To see the difference, we consider a quantum harmonic oscillator. The
Hamiltonian is given as $H=\hat{p}^{2}/2m+m\omega^{2}\hat{q}^{2}/2$. Note that
the quantum amplitude of a quantum harmonic oscillator is given as
$\bra{q_{2}}U(t_{2},t_{1})\ket{q_{1}}=\left(\frac{m\omega}{2\pi\hbar\sinh\omega\tau}\right)^{1/2}\exp\left\\{-\frac{m\omega}{2\hbar\sinh\omega\tau}[(q_{1}^{2}+q_{2}^{2})\cosh\omega\tau-2q_{1}q_{2}]\right\\},$
(4.130)
where $\tau=t_{2}-t_{1}$. In the Gaussian representation of pseudo-density
matrices, temporal correlations are represented as
$\langle\\{q_{1},q_{2}\\}\rangle=\int\textrm{d}q_{1}\textrm{d}q_{2}q_{1}q_{2}|\bra{q_{2}}U\ket{q_{1}}|^{2}=\frac{\hbar}{8m\omega\sinh^{2}\omega\tau}.$
(4.131)
However, in the path integral formalism, we consider
$\Tr
U_{G}(\tau/2,-\tau/2;b)=\int[\textrm{d}q(t)]\exp[-\mathcal{S}_{G}(q,b)/\hbar]$
(4.132)
with
$\mathcal{S}_{G}(q,b)=\int_{-\tau/2}^{\tau/2}\textrm{d}t[\frac{1}{2}m\dot{q}^{2}(t)+\frac{1}{2}m\omega^{2}q^{2}(t)-b(t)q(t)]$
(4.133)
and periodic boundary conditions $q(\tau/2)=q(-\tau/2)$. We have
$\mathcal{Z}_{G}(\beta,b)=\Tr
U_{G}(\hbar\beta/2,-\hbar\beta/2;b)=\mathcal{Z}_{0}(\beta)\left\langle\exp\left[\frac{1}{\hbar}\int_{-\hbar\beta/2}^{\hbar\beta/2}\textrm{d}tb(t)q(t)\right]\right\rangle_{0}$
(4.134)
where $\langle\cdot\rangle_{0}$ denotes the Gaussian expectation value in
terms of the distribution $e^{-\mathcal{S_{0}}/\hbar}/\mathcal{Z}_{0}(\beta)$
and periodic boundary conditions. Here $\mathcal{Z}_{0}(\beta)$ is the
partition function of the harmonic oscillator as
$\mathcal{Z}_{0}(\beta)=\frac{1}{2\sinh(\beta\omega/2)}=\frac{e^{-\beta\hbar\omega/2}}{1-e^{-\beta\hbar\omega}}.$
(4.135)
Then two-point correlations functions are given as
$\langle
q(t_{1})q(t_{2})\rangle=\mathcal{Z}_{0}^{-1}(\beta)\hbar^{2}\left.\frac{\delta^{2}}{\delta
b(t)\delta
b(u)}\mathcal{Z}_{G}(\beta,b)\right|_{b=0}=\frac{\hbar}{2\omega\tanh(\omega\tau/2)}.$
(4.136)
It is no surprise that the temporal correlations are different in path
integrals and in pseudo-density matrices.
###### Claim 5.
In general, temporal correlations in path integrals do not have the same
operational meaning as those in pseudo-density matrices since they use
different measures, with exception of path-integral representation for
spacetime states and decoherence functionals.
This indicates a fundamental difference of temporal correlations in path
integrals and other spacetime approaches, and raises again the question
whether the probability or the amplitude serves as the measure in quantum
theory. It is natural that amplitudes interferes with each other in field
theory and expectation values of operators are defined with the amplitude
interference. Thus temporal correlations in path integrals cannot be
operationally represented as pseudo-density matrices. However, spacetime
states defined via position measurements and weak measurements in pseudo-
density matrix formulation [85] are motivated by the path integral formalism
and have a path-integral representation naturally. In addition, consistent
histories also have a path-integral representation of decoherence functionals
as we mentioned earlier.
### 4.7 Conclusion and discussion
In this section, we unify these spacetime approaches in non-relativistic
quantum mechanics and summarise all the claims.
We consider a unified picture in which temporal correlations serve as a
connection for indefinite causal structures, consistent histories, generalised
quantum games and OTOCs. Given a tripartite pseudo-density matrix, a qubit in
the state $\rho$ evolves in time under the unitary evolution $U$ and then back
in time under $U^{{\dagger}}$. The correlations in the pseudo-density matrix
are given as
$\langle\sigma_{i},\sigma_{j},\sigma_{k}\rangle=\sum_{\alpha,\beta,\gamma=\pm
1}\alpha\beta\gamma\Tr[P^{\gamma}_{k}U^{{\dagger}}P^{\beta}_{j}UP^{\alpha}_{i}\rho
P^{\alpha}_{i}U^{{\dagger}}P^{\beta}_{j}U]$ (4.137)
where $P^{\alpha}_{i}=\frac{1}{2}(\mathbbm{1}+\alpha\sigma_{i})$,
$P^{\beta}_{j}=\frac{1}{2}(\mathbbm{1}+\beta\sigma_{j})$ and
$P^{\gamma}_{k}=\frac{1}{2}(\mathbbm{1}+\gamma\sigma_{k})$. As the pseudo-
density matrix belongs to indefinite causal structures, we won’t discuss the
transformation for other formalisms of indefinite causal structures.
For consistent histories, we assume the state in $\rho$ at the initial time
and construct a set of histories
$[\chi]=[\alpha\rightarrow\beta\rightarrow\gamma]$ with projections
$\\{P^{\alpha}_{i},P^{\beta}_{j},P^{\gamma}_{k}\\}$. Then the decoherence
functional is given as
$D([\xi],[\xi^{\prime}])=\Tr[P^{\gamma}_{k}U^{{\dagger}}P^{\beta}_{j}UP^{\alpha}_{i}\rho
P^{\alpha^{\prime}}_{i}U^{{\dagger}}P^{\beta^{\prime}}_{j}UP^{\gamma^{\prime}}_{k}]$
(4.138)
When we apply the consistency conditions, it is part of Eqn. (4.137) as
$D([\xi],[\xi])=\Tr[P^{\gamma}_{k}U^{{\dagger}}P^{\beta}_{j}UP^{\alpha}_{i}\rho
P^{\alpha}_{i}U^{{\dagger}}P^{\beta}_{j}UP^{\gamma}_{k}],$ (4.139)
$\langle\sigma_{i},\sigma_{j},\sigma_{k}\rangle=\sum_{\alpha,\beta,\gamma=\pm
1}\alpha\beta\gamma D([\xi],[\xi]).$ (4.140)
A quantum-classical signalling game is described in terms of one player Abby
at two times in a loop, or one player Abby at three times with evolution $U$
and $U^{{\dagger}}$. The quantum-classical signalling game is formulated by
$(\pi(x,y),l(a,b|x,y))$ on
$\overrightarrow{qcsg}=\langle\\{\tau^{x}\\},\\{\omega^{y}\\},\\{\eta^{z}\\};\mathcal{A},\mathcal{B},\mathcal{C};l\rangle.$
(4.141)
The referee associates three quantum systems in the states $\tau^{x}$,
$\omega^{y}$ and $\eta^{z}$ with the questions chosen from the question spaces
$x\in\mathcal{X}$, $y\in\mathcal{Y}$, and $z\in\mathcal{Z}$. Suppose Abby at
$t_{1}$ receives $\tau^{x}_{X}$ and makes a measurement of instruments
$\\{M_{i}^{a}\\}_{i}$ with the outcome $a$. From $t_{1}$ to $t_{2}$, the
quantum output evolves under the unitary quantum memory $U:A\rightarrow B$.
After that, Abby receives the output of the channel and $\omega^{y}$, and
makes a measurement of instruments $\\{N_{j}^{b}\\}_{j}$ with the outcome $b$.
Then, we can consider that either the quantum memory goes backwards to $t_{1}$
or evolves under $U^{{\dagger}}:B\rightarrow C$ to $t_{3}$. Abby receives the
output of the channel again and $\eta^{z}$, and makes a measurement of
instruments $\\{O_{k}^{c}\\}_{k}$ with the outcome $c$. Then we have
$p_{q}(a,b,c|x,y,z)=\sum_{\lambda,i,j,k}\pi(\lambda)\Tr[O_{k}^{c}U^{{\dagger}}N_{j}^{b}UM_{i}^{a}\rho
M_{i}^{a}U^{{\dagger}}N_{j}^{b}UO_{k}^{c}].$ (4.142)
If we properly choose the measurements, we will have the decoherence
functionals and the probabilities in the correlations of pseudo-density
matrix.
What is more, the tripartite pseudo-density matrix we describe is just the one
we used to construct OTOC. Thus, through this tripartite pseudo-density
matrix, we gain a unified picture for indefinite causal order, consistent
histories, generalised quantum games and OTOCs in which temporal correlations
are the same or operationally equivalent. Thus all these approaches are
mapping into each other directly in this particular case via temporal
correlations. Generalisation to more complicated scenarios are
straightforward.
Now we conclude that there is not much difference in different spacetime
approaches for non-relativistic quantum mechanics under this comparison of
temporal correlations except path integrals. They are closely related compared
with pseudo-density matrices and formulate temporal correlations in the same
way or operationally equivalent. However, the path integral approach of
quantum mechanics give temporal correlation in a different way. Via the
pseudo-density matrix formalism, we establish the relations among different
spacetime formulations like indefinite causal structures, consistent
histories, generalised nonlocal games, out-of-time-order correlation
functions, and path integrals. As we can see, all these relations are rather
simple. The big surprise we learn from these relations is that almost
everything we know about space-time in non-relativistic quantum mechanics so
far is connected with each other but path integrals are not. Thus, it shows
the possibility of a unified picture of non-relativistic quantum mechanics in
spacetime and a gap to relativistic quantum field theory. We claim:
(1) A process matrix and the corresponding pseudo-density matrix allow the
same correlations or probabilities in three different mappings.
(2) The decoherence functional in consistent histories is the probabilities in
temporal correlations of pseudo-density matrices.
(3) The probability in a quantum-classical signalling game with a trivial
input at later time corresponds to the probability in a pseudo-density matrix
with quantum channels as measurements.
(4) OTOCs can be represented as temporal correlations in pseudo-density
matrices with half numbers of steps for calculation; for example, a four-point
OTOC, usually calculated by evolving forwards and backwards twice, is
represented by a tripartite pseudo-density matrix with only once evolving
forwards and backwards.
(5) In general, temporal correlations in path integrals do not have the
operational meaning as those in pseudo-density matrices since they use
different measures, with exception of path-integral representation for
spacetime states and decoherence functionals.
A unified theory for non-relativistic quantum mechanics is suggested;
nevertheless, how to move on to relativistic quantum information, or further
to quantum gravity, is still a big gap worth exploring.
## Chapter 5 Time crystals as long-range order in time
### 5.1 Literature review for time crystals
In this section we review spontaneous symmetry breaking, time translation
symmetry breaking, and a few mathematical definitions for time crystals.
#### 5.1.1 Spontaneous symmetry breaking
Spontaneous symmetry breaking [131] occurs when the ground state does not hold
the symmetry which the equation of motion or the Lagrangian holds. Phases of
matter are described by spontaneous symmetry breaking. For example, the
spontaneous breaking of continuous space translation symmetry gives a normal
spatial crystal with periodic structures; spin rotational symmetry is
spontaneously broken with a net magnetisation along certain direction in
ferromagnets, in contrast that spins are uncorrelated in a paramagnetic phase
without a net magnetisation.
There are two diagnostics for spontaneous symmetry breaking [132]. Note that
in equilibrium the expectation values of the order parameter are zero and
cannot serve as a measure for spontaneous symmetry breaking. However, we can
use two-point correlation functions, when taken the long distance range, to be
long-range order.
$\lim_{|\bm{r}-\bm{r^{\prime}}|\rightarrow\infty}\lim_{V\rightarrow\infty}\langle
C(\bm{r},\bm{r^{\prime}})\rangle=\lim_{|\bm{r}-\bm{r^{\prime}}|\rightarrow\infty}\lim_{V\rightarrow\infty}\langle
O(\bm{r})O(\bm{r^{\prime}})\rangle-\langle O(\bm{r})\rangle\langle
O(\bm{r^{\prime}})\rangle\neq 0,$ (5.1)
where $O(\bm{r})$ is a local order parameter and $\langle\cdot\rangle$ is the
expectation value in the equilibrium Gibbs states (or eigenstates). This is
the standard diagnostic for spontaneous symmetry breaking. Another diagnostic
is to add a small symmetry breaking field with strength $h$ and compute the
expectation value of the global order parameter $\langle O\rangle_{h}$ which
turns into non-zero.
$\lim_{h\rightarrow 0}\lim_{V\rightarrow\infty}\frac{1}{V}\langle
O\rangle_{h}\neq 0.$ (5.2)
One variant is to apply a small symmetry breaking field at the boundaries and
evaluate how the expectation value of order parameter have influence on the
bulk.
The Goldstone theorem [133, 134, 135] states that at least one massless
bosonic state exists in the spectrum when the theory allows a universal
symmetry to be spontaneously broken. The Mermin-Wagner theorem [136, 137, 138]
concludes that, in one or two dimensions, continuous symmetries cannot be
spontaneously broken at finite temperature in systems with sufficiently short-
range interaction. Hohenberg [137] shows no phase transition at finite
temperature for one- and two-dimensional superfluid systems; Mermin and Wagner
[136] further exclude the possibility for spontaneous magnetisation in the
Heisenberg model.
#### 5.1.2 Time translation symmetry breaking
Time translation symmetry breaking is associated with the emergence of time
crystals, as an analogue to ordinary spatial crystals. In the following
context, we only focus on quantum time crystals. We know that time-independent
systems preserve the continuous time translation symmetry, and when the
continuous time translation symmetry is broken, the system displays certain
time-dependent properties. For an operator $O$ without the intrinsic time
dependence, we have
$\bra{\Psi}\dot{O}\ket{\Psi}=i\bra{\Psi}[H,O]\ket{\Psi}=0\qquad\text{for}\
\Psi=\Psi_{E}$ (5.3)
where $\Psi_{E}$ is the eigenstate of the Hamiltonian $H$ of the system. It
seems impossible for the breaking of even an infinitesimal time translation
symmetry. However, in the spatial analogue, one-point expectation values do
not serve as a proper diagnostic for spontaneous symmetry breaking as well.
Wilczek initially proposes a model with periodic motion in the ground state
[139]. Later it is pointed out that periodic motion are exhibited in some
excites state instead and the actual ground state does not show any time
crystallinity [140]. Further the possibilities of any spontaneous rotating
time crystals are excluded [141].
In general, continuous time crystals are proved to be impossible in the ground
state and in the equilibrium [142]. More specifically from Ref. [142], two-
point temporal correlation functions do not have a period to break the
continuous time translation symmetry but tend to be time independent under the
large volume limit for the system. Note that Ref. [132] pointed out some
errors in the original proofs.
Instead of continuous time translation symmetry, we may also consider whether
discrete time translation symmetry can be broken down in periodically driven
systems. These are referred to Floquet time crystals [143], or discrete time
crystals [144]. In many-body localised Floquet systems with a period-$T$
driving, the temporal correlations exhibit a period of $nT(n>1)$ or a Fourier
peak at $k/n$-frequency $(k=1,2,\dots,n)$ and show robustness of the
perturbation. Experimental verification for discrete time crystals has been
conducted in trapped ions [145], nitrogen-vacancy centres in diamond [146],
and NMR [147, 148].
There are also other variants of time crystals, like prethermal continuous
time crystals [149], boundary time crystals [150], cosmological time crystals
[151, 152], time quasi-crystals [153], and so on.
#### 5.1.3 Mathematical definitions of time crystals
There are a few mathematical definitions for time crystals. They are
consistent with each other but exhibit in different forms. As the temporal
analogue of crystals, time crystals are expected to break time translation
symmetry and exhibit long-range correlations in time.
These definition use two-point correlations functions in space and time, take
the large volume limit and show symmetry-breaking properties in time. We
introduce Watanabe and Oshikawa’s definitions first via local and integrated
order parameters respectively for continuous time translation symmetry. Then
we offer the corresponding definitions for discrete time translation symmetry.
We also give a practical definition for experimental use. We further
illustrate the definitions in the representation theory.
The mathematical definition of time crystals is firstly given by Watanabe and
Oshikawa [142] via time-dependent long-range order. In this way, they argue
that time crystals cannot exist in the ground state or in the equilibrium.
Long-range order exists if the spatial correlation of a local order parameter
$\hat{\phi}(\vec{x},t)$ has a non-zero limit
$\lim_{V\rightarrow\infty}\langle\hat{\phi}(\vec{x},0)\hat{\phi}(\vec{x}^{\prime},0)\rangle\rightarrow
c\neq 0,$ (5.4)
for $|\vec{x}-\vec{x}^{\prime}|$ very large compared to microscopic scales we
are considering. In the representation of integrated order parameter
$\hat{\Phi}\equiv\int_{V}\mathrm{d}^{d}x\hat{\phi}(\vec{x},0)$, the long-range
order is defined to exist when
$\lim_{V\rightarrow\infty}\langle\hat{\Phi}^{2}\rangle/V^{2}=c\neq 0$. As
crystals are characterised by long-range order, time crystals are defined
analogously in terms of the temporal version of long-range correlations; that
is,
$\lim_{V\rightarrow\infty}\langle\hat{\phi}(\vec{x},t)\hat{\phi}(0,0)\rangle\rightarrow
f(t),$ (5.5)
$f(t)$ is a non-vanishing periodic function for large $|\vec{x}|$, or,
$\lim_{V\rightarrow\infty}\langle
e^{i\hat{H}t}\hat{\Phi}e^{-i\hat{H}t}\hat{\Phi}\rangle/V^{2}\rightarrow f(t).$
(5.6)
In the above we consider continuous time translation symmetry. For discrete
time translation breaking [154], we denote the local operator $O_{i}$ with the
subscript $i$ for the position. Then we have the long-range order in time as
$\lim_{|i-j|\rightarrow\infty}\lim_{L\rightarrow\infty}\langle
O_{i}(t)O_{j}\rangle=f(t),$ (5.7)
where $L$ is the system size. Or consider the superposition of local operators
$O=\frac{1}{L}\sum_{i}c_{i}O_{i}$, then long-range order in time can be
written as
$\lim_{L\rightarrow\infty}\langle O^{{\dagger}}(t)O\rangle=f(t).$ (5.8)
With a Floquet unitary $U(T)$, the system exhibits the temporal correlations
in the limit of large system size when $f(t)$ has a period $t=nT$,
$n\in\mathbb{Z}$; this is a special case for the so-called discrete time
crystals. In particular, time translation symmetry breaking is defined in Ref.
[143] when the expectation values of a local operator are different in a
period of the Floquet system for every state with short-range correlations.
The short-range correlations exist in a state $\ket{\psi}$ when
$\bra{\psi}\phi(x)\phi(x^{\prime})\ket{\psi}-\bra{\psi}\phi(x)\ket{\psi}\bra{\psi}\phi(x^{\prime})\ket{\psi}\rightarrow
0$ for any local operator $\phi(x)$.
However, in practice, it is hard to measure this long-range order in time for
experiments. Thus, a adapted definition frequently used in experiments is
given by
$\lim_{t\rightarrow\infty}\lim_{V\rightarrow\infty}\bra{\psi_{0}}O_{i}(t)\ket{\psi_{0}}=f(t)$
(5.9)
with $\ket{\psi_{0}}$ as a generic short-range correlated initial state.
Another definition uses the representation theory [154]. Suppose that a family
of local order parameters $\Phi_{i,\alpha}$, labeled by the position $i$ and
the irreducible representation $\alpha$ of the time translation symmetry
(either continuous $\mathbb{R}$ or discrete $\mathbb{Z}$), transform under
nontrivial irreducible representations as
$U^{{\dagger}}(t)\Phi_{i,\alpha}U(t)=e^{i\Delta_{\alpha}t}\Phi_{i,\alpha}$. By
$\alpha$ nontrivial, we mean that $\bra{n}\Phi_{i,\alpha}\ket{n}=0$ for all
eigenstates $\ket{n}$ of either the Hamiltonian $H$ or the Floquet unitary
$U(T)$. Then continuous or discrete symmetry $\mathbb{R}$ or $\mathbb{Z}$ is
spontaneously broken into a discrete subgroup $H$, if (1)
$\lim_{|i-j|\rightarrow\infty}\lim_{L\rightarrow\infty}|\bra{n}\Phi_{i,\alpha}\Phi_{j,\bar{\alpha}}\ket{n}-\bra{n}\Phi_{i,\alpha}\ket{n}\bra{n}\Phi_{j,\bar{\alpha}}\ket{n}|=c_{0}\neq
0$ (5.10)
for $\Phi_{i,\alpha}$ transforming trivially under $H$ but nontrivially under
$\mathbb{R}$ or $\mathbb{Z}$; and (2)
$\lim_{|i-j|\rightarrow\infty}\lim_{L\rightarrow\infty}|\bra{n}\Phi_{i,\alpha}\Phi_{j,\bar{\alpha}}\ket{n}-\bra{n}\Phi_{i,\alpha}\ket{n}\bra{n}\Phi_{j,\bar{\alpha}}\ket{n}|=0$
(5.11)
for $\Phi_{i,\alpha}$ transforming nontrivially under $H$.
### 5.2 Definition: time crystals as long-range order in time
In this section, we propose a definition for time crystals in the pseudo-
density matrix formalism. It is consistent with all other definitions proposed
so far. Before that, we add a bit more discussion for long-range order.
#### 5.2.1 Long-range order
A crystal or crystalline solid is defined as a solid material whose
constituents are arranged in a periodic array on the microscopic level.
Specifically, in a unit cell, the arrangement of atoms or other constituents
is repeated again and again under the translation invariance. This lattice
periodicity as the defining property of a crystal, implies long-range order:
the orderliness over long distances can be predicted with the knowledge of one
cell and the translation symmetry. In Ref. [155], it is explained that the
solid phase is characterised by the existence of a long-range correlation. In
other words, it is known that a solid is crystalline if it has long-range
order. Thus a crystal is characterised by the existence of long-range order. A
crystal has long-range order in space; a time crystal, however it is defined,
should have long-range order in time.
To define a time crystal, the off-diagonal long-range order might be
interesting as well. The long-range order in the solid is exhibited in the
quantum mechanics in the diagonal element of the reduced density matrix
$\rho_{2}$ in the coordinate space [155]. For a density matrix $\rho$ with
$\Tr\rho=1$, reduced density matrices $\rho_{1}$, $\rho_{2}$, $\cdots$ are
defined as $\bra{j}\rho_{1}\ket{i}=\Tr(a_{j}\rho a_{i}^{{\dagger}})$,
$\bra{kl}\rho_{2}\ket{ij}=\Tr(a_{k}a_{l}\rho
a_{j}^{{\dagger}}a_{i}^{{\dagger}})$, etc., where $a_{i}$, $a_{j}$ represent
annihilation operators for the one-particle states $\ket{i}$, $\ket{j}$. Then
the off-diagonal long-range order exists if
$\bra{\mathbf{x^{\prime}}}\rho_{1}\ket{\mathbf{x}}$ does not vanish as
$|\mathbf{x}-\mathbf{x^{\prime}}|\rightarrow\infty$. Yang [155] also defines
the off-diagonal long-range order in a many-body system of bosons or fermions
with annihilation operators on different particles; the order characterises
the existence of a Bose-Einstein condensation in the phases He II and
superconductors. Thus, we expect that a time crystal is characterised by the
existence of long-range order in time and take it as the definition of a time
crystal.
#### 5.2.2 Time crystals in terms of temporal correlations
In the pseudo-density matrix formulation, the measure of long-range order in
time [44] is expressed as the two-point temporal correlation at times $t_{1}$
and $t_{N}$:
$\langle\sigma^{(1)},\sigma^{(N)}\rangle=\Tr\\{(\sigma^{(1)}\otimes\sigma^{(N)})R_{1,N}[\rho,\Phi]\\},$
(5.12)
where $R_{1,N}$ is the pseudo-density matrix between $t_{1}$ and $t_{N}$,
$\rho$ is the initial state and $\Phi$ is the channel evolution between
different times. $\sigma^{(1)}$ and $\sigma^{(N)}$, for example, can be chosen
as Pauli operators measured at $t_{1}$ and $t_{N}$ in the spin chains. For
$\rho$ with multiple qubits, $\sigma^{(1)}$ and $\sigma^{(N)}$ are usually
acted on different qubits with a large separation in space.
As a simple example, we consider the long-range order in time for a single
spin under the unitary evolution. The temporal correlation is always preserved
and no symmetry is broken here. Consider a qubit evolving unitarily in time.
Suppose the qubit is in an arbitrary state
$\rho=\frac{1}{2}\mathbbm{1}+\sum_{i=1,2,3}c_{i}\sigma_{i}$. From the time
$t_{k}$ to $t_{k+1}$, $k=1,2,\dots$, the qubit evolves under the same unitary
evolution $U$. Consider the temporal correlation from $t_{1}$ to $t_{n}$,
$\displaystyle\langle\sigma_{i},\sigma_{j}\rangle$
$\displaystyle=\Tr[\sigma_{i}\otimes\sigma_{j}R_{1n}]$
$\displaystyle=\sum_{\alpha,\beta=\pm
1}\Tr[P^{\beta}_{j}U^{n-1}P^{\alpha}_{i}\rho
P^{\alpha}_{i}(U^{{\dagger}})^{n-1}]$
$\displaystyle=\frac{1}{2}(\Tr[\sigma_{j}U^{n-1}\rho\sigma_{i}(U^{{\dagger}})^{n-1}]+\Tr[\sigma_{j}U^{n-1}\sigma_{i}\rho(U^{{\dagger}})^{n-1}])$
$\displaystyle=\frac{1}{2}\Tr[\sigma_{j}U^{n-1}\sigma_{i}(U^{{\dagger}})^{n-1}].$
(5.13)
Here $P^{\alpha}_{i}=\frac{1}{2}(\mathbbm{1}+\alpha\sigma_{i})$. For the last
equality, we use $\rho\sigma_{i}+\sigma_{i}\rho=2c_{i}I+\sigma_{i}$ with
$\rho=\frac{1}{2}I+\sum_{i=1,2,3}c_{i}\sigma_{i}$. Take $i=j$, Eqn. (5.2.2) is
equivalent to
$\langle\sigma_{i},(U^{{\dagger}})^{n-1}\sigma_{i}U^{n-1}\rangle=1.$ (5.14)
We conclude that long-range order in time is preserved for the unitary
evolution.
### 5.3 Continuous time translation symmetry
In this section, we discuss continuous time translation symmetry in terms of
general decoherent processes and the Mermin-Wagner theorem.
#### 5.3.1 General decoherent process
We have considered a single spin under the unitary evolution in the previous
section. In practice, interaction with the environment is unavoidable and
noise is always present. For a qubit evolving under a generic decohering
channel evolution, $\Phi$, we prove that there is no long-range order in time
for whatever strength of the decoherence. Specifically, there exists an
effective rate $\gamma<1$ from one time to the next 111Note that in the case
where only ($Z$) dephasing noise is present in the system, the (unrealistic)
exact initial state preparation of $\rho=(I+rZ)/2$, could lead to long-range
order in time., for which the long-range order in time is bounded by
$\Tr\\{(X^{(1)}\otimes X^{(N)})R_{1,N}[\rho,\Phi]\\}\leq\gamma^{N-1},$ (5.15)
which tends to $0$ exponentially as $N\rightarrow\infty$.
For the depolarising noise [12], suppose the evolution between two times
$t_{k}$ and $t_{k+1}$ (k = 1, 2, … , N-1) is
$\Phi:\rho\rightarrow(1-p)\rho+p\frac{I}{2}.$ (5.16)
For an arbitrary initial state, the two-time correlation function $\langle
XX\rangle$ between $t_{1}$ and $t_{N}$ is
$\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes X)R_{1N}]=(1-p)^{N-1}.$ (5.17)
It goes down exponentially with N, which suggests that the temporal long-range
order vanishes and no possible existence for time crystals.
Dephasing corresponds to the Bloch vector transformation
$\Phi:\vec{r}=(r_{x},r_{y},r_{z})\rightarrow(r_{x}\sqrt{1-\lambda},r_{y}\sqrt{1-\lambda},r_{z}),$
(5.18)
where $\vec{r}$ is a three component real vector and the state of a single
qubit is written in the Bloch representation
$\rho=\frac{I+\vec{r}\cdot\vec{\sigma}}{2}$, $\vec{\sigma}=(X,Y,Z)$;
$e^{-t/2T_{2}}=\sqrt{1-\lambda}$ with the dephasing as a ‘$T_{2}$’ (or ‘spin-
spin’) relaxation process [12]. For an arbitrary initial state, suppose the
evolution between two times $t_{k}$ and $t_{k+1}$ (k = 1, 2, … , N-1) is
$\Phi$, the two-time correlation function $\langle XX\rangle$ between $t_{1}$
and $t_{N}$ is
$\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes
X)R_{1N}]=(\sqrt{1-\lambda})^{N-1}.$ (5.19)
It also goes down exponentially with N, so that the temporal long-range order
vanishes and no time-crystalline phase can exist.
Another example may be a spin-echo unitary in an open system. Suppose the
Hamiltonian is given by $H=\frac{1}{2}\omega\sigma_{z}$. Putting it into the
Lindblad master equation, we have
$\frac{\partial\rho}{\partial
t}=\frac{i\omega}{2}[\sigma_{z},\rho]+\frac{\gamma}{2}(\sigma_{z}\rho\sigma_{z}-\rho)$
(5.20)
The solution $\Phi:\rho(0)\rightarrow\rho(t)$ is given in terms of the matrix
elements
$\begin{split}\rho_{00}(t)&=\rho_{00}(0)\\\
\rho_{01}(t)&=\rho_{01}(0)e^{-i\omega t-\gamma t}\\\
\rho_{10}(t)&=\rho_{10}(0)e^{i\omega t-\gamma t}\\\
\rho_{11}(t)&=\rho_{11}(0)\end{split}$ (5.21)
For $\rho=\frac{1}{2}(I+X)$, So
$\langle\sigma_{i},\sigma_{j}\rangle=\sum_{\alpha,\beta=\pm
1}\alpha\beta\Tr(\frac{1+\beta\sigma_{j}}{2}\Phi(\frac{1+\alpha\sigma_{i}}{2}))\Tr(\frac{1+\alpha\sigma_{i}}{2}\rho)=\cos(\omega
t)e^{-\gamma t}.$ (5.22)
Similarly, long-range order in time vanishes under the dephasing noise.
Now we consider a general decohering evolution. Instead of a particular kind
of noise, we assume the evolution under a completely positive trace-preserving
map $\mathcal{E}:\rho\rightarrow\sum_{k}E_{k}\rho E_{k}^{{\dagger}}$ with
$\sum_{k}E_{k}E_{k}^{{\dagger}}=I$. For every $E_{k}$, $E_{k}<I$ in the
decohering case; then there exists $\gamma<1$ such that $E_{k}\leq\gamma I$
for all $k$. Then, for one round of evolution,
$Tr[(\sigma_{i}\otimes\sigma_{j})R]=\sum_{k}\Tr[\sigma_{j}E_{k}\sigma_{i}E_{k}^{{\dagger}}]$
(5.23)
For $n$ rounds of evolution,
$\displaystyle Tr[(\sigma_{i}^{(1)}\otimes\sigma_{i}^{(n+1)})R]$
$\displaystyle=\sum_{k_{1},\dots,k_{n}}\Tr[\sigma_{i}E_{k_{n}}\cdots
E_{k_{1}}\sigma_{i}E_{k_{1}}^{{\dagger}}\cdots E_{k_{n}}^{{\dagger}}]$
$\displaystyle=\sum_{k_{1},\dots,k_{n}}\Tr[E_{k_{n}}\cdots
E_{k_{1}}\sigma_{i}(E_{k_{n}}\cdots E_{k_{1}}\sigma_{i})^{{\dagger}}]$
$\displaystyle\leq\gamma^{2n}.$ (5.24)
The long-range temporal correlations decay exponentially in time, suggesting
that the order vanishes. Thus under arbitrary decoherent evolutions in terms
of CPTP maps, a single spin has no long-range order in time. This result and
the discussion on unitary evolutions, exclude the possibility of time crystals
in 0+1 dimension unless we take the definition too trivial.
#### 5.3.2 Generalised Mermin-Wagner theorem
We mentioned the Goldstone theorem and the Mermin-Wagner theorem in the
literature review part. Here we discuss how to apply them to continuous time
translation symmetry breaking and time crystals.
In one of the early papers on the Goldstone theorem, it states that if the
Lagrangian of the system is invariant under the continuous symmetry
transformation, then either spinless particles with zero mass exist, or the
vacuum state is invariant [135]. That is, for a local scalar field $\phi$ and
a local conserved vector current $j_{\nu}$ with $\partial^{\nu}j_{\nu}=0$,
either Goldstone bosons exist, or the expectation value of $\delta\phi$
vanishes in the vacuum state, where the scalar field $\delta\phi$ is defined
by $\delta\phi(y)=i\int\textrm{d}^{3}\bm{x}[j_{0}(x_{0},\bm{x}),\phi(y)].$ In
Ref. [156], it is argued that the vacuum expectation value of $\delta\phi$
always vanishes for two-dimensional spacetime:
$\bra{0}\delta\phi(0)\ket{0}=i\bra{0}\int\textrm{d}x_{1}[j_{0}(x_{0},x_{1}),\phi(0)]\ket{0}=0.$
(5.25)
This suggests continuous symmetry cannot be broken in 1+1 dimensions for the
ground state. The proof is straightforward for continuous time translation
symmetry as the Hamiltonian $H$ is conserved. In 1+1 dimensions, $H\delta
t=\int\textrm{d}xj_{0}(t,x)$; thus, the expectation value of $\delta\phi$
vanishes in the vacuum state. For a general continuous symmetry, consider the
integrals
$F_{\nu}(k_{0},k_{1})=\iint\textrm{d}x_{0}\textrm{d}x_{1}e^{i(k_{0}x_{0}+k_{1}x_{1})}\bra{0}j_{\nu}(x_{0},x_{1})\phi(0)\ket{0}$
in the momentum space. After solving the integrals from conservation
conditions, we find the only contribution to $\bra{0}\delta\phi(0)\ket{0}$
vanishes to avoid a singularity. The proof is given similarly as in Ref.
[156].
Further, we apply the Mermin-Wagner theorem [136, 137] to 1+1 dimensional
spacetime and argue that no continuous time translation symmetry breaking
occurs for finite temperature due to lack of long-range temporal order. More
specifically, consider a system in the equilibrium, that is, in the thermal
state $\rho=e^{-\beta H}/\Tr(e^{-\beta H})$. The expectation value of an
operator $A$ is given by $\langle
A\rangle=\lim_{V\rightarrow\infty}\Tr(e^{-\beta H}A)/\Tr(e^{-\beta H})$. Under
the continuous time translation symmetry, the Hamiltonian $H$ serves as the
generator and is invariant. From statistical mechanics, we learn that even for
an operator $B$ that does not commute with the generator $H$ ($[B,H]=C\neq
0$), the expectation value of $[B,H]=C$ is still 0. However, in the
spontaneous symmetry breaking, when we add a small perturbation to the
Hamiltonian, the expectation value of $[B,H]=C$ does not vanish anymore. Here
we use the long-range temporal correlations as the indicator and argue that
they vanish as the perturbation parameter goes smaller to exclude the
possibility of spontaneous continuous time translation symmetry breaking. We
take the experimental definition of time crystals and investigate the temporal
correlations in the Heisenberg model. We prove that under the finite
temperature, when we add a small perturbation to the Hamiltonian, the long-
range temporal correlation vanishes for the perturbation goes smaller and
smaller. Thus, we conclude that there can be no spontaneous breaking for
continuous time translation symmetry in 1+1 dimensions for finite temperature.
The proof is given via the Bogoliubov inequality in Appendix C.
In this subsection, we apply the Goldstone theorem and the Mermin-Wagner
theorem to 1+1 dimensional spacetime, and argue that there is no continuous
time translation symmetry breaking for the ground states and the equilibrium.
This result is consistent with general absence of continuous time crystals and
provides a different understanding which might be useful to discuss space-time
crystals in relativistic field theory. We leave it for the future work.
So far, we investigate the possibilities of continuous time translation
symmetry breaking in 0+1 and 1+1 dimensions. As a result of the lack of long-
range order in time, continuous time translation symmetry cannot be
spontaneously broken in these cases.
### 5.4 Discrete time translation symmetry
In this section, we investigate discrete time translation symmetry. One
possible suggestion from quantum information is to apply periodic
stabilisation of quantum computation and quantum error correction to
counteract the decoherence to preserve long-range order in time. In this case,
discrete time translation symmetry is preserved for the single-qubit case. We
further turn on to one-dimensional spin chains under many-body localisation
and Floquet driving. That is the usual model considered for discrete time
crystals. We apply the pseudo-density matrix formulation to simplify the
calculation for temporal correlations and use group theory to gain a better
understanding of how the subharmonic periodicity emerges.
#### 5.4.1 Stabilisation of quantum computation
In this subsection we discuss temporal correlations under the stabilisation of
quantum computation. For simplicity, only single-qubit error is considered
here.
Figure 5.1: Quantum circuit for symmetrisation error correction as a time
crystal. If the auxiliary qubit is found in state $\ket{0}$, the
symmetrisation has been successful.
Let us recall the principle of stabilisation of quantum computation via the
projection onto the symmetric subspace [157]. The key idea is that a pure
state $\ket{\phi}$ can be protected against decoherence by encoding it
redundantly in $N$ qubits and projecting their overall state onto the
symmetric subspace (i.e., the minimal subspace containing all the states
$\ket{\phi}^{\otimes N}$). For the sake of simplicity, let us use two qubits
(see Fig. 5.1). This can be generalised to $N$ qubits easily. Suppose the two
qubits, initialised in an arbitrary pure state $\ket{\psi}\otimes\ket{\psi}$,
undergo the noisy channel evolution $\Phi\otimes\Phi$, where $\Phi$ is the
depolarising noise [12]. Suppose the evolution between two times $t_{k}$ and
$t_{k+1}$ (k = 1, 2, … , N-1) is
$\Phi:\rho\rightarrow\rho_{p}(1-p)\rho+p\frac{I}{2},$ (5.26)
so that the two qubits evolve into some mixed state $\rho_{p}$. Now, project
each of them onto the symmetric subspace by measuring the auxiliary qubit and
discarding outcomes of 1. This is represented by the operator:
$\Sigma_{12}=\frac{1}{2}(I_{12}+S_{12}),$ (5.27)
where $S_{12}$ is the SWAP operator acting on the two qubits. It completes the
effective evolution caused by the error correction protocol (which is a
probabilistic procedure). The action of the projection on a single qubit
starting in the state $\rho_{p}$ is
$\Sigma:\rho_{p}\rightarrow\Tr_{2}\left[\frac{\Sigma_{12}(\rho_{p}\otimes\rho_{p})\Sigma_{12}^{{\dagger}}}{\Tr(\Sigma_{12}(\rho_{p}\otimes\rho_{p})\Sigma_{12}^{{\dagger}})}\right]=\frac{\rho_{p}+\rho_{p}^{2}}{\Tr(\rho_{p}+\rho_{p}^{2})}\equiv\rho^{\prime},$
(5.28)
where
$\Tr(\rho^{\prime 2})>\Tr(\rho_{p}^{2}).$ (5.29)
Thus, error correction by symmetrisation makes the state purer. The
convergence to a pure state is improved by acting on a larger number of
qubits.
We can apply the pseudo-density matrix description to the evolution outlined
above. For an arbitrary initial state, only with depolarising noise $\Phi$ and
without symmetrisation error correction $\Sigma$, recall that the two-time
correlation function $\langle XX\rangle$ between $t_{1}$ and $t_{N}$ is
$\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes X)R_{1N}]=(1-p)^{N-1}.$ (5.30)
With both of depolarising noise $\Phi$ and symmetrisation error correction
$\Sigma$,
$\displaystyle\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes X)R_{1N}]=a_{N},$
$\displaystyle\text{where}\ a_{n+1}=\frac{4a_{n}(1-p)}{3+a_{n}^{2}(1-p)^{2}},\
a_{1}=1.$ (5.31)
For $p\leq 1/4$, $\langle X^{(1)}X^{(N)}\rangle$ converges to a constant
$\frac{\sqrt{1-4p}}{1-p}$ as $N$ becomes large. For $1/4<p<1$, it decays to 0
with a smaller rate than in the case of uncorrected noise (cf. Fig. 5.2).
Figure 5.2: $\langle X^{(1)}X^{(N)}\rangle$ vs. N for depolarising noise with
and without error correction (solid and dashed lines, resp.).
The analysis is similar for dephasing noise mentioned before. For an arbitrary
initial state, suppose the evolution between two times $t_{k}$ and $t_{k+1}$
(k = 1, 2, … , N-1) is $\Phi$, the two-time correlation function $\langle
X,X\rangle$ between $t_{1}$ and $t_{N}$ is
$\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes
X)R_{1N}]=(\sqrt{1-\lambda})^{N-1}.$ (5.32)
With the full protocol applied, the two-time correlation function for the
first qubit reads222As mentioned in footnote 1, initial states of form
$\rho=(I+rZ)/2$ already exhibit long-range order in time and do not require
the protocol for dephasing noise only.
$\displaystyle\langle X^{(1)},X^{(N)}\rangle=\Tr[(X\otimes X)R_{1N}]=b_{N},$
$\displaystyle\text{where}\
b_{n+1}=\frac{4b_{n}\sqrt{1-\lambda}}{3+b_{n}^{2}(1-\lambda)},\ b_{1}=1.$
(5.33)
Comparing the two results with and without error correction, we find that the
two-time correlation function with error correction converges to a finite
value for $p<1/4$ or $\lambda<7/16$ and decays at a slower rate otherwise;
this implies that (finite) long-range order can in principle be restored.
Furthermore, if we keep applying the error correction scheme on a larger
number $N$ of qubits, the long-range order in time will be fully restored.
#### 5.4.2 Quantum error correction of phase flip codes
Figure 5.3: Quantum circuit of the phase flip code. The noise
$E_{\text{phase}}$ flips $\ket{+}$ to $\ket{-}$ for one qubit and vice versa.
In the error model, a qubit is left alone with probability $1-p$, and with
probability $p$ the relative phase of the $\ket{0}$ and $\ket{1}$ states is
flipped. That is, the initial state $\alpha\ket{0}+\beta\ket{1}$ goes to the
state $\alpha\ket{0}-\beta\ket{1}$ after the phase flip $Z$.
Now we consider the quantum error correction of phase flip codes. Let
$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$ be an arbitrary qubit. Suppose the
only noise is one single phase flip $Z$ on one of the three qubits in Figure
5.3. This occurs with probability $(1-p)^{3}+3p(1-p)^{2}=1-3p^{2}+2p^{3}$.
For single flip and no flip, after the error correction, the state remain
unchanged as $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$ with probability
$1-3p^{2}+2p^{3}=1-q$. For two or three flips, after the error correction, the
state becomes $\ket{\psi^{\prime}}=\alpha\ket{1}+\beta\ket{0}$ with
probability $3p^{2}-2p^{3}=q$. Now apply the protocol for $N$ times. Consider
the two-time correlation functions for the first qubit at the initial time
$t_{1}$ and the final time $t_{N}$333Consider small $p$. For $N$ odd, the
probability $p_{c}$ to change the state is
$C_{N}^{0}q^{N}+C_{N}^{2}q^{N-2}(1-p)^{2}+\cdots+C_{N}^{N-1}q(1-q)^{N-1}=Nq+O(q^{2})$;
for N even,
$p_{c}=C_{N}^{1}q^{N-1}(1-q)+C_{N}^{3}q^{N-3}(1-q)^{3}+\cdots+C_{N}^{N-1}q(1-q)^{N-1}=Nq+O(q^{2})$.
$\langle Z^{(1)}Z^{(N)}\rangle=1-2p_{c}$.:
$\displaystyle\langle X^{(1)},X^{(N)}\rangle$ $\displaystyle=1,$
$\displaystyle\langle Z^{(1)},Z^{(N)}\rangle$
$\displaystyle=1-2N(3p^{2}-2p^{3})+O(p^{4}).$ (5.34)
In this case, the $\langle XX\rangle$ correlation is always 1 and long-range
correlation in time along the $X$ direction is preserved. For small $p$ and
finite $N$, long-range correlation in time along the $X$ direction remains
close to 1 and is almost preserved. We will not discuss a full quantum error
correction scheme here as it is very similar with the $X$ direction here, or
the $Z$ direction with $p=0$. In principle, long-range order in time can be
fully restored under a full error correction scheme.
#### 5.4.3 Floquet many-body localisation
In this subsection, we consider many-body localised systems with Floquet
driving. Discrete time translation symmetry is broken and thus the model
constitutes a discrete time crystal. We formulate these discrete time crystals
in the language of pseudo-density matrices and group theory.
##### Temporal correlations in pseudo-density matrix formulation
Here we calculate temporal correlations from the pseudo-density matrix
formulation. In such a particular Floquet many-body localised system, discrete
time translation symmetry of a period $T$ is broken to discrete time
translation symmetry of a period $nT(n\in\mathbb{Z},n>1)$. In particular, we
consider a one-dimensional spin-$\frac{1}{2}$ chain under the binary
stroboscopic Floquet Hamiltonian for a period $T=T_{1}+T_{2}$.
$H_{f}(t)=\left\\{\begin{array}[]{lcl}H_{1}=(g-\epsilon)\sum_{i}\sigma_{i}^{x}&&{0<t<T_{1}}\\\
H_{2}=\sum_{i}J_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{i}^{z}\sigma_{i}^{z}+h_{i}^{x}\sigma_{i}^{x}&&{T_{1}<t<T}\end{array}\right.$
(5.35)
Without the loss of generality, we assume that $T_{1}=T_{2}=1$. Then the
Floquet unitary is given by
$U_{f}=U_{2}U_{1}=e^{-iH_{2}}e^{-iH_{1}}$ (5.36)
Take $g=\pi/2$. For small perturbations with $\epsilon>0$, the periodicity
does not hold.
The simplest case takes $\epsilon=J_{i}=h_{i}^{x}=0$. We take an arbitrary
state in $z$-basis $\ket{\psi_{0}}=\ket{\\{s_{i}\\}}$ with $s_{i}=\pm 1$ and
$\sigma_{k}^{z}\ket{\\{s_{i}\\}}=s_{k}\ket{\\{s_{i}\\}}$. After the spin-echo
unitary $U_{1}=e^{i\pi/2\sum_{i}\sigma_{i}^{x}}=\prod_{i}i\sigma_{i}^{x}$, the
state evolves to $\ket{\psi_{1}}=\ket{\\{-s_{i}\\}}$. Then
$U_{2}=\sum_{i}h_{i}^{z}\sigma_{i}^{z}$ gives a global phase that
$\ket{\psi_{2}}=e^{i\phi}\ket{\\{-s_{i}\\}}$. In the pseudo-density matrix
formulation, the temporal correlation for odd periods is $-e^{i\phi}$. For
even periods, temporal correlation remains equal to 1. However, decoupled
spins under spin echos cannot be taken as a discrete time crystal. The reason
is that for small $\epsilon>0$, the $\omega/2$ Fourier peak is split and
$2T$-periodicity is broken down.
Now we turn on the interaction $J_{i}>0$. Take $h_{i}^{x}=0$. Eigenstates of
$H_{2}$ are eigenstates of $\sigma_{i}^{z}$ in the form of $\ket{\\{s_{i}\\}}$
as before:
$H\ket{\\{s_{i}\\}}=[E^{+}(\\{s_{i}\\})+E^{-}(\\{s_{i}\\})]\ket{\\{s_{i}\\}}$
(5.37)
with $E^{+}(\\{s_{i}\\})=\sum_{i}J_{i}s_{i}s_{i+1}$ and
$E^{-}(\\{s_{i}\\})=\sum_{i}h_{i}^{z}s_{i}$. Consider $\epsilon=0$ first.
Again $U_{1}=e^{i\pi/2\sum_{i}\sigma_{i}^{x}}=\prod_{i}i\sigma_{i}^{x}$. Then
the Floquet eigenstates of the Floquet unitary $U_{f}$ are
$e^{iE^{-}(\\{s_{i}\\})/2}\ket{\\{s_{i}\\}}\pm
e^{-iE^{-}(\\{s_{i}\\})/2}\ket{\\{-s_{i}\\}}$. The Floquet eigenvalues are
$\pm\exp[iE^{+}(\\{s_{i}\\})]$. In the pseudo-density matrix formulation, for
the arbitrary initial state $\ket{\\{s_{i}\\}}$, the temporal correlation on
$\sigma_{z}^{k}$ of a particular spin $s_{k}$ is
$-e^{iE^{+}(\\{s_{i}\\})-iE^{-}(\\{s_{i}\\})}$ in one period. For double
periods, it will be $e^{2iE^{+}(\\{s_{i}\\})}$ with the absolute value 1. For
all even periods, the absolute value of the temporal correlation remains equal
to 1. Note here we consider for temporal correlations for a single spin
instead of a superposition of all spins. An arbitrary superposition will give
no correlations; for certain particular superpositions, the temporal
correlations are the same as single-spin temporal correlations. When
$\epsilon>0$,
$U_{1}=e^{i(\pi/2-\epsilon)\sum_{i}\sigma_{i}^{x}}=\prod_{i}I\sin\epsilon+i\sigma_{i}^{x}\cos\epsilon$.
In the pseudo-density matrix formulation, the temporal correlation is a bit
complicated but the absolute value still converges to 1 for even periods
without the half-frequency peak splitting. The robustness guarantees the model
to be taken as a discrete time crystal.
Consider $h_{i}^{x}\neq 0$. For simplicity, assume that $h_{i}^{z}=0$. In this
particular case, the model exhibits a hidden emergent Ising symmetry
$\tilde{S}=U^{{\dagger}}_{FD}\prod_{i}\sigma_{i}^{x}U_{FD}=\prod_{i}\sigma_{i}^{x}$.
Here a finite depth unitary transformation $U_{FD}$ satisfies
$U_{FD}U(T)U_{FD}^{{\dagger}}=e^{-i\tilde{H}T}\prod_{i}\sigma_{i}^{x},$ (5.38)
with
$\tilde{H}=\sum_{i}\tilde{J}_{i}^{z}\sigma_{i}^{z}\sigma_{i+1}^{z}+\tilde{h}_{i}^{x}\sigma_{i}^{x}$.
Then we have
$U(2T)=U_{FD}^{{\dagger}}e^{-2i\tilde{H}T}U_{FD}=e^{-2iU_{FD}^{{\dagger}}\tilde{H}U_{FD}T}=e^{-2iH_{eff}T}.$
(5.39)
Referring back to the results in unitary evolution part, this suggests the
$2T$-periodicity of temporal correlations in the model. Noisy perturbations
won’t split the half-frequency peak. Thus, the model constitutes a discrete
time crystal. With $h_{i}^{z}\neq 0$, the results are similar but with a
different hidden Ising symmetry $\tilde{S}$.
##### Group representation
Here we consider discrete time translation symmetry breaking in terms of group
representation. It is more clear how multiple periods come into existence in
the Floquet many-body localisation.
Consider the Hamiltonians $H(t)$ have an onsite symmetry group $G$ and a
discrete time translation symmetry $\mathbb{Z}$ that $H(t+T)=H(t)$. Based on
the discussion in Ref. [158], the Floquet phases are characterised by a
central element of the group, that is, Floquet unitary takes the form
$U_{f}=u_{\\{B\\}}(z_{0})V(z_{0}),$ (5.40)
where $z_{0}$ is an element of the centre of the group denoted $Z(G)$. The
onsite symmetry group $G$ has, for example, an irreducible representation
$\chi$ with operators $\mathfrak{g}_{ij}^{\xi}$. An initial state in a singlet
evolves under the global symmetry in this irreducible representation $\chi$.
Remember that in any irreducible representation of a finite group G, all the
elements of $Z(G)$ are represented by $\lambda I$ where $\lambda$ is a
constant and $I$ is unit matrix [159]. Then we have
$U_{f}\mathfrak{g}_{ij}^{\chi}U_{f}^{{\dagger}}=\frac{\chi(z)}{\chi(1)}\mathfrak{g}_{ij}^{\chi}$
(5.41)
where $\chi(z)$ is the shifted constant at $z\in Z(G)$ under the irreducible
representation $\chi$. Apply $U_{f}$ for $n$ times. For $z\neq 1$,
$\mathfrak{g}_{ij}^{\chi}(nT)=\left[\frac{\chi(z)}{\chi(1)}\right]^{n}\mathfrak{g}_{ij}^{\chi}(0).$
(5.42)
For a one-dimensional spin chain under Floquet many-body localisation as in
the previous subsection, $Z(nT)=(-1)^{n}Z(0)$ shows a period of $2T$ for the
order parameter. Thus, it constitutes a discrete time crystal.
#### 5.4.4 Possible sufficient conditions for general open systems
A straightforward generalisation to time crystals in open systems is to
formulate the Hamiltonian in terms of annihilation and creation operators and
solve the Lindblad equation. The difficulty lies in the exact solution of
Lindblad equations. Here, we attempt to reformulate the evolution into Kraus
operators. In general, it is hard to see what kind of Kraus operators will
work for arbitrary initial states and arbitrary periods, as it is unknown what
kind of physical evolution a general Kraus operator is represented for and its
physical meaning. We only give a simple illustration on the mathematical
conditions for the initial state $\ket{\\{s_{i}\\}}$ and $2T$-periodicity in
the pseudo-density matrix formulation. The general cases work in the similar
way.
For an initial state $\ket{\\{s_{j}\\}}$, we measure $\sigma_{i}^{z}$ to gain
the eigenvalue $s_{i}$ with probability $1$ and leave the state unchanged.
Assume the evolution is given by a set of Kraus operators $\\{E_{k}\\}$, then
the state evolves to
$\sum_{k}E_{k}\ket{\\{s_{j}\\}}\bra{\\{s_{j}\\}}E_{k}^{{\dagger}}$. We measure
for $\sigma_{i}^{z}$ again. The temporal correlation given by the pseudo-
density matrix formulation is
$\langle\sigma_{i}^{z},\sigma_{i}^{z}\rangle=\Tr[\sigma_{i}^{z}\otimes\sigma_{i}^{z}R]=s_{i}\Tr[\sigma_{i}^{z}\sum_{k}E_{k}\ket{\\{s_{j}\\}}\bra{\\{s_{j}\\}}E_{k}^{{\dagger}}]$
(5.43)
For $\langle\sigma_{i}^{z},\sigma_{i}^{z}\rangle$ to have $2T$-periodicity, a
sufficient condition might be
$\sum_{k}E_{k}\ket{\\{s_{j}\\}}\bra{\\{s_{j}\\}}E_{k}^{{\dagger}}=-\ket{\\{-s_{j}\\}}\bra{\\{-s_{j}\\}}.$
(5.44)
For $k=1$, it reduces to the temporal correlation which is the same as in one-
dimensional spin chain under Floquet many-body localisation.
### 5.5 An algebraic point of view
In this section, we apply the algebraic tools to analyse spontaneous time
translation symmetry breaking. The algebraic approach of symmetry breaking
offer a representation with clear mathematics. Note that spontaneous time
translation symmetry breaking can only be exhibited in the thermodynamic limit
where the number of particles $N\rightarrow\infty$, the volume of the system
$V\rightarrow\infty$ and the ratio $n=N/V$ fixed. For infinite degrees of
freedom, the algebraic approach does not distinguish relativistic quantum
field theory and quantum mechanics for continuous variables and will naturally
offer a relativistic treatment. Here we attempt to treat space and time more
equally in the pseudo-density matrix formalism, and it is interesting to
investigate the algebraic approach of symmetry breaking for further
generalisation to the relativistic context.
In the following context, we review the algebraic criterion on spontaneous
symmetry breaking and later apply them to explore time crystals. We further
discuss the possibility to classify and understand temporal correlations from
operator algebra.
#### 5.5.1 Preliminaries
In this subsection we review the preliminaries for the algebraic approach.
Specifically, we introduce the Weyl algebra, the concept of states, the
Gelfand-Naimark-Segal(GNS) construction, and the algebraic symmetry of an
algebra. This part is based on Ref. [131].
Instead of the canonical variables $q$, $p$ and the Heisenberg algebra
$\mathcal{A}_{H}$, we construct the Weyl operators $U(\alpha)\equiv e^{i\alpha
q}$, $V(\beta)\equiv e^{i\beta p}$, where $\alpha
q\equiv\sum_{i}\alpha_{i}q_{i}$, $\beta p\equiv\sum_{i}\beta_{i}p_{i}$,
$\alpha_{i},\beta_{i}\in\mathbb{R}$, and the corresponding Weyl algebra
$\mathcal{A}_{W}$. The Heisenberg commutation relations ($\hbar=1$) given as
$[q_{i},p_{j}]=i\delta_{ij}$, $[q_{i}.q_{j}]=0=[p_{i},p_{j}]$,
$i,j=1,2,\cdots,N,$ turns into
$\displaystyle U(\alpha)U(\alpha^{\prime})=U(\alpha+\alpha^{\prime}),$
$\displaystyle\qquad V(\beta)V(\beta^{\prime})=V(\beta+\beta^{\prime})$
$\displaystyle U(\alpha)V(\beta)=$ $\displaystyle
e^{-i\alpha\beta}V(\beta)U(\alpha).$ (5.45)
The conditions of $q=q^{{\dagger}}$ and $p=p^{{\dagger}}$ give
$U(\alpha)^{*}=U(-\alpha)$, $V(\beta)^{*}=V(-\beta)$. We introduce a norm
$\norm{\cdot}$ for elements in $\mathcal{A}_{W}$ that
$\norm{A^{*}A}=\norm{A}^{2}$, $\forall A\in\mathcal{A}_{W},$ then the Weyl
algebra $\mathcal{A}_{W}$ becomes a $C^{*}$-algebra.
A state $\Omega$ of the system is characterised by the set of expectation
values $\\{\Omega(A),A\in\mathcal{A}\\}$ where $\Omega(A)\equiv\langle
A\rangle_{\Omega}$. That is, $\Omega$ is a functional
$\Omega:\mathcal{A}\rightarrow\mathbb{C}$ satisfying the linearity
$\Omega(\alpha A+\beta B)=\alpha\Omega(A)+\beta\Omega(B)$, the positivity
$\Omega(A^{*}A)\geq 0,\forall A\in\mathcal{A}$, and the normalisation
$\Omega(1)=1$. In $C^{*}$-algebra, any state which cannot be decomposed into
any other two states as
$\Omega=\lambda\Omega_{1}+(1-\lambda)\Omega_{2},0<\lambda<1$ is a pure state;
otherwise, it is mixed. In a Hilbert space $\mathcal{H}$, a representation
$\pi$ of a $C^{*}$-algebra is a ∗-homomorphism $\pi$ of $\mathcal{A}$
preserving all the algebraic operations, into the $C^{*}$-algebra of bounded
linear operators in $\mathcal{H}$ . The Gelfand-Naimark-Segal(GNS)
construction uses a representation $\pi_{\Omega}$ of $A(A\in\mathcal{A})$
which is uniquely determined by the state $\Omega$ in terms of its
expectations on $\mathcal{A}$ up to isometries:
$(\Psi_{\Omega},\pi_{\Omega}(A)\Psi_{\Omega})=\Omega(A),\forall
A\in\mathcal{A},$ (5.46)
where $\Psi_{\Omega}$ is a reference vector in the Hilbert space
$\mathcal{H}_{\Omega}$.
The algebraic symmetry of an algebra $\mathcal{A}$ is then defined by an
invertible mapping $\beta$ of the algebra into itself, preserving all the
algebraic relations including the $*$-automorphism of $\mathcal{A}$. For a
state $\omega$ on $\mathcal{A}$,
$(\beta^{*}\omega)(A)\equiv\omega(\beta(A))$ (5.47)
is a state on $\mathcal{A}$ as well. The algebraic symmetry $\beta$, under a
representation $\pi_{\omega}$ of $\mathcal{A}$, has a Wigner symmetry in
$\mathcal{H}_{\omega}$ under a unitary operator $U_{\beta}$ such that
$U_{\beta}\pi_{\omega}(A)U_{\beta}^{{\dagger}}=\pi_{\omega}(\beta(A))=\pi_{\beta^{*}\omega}(A).$
(5.48)
This is, $\pi_{\beta^{*}\omega}$ is unitarily equivalent to $\pi_{\omega}$. We
will say $\\{\pi_{\omega},\mathcal{H}_{\omega}\\}$ is $\beta$-symmetric.
However, when $\pi_{\beta^{*}\omega}$ is not unitarily equivalent to
$\pi_{\omega}$, the symmetry $\beta$ is spontaneously broken.
#### 5.5.2 Existence of time crystals
In this subsection we review the criteria on spontaneous symmetry breaking and
apply them to time crystals.
##### Criteria on spontaneous symmetry breaking
Here we review the criteria on spontaneous symmetry breaking for the ground
state [131] and the equilibrium [160].
Given the following conditions for a representation $\pi$ of the algebra
$\mathcal{A}$:
(1) The space and time translations, represented by different continuous
groups for unitary operators, guarantee the existence of energy and momentum
in the representation space $\mathcal{H}_{\pi}$;
(2) The energy remains nonnegative; that is, all the possible values for the
Hamiltonian are bounded;
(3) The ground state $\omega$ is uniquely invariant under translations in
$\mathcal{H}_{\pi}$, and it is represented by a cyclic vector locally.
For an algebraic symmetry $\beta$ which commutes with space translations and
time translations, $\beta$ is unbroken in $\pi$ if and only if correlation
functions for all the ground states are invariant under $\beta$:
$\omega(\beta(A))\equiv\langle\beta(A)\rangle_{0}=\langle
A\rangle_{0}=\omega(A),\forall A\in\mathcal{A},$ (5.49)
where $\omega$ is the ground state. Conditions (1)-(3) imply the cluster
property: the correlations of two operators factorise when one of them goes to
the spacial infinity
$\lim_{|\mathbf{x}\rightarrow\infty|}[\langle
AB_{\mathbf{x}}\rangle_{0}-\langle A\rangle_{0}\langle B\rangle_{0}]=0.$
(5.50)
Similar for the equilibrium. If we change the condition (3) into
(3’) The state $\omega$ is invariant under a subgroup $\mathcal{T}$ of spatial
translations in $\mathcal{H}_{\pi}$, and it satisfies $\mathcal{T}$-asymptotic
abelianess:
$\lim_{n\rightarrow\infty}[T^{n}(A),B]=0,\forall
A,B\in\mathcal{A},T\in\mathcal{T},$ (5.51)
and calculate correlation functions for the thermal states, then we have the
criteria for the equilibrium as Ref. [160]. These criteria for spontaneous
symmetry breaking are equivalent to the existence of long-range order in
infinitely extended systems with local structures and asymptotic abelianess;
that is,
$\lim_{n\rightarrow\infty}\omega(T^{n}(\Delta A)B)=\omega(\Delta
A)\omega(B)\neq 0$ (5.52)
where $\Delta A\equiv\beta(A)-A$.
##### Example: time translation symmetry breaking
Now we apply the above criteria to time translation symmetry breaking. First
we consider the continuous time translation symmetry group denoted by
$U(t),t\in\mathbb{R}$. Then $\beta(A)=U(t)AU^{{\dagger}}(t)$. And the
continuous time translation symmetry group is denoted by
$U(a),a\in\mathbb{R}^{s}$. It is easy to see that the long-range order does
not exist:
$\displaystyle\lim_{n\rightarrow\infty}\omega(T^{n}(\Delta A)B)$
$\displaystyle=\lim_{n\rightarrow\infty}\omega(T^{n}(U(t)AU^{{\dagger}}(t)-A)B)$
$\displaystyle=\lim_{n\rightarrow\infty}\omega(T^{n}(U(t)AU^{{\dagger}}(t)-A)\omega(B)$
$\displaystyle=\lim_{n\rightarrow\infty}\omega(U(a_{n})\cdots
U(a_{1})U(t)AU^{{\dagger}}(t)U^{{\dagger}}(a_{1})\cdots
U^{{\dagger}}(a_{n})-A)\omega(B)$ $\displaystyle=0$ (5.53)
The limit goes to 0 as we can always choose infinite runs of space
translations to mimic a time evolution such that after time translations and
space translation the operator goes back to itself under the average of the
ground state or the thermal state.
For discrete time translation symmetry, we consider a one-dimensional Floquet
many-body localised spin chain again. Recall that the Floquet evolution is
given by $U_{f}=U_{2}U_{1}$, where $U_{1}=\exp[it_{1}\sum_{i}\sigma_{i}^{x}]$
and $U_{2}=\exp[-iH_{MBL}t_{2}]$ where
$H_{MBL}=\sum_{i}J_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{i}^{z}\sigma_{i}^{z}$
for simplicity here. For $t_{1}\approx\pi/2$,
$U_{1}=\prod_{i}i\sigma_{i}^{x}$. Take $A=\sigma_{i}^{z}$ and
$B=\sigma_{j}^{z}$. After a period of $T$,
$\beta_{1}(A)=U_{f}\sigma_{i}^{z}U^{{\dagger}}_{f}=U_{2}U_{1}\sigma_{i}^{z}U_{1}^{{\dagger}}U_{2}^{{\dagger}}=-\sigma_{i}^{z}$,
then
$\lim_{n\rightarrow\infty}\omega(T^{n}(\Delta
A)B)=\lim_{n\rightarrow\infty}\omega(T^{n}(-2\sigma_{i}^{z})\sigma_{j}^{z})=0,$
(5.54)
where $n$ and $|i-j|$ go to infinity. The long-range order does not exist for
a single period. After two periods of $T$,
$\beta_{2}(A)=U_{f}U_{f}\sigma_{i}^{z}U^{{\dagger}}_{f}U^{{\dagger}}_{f}=\sigma_{i}^{z}$
$\lim_{n\rightarrow\infty}\omega(T^{n}(\Delta
A)B)=\lim_{n\rightarrow\infty}\omega(T^{n}(\sigma_{i}^{z}-\sigma_{i}^{z})\sigma_{j}^{z})=\lim_{n\rightarrow\infty}\omega(T^{n}(0)\sigma_{j}^{z})\neq
0,$ (5.55)
here $T^{n}(0)$ gives a constant when $n$ goes to infinity. This suggests the
existence of long-range order after $2T$, showing the $2T$-periodicity of
discrete time crystals. For a general case, we may choose an arbitrary
rotational invariant $H_{MBL}$ but with the same $U_{1}$. Since
$\beta(A)=U_{f}\sigma_{i}^{z}U^{{\dagger}}_{f}=U_{2}U_{1}\sigma_{i}^{z}U_{1}^{{\dagger}}U_{2}^{{\dagger}}=-U_{2}\sigma_{i}^{z}U_{2}^{{\dagger}}$,
any rotational invariant $U_{2}$ will give the expectation values under
$T^{n}$ for $T\in\mathcal{T}$. This result is similar as in Ref. [160].
#### 5.5.3 Temporal correlations
In the previous sections the long-range order in time has different
representations in terms of the mixture of spatial and temporal correlations.
For time translation symmetry breaking, these representations do not make much
difference in the algebraic language as algebraic symmetries are already
assumed to commute with both of space translations and time translations. Here
we discuss the possibility of a measure of temporal correlations based on
operator algebra in which the study of spatial correlations is nicely
formulated in terms of the hierarchy in the Tsirelson’s problem.
One possibility is to use generalised non-local games in the time domain. As
we already discussed in the last chapter, quantum-classical signalling games
give temporal correlations formulated by pseudo-density matrices. We may
consider other variants of signalling games as an analogue of finite input-
output games and synchronous games. What is more, even for a particular kind
of signalling games, we may discuss different possibilities for strategies.
Another possibility is to generalise the pseudo-density matrix beyond the
tensor product structure and projective measurements. It is known that, for
spatial correlations, the hierarchy from the smallest set to the largest set
is classical correlations, correlations of tensor product structures in finite
dimensional Hilbert spaces, correlations of tensor product structures in
infinite dimensional Hilbert spaces, the closure of correlations of tensor
product structures in infinite dimensional Hilbert spaces, correlations of
commutative structures in arbitrary dimensional Hilbert spaces. We may have
generalised temporal correlations in terms of commutative structures.
Indefinite causal structures may be involved in such representation.
We will leave the formal establishment as a future work. It is interesting to
ask what kind of temporal correlations current time crystals hold in the
hierarchy and whether generalised temporal correlations may lead to different
understanding for time crystals. And it might be possible to generalise all
these discussion to the relativistic setting via algebraic quantum field
theory.
## Chapter 6 Conclusion and outlook
We conclude this thesis with the summary of results in the main chapters and
provide the outlook for future possible work.
The results are summarised as follows.
* •
In Chapter 3, we generalise the pseudo-density matrix formalism to continuous
variables and general measurement processes. First we define spacetime
Gaussian states from the first two statistical moments which fully
characterise Gaussian states, and compare temporal Gaussian states with
spatial Gaussian state to show a similar correlation relationship as the qubit
case. Via the Wigner function representation, we define spacetime density
matrices in continuous variables in general, and show that spacetime Wigner
functions hold the similar properties which uniquely determine spatial Wigner
functions. We further discuss the possibilities of defining spacetime states
via position measurements and weak measurements, and generalise the pseudo-
density matrix formulation to more general measurement processes. An
experimental tomography based on quantum optics is proposed to verify the
operational meaning for the generalised pseudo-density matrix formalism from
measurement correlations.
* •
In Chapter 4, we use quantum correlation in time to compare the pseudo-density
matrix formalism with indefinite causal structures, consistent histories,
generalised non-local games, and out-of-time-order correlation functions, and
path integrals. We aim to argue that spacetime formulations in non-
relativistic quantum mechanics are remarkably similar. In the section of
indefinite causal structures, we use the process matrix formalism in
particular, compare it with the pseudo-density matrix formalism via
correlations, formulate causal inequalities, and discuss the role of post-
selection in indefinite causal structures. In consistent histories, the
consistency conditions give the generalised pseudo-density matrix a better
argument for its existence. Pseudo-density matrices can be formulated in terms
of quantum-classical signalling games as well. We also provide a simple
calculation for out-of-time-order correlation functions and apply it to black
hole information paradox. Nevertheless, the path integral formalism has a
different representation of quantum correlations from the pseudo-density
matrix approach, suggesting interesting directions for quantum measure and
relativistic quantum theory.
* •
In Chapter 5, we apply the temporal correlations in the pseudo-density matrix
formalism to time crystals. We define time crystals as long-range order in
time, a particular kind of temporal correlations which do not vanish after a
long time. Then we analyse continuous time translation symmetry in terms of
general decoherent processes and a generalised version of Mermin-Wagner
theorem. We also discuss discrete time translation symmetry via a
stabilisation protocol of quantum computation, phase flip codes of quantum
error correction and Floquet many-body localisation. Finally we explore the
possibility of time crystals from the algebraic point of view.
Some of the possible future directions for work are listed as below.
* •
_Mutual information in time._ Mutual information of two random variables $X$
and $Y$ measures how much information $X$ and $Y$ have in common [12]. It is
also a measure of the total correlations between two subsystems of a bipartite
quantum system [161]. Classically, the mutual information for two systems at
different times is defined as the same as two systems at different positions.
However, quantifying the mutual information for two quantum systems evolving
in time is still a difficult open problem. Note that a basic fact of quantum
mutual information between two entangled systems of a pure state is that, it
is equal to twice the von Neumann entropy of a reduced subsystem, while it can
only be at most the same as the Shannon entropy of a single subsystem in the
classical case. On the one hand, it shows that quantum correlations are
stronger than classical ones; on the other hand, the quantum mutual
information in time is supposed to show its quantum advantages over the
classical mutual information. So far, we investigate different proposals for
quantum mutual information in time but none of them could be called quantum.
Difficulties of defining mutual information in time in the pseudo-density
matrix formalism come from the negativity of temporal pseudo-density matrices.
One possible solution is to purify pseudo-density matrices and make them to be
positive semi-definite, then we may define the mutual information in time for
subsystems at different times.
* •
_Tripartite correlations in spacetime._ Bipartite quantum correlations in
space-time are well-studied in the pseudo-density matrix formalism [40]. A
symmetric structure has been shown in two-point quantum correlations in space
and time. Specifically, two-point spatial correlations in arbitrary bipartite
quantum states and two-point temporal correlations for a single qubit evolving
under a unitary quantum channel are mapped to each other under the operation
of partial transposition. This suggests an interesting relationship between
spatial and temporal correlations in the bipartite case. We further analyse
tripartite correlations. One question remaining unknown is that given a
tripartite correlation, how can we distinguish it from a qubit at three times,
one qubit at one time and another at two time, or three qubits at a single
time? Another interesting question may be the spatial-temporal analogue of
monogamy of entanglement. As we know that the subsystem of a maximally
entangled state cannot be entangled with a third system, a maximally
temporally correlated system, that is a system under the identity evolution,
may still be maximally temporally correlated with the system under the
identity evolution at a later time. What will be the temporal analogue of
monogamy of entanglement? Or is it a fundamental difference between spatial
and temporal correlations?
* •
_Spacetime from spatial-temporal correlations rather than entanglement._ It is
claimed among AdS/CFT community that it will be possible to build up spacetime
with quantum entanglement [162]. However, quantum entanglement is only a
particular kind of spatial correlation. A better argument may be to build
spacetime from spatial-temporal correlations rather than entanglement. We are
discussing the possibilities of deriving the Einstein field equation from a
spacetime area law of quantum correlations. The Einstein field equation can be
derived through the area law for entanglement entropy [163] as well as the
quantum geometrical limit for the energy density of clocks and signals [164].
We want to argue that quantum correlations are much more than entanglement,
and temporal correlations in quantum mechanics may provide better insights for
understanding spacetime or gravity in the quantum sense.
* •
_Application in black hole information paradox._ In Chapter 4, we already
applied the pseudo-density matrix formalism to the out-of-time-order
correlation functions in the black hole final state proposal. We are looking
for further applications in black hole information paradox. One possibility
still lies in the out-of-time-order correlation functions. We may use the out-
of-time-order correlation functions as a tool to analyse the behaviours in the
black hole formation and evaporation. We may also understand the information
loss via temporal correlations. Spatial correlations like entanglement have
been discussed in the black hole scenarios. Will temporal correlations between
early radiation and late radiation help to understand the information loss?
These questions are worth exploring.
As we have asked in the introductory chapter, “what is time”, we briefly
report on our little lessons from quantum correlations.
The thesis is based on the assumption that space and time should be treated on
an equal footing. The pseudo-density matrix formulation treats temporal
correlations equally in form as spatial correlations. We are a bit concerned
about this assumption under a simple argument on monogamy. As we mentioned
before, monogamy of entanglement cannot find a temporal analogue. Entanglement
is a kind of spatial correlation; nevertheless, we cannot observe the monogamy
of any temporal correlation. The maximally temporal correlated states are
under the identity evolution and we can make as many copies as we want. Thus
temporal correlations have no monogamy constraint; this suggests intrinsic
difference between spatial correlations and temporal correlations. Another
example is from time crystals. We have continuous space translation symmetry
breaking but no continuous time translation symmetry breaking. One deep
concern from Ref. [132] is that “causality distinguishes between spacelike and
timelike separations”. While generators of space translations are the momenta,
generators of time translations are the Hamiltonians which is much more system
dependent. We suspect the assumption on the equal treatment of space and time
to be too strong. It is a possible route to learn about temporal correlations
by taking them operationally equal as spatial correlations; but we would
carefully keep in mind that, space is space, time is time.
One possible link between spatial and temporal correlations is the partial
transpose. We cannot see exactly why this operation is so important in space-
time inversion; a simple understanding might be that for two systems in space
converting to two systems in time, one evolves forwards under normal evolution
while the other evolves backwards under the transpose. Path integrals are
important to understand spacetime. They have shown the difference in the
operational meaning of quantum correlations. Further investigation in terms of
quantum measure and relativistic quantum information are ongoing. We are also
concerned about indefinite causal structures, as it might not be enough to
quantising gravity as a linear superposition of causal structures. It is
interesting to explore further on algebraic field theory in search for the
relativistic version for quantum correlations in space and time.
Anyway, the long long journey towards time just started.
## Chapter 7 Choi-Jamiołkowski isomorphism
Here we introduce the Choi-Jamiołkowski isomorphism based on Ref. [68].
The set of linear operators on the finite dimensional Hilbert space
$\mathcal{H}$ is denoted as $\mathcal{L}(\mathcal{H})$. The set of linear
operators from $\mathcal{H}_{0}$ to $\mathcal{H}_{1}$ is denoted as
$\mathcal{L}(\mathcal{H}_{0},\mathcal{H}_{1})$. An operator
$X\in\mathcal{L}(\mathcal{H}_{0},\mathcal{H}_{1})$ has a one-to-one
correspondence with a vector
$|X\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\in\mathcal{H}_{1}\otimes\mathcal{H}_{0}$
as
$|X\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=(X\otimes
I_{\mathcal{H}_{0}})|I_{\mathcal{H}_{0}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=(I_{\mathcal{H}_{1}}\otimes
X^{T})|I_{\mathcal{H}_{1}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
(7.1)
where $\mathcal{I}_{\mathcal{H}}$ is the identity operator in $\mathcal{H}$,
$|I_{\mathcal{H}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\in\mathcal{H}\otimes\mathcal{H}$
is the maximally entangled vector
$|I_{\mathcal{H}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=\sum_{n}\ket{n}\ket{n}$
($\ket{n}$ is the orthonormal basis in $\mathcal{H}$),
$X^{T}\in\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{0})$ is the transpose of $X$
with respect to two given bases in $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$.
The set of linear maps from $\mathcal{L}(\mathcal{H}_{0})$ to
$\mathcal{L}(\mathcal{H}_{1})$ is denoted by
$\mathcal{L}(\mathcal{L}(\mathcal{H}_{0}),\mathcal{L}(\mathcal{H}_{1}))$. A
linear map
$\mathcal{M}\in\mathcal{L}(\mathcal{L}(\mathcal{H}_{0}),\mathcal{L}(\mathcal{H}_{1}))$
has a one-to-one correspondence with a linear operator
$M\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{0})$ as
$M=\mathcal{M}\otimes\mathcal{I}_{\mathcal{L}(\mathcal{H}_{0})}(|I_{\mathcal{H}_{0}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}I_{\mathcal{H}_{0}}|)$
(7.2)
where $\mathcal{I}_{\mathcal{L}(\mathcal{H}_{0})}$ is the identity map on
$\mathcal{L}(\mathcal{H}_{0})$. This is called Choi-Jamiołkowski isomorphism.
The operator $M$ is called Choi-Jamiołkowski operator of $\mathcal{M}$. Its
inverse transforms $M\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{0})$ to
a map
$\mathcal{M}\in\mathcal{L}(\mathcal{L}(\mathcal{H}_{0}),\mathcal{L}(\mathcal{H}_{1}))$
that acts on an operator $X\in\mathcal{L}(\mathcal{H}_{0})$ as
$\mathcal{M}(X)=\Tr_{\mathcal{H}_{0}}[(I_{\mathcal{H}_{1}}\otimes X^{T})M]$
(7.3)
A linear map $\mathcal{M}$ is trace preserving if and only if its Choi-
Jamiołkowski operator $M$ satisfies
$\Tr_{\mathcal{H}_{1}}[M]=I_{\mathcal{H}_{0}}.$ (7.4)
A linear map $\mathcal{M}$ is Hermitian preserving if and only if its Choi-
Jamiołkowski operator $M$ is Hermitian. A linear map $\mathcal{M}$ is
completely positive if and only if its Choi-Jamiołkowski operator $M$ is
positive semi-definite.
## Chapter 8 Proofs for the properties for spacetime Wigner functions
Here we provide the proof for six properties for spacetime Wigner functions.
The additional one is listed before the five properties in the main text,
about the expectation value of an arbitrary operator $\hat{A}$. Before that,
we introduce the Wigner representation in Liouville Space [165].
### 8.1 Wigner Representation in Liouville Space
Ref. [165] gives an introduction to the Wigner representation in Liouville
Space. In Liouville space, operators are treated as vectors in a superspace.
For a bra-ket notation, we call $|A\\}$ a L-ket and $\\{A|$ a L-bra for an
operator $A$, with the scalar product as
$\\{B|A\\}=\Tr\\{B^{{\dagger}}A\\}.$ (8.1)
Different from Ref. [165], we take $\hbar=1$. Define a Liouville basis
$|qp\\}=\left(\frac{2}{\pi}\right)^{1/2}|\Pi_{qp}\\},$ (8.2)
where $\Pi_{qp}$ is given by
$\displaystyle\Pi_{qp}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\textrm{d}se^{isp}\ket{q+\frac{\hbar}{2}s}\bra{x-\frac{\hbar}{2}s}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\textrm{d}ke^{-ikq}\ket{p+\frac{\hbar}{2}k}\bra{p-\frac{\hbar}{2}k}$
$\displaystyle=\frac{1}{4\pi}\int_{-\infty}^{\infty}\textrm{d}k\int_{-\infty}^{\infty}\textrm{d}se^{ik(\hat{q}-q)-is(\hat{p}-p)}.$
(8.3)
In fact $\Pi_{qp}$ is the parity operator about the phase point $(x,p)$:
$\Pi_{qp}(\hat{q}-q)\Pi_{qp}=-(\hat{q}-q),\
\Pi_{qp}(\hat{p}-p)\Pi_{qp}=-(\hat{p}-p).$ (8.4)
It is the same as the displaced parity operator $U(\alpha)$ with the mapping
$\alpha=\frac{1}{\sqrt{2}}(q+ip)$.
$|qp\\}$ forms an orthogonal and complete basis:
$\displaystyle\\{q^{\prime}p^{\prime}|qp\\}=\delta(q^{\prime}-q)\delta(p^{\prime}-p)$
(8.5)
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\textrm{d}q\textrm{d}p|qp\\}\\{qp|=\hat{\hat{1}},$
(8.6)
where $\hat{\hat{1}}$ is a unit L-operator. However, we need to remember that
$|qp\\}$ is not a valid quantum state because $\Pi_{qp}$ is not positive
definite.
The Weyl form of an operator $\hat{A}$ is defined as
$A(q,p)\equiv(2\pi)^{1/2}\\{qp|A\\}=2\Tr[\Pi_{qp}\hat{A}].$ (8.7)
Then the Wigner function of a state $\hat{\rho}$ is given by
$W(q,p)\equiv(2\pi)^{-1/2}\\{qp|\rho\\}=(2\pi)^{-1}\int\textrm{d}se^{-isp}\bra{q+\frac{1}{2}\hbar
s}\rho\ket{q-\frac{1}{2}\hbar s},$ (8.8)
where the normalisation holds for $\iint\textrm{d}q\textrm{d}pW(q,p)=1$. For
an operator $\hat{A}$ measured in the state $\hat{\rho}$, its expectation
value is given as
$\langle\hat{A}\rangle_{\rho}=\\{A|\rho\\}=\iint\textrm{d}q\textrm{d}p\\{A|qp\\}\\{qp|\rho\\}=\iint\textrm{d}q\textrm{d}pA^{*}(q,p)W(q,p).$
(8.9)
### 8.2 Proofs for the properties
We prove all the six properties listed as (0) to (5) in this subsection.
Following the notation in the previous subsection, we have the bipartite
spacetime Wigner function
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=(2\pi)^{-1}\\{q_{1}p_{1},q_{2}p_{2}|R\\}=4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})\hat{R}],$
(8.10)
for a bipartite spacetime density matrix in continuous variables $\hat{R}$.
(0) For bipartite case,
$\langle\hat{A}\rangle_{R}=\Tr[\hat{A}\hat{R}]=\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}A^{*}(q_{1},p_{1},q_{2},p_{2})\mathcal{W}(q_{1},p_{1},q_{2},p_{2}),$
(8.11)
where
$A(q_{1},p_{1},q_{2},p_{2})=(2\pi)\\{qp|A\\}=4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})\hat{A}].$
(8.12)
Note that $T(\alpha)=2U(\alpha)=2\Pi(q_{1},p_{1})$ and
$T(\beta)=2U(\beta)=2\Pi(q_{2},p_{2})$. The above statement is equivalent to
$\langle\hat{A}\rangle_{R}=\Tr[\hat{A}\hat{R}]=\iint\textrm{d}^{2}\alpha\textrm{d}^{2}\beta
A^{*}(\alpha,\beta)\mathcal{W}(\alpha,\beta),$ (8.13)
where
$A(\alpha,\beta)=\Tr\\{[T(\alpha)\otimes T(\beta)]\hat{A}\\}.$ (8.14)
###### Proof.
Compared to Eqn. (8.9),
$\displaystyle\langle\hat{A}\rangle_{R}=$ $\displaystyle\\{A|R\\}$
$\displaystyle=$
$\displaystyle\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}\\{A|q_{1}p_{1},q_{2}p_{2}\\}\\{q_{1}p_{1},q_{2}p_{2}|R\\}$
$\displaystyle=$
$\displaystyle\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}A^{*}(q_{1},p_{1},q_{2},p_{2})\mathcal{W}(q_{1},p_{1},q_{2},p_{2}).$
(8.15)
∎
Generalisation to $n$ events is straightforward.
(1) $\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ is given by
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ $=$ $\Tr[M(q_{1},p_{1},q_{2},p_{2})R]$
for $M(q_{1},p_{1},q_{2},p_{2})$ $=$ $M^{{\dagger}}(q_{1},p_{1},q_{2},p_{2})$.
Therefore, it is real.
###### Proof.
Compared to Eqn. (3.31),
$M(q_{1},p_{1},q_{2},p_{2})=4\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}}$, thus it
is obvious that $M(q_{1},p_{1},q_{2},p_{2})$ $=$
$M^{{\dagger}}(q_{1},p_{1},q_{2},p_{2})$.
Because a spacetime density matrix is Hermitian, the spacetime Wigner function
is real. ∎
Note that we prove the Hermicity of a spacetime density matrix from the
property that spacetime Wigner function is real.
(2)
$\displaystyle\iint\textrm{d}p_{1}\textrm{d}p_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\bra{q_{1},q_{2}}\hat{R}\ket{q_{1},q_{2}},$
$\displaystyle\iint\textrm{d}q_{1}\textrm{d}q_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\bra{p_{1},p_{2}}\hat{R}\ket{p_{1},p_{2}},$
$\displaystyle\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\Tr\hat{R}=1.$
(8.16)
###### Proof.
Taking $\hat{A}$ in the property (0) to be
$\hat{A}=\delta(\hat{q}_{1}-q_{1})\delta(\hat{q}_{2}-q_{2}),$ (8.17)
then
$A^{*}(q_{1},p_{1},q_{2},p_{2})=\delta(\hat{q}_{1}-q_{1})\delta(\hat{q}_{2}-q_{2}).$
(8.18)
Thus
$\Tr[\hat{A}\hat{R}]=\bra{q_{1},q_{2}}\hat{R}\ket{q_{1},q_{2}},$ (8.19)
and
$\iiiint\textrm{d}q_{1}\textrm{d}q_{2}\textrm{d}p_{1}\textrm{d}p_{2}A^{*}(q_{1},p_{1},q_{2},p_{2})\mathcal{W}(q_{1},p_{1},q_{2},p_{2})=\iint\textrm{d}p_{1}\textrm{d}p_{2}\mathcal{W}(q_{1},p_{1},q_{2},p_{2}).$
(8.20)
Via Eqn. (8.11), the first equality holds.
Similar for the second equality. The normalisation property is already proven
before. ∎
(3) $\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ is Galilei covariant, that is, if
$\bra{q_{1},q_{2}}R\ket{q^{\prime}_{1},q^{\prime}_{2}}$ $\rightarrow$
$\bra{q_{1}+a,q_{2}+b}R\ket{q^{\prime}_{1}+a,q^{\prime}_{2}+b}$, then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ $\rightarrow$
$\mathcal{W}(q_{1}+a,p_{1},q_{2}+b,p_{2})$ and if
$\bra{q_{1},q_{2}}R\ket{q^{\prime}_{1},q^{\prime}_{2}}$ $\rightarrow$
$\exp\\{[ip^{\prime}_{1}(-q_{1}+q^{\prime}_{1})+ip^{\prime}_{2}(-q_{2}+q^{\prime}_{2})]/\hbar\\}\bra{q_{1},q_{2}}R\ket{q^{\prime}_{1},q^{\prime}_{2}}$,
then $\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ $\rightarrow$
$\mathcal{W}(q_{1},p_{1}-p^{\prime}_{1},q_{2},p_{2}-p^{\prime}_{2})$.
###### Proof.
If
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{q_{1}+a,q_{2}+b}\hat{R}\ket{q^{\prime}_{1}+a,q^{\prime}_{2}+b},$
that is,
$\hat{R}\rightarrow D^{{\dagger}}_{a0}\otimes
D^{{\dagger}}_{b0}\hat{R}D_{a0}\otimes D_{b0},$
then
$\displaystyle\mathcal{W}(q_{1},p_{1},$ $\displaystyle
q_{2},p_{2})=4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})\hat{R}]\rightarrow$
$\displaystyle
4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})(D^{{\dagger}}_{a0}\otimes
D^{{\dagger}}_{b0}\hat{R}D_{a0}\otimes
D_{b0})]=\mathcal{W}(q_{1}+a,p_{1},q_{2}+b,p_{2}).$
If
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\exp\\{[ip^{\prime}_{1}(-q_{1}+q^{\prime}_{1})+ip^{\prime}_{2}(-q_{2}+q^{\prime}_{2})]/\hbar\\}\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}},$
that is,
$\hat{R}\rightarrow D^{{\dagger}}_{0,-p^{\prime}_{1}}\otimes
D^{{\dagger}}_{0,-p^{\prime}_{2}}\hat{R}D_{0,-p^{\prime}_{1}}\otimes
D_{0,-p^{\prime}_{2}},$
then
$\displaystyle\mathcal{W}$
$\displaystyle(q_{1},p_{1},q_{2},p_{2})=4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})\hat{R}]\rightarrow$
$\displaystyle
4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})(D^{{\dagger}}_{0,-p^{\prime}_{1}}\otimes
D^{{\dagger}}_{0,-p^{\prime}_{2}}\hat{R}D_{0,-p^{\prime}_{1}}\otimes
D_{0,-p^{\prime}_{2}})\big{]}=\mathcal{W}(q_{1},p_{1}-p^{\prime}_{1},q_{2},p_{2}-p^{\prime}_{2}).$
∎
(4) $\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ has the following property under
space and time reflections: if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}$ $\rightarrow$
$\bra{-q_{1},-q_{2}}\hat{R}\ket{-q^{\prime}_{1},-q^{\prime}_{2}}$, then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ $\rightarrow$
$\mathcal{W}(-q_{1},-p_{1},-q_{2},-p_{2})$ and if
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}$ $\rightarrow$
$\bra{q^{\prime}_{1},q^{\prime}_{2}}\hat{R}\ket{q_{1},q_{2}}$, then
$\mathcal{W}(q_{1},p_{1},q_{2},p_{2})$ $\rightarrow$
$\mathcal{W}(q_{1},-p_{1},q_{2},-p_{2})$.
###### Proof.
If
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{-q_{1},-q_{2}}\hat{R}\ket{-q^{\prime}_{1},-q^{\prime}_{2}}$,
that is,
$\hat{R}\rightarrow\Pi_{00}\hat{R}\Pi_{00},$
then
$\displaystyle\mathcal{W}(q_{1},p_{1},$ $\displaystyle
q_{2},p_{2})=4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})\hat{R}]\rightarrow$
$\displaystyle
4\Tr[(\Pi_{q_{1}p_{1}}\otimes\Pi_{q_{2}p_{2}})(\Pi_{00}\hat{R}\Pi_{00})]\mathcal{W}(-q_{1},-p_{1},-q_{2},-p_{2}).$
For
$\bra{q_{1},q_{2}}\hat{R}\ket{q^{\prime}_{1},q^{\prime}_{2}}\rightarrow\bra{q^{\prime}_{1},q^{\prime}_{2}}\hat{R}\ket{q_{1},q_{2}}$,
it is similar to transpose. Consider $\hat{q}^{T}=q$ and $\hat{p}^{T}=-p$,
$\displaystyle\mathcal{W}(q_{1},p_{1},$ $\displaystyle
q_{2},p_{2})\rightarrow\mathcal{W}(q_{1},-p_{1},q_{2},-p_{2}).$
∎
(5) Take $\hbar=1$.
$\Tr(\hat{R}_{1}\hat{R}_{2})=(2\pi)\iint\textrm{d}q\textrm{d}p\mathcal{W}_{R_{1}}(q,p)\mathcal{W}_{R_{2}}(q,p),$
(8.21)
for $\mathcal{W}_{R_{1}}(q,p)$ and $\mathcal{W}_{R_{2}}(q,p)$ are pseudo-
Wigner functions for pseudo-density matrices $\hat{R}_{1}$ and $\hat{R}_{2}$
respectively.
###### Proof.
$\Tr(\hat{R}_{1}\hat{R}_{2})=\\{R_{1}|R_{2}\\}=\iint\textrm{d}q\textrm{d}p\\{R_{1}|qp\\}\\{qp|R_{2}\\}=(2\pi)\iint\textrm{d}q\textrm{d}p\mathcal{W}_{R_{1}}(q,p)\mathcal{W}_{R_{2}}(q,p).$
(8.22)
∎
## Chapter 9 Proof for continuous time translation symmetry in 1+1 dimensions
Now we prove that there is no continuous time translation symmetry breaking in
the Heisenberg model at finite temperature in 1+1 dimensions. As the original
Mermin-Wagner theorem, we use the Bogoliubov inequality:
$\frac{1}{2}\beta\langle[A,A^{{\dagger}}]_{+}\rangle\langle[[C,H]_{-},C^{{\dagger}}]_{-}\rangle\geq|\langle[C,A]_{-}\rangle|^{2}$
(9.1)
where $\beta=1/k_{B}T$ is the inverse temperature, $A$ and $C$ are arbitrary
operators and $H$ is the Hamiltonian of the system. $\langle\cdots\rangle$
gives the expectation value in the thermal state. In the one-dimensional
Heisenberg model, the Hamiltonian is given as
$H=-\sum_{ij}J_{ij}S_{i}^{z}S_{j}^{z}-b\sum_{i}S_{i}^{z}$ (9.2)
where $S_{i}^{z}$ is the spin $i$ along the $z$-direction and $b$ is the
parameter for a small perturbation. We assume that
$Q=\frac{1}{N}\sum_{i,j}|R_{i}-R_{j}|^{2}|J_{ij}|$ remains finite where
$R_{i}$ denotes the position of spin $i$. We assign $A$ and $C$ to be
$\displaystyle A$ $\displaystyle=e^{iHt}S^{-}(-k)e^{-iHt}$ (9.3)
$\displaystyle C$ $\displaystyle=e^{iHt}S^{+}(k)e^{-iHt}$ (9.4)
where $S^{\alpha}(k)=\sum_{i}S_{i}^{\alpha}e^{-ikR_{i}}$ and $S^{\pm}=S_{x}\pm
iS_{y}$. Then
$\langle[C,A]_{-}\rangle=2\hbar\sum_{i}\langle
e^{iHt}S_{i}^{z}e^{-iHt}\rangle=2\hbar NZ(t),$ (9.5)
where $Z(t)=\langle e^{iHt}S_{i}^{z}e^{-iHt}\rangle$ is the temporal
correlation in the model.
$\sum_{k}\langle[A,A^{{\dagger}}]_{+}\rangle\leq 2\hbar^{2}N^{2}S(S+1),$ (9.6)
and
$\langle[[C,H]_{-},C^{{\dagger}}]_{-}\rangle\leq
4\hbar^{2}bNZ(t)+4N\hbar^{2}k^{2}QS(S+1)$ (9.7)
Substituting the above inequalities into the Bogoliubov inequality and summing
over all the wavevectors, we have
$S(S+1)\geq\frac{C(t)^{2}v}{2\pi\beta\hbar^{2}}\int_{0}^{k_{0}}\frac{\textrm{d}k}{bZ(t)+k^{2}QS(S+1)}=\frac{Z(t)^{2}v}{2\pi\beta\hbar^{2}}\frac{\arctan(k_{0}\sqrt{\frac{QS(S+1)}{bZ(t)}})}{\sqrt{QS(S+1)bZ(t)}}.$
(9.8)
Thus
$Z(t)\leq\text{const}\cdot\frac{b^{1/3}}{T^{2/3}}\quad\text{as}\ b\rightarrow
0.$ (9.9)
The temporal correlation vanishes as the perturbation parameter goes to 0
under finite temperature; thus, there is no spontaneous continuous time
translation symmetry breaking in this case.
## References
* [1] Don N. Page and William K. Wootters “Evolution without evolution: Dynamics described by stationary observables” In _Phys. Rev. D_ 27 American Physical Society, 1983, pp. 2885–2892 DOI: 10.1103/PhysRevD.27.2885
* [2] Julian Barbour “The end of time: The next revolution in physics” Oxford University Press, 2001
* [3] R. Arnowitt, S. Deser and C. W. Misner “Dynamical Structure and Definition of Energy in General Relativity” In _Phys. Rev._ 116 American Physical Society, 1959, pp. 1322–1330 DOI: 10.1103/PhysRev.116.1322
* [4] J. B. Hartle and S. W. Hawking “Wave function of the Universe” In _Phys. Rev. D_ 28 American Physical Society, 1983, pp. 2960–2975 DOI: 10.1103/PhysRevD.28.2960
* [5] Bryce S. DeWitt “Quantum Theory of Gravity. I. The Canonical Theory” In _Phys. Rev._ 160 American Physical Society, 1967, pp. 1113–1148 DOI: 10.1103/PhysRev.160.1113
* [6] Lee Smolin “Time reborn: From the crisis in physics to the future of the universe” HMH, 2013
* [7] Marina Cortês and Lee Smolin “Quantum energetic causal sets” In _Phys. Rev. D_ 90 American Physical Society, 2014, pp. 044035 DOI: 10.1103/PhysRevD.90.044035
* [8] Lee Smolin “Temporal naturalism” Cosmology and Time: Philosophers and Scientists in Dialogue In _Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics_ 52, 2015, pp. 86 –102 DOI: https://doi.org/10.1016/j.shpsb.2015.03.005
* [9] Ross Sheldon “A first course in probability” Pearson Education India, 2002
* [10] James Sethna “Statistical mechanics: entropy, order parameters, and complexity” Oxford University Press, 2006
* [11] Steven Weinberg “The Quantum theory of fields. Vol. 1: Foundations” Cambridge University Press, 2005
* [12] Michael A Nielsen and Isaac Chuang “Quantum computation and quantum information” AAPT, 2002
* [13] John Preskill “Lecture notes for physics 229: Quantum information and computation” In _California Institute of Technology_ 16, 1998
* [14] Masanori Ohya and Dénes Petz “Quantum entropy and its use” Springer Science & Business Media, 2004
* [15] Branko Grünbaum, Victor Klee, Micha A Perles and Geoffrey Colin Shephard “Convex polytopes” Springer, 1967
* [16] John Von Neumann “Mathematical Foundations of Quantum Mechanics: New Edition” Princeton university press, 2018
* [17] Benjamin Schumacher “Quantum coding” In _Phys. Rev. A_ 51 American Physical Society, 1995, pp. 2738–2747 DOI: 10.1103/PhysRevA.51.2738
* [18] Roope Uola, Ana Costa, H Chau Nguyen and Otfried Gühne “Quantum Steering” In _arXiv preprint arXiv:1903.06663_ , 2019
* [19] John S Bell “On the Einstein-Podolsky-Rosen paradox” In _Physics Physique Fizika_ 1.3 APS, 1964, pp. 195
* [20] Nicolas Brunner et al. “Bell nonlocality” In _Rev. Mod. Phys._ 86 American Physical Society, 2014, pp. 419–478 DOI: 10.1103/RevModPhys.86.419
* [21] Kavan Modi et al. “The classical-quantum boundary for correlations: Discord and related measures” In _Rev. Mod. Phys._ 84 American Physical Society, 2012, pp. 1655–1707 DOI: 10.1103/RevModPhys.84.1655
* [22] L Henderson and V Vedral “Classical, quantum and total correlations” In _Journal of Physics A: Mathematical and General_ 34.35 IOP Publishing, 2001, pp. 6899–6905 DOI: 10.1088/0305-4470/34/35/315
* [23] Harold Ollivier and Wojciech H. Zurek “Quantum Discord: A Measure of the Quantumness of Correlations” In _Phys. Rev. Lett._ 88 American Physical Society, 2001, pp. 017901 DOI: 10.1103/PhysRevLett.88.017901
* [24] Michał Horodecki et al. “Local Information as a Resource in Distributed Quantum Systems” In _Phys. Rev. Lett._ 90 American Physical Society, 2003, pp. 100402 DOI: 10.1103/PhysRevLett.90.100402
* [25] Igor Devetak and Andreas Winter “Distilling common randomness from bipartite quantum states” In _IEEE Transactions on Information Theory_ 50.12 IEEE, 2004, pp. 3183–3196
* [26] Shunlong Luo “Using measurement-induced disturbance to characterize correlations as classical or quantum” In _Phys. Rev. A_ 77 American Physical Society, 2008, pp. 022301 DOI: 10.1103/PhysRevA.77.022301
* [27] Shengjun Wu, Uffe V. Poulsen and Klaus Mølmer “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality” In _Phys. Rev. A_ 80 American Physical Society, 2009, pp. 032319 DOI: 10.1103/PhysRevA.80.032319
* [28] Kavan Modi et al. “Unified View of Quantum and Classical Correlations” In _Phys. Rev. Lett._ 104 American Physical Society, 2010, pp. 080501 DOI: 10.1103/PhysRevLett.104.080501
* [29] William Slofstra “The set of quantum correlations is not closed” In _Forum of Mathematics, Pi_ 7 Cambridge University Press, 2019, pp. e1 DOI: 10.1017/fmp.2018.3
* [30] Volkher B Scholz and Reinhard F Werner “Tsirelson’s problem” In _arXiv preprint arXiv:0812.4305_ , 2008
* [31] Boris Tsirelson “Bell inequalities and operator algebras” Citeseer, 2006
* [32] Kenneth J Dykema and Vern Paulsen “Synchronous correlation matrices and Connes’ embedding conjecture” In _Journal of Mathematical Physics_ 57.1 AIP Publishing, 2016, pp. 015214
* [33] William Slofstra “Tsirelson’s problem and an embedding theorem for groups arising from non-local games” In _Journal of the American Mathematical Society_ , 2019
* [34] Zhengfeng Ji et al. “MIP*=RE”, 2020 arXiv:2001.04383 [quant-ph]
* [35] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki and Karol Horodecki “Quantum entanglement” In _Rev. Mod. Phys._ 81 American Physical Society, 2009, pp. 865–942 DOI: 10.1103/RevModPhys.81.865
* [36] He Lu et al. “Entanglement Structure: Entanglement Partitioning in Multipartite Systems and Its Experimental Detection Using Optimizable Witnesses” In _Phys. Rev. X_ 8 American Physical Society, 2018, pp. 021072 DOI: 10.1103/PhysRevX.8.021072
* [37] Michael E Peskin “An introduction to quantum field theory” CRC Press, 2018
* [38] Kun Il Park and Park “Fundamentals of Probability and Stochastic Processes with Applications to Communications” Springer, 2018
* [39] Joseph F Fitzsimons, Jonathan A Jones and Vlatko Vedral “Quantum correlations which imply causation” In _Scientific reports_ 5 Nature Publishing Group, 2015, pp. 18281
* [40] Zhikuan Zhao et al. “Geometry of quantum correlations in space-time” In _Phys. Rev. A_ 98 American Physical Society, 2018, pp. 052312 DOI: 10.1103/PhysRevA.98.052312
* [41] Robert Pisarczyk et al. “Causal limit on quantum communication” In _arXiv preprint arXiv:1804.02594_ , 2018
* [42] Tian Zhang, Oscar Dahlsten and Vlatko Vedral “Constructing continuous-variable spacetime quantum states from measurement correlations” In _arXiv preprint arXiv:1903.06312_ , 2019
* [43] Tian Zhang “Pseudo-density matrix: the relation with indefinite causal structures, consistent histories and generalised non-local games and out-of-time-order correlations” In _In Preparation_ , 2019
* [44] Tian Zhang et al. “Long-range temporal correlations in quantum error correction and time crystals” In _In Preparation_ , 2019
* [45] Ryszard Horodecki and Michal/ Horodecki “Information-theoretic aspects of inseparability of mixed states” In _Phys. Rev. A_ 54 American Physical Society, 1996, pp. 1838–1843 DOI: 10.1103/PhysRevA.54.1838
* [46] E. P. Wigner “Epistemological Perspective on Quantum Theory” In _Philosophical Reflections and Syntheses_ Berlin, Heidelberg: Springer Berlin Heidelberg, 1995, pp. 55–71 DOI: 10.1007/978-3-642-78374-6_5
* [47] Carlton M. Caves “Quantum mechanics of measurements distributed in time. A path-integral formulation” In _Phys. Rev. D_ 33 American Physical Society, 1986, pp. 1643–1665 DOI: 10.1103/PhysRevD.33.1643
* [48] Carlton M. Caves “Quantum mechanics of measurements distributed in time. II. Connections among formulations” In _Phys. Rev. D_ 35 American Physical Society, 1987, pp. 1815–1830 DOI: 10.1103/PhysRevD.35.1815
* [49] Carlton M. Caves and G. J. Milburn “Quantum-mechanical model for continuous position measurements” In _Phys. Rev. A_ 36 American Physical Society, 1987, pp. 5543–5555 DOI: 10.1103/PhysRevA.36.5543
* [50] A. Barchielli, L. Lanz and G. M. Prosperi “A model for the macroscopic description and continual observations in quantum mechanics” In _Il Nuovo Cimento B (1971-1996)_ 72.1, 1982, pp. 79–121 DOI: 10.1007/BF02894935
* [51] Dominic Horsman et al. “Can a quantum state over time resemble a quantum state at a single time?” In _Proc. R. Soc. A_ 473.2205 The Royal Society, 2017, pp. 20170395
* [52] Christian Weedbrook et al. “Gaussian quantum information” In _Rev. Mod. Phys._ 84 American Physical Society, 2012, pp. 621–669 DOI: 10.1103/RevModPhys.84.621
* [53] Xiang-Bin Wang, Tohya Hiroshima, Akihisa Tomita and Masahito Hayashi “Quantum information with Gaussian states” In _Physics Reports_ 448.1, 2007, pp. 1 –111 DOI: https://doi.org/10.1016/j.physrep.2007.04.005
* [54] Gerardo Adesso, Sammy Ragy and Antony R Lee “Continuous variable quantum information: Gaussian states and beyond” In _Open Systems & Information Dynamics_ 21.01n02 World Scientific, 2014, pp. 1440001
* [55] R. Simon, N. Mukunda and Biswadeb Dutta “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms” In _Phys. Rev. A_ 49 American Physical Society, 1994, pp. 1567–1583 DOI: 10.1103/PhysRevA.49.1567
* [56] E. Wigner “On the Quantum Correction For Thermodynamic Equilibrium” In _Phys. Rev._ 40 American Physical Society, 1932, pp. 749–759 DOI: 10.1103/PhysRev.40.749
* [57] K. E. Cahill and R. J. Glauber “Ordered Expansions in Boson Amplitude Operators” In _Phys. Rev._ 177 American Physical Society, 1969, pp. 1857–1881 DOI: 10.1103/PhysRev.177.1857
* [58] K. E. Cahill and R. J. Glauber “Density Operators and Quasiprobability Distributions” In _Phys. Rev._ 177 American Physical Society, 1969, pp. 1882–1902 DOI: 10.1103/PhysRev.177.1882
* [59] Antoine Royer “Wigner function as the expectation value of a parity operator” In _Phys. Rev. A_ 15 American Physical Society, 1977, pp. 449–450 DOI: 10.1103/PhysRevA.15.449
* [60] Konrad Banaszek and Krzysztof Wódkiewicz “Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation” In _Phys. Rev. A_ 58 American Physical Society, 1998, pp. 4345–4347 DOI: 10.1103/PhysRevA.58.4345
* [61] M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner “Distribution functions in physics: Fundamentals” In _Physics Reports_ 106.3, 1984, pp. 121 –167 DOI: https://doi.org/10.1016/0370-1573(84)90160-1
* [62] R. F. O’Connell and E. P. Wigner “Quantum-Mechanical Distribution Functions: Conditions for Uniqueness” In _Part I: Physical Chemistry. Part II: Solid State Physics_ Berlin, Heidelberg: Springer Berlin Heidelberg, 1997, pp. 263–266 DOI: 10.1007/978-3-642-59033-7_26
* [63] Karl Kraus “States, effects and operations: fundamental notions of quantum theory” Springer, 1983
* [64] Sebastian Steinlechner et al. “Quantum-dense metrology” In _Nature Photonics_ 7.8 Nature Publishing Group, 2013, pp. 626
* [65] Tian Zhang, Oscar Dahlsten and Vlatko Vedral “Quantum correlations in time”, 2020 arXiv:2002.10448 [quant-ph]
* [66] Edward Anderson “The Problem of Time in Quantum Gravity”, 2010 arXiv:1009.2157 [gr-qc]
* [67] G. Chiribella, G. M. D’Ariano and P. Perinotti “Quantum Circuit Architecture” In _Phys. Rev. Lett._ 101 American Physical Society, 2008, pp. 060401 DOI: 10.1103/PhysRevLett.101.060401
* [68] Giulio Chiribella, Giacomo Mauro D’Ariano and Paolo Perinotti “Theoretical framework for quantum networks” In _Phys. Rev. A_ 80 American Physical Society, 2009, pp. 022339 DOI: 10.1103/PhysRevA.80.022339
* [69] Lucien Hardy “The operator tensor formulation of quantum theory” In _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 370.1971 The Royal Society Publishing, 2012, pp. 3385–3417
* [70] Ognyan Oreshkov, Fabio Costa and Časlav Brukner “Quantum correlations with no causal order” In _Nature communications_ 3 Nature Publishing Group, 2012, pp. 1092
* [71] Felix A. Pollock et al. “Non-Markovian quantum processes: Complete framework and efficient characterization” In _Phys. Rev. A_ 97 American Physical Society, 2018, pp. 012127 DOI: 10.1103/PhysRevA.97.012127
* [72] Jordan Cotler, Chao-Ming Jian, Xiao-Liang Qi and Frank Wilczek “Superdensity operators for spacetime quantum mechanics” In _Journal of High Energy Physics_ 2018.9, 2018, pp. 93 DOI: 10.1007/JHEP09(2018)093
* [73] Robert B. Griffiths “Consistent histories and the interpretation of quantum mechanics” In _Journal of Statistical Physics_ 36.1, 1984, pp. 219–272 DOI: 10.1007/BF01015734
* [74] Robert B Griffiths “Consistent quantum theory” Cambridge University Press, 2003
* [75] Murray Gell-Mann and James B. Hartle “Quantum Mechanics in the Light of Quantum Cosmology”, 1989 arXiv:1803.04605 [gr-qc]
* [76] Murray Gell-Mann and James B. Hartle “Classical equations for quantum systems” In _Phys. Rev. D_ 47 American Physical Society, 1993, pp. 3345–3382 DOI: 10.1103/PhysRevD.47.3345
* [77] Roland Omnés “From Hilbert space to common sense: A synthesis of recent progress in the interpretation of quantum mechanics” In _Annals of Physics_ 201.2 Elsevier, 1990, pp. 354–447 URL: https://www.sciencedirect.com/science/article/pii/000349169090045P
* [78] Francesco Buscemi “All Entangled Quantum States Are Nonlocal” In _Phys. Rev. Lett._ 108 American Physical Society, 2012, pp. 200401 DOI: 10.1103/PhysRevLett.108.200401
* [79] Denis Rosset, Francesco Buscemi and Yeong-Cherng Liang “Resource Theory of Quantum Memories and Their Faithful Verification with Minimal Assumptions” In _Phys. Rev. X_ 8 American Physical Society, 2018, pp. 021033 DOI: 10.1103/PhysRevX.8.021033
* [80] Juan Maldacena, Stephen H. Shenker and Douglas Stanford “A bound on chaos” In _JHEP_ 08, 2016, pp. 106 DOI: 10.1007/JHEP08(2016)106
* [81] Daniel A. Roberts and Beni Yoshida “Chaos and complexity by design” In _JHEP_ 04, 2017, pp. 121 DOI: 10.1007/JHEP04(2017)121
* [82] Richard P Feynman, Albert R Hibbs and Daniel F Styer “Quantum mechanics and path integrals” Courier Corporation, 2010
* [83] Jean Zinn-Justin “Path integrals in quantum mechanics” Oxford University Press, 2010
* [84] Robert Pisarczyk et al. “Causal Limit on Quantum Communication” In _Phys. Rev. Lett._ 123 American Physical Society, 2019, pp. 150502 DOI: 10.1103/PhysRevLett.123.150502
* [85] Tian Zhang, Oscar Dahlsten and Vlatko Vedral “Different instances of time as different quantum modes: quantum states across space-time for continuous variables” In _New Journal of Physics_ 22.2 IOP Publishing, 2020, pp. 023029 DOI: 10.1088/1367-2630/ab6b9f
* [86] Lucien Hardy “Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure” In _Journal of Physics A: Mathematical and Theoretical_ 40.12 IOP Publishing, 2007, pp. 3081–3099 DOI: 10.1088/1751-8113/40/12/s12
* [87] Lucien Hardy “Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure” In _Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony_ Dordrecht: Springer Netherlands, 2009, pp. 379–401 DOI: 10.1007/978-1-4020-9107-0_21
* [88] Lucien Hardy “The construction interpretation: a conceptual road to quantum gravity” In _arXiv preprint arXiv:1807.10980_ , 2018
* [89] Mateus Araújo et al. “Witnessing causal nonseparability” In _New Journal of Physics_ 17.10 IOP Publishing, 2015, pp. 102001 DOI: 10.1088/1367-2630/17/10/102001
* [90] Simon Milz, Felix A Pollock and Kavan Modi “An introduction to operational quantum dynamics” In _Open Systems & Information Dynamics_ 24.04 World Scientific, 2017, pp. 1740016
* [91] Jordan Cotler, Xizhi Han, Xiao-Liang Qi and Zhao Yang “Quantum causal influence” In _Journal of High Energy Physics_ 2019.7, 2019, pp. 42 DOI: 10.1007/JHEP07(2019)042
* [92] Fabio Costa et al. “Unifying framework for spatial and temporal quantum correlations” In _Phys. Rev. A_ 98 American Physical Society, 2018, pp. 012328 DOI: 10.1103/PhysRevA.98.012328
* [93] Dennis Kretschmann and Reinhard F. Werner “Quantum channels with memory” In _Phys. Rev. A_ 72 American Physical Society, 2005, pp. 062323 DOI: 10.1103/PhysRevA.72.062323
* [94] Gus Gutoski and John Watrous “Toward a general theory of quantum games” In _Proceedings of the thirty-ninth annual ACM symposium on Theory of computing_ , 2007, pp. 565–574 ACM
* [95] Yakir Aharonov, Peter G. Bergmann and Joel L. Lebowitz “Time Symmetry in the Quantum Process of Measurement” In _Phys. Rev._ 134 American Physical Society, 1964, pp. B1410–B1416 DOI: 10.1103/PhysRev.134.B1410
* [96] Yakir Aharonov, Sandu Popescu, Jeff Tollaksen and Lev Vaidman “Multiple-time states and multiple-time measurements in quantum mechanics” In _Phys. Rev. A_ 79 American Physical Society, 2009, pp. 052110 DOI: 10.1103/PhysRevA.79.052110
* [97] Ralph Silva et al. “Connecting processes with indefinite causal order and multi-time quantum states” In _New Journal of Physics_ 19.10 IOP Publishing, 2017, pp. 103022 DOI: 10.1088/1367-2630/aa84fe
* [98] Robert Oeckl “A “general boundary” formulation for quantum mechanics and quantum gravity” In _Physics Letters B_ 575.3, 2003, pp. 318 –324 DOI: https://doi.org/10.1016/j.physletb.2003.08.043
* [99] Fabio Costa and Sally Shrapnel “Quantum causal modelling” In _New Journal of Physics_ 18.6 IOP Publishing, 2016, pp. 063032 DOI: 10.1088/1367-2630/18/6/063032
* [100] John-Mark A. Allen et al. “Quantum Common Causes and Quantum Causal Models” In _Phys. Rev. X_ 7 American Physical Society, 2017, pp. 031021 DOI: 10.1103/PhysRevX.7.031021
* [101] Mateus Araújo, Adrien Feix, Miguel Navascués and Časlav Brukner “A purification postulate for quantum mechanics with indefinite causal order” In _Quantum_ 1 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2017, pp. 10
* [102] A. Jamiołkowski “Linear transformations which preserve trace and positive semidefiniteness of operators” In _Reports on Mathematical Physics_ 3.4, 1972, pp. 275 –278 DOI: https://doi.org/10.1016/0034-4877(72)90011-0
* [103] Man-Duen Choi “Completely positive linear maps on complex matrices” In _Linear Algebra and its Applications_ 10.3, 1975, pp. 285 –290 DOI: https://doi.org/10.1016/0024-3795(75)90075-0
* [104] Cyril Branciard et al. “The simplest causal inequalities and their violation” In _New Journal of Physics_ 18.1 IOP Publishing, 2015, pp. 013008 DOI: 10.1088/1367-2630/18/1/013008
* [105] Kurt Gödel “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation” In _Rev. Mod. Phys._ 21 American Physical Society, 1949, pp. 447–450 DOI: 10.1103/RevModPhys.21.447
* [106] David Deutsch “Quantum mechanics near closed timelike lines” In _Phys. Rev. D_ 44 American Physical Society, 1991, pp. 3197–3217 DOI: 10.1103/PhysRevD.44.3197
* [107] D. Ahn, C. R. Myers, T. C. Ralph and R. B. Mann “Quantum-state cloning in the presence of a closed timelike curve” In _Phys. Rev. A_ 88 American Physical Society, 2013, pp. 022332 DOI: 10.1103/PhysRevA.88.022332
* [108] Todd A. Brun, Mark M. Wilde and Andreas Winter “Quantum State Cloning Using Deutschian Closed Timelike Curves” In _Phys. Rev. Lett._ 111 American Physical Society, 2013, pp. 190401 DOI: 10.1103/PhysRevLett.111.190401
* [109] CH Bennett and B Schumacher “Teleportation, Simulated Time Travel, and How to Flirt with Someone Who Has Fallen into a Black Hole” In _QUPON, Wien_ , 2005
* [110] George Svetlichny “Time Travel: Deutsch vs. Teleportation” In _International Journal of Theoretical Physics_ 50.12, 2011, pp. 3903–3914 DOI: 10.1007/s10773-011-0973-x
* [111] Todd A. Brun and Mark M. Wilde “Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves” In _Foundations of Physics_ 42.3, 2012, pp. 341–361 DOI: 10.1007/s10701-011-9601-0
* [112] Seth Lloyd et al. “Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency” In _Phys. Rev. Lett._ 106 American Physical Society, 2011, pp. 040403 DOI: 10.1103/PhysRevLett.106.040403
* [113] Mateus Araújo, Philippe Allard Guérin and Ämin Baumeler “Quantum computation with indefinite causal structures” In _Phys. Rev. A_ 96 American Physical Society, 2017, pp. 052315 DOI: 10.1103/PhysRevA.96.052315
* [114] H. F. Dowker and J. J. Halliwell “Quantum mechanics of history: The decoherence functional in quantum mechanics” In _Phys. Rev. D_ 46 American Physical Society, 1992, pp. 1580–1609 DOI: 10.1103/PhysRevD.46.1580
* [115] Fay Dowker and Adrian Kent “On the consistent histories approach to quantum mechanics” In _Journal of Statistical Physics_ 82.5-6 Springer, 1996, pp. 1575–1646 URL: https://link.springer.com/article/10.1007/BF02183396
* [116] Roger B Myerson “Game theory” Harvard University Press, 2013
* [117] R. Cleve, P. Hoyer, B. Toner and J. Watrous “Consequences and limits of nonlocal strategies” In _Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004._ , 2004, pp. 236–249 DOI: 10.1109/CCC.2004.1313847
* [118] John F. Clauser, Michael A. Horne, Abner Shimony and Richard A. Holt “Proposed Experiment to Test Local Hidden-Variable Theories” In _Phys. Rev. Lett._ 23 American Physical Society, 1969, pp. 880–884 DOI: 10.1103/PhysRevLett.23.880
* [119] S. W. Hawking “Particle creation by black holes” In _Communications in Mathematical Physics_ 43.3, 1975, pp. 199–220 DOI: 10.1007/BF02345020
* [120] S. W. Hawking “Breakdown of predictability in gravitational collapse” In _Phys. Rev. D_ 14 American Physical Society, 1976, pp. 2460–2473 DOI: 10.1103/PhysRevD.14.2460
* [121] John Preskill “Do black holes destroy information” In _Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings, S. Kalara and DV Nanopoulos, eds.(World Scientific, Singapore, 1993) pp_ , 1992, pp. 22–39 World Scientific
* [122] T. Banks, A. Dabholkar, M. R. Douglas and M. O’Loughlin “Are horned particles the end point of Hawking evaporation?” In _Phys. Rev. D_ 45 American Physical Society, 1992, pp. 3607–3616 DOI: 10.1103/PhysRevD.45.3607
* [123] S. W. Hawking “Wormholes in spacetime” In _Phys. Rev. D_ 37 American Physical Society, 1988, pp. 904–910 DOI: 10.1103/PhysRevD.37.904
* [124] Stephen W Hawking “Baby Universes II” In _Mod. Phys. Lett. A_ 5, 1990, pp. 453–466
* [125] Don N. Page “Average entropy of a subsystem” In _Phys. Rev. Lett._ 71 American Physical Society, 1993, pp. 1291–1294 DOI: 10.1103/PhysRevLett.71.1291
* [126] Don N. Page “Information in black hole radiation” In _Phys. Rev. Lett._ 71 American Physical Society, 1993, pp. 3743–3746 DOI: 10.1103/PhysRevLett.71.3743
* [127] Gary T. Horowitz and Juan Martin Maldacena “The Black hole final state” In _JHEP_ 02, 2004, pp. 008 DOI: 10.1088/1126-6708/2004/02/008
* [128] Daniel Gottesman and John Preskill “Comment on ‘The Black hole final state”’ In _JHEP_ 03, 2004, pp. 026 DOI: 10.1088/1126-6708/2004/03/026
* [129] Seth Lloyd and John Preskill “Unitarity of black hole evaporation in final-state projection models” In _JHEP_ 08, 2014, pp. 126 DOI: 10.1007/JHEP08(2014)126
* [130] Raphael Bousso and Douglas Stanford “Measurements without probabilities in the final state proposal” In _Phys. Rev. D_ 89 American Physical Society, 2014, pp. 044038 DOI: 10.1103/PhysRevD.89.044038
* [131] Franco Strocchi “Symmetry breaking” Springer, 2005
* [132] Vedika Khemani, Roderich Moessner and S.L. Sondhi “A Brief History of Time Crystals”, 2019 arXiv:1910.10745 [cond-mat.str-el]
* [133] Yoichiro Nambu “Quasi-Particles and Gauge Invariance in the Theory of Superconductivity” In _Phys. Rev._ 117 American Physical Society, 1960, pp. 648–663 DOI: 10.1103/PhysRev.117.648
* [134] J. Goldstone “Field theories with « Superconductor » solutions” In _Il Nuovo Cimento (1955-1965)_ 19.1, 1961, pp. 154–164 DOI: 10.1007/BF02812722
* [135] Jeffrey Goldstone, Abdus Salam and Steven Weinberg “Broken Symmetries” In _Phys. Rev._ 127 American Physical Society, 1962, pp. 965–970 DOI: 10.1103/PhysRev.127.965
* [136] N. D. Mermin and H. Wagner “Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models” In _Phys. Rev. Lett._ 17 American Physical Society, 1966, pp. 1133–1136 DOI: 10.1103/PhysRevLett.17.1133
* [137] P. C. Hohenberg “Existence of Long-Range Order in One and Two Dimensions” In _Phys. Rev._ 158 American Physical Society, 1967, pp. 383–386 DOI: 10.1103/PhysRev.158.383
* [138] Axel Gelfert and Wolfgang Nolting “The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results” In _Journal of Physics: Condensed Matter_ 13.27 IOP Publishing, 2001, pp. R505–R524 DOI: 10.1088/0953-8984/13/27/201
* [139] Frank Wilczek “Quantum Time Crystals” In _Phys. Rev. Lett._ 109 American Physical Society, 2012, pp. 160401 DOI: 10.1103/PhysRevLett.109.160401
* [140] Patrick Bruno “Comment on “Quantum Time Crystals”” In _Phys. Rev. Lett._ 110 American Physical Society, 2013, pp. 118901 DOI: 10.1103/PhysRevLett.110.118901
* [141] Patrick Bruno “Impossibility of Spontaneously Rotating Time Crystals: A No-Go Theorem” In _Phys. Rev. Lett._ 111 American Physical Society, 2013, pp. 070402 DOI: 10.1103/PhysRevLett.111.070402
* [142] Haruki Watanabe and Masaki Oshikawa “Absence of Quantum Time Crystals” In _Phys. Rev. Lett._ 114 American Physical Society, 2015, pp. 251603 DOI: 10.1103/PhysRevLett.114.251603
* [143] Dominic V. Else, Bela Bauer and Chetan Nayak “Floquet Time Crystals” In _Phys. Rev. Lett._ 117 American Physical Society, 2016, pp. 090402 DOI: 10.1103/PhysRevLett.117.090402
* [144] N. Y. Yao, A. C. Potter, I.-D. Potirniche and A. Vishwanath “Discrete Time Crystals: Rigidity, Criticality, and Realizations” In _Phys. Rev. Lett._ 118 American Physical Society, 2017, pp. 030401 DOI: 10.1103/PhysRevLett.118.030401
* [145] J Zhang et al. “Observation of a discrete time crystal” In _Nature_ 543.7644 Nature Publishing Group, 2017, pp. 217
* [146] Soonwon Choi et al. “Observation of discrete time-crystalline order in a disordered dipolar many-body system” In _Nature_ 543.7644 Nature Publishing Group, 2017, pp. 221
* [147] Jared Rovny, Robert L. Blum and Sean E. Barrett “Observation of Discrete-Time-Crystal Signatures in an Ordered Dipolar Many-Body System” In _Phys. Rev. Lett._ 120 American Physical Society, 2018, pp. 180603 DOI: 10.1103/PhysRevLett.120.180603
* [148] Jared Rovny, Robert L. Blum and Sean E. Barrett “${}^{31}\mathrm{P}$ NMR study of discrete time-crystalline signatures in an ordered crystal of ammonium dihydrogen phosphate” In _Phys. Rev. B_ 97 American Physical Society, 2018, pp. 184301 DOI: 10.1103/PhysRevB.97.184301
* [149] Dominic V. Else, Bela Bauer and Chetan Nayak “Prethermal Phases of Matter Protected by Time-Translation Symmetry” In _Phys. Rev. X_ 7 American Physical Society, 2017, pp. 011026 DOI: 10.1103/PhysRevX.7.011026
* [150] F. Iemini et al. “Boundary Time Crystals” In _Phys. Rev. Lett._ 121 American Physical Society, 2018, pp. 035301 DOI: 10.1103/PhysRevLett.121.035301
* [151] Praloy Das, Supriya Pan, Subir Ghosh and Probir Pal “Cosmological time crystal: Cyclic universe with a small cosmological constant in a toy model approach” In _Phys. Rev. D_ 98 American Physical Society, 2018, pp. 024004 DOI: 10.1103/PhysRevD.98.024004
* [152] Xing-Hui Feng et al. “Cosmological time crystals from Einstein-cubic gravities” In _arXiv preprint arXiv:1807.01720_ , 2018
* [153] S. Autti, V. B. Eltsov and G. E. Volovik “Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal” In _Phys. Rev. Lett._ 120 American Physical Society, 2018, pp. 215301 DOI: 10.1103/PhysRevLett.120.215301
* [154] Vedika Khemani, C. W. Keyserlingk and S. L. Sondhi “Defining time crystals via representation theory” In _Phys. Rev. B_ 96 American Physical Society, 2017, pp. 115127 DOI: 10.1103/PhysRevB.96.115127
* [155] C. N. Yang “Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors” In _Rev. Mod. Phys._ 34 American Physical Society, 1962, pp. 694–704 DOI: 10.1103/RevModPhys.34.694
* [156] Sidney Coleman “There are no Goldstone bosons in two dimensions” In _Communications in Mathematical Physics_ 31.4, 1973, pp. 259–264 DOI: 10.1007/BF01646487
* [157] Adriano Barenco et al. “Stabilization of quantum computations by symmetrization” In _SIAM Journal on Computing_ 26.5 SIAM, 1997, pp. 1541–1557
* [158] C. W. Keyserlingk and S. L. Sondhi “Phase structure of one-dimensional interacting Floquet systems. II. Symmetry-broken phases” In _Phys. Rev. B_ 93 American Physical Society, 2016, pp. 245146 DOI: 10.1103/PhysRevB.93.245146
* [159] Derek JS Robinson “A Course in the Theory of Groups” Springer Science & Business Media, 2012
* [160] Carlo Heissenberg and Franco Strocchi “Generalized criteria of symmetry breaking. A strategy for quantum time crystals”, 2019 arXiv:1906.12293 [cond-mat.stat-mech]
* [161] Vlatko Vedral “Introduction to quantum information science” Oxford University Press on Demand, 2006
* [162] Mark Van Raamsdonk “Building up spacetime with quantum entanglement” In _General Relativity and Gravitation_ 42.10, 2010, pp. 2323–2329 DOI: 10.1007/s10714-010-1034-0
* [163] Ted Jacobson “Thermodynamics of Spacetime: The Einstein Equation of State” In _Phys. Rev. Lett._ 75 American Physical Society, 1995, pp. 1260–1263 DOI: 10.1103/PhysRevLett.75.1260
* [164] Seth Lloyd “The quantum geometric limit” In _arXiv preprint arXiv:1206.6559_ , 2012
* [165] Antoine Royer “Measurement of quantum states and the Wigner function” In _Foundations of Physics_ 19.1, 1989, pp. 3–32 DOI: 10.1007/BF00737764
|
# Multi-robot energy autonomy with wind and constrained resources
Hassan Fouad and Giovanni Beltrame *This work was supported by the NSERC
Discovery Grant 2019-05165.Hassan Fouad and Giovanni Beltrame are with
Computer and Software Engineering department, Ecole polytechnique de Montreal
<EMAIL_ADDRESS>appendices of this paper can be found at
https://mistlab.ca/papers/energy2020
###### Abstract
One aspect of the ever-growing need for long term autonomy of multi-robot
systems, is ensuring energy sufficiency. In particular, in scenarios where
charging facilities are limited, battery-powered robots need to coordinate to
share access. In this work we extend previous results by considering robots
that carry out a generic mission while sharing a single charging station,
while being affected by air drag and wind fields. Our mission-agnostic
framework based on control barrier functions (CBFs) ensures energy sufficiency
(i.e., maintaining all robots above a certain voltage threshold) and proper
coordination (i.e., ensuring mutually exclusive use of the available charging
station). Moreover, we investigate the feasibility requirements of the system
in relation to individual robots’ properties, as well as air drag and wind
effects. We show simulation results that demonstrate the effectiveness of the
proposed framework.
## I Introduction
The continuous advances in multi-robot systems gave rise to many new
applications like patrolling [1], coverage [2], exploration [3] and
construction [4] to give a few examples. This has drawn many researchers’
attention in recent years to long term autonomy and resilience of multi-robot
systems, with the aim of providing more practical and robust systems.
Energy autonomy, the ability of the robots in a multi-robot system to
replenish their energy reserves, is particularly important to extend mission
duration and general survivability.
Earlier interest in optimizing energy consumption in a multi-robot system can
be traced back to energy aware path planning [5] and node scheduling in
wireless sensor networks [6]. These ideas have been applied to multi-robot
systems as in [7], where the mission tasks are divided among robots according
to their energy content.
One option for tackling the issue of limited energy is through the
introduction of stationary or mobile charging stations. Ding et al. [8]
propose a method for planning routes of charging robots that deposit batteries
along the trajectories of other robots carrying out a surveillance mission.
Notomista et al. [9] use a control barrier function framework that allows each
robot in a multi-robot system to recharge from a dedicated static charging
station in a mission agnostic and minimally invasive manner.
In this work we extend [10], which is in turn inspired from [9], by
considering a group of robots affected by air drag that perform a generic
mission (e.g. coverage or patrolling) in a known wind field. These robots need
to share a single charging station.
The contributions of this paper are: 1) We extend the results in [10] so that
the CBF-based coordination framework proposed can account for the effect of
air drag and winds, while ensuring mutually exclusive use of the charging
station, and 2) we extend the sufficient feasibility conditions proposed in
[10] to express the system’s capacity in case of wind and air drag effects and
ensure the feasibility of coordination.
## II Preliminaries
### II-A Control Barrier Functions (CBF)
A control barrier function (CBF) [11] is a tool that is mainly used to ensure
set invariance of control affine systems, having the form
$\dot{x}=f(x)+g(x)u.$ (1)
This is often used for ensuring system’s safety by enforcing forward
invariance of a desired safe set.
The safe set is defined to be the superlevel set of a continuously
differentiable function $h(x)$ such that [11]:
$\begin{split}\mathcal{C}&=\\{x\in\mathbb{R}^{n}:h(x)\geq 0\\}\\\
\partial\mathcal{C}&=\\{x\in\mathbb{R}^{n}:h(x)=0\\}\\\
Int(\mathcal{C})&=\\{x\in\mathbb{R}^{n}:h(x)>0\\}\\\ \end{split}$ (2)
where ensuring $h(x)>0,\forall t\geq 0$ implies the safe set $\mathcal{C}$ is
positively invariant. For a control affine system, having a control action $u$
that achieves
$\underbrace{L_{f}h(x)+L_{g}h(x)u}_{\dot{h}(x)}\geq-\alpha(h(x))$ (3)
where $\alpha(h(x))$ is an extended type $\mathcal{K}$ function, ensures
positive invariance of $\mathcal{C}$.
One popular type of CBFs that we use in this paper is the zeroing control
barrier function (ZCBF) [12], as they have favourable robustness and
asymptotic stability properties [13].
###### Definition 1
[12] For a region $\mathcal{D}\in\mathcal{C}$ a continuously differentiable
function $h(x)$ is called a ZCBF if there exists an extended class
$\mathcal{K}$ function $\alpha(h(x))$ such that
$\sup_{u\in U}\left(L_{f}h(x)+L_{g}hu+\alpha(h(x))\geq 0\right)$ (4)
The set $K_{zcbf}$[12] is defined as
$K_{zcbf}=\\{u\in U:L_{f}h(x)+L_{g}hu+\alpha(h(x))\geq 0\\}$
and it is the set that contains all the safe control inputs, thus choosing a
Lipschitz continuous controller $u\in K_{zcbf}$ ensures forward invariance of
$\mathcal{C}$ and system’s safety.
To mix the safety control input with an arbitrary mission’s control input
$u_{nom}$, we use a quadratic program: [9]
$\displaystyle u^{*}=\underset{u}{\text{min}}$
$\displaystyle||u-u_{nom}||^{2}$ (5) s.t. $\displaystyle
L_{f}h(x)+L_{g}h(x)u\geq-\alpha(h(x)).$
### II-B Higher order control barrier functions (HOCBF)
If $h(x)$ is of a higher relative degree (the control action $u$ doesn’t
appear after differentiating once, i.e. $L_{g}h(x)=0$), using (3) to find an
appropriate control action becomes invalid. HOCBFs [14] are an effective
solution of this problem. To define a HOCBF, we first need to define the
following set of functions for an $m^{th}$ order differentiable function
$h(x)$
$\begin{split}\psi_{0}(x)&=h(x)\\\
\psi_{1}(x)&=\dot{\psi}_{0}(x)+\alpha_{1}(\psi_{0}(x))\\\ \vdots\\\
\psi_{m}(x)&=\dot{\psi}_{m-1}(x)+\alpha_{m}(\psi_{m-1}(x))\end{split}$ (6)
where $\alpha_{1},\dots,\alpha_{m-1}$ are class $\mathcal{K}$ functions. Also
we define the following series of sets
$\begin{split}\mathcal{C}_{1}\vcentcolon&=\\{x\in\mathbb{R}^{n}:\psi_{0}(x)\geq
0\\}\\\ \vdots\\\
\mathcal{C}_{m}\vcentcolon&=\\{x\in\mathbb{R}^{n}:\psi_{m}(x)\geq
0\\}\end{split}$ (7)
###### Definition 2
[14] Let $\mathcal{C}_{1},\mathcal{C}_{2},\dots,\mathcal{C}_{m}$ be defined by
(7) and $\psi_{0}(x),\psi_{1}(x),\dots,\psi_{m}(x)$ be defined by (6). A
function $h(x)$ is a HOCBF of relative degree $m$ for system (1) if there
exists differentiable class $\mathcal{K}$ functions
$\alpha_{1},\alpha_{2},\dots,\alpha_{m}$ such that
$L_{f}^{m}h(x)+L_{g}L_{f}^{m-1}h(x)+O(h(x))+\alpha_{m}(\psi_{m-1}(x))\geq 0$
(8)
for all $x\in\mathcal{C}_{1}\cap\mathcal{C}_{2}\dots\cap\mathcal{C}_{m}$. Here
$O(h(x))$ denotes the remaining Lie derivatives along $f$ with degrees less
than or equal $m-1$.
Xiao et al. show in [14, Theorem 5] that choosing a control action that
satisfies (8) renders the set
$\mathcal{C}_{1}\cap\mathcal{C}_{2}\dots\cap\mathcal{C}_{m}$ forward invariant
for system (1).
## III Problem formulation
We assume $n$ robots moving in a given wind field with the following dynamics:
$\begin{split}\dot{\mathbf{x}}&=\mathbf{v}\\\
\dot{\mathbf{v}}&=u-C_{d}(\mathbf{v}-\mathbf{v}_{w})\\\
\dot{E}&=\begin{cases}-k_{e}-k_{v}||\mathbf{v}-\mathbf{v}_{w}||\quad,\text{if}||\mathbf{x}-\mathbf{x}_{c}||>\delta\\\
k_{ch}\quad,\text{Otherwise}\end{cases}\end{split}$ (9)
where $\mathbf{x}\in\mathbb{R}^{2}$ is the robot’s position,
$\mathbf{v}\in\mathbb{R}^{2}$ is its velocity,
$C_{d}>0,k_{ch}>0,k_{e}>0,k_{v}>0$ are coefficients of linear drag, recharge,
static and dynamic discharge respectively. Also $E>0$ is the robot’s voltage,
$u\in\mathbb{R}^{2}$ is the control input (no constraints on the control
input), and $\mathbf{v}_{w}$ is a known wind vector. Moreover we suppose that
all the robots operate in a certain known operational range
$\mathbf{x}\in\mathcal{R}\subset\mathbb{R}^{2}$, where $\mathcal{R}$ is a
closed set, and the size of this operational range is described by the
operational radius $R_{0}$. The robots are carrying out a mission specified by
$u_{nom}$ and they have one charging station at a known location $x_{c}$ (in
the origin without loss of generality), and this station can only serve one
robot at a time, and has an effective charging range of $\delta>0$.
We point out that in our model we use a linear drag term to account for the
air drag effect, which is a reasonable approximation for bodies moving at low
speeds. The main assumptions we are adopting in this work are: 1. all robots
have the same properties 2. robots have a complete communication graph 3.
robots start discharging from the maximum voltage 4. the charging rate is
faster than the discharge rate111E.g. battery swapping or high power wireless
charging. 5. An upper bound of the average relative velocity w.r.t. wind
velocity(we call it $\tilde{V}$) of all robots is known at the beginning of
the mission. We propose a CBF framework that:
* •
Ensures no robot runs out of energy during the mission
* •
Coordinates the times of arrival to the charging station so they are mutually
exclusive.
Additionally, we describe the system’s capacity as the relationship between
number of robots and robot properties with feasible separation in arrival
times at the charging station.
_It is worth mentioning that for the rest of the paper we are omitting the
proofs due to space constraints, and putting them all in the appendix._
## IV Energy sufficiency
We provide a CBF that ensures that the voltage of all robots does not go below
a certain desired minimum voltage $E_{min}$. We take inspiration from [9], but
we extend it to accommodate the system dynamics in (9). The candidate CBF is
$h_{e}=E-E_{min}-k_{c}\log\frac{D}{\delta}$ (10)
where $D=||x-x_{c}||$222The choice of $k_{c}$ is explained in the appendix.
The first derivative of this function is
$\dot{h}_{e}=-k_{e}-k_{v}||\mathbf{v}-\mathbf{v}_{w}||-\frac{k_{v}}{D^{2}}(\mathbf{x}-\mathbf{x}_{c})^{T}\mathbf{v}$
(11)
so we need to differentiate twice for the control input $u$ to appear
$\begin{split}&\ddot{h}_{e}=\overbrace{-\left[kv\frac{(\mathbf{v}-\mathbf{v}_{w})^{T}}{||\mathbf{v}-\mathbf{v}_{w}||}+k_{c}\frac{(\mathbf{x}-\mathbf{x}_{c})^{T}}{D^{2}}\right]}^{L_{g}L_{f}h_{e}(x)}u\\\
&+k_{v}C_{d}||\mathbf{v}-\mathbf{v}_{w}||\\\
&+\frac{k_{c}}{D^{2}}\left[\frac{2((\mathbf{x}-\mathbf{x}_{c})^{T}\mathbf{v})^{2}}{D^{2}}-\mathbf{v}^{T}\mathbf{v}+C_{d}\mathbf{x}-\mathbf{x}_{c})^{T}(\mathbf{v}-\mathbf{v}_{w})\right]\end{split}$
(12)
with the second and third expressions being $L_{f}^{2}h_{e}(x)$. We can then
create an inequality similar to (8) using $\alpha_{1}(h)=p_{1}h$ and
$\alpha_{2}(h)=p_{2}h$
$L_{f}^{2}h_{e}(x)+L_{g}L_{f}h_{e}(x)u+(p_{1}+p_{2})\dot{h}_{e}+p_{1}p_{2}h_{e}\geq
0$ (13)
and $p_{1}>$ and $p_{2}>$ are chosen in such a way that lends the
characteristic equation of the left side of (13) with distinct real roots.
###### Theorem 1
For a robot described by dynamics in (9), and provided that the robot is out
of the charging region, and that $k_{c}>k_{v}R_{0}$, then $h_{e}$ is a HOCBF.
###### Lemma 1
For a robot with dynamics described by (9) applying a QP as in (5) with (13)
as being the constraint, then the quantity $E-E_{min}$ at the time of arrival
to the charging station is upper bounded with a quantity exponentially
decaying with a rate of $\frac{1}{2}\left(-(p_{1}+p_{2})+|p_{1}-p_{2}|\right)$
and lower bounded by zero.
## V Coordination
The second component in our framework ensures that the difference in arrival
times of any two robots to the charging station is above a desired limit. The
main idea is that if two robots have different values of $E_{min}$, they
arrive to the charging station at different times. We propose a method for
changing the values of $E_{min}$ to achieve the aforementioned coordination.
To get this expression, we integrate the voltage relation in (9) to get
$\begin{split}\int_{E_{max}}^{E_{min}}\dot{E}dt&=-\int_{0}^{T_{L}}(k_{e}+k_{v}||\mathbf{v}-\mathbf{v}_{w}||)dt\\\
E_{max}-E_{min}&=k_{e}T_{L}+k_{v}\int_{0}^{T_{L}}||\mathbf{v}-\mathbf{v}_{w}||dt.\end{split}$
(14)
Supposing we have the average relative speed
$\bar{z}=\frac{1}{T_{L}}\int_{0}^{T_{L}}||\mathbf{v}-\mathbf{v}_{w}||dt$, the
last integral can be replaced and the arrival time becomes
$T_{L}=\frac{E_{max}-E_{min}}{k_{e}+k_{v}\bar{z}}.$ (15)
We then replace $E_{max}$ in the last expression by $E(t)$ to get an
expression for $T_{L}(t)$ that changes with time
$T_{L}(t)=\frac{E(t)-E_{min}}{k_{e}+k_{v}\bar{z}}.$ (16)
In this work, we use a moving average $\bar{V}$ to estimate the average
velocity relative to wind defined as
$\bar{V}=\frac{1}{w}\int_{0}^{w}||\mathbf{v}-\mathbf{v}_{w}||dt$ (17)
where $w>0$ is the width of the integration window. The larger the window, the
closer the estimate is to the true average. The approximate value of the
arrival time is
$T_{L}(t)\approx\frac{E(t)-E_{min}}{k_{e}+k_{v}\bar{V}}.$ (18)
To be able to change $E_{min}$ to achieve coordination, we propose a simple
single integrator model for $E_{min}$ as follows
$\displaystyle\dot{E}_{min}=\eta$ (19)
where $\eta\in\Theta\subset\mathbb{R}$ is a control input to manipulate
$E_{min}$,and $\Theta$ is being the set of all possible values of $\eta$. It
is useful to point out that $\eta$ has a default value of $\eta_{nom}=0$
unless modified by the proposed coordination framework.
### V-A Coordination CBF
We propose a CBF approach to change the values of $E_{min}$ to ensure mutually
exclusive use of the charging station. We define a coordination CBF
$h_{c_{ij}}$ between robots $i$ and $j$, as well as an associated pairwise
safe set $\mathcal{C}_{ij}$
$\mathcal{C}_{ij}=\\{(E_{min_{i}},E_{min_{j}})\in\mathbb{R}^{2}:h_{c_{ij}}\geq
0\\}.$ (20)
We use the same coordination CBF as in [10]
$h_{c_{ij}}=\log\frac{|T_{L_{i}}-T_{L_{j}}|}{\delta_{t}}$ (21)
and to get a constraint similar to (3)
$\frac{T_{L_{i}}-T_{L_{j}}}{|T_{L_{i}}-T_{L_{j}}|^{2}}(\theta_{i}\eta_{i}+\beta_{i}-\beta_{j})\geq\alpha(h_{c_{ij}})$
(22)
where
$\begin{split}&\theta_{i}=-\frac{1}{k_{e}+k_{v}\bar{V}}\\\
&\beta_{i}=\frac{-k_{e}-k_{v}V}{k_{e}+k_{v}\bar{V}}-\frac{k_{v}}{w}\frac{(E-E_{min})(V(t)-V(t-w))}{k_{e}+k_{v}\bar{V}}\\\
&\dot{T}_{L_{i}}=\theta_{i}\eta_{i}+\beta_{i}\end{split}$
where $V=||\mathbf{v}-\mathbf{v}_{w}||$. For decentralized implementation, we
dropped out the term $\theta_{j}$ so the constraint equation is independent of
$\eta_{j}$, and provided both robots are adopting the constraint (22), each
will try to stay in the safe set $\mathcal{C}_{ij}$. For the right hand side
of (22) we use the following
$\begin{split}&\alpha(h_{c_{ij}})=\gamma_{ij}.\text{sign}(h_{c_{ij}}).|h_{c_{ij}}|^{\rho}\quad,\rho\in\left[0,1\right)\\\
&\gamma_{ij}=\begin{cases}\gamma_{h}\quad,\text{if}\hskip
5.69054ptD_{i}>\delta\text{ and }D_{j}>\delta\\\
0\quad,\text{otherwise}\end{cases}\end{split}$ (23)
which is inspired from [15] and leads to the favourable quality of converging
to the safe set in a finite time, in case the initial condition is out of the
safe set.
###### Theorem 2
[10, Theorem 2] For a pair of robots $(i,j)$ that belongs to a multi-robot
system and satisfying $D_{i}>\delta$ and $D_{j}>\delta$, and provided that
$\eta\in\Theta=\mathbb{R}$ then $h_{c_{ij}}$ is a ZCBF. Moreover, if
$(E_{min_{i}}(t_{0}),E_{min_{j}}(t_{0}))\notin\mathcal{C}_{ij}$, then the
constraint (22) leads $(E_{min_{i}}(t),E_{min_{j}}(t))$ to converge to
$\partial\mathcal{C}_{ij}$ in finite time.
### V-B Lower bound on $E_{min}$
Since $E_{min}$ is supposed to be the voltage at which the robot arrives to
the charging station, then it is necessary to enforce a lower bound on its
value to avoid any potential damage to the batteries or the loss of a robot
with excessively low voltage. For this reason, we propose another CBF:
$h_{L}=k_{s}(E_{min}-E_{lb})$ (24)
where $E_{lb}>0$ is the desired lower bound voltage and $k_{s}>0$ is a scaling
gain. Differentiating $h_{L}$ and obtaining the QP constraint gives
$k_{s}\eta\geq-\alpha(h_{L})$ (25)
where $\alpha(h_{L})=p_{L}h_{L}$ for $p_{L}>0$. It can be easily shown that
$h_{L}$ is a ZCBF, since $\eta\in\Theta=\mathbb{R}$ (no constraint on $\eta$)
then there exists a control input $\eta$ that satisfies (25).
### V-C System capacity description
To successfully apply the coordination CBF in a pairwise manner, the value of
the desired $\delta_{t}$ should be reasonable with respect to individual
robot’s properties and the number of robots in the system (e.g. we can’t ask
for $\delta_{t}$ that is longer than the total discharge time of a battery).
We consider the relation between the robots’ parameters, their number and the
feasible limits on $\delta_{t}$ as being an expression of the system’s
capacity.
We propose a sufficient condition on the upper and lower limits of
$\delta_{t}$, in relation to properties like maximum and minimum battery
voltages, discharge and recharge rates, and the number of robots in the
system.
$\bar{E}_{M}$
---
$E_{max}$
---
$E_{min}$
---
$E_{lb}$
---
$t_{1}$
---
$t_{2}$
---
$\delta_{t}$
---
$t_{3}$
---
$\delta_{t}$
---
Figure 1: Schematic of a charging cycle with three robots. The red line
represents the more needy robot, while the blue one is the least needy one,
and it defines one recharging cycle.
For the sake of being conservative, we derive this capacity relation assuming
that the system is pushed to its limits, meaning that all robots operate with
the maximum average relative velocity w.r.t. wind $\tilde{V}$. Suppose we have
a group of $n$ robots, each has its own $E_{min}$ value, and one of them is
the “neediest” robot that recharges first and most often, while another is the
least needy one (represented in Figure 1 as the red and blue lines
respectively). We want $t_{2}-t_{1}\geq\delta_{t}$, which means
$\begin{split}&t_{1}=\frac{E_{max}-E_{lb}}{k_{e}+k_{v}\tilde{V}}\\\
&t_{2}=\frac{E_{max}-\bar{E}_{M}}{k_{e}+k_{v}\tilde{V}}+\frac{E_{max}-\bar{E}_{M}}{k_{ch}}+\frac{E_{max}-\bar{E}_{M}}{k_{e}+k_{v}\tilde{V}}.\end{split}$
(26)
Calculating $t_{2}-t_{1}$ and considering that
$\bar{E}_{M}=E_{M}+\varepsilon$, where $E_{M}$ being _the actual value of
$E_{min}$ of the neediest robot during the coordination_, and $\varepsilon\geq
0$ being an additional increment of voltage to $E_{M}$ that is caused by the
dependence of discharge rate on robot’s speed, we have:
$E_{M}\leq\frac{(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})E_{max}+E_{lb}-\delta_{t}(k_{e}+k_{v}\tilde{V})-\kappa\varepsilon}{\kappa}$
(27)
where $\kappa=2+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}}$. We can then calculate
$\Delta E_{M}=\tfrac{E_{M}-E_{lb}}{n-1}$ which is a uniform increment of
$E_{min}$ between $E_{lb}$ and $E_{M}$ to create the desired separation of
arrival times
$\Delta
E_{M}=\frac{(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})(E_{max}-E_{lb})-\delta_{t}(k_{e}+k_{v}\tilde{V})-\kappa\varepsilon}{\kappa(n-1)}$
(28)
What we want then is to have $t_{1}-t_{3}\geq\delta_{t}$, meaning that the
arrival times of the last two robots (or any two consecutive robots) to be at
least $\delta_{t}$
$\frac{E_{max}-E_{lb}}{k_{e}+k_{v}\tilde{V}}-\frac{E_{max}-(E_{lb}+\Delta
E_{M})}{k_{e}+k_{v}\tilde{V}}\geq\delta_{t}$ (29)
and substituting 28 into the last equation we get
$\begin{split}&\frac{(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})(E_{max}-E_{lb})-\delta_{t}(k_{e}+k_{v}\tilde{V})-\kappa\varepsilon}{\kappa(n-1)}\\\
&-\delta_{t}(k_{e}+k_{v}\tilde{V})\geq 0\end{split}$ (30)
then we obtain a critical value of $\delta_{t}$ at which the inequality
becomes an equality
$\delta_{t_{cr}}=\frac{\left(1+\frac{k_{e}+k_{v}\tilde{V}}{k_{ch}}\right)(E_{max}-E_{lb})-\kappa\varepsilon}{(k_{e}+k_{v}\tilde{V})\left[1+\kappa(n-1)\right]}.$
(31)
One final requirement on $\delta_{t_{cr}}$ is to be greater than half the time
taken to recharge a battery from $E_{lb}$ to $E_{max}$
$\delta_{t_{cr}}\geq\frac{E_{max}-E_{lb}}{2k_{ch}}$ (32)
The value of $\delta_{t_{cr}}$ represents in this case an upper bound on the
feasible $\delta_{t}$ that can be achieved by the system. To motivate the need
for $\varepsilon$333 More details on its derivation can be found in the
appendix, we consider the critical case when
$(E_{min_{i}},E_{min_{j}})\notin\mathcal{C}_{ij}$, in which case the QP
produces $\eta_{i}$ that renders (22) an equality, thus
$\dot{h}_{c_{ij}}=\alpha(h_{c_{ij}})$ that reaches steady state in finite time
(i.e. coordination achieved) at which $\dot{h}_{c_{ij}}=0$. Considering the
case where all robots have same $\bar{V}=\tilde{V}$ (which we already supposed
when deriving $\delta_{t_{cr}}$) then from the LHS of (22) we have
$\eta_{i}=k_{v}(V_{j}-V_{i})$. So if $V_{i}$ decreases (robot $i$ going to
recharge for example), $\eta_{i}$ increases and so $\varepsilon=\Delta
E_{min_{i}}$ resulting from the increase in $\eta$. The value of $\varepsilon$
can be estimated by approximating the integration of $\eta_{i}$ over the time
it takes the robot to go back to the charging station. One approximation for
$\varepsilon$ is
$\varepsilon=\frac{k_{v}V_{n}\left(T_{end}-\tfrac{n(E_{max}-E_{lb})-k_{c}\log\tfrac{R_{0}}{\delta}(1+\kappa(n-1))}{(k_{e}+k_{v}\tilde{V})(1+\kappa(n-1))}\right)}{1+\tfrac{k_{v}V_{n}}{k_{e}+k_{v}\tilde{V}}\left(1-\tfrac{1}{1+\kappa(n-1)}\right)}$
(33)
where
$T_{end}=\frac{E_{max}-\tfrac{E_{max}+E_{lb}}{2}}{k_{e}+k_{v}\tilde{V}}$, and
$V_{n}$ is the magnitude of the mission’s nominal velocity w.r.t. the wind
velocity vector.
###### Lemma 2
For a group of $n$ robots that have distinct values of $E_{min}$ that satisfy
(31), (32) and (28), and provided they all operate such that their average
relative velocity (w.r.t. wind) is equal to its upper bound, i.e.
$\bar{V}=\tilde{V}$, let $z_{i}$ be the number of recharges that one robot can
have in one charging cycle, then the maximum number of recharges for any robot
is $\bar{z}_{i}=2$. Moreover, $\bar{E}_{M}\leq\tfrac{E_{max}+E_{lb}}{2}$.
###### Lemma 3
For a group of $n$ robots, if $\delta_{t}$ satisfies
$\frac{E_{max}-E_{lb}}{2k_{ch}}\leq\delta_{t}\leq\delta_{t_{cr}}$ (34)
as well as equation (28), then there exists
$\mathbf{E}_{m}=\\{E_{min_{1}},\dots,E_{min_{n}}\\}$ such that the difference
in arrival times between any two robots is at least $\delta_{t}$ (i.e. the
scheduling problem is feasible).
### V-D Feasibility of QP
In the proposed coordination framework so far $E_{min}$, which is a 1-D value,
is being manipulated to vary the arrival times of robots to the charging
station. However, this can potentially cause a QP infeasibility problem. For
example, a robot might need to use a negative $\eta$ to evade a neighbour’s
arrival time, but at the same time it may need $\eta$ to be positive so as not
to go below $E_{lb}$. Some methods have been proposed to deal with this issue
as in [16] and [17], and we adapt the core idea of the latter. To avoid the
infeasibility problem, each agent carries out coordination only with its
neighbour with the closest arrival time. Moreover, it gives higher priority
for maintaining $E_{min}\geq E_{lb}$ over coordination. This way $\eta$ has to
change to adapt one thing at a time and avoid potential infeasibility (see
Algorithm 1).
Input: $T_{L_{k}}\quad,\forall k\in\mathcal{N}_{i}$
Result: $A_{c}$ and $B_{c}$ for QP constraints
$h_{c_{min}}=h_{0}$
$h_{L_{i}}=E_{min_{i}}-E_{lb}$
while _j in $\mathcal{N}_{i}$_ do
$h_{c_{ij}}=\log\tfrac{T_{L_{i}}-T_{L_{j}}}{\delta}$
if _$h_{c_{ij}} <h_{c_{min}}$ and $D_{j}>\delta$_ then
$h_{c_{min}}=h_{c_{ij}}$
end if
end while
if _$h_{c_{min}} <h_{L_{i}}$_ then
$A_{c}=L_{g}h_{c_{min}}$
$B_{c}=-L_{f}h_{c_{min}}-\alpha(h_{c_{min}})$ …(eqn. 22)
else
$A_{c}=L_{g}h_{L_{i}}$
$B_{c}=-L_{f}h_{L_{i}}-\alpha(h_{L_{i}})$ …(eqn. 25)
end if
Algorithm 1 Coordination algorithm
The final QP is
$\displaystyle\mathbf{u}^{*}=\underset{\mathbf{u}\in\mathbb{R}^{3}}{\text{min}}$
$\displaystyle||\mathbf{u}-\mathbf{u}_{nom}||^{2}$ (35) s.t. $\displaystyle
A\mathbf{u}\geq B$
where
$\begin{split}A&=\begin{bmatrix}A_{e}^{T}\\\
A_{c}^{T}\end{bmatrix}=\begin{bmatrix}L_{g}L_{f}h_{e}&0\\\ \mathbf{0}_{1\times
2}&A_{c}^{T}\end{bmatrix},\\\ B&=\begin{bmatrix}B_{e}\\\
B_{c}\end{bmatrix}=\begin{bmatrix}-L_{f}^{2}h_{e}-(p_{1}+p_{2})\dot{h}_{e}-p_{1}p_{2}h_{e}\\\
B_{c}\end{bmatrix}\end{split}$
while $A_{c}$ and $B_{c}$ are determined from Algorithm (1).
###### Theorem 3
[10, Theorem 3] For a multi robot system of $n$ robots, with dynamics defined
in (9), and each robot applying energy sufficiency, coordination and lower
bound constraints defined in (13), (22) and (25), and provided that the
inequalities (27) and (34) are satisfied, then Algorithm (1) ensures mutual
exclusive use of the charging station.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2: Evolution of values of $E(t)$ and $E_{min}(t)$ for the three
different scenarios under consideration
## VI Results
In this section we present Matlab simulation results of the proposed
framework, aiming to highlight its effectiveness, as well as the utility of
the capacity estimation we propose. We tackled three different scenarios of a
simple patrolling mission for a group of robots spinning around the charging
station at a certain distance with a desired nominal mission speed. In these
scenarios we have $\delta=0.2$\mathrm{m}$$, $R_{0}=9$\mathrm{m}$$ and we want
to achieve $\delta_{t}=35$\mathrm{s}$$. The results are depicted in Figure 2,
and the main parameters used are presented in Table I. For all the cases
discussed, the $\varepsilon$ is calculated using (33).
TABLE I: Values of parameters used in simulation Parameter | $k_{e}$ | $k_{v}$ | $k_{ch}$ | $E_{max}$ | $E_{lb}$
---|---|---|---|---|---
Value | 0.005$\mathrm{V}\mathrm{/}\mathrm{s}$ | 0.015$\mathrm{V}\mathrm{/}\mathrm{m}$ | 0.2$\mathrm{V}\mathrm{/}\mathrm{s}$ | 14.8$\mathrm{V}$ | 12$\mathrm{V}$
### VI-A Base scenario
In this scenario we have a group of five robots that revolve around the
charging station, with an upper bound of average relative velocity
$\tilde{V}=0.15$\mathrm{m}\mathrm{/}\mathrm{s}$$.
Each robot applies a proportional control on the speed to produce a nominal
control input $u_{nom}=-k_{d}(\mathbf{v}-\mathbf{v}_{n})$, where
$\mathbf{v}_{n}$ is a nominal mission velocity and $k_{d}>0$ is a gain. The
value of $u_{nom}$ is the one that goes into the QP (35). To generate
$\mathbf{v}_{n}$ for patrolling, we specify a desired magnitude
$V_{n}=||\mathbf{v}_{n}||$ then we use potential flow theory to specify the
direction. We calculate a potential function $\phi$ of a source near the
charging station, and of a vortex near the boundary
$\phi=\begin{cases}m\log D\quad,\text{if }\hskip
2.84544ptD<\delta+\Delta_{tol}\\\ \frac{m}{2\pi}\theta_{p},\text{if }\hskip
2.84544ptD>R_{0}+\Delta_{tol}\end{cases}$ (36)
where $m>0$, $\Delta_{tol}>0$, $\theta_{p}=\angle(\mathbf{x}-\mathbf{x}_{c})$
and $\mathbf{v}_{n}=V_{n}\tfrac{\nabla\phi}{||\nabla\phi||}$.
The requirement is to have a $\delta_{t}=35$\mathrm{s}$$. The value of
$\delta_{t_{cr}}$ from (31) is $\delta_{t_{cr}}=36.39$\mathrm{s}$$ for
$\varepsilon=0.24$, and $\tfrac{E_{max}-E_{lb}}{2k_{ch}}=7$\mathrm{V}$$, thus
(34) is satisfied. The evolution of voltages and $E_{min}$ values is depicted
in figures 2(a) and 2(d).
### VI-B Base scenario with wind
Here we add a constant wind field of
$\mathbf{v}_{w}=(0.08,0.08)$\mathrm{m}\mathrm{/}\mathrm{s}$$ and we have an
upper bound $\tilde{V}=0.2$\mathrm{m}\mathrm{/}\mathrm{s}$$. In this case
$\delta_{t_{cr}}=31.9$\mathrm{s}$<\delta_{t}$ for $\varepsilon=0.28$. Choosing
to use 5 robots causes $E_{min}$ for some robots to go over
$\tfrac{E_{max}+E_{lb}}{2}$ as shown in Figure 3 (when it should be less if it
abides by the capacity condition (34), according to lemma 2), which is a sign
of overloading the system. Reducing the robots to 4 gives
$\delta_{t_{cr}}=41.1$\mathrm{s}$>\delta_{t}$ for $\varepsilon=0.27$. $E(t)$
and $E_{min}(t)$ are depicted in Figs. 2(b) and 2(e).
### VI-C Base scenario with wind and less $k_{v}$
Here we consider the same previous scenario, but with robots having
$k_{v}=0.0045$. Here for 5 robots $\delta_{t_{cr}}=48.3$\mathrm{s}$$ for
$\varepsilon=0.14$, which alludes to the possibility of adding a robot.
Indeed, for 6 robots $\delta_{t_{cr}}=39.5$\mathrm{s}$>\delta_{t}$ for
$\varepsilon=0.14$. $E(t)$ and $E_{min}(t)$ are depicted in figures 2(c) and
2(f).
Figure 3: Plot of $E_{min}$ when (34) is not satisfied, as described in the
second scenario.
## VII Conclusions
In this paper we propose a CBF based framework for ensuring energy sufficiency
of a multi-robot system while sharing one charging station in a mutually
exclusive manner, given that the robots are affected by air drag and known
constant wind fields.
As a future work we consider extending the current framework to allow for
sharing multiple charging stations, as well as exploring the possibility of
relaxing the assumption of having complete communication graph.
## References
* [1] D. Portugal and R. Rocha, “A survey on multi-robot patrolling algorithms,” in _Doctoral conference on computing, electrical and industrial systems_. Springer, 2011, pp. 139–146.
* [2] J. Cortes, S. Martinez, T. Karatas, and F. Bullo, “Coverage control for mobile sensing networks,” _IEEE Transactions on robotics and Automation_ , vol. 20, no. 2, pp. 243–255, 2004.
* [3] W. Burgard, M. Moors, C. Stachniss, and F. E. Schneider, “Coordinated multi-robot exploration,” _IEEE Transactions on robotics_ , vol. 21, no. 3, pp. 376–386, 2005.
* [4] K. H. Petersen, R. Nagpal, and J. K. Werfel, “Termes: An autonomous robotic system for three-dimensional collective construction,” _Robotics: science and systems VII_ , 2011.
* [5] Z. Sun and J. Reif, “On energy-minimizing paths on terrains for a mobile robot,” in _2003 IEEE International Conference on Robotics and Automation (Cat. No. 03CH37422)_ , vol. 3. IEEE, 2003, pp. 3782–3788.
* [6] S. Slijepcevic and M. Potkonjak, “Power efficient organization of wireless sensor networks,” in _ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No. 01CH37240)_ , vol. 2. IEEE, 2001, pp. 472–476.
* [7] A. Kwok and S. Martinez, “Energy-balancing cooperative strategies for sensor deployment,” in _2007 46th IEEE Conference on Decision and Control_. IEEE, 2007, pp. 6136–6141.
* [8] Y. Ding, W. Luo, and K. Sycara, “Decentralized multiple mobile depots route planning for replenishing persistent surveillance robots,” in _2019 International Symposium on Multi-Robot and Multi-Agent Systems (MRS)_. IEEE, 2019, pp. 23–29.
* [9] G. Notomista, S. F. Ruf, and M. Egerstedt, “Persistification of robotic tasks using control barrier functions,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 2, pp. 758–763, 2018.
* [10] H. Fouad and G. Beltrame, “Energy autonomy for resource-constrained multi robot missions,” in _International Conference on Intelligent Robots and Systems (IROS)_ , 2020.
* [11] A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” in _2019 18th European Control Conference (ECC)_. IEEE, 2019, pp. 3420–3431.
* [12] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 8, pp. 3861–3876, 2016.
* [13] X. Xu, P. Tabuada, J. W. Grizzle, and A. D. Ames, “Robustness of control barrier functions for safety critical control,” _IFAC-PapersOnLine_ , vol. 48, no. 27, pp. 54–61, 2015.
* [14] W. Xiao and C. Belta, “Control barrier functions for systems with high relative degree,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_. IEEE, 2019, pp. 474–479.
* [15] A. Li, L. Wang, P. Pierpaoli, and M. Egerstedt, “Formally correct composition of coordinated behaviors using control barrier certificates,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2018, pp. 3723–3729.
* [16] L. Wang, A. D. Ames, and M. Egerstedt, “Multi-objective compositions for collision-free connectivity maintenance in teams of mobile robots,” in _2016 IEEE 55th Conference on Decision and Control (CDC)_. IEEE, 2016, pp. 2659–2664.
* [17] M. Egerstedt, J. N. Pauli, G. Notomista, and S. Hutchinson, “Robot ecology: Constraint-based control design for long duration autonomy,” _Annual Reviews in Control_ , vol. 46, pp. 1–7, 2018.
## Appendix A Proof of theorem 1
###### Proof:
Since $u\in\mathbb{R}^{2}$ then there should be a value of control input that
satisfies (13) provided that $L_{g}L_{f}h(x)\neq 0$. $L_{g}L_{f}h(x)$ can be
written as
$L_{g}L_{f}h(x)=-kv\frac{(\mathbf{v}-\mathbf{v}_{w})^{T}}{||\mathbf{v}-\mathbf{v}_{w}||}-\frac{k_{c}}{D}\frac{(\mathbf{x}-\mathbf{x}_{c})^{T}}{D}$
(37)
both expressions are basically unit vectors multiplied by an expression or a
factor. If we want the second expression to dominate the first one (so even if
both vectors are opposite, the summation will not be equal to zero), we pick
$k_{c}$ so that the least possible value of $\frac{k_{c}}{D}$ be greater than
$k_{v}$ so $\frac{k_{c}}{R_{0}}>k_{v}\Rightarrow k_{c}>R_{0}k_{v}$, meaning
$L_{g}L_{f}h(x)\neq 0$. ∎
## Appendix B Proof of lemma 1
###### Proof:
To show this, it is useful to point out to the fact that the minimum value of
the quadratic cost function of the QP (5) would be $u=u_{nom}$ in case
$u_{nom}$ doesn’t violate the constraints on the QP. Otherwise, the QP
produces a value of $u$ that abides with the constraint in the equality sense
(produces a control input that renders the constraint as an equality).
Provided that the system starts in the safe set for $h_{e}>0$, then at some
time $T_{b}$ the nominal control action will cause inequality (13) to be
violated, in which case the QP produces a safe control input $u$ that follows
the constraint in the sense of equality. Therefore the produced control input
causes $h_{e}$ to vary in the following way
$\ddot{h}_{e}+(p_{1}+p_{2})\dot{h}_{e}+p_{1}p_{2}h_{e}=0$ (38)
for which the solution is
$h(t)=Ae^{\lambda_{1}t}+Be^{\lambda_{2}t}$ (39)
where $\lambda_{1}=\frac{1}{2}\left(-(p_{1}+p_{2})+|p_{1}-p_{2}|\right)$ (the
dominant
mode),$\lambda_{2}=\frac{1}{2}\left(-(p_{1}+p_{2})-|p_{1}-p_{2}|\right)$, and
the constants $A$ and $B$ are determined from the initial conditions on
$h_{e}$ and $\dot{h}_{e}$ at the time $T_{b}$.
When the robot arrives on the boundary of the charging station at time $t_{a}$
we have
$h_{e}(t_{a})=E(t_{a})-E_{min}=Ae^{\lambda_{1}t_{a}}+Be^{\lambda_{2}t_{a}}$
(40)
thus by properly choosing $p_{1}$ and $p_{2}$ we can gauge how closely the
robot tracks $E_{min}$ on arrival to the charging station. We also point out
to the fact that $h_{e}=0$ only at the boundary of the charging region,
because if $E=E_{min}$ (which is the boundary of our safe set so to speak) we
want this to be at $h_{e}=0$, which happens if
$\log\frac{D}{\delta}=0\Rightarrow D=\delta$.
Since $h_{e}$ is a HOCBF, then any control input satisfying (13) renders the
safe set forward invariant ($h_{e}\geq 0$) so in case if $h_{e}=0$ and being
on the boundary at the same time, then $E-E_{min}=0$. ∎
## Appendix C A method for choosing $k_{c}$
In this discussion we provide a heuristic to choose the value of $k_{c}$ in
the definition of the energy sufficiency CBF. The third term in the definition
of $h_{e}$ signifies the voltage change that a robot needs to go back to the
charging station [9]. The basic idea of choosing $k_{c}$ starts by supposing
that a robot can use a PD controller to go back to the charging station,
starting on the boundary of the operating range (i.e.
$||\mathbf{x}_{0}||=R_{0}$). For more conservatism, we suppose that there is a
headwind with a magnitude of $||\mathbf{v}_{w}||$ opposing the robot’s motion.
Without loss of generality, we suppose that the robot is moving on a line so
the robot’s motion is 1-D, and that the charging station is in the origin. In
this case the system’s model will be
$\begin{split}\dot{x}&=v\\\
\dot{v}&=-k_{p}x-k_{d}v-C_{d}(v-||\mathbf{v}_{w}||)\\\ \end{split}$ (41)
which is a second order ordinary differential equation, the solution of which
is 444The solution has been obtained using symbolic manipulation in Matlab.
$\begin{split}x(t)&=\frac{R_{0}}{G}\left[\left(L_{2}-c\right)e^{-L_{2}t}-\left(L_{1}-c\right)e^{-L_{1}t}\right]\\\
&+\frac{C_{d}||\mathbf{v}_{w}||}{Gk_{p}}\left[L_{1}\left(e^{-L_{2}t}-1\right)-L_{2}\left(e^{-L_{1}t}-1\right)\right]\\\
v(t)&=\frac{C_{d}||\mathbf{v}_{w}||-R_{0}k_{p}}{D}\left(e^{-L_{1}t}-e^{-L_{2}t}\right)\end{split}$
(42)
where $c=k_{d}+C_{d}$,
$G=\sqrt{c^{2}-4k_{p}}$,$L_{1}=\frac{c-G}{2}$,$L_{2}=\frac{c+G}{2}$. We then
approximate the time needed to go from the initial position to a distance
$\delta$ from the center (arriving at the boundary of the charging region) by
taking only the dominant terms in consideration, so the position equation will
be
$x(t)=\left(\frac{R_{0}}{G}(c-L_{1})-\frac{C_{d}||\mathbf{v}_{w}||}{Gk_{p}}L_{2}\right)e^{-L_{1}t}+\frac{C_{d}||\mathbf{v}_{w}||}{Gk_{p}}L_{2}$
(43)
thus the time at which $x(t)=\delta$ is
$\Delta
T=\frac{1}{L_{1}}\log\left(\frac{-C_{d}||\mathbf{v}_{w}||L_{2}+(c-L_{1})k_{p}R_{0}}{-C_{d}||\mathbf{v}_{w}||L_{2}+k_{p}G\delta}\right)$
(44)
then in order to consider the voltage change during this trip back to the
charging station, we can integrate the voltage rate
$\dot{E}=-k_{e}-k_{v}||v-v_{w}||$, however, to increase the conservatism in
the estimate we choose to consider that the robot is moving on a constant
speed equal to the maximum peak speed of $v(t)$, which can be obtained by
differentiating $v(t)$ and getting the time at which the differential is equal
to zero and use the velocity at this time, and we call it $v^{*}$ which is
expressed as
$v^{*}=\frac{-R_{0}k_{p}+C_{d}||\mathbf{v}_{w}||}{G}\left(\left(\frac{L_{2}}{L_{1}}\right)^{\frac{-L_{2}}{L_{2}-L_{1}}}-\left(\frac{L_{2}}{L_{1}}\right)^{\frac{-L_{1}}{L_{2}-L_{1}}}\right)$
(45)
and the voltage change needed becomes
$\begin{split}&\Delta E=\dot{E}\Delta T\\\
&=-\underbrace{\frac{k_{e}+k_{v}(|v^{*}|+||\mathbf{v}_{w}||)}{L_{1}}}_{k_{c}}\log\left(\frac{-C_{d}||\mathbf{v}_{w}||+(c-L_{1})k_{p}GR_{0}}{-C_{d}||\mathbf{v}_{w}||+k_{p}G\delta}\right)\end{split}$
(46)
however the current form of $\Delta E$ does not necessarily satisfy the
condition that $E=E_{min}$ only on the boundary of the charging region, so we
choose
$\Delta E=-k_{c}\log\left(\frac{D}{\delta}\right)$ (47)
## Appendix D Proof of Theorem 2
###### Proof:
Since $\eta\in\Theta=\mathbb{R}$ there exists a control action $\eta$ that
satisfies (22)) (and keeps $\mathcal{C}_{ij}$ invariant), then to show that
$h_{c_{ij}}$ is a ZCBF, we need to ensure that $|T_{L_{i}}-T_{L_{j}}|\neq 0$.
The only chance that this difference can be equal to zero is when one of the
robots enters to the charging region. To show this, consider having two robots
$(i,j)$ applying (22)and without loss of generality suppose that robot $j$
arrives at the charging station, so the difference in arrival times is
$\Delta
T_{L_{ij}}=\frac{E_{i}-E_{min_{i}}}{k_{e}+k_{v}\bar{V}_{i}}-\frac{E_{j}-E_{min_{j}}}{k_{e}+k_{v}\bar{V}_{j}}>0$
(48)
it suffices to show that by the end of the charging process $\Delta T_{ij}<0$
which means that $\Delta T_{ij}=0$ at some point. To show this, we first point
out that when the QP manipulates $\eta$ for coordination, then (22) becomes an
equality, and due to the choice of $\gamma$ in (23), the right hand side will
be equal to zero. Thus
$\eta_{i}\approx-(k_{e}+k_{v}\bar{V}_{i})\left(-\frac{k_{e}+k_{v}V_{j}}{k_{e}+k_{v}\bar{V}_{j}}+\frac{k_{e}+k_{v}V_{i}}{k_{e}+k_{v}\bar{V}_{i}}\right)$
(49)
notice that we neglected the expression
$\frac{k_{v}}{w}\frac{(E-E_{min})(V(t)-V(t-w))}{k_{e}+k_{v}\bar{V}}$ because
it can be significantly less than
$\frac{k_{e}+k_{v}V_{i}}{k_{e}+k_{v}\bar{V}_{i}}$ for most cases of $w$. An
extreme case for (49) can be anticipated if we neglect
$\frac{k_{e}+k_{v}V_{j}}{k_{e}+k_{v}\bar{V}_{j}}$ alltogether and take
$V_{i}=\max(V_{n},\tilde{V})$, so an extreme case for $\eta_{i}$ can be
$\eta_{i}\approx-(k_{e}+k_{v}\max\\{V_{n},\tilde{V}\\})$ (50)
however if $|\eta_{i}|$ is less than $k_{ch}$ (by assumption) it means that
$E_{j}$ increases faster than $E_{min}$ changes for both robots $(i,j)$
(noticing that for $\eta_{j}$ the bracket in (49) will be of reversed sign).
Without loss of generality, for cases where
$k_{ch}\gg(k_{e}+k_{v}\max\\{V_{n},\tilde{V}\\})$ we can consider the change
in $E_{min}$ values is sufficiently slow that they can be considered constant,
so from (48) the value of $T_{L_{j}}$ is increasing in a faster rate than
$T_{L_{i}}$ is decreasing, and at some point
$T_{L_{j}}=\frac{E_{max}-E_{min_{j}}}{k_{e}+k_{v}\bar{V}_{j}}>\frac{E_{i}-E{min_{i}}}{k_{e}+k_{v}\bar{V}_{i}}=T_{L_{i}}$,
which means that $\Delta T_{L_{ij}}=0$ at some point during the recharge.
The proof of the second part is the same as that of proposition III.1 in [15]
and is omitted for brevity. ∎
###### Remark 1
To demonstrate the fact that $k_{ch}$ is sufficiently big, a minimum threshold
on $k_{ch}$ can be obtained by equating $\delta_{t_{cr}}$ with
$\frac{E_{max}-E_{lb}}{2}$ in (34). In other words, by doing so we can get a
minimum acceptable value of $k_{ch}$ so that the capacity constraint (34) is
technically satisfied. Doing so we get
$k_{ch}=(k_{e}+k_{v}\tilde{V})\left[\tfrac{(2(n-1)\Delta
E+\varepsilon)\pm\sqrt{(2(n-1)\Delta E+\varepsilon)^{2}+4(n-1)(\Delta
E-2\varepsilon)\Delta E}}{2(\Delta E-2\varepsilon)}\right]$ (51)
where $\Delta E=E_{max}-E_{lb}$. If we set $\varepsilon=0$ for simplicity, and
for $n=2$ we get $k_{ch}=(1+\sqrt{2})(k_{e}+k_{v}\tilde{V})$. In practice, the
value of $\delta_{t_{cr}}$ is usually significantly larger than
$\frac{E_{max}-E_{lb}}{2k_{ch}}$. Moreover the value of $n$ is bigger than
two, which means that $k_{ch}$ is in practice significantly larger than
$(k_{e}+k_{v}\tilde{V})$.
## Appendix E Proof of lemma 2
###### Proof:
We start by showing that
$\bar{E}_{M}\leq\frac{E_{max}+E_{lb}}{2}$
To do this, we calculate the difference
$\begin{split}&\tfrac{E_{max}+E_{lb}}{2}-\bar{E}_{M}=\tfrac{E_{max}+E_{lb}}{2}-\tfrac{(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})E_{max}+E_{lb}-\delta_{t}(k_{e}+k_{v}\tilde{V})}{\kappa}\end{split}$
(52)
where $\kappa=2+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}}$. This gives
$\tfrac{E_{max}+E_{lb}}{2}-\bar{E}_{M}=\tfrac{(k_{e}+k_{v}\tilde{V})}{2\kappa}\left(2\delta_{t}-\tfrac{E_{max}-E_{min}}{k_{ch}}\right)$
(53)
but due to the choice (32) then $\tfrac{E_{max}+E_{lb}}{2}-\bar{E}_{M}\geq 0$
This sets an upper bound on the value of $E_{min}$ of the most needy agent
(with which it arrives to the charging station). Now the number of arrivals of
a robot in a cycle is
$\zeta_{i}=1+\left\lfloor\frac{\tfrac{(E_{max}-E_{lb})\left(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}}\right)}{k_{e}+k_{v}\tilde{V}}}{\tfrac{(E_{max}-\bar{E}_{M})\left(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}}\right)}{k_{e}+k_{v}\tilde{V}}}\right\rfloor$
(54)
where $\lfloor.\rfloor$ is the floor operator. Here the numerator represents
the time the least needy robot (that defines the cycle) takes to discharge and
recharge once, while the denominator expresses the same thing for the most
needy robot. The ratio represents how many whole sections to which a cycle can
be divided, or in other words, how many small cycles can we fit in the large
one (i.e. how many visits the most needy robot can do in a cycle). Since we
are considering the case where $\bar{V}=\tilde{V}$ for all robots to be more
conservative, then $\zeta_{i}$ can be reduced to
$\zeta_{i}=1+\left\lfloor\frac{E_{max}-E_{lb}}{E_{max}-\bar{E}_{M}}\right\rfloor$
(55)
substituting the upper bound of $\bar{E}_{M}$ in the last equation
$\begin{split}\zeta_{i}&=1+\left\lfloor\frac{E_{max}-E_{lb}}{E_{max}-\tfrac{E_{max}+E_{lb}}{2}}\right\rfloor=1+\left\lfloor\frac{E_{max}-E_{lb}}{E_{max}-E_{lb}}\right\rfloor\\\
&=2\end{split}$ (56)
since this has been considered for the most needy robot, this means that all
other robots, which have less $E_{min}$ values, visit the charging station at
most two times per cycle. Notice that in this proof we used the more critical
value of $\bar{E}_{M}$ at which the robot arrives to the charging station. ∎
## Appendix F Proof of lemma 3
###### Proof:
Since from lemma 2 we know that for any robot the maximum number of visits to
the charging station is at most two, then the maximum number of total visits
to the charging station _within_ one cycle is $2(n-1)$. Moreover, the total
number of spaces between these visits (taking the start and end of the cycle
into account) is $M=2(n-1)+1=2n-1$. We then calculate the amount of available
time between visits $\delta_{av}$ by dividing the cycle length (while still
assuming that all robots operate such that $\bar{V}=\tilde{V}$) and compare
this quantity to $\delta_{t_{cr}}$
$\delta_{av}=\frac{(E_{max}-E_{lb})(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})}{(2n-1)(k_{e}+k_{v}\tilde{V})}$
(57)
To check that $\delta_{av}>\delta_{t_{cr}}$ we calculate the difference
$\begin{split}\delta_{av}-\delta_{t_{cr}}&=\tfrac{(E_{max}-E_{lb})(1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}})}{(k_{e}+k_{v}\tilde{V})}\left(\tfrac{1}{(2n-1)}-\tfrac{1}{1+\kappa(n-1)}\right)\\\
&+\tfrac{\kappa\varepsilon}{k_{e}+k_{v}\tilde{V}}\end{split}$ (58)
but since
$1+1+\kappa(n-1)=2n-1+\tfrac{k_{e}+k_{v}\tilde{V}}{k_{ch}}(n-1)>2n-1$, and
that $\kappa\varepsilon>0$, then $\delta_{av}-\delta_{t_{cr}}>0$, meaning that
the available time is bigger than $\delta_{t_{cr}}$, which means a
$\delta_{t}$ satisfying (34) can be accommodated(since $\delta_{t_{cr}}$ can
be accommodated). ∎
## Appendix G Proof of Theorem 3
###### Proof:
[10, Theorem 3] From Algorithm (1), each robot is either applying the
coordination CBF $h_{c_{ij}}$ or the lower bound CBF $h_{L}$. For the robots
which don’t apply $h_{L}$, the value of the control input $\eta_{i}$ that
respects (22) leads $E_{min_{i}}$ into safe set $\mathcal{C}_{ij}$ with
respect to its neighbour with the closest landing time (by virtue of theorem
2). Each robot applies this to its neighbour with the closest landing time
$\\{(i,j)|j\in\mathcal{N}_{i}\text{ and
}h_{c_{ij}}=\min_{k\in\mathcal{N}_{i}}h_{c_{ik}}\\}$, eventually leading to
$E_{min_{i}}\in\mathcal{C}=\bigcap\limits_{\forall i\neq
j}\mathcal{C}_{ij}\quad,\forall i$. Moreover, since we have established the
feasibility of the scheduling problem in Lemma (3), then we know that the sets
$\mathcal{C}_{ij}$ are nonempty and that a solution exists.
If a robot $i$ is applying the lower bound $h_{L}$, then it can’t push its
arrival time any further. In this case The nearest robot $j$ that applies the
coordination CBF will have a control action $\eta_{j}$ that will lead
$E_{min_{j}}$ to $\mathcal{C}_{ij}$ (noticing that $\mathcal{C}_{ij}$ is non
empty), and then all other robots applying coordination CBF will coordinate in
a pairwise fashion based on the neighbour of closest landing time as discussed
in the previous point. If we add to the previous points the ability of each
robot to arrive at the charging station at almost $E_{min}$ (by virtue of
lemma 1), then mutual exclusive use of the charging station is satisfied. ∎
## Appendix H Estimation of $\varepsilon$ parameter
To motivate the need for $\varepsilon$ , let’s consider a pair of consecutive
robots in the charging schedule which are manipulating $\eta_{i}$ so that
their values of $E_{min}$ stay inside $\mathcal{C}_{ij}$ or on its boundary.
We are interested in the critical case when (22) is violated (when both
$E_{min}$ values start outside $\mathcal{C}_{ij}$ or approach to the boundary
from the inside), in which case the QP produces values of $\eta$ that renders
(22) an equality, hence $\dot{h}_{c_{ij}}=-\alpha(h_{c_{ij}})$, which reaches
a steady state in finite time [15], i.e. $\dot{h}_{c_{ij}}=0$. Thus $\eta$ in
(22) changes such that $\dot{h}_{c_{ij}}=0$ after reaching the steady state.
Supposing that both robots operate on the maximum nominal speed of the mission
$V_{n}$(relative w.r.t. wind) such that they have an equal average relative
speed w.r.t. wind $\bar{V}_{i}=\bar{V}_{j}$, then as $\dot{h}_{c_{ij}}=0$,
from LHS of (22) we have $\eta_{i}=k_{v}(V_{j}-V_{i})$. Suppose that robot $i$
goes back to the charging station and that its speed decreases exponentially
from $V_{n}$ to $V_{f}\ll V_{n}$ with a rate $a$, then
$\dot{E}_{min}=\eta=k_{v}V_{n}(1-e^{-at})$ (59)
which means that as the exponential term decreases, $\eta$ increases and thus
$E_{min}$ increases. In order to be able to estimate this increase, we need to
integrate (59) from the time the robot starts moving towards the charging
station till it arrives.
Considering the most needy robot as this robot $i$555Since we considered
$\bar{E}_{M}=E_{M}+\varepsilon$ for the most needy agent and defined
$\delta_{t_{cr}}$ based on that, then we can say that the arrival time
$T_{end}$ in the most critical case is
$T_{end}=\frac{E_{max}-\bar{E}_{M}}{k_{e}+k_{v}\tilde{V}}=\frac{E_{max}-\tfrac{E_{max}+E_{min}}{2}}{k_{e}+k_{v}\tilde{V}}$
(60)
notice here that for agent $i$ in the above equation, the average speed
expression may include the mission segment (at which the robot operates at a
speed equal to $V_{n}$), and the approach where the speed decreases, so for
conservativeness we suppose that $\bar{V}_{i}=\tilde{V}$, which is the same
thing we did on deriving $\delta_{t_{cr}}$. An example demonstration for the
aforementioned velocities is in Figure 4.
Figure 4: Demonstration of the maximum average relative velocity w.r.t. wind
$\tilde{V}$, the average velocity $\bar{V}$ and the nominal mission velocity
$V_{n}$ for a robot revolving around the charging station.
To estimate the time at which the neediest robot starts approaching the
charging station $T_{start}$, we can approximate it as being the time at which
$h_{e}=0$ for this robot, while supposing it is operating at the boundary of
the operating range $\mathcal{R}$666This approximation is based on the idea
that the safe control input described by the constraints of the QP start
taking over when the states of the system are close to the boundary of the
safe set ($h_{e}=\epsilon$, where $\epsilon\ll$).. This means
$E(T_{start})-E_{min}(T_{start})-k_{c}\log\frac{R_{0}}{\delta}=0$ (61)
we can take $E(t)=E_{max}-(k_{e}+k_{v}\tilde{V})(t-t_{0})$ where $t_{0}=0$
(considering the first cycle) and we can take $E_{min}(T_{start})=E_{M}$.
Substituting in the last equation we get
$T_{start}=\frac{E_{max}-E_{M}-k_{c}\log\frac{R_{0}}{\delta}}{k_{e}+k_{v}\tilde{V}}$
(62)
substituting (27) in the last equation we get
$\begin{split}T_{start}&=\frac{\varepsilon}{k_{e}+k_{v}\tilde{V}}\left(1-\frac{1}{1+\kappa(n-1)}\right)\\\
&+\frac{n(E_{max}-E_{lb})-k_{c}(1+\kappa(n-1))\log\tfrac{R_{0}}{\delta}}{(1+\kappa(n-1))(k_{e}+k_{v}\tilde{V})}\end{split}$
(63)
Supposing that the robot decreases its velocity from $V_{n}$ to $v_{f}$ in an
amount of time equal to $T_{end}-T_{start}$, then
$\begin{split}v_{f}=V_{n}e^{-a(T_{end}-T_{start})}\end{split}$ (64)
and
$a=-\frac{1}{T_{end}-T_{start}}\log\frac{v_{f}}{V_{n}}$ (65)
Now in order to calculate the increase in $E_{min}$
$\begin{split}\varepsilon&=k_{v}V_{n}\int_{0}^{T_{end}-T_{start}}(1-e^{-at})dt\\\
&=k_{v}V_{n}\left[t+\frac{1}{a}e^{-at}\right]_{0}^{T_{end}-T_{start}}\\\
&=k_{v}V_{n}\left[T_{end}-T_{start}+\frac{1}{a}\left(\frac{v_{f}}{V_{n}}-1\right)\right]\\\
&=k_{v}V_{n}\underbrace{\left[1+\frac{1}{\log\frac{V_{n}}{v_{f}}}\left(\frac{v_{f}}{V_{n}}-1\right)\right]}_{\Gamma}(T_{end}-T_{start})\end{split}$
(66)
the smaller the choice of $v_{f}$, the closer $\Gamma$ approaches one, and the
more conservative the estimate of $\varepsilon$ will be. Substituting (60) and
(63) in the above equation we get
$\varepsilon=\frac{\Gamma
k_{v}V_{n}\left(T_{end}-\tfrac{n(E_{max}-E_{lb})-k_{c}(1+\kappa(n-1)\log\tfrac{R_{0}}{\delta}}{(k_{e}+k_{v}\tilde{V})(1+\kappa(n-1)}\right)}{1+\tfrac{\Gamma
k_{v}V_{n}}{k_{e}+k_{v}\tilde{V}}\left(1-\tfrac{1}{1+\kappa(n-1)}\right)}$
(67)
|
# An empirical evaluation of active inference in multi-armed bandits
Dimitrije Marković111These authors contributed equally dimitrije.markovic@tu-
dresden.de Hrvoje Stojić222These authors contributed equally Sarah Schwöbel
Stefan J. Kiebel Faculty of Psychology, Technische Universität Dresden, 01062
Dresden, Germany Centre for Tactile Internet with Human-in-the-Loop (CeTI),
Technische Universität Dresden, 01062 Dresden, Germany Max Planck UCL Centre
for Computational Psychiatry and Ageing Research, University College London,
10-12 Russell Square, London, WC1B 5EH, United Kingdom Secondmind, 72 Hills
Rd, Cambridge, CB2 1LA, United Kingdom
###### Abstract
A key feature of sequential decision making under uncertainty is a need to
balance between exploiting–choosing the best action according to the current
knowledge, and exploring–obtaining information about values of other actions.
The multi-armed bandit problem, a classical task that captures this trade-off,
served as a vehicle in machine learning for developing bandit algorithms that
proved to be useful in numerous industrial applications. The active inference
framework, an approach to sequential decision making recently developed in
neuroscience for understanding human and animal behaviour, is distinguished by
its sophisticated strategy for resolving the exploration-exploitation trade-
off. This makes active inference an exciting alternative to already
established bandit algorithms. Here we derive an efficient and scalable
approximate active inference algorithm and compare it to two state-of-the-art
bandit algorithms: Bayesian upper confidence bound and optimistic Thompson
sampling. This comparison is done on two types of bandit problems: a
stationary and a dynamic switching bandit. Our empirical evaluation shows that
the active inference algorithm does not produce efficient long-term behaviour
in stationary bandits. However, in the more challenging switching bandit
problem active inference performs substantially better than the two state-of-
the-art bandit algorithms. The results open exciting venues for further
research in theoretical and applied machine learning, as well as lend
additional credibility to active inference as a general framework for studying
human and animal behaviour.
###### keywords:
Decision making, Bayesian inference, Multi-armed bandits, Active Inference,
Upper confidence bound, Thompson sampling
## 1 Introduction
When we are repeatedly deciding between alternative courses of action – about
whose outcomes we are uncertain – we have to strike a trade-off between
exploration and exploitation. Do we exploit and choose an option that we
currently expect to be the best, or do we sample more options with uncertain
outcomes in order to learn about them, and potentially find a better option?
This trade-off is one of the fundamental problems of sequential decision
making and it has been extensively studied both in the context of neuroscience
[1, 2, 3] as well as machine learning [4, 5, 6, 7]. Here we propose active
inference – an approach to sequential decision making developed recently in
neuroscience [8, 9, 10, 11, 12] – as an attractive alternative to established
algorithms in machine learning. Although the exploration-exploitation trade-
off has been described and analysed within the active inference framework [13,
14, 15], the focus was on explaining animal and human behaviour rather than
the algorithm performance on a given problem. What is lacking for a convincing
machine learning application is the evaluation on multi-armed bandit problems
[4], a set of standard problems that isolate the exploration-exploitation
trade-off, thereby enabling a focus on best possible performance and the
comparison to state-of-the-art bandit algorithms from machine learning.
Conversely, these analyses will also feed back into neuroscience research,
giving rational foundations to active inference explanations of animal and
human behaviour.
When investigating human and animal behaviour in stochastic (uncertain)
environments, it has become increasingly fruitful to model and describe
behaviour based on principles of Bayesian inference [16, 17, 18], both when
describing perception, and decision making and planning [19]. The approach to
describing sequential decision making and planning as probabilistic inference
is jointly integrated within active inference [8, 9, 10, 11, 12, 15], a
mathematical framework for solving partially observable Markov decision
processes, derived from the general self-organising principle for biological
systems – the free energy principle [20, 21]. Recent work has demonstrated
that different types of exploratory behaviour – directed and random
exploration – naturally emerge within active inference [22]. This makes active
inference a useful approach for modelling how animals and humans resolve the
exploration-exploitation trade-off, but also points at its potential
usefulness for bandit and reinforcement learning problems in machine learning
where the exploration-exploitation trade-off plays a prominent role [4, 23].
Active inference in its initial form was developed for small state spaces and
toy problems without consideration for applications to typical machine
learning problems. This has recently changed and various scalable solutions
have been proposed [24, 25], in addition to complex sequential policy
optimisation that involves sophisticated (deep tree) searches [26, 27].
Therefore, to make the active inference approach practical and scalable to
bandit problems typically used in machine learning, we introduce here an
approximate active inference (A-AI) algorithm.
Here we examine how well the exact and A-AI algorithms perform in multi-armed-
bandit problems that are traditionally used as benchmarks in the research on
the exploration-exploitation trade-off [4]. Although originally formulated for
improving medical trials [28], multi-armed bandits have become an essential
tool for studying human learning and decision making early on [29], and later
on attracted the attention of statisticians [30, 31] and machine learning
researchers [4] for studying the nature of sequential decision making more
generally. We consider two types of bandit problems in our empirical
evaluation: a stationary bandit as a classical machine learning problem [31,
6, 7, 32] and a switching bandit commonly used in neuroscience [33, 34, 35,
36, 37, 38]. This will make the presented results directly relevant not only
for the machine learning community, but also for learning and decision making
studies in neuroscience, which are often utilising the active inference
framework for a wide range of research questions.
Using these two types of bandit problems we empirically compare the active
inference algorithm to two state-of-the-art bandit algorithms from machine
learning: a variant of the upper confidence bound (UCB) [6] algorithm – the
Bayesian UCB algorithm [39, 32, 7] – and a variant of Thompson sampling –
optimistic Thompson sampling [40]. Both types of algorithms keep track of
uncertainty about the values of actions, in the form of posterior beliefs
about reward probabilities, and leverage these to balance between exploration
and exploitation, albeit in a different way. These two algorithms reach state-
of-the-art performance on various types of stationary bandit problems [6, 5,
39, 32], achieving regret (the difference between actual and optimal
performance) that is close to the best possible logarithmic regret [31]. In
switching bandits, learning is more complex, but once this is properly
accounted for, both the optimistic Thompson sampling and Bayesian UCB exhibit
the state-of-the-art performance [41, 42, 43, 40, 44].
We use a Bayesian approach to the bandit problem, also known as Bayesian
bandits [45], for all algorithms – active inference, Bayesian UCB and
optimistic Thompson sampling. The Bayesian treatment allows us to keep the
learning rules equivalent, thus facilitating the comparison of different
action selection strategies. In other words, belief updating and learning of
the hidden reward probabilities exclusively rests on the learning rules
derived from an (approximate) inference scheme, and are independent on the
specific action selection principle [40]. Furthermore, learning algorithms
derived from principles of Bayesian inference can be made domain-agnostic and
fully adaptive to a wide range of unknown properties of the underlying bandit
dynamics, such as the frequency of changes of choice-reward contingencies.
Therefore, we use the same inference scheme for all algorithms – variational
surprise minimisation learning (SMiLE), an algorithm inspired by recent work
in the field of human and animal decision making in changing environments [46,
47]. The variational SMiLE algorithm corresponds to online Bayesian inference
modulated by surprise, which can be expressed in terms of simple delta-like
learning rules operating on the sufficient statistics of posterior beliefs.
In what follows, we will first introduce in detail the two types of bandit
problems we focus on: the stationary and the dynamic bandit problem. We first
describe each bandit problem formally in an abstract way and then specify the
particular instantiation we use in our computational experiments. We will
constrain ourselves to a well-studied version of bandits, the so-called
Bernoulli bandits. For Bernoulli bandits, choice outcomes are drawn from an
arm-specific Bernoulli distribution. Bernoulli bandits together with Gaussian
bandits are the most commonly studied variant of multi armed bandits, both in
theoretical and applied machine learning [5, 40, 48, 32] and experimental
cognitive neuroscience [36, 49, 38]. This is followed by an introduction of
three algorithms: we start with the derivation of the learning rules based on
variational SMiLE, and introduce different action selection algorithms.
Importantly, in active inference we will derive an approximate action
selection scheme comparable in form to the well known UCB algorithm. Finally,
we empirically evaluate the performance of different algorithms, and discuss
the implications of the results for the fields of machine learning and
cognitive neuroscience.
## 2 The multi-armed bandit problem
The bandit problem is a sequential game between an agent and an environment
[4]. The game is played in a fixed number of rounds (a horizon), where in each
round the agent chooses an action (commonly referred to as a bandit arm). In
response to the action, the environment delivers an outcome (e.g. a reward,
punishment, or null). The goal of the agent is to develop a policy that
allocates choices so as to maximise cumulative reward over all rounds. Here,
we will be concerned with a bandit problem where the agent chooses between
multiple arms (actions), a so-called multi-armed bandit (MAB). A well-studied
canonical example is the stochastic stationary bandit, where rewards are drawn
from arm-specific and fixed (stationary) probability distributions [50].
Here, the exploration-exploitation trade-off stems from the uncertainty of the
agent about how the environment is delivering the rewards, and from the fact
that the agent observes outcomes only for the chosen arms, that is, it has
only incomplete information about the environment. Hence, the agent obtains
not only rewards from outcomes but also learns about the environment by
observing the relation between an action and its outcome. Naturally, more
information can be obtained from arms that have been tried fewer times, thus
creating a dilemma between obtaining information, about an unknown reward
probability of an arm, or trying to obtain a reward from a familiar arm.
Importantly, in bandit problems there is no need to plan ahead because
available choices and rewards in the next run are not affected by current
choices 333Note that this type of dependence between current and future choice
sets, or rewards, would convert the bandit problem into a reinforcement
learning problem. It makes the exploration-exploitation trade-off more complex
and optimal solutions cannot be derived beyond trivial problems.. The lack of
need for planning simplifies the problem substantially and puts a focus on the
exploration-exploitation trade-off, making the bandit problem a standard test-
bed for any algorithm that purports to address the trade-off [51].
Bandit problems were theoretically developed largely in statistics and machine
learning, usually focusing on the canonical stationary bandit problem [4, 31,
50, 6, 39, 32]. However, they also play an important role in cognitive
neuroscience and psychology, where they have been applied in a wide range of
experimental paradigms, investigating human learning and decision-making
rather than optimal performance. Here dynamic or non-stationary variants have
been used more often, as relevant changes in choice-reward contingencies in
everyday environments of humans and other animals are typically hidden and
stochastic [52, 53, 36, 3, 38]. We focus on a switching bandit, a particularly
popular variant of the dynamic bandit where contingencies change periodically
and stay fixed for some time between switches [33, 34, 35, 36, 37, 38]. The
canonical stationary bandit has been influential in cognitive neuroscience and
psychology as well [49, 54, 55, 56], in particular when combined with side
information or context to investigate structure or function learning [57, 58,
59, 60].
Note that many experimental tasks, even if not explicitly referred to as
bandit problems, can be in fact reformulated as an equivalent bandit problem.
The often used reversal learning task [33], for example, corresponds to a
dynamic switching two-armed bandit [61], and the popular go/no-go task can be
expressed as a four-armed stationary bandit [62], as another example.
Furthermore, various variants of the well-established multi-stage task [63]
can be mapped to a multi-armed bandit problem, where the choice of arm
corresponds to a specific sequence of choices in the task [64].
In summary, we will perform a comparative analysis on two types of bandit
problems: stationary stochastic and switching bandit. In this section, we
first describe each bandit problem formally in an abstract way and then
specify the particular instantiations we use in our computational experiments.
### 2.1 Stationary stochastic bandit
A stationary stochastic bandit with finitely many arms is defined as follows:
in each round $t\in\mathinner{\left\\{1,\ldots,T\right\\}}$ the agent chooses
an arm or action $k$ from a finite set of $K$ arms, and the environment then
reveals an outcome $o_{t}$ (e.g. reward or punishment). The stochasticity of
the bandit implies that outcomes $o_{t}$ are i.i.d. random variables drawn
from a probability distribution $o_{t}\sim p(o_{t}|\vec{\theta},a_{t})$. In
Bernoulli bandits, these are draws specifically from a Bernoulli distribution
for which outcomes are binary, that is,
$o_{t}\in\mathinner{\left\\{0,1\right\\}}$, where each arm $k$ has a reward
probability $\theta_{k}$ that parametrises the Bernoulli distribution. Hence,
we can express the observation likelihood as
$p\mathinner{\left(o_{t}|\vec{\theta},a_{t}=k\right)}=\theta_{k}^{o_{t}}\left(1-\theta_{k}\right)^{1-o_{t}}$
(1)
where $a_{t}$ denotes the chosen arm on trial $t$. In stationary bandits
reward probabilities of individual arms $\theta_{k}$ are fixed for all trials
$t$. We use $k^{*}$ to denote an optimal arm associated with the maximal
expected reward $\theta_{k^{*}}$.
In our computational experiments we follow a setup that has been used in
previous investigations of stationary stochastic bandits [5]: We consider the
variant of the problem in which all but the best arm $k^{*}$ have the same
reward probability $\theta_{k}=p=\frac{1}{2},\forall
k\in\\{1,\ldots,K\\}\setminus\\{k^{*}\\}$. The probability of the best arm is
set to $\theta_{k^{*}}=p+\epsilon$, where $0<\epsilon<\frac{1}{2}$. The number
of arms $K$ and the mean outcome difference $\epsilon$ modulate the task
difficulty. The more arms and the lower the reliability, the more difficult is
the problem. To understand how task difficulty influences the performance of
different action selection algorithms, in the experiments we systematically
vary $K\in\\{10,20,40,80\\}$ and $\epsilon\in\\{0.05,0.10,0.20\\}$ steps. Note
that the larger number of arms ($K>10$) is a standard setting in machine
learning benchmarks, as many industrial applications of multi-armed bandits
contain a large number of options [50]. In contrast, in experimental cognitive
neuroscience one typically considers only a small number of options (e.g. two
or three), to reduce the task complexity, thus, the training time and the
experiment duration. Interestingly, when humans are exposed to a large number
of options it appears that they are a priori discounting a number of options,
thus simplifying the tasks for themselves. The exact neuronal and
computational mechanisms of option discounting in complex problems are still a
topic of extensive research [65, 66, 67, 54] and go beyond the the scope of
this paper.
### 2.2 Switching bandit
A switching bandit is a dynamic multi-armed bandit, which, as the stationary
bandit, is characterised by a set of $K$ arms, where each arm
$k\in\mathinner{\left\\{1,\ldots,K\right\\}}$ is associated with an i.i.d.
random variable $o_{t}$ at a given time step
$t\in\mathinner{\left\\{1,\ldots,T\right\\}}$. However, in contrast to the
stationary bandit problem, outcomes are drawn from a time-dependent Bernoulli
probability distribution
$p(o_{t}|\vec{\theta}_{t},a_{t}=k)=\theta_{t,k}^{o_{t}}\mathinner{\left(1-\theta_{t,k}\right)}^{1-o_{t}}.$
(2)
We use $k_{t}^{*}$ to denote the optimal arm associated with the maximal
expected reward $\theta_{t,k^{*}_{t}}$ at trial $t$; hence,
$k_{t}^{*}=\operatorname*{arg\,max}_{k}\theta_{t,k}$.
In the switching bandit [68, 69] the reward probability $\theta_{t,k}$ changes
suddenly but is otherwise constant. Here we use the same reward probability
structure as in the stationary bandits, but change the optimal arm $k_{t}^{*}$
with probability $\rho$ as follows
$\begin{split}j&\sim
p(j_{t})=\rho^{j_{t}}\mathinner{\left(1-\rho\right)}^{1-j_{t}}\\\
k_{t}^{*}&\sim\left\\{\begin{array}[]{cc}\delta_{k^{*}_{t-1},k_{t}^{*}},&\textrm{
if }j_{t}=0,\\\ \frac{1-\delta_{k^{*}_{t-1},k_{t}^{*}}}{K-1},&\textrm{ if
}j_{t}=1,\end{array}\right.\end{split}$ (3)
where $\delta_{i,j}$ denotes the Kronecker delta, and $j_{t}$ denotes an
auxiliary Bernoulli random variable representing the presence or absence of a
switch on trial $t$. The optimal arm is always associated with the same reward
probability $\theta_{t,k_{t}^{*}}=p+\epsilon$ and the probability of all other
arms is set to the same value $\theta_{t,k}=p=\frac{1}{2},\forall k\neq
k_{t}^{*}$. In the experiments with the switching bandit problem we
systematically vary $K\in\\{10,20,40,80\\}$ and
$\rho\in\\{0.005,0.01,0.02,0.04\\}$, and $\epsilon\in\\{0.05,0.10,0.20\\}$
sets.
In addition, we will consider the possibility that the task difficulty changes
over time. Specifically, we will consider a setup in which the mean outcome
difference $\epsilon$ is not fixed, and changes over time. We obtain an
effective non-stationary $\epsilon$ by introducing a time evolution of the
reward probabilities $\theta_{t,k}$. At each switch ($j_{t}=1$) point, we
generate the reward probabilities from a uniform distribution. Hence, the
dynamics of the switching bandit with non-stationary difficulty can be
expressed with the following transition probabilities
$\begin{split}p(\theta_{t,k}|\theta_{t-1,k},j_{t})&=\left\\{\begin{array}[]{cc}\delta(\theta_{t,k}-\theta_{t-1,k}),&\textrm{for
}j_{t}=0,\\\ \mathcal{B}e\mathinner{\left(1,1\right)},&\textrm{for
}j_{t}=1,\end{array}\right.\end{split}$ (4)
where $\delta(x)$ denotes Dirac’s delta function, and
$\mathcal{B}e\mathinner{\left(1,1\right)}$ a uniform distribution on
$\mathinner{\left[0,1\right]}$ interval, expressed as the special case of a
Beta distribution.
### 2.3 Evaluating performance in bandit problems
A standard approach to evaluate the performance of different decision making
algorithms in bandit problems is regret analysis [4, 70], and we will
therefore use it here as a primary measure. Regret is typically defined as an
external measure of performance which computes a cumulative expected loss of
an algorithm relative to an oracle which knows the ground truth and always
selects the optimal arm $k*$. If we define the cumulative expected reward of
an agent, up to trial $T$, that chose arm $a_{t}$ on trial $t$ as
$\sum_{t=1}^{T}\theta_{t,a_{t}}$ then the (external) cumulative regret is
defined as
$R_{T}=T\theta_{t,k^{*}}-\sum_{t=1}^{T}\theta_{t,a_{t}}.$ (5)
The cumulative regret can also be viewed as a retrospective loss, which an
agent playing the bandit game can estimate after it learns which arm was
optimal. This definition makes sense for stationary stochastic bandits and in
the limit of $T\rightarrow\infty$. In practice, the cumulative regret $R_{T}$
of a specific agent playing the game will be a function of the sequence of
observed outcomes $o_{1\mathrel{\mathop{\mathchar 58\relax}}T}$, the sequence
of chosen arms $a_{1\mathrel{\mathop{\mathchar 58\relax}}T}$, and a selection
strategy of the given agent.
We additionally introduce a regret rate measure, a time average of the
cumulative regret
$\tilde{R}_{T}=\frac{1}{T}R_{T}=\theta_{k^{*}}-\frac{1}{T}\sum_{t=1}^{T}\theta_{t,a_{t}}.$
(6)
In the case of stationary bandits a decision making algorithm is considered
consistent if $\lim_{T\rightarrow\infty}\tilde{R}_{T}=0$ and asymptotically
efficient if its cumulative regret approaches the following lower bound as
$T\rightarrow\infty$ [31]
$\begin{split}\tilde{R}_{T}&\geq\underline{R}_{T}\\\
\underline{R}_{T}&=\ln(T)\sum_{i\neq
k^{*}}\frac{\theta_{k^{*}}-\theta_{i}}{D_{KL}\mathinner{\left(p_{\vec{\theta}}(o_{t}|i)||p_{\vec{\theta}}(o_{t}|k^{*})\right)}}+const.\equiv\omega(K,\epsilon)\ln
T+const.\end{split}$ (7)
In our case of Bernoulli bandits and specifically structured reward
probabilities (see Stationary stochastic bandit subsection) the Kullback-
Leibler divergence between outcome likelihoods of any arm $i\neq k^{*}$ and
the arm $k^{*}$ associated with highest reward probability becomes
$D_{KL}\mathinner{\left(p_{\vec{\theta}}(o_{t}|k^{*})||p_{\vec{\theta}}(o_{t}|i)\right)}=-\frac{1}{2}\ln\mathinner{\left(1-4\epsilon^{2}\right)}\approx
2\epsilon^{2}.$ (8)
Hence, the lower bound to the cumulative regret becomes approximately
$\underline{R}_{T}=2\epsilon\frac{K-1}{\ln\mathinner{\left(1+4\epsilon^{2}\right)}}\ln
T\approx\frac{K-1}{2\epsilon}\ln T.$ (9)
In addition, we can define an upper bound in terms of a random choice
algorithm, which selects any arm with same probability on every trial. In the
case of random and uniform action selection the cumulative regret becomes
$\bar{R}_{T}=T\epsilon\frac{K-1}{K}$ (10)
Note that the cumulative regret is an external quantity not accessible to an
agent, which has uncertain beliefs about the reward probabilities of different
arms. Although, in stationary bandits the cumulative regret can reveal how
efficient an algorithm is in accumulating reward in the long term, it tells us
little about how efficient an algorithm is in reducing regret in the short-
term. This short-term efficiency is particularly important for dynamic bandits
as an agent has to switch constantly between exploration and exploitation.
Therefore, to investigate short-term efficiency of the algorithm, specifically
in the dynamic context, we will analyse the regret rate, instead of the
commonly used cumulative regret (see [71]).
## 3 Algorithms
Bandit algorithms can be thought of as consisting of two parts: (i) a learning
rule that estimates action values, and (ii) an action-selection strategy that
uses the estimates to choose actions and effectively balance between
exploration and exploitation. As described in the previous section, for the
canonical stationary problem a good bandit algorithm achieves a regret that
scales sub-linearly with the number of rounds (see Eq. 9). Intuitively, this
means that the algorithm should be reducing exploration and allocating more
choices over time to arms with high expected value. The relevant question is
how to reduce exploration concretely? Naturally, this is a fine balancing act:
reducing exploration too quickly would potentially result in false beliefs
about the best arm, hence repeatedly choosing sub-optimal arms and
accumulating regret. In contrast, reducing exploration too slowly would result
in wasting too many rounds exploring sub-optimal arms and again accumulating
regret.
For comparison with the algorithm based on active inference, we focus on two
popular classes of bandit algorithms that are known to hit the right balance:
the (Bayesian) upper confidence bound (B-UCB) [6, 39] and (optimistic)
Thompson sampling (O-TS) [5, 28, 32, 71] algorithms. Our aim is not to
exhaustively test bandit algorithms, but to provide a proof-of-concept and
evaluate whether active inference based algorithms are viable bandit
algorithms. Hence, we have to necessarily ignore a multitude of other bandit
algorithms that would also be interesting competitors, but are in our
judgement less popular. For example, there are other interesting information-
directed algorithms for the stationary case [72, 73], or algorithms that are
more finely tuned for the switching bandits [74, 75, 76]. Note that
$\epsilon$-greedy or Softmax action-selection strategies [23], frequently used
in reinforcement learning, have fixed exploration, and consequently poor
regret performance in open ended bandit problems (i.e. problems with an
unknown time horizon). There are variants of these strategies where
exploration parameters, $\epsilon$ in $\epsilon$-greedy and $\tau$ in Softmax,
are reduced with specific schedules [6]. However, choosing a schedule is based
on heuristics and parameters are difficult to tune. Hence, we do not include
these types of strategies in our comparisons.
In what follows we decompose active inference and the other two bandit
algorithms into two components: the learning rule and the action selection
strategy. We derive learning rules from an approximate Bayesian inference
scheme and keep the rules fixed across action selection strategies, and modify
only the action selection strategy. This setup allows us to have a fair
comparison between active inference and the competing bandit algorithms.
Finally, we will use the same action selection strategies for both stationary
and dynamic bandit problem, and derive parameterised learning rules e that can
account for the presence or absence of changes.
### 3.1 Shared learning rule - variational SMiLe
To derive the belief update equations we start with a hierarchical generative
model described here and apply variational inference to obtain approximate
learning rules. The obtained belief update equations correspond to the
variational surprise minimisation learning (SMiLe) rule [46, 47]. Importantly,
we recover the learning rules for the stationary bandit (see 21) as a special
case when changes are improbable.
Figure 1: Graphical representation of the generative model. Shaded circles
denote observables and transparent circles denote latent random variables.
Note that unlike the outcomes $o_{t}$, which depend on latent states
$\vec{\theta}_{t}$ actions are generated from beliefs (probability
distribution) $p(\vec{\theta}_{t},j_{t})$ about current reward probabilities
$\vec{\theta}_{t}$ and change probability $j_{t}$. Hence, we use dashed red
arrows to underline the causal dependence on beliefs, in contrast to the
causal dependence on latent states marked with black arrows. In practice,
beliefs about latent states are fully described with parameters of a Beta
distribution $(\alpha_{k,t-1},\beta_{k,t-1})$ associated with each arm $k$ and
the explicit knowledge of the change probability $\rho$. Hence, Bayesian
bandit algorithms will differ only in the way they map the beliefs into
actions.
We will express the hierarchical generative model of choice outcomes
$o_{1\mathrel{\mathop{\mathchar
58\relax}}T}=\mathinner{\left(o_{1},\ldots,o_{T}\right)}$ as the following
joint distribution
$p(o_{1\mathrel{\mathop{\mathchar
58\relax}}T},\vec{\theta}_{1\mathrel{\mathop{\mathchar
58\relax}}T},j_{1\mathrel{\mathop{\mathchar
58\relax}}T}|a_{1\mathrel{\mathop{\mathchar
58\relax}}T})=\prod_{t=1}^{T}p(o_{t}|\vec{\theta}_{t},a_{t})p(\vec{\theta}_{t}|\vec{\theta}_{t-1},j_{t})p(j_{t}),$
(11)
where the observation likelihood corresponds to the Bernoulli distribution.
Hence,
$p(o_{t}|\vec{\theta}_{t},a_{t})=\prod_{k=1}^{K}\mathinner{\left[\theta_{t,k}^{o_{t}}\mathinner{\left(1-\theta_{t,k}\right)}^{1-o_{t}}\right]}^{\delta_{a_{t},k}}.$
(12)
If no change ($j_{t}=0$) occurs on a given trial $t$ the reward probabilities
are fixed, $\vec{\theta}_{t}=\vec{\theta}_{t-1}$. Otherwise, if a change
occurs ($j_{t}=1$), a new value is generated for each arm from some prior
distribution $\mathcal{B}e\mathinner{\left(\alpha_{0},\beta_{0}\right)}$.
Formally, we can express this process as
$p(\vec{\theta}_{t}|\vec{\theta}_{t-1},j_{t})=\left\\{\begin{array}[]{ll}\delta\mathinner{\left(\vec{\theta}_{t}-\vec{\theta}_{t-1}\right)},&\textrm{
if }j_{t}=0\\\
\prod_{i=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{0},\beta_{0}\right)}&\textrm{
if }j_{t}=1\end{array}\right.$ (13)
Similarly, the probability that a change in reward probabilities occurs on a
given trial is $\rho$, hence we f express the probability of change occurring
on trial $t$ as the following Bernoulli distribution
$p\mathinner{\left(j_{t}\right)}=\rho^{j_{t}}\mathinner{\left(1-\rho\right)}^{1-j_{t}}$
(14)
The Bayesian approach requires us to specify a prior. The prior over reward
probabilities associated with each arm
$p(\vec{\theta}_{0})=p(\theta_{0,1},\ldots,\theta_{0,K})$ is given as the
product of conjugate priors of the Bernoulli distribution, that is, the Beta
distribution
$p(\vec{\theta}_{0})=\prod_{k=1}^{K}\mathcal{B}e(\alpha_{0,k},\beta_{0,k}),$
(15)
where we initially set the prior to a uniform distribution,
$\alpha_{0,k},\beta_{0,k}=1,\forall\>k$. In Fig. 1 we show the graphical
representation of the generative model.
Hence, given the Bayes rule at time step $t$
$p(\vec{\theta}_{t},j_{t}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t},a_{1\mathrel{\mathop{\mathchar 58\relax}}t})\propto
p(o_{t}|\vec{\theta}_{t},a_{t})p(\vec{\theta}_{t},j_{t}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}),$ (16)
we can express the exact marginal posterior beliefs over reward probabilities
$\vec{\theta}_{t}$ as
$\begin{split}p\mathinner{\left(\vec{\theta}_{t}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t},a_{1\mathrel{\mathop{\mathchar
58\relax}}t}\right)}&=\mathinner{\left(1-\gamma_{t}\right)}p\mathinner{\left(\vec{\theta}_{t}|j_{t}=0,o_{1\mathrel{\mathop{\mathchar
58\relax}}t},a_{1\mathrel{\mathop{\mathchar 58\relax}}t}\right)}\\\
&+\gamma_{t}p\mathinner{\left(\vec{\theta}_{t}|j_{t}=1,o_{t},a_{t}\right)}\end{split}$
(17)
where and $a_{1\mathrel{\mathop{\mathchar 58\relax}}t}$ corresponds to the
sequence of chosen arms, and $\gamma_{t}$ corresponds to the marginal
posterior probability that a change occurred on trial $t$. g We obtain the
posterior change probability as follows
$\begin{split}\gamma_{t}&=\gamma\mathinner{\left(S_{BF}^{t},m\right)}\\\
\gamma(S,m)&=\frac{mS}{1+mS}\\\
S_{BF}^{t}&=\frac{p\mathinner{\left(o_{t}|j_{t}=0,a_{t},o_{t\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)}}{p\mathinner{\left(o_{t}|j_{t}=1,a_{t},o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)}}\\\ m&=\frac{\rho}{1-\rho}\end{split}$ (18)
The exact marginal posterior in Eq. 17 will not belong to the Beta
distribution family, making the exact inference analytically intractable, as
each iteration of the belief update results in a novel distribution family
with ever increasing complexity. In practice, there are numerous ways one can
perform approximate inference in dynamic Bernoulli bandits [77, 78, 40, 79].
Here we will focus on the method based on variational inference due to its
simplicity and efficiency. Although, more optimal inference methods do exist,
we do not expect them to change the relative performance of different decision
algorithms as for Bayesian bandits we can always use the same (most optimal)
learning rule for all Bayesian decision algorithms.
To obtain the learning rule we constrain the joint posterior to an
approximate, fully factorised, form, expressed as
$p(\vec{\theta}_{t},j_{t}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t},a_{1\mathrel{\mathop{\mathchar 58\relax}}t})\approx
Q(j_{t})\prod_{k=1}^{K}Q(\theta^{k}_{t}).$ (19)
Applying the variational calculus results in the following variational SMiLe
rule (for more details on derivations of the SMiLe rule see [46])
$\begin{split}\alpha_{t,k}&=(1-\gamma_{t})\alpha_{t-1,k}+\gamma_{t}\alpha_{0}+\delta_{a_{t},k}o_{t}\\\
\beta_{t,k}&=(1-\gamma_{t})\alpha_{t-1,k}+\gamma_{t}\alpha_{0}+\delta_{a_{t},k}\mathinner{\left(1-o_{t}\right)}\end{split}$
(20)
for the parameters of the Beta distributed factors
$Q(\theta_{t,k})=\mathcal{B}e\mathinner{\left(\alpha_{t,k},\beta_{t,k}\right)}$.
The categorically distributed change probability $Q(j_{t}=1)=\gamma_{t}$ is
update based on Eq. 18.
Note that for a stationary environment the changes are improbable, hence
$\rho=0$ and consequently $\gamma_{t}=0$ for every $t$. This implies that for
the stationary bandit we recover the following learning rules
$\begin{split}\alpha_{t,k}&=\alpha_{t-1,k}+o_{t}\cdot\delta_{a_{t},k},\\\
\beta_{t,k}&=\beta_{t-1,k}+(1-o_{t})\delta_{a_{t},k},\end{split}$ (21)
that correspond to the exact Bayesian inference over the stationary Bernoulli
bandit problem, as in absence of changes the Beta prior corresponds to a
conjugate prior of a Bernoulli likelihood.
### 3.2 Action selection
#### 3.2.1 Active inference
One view on the exploration-exploitation trade-off is that it can be
formulated as an uncertainty-reduction problem [22], where choices aim to
resolve expected and unexpected uncertainty about hidden properties of the
environment [80]. This leads to casting choice behaviour and planning as a
probabilistic inference problem [8, 9, 10, 11, 12], as expressed by active
inference. Using this approach, different types of exploitative and
exploratory behaviour naturally emerge [22]. In active inference, decision
strategies (behavioural policies) are chosen based on a single optimisation
principle: minimising expected surprisal about observed and future outcomes,
that is, the expected free energy [10]. Formally, we express the expected free
energy of a choice $a$ on trial $t$ as
$\begin{split}G_{t}(a)&=\underbrace{D_{KL}\mathinner{\left(Q(o_{t}|a_{t}=a)||P(o_{t})\right)}}_{\textrm{Risk}}+\underbrace{E_{Q(\vec{\theta})}\left[H\mathinner{\left[o_{t}|\vec{\theta},a_{t}=a\right]}\right]}_{\textrm{Ambiguity}}\\\
&=-\underbrace{E_{Q(o_{t}|a_{t}=a)}\left[\ln
P(o_{t})\right]}_{\textrm{Extrinsic value}}\\\
&\quad\>-\underbrace{E_{Q(o_{t}|a_{t}=a)}\left[D_{KL}\mathinner{\left(Q\mathinner{\left(\vec{\theta},j_{t}|o_{t},a_{t}=a\right)}||Q\mathinner{\left(\vec{\theta},j_{t}\right)}\right)}\right]}_{\textrm{Intrinsic
value/Novelty}}\end{split}$ (22)
where
$Q\mathinner{\left(\vec{\theta},j_{t}\right)}=p\mathinner{\left(\vec{\theta},j_{t}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)}$,
$Q(o_{t}|a_{t})=\int\operatorname{d\\!}\vec{\theta}p(o_{t}|\vec{\theta},a_{t})Q\mathinner{\left(\vec{\theta}\right)}$,
$P(o_{t})$ denotes prior preferences over outcomes,
$H\mathinner{\left[o_{t}|\vec{\theta},a_{t}\right]}$ denotes the conditional
entropy of observation likelihood
$p\mathinner{\left(o_{t}|\vec{\theta},a_{t}\right)}$, and $D_{KL}(p||q)$,
stands for the Kullback-Leibler divergence. Then, a choice $a_{t}$ is made by
selecting the action with the smallest expected free energy444In usual
applications of active inference for understanding human behaviour, rather
than minimising the expected free energy one would sample actions from
posterior beliefs about actions (cf. planning as inference [19, 81]). This
becomes useful when fitting empirical choice behaviour in behavioural
experiments [34, 82].
$a_{t}=\operatorname*{arg\,min}_{a}G_{t}(a),$ (23)
where we consider the simplest form of active inference, as in other bandit
algorithms, one-step-ahead beliefs about actions.
Note that in active inference, the most likely action has dual imperatives,
implicit within the expected free energy acting as the loss function (see the
different decomposition in Eq. 22): The expected free energy can, on one hand,
be decomposed into ambiguity and risk. On the other hand, it can be understood
as a combination of intrinsic and extrinsic value, where intrinsic value
corresponds to the expected information gain, and the extrinsic value to the
expected value. The implicit information gain or uncertainty reduction
pertains to beliefs about the parameters of the likelihood mapping, which has
been construed as novelty [83, 15]. efe Therefore, selecting actions that
minimise the expected free energy dissolves the exploration-exploitation
trade-off, as every selected action tries to maximise the expected value and
the expected information gain at the same time.
To express the expected free energy, $G_{t}(a)$, in terms of beliefs about
arm-specific reward probabilities, we will first constrain the prior
preference to the following Bernoulli distribution
$P(o_{t})=\frac{1}{Z(\lambda)}e^{o_{t}\lambda}e^{-(1-o_{t})\lambda}.$ (24)
In active inference, prior preferences determine whether a particular outcome
is attractive or rewarding. Here we assume that agents prefer outcome
$o_{t}=1$ over outcome $o_{t}=0$. Hence, we specify payoffs or rewards with
prior preferences over outcomes that have an associated precision $\lambda$,
where $\lambda\geq 0$. The precision parameter $\lambda$ determines the
balance between epistemic and pragmatic imperatives. When prior preferences
are very precise, corresponding to large $\lambda$, the agent becomes risk
sensitive and will tend to forgo exploration if the risk (i.e., the divergence
between predicted and preferred outcomes, see Eq. 22) is high. Conversely, a
low lambda corresponds to an agent which is less sensitive to risk and will
engage in exploratory, epistemic behaviour, until it has familiarised itself
with the environment (i.e., the latent reward probabilities h of different
arms).
Given the following expressions for the marginal predictive likelihood,
obtained as,
$\begin{split}Q\mathinner{\left(o_{t}|a_{t}\right)}&=\int\operatorname{d\\!}\vec{\theta}_{t}p\mathinner{\left(o_{t}|\vec{\theta}_{t},a_{t}\right)}Q\mathinner{\left(\vec{\theta}_{t}\right)}=\prod_{k=1}^{K}\mathinner{\left[\mathinner{\left[\tilde{\mu}_{t,k}\right]}^{o_{t}}\mathinner{\left[1-\tilde{\mu}_{t,k}\right]}^{1-o_{t}}\right]}^{\delta_{a_{t},k}}\\\
Q\mathinner{\left(\vec{\theta}\right)}&=p\mathinner{\left(\vec{\theta}_{t}|o_{t-1\mathrel{\mathop{\mathchar
58\relax}}1}\right)}=(1-\rho)\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{t-1,k},\beta_{t-1,k}\right)}+\rho\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{0},\beta_{0}\right)}\\\
\tilde{\mu}_{t,k}&=\mu_{t-1,k}+\rho\mathinner{\left(\frac{1}{2}-\mu_{t-1,k}\right)}\\\
\mu_{t-1,k}&=\frac{\alpha_{t-1,k}}{\nu_{t-1,k}}\\\
\nu_{t-1,k}&=\alpha_{t-1,k}+\beta_{t-1,k}\end{split}$ (25)
we get the following expressions for the expected free energy
$\begin{split}G_{t}(a)&=-2\lambda(1-\rho)\mu_{t-1,a}+\tilde{\mu}_{t,a}\ln\tilde{\mu}_{t,a}+(1-\tilde{\mu}_{t,a})\ln(1-\tilde{\mu}_{t,a})\\\
&-(1-\rho)\mathinner{\left[\mu_{t-1,a}\psi\mathinner{\left(\alpha_{t-1,a}\right)}+\mathinner{\left(1-\mu_{t-1,a}\right)}\psi\mathinner{\left(\beta_{t-1,a}\right)}\right]}\\\
&+(1-\rho)\mathinner{\left[\psi\mathinner{\left(\nu_{t-1,a}\right)}-\frac{1}{\nu_{t-1,a}}\right]}+const.\end{split}$
(26)
More details on how we derive Eq. 26 is available in the Appendix Section A.
If we approximate digamma function as $\psi(x)\approx\ln x-\frac{1}{2x}$ i
(which is valid for $x\gg 1$), and note that for all relevant use cases
$\rho\ll 1$; then by substituting the approximate digamma expression into Eq.
(26) we get the following action selection algorithm
$a_{t}=\operatorname*{arg\,max}_{k}\mathinner{\left[2\lambda\mu_{t-1,k}+\frac{1}{2\nu_{t-1,k}}\right]}.$
(27)
More details on how we arrive at Eq. 26 is available in the Appendix Section
B.
Note that a similar exploration bonus – inversely proportional to the number
of observations – was proposed in the context of Bayesian reinforcement
learning [84] when working with Dirichlet prior and posterior distributions.
We will denote active inference agents which make choices based on the
approximate expected free energy, Eq. 27, with A-AI, and agents which minimise
directly the exact expected free energy, Eq. 23, with G-AI.
#### 3.2.2 Bayesian upper confidence bound
The upper confidence bound (UCB) is a classical action selection strategy for
resolving the exploration-exploitation dilemma [6]. ucb-bernoulli When fine-
tuned for Bernoulli bandits, the action selection strategy can be defined as
$a_{t}=\left\\{\begin{array}[]{cc}\operatorname*{arg\,max}_{k}\left(m_{t,k}+\frac{\ln
t}{n_{t,k}}+\sqrt{\frac{m_{t,k}\ln t}{n_{t,k}}}\right)&\textrm{for }t>K\\\
t&\textrm{otherwise}\end{array},\right.$ (28)
where $m_{t,k}$ is the expected reward of $k$-th arm and $n_{t,k}$ the number
of times the $k$-th arm was selected (see [5] for more details).
However, we consider a more recent variant called Bayesian UCB [39], grounded
in Bayesian bandits. In Bayesian UCB the best arm is selected as the one with
the highest $z$-th percentile of posterior beliefs, where the percentile
increases over time as $z_{t}=1-\frac{1}{t}$. Hence, we can express the action
selection rule as
$a_{t}=\operatorname*{arg\,max}_{k}CDF^{-1}(z_{t},\bar{\alpha}_{t}^{k},\bar{\beta}_{t}^{k})$
(29)
where $CDF(\cdot)$ denotes cumulative distribution function of Beta
distributed posterior beliefs, and the parameters ($\bar{\alpha}_{t}^{k}$,
$\bar{\beta}_{t}^{k}$) denote approximate sufficient statistics of the Beta
distributed prior beliefs on trial $t$. Note that the exact predictive prior
on trial $t$ corresponds to a mixture of two Beta distributions
$p\mathinner{\left(\theta_{t}^{k}|o_{t-1\mathrel{\mathop{\mathchar
58\relax}}1}\right)}=(1-\rho)\mathcal{B}e\mathinner{\left(\alpha_{t-1}^{k},\beta_{t-1}^{k}\right)}+\rho\mathcal{B}e\mathinner{\left(\alpha_{0},\beta_{0}\right)}.$
(30)
As the inverse of a cumulative distribution function of the above mixture
distribution is analytically intractable we will assume the following
approximation
$\begin{split}p\mathinner{\left(\theta_{t}^{k}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)}&\approx\mathcal{B}e\mathinner{\left(\bar{\alpha}_{t}^{k},\bar{\beta}_{t}^{k}\right)}\\\
\bar{\alpha}_{t}^{k}&=(1-\rho)\alpha_{t-1}^{k}+\rho\alpha_{0}\\\
\bar{\beta}_{t}^{k}&=(1-\rho)\beta_{t-1}^{k}+\rho\beta_{0}\end{split}$ (31)
Thus, in the case of the Beta distributed prior beliefs, the inverse
cumulative distribution function corresponds to the inverse incomplete
regularised beta function. Hence, we can write
$CDF^{-1}(z,\alpha,\beta)=I_{z}^{-1}(\alpha,\beta),$ (32)
j where $I_{z}^{-1}(\alpha,\beta)$ corresponds to the solution of the
following equation with respect to $x$
$z=\frac{\Gamma\mathinner{\left(\alpha+\beta\right)}}{\Gamma\mathinner{\left(\alpha\right)}\Gamma\mathinner{\left(\beta\right)}}\int_{0}^{x}u^{\alpha-1}\mathinner{\left(1-u\right)}^{\beta-1}\operatorname{d\\!}u\leavevmode\nobreak\
.$ (33)
#### 3.2.3 Thompson sampling
Thompson sampling is traditionally associated with Bayesian bandits [85, 5,
28], where the action selection is derived from the i.i.d samples from the
posterior beliefs about the reward probability. The standard algorithm
corresponds to
$a_{t}=\operatorname*{arg\,max}_{k}\theta^{*}_{t,k},\qquad\theta^{*}_{t,k}\sim
p\mathinner{\left(\theta_{t,k}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)},$ (34)
where $\theta^{*}_{t,k}$ denotes a single sample from the current beliefs
about reward probabilities associated with the $k$-th arm.
An extension of the standard algorithm, proposed in the context of dynamic
bandits, is called optimistic Thompson sampling [71], defined as
$a_{t}=\operatorname*{arg\,max}_{k}\mathinner{\left[\max\mathinner{\left(\theta^{*}_{t,k},\tilde{\mu}_{t,k}\right)}\right]},\qquad\theta^{*}_{t,k}\sim
p\mathinner{\left(\theta_{t,k}|o_{1\mathrel{\mathop{\mathchar
58\relax}}t-1}\right)},$ (35)
where the expected reward probability at current trial $t$,
$\tilde{\mu}_{t,k}=\mu_{t-1,k}+\rho\mathinner{\left(\frac{1}{2}-\mu_{t-1,k}\right)},$
constrains the minimal accepted value of the sample from the prior, hence
biasing the sampling towards optimistic larger values.
code-and-data-availability
### 3.3 Code and data availability
The code accompanying the paper is available at github.com/dimarkov/aibandits.
The repository contains the implementation of all algorithms and scripts for
execution of the simulations. The folder with jupyter notebooks contains the
scripts used to generate the figures. The results of simulations, which can be
used to reproduce the figures, are available at osf.io/85ek4/. All the
simulations are controlled with a manually set seed and it should be possible
to reproduce the results exactly.
## 4 Results
In what follows, we first examine the performance of active inference based
agents, A-AI (minimising approximated estimate of the expected free energy)
and G-AI (minimising exact expected free energy) in the stationary Bernoulli
bandits. Using the regret rate as performance criterion we analyse the
dependence of agent’s performance on the precision of prior preferences
($\lambda$) parameter and simultaneously verify that our approximation is good
enough. After illustrating the effectiveness of A-AI (Eq. 27), in comparison
to G-AI (Eq. 23), we empirically compare only the A-AI algorithm – now in
terms of the cumulative regret – with agents using the optimistic Thompson
sampling (O-TS; Eq. 35) and Bayesian upper confidence bound (B-UCB; Eq. 29)
algorithms, in the same stationary Bernoulli bandit. k Finally, we provide an
empirical comparison of the algorithms in the case of switching Bernoulli
bandit, both in scenarios with fixed and varying difficulty.
### 4.1 The stationary Bernoulli bandit
The precision parameter $\lambda$ acts as a balancing parameter between
exploitation and exploration (Eq. 27). Hence, it is paramount to understand
how $\lambda$ impacts the performance across different difficulty conditions.
We expect that there will be a $\lambda^{*}(\epsilon,K)$ for which the active
inference algorithm achieves minimal cumulative regret after a fixed number of
trials $T$, for each mean outcome difference $\epsilon$ and each number of
arms $K$. When the AI agent has l weak preferences ($\lambda\rightarrow 0$),
it would engage in exploration for longer, thereby reducing its free energy
(i.e., uncertainty about the likelihood mappings), at the expense of
accumulating reward. Conversely, an AI agent with m strong preferences
($\lambda\rightarrow\infty$) would commit to a particular arm as soon as it
had inferred that this was the arm with highest likelihood of payoffs.
However, the ensuing ‘superstitious’ behaviour would prevent it from finding
the best arm. To illustrate this, in Fig. 2 we report regret rate averages
over a $N=10^{3}$ simulations, and compare the agents using either the
approximate (A-AI) or the exact (G-AI) expected free energy for action
selection. Using the regret rate simplifies the comparison, as unlike
cumulative regret, the regret rate stays on the same range of values
independent of trial number $T$.
Figure 2: Regret rate analysis for active inference based agents in the
stationary Bernoulli bandit. The regret rate $\tilde{R}_{T}$, Eq. 6, for the
approximate (A-AI) and the exact (G-AI) variants of active inference as a
function of the precision over prior preferences $\lambda$. The coloured lines
show numeric estimates obtained as an average over $N=10^{3}$ runs. Different
line styles denote $\tilde{R}_{T}$ values estimated after different numbers of
trials $T$: Dotted lines correspond to $T=10^{2}$, dotted dashed lines to
$T=10^{3}$ and solid lines to $T=10^{4}$, as annotated in the top left plot.
The dashed black line denotes the upper bound on the regret rate corresponding
to the random (RC) agent which gains no information from the choice outcomes.
The vertical doted line (purple) corresponds to $\lambda=0.1$ level, which we
find to be sufficiently close to the minimum or regret rate in a range of
conditions. Each column and row of the plot corresponds to different task
difficulties, characterised by the number of arms $K$, and the mean outcome
difference $\epsilon$, respectively.
differences Surprisingly, the A-AI algorithm achieves slightly lower regret
rate in certain ranges of $\lambda$ values depending on the problem
difficulty. The reason for this is that the approximate information gain used
in A-AI algorithm Eq. 27 is always larger or equal than the exact information
gain Eq. 26. Hence, for sufficiently low values of $\lambda$ (e.g.
$\lambda<0.25$) the behaviour is initially strongly dominated by the
exploratory part of AI algorithms, and both algorithms exhibit similar regret
rates. As we increase the value of $\lambda$ the exploitative part becomes
more dominant in action selection. However, a higher value of $\lambda$ is
required in the A-AI algorithm for the exploitative part to become dominant,
as the approximate information gain is initially larger and converges slower
to zero than the exact information gain. As $\lambda$ becomes sufficiently
large both algorithms become equally bad (action selection start depending
only on the expected value) hence the difference in regret rate disappears
again. Interestingly, this performance differences are not visible in the case
of switching bandits analysed later in this section.
lambda Using a visual inspection of Fig. 2 we find the minimal regret rate –
at the asymptotic limit of large number of trials $T=10^{4}$, see solid lines
in Fig. 2 – is close to the value $\lambda=0.1$ for a range of problem
difficulties555For the hardest considered setting, corresponding to
$\epsilon=0.05$, the minimum is sharp and corresponds to the value
$\lambda=0.06$.. Hence, for the n subsequent between-agent comparisons we
restrict the active inference agents to a fixed precision of prior
preferences, $\lambda=0.1$. As both G-AI (red lines) and A-AI (green lines)
achieve very similar regret rates as a function of precision $\lambda$ and
number of trials $T$, we will only consider the A-AI variant for the between-
agent comparison. We anticipated that even this approximate form of active
inference would outperform bandit algorithms; most notably when considering
short sessions in the stationary scenario: i.e., when exploration gives way to
exploitation after the agent becomes familiar with the payoffs afforded by the
multi-armed options. The reason for this expectation is the exact computation
of the information gain implicit within the expected free energy (see 22).
Next we compare and contrast the cumulative regret, as a function of trial
number $t$, of the A-AI agents with agents based on the optimistic Thompson
sampling (O-TS) and the Bayesian UCB (B-UCB) algorithms (Fig. 3). The dotted
lines mark the corresponding asymptotic limit (see Eq. 9) of the corresponding
problem difficulty ($\epsilon,K$). The asymptotic limit scales as $\ln t$ and
defines long term behaviour of the asymptotically efficient algorithm. Note
that the limit behaviour can be offset by an arbitrary constant to form a
lower bound [31, 5]. For convenience we fix the constant to zero, and show the
asymptotic curve only as a reference for long term behaviour of cumulative
regret for different algorithms.
Figure 3: Between-agent comparison in the stationary Bernoulli bandit.
Comparison of cumulative regret trajectories for the approximate active
inference (A-AI), the optimistic Thompson sampling (O-TS), and Bayesian upper
confidence bound (B-UCB) based agents. For the A-AI based agent the prior
precision is set to $\lambda=0.1$, that is, to the near optimal value for a
range of difficulty conditions. Solid coloured lines denote the ensemble
cumulative regret average and shaded regions (not visible in every subplot)
mark the 95% confidence interval of the mean estimate. All the values are
estimated as ensemble averages over $N=10^{3}$ simulations.
The comparison reveals that the A-AI agent o on average outperforms the bandit
algorithms, but only up until some trial $t$ that depends on the task
difficulty – in the asymptotic limit the regret grows faster than logarithmic
with trial number. For example, for $K=10$, A-AI outperforms bandit algorithms
only up to $T=10^{4}$. The divergence in cumulative regret is driven by a
percentage of the $N=10^{3}$ agents in the ensemble that did not not find the
optimal solution and are over-confident in their estimate of the arm with the
highest reward probability. histogram We illustrate this in Fig. S1 in the
form of histogram of the logarithm of cumulative regret at $T=10^{6}$
estimated over the ensemble of $N=10^{3}$ agents. It might appear surprising,
that the divergence is p more prominent for the smaller number of arms.
However, the reason for this is, that the smaller the number of arms is, the
more chance an agent has to explore each individual arm, for a limited trial
number. Hence, the agent will commit faster to a wrong arm and stay with that
choice longer. Therefore, we found that our initial expectation about the
performance of active inference algorithms is only partially correct. Although
one could set $\lambda$ for any task difficulty in a way that active inference
initially outperforms the alternative algorithms, in the asymptotic limit the
high performance level will not hold. The reason for this can be seen already
in Fig. 2, if one notes that maximal performance (minimal regret rate) depends
both on preference precision $\lambda$ and trial number $T$, for every
$K,\epsilon$ tuple.
execution-times We also timed the execution of all algorithms, to provide an
additional measure of practicality of our approximate A-AI algorithm. All
algorithms use the same learning rule, hence the only difference in execution
time would come from action selection part of the algorithm. Results show that
A-AI obtains the lowest time, on par with classical UCB (see Table 1 in the
appendix). Usual caveats with timing apply, results depend on implementation
details and hardware used, as well as on the specifics of our bandit problem,
hence one should be careful with generalizing from these results.
Although active inference based agents behave poorly in the asymptotic limit,
the fact that they achieve higher performance on a short time scale suggests
that in dynamic environments – if changes occur sufficiently often – one would
get higher performance on average when compared to considered alternatives.
### 4.2 The switching bandit problem
In the case of our switching bandit problem, the change probability $\rho$
acts as an additional difficulty parameter, besides the number of arms $K$ and
the mean outcome difference $\epsilon$. Therefore, for the between-algorithm
comparison we will first keep $\epsilon$ fixed at its medial value,
$\epsilon=0.1$ and vary number of arms in Fig. 4, and then keep the number of
arms fixed at $K=40$ and vary the expected outcome difference in Fig. 5.
lambda2 For the algorithm comparison in switching bandits we fix the precision
parameter $\lambda$, to $\lambda=0.5$, based on a similar visual inspection of
regret rate dependence on $\lambda$ (see Fig. S2). Interestingly, in the case
of switching bandits the minimum of the regret rate stabilises after certain
trial number and is not dependent on $T$, like in stationary case. Note that
in stationary environments small values of $\lambda$ are desirable to achieve
low cumulative regret for large $T$, in switching environments larger values
of $\lambda$ are preferable. q Furthermore, we find that the larger the arm
number $K$ is, the larger would be the preferable $\lambda$ value. However,
here we will not optimise $\lambda$ for different difficulty settings but use
the same value in all examples. For between-algorithm comparison in switching
bandits we will use regret rate, instead of cumulative regret, as a reference
performance measure. The reason for this is that in dynamic environments
cumulative regret increases linearly with trial number $t$, and regret rate
provides visually more accessible gauge of performance differences [71].
In Fig. 4 we illustrate the regret rate for each agent type over the course of
the experiment for a range of different values of change probability $\rho$
and number of arms $K$, and a fixed mean outcome difference $\epsilon=0.1$.
Importantly, when estimating the mean regret rate over an ensemble of
$N=10^{3}$ agents, for each agent
$n\in\mathinner{\left\\{1,\ldots,N\right\\}}$ we simulate a distinct switching
schedule with the same change probability $\rho$. Hence, the average is
performed not only over different choice outcome trajectories but also over
different hidden trajectories of changes. This ensures that comparison is
based on environmental properties, and not on specific realisation of the
environment. We find better performance for the active inference agents
compared to other bandit algorithms in all conditions. However, we observe
that the more difficult the task is (in terms of higher change probability
$\rho$ and larger number of arms $K$) the less pronounced is the performance
advantage of the active inference based agents.
Figure 4: Between-agent comparison in switching Bernoulli bandits with a fixed
mean outcome difference ($\epsilon=0.1$). Comparison of the regret rate of
approximate active inference (A-AI), optimistic Thompson sampling (O-TS), and
Bayesian upper-confidence-bound (B-UCB) based agents in the switching bandit
problem (see Switching bandit subsection). Each column and row of the plot
corresponds to different task difficulties, characterised by the change
probability $\rho$, and the number of arms $K$. For the A-AI agents the prior
precision over outcome preferences is fixed to $\lambda=0.5$. All the values
are estimated as ensemble averages over $N=10^{3}$ simulations, where the
switching schedule is also generated randomly for each agent instance within
the ensemble. The 95% confidence intervals, although plotted, are hardly
visible, implying a statistically robust comparison.
In Fig. 5 we show the regret rate for each agent type, however with a fixed
number of arms, $K=40$, but varying mean outcome difference $\epsilon$. Here,
the picture is very similar, where for increasing task difficulty the A-AI
agent type exhibits a diminishing performance advantage relative to the bandit
algorithms. Importantly, although we present here the regret analysis only up
to $T=5\cdot 10^{3}$, unlike in the stationary bandit problem, the results do
not change after a further increase in the number of trials. When we simulate
longer experiments we find a convergent performance for all algorithms towards
a non-zero regret rate; implying a linear increase in cumulative regret with
trial number $t$.
Figure 5: Between-agent comparison in the switching Bernoulli bandit with a
fixed number of arms ($K=40$). Comparison of the regret rate of approximate
active inference (A-AI), optimistic Thompson sampling (O-TS), and Bayesian
upper-confidence-bound (B-UCB) based agents in the switching bandit (see
Switching bandit subsection). Each column and row of the plot corresponds to
different task difficulties, characterised by the change probability $\rho$,
and the mean outcome difference $\epsilon$. For the A-AI agents the prior
precision over outcome preferences is fixed to $\lambda=0.5$. As in the
previous figure, all the values are estimated as ensemble averages over
$N=10^{3}$ simulations, with instance specific switching schedule within the
ensemble. The 95% confidence intervals, although plotted, are in most cases
not larger then the line thickness.
Finally, we further illustrate the dependence of performance on mean outcome
preference, using the switching bandit with non-stationary task difficulty,
where $\epsilon$ is not fixed but changes stochastically over the course of
experiment (see Switching bandit for more details). r Importantly, in the case
of non-stationary difficulty we will include the G-AI algorithm into
comparison. The regret rate based comparison between A-AI and G-AI algorithms
(shown in Fig. S3) reveals, for the first time, a noticeable difference
between the two algorithms. Furthermore, the G-AI algorithm shows a more
stable minimum of the regret rate as a function of $\lambda$ in a range of
conditions, corresponding to $\lambda=0.25$, suggesting potential benefits of
exact form of active inference over the approximate one. As shown in Fig. 6,
we find an increasing advantage of G-AI (and similarly A-AI algorithm) over
B-UCB and O-TS algorithms (Fig. 4) in more difficult problems – with either
larger number of arms $K$ or larger change probability $\rho$. However, the
opposite is the case for lowered task difficulty; e.g. for $\rho=0.005$ and
$K=10$, where B-UCB achieves higher performance then A-AI algorithm, but is
matched with the G-AI algorithm. Notably, we would expect that for small
number of arm ($K<10$) and slower changing environments ($\rho<0.001$) the
drop in performance of the AI agents becomes even more pronounced, as we are
approaching the stationary limit.
As a final remark, we find it interesting that the B-UCB algorithm
consistently outperforms the O-TS algorithm, in almost all non-stationary
problems we examined. This is in contrast to the previous asymptotic analysis
in the stationary bandit problem, which concluded that Thompson sampling
exhibits better asymptotic scaling than B-UCB [32, 7]. We are not aware of
previous works comparing these two algorithms in the context of the switching
bandit problem. bucb-guess However, the two papers which we found to compare
B-UCB and TS in stationary bandits [7, 32] show similar patterns in cumulative
regret to what we found. For the initial $T=1,000$ trials B-UCB achives lower
regret, and TS outperforms B-UCB only at later stages. Hence, we would infer
from these findings that in the switching case B-UCB achieves a lower regret
rate as changes occur on a shorter time scale, similar to the advantage we
find for the A-AI and G-AI algorithms.
Figure 6: Between-agent comparison in the switching bandit with non-stationary
difficulty. Comparison of the regret rate of exact (G-AI) and approximate
active inference (A-AI), optimistic Thompson sampling (O-TS), and Bayesian
upper-confidence-bound (B-UCB) agents in the switching bandit (see Switching
bandit subsection) when the reward probabilities are sampled from uniform
distribution after every switch. The re-sampling of latent reward
probabilities makes the difficulty of the problem non-stationary, as the
advantage of the best arm over the second best arm changes with time. For the
A-AI agent we fixed the prior precision over outcome preferences to
$\lambda=0.5$, as in the previous examples. However, for the G-AI agent we
fixed the lambda to $\lambda=0.25$ based on a visual inspection of dependency
of regret rate on $\lambda$ shown in Fig. S3. All the values are estimated as
ensemble averages over $N=10^{3}$ simulations, and result in tight confidence
intervals.
## 5 Discussion
In this paper we provide an empirical comparison between active inference, a
Bayesian information-theoretic framework [10], and two state-of-the-art
machine learning algorithms – Bayesian UCB and optimistic Thompson sampling –
in stationary and non-stationary stochastic multi-armed bandits. We introduced
an approximate active inference algorithm, for which our checks on the
stationary bandit problem showed that its performance closely follows that of
the exact version. Hence, we derived an active inference algorithm that is
efficient and easily scalable to high-dimensional problems.
To our surprise, the empirical algorithm comparison in the stationary bandit
problem showed that the active inference algorithm is not asymptotically
efficient – the cumulative regret increased faster than logarithmic in the
limit of large number of trials. The cause for this behaviour seems to be the
fixed prior precision over preferences $\lambda$, which acts as a balancing
parameter between exploration and exploitation. An analysis of how the
performance depends on this parameter showed that parameter values that give
the best performance decrease over time, suggesting that this parameter should
be adaptive and decay over time as the need for exploration decreases.
Attempts to remedy the situation with a simple and widely used decay scheme
were not successful (for example logarithm of time, not reported here).
adaptive-lambda Similarly, introducing a hyper-prior over $\lambda$ and
deriving learning rules for the precision parameter [86, 87] did not result in
the desired asymptotic behaviour. This indicates it is not a simple
relationship and a proper theoretical analysis will be needed to identify
whether such a scheme exists.
In the non-stationary switching bandit problem the active inference algorithm
generally outperformed Bayesian UCB and optimistic Thompson sampling. This
provides evidence that the active inference framework may provide a good
solution for optimisation problems that require continuous adaptation. Active
inference provides the most efficient way of gaining information and this
property of the algorithm pays off in the non-stationary setting. Such dynamic
settings are also relevant in neuroscience, as relevant changes in choice-
reward contingencies are typically hidden and stochastic in everyday
environments of humans and other animals [52, 53, 36, 3, 38]. In contrast to
previous neuroscience research that showed that active inference is a good
description of human learning and decision making [88, 89, 90, 91, 92, 93],
our results on the dynamic switching bandit show that active inference also
performs well in objective sense. Such explanations of cognitive mechanisms
that are grounded in optimal solutions are arguably more plausible [94].
Hence, this result lends additional credibility to active inference as a
generalised framework for understanding human behaviour, not only in the
behavioural experiments inspired by multi-armed bandits [33, 34, 35, 36, 37,
38], but in a range of related investigations of human and animal decision
making in complex dynamic environments under uncertainty [89, 95, 96, 97].
An important next step in examining active inference in the context of multi-
armed bandits is to establish theoretical bounds on the cumulative regret for
the stationary bandit problem. A key part of these theoretical studies will be
to investigate whether it is possible to devise a sound decay scheme for the
$\lambda$ parameter (see Eq. 27), that provably works for all instances of the
canonical stationary bandit. This would lead to the development of new active
inference inspired algorithms which can achieve asymptotic efficiency. These
theoretical bounds would allow us to more rigorously compare active inference
algorithms to the already established bandit algorithms for which regret
bounds are known. Moreover, we would potentially be able to generalise beyond
the settings we have empirically tested here. Future work may also consider an
information-theoretic analysis of active inference, which might be more
appropriate than regret analysis [43]. For example, the Bayesian exploration
bonus previously considered in Bayesian reinforcement learning was analysed
with respect to sample complexity of identifying a good policy [84].
Similarly, in [98] the authors introduced a new measure of regret weighted by
the inverse information gain between actions and outcomes, and provided
expected bounds for this measure for several Bayesian algorithms, such as
Thompson sampling and Bayesian UCB. s Finally, future work should contrast the
active inference framework with alternative approaches that can generate
directed exploration [99, 73].
As optimal behaviour is always defined with respect to a chosen objective
function, a different objective function will lead to different behaviour, and
the appropriateness of the objective function for the specific problem
determines the performance of the algorithm on a given task. In other words,
behaviour is determined not only by the beliefs about the hidden structure
about the states of the world but also by the beliefs about useful preferences
and objectives one should take into account in that environment. Therefore,
although one can consider the sensitivity of the introduced active inference
algorithm on the prior precision over preferences $\lambda$ as a limitation of
the algorithm in comparison to the other two algorithms, we believe that it is
possible to introduce various adaptations to the algorithm to improve
asymptotic behaviour. Fore example, one can consider learning rules for prior
outcome preferences, as illustrated in [87, 95]. This would introduce a way to
adapt an objective function to different environments achieving high
performance in a wide-range of multi-armed bandit problems. Alternatively,
instead of basing action selection on the expected free energy, one can define
a stochastic counterpart, which is estimated based on samples from the
posterior, akin to Thompson sampling. This would enable the algorithm to
better leverage directed and random exploration.
Despite of the poor asymptotic performance in the stationary bandit problem
there are some advantages of active inference over classical bandit
algorithms, both for artificial intelligence and neuroscience. Unlike the
Thompson sampling and UCB algorithms, active inference is easily extendable to
more complex settings where actions affect future states and actions
available. Such settings are usually formalised as a (partially observable)
Markov decision process, which require the combination of adaptive decision
making with complex planning mechanism [25, 24, 27]. In these settings
learning is non-stationary because changes in policy cause a shift in state
value distributions [23]. Given our finding that active inference algorithm
has an advantage in non-stationary settings, it seems promising to apply the
framework to Markov decision processes. Reinforcement learning algorithms is a
popular choice for tackling Markov decision processes, in particular it would
be interesting to compare active inference to Bayesian reinforcement learning
approaches [100, 101, 102].
The generative modelling approach integral to active inference allows several
improvements to the presented algorithm, which also holds for related Bayesian
approaches. For example, we have considered here only one learning algorithm,
variational SMiLE [46], which we have chosen based on its simplicity and
efficiency. A potential drawback of variational SMiLE is that it might not be
optimal (in terms of inference) for t switching bandits or a generic problem
of dynamic bandits (e.g. different mechanisms for generating changes and
different reward distribution). For example, alternative-switching-learning
for switching bandits several candidates come closer to the exact inference
and would likely improve performance [77, 78, 40, 79]. For restless bandits,
which follow a random walk process, recently published alternative efficient
learning algorithms derived from different generative models are likely to
provide a better performance [103, 104]. Employing a good learning algorithm
is especially important in dynamic settings, where exact inference is not
tractable, and the performance of learning rules is tightly coupled to the
overall performance of the algorithm. In practice, one would expect that the
better the generative model and the corresponding approximate inference
algorithm, the better the performance will be on a given multi-armed bandit
problem. Furthermore, one can easily extend the learning algorithms with deep
hierarchical variants, which can infer a wide range of unknown dynamical
properties of the environment [103] and learn higher order temporal statistics
[34, 95].
## 6 Conclusion
We have derived an approximate active inference algorithm, based on a Bayesian
information-theoretic framework recently developed in neuroscience, proposing
it as a novel machine learning algorithm for bandit problems that can compete
with state-of-the-art bandit algorithms. Our empirical evaluation has shown
that the active inference framework can indeed be used to derive a promising
bandit algorithm. We consider the present work as a first step, where two
important next steps are the development of a decay schedule for the outcome
preference precision parameter $\lambda$ and a theoretical regret analysis for
the stationary bandit. The fact that the active inference algorithm achieves
excellent performance in switching bandit problems, commonly used in cognitive
neuroscience, provides rational grounds for using active inference as a
generalised framework for understanding human and animal learning and decision
making.
## 7 Acknowledgements
We thank Karl Friston and Gergely Neu for valuable feedback and constructive
discussions. DM and SS were funded by the German Research Foundation (DFG,
Deutsche Forschungsgemeinschaft), SFB 940/3 - Project ID 178833530, A09,TRR
265/1 - Project ID 402170461, B09 and partially supported by Germany’s
Excellence Strategy – EXC 2050/1 – Project ID 390696704 – Cluster of
Excellence “Centre for Tactile Internet with Human-in-the-Loop” (CeTI) of
Technische Universität Dresden. The Max Planck UCL Centre for Computational
Psychiatry and Ageing Research is funded by the Max Planck Society, Munich,
Germany, URL: https://www.mpg.de/en, grant number: 647070403019.
## References
* [1] R. C. Wilson, E. Bonawitz, V. D. Costa, R. B. Ebitz, Balancing exploration and exploitation with information and randomization, Current Opinion in Behavioral Sciences 38 (2020) 49–56.
* [2] K. Mehlhorn, B. R. Newell, P. M. Todd, M. D. Lee, K. Morgan, V. A. Braithwaite, D. Hausmann, K. Fiedler, C. Gonzalez, Unpacking the exploration–exploitation tradeoff: A synthesis of human and animal literatures., Decision 2 (3) (2015) 191\.
* [3] J. D. Cohen, S. M. McClure, A. J. Yu, Should i stay or should i go? how the human brain manages the trade-off between exploitation and exploration, Philosophical Transactions of the Royal Society B: Biological Sciences 362 (1481) (2007) 933–942.
* [4] T. Lattimore, C. Szepesvári, Bandit algorithms, Cambridge University Press, 2020\.
* [5] O. Chapelle, L. Li, An empirical evaluation of thompson sampling, in: Advances in neural information processing systems, 2011, pp. 2249–2257.
* [6] P. Auer, N. Cesa-Bianchi, P. Fischer, Finite-time analysis of the multiarmed bandit problem, Machine learning 47 (2-3) (2002) 235–256.
* [7] E. Kaufmann, et al., On bayesian index policies for sequential resource allocation, The Annals of Statistics 46 (2) (2018) 842–865.
* [8] R. Kaplan, K. Friston, Planning and navigation as active inference, bioRxiv (2017). arXiv:https://www.biorxiv.org/content/early/2017/12/07/230599.full.pdf, doi:10.1101/230599.
URL https://www.biorxiv.org/content/early/2017/12/07/230599
* [9] K. J. Friston, R. Rosch, T. Parr, C. Price, H. Bowman, Deep temporal models and active inference, Neuroscience & Biobehavioral Reviews 77 (Supplement C) (2017) 388 – 402. doi:https://doi.org/10.1016/j.neubiorev.2017.04.009.
URL http://www.sciencedirect.com/science/article/pii/S0149763416307096
* [10] K. Friston, T. FitzGerald, F. Rigoli, P. Schwartenbeck, G. Pezzulo, Active inference: A process theory, Neural Computation 29 (1) (2017) 1–49, pMID: 27870614\. doi:10.1162/NECO\\_a\\_00912.
* [11] M. B. Mirza, R. A. Adams, C. D. Mathys, K. J. Friston, Scene construction, visual foraging, and active inference, Frontiers in computational neuroscience 10 (2016).
* [12] K. Friston, T. FitzGerald, F. Rigoli, P. Schwartenbeck, J. O’Doherty, G. Pezzulo, Active inference and learning, Neuroscience & Biobehavioral Reviews 68 (Supplement C) (2016) 862 – 879. doi:https://doi.org/10.1016/j.neubiorev.2016.06.022.
URL http://www.sciencedirect.com/science/article/pii/S0149763416301336
* [13] T. H. FitzGerald, P. Schwartenbeck, M. Moutoussis, R. J. Dolan, K. Friston, Active inference, evidence accumulation, and the urn task, Neural computation 27 (2) (2015) 306–328.
* [14] K. Friston, F. Rigoli, D. Ognibene, C. Mathys, T. Fitzgerald, G. Pezzulo, Active inference and epistemic value, Cognitive neuroscience 6 (4) (2015) 187–214.
* [15] P. Schwartenbeck, T. FitzGerald, R. Dolan, K. Friston, Exploration, novelty, surprise, and free energy minimization, Frontiers in psychology 4 (2013) 710.
* [16] K. Friston, The history of the future of the bayesian brain, NeuroImage 62 (2) (2012) 1230–1233.
* [17] K. Doya, S. Ishii, A. Pouget, R. P. Rao, Bayesian brain: Probabilistic approaches to neural coding, MIT press, 2007.
* [18] D. C. Knill, A. Pouget, The bayesian brain: the role of uncertainty in neural coding and computation, TRENDS in Neurosciences 27 (12) (2004) 712–719.
* [19] M. Botvinick, M. Toussaint, Planning as inference, Trends in cognitive sciences 16 (10) (2012) 485–488.
* [20] F. Karl, A free energy principle for biological systems, Entropy 14 (11) (2012) 2100–2121.
* [21] K. Friston, J. Kilner, L. Harrison, A free energy principle for the brain, Journal of Physiology-Paris 100 (1-3) (2006) 70–87.
* [22] P. Schwartenbeck, J. Passecker, T. U. Hauser, T. H. FitzGerald, M. Kronbichler, K. J. Friston, Computational mechanisms of curiosity and goal-directed exploration, Elife 8 (2019) e41703.
* [23] R. S. Sutton, A. G. Barto, Reinforcement learning: An introduction, MIT press, 2018\.
* [24] K. Ueltzhöffer, Deep active inference, Biological cybernetics 112 (6) (2018) 547–573.
* [25] B. Millidge, Deep active inference as variational policy gradients, Journal of Mathematical Psychology 96 (2020) 102348.
* [26] K. Friston, L. Da Costa, D. Hafner, C. Hesp, T. Parr, Sophisticated inference, arXiv preprint arXiv:2006.04120 (2020).
* [27] Z. Fountas, N. Sajid, P. A. Mediano, K. Friston, Deep active inference agents using monte-carlo methods, arXiv preprint arXiv:2006.04176 (2020).
* [28] W. R. Thompson, On the likelihood that one unknown probability exceeds another in view of the evidence of two samples, Biometrika 25 (3/4) (1933) 285–294.
* [29] R. R. Bush, F. Mosteller, A stochastic model with applications to learning, The Annals of Mathematical Statistics (1953) 559–585.
* [30] P. Whittle, Multi-armed bandits and the gittins index, Journal of the Royal Statistical Society: Series B (Methodological) 42 (2) (1980) 143–149.
* [31] T. L. Lai, H. Robbins, Asymptotically efficient adaptive allocation rules, Advances in applied mathematics 6 (1) (1985) 4–22.
* [32] E. Kaufmann, N. Korda, R. Munos, Thompson sampling: An asymptotically optimal finite-time analysis, in: International conference on algorithmic learning theory, Springer, 2012, pp. 199–213.
* [33] A. Izquierdo, J. L. Brigman, A. K. Radke, P. H. Rudebeck, A. Holmes, The neural basis of reversal learning: an updated perspective, Neuroscience 345 (2017) 12–26.
* [34] D. Marković, A. M. Reiter, S. J. Kiebel, Predicting change: Approximate inference under explicit representation of temporal structure in changing environments, PLoS computational biology 15 (1) (2019) e1006707.
* [35] S. Iglesias, C. Mathys, K. H. Brodersen, L. Kasper, M. Piccirelli, H. E. den Ouden, K. E. Stephan, Hierarchical prediction errors in midbrain and basal forebrain during sensory learning, Neuron 80 (2) (2013) 519–530.
* [36] R. C. Wilson, Y. Niv, Inferring relevance in a changing world, Frontiers in human neuroscience 5 (2012) 189.
* [37] D. Racey, M. E. Young, D. Garlick, J. N.-M. Pham, A. P. Blaisdell, Pigeon and human performance in a multi-armed bandit task in response to changes in variable interval schedules, Learning & behavior 39 (3) (2011) 245–258.
* [38] T. E. Behrens, M. W. Woolrich, M. E. Walton, M. F. Rushworth, Learning the value of information in an uncertain world, Nature neuroscience 10 (9) (2007) 1214–1221.
* [39] E. Kaufmann, O. Cappé, A. Garivier, On bayesian upper confidence bounds for bandit problems, in: Artificial intelligence and statistics, 2012, pp. 592–600.
* [40] X. Lu, N. Adams, N. Kantas, On adaptive estimation for dynamic bernoulli bandits, Foundations of Data Science 1 (2) (2019) 197.
* [41] Y. Cao, W. Zheng, B. Kveton, Y. Xie, Nearly optimal adaptive procedure for piecewise-stationary bandit: a change-point detection approach, arXiv preprint arXiv:1802.03692 (2018).
* [42] R. Alami, O. Maillard, R. Féraud, Memory bandits: a bayesian approach for the switching bandit problem, in: NIPS 2017-31st Conference on Neural Information Processing Systems, 2017.
* [43] D. J. Russo, B. Van Roy, A. Kazerouni, I. Osband, Z. Wen, et al., A tutorial on thompson sampling, Foundations and Trends® in Machine Learning 11 (1) (2018) 1–96.
* [44] D. M. Roijers, L. M. Zintgraf, A. Nowé, Interactive thompson sampling for multi-objective multi-armed bandits, in: International Conference on Algorithmic DecisionTheory, Springer, 2017, pp. 18–34.
* [45] Y. Wang, J. Gittins, Bayesian bandits in clinical trials: Clinical trials, Sequential analysis 11 (4) (1992) 313–325.
* [46] V. Liakoni, A. Modirshanechi, W. Gerstner, J. Brea, Learning in volatile environments with the bayes factor surprise, Neural Computation 33 (2) (2021) 269–340.
* [47] D. Marković, S. J. Kiebel, Comparative analysis of behavioral models for adaptive learning in changing environments, Frontiers in Computational Neuroscience 10 (2016) 33.
* [48] F. Liu, J. Lee, N. Shroff, A change-detection based framework for piecewise-stationary multi-armed bandit problem, in: Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
* [49] M. Steyvers, M. D. Lee, E.-J. Wagenmakers, A bayesian analysis of human decision-making on bandit problems, Journal of Mathematical Psychology 53 (3) (2009) 168–179.
* [50] A. Slivkins, et al., Introduction to multi-armed bandits, Foundations and Trends® in Machine Learning 12 (1-2) (2019) 1–286.
* [51] D. I. Mattos, J. Bosch, H. H. Olsson, Multi-armed bandits in the wild: pitfalls and strategies in online experiments, Information and Software Technology 113 (2019) 68–81.
* [52] E. Schulz, S. J. Gershman, The algorithmic architecture of exploration in the human brain, Current opinion in neurobiology 55 (2019) 7–14.
* [53] J. Gottlieb, P.-Y. Oudeyer, M. Lopes, A. Baranes, Information-seeking, curiosity, and attention: computational and neural mechanisms, Trends in cognitive sciences 17 (11) (2013) 585–593.
* [54] H. Stojić, J. L. Orquin, P. Dayan, R. J. Dolan, M. Speekenbrink, Uncertainty in learning, choice, and visual fixation, Proceedings of the National Academy of Sciences 117 (6) (2020) 3291–3300.
* [55] R. C. Wilson, A. Geana, J. M. White, E. A. Ludvig, J. D. Cohen, Humans use directed and random exploration to solve the explore–exploit dilemma., Journal of Experimental Psychology: General 143 (6) (2014) 2074.
* [56] P. B. Reverdy, V. Srivastava, N. E. Leonard, Modeling human decision making in generalized gaussian multiarmed bandits, Proceedings of the IEEE 102 (4) (2014) 544–571.
* [57] D. Acuna, P. Schrater, Bayesian modeling of human sequential decision-making on the multi-armed bandit problem, in: Proceedings of the 30th annual conference of the cognitive science society, Vol. 100, Washington, DC: Cognitive Science Society, 2008, pp. 200–300.
* [58] H. Stojić, E. Schulz, P. P Analytis, M. Speekenbrink, It’s new, but is it good? how generalization and uncertainty guide the exploration of novel options., Journal of Experimental Psychology: General (2020).
* [59] E. Schulz, E. Konstantinidis, M. Speekenbrink, Putting bandits into context: How function learning supports decision making., Journal of Experimental Psychology: Learning, Memory, and Cognition 44 (6) (2018) 927.
* [60] E. Schulz, N. T. Franklin, S. J. Gershman, Finding structure in multi-armed bandits, Cognitive Psychology 119 (2020) 101261.
* [61] L. Clark, R. Cools, T. Robbins, The neuropsychology of ventral prefrontal cortex: decision-making and reversal learning, Brain and cognition 55 (1) (2004) 41–53.
* [62] M. Guitart-Masip, L. Fuentemilla, D. R. Bach, Q. J. Huys, P. Dayan, R. J. Dolan, E. Duzel, Action dominates valence in anticipatory representations in the human striatum and dopaminergic midbrain, Journal of Neuroscience 31 (21) (2011) 7867–7875.
* [63] N. D. Daw, S. J. Gershman, B. Seymour, P. Dayan, R. J. Dolan, Model-based influences on humans’ choices and striatal prediction errors, Neuron 69 (6) (2011) 1204–1215.
* [64] A. Dezfouli, B. W. Balleine, Habits, action sequences and reinforcement learning, European Journal of Neuroscience 35 (7) (2012) 1036–1051.
* [65] A. Tversky, Elimination by aspects: A theory of choice., Psychological review 79 (4) (1972) 281.
* [66] E. Reutskaja, R. Nagel, C. F. Camerer, A. Rangel, Search dynamics in consumer choice under time pressure: An eye-tracking study, American Economic Review 101 (2) (2011) 900–926.
* [67] F. Lieder, T. L. Griffiths, Strategy selection as rational metareasoning., Psychological review 124 (6) (2017) 762.
* [68] W. C. Cheung, D. Simchi-Levi, R. Zhu, Hedging the drift: Learning to optimize under non-stationarity, arXiv preprint arXiv:1903.01461 (2019).
* [69] L. Besson, E. Kaufmann, The generalized likelihood ratio test meets klucb: an improved algorithm for piece-wise non-stationary bandits, arXiv preprint arXiv:1902.01575 (2019).
* [70] A. Blum, Y. Mansour, Learning, Regret Minimization, and Equilibria, Cambridge University Press, 2007, Ch. 4, p. 79–102. doi:10.1017/CBO9780511800481.006.
* [71] V. Raj, S. Kalyani, Taming non-stationary bandits: A bayesian approach, arXiv preprint arXiv:1707.09727 (2017).
* [72] D. Russo, B. Van Roy, Learning to optimize via information-directed sampling, Advances in Neural Information Processing Systems 27 (2014) 1583–1591.
* [73] P. I. Frazier, W. B. Powell, S. Dayanik, A knowledge-gradient policy for sequential information collection, SIAM Journal on Control and Optimization 47 (5) (2008) 2410–2439.
* [74] A. Garivier, E. Moulines, On upper-confidence bound policies for switching bandit problems, in: International Conference on Algorithmic Learning Theory, Springer, 2011, pp. 174–188.
* [75] O. Besbes, Y. Gur, A. Zeevi, Stochastic multi-armed-bandit problem with non-stationary rewards, Advances in neural information processing systems 27 (2014) 199–207.
* [76] R. Allesiardo, R. Féraud, O.-A. Maillard, The non-stationary stochastic multi-armed bandit problem, International Journal of Data Science and Analytics 3 (4) (2017) 267–283.
* [77] R. P. Adams, D. J. MacKay, Bayesian online changepoint detection, arXiv preprint arXiv:0710.3742 (2007).
* [78] J. Mellor, J. Shapiro, Thompson sampling in switching environments with bayesian online change detection, in: Artificial Intelligence and Statistics, PMLR, 2013, pp. 442–450.
* [79] R. Alami, O. Maillard, R. Féraud, Restarted bayesian online change-point detector achieves optimal detection delay, in: International Conference on Machine Learning, PMLR, 2020, pp. 211–221.
* [80] A. Soltani, A. Izquierdo, Adaptive learning under expected and unexpected uncertainty, Nature Reviews Neuroscience 20 (10) (2019) 635–644.
* [81] H. Attias, Planning by probabilistic inference., in: AISTATS, Citeseer, 2003.
* [82] P. Schwartenbeck, K. Friston, Computational phenotyping in psychiatry: a worked example, ENeuro 3 (4) (2016).
* [83] R. Kaplan, K. J. Friston, Planning and navigation as active inference, Biological cybernetics 112 (4) (2018) 323–343.
* [84] J. Z. Kolter, A. Y. Ng, Near-bayesian exploration in polynomial time, in: Proceedings of the 26th annual international conference on machine learning, 2009, pp. 513–520.
* [85] K. Kandasamy, A. Krishnamurthy, J. Schneider, B. Póczos, Parallelised bayesian optimisation via thompson sampling, in: International Conference on Artificial Intelligence and Statistics, 2018, pp. 133–142.
* [86] L. Da Costa, T. Parr, N. Sajid, S. Veselic, V. Neacsu, K. Friston, Active inference on discrete state-spaces: a synthesis, Journal of Mathematical Psychology 99 (2020) 102447.
* [87] N. Sajid, P. J. Ball, K. J. Friston, Active inference: demystified and compared, arXiv preprint arXiv:1909.10863 (2019) 2–3.
* [88] J. Limanowski, K. Friston, Active inference under visuo-proprioceptive conflict: Simulation and empirical results, Scientific reports 10 (1) (2020) 1–14.
* [89] R. A. Adams, M. Moutoussis, M. M. Nour, T. Dahoun, D. Lewis, B. Illingworth, M. Veronese, C. Mathys, L. de Boer, M. Guitart-Masip, et al., Variability in action selection relates to striatal dopamine 2/3 receptor availability in humans: A pet neuroimaging study using reinforcement learning and active inference models, Cerebral Cortex 30 (6) (2020) 3573–3589.
* [90] R. Smith, P. Schwartenbeck, J. L. Stewart, R. Kuplicki, H. Ekhtiari, M. P. Paulus, T. . Investigators, et al., Imprecise action selection in substance use disorder: Evidence for active learning impairments when solving the explore-exploit dilemma, Drug and Alcohol Dependence 215 (2020) 108208.
* [91] M. Cullen, B. Davey, K. J. Friston, R. J. Moran, Active inference in openai gym: a paradigm for computational investigations into psychiatric illness, Biological psychiatry: cognitive neuroscience and neuroimaging 3 (9) (2018) 809–818.
* [92] P. Schwartenbeck, T. H. FitzGerald, C. Mathys, R. Dolan, M. Kronbichler, K. Friston, Evidence for surprise minimization over value maximization in choice behavior, Scientific reports 5 (2015) 16575.
* [93] P. Schwartenbeck, T. H. FitzGerald, C. Mathys, R. Dolan, K. Friston, The dopaminergic midbrain encodes the expected certainty about desired outcomes, Cerebral cortex 25 (10) (2015) 3434–3445.
* [94] N. Chater, M. Oaksford, Ten years of the rational analysis of cognition, Trends in Cognitive Sciences 3 (2) (1999) 57–65.
* [95] D. Marković, T. Goschke, S. J. Kiebel, Meta-control of the exploration-exploitation dilemma emerges from probabilistic inference over a hierarchy of time scales, Cognitive, Affective, & Behavioral Neuroscience (2020) 1–25.
* [96] R. A. Adams, S. Shipp, K. J. Friston, Predictions not commands: active inference in the motor system, Brain Structure and Function 218 (3) (2013) 611–643.
* [97] G. Pezzulo, An active inference view of cognitive control, Frontiers in psychology 3 (2012) 478.
* [98] D. Russo, B. Van Roy, An information-theoretic analysis of thompson sampling, The Journal of Machine Learning Research 17 (1) (2016) 2442–2471.
* [99] D. Russo, B. Van Roy, Learning to optimize via information-directed sampling, Operations Research 66 (1) (2018) 230–252.
* [100] M. Ghavamzadeh, S. Mannor, J. Pineau, A. Tamar, et al., Bayesian reinforcement learning: A survey, Foundations and Trends® in Machine Learning 8 (5-6) (2015) 359–483.
* [101] A. Guez, D. Silver, P. Dayan, Scalable and efficient bayes-adaptive reinforcement learning based on monte-carlo tree search, Journal of Artificial Intelligence Research 48 (2013) 841–883.
* [102] A. Guez, N. Heess, D. Silver, P. Dayan, Bayes-adaptive simulation-based search with value function approximation., in: NIPS, Vol. 27, 2014, pp. 451–459.
* [103] P. Piray, N. D. Daw, A simple model for learning in volatile environments, PLOS Computational Biology 16 (7) (2020) 1–26. doi:10.1371/journal.pcbi.1007963.
URL https://doi.org/10.1371/journal.pcbi.1007963
* [104] V. Moens, A. Zénon, Learning and forgetting using reinforced bayesian change detection, PLoS computational biology 15 (4) (2019) e1006713.
* [105] J. M. Bernardo, Algorithm as 103: Psi (digamma) function, Journal of the Royal Statistical Society. Series C (Applied Statistics) 25 (3) (1976) 315–317.
* [106] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018).
URL http://github.com/google/jax
## Appendix
### A Deriving the expression for the expected free energy
We write the approximate (exact in the case of stationary Bernoulli bandits)
posterior beliefs about reward probabilities $\vec{\theta}_{t-1}$ at trial
$t-1$ as
$Q\mathinner{\left(\vec{\theta}_{t-1}|\vec{\eta}_{t}\right)}=\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{t-1,k},\beta_{t-1,k}\right)},$
(A.36)
where
$\vec{\eta}_{t-1}=(\alpha_{t-1,1},\beta_{t-1,1},\ldots,\alpha_{t-1,K},\beta_{t-1,K})$,
contains the information about the history of choices
$a_{1\mathrel{\mathop{\mathchar 58\relax}}t-1}$ and outcomes
$o_{1\mathrel{\mathop{\mathchar 58\relax}}t-1}$ up to trial $t-1$. Next, we
obtain the predictive prior distribution at trial $t$ as
$\begin{split}p\mathinner{\left(\vec{\theta}_{t}|j_{t},\vec{\eta}_{t-1}\right)}&=\int\operatorname{d\\!}\vec{\theta}_{t-1}p\mathinner{\left(\vec{\theta}_{t}|\vec{\theta}_{t-1},j_{t}\right)}Q\mathinner{\left(\vec{\theta}_{t-1}|\vec{\eta}_{t}\right)}\\\
&=\left\\{\begin{array}[]{ll}\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{t-1,k},\beta_{t-1,k}\right)}&\text{for
}j_{t}=0\\\
\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{0,k},\beta_{0,k}\right)}&\text{for
}j_{t}=1\end{array}\right.\end{split}$ (A.37)
where $\alpha_{0,k}=\beta_{0,k}=1$. Marginalising out $j_{t}$ from the joint
predictive distribution
$p\mathinner{\left(\vec{\theta}_{t}|j_{t},\vec{\eta}_{t-1}\right)}$ leads to
the following marginal predictive probability
$p\mathinner{\left(\vec{\theta}_{t}|\vec{\eta}_{t-1}\right)}=(1-\rho)\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{t-1,k},\beta_{t-1,k}\right)}+\rho\prod_{k=1}^{K}\mathcal{B}e\mathinner{\left(\alpha_{0,k},\beta_{0,k}\right)}.$
(A.38)
Finally we compute the probability of observing outcome $o_{t}$ given action
$a_{t}$ on current trial $t$ by marginalising the joint distribution
$p\mathinner{\left(o_{t},\vec{\theta}_{t}|a_{t},\vec{\eta}_{t-1}\right)}$ over
latent states $\vec{\theta}_{t}$. Hence,
$\begin{split}Q\mathinner{\left(o_{t}|a_{t},\vec{\eta}_{t-1}\right)}&=\int\operatorname{d\\!}\vec{\theta}_{t}p\mathinner{\left(o_{t}|a_{t},\vec{\theta}_{t}\right)}p\mathinner{\left(\vec{\theta}_{t}|\vec{\eta}_{t-1}\right)}\\\
&=\prod_{k=1}^{K}\mathinner{\left[\tilde{\mu}_{t,k}^{o_{t}}\mathinner{\left(1-\tilde{\mu}_{t,k}\right)}^{1-o_{t}}\right]}^{\delta_{a_{t},k}},\end{split}$
(A.39)
where $\tilde{\mu}_{t,k}$ corresponds to the expression used in Eq. 25. To
compute the expected free energy $G(a_{t})$ we will split the full expression
on two terms the risk, denoted with $G_{R}(a_{t})$, and the ambiguity, denoted
with $G_{A}(a_{t})$. Hence,
$G(a_{t})=G_{R}(a_{t})+G_{A}(a_{t}).$ (A.40)
Combining Eq. 24 and Eq. 25 we compute the risk using the following relation
$\begin{split}G_{R}(a)&=D_{KL}\mathinner{\left(Q(o_{t}|a_{t}=a)||P(o_{t})\right)}\\\
&=\sum_{o_{t}}Q(o_{t}|a)\ln\frac{Q(o_{t}|a)}{P(o_{t})}\\\
&=const.+\sum_{o_{t}}Q(o_{t}|a)\mathinner{\left[o_{t}\mathinner{\left(\ln\tilde{\mu}_{t,a}-\lambda\right)}+(1-o_{t})\mathinner{\left(\ln\mathinner{\left(1-\tilde{\mu}_{t,a}\right)}+\lambda\right)}\right]}\\\
&=const.+\mathinner{\left[\tilde{\mu}_{t,a}\mathinner{\left(\ln\tilde{\mu}_{t,a}-\lambda\right)}+(1-\tilde{\mu}_{t,a})\mathinner{\left(\ln\mathinner{\left(1-\tilde{\mu}_{t,a}\right)}+\lambda\right)}\right]}\\\
&=-\lambda\mathinner{\left(2\tilde{\mu}_{t,a}-1\right)}+\tilde{\mu}_{t,a}\ln\tilde{\mu}_{t,a}+(1-\tilde{\mu}_{t,a})\ln\mathinner{\left(1-\tilde{\mu}_{t,a}\right)}+const.\end{split}$
(A.41)
To derive the expression for the ambiguity part $G_{A}(a)$ of the expected
free energy we will start by computing conditional entropy of outcomes
$o_{t}$, defined as
$\begin{split}H\mathinner{\left[o_{t}|\vec{\theta}_{t},a_{t}\right]}&=-\sum_{o_{t}}p(o_{t}|\vec{\theta}_{t},a_{t})\ln
p(o_{t}|\vec{\theta}_{t},a_{t})\\\
&=-\sum_{o_{t}}p(o_{t}|\vec{\theta}_{t},a_{t})\sum_{k=1}^{K}\delta_{a_{t},k}\mathinner{\left[o_{t}\ln\theta_{t,k}+(1-o_{t})\ln\mathinner{\left(1-\theta_{t,k}\right)}\right]}\\\
&=-\theta_{t,a_{t}}\ln\theta_{t,a_{t}}-\mathinner{\left(1-\theta_{t,a_{t}}\right)}\ln\mathinner{\left(1-\theta_{t,k}\right)}.\end{split}$
(A.42)
As the ambiguity term corresponds to expectation over latent states of the
conditional entropy we obtain the following expression
$G_{A}(a)=E_{p(\vec{\theta}_{t}|\vec{\eta}_{t-1})}\mathinner{\left[H\mathinner{\left[o_{t}|\vec{\theta}_{t},a_{t}=a\right]}\right]}\\\
=-(1-\rho)\mathinner{\left[\mu_{t-1,a}\psi(\alpha_{t-1,a})+(1-\mu_{t-1,a})\psi(\beta_{t-1,a})-\psi(\nu_{t-1,a})+\frac{1}{\nu}\right]}+const.$
(A.43)
Where we used the following relations
$\begin{split}\int dx\mathcal{B}e\mathinner{\left(x;\alpha,\beta\right)}x\ln
x\\\ &=\frac{B(\alpha+1,\beta)}{B(\alpha,\beta)}\int
dxB\mathinner{\left(x;\alpha+1,\beta\right)}\ln x\\\
&=\frac{B(\alpha+1,\beta)}{B(\alpha,\beta)}\mathinner{\left[\psi(\alpha+1)-\psi\mathinner{\left(\nu+1\right)}\right]}\\\
&=\mu\mathinner{\left[\psi(\alpha+1)-\psi\mathinner{\left(\nu+1\right)}\right]}\\\
&=\mu\mathinner{\left[\psi(\alpha)+\frac{1}{\alpha}-\psi\mathinner{\left(\nu\right)}-\frac{1}{\nu}\right]}\\\
&=\mu\psi(\alpha)+\frac{1}{\nu}-\mu\mathinner{\left[\psi\mathinner{\left(\nu\right)}+\frac{1}{\nu}\right]}\end{split}$
(A.44)
and
$\begin{split}\int
dx\mathcal{B}e\mathinner{\left(x;\alpha,\beta\right)}(1-x)\ln(1-x)\\\
&=\frac{B(\alpha,\beta+1)}{B(\alpha,\beta)}\int
dxB\mathinner{\left(x;\alpha,\beta+1\right)}\ln(1-x)\\\
&=\frac{B(\alpha,\beta+1)}{B(\alpha,\beta)}\mathinner{\left[\psi(\beta+1)-\psi\mathinner{\left(\nu+1\right)}\right]}\\\
&=(1-\mu)\mathinner{\left[\psi(\beta+1)-\psi\mathinner{\left(\nu+1\right)}\right]}\\\
&=(1-\mu)\mathinner{\left[\psi(\beta)+\frac{1}{\beta}-\psi\mathinner{\left(\nu\right)}-\frac{1}{\nu}\right]}\\\
&=(1-\mu)\psi(\beta)+\frac{1}{\nu}-(1-\mu)\mathinner{\left[\psi\mathinner{\left(\nu\right)}+\frac{1}{\nu}\right]}\end{split}$
(A.45)
to obtain the expectations of conditional entropy. Combining Eqs. A.41 and
A.43 we get the expression for the expected free energy of action $a$ on trial
$t$ shown in Eq. 26.
### B Deriving the approximate expression of the expected free energy
To derive the approximate expression for the expected free energy shown in Eq.
27 we note that for a sufficiently large $x$ the following relation holds
[105]
$\psi(x)=\ln x-\frac{1}{2x}.$ (B.46)
As parameters of the Beta distribution are monotonically increasing with each
update, we can assume that they reach large enough value after certain number
of trials. Hence, instead of using the exact expression for ambiguity derived
in Eq. A.43, we can used a simplified expression based on the mentioned
approximation of the digamma function. Therefore, the approximate ambiguity
term becomes
$\begin{split}G_{A}(a)&\approx-(1-\rho)\mu_{t-1,a}\mathinner{\left(\ln\alpha_{t-1,a}-\frac{1}{2\alpha_{t-1,a}}\right)}\\\
&\quad-(1-\rho)(1-\mu_{t-1,a})\mathinner{\left(\ln\beta_{t-1,a}-\frac{1}{2\beta{t-1,a}}\right)}\\\
&\quad+(1-\rho)\mathinner{\left(\ln\nu_{t-1,a}-\frac{3}{2\nu}\right)}+const.\\\
&=-(1-\rho)\mathinner{\left[\mu_{t-1,a}\ln\mu_{t-1,a}+\mathinner{\left(1-\mu_{t-1,a}\right)}\ln\mathinner{\left(1-\mu_{t-1,a}\right)}+\frac{1}{2\nu_{t-1,a}}\right]}\end{split}$
(B.47)
As we are interested only in cases for which change probability is small,
namely $\rho\leq 0.1$ we can further approximate the risk term of the expected
free energy as
$\begin{split}G_{R}(a)&\approx-(1-\rho)\mathinner{\left[2\lambda\mu_{t-1,a}-\mu_{t-1,a}\ln\tilde{\mu}_{t,a}-\mathinner{\left(1-\mu_{t-1,a}\right)}\ln\mathinner{\left(1-\tilde{\mu}_{t,a}\right)}\right]}\\\
&\approx-(1-\rho)\mathinner{\left[2\lambda\mu_{t-1,a}-\mu_{t-1,a}\ln\mu_{t-1,a}-\mathinner{\left(1-\mu_{t-1,a}\right)}\ln\mathinner{\left(1-\mu_{t-1,a}\right)}\right]}\end{split}$
(B.48)
where we set $\ln\tilde{\mu}_{t,a}\approx\ln\mu_{t-1,a}$ and
$\ln\mathinner{\left(1-\tilde{\mu}_{t,a}\right)}\approx\ln\mathinner{\left(1-\mu_{t-1,a}\right)}$.
Adding together the two approximate terms leads to the following expression
for the approximate free energy
$G_{t}(a)\approx-(1-\rho)\mathinner{\left[2\lambda\mu_{t-1,a}+\frac{1}{2\nu_{t-1,a}}\right]}.$
(B.49)
Thus minimising approximate expected free energy corresponds the expression
shown in Eq. 27, where cancelling the negative sign turns minimisation into
maximisation process.
### C Benchmark of action selection algorithms
All multi-armed bandits algorithms used in this paper have been implemented
using JAX (Autorgrad and XLA) [106] and executed in Python 3.9.4 environment.
JAX uses XLA (Accelerated Linear algebra) to just-in-time compile Python
functions into XLA-optimized kernels and run them on GPUs and TPUs. For the
benchmarks presented in Table 1 we have used a Lenovo Workstation with AMD
Threadripper 3955WX CPU, NVIDIA Quadro RTX 4000 GPU, and 64 GB RAM.
Algorithm | $K=10$ | $K=20$ | $K=40$ | $K=80$
---|---|---|---|---
UCB | 0.038 | 0.039 | 0.042 | 0.042
B-UCB | 0.963 | 0.980 | 1.052 | 1.185
TS | 0.992 | 0.967 | 1.207 | 1.864
O-TS | 0.968 | 0.939 | 1.034 | 1.609
G-AI | 0.041 | 0.041 | 0.054 | 0.072
A-AI | 0.034 | 0.036 | 0.039 | 0.040
Table 1: Compute times per decision presented in milliseconds. The compute
times were estimated as an average over ten repetitions of a $T=10000$ long
loop, consisting of only action selection and learning algorithm. Outcomes
were kept fixed on all trials. Each algorithm was executed with $N=1000$
parallel simulations. For comparison, we show run times of classical variants
of UCB and TS [5].
Note that the presented compute times can hardly be generalised to other
multi-armed bandit problems, and are highly dependent on the efficiency of JAX
framework to optimise various functions, and sampling algorithms.
### D Supplementary Figures
Figure S1 : Histogram of the logarithm of cumulative regret. The ensemble
based distribution the cumulative regret at $T=10^{6}$ for different
algorithms estimated from $N=10^{3}$ simulations. Note that the peak in the
tail of the distribution for A-AI algorithm, proportional to random choices,
implies that percentage of agents in the ensemble never found a correct
solution. Hence, as number of trials increases the average cumulative regret
over the ensemble is pulled towards values which grow linearly with the trial
number $t$. Figure S2 : Regret rate analysis for active inference based agents
in the switching Bernoulli bandits with fixed difficulty. The regret rate
$\tilde{R}_{T}$, Eq. 6, for the approximate (A-AI) and the exact (G-AI)
variants of active inference as a function of the precision over prior
preferences $\lambda$. The coloured lines show numeric estimates obtained as
an average over $N=10^{3}$ runs. Different line styles denote $\tilde{R}_{T}$
values estimated after different numbers of trials $T$. The dashed black line
denotes the upper bound on the regret rate corresponding to the random (RC)
agent which gains no information from the choice outcomes. The vertical dotted
line (purple) corresponds to $\lambda=0.5$, which we find to be sufficiently
close to the minimum or regret rate in a range of conditions. Each column and
row of the plot corresponds to different task difficulties, characterised by
the change probability $\rho$, and the mean outcome difference $\epsilon$,
respectively. We have fixed the arm number to $K=10$. Figure S3 : Regret rate
analysis for active inference based agents in the switching Bernoulli bandits
with varying difficulty. The regret rate $\tilde{R}_{T}$, Eq. 6, for the
approximate (A-AI) and the exact (G-AI) variants of active inference as a
function of the precision over prior preferences $\lambda$. The coloured lines
show numeric estimates obtained as an average over $N=10^{3}$ runs. Different
line styles denote $\tilde{R}_{T}$ values estimated after different numbers of
trials $T$. The dashed black line denotes the upper bound on the regret rate
corresponding to the random (RC) agent which gains no information from the
choice outcomes. The vertical dotted line (purple) corresponds to
$\lambda=0.25$, which we find to be sufficiently close to the minimum or
regret rate of the G-AI algorithm in a range of conditions. Each column and
row of the plot corresponds to different task difficulties, characterised by
the change probability $\rho$, and the arm number $K$, respectively.
|
# Quantum-Mechanical Force Balance Between Multipolar Dispersion and
Pauli Repulsion in Atomic van der Waals Dimers
Ornella Vaccarelli<EMAIL_ADDRESS>Department of Physics and
Materials Science, University of Luxembourg, L-1511 Luxembourg City,
Luxembourg Dmitry V. Fedorov Department of Physics and Materials Science,
University of Luxembourg, L-1511 Luxembourg City, Luxembourg Martin Stöhr
Department of Physics and Materials Science, University of Luxembourg, L-1511
Luxembourg City, Luxembourg Alexandre Tkatchenko<EMAIL_ADDRESS>Department of Physics and Materials Science, University of Luxembourg, L-1511
Luxembourg City, Luxembourg
###### Abstract
The structure and stability of atomic and molecular systems with van der Waals
(vdW) bonding are often determined by the interplay between attractive
dispersion interactions and repulsive interactions caused by electron
confinement. Arising due to different mechanisms — electron correlation for
dispersion and the Pauli exclusion principle for exchange-repulsion — these
interactions do not appear to have a straightforward connection. In this
paper, we use a coarse-grained approach for evaluating the exchange energy for
two coupled quantum Drude oscillators and investigate the mutual compensation
of the attractive and repulsive forces at the equilibrium distance within the
multipole expansion of the Coulomb potential. This compensation yields a
compact formula relating the vdW radius of an atom to its multipole
polarizabilities, $R_{\rm
vdW}=A_{l}^{\,}\,\alpha_{l}^{\nicefrac{{2}}{{7(l+1)}}}$, where $l$ is the
multipole rank and $A_{l}$ is a conversion factor. Such a relation is
compelling because it connects an electronic property of an isolated atom
(atomic polarizability) with an equilibrium distance in a dimer composed of
two closed-shell atoms. We assess the accuracy of the revealed formula for
noble-gas, alkaline-earth, and alkali atoms and show that the $A_{l}$ can be
assumed to be universal constants. Besides a seamless definition of vdW radii,
the proposed relation can also be used for the efficient determination of
atomic multipole polarizabilities solely based on the corresponding dipole
polarizability and the vdW radius. Finally, our work provides a basis for the
construction of efficient and minimally-empirical interatomic potentials by
combining multipolar interatomic exchange and dispersion forces on an equal
footing.
## I Introduction
Noncovalent interatomic and intermolecular interactions represent one of the
key factors that determine the physicochemical properties of molecules and
materials across chemistry, biology and materials science Stone2016 ;
Hirschfelder1967 ; Margenau1971 ; Kaplan2006 . Noncovalent interactions are
traditionally classified in a perturbative formalism, from which
electrostatics, induction, Pauli (exchange) repulsion and van der Waals (vdW)
dispersion arise as the leading contributions from the first two orders of
perturbation theory. From the perspective of computational modeling, the
individual terms are usually treated with different effective approaches.
Especially the methods used to describe Pauli repulsion and vdW dispersion
typically rely on fundamentally different physical models. The vdW dispersion
represents a major part of long-range electron correlation forces arising from
Coulomb-coupled instantaneous quantum fluctuations of the electronic charge
distribution Parsegian2005 ; Tkatchenko2015 ; Woods2016 ; Mahanty1973 ;
Richardson1828 ; Paranjape1979 . Common (semi-)local approximations to
density-functional theory (DFT), representing one of the main workhorse
methods in atomistic modeling, neglect long-range correlation forces and thus
do not account for vdW interactions. In recent years, an intense effort has
been devoted to develop robust approaches to address this challenge Klimes2012
; Bjorkman2012 ; Berland2015 ; Grimme2016 ; Hermann2017 ; Stoehr_CSR_2019 .
Although a unified vdW functional valid for all kinds of systems is still
under construction Hermann2020 , significant progress has been achieved to
include dispersion interactions in the form of non-local (vdW) density
functionals Dobson1999 ; Dion2004 ; Vydrov2009 ; Sabatini2013 . Furthermore,
coarse-grained vdW models have shown great success in describing dispersion
interactions at lower computational costs Becke2005 ; Tkatchenko2009 ;
Grimme2010 ; Odbadrakh2015 ; Caldeweyher2017 . Among them, the quantum Drude
oscillator (QDO) model Bade1957 ; Wang2001 ; Lamoureux_2003 ; Lamoureux2003 ;
Sommerfeld2005 ; Whitfield2006 ; Jones2013 has been firmly established as an
efficient and accurate approach for modeling and understanding vdW
interactions Whitfield2006 ; Tkatchenko2012 ; Jones2013 ; Reilly2015 ;
Sadhukhan2016 ; Sadhukhan2017 ; Hermann_NComm_2017 . Within this approach,
each QDO models an atom or a molecule, representing the effective, localized
response and polarization fluctuation of its valence electrons. The success of
the coupled-oscillator model is exemplified by its excellent description of
the electronic response properties of atoms and molecules. In a continuous
formalism, with one oscillator at every point in space, coupled oscillators
can describe any response allowed by quantum field theory and thus model the
response of arbitrary molecules or materials Gobre2016 ; Ambrosetti2014 . In
the common practical coarse-grained formalism, with each oscillator
representing one atom, the QDO framework reproduces the leading-order behavior
of the electronic polarizability of atoms Tang1968 , providing an accurate and
reliable description of polarization effects in molecules and materials
Thole1981 ; DiStasio2014 . Moreover, the QDO model allows one to describe
excess electrons in matter Wang2001 and to reproduce dispersion-polarized
electron densities Hermann_NComm_2017 as well as Coulomb interactions between
dipolar quantum fluctuations Stoehr_NComm_2021 .
Extending the applicability of the QDO framework towards a more complete and
systematic description of noncovalent interactions necessitates the
incorporation of the exchange-induced repulsion VanVleet2016 . Recently, we
made a first step in this direction by evaluating the exchange energy between
two QDOs within the dipole approximation of the Coulomb potential Fedorov2018
. Here, we take the next natural step by constructing a common coarse-grained
approach for the multipolar dispersion and exchange interactions in vdW-bonded
atomic or molecular dimers.
It is important to embed our developments of coarse-grained models into the
broader context given by the theory of intermolecular interactions for systems
composed of nuclei and electrons Stone2016 , which states that the equilibrium
geometry of two vdW-bonded atoms or molecules is governed by an interplay of
several interactions. The generalized Heitler-London (GHL) theory Tang1998
offers one of the most compact schemes for the interatomic energy
decomposition. In the GHL approach, isotropic closed-shell atoms only
experience mutual exchange-repulsion and dispersion forces. Another very
successful scheme to describe intermolecular interactions and analyze their
complex interplay is based on the symmetry-adapted perturbation theory (SAPT)
decomposition Jeziorski1994 ; Hesselmann2005 ; Szalewicz2012 . The higher-
level SAPT methods, while being computationally expensive, approach a “gold
standard” accuracy Parker2014 comparable to the coupled-cluster method with
single, double and perturbative triple excitations [CCSD(T)] for small
molecules. Within second-order SAPT, which is the most practical approach, one
obtains six contributions Stone2016 ; VanVleet2016 : (i) electrostatics,
$E_{\rm elst}^{(1)}$; (ii) exchange, $E_{\rm ex}^{(1)}$; (iii) induction,
$E_{\rm ind}^{(2)}$; (iv) exchange-induction, $E_{\rm ex\shortminus
ind}^{(2)}$; (v) dispersion, $E_{\rm disp}^{(2)}$; (vi) exchange-dispersion,
$E_{\rm ex\shortminus disp}^{(2)}$. Here, the superscripts $(1)$ and $(2)$
denote the order of the perturbation theory required to derive the
corresponding term. In the case of neutral and isotropic fragments, the two
induction contributions, $E_{\rm ind}^{(2)}$ and $E_{\rm ex\shortminus
ind}^{(2)}\,$, practically compensate each other Hesselmann2014 . Then, the
problem reduces to four remaining terms, which still yield significant
contributions to the interaction energy for noble gas dimers Shirkov2017 . On
the other hand, the Tang-Toennies (TT) model Tang1995 , relying just on the
exchange-repulsion and dispersion-attraction interactions, is known to
reproduce the binding energy curves of closed-shell dimers with high accuracy
and efficiency Tang2003 . Recently, an extension (TT2) of this model was
proposed Sheng2020 to accurately describe noble gas dimers also at relatively
short internuclear distances. Based on the concepts of the GHL theory for
interatomic interactions Tang1998 , the TT model can be considered as one of
the most compact yet accurate models for closed-shell vdW dimers. According to
the discussion in Ref. Tang1998 , the simplicity of the TT potential arises
due to the used analytical asymptotic form of the exchange energy obtained by
the surface integral method Tang1989 ; Tang1992 . Since this method is known
to deliver the same asymptotic result Tang1991 as the approach based on the
multipole expansion of the perturbing potential Dalgarno1956 ; Morse1953 , the
latter can be used as an alternative way to construct compact TT-like
potentials. This idea is supported by our recent study Fedorov2018 , which
established a quantum-mechanical scaling law, $\alpha_{1}\propto R_{\rm
vdW}^{7}$, between the atomic dipole polarizability and the vdW radius from
the force balance between exchange-repulsion and dispersion attraction at the
equilibrium distance. The corresponding analysis in Ref. Fedorov2018 was
based on the consideration of these two forces stemming from the dipolar term
in the multipole expansion Hirschfelder1967 ; Margenau1971 of the interatomic
Coulomb potential. Subsequently, we have derived Tkatchenko2020 the
proportionality coefficient, which finally led to the relation
$\alpha_{1}=(4\pi\epsilon_{0}/a_{0}^{4})\alpha_{f}^{4/3}R_{\rm vdW}^{7}$, as
expressed in terms of the vacuum permittivity $\epsilon_{0}$, the Bohr radius
$a_{0}$ and the fine-structure constant $\alpha_{f}$. Such a relation is not
trivial because it connects an electronic polarizability of an atom with an
equilibrium distance in a dimer composed of two closed-shell atoms.
In this work, we build on our previous study by going beyond the dipole
approximation and considering further terms in the multipole expansion of the
interatomic Coulomb potential. This is performed for both exchange and
dispersion interactions between closed-shell systems described within the QDO
model. To this end, we investigate the balance between the two types of
forces, which yield the dominant contributions in vdW-bonded systems. For
atomic dimers at the vdW equilibrium distance, this allows us to study a term-
by-term compensation of the attractive (dispersion) and repulsive (exchange)
forces for each contribution in the multipole expansion of the full Coulomb
interaction between the QDOs. This mutual compensation yields a relation
between atomic multipole polarizabilities and the vdW radius as first
empirically obtained in Ref. Fedorov2018 . The presented relation enables a
practical and seamless determination of vdW radii as an effective atomic
length scale from atomic polarizabilities. From the opposite perspective, the
generalized relation also allows one to obtain atomic polarizabilities across
the periodic table and at arbitrary multipole rank based on (pre-tabulated)
vdW radii and without the need to resort to the otherwise challenging direct
computation of electronic response properties. Altogether, our results deliver
deeper insights into the connection between Pauli repulsion and dispersion
attraction — two forces which appear at different orders of SAPT. The
existence of a quantum-mechanical relation between the two main contributions
to the vdW interaction energy at the equilibrium distance reveals a strong
connection between exchange and correlation effects and should have
implications for achieving an improved understanding of the stability of vdW-
bonded matter.
## II Method: Quantum Drude Oscillators
Let us consider two vdW-bonded atoms, $A$ and $B$, separated by a distance $R$
and describe them within the QDO model, as illustrated in Fig. 1. Each of the
two QDOs representing atoms has three effective parameters — mass $\mu$,
charge $q$ and characteristic frequency $\omega$ — which are parametrized to
reproduce three atomic observables $\\{\alpha_{1},{\rm C}_{6},{\rm C}_{8}\\}$
Jones2013 :
$\omega=\frac{4\,{\rm C}_{6}}{3\,\hbar\,\alpha_{1}^{2}}\ ,\ \ \
\mu=\frac{5\,\hbar\,{\rm C}_{6}}{\omega\,{\rm C}_{8}}\ ,\ \ \
q=\sqrt{\mu\omega^{2}\alpha_{1}}\ ,$ (1)
where the Drude (quasi-)particle and the related nucleus have charges $(-q)$
and $q$, respectively. The conditions of Eq. (1) use the dipole polarizability
$\alpha_{1}$ and the dominant dispersion coefficients ${\rm C}_{6}$ (induced-
dipole–induced-dipole interaction) and ${\rm C}_{8}$ (induced-dipole–induced-
quadrupole interactions) in order to parametrize this powerful model, able to
efficiently reproduce long-range forces and electronic response properties of
atoms and molecules.
Figure 1: Schematic representation of two QDOs separated by the distance
$R=|\mathbf{R}|$. The black and white spheres represent the two Drude
particles and fixed nuclei, respectively. The Coulomb interactions between the
two QDOs (gray arrows) and the harmonic intra-QDO potentials (blue arrows) are
explicitly highlighted with their connection to Eqs. (2) and (4).
The Hamiltonian of the interacting QDOs is given by $H=H_{0}+V$, where $V$ is
the interaction and $H_{0}=h_{A}+h_{B}$ consists of the unperturbed QDO
Hamiltonians
$\begin{split}h_{A}(\textbf{r})&=-({\hbar^{2}}/{2\mu_{A}})\bm{\nabla}_{\textbf{r}}^{2}+({\mu_{A}\omega_{A}^{2}}/{2})\textbf{r}^{2}\
,\\\
h_{B}(\textbf{r})&=-({\hbar^{2}}/{2\mu_{B}})\bm{\nabla}_{\textbf{r}}^{2}+({\mu_{B}\omega_{B}^{2}}/{2})(\textbf{r}-\textbf{R})^{2}\
.\end{split}$ (2)
The corresponding wave functions are given by
$\begin{split}\psi_{A}(\textbf{r})&=({\mu_{A}\omega_{A}}/{\pi\hbar})^{\nicefrac{{3}}{{4}}}\,e^{-\frac{\mu_{A}\omega_{A}}{2\hbar}\textbf{r}^{2}}\
,\\\
\psi_{B}(\textbf{r})&=({\mu_{B}\omega_{B}}/{\pi\hbar})^{\nicefrac{{3}}{{4}}}\,e^{-\frac{\mu_{B}\omega_{B}}{2\hbar}(\textbf{r}-\textbf{R})^{2}}\
.\end{split}$ (3)
The full Coulomb interaction between the two QDOs is
$V=\tfrac{q_{A}q_{B}}{4\pi\epsilon_{0}}\left\\{\tfrac{1}{R}+\tfrac{1}{\absolutevalue{\textbf{R}-\textbf{r}_{1}+\textbf{r}_{2}}}-\tfrac{1}{\absolutevalue{\textbf{R}-\textbf{r}_{1}}}-\tfrac{1}{\absolutevalue{\textbf{R}+\textbf{r}_{2}}}\right\\}\
,$ (4)
where $\textbf{r}_{1}$ and $\textbf{r}_{2}$ are the coordinates of the Drude
particles measured from the corresponding fixed nuclei. The Coulomb
interaction can be written as a multipole expansion as Margenau1938
$V=\textstyle\sum\limits_{n=1,2,...}V_{n}=V_{1}+V_{2}+V_{3}+V_{4}+V_{5}+...\
,$ (5)
where $V_{n}\propto R^{-(n+2)}$ with $R=\absolutevalue{\textbf{R}}$.
Furthermore, $n=l_{A}+l_{B}-1$, where $l_{A}$ and $l_{B}$ refer to the rank of
the multipole moments of the two interacting QDOs. Here, we restrict our
consideration to the first five terms in the multipole expansion of Eq. (5).
The first term, $V_{1}\propto R^{-3}$, corresponds to the dipole approximation
of the Coulomb potential, $V_{1}\equiv V_{{\rm
dip}}=\frac{q_{A}q_{B}}{4\pi\epsilon_{0}}\left(\tfrac{(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})}{R^{3}}\\!-\\!\tfrac{3(\textbf{r}_{1}{{\cdot}}\textbf{R})(\textbf{r}_{2}{{\cdot}}\textbf{R})}{R^{5}}\right)$,
describing the dipole-dipole (d-d) electrostatic interaction
($l_{A}=l_{B}=1$). The higher terms arise from the dipole-quadrupole (d-q) for
$V_{2}\propto R^{-4}$, dipole-octupole (d-o) + quadrupole-quadrupole (q-q) for
$V_{3}\propto R^{-5}$, dipole-hexadecapole (d-h) + quadrupole-octupole (q-o)
for $V_{4}\propto R^{-6}$ and dipole-triakontadipole (d-t) + quadrupole-
hexadecapole (q-h) + octupole-octupole (o-o) interactions for $V_{5}\propto
R^{-7}\,$. The formulas for $V_{n}\,$, with $n$ = 2, 3, 4 and 5, are given in
Appendix A. Within the next section, we consider the multipolar contributions
to the dispersion and exchange interaction between two QDOs. The analytical
formulas are derived in the most general form valid in any system of units,
whereas we employ (a.u.) atomic units (with $4\pi\epsilon_{0}=\hbar=1$), to
present our numerical results in Section III.4.
## III Results
### III.1 Dispersion interaction
The multipole expansion has been the starting point for quantum-mechanical
perturbation calculations of the vdW dispersion interactions of Coulomb-
coupled Drude oscillators Wang2001 ; Hermann2017 ; Crosse2009 . Owing to this
approach, the vdW dispersion energies can be expressed in terms of the atomic
multipole polarizabilities (with $l=1,2,...$)
$\alpha_{l}\equiv\alpha_{l,{\rm
QDO}}=\left(\tfrac{q^{2}}{\mu\omega^{2}}\right)\tfrac{(2l-1)!!}{l}\left(\tfrac{\hbar}{2\mu\omega}\right)^{l-1}$
(6)
by using the series expansion Jones2013
$E^{{\rm
AB,disp}}\\!=\\!-\sum_{l_{A}l_{B}}\absolutevalue{T_{l_{A},\,l_{B}}^{AB}}^{2}\,\tfrac{\alpha_{l_{A}}^{A}\alpha_{l_{B}}^{B}}{(4\pi\epsilon_{0})^{2}}\left[\tfrac{\hbar}{4}\tfrac{l_{A}l_{B}\omega_{A}\omega_{B}}{(l_{A}\omega_{A}+l_{B}\omega_{B})}\right]\
,$ (7)
where $T_{l_{A},\,l_{B}}^{AB}$ represents the multipole–multipole interaction-
tensor. We remark that $T_{l_{A},\,l_{B}}^{AB}$ above has been obtained using
a spherical harmonic expansion of the Coulomb potential instead of the
Cartesian multipolar potential described in Appendix A. Both expansions yield
equivalent results Smith1998 ; Lin2015 . In the Supplemental Material of Ref.
Jones2013 , the following spherical components of this tensor were given
$\begin{split}|T_{1,1}|^{2}&=6R^{-6}\,\quad,~{}~{}~{}\,|T_{1,2}|^{2}=15R^{-8}~{}~{}\,,\\\
|T_{1,3}|^{2}&=28R^{-10}\ ,~{}~{}~{}\,|T_{2,2}|^{2}=70R^{-10}\ .\end{split}$
(8)
For our derivations here, we further introduce the higher-order coupling
components via a generalized expression of the multipolar interaction tensor,
$\absolutevalue{T_{l_{A},\,l_{B}}^{AB}}^{2}\equiv\sum\limits_{m_{A}=-l_{A}}^{l_{A}}\,\sum\limits_{m_{B}=-l_{B}}^{l_{B}}|T_{l_{A}m_{A},\,l_{B}m_{B}}(\mathbf{R})|^{2}\
.$ (9)
This tensor is derived based on the approach of Ref. Popelier2001 used by
some of us in Ref. Kleshchonok2018 as well. Popelier _et al._ Popelier2001
employed the relation
$\begin{split}T&{}_{l_{A}m_{A},\,l_{B}m_{B}}(\mathbf{R})=(-1)^{l_{A}}\,\sqrt{\tfrac{(2l_{A}+2l_{B}+1)!}{(2l_{A})!(2l_{B})!}}\\\
&\times\scalebox{0.8}{\mbox{$\displaystyle\begin{pmatrix}l_{A}&l_{B}&l_{A}+l_{B}\\\
m_{A}&m_{B}&-(m_{A}+m_{B})\end{pmatrix}$}}\,I_{l_{A}+l_{B}\,,\,-(m_{A}+m_{B})}(\mathbf{R})\
,\end{split}$ (10)
where the expression in the large parentheses is a Wigner $3j$-symbol
Varshalovich1988 and the irregular normalized spherical harmonics are
$I_{l,m}(\mathbf{r})=\sqrt{\tfrac{4\pi}{2l+1}}\,r^{-l-1}\,Y_{l,m}(\theta,\phi)\,.$
(11)
The spherical harmonics are defined as Varshalovich1988
$Y_{l,m}(\theta,\phi)=\sqrt{\tfrac{2l+1}{4\pi}}\sqrt{\tfrac{(l-m)!}{(l+m)!}}\,P_{l}^{m}(\cos\theta)\,e^{im\phi}\
,$ (12)
where $P_{l}^{m}(\cos\theta)$ are the associated Legendre polynomials. If we
assume now that the distance between two atoms is along the $z$ axis,
$\mathbf{R}=(0,0,R)$, then $\cos\theta=1$. Due to
$P_{l}^{m}(1)=\delta_{m,0}\,$, one can then easily obtain
$Y_{l,m}^{*}(0,\phi)\,Y_{l,m}(0,\phi)=\tfrac{2l+1}{4\pi}\tfrac{(l-m)!}{(l+m)!}\delta_{m,0}\
.$ (13)
Consequently, we have
$\absolutevalue{T_{l_{A},\,l_{B}}^{AB}}^{2}=~{}\tfrac{(2l_{A}+2l_{B}+1)!\sum\limits_{m=-l_{\leq}}^{m=+l_{\leq}}\scalebox{0.8}{\mbox{$\displaystyle\begin{pmatrix}l_{A}&l_{B}&l_{A}+l_{B}\\\
m&-m&0\end{pmatrix}^{2}$}}}{(2l_{A})!\,(2l_{B})!\,R^{2(l_{A}+l_{B}+1)}}\ ,$
(14)
where $l_{\leq}=\\{l_{A}\,,\ {\rm if}\ l_{A}\leq l_{B}\,;\ l_{B}\,,\ {\rm if}\
l_{B}\leq l_{A}\\}$. Now we use the following property of the Wigner
$3j$-symbols Varshalovich1988
$\begin{split}&\scalebox{0.9}{\mbox{$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{1}+j_{2}\\\
m_{1}&m_{2}&-(m_{1}+m_{2})\end{pmatrix}$}}=(-1)^{j_{1}-j_{2}+m_{1}+m_{2}}\\\
&\quad\sqrt{\tfrac{(2j_{1})!(2j_{2})!(j_{1}+j_{2}+m_{1}+m_{2})!(j_{1}+j_{2}-m_{1}-m_{2})!}{(2j_{1}+2j_{2}+1)!(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!}}\
\ ,\end{split}$ (15)
which gives us
$\absolutevalue{T_{l_{A},\,l_{B}}^{AB}}^{2}=\tfrac{\sum\limits_{m=-l_{\leq}}^{m=+l_{\leq}}\frac{(l_{A}+l_{B})!(l_{A}+l_{B})!}{(l_{A}+m)!(l_{A}-m)!(l_{B}+m)!(l_{B}-m)!}\
}{R^{2(l_{A}+l_{B}+1)}}\ .$ (16)
Obviously, $|T_{l_{A},\,l_{B}}^{AB}|^{2}=|T_{l_{B},\ l_{A}}^{AB}|^{2}$.
Therefore, it is enough to derive the components with $l_{A}\leq l_{B}$. This
means
$\displaystyle\absolutevalue{T_{l_{A},\,l_{B}}^{AB}}^{2}=\frac{\sum\limits_{m=0}^{m=+l_{A}}(2-\delta_{m,0})\scalebox{0.8}{\mbox{$\displaystyle\begin{pmatrix}l_{A}+l_{B}\\\
l_{A}-m\end{pmatrix}$}}\scalebox{0.8}{\mbox{$\displaystyle\begin{pmatrix}l_{A}+l_{B}\\\
l_{B}-m\end{pmatrix}$}}\ }{R^{2(l_{A}+l_{B}+1)}}\ ,$ (17)
where the factorials have been rewritten in terms of the binomial coefficients
$\scalebox{0.62}{\mbox{$\displaystyle\begin{pmatrix}n\\\
k\end{pmatrix}$}}=\frac{n!}{(n-k)!\,k!}$ . Then, we obtain
$\displaystyle\begin{split}|T_{1,4}|^{2}&=45R^{-12}\,\quad,\ \ \
|T_{1,5}|^{2}=66R^{-14}\quad,\\\ |T_{2,3}|^{2}&=210R^{-12}~{}\,\,,\ \ \
|T_{2,4}|^{2}=495R^{-14}~{}\,,\\\ |T_{3,3}|^{2}&=924R^{-14}\ \ ,\end{split}$
(18)
in addition to the results of Eq. (8). With the above expressions, the first
few multipolar contributions to the dispersion energy between two oscillators
become
$\displaystyle E_{1({\rm d\shortminus d})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{3k_{e}^{2}}{2R^{6}}\,\alpha_{1}^{A}\alpha_{1}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+\omega_{B}}\
,$ $\displaystyle E_{2({\rm d\shortminus q})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{15k_{e}^{2}}{2R^{8}}\left[\alpha_{1}^{A}\alpha_{2}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+2\omega_{B}}+\alpha_{2}^{A}\alpha_{1}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{2\omega_{A}+\omega_{B}}\right]\
,$ $\displaystyle E_{3({\rm d\shortminus o})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{21k_{e}^{2}}{R^{10}}\left[\alpha_{1}^{A}\alpha_{3}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+3\omega_{B}}+\alpha_{3}^{A}\alpha_{1}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{3\omega_{A}+\omega_{B}}\right]\
,$ $\displaystyle E_{3({\rm q\shortminus q})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{35k_{e}^{2}}{R^{10}}\,\alpha_{2}^{A}\alpha_{2}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+\omega_{B}}\
,$ $\displaystyle E_{4({\rm d\shortminus h})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{45k_{e}^{2}}{R^{12}}\left[\alpha_{1}^{A}\alpha_{4}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+4\omega_{B}}+\alpha_{4}^{A}\alpha_{1}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{4\omega_{A}+\omega_{B}}\right]\
,$ $\displaystyle E_{4({\rm q\shortminus o})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{315k_{e}^{2}}{R^{12}}\left[\alpha_{2}^{A}\alpha_{3}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{2\omega_{A}+3\omega_{B}}+\alpha_{3}^{A}\alpha_{2}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{3\omega_{A}+2\omega_{B}}\right]\
,$ $\displaystyle E_{5({\rm d\shortminus t})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{165k_{e}^{2}}{2R^{14}}\left[\alpha_{1}^{A}\alpha_{5}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+5\omega_{B}}+\alpha_{5}^{A}\alpha_{1}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{5\omega_{A}+\omega_{B}}\right]\
,$ $\displaystyle E_{5({\rm q\shortminus h})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{495k_{e}^{2}}{R^{14}}\left[\alpha_{2}^{A}\alpha_{4}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+2\omega_{B}}+\alpha_{4}^{A}\alpha_{2}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{2\omega_{A}+\omega_{B}}\right]\
,$ $\displaystyle E_{5({\rm o\shortminus o})}^{{\rm AB,disp}}$
$\displaystyle=-\tfrac{693k_{e}^{2}}{R^{14}}\,\alpha_{3}^{A}\alpha_{3}^{B}\tfrac{\hbar\omega_{A}\omega_{B}}{\omega_{A}+\omega_{B}}\
\ ,$ (19)
where $k_{e}=(4\pi\epsilon_{0})^{-1}$ is the Coulomb constant.
Based on the above formulas, we can now rewrite the dispersion energy in its
conventional expansion Stone2016 ; Jones2013
$E^{{\rm
AB,disp}}=\\!\\!\\!\textstyle\sum\limits_{n=1,2,...}\\!\\!\\!E_{n}^{{\rm
AB,disp}}=\\!\\!\\!\textstyle\sum\limits_{n=1,2,...}\\!\\!\\!-\tfrac{{\rm
C}_{(2n+4)}^{{\rm AB}}}{R^{(2n+4)}}\ \ ,$ (20)
where ${\rm C}_{(2n+4)}^{{\rm AB}}$ are the dispersion coefficients and all
the contributions to $E_{n}^{{\rm AB,disp}}$, with $n$ up to 5, are given by
Eq. (III.1). Equation (20) arises from second-order perturbation theory with
the multipole expansion of the Coulomb potential, as an interaction potential
between spherically symmetric atoms. The leading term is the dipole-dipole
(d-d) interaction, $E_{1}^{{\rm AB,disp}}\propto R^{-6}$, stemming from the
dipolar potential, $V_{{\rm dip}}\propto R^{-3}$. The higher-order terms in
the multipole expansion of the Coulomb interaction yield the dispersion
energies $E_{2}^{{\rm AB,disp}}\propto R^{-8}$, $E_{3}^{{\rm AB,disp}}\propto
R^{-10}$, $E_{4}^{{\rm AB,disp}}\propto R^{-12}$ and $E_{5}^{{\rm
AB,disp}}\propto R^{-14}$ coming, respectively, from the instantaneous dipole-
quadrupole (d-q), dipole-octupole (d-o) and quadrupole-quadrupole (q-q),
dipole-hexadecapole (d-h) and quadrupole-octupole (q-o) interactions, and
dipole-triakontadipole (d-t), quadrupole-hexadecapole (q-h) and octupole-
octupole (o-o) interactions. For non-centrosymmetric molecules, Eq. (20) would
have terms with odd powers in $R$ starting with $\propto R^{-7}$ Stone2016 .
However, here we restrict our consideration to vdW-bonded atoms assumed to
possess closed valence-electron shells with a spherically-symmetric charge
density, for which the dispersion terms proportional to $1/R^{2i+1}$, with
$i\in\mathbb{N}$, vanish.
### III.2 Exchange-repulsion interaction
The above derivation of the dispersion energy was performed for the general
case of two QDOs with arbitrary parameters, however the description of the
exchange-repulsion between two QDOs is more subtle. The exchange interaction
should obviously be present for two different QDOs, as caused by the Pauli
repulsion between electrons constituting the two Drude particles. Nonetheless,
in order to construct the exchange interaction, one needs to deal with
indistinguishable particles, a concept that requires generalization for two
Drude particles possessing different parameters. Our starting assumption is
that the exchange energy should be proportional to the overlap integral $S$
between the wave functions of two different QDOs, similar to the case of two
identical Drude particles Fedorov2018 . This assumption was recently employed
Silvestrelli2019 for a simplified generalization of the coarse-grained
dipole-dipole exchange energy of a homonuclear dimer, $E^{{\rm ex}}_{({\rm
d\shortminus d})}\approx k_{e}q^{2}S/2R$, derived in Ref. Fedorov2018 . The
authors of Ref. Silvestrelli2019 have simply replaced the overlap integral
$S$ of two identical QDOs by its counterpart obtained for different QDOs and
shown that already such a simplified treatment improves their computational
scheme for vdW dispersion interactions. However, due to the coarse-grained
treatment of valence electrons within the QDO model, care needs to be taken
for the most general definition of the exchange energy between QDOs. This is a
subject of our ongoing studies. Here, we follow the approach of Ref.
Fedorov2018 and derive multipole contributions to the exchange energy of two
identical QDOs.
Formally, we consider two indistinguishable Drude particles
($\mu=\mu_{A}=\mu_{B}\,$, $\omega=\omega_{A}=\omega_{B}\,$ and
$q=q_{A}=q_{B}$) as bosons assuming that they represent closed valence shells
with vanishing total spin. Therefore, the total wave function of a dimer
should be written as a permanent
$\Psi(\textbf{r}_{1},\textbf{r}_{2})=\tfrac{1}{\sqrt{2}}\big{(}\psi_{A}(\textbf{r}_{1})\psi_{B}(\textbf{r}_{2})+\psi_{A}(\textbf{r}_{2})\psi_{B}(\textbf{r}_{1})\big{)}\
.$ (21)
By employing the Heitler-London perturbation theory Heitler1927 ; Slater1965 ,
the exchange energy for two identical vdW-bonded QDOs at their equilibrium
distance becomes well approximated with its exact asymptotic result given by
the exchange integral Fedorov2018
$\displaystyle J^{{\rm
ex}}=\matrixelement{\psi_{A}(\textbf{r}_{1})\psi_{B}(\textbf{r}_{2})}{V}{\psi_{A}(\textbf{r}_{2})\psi_{B}(\textbf{r}_{1})}\
.$ (22)
The evaluation of Eq. (22) with the expansion of Eq. (5) results in multipole
contributions to the exchange energy, where each of them is directly
proportional to the overlap integral defined as
$S=\absolutevalue{\innerproduct{\psi_{A}}{\psi_{B}}}^{2}=e^{-\frac{\mu\omega}{2\hbar}R^{2}}\
.$ (23)
For the dipole-dipole contribution, $V_{1}\propto R^{-3}$, we obtain
$J_{1({\rm d\shortminus d})}^{{\rm ex}}=\tfrac{k_{e}q^{2}S}{2R}\ ,$ (24)
which reproduces the result of Ref. Fedorov2018 . Now, we evaluate further
contributions going beyond the dipole approximation. For the dipole-quadrupole
interaction, described by the second term, $V_{2}\propto R^{-4}$, in the
multipole expansion of the Coulomb potential, we derive
$J^{{\rm ex}}_{2({\rm d\shortminus q})}=\tfrac{3k_{e}q^{2}S}{4R}\ .$ (25)
Then, the next term, $V_{3}\propto R^{-5}$, has two contributions, $V_{3({\rm
d\shortminus o})}$ and $V_{3({\rm q\shortminus q})}\,$, related to the dipole-
octupole and the quadrupole-quadrupole interaction Margenau1971 ; Merwe1967 ,
respectively. The corresponding exchange integrals are obtained as
$\begin{split}J^{{\rm ex}}_{3({\rm d\shortminus o})}&=\tfrac{k_{e}q^{2}S}{2R}\
,\\\ J^{{\rm ex}}_{3({\rm q\shortminus
q})}&=\tfrac{3k_{e}q^{2}S}{8R}\left(1\\!-\\!\tfrac{\hbar}{R^{2}\mu\omega}\\!-\\!\tfrac{\hbar^{2}}{R^{4}\mu^{2}\omega^{2}}\right).\end{split}$
(26)
Further on, we have two contributions from $V_{4}\propto R^{-6}$, the dipole-
hexadecapole (d-h) and the quadrupole-octupole (q-o) interaction. The related
exchange integrals are
$\begin{split}J^{{\rm ex}}_{4({\rm d\shortminus
h})}&=\tfrac{5k_{e}q^{2}S}{16R}\left(1\\!-\\!\tfrac{3\hbar}{R^{2}\mu\omega}\\!-\\!\tfrac{9\hbar^{2}}{R^{4}\mu^{2}\omega^{2}}\right)\
,\\\ J^{{\rm ex}}_{4({\rm q\shortminus o})}&=\tfrac{5k_{e}q^{2}S}{8R}\
.\end{split}$ (27)
Finally, for the dipole-triakontadipole (d-t), quadrupole-hexadecapole (q-h)
and octupole-octupole (o-o) interactions, from $V_{5}\propto R^{-7}$, we
obtain
$\displaystyle J^{{\rm ex}}_{5({\rm d\shortminus t})}$
$\displaystyle=\tfrac{3k_{e}q^{2}S}{16R}\left(1\\!-\\!\tfrac{15\hbar}{R^{2}\mu\omega}\\!-\\!\tfrac{105\hbar^{2}}{R^{4}\mu^{2}\omega^{2}}\right)\,$
$\displaystyle J^{{\rm ex}}_{5({\rm o\shortminus o})}$
$\displaystyle=\tfrac{5k_{e}q^{2}S}{16R}\,,$ (28) $\displaystyle J^{{\rm
ex}}_{5({\rm q\shortminus h})}$
$\displaystyle=\tfrac{15k_{e}q^{2}S}{32R}\left(1\\!-\\!\tfrac{7\hbar}{4R^{2}\mu\omega}\\!-\\!\tfrac{35\hbar^{2}}{4R^{4}\mu^{2}\omega^{2}}\\!-\\!\tfrac{21\hbar^{3}}{2R^{6}\mu^{3}\omega^{3}}\right).$
According to Eqs. (24) and (25), $J^{{\rm ex}}_{2({\rm d\shortminus q})}$ is
larger than $J^{{\rm ex}}_{1({\rm d\shortminus d})}$ for all interatomic
distances. This is in contrast to the dispersion contributions, where $E^{{\rm
disp}}_{1({\rm d\shortminus d})}$ clearly dominates at large distances. Such a
nonmonotonic behavior of the multipole contributions, as we obtain here for
the exchange energy, was also found in Ref. Chalasinski1977 for the multipole
expansion of the exchange-dispersion energy.
Now, we will use the derived dominant multipole contributions to the
dispersion and exchange energies, in order to study the balance of the
corresponding forces at the equilibrium distance in homonuclear dimers.
### III.3 Force balance between multipolar dispersion and exchange
contributions
The equilibrium geometry of atomic or molecular systems is dictated by the
condition that the net forces acting on each atom vanish. Therefore, for two
atoms or molecules separated by a distance $R$, this condition is determined
by $\mathbf{F}_{\rm net}(R_{\rm eq})=-\bm{\nabla}_{\mathbf{R}}E_{\rm
tot}(R)|_{R=R_{\rm eq}}=0$, where $R_{\rm eq}$ represents the equilibrium
distance and $E_{\rm tot}$ is the total interaction energy. The structure,
stability and dynamics of vdW-bonded atomic dimers are governed by the
interplay between the dispersion and exchange interactions Tang1998 . This
means that at $R=R_{\rm eq}\,$ the two respective forces have to mutually
compensate each other. In what follows, we consider such a compensation by
going beyond the dipole approximation for the interaction, in order to obtain
higher-order multipole contributions to the attractive and repulsive forces.
At the equilibrium distance, $R_{{\rm eq}}=2R_{\rm vdW}$, in homonuclear
dimers composed of two identical Drude particles ($\mu=\mu_{A}=\mu_{B}$,
$\omega=\omega_{A}=\omega_{B}$ and $q=q_{A}=q_{B}$), the exchange force can be
well approximated by the expression $F^{{\rm ex}}\approx-\nabla_{R}\,J^{{\rm
ex}}$, according to Ref. Fedorov2018 and our discussion above. In addition,
at the internuclear distances comparable to or larger than the equilibrium
one, $R\gg\sqrt{\hbar/\mu\omega}$ Fedorov2018 . Then, the corresponding
multipole contributions to the exchange force are obtained as
$\displaystyle F_{1({\rm d\shortminus d})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{\alpha_{1}\hbar\omega
S}{2(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $
$\displaystyle,\ \ F_{2({\rm d\shortminus q})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{3\alpha_{1}\hbar\omega
S}{4(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $ ,
$\displaystyle F_{3({\rm d\shortminus o})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{\alpha_{1}\hbar\omega
S}{2(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $
$\displaystyle,\ \ F_{3({\rm q\shortminus q})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{3\alpha_{1}\hbar\omega
S}{8(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $ ,
$\displaystyle F_{4({\rm d\shortminus h})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{5\alpha_{1}\hbar\omega
S}{16(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $
$\displaystyle,\ \ F_{4({\rm q\shortminus o})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{5\alpha_{1}\hbar\omega
S}{8(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $ ,
$\displaystyle F_{5({\rm d\shortminus t})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{3\alpha_{1}\hbar\omega
S}{16(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $
$\displaystyle,\ \ F_{5({\rm q\shortminus h})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{15\alpha_{1}\hbar\omega
S}{32(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $ ,
$\displaystyle F_{5({\rm o\shortminus o})}^{{\rm ex}}$
$\displaystyle\approx\tfrac{5\alpha_{1}\hbar\omega
S}{16(4\pi\epsilon_{0})}\left(\tfrac{\mu\omega}{\hbar}\right)^{2}\ $ . (29)
From Eq. (III.1) we calculate the multipole contributions to the dispersion
force (for homonuclear dimers)
$\displaystyle F_{1({\rm d\shortminus d})}^{{\rm disp}}$
$\displaystyle=-\tfrac{9\alpha_{1}\alpha_{1}\hbar\omega}{2R^{7}(4\pi\epsilon_{0})^{2}}\
$ $\displaystyle,\ \ F_{2({\rm d\shortminus q})}^{{\rm disp}}$
$\displaystyle=-\tfrac{40\alpha_{1}\alpha_{2}\hbar\omega}{R^{9}(4\pi\epsilon_{0})^{2}}\
$ , $\displaystyle F_{3({\rm d\shortminus o})}^{{\rm disp}}$
$\displaystyle=-\tfrac{105\alpha_{1}\alpha_{3}\hbar\omega}{R^{11}(4\pi\epsilon_{0})^{2}}\
$ $\displaystyle,\ \ F_{3({\rm q\shortminus q})}^{{\rm disp}}$
$\displaystyle=-\tfrac{175\alpha_{2}\alpha_{2}\hbar\omega}{R^{11}(4\pi\epsilon_{0})^{2}}\
$ , $\displaystyle F_{4({\rm d\shortminus h})}^{{\rm disp}}$
$\displaystyle=-\tfrac{216\alpha_{1}\alpha_{4}\hbar\omega}{R^{13}(4\pi\epsilon_{0})^{2}}\
$ $\displaystyle,\ \ F_{4({\rm q\shortminus o})}^{{\rm disp}}$
$\displaystyle=-\tfrac{1512\alpha_{2}\alpha_{3}\hbar\omega}{R^{13}(4\pi\epsilon_{0})^{2}}\
$ , $\displaystyle F_{5({\rm d\shortminus t})}^{{\rm disp}}$
$\displaystyle=-\tfrac{385\alpha_{1}\alpha_{5}\hbar\omega}{R^{15}(4\pi\epsilon_{0})^{2}}\
$ $\displaystyle,\ \ F_{5({\rm q\shortminus h})}^{{\rm disp}}$
$\displaystyle=-\tfrac{4620\alpha_{2}\alpha_{4}\hbar\omega}{R^{15}(4\pi\epsilon_{0})^{2}}\
$ , $\displaystyle F_{5({\rm o\shortminus o})}^{{\rm disp}}$
$\displaystyle=-\tfrac{4851\alpha_{3}\alpha_{3}\hbar\omega}{R^{15}(4\pi\epsilon_{0})^{2}}\
$ . (30)
At $R=R_{\rm eq}$, the attractive and repulsive forces should cancel each
other. Within the dipole approximation, from the force balance, $F_{1({\rm
d\shortminus d})}^{{\rm disp}}+F_{1({\rm d\shortminus d})}^{{\rm ex}}=0$, one
obtains
$\displaystyle\frac{9\alpha_{1}}{(4\pi\epsilon_{0})R_{\rm
eq}^{7}}=\left(\frac{\mu\omega}{\hbar}\right)^{2}e^{-\frac{\mu\omega}{2\hbar}R_{\rm
eq}^{2}}\ .$ (31)
This formula not only expresses a relation between $\alpha_{1}$ and $R_{\rm
vdW}=R_{\rm eq}/2$, but also contains the QDO parameters $\mu$ and $\omega$,
which are not uniquely defined for atoms. To obtain a formula connecting
atomic parameters $R_{\rm vdW}$ and $\alpha_{1}$, we rewrite Eq. (31) as
$\displaystyle\frac{9\alpha_{1}}{2^{5}(4\pi\epsilon_{0})R_{\rm
vdW}^{7}}=\frac{e^{-(R_{\rm vdW}/\sigma_{\rm QDO})^{2}}}{\sigma_{\rm
QDO}^{4}}\ ,$ (32)
where $\sigma_{\rm QDO}=\sqrt{\hbar/2\mu\omega}$ is the spatial variance or
spread of a QDO. Within the QDO model, $\sigma_{\rm QDO}$ describes an
effective atomic length, which corresponds to the Bohr radius in case of the
hydrogen atom Szabo2020 . According to Ref. Fedorov2018 , the ratio $R_{\rm
vdW}/\sigma_{\rm QDO}$ decreases with increasing $\sigma_{\rm QDO}$ and the
factors $\sigma_{\rm QDO}^{4}$ and $e^{-(R_{\rm vdW}/\sigma_{\rm QDO})^{2}}$
in Eq. (32) compensate each other. This compensation allows the QDO model to
approximately capture the constant behavior of the ratio $\alpha_{1}/R_{\rm
vdW}^{7}$ confirmed empirically for many atoms Fedorov2018 . Therefore, within
the QDO model, the relation between the vdW radius and the dipole
polarizability can be expressed as
$R_{\rm vdW}=A_{1}(\mu\omega,R_{\rm vdW})\,\alpha_{1}^{\nicefrac{{1}}{{7}}}\
,$ (33)
where the proportionality coefficient, as a function of the product
$\mu\omega$ and the vdW radius, is given by
$A_{1}^{\mu\omega}\equiv A_{1}(\mu\omega,R_{\rm
vdW})=\frac{3^{\nicefrac{{2}}{{7}}}}{2(4\pi\epsilon_{0})^{\nicefrac{{1}}{{7}}}}\left(\frac{\hbar}{\mu\omega}\,e^{\frac{\mu\omega
R^{2}_{\rm vdW}}{\hbar}}\right)^{\nicefrac{{2}}{{7}}}\,.$ (34)
As was discussed in Ref. Fedorov2018 , this coefficient can be also written in
terms of the radial volume
$\displaystyle V_{r}=\int
r^{3}\,n_{0}(\mathbf{r})\,d\mathbf{r}=\frac{4}{\sqrt{\pi}}\left(\frac{\hbar}{\mu\omega}\right)^{\nicefrac{{3}}{{2}}}$
(35)
occupied by the ground-state charge density of the QDO,
$n_{0}(\mathbf{r})\equiv|\Psi_{0}(\mathbf{r})|^{2}=(\frac{\mu\omega}{\pi\hbar})^{\nicefrac{{3}}{{2}}}e^{-\frac{\mu\omega}{\hbar}r^{2}}$,
and its value at the vdW radius, $n_{0}(R_{\rm vdW})$, as
$\displaystyle
A_{1}^{\mu\omega}=\frac{3^{\nicefrac{{2}}{{7}}}}{2(4\pi\epsilon_{0})^{\nicefrac{{1}}{{7}}}}\left[\left(\frac{4}{\pi^{5}}\right)^{\nicefrac{{1}}{{3}}}\frac{1}{n_{0}(R_{\rm
vdW})V_{r}^{\nicefrac{{1}}{{3}}}}\right]^{\nicefrac{{2}}{{7}}}\ .$ (36)
Taking into account that $A_{1}$ was found to be essentially a constant for 72
atoms in the periodic table Fedorov2018 , Eq. (36) suggests a relation between
an atomic volume and the electron charge density at the vdW radius.
The results of Eqs. (34) and (36) are based on taking into account only the
first term, $V_{1}\propto R^{-3}$, in the expansion of Eq. (5) for the Coulomb
potential. However, it is well-known that at least two further terms,
$V_{2}\propto R^{-4}$ and $V_{3}\propto R^{-5}$, are important to properly
describe the binding curves of vdW-bonded atomic dimers Tang2003 . Therefore,
here we consider an extension of Eq. (33) by including the higher-order
multipole terms from the expansion of the Coulomb potential. To this end, we
evaluate the individual multipole contributions to the dispersion and exchange
forces given by Eqs. (III.3) and (III.3), respectively, at the equilibrium
distance of vdW-bonded dimers. Based on the empirical findings of Ref.
Fedorov2018 , the mutual compensation of multipolar dispersion and exchange
forces can ultimately be represented by the general expression
$R_{\rm vdW}=A_{l}^{\mu\omega}\,\alpha_{l}^{\nicefrac{{2}}{{7(l+1)}}}\ ,$ (37)
which extends Eq. (33) to the multipole polarizabilities, $\alpha_{l}$. One
can also rewrite Eq. (37) in the following way
$\displaystyle\alpha_{l}=(R_{\rm
vdW}/A_{l}^{\mu\omega})^{\nicefrac{{7(l+1)}}{{2}}}\ \ ,$ (38)
where each multipole polarizability is expressed in terms of the vdW radius.
This allows one to obtain $\alpha_{l}$ either from $R_{\rm vdW}$ or
$\alpha_{1}$, for an arbitrary $l$. Since first-principles calculations of
higher-order polarizabilities are computationally demanding Lao2020 , our
finding provides an alternative way to approximate multipole polarizabilities.
Based on the expressions for the multipolar dispersion and exchange forces, we
can now also explicitly calculate the proportionality coefficients
$A_{l}^{\mu\omega}\equiv A_{l}(\mu\omega,R_{\rm vdW})$. As a particularly
helpful example, we can consider the force balance condition for the dipole-
multipole interaction, i.e., the $F_{l({\rm d}\shortminus z(l))}$-terms of
Eqs. (III.3) and (III.3) with $z(1)={\rm d}$, $z(2)={\rm q}$, $z(3)={\rm o}$,
$z(4)={\rm h}$ and $z(5)={\rm t}$. The resulting proportionality coefficients
of this series can be cast into a compact generalized formula
$A_{l}^{\mu\omega}=\left(\tfrac{D_{l}}{4\pi\epsilon_{0}}\right)^{\frac{2}{7(l+1)}}R_{\rm
vdW}^{\frac{3(l-1)}{7(l+1)}}\left(\frac{\hbar}{\mu\omega}\,e^{\frac{\mu\omega
R^{2}_{\rm vdW}}{\hbar}}\right)^{\frac{4}{7(l+1)}}\ ,$ (39)
with the $l$-dependent rational constants
$D_{l}=\frac{l(l+2)(2l+1)}{2^{l+5}(l+1)}$. Equation (39) generalizes Eq. (34)
and shows an additional factor of $R_{\rm vdW}^{{3(l-1)}/{7(l+1)}}$ arising
for $l>1$. It is worth noting that the derived $A_{l}^{\mu\omega}$ within the
QDO model formally still contain $R_{\rm vdW}$. The values for
$A_{l}^{\mu\omega}$ calculated from Eq. (39), however, remain almost constant
for any choice of realistic parameters (cf. Table 1) as was also observed for
the corresponding empirical proportionality factors Fedorov2018 . Moreover, in
terms of the quantities related to $l=1$, the above expression can be
simplified even further
$\displaystyle
A_{l}^{\mu\omega}=\left(\tfrac{D_{l}}{D_{1}}\right)^{\frac{2}{7(l+1)}}R_{\rm
vdW}^{\frac{3(l-1)}{7(l+1)}}\left(A_{1}^{\mu\omega}\right)^{\frac{2}{l+1}}\ .$
(40)
The general formula given by Eq. (39) allows us to obtain the proportionality
coefficients $A_{l}^{\mu\omega}$ for every order in the dipole-multipole
interactions, even without deriving further multipolar contributions to the
dispersion and exchange forces.
The presented findings based on the dipole-multipole interaction can be
generalized via the force balance at each order of the multipole expansion as
we highlight for the quadrupole-quadrupole and octupole-octupole interactions
in Appendix B. Alternatively, one can use the general expression for the QDO
multipole polarizabilities given by Eq. (6), in order to derive $A_{l}^{{\rm
QDO}}=R_{\rm vdW}/\alpha_{l,{\rm QDO}}^{2/7(l+1)}$ by means of Eq. (37). A
comparison between the two approaches to the proportionality coefficients
$A_{l}$ is given in the following section.
### III.4 Assessment of our formalism for atoms
Table 1: Comparison between the reference ratios $A_{l}^{\rm ref}=R_{\rm vdW}^{\rm ref}/\alpha_{l,{\rm ref}}^{2/7(l+1)}$ for $l=\\{1,2,3\\}$ and their QDO counterparts $A_{l}^{\rm QDO}=R_{\rm vdW}^{\rm ref}/\alpha_{l,{\rm QDO}}^{2/7(l+1)}$ for $l=\\{1,2,3,4,5\\}$ versus the proportionality coefficients $A_{l}^{\mu\omega}$ for $l=\\{1,2,3,4,5\\}$ given by Eq. (39). For alkali and alkaline-earth elements, we use $R_{\rm vdW}^{\rm ref}$ from the recent database of Batsanov Batsanov2001 . For noble gas atoms and hydrogen, missing in Ref. Batsanov2001 , the reference vdW radii are taken from Refs. Bondi1964 and Tkatchenko2009 , respectively. The QDO parameters $\\{q,\mu,\omega\\}$ are set according to Eq. (1), to reproduce $\alpha_{1}^{\rm ref}$ as well as the homoatomic dispersion coefficients ${\rm C}_{6}$ and ${\rm C}_{8}$. The three fitted quantities together with the reference quadrupole ($\alpha_{2}^{\rm ref}$) and octupole ($\alpha_{3}^{\rm ref}$) polarizabilities are taken from Refs. Fedorov2018 ; Gould2016 for the noble gases (He, Ne, Ar, Kr, Xe), from Refs. Derevianko1999 ; Porsev2003 for the elements in Group I (H, Li, Na, K, Rb, Cs), and from Ref. Porsev2006 for the elements in Group II (Be, Mg, Ca, Sr, Ba). The QDO multipole polarizabilities are obtained from Eq. (6), where $\alpha_{1}^{\rm QDO}\equiv\alpha_{1}^{\rm ref}$ due to the QDO fitting procedure Jones2013 leading to $A_{1}^{\rm QDO}\equiv A_{1}^{\rm ref}$. The average values $\langle A_{l}\rangle$ are calculated based on the results of the noble gas atoms. The standard deviation $\sigma=\sqrt{1/N\,\sum_{X}(A_{l}[X]-\langle A_{l}\rangle)^{2}}$ and its mean absolute relative deviation (MARD), $1/N\,\sum_{X}\absolutevalue{A_{l}[X]-\langle A_{l}\rangle}/\langle A_{l}\rangle$, are also reported. All quantities are in atomic units (MARD in %). Atom | $A_{1}^{\rm ref}$ | $A_{2}^{\rm ref}$ | $A_{3}^{\rm ref}$ | $A_{1}^{\rm QDO}$ | $A_{2}^{\rm QDO}$ | $A_{3}^{\rm QDO}$ | $A_{4}^{\rm QDO}$ | $A_{5}^{\rm QDO}$ | $A_{1}^{\mu\omega}$ | $A_{2}^{\mu\omega}$ | $A_{3}^{\mu\omega}$ | $A_{4}^{\mu\omega}$ | $A_{5}^{\mu\omega}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
He | 2.53 | 2.43 | 2.24 | 2.53 | 2.48 | 2.32 | 2.17 | 2.05 | 2.33 | 2.10 | 1.93 | 1.82 | 1.74
Ne | 2.53 | 2.44 | 2.26 | 2.53 | 2.53 | 2.38 | 2.24 | 2.12 | 2.57 | 2.27 | 2.07 | 1.94 | 1.85
Ar | 2.52 | 2.44 | 2.27 | 2.52 | 2.51 | 2.37 | 2.23 | 2.11 | 2.33 | 2.18 | 2.06 | 1.96 | 1.89
Kr | 2.55 | 2.47 | 2.29 | 2.55 | 2.55 | 2.40 | 2.26 | 2.14 | 2.35 | 2.22 | 2.10 | 2.01 | 1.94
Xe | 2.54 | 2.45 | 2.27 | 2.54 | 2.52 | 2.37 | 2.23 | 2.10 | 2.28 | 2.20 | 2.10 | 2.02 | 1.96
H | 2.50 | 2.40 | 2.19 | 2.50 | 2.43 | 2.26 | 2.11 | 2.09 | 2.06 | 1.97 | 1.88 | 1.80 | 1.75
Li | 2.40 | 2.49 | 2.33 | 2.40 | 2.48 | 2.38 | 2.26 | 1.99 | 2.50 | 2.41 | 2.29 | 2.20 | 2.14
Na | 2.53 | 2.55 | 2.40 | 2.53 | 2.56 | 2.43 | 2.30 | 2.17 | 2.50 | 2.42 | 2.32 | 2.23 | 2.17
K | 2.54 | 2.54 | 2.41 | 2.54 | 2.55 | 2.42 | 2.28 | 2.20 | 2.53 | 2.47 | 2.38 | 2.29 | 2.23
Rb | 2.61 | 2.58 | 2.46 | 2.61 | 2.60 | 2.46 | 2.31 | 2.22 | 2.58 | 2.57 | 2.42 | 2.34 | 2.27
Cs | 2.65 | 2.58 | 2.48 | 2.65 | 2.62 | 2.46 | 2.31 | 2.22 | 2.62 | 2.56 | 2.46 | 2.38 | 2.31
Be | 2.51 | 2.45 | 2.34 | 2.51 | 2.47 | 2.32 | 2.17 | 2.05 | 2.22 | 2.17 | 2.08 | 2.01 | 1.96
Mg | 2.49 | 2.42 | 2.32 | 2.49 | 2.45 | 2.29 | 2.15 | 2.03 | 2.25 | 2.21 | 2.13 | 2.07 | 2.01
Ca | 2.55 | 2.44 | 2.39 | 2.55 | 2.50 | 2.33 | 2.18 | 2.06 | 2.38 | 2.34 | 2.26 | 2.19 | 2.13
Sr | 2.61 | 2.49 | 2.43 | 2.61 | 2.55 | 2.37 | 2.22 | 2.09 | 2.45 | 2.41 | 2.32 | 2.25 | 2.19
Ba | 2.59 | 2.42 | 2.41 | 2.59 | 2.51 | 2.34 | 2.18 | 2.05 | 2.48 | 2.44 | 2.36 | 2.28 | 2.22
${\langle A_{l}\rangle}$ | 2.54 | 2.45 | 2.27 | 2.54 | 2.52 | 2.37 | 2.23 | 2.10 | 2.37 | 2.19 | 2.05 | 1.95 | 1.88
${\sigma}$ | 0.05 | 0.06 | 0.08 | 0.05 | 0.05 | 0.05 | 0.06 | 0.06 | 0.15 | 0.16 | 0.17 | 0.17 | 0.18
MARD [%] | 1.60 | 1.80 | 3.90 | 1.60 | 1.61 | 1.87 | 2.20 | 2.47 | 5.41 | 7.13 | 8.88 | 10.14 | 11.05
In the previous sections, we have presented a coarse-grained approach to
describe dispersion and exchange interactions between two closed-shell atoms
within the QDO model. Here, we examine the applicability of the presented
formulas and apply them to analyze the ratio between the vdW radius and
multipole polarizabilities for atoms, thus demonstrating the validity of the
scaling law of Eq. (37) obtained within the QDO model. Our analysis will be
focused on hydrogen, noble gases from He to Xe, alkali atoms from Li to Cs,
and alkaline-earth elements from Be to Ba. To this end, the atomic multipole
polarizabilities, $\alpha_{l}$, are either taken from high-level _ab initio_
calculations in the literature Derevianko1999 ; Porsev2003 ; Porsev2006 ,
$\alpha_{l}^{\rm ref}$, or calculated by means of Eq. (6), $\alpha_{l}^{\rm
QDO}$. We determine $q$, $\mu$ and $\omega$ for each atom by means of Eq. (1),
using accurate reference data for the set $\\{\alpha_{1}^{\rm ref},{\rm
C}_{6},{\rm C}_{8}\\}$ Fedorov2018 ; Gould2016 ; Derevianko1999 ; Porsev2003 ;
Porsev2006 , as explained in Section II. Here, due to the fact that the QDO
parameters are set to reproduce the dipole polarizability, we have
$\alpha_{1}^{\rm QDO}\equiv\alpha_{1}^{\rm ref}$, for all considered atoms.
While eventually it would be interesting and important to extend our analysis
to a broader set of atoms and small molecules, we are not aware of a
comprehensive set of accurate data for atomic and molecular multipole
polarizabilities. Accurate ab initio reference calculations of $\alpha_{l}$ in
general require demanding computational approaches with sophisticated
treatment of electron correlation effects and, especially with increasing
order $l$, large and diffuse basis sets Woon1994 ; Lao2018 ; Lao2020 . As a
result, calculating converged multipolar polarizabilities is difficult, a
problem which is further enhanced by the numerical aspects associated with
finite-field derivative techniques as used in such calculations. Experimental
determination, on the other hand, is subject to origin and orientational
dependencies as well as a strong influence of thermal effects Bishop1990 ;
Kuemmel2000 ; Lao2020 , which can introduce considerable uncertainties — in
particular with increasing system size or multipole order.
To apply the derived formulas, a set of reference vdW radii, $R_{\rm vdW}^{\rm
ref}$, is required. In the case of alkali as well as alkaline-earth elements,
these radii are taken from the recent database of Batsanov Batsanov2001 . For
noble-gas atoms, missing in Ref. Batsanov2001 , we use the database of Bondi
Bondi1964 , which provides often used values of vdW radii for Group 18 of the
periodic table. In addition, for hydrogen, we use $R_{\rm vdW}^{\rm ref}=3.1$
a.u. from Ref. Tkatchenko2009 , where it was theoretically estimated based on
the atomic charge density. This value was shown to work well for the relation
between the atomic dipole polarizability and vdW radius in Ref. Fedorov2018 .
Both, the vdW radii of Batsanov and Bondi, are extracted from experimental
crystallographic structural data. However, it is important to mention that a
straightforward definition of the vdW radius is only possible for noble-gas
atoms as inert elements with closed valence shells. For other atoms, $R_{\rm
vdW}$ is evaluated by considering a variety of different molecular crystal
structures and extracting neighboring atom–atom distances, where each atom
belongs to a different closed-shell molecule. This definition is especially
subtle for chemical elements with spin-polarized valence shells, such as
alkali atoms, which can form bonds with different spin states. Therefore, one
has to keep in mind that existing vdW radii are just statistical quantities
for most chemical elements.
Figure 2: The ratios $R_{\rm vdW}/\alpha_{l,{\rm ref}}^{2/7(l+1)}$ listed in
Table 1 are shown (filled symbols) with respect to the “universal” values
$A_{1}=2.54$ a.u., $A_{2}=2.45$ a.u. and $A_{3}=2.27$ a.u., represented with
blue, red and green solid lines, respectively. By contrast, the ratios $R_{\rm
vdW}/\alpha_{l,{\rm QDO}}^{2/7(l+1)}$ (open symbols) are plotted with respect
to the constant values obtained from the QDO polarizabilities, $A_{2}^{{\rm
QDO}}=2.52$ a.u., $A_{3}^{{\rm QDO}}=2.37$ a.u., $A_{4}^{{\rm QDO}}=2.23$ a.u.
and $A_{5}^{{\rm QDO}}=2.10$ a.u., shown in red, green, fuchsia and light blue
dashed lines, respectively. The noble gases are shown in the yellow box, the
elements of the Group I in the light blue box and the elements of the Group II
in the light green box. The error bars represent the relative errors ${\rm
R.E.}=\absolutevalue{A_{l}[X]-A_{l}}/A_{l}\,$ of each species $X$, where
$A_{l}$ are the “universal” values expressed in Eqs. (42) for the ratios
$R_{\rm vdW}/\alpha_{l,{\rm ref}}^{2/7(l+1)}$ and in Eqs. (43) for $R_{\rm
vdW}/\alpha_{l,{\rm QDO}}^{2/7(l+1)}$.
First, we analyze the empirical proportionality constants
$A_{l}^{\rm ref}=R_{\rm vdW}^{{\rm ref}}/\alpha_{l,{\rm
ref}}^{\nicefrac{{2}}{{7(l+1)}}}$ (41)
based on the reference data of $R_{\rm vdW}$ and the atomic dipole
($\alpha_{1}^{{\rm ref}}$), quadrupole ($\alpha_{2}^{{\rm ref}}$) and octupole
($\alpha_{3}^{{\rm ref}}$) polarizabilities Fedorov2018 ; Gould2016 ;
Derevianko1999 ; Porsev2003 ; Porsev2006 . The results are shown in Table 1,
for the chosen test set of 16 chemical elements.
For noble gases, $A_{l}^{\rm ref}$ is essentially constant, and, in line with
Ref. Fedorov2018 , we find $\langle A_{1}^{\rm ref}\rangle=2.54$ a.u.
Moreover, this result is further specified in form of the unified formula,
$R_{\rm
vdW}(\alpha_{1})=(a_{0}^{4}/4\pi\epsilon_{0})^{1/7}\,(1/\alpha_{f})^{4/21}\,\alpha_{1}^{1/7}$
with $a_{0}$ and $\alpha_{f}$ denoting the Bohr radius and the fine-structure
constant, respectively Tkatchenko2020 . Using Hartree atomic units, this can
be further simplified to $R_{\rm
vdW}(\alpha_{1})\approx(137)^{4/21}\,\alpha_{1}^{1/7}$, which gives an
appropriate proportionality factor of $2.55$ a.u., in excellent agreement with
the empirical value of $\langle A_{1}^{\rm ref}\rangle$. For the higher-order
multipoles, we obtain $\langle A_{2}^{\rm ref}\rangle=2.45$ a.u. and $\langle
A_{3}^{\rm ref}\rangle=2.27$ a.u.. The resulting values for $A_{l}^{\rm ref}$
remain close to the average values determined for noble-gas atoms for all the
atoms shown in Table 1, where as a general trend alkali metal atoms show the
largest deviation with an average relative deviation of 4.3 % (compared to 2.4
% for alkaline earth metals and 0.5 % for noble gases). The alkali atoms
possess a relatively weakly bound and, therefore, highly polarizable single
valence electron. This feature and the different possible spin states of
alkali atoms in molecular solid-state systems arguably allow the vdW radii
observed for alkali metals to change widely, causing the largest deviation for
the empirical constants $A_{l}^{\rm ref}$ from their average values, as also
illustrated in Fig. 2.
Overall, the observed deviations in $A_{l}^{\rm ref}$ stay within 4 %
($\hat{=}$ 0.1 a.u.) for all considered species except for Cs and Rb, which
show a slightly higher average relative deviation of 6.3 % and 5.5 %,
respectively. Hence, we suggest that the coefficients $A_{l}^{\rm ref}$ can be
considered as universal constants for the studied atomic species. Taking the
average values $A_{l}=\langle A_{l}^{\rm ref}\rangle$ for the noble-gas atoms
we can thus write the unified relations between the vdW radius and the dipole,
quadrupole and octupole polarizabilities,
$\displaystyle R_{\rm vdW}(\alpha_{1})$
$\displaystyle=A_{1}\,\alpha_{1}^{\nicefrac{{1}}{{7}}}\ \ $ $\displaystyle,\ \
\ \ A_{1}$ $\displaystyle=2.54\ {\rm a.u.}$ (42) $\displaystyle R_{\rm
vdW}(\alpha_{2})$ $\displaystyle=A_{2}\,\alpha_{2}^{\nicefrac{{2}}{{21}}}\ \ $
$\displaystyle,\ \ \ \ A_{2}$ $\displaystyle=2.45\ {\rm a.u.}$ $\displaystyle
R_{\rm vdW}(\alpha_{3})$
$\displaystyle=A_{3}\,\alpha_{3}^{\nicefrac{{1}}{{14}}}\ \ $ $\displaystyle,\
\ \ \ A_{3}$ $\displaystyle=2.27\ {\rm a.u.}$
which are equivalent to the empirical relations reported in Ref. Fedorov2018 .
The relations obtained above can be used for at least three different
purposes. First, the vdW radius of atoms can now be calculated given any
single multipolar atomic polarizability. This polarizability can correspond to
a free atom or an atom in a molecule or material Tkatchenko2009 . The vdW
radius can then be used for a conceptual understanding of an atom in its
environment or for practical calculations of the vdW energy Tkatchenko2009 ;
Hermann2020 . Second, given the dipole polarizability, one can accurately
determine multipole polarizabilities (at least up to octupole) from Eq. (42).
In fact, this approach is substantially more reliable for atoms than using the
QDO model for multipole polarizabilities. A third application would be the
possibility to determine atomic multipole polarizabilities from calculated or
measured atomic vdW radii. A potential downside of this application is that a
small error in the vdW radius would result in a large error for multipole
polarizabilities, according to Eq. (42).
Figure 3: The vdW radius as a function of the multipole polarizabilities,
$R_{\rm vdW}(\alpha_{l})=A_{l}\,\alpha_{l}^{\nicefrac{{2}}{{7(l+1)}}}$,
obtained by means of (a) Eqs. (42) and (b) Eqs. (43) is presented, taking into
account that $R_{\rm vdW}^{\rm QDO}(\alpha_{1,{\rm QDO}})\equiv R_{\rm
vdW}(\alpha_{1,{\rm ref}})$, in comparison to the reference values $R_{\rm
vdW}^{{\rm ref}}$ Tkatchenko2009 ; Batsanov2001 ; Bondi1964 for three classes
of species: noble gases (in yellow), H + alkali-metal atoms (Group I, in blue)
and alkaline-earth atoms (Group II, in green). The vdW radii are shown (a) for
the reference Jones2013 ; Derevianko1999 ; Porsev2003 ; Porsev2006 multipole
polarizabilities $\alpha_{1,{\rm ref}}$, $\alpha_{2,{\rm ref}}$ and
$\alpha_{3,{\rm ref}}$ (blue, red and green filled symbols) and (b) for the
QDO multipole polarizabilities $\alpha_{2,{\rm QDO}}$, $\alpha_{3,{\rm QDO}}$,
$\alpha_{4,{\rm QDO}}$ and $\alpha_{5,{\rm QDO}}$ given by Eq. (6) (red,
green, fuchsia and light blue open symbols). The insets show the relative
errors, ${\rm R.E.}=\absolutevalue{R_{\rm vdW}(\alpha_{1})-R_{\rm vdW}^{{\rm
ref}}}/R_{\rm vdW}^{{\rm ref}}$, for the noble gases (yellow box), the alkali-
metals + hydrogen (light blue box) and the alkaline-earth elements (light
green box). In addition, the mean values of the relative errors, $\langle{\rm
R.E.}\rangle$, are reported in the legens, for each considered multipole
order.
The quantity $R_{\rm vdW}(\alpha_{l})$ defined in Eq. (42) represents an
effective vdW radius expressed in terms of the multipolar polarizability. To
demonstrate the validity of this new definition for the vdW radius, Fig. 3(a)
compares the results for $R_{\rm
vdW}(\alpha_{l})=A_{l}\,\alpha_{l}^{2/7(l+1)}$ with $l=\\{1,2,3\\}$ to the
reference values, $R_{\rm vdW}^{{\rm ref}}$. There is an excellent correlation
between $R_{\rm vdW}(\alpha_{1})$ and its reference value for all considered
elements, with a maximum relative error ${\rm R.E.}=\absolutevalue{R_{\rm
vdW}(\alpha_{1})-R_{\rm vdW}^{{\rm ref}}}/R_{\rm vdW}^{{\rm ref}}$ of 0.91 %
for the noble gases, 2.74 % for alkaline-earth (Group II) elements and 5.90 %
for hydrogen and alkali metals (Group I). The increasing errors when going
from alkaline-earth to alkali metals are related to the increase in the
statistical errors of $R_{\rm vdW}^{\rm ref}$ stemming from the increasingly
complicated evaluation of the vdW radii based on experimental crystal-
structure data. Indeed, this evaluation becomes less accurate for elements
with more pronounced metallic properties Batsanov2012 ; Batsanov2001 .
Comparing Groups I and II of the periodic table, the statistical errors in
$R_{\rm vdW}^{\rm ref}$ of the alkaline-earth elements are smaller since they
have closed $s$-electron shell, which makes their behavior closer to that of
the noble gases with completely closed valence shells.
Although the dipole polarizability $\alpha_{1,{\rm ref}}$ is known with high
accuracy for many chemical elements in the periodic table, the accurate
determination of higher-order multipolar polarizabilities is more involved.
Indeed, Fig. 3(a) shows an increase in the maximum R.E. for $R_{\rm
vdW}(\alpha_{2})$ (within 0.99 % for noble gases, 1.48 % for Group I and 5.14
% for Group II) and, subsequently, for $R_{\rm vdW}(\alpha_{3})$ (within 1.39
% for noble gases, 6.60 % for Group I and $8.62$ % for Group II). This
analysis is validated in Fig. 2 as well, where we compare the average values
$A_{1}=2.54$ a.u., $A_{2}=2.45$ a.u. and $A_{3}=2.27$ a.u. with the relative
ratios $R_{\rm vdW}/\alpha_{l}^{2/7(l+1)}$, for $l=\\{1,2,3\\}$.
It is also noteworthy that reference values for higher-order polarizabilities
are rather limited in literature, with the exception of hydrogen, the only
element in the periodic table for which the multipole polarizability
$\alpha_{l}^{\rm H}$ is known analytically Kharchenko2015 . Therefore, we
employ the multipole polarizabilities obtained within the QDO model by means
of Eq. (6). Notably, the QDO coefficients $A_{l}^{{\rm QDO}}=R_{\rm
vdW}/\alpha_{l,{\rm QDO}}^{2/7(l+1)}$, for $l=\\{1,2,3,4,5\\}$, are
practically constant for all noble-gas atoms, which leads to the QDO set of
relations
$\displaystyle R_{\rm vdW}^{\rm QDO}(\alpha_{1})$ $\displaystyle=\langle
A_{1}^{{\rm QDO}}\rangle\,\alpha_{1}^{\nicefrac{{1}}{{7~{}}}}\ \ ,\ \ \langle
A_{1}^{{\rm QDO}}\rangle=2.54\ {\rm a.u.}$ (43) $\displaystyle R_{\rm
vdW}^{\rm QDO}(\alpha_{2})$ $\displaystyle=\langle A_{2}^{{\rm
QDO}}\rangle\,\alpha_{2}^{\nicefrac{{2}}{{21}}}\ \ ,\ \ \langle A_{2}^{{\rm
QDO}}\rangle=2.52\ {\rm a.u.}$ $\displaystyle R_{\rm vdW}^{\rm
QDO}(\alpha_{3})$ $\displaystyle=\langle A_{3}^{{\rm
QDO}}\rangle\,\alpha_{3}^{\nicefrac{{1}}{{14}}}\ \ ,\ \ \langle A_{3}^{{\rm
QDO}}\rangle=2.37\ {\rm a.u.}$ $\displaystyle R_{\rm vdW}^{\rm
QDO}(\alpha_{4})$ $\displaystyle=\langle A_{4}^{{\rm
QDO}}\rangle\,\alpha_{4}^{\nicefrac{{2}}{{35}}}\ \ ,\ \ \langle A_{4}^{{\rm
QDO}}\rangle=2.23\ {\rm a.u.}$ $\displaystyle R_{\rm vdW}^{\rm
QDO}(\alpha_{5})$ $\displaystyle=\langle A_{5}^{{\rm
QDO}}\rangle\,\alpha_{5}^{\nicefrac{{1}}{{21}}}\ \ ,\ \ \langle A_{5}^{{\rm
QDO}}\rangle=2.10\ {\rm a.u.}$
These results are shown in Fig. 3(b), where a good agreement between $R_{\rm
vdW}^{\rm QDO}(\alpha_{l})$ and $R_{\rm vdW}^{\rm ref}$ is observed for
$l=\\{2,3,4,5\\}$, in addition to the case of $l=1$ for which we have $R_{\rm
vdW}^{\rm QDO}(\alpha_{1,{\rm QDO}})\equiv R_{\rm vdW}(\alpha_{1,{\rm ref}})$.
We note that the QDO model is constructed, by definition, on the dispersion
coefficients and the QDO polarizabilities (with $l>1$) are underestimated for
the noble-gas atoms with respect to the reference data Jones2013 .
Consequently, the QDO proportionality coefficients $\langle A_{2}^{{\rm
QDO}}\rangle=2.52$ a.u. and $\langle A_{3}^{{\rm QDO}}\rangle=2.37$ a.u. are
overestimated with respect to the determined “universal” values $A_{2}=2.45$
a.u. and $A_{3}=2.27$ a.u., as also shown in Fig. 2. Therefore, one can expect
that the higher-order QDO coefficients, $\langle A_{4}^{{\rm
QDO}}\rangle=2.23$ a.u. and $\langle A_{5}^{{\rm QDO}}\rangle=2.10$ a.u., are
also overestimated.
To further assess the scaling law of Eq. (37), we compare the resulting
empirical constants with the proportionality coefficients $A_{l}^{\mu\omega}$,
obtained by means of Eq. (39). Table 1 summarizes the results for
$A_{l}^{\mu\omega}$ compared with $A_{l}^{\rm ref}$ and $A_{l}^{\rm QDO}$.
Considering the noble-gas atoms only, we find the following averaged values
$\displaystyle\langle A_{1}^{\mu\omega}\rangle=2.37\ {\rm a.u.},\ \langle
A_{2}^{\mu\omega}\rangle=2.19\ {\rm a.u.},\ \langle
A_{3}^{\mu\omega}\rangle=2.05\ {\rm a.u.}$ $\displaystyle\langle
A_{4}^{\mu\omega}\rangle=1.95\ {\rm a.u.},\ \langle
A_{5}^{\mu\omega}\rangle=1.88\ {\rm a.u.}\ \ ,$ (44)
which shall serve as the suggested values of $A_{l}^{\mu\omega}$. The mean
absolute relative deviations from this reference across all considered
elements are 0.15 a.u. (5.41 %) for the dipole-dipole term, 0.16 a.u. (7.13 %)
for the dipole-quadrupole term, 0.17 a.u. (8.88 %) for the dipole-octupole
term, 0.17 a.u. (10.14 %) for the dipole-hexadecapole term and 0.18 a.u.
(11.05 %) for the dipole-triakontadipole term in the multipole expansion.
Hence, $A_{l}^{\mu\omega}$ do not remain constant among all considered
elements, in contrast to the respective atomic proportionality constants
$A_{l}^{\rm ref}$. The deviations between the two sets of coefficients amount
to $0.17$ a.u. (6.6 %) for $A_{1}$, $0.25$ a.u. (10.3 %) for $A_{2}$, and
$0.21$ a.u. (9.4 %) for $A_{3}$, where the derived $A_{l}^{\mu\omega}$ are
always lower than the empirical reference values $A_{l}^{\rm ref}$. In both
cases, we consistently observe a decrease of the average values and an
increase in the standard deviations with increasing multipole order.
Interestingly, the standard deviations among $A_{l}^{\rm ref}$ and $A_{l}^{\rm
QDO}$ are comparable to each other but considerably smaller than $\sigma$
shown in Table 1 for the corresponding $A_{l}^{\mu\omega}$. Hence, the
simplifications in the coarse-grained description of valence electrons within
the QDO model and its parametrization lead to a less accurate determination of
the proportionality coefficients, based on Eq. (39), than their indirect
evaluation based on the QDO multipole polarizabilities calculated by means of
Eq. (6). However, even $A_{l}^{\rm QDO}$ obtained for noble gases still show
noticeable deviations from a constant behavior. It is unclear yet what aspect
of atoms makes $A_{l}^{\rm ref}$ behave as “universal” constants for different
chemical elements. Nevertheless, we expect the clarification of this question
to be crucial for the eventual improvement of the QDO model.
Figure 4: The coefficients $A_{l}=R_{\rm vdW}/\alpha_{l}^{2/7(l+1)}$ for
hydrogen: the analytical solution $A_{l}^{{\rm H}}$, from the multipole
polarizabilities $\alpha_{l}^{{\rm
H}}=(4\pi\epsilon_{0})a_{0}^{2l+1}(2l+1)!(l+2)/2^{2l}l$ Kharchenko2015 , where
$a_{0}$ is the Bohr radius, is shown (blue filled circles) in comparison to
the results of the QDO model, $A_{l}^{{\rm QDO}}$ (blue open circles), which
are obtained from $\alpha_{l}^{{\rm QDO}}$ calculated by means of Eq. (6).
Finally, we calculate the ratio $R_{\rm vdW}/\alpha_{l}^{2/7(l+1)}$ for the
hydrogen atom, taking into account the known analytical expression of its
multipole polarizabilities given in Ref. Kharchenko2015 as $\alpha_{l}^{{\rm
H}}=(4\pi\epsilon_{0})\,a_{0}^{2l+1}(2l+1)!(l+2)/2^{2l}l$, and compare it with
the proportionality coefficients $A_{l}^{\rm QDO}$, which are obtained from
$\alpha_{l,{\rm QDO}}$ provided by Eq. (6). As shown in Fig. 4, the current
QDO parametrization yields quite accurate results for dipole, quadrupole and
octupole orders, but then exhibits an increasing overestimation of the
proportionality constant with increasing multipole rank $l$. Given that the
vdW radius is fixed at its reference value Tkatchenko2009 , this reflects the
fact that the QDO model predicts underestimated multipolar polarizabilities
Jones2013 . In order to improve the model, a possible future step is to use
highly accurate theoretical or experimental reference data for the quadrupole
polarizability instead of the dispersion coefficient ${\rm C}_{8}$ in the
parametrization scheme, which would increase the accuracy of the higher-order
$\alpha_{l,\mathrm{QDO}}$ values. Consequently, such new parametrization would
yield smaller values of $A_{l}^{\rm QDO}$, providing better agreement with the
reference data and improving the relations between multipole polarizabilities
and the equilibrium distance in vdW-bonded atomic dimers. Figure 4 also shows
that the $A_{l}^{\rm QDO}$ curve produces a slight deviation in the general
trend of the reference proportionality coefficients at $l=2$. This kink stems
from the nonmonotonic behavior of the multipole contributions to the exchange
energy, where $J^{{\rm ex}}_{2({\rm d\shortminus q})}$ is larger than $J^{{\rm
ex}}_{1({\rm d\shortminus d})}$ (see our discussion at the end of the
subsection B).
One possible explanation for the difference between the polarizabilities of
the QDO model and the hydrogen atom is the contribution of excitations to
continuum states in the latter case. The QDO model has no continuum states and
can only effectively describe such excitations. Despite the observed
deviations in the higher-order multipole polarizabilities and the
corresponding proportionality coefficients, the QDO approach allowed us to
verify the scaling relation $R_{\rm vdW}\propto\alpha_{l}^{2/7(l+1)}$, which
is remarkably valid for atoms.
## IV Discussion and Summary
We have presented a coarse-grained description of the repulsive force due to
the Pauli principle and attractive force due to dispersive fluctuations
between two closed-shell atoms or molecules. Our formalism is based on two
interacting quantum Drude oscillators, for which the dispersion and exchange-
repulsion energies up to an arbitrary order in the multipole expansion of the
Coulomb potential were derived. The obtained formulas can be employed for
constructing and rationalizing effective interaction potential models, as well
as for finding new scaling laws between electronic and geometric properties of
atoms and molecules.
As a practical illustration of our theory, we investigated a mutual
compensation between the repulsive exchange and attractive dispersion forces
for each term in the multipole expansion. The results confirm and extend the
recently proposed relations Fedorov2018 between atomic multipole
polarizabilities, $\alpha_{l}$, and the van der Waals radius, $R_{\rm vdW}$.
The generalized scaling law, $R_{\rm vdW}=A_{l}\,\alpha_{l}^{2/7(l+1)}$, is
compelling because it connects an electronic response property of a single
atom (atomic multipole polarizability) with the equilibrium distance in a
homonuclear dimer.
Let us enumerate some of the potential applications of the formulas presented
in this paper and possible future research directions:
* •
First and foremost, the relation between atomic vdW radius and atomic
polarizabilities, $R_{\rm vdW}=A_{l}\,\alpha_{l}^{2/7(l+1)}$, dispenses with
the need to indirectly measure vdW radii. Once the polarizability is
calculated for a free atom or an atom in a molecule/material, the vdW radius
can be computed from the formula above. Subsequently, the vdW radius can be
used as a proxy for an atomic size, as an effective radius in interatomic vdW
potentials or in damping functions for vdW-inclusive electronic-structure
calculations. We remark that in quantum mechanics, many possible definitions
can be made for an effective atomic size. Our derivations provide a novel
definition of the atomic vdW radius in terms of observable quantities — atomic
multipole polarizabilities. Obviously, a more detailed comparison of
calculated vdW radii to experiment would be welcome by measuring effective vdW
radii in a wide set of systems and comparing the measured radii to first-
principles calculations and our formulas.
* •
Our formula allows a straightforward and accurate calculation of atomic
multipole polarizabilities from the dipole polarizability. Given $\alpha_{1}$
and the universal values of $A_{l}^{\rm{ref}}$ obtained in this work, any
multipole polarizability can now be calculated as a function of these two
parameters. This is especially important given the high computational cost of
calculating multipole polarizabilities from first principles of quantum
mechanics. Going further, it would be interesting to assess different
recursive relations between $\alpha_{l}$ and $\alpha_{l+1}$ polarizabilities
based on the QDO model and the definition of the vdW radius.
* •
Our analytical results allow calculating multipole polarizabilities
$\alpha_{l}$ for an arbitrary value of $l$. Such data become increasingly
important in coarse-grained models, which describe molecular response by
increasingly larger fragments. For example, one might want to describe the
response of a protein, where one QDO models the response of each amino acid.
Similar to electrostatics, where higher multipoles become of growing
importance when increasing the fragment size, polarization response follows
the same trend. Hence, we expect our formulas to play a key role in the
development of coarse-grained models for chemical and biological matter.
* •
While most of the results in this paper were presented for homonuclear dimers,
an accurate combination rule is already known for computing equilibrium
distances, $R_{\rm eq}^{\rm AB}$, in heteronuclear dimers Fedorov2018
$\displaystyle R_{\rm eq}^{\rm AB}=2\times A_{l}[(\alpha_{\rm A}+\alpha_{\rm
B})/2]^{\nicefrac{{2}}{{7(l+1)}}}\quad.$ (45)
This formula allows the calculation of equilibrium distances in heteronuclear
closed-shell dimers solely based on the knowledge of atomic polarizabilities
of each atom. The derivation of such combination rules from first principles
requires generalizing the Pauli principle to QDOs with different parameters
and will be a subject of future work.
* •
The determination of $R_{\rm eq}^{\rm AB}$ for two atoms A and B from their
dipole polarizabilities provides a way to construct generalized Tang-Toennies-
type potentials Tang1995 ; Tang2003 ; Sheng2020 ; Tang1998 that require only
one adjustable parameter: the equilibrium interaction energy $E_{\rm eq}$. It
remains to be investigated whether the asymptotic dispersion coefficients
could be connected to $E_{\rm eq}$, allowing one to construct parameter-free
Tang-Toennies-type interatomic potentials for closed-shell systems.
* •
Last but not least, the relation between $R_{\rm vdW}$ and the polarizability
could be used to develop a more general and more accurate parametrization of
the QDO model. Namely, the universality of $A_{l}^{\rm ref}$ coefficients
holds for atoms but it is not such a good approximation within the QDO model
itself. One could enforce the obtained relation, $R_{\rm
vdW}=A_{l}\,\alpha_{l}^{2/7(l+1)}$ using “universal” values $A_{l}$, to hold
on average for the QDO model during the parametrization procedure. This is a
direction of our current study.
Ultimately, the close connection between vdW attraction and Pauli repulsion
unveiled in our work paves the way for the construction of efficient coarse-
grained models for the description of the exchange-repulsion interaction in
atomic and molecular systems. Together with the well established success of
the QDO model in describing vdW dispersion, our results also provide the basis
for constructing consistent and minimally-empirical models for interatomic and
intermolecular forces.
## ACKNOWLEDGMENTS
The authors acknowledge financial support from the European Research Council
via the ERC Consolidator Grant “BeStMo (725291)”as well as the Luxembourg
National Research Fund via the FNR CORE Jr project “PINTA(C17/MS/11686718)”
and the AFR PhD Grant “CNDTEC(11274975)”.
## Appendix A Multipole Expansion of Coulomb Potential
Here, we present the contributions $V_{n}$ to the multipole expansion of the
Coulomb potential given by Eq. (5).
$\displaystyle V_{2({\rm d\shortminus q})}$
$\displaystyle=\tfrac{3q_{A}q_{B}}{2R^{4}}\big{(}{{-}}5(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{+}}(\textbf{r}_{1}^{2}{{-}}2\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{+}}\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}({{-}}\textbf{r}_{2}^{2}{{+}}2\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}$
$\displaystyle\,{{+}}5(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})\big{)}$
$\displaystyle V_{3({\rm d\shortminus o})}$
$\displaystyle=\tfrac{q_{A}q_{B}}{2R^{5}}\big{(}5\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(3(\textbf{r}_{1}^{2}{{+}}\textbf{r}_{2}^{2}){{-}}7(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{-}}3\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}(\textbf{r}_{1}^{2}$
$\displaystyle\,{{+}}\textbf{r}_{2}^{2}{{-}}5(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}{{-}}5(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})\big{)}$
$\displaystyle V_{3({\rm q\shortminus q})}$
$\displaystyle=\tfrac{3q_{A}q_{B}}{4R^{5}}\big{(}2(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{2}{{-}}20\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{-}}5(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}(\textbf{r}_{2}^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})$
$\displaystyle\,{{+}}\textbf{r}_{1}^{2}(\textbf{r}_{2}^{2}{{-}}5(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})\big{)}$
$\displaystyle V_{4({\rm d\shortminus h})}$
$\displaystyle=\tfrac{5q_{A}q_{B}}{8R^{6}}\big{(}28(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{3}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}{{+}}42\textbf{r}_{1}^{2}(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{-}}63(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{4}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}$
$\displaystyle\,{{+}}3\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{2}^{4}{{-}}4\textbf{r}_{1}^{2}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}{{-}}14\textbf{r}_{2}^{2}(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}{{+}}21(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{4}){{+}}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}({{-}}3\textbf{r}_{1}^{4}$
$\displaystyle\,{{+}}4\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}(3\textbf{r}_{2}^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}))\big{)}$
$\displaystyle V_{4({\rm q\shortminus o})}$
$\displaystyle=\tfrac{5q_{A}q_{B}}{4R^{6}}\big{(}{{-}}7(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{3}(\textbf{r}_{2}^{2}{{-}}9(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{+}}21(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{2}^{2}{{-}}2\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}$
$\displaystyle\,{{-}}3(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{+}}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(6\textbf{r}_{1}^{2}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}{{-}}6(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{2}{{+}}\textbf{r}_{1}^{2}({{-}}3\textbf{r}_{2}^{2}{{+}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}))$
$\displaystyle\,{{+}}3\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{1}^{2}(\textbf{r}_{2}^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{+}}2\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}({{-}}\textbf{r}_{2}^{2}{{+}}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}{{+}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}))\big{)}$
$\displaystyle V_{5({\rm d\shortminus t})}$
$\displaystyle=\tfrac{3q_{A}q_{B}}{8R^{7}}\big{(}5\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}(\textbf{r}_{1}^{4}{{+}}\textbf{r}_{2}^{4}{{-}}14\textbf{r}_{1}^{2}(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}{{+}}21(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{4}{{-}}14\textbf{r}_{2}^{2}(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}$
$\displaystyle\,{{+}}21(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{4}){{-}}7\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(5(\textbf{r}_{1}^{4}{{+}}\textbf{r}_{2}^{4}){{-}}30\textbf{r}_{1}^{2}(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}{{+}}33(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{4}$
$\displaystyle\,{{-}}30\textbf{r}_{2}^{2}(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}{{+}}33(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{4})\big{)}$
$\displaystyle V_{5({\rm q\shortminus h})}$
$\displaystyle=\tfrac{15q_{A}q_{B}}{16R^{7}}\big{(}{{-}}168(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{3}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{-}}21(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{4}(\textbf{r}_{2}^{2}{{-}}11(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})$
$\displaystyle\,{{-}}4(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{2}(\textbf{r}_{1}^{2}{{+}}\textbf{r}_{2}^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{+}}56\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{1}^{2}{{+}}\textbf{r}_{2}^{2}$
$\displaystyle\,{{-}}3(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{-}}\textbf{r}_{1}^{2}(\textbf{r}_{2}^{2}(\textbf{r}_{1}^{2}{{+}}\textbf{r}_{2}^{2}){{-}}7(\textbf{r}_{1}^{2}{{+}}2\textbf{r}_{2}^{2})(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}{{+}}21(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{4})$
$\displaystyle\,{{+}}7(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}(2\textbf{r}_{1}^{2}\textbf{r}_{2}^{2}{{+}}\textbf{r}_{2}^{4}{{+}}4(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{2}{{-}}18(\textbf{r}_{1}^{2}{{+}}\textbf{r}_{2}^{2})(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}{{+}}33(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{4})\big{)}$
$\displaystyle V_{5({\rm o\shortminus o})}$
$\displaystyle=\tfrac{5q_{A}q_{B}}{4R^{7}}\big{(}2(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{3}{{-}}42\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{1}{{\cdot}}\textbf{r}_{2})^{2}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}{{+}}3\textbf{r}_{1}{{\cdot}}\textbf{r}_{2}({{-}}7(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}(\textbf{r}_{2}^{2}$
$\displaystyle\,{{-}}9(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}){{+}}\textbf{r}_{1}^{2}(\textbf{r}_{2}^{2}{{-}}7(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})){{-}}21\textbf{r}_{1}{{\cdot}}\textbf{R}_{0}\textbf{r}_{2}{{\cdot}}\textbf{R}_{0}(\textbf{r}_{1}^{2}(\textbf{r}_{2}^{2}{{-}}3(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2})$
$\displaystyle\,{{+}}(\textbf{r}_{1}{{\cdot}}\textbf{R}_{0})^{2}({{-}}3\textbf{r}_{2}^{2}{{+}}11(\textbf{r}_{2}{{\cdot}}\textbf{R}_{0})^{2}))\big{)}$
## Appendix B Force balance for the quadrupole-quadrupole and octupole-
octupole interactions
Table 2: The proportionality coefficients $A_{2}^{\mu\omega}$, $A_{2({{\rm q-q})}}^{\mu\omega}$ and $A_{3}^{\mu\omega}$, $A_{3({{\rm o-o})}}^{\mu\omega}$ in comparison to the empirical reference values $A_{l}^{\rm ref}=R_{{\rm vdW}}^{{\rm ref}}/\alpha_{l,{\rm ref}}^{2/7(l+1)}$. The used values of $R_{{\rm vdW}}^{{\rm ref}}$, $\\{\mu,\omega\\}$ and $\\{\alpha_{2,{\rm ref}},\alpha_{3,{\rm ref}}\\}$ are the same as in Table 1. The MARD is given in %, whereas all the other quantities are in atomic units. Atom | $A_{2}^{\mu\omega}$ | $A_{2({{\rm q-q})}}^{\mu\omega}$ | $A_{2}^{{\rm ref}}$ | $A_{3}^{\mu\omega}$ | $A_{3({{\rm o-o})}}^{\mu\omega}$ | $A_{3}^{{\rm ref}}$
---|---|---|---|---|---|---
He | 2.10 | 1.95 | 2.43 | 1.93 | 1.82 | 2.24
Ne | 2.27 | 2.09 | 2.44 | 2.07 | 1.94 | 2.26
Ar | 2.18 | 2.08 | 2.44 | 2.06 | 2.01 | 2.27
Kr | 2.22 | 2.13 | 2.47 | 2.10 | 2.08 | 2.29
Xe | 2.20 | 2.14 | 2.45 | 2.10 | 2.13 | 2.27
H | 1.97 | 1.89 | 2.40 | 1.88 | 1.87 | 2.19
Li | 2.41 | 2.36 | 2.49 | 2.29 | 2.34 | 2.33
Na | 2.42 | 2.40 | 2.55 | 2.32 | 2.41 | 2.40
K | 2.47 | 2.49 | 2.54 | 2.38 | 2.54 | 2.41
Rb | 2.57 | 2.55 | 2.58 | 2.42 | 2.60 | 2.46
Cs | 2.56 | 2.62 | 2.58 | 2.46 | 2.69 | 2.48
Ba | 2.17 | 2.14 | 2.45 | 2.08 | 2.16 | 2.34
Mg | 2.21 | 2.22 | 2.42 | 2.13 | 2.27 | 2.32
Ca | 2.34 | 2.39 | 2.44 | 2.26 | 2.46 | 2.39
Sr | 2.41 | 2.47 | 2.49 | 2.32 | 2.54 | 2.43
Ba | 2.44 | 2.54 | 2.42 | 2.36 | 2.63 | 2.41
${\langle A_{l}\rangle}$ | 2.19 | 2.08 | 2.45 | 2.05 | 2.00 | 2.27
${\sigma}$ | 0.16 | 0.22 | 0.06 | 0.17 | 0.40 | 0.08
MARD | 7.13 | 11.49 | 1.80 | 8.88 | 13.74 | 3.90
In order to demonstrate that the force balance is valid for each term in the
multipole expansion, we calculate the proportionality coefficients $A_{2({\rm
q\shortminus q})}\equiv A_{2({\rm q\shortminus q})}(\mu\omega,R_{\rm vdW})\,$
and $A_{3({\rm o\shortminus o})}\equiv A_{3({\rm o\shortminus
o})}(\mu\omega,R_{\rm vdW})\,$ for the quadrupole-quadrupole and octupole-
octupole interactions, which correspond to the ratios $R_{\rm
vdW}/\alpha_{2}^{2/21}\,$ and $R_{\rm vdW}/\alpha_{3}^{1/14}\,$, respectively.
According to Eq. (6), one has $\alpha_{2}=(3/4)(\hbar/\mu\omega)\alpha_{1}$
and $\alpha_{3}=(5/4)(\hbar/\mu\omega)^{2}\alpha_{1}$. By employing the
aforementioned two relations and considering the quadrupole-quadrupole and
octupole-octupole terms in Eqs. (III.3) and (III.3), we derive
$A_{2({\rm q\shortminus
q})}^{\mu\omega}=\left(\tfrac{175}{1024}\right)^{\nicefrac{{2}}{{21}}}R_{\rm
vdW}^{-\nicefrac{{1}}{{21}}}\left(\tfrac{\hbar}{\mu\omega}\right)^{\nicefrac{{2}}{{7}}}e^{\frac{4\mu\omega
R_{\rm vdW}^{2}}{21\hbar}}$ (46)
and
$\begin{split}A_{3({\rm o\shortminus
o})}^{\mu\omega}=\left(\tfrac{4851}{8192}\right)^{\nicefrac{{1}}{{14}}}R_{\rm
vdW}^{-\nicefrac{{1}}{{14}}}\left(\tfrac{\hbar}{\mu\omega}\right)^{\nicefrac{{2}}{{7}}}e^{\frac{\mu\omega
R_{\rm vdW}^{2}}{7\hbar}}\ .\end{split}$ (47)
These two expressions can be compared to the coefficients $A_{2}^{\mu\omega}$
and $A_{3}^{\mu\omega}$, expressed by Eq. (39). The results are shown in Table
2, where we compare the results obtained from Eq. (39) to those obtained from
quadrupole-quadrupole and octupole-octupole interactions, expressed by Eqs.
(46) and (47), respectively. Remarkably, we found a good agreement between the
proportionality coefficients $A_{2}^{\mu\omega}$ and $A_{2({\rm q\shortminus
q})}^{\mu\omega}$ with respect to the ratio of the vdW radius over the
quadrupole polarizability; and between $A_{3}^{\mu\omega}$ and $A_{3({\rm
o\shortminus o})}^{\mu\omega}$ with respect to $R_{{\rm vdW}}^{{\rm
ref}}/\alpha_{3,{\rm ref}}^{1/14}$. For the noble gases, the mean values are
$\langle A_{2({\rm q\shortminus q})}^{\mu\omega}\rangle=2.08$ a.u. and
$\langle A_{3({\rm o\shortminus o})}^{\mu\omega}\rangle=2.00$ a.u. with a
deviation of $6.3\%$ for the quadrupole-quadrupole interaction and $8.7\%$ for
the octupole-octupole interaction. In both cases, the mean values obtained
form the high multipole terms (quadrupole-quadrupole and octupole-octupole)
differ most from the “universal” values $A_{2}=2.45$ a.u. and $A_{3}=2.27$
a.u. with respect to the ones obtained from dipole-quadrupole and dipole-
octupole interactions. Moreover, the error for $\langle A_{3({\rm o\shortminus
o})}^{\mu\omega}\rangle$ is bigger than that for $\langle A_{2({\rm
q\shortminus q})}^{\mu\omega}\rangle$. This means that, as expected, the
lower-order contributions in the multipole expansion are more accurate with
respect to the higher-order terms.
## References
* (1) A. Stone, _The Theory of Intermolecular Forces_ (Oxford University Press, Oxford, 2013).
* (2) J. O. Hirschfelder and W. J. Meath, _The Nature of Intermolecular Forces_ , Adv. Chem. Phys. 12, 3 (1967).
* (3) H. Margenau and N. R. Kestner, _Theory of intermolecular forces_ (Pergamon Press, Oxford, 1971).
* (4) I. G. Kaplan, _Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials_ (John Wiley & Sons, 2006).
* (5) V. A. Parsegian, _Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists_ (Cambridge University Press, Cambridge, 2005).
* (6) A. Tkatchenko, _Current understanding of van der Waals effects in realistic materials_ , Adv. Func. Mat. 25, 2054 (2015).
* (7) L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, _Materials perspective on Casimir and van der Waals interactions_ , Rev. Mod. Phys. 88, 045003 (2016).
* (8) J. Mahanty and B. W. Ninham, _Dispersion contributions to surface energy_ , J. Chem. Phys. 59, 6157-6162 (1973).
* (9) D. D. Richardson, _Dispersion contribution of two-atom interaction energy: Multipole interactions_ , J. Phys. A: Math. Gen. 8 (1828).
* (10) V. V. Paranjape and J. Mahanty, _Van der Waals interaction between atoms: Finite-size effects_ , Phys. Rev. A 19, 6 (1979).
* (11) J. Klimes̆ and A. Michaelides, _Perspective: Advances and challenges in treating van der Waals dispersion forces in density functional theory_ , J. Chem. Phys. 137, 120901 (2012).
* (12) T. Björkman, A. Gulans, A. V. Krasheninnikov, and R. M. Nieminen, _Are we van der Waals ready?_ , J. Phys.: Condens. Matter 24, 424218 (2012).
* (13) K. Berland, V. R. Cooper, K. Lee, E. Schröder, T. Thonhauser, P. Hyldgaard, and B. I. Lundqvist, _van der Waals forces in density functional theory: a review of the vdW-DF method_ , Rep. Prog. Phys. 78, 066501 (2015).
* (14) S. Grimme, A. Hansen, J. G. Brandenburg, and C. Bannwarth, _Dispersion-Corrected Mean-Field Electronic Structure Methods_ , Chem. Rev. 116, 5105 (2016).
* (15) J. Hermann, R. A. DiStasio Jr., and A. Tkatchenko, _First-Principles Models for van der Waals Interactions in Molecules and Materials: Concepts, Theory, and Applications_ , Chem. Rev. 117, 4714 (2017).
* (16) M. Stöhr, T. Van Voorhis, and A. Tkatchenko, _Theory and practice of modeling van der Waals interactions in electronic-structure calculations_ , Chem. Soc. Rev. 48, 4118 (2019).
* (17) J. Hermann and A. Tkatchenko, _Density Functional Model for van der Waals Interactions: Unifying Many-Body Atomic Approaches with Nonlocal Functionals_ , Phys. Rev. Lett. 124, 146401 (2020).
* (18) J. F. Dobson and J. Wang, _Successful Test of a Seamless van der Waals Density Functional_ , Phys. Rev. Lett. 82, 2123 (1999).
* (19) M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, _Van der Waals Density Functional for General Geometries_ , Phys. Rev. Lett. 92, 246401 (2004).
* (20) O. A. Vydrov and T. Van Voorhis, _Nonlocal van der Waals Density Functional Made Simple_ , Phys. Rev. Lett. 103, 063004 (2009).
* (21) R. Sabatini, T. Gorni, and S. de Gironcoli, _Nonlocal van der Waals density functional made simple and efficient_ , Phys. Rev. B 87, 041108 (2013).
* (22) T. T. Odbadrakh, V. Voora, and K. D. Jordan, _Application of electronic structure methods to coupled Drude oscillators_ , Chem. Phys. Lett. 630, 76 (2015).
* (23) A. D. Becke and E. R. Johnson, _A density-functional model of the dispersion interaction_ , J. Chem. Phys. 123, 154101 (2005).
* (24) A. Tkatchenko and M. Scheffler, _Accurate Molecular van der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data_ , Phys. Rev. Lett. 102, 073005 (2009).
* (25) S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, _A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu_ , J. Chem. Phys. 132, 154104 (2010).
* (26) E. Caldeweyher, C. Bannwarth, and S. Grimme, _Extension of the D3 dispersion coefficient model_ , J. Chem. Phys. 147, 034112 (2017).
* (27) W. L. Bade, _Drude‐Model Calculation of Dispersion Forces. I. General Theory_ , J. Chem. Phys. 27, 1280 (1957).
* (28) F. Wang and K. D. Jordan, _A Drude-model approach to dispersion interactions in dipole-bound anions_ , J. Chem. Phys. 114, 10717 (2001).
* (29) G. Lamoureux and B. Roux, _Modeling induced polarization with classical Drude oscillators: Theory and molecular dynamics simulation algorithm_ , J. Chem. Phys. 119, 3025 (2003).
* (30) G. Lamoureux, A. D. MacKerell Jr., and B. Roux, _A simple polarizable model of water based on classical Drude oscillators_ , J. Chem. Phys. 119, 5185 (2003).
* (31) T. Sommerfeld and K. D. Jordan, _Quantum Drude Oscillator Model for Describing the Interaction of Excess Electrons with Water Clusters: An Application to (H 2O)${}_{13}^{~{}~{}-}$_, J. Phys. Chem. A 109, 11531 (2005).
* (32) T. W. Whitfield and G. J. Martyna, _A unified formalism for many-body polarization and dispersion: The quantum Drude model applied to fluid Xenon_ , Chem. Phys. Lett. 424, 409 (2006).
* (33) A. P. Jones, J. Crain, V. P. Sokhan, T. W. Whitfield, and G. J. Martyna, _Quantum Drude oscillator model of atoms and molecules: Many-body polarization and dispersion interactions for atomistic simulation_ , Phys. Rev. B 87, 144103 (2013).
* (34) A. Tkatchenko, R. A. DiStasio Jr., R. Car, and M. Scheffler, _Accurate and Efficient Method for Many-Body van der Waals Interactions_ , Phys. Rev. Lett. 108, 236402 (2012).
* (35) A. M. Reilly and A. Tkatchenko, _van der Waals dispersion interactions in molecular materials: Beyond pairwise additivity_ , Chem. Sci. 6, 3289 (2015).
* (36) M. Sadhukhan and F. R. Manby, _Quantum mechanics of Drude oscillators with full Coulomb interaction_ , Phys. Rev. B 94, 115106 (2016).
* (37) M. Sadhukhan and A. Tkatchenko, _Long-Range Repulsion Between Spatially Confined van der Waals Dimers_ , Phys. Rev. Lett. 118, 210402 (2017).
* (38) J. Hermann, D. Alfè, and A. Tkatchenko, _Nanoscale $\pi$–$\pi$ stacked molecules are bound by collective charge fluctuations_, Nat. Commun. 8, 14052 (2017).
* (39) V. V. Gobre, _Efficient modelling of linear electronic polarization in materials using atomic response functions_ , PhD thesis, Fritz Haber Institute Berlin (2016).
* (40) A. Ambrosetti, D. Alfè, R. A. DiStasio Jr., and A. Tkatchenko, _Hard Numbers for Large Molecules: Toward Exact Energetics for Supramolecular Systems_ , J. Phys. Chem. Lett. 5, 849 (2014).
* (41) K. T. Tang and M. Karplus, _Padé-Approximant Calculation of the Nonretarded van der Waals Coefficients for Two and Three Helium Atoms_ , Phys. Rev. 171, 70 (1968).
* (42) B. T. Thole, _Molecular polarizabilities calculated with a modified dipole interaction_ , Chem. Phys. 59, 341 (1981).
* (43) R. A. DiStasio, Jr., V. V. Gobre, and A. Tkatchenko, _Many-body van der Waals interactions in molecules and condensed matter_ , J. Phys.: Condens. Matter 26, 213202 (2014).
* (44) M. Stöhr, M. Sadhukhan, Y. S. Al-Hamdani, J. Hermann, and A. Tkatchenko, _Coulomb Interactions between Dipolar Quantum Fluctuations in van der Waals Bound Molecules and Materials_ , Nat. Commun. 12, 137 (2021).
* (45) M. J. Van Vleet, A. J. Misquitta, A. J. Stone, and J. R. Schmidt, _Beyond Born–Mayer: Improved Models for Short-Range Repulsion in ab Initio Force Fields_ , J. Chem. Theory Comput. 12, 3851–3870 (2016).
* (46) D. V. Fedorov, M. Sadhukhan, M. Stöhr, and A. Tkatchenko, _Quantum-Mechanical Relation between Atomic Dipole Polarizability and van der Waals Radius_ , Phys. Rev. Lett. 121, 183401 (2018).
* (47) K. T. Tang, J. P. Toennies, and C. L. Yiu, _The generalized Heitler-London theory for interatomic interaction and surface integral method for exchange energy_ , Int. Rev. Phys. Chem. 17, 363 (1998).
* (48) B. Jeziorski, R. Moszynski, and K. Szalewicz, _Perturbation Theory Approach to Intermolecular Potential Energy Surfaces of van der Waals Complexes_ , Chem. Rev. 94, 1887 (1994).
* (49) A. Heßelmann, G. Jansen, and M. Schütz, _Density-functional theory-symmetry-adapted intermolecular perturbation theory with density fitting: A new efficient method to study intermolecular interaction energies_ , J. Chem. Phys. 122, 014103 (2005).
* (50) K. Szalewicz, _Symmetry-adapted perturbation theory of intermolecular forces_ , WIREs Comput. Mol. Sci. 2, 254 (2012).
* (51) T. M. Parker, L. A. Burns, R. M. Parrish, A. G. Ryno, and C. D. Sherrill, _Levels of symmetry adapted perturbation theory (SAPT). I. Efficiency and performance for interaction energies_ , J. Chem. Phys. 140, 094106 (2014).
* (52) A. Heßelmann and T. Korona, _Intermolecular symmetry-adapted perturbation theory study of large organic complexes_ , J. Chem. Phys. 141, 094107 (2014).
* (53) L. Shirkov and V. Sladek, _Benchmark CCSD-SAPT study of rare gas dimers with comparison to MP-SAPT and DFT-SAPT_ , J. Chem. Phys. 147, 174103 (2017).
* (54) K. T. Tang, J. P. Toennies, and C. L. Yiu, _Accurate Analytical He-He van der Waals Potential Based on Perturbation Theory_ , Phys. Rev. Lett. 74, 1546 (1995).
* (55) K. T. Tang and J. P. Toennies, _The van der Waals potentials between all the rare gas atoms from He to Rn_ , J. Chem. Phys. 118, 4976 (2003).
* (56) X. Sheng, J. P. Toennies, and K. T. Tang, _Conformal Analytical Potential for All the Rare Gas Dimers over the Full Range of Internuclear Distances_ , Phys. Rev. Lett. 125, 253402 (2020).
* (57) K. T. Tang, J. P. Toennies, and C. L. Yiu, _A simple method for calculating the exchange energy for H ${}_{2}^{+}$ from polarization perturbation theory_, Chem. Phys. Lett. 162, 170 (1989).
* (58) K. T. Tang, J. P. Toennies, M. Wanschura, and C. L. Yiu, _Exchange energy of alkali-metal dimer cations calculated from the atomic polarizability with the Holstein-Herring method_ , Phys. Rev. A 46, 3746 (1992).
* (59) K. T. Tang, J. P. Toennies, and C. L. Yiu, _The exchange energy of H ${}_{2}^{+}$ calculated from polarization perturbation theory_, J. Chem. Phys. 94, 7266 (1991).
* (60) A. Dalgarno and A. L. Stewart, _On the perturbation theory of small disturbances_ , Proc. Roy. Soc. A 238, 269 (1956).
* (61) P. M. Morse and H. Feshbach, _Methods of Theoretical Physics. Part 1 and 2_ (Mc-Graw-Hill Book Company, Inc., New York, 1953)
* (62) A. Tkatchenko and D. V. Fedorov, _Fine-Structure Constant Connects the Polarizability of Atoms and Vacuum_ , arXiv:2007.02992 (2020).
* (63) H. Margenau, _Quadrupole Contributions to London’s Dispersion Forces_ , J. Chem. Phys. 6, 896 (1938).
* (64) J. A. Crosse and S. Scheel, _Atomic multipole relaxation rates near surfaces_ , Phys. Rev. A 79, 062902 (2009).
* (65) W. Smith, _Point Multipoles in the Ewald Summation (Revisited)_ , CCP5 Newsletter 46, 18 (1998).
* (66) D. Lin, _Generalized and Efficient Algorithm for Computing Multipole Energies and Gradients Based on Cartesian Tensors_ , J. Chem. Phys. 143, 114115 (2015).
* (67) P. L. A. Popelier, L. Joubert, and D. S. Kosov, _Convergence of the Electrostatic Interaction Based on Topological Atoms_ , J. Phys. Chem. A 105, 8254 (2001).
* (68) A. Kleshchonok and A. Tkatchenko, _Tailoring van der Waals dispersion interactions with external electric charges_ , Nat. Commun. 9, 3017 (2018).
* (69) D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, _Quantum theory of angular momentum_ (World Scientific, 1988).
* (70) P. L. Silvestrelli and A. Ambrosetti, _van der Waals interactions in DFT using Wannier functions without empirical parameters_ , J. Chem. Phys. 150, 164109 (2019).
* (71) W. Heitler and F. London, _Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik_ , Z. Phys. 44, 455 (1927).
* (72) J. C. Slater, _Molecular Orbital and Heitler-London Methods_ , J. Chem. Phys. 43, S11 (1965).
* (73) A. J. Van Der Merwe, _Quadrupole-Quadrupole and Dipole-Octupole Dispersion Forces Between Axially Symmetric Molecules_ , Z. Naturforschung 22a, 593 (1967).
* (74) G. Chałasiński, B. Jeziorski, J. Andzelm, and K. Szalewicz, _On the multipole structure of exchange dispersion energy in the interaction of two helium atoms_ , Mol. Phys. 33, 971 (1977).
* (75) P. Szabó, S. Góger, J. Charry, M. R. Karimpour, D. V. Fedorov, and A. Tkatchenko, _Four-Dimensional Scaling of Dipole Polarizability in Quantum Systems_ , arXiv:2010.11809 (2020).
* (76) K. U. Lao, Y. Yang, and R. A. DiStasio Jr., _On the Higher-Order Static Polarizabilities and Dispersion Coefficients of the Fullerenes: An Ab Initio Study_ , ChemRxiv (2020).
* (77) A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, _High-Precision Calculations of Dispersion Coefficients, Static Dipole Polarizabilities, and Atom-Wall Interaction Constants for Alkali-Metal Atoms_ , Phys. Rev. Lett. 82, 3589 (1999).
* (78) S. G. Porsev and A. Derevianko, _Accurate relativistic many-body calculations of van der Waals coefficients $C_{8}$ and $C_{10}$ for alkali-metal dimers_, J. Chem. Phys. 119, 844 (2003).
* (79) S. G. Porsev and A. Derevianko, _High-Accuracy Calculations of Dipole, Quadrupole, and Octupole Electric Dynamic Polarizabilities and van der Waals Coefficients $C_{6}$ , $C_{8}$ , and $C_{10}$ for Alkaline-Earth Dimers_, J. Exp. Theor. Phys. 102, 195–205 (2006).
* (80) T. Gould and T. Buc̆ko, _C6 Coefficients and Dipole Polarizabilities for All Atoms and Many Ions in Rows 1-6 of the Periodic Table_ , J. Chem. Theory Comput. 12, 3603–3613 (2016).
* (81) D. E. Woon and T. H. Dunning, _Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties_ , J. Chem. Phys. 100, 2975–2988 (1994).
* (82) K. U. Lao, J. Jia, R. Maitra, and R. A. DiStasio Jr., _On the geometric dependence of the molecular dipole polarizability in water: A benchmark study of higher-order electron correlation, basis set incompleteness error, core electron effects, and zero-point vibrational contributions_ , J. Chem. Phys. 149, 204303 (2018).
* (83) S. Kümmel, J. Akola, and M. Manninen, _Thermal Expansion in Small Metal Clusters and its Impact on the Electric Polarizability_ , Phys. Rev. Lett. 84, 3827–3830 (2000).
* (84) D. M. Bishop, _Molecular vibrational and rotational motion in static and dynamic electric fields_ , Rev. Mod. Phys. 62, 343–374 (1990).
* (85) S. S. Batsanov, _Van der Waals radii of elements_ , Inorg. Mater. 37, 871 (2001).
* (86) A. Bondi, _van der Waals Volumes and Radii_ , J. Phys. Chem. 68, 441 (1964).
* (87) S. S. Batsanov and A .S. Batsanov, _Introduction to Structural Chemistry_ (Springer Netherlands, 2012).
* (88) V. F. Kharchenko, _Analytical transition-matrix treatment of electric multipole polarizabilities of hydrogen-like atoms_ , Ann. Phys. 355, 153–169 (2015).
|
# Real-fibered morphisms of del Pezzo surfaces and conic bundles
Mario Kummer Technische Universität Dresden, Germany mario.kummer@tu-
dresden.de , Cédric Le Texier Universitetet i Oslo, Norway
<EMAIL_ADDRESS>and Matilde Manzaroli Universitetet i Oslo, Norway
<EMAIL_ADDRESS>
###### Abstract.
It goes back to Ahlfors that a real algebraic curve admits a real-fibered
morphism to the projective line if and only if the real part of the curve
disconnects its complex part. Inspired by this result, we are interested in
characterising real algebraic varieties of dimension $n$ admitting real-
fibered morphisms to the $n$-dimensional projective space. We present a
criterion to classify real-fibered morphisms that arise as finite surjective
linear projections from an embedded variety which relies on topological
linking numbers. We address special attention to real algebraic surfaces. We
classify all real-fibered morphisms from real del Pezzo surfaces to the
projective plane and determine which such morphisms arise as the composition
of a projective embedding with a linear projection. Furthermore, we give some
insights in the case of real conic bundles.
###### 2010 Mathematics Subject Classification:
Primary: 14P25, 14J26
The research of the last two authors is funded by the Trond Mohn Stiftelse
(TMS) project “Algebraic and topological cycles in complex and tropical
geometry”
###### Contents
1. 1 Introduction
2. 2 Preliminaries and notation
3. 3 Necessary conditions for surfaces
4. 4 Linking lemma and varieties of higher dimension
5. 5 Hyperbolic del Pezzo surfaces
6. 6 Explicit constructions and examples
7. 7 Conic bundles
## 1\. Introduction
This work concerns the study of real-fibered morphisms from real algebraic
varieties to projective spaces of same dimension. Apart from the case of real
algebraic curves, the topology of the real part of higher dimensional real
algebraic varieties admitting real-fibered morphisms is bound to be a disjoint
union of spheres and real projective spaces. We mainly focus on real del Pezzo
surfaces and real conic bundles whose real classification is well known. The
study of real algebraic varieties dates back to the 19th century. One of the
first significant results was the classification of real cubic surfaces
presented in [Sch63]. Then, in [Zeu74], the study of real plane algebraic
curves of degree $4$ and their bitangents was carried out (which is equivalent
to the study of real del Pezzo surfaces of degree $2$). The first systematic
study of real algebraic varieties was pursued by Harnack, Klein, Hilbert and
Comessatti [Har76, Hil, Com13, Com14, Kle73]. In particular, Comessatti
classified real rational algebraic $\mathbb{R}$-minimal surfaces. Moreover,
since we can obtain any real rational surface as a sequence of real blow-ups
of a real rational $\mathbb{R}$-minimal surface, Comesatti’s approach in
[Com14] leads to a complete classification of real del Pezzo surfaces.
Let $X$ be a non-singular algebraic variety of dimension $n$ (by variety we
will always mean an integral and separated scheme of finite type over
${\mathbb{R}}$). In this article, we suppose that all varieties have non-empty
real part unless otherwise stated.
###### Definition 1.1.
Let $X$ and $Y$ be non-singular algebraic varieties of dimension $n$. We say
that a real morphism $f:X\rightarrow Y$ is _real-fibered_ , if
$X({\mathbb{R}})$ is non-empty and $f^{-1}(Y(\mathbb{R}))=X(\mathbb{R})$.
As already mentioned, we are particularly interested in real-fibered morphisms
$X\to\mathbb{P}^{n}$ where $\mathbb{P}^{n}$ is the scheme
$\mathbb{P}^{n}_{\mathbb{R}}=\textrm{Proj}({\mathbb{R}}[x_{0},\ldots,x_{n}])$.
According to a result by Ahlfors [Ahl50, §4.2], it is known which projective
irreducible smooth curves $C$ admit a real-fibered morphism
$C\to\mathbb{P}^{1}$. This is the case if and only if $C$ is of _Type I_ or
_separating_ in the sense that its real points $C(\mathbb{R})$ disconnect its
set of complex points $C(\mathbb{C})$. Note that $C$ is separating whenever
$C$ is an _$M$ -curve_, i.e. the number $r$ of connected components of
$C({\mathbb{R}})$ equals $g+1$ where $g$ is the genus of $C$. On the other
hand, if $C$ is separating then $r$ has the same parity as $g+1$. Separating
curves and their real-fibered morphisms to $\mathbb{P}^{1}$ have been studied
by several authors, see for example [Gab06, CH13, Cop13, Ore19, KS20b]. While
any separating curve $C$ admits real-fibered morphisms to $\mathbb{P}^{1}$ of
arbitrary large degree, the situation is much more rigid for varieties of
higher dimension. This is mainly due to the fact that for any real-fibered
morphism $X\to Y$ of smooth varieties the restriction to the real parts is an
unramified covering map [KS20a, Thm. 2.19]. Among others, this implies that
for any smooth variety $X$ of dimension $n\geq 2$, the topology of
$X({\mathbb{R}})$ already determines the degree of any real-fibered morphism
$X\to\mathbb{P}^{n}$. More precisely, if $X$ is a smooth variety of dimension
$n\geq 2$ and $f:X\to\mathbb{P}^{n}$ a real-fibered morphism, then
$X({\mathbb{R}})$ is homeomorphic to a disjoint union of $s$ spheres and $r$
real projective spaces such that $\deg(f)=2s+r$ (or $X({\mathbb{R}})$ is
empty). However, there is no topological characterisation known, similar to
Ahlfors’ above result on curves of Type I, of those $n$-dimensional varieties
that admit a real-fibered morphism $X\to\mathbb{P}^{n}$. A possible approach
to extend the notion of being Type I to all varieties is presented in [Vir94],
where Viro introduces the definition of bound in complexification for a smooth
$n$-dimensional variety $X$: this is the case if $X({\mathbb{R}})$ realises
the trivial element in the homology group
$H_{n}(X_{\mathbb{C}};\mathbb{Z}/2\mathbb{Z})$. In this article, we go towards
a different direction and we give criteria to characterise smooth
$n$-dimensional varieties admitting real-fibered morphisms
$X\to\mathbb{P}^{n}$. Moreover, we completely characterise _del Pezzo
surfaces_ which admit real-fibered morphisms to $\mathbb{P}^{2}$.
In the following, for a non-singular algebraic variety $X$ of dimension $n$,
we write _real Picard group_ for the Picard group of $X_{{\mathbb{R}}}$,
respectively Picard group for the Picard group of $X_{{\mathbb{C}}}$. We say
that a morphism of real varieties is a _finite_ real-fibered morphism if it is
real-fibered and a finite morphism in the sense of [Har77, p. 84]. We first
present the following characterisation of finite real-fibered morphisms
$X\to\mathbb{P}^{2}$ from a del Pezzo surface $X$.
###### Theorem 1.2.
Let $X$ be a del Pezzo surface such that each connected component of
$X({\mathbb{R}})$ is homeomorphic to either the sphere or the real projective
plane. There is a finite real-fibered morphism $X\to\mathbb{P}^{2}$ if and
only if we have one of the following:
1. (1)
$X$ has real Picard rank $1$;
2. (2)
$X$ is a conic bundle of real Picard rank $2$;
3. (3)
$X$ is the blow-up of one of the above surfaces at one or two real points.
Since the blow-up at a pair of complex conjugate points is always real-
fibered, we obtain the following.
###### Corollary 1.3.
Let $X$ be a del Pezzo surface with $X({\mathbb{R}})\neq\emptyset$. There is a
(possibly non-finite) real-fibered morphism $X\to\mathbb{P}^{2}$ if and only
if each connected component of $X({\mathbb{R}})$ is homeomorphic to either the
sphere or the real projective plane.
A concept closely related to real-fibered morphisms is the notion of
hyperbolic varieties.
###### Definition 1.4.
Let $X\subset\mathbb{P}^{N}$ be an embedded variety and
$E\subset\mathbb{P}^{N}$ a linear subspace of dimension
$d=\operatorname{codim}(X,\mathbb{P}^{N})-1$ with $E\cap X=\emptyset$. Then
$X$ is _hyperbolic_ with respect to $E$ if for all linear subspaces
$E^{\prime}\supset E$ of dimension $d+1$ we have that $E^{\prime}\cap X$
consists only of real points.
Note that $X$ is hyperbolic with respect to $E$ if and only if the linear
projection $\pi_{E}:X\to\mathbb{P}^{\dim X}$ from center $E$ is real-fibered.
Hyperbolic embeddings of curves were studied for instance in [Ore19, KS20b].
For example it was shown in [KS20b] that any embedding of a separating curve
via a complete linear system of large enough degree is hyperbolic. Hyperbolic
curves also played an important role in the recent classification of maximally
writhed real algebraic links [MO19]. For higher dimensional varieties we give
the following characterisation of hyperbolic varieties that allows us to
reduce the problem to smaller dimensions. From now on, we will write
$X({\mathbb{R}})\simeq sS^{k}\sqcup r{\mathbb{R}}\mathbb{P}^{k}$ in order to
express that the real part is homeomorphic to the disjoint union of $s$
$k$-spheres and $r$ real projective spaces of dimension $k$.
###### Theorem 1.5.
Let $X\subset\mathbb{P}^{n}$ be a smooth variety of dimension $k\geq 2$. Let
$H\subset\mathbb{P}^{n}$ be a hyperplane such that $C=X\cap H$ is a smooth
$(k-1)$-variety. Assume that each connected component of $X(\mathbb{R})$
contains exactly one connected component of $C(\mathbb{R})$. Moreover, let
$E\subset H$ be a linear space of dimension $n-k-1$ with $X\cap E=\emptyset$.
Then the following are equivalent:
1. (1)
$X$ is hyperbolic with respect to $E$.
2. (2)
$X$ satisfies $X({\mathbb{R}})\simeq sS^{k}\sqcup r{\mathbb{R}}\mathbb{P}^{k}$
such that $\deg(X)=2s+r$. The class of each connected component that is
homeomorphic to a real projective space is nontrivial in
$H_{k}(\mathbb{P}^{n}(\mathbb{R});{\mathbb{Z}}_{2})$ and $C\subset
H=\mathbb{P}^{n-1}$ is hyperbolic with respect to $E$.
This allows us to characterise del Pezzo surfaces which can be embedded in
some projective space as a hyperbolic variety.
###### Theorem 1.6.
Let $X$ be a del Pezzo surface such that each connected component of
$X({\mathbb{R}})$ is homeomorphic to either the sphere or the real projective
plane. There is an embedding $X\hookrightarrow\mathbb{P}^{n}$ such that the
image is a hyperbolic variety if and only if we have one of the following:
1. (1)
$X$ has real Picard rank $1$;
2. (2)
$X$ is a conic bundle of real Picard rank $2$;
3. (3)
$X$ is the blow-up of one of the above surfaces at one real point.
Furthermore, we characterise these embeddings.
###### Theorem 1.7.
Let $X\subset\mathbb{P}^{n}$ be a smooth nondegenerate real del Pezzo surface
embedded via a complete linear system. There exists a linear subspace
$E\subset\mathbb{P}^{n}$ of codimension $3$ such that $X$ is hyperbolic with
respect to $E$ if and only if:
1. (1)
$X({\mathbb{R}})\simeq sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$;
2. (2)
$\deg(X)=2s+r$; and
3. (3)
the genus of a hyperplane section on $X$ equals $s+r-1$.
In this case we further have $r\in\\{0,1\\}$ and $n=s+2$.
Part $(2)$ of the Theorems 1.2 and 1.6 motivates the question of which real
conic bundles $X$ (over $\mathbb{P}^{1}$) with real Picard rank 2 admit a
real-fibered morphism to $\mathbb{P}^{2}$, respectively which ones admit a
hyperbolic embedding. In order to treat this question, we will consider those
surfaces $X$ as the zero set of a section of
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$, for $\mathbb{P}(\mathcal{E})$ a
projective plane bundle over $\mathbb{P}^{1}$. This will allow us to construct
hyperbolic conic bundles with an arbitrary large number of components
homeomorphic to a sphere. Finally, for all pairs $(s,r)$ of nonnegative
integers we decide whether there exist a smooth hyperbolic surface with real
part homeomorphic to $sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$, except for
the case $s=2,r\geq 2$. We provide a construction when $s\geq 3,r\geq 0$ using
the tautological embedding of the projective bundle $\mathbb{P}(\mathcal{E})$.
The paper will be organised as follows. We start by recalling several facts
and notations about real del Pezzo surfaces in Section 2. We then give in
Section 3 a classification of ample divisors $D$ on real del Pezzo surfaces
$X$ satisfying some necessary conditions for the associated morphism
$f:X\rightarrow\mathbb{P}^{2}$ to be finite real-fibered. In Section 4, we
give a criterion for a real variety $X\subset\mathbb{P}^{n}$ of dimension $k$
to be hyperbolic with respect to a given linear subspace $E$ of dimension
$n-k-1$, in terms of linking numbers (generalising the criterion for real
curves given in [KS20b]). This will allow us to prove Theorem 1.5. In the
subsequent Section 5 we apply this to prove the Theorems 1.2, 1.6 and 1.7 as
well as Corollary 1.3. For some examples of real del Pezzo surfaces $X$ we
construct in Section 6 an explicit linear subspace $E\subset\mathbb{P}^{n}$ of
codimension 3 such that $X$ is hyperbolic with respect to $E$. Finally, in
Section 7, we treat the case of hyperbolic minimal conic bundles and the
question of which topological types are realisable as real part of a
hyperbolic surface.
## 2\. Preliminaries and notation
Let $X$ be a real smooth irreducible surface. By a surface we always mean a
projective and irreducible variety of dimension $2$. We will denote by
$-K_{X}$, or simply $-K$ if there is no ambiguity, the anti-canonical class of
$X$.
###### Definition 2.1.
If $-K$ is ample, then $X$ is a del Pezzo surface of degree $K^{2}$.
Let $X$ be a del Pezzo surface. From the complex view point $X_{\mathbb{C}}$
is either $\mathbb{P}^{2}_{\mathbb{C}}$ blown up in $r$ points in general
position, where $r\leq 8$, or
$\mathbb{P}^{1}_{\mathbb{C}}\times\mathbb{P}^{1}_{\mathbb{C}}$. In the former
case one has $K^{2}=9-r$ and $K^{2}=8$ in the latter. In particular $K^{2}\leq
9$.
###### Definition 2.2.
Let $X$ be a smooth irreducible surface.
1. (1)
If every (real) birational morphism from $X$ into a smooth surface is an
isomorphism, then we say that $X$ is _minimal (over ${\mathbb{R}}$)_.
2. (2)
If $X$ is a conic bundle of real Picard rank two, we say that $X$ is a
_minimal conic bundle_.
Every real del Pezzo surface is one of the following or a blow-up of one of
the following at a zero dimensional real subvariety:
* •
The projective plane $\mathbb{P}^{2}$ which is a del Pezzo surface of degree
$9$ whose real part is the real projective plane ${\mathbb{R}}\mathbb{P}^{2}$.
* •
The quadric hypersurface
$Q^{n,4-n}={\mathcal{V}}(\sum\limits_{i=1}^{n}x_{i}^{2}-\sum\limits_{j=n+1}^{4}x_{i}^{2})$,
$n\in\\{0,1,2\\}$, in $\mathbb{P}^{3}$ which is a del Pezzo surface of degree
$8$. Its real part is empty when $n=0$ and homeomorphic to the sphere $S^{2}$
resp. the torus $S^{1}\times S^{1}$ when $n=1$ or $n=2$ respectively.
* •
The direct product $\mathbb{P}^{1}\times C$ where $C$ is a smooth rational
curve without real points. This is a del Pezzo surface of degree $8$ whose
real part is empty.
* •
A minimal conic bundle ${\mathbb{D}}_{4}$ which is a del Pezzo surface of
degree $4$ whose real part is homeomorphic to a disjoint union of $2$ spheres.
* •
A minimal conic bundle ${\mathbb{D}}_{2}$ which is a del Pezzo surface of
degree $2$ whose real part is homeomorphic to a disjoint union of $3$ spheres.
* •
A minimal surface ${\mathbb{G}}_{2}$ which is a del Pezzo surface of degree
$2$ whose real part is homeomorphic to a disjoint union of $4$ spheres.
* •
A minimal surface ${\mathbb{B}}_{1}$ which is a del Pezzo surface of degree
$1$ whose real part is homeomorphic to a disjoint union of $4$ spheres and a
real projective plane.
Let $X$ be a real surface such that all connected components of
$X({\mathbb{R}})$ are homeomorphic to each other. We indicate by $X(a,2b)$ the
real surface obtained by blowing up the real surface $X$ in $a$ real points
and $b$ pairs of complex conjugated points. If $a=1$, the topology of
$X(a,2b)({\mathbb{R}})$ does not depend on the real point at which we blow up.
If $a=2$, we denote by $X(2,2b)^{2}_{0}$ resp. $X(2,2b)^{1}_{1}$ the surfaces
obtained by blowing up two real points from the same or two points from
different connected components of $X({\mathbb{R}})$ respectively. The case
$a\geq 3$ will not occur in this work.
## 3\. Necessary conditions for surfaces
We first derive some necessary conditions on the ample divisor classes on a
surface that arise from a finite real-fibered morphism to $\mathbb{P}^{2}$
(3.1, 3.3). Then, we specify such conditions to the case of del Pezzo surfaces
(3.4). The case of conic bundles is treated in Section 7.
###### Lemma 3.1.
Let $X$ be a smooth irreducible surface and $f:X\to\mathbb{P}^{2}$ a real-
fibered morphism. Further let $X({\mathbb{R}})\neq\emptyset$. Then $f$ is
generically finite to one. Let us denote its degree by $d$. Then
$X({\mathbb{R}})\simeq sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$ such that
$d=r+2s$. The preimage of a general real line in $\mathbb{P}^{2}$ is a smooth
irreducible separating curve $C$ such that $C({\mathbb{R}})$ has $s+r$
connected components.
###### Proof.
Since $f$ is real-fibered, the fiber $f^{-1}(f(p))$ for any $p\in
X({\mathbb{R}})$ must be finite as any nonfinite variety has nonreal points.
Since the dimension of a fiber is upper-semicontinuous [Har95, Thm. 11.12],
this shows that $f$ is generically finite to one. Let $U\subset\mathbb{P}^{2}$
be the open subset of all $p\in\mathbb{P}^{2}$ such that $f^{-1}(p)$ is finite
and let $X^{\prime}=f^{-1}(U)$. The restriction $f^{\prime}:X^{\prime}\to U$
of $f$ to $X^{\prime}$ is _quasi-finite_ in the sense that each fiber is
finite. Moreover, it is proper because $f$ is proper as a morphism between
projective varieties. Thus $f^{\prime}$ is finite by [Gro66, Thm. 8.11.1].
Furthermore, we have $X({\mathbb{R}})\subset X^{\prime}$ and
$\mathbb{P}^{2}({\mathbb{R}})\subset U$. Thus by [KS20a, Thm. 2.19] the map
$X({\mathbb{R}})\to\mathbb{P}^{2}({\mathbb{R}})$ obtained by restricting $f$
to the real part of $X({\mathbb{R}})$ is an unramified covering map. This
implies that the real part $X({\mathbb{R}})$ is homeomorphic to the disjoint
union of $s$ spheres and $r$ projective planes such that $d=r+2s$, see also
[KS20a, Cor. 2.20]. By Bertini’s lemma [Jou79, Thm. 6.10], the preimage of a
general real line in $\mathbb{P}^{2}$ is a smooth irreducible curve $C$ with
$C({\mathbb{R}})$ having $r+s$ connected components, one for each connected
component of the covering space $X({\mathbb{R}})$. Furthermore, the
restriction of $f$ to $C$ is again real-fibered. Thus $C$ is separating, see
e.g. [KS20a, Thm. 2.8]. ∎
###### Example 3.2.
By the Noether–Lefschetz Theorem we can approximate the polynomial
$x_{1}^{4}+x_{2}^{4}+x_{3}^{4}-x_{0}^{4}$ arbitrarily close by a homogeneous
polynomial $p\in{\mathbb{R}}[x_{0},x_{1},x_{2},x_{3}]$ of degree $4$ such that
the zero set $X={\mathcal{V}}(p)\subset\mathbb{P}^{3}$ is a smooth surface
whose Picard group is generated by the class of a hyperplane section. In
particular, there is no morphism $f:X\to\mathbb{P}^{2}$ of degree $2$. Further
we can choose $p$ in such a way that $X({\mathbb{R}})$ is homeomorphic to a
sphere as this is the case for the real zero set of
$x_{1}^{4}+x_{2}^{4}+x_{3}^{4}-x_{0}^{4}$. Thus by 3.1 there is no real-
fibered morphism $f:X\to\mathbb{P}^{2}$. In particular, for an abstract smooth
surface $X$ the criterion $X({\mathbb{R}})\simeq sS^{2}\sqcup
r{\mathbb{R}}\mathbb{P}^{2}$, $r,s\in{\mathbb{Z}}_{\geq 0}$, is necessary but
not sufficient for admitting a real-fibered morphism $f:X\to\mathbb{P}^{2}$.
###### Theorem 3.3.
Let $f:X\to\mathbb{P}^{2}$ be a finite real-fibered morphism from a smooth
surface $X$ and $D$ the corresponding ample divisor class. Then we have the
following:
1. (1)
$X({\mathbb{R}})\simeq sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$;
2. (2)
$D.D=r+2s$;
3. (3)
$r\leq D.K+4$;
4. (4)
$D.K\equiv r\mod 4$;
5. (5)
$D.L>0$ for all effective divisors $L\subset X_{\mathbb{C}}$.
###### Proof.
Part $(1)$ and $(2)$ are part of the previous lemma. In order to prove the
inequality of $(3)$, we note that by the preceding lemma the preimage of a
general line in $\mathbb{P}^{2}$ is a smooth separating curve whose real part
has $r+s$ connected components. Its genus $g$ is according to the Adjunction
Formula [Har77, V, Prop. 1.5]:
$g=\frac{1}{2}(D.(D+K))+1=\frac{1}{2}(r+D.K)+s+1.$
Now Harnack’s inequality says that
$r+s\leq g+1=\frac{1}{2}(r+D.K)+s+2$
which implies $r\leq D.K+4$. Since the curve is separating, we furthermore
have that $g+r+s=\frac{1}{2}(r+D.K)+2s+r+1$ is odd which implies claim $(4)$.
Part $(5)$ is clear because $f$ is finite and $D$ therefore ample. This
remains true under a base change to ${\mathbb{C}}$. ∎
From 3.3 and the Kodaira Vanishing Theorem, one obtains the following
statement for del Pezzo surfaces. The Kodaira Vanishing Theorem is used to
prove the right-hand inequality in $(3)$ of 3.4. This allows us to narrow down
the ample divisor classes that can possibly arise as the pull-back of a
hyperplane section under a finite real-fibered morphism $X\to\mathbb{P}^{2}$
to a finite list.
###### Corollary 3.4.
Let $f:X\to\mathbb{P}^{2}$ be a finite real-fibered morphism from a del Pezzo
surface $X$ and $D$ the corresponding ample divisor class. Then we have the
following:
1. (1)
$X({\mathbb{R}})\simeq sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$;
2. (2)
$D.D=r+2s$;
3. (3)
$r\leq D.K+4\leq r+2s$;
4. (4)
$D.K\equiv r\mod 4$;
5. (5)
$D.L>0$ for all lines $L\subset X_{\mathbb{C}}$.
In particular, there are only finitely many possibilities for such $D$.
###### Proof.
In order to prove the missing inequality of $(3)$, we will compute $\ell(D)$
using the Riemann–Roch Theorem for surfaces [Har77, V, Thm. 1.6]. For this we
note that by [Har77, V, Cor. 3.5] the arithmetic genus of $X$ is zero.
Furthermore, because $D$ and $-K$ are ample, we have that $D-K$ is ample as
well and thus the Kodaira Vanishing Theorem [Har77, V, Rem. 7.15] implies that
$h^{i}(D)=0$ for $i>0$. Therefore, we have
$\ell(D)=\frac{1}{2}D.(D-K)+1$
by Riemann–Roch. Since $D$ comes from a morphism to $\mathbb{P}^{2}$, we have
$\ell(D)\geq 3$. Together with $(2)$ this implies $D.K+4\leq r+2s$. Finally,
it follows from the Hodge Index Theorem [Har77, Rem 1.9.1] and the fact that
$-K$ is ample that there can be only finitely many $D$ satisfying $(2)$ and
$(3)$. ∎
For each real del Pezzo surface $X$ whose real part $X({\mathbb{R}})$ consists
of connected components that are homeomorphic to spheres and real projective
planes only, we determined all divisor classes on $X$ that satisfy all
requirements of 3.4 via a brute-force search. The result is listed in Table 1.
In this table, we use the following notation regarding generators of the real
Picard group of real del Pezzo surface:
* •
$\textrm{Pic}(\mathbb{D}_{2})=\langle-K,F\rangle$, where $F$ denotes the class
of a fiber (when regarding ${\mathbb{D}}_{2}$ as a conic bundle
${\mathbb{D}}_{2}\to\mathbb{P}^{1}$) and $-K.F=2$;
* •
$\textrm{Pic}(\mathbb{D}_{2}(1,0))=\langle-K,\tilde{F},E\rangle$, where $E$
and $\tilde{F}$ are $(-1)$-curves and any two distinct generators have
intersection equal to one;
* •
$\textrm{Pic}(\mathbb{G}_{2}(1,0))=\langle-K,E\rangle$, where $E$ is a
$(-1)$-curve and $-K.E=1$.
In the following example, we carry this computation out for the real del Pezzo
surface $\mathbb{D}_{2}$. The other cases can be treated analogously but we do
not write down the explicit calculations here.
###### Example 3.5.
The real part of the del Pezzo surface $\mathbb{D}_{2}$ of degree two consists
of three connected components that are homeomorphic to a sphere [Rus02, Cor.
4.3]. Thus we have $s=3$ and $r=0$. The complexification
$(\mathbb{D}_{2})_{\mathbb{C}}$ of $\mathbb{D}_{2}$ is the blow-up of
$\mathbb{P}^{2}$ at seven points. Thus the Picard group of
$(\mathbb{D}_{2})_{\mathbb{C}}$ is the free abelian group generated by the
pull-back of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)$ and the classes of the seven
exceptional divisors $E_{1},..,E_{7}$. In this basis the complex conjugation
on $\operatorname{Pic}(\mathbb{D}_{2})_{\mathbb{C}}$ is given by the matrix
$\begin{pmatrix}4&3&1&1&1&1&1&1\\\ -3&-2&-1&-1&-1&-1&-1&-1\\\
-1&-1&-1&0&0&0&0&0\\\ -1&-1&0&-1&0&0&0&0\\\ -1&-1&0&0&-1&0&0&0\\\
-1&-1&0&0&0&-1&0&0\\\ -1&-1&0&0&0&0&-1&0\\\ -1&-1&0&0&0&0&0&-1\end{pmatrix}$
where the first coordinate corresponds to the pull-back of
${\mathcal{O}}_{\mathbb{P}^{2}}(1)$, see [Rus02, Exp. 2]. The real Picard
group of $\mathbb{D}_{2}$ consists of those divisor classes of
$(\mathbb{D}_{2})_{\mathbb{C}}$ that are fixed under this involution. Thus it
is generated by the two divisor classes $F=(1,-1,0,0,0,0,0,0)$ and
$K=(-3,1,1,1,1,1,1,1)$ where the latter is the canonical divisor class. We
observe that $F.F=0$, $F.K=-2$ and $K.K=2$. Assume that the divisor $D=hF-lK$
for $h,l\in{\mathbb{Z}}$ satisfies the conditions from 3.4. Condition $(2)$
says that $2(2h+l)l=6$. Here are the only pairs $(h,l)$ of integers that
satisfy this condition:
$(1,-3),(-1,-1),(1,1),(-1,3).$
Finally, condition $(5)$ applied to $L=E_{6}$ for instance rules out the
possibility of $l<0$, so we are left with $(h,l)=(1,1)$ and $(h,l)=(-1,3)$.
Alternatively, this is also implied by conditions $(3)$ and $(4)$ which show
that $h+l\in\\{0,2\\}$.
$X$ | degree | $s$ | $r$ | $D$ | $\ell(D)$ | $g$ | very ample? | See also:
---|---|---|---|---|---|---|---|---
$\mathbb{P}^{2}$ | $9$ | $0$ | $1$ | ${\mathcal{O}}_{\mathbb{P}^{2}}(1)$ | 3 | $0$ | yes | 3.7
$Q^{3,1}$ | $8$ | $1$ | $0$ | ${\mathcal{O}}_{\mathbb{P}^{2}}(1,1)$ | 4 | $0$ | yes | 3.7
$\mathbb{P}^{2}(0,2)$ | $7$ | $0$ | $1$ | — | - | - | - | 1.3,5.6
$Q^{3,1}(0,2)$ | $6$ | $1$ | $0$ | — | - | - | - | 1.3 ,5.6
$\mathbb{P}^{2}(0,4)$ | $5$ | $0$ | $1$ | — | - | - | - | 1.3,5.6
$Q^{3,1}(0,4)$ | $4$ | $1$ | $0$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{4}$ | $4$ | $2$ | $0$ | $-K$ | 5 | 1 | yes | 6.4
$\mathbb{P}^{2}(0,6)$ | $3$ | $0$ | $1$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{4}(1,0)$ | $3$ | $1$ | $1$ | $-K$ | 4 | 1 | yes | 3.7
$\mathbb{D}_{4}(2,0)^{1}_{1}$ | $2$ | $0$ | $2$ | $-K$ | 3 | 1 | no | 3.7
$Q^{3,1}(0,6)$ | $2$ | $1$ | $0$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{4}(0,2)$ | $2$ | $2$ | $0$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{2}$ | $2$ | $3$ | $0$ | $F-K$ | 6 | 2 | yes | 5.5, 5.7
| | | | $-F-3K$ | 6 | 2 | yes | 5.5, 5.7
$\mathbb{G}_{2}$ | $2$ | $4$ | $0$ | $-2K$ | 7 | 3 | yes | 6.5
$\mathbb{P}^{2}(0,8)$ | $1$ | $0$ | $1$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{4}(1,2)$ | $1$ | $1$ | $1$ | — | - | - | - | 1.3,5.6
$\mathbb{D}_{2}(1,0)$ | $1$ | $2$ | $1$ | $-3K-\tilde{F}+E$ | 5 | 2 | yes | 5.5, 5.7
| | | | $-5K-\tilde{F}-E$ | 5 | 2 | yes | 5.5, 5.7
| | | | $-K+\tilde{F}+E$ | 5 | 2 | yes | 5.5, 5.7
| | | | $-3K+\tilde{F}-E$ | 5 | 2 | yes | 5.5, 5.7
$\mathbb{G}_{2}(1,0)$ | $1$ | $3$ | $1$ | $-2K+E$ | 6 | 3 | yes | 6.6
| | | $1$ | $-4K-E$ | 6 | 3 | yes | 6.6
$\mathbb{B}_{1}$ | $1$ | $4$ | $1$ | $-3K$ | 7 | 4 | yes | 6.1
Table 1. A list of all del Pezzo surfaces $X$ whose real part consists of
spheres and real projective planes together with all divisor classes $D$ that
satisfy the conditions of 3.4. The $7$th column keeps track of the genus of
the divisor $D$. In the last column, the references are 3.7, 1.3, 5.6, 6.4,
5.5, 6.6 and 6.1.
In the next section we will prove that in fact all these divisors come from a
real-fibered morphism $f:X\to\mathbb{P}^{2}$ (and a hyperbolic embedding of
$X$ in the very ample cases). The last column of Table 1 indicates where a
detailed treatment of these divisors can be found. Note that for determining
which divisors are very ample we can employ [DR96]. For now we extract from
Table 1 the following.
###### Corollary 3.6.
Let $X$ be a smooth del Pezzo surface such that $X({\mathbb{R}})\simeq
sS^{2}\sqcup r{\mathbb{R}}\mathbb{P}^{2}$. If there is a finite and real-
fibered morphism $f:X\to\mathbb{P}^{2}$, then the preimage of a generic real
line is an $M$-curve, i.e., has genus $r+s-1$. Furthermore, the space of
global sections of $f^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)$ has dimension
$s+3$ and if $f^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)$ is very ample, then
$r\in\\{0,1\\}$.
###### Remark 3.7.
In some cases it is rather easy to verify that a morphism is real-fibered. For
example, it is clear that any real automorphism
$\mathbb{P}^{2}\to\mathbb{P}^{2}$ is real-fibered. Denote by
$\mathbb{D}_{4}(2,0)^{1}_{1}$ the blow-up of $\mathbb{D}_{4}$ at two points
belonging to different connected components. The anti-canonical map
$\mathbb{D}_{4}(2,0)^{1}_{1}\to\mathbb{P}^{2}$ is a double cover of
$\mathbb{P}^{2}$ ramified along a plane quartic curve without real points.
This is clearly real-fibered. Finally, the hypersurfaces $Q^{3,1}$ and
$\mathbb{D}_{4}(1,0)$ in $\mathbb{P}^{3}$ are hyperbolic with respect to any
point in the interior of the $2$-sphere as each $2$-sphere disconnects
$\mathbb{P}^{3}(\mathbb{R})$, and the intersection of any
$\mathbb{P}^{1}(\mathbb{R})$ with any $\mathbb{P}^{2}(\mathbb{R})$ is odd (see
also [HV07, Theorem 5.2]).
## 4\. Linking lemma and varieties of higher dimension
In this section, we give a criterion to determine whether an embedded variety
of dimension $k$ is hyperbolic in terms of linking numbers.
###### Definition 4.1 ([Pra07, §2.5]).
Let $X,Y$ be disjoint embedded oriented spheres in $S^{n}$ of dimensions $l$
and $m$ respectively, where $n=l+m+1$. Consider the fundamental cycles $[X]$
and $[Y]$ as cycles in the integral homology of $S^{n}$. There exists a chain
$W$ whose boundary is $[X]$. The _linking number_ $\operatorname{lk}(X,Y)$ is
defined to be the intersection number of $W$ and $[Y]$.
###### Remark 4.2.
For $l$ and $m$ non zero, the linking number does not depend on the choice of
$W$. Indeed, given another chain $W^{\prime}$ satisfying $\partial
W^{\prime}=[X]$, the chain $W^{\prime}-W$ is a cycle, and hence the boundary
of a chain $e$. Then the intersection number between $W^{\prime}-W$ and $[Y]$
is the intersection number between $\partial e$ and $[Y]$, which is equal to
the intersection number between $e$ and $\partial[Y]=0$, hence it is equal to
zero. If one of $l,m$ is zero (say $m$), then the linking number must be
computed as the intersection number of a 1-dimensional chain $W$ with boundary
$[Y]$ with the cycle $[X]$ (and it will not depend on the choice of $W$ in
that case), as otherwise the intersection number would depend on the choice of
chain $W$ with boundary $[X]$.
We extend the definition of linking numbers to spheres and linear subspaces
inside a real projective space, following the idea in [KS20b, §2] of linking
numbers of 1-dimensional spheres and linear subspaces inside a real projective
space.
###### Definition 4.3.
Let $K\subset\mathbb{P}^{n}(\mathbb{R})$ be an embedded $k$-sphere in
$\mathbb{P}^{n}(\mathbb{R}$) and $L\subset\mathbb{P}^{n}(\mathbb{R})$ be a
linear subspace of dimension $n-k-1$. Let
$p:S^{n}\rightarrow\mathbb{P}^{n}(\mathbb{R})$ be an unramified double cover.
The _linking number_ $\operatorname{lk}(K,L)$ is defined as the linking number
of $K_{1}\sqcup K_{2}$ with $p^{-1}(L)$ in $S^{n}$, where $K_{1}\sqcup K_{2}$
is the preimage of $K$ via $p$ (one of the $K_{i}$ may be empty).
The following proposition is a generalisation of [KS20b, Prop. 2.12] to the
case of projective subvarieties of any dimension.
###### Proposition 4.4.
Let $X\subset\mathbb{P}^{n}$ be a smooth subvariety of dimension $k$ and
degree $2s+r$ such that the set of real points $X({\mathbb{R}})$ is
homeomorphic to $sS^{k}\sqcup r{\mathbb{R}}\mathbb{P}^{k}$. Let
$E\subset\mathbb{P}^{n}$ be a linear subspace of dimension $n-k-1$ with $X\cap
E=\emptyset$. Let $X_{1},\ldots,X_{r+s}$ be the connected components of
$X({\mathbb{R}})$. Then $X$ is hyperbolic with respect to $E$ if and only if
$\sum\limits_{i=1}^{r+s}|\operatorname{lk}(X_{i},E({\mathbb{R}}))|=2s+r.$
###### Proof.
The variety $X$ is hyperbolic with respect to $E$ if and only if every
dimension $n-k$ real linear space $L$ that contains $E$ intersects $X$ in
$\deg X$ many (distinct) real points. Let
$p:S^{n}\rightarrow\mathbb{P}^{n}({\mathbb{R}})$ be an unramified double
cover. For any choice of such an $L\subset\mathbb{P}^{n}$ containing $E$, the
preimage $p^{-1}(E({\mathbb{R}}))$ is a sphere of dimension $n-k-1$ inside
$p^{-1}(L({\mathbb{R}}))$ which in turn is a sphere of dimension $n-k$. Let
$W\subset p^{-1}(L({\mathbb{R}}))$ be a hemisphere whose boundary is
$p^{-1}(E({\mathbb{R}}))$. If $X$ is hyperbolic with respect to $E$, then the
absolute values of the linking numbers
$\operatorname{lk}(X_{i},E({\mathbb{R}}))$, which are the intersection numbers
of the $p^{-1}(X_{i})$ with $W$, sum up to $\deg X$. Conversely, if the
intersection number of $W$ with the preimage of $X({\mathbb{R}})$ is $\deg X$,
then $L$ has (at least) $\deg X$ many real intersection points with $X$. ∎
The following proposition is a direct application of 4.4 and is used in
Section 6.
###### Proposition 4.5.
Let $X\subset\mathbb{P}^{n}$, for some $n\geq 3$, be a smooth surface and let
$X_{0}$ be a connected component of $X(\mathbb{R})$ homeomorphic to $S^{2}$.
Let $E\subset\mathbb{P}^{n}$ be a linear subspace of codimension 3 with $E\cap
X=\emptyset$. If $E({\mathbb{R}})$ intersects a 3-dimensional disc
$W\subset\mathbb{P}^{n}({\mathbb{R}})$, whose boundary is $X_{0}$, in exactly
one point, then $|\operatorname{lk}(X_{0};E({\mathbb{R}}))|=2$.
###### Proof.
Let $p:S^{n}\rightarrow\mathbb{P}^{n}({\mathbb{R}})$ be an unramified double
cover, and let $f:S^{2}\rightarrow\mathbb{P}^{n}({\mathbb{R}})$ be a
continuous map. Let $p_{*},f_{*}$ be the corresponding induced homomorphisms
between fundamental groups. Since $\pi_{1}(S^{k})=0$ for $k\geq 2$, we have
that $f_{*}(\pi_{1}(S^{2}))\subseteq p_{*}(\pi_{1}(S^{n}))$ and, by the
Lifting Property ([Hat02, Proposition 1.33]), there exists a lift
$\widetilde{f}:S^{2}\rightarrow S^{n}$ of $f$. As homology is functorial (and
covariant), it preserves compositions, hence we have
$f_{*}=p_{*}\circ\widetilde{f}_{*}$ for the induced homomorphisms between
${\mathbb{Z}}_{2}$-homology groups. As $H_{2}(S^{n};{\mathbb{Z}}_{2})=0$ for
$n\geq 3$, the homomorphism $f_{*}$ factors through zero, therefore it is the
zero homomorphism. In particular, if $[S^{2}]\in
H_{2}(S^{2};{\mathbb{Z}}_{2})$ is the fundamental class of $S^{2}$, then
$f_{*}[S^{2}]=0$. From this, we get that the linking number between $X_{0}$
and $E(\mathbb{R})$ must be even. By assumption, the real part
$E({\mathbb{R}})$ intersects a 3-dimensional disc
$W\subset\mathbb{P}^{n}({\mathbb{R}})$ of boundary $X_{0}$ in exactly one
point, therefore $\operatorname{lk}(X_{0},E({\mathbb{R}}))=\pm 2$. ∎
We conclude this section by giving a proof of 1.5.
###### Proof of 1.5.
The variety $C$ is hyperbolic with respect to $E$ if and only if every
dimension $n-k$ linear space $L$ that contains $E$ intersects $C$ in $\deg C$
many (distinct) real points. Let $p:S^{n-1}\rightarrow H({\mathbb{R}})$ be an
unramified double cover. For any choice of such a $L\subset H$ containing $E$,
the preimage $p^{-1}(E({\mathbb{R}}))$ is a sphere of dimension $n-k-1$ inside
$p^{-1}(L({\mathbb{R}}))$ which in turn is a sphere of dimension $n-k$. Let
$W\subset p^{-1}(L({\mathbb{R}}))$ be a hemisphere whose boundary is
$p^{-1}(E({\mathbb{R}}))$. If $C$ is hyperbolic with respect to $E$, then the
absolute values of the linking numbers
$\operatorname{lk}(C_{i},E({\mathbb{R}}))$, which are the intersection numbers
of the $p^{-1}(C_{i})$ with $W$, sum up to $\deg C$ (4.4). Now, let
$j:S^{n-1}\hookrightarrow S^{n}$,
$i:H({\mathbb{R}})\hookrightarrow\mathbb{P}^{n}({\mathbb{R}})$ and
$q:S^{n}\rightarrow\mathbb{P}^{n}({\mathbb{R}})$ respectively be two
inclusions and a double unramified cover. One has that $j\circ
p^{-1}=q^{-1}\circ i$.
Denote by $\tilde{E}\subset S^{n}$ the image of $p^{-1}(E({\mathbb{R}}))$ via
$j$, which is still a sphere of dimension $n-k-1$ in $S^{n}$. The $n-k$ sphere
$\tilde{L}:=j(p^{-1}(L({\mathbb{R}}))$ has $\tilde{E}$ as an equator, and
$\tilde{W}:=j(W)$ is one of its two hemispheres of boundary $\tilde{E}$. It
follows that $q^{-1}(X_{i})$ has to intersect $\tilde{W}$ in at least
$|\operatorname{lk}(C_{i},E({\mathbb{R}}))|$ number of points. Therefore,
(1) $\sum\limits_{i=1}^{r+s}|\operatorname{lk}(X_{i},i(E({\mathbb{R}})))|\geq
2s+r,$
as $\tilde{E}=j(p^{-1}(E({\mathbb{R}}))=q^{-1}(i(E({\mathbb{R}}))$. Moreover
the inequality (1) is an equality, since the sum on the left-hand side is at
most $\deg(X)=2s+r$. Thus by Proposition 4.4, the variety $X$ is hyperbolic
with respect to $i(E)=E$.
The converse can be seen by taking the restriction of the set of linear spaces
of dimension $n-k-1$ through $E$ to the hyperplane $H$. Every such linear
space intersects $C=X\cap H$ in $2s+r$ real points. The hyperbolicity of $X$
with respect to $E$ implies that $C$ is hyperbolic with respect to $E$. ∎
## 5\. Hyperbolic del Pezzo surfaces
In this section we will prove Theorems 1.2, 1.6 and 1.7.
###### Lemma 5.1.
Let $X\subset\mathbb{P}^{n}$ be a smooth surface such that $X({\mathbb{R}})$
is homeomorphic to $sS^{2}\sqcup{\mathbb{R}}\mathbb{P}^{2}$ with
$\deg(X)=2s+1$. The connected component that is homeomorphic to a real
projective plane realises the nontrivial homology class in
$H_{2}(\mathbb{P}^{n}({\mathbb{R}});{\mathbb{Z}}_{2})$.
###### Proof.
Since $X$ has odd degree, its real part $X({\mathbb{R}})$ realises the
nontrivial homology class in
$H_{2}(\mathbb{P}^{n}({\mathbb{R}});{\mathbb{Z}}_{2})$. Every sphere embedded
to $\mathbb{P}^{n}({\mathbb{R}})$ is homologous to zero (see proof of 4.5).
Thus the remaining connected component must realise the nontrivial class. ∎
###### Lemma 5.2.
Let $X\subset\mathbb{P}^{n}$ be a smooth nondegenerate surface such that
$X({\mathbb{R}})$ is homeomorphic to $sS^{2}\sqcup
r{\mathbb{R}}\mathbb{P}^{2}$ with $\deg(X)=2s+r$, $n=s+2$ and $r\in\\{0,1\\}$.
Assume that the sectional genus of $X$ is $s+r-1$. There is a hyperplane
$H\subset\mathbb{P}^{n}$ such that $C=X\cap H$ is a smooth and nondegenerate
$M$-curve with the property that each connected component of $X({\mathbb{R}})$
contains exactly one connected component of $C({\mathbb{R}})$.
###### Proof.
For any choice of one point $p_{i}$ on each connected component of
$X({\mathbb{R}})$ that is homeomorphic to a sphere, there is a hyperplane of
$\mathbb{P}^{n}$ that contains these points since $n>s$. If we choose these
points general enough, then by Bertini’s lemma there is such a hyperplane $H$
whose intersection with $X$ is smooth and nondegenerate in $H$. Let $C$ be
$X\cap H$. By construction and 5.1, each connected component of
$X({\mathbb{R}})$ contains at least one connected component of
$C({\mathbb{R}})$. But since the genus of $C$ is $s+r-1$, it must be exactly
one connected component of $C({\mathbb{R}})$ on each connected component of
$X({\mathbb{R}})$. ∎
###### Lemma 5.3.
Let $C\subset\mathbb{P}^{s+1}$ be a smooth nondegenerate $M$-curve of genus
$g=s+r-1$ and degree $2s+r$ such that $r$ components of $C({\mathbb{R}})$
realise the nontrivial homology class in
$H_{1}(\mathbb{P}^{s+1}({\mathbb{R}});{\mathbb{Z}}_{2})$. Then $C$ is
hyperbolic.
###### Proof.
First we note that $C$ has $s+r$ connected components. A general enough
hyperplane $H$ that intersects each of the $s$ connected components
$C_{1},\ldots,C_{s}$ of $C({\mathbb{R}})$ that realise the trivial homology
class will intersect $C$ in $2s+r$ distinct real points. Let $D$ be the
divisor corresponding to this hyperplane section. It is of the form
$D=D_{0}+\sum_{i=1}^{s}P_{i}$
for some effective divisor $D_{0}$ and points $P_{i}\in C_{i}$ that are not in
the support of $D_{0}$. Note that $D_{0}$ is the sum of one point from each
connected component of $C({\mathbb{R}})$. The divisor
$D_{k}=D_{0}+\sum_{i=1}^{k}P_{i}$ is nonspecial by [Hui03, Thm. 2.5] for all
$k=1,\ldots,s$. The complete linear system $|D_{0}|$ has dimension $2$ and the
corresponding morphism $C\to\mathbb{P}^{1}$ is real-fibered by [Gab06, Prop.
4.1]. Therefore, an iterated application of [KS20b, Prop. 3.2] shows together
with [KS20b, Lem. 2.10] that the embedding of $C$ via the complete linear
system $|D|$ is hyperbolic. But since $D$ is nonspecial, Riemann–Roch shows
that this is exactly our embedding $C\subset\mathbb{P}^{s+1}$ we started with.
∎
###### Theorem 5.4.
Let $X\subset\mathbb{P}^{n}$ be a smooth nondegenerate surface such that
$X({\mathbb{R}})$ is homeomorphic to $sS^{2}\sqcup
r{\mathbb{R}}\mathbb{P}^{2}$ with $\deg(X)=2s+r$, $r\in\\{0,1\\}$ and $n=s+2$.
Assume that the genus of a hyperplane section on $X$ is $s+r-1$. Then $X$ is
hyperbolic.
###### Proof.
By 5.2 there is a hyperplane $H\subset\mathbb{P}^{n}$ such that $C=X\cap H$ is
a smooth and nondegenerate $M$-curve with the property that each connected
component of $X({\mathbb{R}})$ contains exactly one connected component of
$C({\mathbb{R}})$. Note that by 5.1 exactly $r$ connected components of
$C({\mathbb{R}})$ realise the nontrivial homology class. This curve $C$ is
hyperbolic by 5.3. Thus $X$ is hyperbolic by 1.5. ∎
###### Proposition 5.5.
All the divisors listed in Table 1 correspond to a real-fibered morphism. In
addition, those divisors which are very ample correspond to a hyperbolic
embedding.
###### Proof.
This follows from 5.4 together with 3.6 and 3.7. ∎
###### Proof of Theorem 1.2.
All del Pezzo surfaces admitting a real-fibered morphism (i.e. those given by
5.5) satisfy one of the conditions $(1),(2),(3)$. Namely, the surfaces
$\mathbb{P}^{2},\mathbb{G}_{2}$ and $\mathbb{B}_{1}$ have real Picard rank 1,
the surfaces $\mathbb{D}_{4}$ and $\mathbb{D}_{2}$ are conic bundles of real
Picard rank 2, and the other surfaces listed are blow-ups of one of the
previous surfaces at one or two real points. ∎
###### Proof of 1.3.
Excluding the cases already treated in 1.2 and those where $X({\mathbb{R}})$
is not homeomorphic to $sS^{2}\sqcup r\mathbb{R}P^{2}$, we get that each
remaining del Pezzo surface $X$ is a blow-up at pairs of complex conjugate
points of another del Pezzo surface which admits finite real-fibered
morphisms. Therefore $X$ does admit real-fibered morphisms to
$\mathbb{P}^{2}$. ∎
###### Proof of Theorem 1.6.
All del Pezzo surfaces admitting a hyperbolic embedding (i.e. those given by
5.5 with the very ampleness condition) satisfy one of the conditions
$(1),(2),(3)$. Those satisfying $(1)$ or $(2)$ are the same as for Theorem
1.2, and all other surfaces listed satisfy $(3)$, except for
$\mathbb{D}_{4}(2,0)^{1}_{1}$, whose divisor was the only one not satisfying
very ampleness. ∎
###### Proof of Theorem 1.7.
First assume that we have $(1)$—$(3)$ from 1.7. Then the divisor $D$ given by
a generic hyperplane section satisfies $(1)$—$(5)$ from 3.4. The conditions
$(1)$, $(2)$ and $(5)$ are clear and $(3)$, $(4)$ follow from our assumption
on the genus of a hyperplane section. Indeed, the Adjunction Formula implies
that $r=D.K+4$. Thus by Table 1 we also have $r\in\\{0,1\\}$ and $n=s+2$. Now
5.4 implies that $X$ is hyperbolic. Conversely, if $X$ is hyperbolic, then
$(1)$—$(3)$ follow from 3.6. ∎
###### Remark 5.6.
Let $X$ be either $Q^{3,1}(0,2h)$ with $1\leq h\leq 3$, or
$\mathbb{P}^{2}(0,2j)$ with $1\leq j\leq 4$, or $\mathbb{D}_{4}(0,2)$, or
$\mathbb{D}_{4}(1,2)$. Each $X$ admits a real-fibered morphism (1.3) but no
finite real-fibered morphism (1.2). Therefore, each such $X$ can only admit
non-finite real-fibered morphisms.
###### Remark 5.7.
Let us consider the surface $\mathbb{D}_{2}$ and the divisors $D_{1}=F-K$ and
$D_{2}=-F-3K$. Thanks to 5.4, both divisors correspond to hyperbolic
embeddings in $\mathbb{P}^{5}$. Observe that $D_{i}$ is obtained by applying
the Geiser involution to $D_{j}$, where $\\{i,j\\}=\\{1,2\\}$ (see [Rus02, §4,
Example 3] for a description of Geiser involution). A similar approach works
for the surface $\mathbb{D}_{2}(1,0)$ and the divisors
$D_{1}=-3K-\tilde{F}+E$, $D_{2}=-3K+\tilde{F}-E$, $D_{3}=-5K-\tilde{F}-E$ and
$D_{4}=-K+\tilde{F}+E$, one has that all divisors correspond to hyperbolic
embeddings in $\mathbb{P}^{4}$. Moreover, the Bertini involution sends $D_{j}$
to $D_{j+1}$, for $j=1,3$ (see [Rus02, §5 , Example 4]).
## 6\. Explicit constructions and examples
Here we construct for most of the embeddings in Table 1 explicit linear
subspaces with respect to which the del Pezzo surfaces under consideration are
hyperbolic.
###### Example 6.1.
Consider the embedding $h:\mathbb{B}_{1}\rightarrow\mathbb{P}^{6}$ associated
to $|-3K|$. We will explicitly construct linear subspaces $E\subset
H\subset\mathbb{P}^{6}$ of dimension $3$ and $5$ to which we can apply Theorem
1.5. The anti-bicanonical map $\phi$ of $\mathbb{B}_{1}$ is the double cover
of the quadratic cone $Q$ in $\mathbb{P}^{3}$ ramified along the vertex $V$ of
$Q$ and a real non-singular cubic section $S\subset Q$ disjoint from $V$ that
is an $M$-curve of genus four. Via the map $\phi$, we want to construct three
smooth curves $C_{1}$, $C_{2}$ and $C_{3}$ on $X$ that are linearly equivalent
to $-3K$. Let us pick $C_{3}$ as $\phi^{-1}(S)$ (set-theoretical preimage). We
observe that each connected component of $\mathbb{B}_{1}({\mathbb{R}})$
contains exactly one connected component of $C_{3}({\mathbb{R}})$. We
construct $C_{1}$ and $C_{2}$ as follows. Choose a point $p_{i}$ on each
connected component of $Q(\mathbb{R})\setminus S(\mathbb{R})$ homeomorphic to
a disk for $i=1,2,3,4$. Pick two curves $C_{ijk}$ and $C_{jkt}$ on $Q$ as the
intersection of $Q$ with the hyperplane passing through $p_{i},p_{j},p_{k}$
and $p_{j},p_{k},p_{t}$ respectively, where $\\{i,j,k,t\\}=\\{1,2,3,4\\}$.
Moreover, pick the two generatrices $L_{t}$ and $L_{i}$ of $Q$ passing through
$p_{t}$ and $p_{i}$ respectively (see Fig. 1). One can perturb the union of
$C_{ijk}$ and $L_{t}$ resp. the union of $C_{jkt}$ and $L_{i}$ to a smooth
curve $X_{1}$ resp. $X_{2}$ (see [Man20, Section 4] for details) such that
$S(\mathbb{R})\cap X_{i}(\mathbb{R})$ consists of nine distinct real points
and $X_{i}(\mathbb{R})$ intersects each connected component of
$S(\mathbb{R})$, for $i=1,2$. Then we let $C_{i}=\phi^{-1}(X_{i})$ for
$i=1,2$. We further choose hyperplanes $H_{i}\subset\mathbb{P}^{6}$ such that
$C_{i}=X\cap H_{i}$ for $i=1,2,3$. The linear subspace $E=H_{1}\cap H_{2}\cap
H_{3}$ has dimension $3$ and the divisors $H_{i}\cap C_{3}$ on $C_{3}$ for
$i=1,2$ interlace on $C_{3}$ in the sense of [KS20b, §2.1]. Thus [KS20b, Lemma
2.1] shows that $C_{3}$ is hyperbolic with respect to $E$. Therefore, by
Theorem 1.5 applied to $E$ and $H=H_{3}$ we find that $\mathbb{B}_{1}$ is
hyperbolic with respect to $E$ as well.
Figure 1. The quadrangle, whose vertical sides are identified accordingly with
the arrows and horizontal sides represent the vertex $V$, is
$Q({\mathbb{R}})$. The cubic section $S({\mathbb{R}})$ is in thick black,
while $C_{ijk}({\mathbb{R}})$ and $C_{jkt}({\mathbb{R}})$ in dashed. The
generatrices $L_{i}({\mathbb{R}})$ and $L_{t}({\mathbb{R}})$ are in grey.
###### Remark 6.2.
Let $X$ be a real non-singular del Pezzo surface and $D$ a real very ample
divisor on $X$. An analogue construction to 6.1 can be applied to the
embedding associated to $|D|$, where $X=\mathbb{G}_{2}$ and $D=-2K$.
Now, we study in more details the cases of real del Pezzo surfaces obtained as
a double cover of some real surfaces ramified along a real curve. In
particular, the real part of these surfaces consist of spheres. The double
cover assumption allows us to talk about the “interior” of these spheres,
enabling us to choose a suitable linear space $E$ in order to apply
Proposition 4.4.
###### Proposition 6.3.
Let $Y\subset\mathbb{P}^{n}$ be a smooth surface of degree $n-2$ contained in
some hyperplane $H\subset\mathbb{P}^{n}$. Let $p\in\mathbb{P}^{n}$ be a real
point that is not in $H$ and let $\tilde{Y}\subset\mathbb{P}^{n}$ be the cone
over $Y$ with apex $p$. Finally, let $X\subset\tilde{Y}$ be a smooth surface
with $X({\mathbb{R}})\neq\emptyset$ that does not contain $p$ such that the
projection $\pi_{p}:X\to Y$ is a double cover branched along the intersection
$C$ of $Y$ with a quadratic hypersurface.
1. (1)
If $C({\mathbb{R}})=\emptyset$ and $Y$ is hyperbolic, then $X$ is hyperbolic.
2. (2)
If $C({\mathbb{R}})\neq\emptyset$ and $X({\mathbb{R}})$ is homeomorphic to the
disjoint union of $n-2$ spheres, then $X$ is hyperbolic. Furthermore, for any
$v\in X({\mathbb{R}})$ the embedding of $\operatorname{Bl}_{v}X$ to
$\mathbb{P}^{n-1}$ obtained by projecting $X$ from $v$ is also hyperbolic.
###### Proof.
Without loss of generality, we can assume that $H={\mathcal{V}}(x_{0})$ and
$p=[1:0:\cdots:0]$. Then there is a quadratic polynomial
$f\in{\mathbb{R}}[x_{1},\ldots,x_{n}]$ such that $X$ is the intersection of
$\tilde{Y}$ with ${\mathcal{V}}(x_{0}^{2}-f)$. In case $(1)$ the quadratic
polynomial $f$ is strictly positive on the real part of $Y$ and the map
$\pi_{p}$ is real-fibered. Thus if $Y$ is hyperbolic with respect to a linear
subspace $E\subset H$, then $X$ is hyperbolic with respect to the subspace
spanned by $E$ and $p$. Now assume that we are in case $(2)$. Consider the set
$W\subset\tilde{Y}({\mathbb{R}})$ of all points $x$ with $x_{0}^{2}\leq f(x)$.
This set has $n-2$ connected components $W_{1},\ldots,W_{n-2}$ and the
boundary of each $W_{i}$ is a connected component $X_{i}$ of
$X({\mathbb{R}})$. Let $E$ be a linear space of dimension $n-3$ that
intersects each $W_{i}\setminus X_{i}$ in (at least) one point $p_{i}$. Then
since $\deg(\tilde{Y})=n-2$ there are no further intersection point of $E$ and
$\tilde{Y}$ (note that this implies that $E$ is spanned by the $p_{i}$ and
that $E\cap X=\emptyset$). Thus $E$ intersects each $W_{i}$ in exactly one
point, so by 4.5 we have $|\operatorname{lk}(X_{i};E({\mathbb{R}}))|=2$ for
$i=1,\ldots,n-2$. Therefore, by 4.4 the variety $X$ is hyperbolic with respect
to $E$. For the additional statement assume without loss of generality that
$v\in X_{1}$ and take $v_{j}\in W_{1}\setminus X_{1}$ a sequence of points
that converges to $v$. Let $E_{j}$ be the linear subspace of dimension $n-3$
that is spanned by $v_{j}$ and $p_{2},\ldots,p_{n-2}$. This sequence converges
to the linear subspace $\tilde{E}$ that is spanned by $v$ and
$p_{2},\ldots,p_{n-2}$. Thus, since $X$ is hyperbolic with respect to each
$E_{j}$, every linear subspace $E^{\prime}$ of dimension $n-2$ that contains
$\tilde{E}$ intersects $X$ only in real points. This implies that the image of
$X$ under the projection from $v$ is hyperbolic with respect to the image of
$\tilde{E}$ under this projection. ∎
###### Example 6.4.
The anticanonical divisor on $X=\mathbb{D}_{4}$ gives an embedding into
$\mathbb{P}^{4}=\mathbb{P}^{n+2}$ where $n=2$ is the number of spheres in
$X({\mathbb{R}})$. The image is cut out by a pencil of quadrics. The
corresponding complex pencil contains (counted with multiplicity) five
singular quadrics. Thus the image of $\mathbb{D}_{4}$ is contained in at least
one real singular quadric $Q\subset\mathbb{P}^{4}$. The projection from the
vertex of $Q$ realises $\mathbb{D}_{4}$ as a double cover of a quadratic
hypersurface $Y\subset\mathbb{P}^{3}$ ramified along a smooth curve $C$ of
bidegree $(2,2)$. If $C({\mathbb{R}})=\emptyset$, then $Y=Q^{3,1}$ and we can
apply part $(1)$ of 6.3 to obtain a plane $E$ with respect to which the image
of $\mathbb{D}_{4}$ is hyperbolic. Otherwise, we can apply part $(2)$ of 6.3.
###### Example 6.5.
Consider the embedding $\mathbb{G}_{2}\rightarrow\mathbb{P}^{6}$ associated to
$|-2K|$. We have $l(-2K)=7$, hence $|-2K|$ embeds ${\mathbb{G}}_{2}$ into
$\mathbb{P}^{6}=\mathbb{P}^{n+2}$ for $n$ the number of spheres in the real
part of $\mathbb{G}_{2}$. The canonical map
${\mathbb{G}}_{2}\to\mathbb{P}^{2}$ is a double cover of $\mathbb{P}^{2}$
ramified along a smooth quartic curve with four connected components in its
real part. Therefore, we can apply part $(2)$ of 6.3 to
$Y\subset\mathbb{P}^{5}$ being the image of $\mathbb{P}^{2}$ under second
Veronese map.
###### Example 6.6.
Now consider the embeddings $\mathbb{G}_{2}(1,0)\rightarrow\mathbb{P}^{5}$
associated to $|D_{i}|$ for $i=1,2$ with $D_{1}=-2K+E$ and $D_{2}=-4K-E$. By
applying part $(2)$ of 6.3 to the situation considered in 6.5, we obtain a
hyperbolic embedding of $\mathbb{G}_{2}(1,0)$ to $\mathbb{P}^{5}$. This
correspond to the divisors $D_{1}$. The other divisor is obtained by applying
the Bertini involution $\tau$ on $D_{1}$ ( [Rus02, §5 , Example 4]).
## 7\. Conic bundles
In this section $\pi:X\to\mathbb{P}^{1}$ will denote a geometrically
irreducible smooth minimal conic bundle, i.e. each fiber of $\pi$ is
isomorphic to a plane conic and the real Picard rank of $X$ is $2$. Assume
that $X({\mathbb{R}})$ consists of $s$ spheres. Let $F$ denote a fiber of
$\pi$. Then $\textrm{Pic}(X)$ is generated by $-K$ and $F$. We clearly have
$F.F=0$. We further have $K.K=8-2s$ and $F.K=-2$, see [Kol17, p. 5]. Applying
the necessary criteria for the existence of a real-fibered morphism
$X\to\mathbb{P}^{2}$ from 3.3 to this situation, gives the following.
###### Corollary 7.1.
Let $f:X\to\mathbb{P}^{2}$ be a finite real-fibered morphism and $D=aF-bK$,
$a,b\in{\mathbb{Z}}$, the corresponding ample divisor class. Then we have the
following:
1. (1)
$b\geq 1$;
2. (2)
$a\geq-1$;
3. (3)
$s=b\cdot((4-s)\cdot b+2a)$;
4. (4)
$a+(4-s)\cdot b\leq 2$;
5. (5)
$a\equiv s\cdot b\mod 2$;
6. (6)
$2a>b(s-4)$;
###### Proof.
Since $F$ is effective and $D$ ample we must have $2b=F.D>0$. This shows
$(1)$. By 3.3(2) we have
$2s=D.D=4ab+b^{2}(8-2s)$
which shows $(3)$. By 3.3(3) we have
$0\leq D.K+4=-2a-b\cdot(8-2s)+4$
which shows $(4)$. Part $(5)$ follows from 3.3(4). Finally, part $(4)$ implies
$2a+(4-s)\cdot b\leq 2+a.$
Multiplying this with $b$ and using part $(3)$ we obtain
$s=(2a+(4-s)\cdot b)\cdot b\leq(2+a)\cdot b.$
Since $b,s>0$, we obtain $2+a>0$. For part $(6)$ we observe that since $D$ is
ample we must have
$0<D.D=4ab+b^{2}(8-2s).$
Since $b>0$, this shows $0<2a+b(4-s)$. ∎
###### Lemma 7.2.
Let $D=aF-bK$ be an ample divisor class satisfying $(1)-(6)$ of 7.1. Then
$\ell(D)\geq s+3$.
###### Proof.
By Riemann–Roch and since $X$ is rational we have
$\ell(D)+\ell(K-D)\geq\frac{1}{2}D.(D-K)+1=s+3.$
Thus it suffices to show that $\ell(K-D)=0$. We note that $F$ is nef as the
pullback of a nef divisor [Laz04, Exp. 1.4.4]. But since we have
$(K-D).F=-2(1+b)$, the divisor $K-D$ cannot be effective. ∎
This gives us a good candidate for a linear system giving rise to a hyperbolic
embedding. Namely, for any fixed $s$, the divisor $D=(s-2)F-K$ satisfies all
of the necessary conditions in 7.1 and we have the following.
###### Corollary 7.3.
If $D=(s-2)F-K$ is very ample, then it gives rise to a hyperbolic embedding of
$X$.
###### Proof.
By the Adjunction Formula we compute the sectional genus of $D$ as
$\frac{1}{2}D.(D+K)+1=\frac{1}{2}((s-2)F-K).(s-2)F+1=s-1.$
Thus by 7.2 we can apply 5.4. ∎
###### Remark 7.4.
The divisor $D=(s-2)F-K$ from 7.3 is the only divisor for which equality holds
in 3.3(3).
###### Remark 7.5.
Observe that $\mathbb{D}_{4}$ and $\mathbb{D}_{2}$ are minimal conic bundles
with $s=2$ and $s=3$ respectively. The only divisors of these surfaces
corresponding to real-fibered morphisms are of the form $(s-2)F-K$ (Table 1).
Note that in the case of ${\mathbb{D}}_{2}$ we have two different divisors
which is due to the fact that ${\mathbb{D}}_{2}$ can be equipped with two
different structures of a conic bundle. In both cases, these divisors are very
ample.
We focus on the divisor class $D=(s-2)F-K$ on a minimal conic bundle $X$. We
first observe the existence of a particular rank $3$ bundle $\mathcal{E}$ for
any given $X$ by 7.6. Then we look at some minimal conic bundles for which $D$
is always very ample.
###### Proposition 7.6.
There is a vector bundle $\mathcal{E}$ of rank 3 on $\mathbb{P}^{1}$ with
first Chern class $s$ and a section $t$ of
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$ such that $X$ is the zero set of
$t$ and $\pi$ is the restriction of the natural projection
$f:\mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^{1}$ to $X$. Furthermore, the
restriction $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)|_{X}$ corresponds to the
class $(s-2)F-K$.
###### Proof.
By [KSC04, Ex. 3.13.4], there is a rank 3 vector bundle $\mathcal{E}^{\prime}$
on $\mathbb{P}^{1}$ and an embedding $X\rightarrow\mathbb{P}^{1}$ where $X$ is
realised as a family of conics in the projective plane fibers of
$\mathbb{P}(\mathcal{E}^{\prime})$. From [EH16, §9.3], the Chow ring of
$\mathbb{P}(\mathcal{E}^{\prime})$ is given by
${\mathbb{Z}}[H,E]/(E^{2},H^{3}-cEH^{2})$, where $H$ is the class of the line
bundle $\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\prime})}(1)$, the class $E$ is a
fiber of the map $\mathbb{P}(\mathcal{E}^{\prime})\rightarrow\mathbb{P}^{1}$
and $c$ is the first Chern class of $\mathcal{E}^{\prime}$. The class of a
point is $EH^{2}$. Because $X$ has codimension $1$ in
$\mathbb{P}({\mathcal{E}}^{\prime})$, its class of $X$ in the Chow ring of
$\mathbb{P}(\mathcal{E}^{\prime})$ must be of the form $aE+a^{\prime}H$ for
some integers $a,a^{\prime}$. Because the intersection of $X$ with a fiber $E$
is a plane conic, we must have $a^{\prime}=2$. The canonical class on
$\mathbb{P}(\mathcal{E}^{\prime})$ is $(c-2)E-3H$. Thus by the Adjunction
Formula [EH16, Prop. 1.33] the canonical class of $X$ can be obtained by
intersecting $((a+c-2)E-H)$ with $X$. This implies that we have
$K.K=((a+c-2)E-H)^{2}.(2H+aE)=8-3a-2c.$
On the other hand we know that $K.K=8-2s$ which implies that $s=3b+c$ where
$b$ is an integer satisfying $2b=a$. Consider the vector bundle
$\mathcal{E}=\mathcal{E}^{\prime}(b)$ on $\mathbb{P}^{1}$. The first Chern
class of $\mathcal{E}$ is $c+3b=s$ [EH16, Prop. 5.17] and
$\mathbb{P}(\mathcal{E})$ is isomorphic to $\mathbb{P}(\mathcal{E}^{\prime})$
as scheme over $\mathbb{P}^{1}$ [EH16, Cor. 9.5]. Finally, we have that
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)=\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\prime})}(1)\otimes
f^{*}\mathcal{O}_{\mathbb{P}^{1}}(b)$ [EH16, Cor. 9.5], i.e., the zero set of
a nonzero section in $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ has the class
$H+bE$. This shows that the class corresponding to
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$ is $2H+2bE$, the class of $X$. In
order to prove the additional statement we can, after replacing
$\mathcal{E}^{\prime}$ by $\mathcal{E}$, assume without loss of generality
that $c=s$ and $a=0$. Let $xF+yK$ be the class of
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)|_{X}$. On one hand, we have
$(xF+yK).F=-2y$ in the Chow ring of $X$. We can compute the same number in the
Chow ring of $\mathbb{P}(\mathcal{E})$ as
$2H.H.E=2$
which shows that $y=-1$. Similarly, we have in the Chow ring of $X$ that
$(xF-K).K=-2x-8+2s$. This can be computed in the Chow ring of
$\mathbb{P}(\mathcal{E})$ as
$2H.H.((s-2)E-H)=-4$
which implies $x=s-2$. ∎
###### Remark 7.7.
Conversely, let $\mathcal{E}$ be a vector bundle of rank 3 on $\mathbb{P}^{1}$
with first Chern class $s$ and $X$ the smooth zero set of a section of
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$. Then clearly the restriction of
the natural projection $\mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^{1}$ to
$X$ gives $X$ the structure of a conic bundle. A direct computation as in the
proof of Proposition 7.6 shows that $X$ has $2s$ singular fibers and that the
restriction of ${\mathcal{O}}_{\mathbb{P}({\mathcal{E}})}(1)$ to $X$
corresponds to the divisor class $(s-2)F-K$.
Let $\mathcal{E}$ be a vector bundle as in 7.6. By Grothendieck’s splitting
theorem ([EH16, Theorem 6.29]) we have
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}(a_{1})\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{2})\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{3})$
where the integers $a_{i}$ sum up to $s$. If each $a_{i}>0$, then
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is very ample [EH16, Proof of Cor.
9.9], and therefore $D$ is also very ample on $X$. It follows that one can
construct explicit examples in which $D$ is very ample and embeds $X$ in some
projective space as a hyperbolic variety, see 7.8.
###### Example 7.8 (Very ample).
Consider the zero set $X$ of the bihomogeneous polynomial
$G=uvx_{0}^{2}+(u^{2}-v^{2})x_{1}^{2}+(u^{2}-4v^{2})x_{2}^{2}$
inside $\mathbb{P}^{1}\times\mathbb{P}^{2}$. Clearly $X$ is a conic bundle
with $2s=6$ singular fibers. Letting
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}(1)^{3}$ we have
$\mathbb{P}(\mathcal{E})=\mathbb{P}^{1}\times\mathbb{P}^{2}$ and $G$ is a
global section of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$. The Segre
embedding of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ to $\mathbb{P}^{5}$ is the
embedding associated to $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$. Via this
embedding $X$ is hyperbolic.
Here is an example in which the very ampleness fails.
###### Example 7.9 (Not very ample).
Consider the vector bundle
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}^{2}\oplus\mathcal{O}_{\mathbb{P}^{1}}(2)$
and let
$\mathbb{P}(\mathcal{E})=\operatorname{Proj}(\operatorname{Sym}\mathcal{E})$
the associated projective plane bundle over $\mathbb{P}^{1}$. We define
$X\subset\mathbb{P}(\mathcal{E})$ to be the Zariski closure of the zero set of
$(t^{3}-t)x_{0}^{2}+x_{1}^{2}+x_{2}^{2}$
in a chart $\mathbb{A}^{1}\times\mathbb{P}^{2}\subset\mathbb{P}(\mathcal{E}).$
One can check that $X$ is a smooth conic bundle with singular fibers at
$t=-1,0,1,\infty$. It is a desingularisation of the intersection of
$S(0,0,2)=\mathcal{V}(y_{0}y_{2}-y_{1}^{2})\subset\mathbb{P}^{4}$
with the quadratic hypersurface defined by
$y_{3}^{2}+y_{4}^{2}-y_{0}y_{1}+y_{1}y_{2}=0.$
For example using the package Divisor [SY, SY18] for the computer algebra
system Macaulay2 [GS] one can show that in this case $-K$ is not very ample.
If not all of the $a_{i}$ are positive, we can still get a hyperbolic
embedding under some conditions.
###### Proposition 7.10.
Let
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{1})\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{2})$
be the vector bundle from Proposition 7.6 with $0<a_{1}\leq a_{2}$ (and
$a_{1}+a_{2}=s$ by Proposition 7.6). Then we have an hyperbolic embedding
$X\rightarrow\mathbb{P}^{s+2}$ if the image of $X$ does not contain the vertex
of $S(0,a_{1},a_{2})$.
###### Proof.
The morphism induced by $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is an
immersion
$\varphi:\mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^{s+2}=\mathbb{P}^{a_{1}+a_{2}+2}.$
The image of $\mathbb{P}(\mathcal{E})$ via $\varphi$ is the rational normal
scroll $S(0,a_{1},a_{2})$, which is a cone over the rational normal scroll
$S(a_{1},a_{2})$ [EH16, §9.1]. The immersion $\varphi$ is an embedding except
for the vertex of the cone. Since $X$ is given as the zero set of a section of
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(2)$, the restriction of $\varphi$ to
$X$ is an immersion, which is an embedding if $\varphi(X)$ does not contain
the vertex of $S(0,a_{1},a_{2})$. Since the restriction
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)|_{X}$ is associated to the divisor
$D=(s-2)F-K$ on $X$ by Proposition 7.6, and by Corollary 7.3, the embedding
$\varphi|_{X}$ is hyperbolic. ∎
We conclude this section with the following question on the topology of
hyperbolic surfaces.
###### Question.
For which pairs $(s,r)$ does a smooth irreducible hyperbolic surface
$X\subset\mathbb{P}^{n}$ exist such that $X({\mathbb{R}})$ is homeomorphic to
the disjoint union of $s$ spheres and $r$ real projective planes?
From [HV07, Theorem 5.2] it is known that for arbitrary
$s\in{\mathbb{Z}}_{\geq 0}$ and $r\in\\{0,1\\}$ there is a smooth irreducible
hyperbolic hypersurface $X\subset\mathbb{P}^{3}$ such that $X({\mathbb{R}})$
is homeomorphic to the disjoint union of $s$ spheres and $r$ real projective
planes. Moreover, when $s\in\\{0,1\\}$, then we must have $r\in\\{0,1\\}$ as
well. This follows from [KS20b, Lemmas 2.16 and 2.17] by intersecting $X$ with
a generic hyperplane containing the space of hyperbolicity. We want to show
that for $s\geq 3$, we can have arbitrary $r\in{\mathbb{Z}}_{\geq 0}$.
###### Proposition 7.11.
Let $s\geq 3$ and $r\geq 0$. There exists a smooth irreducible hyperbolic
surface $X\subset\mathbb{P}^{n}$ such that $X({\mathbb{R}})$ is homeomorphic
to the disjoint union of $s$ spheres and $r$ real projective planes.
###### Proof.
Let $a_{1},a_{2},a_{3}>0$ such that $s=a_{1}+a_{2}+a_{3}$ and let
${\mathcal{E}}=\mathcal{O}_{\mathbb{P}^{1}}(a_{1})\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{2})\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{3})$.
We denote by $H$ the class of the line bundle
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and by $E$ the class of a fiber of
the map $\mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^{1}$. For $i=1,2,3$, let
$p_{i}\in{\mathbb{R}}[u,v]_{2a_{i}}$ be a binary form of degree $2a_{i}$ which
has only simple, real zeros and assume that the $p_{i}$ are pairwise coprime.
Then
$t=p_{1}x_{1}^{2}+p_{2}x_{2}^{2}+p_{3}x_{3}^{2}$
is a global section of ${\mathcal{O}}_{\mathbb{P}({\mathcal{E}})}(2)$. Here
the $x_{i}$’s are the coordinates on the projective plane fibers. After a
small perturbation if necessary, the zero set $X$ of $t$ is smooth and
irreducible by Bertini’s Theorem [Jou79, Thm. 6.10]. It is thus a minimal
conic bundle which has by construction $2s$ singular fibers. We also note that
$X({\mathbb{R}})$ is homeomorphic to the disjoint union of $s$ spheres. Now
let $X^{\prime}$ be the union of $X$ with $r$ different fibers of
$\mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^{1}$ all of whose real parts are
disjoint from $X({\mathbb{R}})$. This guarantees that
$X^{\prime}({\mathbb{R}})$ is smooth and $X^{\prime}\subset
S(a_{1},a_{2},a_{3})\subset\mathbb{P}^{s+2}$ is hyperbolic as the union of
hyperbolic varieties. The divisor class of $X^{\prime}$ on
$\mathbb{P}({\mathcal{E}})$ is $D=2H+rE$. Since $H$ is very ample and $E$ is
base-point free, we have that $D$ is also very ample. Thus by Bertini’s
theorem, a general member of the linear system $|D|$ is smooth and irreducible
[Jou79, Thm. 6.10]. In particular, a sufficiently small perturbation of
$X^{\prime}$ in $|D|$ gives a surface with the desired properties. ∎
###### Remark 7.12.
The above construction does not work for $s=2$ and $r\geq 2$ as in this case
we must have $a_{1}=0$ and $a_{2}=a_{3}=1$. The image of
$\mathbb{P}({\mathcal{E}})$ in $\mathbb{P}^{4}$ under the map associated to
the linear system $|H|$ is then $S(0,1,1)$, i.e. the cone over a quadratic
hypersurface in $\mathbb{P}^{3}$. Letting $F\subset\mathbb{P}({\mathcal{E}})$
be the preimage of the vertex of $S(0,1,1)$, we find that
$X^{\prime}.F=(2H+rE).F>1$ which implies that the image of $X^{\prime}$ in
$\mathbb{P}^{4}$ cannot be smooth. We do not know the possible values for $r$
when $s=2$.
Acknowledgements. We would like to thank Kristin Shaw for supporting and
encouraging this project and for interesting mathematical discussions. Also,
we thank John Christian Ottem for helpful insights into very ampleness
criteria.
## References
* [Ahl50] L. Ahlfors. Open riemann surfaces and extremal problems on compact subregions. Comment. Math. Helv. 24, 100-134., 1950.
* [CH13] Marc Coppens and Johannes Huisman. Pencils on real curves. Math. Nachr., 286(8-9):799–816, 2013.
* [Com13] Annibale Comessatti. Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale. Math. Ann., 73:1–72, 1913.
* [Com14] Annibale Comessatti. Sulla connessione delle superfizie razionali reali. Math. Ann., 23:215–283, 1914.
* [Cop13] Marc Coppens. The separating gonality of a separating real curve. Monatsh. Math., 170(1):1–10, 2013.
* [DR96] Sandra Di Rocco. $k$-very ample line bundles on del Pezzo surfaces. Math. Nachr., 179:47–56, 1996.
* [EH16] David Eisenbud and Joe Harris. 3264 and all that—a second course in algebraic geometry. Cambridge University Press, Cambridge, 2016.
* [Gab06] Alexandre Gabard. Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes. Comment. Math. Helv., 81(4):945–964, 2006.
* [Gro66] A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math., (28):255, 1966.
* [GS] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/.
* [Har76] A. Harnack. Über vieltheiligkeit der ebenen algebraischen curven. Math. Ann., 10(2):189–198, 1876.
* [Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52.
* [Har95] Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. A first course, Corrected reprint of the 1992 original.
* [Hat02] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
* [Hil] D. Hilbert. Mathematical problems. Bull. Amer. Math. Soc., 8(10):437–479.
* [Hui03] J. Huisman. Non-special divisors on real algebraic curves and embeddings into real projective spaces. Ann. Mat. Pura Appl. (4), 182(1):21–35, 2003.
* [HV07] J. William Helton and Victor Vinnikov. Linear matrix inequality representation of sets. Comm. Pure Appl. Math., 60(5):654–674, 2007.
* [Jou79] Jean-Pierre Jouanolou. Théorèmes de Bertini et applications. Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1979.
* [Kle73] F. Klein. Ueber flächen dritter ordnung. Math. Ann., 6(3):398–416, 1873.
* [Kol17] János Kollár. Conic bundles that are not birational to numerical Calabi-Yau pairs. Épijournal Geom. Algébrique, 1:Art. 1, 14, 2017.
* [KS20a] Mario Kummer and Eli Shamovich. Real fibered morphisms and Ulrich sheaves. J. Algebraic Geom., 29(1):167–198, 2020.
* [KS20b] Mario Kummer and Kristin Shaw. The separating semigroup of a real curve. Ann. Fac. Sci. Toulouse Math. (6), 29(1):79–96, 2020.
* [KSC04] János Kollár, Karen E. Smith, and Alessio Corti. Rational and nearly rational varieties, volume 92 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2004.
* [Laz04] Robert Lazarsfeld. Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series.
* [Man20] Matilde Manzaroli. Real algebraic curves on real del Pezzo surfaces. International Mathematics Research Notices, 2020. rnaa169.
* [MO19] Grigory Mikhalkin and Stepan Orevkov. Maximally writhed real algebraic links. Invent. Math., 216(1):125–152, 2019.
* [Ore19] S. Yu. Orevkov. A separating semigroup of hyperelliptic curves and curves of genus 3. Algebra i Analiz, 31(1):108–113, 2019.
* [Pra07] V. V. Prasolov. Elements of homology theory, volume 81 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2007. Translated from the 2005 Russian original by Olga Sipacheva.
* [Rus02] Francesco Russo. The antibirational involutions of the plane and the classification of real del Pezzo surfaces. In Algebraic geometry, pages 289–312. de Gruyter, Berlin, 2002\.
* [Sch63] L. Schläfli. On the distribution of surfaces of the third order into species. Phil. Trans. Roy. Soc. London, 153:193–241, 1863.
* [SY] Karl Schwede and Zhaoning Yang. Divisor: Weil divisors. Version 0.3. A _Macaulay2_ package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/pack- ages.
* [SY18] Karl Schwede and Zhaoning Yang. Divisor package for Macaulay2. J. Softw. Algebra Geom., 8:87–94, 2018.
* [Vir94] O. Viro. Complex orientations of real algebraic surfaces. In Topology of manifolds and varieties, volume 18 of Adv. Soviet Math., pages 261–284. Amer. Math. Soc., Providence, RI, 1994.
* [Zeu74] H. G. Zeuthen. Sur les différentes formes des courbes du quatrième ordre. Math. Ann., 7:410–432, 1874.
|
11institutetext: Instituto Superior Técnico and INESC-ID, University of
Lisbon, Portugal 22institutetext: Carnegie Mellon University, USA
# Assessing the Benefits of Model Ensembles in Neural Re-Ranking for Passage
Retrieval ††thanks: This research was supported by Fundação para a Ciência e
Tecnologia (FCT), through the Ph.D. scholarship with reference
SFRH/BD/150497/2019, and the INESC-ID multi-annual funding from the PIDDAC
programme (UIDB/50021/2020).
Luís Borges 1122 Bruno Martins 11 Jamie Callan 22
###### Abstract
Our work aimed at experimentally assessing the benefits of model ensembling
within the context of neural methods for passage re-ranking. Starting from
relatively standard neural models, we use a previous technique named Fast
Geometric Ensembling to generate multiple model instances from particular
training schedules, then focusing or attention on different types of
approaches for combining the results from the multiple model instances (e.g.,
averaging the ranking scores, using fusion methods from the IR literature, or
using supervised learning-to-rank). Tests with the MS-MARCO dataset show that
model ensembling can indeed benefit the ranking quality, particularly with
supervised learning-to-rank although also with unsupervised rank aggregation.
###### Keywords:
Model Ensembling Rank Fusion Passage Re-Ranking
## 1 Introduction
Ensemble methods are known to typically perform better than individual
systems. In the field of information retrieval, several rank aggregation
techniques have for instance been proposed to combine the results of different
ranking methods [1, 7, 13, 12], with previous studies showing that ensembles
indeed lead to superior results. Ensemble methods are also common in the
machine learning literature. Specifically within the context of learning with
deep neural networks, ensembling algorithms such as Fast Geometric Ensembling
(FGE) have recently been proposed and successfully applied to multiple tasks
[11], using particular learning rate updating schedules to create multiple
neural networks with no additional training cost, which can afterwards be
combined (e.g., by averaging the scores from the resulting models) for
improved performance.
In this paper, we assess the benefits of ensemble approaches within the
context of neural models for passage re-ranking. We specifically leverage the
FGE approach together with relatively standard neural retrieval models [14],
corresponding to re-ranking approaches based on recurrent neural networks, or
instead based on Transformer-based models like RoBERTa [15]. Leveraging model
ensembles, we focused our attention on different approaches for combining the
results, and we compared strategies based on (a) averaging the scores (i.e.,
the relevance estimates) produced by the multiple models, (b) combining the
rankings from the multiple models with rank fusion approaches, or (c) using
supervised learning-to-rank as a meta-learning strategy to combine the model
scores.
We evaluated the different approaches on the well-known MS-MARCO passage re-
ranking task [2]. The obtained results show that model ensembling indeed leads
to improvements over individual neural ranking models, particularly with
supervised learning-to-rank and/or in the case of RoBERTa models.
## 2 Passage Re-Ranking with Neural Ensembles
Within our general approach, we first train a neural ranking model with the
Fast Geometric Ensembling (FGE) technique, which outputs $N$ different model
checkpoints, saved at different phases of the training process. The different
checkpoints are used to re-rank initial lists with the top 1000 passages for
each test query, resulting in the generation of $N$ ranked lists. For the
initial rankings, we used the DeepCT first-stage retrieval algorithm, which
extends BM25 with context-aware term weights derived from a BERT model [8].
Finally, the $N$ different ranked lists are used as input to a fusion method,
which combines the scores to produce a final re-ranked list. The following
sub-sections describe the FGE technique and the different fusion methods that
were considered.
### 2.1 Fast Geometric Model Ensembling
Fast Geometric Ensembling (FGE) consists of an ensembling technique for deep
neural networks that generates multiple points in the weight space (i.e.,
multiple model instances, resulting from different checkpoints during
training), that share a similar low test error [11]. The approach is inspired
on the observation that the optima for the loss functions being optimized
while training neural models are often connected by simple curves, over which
the training/test accuracy are nearly constant. FGE uses a training procedure
that leverages this geometric intuition, discovering points (i.e., model
checkpoints) within the high-accuracy pathways through a particular learning
rate update schedule.
The FGE algorithm starts with model weights corresponding to an initial
training of the neural network, and resumes the training with a cyclical
learning rate defined as follows, where $\alpha_{1}$ and $\alpha_{2}$ are the
minimum and maximum values for the learning rate, while $\alpha(i)$ represents
the learning rate at iteration $i$.
$\alpha(i)=\begin{cases}\left(1-2\times\mathrm{t}(i)\right)\times\alpha_{1}+2\times\mathrm{t}(i)\times\alpha_{2}&0<\mathrm{t}(i)\leq
0.5\\\
\left(2-2\times\mathrm{t}(i)\right)\times\alpha_{2}+\left(2\times\mathrm{t}(i)-1\right)\times\alpha_{1}&0.5<\mathrm{t}(i)\leq
1\end{cases}$ (1)
Each iteration corresponds to processing one mini-batch. The parameter
$\mathrm{t}(i)$ can be defined with basis on the number of iterations $c$
corresponding to a cycle.
$\mathrm{t}(i)=\frac{1}{c}\times\left(\mathrm{mod}(i-1,c)+1\right)$ (2)
In the middle of each cycle, when the learning rate reaches its minimum value
$\alpha_{2}$, the model weights are collected to form a checkpoint. After
training, the checkpoints can be individually evaluated on a test set, and the
corresponding results can afterwards by combined to form ensemble predictions.
### 2.2 Rank Fusion Methods
Multiple methods for fusing lists into a final consensus ranking have been
proposed in the information retrieval literature [1]. As a simple approach,
one can for instance rank instances according to the average of the scores
associated to the different lists. Other approaches often leverage instead the
ranking positions.
One example is Reciprocal Rank Fusion [7], which is based on summing the
multiplicative inverse of the original rankings. Given a set of instances $P$
(i.e., the passages to be retrieved) and multiple rankings $R$ for a given
query, the instances can be sorted according to the following score:
$\mathrm{RRFscore}(p\in P)=\sum_{\mathrm{r}\in R}\frac{1}{k+\mathrm{r}(p)}$
(3)
In Equation 3, $k$ is a smoothing constant often set to the constant value of
60 [7], and $\mathrm{r}(p)$ is the rank of passage $p$ in the ranked list
$\mathrm{r}()$. A simple variation, named MAP Fusion, was proposed by Lillis
et al. [13] and involves weighting the contribution of each ranked list
according its Mean Average Precision (MAP) score, as measured over a held-out
set of queries:
$\mathrm{MAPFscore}(p\in P)=\sum_{\mathrm{r}\in
R}\frac{1\times\mathrm{MAP}_{\mathrm{r}}}{k+\mathrm{r}(p)}$ (4)
Previous studies have also advanced probabilistic data fusion techniques,
using training queries to estimate the probability that a resource is relevant
to a given query, and leveraging those probabilities in order to create new
ranking scores. One of those probabilistic techniques is SlideFuse [12], which
first estimates the probability that a passage $p$, occurring in position $i$
of a ranked list produced through a procedure $r$, is relevant. This can be
computed according to the following equation, where $Q_{p}$ is the set of
training queries for which at least $i$ instances were returned in lists
produced through procedure $r$, and where $\mathrm{Rel}(p_{i},q)$ is 1 if
$p_{i}$ is relevant to query $q$, and 0 otherwise.
$\mathrm{P}(p_{i}|r)=\frac{\sum_{q\in Q_{i}}\mathrm{Rel}(p_{i},q)}{Q_{i}}$ (5)
The final aggregated score for each document also considers a sliding window
around each position of the rankings to be merged:
$\mathrm{SlideFscore}(p\in P)=\sum_{r\in R}\mathrm{P}(p_{i,w}|r)$ (6)
In the previous equation, $\mathrm{P}(p_{i,w}|r)$ is the probability of
relevance of passage $p$ in position $i$, this time considering a window of
$w$ documents around each side of $i$. This can be estimated as follows, where
the values $a$ and $b$ correspond to the window limits for every position $i$,
considering $N$ as the total number of documents for each query.
$\begin{gathered}\mathrm{P}(p_{i,w}|r)=\frac{\sum_{j=a}^{b}\mathrm{P}(p_{j}|r)}{b-a+1},\text{~{}with~{}}\\\
a=\begin{cases}i-w&i-w\geq 0\\\
0&i-w<0\end{cases}\text{~{}~{}~{}and~{}~{}~{}}b=\begin{cases}i+w&i+w<N\\\
N-1&i+w\geq N\end{cases}\end{gathered}$ (7)
Variations on SlideFuse, weighting the contribution of individual ranked
lists, are also possible. For instance Equation 6 can be adapted in the same
way as Equation 4 extends from Equation 3, weighting each system by the
corresponding MAP score, and resulting in a MAP SlideFuse approach.
Besides rank aggregation methods we also experimented with a supervised
learning-to-rank approach, specifically the LambdaRank [5] implementation from
the XGBoost111https://github.com/dmlc/xgboost package. In this case, for each
training query, we collected the relevant passage and two other passages in
the top 1000 list, ranked according to DeepCT. The LambdaRank model was
trained on this data, using as features the DeepCT scores plus those from the
FGE snapshots, together with the average and standard deviation, and
attempting to optimize the MAP metric.
Still on what regards experimental settings, the SlideFuse method considered a
window size of 6, and the LambdaRank algorithm used the default parameters
from the XGBoost library, except in the choice of MAP as the optimized metric.
## 3 Neural Ranking Models
We experimented with two distinct types of neural ranking models, respectively
leveraging recurrent neural networks, and Transformer-based language models.
The first model is inspired on a previous proposal for encoding and matching
textual contents [4]. A sentence encoder is used to compute fixed-size vector
representations for input sequences, leveraging pre-trained FastText [3] word
embeddings together with two layers of bi-directional LSTM units with shortcut
connections between them, and a max-pooling operation over the sequence
produced by the second bi-LSTM. The query is processed through the
aforementioned encoder, which outputs the corresponding representation. In
turn, each sentence that composes the passage is also processed through the
same encoder, generating a sequence of representations. This sequence of
sentence representations is then fed as input to a different encoder, using a
similar structure (except for the initial FastText embedding layer) to produce
a single fixed-size representation for the passage. The representations for
the query and the passage are combined through different operations (i.e.,
vector concatenation, difference, and element-wise product), and the result is
feed into a final feed-forward layer, which outputs the relevance score of the
passage towards the query.
For the second neural ranking approach, we fine-tune RoBERTa-base [15] to our
ranking problem, passing as input to the model the concatenation of the query
and the passage text, separated by a special [SEP] token. We concatenate the
vector representation of the special [CLS] token, together with the result of
a max-pooling operation over the last sequence of hidden states output by
RoBERTa-base, feeding the result to a final feed-forward layer which outputs
the relevance score of the passage towards the query.
When training our models, we first use a fast approach (i.e., DeepCT [8]) to
retrieve the top 1000 passages for the provided training queries. The loss
function takes as input the scores between a query and a relevant passage, a
non-relevant passage sampled from the top 25 passages retrieved for the query,
and a negative passage sampled from the remaining 975 passages in the top
1000. The loss is formally defined as follows, where $p$ is the score between
the query and a positive passage, $n_{25}$ is the score between the query and
the passage sampled from the top 25, and $n_{975}$ is the score between the
query and the passage sampled from the remaining 975 passages.
$\begin{gathered}\mathrm{loss}=\mathrm{hinge}(p,n_{25})+\mathrm{hinge}(p,n_{975})+0.25\times\mathrm{hinge}(n_{25},n_{975})\text{,
with}\\\ \mathrm{hinge}(p,n)=\max(0,1-p+n)\end{gathered}$ (8)
For our RNN-based model, we used a dimensionality of 300 in the
representations produced by the recurrent units. For RoBERTa-base, we used the
default base parameters as defined in the Huggingface Transformers
library222https://github.com/huggingface/transformers. We trained our models
for a total of 15 epochs with the AdaMod [10] optimizer. The first five epochs
produced the initial weights for the Fast Geometric Ensembling (FGE)
technique. In the remaining ten epochs with FGE, we used cycles of $c=4$
epochs, with a cyclic learning rate between $\alpha_{1}=2\cdot 10^{-5}$ and
$\alpha_{2}=2\cdot 10^{-7}$, hence generating five different checkpoints.
## 4 Experimental Evaluation
Our experiments relied on the passage ranking data from MS-MARCO [2]. For each
test query, a first-stage ranker (in our case, DeepCT [8]) retrieves a set of
possibly relevant passages from the whole collection, and the top $k$ results
are then re-ranked through a second more expensive model.
Table 1 presents a comparison between the different alternatives described in
Sections 2 and 3, with results measured over the development portion of the
MS-MARCO dataset. We specifically measured the Mean Average Precision (MAP),
Mean Reciprocal Rank (MRR), and MRR@10. The first two lines of Table 1 compare
two first-stage retrieval approaches, returning 1000 possibly relevant
passages for each development query. DeepCT outperformed BM25 in this initial
task, and the remaining experiments focused on re-ranking the top 100 passages
retrieved by DeepCT. A separate round of tests, not detailed in this paper,
showed that re-ranking the top 100 passages lead to consistently better
results than re-ranking the entire set of 1000 passages per query.
The second group of rows in Table 1 compares the results for both types of
neural models, trained for a total of 15 epochs. The model based on RoBERTa-
base clearly outperformed the RNN-based model, which even failed to outperform
DeepCT. We also attempted to combine the rankings from each of these models
and DeepCT, through the MAPFuse strategy. The results, given in the third
group of rows, showed that the combination improved results for the RNN model,
but not for the RoBERTa-base model.
The remaining rows from Table 1 show the results achieved with FGE ensembles,
leveraging different types of techniques for combining the rankings. The
results show that model ensembling has clear benefits for RoBERTa-base models,
with mixed results for RNN models. Few differences were measured between the
alternative rank aggregation approaches, and significantly better results were
obtained with learning-to-rank. We expect that similar benefits from
ensembling can be expected for larger models than RoBERTa-base.
## 5 Conclusions and Future Work
Table 1: Results over the MS-MARCO development dataset. Statistical significance tests were used to compare ensembles against individual models for re-ranking the DeepCT results, both for RNN (†) and RoBERTa-base (‡) models, as well as to compare the learning-to-rank ensembles against the second best ensemble (*). The methods whose difference is statistically significant, for a $p$-value of 0.05, are marked on the table. Although this is not reported on the table, not including DeepCT scores in the FGE ensembles is consistently worse (i.e., approx. 0.01 points lower in terms of MRR@10 for RoBERTa-base ensembles, and up to 0.1 points lower for RNN ensembles). Method | MAP | MRR | MRR@10
---|---|---|---
BM25 | 0.1835 | 0.1867 | 0.1758
DeepCT | 0.2506 | 0.2546 | 0.2425
RNN | 0.2127 | 0.2160 | 0.2010
RoBERTa-base | 0.3356 | 0.3403 | 0.3311
RNN + DeepCT | 0.2888 | 0.2936 | 0.2821
RoBERTa-base + DeepCT | 0.3326 | 0.3378 | 0.3285
RNN FGE + DeepCT + Average† | 0.3000 | 0.3056 | 0.2952
RNN FGE + DeepCT + RRFuse | 0.2845 | 0.2891 | 0.2769
RNN FGE + DeepCT + MAPFuse | 0.2847 | 0.2893 | 0.2771
RNN FGE + DeepCT + SlideFuse | 0.2738 | 0.2781 | 0.2645
RNN FGE + DeepCT + MAPSlideFuse† | 0.2741 | 0.2784 | 0.2649
RNN FGE + DeepCT + Learning-to-Rank†∗ | 0.3131 | 0.3181 | 0.3080
RoBERTa-base FGE + DeepCT + Average‡ | 0.3354 | 0.3411 | 0.3324
RoBERTa-base FGE + DeepCT + RRFuse‡ | 0.3819 | 0.3879 | 0.3813
RoBERTa-base FGE + DeepCT + MAPFuse‡ | 0.3818 | 0.3874 | 0.3806
RoBERTa-base FGE + DeepCT + SlideFuse‡ | 0.3787 | 0.3844 | 0.3774
RoBERTa-base FGE + DeepCT + MAPSlideFuse‡ | 0.3789 | 0.3844 | 0.3774
RoBERTa-base FGE + DeepCT + Learning-to-Rank‡∗ | 0.3856 | 0.3913 | 0.3846
We tested the use of Fast Geometric Ensembling (FGE) with neural passage re-
ranking models, comparing different fusion methods to combine the rankings
from FGE checkpoints. Results over MS-MARCO show that model ensembling indeed
leads to consistent improvements over individual models, thus constituting a
viable approach to further improve state-of-the-art approaches.
For future work, we plan to conduct similar tests with other datasets,
including TREC CAR [9] and WikiPassageQA [6], in addition to testing different
ensembling methods, such as the Auto-Ensembling approach from Jun et al. [16].
## References
* [1] Anava, Y., Shtok, A., Kurland, O., Rabinovich, E.: A probabilistic fusion framework. In: Proceedings of the ACM International on Conference on Information and Knowledge Management (2016)
* [2] Bajaj, P., Campos, D., Craswell, N., Deng, L., Gao, J., Liu, X., Majumder, R., McNamara, A., Mitra, B., Nguyen, T., et al.: MS-MARCO: A human generated machine reading comprehension dataset. arXiv preprint 1611.09268 (2016)
* [3] Bojanowski, P., Grave, E., Joulin, A., Mikolov, T.: Enriching word vectors with subword information. Transactions of the Association for Computational Linguistics 5 (2017)
* [4] Borges, L., Martins, B., Calado, P.: Combining similarity features and deep representation learning for stance detection in the context of checking fake news. Journal of Data and Information Quality 11(3) (2019)
* [5] Burges, C.J.: From RankNet to LambdaRank to LambdaMART: An overview. Learning 11(23-581) (2010)
* [6] Cohen, D., Yang, L., Croft, W.B.: WikiPassageQA: A benchmark collection for research on non-factoid answer passage retrieval. In: Proceedings of the International ACM SIGIR Conference on Research and Development in Information Retrieval (2018)
* [7] Cormack, G.V., Clarke, C.L.A., Buettcher, S.: Reciprocal rank fusion outperforms Condorcet and individual rank learning methods. In: Proceedings of the International ACM SIGIR Conference on Research and Development in Information Retrieval (2009)
* [8] Dai, Z., Callan, J.: Context-aware sentence/passage term importance estimation for first stage retrieval. arXiv preprint 1910.10687 (2019)
* [9] Dietz, L., Gamari, B., Dalton, J., Craswell, N.: TREC complex answer retrieval overview. In: Proceedings of the Text REtrieval Conference (2018)
* [10] Ding, J., Ren, X., Luo, R., Sun, X.: An adaptive and momental bound method for stochastic learning. arXiv preprint 1910.12249 (2019)
* [11] Garipov, T., Izmailov, P., Podoprikhin, D., Vetrov, D.P., Wilson, A.G.: Loss surfaces, mode connectivity, and fast ensembling of DNNs. In: Proceedings of the Annual Conference on Neural Information Processing Systems (2018)
* [12] Lillis, D., Toolan, F., Collier, R., Dunnion, J.: Extending probabilistic data fusion using sliding windows. In: Proceedings of the European Conference on Information Retrieval (2008)
* [13] Lillis, D., Zhang, L., Toolan, F., Collier, R.W., Leonard, D., Dunnion, J.: Estimating probabilities for effective data fusion. In: Proceedings of the International ACM SIGIR Conference on Research and Development in Information Retrieval (2010)
* [14] Lin, J., Nogueira, R., Yates, A.: Pretrained transformers for text ranking: BERT and beyond. arXiv preprint arXiv:2010.06467 (2020)
* [15] Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D., Levy, O., Lewis, M., Zettlemoyer, L., Stoyanov, V.: RoBERTa: A robustly optimized BERT pretraining approach. arXiv preprint 1907.11692 (2019)
* [16] Yang, J., Wang, F.: Auto-ensemble: An adaptive learning rate scheduling based deep learning model ensembling. arXiv preprint 2003.11266 (2020)
|
# Robust Output Regulation and Reinforcement Learning-based Output Tracking
Design for Unknown Linear Discrete-Time Systems
Ci Chen, Lihua Xie, Yi Jiang, Kan Xie, and Shengli Xie This work was supported
in part by the Wallenberg-NTU Presidential Postdoctoral Fellowship and in part
by National Natural Science Foundation of China under Grants 61703112,
61973087, and 61727810. (Corresponding author: Lihua Xie).C. Chen and L. Xie
are with School of Electrical and Electronic Engineering, Nanyang
Technological University, Singapore 639798. (e-mail<EMAIL_ADDRESS>ELHXIE@ntu.edu.sg). Y. Jiang is with State Key Laboratory of Synthetical
Automation for Process Industries, Northeastern University, Shenyang, China.
(e-mail: yijiang.ezhou@gmail.com).K. Xie and S. Xie are with School of
Automation, Guangdong University of Technology, Guangdong Key Laboratory of
IoT Information Technology, Guangzhou, 510006 China (e-mail:
shlxie@gdut.edu.cn).
###### Abstract
In this paper, we investigate the optimal output tracking problem for linear
discrete-time systems with unknown dynamics using reinforcement learning and
robust output regulation theory. This output tracking problem only allows to
utilize the outputs of the reference system and the controlled system, rather
than their states, and differs from most existing tracking results that depend
on the state of the system. The optimal tracking problem is formulated into a
linear quadratic regulation problem by proposing a family of dynamic discrete-
time controllers. Then, it is shown that solving the output tracking problem
is equivalent to solving output regulation equations, whose solution, however,
requires the knowledge of the complete and accurate system dynamics. To remove
such a requirement, an off-policy reinforcement learning algorithm is proposed
using only the measured output data along the trajectories of the system and
the reference output. By introducing re-expression error and analyzing the
rank condition of the parameterization matrix, we ensure the uniqueness of the
proposed RL based optimal control via output feedback.
###### Index Terms:
Reinforcement learning, robust output regulation, output tracking, adaptive
optimal control.
## I Introduction
Output tracking, whose objective is to make the system output follow a desired
reference trajectory, is a fundamental research topic of practical importance
(see examples in [1]). One systematic way to approach the output tracking is
to transform it into an output regulation problem, whose solution and
corresponding control design date back to [2], where both the closed-loop
stability and the asymptotic tracking of even an unbounded reference are
achieved. Though elegant, such a solution is built on the knowledge of the
complete and accurate system dynamics, making the output tracking design model
dependent.
Reinforcement learning (RL) features making sequential decisions through
interactions between the agent’s actions and unknown environment [3, 4]. RL
algorithms have been applied in the control field to solve optimal control
problems for both discrete-time (DT) (see, e.g., [4, 5]) and continuous-time
(CT) systems (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) without any
knowledge of the system dynamics. RL-based methods in Chapter 11 of [4] were
used in [15, 16] to handle the optimal tracking control of linear and
nonlinear DT systems. However, most RL algorithms are classified as on-policy
learning, which assumes that the behaviour policy for generating the data for
learning is the same as the target policy. Off-policy RL differs from the on-
policy learning in its separating the target and behaviour policies. Off-
policy RL for linear CT systems with unknown dynamics was given in Chapter 2
of [8] to solve the optimal regulation problem. A solution to the zero-sum
game problem for the regulation of DT systems was given in [17] using off-
policy RL. Together with RL and the output regulation theory, [18] and [19]
gave tracking controllers for single and multiple DT systems and established
the asymptotic stability of the tracking error, which is the DT version of
[9]. Note that one essential assumption for [18, 19], as well as [9], is that
not only the outputs of the reference and the controlled system but also their
states are required in the optimal control design process. Therefore, a key
challenge to be addressed is how to achieve the optimal output tracking of DT
systems without using the reference state.
In contrast to full states feedback, output feedback utilizes the system
output and adds extra flexibilities to control design. As for the output-
feedback control, [20] utilized the state reconstruction to form an output-
feedback RL for regulating the states of DT systems. Based on [20], off-policy
$H_{\infty}$ control of DT systems was given based on input and output data in
the literature such as [21, 22]. The optimal tracking control using output-
feedback RL algorithm was given in [23]. [24] gave a parameterization of the
state of a DT system in terms of the input and output data for the output
feedback LQR by $Q$-learning. Based on [24], [25] considered an output-
feedback tracking control for DT systems with the assumption that the
reference state is available during the learning and control processes.
Utilizing the state reconstruction in [20], [26, 27] considered tracking-based
RL algorithms for disturbance rejection with the augmented system being
observable. It turns out that, in the state parametrization, the full row rank
of the parametrization matrix is required, see [28, 7, 12, 29, 30, 31] for
example. In [29, 30, 31], sufficient conditions were established that
guarantee such a rank condition in both the CT and DT settings. Within the
framework of output-feedback RL, most of the existing output tracking results
for DT systems are based on the linear output regulation theory, which,
however, remains challenging for its extension to robust output regulation.
Motivated by the analysis above, we investigate the output tracking problem of
DT systems using RL and robust output regulation theory based on the reference
output. We use real-time data collected along the trajectories of the DT
system and propose an RL algorithm that ensures not only the stability of the
closed-loop system, but also the predefined system performance. The
contributions of this paper are given as follows.
1. 1.
Compared to the linear output regulation based RL controllers for tracking DT
systems [18, 19, 26, 25, 27], we amend the standard robust output regulation
theory to propose a new family of robust DT controllers, based on which an
optimal output tracking problem is formulated.
2. 2.
We invoke the CT work in [12] to propose a criterion for guaranteeing the
parameterization matrix in the DT state reconstruction to be full row rank,
which leads to the uniqueness of the data-based optimal output-feedback DT
controller. Inspired by [12], we establish the rank of the parameterization
matrix for DT systems, and based on which we show that the controllability is
a sufficient condition for ensuring the full row rank of the parameterization
matrix for DT systems.
3. 3.
We do not need to compute analytic solutions to the output regulation
equations. We take the DT state re-expression error into account during the
optimal control learning. We add a phase of model-free pre-collection into the
off-policy RL for reducing the adverse impact of the re-expression error.
Notation: Throughout this paper, given any square matrix $\mathcal{M}$, the
notation $\lambda(\mathcal{M})$ indicates the spectrum of $\mathcal{M}$,
$\rho(\mathcal{M})$ is its spectral radius, $\det(\mathcal{M})$ denotes its
determinant, ${\rm{adj}}(\mathcal{M})$ is its adjugate, $\mathcal{M}>0$
$(\mathcal{M}\geq 0)$ means that the matrix is positive definite (positive
semi-definite),
${\rm{vec}}(\mathcal{M})=[\mathcal{M}_{1}^{T},\mathcal{M}_{2}^{T},\cdots,\mathcal{M}_{n}^{T}]^{T}$
is a vector with $\mathcal{M}_{i}$ being the $i$th column of $\mathcal{M}$,
and
${{\rm{vecs}}(\mathcal{W})}=[\mathcal{W}_{11},2\mathcal{W}_{12},...,2\mathcal{W}_{1n},\mathcal{W}_{22},2\mathcal{W}_{23},...,2\mathcal{W}_{n-1,n},\mathcal{W}_{n,n}]^{T}\in\mathbb{R}^{\frac{1}{2}n(n+1)}$
with $\mathcal{W}_{ij}$ being the entry at the $i$th row and $j$th column of
$\mathcal{W}$. Given any vector $\mathcal{T}\in\mathbb{R}^{n}$,
${{\rm{vecv}}(\mathcal{T})}=[\mathcal{T}_{1}^{2},\mathcal{T}_{1}\mathcal{T}_{2},...,\mathcal{T}_{1}\mathcal{T}_{n},\mathcal{T}_{2}^{2},\mathcal{T}_{2}\mathcal{T}_{3},...,\mathcal{T}_{n-1}\mathcal{T}_{n},\mathcal{T}_{n}^{2}]^{T}\in\mathbb{R}^{\frac{1}{2}n(n+1)}$.
Let the controllability matrix $\mathcal{C}(A_{1},A_{2})=[A_{2},{A_{1}}A_{2}$,
${A_{1}^{2}}A_{2},\cdots,A_{1}^{n-1}A_{2}]$ with the dimension of $A_{1}$
being $n\times n$. The notation $\otimes$ denotes the Kronecker product. The
notations $0$ and $I$, respectively, indicate a zero matrix and an identity
matrix with appropriate dimensions.
## II Problem Formulation and Preliminaries
### II-A Problem Formulation
We aim to study a class of DT dynamical systems modeled by
$\displaystyle x(k+1)$ $\displaystyle=Ax(k)+Bu(k),$ (1) $\displaystyle y(k)$
$\displaystyle=Cx(k),$ (2)
where $x(k)\in\mathbb{R}^{r_{n}}$, $u(k)\in\mathbb{R}^{r_{m}}$, and
$y(k)\in\mathbb{R}^{r_{p}}$ denote the system state, input, and output,
respectively, and, $A\in\mathbb{R}^{r_{n}\times r_{n}}$,
$B\in\mathbb{R}^{r_{n}\times r_{m}}$, and $C\in\mathbb{R}^{r_{p}\times
r_{n}}$, involved in the system dynamics (1) and (2), are unknown constant
matrices. The control objective is to drive $y(k)$ in (2) to follow a
reference signal $y_{d}(k)$ given by
$\displaystyle x_{d}(k+1)$ $\displaystyle={S}x_{d}(k),$ (3) $\displaystyle
y_{d}(k)$ $\displaystyle=Rx_{d}(k),$ (4)
where $x_{d}(k)\in\mathbb{R}^{r_{q_{m}}}$ and $y_{d}(k)\in\mathbb{R}^{r_{p}}$
respectively denote the reference state and output. Similar to the $(A,B,C)$
in (1) and (2), ${S}\in\mathbb{R}^{r_{q_{m}}\times r_{q_{m}}}$ and
${R}\in\mathbb{R}^{r_{p}\times r_{q_{m}}}$ are unknown constant matrices. The
output tracking error between (2) and (4) is defined as
$\displaystyle y_{e}(k)$ $\displaystyle=y(k)-y_{d}(k).$ (5)
For the purpose of control design, some assumptions on the dynamics of DT
system (1)–(4) are made as follows.
###### Assumption 1
For system (1)–(2), $(A,B)$ is controllable and $(A,C)$ is observable.
###### Assumption 2
The eigenvalues of the matrix $S$ are on or outside the unit circle.
###### Assumption 3
The minimal polynomial of $S$ is known.
###### Assumption 4
rank$\left(\left[\begin{array}[]{cc}A-\lambda_{i}I&\ B\\\ C&\
0\end{array}\right]\right)=r_{n}+r_{p}$, $\forall\lambda_{i}\in\lambda({S})$.
Assumption 1 is standard in the optimal control[4]. Assumption 2 is made to
rule out the trivial case wherein the matrix $S$ in (3) is Schur (see Chapter
1 of [32]), namely, the reference state is asymptotically stable. Assumptions
2–4 are standard in the output regulation literature.
Based on the system descriptions above, we focus on formulating an optimal
tracking control problem that is solvable. To do this, we extend the work of
CT systems [12] and construct a DT control protocol
$\displaystyle z(k+1)$ $\displaystyle=Fz(k)-Gy_{e}(k),\hfill$ (6a)
$\displaystyle u(k)$ $\displaystyle=-Kx(k)-Hz(k)-Tz(k),\hfill$ (6b)
where $z(k)\in\mathbb{R}^{r_{p}r_{q_{m}}}$ is a dynamical signal driven by
output error $y_{e}(k)$, $(F,G)$ is an $r_{p}$-copy internal model of $S$
111The design of $(F,G)$ is available under Assumption 3; see Chapter 1 of
[32]. with $F\in\mathbb{R}^{r_{p}r_{q_{m}}\times r_{p}r_{q_{m}}}$ and
$G\in\mathbb{R}^{r_{p}r_{q_{m}}\times r_{p}}$, ${T\in\mathbb{R}^{r_{m}\times
r_{p}r_{q_{m}}}}$ is a newly proposed feedforward gain matrix compared to the
standard DT controller by the output regulation theory [32], and
$K\in\mathbb{R}^{r_{m}\times r_{n}}$ and $H\in\mathbb{R}^{r_{m}\times
r_{p}r_{q_{m}}}$ are gain matrices for solving the tracking problem to be
specified later.
Substituting the dynamical controller (6) into the DT system (1) yields
$\displaystyle{\bar{x}}(k+1)$
$\displaystyle={\bar{A}}(k){\bar{x}}(k)+{{\bar{G}}}R{x_{d}}(k),$ (7a)
$\displaystyle{y_{e}}(k)$
$\displaystyle={{\bar{C}}}{{\bar{x}}}(k)-R{x_{d}}(k),$ (7b)
where ${\bar{x}(k)}=[{x}(k)^{T},{z}(k)^{T}]^{T}\in\mathbb{R}^{n_{z}}$ with
$n_{z}=r_{n}+r_{p}r_{q_{m}}$, and the system matrices
$\bar{A}\in\mathbb{R}^{n_{z}\times n_{z}}$, $\bar{G}\in\mathbb{R}^{n_{z}\times
r_{p}}$, and $\bar{C}\in\mathbb{R}^{r_{p}\times n_{z}}$ are, respectively,
given as $\bar{A}=\left[{\begin{array}[]{*{20}{c}}{{A}-{B}{K}}&{-{B}{H}}-BT\\\
{-{G}{C}}&{{F}}\end{array}}\right]$,
$\bar{G}=\left[{\begin{array}[]{*{20}{c}}0\\\ {{G}}\end{array}}\right],\ \
\mbox{and}\ \
\bar{C}=\left[{\begin{array}[]{*{20}{c}}{{C}},&0\end{array}}\right]$. Note
that
$\displaystyle{\bar{A}}=\underline{A}-\bar{B}[K,H],$ (8)
where $\underline{A}=\left[{\begin{array}[]{cc}{A}&\ -BT\\\ -{G}{C}&\
F\end{array}}\right]{\in\mathbb{R}^{n_{z}\times n_{z}}}$ and
$\bar{B}=\left[{\begin{array}[]{c}B\\\
0\end{array}}\right]{\in\mathbb{R}^{n_{z}\times r_{m}}}$. By [12], the
following properties of the system dynamics hold.
###### Lemma 1
Under Assumption 4 and when $(F,G)$ incorporates an $r_{p}$-copy internal
model of $S$, we have that
1. 1.
given any matrix $T$, if $(A,B)$ is stabilizable (controllable), then
$(\underline{A},\bar{B})$ is stabilizable (controllable).
2. 2.
given the matrix $T$ such that $(F,T)$ is observable with $r_{p}\geq r_{m}$,
if $(A,C)$ is detectable (observable), then the pair $(\underline{A},\bar{C})$
is detectable (observable). In addition, if $(A,B)$ is stabilizable
(controllable), then $(\underline{A},\bar{B})$ is stabilizable (controllable).
Given a Schur matrix $\bar{A}$, under Assumptions 1–2, the equations
${{\bar{A}}}{X}+{{\bar{G}}}R={X}S$ and ${{\bar{C}}}{X}=R$ have a common
solution $X$ [32]. Based on the matrix ${X}$, let
$\displaystyle{e}(k)={\bar{x}}(k)-{X}{x_{d}}(k),$ (9)
which leads to
$\displaystyle e(k+1)=$ $\displaystyle{{\bar{A}}}{{e(k)}},$ (10)
where (7) and (9) are used. Substituting (8) into (10) yields
$\displaystyle e(k+1)$ $\displaystyle=\underline{A}{{e(k)}}+\bar{B}u_{e}(k),$
(11a) $\displaystyle{y_{e}}(k)$ $\displaystyle={{\bar{C}}}{{e}}(k),$ (11b)
$\displaystyle u_{e}(k)$ $\displaystyle=-\bar{K}{e}(k),$ (11c)
where $\bar{K}=[K,H]\in\mathbb{R}^{r_{m}\times(r_{m}+r_{p}r_{q_{m}})}$ is the
feedback gain matrix with $K$ and $H$ being from (6). Note that if $e(k)$
decays to zero, so does ${y_{e}}(k)$. Here, the matrix $\bar{K}$ in (11c) is
designed based on the following optimization problem.
###### Problem 1
$\displaystyle\min_{u_{e}}\sum_{i=k}^{\infty}({y_{e}^{T}}(i)Q{y_{e}}(i)+u_{e}^{T}(i)\bar{R}u_{e}(i))$
(12) $\displaystyle\rm{subject\ \ to\ \ }(\ref{eq1901pre})$
with $Q>0$ and $\bar{R}>0$.
### II-B Preliminaries on Optimal Control
The optimal tracking problem is solved if $\bar{K}$ in (11c) is designed such
that $e(k)$ in (11b) is stabilized to zero and, meanwhile, the performance
index in (12) is minimized.
The optimal feedback gain matrix that solves the optimization problem (12) is
labelled as $\bar{K}^{*}$ and is given as follows (see Chapter 2 of [4]),
$\displaystyle\bar{K}^{*}=(\bar{R}+\bar{B}^{T}P^{*}\bar{B})^{-1}\bar{B}^{T}P^{*}\underline{A},$
(13)
where $P^{*}$ satisfies the DT algebraic Riccati equation (ARE)
$\displaystyle\underline{A}^{T}P^{*}\bar{B}(\bar{R}+\bar{B}^{T}P^{*}\bar{B})^{-1}\bar{B}^{T}P^{*}\underline{A}$
$\displaystyle=\bar{C}^{T}\bar{Q}\bar{C}+\underline{A}^{T}P^{*}\underline{A}-P^{*}$
(14)
with $\bar{Q}=\mbox{diag}\\{Q,0\\}$ and assuming that the stabilizability
condition of $(\underline{A},\bar{B})$ and the observability condition of
$(\underline{A},{\bar{C}})$ hold. Note that the detectability condition of
$(\underline{A},{\bar{C}})$ may not hold through the design of the output
regulation-based standard DT controller (see Chapter 1 of [32]).
As for solving the DT ARE (II-B), a model-based algorithm with the knowledge
of $\underline{A}$ and $\bar{B}$ was given in [33], and is recalled below.
###### Lemma 2
([33]) Let $\bar{K}^{0}$ be a stabilizing gain matrix such that
$\underline{A}-\bar{B}{\bar{K}}^{0}$ is Schur. Solve $P^{j}$ from
$\displaystyle
P^{j}=\bar{Q}+(\bar{K}^{j})^{T}\bar{R}\bar{K}^{j}+(\underline{A}-\bar{B}\bar{K}^{j})^{T}P^{j}(\underline{A}-\bar{B}\bar{K}^{j}).$
(15)
Update the policy as
$\displaystyle\bar{K}^{j+1}=(\bar{R}+\bar{B}^{T}P^{j}\bar{B})^{-1}\bar{B}^{T}P^{j}\underline{A}.$
(16)
Then,
1. 1.
$P^{*}\leq P^{j+1}\leq P^{j}$ for each $j=1,2,\ldots$;
2. 2.
$\lim_{j\rightarrow\infty}\bar{K}^{j}=\bar{K}^{*}$,
$\lim_{j\rightarrow\infty}P^{j}=P^{*}$.
## III Output-Feedback RL for Optimal Output Tracking Control of DT Systems
This section is to solve Problem 1 using the data collected along the
trajectories of the controlled system and the reference output in the absence
of any knowledge of the system dynamics. To achieve this, we use the following
behavior policy to excite the DT system (1)
$\displaystyle\bar{u}(k)=-{\bar{K}}^{0}r(k)+\xi(k)-T\bar{z}(k),$ (17)
where
${r}(k)={\left[{{x}^{T}(k),{\bar{z}}^{T}(k)}\right]^{T}}\in\mathbb{R}^{n_{z}}$,
${\bar{K}}^{0}$ is an initial stabilizing gain, $\xi(k)$ is an exploration
noise, and $\bar{z}(k)\in\mathbb{R}^{r_{p}r_{q_{m}}}$ is an alternative
dynamical signal driven by the output error $y_{e}(k)$ as defined by
${\bar{z}}(k+1)=F\bar{z}(k)-Gy(k)+G\vartheta(k)$ with $F$ and $G$ being given
from (6) and $\vartheta(k)$ being either the exploration noise or the
reference’s output $y_{d}(k)$. Here, $\bar{z}(k)$ is defined to differ from
$z(k)$ in (6). We will specify $\vartheta(k)$ in our design later. Note that,
if $\vartheta(k)=y_{d}(k)$ and $\xi(k)=0$, then (17) is equivalent to (6). It
follows from (1)–(4) and (17) that
$\displaystyle r(k+1)$
$\displaystyle={\underline{A}}{{r}}(k)+\bar{B}\bar{u}(k)+\bar{G}\vartheta(k),\hfill$
(18a) $\displaystyle{y}(k)$ $\displaystyle={{\bar{C}}}{{r}}(k).\hfill$ (18b)
Based on the policy iteration of (15) and (16), (18) results in the off-policy
Bellman equation for DT systems in the state-feedback form as
$\displaystyle r^{T}(k+1)P^{j+1}r(k+1)-r^{T}(k)P^{j+1}r(k)$ $\displaystyle=$
$\displaystyle-r^{T}(k)(\bar{Q}+(\bar{K}^{j})^{T}\bar{R}\bar{K}^{j})r(k)+\vartheta^{T}(k)\bar{G}^{T}P^{j+1}\bar{G}\vartheta(k)$
$\displaystyle+(-\bar{K}^{j}r(k)+\bar{u}(k))^{T}\bar{B}^{T}P^{j+1}\bar{B}(\bar{K}^{j}r(k)+\bar{u}(k))$
$\displaystyle+2\vartheta^{T}(k)\bar{G}^{T}P^{j+1}\bar{B}u(k)+2r^{T}(k)\underline{A}^{T}P^{j+1}\bar{G}\vartheta(k)$
$\displaystyle+2r^{T}(k)\underline{A}^{T}P^{j+1}\bar{B}(\bar{K}^{j}r(k)+\bar{u}(k)),$
(19)
where $\underline{A}^{j}=\underline{A}-\bar{B}\bar{K}^{j}$.
In (III), the system state $x(k)$ should be known. However, a variety of
physical systems, including aircraft and power networks, only allow the
measurement of the system output. Note that, within the output-feedback
framework, the unknown term of $r(k)$ (or $x(k)$) prevents us from directly
obtaining $\bar{K}^{k+1}$ from (III). The next four subsections describe our
output-feedback RL for solving Problem 1.
### III-A State Reconstruction
Since the system state $r(k)$ in (18) is not available for feedback control,
this subsection provides a method to reconstruct $r(k)$ using input-output
data from DT systems.
To reconstruct the DT state, we first generate the DT states
$\zeta_{\bar{u}}(k)\in\mathbb{R}^{n_{z}r_{m}}$,
$\zeta_{y}(k)\in\mathbb{R}^{n_{z}r_{p}}$, and
$\zeta_{\vartheta}(k)\in\mathbb{R}^{n_{z}r_{p}}$ through three difference
equations with zero initial conditions as
$\displaystyle{\zeta}_{\bar{u}}(k+1)$ $\displaystyle=(I_{m}\otimes
A_{\zeta})\zeta_{\bar{u}}(k)+\bar{u}(k)\otimes b,$ (20)
$\displaystyle{\zeta}_{y}(k+1)$ $\displaystyle=(I_{p}\otimes
A_{\zeta})\zeta_{y}(k)+y(k)\otimes b,$ (21)
$\displaystyle{\zeta}_{\vartheta}(k+1)$ $\displaystyle=(I_{p}\otimes
A_{\zeta})\zeta_{\vartheta}(k)+\vartheta(k)\otimes b,$ (22)
where $A_{\zeta}$ is a companion matrix with $-d_{j}$ for $j=1,2,\cdots,n_{z}$
at the last row designed to make $A_{\zeta}$ Schur and
$b=[0,0,\cdots,0,1]^{T}\in\mathbb{R}^{n_{z}}$.
The following result is attained by using the Luenberger observer for DT
systems and extending [12], Chapters 3.6.1 and 4.5.4 of [34], [24].
###### Lemma 3
Consider the DT system (18). There exists a constant matrix
$\bar{M}\in\mathbb{R}^{n_{z}\times r_{\bar{\zeta}}}$ satisfying
$\displaystyle r(k)=\bar{M}\bar{\zeta}(k)+\omega(k),$ (23)
where
$\bar{\zeta}^{T}(k)=[\zeta_{\bar{u}}^{T}(k),\zeta_{y}^{T}(k),\zeta_{\vartheta}^{T}(k)]^{T}\in\mathbb{R}^{r_{\bar{\zeta}}}$
with $r_{\bar{\zeta}}={n_{z}r_{m}}+{n_{z}r_{p}}+{n_{z}r_{p}}$ and
$\omega(k)=({\underline{A}-\bar{L}\bar{C}})^{k}r(0)$.
The matrix $\bar{M}$ in (23) is termed as a parameterization matrix as in
[12]. The following result shows that the DT state reconstruction in (23) has
a structural property that ${\rm{rank}}(\bar{M})$ is linked to the three
controllability matrices $\mathcal{C}(\underline{A},\bar{B})$,
$\mathcal{C}(\underline{A},\bar{L})$, and
$\mathcal{C}(\underline{A},\bar{G})$. This is a necessary and sufficient
condition for measuring the rank of the parameterization matrix for DT state
reconstruction.
###### Lemma 4
rank$(\bar{M})=$rank$([\mathcal{C}(\underline{A},\bar{B}),\mathcal{C}(\underline{A},\bar{L}),\mathcal{C}(\underline{A},\bar{G})])$.
Proof: See Appendix A. $\square$
This lemma reveals that the parameterization matrix $\bar{M}$ in the DT
version of (23) has the same structural property as that in the CT version of
[12]. Note that the state reconstruction in (23) is for a tracking problem,
which is thus applicable to a regulation problem. Based on the property in
Lemma 4, one has the following convergence result for the DT system (18).
###### Theorem 1
Consider the DT system (18) satisfying the controllability condition of
$(\underline{A},\bar{B})$ and the observability condition of
$(\underline{A},\bar{C})$, $r(k)-\bar{M}\bar{\zeta}(k)$ asymptotically decays
to zero.
Proof: Under the condition that $(\underline{A},\bar{B})$ is controllable, it
follows from Lemma 4 that rank$(\bar{M})=n_{z}$. The matrix $A_{\zeta}$ in
(20)–(22) is designed to be Schur by choosing appropriate coefficients $d_{j}$
for $j=1,2,\cdots,n_{z}$. Thus, the vector $\bar{\zeta}(k)$ in (23), formed by
the difference equations (20)–(22), is known. In addition, since
$(\underline{A},\bar{C})$ is observable, then the eigenvalues of
$\underline{A}-\bar{L}\bar{C}$ can be designed to be equal to those of
$A_{\zeta}$ through choosing an appropriate matrix $\bar{L}$. By doing this,
$\underline{A}-\bar{L}\bar{C}$ is Schur, based on which
$r(k)-\bar{M}\bar{\zeta}(k)$ asymptotically decays to zero. This completes the
proof. $\square$
The reason for seeking rank$(\bar{M})=n_{z}$ is that the uniqueness of the
approximate solution to the Bellman equation will depend on it (see Proof of
Lemma 5 in the next subsection).
In the next two subsections, we will focus on how to use the input-output data
to learn the optimal control gain matrix after taking the state reconstruction
into account.
### III-B Off-Policy Bellman Equation in Output-Feedback Form
In this subsection, both $\bar{M}$ and $\omega(k)$ are introduced to change
the state-feedback Bellman equation (III) into an output-feedback form as
$\displaystyle\bar{\zeta}^{T}(k+1)\bar{P}^{j+1}\bar{\zeta}(k+1)-\bar{\zeta}^{T}(k)\bar{P}^{j+1}\bar{\zeta}(k)$
$\displaystyle=$
$\displaystyle-y^{T}Qy-\bar{\zeta}^{T}(k)\bar{M}^{T}(\bar{K}^{j})^{T}\bar{R}\bar{K}^{j}\bar{M}\bar{\zeta}(k)$
$\displaystyle+(-\bar{K}^{j}\bar{M}\bar{\zeta}(k)+\bar{u}(k))^{T}\bar{B}^{T}P^{j+1}\bar{B}(\bar{K}^{j}\bar{M}\bar{\zeta}(k)+\bar{u}(k))$
$\displaystyle+2\vartheta^{T}(k)\bar{G}^{T}P^{j+1}\bar{B}u(k)+2\bar{\zeta}^{T}(k)\bar{M}^{T}\underline{A}^{T}P^{j+1}\bar{G}\vartheta(k)$
$\displaystyle+2\bar{\zeta}^{T}(k)\bar{M}^{T}\underline{A}^{T}P^{j+1}\bar{B}(\bar{K}^{j}\bar{M}\bar{\zeta}(k)+\bar{u}(k))$
$\displaystyle+\vartheta^{T}(k)\bar{G}^{T}P^{j+1}\bar{G}\vartheta(k)+\bar{\chi}^{j+1}(t),$
(24)
where
$\displaystyle\bar{\chi}^{j+1}(k)$ $\displaystyle=$
$\displaystyle-2\omega^{T}(k+1)P^{j+1}\bar{M}\bar{\zeta}(k+1)-\omega^{T}(k+1)P^{j+1}\omega(k+1)$
$\displaystyle-2\omega^{T}(k)P^{j+1}\bar{M}\bar{\zeta}(k)-\omega^{T}(k)P^{j+1}\omega(k)$
$\displaystyle-2\omega^{T}(k)(\bar{K}^{j})^{T}\bar{R}\bar{K}^{j}\bar{M}\bar{\zeta}(k)-\omega^{T}(k)(\bar{K}^{j})^{T}\bar{R}\bar{K}^{j}\omega(k)$
$\displaystyle-2\omega^{T}(k)(\bar{K}^{j})^{T}\bar{B}^{T}P^{j+1}\bar{B}\bar{K}^{j}\bar{M}\bar{\zeta}(k)$
$\displaystyle-\omega^{T}(k)(\bar{K}^{j})^{T}\bar{B}^{T}P^{j+1}\bar{B}\bar{K}^{j}\omega(k)$
$\displaystyle+2\omega^{T}(k)\underline{A}^{T}P^{j+1}\bar{G}\vartheta(k)+2\omega^{T}(k)\underline{A}^{T}P^{j+1}\bar{B}\bar{K}^{j}\bar{M}\bar{\zeta}(k)$
$\displaystyle+2\omega^{T}(k)\underline{A}^{T}P^{j+1}\bar{B}\bar{K}^{j}\omega(k)+2\omega^{T}(k)\underline{A}^{T}P^{j+1}\bar{B}u(k)$
$\displaystyle+2\bar{\zeta}^{T}(k)\bar{M}^{T}\underline{A}^{T}P^{j+1}\bar{B}\bar{K}^{j}\omega(k).$
(25)
Let
$\mathcal{C}_{\bar{\zeta}}=[{\rm{vecv}}(\bar{\zeta}(k_{1}))-{\rm{vecv}}(\bar{\zeta}(k_{0})),{\rm{vecv}}(\bar{\zeta}(k_{2}))-{\rm{vecv}}(\bar{\zeta}(k_{1})),\cdots,{\rm{vecv}}(\bar{\zeta}(k_{f}))-{\rm{vecv}}(\bar{\zeta}(k_{f-1}))]^{T}$,
$\mathcal{D}_{{\bar{K}}_{o}^{j}\bar{\zeta}}=[{\rm{vecv}}({\bar{K}}_{o}^{j}\bar{\zeta}(k_{0})),{\rm{vecv}}({\bar{K}}_{o}^{j}\bar{\zeta}(k_{1})),\cdots,{\rm{vecv}}({\bar{K}}_{o}^{j}\bar{\zeta}(k_{f-1}))]^{T}$,
$\mathcal{D}_{\vartheta\bar{\zeta}}=[\vartheta(k_{0})\otimes\bar{\zeta}(k_{0}),\vartheta(k_{1})\otimes\bar{\zeta}(k_{1}),\cdots,\vartheta(k_{f-1})\otimes\bar{\zeta}(k_{f-1})]^{T}$,
$\mathcal{D}_{\bar{\zeta}\bar{\zeta}}=[\bar{\zeta}(k_{0})\otimes\bar{\zeta}(k_{0}),\bar{\zeta}(k_{1})\otimes\bar{\zeta}(k_{1}),\cdots,$
$\bar{\zeta}(k_{f-1})\otimes\bar{\zeta}(k_{f-1})]^{T}$,
$\mathcal{D}_{\bar{u}\vartheta}=[\bar{u}(k_{0})\otimes\vartheta(k_{0}),\bar{u}(k_{1})\otimes\vartheta(k_{1}),\cdots,\bar{u}(k_{f-1})\otimes\vartheta(k_{f-1})]^{T}$,
and $\mathcal{D}_{\bar{\chi}^{j+1}}=[\bar{\chi}^{j+1}(t_{0}),$
$\bar{\chi}^{j+1}(t_{1}),\cdots,\bar{\chi}^{j+1}(t_{s})]^{T}$. Besides, let
$\bar{L}_{P}^{j+1}=\bar{M}^{T}P^{j+1}\bar{M}$,
$\bar{L}_{1}^{j+1}=\bar{M}^{T}\underline{A}^{T}P^{j+1}\bar{B}$,
$\bar{L}_{2}^{j+1}=\bar{B}^{T}P^{j+1}\bar{B}$,
$L_{3}^{j+1}=\bar{M}^{T}\underline{A}^{T}P^{j+1}\bar{G}$,
$L_{4}^{j+1}=\bar{G}^{T}P^{j+1}\bar{B}$,
$L_{5}^{j+1}=\bar{G}^{T}P^{j+1}\bar{G}$, and
${\bar{K}}_{o}^{j}={\bar{K}}^{j}\bar{M}$. Now, (III-B) is rewritten as
$\displaystyle{\varrho_{o}^{j}}\bar{L}_{vec}={\nu_{o}^{j}}+\mathcal{D}_{\bar{\chi}^{j+1}},$
(26)
where
$\displaystyle\bar{L}_{vec}=$ $\displaystyle\
[{\rm{vecs}}^{T}(\bar{L}_{P}^{j+1}),{\rm{vec}}^{T}(\bar{L}_{1}^{j+1}),{\rm{vecs}}^{T}(\bar{L}_{2}^{j+1}),$
$\displaystyle\ \
{\rm{vec}}^{T}(\bar{L}_{3}^{j+1}),{\rm{vec}}^{T}(\bar{L}_{4}^{j+1}),{\rm{vecs}}^{T}(\bar{L}_{5}^{j+1})]^{T},$
(27) $\displaystyle{\varrho_{o}^{j}}=$ $\displaystyle\
[\mathcal{C}_{\bar{\zeta}},-2\mathcal{D}_{\bar{\zeta}\bar{\zeta}}(I\otimes({\bar{K}}_{o}^{j})^{T})-2\mathcal{D}_{\bar{u}\bar{\zeta}},$
$\displaystyle-\mathcal{D}_{\bar{u}}+\mathcal{D}_{{\bar{K}}_{o}^{j}\bar{\zeta}},-2\mathcal{D}_{\vartheta\bar{\zeta}},-2\mathcal{D}_{\bar{u}\vartheta},-\mathcal{D}_{\vartheta}],$
(28) $\displaystyle\nu_{o}^{j}=$
$\displaystyle-\mathcal{D}_{\bar{\zeta}\bar{\zeta}}{\rm{vec}}(({\bar{K}}_{o}^{j})^{T}\bar{R}{\bar{K}}_{o}^{j})-\mathcal{D}_{yy}{\rm{vec}}(Q).$
(29)
From (23), if the initial state of $r(k)$ satisfies $r(0)=0$, then $\omega(k)$
in (23) and $\mathcal{D}_{\bar{\chi}^{j+1}}$ in (26) are zeros. In the next
subsection, we will take the non-zero initial state $r(0)\neq 0$ into account,
and seek for approximating (26).
### III-C Solution to Output-Feedback Off-Policy Bellman Equation
In this subsection, we are to reduce the influence from non-zero initials and
to give a sufficient condition for approximately solving the off-policy
Bellman equation in the output-feedback form.
Here, $\mathcal{D}_{\bar{\chi}^{j+1}}$ is a nonlinear function of unknown
terms $\omega(t)$, $\bar{L}_{1}^{j+1}$, and $\bar{L}_{2}^{j+1}$. It thus
becomes difficult to obtain an accurate analytical solution from (26). Instead
of directly solving (26), we turn to computing the following linear equation
$\displaystyle{\varrho_{o}^{j}}\hat{\bar{L}}_{vec}={\nu_{o}^{j}},$ (30)
where
$\hat{\bar{L}}_{vec}=[{\rm{vecs}}^{T}(\hat{\bar{L}}_{P}^{j+1}),{\rm{vec}}^{T}(\hat{\bar{L}}_{1}^{j+1}),{\rm{vecs}}^{T}(\hat{\bar{L}}_{2}^{j+1})$,
${\rm{vec}}^{T}(\hat{\bar{L}}_{3}^{j+1}),{\rm{vec}}^{T}(\hat{\bar{L}}_{4}^{j+1}),{\rm{vecs}}^{T}(\hat{\bar{L}}_{5}^{j+1})]^{T}$,
and the notation $\hat{(\cdot)}$ is employed to differ the computed solution
in (30) from the analytical one in (26). We have seen that (30) equals (26) if
$r(0)=0$. In what follows, we focus on handling the non-zero case and give the
following result on reducing the solution error between (26) and (30).
###### Theorem 2
Suppose that the non-zero matrix $\bar{\varrho}^{j}$ in (28) is collected over
the time interval $[k_{0},k_{f}]$. If the starting time for the data
collection $k_{0}$ is sufficiently large, then the computed solution from (30)
can be considered as an approximate solution of (26) with the solution error
being sufficiently small.
Proof: Consider two difference equations
$\displaystyle v(s+1)=$ $\displaystyle
v(s)-\varepsilon{(\varrho_{o}^{j})}^{T}({\varrho_{o}^{j}}v(s)-\nu_{o}^{j}-\mathcal{D}_{\bar{\chi}^{j+1}}),$
(31) $\displaystyle\hat{v}(s+1)=$
$\displaystyle\hat{v}(s)-\varepsilon{(\varrho_{o}^{j})}^{T}({\varrho_{o}^{j}}\hat{v}(s)-\nu_{o}^{j}),$
(32)
to solve (26) and (30) with $v(0)=\hat{v}(0)=0$ and the constant $\varepsilon$
satisfying
$\displaystyle
0<\varepsilon<{2\rho^{-1}({(\varrho_{o}^{j})}^{T}({\varrho_{o}^{j}}))}.$ (33)
Bringing the algorithmic time $s$ into (31) and (32) is to distinguish it from
the system evolution time $k$ used in (1)–(4).
It follows from the singular value decomposition that there exist matrices $W$
and $\varrho_{w}^{j}$ with $W^{T}W=WW^{T}=I$ and
$(\varrho_{w}^{j})^{T}\varrho_{w}^{j}>O$ such that[12, 35]
$\varrho_{o}^{j}W=[\varrho_{w}^{j}\ O]$. Define
$\displaystyle\bar{v}(s)=W^{T}v(s)=[{\bar{v}}_{1}^{T}(s),{\bar{v}}_{2}^{T}(s)]^{T}.$
(34)
From (34), (31) is rewritten into
$\displaystyle{\bar{v}}_{1}(s+1)=$
$\displaystyle{\bar{v}}_{1}(s)-\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j}{\bar{v}}_{1}(s)$
$\displaystyle+\varepsilon{(\varrho_{w}^{j})}^{T}\nu_{o}^{j}+\varepsilon{(\varrho_{w}^{j})}^{T}\mathcal{D}_{\bar{\chi}^{j+1}},$
(35) $\displaystyle{\bar{v}}_{2}(s+1)=$ $\displaystyle{\bar{v}}_{2}(s),$ (36)
where ${\bar{v}}_{2}(s)$ is zero for any $s$. Hence, (III-C) is changed into
$\displaystyle||{\bar{v}}_{1}(s+1)||$ $\displaystyle\leq$
$\displaystyle\sqrt{|\lambda_{\max}(I-\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j})|}||{\bar{v}}_{1}(s)||+||\varepsilon{(\varrho_{w}^{j})}^{T}\nu_{o}^{j}||$
$\displaystyle+||\varepsilon{(\varrho_{w}^{j})}^{T}\mathcal{D}_{\bar{\chi}^{j+1}}||.$
(37)
From (33), one has that
$0<\rho(\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j})<2$ such that
$\rho(I-\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j})<1$. Therefore,
given the constant $\varepsilon$ and the matrix
${(\varrho_{w}^{j})}^{T}\varrho_{w}^{j}$, there exists a constant $\epsilon$
so that
$|\lambda_{\max}(I-\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j})|<\epsilon<1$.
As a result, one changes (III-C) into
$\displaystyle||{\bar{v}}_{1}(s+1)||\leq$
$\displaystyle\epsilon_{1}||{\bar{v}}_{1}(s)||+||\varepsilon{(\varrho_{w}^{j})}^{T}\nu_{o}^{j}||$
$\displaystyle+||\varepsilon{(\varrho_{w}^{j})}^{T}\mathcal{D}_{\bar{\chi}^{j+1}}||,$
(38)
where
$\epsilon_{1}=\sqrt{|\lambda_{\max}(I-\varepsilon{(\varrho_{w}^{j})}^{T}\varrho_{w}^{j})|}<\sqrt{\epsilon}<1$.
It is noted that the term $||\varepsilon{(\varrho_{w}^{j})}^{T}\nu_{o}^{j}||$
in (III-C) is bounded.
Now, we focus on the boundedness of $\mathcal{D}_{\bar{\chi}^{j+1}}$ in
(III-C). Let $\alpha_{i}$ for $i=0,1,2,3,4$ and $\epsilon_{l}$ for $l=1,2$ be
certain positive constants. Since $\underline{A}-\bar{L}\bar{C}$ is Schur,
$0<|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|<1$ holds. From (25), one has
$\displaystyle||\mathcal{D}_{\bar{\chi}^{j+1}}||\leq$ $\displaystyle\
||{\bar{v}(s)}||\alpha_{2}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}.$
(39)
Substituting (39) into (III-C) yields
$\displaystyle||{\bar{v}}_{1}(s+1)||\leq\ $
$\displaystyle(\epsilon_{1}+\alpha_{3}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}})||{\bar{v}}_{1}(s)||+\bar{b}^{j},$
where $\bar{b}^{j}$ is defined as an upper bound of
$||\varepsilon{(\varrho_{w}^{j})}^{T}\mathcal{D}_{\bar{\chi}^{j+1}}||$. Thus,
with sufficiently large $k_{0}$,
$\alpha_{3}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}$ is thus
sufficiently small. There exists a positive constant $\epsilon_{2}$ satisfying
$0<\epsilon_{1}<\epsilon_{2}<1$ and
$\epsilon_{2}-\epsilon_{1}>\alpha_{3}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}$
such that
$\displaystyle||{\bar{v}}_{1}(s+1)||\leq$
$\displaystyle\epsilon_{2}||{\bar{v}}_{1}(s)||+\bar{b}^{j},$ (40)
which leads to
$\displaystyle||{\bar{v}}_{1}(s)||\leq$
$\displaystyle\Big{(}||{\bar{v}}_{1}(0)||-\frac{\bar{b}^{j}}{1-\epsilon_{2}}\Big{)}\epsilon_{2}^{s}+\frac{\bar{b}^{j}}{1-\epsilon_{2}}.$
(41)
Thus, $\bar{v}(s)$, $v(s)$, and $\mathcal{D}_{\bar{\chi}^{j+1}}$ are bounded
for any $s$.
With the boundedness of $v(s)$, we now prove the convergence of the solution
error between (26) and (30) in the remaining analysis. Similar to (34), we
define
$\displaystyle\hat{\bar{v}}(s)=W^{T}\hat{v}(s)=[\hat{\bar{v}}_{1}^{T}(s),\hat{\bar{v}}_{2}(s)]^{T},$
(42)
based on which one has
$\displaystyle\hat{\bar{v}}_{1}(s+1)$
$\displaystyle=\hat{\bar{v}}_{1}(s)-{(\varrho_{w}^{j})}^{T}{\varrho_{w}^{j}}\hat{\bar{v}}_{1}(s)+{(\varrho_{w}^{j})}^{T}\nu_{o}^{j},$
(43) $\displaystyle\hat{\bar{v}}_{2}(s+1)$
$\displaystyle=\hat{\bar{v}}_{2}(s),$ (44)
where $\hat{\bar{v}}(s)$ is zero. Thus, from (44), $\hat{\bar{v}}_{2}(s)=0$
holds. Let the error be
$\tilde{\bar{v}}_{1}(s)=\bar{v}_{1}(s)-\hat{\bar{v}}_{1}(s)$. From (III-C) and
(43), one has
$\displaystyle||\tilde{\bar{v}}_{1}(s+1)||\leq\ $
$\displaystyle\epsilon_{1}||\tilde{\bar{v}}_{1}(s)||+\alpha_{4}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}},$
(45)
where the boundedness of $\mathcal{D}_{\bar{\chi}^{j+1}}$ is employed. By
(45), one has
$\displaystyle||\tilde{\bar{v}}_{1}(s)||\leq\ $
$\displaystyle\Big{(}||{\bar{v}}_{e_{1}}(0)||-\frac{\alpha_{4}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}}{1-\epsilon_{1}}\Big{)}|\epsilon_{1}|^{s}$
$\displaystyle+\frac{\alpha_{4}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}}{1-\epsilon_{1}},$
(46)
where the constant $\epsilon_{1}$ has been defined in (III-C) satisfying
$0<\epsilon_{1}<1$ and the term
$\alpha_{4}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}$ has been
tuned sufficiently small under sufficiently large $k_{0}$. From (34) and (42),
one changes (III-C) into
$\displaystyle\lim_{s\rightarrow\infty}||\hat{v}(s)-v(s)||\leq$
$\displaystyle\lim_{s\rightarrow\infty}||W||||\hat{\bar{v}}(s)-\bar{v}(s)||$
$\displaystyle=$
$\displaystyle\lim_{s\rightarrow\infty}||\hat{\bar{v}}(s)-\bar{v}(s)||$
$\displaystyle=$ $\displaystyle\
\frac{\alpha_{4}|\lambda_{\max}(\underline{A}-\bar{L}\bar{C})|^{k_{0}}}{1-\epsilon_{1}},$
(47)
which implies that the solution of (26) converges to that of (30) with the
error being sufficiently small by increasing $k_{0}$. Therefore, the proof is
completed. $\square$
###### Remark 1
Theorem 2 shows that the solution error between (26) and (30) can be made
smaller by choosing a larger starting time $k_{0}$ for the data collection.
This prompts us to introduce an additional Model-Free Pre-Collection Phase.
The necessity of the additional phase roots in the requirement of the
convergence in the DT state reconstruction and the idea is inspired by the CT
work of [12]. Details on how to coordinately execute the DT off-policy
learning will be presented in Algorithm 1. $\bullet$
###### Remark 2
In contrast to [12] wherein differential equations are used to solve the
linear equation, Theorem 2 uses difference equations. In addition, we find
that our result in Theorem 2 and that in [36] are dual to each other in a
certain sense. To be specific, Theorem 2 shows that a class of unknown
nonlinear equations are approximately solved by a known linear difference
equation, while [36] shows that an unknown linear difference equation is
approximately solved by a class of nonlinear equations. $\bullet$
The following result shows that how much data should we collect to achieve the
optimal output tracking control within the output-feedback RL framework.
###### Lemma 5
The off-policy Bellman equation in the output-feedback form (30) over the time
interval $[k_{0},k_{f}]$, $\hat{\bar{L}}_{i}^{j+1}$, for $i=1,2,\ldots,5$, can
be uniquely solved, if
1. 1.
the collected input-output data at the time $k_{f}$ satisfy
$\displaystyle{\rm{rank}}([\mathcal{D}_{\bar{\zeta}\bar{\zeta}},\mathcal{D}_{\bar{u}\bar{\zeta}},\mathcal{D}_{\bar{u}},\mathcal{D}_{\vartheta\bar{\zeta}},\mathcal{D}_{\bar{u}\vartheta},\mathcal{D}_{\vartheta}])$
$\displaystyle=\frac{1}{2}(n_{\bar{\zeta}}(n_{\bar{\zeta}}+1)+r_{m}(r_{m}+1)+r_{p}(r_{p}+1))$
$\displaystyle\quad\ +n_{\bar{\zeta}}r_{m}+r_{p}n_{\bar{\zeta}}+r_{p}r_{m};$
(48)
2. 2.
rank$(\bar{M})=n_{z}$;
3. 3.
$r(0)=0$.
Proof: See Appendix B. $\square$
In Lemma 5, the condition $r(0)=0$ is required. We now extend it to the
condition $r(0)\neq 0$ by choosing the starting time $k_{0}$ to be
sufficiently large. This is because the solution $\hat{\bar{L}}_{i}^{j+1}$,
for $i=1,2,\ldots,5$ in (30) under the condition $r(0)\neq 0$ converges to the
unique solution in Lemma 5 under the condition $r(0)=0$ after recalling the
result in Theorem 2.
### III-D Optimal Output Tracking Design via Output-Feedback RL
Our output-feedback off-policy learning algorithm is summarized in Algorithm
1, where the successive error of the control matrix
$\|{\hat{\bar{K}}_{o}^{j+1}-\hat{\bar{K}}_{o}^{j}}\|$ is used for the stopping
indicator since $\hat{\bar{K}}_{o}^{j+1}$ is solved uniquely under conditions
in Lemma 5.
Based on the learned optimal control gain matrix $\hat{\bar{K}}_{o}^{*}$ from
Algorithm 1, we achieve optimal output tracking control of DT systems via
output feedback as below.
Algorithm 1 Output-Feedback Off-Policy RL for Optimal Output Tracking Control
of DT Systems
1:Initialize: Let $j=0$ and ${\bar{K}}_{o}^{j}$ be a stabilizing gain. Let the
pair $(F,G)$ be an $r_{p}$-copy internal model of $S$.
2:Model-Free Pre-Collection Phase: From (20)–(22), compute ${\zeta}_{u}$,
${\zeta}_{y}$, and $\zeta_{\vartheta}$ over the time interval $[0,k_{0})$,
where $k_{0}$ is set sufficiently large.
3:Data-Collection Phase: Apply the behavior policy $\bar{u}(k)$ in (17) with
$(F,T)$ being observable to the system (1) with $r_{p}\geq r_{m}$ over the
time interval $[k_{0},k_{f}]$ for collecting the input-output data
$\mathcal{D}_{\bar{\zeta}\bar{\zeta}}$, $\mathcal{D}_{\bar{u}\bar{\zeta}}$,
$\mathcal{D}_{\bar{u}}$, $\mathcal{D}_{\vartheta\bar{\zeta}}$,
$\mathcal{D}_{\bar{u}\vartheta}$, and $\mathcal{D}_{\vartheta}$.
4:if (1) holds then
5: while the stopping indicator
$\|{\hat{\bar{K}}_{o}^{j+1}-\hat{\bar{K}}_{o}^{j}}\|\leqslant{\varepsilon}$ is
not satisfied with ${\varepsilon}>0$ being a small constant do
6: Solve (30) and update the feedback gain as
$\displaystyle\hat{\bar{K}}_{o}^{j+1}=(\bar{R}+\hat{\bar{L}}_{2}^{j+1})^{-1}(\hat{\bar{L}}_{1}^{j+1})^{T}.$
(49)
7: end while
8:end if
9:Optimal Control Phase: The learned optimal control gain matrix
$\hat{\bar{K}}_{o}^{*}$ is given as
$\hat{\bar{K}}_{o}^{*}=\hat{\bar{K}}_{o}^{j+1}$.
###### Theorem 3
Let the DT system (1)–(4) satisfy Assumptions 1–4 and the output-feedback
adaptive optimal output tracking DT controller be designed as
$\displaystyle u(k)=$
$\displaystyle-\hat{\bar{K}}_{o}^{*}\bar{\zeta}(k)-T{\bar{z}(k)},$ (50)
where $\bar{\zeta}$ is given in (23), $\bar{z}(k)$ is given in (17) with
$\vartheta(k)$ being assigned as $y_{d}(k)$, and $\hat{\bar{K}}_{o}^{*}$ is
the learnt optimal control gain matrix by Algorithm 1. Therefore, the state-
oriented tracking optimization problem, Problem 1, is solved with the output
tracking error $y_{e}(k)$ in (5) decaying to zero, asymptotically.
Proof: We first show that the learnt matrix $\hat{\bar{K}}_{o}^{j+1}$ in (49)
converges to the ideal matrix $\bar{K}^{*}\bar{M}$ with $\bar{K}^{*}$ and
$\bar{M}$ being defined in (13) and (23), respectively. With the condition in
Theorem 2 satisfied, $\hat{\bar{L}}_{1}^{j+1}$ and $\hat{\bar{L}}_{2}^{j+1}$
in (30) can be made to converge to ${\bar{L}}_{1}^{j+1}$ and
${\bar{L}}_{2}^{j+1}$ in (26), respectively. This, together with the
uniqueness in Lemma 5, leads that the learned matrix
$(\bar{R}+\hat{\bar{L}}_{2}^{j+1})^{-1}(\hat{\bar{L}}_{1}^{j+1})^{T}$ in (49)
uniquely converges to $\bar{K}^{j+1}\bar{M}$, where $\bar{K}^{j+1}$ is defined
in (16). It follows from Lemma 2 that $\bar{K}^{j+1}$ converges to
$\bar{K}^{*}$ as the integer $j$ gets larger. Note that $\bar{K}^{*}$ is the
unique solution satisfying the Problem 1 under the controllability of
$(\underline{A},\bar{B})$ and the observability of $(\underline{A},\bar{C})$.
Therefore, if $\hat{\bar{K}}_{o}^{j+1}$ converges, then the unique solution
$\hat{\bar{K}}^{j+1}$ from (30) converges to $\bar{K}^{*}\bar{M}$. It reveals
that the ideal matrix $\bar{K}^{*}\bar{M}$ is learned by the matrix
$\hat{\bar{K}}_{o}^{*}$ from Algorithm 1. We then show the convergence of the
output tracking error $y_{e}(k)$. With the learned optimal control gain matrix
$\hat{\bar{K}}_{o}^{*}$, the closed-loop system becomes
$e(k+1)={{\bar{A}}_{o}^{*}}e(k)+\bar{B}{(\underline{A}-\bar{L}\bar{C})^{k}}r(0)$,
where $e(k)$ is given in (9) and both ${\bar{A}_{o}^{*}}{\text{ =
}}\underline{A}-\bar{B}\hat{\bar{K}}_{o}^{*}$ and
$\underline{A}-\bar{L}\bar{C}$ are Schur. Considering that
$\lim_{k\rightarrow\infty}\bar{B}{(\underline{A}-\bar{L}\bar{C})^{k}}r(0)=0$,
one obtains that $\lim_{k\rightarrow\infty}e(k)=0$[36], based on which the
output tracking error satisfies $\lim_{k\rightarrow\infty}y_{e}(k)=0$ from
(11b). This completes the proof. $\square$
###### Remark 3
The RL-based controller in (50) is robust to system uncertainties, which
corresponds to the linear robust output regulation in the literature such as
[32]. That is, after the learning is completed, the proposed controllers are
robust to some model uncertainties in the system dynamics matrices $A+\Delta
A$, $B+\Delta B$, and $C+\Delta C$, where $A$, $B$, and $C$ denote the nominal
part of the plant; $\Delta A$, $\Delta B$, and $\Delta C$ represent the model
uncertainties. $\bullet$
## IV Conclusion
This paper investigated the output-feedback optimal output tracking problem
for DT systems with unknown dynamics using the off-policy RL and robust output
regulation theory. We have formulated the output tracking optimization problem
based on the newly proposed dynamical DT controller in contrast to the
standard DT controller by linear output regulation theory. We have shown that,
by making use of the collected data along with the controlled system and the
reference output, we are able to approximate the optimal output-feedback
controller. We have studied the parameterization matrix and re-expression
error so that the learned optimal controller has a satisfactory performance.
## Appendix A Proof of Lemma 4
Let us consider a standard Luenberger observer as
$\displaystyle{\hat{r}}(k+1)=$
$\displaystyle\underline{A}\hat{r}(k)+\bar{B}\bar{u}(k)+\bar{G}\vartheta(k)$
$\displaystyle+\bar{L}(y(k)-\bar{C}\hat{r}(k))$ (51)
with $\bar{L}$ being a $n_{z}\times r_{p}$ matrix so that $\hat{r}(k)-r(k)$
decays to zero with $r(k)$ given by (18). Similar to [24], (A) is rewritten as
$\displaystyle\hat{r}(k)=$ $\displaystyle\
G_{u}(z)[\bar{u}(k)]+G_{y}(z)[y(k)]$
$\displaystyle+G_{\vartheta}(z)[\vartheta(k)]+{(\mathcal{A}_{o})^{k}}\hat{r}(0),$
(52)
where $\mathcal{A}_{o}=\underline{A}-\bar{L}\bar{C}$,
$G_{\bar{u}}(z)[\bar{u}(k)]$ is a time-domain DT signal represented by the
frequency-domain representation
$G_{\bar{u}}(z)=\left[{\begin{array}[]{*{20}{cc}}G_{1,1}^{\bar{u}}(z)&\cdots&G_{1,r_{m}}^{\bar{u}}(z)\\\\[-5.69046pt]
\vdots&\ddots&\vdots\\\\[-5.69046pt]
G_{n_{z},1}^{\bar{u}}(z)&\cdots&G_{n_{z},r_{m}}^{\bar{u}}(z)\\\
\end{array}}\right]\in\mathbb{R}^{n_{z}\times r_{m}}$. The entry
$G_{i,j}^{\bar{u}}(z)$ at the $i$th row and $j$th column of $G_{\bar{u}}(z)$
is extended to
$\displaystyle G_{i,j}^{\bar{u}}(z)=$
$\displaystyle\frac{1}{\det(zI-\mathcal{A}_{o})}\Big{[}g_{i,j,n_{z}-1}^{\bar{u}}z^{n_{z}-1}+g_{i,j,n_{z}-2}^{\bar{u}}z^{n_{z}-2}$
$\displaystyle\quad\quad\quad+\cdots+g_{i,j,1}^{\bar{u}}z+g_{i,j,0}^{\bar{u}}\Big{]}.$
(53)
Here, $G_{\bar{u}}(z)[\bar{u}(k)]$ is interpreted as
$\displaystyle G_{\bar{u}}(z)[\bar{u}(k)]=$
$\displaystyle\underbrace{\left[{\begin{array}[]{*{20}{cc}}g_{1,1,0}^{\bar{u}}&\cdots&g_{1,r_{m},n_{z}-1}^{\bar{u}}&g_{1,r_{m}+1,0}^{\bar{u}}&\cdots&g_{1,n_{z}r_{m},n_{z}-1}^{\bar{u}}\\\\[-5.69046pt]
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\\[-5.69046pt]
g_{n_{z},1,0}^{\bar{u}}&\cdots&g_{n_{z},r_{m},n_{z}-1}^{\bar{u}}&g_{1,r_{m}+1,0}^{\bar{u}}&\cdots&g_{n_{z},n_{z}r_{m},n_{z}-1}^{\bar{u}}\\\
\end{array}}\right]}_{M_{\bar{u}}}$ (57)
$\displaystyle\quad\quad\times\underbrace{[u(k)]\otimes\left[1,z,\cdots,z^{n_{z}-1}\right]^{T}\frac{1}{\det(zI-\mathcal{A}_{o})}}_{\zeta_{\bar{u}}(k)}$
$\displaystyle\triangleq M_{\bar{u}}\zeta_{\bar{u}}(k).$ (58)
Similar definitions of $M_{y}\zeta_{y}(k)$ and
$M_{\vartheta}\zeta_{\vartheta}(k)$ apply to $G_{y}(z)[y(k)]$ and
$G_{\vartheta}(z)[\vartheta(k)]$, respectively. Thus, one obtains that
$\displaystyle\hat{r}(k)$
$\displaystyle=\bar{M}\bar{\zeta}(k)+{(\mathcal{A}_{o})^{k}}\hat{r}(0).$ (59)
From (A) and the z-transform operator, one has
$\displaystyle\hat{r}(k)=$
$\displaystyle{(\underline{A}-\bar{L}\bar{C})^{k}}\hat{r}(0)+(zI-\mathcal{A}_{o})^{-1}\bar{B}[\bar{u}(k)]$
$\displaystyle+(zI-\mathcal{A}_{o})^{-1}\bar{L}[y(k)]+(zI-\mathcal{A}_{o})^{-1}\bar{G}[\vartheta(k)],$
where
$\displaystyle(zI-\mathcal{A}_{o})^{-1}=$
$\displaystyle\frac{{\rm{adj}}(zI-\mathcal{A}_{o})}{\det(zI-\mathcal{A}_{o})},$
(60) $\displaystyle\det(zI-\mathcal{A}_{o})=$ $\displaystyle
z^{n_{z}}+d_{1}z^{{n_{z}}-1}+d_{2}z^{{n_{z}}-2}+\cdots$
$\displaystyle+d_{{n_{z}}-1}z+d_{n_{z}},$ (61)
$\displaystyle{\rm{adj}}(zI-\mathcal{A}_{o})=$ $\displaystyle
B_{0}z^{n_{z}-1}+B_{1}z^{{n_{z}}-2}+\cdots$
$\displaystyle+B_{{n_{z}}-2}z+B_{{n_{z}}-1}.$ (62)
From (60) and (62), one has
$\displaystyle\det(zI-\mathcal{A}_{o})I$ $\displaystyle=$ $\displaystyle\
B_{0}z^{n_{z}}+(B_{1}-B_{0}\mathcal{A}_{o})z^{{n_{z}}-1}+(B_{2}-B_{1}\mathcal{A}_{o})z^{{n_{z}}-2}$
$\displaystyle+\cdots+(B_{{n_{z}}-1}-B_{{n_{z}}-2}\mathcal{A}_{o})z-B_{{n_{z}}-1}A.$
(63)
Using (61) and (A), the following equations $B_{0}=I$ and
$B_{i+1}=B_{i}\mathcal{A}_{o}+d_{i+1}I$ hold for $i=0,1,2,\cdots,n_{z}-2$. For
$i=0$, one has
$B_{1}=B_{0}\mathcal{A}_{o}+d_{1}I=\mathcal{A}_{o}+d_{1}I=\underline{A}-\bar{L}\bar{C}+d_{1}I$.
This leads to
$\displaystyle{\rm{rank}}([[B_{0},B_{1}]\bar{B},[B_{0},B_{1}]\bar{L},[B_{0},B_{1}]\bar{G}])$
$\displaystyle=\ $
$\displaystyle{\rm{rank}}([[B_{0},\underline{A}-\bar{L}\bar{C}+d_{1}I]\bar{B},[B_{0},\underline{A}-\bar{L}\bar{C}+d_{1}I]\bar{L},$
$\displaystyle\quad\quad[B_{0},\underline{A}-\bar{L}\bar{C}+d_{1}I]\bar{G}])$
$\displaystyle=\ $
$\displaystyle{\rm{rank}}([[\bar{B},(\underline{A}-\bar{L}\bar{C})\bar{B}+d_{1}\bar{B}],[\bar{L},(\underline{A}-\bar{L}\bar{C})\bar{L}+d_{1}\bar{L}],$
$\displaystyle\quad\quad[\bar{G},(\underline{A}-\bar{L}\bar{C})\bar{G}+d_{1}\bar{G}]])$
$\displaystyle=\ $
$\displaystyle{\rm{rank}}([\bar{B},\underline{A}\bar{B},\bar{L},\underline{A}\bar{L},\bar{G},\underline{A}\bar{G}]),$
(64)
where the last equation is obtained using the fact that column operations do
not change the rank of a matrix. Based on analysis in (A) with $i=0$,
proceeding the order $i$ to be higher one by one, one has
$\displaystyle{\rm{rank}}([\mathcal{B}(\bar{B}),\mathcal{B}(\bar{L}),\mathcal{B}(\bar{G})])$
$\displaystyle=\ $
$\displaystyle{{\rm{rank}}}([\mathcal{C}(\underline{A},\bar{B}),\mathcal{C}(\underline{A},\bar{L}),\mathcal{C}(\underline{A},\bar{G})])$
(65)
with $\mathcal{B}(\bar{B})=[B_{0},B_{1},\cdots,B_{n_{z}-1}]\bar{B}$,
$\mathcal{B}(\bar{L})=[B_{0},B_{1},\cdots$, $B_{n_{z}-1}]\bar{L}$, and
$\mathcal{B}(\bar{G})=[B_{0},B_{1},\cdots,B_{n_{z}-1}]\bar{G}$.
Based on (A), we next clarify that
${\rm{rank}}(\bar{M})={\rm{rank}}([\mathcal{B}(\bar{B}),\mathcal{B}(\bar{L}),\mathcal{B}(\bar{G})])$
with $\bar{M}$ being given in (59). Based on the defintions in (A) and (62),
${\rm{rank}}(M_{\bar{u}})={\rm{rank}}(\mathcal{B}(\bar{B}))$. It leads to
${\rm{rank}}(M_{y})={\rm{rank}}(\mathcal{B}(\bar{L}))$ and
${\rm{rank}}(M_{\vartheta})={\rm{rank}}(\mathcal{B}(\bar{G}))$. Hence, one has
${\rm{rank}}(\bar{M})={\rm{rank}}([\mathcal{B}(\bar{B}),\mathcal{B}(\bar{L}),\mathcal{B}(\bar{G})])$.
This, together with (A) leads to
rank$(\bar{M})=$rank$([\mathcal{C}(\underline{A},\bar{B}),\mathcal{C}(\underline{A},\bar{L}),\mathcal{C}(\underline{A},\bar{G})])$.
Therefore, the proof is completed. $\square$
## Appendix B Proof of Lemma 5
In order to show the uniqueness of $\hat{\bar{L}}_{i}^{j+1}$, for
$i=1,2,\ldots,5$, in (30), it is equivalent to proving that
$\displaystyle 0=$ $\displaystyle{\varrho_{o}^{j}}\bar{\Xi}^{v},$ (66)
with
$\bar{\Xi}^{v}=[\bar{W}^{v},\bar{Y}_{1}^{v},\bar{Y}_{2}^{v},\bar{Y}_{3}^{v},\bar{Y}_{4}^{v},\bar{Y}_{5}^{v}]^{T}$
has a unique zero solution
$[\bar{Y}_{1}^{v},\bar{Y}_{2}^{v},\bar{Y}_{3}^{v},\bar{Y}_{4}^{v},\bar{Y}_{5}^{v}]^{T}=0$,
where $\bar{W}^{v}={\rm{vecs}}(\bar{W}^{m})$,
$\bar{Y}_{1}^{v}={\rm{vec}}(\bar{Y}_{1}^{m})$,
$\bar{Y}_{2}^{v}={\rm{vecs}}(\bar{Y}_{2}^{m})$,
$\bar{Y}_{3}^{v}={\rm{vec}}(\bar{Y}_{3}^{m})$,
$\bar{Y}_{4}^{v}={\rm{vec}}(\bar{Y}_{4}^{m})$, and
$\bar{Y}_{5}^{v}={\rm{vecs}}(\bar{Y}_{5}^{m})$ with
$\bar{W}^{m}=(\bar{W}^{m})^{T}$, $\bar{Y}_{2}^{m}=(\bar{Y}_{2}^{m})^{T}$, and
$\bar{Y}_{5}^{m}=(\bar{Y}_{5}^{m})^{T}$. Define
$\displaystyle\bar{Z}^{m}=(\bar{M}\bar{M}^{T})^{-1}\bar{M}\bar{W}^{m}\bar{M}^{T}(\bar{M}\bar{M}^{T})^{-1},$
(67)
where the property of rank$(\bar{M})=n_{z}$ is employed. Under the condition
(3) of Lemma 5, the approximation errors $\bar{\chi}^{j+1}(t)$ in (25) are
zeros. Thus, it follows from (III-B) and (67) that (66) leads to
$\displaystyle 0$ $\displaystyle=\
\mathcal{D}_{\bar{\zeta}}{\rm{vecs}}(\bar{\kappa}_{P})+2\mathcal{D}_{\bar{u}\bar{\zeta}}{\rm{vec}}(\bar{\kappa}_{1})+\mathcal{D}_{\bar{u}}{\rm{vecs}}(\bar{\kappa}_{2})$
$\displaystyle+2\mathcal{D}_{\vartheta\bar{\zeta}}{\rm{vec}}(\bar{\kappa}_{3})+2\mathcal{D}_{\bar{u}\vartheta}{\rm{vec}}(\bar{\kappa}_{4})+\mathcal{D}_{\vartheta}{\rm{vecs}}(\bar{\kappa}_{5}),$
(68)
where
$\bar{\kappa}_{P}=\bar{M}^{T}[(\underline{A}^{j})^{T}\bar{Z}^{m}\underline{A}^{j}-\bar{Z}^{m}]\bar{M}+({\bar{K}}_{o}^{j})^{T}(\bar{B}^{T}\bar{Z}^{m}\bar{B}-\bar{Y}_{2}^{m}){\bar{K}}_{o}^{j}+(\underline{A}^{T}\bar{Z}^{m}\bar{B}-\bar{Y}_{1}^{m}){\bar{K}}_{o}^{j}+({\bar{K}}_{o}^{j})^{T}(\underline{A}^{T}\bar{Z}^{m}\bar{B}-Y_{1}^{m})^{T}$,
$\bar{\kappa}_{1}=\underline{A}^{T}\bar{Z}^{m}\bar{B}-\bar{Y}_{1}^{m}$,
$\bar{\kappa}_{2}=\bar{B}^{T}\bar{Z}^{m}\bar{B}-\bar{Y}_{2}^{m}$
$\bar{\kappa}_{3}=\underline{A}^{T}\bar{Z}^{m}\bar{G}-\bar{Y}_{3}^{m}$,
$\bar{\kappa}_{4}=\bar{G}^{T}\bar{Z}^{m}\bar{B}-\bar{Y}_{4}^{m}$, and
$\bar{\kappa}_{5}=\bar{G}^{T}\bar{Z}^{m}\bar{G}-\bar{Y}_{5}^{m}$.
The matrix
$\big{[}\mathcal{D}_{\bar{\zeta}},2\mathcal{D}_{\bar{u}\bar{\zeta}},\mathcal{D}_{\bar{u}},2\mathcal{D}_{\vartheta\bar{\zeta}},2\mathcal{D}_{\bar{u}\vartheta},\mathcal{D}_{\vartheta}\big{]}$
is full column rank if (1) holds. Thus, the solution to (B) is uniquely
obtained as
$\displaystyle($
$\displaystyle{\rm{vecs}}^{T}(\bar{\kappa}_{P}),{\rm{vec}}^{T}(\bar{\kappa}_{1}),{\rm{vecs}}^{T}(\bar{\kappa}_{2}),$
$\displaystyle{\rm{vec}}^{T}(\bar{\kappa}_{3}),{\rm{vec}}^{T}(\bar{\kappa}_{4}),{\rm{vecs}}^{T}(\bar{\kappa}_{5}))^{T}=0.$
(69)
Recalling rank$(\bar{M})=n_{z}$, one further rewrites $\bar{\kappa}_{P}$ in
(B) as $(\underline{A}^{j})^{T}\bar{Z}^{m}\underline{A}^{j}-\bar{Z}^{m}=0$,
where $\underline{A}^{j}$ is Schur. Therefore, $\bar{Z}^{m}$ must be zeros,
based on which $\bar{Y}_{i}^{v}$ in (66), for $i=1,2,\ldots,5$, are also
zeros. This implies that $\hat{\bar{L}}_{i}^{j+1}$ in (30), for
$i=1,2,\ldots,5$, are unique. Note that since the non-square matrix $\bar{M}$
in (67) is only full row rank, the zero solution of $\bar{Z}^{m}$ does not
ensure the zero solution of $\bar{W}^{m}$ so that $\hat{\bar{L}}_{P}^{j+1}$
may not be unqiue. This completes the proof. $\square$
## Acknowledgement
The authors are thankful to Prof. Zongli Lin for his helpful comments and
suggestions.
## References
* [1] B. L. Stevens, F. L. Lewis, and E. N. Johnson, _Aircraft control and simulation: dynamics, controls design, and autonomous systems_. John Wiley & Sons, 2015.
* [2] B. A. Francis, “The linear multivariable regulator problem,” _SIAM Journal on Control and Optimization_ , vol. 15, no. 3, pp. 486–505, 1977.
* [3] R. S. Sutton and A. G. Barto, _Introduction to reinforcement learning_. MIT Press Cambridge, 1998.
* [4] F. L. Lewis, D. Vrabie, and V. L. Syrmos, _Optimal control_. John Wiley & Sons, 2012.
* [5] H. Zhang, D. Liu, Y. Luo, and D. Wang, _Adaptive dynamic programming for control: algorithms and stability_. Springer Science & Business Media, 2012.
* [6] D. Vrabie, K. G. Vamvoudakis, and F. L. Lewis, _Optimal adaptive control and differential games by reinforcement learning principles_. Institution of Engineering and Technology, 2013.
* [7] H. Modares, F. L. Lewis, and Z.-P. Jiang, “Optimal output-feedback control of unknown continuous-time linear systems using off-policy reinforcement learning,” _IEEE Trans. Cybern._ , vol. 46, no. 11, pp. 2401–2410, 2016\.
* [8] Y. Jiang and Z.-P. Jiang, _Robust Adaptive Dynamic Programming_. John Wiley & Sons, 2017.
* [9] W. Gao and Z.-P. Jiang, “Adaptive dynamic programming and adaptive optimal output regulation of linear systems,” _IEEE Trans. Autom. Control_ , vol. 61, no. 12, pp. 4164–4169, 2016.
* [10] R. Kamalapurkar, P. Walters, J. Rosenfeld, and W. Dixon, _Reinforcement Learning for Optimal Feedback Control: A Lyapunov-Based Approach_. Springer, 2018.
* [11] C. Chen, H. Modares, K. Xie, F. L. Lewis, Y. Wan, and S. Xie, “Reinforcement learning-based adaptive optimal exponential tracking control of linear systems with unknown dynamics,” _IEEE Trans. Autom. Control_ , vol. 64, no. 11, pp. 4423–4438, 2019.
* [12] C. Chen, L. Xie, F. L. Lewis, and S. Xie, “Adaptive optimal output tracking of continuous-time systems via output-feedback-based reinforcement learning,” _under review_ , 2019.
* [13] C. Chen, F. L. Lewis, K. Xie, S. Xie, and Y. Liu, “Off-policy learning for adaptive optimal output synchronization of heterogeneous multi-agent systems,” _Automatica_ , vol. 119, p. 109081, 2020.
* [14] Z.-P. Jiang, T. Bian, W. Gao _et al._ , “Learning-based control: A tutorial and some recent results,” _Foundations and Trends® in Systems and Control_ , vol. 8, no. 3, pp. 176–284, 2020\.
* [15] B. Kiumarsi and F. L. Lewis, “Actor–critic-based optimal tracking for partially unknown nonlinear discrete-time systems,” _IEEE Trans. Neural Netw. Learning Syst._ , vol. 26, no. 1, pp. 140–151, 2015.
* [16] B. Kiumarsi, F. L. Lewis, H. Modares, A. Karimpour, and M.-B. Naghibi-Sistani, “Reinforcement ${Q}$-learning for optimal tracking control of linear discrete-time systems with unknown dynamics,” _Automatica_ , vol. 50, no. 4, pp. 1167–1175, 2014.
* [17] B. Kiumarsi, F. L. Lewis, and Z.-P. Jiang, “${H}_{\infty}$ control of linear discrete-time systems: Off-policy reinforcement learning,” _Automatica_ , vol. 78, pp. 144–152, 2017.
* [18] Y. Jiang, B. Kiumarsi, J. Fan, T. Chai, J. Li, and F. L. Lewis, “Optimal output regulation of linear discrete-time systems with unknown dynamics using reinforcement learning,” _IEEE Trans. Cybern._ , vol. 50, no. 7, pp. 3147–3156, 2020.
* [19] W. Gao, Y. Liu, A. Odekunle, Y. Yu, and P. Lu, “Adaptive dynamic programming and cooperative output regulation of discrete-time multi-agent systems,” _International Journal of Control, Automation and Systems_ , vol. 16, no. 5, pp. 2273–2281, 2018.
* [20] F. L. Lewis and K. G. Vamvoudakis, “Reinforcement learning for partially observable dynamic processes: Adaptive dynamic programming using measured output data,” _IEEE Trans. Syst. Man Cybern., Part B, Cybern._ , vol. 41, no. 1, pp. 14–25, Feb 2011.
* [21] J. Fan, Z. Li, Y. Jiang, T. Chai, and F. L. Lewis, “Model-free linear discrete-time system ${H}_{\infty}$ control using input-output data,” in _2018 International Conference on Advanced Mechatronic Systems_ , Aug 2018, pp. 207–212.
* [22] S. A. A. Rizvi and Z. Lin, “Output feedback ${Q}$-learning for discrete-time linear zero-sum games with application to the H-infinity control,” _Automatica_ , vol. 95, pp. 213–221, 2018.
* [23] B. Kiumarsi, F. L. Lewis, M. Naghibi-Sistani, and A. Karimpour, “Optimal tracking control of unknown discrete-time linear systems using input-output measured data,” _IEEE Trans. Cybern._ , vol. 45, no. 12, pp. 2770–2779, Dec 2015.
* [24] S. A. A. Rizvi and Z. Lin, “Output feedback ${Q}$-learning control for the discrete-time linear quadratic regulator problem,” _IEEE Trans. Neural Netw. Learning Syst._ , vol. 30, no. 5, pp. 1523–1536, 2018.
* [25] ——, “Output feedback optimal tracking control using reinforcement ${Q}$-learning,” in _2018 Annual American Control Conference (ACC)_. IEEE, 2018, pp. 3423–3428.
* [26] W. Gao and Z.-P. Jiang, “Adaptive optimal output regulation via output-feedback: An adaptive dynamic programing approach,” in _2016 IEEE 55th Conference on Decision and Control (CDC)_. IEEE, 2016, pp. 5845–5850.
* [27] ——, “Adaptive optimal output regulation of time-delay systems via measurement feedback,” _IEEE Trans. Neural Netw. Learning Syst._ , vol. 30, no. 3, pp. 938–945, 2018.
* [28] W. Gao, Y. Jiang, Z.-P. Jiang, and T. Chai, “Adaptive and optimal output feedback control of linear systems: An adaptive dynamic programming approach,” in _Intelligent Control and Automation (WCICA), 2014 11th World Congress on_. IEEE, 2014, pp. 2085–2090.
* [29] S. A. A. Rizvi and Z. Lin, “Output feedback adaptive dynamic programming for linear differential zero-sum games,” _Automatica_ , vol. 122, p. 109272, 2020\.
* [30] ——, “A note on state parameterizations in output feedback reinforcement learning control of linear systems,” _under review_.
* [31] ——, “Output feedback reinforcement learning control for linear systems,” _Birkhauser, to appear_.
* [32] J. Huang, _Nonlinear output regulation: theory and applications_. SIAM, 2004.
* [33] G. Hewer, “An iterative technique for the computation of the steady state gains for the discrete optimal regulator,” _IEEE Trans. Autom. Control_ , vol. 16, no. 4, pp. 382–384, 1971.
* [34] G. Tao, _Adaptive control design and analysis_. John Wiley & Sons, 2003.
* [35] T. Liu and J. Huang, “Adaptive cooperative output regulation of discrete-time linear multi-agent systems by a distributed feedback control law,” _IEEE Trans. Autom. Control_ , vol. 63, no. 12, pp. 4383–4390, 2018.
* [36] J. Huang, “The cooperative output regulation problem of discrete-time linear multi-agent systems by the adaptive distributed observer,” _IEEE Trans. Autom. Control_ , vol. 62, no. 4, pp. 1979–1984, 2016.
|
# A variational discrete element method for the computation of Cosserat
elasticity
Frédéric Marazzato Department of Mathematics, Louisiana State University,
Baton Rouge, LA 70803, USA
email<EMAIL_ADDRESS>
###### Abstract
The variational discrete element method developed in [28] for dynamic elasto-
plastic computations is adapted to compute the deformation of elastic Cosserat
materials. In addition to cellwise displacement degrees of freedom (dofs),
cellwise rotational dofs are added. A reconstruction is devised to obtain
$P^{1}$ non-conforming polynomials in each cell and thus constant strains and
stresses in each cell. The method requires only the usual macroscopic
parameters of a Cosserat material and no microscopic parameter. Numerical
examples show the robustness of the method for both static and dynamic
computations in two and three dimensions.
## 1 Introduction
Cosserat continua have been introduced in [13]. They generalize Cauchy
continua by adding a miscroscopic rotation to every infinitesimal element.
Cosserat continua can be considered as a generalization of Timoshenko beams to
two and three-dimensional structures. Contrary to traditional Cauchy continua
of order one, Cosserat continua are able to reproduce some effects of the
micro-structure of a material through the definition of a characteristic
length written $\ell$ [20]. Cosserat media can appear as homogenization of
masonry structures [44, 21] or be used to model liquid crystals [17],
Bingham–Cosserat fluids [43] and localization in faults under shear
deformation in rock mechanics [39], for instance.
Discrete Element methods (DEM) have been introduced in [22] to model
crystalline materials and in [14] for applications to geotechnical problems.
Their use in granular materials and rock simulation is still widespread [36,
41]. Although DEM are able to represent accurately the behaviour of granular
materials, their use to compute elastic materials is more delicate especially
regarding the choice of microscopic material parameters. The macroscopic
parameters like Young modulus and Poisson ratio are typically recovered from
numerical experiments using the set of microscopic parameters [3, 23]. To
remedy this problem couplings of DEM with Finite Element Methods (FEM) have
been devised [31, 6]. Beyond DEM-FEM couplings, attempts to simulate
continuous materials with DEM have been proposed. In [3, 4], the authors used
a stress reconstruction inspired by statistical physics but the method suffers
from the non-convergence of the macroscopic parameters with respect to the
microscopic parameters. In [35] the authors derive a DEM method from a
Lagrange $P^{1}$ FEM but cannot simulate materials with $\nu\geq 0.3$. In
[32], the authors pose the basis of variational DEM by deriving forces from
potentials and link their method to Cosserat continua. Following this work,
[28] proposed a variational DEM that can use polyhedral meshes and which is a
full discretization of dynamic elasto-plasticity equations for a Cauchy
continua. However, in the process of approximating Cauchy continua, the
unknown rotations in elements and torque between elements from [32] had to be
removed. The present paper builds on the achievements of [28] and reintroduces
rotations by adding cellwise rotational degrees of freedom (dofs) to take into
account micro-rotations thus leading to the natural discretization of Cosserat
continua in lieu of Cauchy continua. Consequently it also greatly simplifies
the integration of rotations in a dynamic evolution compared to [32] where a
non-linear problem is solved at every time-step using an explicit RATTLE
scheme. Using the proposed method allows to use the usual tools of FEM applied
to a DEM and thus makes its use and analysis much easier. Also, the
restrictions on meshes are relaxed allowing to use simplicial meshes in lieu
of Voronoi meshes which are used in [32, 2] and are cumbersome to produce.
Cosserat elasticity is usually computed through a $P^{2}$-$P^{1}$ Lagrange
mixed element [37]. However, the coupling with a traditional DEM is not
obvious due to the location of the dofs in the methods (nodes for FEM and cell
barycentres for DEM). In the proposed method, the dofs are located at the cell
barycentres and thus a coupling with traditional DEM would be greatly
simplified. Designing such a coupling is left for future work. DEM can also be
very useful in computing crack propagation due to their natural ability to
represent discontinuous fields. In [29], the authors have built a variational
DEM derived from [28] to compute cracking in elastic Cauchy materials. DEM are
also very efficient in their ability to compute fragmentation [38]. The
proposed DEM can be considered as a first step towards its extension to
compute cracking in elastic Cosserat materials.
In the present method, displacement and rotational dofs are placed at the
barycentre of every cell and the Dirichlet boundary conditions are imposed
weakly similarly to discontinuous Galerkin methods [5]. The general elasto-
dynamic problem in a Cosserat continuum is presented in Section 2. The
discrete setting is then presented in Section 3 and alongside in Section 3.6,
the discrete strain-stress system derived from the continuous equations is
reinterpreted in a DEM fashion as a force-displacement system. In Section 3.9,
validation tests are performed to prove the robustness and the precision of
the chosen approach. In Section 4, the fully discrete system in space and time
is described to compute dynamic evolutions and two and three dimensional test
cases are presented proving the the correct integration of Cosserat
elasticity. Finally, Section 5 draws some conclusions and presents potential
subsequent work.
## 2 Governing equations
An elastic material occupying, in the reference configuration, the domain
$\Omega\subset\mathbb{R}^{d}$, where $d=2,3$, is considered to evolve
dynamically over the time-interval $(0,T)$, where $T>0$, under the action of
volumetric forces and boundary conditions. The strain regime is limited to
small strains and the material is supposed to have a micro-structure
responding to a Cosserat material law. The material is also supposed to be
isotropic and homogeneous. Consequently, the material law is restricted to
isotropic Cosserat elasticity in the following. The displacement field is
written $u\in\mathbb{R}^{3}$ and the rotation $R\in SO_{3}(\mathbb{R})$. Under
the small strain regime, the rotation $R$ can be mapped to a micro-rotation
$\varphi\in\mathbb{R}^{3}$ such that $\varphi=-\frac{1}{2}\epsilon:R$ and
$R=\mathbf{1}-\epsilon\cdot\varphi$, where $\mathbf{1}$ is the unit three
dimensional tensor and $\epsilon$ is a third-order tensor giving the signature
of a permutation $(i,j,k)$. Thus $\epsilon_{ijk}=1$ for an even permutation,
$-1$ for an odd permutation and $0$, otherwise. For instance
$\epsilon_{123}=1$, $\epsilon_{112}=0$ and $\epsilon_{132}=-1$.
###### Remark 1 (2d case).
In two space dimensions, the micro-rotation is just a scalar
$\varphi\in\mathbb{R}$ and $\epsilon$ is a matrix such that
$\epsilon=\begin{pmatrix}0&1\\\ -1&0\\\ \end{pmatrix}$.
The deformation tensor $e$ and the tension-curvature tensor $\kappa$ are
defined as
$\left\\{\begin{aligned} &e(u,\varphi)=\nabla u+\epsilon\cdot\varphi\\\
&\kappa(\varphi)=\nabla\varphi\end{aligned}\right.$ (1)
The force-stress tensor $\sigma$ and the couple-stress tensor $\mu$ are linked
to the strains $(e,\kappa)$ by
$\left\\{\begin{aligned} &\sigma(u,\varphi)=\mathbb{C}:e(u,\varphi)\\\
&\mu(\varphi)=\mathbb{D}:\kappa(\varphi),\end{aligned}\right.$ (2)
where $\mathbb{C}$ and $\mathbb{D}$ are fourth-order tensors translating the
material behaviour. Note that $\sigma(u,\varphi)$ is generally not symmetric
unlike in Cauchy continua. In the three dimensional case $d=3$, we write:
$\displaystyle\sigma(u,\varphi)=K\mathrm{tr}(e)(u,\varphi)\mathbf{1}+2G\left(\mathrm{Sym}(e)(u,\varphi)-\frac{1}{d}\mathrm{tr}(e)(u,\varphi)\mathbf{1}\right)+2G_{c}\mathrm{Skew}(e)(u,\varphi),$
(3)
$\displaystyle\mu(\varphi)=L\mathrm{tr}(\kappa)(\varphi)\mathbf{1}+2M\left(\mathrm{Sym}(\kappa)(\varphi)-\frac{1}{d}\mathrm{tr}(\kappa)(\varphi)\mathbf{1}\right)+2M_{c}\mathrm{Skew}(\kappa)(\varphi),$
where $K,G,G_{c},L,M$ and $M_{c}$ are elastic moduli and $\mathrm{Sym}$ gives
the symmetric part of a rank-two tensor whereas $\mathrm{Skew}$ gives its
skew-symmetric part. Under supplementary assumptions, the number of elastic
moduli can be reduced from six to four in three dimensions [24] but that is
not the path chosen in this paper.
###### Remark 2 (Length scale).
Following [20], a characteristic length $\ell$ can be defined in a Cosserat
elastic medium as
$\max_{ijkl}\mathbb{C}_{ijkl}=\ell^{2}\left(\max_{ijkl}\mathbb{D}_{ijkl}\right)$.
This length can be interpreted as the length of the microstructure of the
material. When $\ell\to 0$, the Cosserat material law can be homogenized into
a Cauchy law [20].
We introduce the volumic mass $\rho\in\mathbb{R}$ and the micro-inertia per
unit mass $I\in\mathbb{R}$. The dynamics equation in strong form write
$\left\\{\begin{aligned} &\mathrm{div}(\sigma(u,\varphi))+f=\rho\ddot{u}\\\
&\mathrm{div}(\mu(\varphi))-\epsilon:\sigma(u,\varphi)+\mathfrak{c}=\rho
I\ddot{\varphi}.\end{aligned}\right.$ (4)
Let $\partial\Omega=\partial\Omega_{N}\cup\partial\Omega_{D}$ be a partition
of the boundary of $\Omega$. By convention $\partial\Omega_{D}$ is a closed
set and $\partial\Omega_{N}$ is a relatively open set in $\partial\Omega$. The
boundary $\partial\Omega_{D}$ has imposed displacements and micro-rotations
$(u_{D},\varphi_{D})$, we thus enforce
$\left\\{\begin{aligned} &u=u_{D}\text{ on }\partial\Omega_{D},\\\
&\varphi=\varphi_{D}\text{ on }\partial\Omega_{D}.\end{aligned}\right.$ (5)
The normal and couple stresses $(g,m)$ are imposed on $\partial\Omega_{N}$,
that is, we enforce
$\left\\{\begin{aligned} &\sigma\cdot n=g\text{ on }\partial\Omega_{N},\\\
&\mu\cdot n=m\text{ on }\partial\Omega_{N}.\end{aligned}\right.$ (6)
To write a variational DEM, we write the dynamics equations in weak form.
Taking $(\tilde{u},\tilde{\varphi})$ as test functions, verifying homogeneous
Dirichlet boundary conditions on $\partial\Omega_{D}$
($\tilde{u}_{|\partial\Omega_{D}}=0=\tilde{\varphi}_{|\partial\Omega_{D}}$),
one has over $(0,T)$
$\int_{\Omega}\rho\ddot{u}\cdot v+\rho
I\ddot{\varphi}\cdot\psi+\int_{\Omega}e(u,\varphi):\mathbb{C}:e(v,\psi)+\kappa(\varphi):\mathbb{D}:\kappa(\psi)\\\
=\int_{\Omega}f\cdot v+\mathfrak{c}\cdot\psi+\int_{\partial\Omega_{N}}g\cdot
v+m\cdot\psi,$ (7)
while still imposing the Dirichlet boundary conditions of Equation (5). Note
that the bilinear form in the left-hand side of (7) is symmetric.
###### Remark 3.
Initial conditions on $(u,\varphi)$ and $(\dot{u},\dot{\varphi})$ need to be
specified to compute the solution of Equation 7.
## 3 Space semidiscretization
The domain $\Omega$ is discretized with a mesh $\mathcal{T}_{h}$ of size $h$
made of polyhedra with planar facets in three space dimensions or polygons
with straight edges in two space dimensions. We assume that $\Omega$ is itself
a polyhedron or a polygon so that the mesh covers $\Omega$ exactly, and we
also assume that the mesh is compatible with the partition of the boundary
$\partial\Omega$ into the Dirichlet and Neumann parts.
### 3.1 Degrees of freedom
Let $\mathcal{C}$ denote the set of mesh cells. Pairs of vector-valued
volumetric degrees of freedom (dofs) for a generic displacement field and a
generic micro-rotation field
$(v_{h},\psi_{h}):=(v_{c},\psi_{c})_{c\in\mathcal{C}}\in\mathbb{R}^{2d\\#(\mathcal{C})}$
are placed at the barycentre of every mesh cell $c\in\mathcal{C}$, where
$\\#(S)$ denotes the cardinality of any set $S$. Figure 1 illustrates the
position of the dofs in the mesh.
$\Omega$$\partial\Omega$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangledown$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$\blacktriangle$$(u_{c})_{c\in\mathcal{C}}$$\blacktriangledown$$(\phi_{c})_{c\in\mathcal{C}}$
Figure 1: Continuum $\Omega$ covered by a polyhedral mesh and vector-valued
degrees of freedom for the displacement.
Let $\mathcal{F}$ denote the set of mesh facets. We partition this set as
$\mathcal{F}=\mathcal{F}^{i}\cup\mathcal{F}^{b}$, where $\mathcal{F}^{i}$ is
the collection of the internal facets shared by two mesh cells and
$\mathcal{F}^{b}$ is the collection of the boundary facets sitting on the
boundary $\partial\Omega$ (such facets belong to the boundary of only one mesh
cell). The sets $\mathcal{F}^{b}_{N}$ and $\mathcal{F}^{b}_{D}$ are defined as
a partition of $\mathcal{F}^{b}$ such that $\forall
F\in\mathcal{F}^{b}_{N},F\subset\partial\Omega_{N}$ and $\forall
F\in\mathcal{F}^{b}_{D},F\subset\partial\Omega_{D}$.
### 3.2 Facet reconstructions
Using the cell dofs introduced above, we reconstruct a collection of
displacements and micro-rotations
$(v_{\mathcal{F}},\psi_{\mathcal{F}}):=(v_{F},\psi_{F})_{F\in\mathcal{F}}\in\mathbb{R}^{2d\\#(\mathcal{F})}$
on the mesh facets. The facet reconstruction operator is denoted $\mathcal{R}$
and we write
$(v_{\mathcal{F}},\psi_{\mathcal{F}}):=\left(\mathcal{R}(v_{h}),\mathcal{R}(\psi_{h})\right).$
(8)
The reconstruction operator $\mathcal{R}$ is constructed in the same way as in
the finite volume methods studied in [18, Sec. 2.2] and the variational DEM
developed in [28]. For a given facet $F\in\mathcal{F}$, we select neighbouring
cells collected in a subset denoted $\mathcal{C}_{F}$, as well as coefficients
$(\alpha_{F}^{c})_{c\in\mathcal{C}_{F}}$ and we set
$\left\\{\begin{aligned}
&\mathcal{R}_{F}(v_{h}):=\sum_{c\in\mathcal{C}_{F}}\alpha_{F}^{c}v_{c},\qquad\forall
v_{h}\in V_{h},\\\
&\mathcal{R}_{F}(\psi_{h}):=\sum_{c\in\mathcal{C}_{F}}\alpha_{F}^{c}\psi_{c},\qquad\forall\varphi_{h}\in
V_{h}.\end{aligned}\right.$ (9)
The reconstruction is based on barycentric coordinates. The coefficients
$\alpha_{F}^{c}$ are chosen as the barycentric coordinates of the facet
barycentre $\mathbf{x}_{F}$ in terms of the location of the barycenters of the
cells in $\mathcal{C}_{F}$. For any facet $F\in\mathcal{F}$, the set
$\mathcal{C}_{F}$ is constructed so as to contain exactly $(d+1)$ points
forming the vertices of a non-degenerate simplex. Thus, the barycentric
coefficients $(\alpha^{c}_{F})_{c\in\mathcal{C}_{F}}$ are computed by solving
the linear system:
$\left\\{\begin{aligned}
&\sum_{c\in\mathcal{C}_{F}}{\alpha_{F}^{c}}=1,\qquad&\forall
F\in\mathcal{F},\\\
&\sum_{c\in\mathcal{C}_{F}}{\alpha_{F}^{c}\mathbf{x}_{c}}=\mathbf{x}_{F},\qquad&\forall
F\in\mathcal{F},\\\ \end{aligned}\right.$ (10)
where $\mathbf{x}_{c}$ is the position of the barycenter of the cell $c$. An
algorithm is presented thereafter to explain the selection of the neighbouring
dofs in $\mathcal{C}_{F}$. This algorithm has to be viewed more as a proof-of-
concept than as an optimized algorithm. For a more involved algorithm, see
[28]. We observe that this algorithm is only used in a preprocessing stage of
the computations. For a given facet $F\in\mathcal{F}$,
1. 1.
Assemble in a set $\mathcal{N}_{F}$ the cell or cells containing the facet
$F$. Then, add to $\mathcal{N}_{F}$ the cells sharing a facet with the cells
already in $\mathcal{N}_{F}$. Repeat the last operation once.
2. 2.
Select a subset $\mathcal{C}_{F}$ of $\mathcal{N}_{F}$ with exactly $(d+1)$
elements and whose cell barycenters form a non-degenerate simplex (tetrahedron
in 3d and triangle in 2d).
This algorithm ensures that the dofs selected for the reconstruction in
Equation (9) remain $\mathcal{O}(h)$ close to the facet $F$.
###### Remark 4.
The last operation of step 1 described above which consists in enlarging the
set $\mathcal{N}_{F}$ is generally not necessary in two space dimensions but
it becomes necessary in three space dimensions as some locally complicated
mesh geometries can cause all barycenters of cells in $\mathcal{N}_{F}$ to
form a degenerate simplex. In that case, step 2 cannot be performed correctly.
###### Remark 5 (Influence of the choice of $\mathcal{C}_{F}$).
The choice of the elements in $\mathcal{C}_{F}$ for $F\in\mathcal{F}$ has an
impact on the eigenvalues of the rigidity matrix and thus on its conditioning.
This impact is explored in [28] regarding the CFL condition in explicit
dynamic computations.
### 3.3 Gradient reconstruction
Using the reconstructed facet displacements and micro-rotations and a discrete
Stokes formula, it is possible to devise a discrete $\mathbb{R}^{d\times
d}$-valued piecewise-constant gradient field for the displacement and the
micro-rotation that we write
$\mathcal{G}_{\mathcal{C}}(v_{\mathcal{F}}):=(\mathcal{G}_{c}(v_{\mathcal{F}}))_{c\in\mathcal{C}}\in\mathbb{R}^{d^{2}\\#(\mathcal{C})}$
and
$\mathcal{G}_{\mathcal{C}}(\psi_{\mathcal{F}}):=(\mathcal{G}_{c}(\psi_{\mathcal{F}}))_{c\in\mathcal{C}}\in\mathbb{R}^{d^{2}\\#(\mathcal{C})}$.
Specifically we set in every mesh cell $c\in\mathcal{C}$,
$\left\\{\begin{aligned} &\mathcal{G}_{c}(v_{\mathcal{F}}):=\sum_{F\in\partial
c}\frac{|F|}{|c|}v_{F}\otimes n_{F,c},\qquad\forall
v_{\mathcal{F}}\in\mathbb{R}^{d\\#(\mathcal{F})},\\\
&\mathcal{G}_{c}(\psi_{\mathcal{F}}):=\sum_{F\in\partial
c}\frac{|F|}{|c|}\psi_{F}\otimes
n_{F,c},\qquad\forall\psi_{\mathcal{F}}\in\mathbb{R}^{d\\#(\mathcal{F})},\end{aligned}\right.$
(11)
where the summation is over the facets $F$ of $c$ and $n_{F,c}$ is the outward
normal to $c$ on $F$. Consequently, the strains are defined for
$c\in\mathcal{C}$ as
$\left\\{\begin{aligned}
&e_{c}(v_{h}):=\mathcal{G}_{c}(v_{h})+\epsilon\cdot\psi_{c}\in\mathbb{R}^{d\times
d},\\\ &\kappa_{c}(\psi_{h}):=\mathcal{G}_{c}(\psi_{h})\in\mathbb{R}^{d\times
d},\end{aligned}\right.$ (12)
where $\mathcal{G}_{c}(v_{h}):=\mathcal{G}_{c}(\mathcal{R}(v_{h}))$ and
$\mathcal{G}_{c}(\psi_{h}):=\mathcal{G}_{c}(\mathcal{R}(\psi_{h}))$.
Consequently, the discrete bilinear form of elastic energies writes
$a_{\text{elas}}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=\int_{\Omega}\left(e_{h}\left((u_{h},\varphi_{h})\right):\mathbb{C}:e_{h}\left((v_{h},\psi_{h})\right)+\kappa_{h}(\varphi_{h}):\mathbb{D}:\kappa_{h}(\psi_{h})\right).$
(13)
Finally, we define two additional reconstructions which will be used to impose
the Dirichlet boundary conditions. The reconstructions are written
$\mathfrak{R}$ and consist in a collection of cellwise nonconforming $P^{1}$
polynomials defined for all $c\in\mathcal{C}$ by
$\left\\{\begin{aligned}
&\mathfrak{R}_{c}(v_{h})(\mathbf{x}):=v_{c}+\mathcal{G}_{c}(v_{h})\cdot(\mathbf{x}-\mathbf{x}_{c}),\\\
&\mathfrak{R}_{c}(\psi_{h})(\mathbf{x}):=v_{c}+\mathcal{G}_{c}(\psi_{h})\cdot(\mathbf{x}-\mathbf{x}_{c}).\end{aligned}\right.$
(14)
where $\mathbf{x}\in c$ and $\mathbf{x}_{c}$ is the barycentre of the cell
$c$.
### 3.4 Mass bilinear form
In the DEM spirit, the reconstruction of functions is chosen as constant in
each cell so as to obtain a diagonal mass matrix. The mass bilinear form is
thus defined as
$m_{h}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=\sum_{c\in\mathcal{C}}\rho|c|\left(u_{c}\cdot
v_{c}+I\varphi_{c}\cdot\psi_{c}\right).$ (15)
### 3.5 Discrete problem
The discrete problem is defined as a lowest-order discontinuous Galerkin
method similar to [18] and [16]. Consequently, penalty terms will be added to
the discrete formulation of Equation (1) for two reasons. The first is that
the gradient reconstruction of Equation (11) cannot by itself control
$(u_{h},\phi_{h})$ and thus a least-square penalty term will be added to the
formulation to ensure the well-posedness of the problem. More details can be
found in [18] and [16]. The second is that the Dirichlet boundary conditions
will not be imposed strongly but weakly through a non-symmetric Nitsche
penalty acting on the boundary facets in $\partial\Omega_{D}$.
#### 3.5.1 Least-square penalty
For an interior facet $F\in\mathcal{F}^{i}$, writing $c_{F,-}$ and $c_{F,+}$
the two mesh cells sharing $F$, that is, $F=\partial c_{F,-}\cap\partial
c_{F,+}$, and orienting $F$ by the unit normal vector $n_{F}$ pointing from
$c_{F,-}$ to $c_{F,+}$, we define
$[\mathfrak{R}(v_{h})]_{F}:=\mathfrak{R}_{c_{F,-}}(v_{h})-\mathfrak{R}_{c_{F,+}}(v_{h}).$
(16)
$[\mathfrak{R}(\psi_{h})]_{F}$ is defined similarly. The interior penalty term
is defined as
$a_{\text{inner\\_pen}}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=\sum_{F\in\mathcal{F}^{i}}\frac{1}{h_{F}}\int_{F}([\mathfrak{R}(u_{h})]_{F}\otimes
n_{F}):\mathbb{C}:([\mathfrak{R}(v_{h})]_{F}\otimes n_{F})\\\
+([\mathfrak{R}(\varphi_{h})]_{F}\otimes
n_{F}):\mathbb{D}:([\mathfrak{R}(\psi_{h})]_{F}\otimes n_{F}).$ (17)
#### 3.5.2 Non-symmetric Nitsche penalty
As the Dirichlet boundary conditions are imposed weakly, the discrete test
functions do not verify $\mathcal{R}(v_{h})=0$ and $\mathcal{R}(\psi_{h})=0$
on $\partial\Omega_{D}$. Thus the following term, called the consistency term,
coming from the integration by parts, leading from (4) to (7), must be taken
into account
$a_{\text{con}}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=-\int_{\partial\Omega_{D}}(\sigma_{h}(u_{h},\varphi_{h})\cdot
n)\cdot\mathcal{R}(v_{h})+(\mu_{h}(\varphi_{h})\cdot
n)\cdot\mathcal{R}(\psi_{h}).$ (18)
Following ideas from [12, 10], we introduce the following non-symmetric term,
to obtain a method that is stable without having to add a least-square penalty
term on $\partial\Omega_{D}$,
$a_{\text{nsym}}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=\int_{\partial\Omega_{D}}(\sigma_{h}(v_{h},\psi_{h})\cdot
n)\cdot\mathcal{R}(u_{h})+(\mu_{h}(\psi_{h})\cdot
n)\cdot\mathcal{R}(\varphi_{h}).$ (19)
A corresponding linear form is added to compensate the term in (19) when the
Dirichlet boundary conditions are verified exactly,
$l_{\text{nsym}}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right):=\int_{\partial\Omega_{D}}(\sigma_{h}(v_{h},\psi_{h})\cdot
n)\cdot u_{D}+(\mu_{h}(\psi_{h})\cdot n)\cdot\varphi_{D}.$ (20)
#### 3.5.3 Discrete problem
The bilinear form $a_{h}$ is defined as
$a_{h}:=a_{\text{elas}}+a_{\text{inner\\_pen}}+a_{\text{con}}+a_{\text{nsym}}$.
The discrete problem may then be written: search for $(u_{h},\varphi_{h})$
such that for all $(v_{h},\psi_{h})$, one has
$m_{h}\left((\ddot{u}_{h},\ddot{\varphi}_{h});(v_{h},\psi_{h})\right)+a_{h}\left((u_{h},\varphi_{h});(v_{h},\psi_{h})\right)=l_{h}(v_{h},\psi_{h})+l_{D}(v_{h},\psi_{h})+l_{\text{nsym}}(v_{h},\psi_{h}),$
(21)
where $l_{h}$ is the linear form that takes into account Neumann boundary
conditions and volumic loads. It is defined as
$l_{h}(v_{h},\psi_{h}):=\sum_{c\in\mathcal{C}}\left(\int_{c}f\right)\cdot
v_{c}+\left(\int_{c}\mathfrak{c}\right)\cdot\psi_{c}+\sum_{F\in\mathcal{F}^{b}_{N}}\left(\int_{F}g\right)\cdot
v_{c_{F}}+\left(\int_{F}m\right)\cdot\psi_{c_{F}}.$ (22)
### 3.6 Interpretation as a DEM
Traditional DEM are force-displacement systems in the sense that the
deformation of the domain $\Omega$ is computed through the displacement of
particles interacting through nearest-neighbours forces [23]. A major
difference appears with the proposed DEM in which the interactions are not
nearest-neighbours but have a larger stencil due to the facet and gradient
reconstructions (9) and (11).
Reference particleNeighbouring particleNeighbour to Neighbour (a)
For facetreconstruction (b)
Figure 2: DEM interpretation: left: traditional DEM. right: proposed
variational DEM.
As can be seen in Figure 2 on the left, in a traditional DEM, a particle
(represented in red) interacts with its closest-neighbours (in blue) but not
the neighbours of its neighbours (in green). On the contrary, such
interactions are present with the the proposed method as can be seen in Figure
2 on the right. In traditional DEM, the force and torque between two particles
can be parametrized, for instance, by elastic springs in tension, shear and
torque, or by beam elements [23]. The parameters of these springs or beams are
called microscopic parameters. Unfortunately, calibrating the microscopic
parameters to simulate a given continuous material is difficult [24]. As the
elastic bilinear form (13) is written in terms of strains and stresses, as in
continuous materials, forces and torques are not explicit. However, it is
possible to rewrite (13) so as to extract forces between neighbouring cells
considered as discrete elements. The major advantage being that the forces
retrieved are parametrized only by the continuous material parameters and not
by microscopic parameters. Let us do so in the following.
The average value in an inner facet $F\in\mathcal{F}^{i}$ of a quantity $a$ is
defined as $\\{a\\}_{F}:=\frac{1}{2}(a_{c_{F,-}}+a_{c_{F,+}})$. Rewriting
Equation (13) and neglecting second order terms, one has
$\displaystyle-a_{\text{elas}}((u_{h},\varphi_{h});(v_{h},\psi_{h}))$
$\displaystyle=$
$\displaystyle\sum_{F\in\mathcal{F}^{i}}|F|(\\{\sigma_{h}(u_{h},\varphi_{h})\\}_{F}\cdot
n_{F})\cdot[v_{h}]_{F}$ (23) $\displaystyle+$
$\displaystyle\sum_{F\in\mathcal{F}^{i}}|F|(\\{\mu_{h}(\varphi_{h})\\}_{F}\cdot
n_{F})\cdot[\psi_{h}]_{F}$ $\displaystyle+$
$\displaystyle\sum_{c\in\mathcal{C}}|c|(\epsilon:\sigma_{c}(u_{h},\varphi_{h}))\cdot\psi_{c}$
$\displaystyle+$
$\displaystyle\sum_{F\in\mathcal{F}^{b}}|F|(\sigma_{c_{F}}(u_{h},\varphi_{h})\cdot
n_{F})\cdot(v_{c_{F}}-\mathcal{R}_{F}(v_{h}))$ $\displaystyle+$
$\displaystyle\sum_{F\in\mathcal{F}^{b}}|F|(\mu_{c_{F}}(\varphi_{h})\cdot
n_{F})\cdot(\psi_{c_{F}}-\mathcal{R}_{F}(\psi_{h}))$ $\displaystyle+$
$\displaystyle\mathcal{O}(h^{2}),$
where $c_{F}$ designates the unique cell containing a boundary facet
$F\in\mathcal{F}^{b}$,
$\sigma_{h}(u_{h},\varphi_{c}):=\mathbb{C}:e_{h}(u_{h},\varphi_{c})$,
$\mu_{h}(\varphi_{h}):=\mathbb{D}:\kappa_{h}(\varphi_{h})$ and for an inner
facet $F$, $[v_{h}]_{F}:=v_{c_{F,-}}-v_{c_{F,+}}$ and
$[\psi_{h}]_{F}:=\psi_{c_{F,-}}-\psi_{c_{F,+}}$. The principle of action-
reaction (or Newton’s third law) can be read in the first two lines of the
equation through the action of jump terms. The third line represents the work
of the momentum coming from the stresses. The fourth and fifth lines represent
the work of the internal forces and momenta but related to boundary facets.
One can write the dynamics equations of an interior cell $c\in\mathcal{C}$ (or
discrete element), with no facet on the boundary, as follows
$\left\\{\begin{aligned}
&\rho|c|\ddot{u}_{c}\simeq\sum_{F\in\mathcal{F}^{i},F\subset\partial
c}\iota_{c,F}|F|\\{\sigma_{h}(u_{h},\varphi_{h})\\}_{F}\cdot
n_{F}+\int_{c}f,\\\
&\rho|c|I\cdot\ddot{\varphi}_{c}\simeq|c|\epsilon:\sigma_{c}(u_{h},\varphi_{h})+\sum_{F\in\mathcal{F}^{i},F\subset\partial
c}\iota_{c,F}|F|\\{\mu_{h}(\varphi_{h})\\}_{F}\cdot
n_{F}+\int_{c}\mathfrak{c},\\\ \end{aligned}\right.$ (24)
up to second order terms and penalty terms and where $\iota_{c,F}=1$ if
$c=c_{F,-}$ and $\iota_{c,F}=-1$ if $c=c_{F,+}$. Each facet thus represents a
link between two discrete elements and the forces and momenta are average
quantities computed from cell-wise stress and momentum reconstructions. To
obtain similar equations for a cell having a facet on $\partial\Omega$, one
can refer to [28].
### 3.7 Implementation
The method has been implemented in Python and is available at
https://github.com/marazzaf/DEM_cosserat.git. A finite element implementation
available in [42] has been used as a foundation for the implementation of the
previous method. FEniCS [26, 27] is used to handle meshes, compute facet
reconstructions and assemble matrices. PETSc [15, 7] is used for matrix
storage, matrix operations and as a linear solver. Because FEniCS only
supports simplicial meshes, the meshes used with the implementation are only
triangular or tetrahedral. However, the method supports general polyhedra.
### 3.8 Validation test cases
The validation test cases are two-dimensional. Cosserat elasticity is
characterized only by four parameters in two space dimensions. Following [37],
we chose the material parameters to be $G$ the shear modulus, $\ell$ the
characteristic length of the microstructure, $a$ (which measures the ratio
$G_{c}$ over $G$) and $\nu$ the Poisson ratio. Let
$\mathfrak{a}=\frac{2(1-\nu)}{1-2\nu}$ and $\mathfrak{b}=\frac{2\nu}{1-2\nu}$.
The material law then writes
$\left(\begin{array}[]{c}\sigma_{xx}\\\ \sigma_{yy}\\\ \sigma_{xy}\\\
\sigma_{yx}\end{array}\right)=G\begin{pmatrix}&\mathfrak{a}&\mathfrak{b}&0&0\\\
&\mathfrak{b}&\mathfrak{a}&0&0\\\ &0&0&1+a&1-a\\\ &0&0&1-a&1+a\\\
\end{pmatrix}\cdot\left(\begin{array}[]{c}e_{xx}\\\ e_{yy}\\\ e_{xy}\\\
e_{yx}\\\ \end{array}\right),$ (25)
and
$\mu=4G\ell^{2}\kappa.$ (26)
The following test has been used in [37] to validate the P2-P1 finite element
in two space dimension. It consists in three computations on the rectangular
domain $[-0.12,0.12]\times[0,0.12]$ with material parameters $G=10^{3}$Pa,
$\ell=0.1$m, $a=0.5$ and $\nu=0.25$. Estimates of the condition number of the
rigidity matrices with and without the terms corresponding to equation (17)
are also computed. The condition number of the rigidity matrix is approximated
using [19] implemented in scipy.sparse.linalg which is a module of Scipy [1].
A single structured mesh containing $2,500$ elements is used to compute the
condition numbers for the three test cases. It corresponds to $7,500$ dofs for
the proposed DEM.
#### 3.8.1 First patch test
The solution to the first test is
$\left\\{\begin{aligned} &u_{x}(x,y)=\frac{1}{G}(x+\frac{y}{2}),\\\
&u_{y}(x,y)=\frac{1}{G}(x+y),\\\
&\varphi(x,y)=\frac{1}{4G}.\end{aligned}\right.$ (27)
Dirichlet boundary conditions are imposed on the entire boundary. The volumic
load is $f_{x}=f_{y}=\mathfrak{c}=0$. The condition number for the DEM with
the penalty term (17) is $83$ and $51$ without it. Table 1 presents the
analytical results for the stresses and compares them to the minimum and
maximum computed value at every dof of the mesh. The maximum relative error is
also given.
stress | $\sigma_{xx}$ | $\sigma_{yy}$ | $\sigma_{xy}$ | $\sigma_{yx}$ | $\mu_{x}$ | $\mu_{y}$
---|---|---|---|---|---|---
analytical | $4$ | $4$ | $1.5$ | $1.5$ | $0$ | $0$
min computed | $4.00$ | $4.00$ | $1.50$ | $1.50$ | ME | ME
max computed | $4.00$ | $4.00$ | $1.50$ | $1.50$ | ME | ME
max relative error | $2.63\cdot 10^{-11}\%$ | $6.21\cdot 10^{-11}\%$ | $4.15\cdot 10^{-11}\%$ | $1.04\cdot 10^{-10}\%$ | |
Table 1: First patch test: Analytical solution, computed stresses and relative
error.
ME means machine error and the relative error on the momenta is not given as
the expected value is zero.
#### 3.8.2 Second patch test
The solution to the second test is
$\left\\{\begin{aligned} &u_{x}(x,y)=\frac{1}{G}(x+\frac{y}{2}),\\\
&u_{y}(x,y)=\frac{1}{G}(x+y),\\\
&\varphi(x,y)=-\frac{1}{4G}.\end{aligned}\right.$ (28)
Dirichlet boundary conditions are imposed on the entire boundary. The volumic
load is $f_{x}=f_{y}=0$ and $\mathfrak{c}=1$. The condition number with the
penalty term (17) is $31$ and $28$ without it. Table 2 presents the analytical
and computed results.
stress | $\sigma_{xx}$ | $\sigma_{yy}$ | $\sigma_{xy}$ | $\sigma_{yx}$ | $\mu_{x}$ | $\mu_{y}$
---|---|---|---|---|---|---
analytical | $4$ | $4$ | $1$ | $2$ | $0$ | $0$
min computed | $3.99$ | $3.98$ | $0.995$ | $1.96$ | $-4.58\cdot 10^{-2}$ | $-5.36\cdot 10^{-2}$
max computed | $4.01$ | $4.01$ | $1.03$ | $2.01$ | $4.23\cdot 10^{-2}$ | $5.42\cdot 10^{-2}$
max relative error | $1.58\%$ | $1.53\%$ | $3.51\%$ | $2.35\%$ | |
Table 2: Second patch test: Analytical solution, computed stresses and
relative error.
#### 3.8.3 Third patch test
The solution to the third test is
$\left\\{\begin{aligned} &u_{x}(x,y)=\frac{1}{G}(x+\frac{y}{2}),\\\
&u_{y}(x,y)=\frac{1}{G}(x+y),\\\
&\varphi(x,y)=\frac{1}{G}(\frac{1}{4}-x+y)).\end{aligned}\right.$ (29)
Dirichlet boundary conditions are imposed on the entire boundary. The volumic
load is $f_{x}=f_{y}=1$ and $\mathfrak{c}=2(x-y)$. The condition number with
the penalty term (17) is $112$ and $124$ without it. Table 3 presents the
analytical and computed results.
stress | $\sigma_{xx}$ | $\sigma_{yy}$ | $\sigma_{xy}$ | $\sigma_{yx}$ | $\mu_{x}$ | $\mu_{y}$
---|---|---|---|---|---|---
analytical | $4$ | $4$ | $1.5-x+y$ | $1.5+x-y$ | $-4\ell^{2}$ | $4\ell^{2}$
min computed | $3.94$ | $3.94$ | | | $-4.80\cdot 10^{-2}$ | $3.63\cdot 10^{-2}$
max computed | $4.06$ | $4.06$ | | | $-3.75\cdot 10^{-2}$ | $4.73\cdot 10^{-2}$
max relative error | $1.58\%$ | $1.53\%$ | $3.51\%$ | $2.35\%$ | $6.22\%$ | $9.29\%$
Table 3: Third patch test: Analytical and computed stresses and relative
error.
The minimum and maximum computed values for $\sigma_{xy}$ and $\sigma_{yx}$
are not given for this test because they are irrelevant, as they vary with the
position in the domain.
#### 3.8.4 Results
Results for finer meshes are not shown here, but the errors decrease with mesh
refinement. The results provided above validate the correct imposition of
Dirichlet boundary conditions with the proposed method. Also, one can notice
that the condition number of the rigidity matrix remains of the same order
when adding the inner penalty term (17).
### 3.9 Numerical results
#### 3.9.1 Square plate with a hole
This test case has been inspired by [37]. The domain is a square plate of
length $32.4$mm with a hole of radius $r$ in its center. For symmetry reasons,
only a quarter of the plate is meshed. A traction with $\Sigma=1N/m^{2}$ is
imposed on the top surface. Figure 3 shows the boundary conditions.
$\sigma\cdot n=0$$\mu\cdot n=0$$u\cdot n=0$$\varphi\cdot n=0$$u\cdot
n=0$$\varphi\cdot n=0$$\sigma\cdot n=\Sigma n$$\mu\cdot n=0$$\sigma\cdot
n=0$$\mu\cdot n=0$ Figure 3: Square plate with a hole: problem setup.
The material parameters are as follows: $G=10^{3}$Pa and $\nu=0.3$. $\ell$ and
$a$ as well as the radius $r$ take several values as presented in the
following. The meshes in Figure 4 are used for the test cases.
(a) (b)
Figure 4: Plate with a hole: left: mesh for first and second test $r=0.216$mm.
right: mesh for third test $r=0.864$mm.
The mesh for the first two tests is made of $225,816$ dofs and the mesh for
the third test is made of $210,867$ dofs. For the first test case, $r=0.216$mm
and $\frac{r}{\ell}=1.063$ and $a$ varies as in Table 4 which gives the
maximal stress at the boundary of the hole as well as the error.
$a$ | analytical | computed | error (%)
---|---|---|---
$0$ | $3.000$ | $2.998$ | $0.1\%$
$0.0667$ | $2.849$ | $2.848$ | $0.0\%$
$0.3333$ | $2.555$ | $2.555$ | $0.0\%$
$1.2857$ | $2.287$ | $2.287$ | $0.0\%$
$4.2632$ | $2.158$ | $2.157$ | $0.0\%$
Table 4: Square plate with a hole: first test, parameter $a$, analytical
maximal stress, computed maximal stress and error.
For the second test case, $r=0.216$mm and $\frac{r}{\ell}=10.63$ and $a$
varies as in Table 5 which gives the maximal stress at the boundary of the
hole as well as the error.
$a$ | analytical | computed | error (%)
---|---|---|---
$0$ | $3.000$ | $2.998$ | $0.1\%$
$0.0667$ | $2.956$ | $2.955$ | $0.0\%$
$0.3333$ | $2.935$ | $2.936$ | $0.0\%$
$1.2857$ | $2.927$ | $2.929$ | $0.1\%$
$4.2632$ | $2.923$ | $2.925$ | $0.1\%$
Table 5: Square plate with a hole: second test, parameter $a$, analytical
maximal stress, computed maximal stress and error.
For the third test case, $r=0.864$mm and $a=0.3333$mm and $\ell$ varies as in
Table 6 which gives the maximal stress at the boundary of the hole as well as
the error.
$r/\ell$ | analytical | computed | error (%)
---|---|---|---
$1.0$ | $2.549$ | $2.566$ | $0.7\%$
$2.0$ | $2.641$ | $2.660$ | $0.7\%$
$3.0$ | $2.719$ | $2.740$ | $0.8\%$
$4.0$ | $2.779$ | $2.801$ | $0.8\%$
$6.0$ | $2.857$ | $2.881$ | $0.8\%$
$8.0$ | $2.902$ | $2.927$ | $0.9\%$
$10.0$ | $2.929$ | $2.955$ | $0.9\%$
Table 6: Square plate with a hole: third test, parameter $\ell$, analytical
maximal stress, computed maximal stress and error.
The results from the three tests show that the proposed method is able to
reproduce stress concentration with a satisfactory accuracy.
#### 3.9.2 Boundary layer effect
This test case is found in [45, p. 342], for the analytical solution and in
[40], for a numerical implementation. The domain is a square of size $h=1$mm.
The material parameters are $\nu=0$, $G=10$GPa, $a=2$ and $\ell=5\cdot
10^{-2}$mm using the material laws (25) and (26). The boundary conditions are
$u_{1}=0$ and $\varphi=0$ on the bottom surface, $u_{1}=-0.1$m and
$\varphi=0.01\cdot h$ on the top surface. Mirror boundary conditions are
imposed on the left and right boundaries. The mesh is a structured triangular
mesh made of 50 elements along the $x_{2}$-direction and 10 along the
$x_{1}$-direction thus leading to $6,000$ dofs. Figure 5 shows the computed
values of $u_{1}$ and $\varphi$ depending on the $x_{2}$ coordinate compared
to the analytical solution available in [40].
(a) (b)
Figure 5: Boundary layer effect: left: displacement in the $x_{1}$ direction.
right: rotation.
A similar computation is performed with mixed Lagrange $P^{2}$-$P^{1}$ FE on a
mesh containing $6,003$ dofs. The maximum relative error on the computed
values ploted in Figure 5 is $9\%$ for the rotation and the vertical
displacement for the DEM. It is of $1\%$ for the vertical displacement and the
rotation with the FE computation.
## 4 Fully discrete scheme
A Crank–Nicholson time-integration [8] is use to integrate in time the system
of ODEs (21).
### 4.1 Space-time discrete system
The time step is chosen as $\Delta t=\frac{T}{2000}$. The time-interval
$(0,T)$ is discretized in $\\{0=t_{0},\dots,t^{n},\dots,t^{N}=T\\}$. For all
$n=1,\ldots,N$, we compute the discrete displacement and rotation $u_{h}^{n}$
and $\varphi_{h}^{n}$ and the discrete velocity $\dot{u}_{h}^{n}$ and rotation
rate $\dot{\varphi}_{h}^{n}$ as well as the corresponding accelerations
$\ddot{u}_{h}^{n}$ and $\ddot{\varphi}_{h}^{n}$. As, the homogeneous Dirichlet
boundary conditions are imposed weakly in the DEM through (18) and (19), a
damping term is added to impose homogeneous Dirichlet boundary conditions on
the velocities:
$c_{h}((\dot{u}_{h}^{n},\dot{\varphi}_{h}^{n});(v_{h},\psi_{h}))):=\sum_{F\in\mathcal{F}^{b}_{D}}\frac{4G}{h_{F}}\int_{F}(\dot{u}^{n}_{h}\cdot
v_{h}+\ell^{2}\dot{\varphi}^{n}_{h}\cdot\psi_{h}),\quad\forall(v_{h},\psi_{h}).$
(30)
The fully discrete scheme reads as follows: for all $n=1,\ldots,N$, given
$(u_{h}^{n},\varphi_{h}^{n})$, $(\dot{u}_{h}^{n},\dot{\varphi}_{h}^{n})$ and
$(\ddot{u}_{h}^{n},\ddot{\varphi}_{h}^{n})$, compute
$(u_{h}^{n+1},\varphi_{h}^{n+1})$,
$(\dot{u}_{h}^{n+1},\dot{\varphi}_{h}^{n+1})$ and
$(\ddot{u}_{h}^{n+1},\ddot{\varphi}_{h}^{n+1})$ such that
$\left\\{\begin{aligned} &u_{h}^{n+1}=u_{h}^{n}+\Delta
t\dot{u}_{h}^{n}+\frac{\Delta
t^{2}}{2}\frac{\ddot{u}_{h}^{n}+\ddot{u}_{h}^{n+1}}{2},\quad\varphi_{h}^{n+1}=\varphi_{h}^{n}+\Delta
t\dot{\varphi}_{h}^{n}+\frac{\Delta
t^{2}}{2}\frac{\ddot{\varphi}_{h}^{n}+\ddot{\varphi}_{h}^{n+1}}{2},\\\
&\dot{u}_{h}^{n+1}=\dot{u}_{h}^{n}+\Delta
t\frac{\ddot{u}_{h}^{n}+\ddot{u}_{h}^{n+1}}{2},\quad\dot{\varphi}_{h}^{n+1}=\dot{\varphi}_{h}^{n}+\Delta
t\frac{\ddot{\varphi}_{h}^{n}+\ddot{\varphi}_{h}^{n+1}}{2},\\\
&\ddot{u}_{h}^{n+1}=\frac{4}{\Delta t^{2}}(u_{h}^{n+1}-u_{h}^{n}-\Delta
t\dot{u}_{h}^{n})-\ddot{u}_{h}^{n},\quad\ddot{\varphi}_{h}^{n+1}=\frac{4}{\Delta
t^{2}}(\varphi_{h}^{n+1}-\varphi_{h}^{n}-\Delta
t\dot{\varphi}_{h}^{n})-\ddot{\varphi}_{h}^{n},\\\
&m_{h}((\ddot{u}_{h}^{n+1},\ddot{\varphi}_{h}^{n+1});(v_{h},\psi_{h}))+c_{h}((\dot{u}_{h}^{n},\dot{\varphi}_{h}^{n});(v_{h},\psi_{h})))\\\
+&a_{h}((u_{h}^{n+1},\phi_{h}^{n+1});(v_{h},\psi_{h})))=L_{h}(t^{n+1};(v_{h},\psi_{h})),\quad\forall(v_{h},\psi_{h}),\\\
\end{aligned}\right.$ (31)
where $m_{h}$ is defined in (15) and $L_{h}$ is the discrete load linear form
(the right-hand side of (21)). The initial displacement and rotation
$(u_{h}^{0},\varphi_{h}^{0})$ and the initial velocity and rotation rate
$(\dot{u}_{h}^{0},\dot{\varphi}_{h}^{0})$ are evaluated by using the values of
the prescribed initial displacements and velocities at the cell barycentres.
Note that, with respect to the DEM developed in [32], there is no need to
solve a nonlinear problem to compute the rotation at each time-step, which
greatly improves numerical efficiency.
### 4.2 Numerical results
#### 4.2.1 Beam in dynamic flexion
This test case has been inspired by a similar from [9]. This test case
consists of computing the oscillations of a beam of length $\mathcal{L}=1$mm
with a square section of $0.04\times 0.04$mm2. The simulation time is
$T=6.3\cdot 10^{-5}$s. The beam is clamped at one end, it is loaded by a
uniform vertical traction $\sigma\cdot n=g(t)$ at the other end, and the four
remaining lateral faces are stress free ($\sigma\cdot n=0$ and $\mu\cdot
n=0$). The load term $g(t)$ is defined as
$g(t):=\begin{cases}-\frac{tE\cdot 10^{-6}}{T_{c}}e_{x}&\text{for $0\leq t\leq
T_{c}$},\\\ 0&\text{for $T_{c}\leq t\leq T$},\end{cases}$ (32)
where $T_{c}=3.2\cdot 10^{-8}$s. Figure 6 displays the problem setup.
$\mathcal{L}$$g(t)$$u=0$$\varphi=0$ Figure 6: Beam in dynamic flexion: problem
setup.
The material parameters have been taken from [39]. The bulk modulus is
$K=16.67$GPa and the shear moduli are $G=10$GPa and $G_{c}=5$GPa. The
characteristic size of the microstructure is taken as
$\ell=\frac{\mathcal{L}}{100}$. We also take $L=G\ell^{2}$ and
$M=M_{c}=\frac{5}{2}Gl^{2}$. The density is
$\rho=2500\mathrm{kg{\cdot}m^{-3}}$ and the inertia is taken as
$I=\frac{2}{5}\ell^{2}$ following [39], with the assumption that the micro-
structure of the material is composed of balls. The reference solution is a
$P^{2}$-$P^{1}$ Lagrange FEM coupled to a Crank–Nicholson time-integration
[8]. The DEM is integrated according to Equation (31). The DEM computation
contains $23,040$ dofs and the FEM computation $3,642$ dofs. Figure 7 shows
the displacement of the beam tip $u_{h}(\mathcal{L},0.05,0.)\cdot e_{y}$ over
$[0,T]$ for the two computations.
Figure 7: Beam in dynamic flexion: displacement of the tip of the beam over
simulation time.
As expected, the two methods deliver similar results.
#### 4.2.2 Lamb’s problem
Lamb’s problem [25] is a classical test case used to assert the capacity of a
numerical method is reproduce the propagation of seismic waves. Following
[30], we consider a rectangular domain of size $2\times 1$km2. A source
modelled by a Ricker pulse of central frequency $14.5$Hz is placed $100$m
below the top surface in the middle of the rectangle. Homogeneous Neumann
boundary conditions are imposed on the entire boundary. Following [32], the
material parameters are taken as $G=G_{c}=7.52$GPa, $\lambda=3.76$GPa and
$\ell=\frac{h}{\sqrt{2}}$, where $h$ is the size of the mesh, and the inertia
is taken as $I=\frac{\ell^{2}}{6}$. Such a choice for $\ell$ can look baffling
if one considers that the computed material is indeed a Cosserat continuum.
However, if one considers a variational DEM approach to compute seismic waves,
then such a choice is backed by the litterature [32]. The DEM computation is
performed with $153,600$ dofs and a time-step $\Delta t=5.0\cdot 10^{-5}$s.
The reference solution is a P2-P1 Lagrange FEM with $361,503$ dofs and a
similar time-step. Figure 8 shows the magnitude of the velocity vector at
$t=0.2$s for the DEM and the FEM computation.
(a) (b)
(c) (d)
Figure 8: Lamb’s problem: velocity magnitude at $t=0.2$s. top: FEM, bottom:
DEM.
One can observe the propagation of three waves. First a compression P-wave and
then a shear S-wave propagate inside the domain. Finally, a Rayleigh wave
propagates on the upper boundary (in red). The results given by the two
methods coincide strongly and confirm the pertinence of using a Cosserat
material law and DEM to compute seismic waves as proposed in [30, 32].
## 5 Conclusion
In this article, a variational DEM has been introduced which features only
cell unknowns for the displacement and the micro-rotation. A cellwise gradient
reconstruction is used to obtain cellwise constant strains and stresses using
the formalism of Cosserat materials. An interpretation of the method as a DEM
is presented in which the forces exerted by every facet (or link) between two
cells (or discrete elements) are explicitly given as functions of the cellwise
constant reconstructed stresses. The method is proved to give satisfactory
results on many different test cases in both two and three space dimensions
and both in statics and dynamics. Further work could include extending the
present formalism to nonlinear material laws in small strains [39] and then
finite strains [33]. It could also include computing nonlinear dynamic
evolutions [34]. Also, the present formalism could be extended to Gyro-
continua [11] to have a real rotation matrix in each cell.
## Acknowledgements
The author would like to thank A. Ern from Inria and Ecole Nationale des Ponts
et Chaussées for stimulating discussions and L. Monasse from Inria for
carefully proof-reading this manuscript. The author would also like to thank
the anonymous reviewers for their contributions which helped substantially
improve this paper.
## Funding
Not applicable.
## Conflicts of interest/Competing interests
The author has no conflict of interest or competing interests.
## Availability of data and material
Not applicable.
## Code availability
https://github.com/marazzaf/DEM_cosserat.git
## References
* [1] SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261–272, 2020.
* [2] D. André, J. Girardot, and C. Hubert. A novel DEM approach for modeling brittle elastic media based on distinct lattice spring model. Computer Methods in Applied Mechanics and Engineering, 350:100–122, 2019.
* [3] D. André, I. Iordanoff, J.-L. Charles, and J. Néauport. Discrete element method to simulate continuous material by using the cohesive beam model. Computer Methods in Applied Mechanics and Engineering, 213:113–125, 2012.
* [4] D. André, M. Jebahi, I. Iordanoff, J.-L. Charles, and J. Néauport. Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter. Computer Methods in Applied Mechanics and Engineering, 265:136–147, 2013.
* [5] D. Arnold. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, 19(4):742–760, 1982.
* [6] B. Avci and P. Wriggers. A DEM–FEM coupling approach for the direct numerical simulation of 3d particulate flows. Journal of Applied Mechanics, 79(1), 2012.
* [7] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, and W. Gropp. Petsc users manual. 2019\.
* [8] T. Belytschko and T. J. R. Hughes. Computational methods for transient analysis, volume 1. Amsterdam, North-Holland(Computational Methods in Mechanics., 1983.
* [9] J. Bleyer. Numerical tours of computational mechanics with fenics. 2018\.
* [10] T. Boiveau and E. Burman. A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity. IMA Journal of Numerical Analysis, 36(2):770–795, 2016.
* [11] M. Brocato and G. Capriz. Gyrocontinua. International journal of solids and structures, 38(6-7):1089–1103, 2001.
* [12] E. Burman. A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM Journal on Numerical Analysis, 50(4):1959–1981, 2012.
* [13] E. Cosserat and F. Cosserat. Théorie des corps déformables. A. Hermann et fils, 1909.
* [14] P. Cundall and O. Strack. A discrete numerical model for granular assemblies. Geotechnique, 29(1):47–65, 1979.
* [15] L. Dalcin, R. Paz, P. Kler, and A. Cosimo. Parallel distributed computing using python. Advances in Water Resources, 34(9):1124–1139, 2011. New Computational Methods and Software Tools.
* [16] D. A. Di Pietro. Cell centered Galerkin methods for diffusive problems. ESAIM: Mathematical Modelling and Numerical Analysis, 46(1):111–144, 2012.
* [17] J.L. Ericksen. Liquid crystals and Cosserat surfaces. The Quarterly Journal of Mechanics and Applied Mathematics, 27(2):213–219, 1974.
* [18] R. Eymard, T. Gallouët, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA Journal of Numerical Analysis, 30(4):1009–1043, 2009.
* [19] David C.-L. Fong and M. Saunders. Lsmr: An iterative algorithm for sparse least-squares problems. SIAM Journal on Scientific Computing, 33(5):2950–2971, 2011.
* [20] S. Forest, F. Pradel, and K. Sab. Asymptotic analysis of heterogeneous Cosserat media. International Journal of Solids and Structures, 38(26-27):4585–4608, 2001.
* [21] M. Godio, I. Stefanou, K. Sab, J. Sulem, and S. Sakji. A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: application to masonry. European Journal of Mechanics-A/Solids, 66:168–192, 2017.
* [22] W.G. Hoover, W.T. Ashurst, and R.J. Olness. Two-dimensional computer studies of crystal stability and fluid viscosity. The Journal of Chemical Physics, 60(10):4043–4047, 1974.
* [23] M. Jebahi, D. André, I. Terreros, and I. Iordanoff. Discrete element method to model 3D continuous materials. John Wiley & Sons, 2015.
* [24] J. Jeong and P. Neff. Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Mathematics and Mechanics of Solids, 15(1):78–95, 2010.
* [25] H. Lamb. On the propagation of tremors over the surface of an elastic solid. Philosophical Transactions of the Royal Society of London. Series A, Containing papers of a mathematical or physical character, 203(359-371):1–42, 1904.
* [26] A. Logg, K.-A. Mardal, and G. Wells. Automated solution of differential equations by the finite element method: The FEniCS book, volume 84. Springer Science & Business Media, 2012.
* [27] A. Logg and G. N. Wells. Dolfin: Automated finite element computing. ACM Trans. Math. Softw., 37(2), April 2010.
* [28] F. Marazzato, A. Ern, and L. Monasse. A variational discrete element method for quasistatic and dynamic elastoplasticity. International Journal for Numerical Methods in Engineering, 121(23):5295–5319, 2020.
* [29] F. Marazzato, A. Ern, and L. Monasse. Quasi-static crack propagation with a Griffith criterion using a discrete element method. 2021\.
* [30] C. Mariotti. Lamb’s problem with the lattice model Mka3D. Geophysical Journal International, 171(2):857–864, 2007.
* [31] M. Michael, F. Vogel, and B. Peters. DEM–FEM coupling simulations of the interactions between a tire tread and granular terrain. Computer Methods in Applied Mechanics and Engineering, 289:227–248, 2015.
* [32] L. Monasse and C. Mariotti. An energy-preserving Discrete Element Method for elastodynamics. ESAIM: Mathematical Modelling and Numerical Analysis, 46:1527–1553, 2012.
* [33] P. Neff. A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations. International Journal of Engineering Science, 44(8-9):574–594, 2006\.
* [34] P. Neff and K. Chelminski. Well-posedness of dynamic Cosserat plasticity. Applied Mathematics and Optimization, 56(1):19–35, 2007.
* [35] H. Notsu and M. Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 9(4), 2014.
* [36] D. Potyondy and P. Cundall. A bonded-particle model for rock. International Journal of Rock Mechanics and Mining Sciences, 41(8):1329–1364, 2004.
* [37] E. Providas and M.A. Kattis. Finite element method in plane Cosserat elasticity. Computers & Structures, 80(27-30):2059–2069, 2002.
* [38] M. A. Puscas, L. Monasse, A. Ern, C. Tenaud, and C. Mariotti. A conservative Embedded Boundary method for an inviscid compressible flow coupled with a fragmenting structure. International Journal for Numerical Methods in Engineering, 103(13):970–995, 2015.
* [39] H. Rattez, I. Stefanou, and J. Sulem. The importance of Thermo-Hydro-Mechanical couplings and microstructure to strain localization in 3D continua with application to seismic faults. Part i: Theory and linear stability analysis. Journal of the Mechanics and Physics of Solids, 115:54–76, 2018\.
* [40] H. Rattez, I. Stefanou, J. Sulem, M. Veveakis, and T. Poulet. Numerical analysis of strain localization in rocks with thermo-hydro-mechanical couplings using Cosserat continuum. Rock Mechanics and Rock Engineering, 51(10):3295–3311, 2018.
* [41] A. Ries, D. Wolf, and T. Unger. Shear zones in granular media: three-dimensional contact dynamics simulation. Physical Review E, 76(5):051301, 2007.
* [42] C. Sautot, S. Bordas, and J. Hale. Extension of 2d FEniCS implementation of Cosserat non-local elasticity to the 3d case. Technical report, Université du Luxembourg, 2014.
* [43] V.V. Shelukhin and M. Ružička. On Cosserat–Bingham fluids. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 93(1):57–72, 2013.
* [44] I. Stefanou, J. Sulem, and I. Vardoulakis. Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotechnica, 3(1):71–83, 2008.
* [45] J. Sulem and I.G. Vardoulakis. Bifurcation analysis in geomechanics. CRC Press, 1995.
|
# Calculating the polarization in bi-partite lattice models: application to an
extended Su-Schrieffer-Heeger model
Balázs Hetényi1,2, Yetkin Pulcu1, and Serkan Doǧan3 1 Department of Physics,
Bilkent University, TR-06800 Bilkent, Ankara, Turkey
2MTA-BME Quantum Dynamics and Correlations Research Group, Department of
Physics, Budapest University of Technology and Economics, H-1111 Budapest,
Hungary
3 Department of Mathematics, Bilkent University, TR-06800 Bilkent, Ankara,
Turkey
###### Abstract
We address the question of different representation of Bloch states for
lattices with a basis, with a focus on topological systems. The
representations differ in the relative phase of the Wannier functions
corresponding to the diffferent basis members. We show that the phase can be
chosen in such a way that the Wannier functions for the different sites in the
basis both become eigenstates of the position operator in a particular band. A
key step in showing this is the extension of the Brillouin zone. When the
distance between sites within a unit cell is a rational number, $p/q$, the
Brillouin extends by a factor of $q$. For irrational numbers, the Brillouin
zone extends to infinity. In the case of rational distance, $p/q$, the Berry
phase “lives” on a cyclic curve in the parameter space of the Hamiltonian, on
the Brillouin zone extended by a factor of $q$. For irrational distances the
most stable way to calculate the polarization is to approximate the distance
as a rational sequence, and use the formulas derived here for rational
numbers. The use of different bases are related to unitary transformations of
the Hamiltonian, as such, the phase diagrams of topological systems are not
altered, but each phase can acquire different topological characteristics when
the basis is changed. In the example we use, an extended Su-Schrieffer-Heeger
model, the use of the diagonal basis leads to toroidal knots in the
Hamiltonian space, whose winding numbers give the polarization.
††preprint: APS/123-QED
## I Introduction
Recently, several papers investigated Bena09 ; Watanabe18 the different
representation of the wave function and physical observables when the
underlying model is a lattice with a basis. This fundamental question is
important per se, moreover, such models have been the subject of intense
research in the last decades, exemplified by graphene, the kagome, or Lieb
lattices, and others. The canonical models for topological insulators
Bernevig13 ; Kane13 ; Asboth15 are all lattices with bases: the Haldane,
Haldane88 Kane-Mele, Kane05a ; Kane05b Su-Schrieffer-Heeger Su79 models are
some of the most fundamental ones.
Bena and Montambaux Bena09 have studied the two commonly used bases of the
graphene lattice. The difference between the two is that in one (basis $I$),
the different position of the sites within a unit cell are not explicit (they
are taken to be the same, the position of the unit cell itself), whereas in
the other (basis $II$), the positions of the different sites within a unit
cell are explicit. What is meant by this is that in $I$, the Wannier functions
corresponding to the different basis members have the same phase, while in
$II$ their relative phase depends on the distance between them. Bena and
Montambaux Bena09 show that in the case of the graphene lattice, although the
Hamiltonian and operators take different forms in the two bases, important
physical quantities, such as the density or the density of states, as well as
the low-energy theory, are unchanged. However, care must be taken in using the
correct construction of observables, once a representation of the wave
functions is chosen.
Watanabe and Oshikawa Watanabe18 have taken up this question more recently,
and they show that, while the precise form of the current operator does depend
on the choice of basis, the quantized charge pumped during an adiabatic cycle
Thouless82 is unchanged. Note that the current operator is intimately related
to the phase of the wave function, in particular the relative phase wave
functions centered on different sites may exhibit. Also note that the charge
pumped is a topological invariant, which remains the same in different bases.
In this paper we reconsider the question of basis dependence in lattice models
in the context of modern polarization theory with a focus on topological
systems. We compare the two bases used in Ref. Bena09 , and show that one
corresponds to a linear combination of Wannier functions which are
eigenfunctions of the position operator. When the distance between the basis
sites is a rational number $p/q$, the Brillouin zone has to be extended by a
factor of $q$ to derive the above result. If the distance is irrational, then
the Brillouin zone has to be extended to the whole real line.
We then turn to the question of topological phase transitions in the two
bases. As an example system we consider an extended Su-Schrieffer-Heeger Su79
(SSH) model. When the $k$-space Hamiltonian is written considering distance
dependence explicitly the three-dimensional curve $d_{x},d_{y},d_{z}$ (see Eq.
7) is not a closed curve. When the distance between basis sites is a rational
number ($p/q$) extending the Brillouin zone by a factor of $q$ closes the
curve, and an expression for the polarization Berry phase Berry84 ; Zak89 ;
King-Smith93 ; Resta98 ; Resta00 can be derived. In the irrational case a
closed $d_{x},d_{y},d_{z}$-curve results only if the Brillouin zone is
extended to infinity. For this reason, calculation of the irrational case is
not straightforward. We show that a stable calculation results if the
irrational distance is approximated as a rational sequence.
Implementing the dependence of the Hamiltonian on the distance between basis
sites proceeds through a unitary transformation. As such, the Hamiltonian can
acquire auxiliary topological features. In the model we study, the
$d_{x},d_{y},d_{z}$-curve traces out a toroidal knot, and the ratio of the
winding numbers of the knot give the polarization. However, the phase diagram
itself is not altered, since these topological characteristics are merely
different representations of the phases of the original extended SSH model.
In the next section we discuss the two different bases used in lattices with a
basis. In section III the extended SSH model is described. In section IV the
change of basis is discussed in the context of the extended SSH model. In
section V our results are presented, and in section VI we conclude our work.
## II Polarization and choice of representation
It is well-known Blount62 ; Kivelson82 that Wannier functions in systems
without a basis are eigenstates of the position operator within a given band.
Here we show that in a lattice with a basis, one of the representations of the
wave-function studied by Bena and Montambaux Bena09 leads to position
eigenstates.
We consider a one-dimensional system whose potential obeys $V(x)=V(x+1)$ (in
other words we take the length of a unit cell as unity). The unit cell
consists of a two-site basis. We chose the distance between the two internal
sites to be $p/q$, a rational number, and we extend the Brillouin zone by a
factor of $q$. This step is justified below.
In this case, the two commonly used bases Bena09 can be written as
$\displaystyle u_{k}^{I}(x)=$
$\displaystyle\sum\limits_{m=-\infty}^{\infty}e^{ik(m-x)}\left[w_{a}(x-m)+w_{b}\left(x-m-\frac{p}{q}\right)\right]$
$\displaystyle u_{k}^{II}(x)=$
$\displaystyle\sum\limits_{m=-\infty}^{\infty}\left[e^{ik(m-x)}w_{a}(x-m)+e^{ik(m+p/q-x)}\times\right.$
(1) $\displaystyle\left.\times w_{b}\left(x-m-\frac{p}{q}\right)\right].$
The functions $u_{k}^{I}(x)$ and $u_{k}^{II}(x)$ are periodic Bloch functions.
$u_{k}^{I}(x)$ is the more standard approach, and it appears Ashcroft76 in
textbooks. $w_{a}(x),w_{b}(x)$ are two different localized Wannier functions.
In the extended SSH model we study below, they differ only in having their
centers displaced with respect to each other.
Following Zak Zak89 we derive the geometric phase which arises upon
integrating the connection across the Brillouin zone. We define a geometric
phase of the Zak type, in the extended Brillouin zone as:
$\gamma=\frac{i}{2\pi q}\int_{-q\pi}^{q\pi}dk\langle
u_{k}|\partial_{k}|u_{k}\rangle.$ (2)
Using $u_{k}^{I}$ the Zak phase becomes
$\gamma_{I}=\int_{-\infty}^{\infty}dxx\left|w_{a}(x)+w_{b}\left(x-\frac{p}{q}\right)\right|^{2},$
(3)
whereas with $u_{k}^{II}$
$\gamma_{II}=\int_{-\infty}^{\infty}dxx\left(|w_{a}(x)|^{2}+|w_{b}(x)|^{2}\right).$
(4)
In deriving $\gamma_{II}$, we made use of the fact that
$\frac{1}{2\pi q}\int_{-\pi q}^{\pi
q}dk\exp\left(ik(n-n^{\prime})/q\right)=\delta_{nn^{\prime}}.$ (5)
Comparing $\gamma_{I}$ and $\gamma_{II}$ we see that the second Bloch basis
function, $u_{k}^{II}$ is a diagonal basis for the position operator in a
periodic system, since the result is a simple sum of Wyckoff positions for the
two different Wannier functions. It is due to Eq. (5) that the cross terms in
$\gamma_{II}$ disappear, but only of the Brillouin zone is extended by a
factor of $q$.
It is instructive to consider the effect of inversion symmetry on either
$\gamma$. In the original work of Zak Zak89 , where the sites are equivalent,
two types of inversion symmetries were considered, site-inversion and bond-
inversion. The former gives a Zak phase of zero, while the latter gives $\pi$,
which corresponds to a polarization of $qa/2$ ($a$ is the length of the unit
cell). In our case, there are also two types of inversion. If the sites of the
model are a distance $p/q$ apart, one can consider inverting around the
midpoint of the $p/q$ bond, or the $1-p/q$ bond. In the former case it holds
that $w_{c}(-x+p/q)=\pm w_{c}(x)$ for both $c=a,b$, which leads to
$\gamma=\frac{p}{2q}$ (for both $I$ and $II$). In deriving this one can follow
the stame steps as ZakZak89 . Inverting around the other bond center leads to
$w_{c}(-x-1+p/q)=\pm w_{c}(x)$, from which it follows that
$\gamma=\frac{p-q}{2q}$ (again for both $I$ and $II$).
In the case of irrational distance, the Brillouin zone has to be extended to
the range $(-\infty,\infty)$. When this is done, Eq. (5) eliminates the cross
terms and leads to $\gamma_{II}$.
## III Model
The full model we consider is an extension of the well-known Su-Schrieffer-
Heeger model Su79 . It is a bi-partite lattice model ($A$ and $B$
sublattices), whose Hamiltonian has the form,
$\hat{H}=-\sum_{i=1}^{L}[Jc_{i}^{\dagger}d_{i}+J^{\prime}d_{i}^{\dagger}c_{i+1}+iKc_{i}^{\dagger}c_{i+1}-iKd_{i}^{\dagger}d_{i+1}+\mbox{H.c.}],$
(6)
where $L$ denotes the number of unit cells, $c_{i}$($c_{i}^{\dagger}$) denote
the annihiliation(creation) operators on the $A$ site of the $i$th unit cell,
and $d_{i}$($d_{i}^{\dagger}$) denote the same for the $B$ site of the $i$th
unit cell. We take the length of a unit cell to be unity. An extended SSH
model of a different kind, with real second nearest neighbor hoppings, was
invesigated by Li et al. Li14 .
When $K=0$ the SSH model is recovered. The third and fourth terms in Eq. (6)
correspond to second nearest neighbor hoppings (hoppings within one
sublattice). A Peierls phase of $\pi/2$ and $-\pi/2$ is applied along these
hoppings, the former for sublattice $A$, the latter for sublattice $B$. These
second nearest neighbor hoppings play a similar role to those in the Haldane
model Haldane88 . There and in Eq. (6) the Peierls phases of second nearest
neighbor hoppings point in opposite directions on the two sublattices. The
consequence of this term is that for gapped phases and finite $K$ there is
persistent current in the system, but the currents on the different
sublattices cancel, and insulation results. If $K$ changes sign, the direction
of the persistent currents on the different sublattices also change sign. In
this sense, the model can be viewed as an example of Kohn’s tenet Kohn64 .
This tenet states that insulation in a quantum system is not a function of
localization of individual charge carriers, but ultimately results from many-
body localization, or the localization of the center of mass of all the charge
carriers. Individually, charge carriers can be quite mobile, as long as the
center of mass is localized.
Figure 1: Curves traced out by the parameters of the Hamiltonian
$d_{x},d_{y},d_{z}$ as $k$ traverses Brillouin zone for the usual SSH and the
extended SSH model. Upper panel: SSH model and extended SSH model in the usual
basis. Lower panel: extended SSH model in the distance dependent basis
comparing the first Brilllouin zone to the extended Brillouin zone.
According to the symmetry classification Altland97 ; Schnyder08 of
topological insulators, the SSH model, which exhibits time reversal, particle-
hole, and chiral symmetries, falls into the BDI class, which in one dimension
is a $\mathbb{Z}$-topological system. After the addition of the second nearest
neighbor hopping term in Eq. (7) only the particle-hole symmetry remains, and
the model falls in the D symmetry class, which is a $\mathbb{Z}_{2}$ system.
The gap closure points of the SSH remain (they occur at $k=0,\pi$), and we
show below that topological edge states still arise.
Fourier transforming Eq. (6) results in
$\hat{H}=\sum_{k}H_{k}=\sum_{k}[d_{x}(k)\sigma_{x}+d_{y}(k)\sigma_{y}+d_{z}(k)\sigma_{z}],$
(7)
where $\sigma_{x},\sigma_{y},\sigma_{z}$ denote the Pauli matrices, and
$\displaystyle d_{x}(k)$ $\displaystyle=$ $\displaystyle-J-J^{\prime}\cos(k),$
(8) $\displaystyle d_{y}(k)$ $\displaystyle=$
$\displaystyle-J^{\prime}\sin(k),$ $\displaystyle d_{z}(k)$ $\displaystyle=$
$\displaystyle 2K\sin(k).$
Written this way, we see that gap closure occurs at $k=\pi$ if $J=J^{\prime}$.
The upper panel Fig. 1 shows the curves traced out by $d_{x},d_{y},d_{z}$ as
$k$ traverses the Brillouin zone for the usual SSH model
($J=1.25,J^{\prime}=1.75$) with $K=0$ and an example of the extended one, with
$K=2$. The former forms a circle in the $xy$-plane, the latter is a circle
with a tilted axis (axis along the $yz$-plane). Both circles cross the
$x$-axis itself in two places at the same points. This means, that by varying
the ratio of $J/J^{\prime}$ it is possible to close the gap in both models,
and encounter topological phase transitions.
We can also develop a continuum approximation, the usual way Jackiw76 ,
$H=it\partial_{x}\sigma_{x}+iK\partial_{x}\sigma_{y}+2\delta t\sigma_{z},$ (9)
where we used the parametrization:
$\displaystyle t$ $\displaystyle=$ $\displaystyle\frac{J+J^{\prime}}{2}$ (10)
$\displaystyle\delta t$ $\displaystyle=$ $\displaystyle\frac{J^{\prime}-J}{2}$
We can solve for zero energy eigenstates of the form
$\begin{pmatrix}\Psi_{A}(x)\\\ \Psi_{B}(x)\end{pmatrix}.$ (11)
The zero energy solution is
$\Psi_{A/B}^{\prime\prime}(x)=\frac{4\delta t^{2}}{t^{2}+K^{2}}\Psi_{A/B}(x).$
(12)
The solutions are exponentials. If the system is extended, these solutions
diverge, hence they are unnormalizable. If the system has open boundary
conditions, exponential solutions can be normalized, since they can start at
one of the boundaries and decay towards the bulk. Therefore zero-energy states
are possile. The sign of $\delta t$ determines the direction of decay, in
other words, the side of the finite system on which the edge state is located.
The signs of $t$ and $K$ are irrelevant.
It is interesting to compare the above results to those of Jackiw and Rebbi
Jackiw76 . There a first order differential equation is solved for a function
multiplying a $\sigma_{z}$ eigenstate, while in our case each component of the
Pauli spinor satisfies the same second order differential equation. Our
calculations below on the lattice model (see Fig. 3) bear out the predictions
of our continuum theory.
## IV Implementation of distance dependence
We take the shortest distance between the two sublattices to be $p/q$, where
$p$ and $q$ are co-prime integers. The Hamiltonian in $k$-space can be shown
to be:
$\displaystyle d_{x}(k)$ $\displaystyle=$
$\displaystyle-J\cos\left[k\left(\frac{p}{q}\right)\right]-J^{\prime}\cos\left[k\left(1-\frac{p}{q}\right)\right]$
(13) $\displaystyle d_{y}(k)$ $\displaystyle=$ $\displaystyle
J\sin\left[k\left(\frac{p}{q}\right)\right]-J^{\prime}\sin\left[k\left(1-\frac{p}{q}\right)\right]$
$\displaystyle d_{z}(k)$ $\displaystyle=$ $\displaystyle 2K\sin(k).$
This Hamiltonian can also be derived from the original one (Eq. (7)) by
$H_{k}\rightarrow\exp\left(ik\frac{\sigma_{z}}{2}\frac{p}{q}\right)H_{k}\exp\left(-ik\frac{\sigma_{z}}{2}\frac{p}{q}\right),$
(14)
a rotation by angle $kp/q$ around the $z$ axis. We also note that the vector
function $[d_{x},d_{y},d_{z}]$ has a period of $2\pi q$, rather than the usual
$2\pi$.
Due to the fact that the distance dependence coincides with a rotation, a
qualitative difference between the case of rational and irrational distances
arises. As is known Asboth15 ; Kane13 , the curve traced out by the SSH model
on the $d_{x}-d_{y}$ plane is an ellipse whose center, in general, lies on the
$d_{x}$ axis. Implementing the distance dependence will generate a different
curve. A closed curve can be obtained if the distance between sublattice sites
is taken to be a rational number ($p/q$ with $p$ and $q$ coprimes), and the
Brillouin zone is extended to $2\pi q$. Observables related to the Berry (Zak)
phase (polarization, topological invariants) depend on the curve being closed,
since these physical quantities are integrals over a connection over a closed
curve (usually the Brillouin zone).
Figure 2: Curves traced out by the parameters of the Hamiltonian
$d_{x},d_{y},d_{z}$ as $k$ traverses the extended Brillouin zone for the
$\frac{p}{q}=\frac{1}{2}$ case. The Hamiltonian parameters are $K=1$ in all
cases, and $J=1,J^{\prime}=2$ (upper panel), $J=1.5,J^{\prime}=1.5$ (middle
panel), $J=2,J^{\prime}=1$ (lower panel). The curves of the upper and lower
panels appear equivalent, but the orientation of the curves changes as the gap
closure point (the figure-8 of the middle panel) is crossed.
Let us rewrite the $d$-curve of Eq. (13) with a scaled wave vector,
$k=q\kappa$, as
$\displaystyle d_{x}(\kappa)$ $\displaystyle=$ $\displaystyle-\cos(\kappa
p)\left[J+J^{\prime}\cos(\kappa q)\right]$ $\displaystyle-\sin(\kappa
p)J\sin(\kappa q),$ $\displaystyle d_{y}(\kappa)$ $\displaystyle=$
$\displaystyle+\sin(\kappa p)\left[J+J^{\prime}\cos(\kappa q)\right]$
$\displaystyle-\cos(\kappa p)J^{\prime}\sin(\kappa q),$ $\displaystyle
d_{z}(\kappa)$ $\displaystyle=$ $\displaystyle+2K\sin(\kappa q).$
This curve exists on the surface of a torus, albeit, not a torus of the usual
parametrization, and the curve itself is a toroidal knot (see examples in
Figs. 2, 4, 5).
The fundamental group Nakahara03 of the torus is
$\pi_{1}(T^{2})=\pi_{1}(S^{1})\oplus\pi_{1}(S^{1})=\mathbb{Z}\oplus\mathbb{Z}$,
meaning that homotopically equivalent classes of loops on the torus can be
characterized by two integers. In Eq. (IV) the two integers are $p,q$, and in
this parametrization they are toroidal winding numbers. A torus knot is
obtained by a trajectory which loops around the hole of the torus an integer
number of times ($w_{1}=p$), while it makes an integer number of rotations
around the “body” of the torus itself ($w_{2}=q$). As argued in the previous
section, the ratio of winding numbers is proportial to the polarization. A
curve parametrized as Eq. (IV) traces out a toroidal knot for a given integer
pair $p,q$, even if $p$ and $q$ are not co-primes. If they are co-primes then
there is only one curve, if they are not, then the same curve is traced more
than once, as many times as the common divisor of $p$ and $q$. In the original
parametrization, Eq. (13), the rescaling leads to the same curves for a given
$p,q$.
Alternatively, one can define $r=q-p$ in Eq. (13), and arrive at a different
curve, similar in form to Eq. (IV). In this case the winding numbers will be
$w_{1}=r$ and $w_{2}=q$. Thus, we can write the two different fractionally
quantized polarization results of the previous section as:
$\mathfrak{P}=-\frac{e}{2}\frac{w_{1}}{w_{2}}.$ (16)
The lower panel of Fig. 1 shows two curves for this extended model, one using
a normal Brillouin zone ($-\pi,\pi$), the other an extended one ($-q\pi,q\pi$)
(model parameters are $J=1.25,J^{\prime}=1.75,K=2$ and $p/q=1/3$ in both
cases). The former curve is open, the latter is a closed curve, a toroidal
knot.
In our calculations below the Brillouin zone is stretched by a factor of $q$,
and we take every $q$th point in $k$ space in this enlarged Brillouin zone.
The starting point for the polarization expression is the scalar product
$Z_{q}=\langle\Psi|\exp\left(i\frac{2\pi\hat{X}}{L}q\right)|\Psi\rangle.$ (17)
Following the steps of Resta Resta98 , in a band insulator we have
$Z_{q}=\prod_{s=0}^{L-1}\mbox{det}S(k_{qs},k_{q(s+1)}).$ (18)
where
$S_{m,m^{\prime}}(k_{qs},k_{q(s+1)})=\int_{0}^{L}dx\hskip
2.84544ptu^{*}_{k_{qs},m}(x)u_{k_{q(s+1)},m^{\prime}}(x),$ (19)
where $u_{k_{qs},m}(x)$ denote the periodic Bloch functions of band $m$, and
$k_{qs}=2\pi qs/L$. In our numerics we calculate the polarization via
$\mathfrak{P}=-\frac{e}{2\pi}\mbox{Im}\ln\prod_{s=0}^{L-1}[\mbox{det}S(k_{qs},k_{q(s+1)})]^{\frac{1}{q}}.$
(20)
This definition is important from the point of view of implementation, because
the $\frac{1}{q}$ power keeps the terms in the Berry phase between $0$ and
$2\pi$, and provides the correct fractional polarization.
Again the question remains, whether irrational distances can be handled. The
difficulty in this case is that the Brillouin zone has to be extended to
infinity. Two methods suggest themselves. One, use the irrational distance
itself in place of $p/q$, and compare larger and larger Brillouin zones. In
this case, the curve in the $d_{x},d_{y},d_{z}$ curve will be open. Another
way would be to approximate the irrational distance as the limit of a sequence
of rational numbers, and investigate the limit of Eq. (20) along that
sequence. Below a comparison is presented for the case when the distance is
the inverse of the golden ratio.
Figure 3: Calculation for a system with open boundary conditions with $L=100$
(meaning also that there are $100$ states). The parameters of the Hamiltonian
are $J=1.25,J^{\prime}=1.75,K=2.0$. The minimum distance between sublattices
is $p/q=1/3$. The figure show: (a) the zero energy state number $50$, (b) the
zero energy state number $51$, (c) the phase of the wave function as a
function of lattice site for state number $50$, and (d) the phase of the wave
function as a function of lattice site for state number $51$.
## V Results
The curves traced out by the parameters of the Hamiltonian
($d_{x},d_{y},d_{z}$) are shown in Fig. 2 for three cases: $J=1,J^{\prime}=2$;
$J=1.5,J^{\prime}=1.5$; $J=2,J^{\prime}=1$ ($K=1$ in all three cases) for
$p=1,q=2$. The topological transition occurs at $J=1.5,J^{\prime}=1.5$, where
the curve traces out a figure$-8$, with a crossing point at the origin. The
curves on the different sides of this phase diagram (the uppermost and
lowermost panels) can be transformed into each other via a rotation by
$\pi/2$. Since the orientation of the curves (the sense of rotation around the
$d_{z}$-axis) does change when crossing the phase boundary the winding number
goes from one to minus one. This corresponds to a polarization reversal from
$\mathfrak{P}=e/4$ to $\mathfrak{P}=-e/4$, as expected.
Figure 4: Curves traced out by the parameters of the Hamiltonian
$d_{x},d_{y},d_{z}$ as $k$ traverses the extended Brillouin zone. The
upper(lower) set of plots show the case
$\frac{p}{q}=\frac{1}{3}$($\frac{p}{q}=\frac{2}{3}$), where the parameters of
the Hamiltonian are, left: $J^{\prime}=1.75,J=1.25$,
center:$J=1.5,J^{\prime}=1.5$, and right:$J^{\prime}=1.25,J=1.75$ ($K=1$ in
all cases).
Fig. 3 presents a study of how edge states arise in a system with open
boundary conditions. The parameters of the Hamiltonian are
$J=1.25,J^{\prime}=1.75,K=2.0$ with $p=1$ and $q=3$. The upper two panels ((a)
and (b)) show the two zero energy wave functions which are localized at the
edges. The system is fairly small $L=100$, and the paramater $K$ is fairly
large, giving rise to edge state wavefunctions with a sizable finite component
even halfway through the lattice, however, as the size is increased and/or if
$K$ is decreased the bulk compoment of the state decreases. These results are
consistent with our previous derivation (Eq. (9)). For a system with
$J^{\prime}<J$ edge-states are not found. The bottom panels ((c) and (d)) show
the phase of the zero energy wave functions. The phases of both edge states
oscillate when going towards the center from the edges. The left and right
sides mirror each other, and a sudden change occurs at and within a few sites
near the center of the system.
In Figs. 4 and 5 the $d_{x},d_{y},d_{z}$ curves are shown for various values
of $p$ and $q$. The Hamiltonian parameters are as follows: left
$J^{\prime}=1.75,J=1.25$, center $J^{\prime}=1.5,J=1.5$, and right
$J^{\prime}=1.25,J=1.75$. $K=1.0$ for all cases. The plots are knot diagrams
Adams01 , two dimensional knot representations, arrived at by projecting the
curve onto the $d_{x}-d_{y}$ plane, but indicating which strand is below the
other at crossings. Other than the plots in the center in both figures,
neither of which are knots, the left and right plots show torus knots. For
example, in the top left figure of Fig. 5, the curve winds around $w_{1}=4$
times while it makes $w_{2}=5$ revolutions around the body of the torus.
Figure 5: Curves traced out by the parameters of the Hamiltonian
$d_{x},d_{y},d_{z}$ as $k$ traverses the extended Brillouin zone. The
upper(lower) set of plots show the case
$\frac{p}{q}=\frac{1}{5}$($\frac{p}{q}=\frac{2}{5}$). The parameters of the
Hamiltonian are, left: $J^{\prime}=1.75,J=1.25$,
center:$J=1.5,J^{\prime}=1.5$, and right:$J^{\prime}=1.25,J=1.75$ ($K=1$ in
all cases).
The center plots in both Figs. 4 and 5 show curves for the topological phase
transition. These curves do not form knots, they cross at the origin, which is
the gap closure point. It is obvious that these center plots “connect” the two
topologically distinct phases on either side of the phase transition. At the
phase transition the torus surface on which the curves “live” is a horn torus.
On either side of the horn torus, the curves reconnect, in a manner
reminiscent of “partner switching” of edge states in the Kane-Mele model
Kane05a ; Kane05b .
Figure 6: The trajectory of $\phi_{k}$ a parameter which defines the wave-
function (Eq. (21)) across the extended Brillouin zone for a trivial (a) and a
topological (b) case. As $k$ sweeps across the extended Brillouin zone
$\phi_{k}$ increases by $2\pi$ in the trivial, and $-4\pi$ in the topological
system.
The upper plots in Fig. 4 are for $p=1$, the lower ones are $p=2$. The upper
left plot shows a trefoil knot with $w_{1}=2$, $w_{2}=-3$. The upper right
plot shows an un-knot with $w_{1}=1$, $w_{2}=3$. Even though there are
crossings on this curve, they can be eliminated by type I Reidemeister moves
Adams01 , one of three modifications on two-dimensional representations of
knots which leave knot invariants unaltered. The knot invariants (in this case
the two winding numbers) are not altered by these modifications (if all three
twists are eliminated).
The topological phase transition connects the two states. In the insulating
phases Eq. (16) is recovered. In the lower set the $J^{\prime}>J$ curve
exhibits the unknot ($w_{1}=1,w_{2}=-3$), and the $J>J^{\prime}$ the trefoil
knot ($w_{1}=2,w_{2}=3$), again consistent with Eq. (16).
In Fig. 5 the $d_{x},d_{y},d_{z}$ curves are shown for various $q=5$ cases.
The upper plots are $p=1$, the lower ones are $p=2$. The phase transition in
the upper set is a topological one connecting a $w_{1}=4$ phase
($J^{\prime}>J$) with a $w_{1}=1$ phase ($J>J^{\prime}$), while the bottom set
shows a transition between a $w_{1}=3$ phase ($J^{\prime}>J$) and a $w_{1}=2$
phase ($J>J^{\prime}$), both accompanied by sign changes in $w_{2}$ whose
absolute value is five. The cases $p=3$ and $p=4$ (not shown) give winding
numbers of $w_{1}=2$, $w_{1}=3$ and $w_{1}=1$, $w_{1}=4$, respectively, again
with changes in sign in $w_{2}$ across the quantum phase transition. In all
cases, our polarization formula, Eq. (16), is recovered.
To summarize the information of Figs. 4 and 5, it is useful to connect the
results to what is known from the “usual” representation Su79 , in which the
Wannier functions corresponding to different basis sites are taken to have the
same phase (basis $I$). There, there are two distinct phases, one with winding
number zero (trivial phase), one with winding number one (topological phase),
separated by gap closure. Performing the unitary transformation in Eq. (14)
leads to a different representation of the same topological phases, and it is
these different representations which are shown in the figures. In a given
system with fixed $p/q$, the insulating phases are knots on ring or spindle
tori. A ring torus can be deformed into a spindle torus via a horn torus. The
horn torus occurs if $J=J^{\prime}$, that is when the gap closes, and where
the phase transition separating different topological phases occurs (center
panels in Figs. 4 and 5). Note that in the $p/q=1/3$ case, the
topological(trivial) phase corresponds to the spindle(ring) torus, whereas the
reverse is true for $p/q=2/3$.
When $K=0$ (the usual SSH model), the above curves remain on the $d_{x},d_{y}$
plane, hence one cannot speak of a torus. In the “usual” representation
Asboth15 ; Kane13 of this model the $J^{\prime}>J$ is is the topological
state with winding number of one, while the $J^{\prime}<J$ is the trivial site
with winding number zero, depending on whether the $d_{x},d_{y}$ curve
encircles the origin. Implementing distance dependence, turns each of those
phases into phases with distinct quantized polarizations. Gradually changing
the sign of $K$ changes the sign of both winding numbers for a given toroidal
knot (which goes into its chiral counterpart), leading to no change in the
polarization (since the polarization is the ratio of winding numbers). This is
consistent with the result derived in section III: the sign of $K$ does not
determine the presence of edge states (nor whether the state is topological or
trival), only the sign of the $J^{\prime}-J$ does.
Figure 7: Semilogarithmic plot of the absolute value of the difference between
the exact value of the polarization $P_{exact}$ and the polarization
calculated via two different methods. The solid like shows the use of an
irrational distance (inverse of the golden ratio), the diamonds show a
calculation based on a rational approximation to the inverse golden ratio.
Another way of seeing how the fractional polarization arises is to look at the
phase variable of the wave function. The wave functions that are the solution
of the Hamiltonian in Eq. (7) with the $d$-components given by Eq. (13) can be
written in the form
$\begin{pmatrix}\sin\left(\frac{\theta_{k}}{2}\right)\\\
\cos\left(\frac{\theta_{k}}{2}\right)\exp(i\phi_{k})\end{pmatrix},$ (21)
where $\theta_{k}$ and $\phi_{k}$ depend on $d_{x},d_{y},d_{z}$ (Eq. (13)).
The phase $\phi_{k}$ is shown in Fig. 6, for a system with $p=1$ and $q=3$. In
the trivial phase, the phase $\phi_{k}$ changes by $2\pi$ as $k$ covers the
extended Brillouin zone ($2\pi q=6\pi$). Accordingly, the polarization is
$\mathfrak{P}=-\frac{e}{2}\frac{1}{3}$. Meanwhile, in the topological phase,
the variable $\phi_{k}$ changes by $-4\pi$ along the Brillouin zone, and the
polarization is $\mathfrak{P}=\frac{e}{2}\frac{2}{3}$.
In Fig. 7 we present calculations for an irrational distance, namely the
inverse of the golden ratio. In one calculation, we used the irrational number
itself to derive a Bloch Hamiltonian (in place of $p/q$ in Eq. (13)), and
Brillouin zones of size $-\pi q,\pi q$ were used. In another calculation
ratios of the Fibonacci sequence were used to approximate the golden ratio,
and Eq. (20) was used. The results in Fig. 7 show that, while the polarization
based on using an irrational distance is reasonably close to the exact result,
using a rational approximation is a much more stable procedure, which
uniformly approaches the exact result.
If an adiabatic charge pumping experiment Atala13 ; Nakajima16 undergoing a
full cycle is considered the amount of charge pumped would be one unit of
charge per cell. This coincides with the results of Watanabe and Oshikawa
Watanabe18 who showed that the charge pumped is independent of how the
Hamiltonian is written. Fig. 8a shows the Fourier transform of $Z_{q}$,
defined as
$P_{x}=\sum_{q}\exp(i2\pi xq/L)Z_{q};\hskip 5.69046ptx=1,...,L;$ (22)
which corresponds to the underlying polarization distribution Souza00 ;
Resta99 ; Yahyavi17 ; Hetenyi19 . The case $p/q=1/3$ is considered. Shown are
two different parameter sets for each ordered state, as well as the phase
transition, where $J=J^{\prime}$. The distributions of the ordered states show
maxima corresponding to the values of the polarization calculated above. The
tuning of the parameters of the Hamiltonian within one phase change the shape
of the distribution (variance and other cumulants), but not the position of
the maximum. At the transition the distribution becomes nearly flat, the
system does not exhibit a well-defined polarization due to gap closure. For a
finite system the curve representing $J=J^{\prime}$ is not entirely flat,
because the $k-$space sampling does not precisely hit the points where the
gaps occur. However, as the thermodynamic limit is taken, the curve
progressively flattens out.
Figure 8: (a) Fourier transform of the polarization amplitude, or polarization
distribution shown for different cases of the system with $p/q=1/3$; two sets
in each ordered state and at the gap closure point. The curves are normalized
to one over the lattice of $L=100$. Legend is valid only for the upper panel.
(b) Reconstructed polarization distributions along an adiabatic charge pumping
process. For further explanation see the text.
In the cases treated heretofore the polarization can be easily determined by
the ratio of the winding numbers, hence the Berry phase formula is not
necessary. If other terms are added to the Hamiltonian, for example, an on-
site potential, then the Berry phase based formula in Eq. (20) is needed. In
the lower panel of Fig. 8b an adiabatic charge pumping process Nakajima16 is
shown. The Hamiltonian is reparametrized according to Eq. (10) and an
alternating on-site potential of strength $\Delta$ is also added. In the
calculations of Fig. 8b $t=K=1$. The plots shown are along total polarization
distribution curves along the circle in the $\delta t-\Delta$ plane. The
equation of the circle is $0.2\cos(\alpha),0.2\sin(\alpha)$, and the curves
labeled $A,B,C,D,E,F,G,H$ correspond to values of
$\alpha=0,\pi/4,\pi/2,3\pi/4,\pi,5\pi/4,3\pi/2,7\pi/4$ respectively. As
$\alpha$ increases, the maximum of the polarization distribution function
shifts towards the right within the unit cell. For the curves $G$ and $H$ the
maxima of the curves is to the left of the $A$-curve. This can be interpreted
as a charge which entered from a neighboring unit cell (to the left of the one
shown), as is expected to occur for adiabatic charge pumping. Note that at
points $A$ and $E$ the maxima correspond to the ones shown in Fig. 8a, and the
maximum of the charge distribution appears to travel smoothly in between. The
part of the adiabatic charge pump process between $\alpha=0$ and $\alpha=\pi$
occurs between systems which exhibit fractionally quantized polarization. If
the $I$ basis was used, the maximu would be located at the edge of and halfway
through the unit cell.
## VI Conclusion
Lattices with a basis can be solved by various representations of the wave
function. The representations differ in the relative phases of the Wannier
functions associated with different sites within the unit cell. The different
representations are related by unitary transformations. The transformation can
be used to diagonalize the position operator, if t he Brillouin zone is
extended. When the nearest distance between basis sites is a rational number
$p/q$, the Brillouin zone has to be extended by a factor of $q$. When this
distance is an irrational number, the Brillouin zone extends to the entire
real line. This latter state of affairs raises the question of how to
calculate the polarization, to which we suggested the use of a rational
approximation to the distance between intra-unit cell sites.
Applying such a transformation to the Hamiltonian can give rise to auxiliary
topological features. In our example calculations, based on an extension of
the SSH model, the $d_{x},d_{y},d_{z}$-curve forms a toroidal knot, and the
ratio of the two winding numbers of the torus is proportional to the
polarization. Since the modification here is a mere unitary transformation of
the Hamiltonian, or a change in the relative phase of the Wannier functions
corresponding to the different sites within a unit cell, no new phases are
found, the phase diagram is unaltered. If the extended SSH model is studied in
the “usual” basis, where the phases of the Wannier functions are the same,
then there is no torus, the $d_{x},d_{y},d_{z}$ curve is an ellipse, and its
winding number around the $d_{z}$ axis defines the polarization (the distinct
phases have winding numbers zero or one). When the Hamiltonian is transformed
into the basis which diagonalizes the position, and the Brillouin zone is
extended, the $d_{x},d_{y},d_{z}$ curve becomes a knot on a torus. Within each
phase the torus on which the “curve” lives is of a different type, a ring or a
spindle torus. The quantum phase transition occurs when the torus is a horn
torus.
These conclusions fit into the recent findings of Bena and Montambaux Bena09
and those of Watanabe and Oshikawa Watanabe18 . The unitary transformation
which relates different representation choices can change quantities which
depend on the phase of the wave fuction. In Ref. Watanabe18 this is shown for
the current, in our case it is shown for the polarization (Berry-Zak phase),
and for the $d_{x},d_{y},d_{z}$ curves of the system. However, as argued in
both Ref. Bena09 and Watanabe18 , phase independent quantities Watanabe18
will not change. The phase independent quantity we investigated was the charge
pumped in an adiabatic process, and indeed it does not depend on the
representation used. The path of the polarization maximum during the process,
however, does depend on it.
As for experimental tests of our work, the model suggested here can be
realized in a setting of cold atoms trapped in optical lattices Atala13 ;
Nakajima16 . We anticipate that a partial change in the polarization, of the
type between maxima in Fig. 8 can be measured in experiments in which the
polarization is changed, as long as the process does not constitute a full
adiabatic cycle, for example, a polarization switch. One possible way to
realize the distance dependence explicitly is the following. For instance a
$p=1$, $q=3$ system could be realized by constructing first a tri-partite
periodic lattice, and approach our bi-partite system as a limit by decreasing
the coupling of one of the sites in each unit cell to zero.
## Acknowledgments
BH would like to thank J. K. Asbóth and B. Dóra for helpful discussions. BH
was supported by the National Research, Development and Innovation Fund of
Hungary within the Quantum Technology National Excellence Program (Project Nr.
2017-1.2.1-NKP-2017-00001) and by the BME-Nanotechnology FIKP grant (BME FIKP-
NAT).
## References
* (1) C. Bena and G. Montambaux, New J. Phys., 11 095003 (2009).
* (2) H. Watanabe and M. Oshikawa, Phys. Rev. X 8 021065 (2018).
* (3) B. A. Bernevig and T. L. Hughes, Topological Insulators and Superconductors, Princeton University Press (2013).
* (4) C. L. Kane, in Topological Insulators, vol. 6 Eds. M. Franz and L. Molenkamp, Chapter 1, Elsevier (2013).
* (5) J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators: Band Structure Topology and Edge states in One and Two Dimensions, Lecture Notes in Physics, vol. 919, Springer (2018).
* (6) F. D. M. Haldane, Phys. Rev. Lett. 61 2015 (1988).
* (7) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95 226801 (2005).
* (8) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95 146802 (2005).
* (9) W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. Lett. 42 1698 (1979).
* (10) D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs Phys. Rev. Lett. 49 405 (1982).
* (11) M. V. Berry, Proc. Roy. Soc. London A392 45 (1984).
* (12) J. Zak, Phys. Rev. 62 2747 (1989).
* (13) R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47 R1651 (1993).
* (14) R. Resta, Phys. Rev. Lett. 80 1800 (1998).
* (15) R. Resta, J. Phys.: Cond. Mat. 12 R107 (2000).
* (16) E. I. Blount, Solid State Physics 13 305 (1962).
* (17) S. Kivelson, Phys. Rev. B 26 4269 (1982).
* (18) N. W. Ashcroft and M. D. Mermin, Solid State Physics (Harcourt, 1976).
* (19) L. Li, Z. Xu, and S. Chen, Phys. Rev. B 89 085111 (2014).
* (20) W. Kohn, Phys. Rev. 133 A171 (1964).
* (21) A. Altland and M. R. Zirnbauer, Phys. Rev. B 55 1142 (1997).
* (22) A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W. Ludwig, Phys. Rev. B 78 195125 (2008).
* (23) R. Jackiw and C. Rebbi, Phys. Rev. D 13 3398 (1976).
* (24) M. Nakahara, Geometry, Topology, and Physics, Graduate Student Series in Physics, CRC Press, 2nd Edition (2003).
* (25) B. Hetényi and B. Dóra, Phys. Rev. B 99 085126 (2019).
* (26) C. C. Adams, The Knot Book, American Mathematical Society, Providence, Rhode Island (2001).
* (27) I. Souza, T. Wilkens, and R. M. Martin, Phys. Rev. B 62 1666 (2000).
* (28) R. Resta and S. Sorella, Phys. Rev. Lett. 82 370 (1999).
* (29) M. Yahyavi and B. Hetényi, Phys. Rev. A 95 062104 (2017).
* (30) M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, I. Bloch, Nat. Phys. 9 795 (2013).
* (31) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, Y. Takahashi, Nat. Phys. 12 296 (2016).
|
# Toroidic and antitoroidic orders in hexagonal arrays of dielectric trimers:
magnetic group approach
Victor Dmitriev1 Silvio Domingos Silva Santos1 Andrey B. Evlyukhin2 Anton S.
Kupriianov3 Vladimir R. Tuz4<EMAIL_ADDRESS>1Electrical Engineering
Department, Federal University of Para, R. Augusto Correa 01, CEP 66075-900
Belem, Para, Brazil 2Institute of Quantum Optics, Leibniz Universität
Hannover, Welfengarten Street 1, 30167 Hannover, Germany 3College of Physics,
Jilin University, 2699 Qianjin Street, Changchun 130012, China 4State Key
Laboratory of Integrated Optoelectronics, College of Electronic Science and
Engineering, International Center of Future Science, Jilin University, 2699
Qianjin Street, Changchun 130012, China
###### Abstract
Herein, we investigate symmetry-protected toroidal dipole resonances and
conditions of their excitation in a new type of electromagnetic metamaterials.
These metamaterials are all-dielectric planar periodic arrays of dielectric
disks disposed on a dielectric substrate. The elementary building blocks of
the array are trimers which are distributed in hexagonal unit supercells. The
highest geometrical symmetry of the unit supercell is $C_{6v}$. The analysis
is fulfilled by using the representation theory of groups with application of
the magnetic group theory, which is a new approach in solving such problems.
We have shown that to get access to the toroidal supermodes of the array,
symmetry of the unit supercell must be broken twice: firstly, the $C_{3v}$
symmetry of the trimer, and secondly, the $C_{6v}$ symmetry of the unit
supercell needs to be reduced. Selection rules for the symmetric and
antisymmetric orders of the toroidal dipole moments in the arrays are defined.
In particular, we have shown that with the reduction of the unit supercell
symmetry to the $C_{2v}$ group, the array exhibits the toroidal dipole
resonance with antitoroidic order. The arrays with the lower $C_{s}$ symmetry
can provide the resonances with both toroidic and antitoroidic orders. It is
also shown that these arrays are always polarization sensitive. Full-wave
simulations and experiments confirm the theoretical predictions. The suggested
metamaterials can provide an enhanced light-matter interaction due to the
spatially and temporally confined light in resonant systems with very high
quality factors.
††preprint: APS/123-QED
## I Introduction
The majority of physical systems possess intrinsic symmetries which define the
areas of possible solutions of the equations governing these systems. With the
use of group-theoretical methods, one can classify the corresponding solutions
defined by the underlying symmetries. In particular, symmetry analysis by the
group-theoretical approach has been successfully applied to simplify the
description of different physical phenomena and used as a general guideline to
design many practical devices in electronics, acoustics, and optics Rousseau
_et al._ (1981); Bak (1985); Sakoda (1995); Ochiai and Sakoda (2001); Hergert
and Däne (2003); Mock _et al._ (2010); Alagappan _et al._ (2008). To date,
there exist many textbooks describing the use of group theory for solving
specific physical problems Hamermesh and Mullin (1962); Tinkham (1964);
Cornwell (1984); Ludwig and Falter (1988); Inui _et al._ (1996); Bradley and
Cracknell (2009); Hergert and Geilhufe (2018).
Besides simplification of numerical calculations, group theory can be applied
to classify promising systems for further investigations, such as in the case
of search for multiferroic materials (ferroics) Schmid (2008); Saxena and
Lookman (2011). Different ferroics are classified in terms of spatial
inversion and time reversal symmetry of their order parameter Spaldin _et
al._ (2008). In general, four primary orders are distinguished in ferroics:
ferroelasticity, ferroelectricity, ferromagnetism, and ferrotoroidicity Loidl
_et al._ (2008); Khomskii (2009); Gnewuch and Rodriguez (2019) (in what
follows, the prefix “ferro” for the ferrotoroidicity is omitted). The
magnetoelectric coupling is a secondary ferroic effect, which is inherent to
toroidic order. It is of special interest in applications Li and Tang (2019).
The occurrence of multiple ferroic properties in one phase is related to
specific symmetry conditions a material has to accomplish.
The lack of natural ferroic materials usable in optics has motivated a search
for structures and systems that may exhibit magnetoelectric coupling arising
from the metamaterial design. In particular, specifically designed particles
(meta-atoms or meta-molecules) can acquire coupled electric and magnetic
polarizabilities associated with the conduction currents that occur when
metamaterials are irradiated by electromagnetic fields Monticone and Alù
(2014). Such particles typically are metallic split-ring resonators (SRRs)
whose dynamic magneto-optical response somewhat resembles the static one of
natural ferroic substances Pendry _et al._ (1999); Zhou _et al._ (2005);
Decker _et al._ (2009); Kaelberer _et al._ (2010); Savinov _et al._ (2014);
Cong _et al._ (2018). To describe the properties of such metamaterials
composed of SRRs, the group-theoretical approach can be used. It allows one to
calculate the electromagnetic modes of a resonator and determine whether this
resonator exhibits a desired magneto-optical response Wongkasem _et al._
(2006); Baena _et al._ (2006, 2007); Wongkasem _et al._ (2006); Padilla
(2007); Dmitriev (2009, 2011, 2013); Reinke _et al._ (2011). Nevertheless, as
far as we know, such an approach has not previously been used to describe the
properties of SRR-based metamaterials exhibiting toroidicity.
Recent progress in the area of metamaterials is related to the investigation
of their all-dielectric implementations Evlyukhin _et al._ (2010); Jahani and
Jacob (2016); Kruk and Kivshar (2017); Babicheva and Evlyukhin (2021) that are
promising candidates to overcome some issues inherent to metallic SRR-based
metamaterials at higher frequencies. In all-dielectric metamaterials, Mie
resonances of dielectric nanoparticles caused by displacement currents provide
an alternative route for achieving dynamic magneto-optical response Evlyukhin
_et al._ (2012); Miroshnichenko _et al._ (2012, 2017); Lepeshov and Kivshar
(2018); Terekhov _et al._ (2019). Symmetry of the dielectric resonators is
tightly related to the mode structure which determines the electromagnetic
features of the resonator. The structure of fields can be derived using the
group-theoretical approach Sadrieva _et al._ (2019); Gladyshev _et al._
(2020).
Group-theoretical methods are especially useful for metamaterials whose
constitutive particles have a complex shape Xiong _et al._ (2020) or possess
an in-plane broken symmetry Khardikov _et al._ (2012); Tuz _et al._ (2018a);
Sayanskiy _et al._ (2019); Kupriiannov _et al._ (2020); Evlyukhin _et al._
(2020). The resonant conditions in such metamaterials are stipulated by
inherently nonradiating symmetry protected (dark) modes existing in the
particles. Such modes become weakly radiative when the symmetry of the
particles is broken (such symmetry protected states are also referred to as
trapped modes Fedotov _et al._ (2007) or, recently, as bound states in the
continuum (BICs) Koshelev _et al._ (2018)). The coupling to these trapped
modes with incoming radiation can be controlled by the strength of asymmetry
introduced in the particle. After a group-theoretical analysis of asymmetric
particles, it is revealed that the dynamic magneto-optical response of the
entire metamaterial can resemble the characteristics of ferroics with the
magnetic dipole arrangement in either ferromagnetic or antiferromagnetic order
when the trapped mode is excited Yu _et al._ (2019); Tuz _et al._ (2020a).
The mechanism of symmetry reduction for the trapped mode excitation covered by
the group-theoretical approach is also applicable to all-dielectric
metamaterials composed of particle clusters Overvig _et al._ (2020). In the
context of ferroics, such metamaterials can possess toroidicity arising from
the collective modes (supermodes) of the clusters Basharin _et al._ (2015);
Tasolamprou _et al._ (2016, 2020); Xu _et al._ (2019); Zhang _et al._
(2019). Due to low inherent losses in constituent materials, the symmetry-
protected toroidal modes in all-dielectric metamaterials demonstrate a very
high quality factor (high-$Q$) resonant response accompanied by the near-
surface confinement of the strong electromagnetic field Tuz _et al._ (2018b);
Kupriianov _et al._ (2019); Tuz _et al._ (2020b). Remarkably, the conditions
for the appearance of toroidal dipole modes in cluster-based all-dielectric
metamaterials can be expressed in the explicit electromagnetic formulation and
described in the group-theoretical language Dmitriev _et al._ (2021).
In the present paper, we aim to introduce a special approach based on the
magnetic group theory to describe characteristics of the toroidal supermodes,
which are implemented in cluster-based all-dielectric metamaterials. The
constitutive cluster of these metamaterials has a hexagonal geometry, which is
a key design in the topological photonics Wu and Hu (2015); Gorlach _et al._
(2018); Yang _et al._ (2018); Jiang _et al._ (2019); Xiong _et al._ (2019);
Xi _et al._ (2020). We demonstrate that in the given metamaterials, it is
possible to realize excitation of the toroidal supermodes possessing either
symmetric order (toroidal order, TO), when the toroidal dipole moments of all
trimers are parallel, or antisymmetric order (antitoroidal order, ATO), when
the toroidal dipole moments in the neighboring trimers are antiparallel having
a staggered distribution. We define the selection rules and show in detail how
these resonances can be excited by a specific breaking of the metamaterial
unit cell symmetry. We perform a set of numerical simulations supported by
microwave experiments to reveal specific physical properties of such
metamaterials.
The rest of the paper is organized as follows. Section II introduces the
concept of a toroidal dipole moment existing in a trimer of dielectric disks.
Then, in Sec. III, a metamaterial composed of hexagonal unit supercells of
trimers is designed to obtain a system supporting a net toroidal dipole
moment. In Sec. IV the symmetry analysis of the hexagonal unit supercell is
performed. We explain what perturbations should be introduced into the
supercell to excite the toroidal dipole moments of the hexagonal cluster by
the field of a normally incident linearly polarized wave. The theory of
magnetic groups applied to the hexagonal structure is presented in Sec. V.
This theory predicts that the toroidal dipole moments of the supercell can
appear either in symmetric or antisymmetric order. The selection rules for
these orders are derived in Sec. VI. Theoretical description and numerical
simulations of excitation of the symmetric and antisymmetric orders in arrays
due to specific perturbations of their hexagonal unit supercell are given in
Secs. VII and VIII, respectively. In Sec. IX, the verification of theory with
a microwave experiment is provided. The presented results are discussed and
summarized in Secs. X and XI, respectively.
## II Toroidal dipole mode of trimer
Clusters of dielectric particles (meta-molecules) possess a set of natural
modes (eigenmodes) that are defined by the cluster symmetry and obey group
theory rules. This is similar to the case of conventional molecules. The
eigenmodes can be related to coefficients of the multipole decomposition of
the corresponding Mie solution derived for the problem of particles
interacting with an electromagnetic field Bohren and Huffman (1998). When the
particles are arranged into a cluster, the eigenmodes of individual particles
strongly interact and form supermodes of the cluster. A particular supermode
can be designated as bonding or antibonding, depending on whether the induced
coupling between the particles in the cluster appears in the low- or high-
energy configuration Rechberger _et al._ (2003); Chuntonov and Haran (2011).
The exact arrangement of the dielectric particles within a cluster therefore
has an essential impact on the symmetry of the resulting supermode.
In the present study, we consider a complex hexagonal cluster consisting of
six trimers of dielectric disk-shaped particles. Thus, we have the following
interaction hierarchy: the eigenmodes of three individual disks are coupled
into the trimer eigenmodes, which are then coupled into the hexagonal cluster
supermode. We are interested in a particular eigenmode of the trimer, which
exhibits a significant contribution from the toroidal dipole moment Xu _et
al._ (2019); Tuz _et al._ (2020b); Dmitriev _et al._ (2021). Then we
consider the coupling of these toroidal eigenmodes within the hexagonal
cluster which can be realized as a bonding (antitoroidal) or antibonding
(toroidal) state.
One should note that the analysis of eigenmodes of a complex system of
dielectric particles does not necessarily require the involvement of the
concept of a toroidal moment. For instance, a mode coupling theory used for
designing dielectric antennas and microwave circuits can be considered as an
alternative approach Wang and Zaki (2007); Trubin (2016); Fesenko _et al._
(2019). Nevertheless, we are convinced that our description based on the
toroidal modes makes it possible to obtain a clear physical picture of the
realization of a specific resonant state in the system under consideration.
In what follows we analyze the conditions of appearance of a toroidal dipole
moment in a single trimer of dielectric disk-shaped particles. Each disk in
the trimer is situated in the vertex of an equilateral triangle, therefore the
trimer symmetry is described by the point group $C_{3v}$. The triangle side
size is $a_{d}$. The radius and thickness of the disks are $r_{d}$ and
$h_{d}$, respectively. The disks are made of a nonmagnetic material with
permittivity $\varepsilon_{d}$ and arranged in a homogeneous ambient space
with permittivity $\varepsilon_{s}=1$.
A toroidal dipole moment of a system with electric current density
distribution ${\bf j}({\bf r},\omega)$ (where $\bf r$ is the radius-vector of
a volume element and $\omega$ is the angular frequency) is governed by the
following equation Dubovik and Tugushev (1990); Marinov _et al._ (2007); Chen
_et al._ (2011); Evlyukhin _et al._ (2016)
${\bf T}=\frac{1}{10}\int_{V}[({\bf r}\cdot{\bf j}){\bf r}-2r^{2}{\bf j}]d{\bf
r},$ (1)
where $V$ is the volume occupied by the current density. Here we omitted the
arguments of the current density ${\bf j}({\bf r},\omega)$ and consider that
the toroidal dipole moment is located at the origin of the chosen coordinate
frame. Equation (1) can be directly obtained from the Cartesian multipole
decomposition of the current density where toroidal dipole moment presents a
third-order multipole term Chen _et al._ (2011); Evlyukhin _et al._ (2016).
This multipole decomposition is based on the Taylor series of the Dirac
$\delta$-function.
Figure 1: Exact electric dipole moment with corresponding contributions of the
LWA electric dipole and toroidal dipole moments of a single trimer for its (a)
lateral and (b) frontal irradiation by a linearly polarized wave. The
designation of the vectors $\bf k$ and $\bf E$ of the incident wave and an
appearance of the trimer toroidal eigenmode are given in the inserts. In the
eigenmode, red and black arrows correspond to the flow of electric
polarization currents and magnetic field respectively, the bold yellow arrow
indicates the toroidal dipole moment $\bf T$, and the bold blue arrows
indicate the magnetic dipole moments $\bf m$ of individual disks. The
eigenmode position on the wavelength scale is marked by a red cross.
Parameters of the trimer are: $\varepsilon_{d}=22$, $h_{d}/D=0.45$, and
$a_{t}/D=1.125$.
In the approach based on the spherical harmonic expansion of radiated
(scattered) waves Alaee _et al._ (2018); Evlyukhin and Chichkov (2019), the
explicit contribution of the toroidal dipole moment in the multipole
decomposition does not appear, and it is associated with the exact electric
dipole moment Fernandez-Corbaton _et al._ (2017):
${\bf p}=\frac{i}{\omega}\int_{V}\left(j_{0}(kr){\bf
j}+\frac{k^{2}}{\omega}\frac{j_{2}(kr)}{(kr)^{2}}[3({\bf r}\cdot{\bf j}){\bf
r}-r^{2}{\bf j}]\right)d{\bf r},$ (2)
where $j_{0}(kr)$ and $j_{2}(kr)$ are the spherical Bessel functions of the
zero- and second-order, respectively, $k$ is the wave number in surrounding
medium. Applying the corresponding Taylor expansions to the Bessel functions
in Eq. (2) and writing out the first three terms explicitly, one obtains
$\displaystyle{\bf p}$ $\displaystyle=$
$\displaystyle\frac{i}{\omega}\int_{V}{\bf j}d{\bf
r}+\frac{ik}{c}\frac{1}{10}\int_{V}[({\bf r}\cdot{\bf j}){\bf r}-2r^{2}{\bf
j}]d{\bf r}$ (3)
$\displaystyle+\frac{ik^{3}}{c}\frac{1}{280}\int_{V}[3r^{4}{\bf j}-2r^{2}({\bf
r}\cdot{\bf j}){\bf r}]d{\bf r}+\ldots$ $\displaystyle=$ $\displaystyle{\bf
p}_{0}+\frac{ik}{c}{\bf T}+\frac{ik^{3}}{c}{\bf T}^{(R)}+\ldots$
where $c$ is the light speed in surrounding medium, ${\bf p}_{0}$ is the
Cartesian electric dipole moment obtained from the $\delta$-function Taylor
expansion corresponding to the long-wavelength approximation (LWA) Alaee _et
al._ (2018), ${\bf T}^{(R)}$ is the mean-square radius of the toroidal moment
Nemkov _et al._ (2018); Gurvitz _et al._ (2019).
From Eq. (3) one can consider the toroidal dipole moment as only a next
correcting term to the Cartesian electric dipole moment ${\bf p}_{0}$ used for
calculation of the exact electric dipole moment $\bf p$, and, thus, the
toroidal dipole moment cannot be considered separately. However, this
statement does not reflect total physical role of the toroidal dipole moment.
Indeed, depending on the frequency of electromagnetic fields, shape, size, and
material parameters of an electric current system, it can appear that the
first term in Eq. (3) is very small, or even equals to zero, so that the main
contribution goes from the toroidal term ($ik{\bf T}/c$). In this case, the
electric current systems can be considered as supporting a purely electric
toroidal response, and, thus, the toroidal dipole moment acquires an
independent physical meaning.
To demonstrate this peculiarity, we consider the trimer irradiation by an
electromagnetic plane wave from both lateral and frontal directions. The
absolute value of the exact electric dipole moment $\bf p$ and the
corresponding contributions of the LWA electric dipole ${\bf p}_{0}$ and the
toroidal dipole moments $\bf T$ together with the mean-square radius ${\bf
T}^{(R)}$ are presented in Fig. 1 for two irradiation conditions of the
linearly polarized plane wave. Calculation procedure corresponds to the
description from Tuz _et al._ (2020b). In our calculations, we normalize the
incident wavelength and all geometrical parameters of the problem on the disk
diameter $D=2r_{d}$.
One can see that at the lateral irradiation condition [Fig. 1(a)], the exact
dipole moment of the trimer is resonantly excited. Decomposition of its value
on the basis of Eq. (3) shows that the toroidal dipole $\bf T$ provides the
main contribution in the resonant exact electric dipole moment $\bf p$ [the
corresponding wavelength is indicated by a red cross in Fig. 1]. In this case,
the electric dipole response of the trimer is associated with excitation of
the trimer toroidal dipole mode [see the inset with the trimer toroidal
eigenmode in Fig. 1(a)]. For the other irradiation conditions [Fig. 1(b)] the
toroidal dipole mode is not excited. In this case, the exact dipole moment
$\bf p$ is basically determined by the interference between the ${\bf p}_{0}$
and $\bf T$ dipole moments. Note that in this condition the significant
suppression of $\bf p$ appears at the region of large wavelengths as a result
of destructive interference. This state can be attributed to the anapole state
Zenin _et al._ (2017); Yang and Bozhevolnyi (2019); Baryshnikova _et al._
(2019); Savinov _et al._ (2019).
Therefore, it is revealed that for the given eigenmode of the trimer, the
toroidal dipole moment provides a dominant contribution, and this eigenmode
can arise only for a certain type of excitation of the incident field. In
particular, this toroidal mode can be considered as a dark and bright state
for the field of frontally and laterally incident wave, respectively. In what
follows, we show the mechanism of access to this indicated dark state under
frontal irradiation conditions in metamaterials composed of hexagonal clusters
of trimers by breaking their symmetry.
## III Metamaterial with hexagonal unit supercell
Figure 2: Coordinate frame and schematic view of a dielectric hexagonal array
composed of trimer-based supercells. The unit supercell is outlined by a red
dashed contour.
We consider a metamaterial composed of a planar array with trimers of
dielectric disks (i.e., the trimer is a minicell of the metamaterial). The six
trimers are arranged in such a way that hexagonal clusters appear (i.e., the
hexagonal cluster is a supercell of the metamaterial). The hexagon lateral
size is $p$. The array of disks is situated in the $x$-$y$ plane as it is
shown schematically in Fig. 2. To form a metamaterial, the dielectric disks
are immersed symmetrically (preserving the plane of symmetry $z=0$) into a
dielectric substrate with relative permittivity $\varepsilon_{s}$ and
thickness $h_{s}$.
A peculiarity of this new type of metamaterial is a highly symmetric hexagonal
geometry of its unit supercells based on dielectric trimers. The optical
response of the entire metamaterial is largely determined by the symmetry of
the supercell, which can be deliberately lowered by introducing certain
perturbations into the hexagonal cluster.
Our goal is to find specific perturbations so that the toroidal dipole mode of
the trimers can be effectively excited in the metamaterial by the field of a
normally incident linearly polarized wave with the wave vector ${\bf
k}=\\{0,0,-k_{z}\\}$ and wavelength $\lambda=2\pi c/\omega$. We define that
the metamaterial is irradiated by either $x$-polarized (${\bf E},{\bf
H}=\\{E_{x},H_{y}$,0}) or $y$-polarized (${\bf E},{\bf
H}=\\{H_{x},E_{y},0\\}$) wave. Due to nature of the toroidal dipole mode, it
is appropriate to consider the problem using the magnetic field ${\bf H}$ of
the incident wave as the exciting source.
## IV Supercell symmetry analysis
For the given planar arrays of dielectric disks, one can consider the problem
in the framework of two-dimensional (2D) symmetry. Since the center of the
equilateral triangle of the underlying trimer and the center of the minicell
are the same, the minicell geometry corresponds to the $C_{3v}$ symmetry. To
get access to the toroidal dipole mode, the minicell symmetry must be reduced
Dmitriev _et al._ (2021). This symmetry reduction can be realized in several
ways. In particular, the thickness of particular disks in the trimers can be
resized (out-of-plane perturbation) or these disks can be shifted aside (in-
plane perturbation).
The geometry of the hexagonal unit supercell corresponds to the $C_{6v}$
symmetry. As a main reference point, the scheme of the supercell with planes
of symmetry and rotational elements of the $C_{6v}$ group are presented in
Fig. 3 (the Schöenflies notation of elements of point symmetry is given in
Appendix A).
Figure 3: Hexagonal supercell described by the $C_{6v}$ group and elements of
this group. Figure 4: Examples of the perturbed hexagonal unit supercells
whose symmetry is reduced to the groups: (a) $C_{6v}$, (b) $C_{6}$, (c)
$C_{3v}$, (d) $C_{3}$, (e) $C_{2v}$, (f) $C_{2}$, (g) $C_{s}^{(1)}$, and (h)
$C_{s}^{(2)}$. Perturbed disks are denoted by red circles.
With such a supercell symmetry, the toroidal dipole mode is a dark state of
the hexagonal cluster and cannot be excited by the field of a normally
incident linearly polarized wave. Therefore, the symmetry $C_{6v}$ must be
reduced. Particular perturbations lead to different manifestations of the
toroidal dipole mode in the metamaterial, which can arise in either symmetric
or antisymmetric order. The relationship between the perturbed supercell
symmetry and the toroidal dipole mode order is the subject of our subsequent
study.
Here we introduce a classification of possible reductions of the supercell
symmetry, assuming that only one disk in each trimer forming the hexagonal
cluster can be perturbed and only one type of perturbation is allowed. With a
set of such allowed perturbations, one can design supercells having the
highest $C_{6v}$ symmetry as well as the symmetries of the $C_{6v}$ subgroups
(the subgroup decomposition of the $C_{6v}$ group with the $C_{6}$, $C_{3v}$,
$C_{3}$, $C_{2v}$, $C_{2}$, and $C_{s}$ subgroups is illustrated in Fig. 11 of
Appendix A).
The unique way to obtain the $C_{6v}$ symmetry in the perturbed supercell is
shown in Fig. 4(a). In this geometry, all six perturbed disks are closest to
the center of the hexagonal supercell. The perturbed supercells with the
$C_{6}$, $C_{3v}$, and $C_{3}$ symmetries are shown in Figs. 4(b)-4(d). Notice
that these geometries are not unique and the positions of the perturbed disks
can be different. The geometries of the supercells with the $C_{2v}$ and
$C_{2}$ symmetries are presented in Figs. 4(e) and 4(f), respectively. These
geometries are also not unique.
The $C_{s}$ symmetry can be realized in two variants, where the plane of
symmetry $\sigma$ passes through an apex [Fig. 4(g)] or between two apexes
[Fig. 4(h)] of the hexagon. These variants are designated as the $C^{(1)}_{s}$
and $C^{(2)}_{s}$ symmetries, respectively. Again, different geometries are
possible in the hexagonal clusters with the $C^{(1)}_{s}$ and $C^{(2)}_{s}$
symmetries.
## V Magnetic group description
We investigate electromagnetic response of the given metamaterial to external
excitation by an alternating electromagnetic field. However, at a certain
point in time, one can get a fixed picture of the field distribution in the
unit supercell. This field pattern is associated with the characteristics of
both the eigenmode that is excited and the magnetic field vector of the
incident wave. Thus one can construct a general theory which combines the
geometric symmetry of the unit supercell and the “dynamic” symmetry of the
alternating magnetic fields and toroidal moment (see Appendix D).
In the eigenmode of unperturbed trimer, the toroidal dipole moment $\bf T$ has
only single component $T_{z}$ Dmitriev _et al._ (2021). In a fixed moment of
time, the normalized vector $\bf T$ can acquire only two states being oriented
either “up” ($T_{z}=+1$) or “down” ($T_{z}=-1$). Accordingly, in a set of two
trimers, two combinations of the toroidal dipole moments are possible when
they are paired in either bonding (contra-directional) or antibonding (co-
directional) fashion. When considering the hexagonal unit supercell, we are
interested in two specific toroidal eigenmodes, whose moments are oriented
either in the staggered order or in the same direction in all six trimers (see
Fig. 5). In what follows, we distinguish these eigenmodes as the ATO and TO
modes, respectively.
Figure 5: Schematics of (a) ATO and (b) TO modes in a hexagonal unit
supercell. Blue and yellow arrows demonstrate the magnetic field flow and
orientation of the toroidal dipole moments, respectively.
In the symmetry analysis, these eigenmodes can be attributed to the one-
dimensional (1D) irreducible representations (IRREPs) of the $C_{6v}$ group.
The TO mode belongs to IRREP $A_{1}$, i.e., it is not changed under all the
symmetry elements. The ATO mode is transformed in accordance with IRREP
$B_{1}$ and the toroidal dipole moment changes its sign under the $C_{2}$,
$C_{6}^{1}$, and $C_{6}^{-1}$ transformations, and three reflections
$\sigma_{d}$ (see Table 2 of Appendix B).
Notice that the in-plane magnetic dipole moments induced in dielectric disks
by the external magnetic field can be used to construct an alternative theory.
In this case, one has to work with 2D representations of the axial vector $\bf
m$ rather than with 1D IRREPs of the polar vector ${\bf T}$. However, this
alternative description involves more complex transformation properties of
$\bf m$ and this complicates the analysis.
Taking advantage of the time reversal symmetry of the toroidal dipole moment
$\bf T$, one can use this quantity to define the magnetic symmetry of the
alternating fields in the array. Magnetic groups based on the $C_{6v}$ group
of symmetry, its subgroups and elements of these groups are summarized in
Table 1. In this table, ${\cal T}$ is the time reversal operator.
Table 1: Possible symmetries of toroidal dipole modes: magnetic groups based on the $C_{6v}$ symmetry, their subgroups and elements 1st category | $C_{6v}+{\cal T}C_{6v}$ | $C_{6}+{\cal T}C_{6}$ | $C_{3v}+{\cal T}C_{3v}$ | $C_{3}+{\cal T}C_{3}$ | $C_{2v}+{\cal T}C_{2v}$ | $C_{2}+{\cal T}C_{2}$ | $C_{s}+{\cal T}C_{s}$
---|---|---|---|---|---|---|---
2nd category | $\bf C_{6v}$ | $\bf C_{6}$ | $\bf C_{3v}$ | $\bf C_{3}$ | $\bf C_{2v}$ | $\bf C_{2}$ | $\bf C_{s}$
Content | $e$, $C_{6}$, $C_{6}^{-1}$, $C_{2}$ | $e$, $C_{6}$, $C_{6}^{-1}$ | $e$, $C_{3}$, $C_{3}^{-1}$ | $e$, $C_{3}$, $C_{3}^{-1}$ | $e$, $C_{2}$ | e, $C_{2}$ | $e$, $\sigma$
| $C_{3}$, $C_{3}^{-1}$, $3\sigma_{v}$, $3\sigma_{d}$ | $C_{3}$, $C_{3}^{-1}$, $C_{2}$ | $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ | | $\sigma_{1}$, $\sigma_{2}$, | |
3rd category | $C_{6v}(C_{6})$ $C_{6v}(C_{3v})$ | $C_{6}(C_{3})$ | $C_{3v}(C_{3})$ | | $C_{2v}(C_{2})$ $C_{2v}(C_{s})$ | $C_{2}(C_{1})$ | $C_{s}(C_{1})$
Content | $e$, $C_{6}$, $C_{6}^{-1}$ $e$, $C_{3}$, $C_{3}^{-1}$ | $e$, $C_{3}$, $C_{3}^{-1}$ | $e$, $C_{3}$, $C_{3}^{-1}$ | | $e$, $C_{2}$ $e$, $\sigma_{1}$ | $e$, ${\cal T}C_{2}$ | $e$, ${\cal T}\sigma$
| $C_{2}$, $C_{3}$, $C_{3}^{-1}$ $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ | ${\cal T}C_{2}$, ${\cal T}C_{6}$, | ${\cal T}\sigma_{1}$, ${\cal T}\sigma_{2}$, | | ${\cal T}\sigma_{1}$, ${\cal T}\sigma_{2}$ ${\cal T}C_{2}$, ${\cal T}\sigma_{2}$ | |
| $3{\cal T}\sigma_{v}$ ${\cal T}C_{2}$, ${\cal T}C_{6}$ | ${\cal T}C_{6}^{-1}$ | ${\cal T}\sigma_{3}$ | | | |
| $3{\cal T}\sigma_{d}$ ${\cal T}C_{6}^{-1}$ | | | | | |
In terms of magnetic groups, the vector $\bf T$ is odd in time. Application of
the time reversal operator ${\cal T}$ to the toroidal dipole moment $T_{z}$
results in the vector $\bf T$ reorientation, ${\cal T}T_{z}={-T_{z}}$.
Table 2 of Appendix B in the first seven columns presents space transformation
properties of the polar vector $\bf T$. This transformation is defined by the
formula ${T_{z}}^{\prime}=\chi T_{z}$, where $T_{z}$ is a given toroidal
moment, ${T_{z}}^{\prime}$ is the mapped moment, and $\chi$ is 1D IRREP of the
rotation-reflection symmetry element $R$. The eighth column of this table
consists of the space-time transformation of $\bf T$ (which is an odd in time
quantity) in terms of magnetic groups. Thus, Table 2 provides a description of
two sides (two physical properties) of the toroidal moment $\bf T$.
It should be emphasized that we apply the theory of magnetic groups only to
the fields and toroidal dipole moments $\bf T$ of the corresponding
eigenmodes. However, the possible magnetic groups are defined by geometrical
symmetry of the unit supercell.
## VI Selection rules and polarization properties of arrays
A detailed description of selection rules for the quasi-bound states in the
case of photonic crystal slabs is presented in Overvig _et al._ (2020), where
the authors intensively used the theory of nonmagnmetic groups. One of our
motivations in this work is to show that the magnetic group theory gives some
advantages in description of toroidal dipole mode resonances.
The problem of the toroidal dipole mode excitation in the array of hexagonal
supercells can be divided into two subproblems: (i) excitation of the toroidal
dipole mode in an individual trimer and (ii) excitation of the whole hexagonal
unit supercell.
To resolve the first subproblem, one should reduce the $C_{3v}$ symmetry of
the trimer to the $C_{s}$ group. This mechanism is studied both theoretically
and experimentally in Refs. Xu _et al._ (2019); Tuz _et al._ (2020b);
Dmitriev _et al._ (2021), so we omit the details here (for a brief
description, see Appendix D). In the present consideration, we concentrate on
solving the second subproblem, i.e., defining the selection rules for
symmetries of the hexagonal supercell which allow or forbid excitation of
toroidal dipole modes in the metamaterial by the field of a normally incident
linearly polarized wave.
### VI.1 Selection rules for ATO mode
The magnetic field $\bf H$ of the incident wave contains the elements $e$,
${\cal T}C_{2}$, ${\cal T}\sigma_{2}$, and $\sigma_{1}$ of magnetic symmetry
(see Appendix C). The element $C_{2}$ is incompatible with ${\cal T}C_{2}$ of
the magnetic field $\bf H$ (these elements are “orthogonal” in the sense that
their application leads to opposite orientations of the toroidal dipole
moment). Therefore, if magnetic symmetry contains the $C_{2}$ element,
excitation of an ATO mode in the metamaterial with such a symmetry is
forbidden. It imposes a very rigid (hard) restriction, which does not depend
on the orientation of the magnetic field $\bf H$. The same is true for the
groups of symmetries containing the $C_{6}$ and $C_{3}$ elements, because the
$C_{6}$ element is always accompanied by the $C_{2}$ element, whereas, from
the viewpoint of polarization properties, the $C_{2}$ element is “hidden” in
the $C_{3}$ group. However, there is no restriction on the presence of the
element ${\cal T}C_{2}$.
Now we apply to the restrictions imposing by planes $\sigma$ and antiplanes
${\cal T}\sigma$ of symmetry. The magnetic groups of ATO mode cannot contain
an element $\sigma$ coinciding with ${\cal T}\sigma$ of the magnetic field.
Analogously, the magnetic groups of the ATO mode cannot contain the element
${\cal T}\sigma$ coinciding with $\sigma$ of the magnetic field. Therefore,
the vector $\bf H$ of the incident wave parallel to $\sigma$, or perpendicular
to ${\cal T}\sigma$ (both $\sigma$ and ${\cal T}\sigma$ are elements of the
toroidal eigenmode of the supercell) cannot be coupled to this mode. However,
a small deviation of orientation of $\bf H$ from the discussed planes leads to
appearance of the field $\bf H$ component which can interact with the
corresponding mode of the hexagonal unit supercell. Therefore, these selection
rules can be called soft restrictions.
The above analysis allows us to exclude from consideration those symmetries
for which excitation of the ATO mode in the hexagonal unit supercell is
forbidden. The magnetic groups which permit excitation of the ATO mode are
subgroups of the group of magnetic field $\bf H$. They are $C_{2v}(C_{s})$
(the group of $\bf H$ itself), $C_{2}(C_{1})$, $C_{s}(C_{1})$, and $\bf C_{s}$
(see Table 1). It means that the corresponding geometrical symmetries of the
unit supercells are $C_{2v}$, $C_{2}$, and $C_{s}$ (see corresponding
perturbations of the hexagonal unit supercell in Fig. 4).
### VI.2 Selection rules for TO mode
Now we apply to the selections rules of the TO mode. After application of any
of the rotations $C_{6}$, $C_{3}$, and $C_{2}$, the sign of $\bf T$ must be
preserved. Therefore, rotations with ${\cal T}$ in the magnetic groups which
change the sign of $\bf T$ are prohibited (they are ${\cal T}C_{2}$ and ${\cal
T}C_{6}$; the ${\cal T}C_{3}$ element does not exist in the magnetic groups).
For the same reason, the ${\cal T}\sigma$ elements are also not allowed.
Summarizing, the antiunitary elements cannot enter in the magnetic groups of
the TO mode.
On the other hand, the elements $C_{2}$, $C_{3}$, and $C_{6}$ are not allowed
due to their incompatibility with the field $\bf H$ symmetry. Therefore, the
plane $\sigma$ is the only permissible element of symmetry, and for the mode
excitation, this element must be oriented perpendicular to the field $\bf H$.
Thus, the toroidal dipole mode can be excited in the unit supercell with
$\bf{C_{s}}$ symmetry of the magnetic groups, which contains the element
$\sigma$. This condition is similar to that defining the excitation of the
toroidal dipole mode in an isolated trimer Dmitriev _et al._ (2021).
Two schemes of perturbation of the unit supercell described by the $C_{s}$
group are shown in Figs. 4(g) and 4(h). Among them, the scheme presented in
Fig. 4(g) permits excitation of both the ATO and TO modes.
### VI.3 Polarization properties of arrays with ATO and TO modes
Polarization properties of the given arrays depend on their symmetry
conditions. As it is known, for the polarization insensitivity the object must
have the rotational symmetry with the axis $C_{3}$, $C_{4},\ldots,C_{\infty}$
Mackay (1989). From the foregoing analysis, it follows that the symmetries
allowing the excitation of the ATO and TO modes do not contain rotational
elements $C_{n}$ with $n>2$. Therefore, polarization insensitivity of the
arrays is not possible for both the ATO and TO modes.
## VII Specific arrays description
### VII.1 ATO mode in $C_{2v}$ supercells
From the preceding consideration of the selection rules it follows that to get
access to the ATO mode, symmetry of the hexagonal unit supercell needs to be
reduced to the $C_{2v}(C_{s})$ groups [the corresponding unit supercell
perturbation is presented in Fig. 4(e)]. The $C_{2v}(C_{s})$ group of the
third category does not contain the isolated time reversal operator ${\cal
T}$. Thus, at an arbitrary fixed moment in time, changing the sign of time
inverts $T_{z}$.
In the group-theoretical language, the ATO mode belongs to the IRREP $B_{1}$
of the $C_{6v}$ group (see Table 2 in Appendix B). In this symmetry, the
toroidal dipole mode is a dark state. The IRREP $B_{1}$ of the $C_{6v}$
degenerates in the IRREP $B_{1}$ of the lower $C_{2v}$ group (see Tables 3 and
5 in Appendix B). The geometrical symmetry described by the $C_{2v}$ group is
consistent with the $C_{2v}(C_{s})$ symmetry of the incident magnetic field
$\bf H$. Therefore, in this case, one can expect an efficient excitation of
the toroidal dipole mode by a normally incident electromagnetic wave with a
proper polarization.
To obtain the high-$Q$ resonant conditions, the introduced perturbation in the
unit supercell should be small. When the perturbation is small, the
characteristics of the local electromagnetic fields in the array with the
$C_{2v}$ symmetry does not differ significally from those of the $B_{1}$ dark
eigenmode in the non-perturbed array with the $C_{6v}$ symmetry. One can
conclude, that the approximate symmetry of the electromagnetic field in the
hexagonal unit supercell with the $C_{2v}$ symmetry corresponds to the IRREP
$B_{1}$ of the $C_{6v}$ group or, in the framework of magnetic groups, to the
$C_{6v}(C_{3v})$ group.
### VII.2 ATO and TO modes in $C_{s}$ supercells
Reduction of the $C_{6v}$ symmetry to the $C_{s}$ group allows one to excite
the ATO and TO modes (see Tables 4 and 5 in Appendix B). Besides the unit
element $e$, the $C_{s}$ group contains only one plane of symmetry $\sigma$.
This plane can either lie between the trimers [Fig. 4(g)] or pass through the
centers of the trimers [Fig. 4(h)]. Recall, that these geometries are not
unique, some other positions of the perturbed disks can also result in the
$C_{s}$ symmetry.
Figure 6: Excitation of (a) ATO mode and (b) TO mode in the same hexagonal
supercell whose symmetry is reduced to the $C_{s}^{(1)}$ group. Perturbed
disks are denoted by red circles. Orientation of the magnetic fields $\bf H$
of the incident wave and flow of the inner magnetic field of the toroidal
dipole eigenmode are given by the dark and light blue arrows, respectively.
The test lines are denoted by grey color.
Noteworthy that if the magnetic field $\bf H$ of the incident wave is
orthogonal to $\sigma$ [see Fig. 6(a)], the ATO mode can be excited, whereas
for the same geometry but with the magnetic field $\bf H$ parallel to $\sigma$
[see Fig. 6(b)], the TO mode arises.
### VII.3 Rule of thumb for mode order
To determine the direction of in-plane rotation of the magnetic field in the
trimers and therefore the orientation of the out-of-plane toroidal dipole
moments in the array, the following simple rule of thumb can be used. One
should draw test lines through the centers of all trimers in the unit
supercell parallel to the magnetic field $\bf H$. If the perturbed disk of a
given trimer is on one side of the test line, it corresponds to a certain
direction of rotation of the local magnetic field of the corresponding mode.
If the perturbed disk is on the other side of the test line, this gives the
opposite direction of rotation of the local magnetic field. This allows one to
define immediately the order of the excited net toroidal dipole mode. If the
test line passes through the center of the perturbed disk, this resonator
cannot be excited directly by the incident field. However, it can be excited
indirectly via the neighboring resonators due to the coupling effect between
particles in the array.
This intuitive picture can be understood from Fig. 6, where the hexagonal unit
supercell with the $C_{s}^{(1)}$ symmetry is presented. When the vector ${\bf
H}$ is parallel to the $x$ axis [Fig. 6(a)], the perturbed disks appear at
different sides of the test lines which results in the staggered orientation
of the toroidal moments in the neighboring trimers, i.e., the ATO mode can be
excited. Contrariwise, for the same unit supercell but when the vector ${\bf
H}$ is parallel to the $y$ axis [Fig. 6(b)], the structure produces parallel
orientation of the toroidal moments, i.e., the TO mode can be excited.
## VIII Numerical results
We now proceed to check our group-theoretical description of the excitation of
toroidal dipole modes in the discussed arrays by performing the full-wave
numerical simulations. To calculate both eigenmodes and transmitted spectra of
the array irradiated by a linearly polarized plane wave, we use the RF module
of the commercial COMSOL Multiphysics finite-element electromagnetic solver.
In the solver, we impose the Floquet-periodic boundary conditions on four
sides of the unit cell to simulate an infinitely extended in the $x$-$y$ plane
arrangement of dielectric disks. The code related to the multipole
decomposition with accounting for the toroidal dipole term defined by Eq. (1)
is incorporated into the solver as a special component. The solver allows us
to plot the distribution of the inner electromagnetic field within the
hexagonal unit supercell at a specified resonant wavelength.
Initially we calculate the eigenmodes that exist in a reference metamaterial
composed of unperturbed supercells. The material losses in the disks are
excluded in these simulations. From the released results, two specific
eigenmodes are selected, which demonstrate the appearance of the parallel or
antiparallel orientation of the toroidal dipoles. Then the transmitted spectra
of the metamaterial are calculated in the wavelength range where the selected
eigenmodes exist. Both $x$-polarized and $y$-polarized waves are under
consideration.
The results of our simulations of characteristics of the reference
metamaterial are presented in Fig. 7. As before, we present our results in the
dimensionless parameters, where all values are related to the disk diameter
$D$. The eigenwave solution shows that the corresponding mode of the trimer
exists in the array in the ATO and TO states, whose resonant wavelengths are
different from the resonant wavelength of the toroidal dipole mode of a single
trimer.
One can conclude, that the spectral characteristics of the metamaterial are
the same for waves of both polarizations, and while the symmetry of the
supercells is preserved, there are no peculiarities in the transmitted spectra
at the wavelengths of the ATO and TO modes. This confirms the dark feature of
these modes.
Further two particular designs of the perturbed unit supercells are chosen to
show a possibility to excite the ATO and TO modes in the metamaterial. They
are the supercells with the $C_{2v}$ and $C_{s}^{(1)}$ symmetries [see Figs.
4(e) and 4(g), respectively]. As a perturbance, we consider the change in the
thickness of a corresponding disk in each trimer forming the unit supercells.
This change is defined by the value $\Delta h_{d}$, and the dimensionless unit
supercell asymmetry parameter is introduced as $\theta=(h_{d}+\Delta
h_{d})/h_{d}$.
Figure 7: (a) Transmitted spectra of the metamaterial with unperturbed unit
supercells excited by the $x$-polarized (solid blue lines) and $y$-polarized
(dashed red lines) wave. The normalized magnitude and flow of the magnetic
near-field of the (b) ATO and (c) TO modes. The material and geometrical
parameters of the trimers are the same as in Fig. 1 and $p/D=3.375$.
### VIII.1 Array with $C_{2v}$ symmetry of unit supercell
High symmetry imposes severe constraints on the possible geometry of the
electromagnetic field in the array. The group-theoretical description suggests
that the $C_{2v}$ group is the highest symmetry that allows excitation of the
ATO mode. This mode can be excited by the wave whose vector ${\bf H}$ is
parallel to the $x$ axis ($y$-polarized wave). This selection rule is
confirmed by our numerical calculations presented in Fig. 8(a), where the
resonance occurs in the transmitted spectra directly at the wavelength of the
ATO mode. Due to the high symmetry of the unit supercell, the spectral
characteristic around the ATO resonance is rather “clean”, i.e., the toroidal
dipole mode appears to be well isolated from other resonances in the spectra,
and the wavelength separation of the ATO mode is relatively large. When the
vector ${\bf H}$ of the incident wave is parallel to the $y$ axis, none of the
trimers is excited, which is in full accordance with the rule of thumb.
Figure 8: (a) Transmitted spectra of the metamaterial with perturbed trimers
excited by the $x$-polarized (solid blue lines) and $y$-polarized (dashed red
lines) wave. The trimers are perturbed by resonators with different height.
The unit supercell symmetry is $C_{2v}$. The normalized magnitude and flow of
the magnetic near-field of the ATO mode are given in the inserts, where the
perturbed resonators are denoted by dashed circles. (b) Evolution of the
quality factor (blue line) and normalized resonant wavelength (red line) of
the ATO mode as functions of the asymmetry parameter $\theta$. All parameters
of the structure are the same as in Fig. 7.
The frequency shift and quality factor of the ATO resonance as functions of
the asymmetry parameter $\theta$ are presented in Fig. 8(b). It is evident
that the less the asymmetry, the higher the quality factor, which reaches
values higher than $10^{5}$. The resonant wavelength shifts up as the
thickness of the perturbed disks increases.
### VIII.2 Array with the $C_{s}$ symmetry of unit supercell
The group-theoretical description predicts that in a metamaterial composed of
unit supercells with the $C_{s}$ symmetry, both ATO and TO modes can be
excited. The results of corresponding calculations are presented in Fig. 9(a).
These results confirm that the ATO and TO modes arise when the metamaterial is
irradiated by the $x$-polarized and $y$-polarized wave, respectively. In
general, these resonances occur at different wavelengths. However, due to the
low symmetry of the unit supercell, the transmitted spectra are filled with
several resonances, so that the ATO and TO resonances appear to be somewhat
masked against their background.
Figure 9: Same as in Fig. 8 but for arrays with the $C_{s}$ symmetry of the
hexagonal unit supercell. The corresponding normalized magnitude and flow of
the magnetic near-field of the ATO and TO modes are given in the middle
planes. Figure 10: Simulated and measured transmitted spectra and resonant
patterns of the real part and phase of the $z$-component of the electric near-
field for an actual array with the $C_{s}^{(1)}$ symmetry of the hexagonal
unit supercell. The array is irradiated by the (a) $x$-polarized and (b)
$y$-polarized wave to excite the ATO and TO mode, respectively. The inset
demonstrates a photo of the metamaterial prototype. Parameters of the
prototype are: $\varepsilon_{d}=22\pm 1$, $\tan\delta\approx 1\times 10^{-3}$,
$r_{d}=4$, $h_{d}=3.5$, $\Delta h_{d}=-1$ ($\theta\approx 0.71$),
$\varepsilon_{s}=1.1$, $h_{s}=10$, $a_{d}=9$, $p=27$. All geometrical
parameters are given in millimeters.
Since the material losses are excluded from the present simulations, the
maximal value of the quality factor of the given resonances depends only on
the asymmetry parameter $\theta$, which defines the coupling degree of the
toroidal dipole mode with free space. The net toroidal dipole moment related
to the ATO mode is almost zero since the antisymmetric toroidal dipole moments
in adjacent trimers compensate each other and reduce the electromagnetic
coupling of this mode with free space. The equally oriented moments of the TO
mode are lack of this property. Comparing quality factors of the ATO and TO
resonances presented in Fig. 9(b), one can conclude that the ATO mode
possesses the lower radiation losses and consequently higher quality factor
than that of the TO mode. In particular, these quality factors differ by one
order of magnitude. Besides, the ATO and TO resonances experience the same
wavelength shift when the parameter $\theta$ changes.
## IX Experimental results
We further validate our theory in neat experiments. To this end, we construct
a metamaterial prototype based on the $C_{s}^{(1)}$ unit supercells since this
design provides the efficient coupling of the metamaterial with a linearly
polarized incident wave via both the ATO and TO modes. We assemble the array
on a dielectric substrate utilizing disks made of a low-loss, high-
permittivity microwave ceramic.
For our experiments, we use a well-established technique that was described in
detail earlier Kupriianov and Tuz (2019). In particular, we perform our
experimental study in the microwave range ($8-15$ GHz) using Rohde & Schwarz
ZVA50 Vector Network Analyzer as the main measurement platform. Our setup also
includes a pair of dielectric-lens antennas, a near-field imaging system, and
all other necessary accessories.
The results of our measurements are summarized in Fig. 10. It includes a photo
of the actual metamaterial prototype, simulated and measured transmitted
spectra and electric near-field patterns plotted at the corresponding resonant
wavelengths. In the simulations supporting our experimental studies, the real
material losses ($\tan\delta$) existing in the metamaterial constituents are
taken into account.
Foremost, one can conclude that our experimental data are in good agreement
with the simulation results. The measurements performed confirm the existence
of polarization-dependent resonances for the ATO and TO modes. Importantly,
both resonances survive in a lossy metamaterial. Since the TO mode has a lower
quality factor, it can be more easily detected in the experiment.
The calculated and measured characteristics of the electric near-field show a
fundamental difference in these two types of toroidal dipole orders. For the
ATO mode, the electric near-field is confined in the centers of minicells,
while it is practically zero at the center of the hexagonal unit supercells.
In the trimers, one can clearly see the staggered distribution of the real
part of the normal ($E_{z}$) component of the electric near-field. This
staggered distribution is additionally confirmed by the phase pattern of the
$E_{z}$ component. Contrariwise, for the TO mode, the electric near-field has
a bright hotspot in the center of the hexagonal unit supercells complemented
by lower intensity hotspots at the center of each minicell. The maximal field
concentration is reached outside the dielectric particles, which can be
considered as a signature of the toroidal dipole mode. Thus, we have two
different conditions of the field concentration in the same metamaterial,
where the form of the field concentration is determined only by the given
polarization of the incident wave. We believe that this peculiarity is a
rather unique feature of the proposed metamaterial, which is very promising
for practical applications.
## X Discussion
We have focused in this paper only on two possible eigenmodes of the hexagonal
unit supercells, namely, the ATO and TO modes which belong to IRREPs $A_{1}$
and $B_{1}$ (see Table 2 in Appendix B). However, there are also possible
resonant modes belonging to other IRREPs, which can be defined using, for
example, the method of symmetry adapted linear combinations (SALC) Dmitriev
_et al._ (2021).
Symmetry breaking in the unit supercell of the proposed metamaterials can be
fulfilled in many different ways. Besides the discussed above methods of the
out-of-plane symmetry breaking, the in-plane symmetry breaking can be
introduced. For instance, one trimer or a cluster of several trimers in a
hexagonal unit supercell can be dislocated or rotated with respect to the
other trimers. Effectiveness of the rotation method in the metamaterial
composed of square unit cells of trimers was demonstrated recently Tuz _et
al._ (2020b). However, the point symmetry of the resulting unit supercell in
this case is usually completely lost, and the symmetry analysis cannot be
applied for such perturbed arrays.
In the framework of a chosen symmetry, the geometry is not always defined
uniquely due to a high number of disks included in the unit supercell and
different kinds of their perturbation. Therefore, the geometry can be
optimized following simple rules introduced here. This optimization can be
performed before the full-wave numerical simulations, which can significantly
reduce the time needed to find the desired configuration of the unit
supercell.
The proposed configurations of hexagonal arrays based on trimers possess a
technological advantage to have a free space in the center of the hexagonal
unit supercell, where an additional element (for example, a nonlinear or
control element) can be deposited. In particular, for the case of the
excitation of the TO mode, a strong concentration of electric near-field
appears in this free space which can be useful in sensoric and lasing
applications.
## XI Conclusions
We have proposed and studied all-dielectric metamaterials composed of
hexagonal trimer-based unit cells which are able to support two types of
toroidal supermodes. These supermodes differ in the net order of the toroidal
dipole moments. The moments distributed in trimers can possess either co-
directional (symmetric, TO) or staggered (antisymmetric, ATO) arrangement.
A new theoretical approach based on the magnetic group theory is developed to
analyze the mechanism of excitation of these supermodes by the field of a
normally incident linearly polarized wave. This mechanism implies symmetry
breaking in the supercell of the metamaterial. It is revealed that the ATO
mode can be excited in metamaterials whose unit supercells possess the
$C_{2v}$, $C_{2}$, and $C_{s}$ symmetries, whereas the TO mode can be excited
only in the supercells with the $C_{s}$ symmetry. All these symmetries can be
realized by different arrangements of perturbed trimers in the unit supercell.
It was demonstrated that the magnetic groups approach simplifies greatly the
analysis of the arrays and provides a deeper physical insight into the
mechanism of the toroidal modes excitation.
We have found out that by adjusting the perturbation parameters of the
trimers, it is possible to excite the ATO and TO modes in the same
metamaterial by the wave with a proper polarization. Excitation of these modes
results in different characteristics of the electric near-field localization,
which can be important for practical applications.
The method of magnetic groups introduced in this paper can be also applied for
the analysis of arrays with square and rectangular unit supercells as well as
to the isolated oligomers with high symmetries and their assemblies. In a more
wide aspect, this method can be used for the analysis of complex systems with
3D symmetries and eigenmodes different from the toroidal ones.
## Acknowledgment
VD thanks the Brazilian Agency National Council of Technological and
Scientific Development (CNPq) for financial support. SDSS acknowledges support
from CNPq (Grant No. 160344/2019-0). ASK and VRT acknowledge financial support
from the National Key R&D Program of China (Project No. 2018YFE0119900). ABE
thanks funding support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy within the Cluster of
Excellence PhoenixD (EXC 2122, Project ID 390833453).
## Appendix A Elements of point symmetry in Schöenflies notation Bradley and
Cracknell (2009)
In Schöenflies system, an $n$-fold rotation through $2\pi/n$ (where $n$ is an
integer; in our case $n=2,3,6$) about the $z$ axis is denoted by the symbol
$C_{n}$ ($C$ means Cyklus). The symbol $\sigma_{v}$ (the subscript $v$ for
vertical) defines reflection in a plane passing through this axis. The symbol
$\sigma_{d}$ (the subscript $d$ for diagonal) designates a mirror plane
containing the axis which is diagonal to the already existing plane
$\sigma_{v}$.
Let us apply now to the group notations. The groups with one axis of symmetry
are denoted by $C_{n}$. Notice that in this case, the notations for the group
elements and groups themselves coincide. For example, the symbol $C_{2}$
denotes the operation of rotation about an axis by $\pi$, and also it may
denote the $C_{2}$ group consisting of two elements: the identity $e$ and the
rotation $C_{2}$. The meaning of the notations can be clearly understood from
the context. The $C_{nv}$ group has an $n$-fold rotational axis $C_{n}$ and a
finite number of planes of symmetry passing through the axis $C_{n}$. The
$C_{s}$ group contains two elements: the identity $e$ and the reflection
$\sigma_{v}$.
The subgroup decomposition (the group tree) of the $C_{6v}$ group, which is
discussed in the main body of this paper, is shown in Fig. 11. The $C_{6v}$
group has the following subgroups: $C_{6}$, $C_{3v}$, $C_{3}$, $C_{2v}$,
$C_{2}$, $C_{s}$, and $C_{1}$.
Figure 11: Group decomposition of $C_{6v}$ group. Thick and dotted lines
indicate that the corresponding subgroup is not invariant and is not of index
2 with respect to the higher group, respectively. It means that these two
groups cannot form a group of the third category Barybin and Dmitriev (2002).
## Appendix B Tables of group theory
Table 2: Character table of $C_{6v}$ group and magnetic groups of the second and third categories, and corresponding possible toroidal dipole mode orders. $C_{6v}$ | $~{}e~{}~{}$ | $C_{2}$ | 2$C_{3}$ | 2$C_{6}$ | $3\sigma_{v}$ | $3\sigma_{d}$ | Magnetic group | Mode order
---|---|---|---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | $\bf C_{6v}$ | dark TO
$A_{2}$ | 1 | 1 | 1 | 1 | $\\!\\!\\!\\!\\!-1$ | $\\!\\!\\!\\!\\!-1$ | $C_{6v}(C_{6})$ |
$B_{1}$ | 1 | $\\!\\!\\!\\!\\!-1$ | 1 | $\\!\\!\\!\\!\\!-1$ | 1 | $\\!\\!\\!\\!\\!-1$ | $C_{6v}(C_{3v})$ | dark ATO
$B_{2}$ | 1 | $\\!\\!\\!\\!\\!-1$ | 1 | $\\!\\!\\!\\!\\!-1$ | $\\!\\!\\!\\!\\!-1$ | 1 | $C_{6v}(C_{3v})$ |
$E_{1}$ | 2 | $\\!\\!\\!\\!\\!-2$ | $\\!\\!\\!\\!\\!-1$ | 1 | $0$ | $0$ | |
$E_{1}$ | 2 | $2$ | $\\!\\!\\!\\!\\!-1$ | $\\!\\!\\!\\!\\!-1$ | $0$ | $0$ | |
Table 3: Irreducible representations of $C_{2v}$ group and their relation to magnetic groups, and corresponding possible toroidal dipole mode orders. $C_{2v}$ | $~{}e~{}$ | $C_{2}$ | $\sigma_{1}$ | $\sigma_{2}$ | Magnetic group | Mode order
---|---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | $\bf C_{2v}$ | dark TO
$A_{2}$ | 1 | 1 | $\\!\\!\\!\\!\\!-1$ | $\\!\\!\\!\\!\\!-1$ | $C_{2v}(C_{2})$ |
$B_{1}$ | 1 | $\\!\\!\\!\\!\\!-1$ | 1 | $\\!\\!\\!\\!\\!-1$ | $C_{2v}(C_{s})$ | bright ATO
$B_{2}$ | 1 | $\\!\\!\\!\\!\\!-1$ | $\\!\\!\\!\\!\\!-1$ | 1 | $C_{2v}(C_{s})$ |
Table 4: Irreducible representations of $C_{s}$ group and their relation to magnetic groups of the second and third categories, and corresponding possible toroidal dipole mode orders. Superscripts in $C_{s}$ denote orientation of the $\sigma$ plane in the hexagonal cluster, [see Figs. 4(g) and 4(h)]. $C_{s}$ | $~{}e~{}$ | $\sigma_{v}$ | Magnetic group | Mode order
---|---|---|---|---
$A$ | $1$ | $1$ | ${\bf C}_{s}^{(1)}$ | bright TO
$B$ | $1$ | $\\!\\!\\!\\!-1$ | $C_{s}^{(1)}(C_{1})$ | bright ATO
Table 5: Symmetry degeneration table of $C_{6v}$ group into $C_{2v}$ and $C_{s}$ groups Overvig _et al._ (2020). Superscripts in $C_{s}$ denote orientation of the $\sigma$ plane in the hexagonal cluster [see Figs. 4(g) and 4(h)]. $C_{6v}$ | $C_{2v}$ | $C_{s}^{(1)}$ | $C_{s}^{(2)}$
---|---|---|---
$A_{1}$ | $A_{1}$ | $A$ | $A$
$A_{2}$ | $A_{2}$ | $B$ | $B$
$B_{1}$ | $B_{1}$ | $A$ | $B$
$B_{2}$ | $B_{2}$ | $B$ | $A$
$E_{1}$ | $B_{1}$, $B_{2}$ | $A,B$ | $A,B$
$E_{2}$ | $A_{1}$, $A_{2}$ | $A,B$ | $A,B$
## Appendix C Brief description of magnetic groups
Time reversal operator. The time reversal operator ${\cal T}$ can be an
element of magnetic groups entering in these groups either separately or in
combination with elements of the geometrical symmetry. The ${\cal T}$ operator
changes the sign of time $t$ ($t\rightarrow-t$), commutes with all elements of
the geometrical symmetry, and has the property ${\cal T}{\cal T}={\cal
T}^{2}=e$, where $e$ is the unit element of the group.
Here we shall discuss only those properties of the ${\cal T}$ operator which
are necessary for our present consideration. In particular, the ${\cal T}$
operator reverses the velocities and changes the current directions, signs of
magnetic fluxes, magnetic fields, toroidal moments, wave vector, and Poynting
vector. All these quantities are odd in time.
Categories of magnetic groups. There exist three categories of discrete and
continuous point magnetic groups. The group of the first category $G$ consists
of a unitary subgroup $H$ (in our case, it contains the usual rotation-
reflection elements) and products of ${\cal T}$ with all the elements of $H$.
The full group is then $H+{\cal T}H$ including ${\cal T}={\cal T}e$ (these
groups are also referred to as nonmagnetic ones, see the first column of Table
6).
In the case of magnetic groups of the second category ${\bf G}$, there is no
point group elements combined with the time reversal operator ${\cal T}$, and
${\cal T}$ itself is not an element of the groups. The nomenclature and
notations of the groups of the first (nonmagnetic) and second (magnetic)
category coincides. Therefore, to distinguish them, we use bold letters for
denoting the groups of the second category (see second column of Table 6).
Table 6: Content of magnetic groups of symmetry 1st category | 2nd category | 3rd category
---|---|---
$G=H+{\cal T}H$ | $\bf G$ | $G(H)=H+{\cal T}H^{\prime},H^{\prime}\neq H$
including ${\cal T}$ | without ${\cal T}$ | ${\cal T}$ only in combination
| | with rotation-reflections
The Schöenflies system is particularly suitable for notation of the magnetic
groups of the third category. In this case, the notation presents explicitly
the structure of the group, i.e., the unitary subgroup (it contains only
elements of the geometrical symmetry) and the antiunitary elements (these
elements are the product of the operator ${\cal T}$ and an element of the
geometrical symmetry).
In addition to the rotation-reflection elements of the subgroup $H$ of pure
geometrical operators, the magnetic groups of the third category $G(H)$ also
contain the elements which are the product of ${\cal T}$ and the usual
geometrical symmetry elements. These combined elements cause the existence of
the so-called antiaxes and antiplanes of symmetry. The full group is $H+{\cal
T}H^{\prime}$ without ${\cal T}$. Notice that the elements of $H^{\prime}$ are
different from those of $H$ (see third column of Table 6).
Geometrical elements of a magnetic group of the third category form a subgroup
of index 2. It means that in each group of the third category, there is an
equal number of elements with and without ${\cal T}$ (see Fig. 11). In
contrast to the groups of the first category, the operator ${\cal T}$ itself
is not an element of the magnetic groups of the third category.
## Appendix D “Dynamic” magnetic symmetry of alternating magnetic field,
magnetic, and toroidal dipole moments
The term dynamic symmetry is used in classical and quantum mechanical systems
when describing dynamical interactions and revealing the relations between
dynamics and geometry. Description of the dynamic symmetry is based on the Lie
groups. In particular, in electromagnetic theory, the dynamic symmetry is
related to the invariance of the Maxwell equations under the conformal group
of transformations Wulfman (2010). In our present consideration, the notion of
“dynamic” symmetry has another meaning, which is discussed below. That is why
we put the term dynamic in quotes.
The theory of nonmagnetic groups is frequently used to describe optical
properties of metamaterials without magnetic inclusions (e.g., see Ref.
Padilla (2007)), while a traditional application of magnetic groups is related
to the description of systems with a static magnetic field and a static
magnetization in magnetic substances. An external static magnetic field
together with geometrical symmetry of the medium constituents defines symmetry
of the whole magnetic system. This is used, for instance, to calculate the
composition of permittivity and permeability tensors (i.e., to impose some
restrictions on the elements of these tensors) and also to define the
structure of the scattering matrices Barybin and Dmitriev (2002).
In our case, we shall consider two alternating magnetic fields with different
symmetries. They are the fields of the incident wave and toroidal dipole mode.
Although these magnetic fields interact in a nonmagnetic environment of the
array of dielectric particles, an approach based on the magnetic groups makes
it possible to get a deep physical insight into the problem of the toroidal
mode excitation.
Let us consider a trimer based on an equilateral triangle illuminated by a
normally incident linearly polarized wave with the wave vector $\bf k$. For
the chosen geometry of the problem, the magnetic field of the incident wave
$\bf H$ lies in the $x$-$y$ plane. We suppose that a toroidal dipole mode is
excited in the trimer. This mode appears from the circular flow of the
magnetic field ${\bf h}$, which also lies in the $x$-$y$ plane. The appearance
of the vectors $\bf H$ and $\bf h$ is schematically given in Fig. 12.
Figure 12: (a) Electric $\bf E$ and magnetic $\bf H$ fields of an incident
wave with the wave vector $\bf k$ and circular magnetic field $\bf h$ of an
idealized toroidal dipole mode. (b) Orientation of the magnetic dipole moments
${\bf m}_{i}$ ($i=1,2,3$) and toroidal dipole moment $\bf T$ in the magnetic
field $\bf H$ of the incident wave for the trimer with plane of symmetry
$\sigma$.
To relate the symmetries of the magnetic fields of the incident wave $\bf H$
and toroidal dipole mode $\bf h$, two prerequisites must be fulfilled: (i) the
loop of the magnetic field ${\bf h}$ is uniform along its circumference, and
(ii) a phase shift between the fields $\bf H$ and ${\bf h}$ is absent. It
means that these two dynamic quantities are in phase being considered at a
fixed point of time. This time can be fixed at an arbitrary moment, for
example, when the fields reach their maximal values during the oscillation
period.
In nonmagnetic symmetry, the magnetic field $\bf H$ has only one element of
symmetry, namely, the plane perpendicular to the vector $\bf H$. In magnetic
symmetry, the magnetic field $\bf H$ can be described by the 2D (in the
$x$-$y$ plane) magnetic symmetry, namely, by the group of the third category
$C_{2v}(C_{s})$, which contains the following elements and antielements:
$\bullet$ the unit element $e$,
$\bullet$ the antiaxis ${\cal T}C_{2}$ along the $z$ axis,
$\bullet$ the vertical plane $\sigma_{1}$ perpendicular to the vector $\bf H$
(the plane $x=0$),
$\bullet$ the vertical antiplane ${\cal T}\sigma_{2}$, parallel to the vector
$\bf H$ (the plane $y=0$).
In this group, the time reversal operator ${\cal T}$ is combined with the
geometrical elements of the $C_{2}$ symmetry and $\sigma_{2}$.
In an ideal case, the magnetic field $\bf h$ of the toroidal mode presents a
circle [see Fig. 12(a)]. Considering the $x$-$y$ plane where this circle is
situated, the 2D group of symmetry of $\bf h$ is $\mbox{\boldmath{\bf
C}}_{\infty v}$ of the second category with the $z$ axis of infinite order
${C}_{\infty}$, and an infinite number of the planes of symmetry $\sigma_{v}$
passing through this axis. The time reversal operator ${\cal T}$ is not the
element of $\mbox{\boldmath{\bf C}}_{\infty v}$ since when $t\rightarrow-t$,
the sign of $\bf h$ changes.
Comparing symmetries of the fields, one can see that the fields $\bf H$ and
$\bf h$ have the antiaxis ${\cal T}C_{2}$ and axis $C_{2}$, respectively.
Moreover, the field $\bf H$ has the antiplane ${\cal T}\sigma$, while the
field $\bf h$ has the plane $\sigma$ which coincides with ${\cal T}\sigma$.
${\cal T}C_{2}$ and $C_{2}$, and also ${\cal T}\sigma$ and $\sigma$ are
incompatible with each other. This incompatibility can be considered as a
selection rule. Therefore, in the given symmetry, excitation of the toroidal
dipole mode having a circular flow of $\bf h$ induced by the field $\bf H$ is
impossible.
Now we apply to a realistic case. In an isolated trimer, there is a particular
eigensolution where three in-plane magnetic dipoles ${\bf m}_{i}$ ($i=1,2,3$)
appears to be arranged in a head-to-tail fashion [Fig. 12(b)]. In the linear
regime, the moment $\bf m$ of the magnetic dipole in a dielectric disk is
proportional to the incident magnetic field $\bf H$:
${\bf m}=\alpha\bf H,$ (4)
where $\alpha$ is the magnetic polarizability. The trimer based on an
equilateral triangle can be described by symmetry $C_{3v}$ with the $C_{3}$
axis directed along the $z$ axis and three planes of symmetry $\sigma$.
Evidently, the cluster of three magnetic dipoles ${\bf m}_{i}$ inherits the
magnetic symmetry $\mbox{\boldmath{\bf C}}_{3v}$. In this case, the excitation
of the toroidal dipole mode in the trimer by the incident field $\bf H$ is
also impossible Dmitriev _et al._ (2021).
The excitation of the toroidal dipole mode in the trimer becomes possible
after removing the $C_{3}$ axis and two planes of its symmetry $\sigma$. It
can be made by introducing some perturbation to the trimer geometry (e.g., by
changing the geometric dimensions or material parameters of one disk in the
trimer Dmitriev _et al._ (2021)). This perturbation reduces the geometric
symmetry of the trimer to the $C_{s}$ group. Notice that the position of the
perturbed disk in the trimer defines the orientation of its symmetry plane
$\sigma$, and, consequently, imposes a constraint on the orientation of the
vector $\bf H$ necessary for the toroidal dipole mode excitation.
## References
* Rousseau _et al._ (1981) D. L. Rousseau, R. P. Bauman, and S. P. S. Porto, “Normal mode determination in crystals,” J. Raman Spectrosc. 10, 253–290 (1981).
* Bak (1985) P. Bak, “Symmetry, stability, and elastic properties of icosahedral incommensurate crystals,” Phys. Rev. B 32, 5764–5772 (1985).
* Sakoda (1995) K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982–7986 (1995).
* Ochiai and Sakoda (2001) T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
* Hergert and Däne (2003) W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi (a) 197, 620–634 (2003).
* Mock _et al._ (2010) A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
* Alagappan _et al._ (2008) G. Alagappan, X. W. Sun, and H. D. Sun, “Symmetries of the eigenstates in an anisotropic photonic crystal,” Phys. Rev. B 77, 195117 (2008).
* Hamermesh and Mullin (1962) M. Hamermesh and A. A. Mullin, _Group Theory and its Applications to Physical Problems_ (Dover Publictions Inc., New York, 1962).
* Tinkham (1964) M. Tinkham, _Group Theory and Quantum Mechanics_ (McGraw-Hill, New York, 1964).
* Cornwell (1984) J. F. Cornwell, _Group Theory in Physics_ , Techniques of Physics (Academic Press, London, 1984).
* Ludwig and Falter (1988) W. Ludwig and C. Falter, _Symmetries in Physics: Group Theory Applied to Physical Problems_ , Springer Series in Solid-State Sciences, Vol. 64 (Springer, Berlin, 1988).
* Inui _et al._ (1996) T. Inui, Y. Tanabe, and Y. Onodera, _Group Theory and its Application in Physics_ (Springer, Berlin, 1996).
* Bradley and Cracknell (2009) C. J. Bradley and A. P. Cracknell, _The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups_ , Oxford Classic Texts in the Physical Sciences (Oxford University Press Inc., New York, 2009).
* Hergert and Geilhufe (2018) W. Hergert and R. M. Geilhufe, _Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica_ (John Wiley & Sons, Weinheim, 2018).
* Schmid (2008) H. Schmid, “Some symmetry aspects of ferroics and single phase multiferroics,” J. Phys. Condens. Matter 20, 434201 (2008).
* Saxena and Lookman (2011) A. Saxena and T. Lookman, “Magnetic symmetry of low-dimensional multiferroics and ferroelastics,” Phase Transitions 84, 421–437 (2011).
* Spaldin _et al._ (2008) N. A. Spaldin, M. Fiebig, and M. Mostovoy, “The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect,” J. Phys. Condens. Matter 20, 434203 (2008).
* Loidl _et al._ (2008) A. Loidl, H. von Loehneysen, and G. M. Kalvius, “Multiferroics,” J. Phys. Condens. Matter 20, 430301 (2008).
* Khomskii (2009) D. Khomskii, “Trend: Classifying multiferroics: Mechanisms and effects,” Physics 2, 20 (2009).
* Gnewuch and Rodriguez (2019) S. Gnewuch and E. E. Rodriguez, “The fourth ferroic order: Current status on ferrotoroidic materials,” J. Solid State Chem. 271, 175–190 (2019).
* Li and Tang (2019) X.-L. Li and J. Tang, “Recent developments in single-molecule toroics,” Dalton Trans. 48, 15358–15370 (2019).
* Monticone and Alù (2014) F. Monticone and A. Alù, “The quest for optical magnetism: From split-ring resonators to plasmonic nanoparticles and nanoclusters,” J. Mater. Chem. C 2, 9059–9072 (2014).
* Pendry _et al._ (1999) J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Techn. 47, 2075–2084 (1999).
* Zhou _et al._ (2005) J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the Magnetic Response of Split-Ring Resonators at Optical Frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
* Decker _et al._ (2009) M. Decker, S. Burger, S. Linden, and M. Wegener, “Magnetization waves in split-ring-resonator arrays: Evidence for retardation effects,” Phys. Rev. B 80, 193102 (2009).
* Kaelberer _et al._ (2010) T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
* Savinov _et al._ (2014) V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89, 205112 (2014).
* Cong _et al._ (2018) L. Cong, V. Savinov, Y. K. Srivastava, S. Han, and R. Singh, “A metamaterial analog of the Ising model,” Adv. Mater. 30, 1804210 (2018).
* Wongkasem _et al._ (2006) N. Wongkasem, A. Akyurtlu, and K. A. Marx, “Group theory based design of isotropic negative refractive index metamaterials,” Prog. Electromagn. Res. 63, 295–310 (2006).
* Baena _et al._ (2006) J. D. Baena, L. Jelinek, R. Marqués, and J. Zehentner, “Electrically small isotropic three-dimensional magnetic resonators for metamaterial design,” Appl. Phys. Lett. 88, 134108 (2006).
* Baena _et al._ (2007) J. D. Baena, L. Jelinek, and R. Marqués, “Towards a systematic design of isotropic bulk magnetic metamaterials using the cubic point groups of symmetry,” Phys. Rev. B 76, 245115 (2007).
* Padilla (2007) W. J. Padilla, “Group theoretical description of artificial electromagnetic metamaterials,” Opt. Express 15, 1639–1646 (2007).
* Dmitriev (2009) V. Dmitriev, “Symmetry principles and group-theoretical methods in electromagnetics of complex media,” in _Metamaterials Handbook. Theory and Phenomena of Metamaterials_ , Vol. 1, edited by F. Capolino (CRC Press, Boca Raton, FL, 2009) Chap. 3, pp. 3–1–3–19.
* Dmitriev (2011) V. Dmitriev, “Symmetry properties of electromagnetic planar arrays: Long-wave approximation and normal incidence,” Metamaterials 5, 141–148 (2011).
* Dmitriev (2013) V. Dmitriev, “Symmetry properties of electromagnetic planar arrays in transfer matrix description,” IEEE Trans. Antennas Propag. 61, 185–194 (2013).
* Reinke _et al._ (2011) C. M. Reinke, T. M. De la Mata Luque, M. F. Su, M. B. Sinclair, and I. El-Kady, “Group-theory approach to tailored electromagnetic properties of metamaterials: An inverse-problem solution,” Phys. Rev. E 83, 066603 (2011).
* Evlyukhin _et al._ (2010) A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of si-nanoparticle arrays,” Physical Review B 82, 045404 (2010).
* Jahani and Jacob (2016) S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
* Kruk and Kivshar (2017) S. Kruk and Y. Kivshar, “Functional meta-optics and nanophotonics governed by Mie resonances,” ACS Photonics 4, 2638–2649 (2017).
* Babicheva and Evlyukhin (2021) V. E. Babicheva and A. B. Evlyukhin, “Multipole lattice effects in high refractive index metasurfaces,” Journal of Applied Physics 129 (2021).
* Evlyukhin _et al._ (2012) A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano letters 12, 3749–3755 (2012).
* Miroshnichenko _et al._ (2012) A. E. Miroshnichenko, B. Luk’yanchuk, S. A. Maier, and Y. S. Kivshar, “Optically induced interaction of magnetic moments in hybrid metamaterials,” ACS Nano 6, 837–842 (2012).
* Miroshnichenko _et al._ (2017) A. E. Miroshnichenko, D. Filonov, D. Lukyanchuk, and Y. Kivshar, “Antiferromagnetic order in hybrid electromagnetic metamaterials,” New J. Phys. 19, 083013 (2017).
* Lepeshov and Kivshar (2018) S. Lepeshov and Y. Kivshar, “Near-field coupling effects in Mie-resonant photonic structures and all-dielectric metasurfaces,” ACS Photonics 5, 2888–2894 (2018).
* Terekhov _et al._ (2019) P. D. Terekhov, V. E. Babicheva, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Multipole analysis of dielectric metasurfaces composed of nonspherical nanoparticles and lattice invisibility effect,” Physical Review B 99, 045424 (2019).
* Sadrieva _et al._ (2019) Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100, 115303 (2019).
* Gladyshev _et al._ (2020) S. Gladyshev, K. Frizyuk, and A. Bogdanov, “Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties,” Phys. Rev. B 102, 075103 (2020).
* Xiong _et al._ (2020) Z. Xiong, Q. Yang, W. Chen, Z. Wang, J. Xu, W. Liu, and Y. Chen, “On the constraints of electromagnetic multipoles for symmetric scatterers: eigenmode analysis,” Opt. Express 28, 3073–3085 (2020).
* Khardikov _et al._ (2012) V. V. Khardikov, E. O. Iarko, and S. L. Prosvirnin, “A giant red shift and enhancement of the light confinement in a planar array of dielectric bars,” J. Opt. 14, 035103 (2012).
* Tuz _et al._ (2018a) V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H.-B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26, 2905–2916 (2018a).
* Sayanskiy _et al._ (2019) A. Sayanskiy, A. S. Kupriianov, S. Xu, P. Kapitanova, V. Dmitriev, V. V. Khardikov, and V. R. Tuz, “Controlling high-$Q$ trapped modes in polarization-insensitive all-dielectric metasurfaces,” Phys. Rev. B 99, 085306 (2019).
* Kupriiannov _et al._ (2020) A. S. Kupriiannov, K. L. Domina, V. V. Khardikov, A. B. Evlyukhin, and V. R. Tuz, “Homogeneous enhancement of near-fields in all-dielectric metasurfaces with cluster-based unit cells,” Opt. Lett. 45, 1527–1530 (2020).
* Evlyukhin _et al._ (2020) A. B. Evlyukhin, V. R. Tuz, V. S. Volkov, and B. N. Chichkov, “Bianisotropy for light trapping in all-dielectric metasurfaces,” Phys. Rev. B 101, 205415 (2020).
* Fedotov _et al._ (2007) V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp Trapped-Mode Resonances in Planar Metamaterials with a Broken Structural Symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
* Koshelev _et al._ (2018) K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-$Q$ Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121, 193903 (2018).
* Yu _et al._ (2019) P. Yu, A. S. Kupriianov, V. Dmitriev, and V. R. Tuz, “All-dielectric metasurfaces with trapped modes: Group-theoretical description,” J. Appl. Phys. 125, 143101 (2019).
* Tuz _et al._ (2020a) V. R. Tuz, P. Yu, V. Dmitriev, and Y. S. Kivshar, “Magnetic dipole ordering in resonant dielectric metasurfaces,” Phys. Rev. Applied 13, 044003 (2020a).
* Overvig _et al._ (2020) A. C. Overvig, S. C. Malek, M. J. Carter, S. Shrestha, and N. Yu, “Selection rules for quasibound states in the continuum,” Phys. Rev. B 102, 035434 (2020).
* Basharin _et al._ (2015) A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5, 011036 (2015).
* Tasolamprou _et al._ (2016) A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
* Tasolamprou _et al._ (2020) A. C. Tasolamprou, O. Tsilipakos, A. Basharin, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal multipoles in metamaterials,” in _Compendium on Electromagnetic Analysis_, Vol. 5, edited by V. A. Markel (World Scientific, Singapore, 2020) Chap. 7, pp. 237–278.
* Xu _et al._ (2019) S. Xu, A. Sayanskiy, A. S. Kupriianov, V. R. Tuz, P. Kapitanova, H.-B. Sun, W. Han, and Y. S. Kivshar, “Experimental observation of toroidal dipole modes in all-dielectric metasurfaces,” Adv. Opt. Mater. 7, 1801166 (2019).
* Zhang _et al._ (2019) G. Zhang, C. Lan, R. Gao, Y. Wen, and J. Zhou, “Toroidal dipole resonances in all-dielectric oligomer metasurfaces,” Adv. Theory Simul. 2, 1900123 (2019).
* Tuz _et al._ (2018b) V. R. Tuz, V. V. Khardikov, and Y. S. Kivshar, “All-dielectric resonant metasurfaces with a strong toroidal response,” ACS Photonics 5, 1871–1876 (2018b).
* Kupriianov _et al._ (2019) A. S. Kupriianov, Y. Xu, A. Sayanskiy, V. Dmitriev, Y. S. Kivshar, and V. R. Tuz, “Metasurface engineering through bound states in the continuum,” Phys. Rev. Applied 12, 014024 (2019).
* Tuz _et al._ (2020b) V. R. Tuz, V. Dmitriev, and A. B. Evlyukhin, “Antitoroidic and toroidic orders in all-dielectric metasurfaces for optical near-field manipulation,” ACS Appl. Nano Mater. 3, 11315–11325 (2020b).
* Dmitriev _et al._ (2021) V. Dmitriev, A. S. Kupriianov, S. D. S. Santos, and V. R. Tuz, “Symmetry analysis of trimer-based all-dielectric metasurfaces with toroidal dipole modes,” J. Phys. D Appl. Phys. 54, 115107 (2021).
* Wu and Hu (2015) Long-Hua Wu and Xiao Hu, “Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material,” Phys. Rev. Lett. 114, 223901 (2015).
* Gorlach _et al._ (2018) M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
* Yang _et al._ (2018) Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120, 217401 (2018).
* Jiang _et al._ (2019) J.-R. Jiang, S.-C. Wu, W.-T. Chen, and R.-L. Chern, “Double Dirac cones in two-dimensional photonic crystals with $C_{6}$ symmetry,” J. Mod. Opt. 66, 2119–2130 (2019).
* Xiong _et al._ (2019) H. Xiong, Q. Wu, Y. Lu, R. Wang, Q. Zhang, J. Qi, J. Yao, and J. Xu, “Polarization-resolved edge states in terahertz topological photonic crystal,” Opt. Express 27, 22819–22826 (2019).
* Xi _et al._ (2020) X. Xi, K.-P. Ye, and R.-X. Wu, “Topological photonic crystal of large valley Chern numbers,” Photon. Res. 8, B1–B7 (2020).
* Bohren and Huffman (1998) C. F. Bohren and D. R. Huffman, _Absorption and Scattering of Light by Small Particles_ (Wiley, New York, 1998).
* Rechberger _et al._ (2003) W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003).
* Chuntonov and Haran (2011) L. Chuntonov and G. Haran, “Trimeric plasmonic molecules: The role of symmetry,” Nano Lett. 11, 2440–2445 (2011).
* Wang and Zaki (2007) C. Wang and K. A. Zaki, “Dielectric resonators and filters,” IEEE Microwave Mag. 8, 115–127 (2007).
* Trubin (2016) A. Trubin, _Lattices of Dielectric Resonators_, Springer Series in Advanced Microelectronics, Vol. 53 (Springer Cham, Heidelberg, 2016).
* Fesenko _et al._ (2019) V. I. Fesenko, A. S. Kupriianov, A. Sayanskiy, V. I. Shcherbinin, A. Trubin, and V. R. Tuz, “Approach to analysis of all-dielectric free-form antenna systems,” Opt. Express 27, 22363–22374 (2019).
* Dubovik and Tugushev (1990) V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187, 145–202 (1990).
* Marinov _et al._ (2007) K. Marinov, A.D. Boardman, V.A. Fedotov, and N. Zheludev, “Toroidal metamaterial,” New J. Phys. 9, 324 (2007).
* Chen _et al._ (2011) J. Chen, J. Ng, and C.T. Lin, Z.and Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
* Evlyukhin _et al._ (2016) A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrarily shaped nanoparticles,” Phys. Rev. B 94, 205434 (2016).
* Alaee _et al._ (2018) R. Alaee, C. Rockstuhl, and I. Fernandez-Corbaton, “An electromagnetic multipole expansion beyond the long-wavelength approximation,” Opt. Commun. 407, 17–21 (2018).
* Evlyukhin and Chichkov (2019) A. B. Evlyukhin and B. N. Chichkov, “Multipole decompositions for directional light scattering,” Phys. Rev. B 100, 125415 (2019).
* Fernandez-Corbaton _et al._ (2017) I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles from localized electric current distributions,” Sci. Rep. 7, 1–8 (2017).
* Nemkov _et al._ (2018) N. A. Nemkov, A. A. Basharin, and V. A. Fedotov, “Electromagnetic sources beyond common multipoles,” Physical Review A 98, 023858 (2018).
* Gurvitz _et al._ (2019) E. A. Gurvitz, K. S. Ladutenko, P. A. Dergachev, A. B. Evlyukhin, A. E. Miroshnichenko, and A. S. Shalin, “The high-order toroidal moments and anapole states in all-dielectric photonics,” Laser Photonics Rev. 13, 1800266 (2019).
* Zenin _et al._ (2017) V. A. Zenin, A. B. Evlyukhin, S. M. Novikov, Y. Yang, R. Malureanu, A. V Lavrinenko, B. N. Chichkov, and S. I. Bozhevolnyi, “Direct amplitude-phase near-field observation of higher-order anapole states,” Nano Lett. 17, 7152–7159 (2017).
* Yang and Bozhevolnyi (2019) Y. Yang and S. I. Bozhevolnyi, “Nonradiating anapole states in nanophotonics: from fundamentals to applications,” Nanotechnology 30, 204001 (2019).
* Baryshnikova _et al._ (2019) K. V. Baryshnikova, D. A. Smirnova, B. S. Luk’yanchuk, and Y. S. Kivshar, “Optical anapoles: Concepts and applications,” Adv. Opt. Mater. 7, 1801350 (2019).
* Savinov _et al._ (2019) V. Savinov, N. Papasimakis, D.P. Tsai, and N.I. Zheludev, “Optical anapoles,” Commun. Phys. 2, 1–4 (2019).
* Mackay (1989) A. Mackay, “Proof of polarisation independence and nonexistence of crosspolar terms for targets presenting $n$-fold ($n>2$) rotational symmetry with special reference to frequency-selective surfaces,” Electron. Lett. 25, 1624–1625 (1989).
* Kupriianov and Tuz (2019) A. S. Kupriianov and V. R. Tuz, “Microwave approach to study resonant features of all-dielectric metasurfaces,” in _2019 Photonics Electromagnetics Research Symposium - Fall (PIERS - Fall)_ (2019) pp. 866–870.
* Barybin and Dmitriev (2002) A. A. Barybin and V. A. Dmitriev, _Modern Electrodynamics and Coupled-Mode Theory: Application to Guided-Wave Optics_ (Rinton Press Princeton, New Jersey, 2002).
* Wulfman (2010) C. E. Wulfman, _Dynamical Symmetry_ (World Scientific Publishing Company, New Jersey, 2010).
|
capbtabboxtable[][1.0] 11institutetext: Dept of Computing, Imperial College
London
180 Queens Gate, London, SW7 2AZ, United Kingdom
11email: {edward.stow16, riku.murai15<EMAIL_ADDRESS>
22institutetext: Dept of Mechanical and Industrial Engineering, Ryerson
University,
350 Victoria Street, Toronto, Ontario, M5B 2K3, Canada
22email<EMAIL_ADDRESS>
# Cain: Automatic Code Generation for Simultaneous Convolutional Kernels on
Focal-plane Sensor-processors
Edward Stow 11 Riku Murai 11 Sajad Saeedi 22 Paul H J Kelly 11
###### Abstract
Focal-plane Sensor-processors (FPSPs) are a camera technology that enable low
power, high frame rate computation, making them suitable for edge computation.
Unfortunately, these devices’ limited instruction sets and registers make
developing complex algorithms difficult. In this work, we present
Cain111Available at https://github.com/ed741/cain – a compiler that targets
SCAMP-5, a general-purpose FPSP – which generates code from multiple
convolutional kernels. As an example, given the convolutional kernels for an
MNIST digit recognition neural network, Cain produces code that is half as
long, when compared to the other available compilers for SCAMP-5.
###### Keywords:
Convolution SIMD Image sensor Analogue computing Edge inference
## 1 Introduction
Real-time computer vision applications are currently bound to traditional
camera sensors that transfer each pixel at each frame to a host where it is
processed. This requires high-performance buses between the sensors and hosts,
especially where high frame-rates are required. A self-driving car may need to
receive new information for every 1cm travelled to be vigilant of unexpected
scenarios, so at 80 km/hr a frame rate of 2222 Hz would be required. A 2 mega-
pixel camera, with 10-bit pixel depth, running at such a frame rate, requires
a bus capable of 45.6 Gbit/s — which is currently only possible with devices
such as a PCI-e x8 Gen3 interface [21]. For many applications, however,
streaming data at such volumes is too demanding – both in power and
computation time – hence requiring an alternative solution.
Codesign of hardware and software for computer vision applications is an
emerging research field to address the limitations of conventional systems
[17]. Focal-plane Sensor-processors (FPSPs) are a promising avenue for
reducing the data transfer between the camera and the processing unit. FPSPs,
often synonymous with Cellular Processor Arrays (CPAs) and Pixel Processor
Arrays (PPAs), perform processing on the sensor chip itself and are often
designed for tasks which require high frame rates or low latency [22]. The
principle behind them is that a small processor is embedded directly with each
pixel of the sensor. While FPSPs come in various forms for specific
applications, we in this paper we explore a general-purpose fine-grain
architecture SCAMP-5 [5], but one can imagine alternatives that could be
designed for various use cases.
One of the most widely used methods for image analysis is convolution kernels.
From edge detection using Sobel filters to document recognition using
Convolutional Neural Networks [13], convolutional kernels are the foundation
for many complex computer vision applications. Traditionally, application of
the convolutional kernels to the image data occurs on a CPU, but more recently
GPUs and FPGAs are used to accelerate the computations in parallel [1], [9].
Several systems have been designed to optimise the processing of convolutional
kernels on GPUs and FPGAs, leading to a vast array of techniques to reduce the
number of operational cycles needed to apply kernels to input data. While this
significantly increased throughput, these methods are still bounded in latency
as the image must make its way from the camera through to the host system. As
for FPSPs, the ability to process the data on the focal plane enables the
kernels to be applied to the image data at very low latency. Furthermore, the
unique ability to select the data which is transferred from the device to the
host reduces the data volume, which allows for high frame rates. However, the
technology is comparatively new. By design, they offer novel ways to interact
with the data, and while work has been done to provide a Domain-Specific-
Language and associated tools to program such hardware [14], there has been
less work done so far to produce code generation systems to make efficient use
of their architectural features when applying convolutional kernels in
particular.
One such system that does exist, however, is AUKE [11]. Given an $N\times N$
convolutional kernel, AUKE’s reverse-split algorithm generates code for
SCAMP-5 which applies the kernel efficiently to the captured image on the
focal-plane using analogue computation. AUKE is, however, limited to compiling
just a single convolutional kernel at a time using a reduced instruction set
that omits the more powerful instructions available in SCAMP-5.
In this work, we present an improved alternative to AUKE, with the ability to
produce code for applying multiple convolutional kernels at a time. The
problem is presented as a dynamic graph search problem in which we must
efficiently generate and traverse possible processor states to find a path
that describes the relevant convolutional computation. By incorporating
instruction selection and instruction scheduling into the core of search
process, we enable the use of more novel features of CPA architectures than
AUKE is able to use. By optimising the code for multiple kernels
simultaneously, common sub-expressions between kernels can be exploited and
produced only once rather than for each kernel. This reduces the computational
expense of applying the kernels, enabling applications to run at a faster
frame rate.
The primary objective of this work is to push the boundary of code generation
for FPSP devices through simultaneous kernel optimisation. We offer the
following contributions:
* •
Cain: A code generation algorithm which effectively makes use of common sub-
expressions across filters consisting of multiple convolutional kernels. Our
graph search strategy – which enables Cain to efficiently search large graphs
– combines instruction scheduling, instruction selection and register-
allocation constraints into the core of the search to make better use of
specific hardware capabilities in SIMD processors.
* •
We show how this search can be tractable for problems of interest through a
problem formulation based on AUKE’s multi-set–of–Atoms problem representation,
combined with a ranking heuristic and a hybrid graph-generator–graph-search
exploration strategy.
* •
We show how this approach allows flexible exploitation of hardware
capabilities (such as three-operand adds and multi-step shifts), and generates
very efficient use of additions to avoid multiplies.
* •
Evaluation of the effectiveness of Cain on the SCAMP-5 Focal-plane Sensor-
processor. We compare against AUKE and test the effectiveness of simultaneous
kernel optimisation. We conclude by exploring how our simultaneous kernel
optimisation extends to future devices with more registers per pixel.
The remainder of the paper is organised as follows. Section 2 describes the
SCAMP-5 and its instruction sets, Section 3 explains our proposed code
generation algorithm Cain, and in Section 4 detailed comparison is made
between Cain and AUKE, together with an evaluation of the effectiveness of
simultaneous kernel optimisation. Section 5 reviews the related work AUKE in
detail. Finally, Section 6 concludes our work, with a discussion about
potential future research.
Input Kernels Goal Approximation Configurable Traversal System Explore Node
Generate Next Goal Pair Apply Instruction In Reverse Register Allocation Code
Generation Matrices of CoefficientsptFinal-Goals (root
node)ptParentNodeptSpecificInstructionsInitial-Goal FoundOtherwiseNode
CulledptParent Node and Child Node
Figure 1: Cain System Overview.
$\left\\{\left\langle\begin{smallmatrix}1\\\ 2\\\
1\end{smallmatrix}\right\rangle\right\\}$pt$\left\\{\left\langle\begin{smallmatrix}1\\\
2\\\ 0\end{smallmatrix}\right\rangle,\left\langle\begin{smallmatrix}0\\\ 0\\\
1\end{smallmatrix}\right\rangle\right\\}$pt$\left\\{\left\langle\begin{smallmatrix}1\\\
2\\\ 0\end{smallmatrix}\right\rangle,\left\langle\begin{smallmatrix}0\\\ 1\\\
0\end{smallmatrix}\right\rangle\right\\}$pt$\left\\{\left\langle\begin{smallmatrix}1\\\
2\\\ 0\end{smallmatrix}\right\rangle,\left\langle\begin{smallmatrix}0\\\ 0\\\
1\end{smallmatrix}\right\rangle\right\\}$ptmov()pt…ptmov()pt…add()pt$\left\\{\left\langle\begin{smallmatrix}0\\\
1\\\ 1\end{smallmatrix}\right\rangle,\left\langle\begin{smallmatrix}1\\\ 1\\\
0\end{smallmatrix}\right\rangle\right\\}$ptpt$\left\\{\left\langle\begin{smallmatrix}0\\\
1\\\
1\end{smallmatrix}\right\rangle\right\\}$pt$\left\\{\left\langle\begin{smallmatrix}0\\\
1\\\ 0\end{smallmatrix}\right\rangle,\left\langle\begin{smallmatrix}0\\\ 0\\\
1\end{smallmatrix}\right\rangle\right\\}$pt$\left\\{\left\langle\begin{smallmatrix}0\\\
1\\\
0\end{smallmatrix}\right\rangle\right\\}$ptmov()pt…ptadd()pt…ptmov()pt…ptadd()pt…
$\begin{array}[]{p{1em}c|p{3cm}}&\text{Step:}&\text{Instruction:}\\\ \hline\cr
1&\leavevmode\hbox to12.98pt{\vbox
to12.98pt{\pgfpicture\makeatletter\hbox{\hskip 6.49223pt\lower-6.49223pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{6.29224pt}{0.0pt}\pgfsys@curveto{6.29224pt}{3.47514pt}{3.47514pt}{6.29224pt}{0.0pt}{6.29224pt}\pgfsys@curveto{-3.47514pt}{6.29224pt}{-6.29224pt}{3.47514pt}{-6.29224pt}{0.0pt}\pgfsys@curveto{-6.29224pt}{-3.47514pt}{-3.47514pt}{-6.29224pt}{0.0pt}{-6.29224pt}\pgfsys@curveto{3.47514pt}{-6.29224pt}{6.29224pt}{-3.47514pt}{6.29224pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.0pt}{-3.86665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{\hfil#\hfil\cr\cr\vskip-3.0pt\cr\hbox{{7}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&{mov(B,A,south)}\\\
2&\leavevmode\hbox to12.98pt{\vbox
to12.98pt{\pgfpicture\makeatletter\hbox{\hskip 6.49223pt\lower-6.49223pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{6.29224pt}{0.0pt}\pgfsys@curveto{6.29224pt}{3.47514pt}{3.47514pt}{6.29224pt}{0.0pt}{6.29224pt}\pgfsys@curveto{-3.47514pt}{6.29224pt}{-6.29224pt}{3.47514pt}{-6.29224pt}{0.0pt}\pgfsys@curveto{-6.29224pt}{-3.47514pt}{-3.47514pt}{-6.29224pt}{0.0pt}{-6.29224pt}\pgfsys@curveto{3.47514pt}{-6.29224pt}{6.29224pt}{-3.47514pt}{6.29224pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.0pt}{-3.86665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{\hfil#\hfil\cr\cr\vskip-3.0pt\cr\hbox{{6}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&{add(A,A,B)}\\\
3&\leavevmode\hbox to12.98pt{\vbox
to12.98pt{\pgfpicture\makeatletter\hbox{\hskip 6.49223pt\lower-6.49223pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{6.29224pt}{0.0pt}\pgfsys@curveto{6.29224pt}{3.47514pt}{3.47514pt}{6.29224pt}{0.0pt}{6.29224pt}\pgfsys@curveto{-3.47514pt}{6.29224pt}{-6.29224pt}{3.47514pt}{-6.29224pt}{0.0pt}\pgfsys@curveto{-6.29224pt}{-3.47514pt}{-3.47514pt}{-6.29224pt}{0.0pt}{-6.29224pt}\pgfsys@curveto{3.47514pt}{-6.29224pt}{6.29224pt}{-3.47514pt}{6.29224pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.0pt}{-3.86665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{\hfil#\hfil\cr\cr\vskip-3.0pt\cr\hbox{{5}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&{mov(B,A,north)}\\\
4&\leavevmode\hbox to12.98pt{\vbox
to12.98pt{\pgfpicture\makeatletter\hbox{\hskip 6.49223pt\lower-6.49223pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{6.29224pt}{0.0pt}\pgfsys@curveto{6.29224pt}{3.47514pt}{3.47514pt}{6.29224pt}{0.0pt}{6.29224pt}\pgfsys@curveto{-3.47514pt}{6.29224pt}{-6.29224pt}{3.47514pt}{-6.29224pt}{0.0pt}\pgfsys@curveto{-6.29224pt}{-3.47514pt}{-3.47514pt}{-6.29224pt}{0.0pt}{-6.29224pt}\pgfsys@curveto{3.47514pt}{-6.29224pt}{6.29224pt}{-3.47514pt}{6.29224pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.0pt}{-3.86665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{\hfil#\hfil\cr\cr\vskip-3.0pt\cr\hbox{{4}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&{add(A,A,B)}\end{array}$
1234567
Figure 2: Graph showing how Cain might search a simplified 1-dimensional
problem using CGDS. Numbered steps show the order that the paths are explored
with child nodes generated the first time a search step starts at a parent
node. Nodes are checked for being the Initial-Goal when pointed too. The red
node, and edge, correspond to a dead-end where a duplicate node has been found
at a higher cost than previously seen and so the node is not traversed
further. We see a path to the Initial-Goal is found after 7 steps, and the
code produced by this path is presented on the right. The mov() instruction in
step 5 exploits a common sub-expression such that the two Goals in its output
Goal-Bag are produced together, thus shortening the code.
## 2 Background: SCAMP-5 Focal-plane Sensor-processor
In this section, we discuss the capabilities of the next generation camera
technology SCAMP-5, and give an overview of the functionality used by Cain.
SCAMP-5 has been demonstrated in many different computer vision applications,
ranging from Visual Odometry systems [16], [3], [10], an end-to-end neural
sensor which performs learnt pixel exposures [15], to Convolutional Neural
Networks [20], [4]. Its distinctive ability to perform computation on the
focal-plane reduces power consumption and data transfers, making the device
promising for edge computation.
The SCAMP-5 architecture is a general-purpose fine-grain SIMD FPSP [6]. It has
a $256\times 256$ pixel array, and along with each pixel is a small Processing
Element (PE). All 65,536 processors execute the same instruction at one time.
In addition to 14 binary registers, each PE has analogue registers A through
to F as well as a NEWS register. Each PE can also address an XN, XE, XS, and
XW register that is actually that PE’s respective neighbours’ NEWS registers.
Each PE uses an analogue bus to link its available analogue registers, and
because values are stored as charge; analogue arithmetic is done directly on
the bus that connects the registers rather than on a separate arithmetic unit.
Instructions in the architecture control how register values are let into and
out of the bus with the caveat that values are inverted due to the nature of
the analogue electronics. Each macro instruction like add, sub, and mov are
made of multiple bus instructions that create the desired behaviour, where the
$\texttt{bus}n(w_{1},..w_{n},r_{0}..r_{k})$ instruction has the general rule
that the values of registers $r_{0}..r_{k}$ are summed up, negated, and
divided equally between the $n$ receiving-registers $w_{1}..w_{n}$. Since a
bus operation directly controls which registers are opened to the PE’s common
analogue bus, a register may only appear once in each bus instruction. Each
bus instruction also incurs significant noise and error factors, especially
for bus2 and bus3 [8].
Macro instruction arguments are written as if they are assignment statements.
For example; the macro instruction add(A, B, C) means $A:=B+C$ and is made up
of two bus instructions: bus(NEWS, B, C) meaning the NEWS register now
contains the value of $-(\textit{B}+\textit{C})$; and then bus(A, NEWS) so
that register A contains $\textit{B}+\textit{C}$. We can see here that the add
instruction has additional constraints, such that the two operands cannot be
the same register, and that the NEWS register is overwritten, and left
containing $-(\textit{B}+\textit{C})$ as a side effect. When using macro
instructions, we restrict the registers to A to F, and allow the macros
themselves to make use of the NEWS and neighbouring NEWS registers for us by
means of a direction value. We use subscripts to denote the registers of
neighbouring PEs. For example: mov2x(A, B, north, east) computes
$A:=B_{\text{north},\text{east}}$ in two bus instructions: bus(XS, B);
bus(A,XE). The first means that
$\textit{XS}_{\text{north},\text{east}}:=B_{\text{north},\text{east}}$ which
is equivalent to $\textit{NEWS}_{\text{east}}:=B_{\text{north},\text{east}}$
and then the second instruction means
$A:=\textit{XE}=\textit{NEWS}_{\text{east}}\implies
A=B_{\text{north},\text{east}}$.
While interesting uses of the bus instructions exist, allowing adding and
subtracting from neighbouring PEs, individual macro instructions are still
highly restricted in comparison to most modern instruction sets. Only
primitive analogue operations are available to each PE such as: Move, Add,
Subtract, Divide by two, and to acquire the value from the sensor [8]. The
lack of a multiplication instruction means the problem of generating
convolutional filter code for SCAMP-5 builds on the theory of multiplier-free
FIR filters [7].
The chip has been shown to be capable of operating at 100,000 FPS, largely
because it is not limited by the speed of an output bus to transfer all the
pixel data [5]. Instead of only offering an analogue or digitally encoded
output of all pixels at a time, like traditional camera sensors, the SCAMP-5
architecture allows binary outputs per pixel, and even event driven outputs.
This allows each PE to come to a judgement on its input pixel data and fire
its own event that sends the coordinates of the PE to the host; allowing
information transfer without divulging the actual image.
The architecture uses an off-chip controller to manage the fetch-decode-
execute cycle, with every pixel’s processor receiving the same instruction,
making it a single-instruction-multiple-data (SIMD) design. This has benefits
in terms of simplicity and efficiency as none of the Processing Elements need
to be able to fetch instructions for themselves. There is also provision for
masking pixels such that only selected PEs execute instructions.
One important consideration to be made when using and designing algorithms
related to the SCAMP-5 chip is noise introduced by the nature of the analogue
computation. Every use of the 7 analogue registers introduces noise to the
values stored. This makes finding optimal code to perform the convolutions
ever more vital for accurate results.
## 3 Cain
Cain is a framework for compiling convolutional filters, designed to search
through a configurable Cellular Processor Array (CPA) instruction set to find
efficient code. A fundamental concept Cain uses is to only consider a single
arbitrary PE in the CPA, and perform everything relative to it. This works for
SIMD architecture like SCAMP-5 because every PE will be executing the same
steps synchronously in parallel. The assumption we make when producing code is
that the neighbours of our arbitrary PE will exist and so will have done the
same work but at a relative offset in the input image. The aim is to search
through the graph of possible Processing Element states in such a way that
common sub-expressions in the given kernels are exploited and used to reduce
the cost of any path from initial to final PE states. To do this Cain searches
backwards, starting with a set of final kernels, these are the convolutional
filter, and applying instructions in reverse to simplify the kernels until
only the identity kernel222Single-entry matrix. Not to be confused with
identity matrix is left. Fig. 1 shows a high level overview of this process.
Searching backwards is a design choice that makes the search more effective
because it means the aim at each step is to make what needs to be solved
simpler than before. This means heuristics can be produced to always direct
the search towards the identity kernel rather than a system of heuristics
trying to accurately predict the path towards an arbitrary set of final Goals.
We present this as a dynamic graph search problem because the size of the
graph is intractable. Given the AnalogNet2 filter in Equation 1, Cain
identifies 37163 potential children nodes in the first step alone. This can be
reduced to 239 if we are willing to accept a less than exhaustive search of
the solution space. This restriction is applied when the computational cost of
computing the full set of children nodes is too high.
### 3.1 Definitions
This section provides an overview of notation and definition used in this
paper. Cain is designed such that different definitions could be used without
changing the fundamental search algorithm but the definitions we use here for
SCAMP-5 are based largely on AUKE’s, which provides an elegant way to
conceptualise the convolutional kernels without multiplication.
###### Example 1
We will look at a simple example of how a convolutional kernel is represented
in Cain. Here we use AnalogNet2 [20][12] which is a CNN designed for SCAMP-5.
$\textit{AnalogNet2}=\begin{Bmatrix}\frac{1}{4}\begin{bmatrix}0&0&0\\\
-3&1&0\\\ -3&0&2\end{bmatrix},&\frac{1}{4}\begin{bmatrix}-4&-1&-1\\\ -1&2&0\\\
1&1&0\end{bmatrix},&\frac{1}{4}\begin{bmatrix}-1&2&0\\\ -1&1&-3\\\
0&-3&0\end{bmatrix}\end{Bmatrix}$ (1)
Since SCAMP-5 does not have multiplication we must approximate the kernel and
because it does have division-by-two instructions the natural approximation to
make is to find the nearest integer multiple of $\frac{1}{2^{d}}$ for each
coefficient in the kernel, given some number of divisions $d$. In our example
we have already extracted the common denominator such that $d=2$ and this
perfectly represents the kernel. The larger $d$ is, the larger the search
space and complexity of the problem, so $d$ can be limited to allow an
acceptable amount of approximation error such that the resulting program is
shorter and computational expense of compiling it is reduced.
###### Definition 1
Let an Atom, denoted as $(x,y,z,\textit{sign})$, be a representation of
$\frac{1}{2^{d}}$ of a pixel value at coordinate $x,y$, on the $z$th channel.
$x,y$ are coordinates relative to the arbitrary PE and so also the centre of
the kernel, and $z$ refers to an image input channel. The sign is used to
negate the value if necessary.
###### Definition 2
Let a Goal, denoted as $\\{atom_{1},atom_{2},...\\}$, be a multi-set of Atoms.
The Goal represents an arbitrary kernel, however, scaled by $2^{d}$. The
aggregate of the values represented by each of the Atoms yields the same
result as applying the scaled kernel.
Representing a convolutional kernel as a Goal is a convenient way to support
multiply-free instruction set, such as SCAMP-5. One can simply view this as
unrolling the multiply instruction into additions. Using Goals simply re-
frames the problem by scaling everything by $2^{d}$, and approximating
coefficients to the nearest number of Atoms.
###### Definition 3
Let a Goal-Bag, denoted as $\\{goal1,goal2,...\\}$, be a multi-set of Goals.
The Goal-Bag is used to capture the state of our arbitrary PE. This includes
defining the Final-Goals, the set of convolution kernels we wish to compute;
and the Initial-Goals, the set of Goals which the computation will start from.
Using these definitions of Goals and Atoms we see that the first kernel from
Example 1 can be represented by $G$
$K=\frac{1}{4}\begin{bmatrix}0&0&0\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-3}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}&0\\\
{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}-3}&0&{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}2}\end{bmatrix},\quad
G=\begin{Bmatrix}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(-1,0,0,-)},&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(-1,0,0,-)},&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(-1,0,0,-)},\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(0,0,0,+)},&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(-1,-1,0,-)},&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(-1,-1,0,-)},\\\
{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(-1,-1,0,-)},&{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(1,-1,0,+)},&{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(1,-1,0,+)}\end{Bmatrix}$
As our Goal notation is verbose, we provide a compact version that
disambiguates Goals from kernels
$\begin{gathered}G=\left\langle\begin{smallmatrix}0&0&0\\\ -3&1&0\\\
-3&0&2\end{smallmatrix}\right\rangle\implies\frac{1}{2^{2}}\begin{bmatrix}0&0&0\\\
-3&1&0\\\ -3&0&2\end{bmatrix}\star\text{Image Input}\\\ \text{\scriptsize
where the $\star$ operator applies the left-hand convolutional kernel to the
right-hand array}\end{gathered}$ (2)
By repeating this for process the rest of the convolutional kernels in the
AnalogNet2 filter, the Final-Goals Goal-Bag FG is produced:
$\textit{{FG}}\,=\begin{Bmatrix}\left\langle\begin{smallmatrix}0&0&0\\\
-3&1&0\\\
-3&0&2\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}-4&-1&-1\\\
-1&2&0\\\
1&1&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}-1&2&0\\\
-1&1&-3\\\ 0&-3&0\end{smallmatrix}\right\rangle\end{Bmatrix}$ (3)
Since, in our example, $d=2$; the Goal representation of the identity kernel
($G_{\textit{ID}}$) that makes up the Initial-Goals, is based on the
approximation of the Final-Goals:
$\displaystyle K_{\textit{ID}}=\frac{1}{4}\begin{bmatrix}0&0&0\\\ 0&4&0\\\
0&0&0\end{bmatrix}\implies
G_{\textit{ID}}=\left\langle\begin{smallmatrix}0&0&0\\\ 0&4&0\\\
0&0&0\end{smallmatrix}\right\rangle$ (4)
Moving a value around the processor array is expressed by translating every
Atom of a Goal. Addition and subtraction can be expressed by combining two
Goals into one, making sure to cancel out positive and negative Atoms with the
same coordinates. Since Cain searches backwards, we apply these operations in
reverse. For 2-operand addition this means we take a Goal, $G$, that we wish
to generate code for, then produce 2 new Goals that when added together
produce $G$. Defining Goals as multi-sets of Atoms makes this process
intuitive as we can simply split the Atoms between two Goals in every possible
permutation (or fewer if we are willing to assume some are non-optimal, or
willing to miss potentially better code for the sake of more efficient code
generation). This definition also restricts the reverse search process since
when splitting a Goal we cannot split an Atom. To compute the red Atoms in $G$
naively, PEs must sum them and read this value from the west thus translating
the Atoms eastward.
### 3.2 Search Strategy
Cain’s reverse search algorithm works iteratively taking the state of an
arbitrary PE, defined as a Goal-Bag:
$\textit{{F}}\,:=\\{G_{1},G_{2},G_{2},G_{3}...\\}$ (5)
This is a node in our search graph and represents the state we aim to achieve
by executing the instructions that form a path from the initial-Goals to this
node. In the search graph, nodes are generated dynamically as the graph is
explored. Fig. 2 shows a simplified view of how a graph might look as it is
generated and searched. We simplify the exploration such that in each
iteration of the search algorithm we produce a Goal-Bag Pair of an Uppers
Goal-Bag and a Lowers Goal-Bag as well as an instruction, with the following
constraints:
$\displaystyle(\textit{{U}}\,,\textit{{L}}\,),inst=\textit{nextPair}(\textit{{F}}\,)\text{
where
}\textit{{U}}\,\subseteq\textit{{F}}\,,\;\textit{{U}}\,=inst(\textit{{L}}\,)$
(6)
This is in contrast to AUKE’s method, shown later in Equation 16. The new
child node, C , is then produced by applying the instruction in reverse using
the following rule, with the instruction becoming an edge in the graph:
$\textit{{C}}\,=(\textit{{F}}\,\setminus\textit{{U}}\,)\cup\textit{{L}}\,$ (7)
Following our AnalogNet2 example from Equation 3, the first iteration of the
search algorithm will start with FG and the Pair of Goal-Bags Cain produces is
as follows:
$\displaystyle\textit{{U}}\,=\begin{Bmatrix}\left\langle\begin{smallmatrix}-1&2&0\\\
-1&1&-3\\\
0&-3&0\end{smallmatrix}\right\rangle\end{Bmatrix},\quad\textit{{L}}\,=\begin{Bmatrix}\left\langle\begin{smallmatrix}-1&2&0\\\
-1&1&0\\\
0&0&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}0&0&0\\\
0&0&-3\\\
0&0&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}0&0&0\\\
0&0&0\\\ 0&-3&0\end{smallmatrix}\right\rangle\end{Bmatrix}$ (8)
$\displaystyle\textit{inst}=\textit{{U}}\,\leftarrow\textit{add}(L_{1},L_{2},L_{3})$
(9)
$\displaystyle\textit{{C}}\,=\begin{Bmatrix}\left\langle\begin{smallmatrix}0&0&0\\\
-3&1&0\\\
-3&0&2\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}-4&-1&-1\\\
-1&2&0\\\
1&1&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}-1&2&0\\\
-1&1&0\\\
0&0&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}0&0&0\\\
0&0&-3\\\
0&0&0\end{smallmatrix}\right\rangle,&\left\langle\begin{smallmatrix}0&0&0\\\
0&0&0\\\ 0&-3&0\end{smallmatrix}\right\rangle\end{Bmatrix}$ (10)
The multi-set semantics here mean that if the Goals in L are all already part
of F then the number of Goals to solve is reduced, and so by applying more
pairs $(\textit{{U}}\,,\textit{{L}}\,)$ we traverse the graph of Goal-Bags,
until we reach the initial-state, where the only Goal in the Goal-Bag is the
identity Goal. In our example (Equation 10) we see that the sub-expression of
3 negative Atoms is reused in $C_{4}$ and $C_{5}$ since applying a mov2x next
could eliminate $C_{5}$ from C . There is also further potential to reuse this
by how we split $C_{1}$. Once the initial Goal-Bag is found the path from the
initial Goal-Bag back to the Final-Goals becomes the list of instructions that
form our generated program.
Input: s
1 $\textit{deque}\leftarrow[(s,\textit{null})]$
2 while _$\textit{deque}\neq[]$_ do
3 $n,g\leftarrow\textit{deque}[0]$
4 $\textit{deque}\leftarrow\textit{deque}[1..]$
5 if _$g=\textit{null}$_ then
6 do node computation on n
7 $g\leftarrow\textit{childGenerator}(n)$
8 end if
9 $c\leftarrow\textit{g.yield()}$
10 if _$c\neq\textit{null}$_ then
11 $\textit{deque}\leftarrow[(c,\textit{null})]+\textit{deque}+[(n,g)]$
12 end if
13
14 end while
Algorithm 1 CGDS Graph Search
After this point Cain continues searching for shorter paths, and can cull any
nodes with longer paths. During the search the same Goal-Bags may be
reproduced in different ways, we cull the current node any time a Goal-Bag is
produced that has already been seen at a lower or equal cost, or if the Goal-
Bag has more Goals than available registers.
The second part of the search strategy defines the search order. Each
invocation of the reverse search algorithm produces one new node, C , and the
input node is incremented to know how many of its children have been produced
so far. Cain uses this simple definition to allow several graph traversal
algorithms to be implemented. Using Depth-First-Search (DFS), Cain can simply
maintain a stack of the nodes. On each cycle the top node is popped off the
stack and given to the reverse search algorithm. Then the incremented parent
node is put back on the stack, followed by the new child node.
While DFS performs well in AUKE, it struggles in Cain because the number of
child nodes at every level is far greater, since each edge is only one
instruction and there are multiple kernels to consider. This means the size of
the graph we would like to search is much larger and we are unable to search
even a small fraction of it. To overcome this we use a graph-traversal
algorithm that, for our purposes, we call Child-Generator-Deque-Search (CGDS).
The aim of this algorithm is to ensure that the search does not end up
‘trapped’ in one small part of the graph, but can effectively search traverse
many children of many of the nodes that are found where DFS will search all of
the children of nodes at the extent of the paths it searches before searching
the second children of nodes earlier in the graph. Algorithm 1 shows a pseudo-
code implementation of CGDS. In each cycle the front of the queue is polled,
if the node has not been seen before, Cain checks to see if it can be directly
transformed from the initial-state Goal-Bag, this is the ‘node computation’.
The node is then passed to the reverse search algorithm to attempt to produce
the next new child node and to increment parent node – this is implicit in
calling ‘yield()’ on g. The child node, if it exists, is put on the front of
the queue and the incremented parent node is put on the back. We do not claim
that CGDS is novel, but we have found it superior to obvious alternatives, and
the strategy used in [2]; for details see [18].
### 3.3 Cost Function
In the reverse search algorithm we see that the pairs of Uppers and Lowers are
produced one at a time. While this simplification allows us to produce more
generic graph traversal implementations; what allows Cain to efficiently find
solutions, are the heuristics that allow us to order the pairs that are
produced for a node from the most promising to the least. This type of
heuristic provides the order of siblings to search so we call it a ‘local
heuristic’. It doesn’t compare nodes in different parts of the graph, which we
would call a ‘global heuristic’. We found that we were unable to find
effective global heuristics because traversal algorithms that take advantage
of such heuristics end up producing huge frontier sets of nodes making the
memory requirements too large. The use of local heuristics drives the SCAMP-5
code generation in Cain instead, though support for best-first-search with
global heuristics is available in Cain. The local heuristics used for SCAMP-5
are based on generating every child node of the parent and then ordering them
based on a cost function. There are 3 main components considered for the cost:
Atom distance, repeated Goals, and divisions. A simplified formula is shown in
Equation 11.
$\displaystyle cost(\textit{{C}}\,)$
$\displaystyle=\textit{dists}(\textit{{C}}\,)+\textit{reps}(\textit{{C}}\,)+\textit{divs}(\textit{{C}}\,)$
(11) $\displaystyle\textit{dists}(\textit{{C}}\,)$
$\displaystyle=\sum_{G\in\textit{{C}}\,}\left(|G|+\sum_{a\in
G}\left(|a.x|+|a.y|\right)\times\begin{Bmatrix}\frac{1}{2}&\text{if}\not\exists
B\in\textit{{C}}\,.G\subset B\\\ 1&\text{otherwise.}\end{Bmatrix}\right)$ (12)
$\displaystyle\textit{reps}(\textit{{C}}\,)$
$\displaystyle=\sum_{\\{G\in\textit{{C}}\,:\text{$G$ is unique wrt any
translations}\\}}\begin{Bmatrix}|G|^{2}&\exists a,b\in G.a\neq b\\\
0&\text{otherwise.}\end{Bmatrix}$ (13)
$\displaystyle\textit{divs}(\textit{{C}}\,)$
$\displaystyle=\frac{2^{d}}{\text{min}(\textit{multiplicity}(a)\forall a\in
G.\forall G\in\textit{{C}}\,)}$ (14)
The Atom distance part counts up how many Atoms every Goal in C has, and how
far from the centre they are, with some relief if the Goal is a sub-Goal of
another Goal in C . The repeated Goals portion of the cost penalises C by the
square of number of Atoms in each Goal, unless that Goal is equal to a
translation of another Goal in C . The divisions component penalises C for the
number of division operations that would be required to produce the Goals from
the identity-kernel Goal, $G_{\textit{ID}}$.
## 4 Evaluation
All performance evaluation is conducted on an Intel Core i7-7700HQ CPU (4
cores, 8 threads) with a base frequency of 2.80GHz. The computer has 16GB of
RAM, runs Ubuntu 18; as well as Java 1.8 (Oracle) and Python 3.6 to run Cain
and AUKE respectively. The implementation of AUKE used, as developed by
Debrunner, can be found on
Github333github.com/najiji/auto_code_cpa/tree/75c017e5ad28c0f3f040fb9f84d7f8727d035baa.
Cain source code can be found at github.com/ed741/cain, and the specific
version and sources for experimental setups presented in this evaluation can
be found at [19].
### 4.1 Performance Evaluation Against AUKE
Comparison of our work Cain against AUKE is performed by comparing resulting
code generated from the respective compilers, given the same input filters.
Both compilers are given 60 seconds to find a solution using all 6 registers.
Note as Cain supports multi-threading, it spawns 4 worker threads to perform
the search.
As shown in Table 1, Cain significantly outperforms AUKE. Cain supports a
wider set of instructions in contrast of AUKE, enabling generation of more
efficient code. Not only this, the search strategy used by Cain is better than
AUKE’s, as shown in $5\times 5$ Gaussian Kernel, were using the same set of
instructions (Basic), code generated by Cain is half in length when compared
to output of AUKE’s. Although, in further testing, AUKE is able to produce
less inefficient code for this kernel given fewer registers. When given
multiple kernels, Cain is able to perform simultaneous kernel optimisation.
For example when combining $3\times 3$ and $5\times 5$ Gaussian, unlike AUKE,
Cain is implemented to utilise the common sub-expressions between the kernels,
thus, generating shorter code than naively concatenating the code for each of
the Gaussian kernels. Neither Cain or AUKE perform a compete exhaustive
search.
The AnalogNet2 filter is the kernels used in AnalogNet2 [20][12], which is a
CNN for SCAMP-5, capable of MNIST digit recognition. Cain requires only 21
instructions whereas AUKE produces kernel code which has in total 49
instructions. Reduced code not only improves the execution time, but also
reduces the noise build up, which is significant problem as discussed in [20].
If the aim of finding sub-expressions is to eliminate redoing work, then the
number of add and subtract operands is a proxy for how effective the search
for sub-expressions is, regardless of how translations are handled. Table 2
shows that AUKE’s code has 40 add or subtract operands whereas Cain’s code has
only 27. We have compared the runtime of AnalogNet2’s convolution kernels,
generated by AUKE and Cain on the physical SCAMP-5. Note, as AUKE produces
code which performs invalid register manipulation, the fixed code as used in
[12], which executes on the device is 81 instructions long. The execution time
of the code produced by AUKE and Cain for the convolution kernels were
$35\mu\text{s}$ and $9\mu\text{s}$ respectively, showing almost 4 times
speedup.
Table 1: Kernels tested in AUKE and Cain. Values on the righthand side of the table refer to the number of SCAMP-5 macro instructions in the programs generated by AUKE and Cain for each filter. AUKE can only use the ’basic’ macro instructions, so Cain is run twice; to compare its effectiveness under the same restrictions as AUKE. Since AUKE does not offer a way to compile multiple kernels at once, values for each kernel are given separately. Name | Approximated Filter | AUKE | Cain
---|---|---|---
| | Basic | All | Basic
3$\times$3 Gauss | $\left\\{\frac{1}{16}\begin{bmatrix}1&2&1\\\ 2&4&2\\\ 1&2&1\\\ \end{bmatrix}\right\\}$ | 12 | 10 | 12
5$\times$5 Gauss | $\left\\{\frac{1}{64}\begin{bmatrix}0&1&2&1&0\\\ 1&4&6&4&1\\\ 2&6&10&6&2\\\ 1&4&6&4&1\\\ 0&1&2&1&0\\\ \end{bmatrix}\right\\}$ | 50 | 19 | 25
5$\times$5 and 3$\times$3 Gauss | $\left\\{\frac{1}{64}\begin{bmatrix}0&1&2&1&0\\\ 1&4&6&4&1\\\ 2&6&10&6&2\\\ 1&4&6&4&1\\\ 0&1&2&1&0\\\ \end{bmatrix},\frac{1}{64}\begin{bmatrix}0&0&0&0&0\\\ 0&4&8&4&0\\\ 0&8&16&8&0\\\ 0&4&8&4&0\\\ 0&0&0&0&0\\\ \end{bmatrix}\right\\}$ | $(50+12)$ | 26 | 39
AnalogNet2 | $\left\\{\begin{array}[]{c}\frac{1}{4}\begin{bmatrix}0&0&0\\\ -3&1&0\\\ -3&0&2\\\ \end{bmatrix},\frac{1}{4}\begin{bmatrix}-4&-1&1\\\ -1&2&0\\\ 1&1&0\\\ \end{bmatrix},\vspace{0.3em}\frac{1}{4}\begin{bmatrix}-1&2&0\\\ -1&1&-3\\\ 0&-3&0\\\ \end{bmatrix}\end{array}\right\\}$ | $\begin{array}[]{c}(13+21\\\ +15)\end{array}$ | 21 | 30
Table 2: Comparison of Code for the AnalogNet2 filter generated by AUKE and Cain. The Input Register is ‘A’ and the output registers for the 3 kernels are ‘A’,‘B’,‘C’ respectively. For AUKE, kernel 2 is run first since testing showed it was longest so this gives AUKE more registers to use. AUKE | Cain
---|---
Kernel 2 ⬇ 1mov(B,A); 2divq(B,B); 3divq(B,B); 4movx(C,B,north); 5neg(C,C); 6neg(D,C); 7movx(E,D,west); 8neg(E,E); 9add(F,B,E); 10movx(B,D,east); 11add(B,B,E); 12movx(D,E,south); 13movx(D,D,south); 14sub(B,B,D); 15add(B,B,F); 16add(B,C,B); 17movx(C,C,west); 18add(B,B,C); 19movx(C,F,south); 20add(B,C,B); 21add(B,B,F); | Kernel 3 ⬇ 22mov(C,A); 23divq(C,C); 24divq(C,C); 25movx(D,C,south); 26neg(D,D); 27movx(E,C,east); 28sub(D,D,E); 29movx(E,C,north); 30add(E,E,D); 31add(D,D,D); 32add(D,E,D); 33movx(E,C,west); 34sub(C,C,E); 35add(D,D,C); 36movx(C,C,north); 37add(C,D,C); | Kernel 1 ⬇ 38divq(A,A); 39divq(A,A); 40movx(D,A,west); 41neg(D,D); 42movx(E,D,south); 43add(D,D,E); 44add(E,A,D); 45movx(A,A,south); 46movx(A,A,east); 47add(A,D,A); 48add(A,A,A); 49add(A,E,A); | ⬇ 1diva(A,D,E); 2div(D,E,C,A); 3movx(E,D,west); 4movx(C,E,north); 5neg(F,E); 6subx(B,F,east,A); 7addx(E,E,D,south); 8add2x(D,F,D,north,north); 9sub2x(F,D,south,south,C); 10add2x(D,C,D,east,south); 11add(E,E,D); 12movx(D,A,north); 13add2x(A,C,A,east,east); 14movx(C,B,east); 15add(D,F,D); 16add2x(F,F,E,east,south); 17movx(E,B,south); 18addx(A,B,A,south); 19addx(A,B,A,west); 20add2x(B,F,B,north,west); 21add(C,D,C,E);
### 4.2 Effectiveness of the Search Strategy
If Cain has an effective heuristic we will quickly see a point of diminishing
returns in code length, as Cain continues to search new nodes and takes more
time. We can track the number of nodes that are explored before finding any
plan in Cain, and so use this as a measure of the search strategy and
heuristics that is more independent of physical compute performance. With this
in mind we test the effectiveness of our heuristic by constructing 100 samples
of randomly generated single kernel filters as in Equation 15. Running Cain as
per the following configuration – Maximum Nodes to Explore: 20000, Maximum
Search Time: 60s, Worker Threads: 1 – allows us to collect as many plans as
can be found in the given time limit. We then ran Cain again, but with Cain’s
SCAMP-5 heuristic disabled and replaced with a random sort. This allows us to
compare Cains heuristics against an unaided benchmark.
$\begin{array}[]{c}\frac{1}{8}\begin{bmatrix}u_{1}&u_{2}&u_{3}\\\
u_{4}&u_{5}&u_{6}\\\ u_{7}&u_{8}&u_{9}\end{bmatrix}\\\ \text{ \footnotesize
Given $u_{1}..u_{9}$ are integers sampled uniformly from the range
$[0..8]$}\end{array}$ (15)
We found that Cain was unable to find any plan for any of the 100 sample
filters without its heuristics, principally demonstrating that effective
heuristics are required in Cain for any tangible progress to be made. We plot
the lengths of the best plans found against the number of nodes expanded
before the plan is found in Fig. 3. We can see that improvements are fewer and
further between after the first 2500 nodes are explored. After this we see
that we can expect at most a reduction equal to the reduction seen at 2500 for
the rest of the nodes explored. This clearly demonstrates a point of
diminishing returns for these filters. If the heuristic is effective we expect
it to direct the search towards short plans first, and try instructions less
likely to be optimal later. This model fits the data well as we see short
plans are found quickly, and while improvements can be made, it is clear that
they are found less often as the search continues.
Figure 3: Left: Graph showing the median number of instructions in the best
plans found before $n$ nodes have been explored by Cain. With 100 samples of
randomly generated singular $3\times 3$ kernel filters. Right: Graph showing
the number of instructions in the shortest programs found by Cain for filters
with 1, 2, 3, and 4 random $3\times 3$ kernels. 25 samples were produced for
each kernel count.
### 4.3 Effectiveness of the Simultaneous Kernel Optimisation
One of the significant features of Cain is to efficiently generate code for
filters with multiple kernels, and do this simultaneously such that shared
common sub-expressions can be reused. As it is possible for Cain to perform
exhaustive searches for plans, given sufficient time, it will find a solution
that simply computes the individual kernels independently, or find a solution
with lower cost – utilising the common sub-expressions.
First, we wish to test whether the length of generated code is sub-linear to
the number of input kernels. To test this, we again generate kernels using the
using the method in Equation 15. For kernel counts from 1 to 4 we generated 25
filters each and test them all using the same configuration as before except
that we remove the maximum nodes explored constraint, and allow 4 worker
threads. We plot the results in Fig. 3 and see that the results appear worse
than linear, suggesting that common sub-expressions are not effectively being
taken advantage of.
We hypothesise that the limited number of registers in the SCAMP-5
architecture is the major limiting factor in producing efficient code. To test
this we increase the number of available registers to 18. For filters with 1
kernel up to 10 kernels we generate 10 samples each. Every kernel in the 100
filters is produced as in Equation 15. For each sample, Cain compiles the
kernels individually, given the appropriate number of registers such that
other kernels in the filter would not be overwritten. Then we compile the
kernels simultaneously using Cain. All compilations are given 60s to run, with
4 worker threads.
Figure 4: Graph comparing the sum of the shortest SCAMP-5 code lengths found
for kernels compiled individually, against the same kernels compiled
simultaneously as one filter. For each filter a total of 18 registers were
made available (more than in SCAMP-5) to reduce register availability as a
limiting factor. In total 100 filters are produced, 10 for each number of
kernels per filter. Each kernel is a randomly generated $3\times 3$ kernel
with coefficients uniformly selected in eighths from 0 to 1 (inclusive).
Fig. 4 shows the results of this test. We see clearly that when register
limitations are not a restricting factor Cain is able to consistently improve
the performance of filter implementations by compiling them simultaneously. We
see that improvements grow with more kernels, and it appears that the length
of code generated for simultaneously compiled kernels increases sub-linearly.
This supports the idea that with more kernels, ever more common-sub
expressions can be exploited.
## 5 Related Work: AUKE
In this section we look at how AUKE operates to provide extra context and
contrast for Cain. Automatic Kernel Code Generation for Analogue SIMD (AUKE)
is an algorithm for generating code given a single convolutional kernel
created by T. Debrunner [11]. It can be characterised by 4 main steps: kernel
approximation; the reverse split algorithm; graph relaxation; and finally
register allocation. First, AUKE approximates the input kernel into the Goal
representation. In this process Cain is similar to AUKE and the reasoning and
mechanics have been discussed in Section 3.1.
Unlike in Cain, multiple instructions are represented by a single elemental
transformation of Goals. These elemental transformations form edges of a graph
that describe the translation, addition, subtraction and division of Goals to
produce the desired convolutions filter. This abstraction allows AUKE to
reduce the effective size of the search space at the cost of granularity in
instruction selection and being extensible to hardware features such as
3-operand addition. Debrunner called this the ‘Reverse-Split Algorithm’.
The graph of elemental transformations is dynamically generated via a
recursive depth-first search that tries to split a Goal $G$, that needs to be
produced, into 3 sub-Goals:
$G=U\cup L\cup R\quad\text{ where
}U=\textit{elementalTransformation}(\textit{L})$ (16)
This recursive algorithm then means that if the search can find solutions for
$L$ and $R$ (two smaller problems) it can trivially create $U$ and therefore
the desired Goal.
In the ideal case $R=\emptyset$ and so only $L$ needs to be produced and we
save one addition. In the worst case $L=U=\emptyset$ and $R$ is a
transformation of $G$ and so less useful work is done in that step. If two
Goals are equal they are merged such that they aren’t calculated twice, to
exploit common sub-expressions in the Goals. This process is repeated until a
single Goal, the initial-Goal, is left. This algorithm is able to entirely
search the relevant problem space, given a couple of assumptions. Most
notably, the assumption that every sub-Goal generated is a subset of the
Final-Goal. This reduces the search space significantly to the most promising
but not necessarily the best solutions, allowing AUKE to find generally
effective solutions.
The algorithm is made efficient and useful by intelligently selecting the
order with which $U$s, $L$s, and $R$s are generated at every recursive step.
By selecting pairs of $U$ and $L$ that are likely to lead to efficient code,
the algorithm can quickly find some path to the initial-Goal. From then on the
recursive search can stop early if a lower cost solution has already be found.
The Graph Relaxation step aims to mitigate missing optimal solutions because
of the assumption that sub-Goals are always subsets of the Final-Goal by using
a ‘retiming’ algorithm used in integrated circuit design. This is not needed
in Cain since Cain searches instruction by instruction, and so any
optimisations found via graph relaxation are already a part of the search
space.
The final step is to perform register allocation on the graph to be able to
generate usable code. A maximum bound of registers is already accounted for in
the search algorithm, since spilling is not an option for the SCAMP-5
architecture. For this task; variable liveness is considered for each node of
the graph representation, and a graph colouring algorithm is used to find a
solution.
## 6 Conclusion
We have presented Cain, a compiler which produces SCAMP-5 instructions from a
set of convolutional kernels. Although the effectiveness of simultaneous
kernel optimisation is limited on the current iteration of the SCAMP-5, we
demonstrate, that with the increased number of registers, the length of the
output of Cain is sub-linear to the number of kernels given. We have conducted
extensive comparison against AUKE, and we demonstrate that the code generated
by Cain is more efficient, and exhibits almost 4x speed up when the generated
kernel is executed on the SCAMP-5 device. We believe that SCAMP-5 is a strong
candidate for edge computation, and by providing easy to use, yet efficient
code generation toolkit, we hope to accelerate the relevant research in this
field.
#### Acknowledgements
We would like to thank Piotr Dudek, Stephen J. Carey, and Jianing Chen at the
University of Manchester for kindly providing access to SCAMP-5, and their
support in our work. This work was partially supported by the EPSRC, grant
reference EP/P010040/1.
## References
* [1] Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al.: Tensorflow: A system for large-scale machine learning. In: 12th USENIX symposium on operating systems design and implementation OSDI 16). pp. 265–283 (2016)
* [2] Barthels, H., Psarras, C., Bientinesi, P.: Linnea: Automatic generation of efficient linear algebra programs (2019), https://arxiv.org/pdf/1912.12924.pdf
* [3] Bose, L., Chen, J., Carey, S.J., Dudek, P., Mayol-Cuevas, W.: Visual Odometry for Pixel Processor Arrays. In: 2017 IEEE International Conference on Computer Vision (ICCV). pp. 4614–4622 (Oct 2017)
* [4] Bose, L., Chen, J., Carey, S.J., Dudek, P., Mayol-Cuevas, W.: A camera that CNNs: Towards embedded neural networks on pixel processor arrays. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV). pp. 1335–1344 (2019)
* [5] Carey, S.J., Barr, D.R.W., Wang, B., Lopich, A., Dudek, P.: Locating high speed multiple objects using a scamp-5 vision-chip. In: 2012 13th International Workshop on Cellular Nanoscale Networks and their Applications. pp. 1–2 (Aug 2012)
* [6] Carey, S.J., Lopich, A., Barr, D.R.W., Wang, B., Dudek, P.: A 100,000 fps vision sensor with embedded 535GOPS/W $256\times 256$ SIMD processor array. In: 2013 Symposium on VLSI Circuits. pp. C182–C183 (2013)
* [7] Chandra, A., Chattopadhyay, S.: Design of hardware efficient fir filter: A review of the state-of-the-art approaches. Engineering Science and Technology, an International Journal 19(1), 212 – 226 (2016). https://doi.org/https://doi.org/10.1016/j.jestch.2015.06.006, http://www.sciencedirect.com/science/article/pii/S2215098615001068
* [8] Chen, J.: scamp5 kernel api macro analog.hpp file reference (Jan 2020), https://personalpages.manchester.ac.uk/staff/jianing.chen/scamp5d˙lib˙doc˙html/scamp5˙˙kernel˙˙api˙˙macro˙˙analog˙8hpp.html
* [9] Chen, Y.H., Krishna, T., Emer, J.S., Sze, V.: Eyeriss: An energy-efficient reconfigurable accelerator for deep convolutional neural networks. IEEE journal of solid-state circuits 52(1), 127–138 (2016)
* [10] Debrunner, T., Saeedi, S., Bose, L., Davison, A.J., Kelly, P.H.J.: Camera Tracking on Focal-Plane Sensor-Processor Arrays (2019)
* [11] Debrunner, T., Saeedi, S., Kelly, P.H.J.: AUKE: Automatic kernel code generation for an Analogue SIMD Focal-Plane Sensor-Processor Array. ACM Trans. Archit. Code Optim. 15(4) (Jan 2019)
* [12] Guillard, B.: Cnns-on-fpsps (May 2019), https://github.com/brouwa/CNNs-on-FPSPs/tree/c6b5c51839e9e3c453681e5b0a3e3ef541ba3cce
* [13] LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998)
* [14] Martel, J.: Unconventional Processing with Unconventional Visual Sensing. Ph.D. thesis, Institut National des Sciences Appliquées de Lyon (2019)
* [15] Martel, J.N.P., Müller, L.K., Carey, S.J., Dudek, P., Wetzstein, G.: Neural sensors: Learning pixel exposures for HDR imaging and video compressive sensing with programmable sensors. IEEE Trans. Pattern Anal. Mach. Intell. 42(7), 1642–1653 (2020)
* [16] Murai, R., Saeedi, S., Kelly, P.H.J.: BIT-VO: Visual Odometry at 300 FPS using Binary Features from the Focal Plane. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (2020)
* [17] Saeedi, S., Bodin, B., Wagstaff, H., et al.: Navigating the landscape for real-time localization and mapping for robotics and virtual and augmented reality. Proceedings of the IEEE 106(11), 2020–2039 (2018)
* [18] Stow, E.: Automatic Code Generation for Simultaneous Convolutional Kernels on Cellular Processor Arrays. Master’s thesis, Imperial College London (2020), http://edstow.co.uk/pub/2020/MEngThesis.pdf
* [19] Stow, E., Murai, R.: ed741/cain: 3.0-experiments.1 (Aug 2020), https://doi.org/10.5281/zenodo.3975615
* [20] Wong, M.Z., Guillard, B., Murai, R., Saeedi, S., Kelly, P.H.J.: AnalogNet: Convolutional Neural Network Inference on Analog Focal Plane Sensor Processors. arXiv preprint arXiv:2006.01765 (2020)
* [21] XIMEA: xib - pci express cameras with high speed and resolution, https://www.ximea.com/pci-express-camera/pci-express-camera
* [22] Zarándy, Á.: Focal-Plane Sensor-Processor Chips. SpringerLink : Bücher, Springer New York (2011), https://books.google.co.uk/books?id=AKwyCZvsXWMC
|
# Spectral properties of a three body atom-ion hybrid system
Daniel J. Bosworth<EMAIL_ADDRESS>Zentrum für Optische
Quantentechnologien, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
The Hamburg Centre for Ultrafast Imaging, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
Maxim Pyzh Zentrum für Optische Quantentechnologien, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
Peter Schmelcher Zentrum für Optische Quantentechnologien, Universität
Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
The Hamburg Centre for Ultrafast Imaging, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
###### Abstract
We consider a hybrid atom-ion system consisting of a pair of bosons
interacting with a single ion in a quasi-one-dimensional trapping geometry.
Building upon a model potential for the atom-ion interaction developed in
earlier theoretical works, we investigate the behaviour of the low-energy
eigenstates for varying contact interaction strength $g$ among the atoms. In
particular, we contrast the two cases of a static and a mobile ion. Our study
is carried out by means of the Multi-Layer Multi-Configuration Time-Dependent
Hartree method for Bosons, a numerically-exact ab initio method for the
efficient simulation of entangled mixtures. We find that repulsive atom
interactions induce locally-distinct modifications of the atomic probability
distribution unique to each eigenstate. Whilst the atoms on average separate
from each other with increasing $g$, they do not necessarily separate from the
ion. The mobility of the ion leads in general to greater separations among the
atoms as well as between the atoms and the ion. Notably, we observe an
exchange between the kinetic energy of the atoms and the atom-ion interaction
energy for all eigenstates, which is both interaction- and mobility-induced.
For the ground state, we provide an intuitive description by constructing an
effective Hamiltonian for each species, which aptly captures the response of
the atoms to the ion’s mobility. Furthermore, the effective picture predicts
enhanced localisation of the ion, in agreement with our results from exact
numerical simulations.
## I Introduction
Over the past decades, our understanding of the physics of neutral ultra-cold
atom and laser-cooled ion systems has seen an unprecedented development, which
has borne deep insights into their underlying and emergent physical phenomena.
The superb degree of control achieved over these two quantum systems enables a
high degree of accuracy and precision at both the single- and many-particle
levels and has established them at the forefront of modern quantum many-body
research. Recently, the two fields have been combined [1, 2, 3, 4], creating a
versatile experimental platform for exploring fundamental interaction
processes between atoms and ions at milli-Kelvin to micro-Kelvin temperatures
[5, 6]. The most prominent experimental and theoretical accomplishments to
date include: studies on atom-ion collisions and reactions [7, 8, 9, 10, 11]
and related phenomena, such as the formation of chemical bonds [12, 13],
sympathetic cooling [4, 14, 15] and charge transport [16, 17]; quantum
simulation of condensed matter physics [18] and polaron models [19, 20];
quantum information investigations in the context of controlled entanglement
generation [21, 22] and decoherence effects [23]; and precision measurements
where the ion acts as a local probe of the host gas’ properties [3, 4, 24].
One of the on-going challenges faced by experimentalists in the field of atom-
ion research is to create hybrid systems at nano-Kelvin temperatures. The
nano-Kelvin scale marks the boundary of the ultra-cold regime, in which
quantum phenomena dominate. The earliest hybrid traps were based on a
straightforward superposition of optically-trapped atoms with ions confined in
a Paul trap. The drawback of this scheme was found to be a heating mechanism
caused by the excess micromotion of the ion [25, 26]. In an effort to overcome
this perceived limitation, several alternative schemes are currently being
pioneered, such as: photoionisation of an atomic cloud using a femto-second
laser [27], optical traps for ions [28, 29, 30, 31, 32], and highly-excited
Rydberg atoms within an atomic cloud [33, 34, 17].
In addition to the endeavours with alternative trapping schemes, proposals
were also made to use the established Paul trap approach with a
${}^{6}\text{Li-}{}^{174}\text{Yb}^{+}$ hybrid mixture [35], whose high mass-
imbalance was predicted to undermine the micro-motion induced heating. This
set-up was realised in a recent experimental breakthrough [36], reaching
temperatures in the s- and p-wave scattering regime. In this regime of few
partial waves, the increasingly-weighty quantum effects lead to deviations
away from classical predictions and may enable the experimental observation of
hitherto-unseen atom–ion Feshbach resonances [37]. The presence of such a
resonance would allow for complete control over the scattering parameters and
thus, over the atom–ion interaction itself.
These recent experimental advances provide fresh impetus to extend the current
theoretical understanding about the nature of the long-range atom-ion
interaction at zero temperature. In this $T=0$ regime, the inelastic processes
dominate the system’s scattering dynamics and atoms can be captured in the
weakly bound states of the atom-ion polarisation potential. These bound states
enable the formation of so-called mesoscopic molecular ions, which are
typically hundreds of nanometres in size [38, 39]. The amount of atoms
captured by the ion is limited by the interaction strength among the atoms.
The kinetic energy released during the capture process is distributed via
phonon excitations among the unbound fraction and a density disturbance is
created at the ion position [40, 41] or even, in the limit of a Tonks-Giradeau
gas, a density bubble [42].
In our previous studies, we performed detailed analysis of the ground-state
properties and dynamical behaviour of an atom–ion hybrid system for a static
ion in a quasi-1D system [43, 41], analogous to the aforementioned
${}^{6}\text{Li-}{}^{174}\text{Yb}^{+}$ mixture. We have also examined the
case of an equal mass system [39], analogous to optically-trapped ions subject
to the same external potential as the neutral atoms. In this work, we present
an extension of both these investigations to the lowest-energy eigenstates of
a few-body system composed of a single ion and two neutral bosonic atoms at
zero temperature, with both species parabolically-confined in a quasi-1D
geometry. The eigenstates of our few-body hybrid mixture are obtained via the
Multi-Layer Multi-Configuration Time-Dependent Hartree method for Bosons (ML-
MCTDHB) [44, 45], a numerically-exact ab initio method that efficiently
accounts for the intra- and inter-species correlations via a time-dependent,
variationally-optimised basis. Previously, ML-MCTDHB has been successfully
applied to solve similar problems in mixtures of neutral bosonic species [46,
47] and there is also an extension for dealing with mixtures of both bosonic
and fermionic species [48, 49, 50]. Our chosen numerical method requires
interactions to be finite-valued at all spatial grid points of the system,
including at distances below the range of validity $R_{0}$ of the atom-ion
interaction’s long distance tail, which varies with the atom-ion separation
$r$ as $-1/r^{4}$. To this end, we employ a model interaction potential whose
parameters can be mapped to the real scattering parameters by means of quantum
defect theory [43]. We characterise the five lowest-energy eigenstates of our
few-body hybrid mixture across regimes of weak to strong atomic interactions
through using the atom-atom and atom-ion distance correlations, number state
composition and distribution of the total energy among the energy components.
This work is organised as follows. In section II, we introduce our model
Hamiltonian, describe the form of the atom-ion interaction and present our
numerical methodology. In section III, we present the eigenstate spectrum of
our three-body molecular ion system, before proceeding to examine the effects
of varying interatomic interactions and ion mobility on the individual
eigenstates in section IV. In section V, we conclude with a summary of our
findings, examine the experimental viability of our model, and discuss
prospective directions for future work.
## II Atom-ion hybrid model and numerical approach
In this section, we first present the Hamiltonian describing the atom-ion
hybrid system in the laboratory frame (section II.1). We then introduce two
alternative coordinate frames, which will prove themselves useful for the
numerical treatment and physical analysis (section II.2). Finally, we provide
a brief overview of the computational approach used throughout this work
(section II.3) and define several physical quantities, which we will use to
characterise the low-energy eigenstates (section II.4).
### II.1 Atom-ion hybrid model
We consider a system comprised of a single ion of mass $m_{I}$ and $N$ neutral
bosonic atoms of mass $m_{A}$ at zero temperature. We assume that both species
are confined within quasi-1D parabolic traps, such that they can only move
along the $z$-direction, with axial trapping frequency $\omega_{A}$ for the
atoms and $\omega_{I}$ for the ion. The atom-atom interactions are of s-wave
character and are described by a contact pseudopotential.
When approaching a charged particle, the neutral atoms become polarised,
resulting in long-range interactions between the ion and the induced dipole
moments of the atoms. At large separations, the interaction between an atom at
position $z_{A}$ and an ion at position $z_{I}$ behaves as $-\alpha
e^{2}/2(z_{A}-z_{I})^{4}$, where $\alpha$ is the polarisability of the atom
and $e$ is the elementary charge. This interaction introduces a new length
$R^{*}=\sqrt{\alpha{e}^{2}m_{A}/\hbar^{2}}$ and energy scale
$E^{*}=\hbar^{2}/2m_{A}{R^{*}}^{2}$ to the system, in addition to those set by
the external traps. In atom-ion hybrid experiments [6], the interaction range
is typically $R^{*}\sim 100\text{nm}$.
To properly account for interactions between the atomic and ionic species at
all distances, whilst also ensuring our model is numerically tractable, we
introduce a short-distance cut-off to the $1/r^{4}$ potential and describe the
interaction at small separations by a repulsive barrier. The explicit model
interaction used was developed previously in earlier works based on quantum
defect theory [43, 41] and can be expressed in units of $E^{*}$ and $R^{*}$ as
$V_{AI}(r)=v_{0}e^{-\gamma r^{2}}-\frac{1}{r^{4}+\frac{1}{\kappa}},$ (1)
where $r=z_{A}-z_{I}$ denotes the atom-ion separation, $v_{0}$ the height and
$\gamma$ the width of the repulsive short-range barrier, whilst $\kappa$ sets
the short-range cut-off to the attractive tail and determines the number of
bound states. It has been shown theoretically that at ultra-cold temperatures
the rate of inelastic atom-ion collisions is larger for states with smaller
binding energy [38]. Accordingly, we choose our model parameters to be
$v_{0}=3\kappa$, $\gamma=4\sqrt{10\kappa}$ and $\kappa=80$, in units of
$E^{*}$, ${R^{*}}^{-2}$ and ${R^{*}}^{-4}$, respectively. This choice accounts
for the two uppermost bound states closest to the continuum $E=0$ [43].
The species Hamiltonians $H_{A}$ and $H_{I}$ take the following form in units
of $E^{*}$ and $R^{*}$:
$\displaystyle H_{A}$
$\displaystyle=\sum_{i=1}^{N}\bigg{(}-\frac{\partial^{2}}{\partial{z^{2}_{Ai}}}+\frac{z^{2}_{Ai}}{l_{A}^{4}}\bigg{)}+\sum_{i<j}^{N}g\delta(z_{Ai}-z_{Aj})$
(2a) $\displaystyle=K_{A}+P_{A}+V_{AA},$ $\displaystyle H_{I}$
$\displaystyle=-\beta\frac{\partial^{2}}{\partial
z_{I}^{2}}+\frac{z_{I}^{2}}{l_{A}^{4}\beta\eta^{2}}=K_{I}+P_{I},$ (2b)
where $z_{Ai}$ denotes the position of the $i^{\text{th}}$
atom,$l_{A}=\sqrt{\hbar/m_{A}\omega_{A}}/R^{*}$ is the oscillator length of
the parabolically-confined atoms re-scaled by $R^{*}$, $g$ is the effective
strength of the atom-atom interaction, $\beta=m_{A}/m_{I}$ is the interspecies
mass ratio, $z_{I}$ is the position of the ion and
$\eta=\omega_{A}/\omega_{I}$ is the ratio of the trapping frequencies. $K_{A}$
and $K_{I}$ abbreviate the kinetic terms, $P_{A}$ and $P_{I}$ the external
potentials and $V_{AA}$ the contact interaction. We fix $l_{A}=0.5$, $\eta=1$
and $N=2$ for the remainder of this work.
Figure 1: The first four lowest-energy solutions to the single-particle
eigenvalue problem
$h_{\text{1b}}\phi_{i}(z_{A})=\varepsilon_{i}\phi_{i}(z_{A})$ (see eq. (3)),
describing a single atom in a harmonic trap interacting with a static ion
localised at $z=0$. The effective potential experienced by the atom is given
by the solid black curve. The eigenstates $\\{\phi_{i}(z_{A})\\}_{i=0}^{3}$
(filled curves) are shown along red dashed lines, which indicate their
eigenenergies $\\{\varepsilon_{i}\\}_{i=0}^{3}$. Energies and lengths are
given in units of $E^{*}$ and $R^{*}$ set by the atom-ion interaction. The
harmonic trap length is $l_{A}=0.5$, in units of $R^{*}$.
To provide some basic intuition for the system at hand, we assume that the ion
is localised at the origin (i.e.: $l_{I}=0$, well-approximated by either a
heavy ion or a tight trap) and that the atoms are non-interacting ($g=0$). In
this case, the ion acts as a one-body potential for the atoms and does not
receive any feedback from them. Our model reduces to a single-particle
problem:
$h_{\text{1b}}=-\frac{\partial^{2}}{\partial
z_{A}^{2}}+\frac{z_{A}^{2}}{{l_{A}}^{4}}+V_{AI}(z_{A}),$ (3)
which describes a single atom in an effective potential, being the
superposition of the harmonic trap and atom-ion potential (see solid black
curve in fig. 1). The Schrödinger equation belonging to the one-body
Hamiltonian $h_{\text{1b}}$ can be solved straightforwardly using exact
diagonalisation. We choose to use a fast Fourier transform (FFT) discrete
variable representation (DVR) basis 111Specifically, we use a FFT DVR basis of
size $n=333$, which ensures the single particle eigenenergies are converged up
to the sixth decimal place.. The four lowest-energy single particle
eigenstates $\\{\phi_{i}(z)\\}_{i=0}^{3}$ of the eigenvalue problem
$h_{\text{1b}}\phi_{i}=\varepsilon_{i}\phi_{i}$ are depicted in fig. 1. As
mentioned above, our choice of parameters for the model interaction (1)
results in two bound states in the atom-ion potential: $\phi_{0}$ and
$\phi_{1}$. Due to the steep $-1/z^{4}$ contribution, these two bound states
share a similar spatial extent ($\sim R^{*}$) and the peaks of their
probability amplitudes coincide with the potential minima at $\approx\pm
0.3R^{*}$. In contrast, the higher-energy eigenstates $\phi_{2}$ and
$\phi_{3}$ are extended across the harmonic trap ($\sim 2R^{*}$) with a
significantly smaller probability amplitude at the potential minima. From now
on, we will refer to $\phi_{0}$ and $\phi_{1}$ as molecular orbitals and
$\phi_{2}$ and $\phi_{3}$ will be called vibrational orbitals.
In this work, we perturb this single-particle picture $h_{\text{1b}}$ two-
fold: firstly, by considering interactions between the trapped pair of bosons
and secondly, by including the motion of the ion. The former is parameterised
by the contact interaction strength $g$ and the latter is parameterised by the
mass ratio $\beta$, which determines the relative localisation between the two
trapped species.
### II.2 Non-laboratory reference frames
We emphasise that in the laboratory frame (LF), the atomic and ionic degrees
of freedom are highly entangled because atoms can be bound to the mobile ion,
which possesses a spatially-extended probability density. This fact makes it
difficult to obtain well-converged numerical results. To account for these
correlations, we replace the atom coordinates $z_{Ai}$ with the relative
distances w.r.t. the ion $r_{i}=z_{Ai}-z_{I}$. The remaining ion coordinate
$z_{I}$ can be retained (ion frame, IF) or replaced by the combined centre of
mass $R=(m_{I}z_{I}+m_{A}\sum_{i}z_{Ai})/M$ of the system, where
$M=m_{I}+Nm_{A}$ is the system’s total mass (centre of mass frame, CMF).
The primary frame used for the numerical simulations was the CMF, whose main
advantage is its numerical stability during eigenstate acquisition and its
more rapid convergence compared to the other frames. The corresponding CMF
Hamiltonian is given by
$\begin{split}H=&\sum_{i=1}^{N}\Bigg{(}-\bigg{(}1+{\beta}\bigg{)}\frac{\partial^{2}}{\partial
r_{i}^{2}}+(1-d)\frac{r_{i}^{2}}{l_{A}^{4}}\Bigg{)}\\\
+&\sum_{i=1}^{N}\Bigg{(}v_{0}\exp{\big{(}-\gamma
r_{i}^{2}\big{)}-\frac{1}{r_{i}^{4}+\frac{1}{\kappa}}}\Bigg{)}\\\
+&\sum_{i<j}\Bigg{(}g\delta(r_{i}-r_{j})-2\beta\frac{\partial}{\partial
r_{i}}\frac{\partial}{\partial
r_{j}}-\frac{2d}{l_{A}^{4}}r_{i}r_{j}\Bigg{)}\\\
-&d\frac{\partial^{2}}{\partial
R^{2}}+\frac{1}{l_{A}^{4}\beta\eta^{2}}(1+N\beta\eta^{2})R^{2}\\\
+&\frac{2d}{l_{A}^{4}\beta\eta^{2}}(\eta^{2}-1)\sum_{i=1}^{N}Rr_{i},\end{split}$
(4)
where the parameters have the same meanings as discussed in section II.1 and
$d=\beta/(1+N\beta)$. For equal trapping frequencies $\eta=1$, the above
Hamiltonian decouples into two sub-Hamiltonians: one for the centre of mass
co-ordinate $R$ and the other for relative co-ordinates $\\{r_{i}\\}$. The
centre of mass sub-Hamiltonian $H_{R}(\eta=1)=-d\frac{\partial^{2}}{\partial
R^{2}}+\frac{R^{2}}{l_{A}^{4}d}$ describes a quantum harmonic oscillator of
mass $M=1/2d$ and frequency $\Omega=2/l_{A}^{2}$ and can be solved
analytically. The atom-ion interaction now takes the form of a one-body
potential, as was also the case for the static ion example discussed in
section II.1. However, the ion’s motion induces two additional interactions
between the relative coordinates, namely the positional ($r_{i}r_{j}$) and the
derivative ($\frac{\partial}{\partial r_{i}}\frac{\partial}{\partial r_{j}}$)
couplings. In the limit of a static ion ($\beta\rightarrow 0$), these
additional interactions vanish and for $g=0$, we recover the one-body
Hamiltonian (3) describing a single boson interacting with a static ion.
The other frame used for analysis was the IF. Whilst the IF is less efficient
than the CMF, it is nonetheless more efficient than the LF since the
entanglement between the interspecies degrees of freedom is reduced. Due to
numerical instabilities however, it is challenging to obtain higher excited
states in the IF, which limits the analysis in this frame solely to the ground
state. Nevertheless, in contrast to the CMF, the IF provides access to the
single particle atomic $\rho_{1}(z_{A})$ and ionic $\rho_{1}(z_{I})$ density
distributions, which are laboratory frame quantities and thus allow for an
easier interpretation (see Supplementary Material in [39]). The IF Hamiltonian
is given by
$\begin{split}H=&\sum_{i=1}^{N}\bigg{(}-(1+\beta)\frac{\partial^{2}}{\partial
r_{i}^{2}}+\frac{r_{i}^{2}}{l_{A}^{4}}\bigg{)}\\\
+&\sum_{i=1}^{N}\bigg{(}v_{0}e^{-\gamma
r^{2}}-\frac{1}{r^{4}+\frac{1}{\kappa}}\bigg{)}\\\
-\beta&\frac{\partial^{2}}{\partial
z_{I}^{2}}+\frac{1}{l_{A}^{4}}\bigg{(}N+\frac{1}{\beta\eta^{2}}\bigg{)}z_{I}^{2}\\\
+&\sum_{i<j}\bigg{(}g\delta(r_{i}-r_{j})-2\beta\frac{\partial}{\partial
r_{i}}\frac{\partial}{\partial r_{j}}\bigg{)}\\\
+2&\sum_{i=1}^{N}\bigg{(}\frac{z_{I}r_{i}}{l_{A}^{4}}+\beta\frac{\partial}{\partial
z_{I}}\frac{\partial}{\partial r_{i}}\bigg{)}.\end{split}$ (5)
Note that the derivative coupling term ($\frac{\partial}{\partial
r_{i}}\frac{\partial}{\partial r_{j}}$) is also present in this frame and that
$z_{I}$ and $r_{i}$ cannot be decoupled for any choice of parameters.
### II.3 Computational approach
To solve for the lowest-energy eigenstates of our three-body problem, we
employ the Multi-Layer Multi-Configuration Time-Dependent Hartree method for
Bosons (ML-MCTDHB) [44, 45]. ML-MCTDHB is a numerically-exact ab initio method
for performing time-dependent simulations of many-body quantum dynamics and it
belongs to a wider family of multi-configuration Hartree-Fock methods [51, 52,
53, 54]. In the same manner as its sibling methods, ML-MCTDHB utilises a
variationally-optimised time-dependent basis which enables us to perform
efficient calculations in a truncated Hilbert space, whilst ensuring that we
fully cover the active sub-space of the complete Hilbert space. The multi-
layer expansion allows for adopting the wavefunction ansatz to system-specific
intra- and inter-species correlations. As a result, it is able to more
efficiently treat mixtures with large numbers of particles in comparison to
approaches that do not utilise multi-layering [55].
The construction of the ML-MCTDHB wavefunction ansatz describing our three-
body system proceeds as follows. In the first step, we group together the
indistinguishable degrees of freedom (DOF) and assign to them
$S_{\sigma}\in\mathbb{N}$ species wave-functions
$\\{\ket{\psi_{i}^{\sigma}(t)}\\}_{i=1}^{S_{\sigma}}$, with $\sigma$ denoting
the distinct species. For our case, there are only two distinct species,
corresponding to the atomic and ionic DOF. Next, the total many-body
wavefunction is written as a linear combination of product states:
$\ket{\psi(t)}=\sum_{i=1}^{S_{I}}\sum_{j=1}^{S_{A}}A^{1}_{ij}(t)\ket{\psi_{i}^{\text{I}}(t)}\ket{\psi_{j}^{\text{A}}(t)},$
(6)
where $A^{1}_{ij}(t)$ are time-dependent top-layer coefficients and
$\sigma=\text{A}$ stands either for $z_{i}$ or $r_{i}$, whilst
$\sigma=\text{I}$ for $z_{I}$ or $R$, depending on the chosen frame. For
$\eta=1$, the CMF DOF ($r_{i}$ and $R$) decouple, such that (6) becomes a
single product state:
$\ket{\psi(t)}=\ket{\psi^{\text{I}}(t)}\ket{\psi^{\text{A}}(t)}$. In such
cases, the step in eq. (6) is usually skipped as solving the sub-Hamiltonians
independently using single-layer MCTDHB is more efficient.
In the second step, the species wavefunctions $\ket{\psi_{i}^{\sigma}}$ for
indistinguishable DOF are expanded in time-dependent number states
$\ket{\textbf{n}}_{t}^{\sigma}$ to incorporate proper, in our case bosonic,
quantum statistics:
$\ket{\psi_{i}^{(\sigma)}(t)}=\sum_{\textbf{n}|N_{\sigma}}A^{2;\sigma}_{i;\textbf{n}}(t)\ket{\textbf{n}}^{\sigma}_{t},$
(7)
with time-dependent species-layer coefficients
$A^{2;\sigma}_{i;\textbf{n}}(t)$. The number states
$\ket{\textbf{n}}^{\sigma}=(n_{1},\ldots,n_{s_{\sigma}})$ are comprised of
$s_{\sigma}\in\mathbb{N}$ time-dependent single-particle functions (SPFs)
$\\{\ket{\phi_{i}^{\sigma}(t)}\\}_{i=1}^{s_{\sigma}}$ and the sum goes over
all possible number state configurations $\textbf{n}|N_{\sigma}$ which fulfil
the constraint of a fixed number of particles
$\sum_{i=1}^{s_{\sigma}}n_{i}=N_{\sigma}$ .
Finally, the time-dependent SPFs are represented on a one-dimensional discrete
variable representation (DVR) basis
$\\{\ket{\chi^{\sigma}_{i}}\\}_{i=1}^{\mathcal{M}_{\sigma}}$ (in our case, a
FFT DVR basis)
$\ket{\phi_{i}^{\sigma}(t)}=\sum_{j=1}^{\mathcal{M}_{\sigma}}A^{3;\sigma}_{ij}(t)\ket{\chi^{\sigma}_{j}},$
(8)
with time-dependent particle-layer coefficients $A^{3;\sigma}_{i;j}(t)$ [56].
The Hilbert space of our system is truncated at each layer and controlled
through the values of $S_{\sigma}$, $s_{\sigma}$ and $\mathcal{M}_{\sigma}$.
This allows us to tailor our ansatz to suit the degree of intra- and inter-
species correlations present in the system. Note that in contrast to standard
approaches, the SPFs are time-dependent, allowing for a considerable boost in
computation time.
The equations of motion for the three layers of coefficients $A^{1}_{ij}$,
$A^{2;\sigma}_{i;\textbf{n}}$ and $A^{3;\sigma}_{i;j}$ outlined above are
derived from the Dirac-Frenkel variational principle [57]:
$\bra{\delta\psi}(i\partial_{t}-\hat{H})\ket{\psi}=0.$ (9)
The ground state and higher excited states are obtained by means of improved
relaxation of an initial input state. Specifically, ML-MCTDHB propagates the
non-top layer coefficients $A^{2;\sigma}_{i;\textbf{n}}$ and
$A^{3;\sigma}_{i;j}$ in imaginary time to a fixed point, at which point it
diagonalises the top-layer $A^{1}_{ij}$ matrix. This process is performed
recursively until the top-layer and non-top layer coefficients become constant
during the diagonalisation and imaginary-time propagation, respectively. This
converged result is a stationary state of the system, which lies in the
truncated Hilbert space given by the ML-MCTDHB ansatz. Different stationary
states can be obtained by carefully selecting the initial input wave function
provided to the improved relaxation routine.
Working in the CMF and IF offers a distinct numerical advantage to working in
the LF since, as discussed in section II.2, position correlations between the
atom and ion in a bound pair are accounted for implicitly within the relative
coordinate $r_{i}=z_{Ai}-z_{I}$. This enables us to further truncate our
active Hilbert space on the top-layer (6) in these frames, resulting in
greater computational efficiency. We emphasise that this additional level of
truncation is only possible due to the multi-layer structure of our
wavefunction ansatz.
### II.4 Observables
Here we introduce several physical quantities used to characterise the
eigenstates in section IV.
#### II.4.1 Inter-atomic and inter-species separation distributions
In the decoupled CMF ($\eta=1$), we focus on the relative sub-Hamiltonian (see
eq. (4)) with bosonic DOF $r_{i}$. For a system of two atoms with relative
positions $r$ and $r^{\prime}$ to the ion, the wavefunction takes the form
$\psi(r,r^{\prime})$ and the probability density for finding the atoms in the
configuration $(r,r^{\prime})$ is given by the density
$\rho_{2}(r,r^{\prime})=\psi(r,r^{\prime})^{*}\psi(r,r^{\prime})$.
We are now able to extract a useful quantity related to LF coordinates, namely
the interatomic separation distribution $\rho_{1}(z_{A}-z_{A}^{\prime})$. To
this end, we note that
$\int drdr^{\prime}\;\rho_{2}(r,r^{\prime})=\int
dXdY\;\tilde{\rho}_{2}(X,Y)=1,$ (10)
where in the second step we do a coordinate transformation $X=r-r^{\prime}$,
$Y=(r+r^{\prime})/2$ and
$\tilde{\rho}_{2}(X,Y)=\rho_{2}(r(X,Y),r^{\prime}(X,Y))$. Now by integrating
out the coordinate $Y$, we obtain the reduced one-body density
$\rho^{AA}_{1}(X)$:
$\rho^{AA}_{1}(X=z_{A}-z^{\prime}_{A})=\int dY\;\tilde{\rho}_{2}(X,Y).$ (11)
Additionally, in both the CMF and IF we can evaluate the expectation values
for the atom-atom and atom-ion separations:
$\braket{d_{AA}}=\int dX\;|X|\rho^{AA}_{1}(X),$ (12) $\braket{d_{AI}}=\int
dr\;|r|\rho_{1}(r),$ (13)
where in the IF $\rho_{1}(r)=\int
dz_{I}dr^{\prime}\;\psi^{*}(z_{I},r,r^{\prime})\psi(z_{I},r,r^{\prime})$.
Whereas in the CMF, $\rho_{1}(r)=\int dr^{\prime}\;\rho_{2}(r,r^{\prime})$ is
the one-body probability density of the relative coordinate $r=z_{A}-z_{I}$,
giving us the interspecies separation distribution.
#### II.4.2 Bunching probability
In this paper, we often refer to the atoms as being ’bunched’ or ’anti-
bunched’. To clarify what is meant by this quantitatively, we define the so-
called bunching probability as the total probability for the atoms to be found
on the same side of the ion, irrespective of their separation. This can be
written explicitly as follows:
$\begin{split}P_{\text{bunched}}=&\int_{-\infty}^{0}\int_{-\infty}^{0}drdr^{\prime}\rho_{2}(r,r^{\prime})\\\
&+\int_{0}^{\infty}\int_{0}^{\infty}drdr^{\prime}\rho_{2}(r,r^{\prime}),\end{split}$
(14)
i.e. the probability to be found in the lower-left or upper-right quadrants of
the two-particle density $\rho_{2}(r,r^{\prime})$. Naturally, $P_{\text{anti-
bunched}}=1-P_{\text{bunched}}$. In the bunched configuration, atoms favour
the same side of the ion $P_{\text{bunched}}>P_{\text{anti-bunched}}$, whereas
in the anti-bunched configuration, the atoms are more likely to be found on
opposite sides of the ion $P_{\text{bunched}}<P_{\text{anti-bunched}}$.
## III Low-Energy Spectrum
In this section, we analyse how the five lowest-lying eigenenergies of our
hybrid model (see section II) change under variation of the inter-atomic
interaction strength $g$ and elaborate on the differences between a static and
a mobile ion.
((a))
((b))
Figure 2: Low-energy spectrum of two bosons coupled to a single ion as a
function of the atom-atom contact interaction strength $g$ for (a) a static
ion $\beta=0$ (dashed lines) and (b) a mobile ion $\beta=1$ (solid lines). The
corresponding energies for two non-interacting fermions are indicated in (a)
by the horizontal solid red lines. Note that the eigenenergies should approach
the fermionic values in the Tonks-Girardeau limit $g\rightarrow\infty$. The
static ion spectrum from (a) is given additionally for reference in (b)
(dashed lines). Note the different range of $g$ values in the subfigures (a)
and (b).
We first discuss the spectrum for the case of a static ion pinned at
$z_{I}=0$, where the atom-ion interaction (see eq. (1)) reduces to an
effective one-body potential. The first five eigenenergies are given by the
blue dashed lines in fig. 2(a). They increase monotonically with $g$ and
approach the Tonks-Girardeau (TG) energies ($g\rightarrow\infty$), equivalent
to those of two non-interacting fermions subject to the same one-body
potential (solid red lines) [58]. The ground state at $g=0$ corresponds to the
bosonic number state $\ket{2,0,0,0}$ built from SPFs of $h_{1b}$ (eq. (3))
(see also fig. 1). It saturates rapidly to the corresponding TG energy of the
fermionic number state $\ket{1,1,0,0}$, tapering off beyond $g=4$.
The first and second excited states at $g=0$ correspond to excitations of one
or both atoms to the second molecular orbital, i.e. $\ket{1,1,0,0}$ and
$\ket{0,2,0,0}$. We observe that with increasing $g$, the energy gap between
these states, given by $\epsilon_{2}-\epsilon_{1}$, first decreases up to
$g=10$ before increasing again and then tapering off at large $g$ as the
system approaches the TG limit ($g\rightarrow\infty$). In this limit, the
interacting bosons which comprise the first and second excited state are
energetically-mapped to pairs of non-interacting fermions with number state
configurations $\ket{1,0,1,0}$ and $\ket{1,0,0,1}$, respectively. The energy
gap between these states is equal to the gap at $g=0$ between the third and
fourth excited states $\epsilon_{4}-\epsilon_{3}$.
The third and fourth excited states at $g=0$ correspond to excitations of a
single atom to one of the vibrational orbitals, i.e. $\ket{1,0,1,0}$ and
$\ket{1,0,0,1}$. They are quite robust to $g$ variation, being a consequence
of the reduced spatial overlap between the molecular and vibrational orbitals.
In the TG limit, they map to the fermionic states $\ket{0,1,1,0}$ and
$\ket{0,1,0,1}$ and as a result, they have the same energy gap as at $g=0$.
The ion’s mobility has two effects on the spectrum (blue solid lines in fig.
2(b)): (i) a positive energy shift for all states and (ii) increased energy
separation among the eigenstates. Aside from this, we still observe a
monotonous increase of the energies with $g$. Interestingly, we also observe a
tapering off of the energy of the ground state at large $g$, which is
reminiscent of the energy mapping between hard-core bosons and non-interacting
fermions. Formally however, the criteria for the TG mapping are not fulfilled
since firstly, we do not have a single- but rather a two-component system and
secondly, the hard-core interaction exists only between the atoms.
## IV Analysis of Eigenstates
In this section, we will examine in detail the individual eigenstates
comprising the low-energy spectrum presented in section III, from the ground
state up to the fourth excited state (sections IV.1, IV.2, IV.3, IV.4 and
IV.5). In particular, we will explore the effect of varying atomic
interactions and ion mobility on the properties of the eigenstates. For each
state considered, we will analyse the distribution of energy among the various
energy components and discuss what implications this has for the inter-atomic
and inter-species separation distributions introduced in section II.4.1.
Moreover, we will also consider to what extent the single-particle picture
$h_{1b}$ based on eq. (3) is modified by exploring the number state
composition of each eigenstate. Each sub-section focuses on a specific
eigenstate and begins with a short summary of the main physical properties of
that state.
### IV.1 Ground state
In the ground state, both atoms bind to the ion in the lowest bound-state and
show no preference for bunching or anti-bunching when they are non-interacting
and the ion is static. Finite interactions between the atoms cause them to
separate to opposite sides of the ion (see section IV.1.1). For an equal mass
system ($\beta=1$), the ion’s mobility results in a slight preference for the
non-interacting atoms to be bunched, which can be understood using an
effective potential model that shows the atom pair clusters at the trap centre
when the ion is mobile (see section IV.1.2). This interspecies correlation
effect competes against the interatomic anti-correlations, delaying the on-set
of complete separation of the atoms. The fully-separated atom pair pins the
mobile ion from either side, such that it becomes increasingly localised at
the trap-centre (see section IV.1.3).
((a))
((b))
((c))
((d))
((e))
Figure 3: Key observables for the ground state. (a): interatomic separation
distribution $\rho^{AA}_{1}(z_{A}-z_{A}^{\prime})$ for different atom-atom
coupling strengths $g$. The inset shows the expectation value
$\braket{d_{AA}}$ for the atom-atom separation as a function of $g$ (eq.
(12)). (b): interspecies separation distribution $\rho^{AI}_{1}(z_{A}-z_{I})$
for varying atom-atom coupling strengths $g$. The inset shows the expectation
values $\braket{d_{AI}}$ for the atom-ion separation as a function of $g$ (eq.
(13)). (c)-(e): the evolution of the laboratory frame energy components with
atom-atom coupling strength $g$. Note for (a) and (b): the grey lines indicate
the distance between the minima of the atom-ion interaction potential (1). Due
to the parity symmetry it is sufficient to show only the positive semi-axis.
All subfigures: the solid curves correspond to a mobile ion, whilst dashed
curves correspond to a static ion.
((a))
((b))
((c))
((d))
Figure 4: Snapshots of the atomic probability density
$\rho_{2}^{\text{AA}}=|\psi(r,r^{\prime})|^{2}$ of the ground state for
different interaction strengths $g$ for a static ion ((a) and (b)) and a
mobile ion ((c) and (d)).
#### IV.1.1 Static ion
The ground state of two non-interacting ($g=0$) atoms coupled to a static
($\beta=0$) ion located at the trap centre $z_{I}=0$ is given by the number
state $\ket{2,0,0,0}$ w.r.t. the single-particle eigenstates of
$h_{\text{1b}}$ (see eq. (3)), i.e., both atoms occupy the lowest molecular
orbital $\phi_{0}$ in fig. 1. The bunched and anti-bunched configurations are
equally probable (see fig. 4(a)), indicated also by the two peaks in the
interatomic separation distribution $\rho_{1}^{AA}(z_{A}-z_{A}^{\prime})$
(light-blue dashed curve in fig. 3(a)).
With increasing $g$, we observe in fig. 3(a) a depletion of the central peak
at $z_{A}=z_{A}^{\prime}$ in favour of the side humps, which smoothly shift
their position to larger separations. As a result, the atom-atom separation
$d_{\text{AA}}$ increases (black dashed line in the inset of fig. 3(a)). The
sharp initial growth in the total energy (see fig. 2(b)) can be mainly
attributed to the behaviour of the intra-atomic interaction $V_{\text{AA}}$,
which increases monotonously for $g<3$ and decreases thereafter as the
probability for the atoms to occupy the same position gradually vanishes
(light-blue dashed curve in fig. 3(e)). In addition, there is a near 1:1
exchange between the atomic kinetic $K_{A}$ and the atom-ion interaction
$V_{AI}$ energies with increasing interaction strength $g$: the increased
probability for the anti-bunched configuration enables the atoms to localise
more around the atom-ion potential minimum (increasing $K_{A}$, black dashed
line in fig. 3(c)) and slide down within the $V_{AI}$ potential (decreasing
$V_{\text{AI}}$, pink dashed line in fig. 3(d)). Thus the distance between the
atoms and the ion $d_{AI}$ (black dashed line in the inset of fig. 3(b)) is
almost unchanged, though it shows a gradual increase due to the increasingly
depleted $z_{A}=z_{I}$ region (see dashed curves in fig. 3(b)) which is
further reflected in the gentle increase of the atomic trap potential energy
$P_{A}$ (green dashed line in fig. 3(e)) since the atoms have a reduced
probability to be found at the origin $z_{A}=0$.
At stronger interactions ($g=10$), the bunched configuration becomes
completely suppressed and the atoms exist on opposite sides of the ion (see
fig. 4(b)). While approaching the TG regime, all energies begin to saturate
and $d_{\text{AA}}$ approaches the corresponding limit of the separation
between two non-interacting fermions $\approx 0.6$, i.e. the distance between
the $V_{AI}$ minima (see fig. 1). $V_{AA}$ decreases for large $g$ and will
approach zero in the TG limit.
#### IV.1.2 Mobile ion: impact on the atoms
((a))
((b))
Figure 5: (a) atomic density $\rho_{1}(z_{A})$ for the case of a heavy ion
($\beta=0.034$, dashed lines) and a mobile ion ($\beta=1$, solid lines),
obtained via: the lab frame species mean field (LF SMF) ansatz (pink curves),
the ion frame SMF (IF SMF) ansatz (blue curves) and the exact IF result from
the full ML-MCTDH ansatz (see eq. (6)) (IF ML-X) (black curves). (b) effective
potential $P_{A}^{eff}(z_{A})$ experienced by the atoms due to the IF ML-X
density $\rho^{IF}_{1}(z_{I})$ (solid black line) and that obtained with the
IF SMF ansatz $\tilde{\rho}^{IF}_{1}(z_{I})$ (dashed blue line), derived from
eq. (16) with $\beta=1$. The filled curves denote the respective ground state
orbitals $|\phi_{0}|^{2}$ calculated from the potential. Note for both
subfigures that the atoms are non-interacting $g=0$.
Let us now consider the impact of the ion’s mobility ($\beta=1$) on the ground
state properties. The two-body density $\rho_{2}(r,r^{\prime})$ becomes more
spread out (compare figs. 4(a) and 4(c)), which results in a decrease of the
kinetic energy of the atoms $K_{A}$ and a positive shift in the atom-ion
interaction energy $V_{AI}$ (see black and pink solid curves in figs. 3(c) and
3(d)). Similarly, the atom-ion separation distribution $\rho_{1}^{AI}(r)$
broadens, leading to an increase in the atom-ion separation $d_{AI}$ (compare
solid and dashed lines in inset of fig. 3(b)). By comparing the interatomic
separation distributions $\rho_{1}^{AA}$ between the static and mobile cases
(dashed and solid lines in fig. 3(a)), we infer that the additional derivative
and positional coupling terms in eq. (4), introduced by the ion mobility,
impede the process of particle separation. Thus when $\beta=1$, the average
atom-atom distance $d_{AA}$ reaches values of the static ion system only at a
stronger coupling $g$ (compare black curves in the inset of fig. 3(a)).
To obtain a better understanding of this mobility-induced bunching effect (see
fig. 4(c)), we now examine the ground state through the lens of species mean-
field (SMF) theory. This will allow us to extract for each species an
effective one-body potential induced by the other component, effectively
decoupling the equations of motion between the distinguishable DOF. As already
discussed in section II.2, the LF is badly-suited for this purpose. On the
other hand, the IF incorporates the correlations of a bound atom-pair
following the ion movement and moreover, allows us to obtain several useful
physical quantities of the LF. In the notation of ML-MCTDHB, the IF SMF ansatz
assumes a single product state on the top layer (6):
$\psi\big{(}z_{I},r_{1},\ldots,r_{N})\approx\psi_{A}\big{(}r_{1},\ldots,r_{N}\big{)}\psi_{I}\big{(}z_{I}\big{)}.$
(15)
In fig. 5(a), we compare the one-body density $\rho_{1}(z_{A})$ of the atoms
obtained with the SMF ansatz (blue lines) to that of the exact ML-MCTDHB
solution (black lines) in the IF for non-interacting atoms $g=0$ bound to a
mobile ($\beta=1$) as well as a heavy, near-static ($\beta=0.034$) ion. The
latter mass ratio corresponds to the species pairing
${}^{6}\text{Li-}{}^{174}\text{Yb}^{+}$. We additionally show the results
obtained via the SMF ansatz in the LF (pink lines) for comparison. For a heavy
ion, the SMF ansatz is well justified in both frames (compare dashed curves in
fig. 5(a)). The atoms are most likely to be found around the minima of the
atom-ion potential at $\approx\pm 0.3R^{*}$. For a mobile ion ($\beta=1$), the
exact result shows that the atoms are now most likely to be found at the trap
centre (see solid black line in fig. 5(a)). Whilst the IF SMF ansatz
approximately captures this feature, it nonetheless shows substantial
quantitative deviations from the exact result. Thus the entanglement between
the DOF $z_{I}$ and $r_{i}$, neglected in the IF SMF, favours increased
bunching of atoms around the centre of the harmonic trap. The LF SMF ansatz,
which neglects entanglement between the DOF $z_{I}$ and $z_{A}^{i}$, still
predicts that the atoms are most likely to be found around the minima of the
static ion potential. The entanglement between the DOF $z_{I}$ and $z_{A}^{i}$
is therefore crucial for capturing the correct form of the probability
density.
In the following, we ignore the entanglement between $z_{I}$ and $r_{i}$ and
aim at understanding the qualitative features of $\rho_{1}(z_{A})$ for a
mobile ion ($\beta=1$) with non-interacting atoms ($g=0$). To this end, we
perturb atomic Hamiltonian in the LF (2a) with an effective atom-ion
interaction potential found by integrating out the ionic degree of freedom in
the interspecies interaction (1). The result is:
$\begin{split}H_{A}^{eff}&=K_{A}+P_{A}+V_{AA}+\sum_{i=1}^{N}\int
dz_{I}\;V_{\text{AI}}(z_{A}^{i},z_{I})\tilde{\rho}^{IF}_{1}(z_{I})\\\
&=K_{A}+P_{A}^{eff}+V_{AA},\end{split}$ (16)
where $\tilde{\rho}^{IF}_{1}(z_{I})$ is the approximate one-body density of
the ion obtained with the IF SMF ansatz. Note, $\tilde{\rho}^{IF}_{1}(z_{I})$
is different to the density $\tilde{\rho}^{LF}_{1}(z_{I})$ one would normally
use based on the LF SMF ansatz. Since $\tilde{\rho}^{IF}_{1}(z_{I})$
incorporates some of the many-body correlations between laboratory DOF, we
expect this effective Hamiltonian to better capture the behaviour of the
atomic species than $\tilde{\rho}^{LF}_{1}(z_{I})$. Indeed, we observe for a
mobile ion ($\beta=1$) that the extended ion density flattens out the $V_{AI}$
minima and central barrier, yielding an effective potential which takes the
form of a harmonic potential with a small modulation around the origin (blue
dashed lines in fig. 5(b)). The corresponding ground state orbital
$|\phi_{0}|^{2}$ (filled dashed blue curve in fig. 5(b)) shows increased
probability for the atom to be found at the centre of the harmonic trap,
though it still displays peaks around the remnants of potential minima. The
effective potential obtained from eq. (16) with the exact ion density
$\rho^{IF}_{1}(z_{I})$ is qualitatively similar, however the modulation around
the origin is considerably weaker and therefore its ground state orbital
$|\phi_{0}|^{2}$ bears closer resemblance to a Gaussian (solid black lines in
fig. 5(b)), in accordance with the shape observed in fig. 5(a). Whilst the
effective picture obtained using the IF SMF ansatz provides a qualitatively
correct description of the mobility-induced atomic bunching, accounting for
entanglement between $z_{I}$ and $r_{i}$ is necessary for quantitative
correctness.
#### IV.1.3 Mobile ion: impact on the ion
Figure 6: Uhlmann fidelity between the exact ground state $\ket{\psi}$ at
$\beta=1$ and the state describing a mobile ion with non-interacting atoms
$\ket{\psi(\beta=1,g=0)}$ (blue line), a static ion with non-interacting atoms
$\ket{\psi(\beta=0,g=0)}$ (orange line), and a static ion with interacting
atoms $\ket{\psi(\beta=0)}$ (green line).
((a))
((b))
Figure 7: (a) the ion density $\rho_{1}(z_{I})$ at various coupling strengths
$g$, obtained via: the ion frame species mean field (IF SMF) ansatz (dashed
black line) and full ML-MCTDH ansatz (solid lines). Note the IF SMF has only a
single solution for all $g$, since the $z_{I}$ and $r_{i}$ degrees of freedom
decouple in the SMF ansatz (see eq. (15)). (b) The effective potential
$P_{I}^{eff}(z_{I})$ experienced by the mobile ion due to the exact atomic
density $\rho^{IF}_{1}(z_{A})$ at various coupling strengths $g$. The filled
curves denote the corresponding ground state orbitals $|\phi_{0}|^{2}$.
We now perform a complementary analysis on how the mobile ion is affected by
the interatomic coupling strength $g$. In fig. 3(e), we observe that the ion’s
potential energy $P_{I}$ slightly decreases with $g$ (red solid line), whilst
its kinetic energy $K_{I}$ slightly increases (blue solid line). From this, we
can infer that with increasing $g$ the ion density $\rho_{1}(z_{I})$ becomes
squeezed and more localised at the trap centre.
This observation is further confirmed by examining the Uhlmann fidelity
$|\braket{\chi}{\psi}|^{2}$ between the numerically-exact ground state
$\ket{\psi}$ for the mobile ion system and several states $\ket{\chi}$
describing different limiting cases (see fig. 6). At weak couplings,
$\ket{\psi}$ bears strongest similarity to the state describing a mobile ion
with non-interacting atoms $\ket{\psi(\beta=1,g=0)}$. Then around $g\approx
6$, the dominant overlap is with the state describing a static ion with non-
interacting atoms $\ket{\psi(\beta=0,g=0)}$, having no correlations at all.
Finally, for $g>9$ it shares the greatest overlap with the state
$\ket{\psi(\beta=0)}$, describing a static ion with interacting atoms. The
latter exhibits only atom-atom correlations.
To develop a intuitive picture of what is happening to the ion, we first
perform a comparison in terms of the SMF ansatz and exact solution in the IF
for the ion density $\rho_{1}(z_{I})$, similar to what was done for the atoms
in IV.1.2. Note that the SMF prediction for $\rho_{1}(z_{I})$ is independent
of $g$, since here the DOF are decoupled (dashed black line in fig. 7(a)).
Thus, the ion localisation phenomenon cannot be captured by the SMF ansatz
alone. For weak coupling $g\leq 1$, the ion density $\rho_{1}(z_{I})$ obtained
via the exact result spreads out subtly (note the dip in probability at
$z_{I}=0$ between $g=0$ and $g=1$), before gradually becoming higher and
narrower (see solid curves in fig. 7(a)), in line with our prior observations
of $K_{I}$ and $P_{I}$ discussed in the paragraph above.
Next, we construct an effective Hamiltonian for the ion due to the atomic
density, in the same manner as we did for the atoms:
$\begin{split}H_{I}^{eff}&=K_{I}+P_{I}+N\int
dz_{A}\;V_{\text{AI}}(z_{A},z_{I})\rho^{IF}_{1}(z_{A})\\\
&=K_{I}+P_{I}^{eff},\end{split}$ (17)
where $\rho^{IF}_{1}(z_{A})$ is the exact one-body density of the atoms
obtained in the IF. The effective potential $P_{I}^{eff}$ is given in fig.
7(b) for various interatomic couplings. The exact one-body densities
$\rho_{1}(z_{I})$ fit well inside $P_{I}^{eff}$. At $g=0$, the effective trap
takes the form of a harmonic trap with a shallow double-well modulation near
the origin. As the intra-species correlations build up, the double-well
structure inverts, making the ion profile narrower due to the new minimum at
the origin.
### IV.2 First excited state
In the first excited state, both of the atom-ion bound states are occupied by
a single atom. The opposite symmetries of the bound states force the atoms to
reside on the same side w.r.t. the ion, such that the atoms share a large
spatial-overlap (see section IV.2.1). The spatial overlap leads to a swift
rise in the total energy of the first excited at finite interatomic
interactions. To minimise their overlap at greater interaction strengths, one
of the atoms is released from the ion and occupies a vibrational trap state.
Whilst the ion’s mobility leads to an overall positive shift in the total
energy of the state, as well as increased inter- and intra-species
separations, it does not qualitatively affect the underlying physics (see
section IV.2.2).
((a))
((b))
((c))
((d))
((e))
Figure 8: Key observables for the first excited state. (a): interatomic
separation distribution $\rho^{AA}_{1}(z_{A}-z_{A}^{\prime})$ for different
atom-atom coupling strengths $g$. The inset shows the expectation value
$\braket{d_{AA}}$ for the atom-atom separation as a function of $g$ (eq.
(12)). (b): interspecies separation distribution $\rho^{AI}_{1}(z_{A}-z_{I})$
for varying atom-atom coupling strengths $g$. The inset shows the expectation
values $\braket{d_{AI}}$ for the atom-ion separation as a function of $g$ (eq.
(13)). (c)-(e): the evolution of the laboratory frame energy components with
atom-atom coupling strength $g$. Note for (a) and (b): the grey lines indicate
the distance between the minima of the atom-ion interaction potential (1). Due
to the parity symmetry it is sufficient to show only the positive semi-axis.
All subfigures: the solid curves correspond to a mobile ion, whilst dashed
curves correspond to a static ion.
((a))
((b))
((c))
((d))
((e))
((f))
Figure 9: Snapshots of the atomic probability density
$\rho_{2}^{\text{AA}}=|\psi(r,r^{\prime})|^{2}$ of the first excited state for
different interaction strengths $g$ for a static ion ((a) to (c)) and a mobile
ion ((d) to (f)). Figure 10: Overlap spectrum
$|\braket{\psi}{\textbf{n}}|^{2}$ between the first excited state $\ket{\psi}$
and the separate SPF number states $\ket{\textbf{n}}$ calculated from the
static ion model in eq. (3) for a static ion (dashed curves) and a mobile ion
(solid curves). For the sake of clarity, we show here only the dominant
overlap coefficients. The complete representation of the mobile ion state in
the static ion basis requires numerous additional small contributions from
higher-order number states.
#### IV.2.1 Static ion
The first excited state for two non-interacting ($g=0$) atoms coupled to a
static ($\beta=0$) ion is given by the number state $\ket{1,1,0,0}$ w.r.t. the
single-particle eigenstates of $h_{\text{1b}}$ (see eq. (3)), i.e. each
molecular orbital $\phi_{0},\;\phi_{1}$ in fig. 1 is occupied by a single
atom. Unlike in the ground state, the atoms here are completely bunched (note
the single peak in fig. 8(a)). This is caused by the fact that
$\phi_{0}(r)\approx-\operatorname{sign}(r)\phi_{1}(r)$ (compare $\phi_{0}$ and
$\phi_{1}$ in fig. 1). The suppression of the anti-bunched probabilities can
be seen explicitly by inserting this relation into the two-body density:
$\begin{split}\rho_{2}(r,r^{\prime})=&\frac{1}{4}\big{(}|\phi_{0}(r)|^{2}|\phi_{1}(r^{\prime})|^{2}+|\phi_{1}(r)|^{2}|\phi_{0}(r^{\prime})|^{2}\\\
&+2\phi_{0}^{*}(r)\phi_{1}(r)\phi_{1}^{*}(r^{\prime})\phi_{0}(r^{\prime})\big{)}+\text{c.c.},\end{split}$
(18)
where the first two terms cancel out the last two terms whenever
$\operatorname{sign}(r)\neq\operatorname{sign}(r^{\prime})$.
With increasing $g$, the interatomic separation distribution $\rho_{1}^{AA}$
spreads and the peak at $z_{A}=z^{\prime}_{A}$ for $g=0$ is shifted to larger
distances (dashed curves in fig. 8(a)). Even though the average distance
$d_{AA}$ between the atoms gradually increases (black dashed curve in the
inset of fig. 8(a)), they insist on staying in the bunched configuration, i.e.
on the same side w.r.t. ion, which results in the formation of a nodal
structure on the diagonal of the probability density (see figs. 9(b) and
9(c)). Thus, the inter-species separation distribution peak broadens (dashed
curves in fig. 8(b)) which leads to a monotonous increase of $d_{AI}$ (black
dashed curve in the inset of fig. 8(b)).
The above observations clarify the rapid increase of the total energy with $g$
(see fig. 2(a)). One major contribution comes from the atom-atom interaction
energy $V_{AA}$ given by $g\rho_{1}^{AA}(r=0)$. Thus, considering a large
initial amplitude $\rho_{1}^{AA}(r=0)>1$ at $g=0$ and that it decreases slowly
with $g$, we identify a fast linear increase of $V_{AA}$ up to approximately
$g=20$ (light blue dashed line in fig. 8(e)). Once the amplitude drops
significantly below the value of $1.0$, $V_{AA}$ starts decreasing. Another
significant contribution stems from the atom-ion interaction potential
$V_{AI}$. Drawing away from the ion requires the atoms to climb the $V_{AI}$
potential, which costs energy (purple dashed line in fig. 8(d)). For $g<10$,
this energy is compensated by decreasing atomic kinetic energy $K_{A}$ (black
dashed line in fig. 8(c)), but $V_{AI}$ keeps increasing even after $K_{A}$
has saturated.
That the atoms stay on one side of the ion while increasing their separation
with $g$ is due to the growing contribution of the number state
$\ket{1,0,0,1}$ (see dashed lines in fig. 10). The eigenstate is continuously
transitioning to a regime in which one atom remains bound to the ion and the
other is released into the harmonic trap. This can be seen clearly from the
additional peak around $\approx\pm 0.7R^{*}$ that emerges in the interspecies
separation distribution (see green dashed curve in fig. 8(b)). This causes a
monotonous increase of the atomic trap potential energy $P_{A}$ (green dashed
line in fig. 8(e)). Beyond $g=40$, the overlap with $\ket{1,0,0,1}$ becomes
dominant. Since the odd vibration orbital $\phi_{3}$ features a smaller
probability density to be found at the minimum of the atom-ion potential than
$\phi_{0}$, we observe an emergence of anti-bunching probability in the upper-
left and lower-right quadrants of $\rho_{2}$ (see fig. 9(c)).
#### IV.2.2 Mobile ion
The ion’s mobility causes a by-mixture of the number state $\ket{0,1,1,0}$
(green solid line in fig. 10), whose contribution to the eigenstate amounts to
$\sim 10\%$, and is approximately unchanged by the atom-atom coupling strength
$g$. Thus, the impact on the physical quantities from the static ion case is
expected to be qualitatively similar at all $g$.
The positive off-set of the total energy of the first excited state (see fig.
2(b)) is mainly due to the energy of the ion itself $K_{I}+P_{I}$ (dark blue
and red solid lines in figs. 8(c) and 8(e)). With increasing $g$, there is an
exchange of energy between $K_{I}$ and $P_{I}$, with $K_{I}$ decreasing and
$P_{I}$ increasing, indicating the delocalisation of the ion. The ion’s
mobility induces an additional energy exchange between $K_{A}$ and $V_{AI}$,
with $V_{AI}$ increasing and $K_{A}$ decreasing (compare black and purple
solid curves in figs. 8(c) and 8(d) to dashed ones), implying that the atoms
separate from each other. This is also evident from the patterns of
$\rho_{2}(r,r^{\prime})$, which remain qualitatively the same, albeit with a
slightly enhanced overall spread (compare rows in fig. 9) that is further
imprinted on the interatomic $\rho_{1}^{AA}$ and interspecies $\rho_{1}^{AI}$
separation distributions (compare dashed and solid lines in figs. 8(a) and
8(b)). Accordingly, the distance among the atoms $d_{AA}$ increases (black
lines in the inset of fig. 8(a)). This is in contrast to the ground state,
where the ion’s mobility decreased $d_{AA}$. Due to the overall decreased
amplitude of $\rho_{1}^{AA}(r=0)$ compared to $\beta=0$, the atom-atom
interaction energy $V_{AA}$ reaches the turning point already at a slightly
weaker coupling $g$ (light blue solid line in fig. 8(e)). In summary, at
$\beta=1$ the atoms separate further from each other and from the ion as
compared to $\beta=0$.
### IV.3 Second excited state
In the non-interacting second excited state, both atoms occupy the upper bound
state in the atom-ion interaction potential and display preference neither for
bunching nor anti-bunching (see section IV.3.1). The response of the second
excited state to finite atomic interactions closely resembles that of the
first excited state, namely that at intermediate interactions, the atoms
become preferentially bunched and at large interaction strengths, one of the
atoms frees itself from the ion. In addition, the ion’s mobility does not
produce significant qualitative differences to the results for the static case
(see section IV.3.2). These similarities between the states are to be
expected, considering that their energy gap initially narrows (see fig. 2(a)).
There are however, two crucial differences between the first and second
excited states: (i) the bunching of the atoms in the latter state does not
stem from an inherent symmetry of the single particle states, but rather is a
consequence of state mixing with other number states, such as $\ket{2,0,0,0}$
and (ii) the atomic kinetic energy is consistently higher in the second
excited state.
((a))
((b))
((c))
((d))
((e))
Figure 11: Key observables for the second excited state. (a): interatomic
separation distribution $\rho^{AA}_{1}(z_{A}-z_{A}^{\prime})$ for different
atom-atom coupling strengths $g$. The inset shows the expectation value
$\braket{d_{AA}}$ for the atom-atom separation as a function of $g$ (eq.
(12)). (b): interspecies separation distribution $\rho^{AI}_{1}(z_{A}-z_{I})$
for varying atom-atom coupling strengths $g$. The inset shows the expectation
values $\braket{d_{AI}}$ for the atom-ion separation as a function of $g$ (eq.
(13)). (c)-(e): the evolution of the laboratory frame energy components with
atom-atom coupling strength $g$. Note for (a) and (b): the grey lines indicate
the distance between the minima of the atom-ion interaction potential (1). Due
to the parity symmetry it is sufficient to show only the positive semi-axis.
All subfigures: the solid curves correspond to a mobile ion, whilst dashed
curves correspond to a static ion.
((a))
((b))
((c))
((d))
((e))
((f))
Figure 12: Snapshots of the atomic probability density
$\rho_{2}^{\text{AA}}=|\psi(r,r^{\prime})|^{2}$ of the second excited state
for different interaction strengths $g$ for a static ion ((a) to (c)) and a
mobile ion ((d) to (f)). Figure 13: Overlap spectrum
$|\braket{\psi}{\textbf{n}}|^{2}$ between the second excited state
$\ket{\psi}$ and the separate SPF number states $\ket{\textbf{n}}$ calculated
from the static ion model in eq. (3) for a static ion (dashed curves) and a
mobile ion (solid curves). For the sake of clarity, we show here only the
dominant overlap coefficients. The complete representation of the mobile ion
state in the static ion basis requires numerous additional small contributions
from higher-order number states.
#### IV.3.1 Static ion
The second excited state for two non-interacting ($g=0$) atoms coupled to a
static ($\beta=0$) ion is given by the number state $\ket{0,2,0,0}$ w.r.t. the
single-particle eigenstates of $h_{\text{1b}}$ (see eq. (3)), with both atoms
occupying the molecular orbital $\phi_{1}$ (see fig. 1). Similarly to the
ground state, the atoms show no preference for either bunched or anti-bunched
configurations (see fig. 12(a)).
As one increases $g$, there is a by-mixture of the number state
$\ket{2,0,0,0}$ up to $g=15$ (blue dashed line in fig. 13). This leads to an
increased bunching of the atoms (red dashed line in fig. 11(a)), decreasing
the average distance $d_{AA}$ among them (black dashed line in the inset of
fig. 11(a)) until finally the anti-bunched configuration is completely
suppressed and a nodal structure emerges on the diagonal of the probability
density (see fig. 12(b)). The increasingly-bunched repulsive atoms create an
increase in the interspecies separation $d_{AI}$ (see dashed black line in
inset of fig. 11(b)). Beyond $g>10$, we observe a by-mixture of the number
states $\ket{1,0,1,0}$ and $\ket{0,1,0,1}$ (orange and red dashed lines in
fig. 13) featuring stronger separations among the atoms, which causes them to
draw away from each other (see the observed increase of $\braket{d_{AA}}$ in
the inset of fig. 11(a)). The number states $\ket{1,0,1,0}$ and
$\ket{0,1,0,1}$ feature one of the atoms unbound from the ion in a trap state.
As such, the interspecies separation distribution broadens significantly (see
dashed curves in fig. 8(b)). At the same time, the contribution of
$\ket{2,0,0,0}$ displays only a slight decay. Consequently at larger $g$, we
anticipate the preservation of the bunched configuration with an increased
spread of the two-body density $\rho_{2}(r,r^{\prime})$ and a depletion of the
diagonal at $r=r^{\prime}$. (see fig. 12(c)).
We note a strong similarity between the atomic probability densities shown in
figs. 12(b) and 9(b), which is due to a small energy gap between the first and
second excited states (see fig. 2(b)). The two eigenstates also display a
similar energy dependence on the coupling $g$ (compare figs. 8(c), 8(d), 8(e),
11(c), 11(d) and 11(e)). The major difference between them is that the second
excited state has a greater atomic kinetic energy $K_{A}$.
#### IV.3.2 Mobile ion
Similarly to the ground state and first excited state, the ion’s mobility
leads to a shift in the energies (compare solid and dashed curves in figs.
2(b), 11(c), 11(d) and 11(e)) and of the interatomic and interspecies
separations (inset of figs. 11(a) and 11(b)). Specifically, the atoms exhibit
an increase in separation between each other of $\sim 0.2R^{*}$ and an
increase in separation to the ion by $\sim 0.1R^{*}$. Contrary to the first
excited state however, which featured a $g$-independent by-mixture of an
additional number state, the number state composition of the second excited
state undergoes substantial structural changes at $\beta=1$ (see fig. 13).
Thus, the role of $\ket{1,0,1,0}$ and $\ket{2,0,0,0}$ is substantially
suppressed in favour of $\ket{0,1,0,1}$. As a result, we observe a strong
enhancement of the anti-bunched off-diagonal probability in
$\rho_{2}(r,r^{\prime})$ at all $g$ (compare rows in fig. 12).
### IV.4 Third excited state
In the third excited state, one atom remains bound to the ion in the lowest-
energy bound state and the other occupies the lowest-energy trap state. As
shown in section III, the total energy of this eigenstate is not markedly
affected by varying atomic interactions. Unsurprisingly, the principle
observables for the state (probability density, energy components and number
state composition) are likewise largely unaffected by these changes (see
section IV.4.1). This robustness of the third excited state stems from the
relatively small spatial-overlap of the two atoms, due to the contrasting
length scales of the bound and trap states (see discussion in section II.1).
The state is further robust to the ion mobility (see section IV.4.2),
displaying only a positive shift in the total energy due to an increased
spread between the atomic and ionic species, in accordance with the weaker
localisation of the ion.
((a))
((b))
((c))
((d))
((e))
Figure 14: Key observables for the third excited state. (a): interatomic
separation distribution $\rho^{AA}_{1}(z_{A}-z_{A}^{\prime})$ for different
atom-atom coupling strengths $g$. The inset shows the expectation value
$\braket{d_{AA}}$ for the atom-atom separation as a function of $g$ (eq.
(12)). (b): interspecies separation distribution $\rho^{AI}_{1}(z_{A}-z_{I})$
for varying atom-atom coupling strengths $g$. The inset shows the expectation
values $\braket{d_{AI}}$ for the atom-ion separation as a function of $g$ (eq.
(13)). (c)-(e): the evolution of the laboratory frame energy components with
atom-atom coupling strength $g$. Note for (a) and (b): the grey lines indicate
the distance between the minima of the atom-ion interaction potential (1). Due
to the parity symmetry it is sufficient to show only the positive semi-axis.
All subfigures: the solid curves correspond to a mobile ion, whilst dashed
curves correspond to a static ion.
((a))
((b))
((c))
((d))
Figure 15: Snapshots of the atomic probability density
$\rho_{2}^{\text{AA}}=|\psi(r,r^{\prime})|^{2}$ of the third excited state for
different interaction strengths $g$ for a static ion ((a) and (b)) and a
mobile ion ((c) and (d)).
#### IV.4.1 Static ion
The third excited state for two non-interacting ($g=0$) atoms coupled to a
static ($\beta=0$) ion is given by the number state $\ket{1,0,1,0}$, where one
atom is bound in the even molecular orbital $\phi_{0}$ and the other occupies
the even vibrational orbital $\phi_{2}$ (see fig. 1).
Previously, we have seen that the total energy of the vibrational eigenstates
depends only weakly on $g$ (see fig. 2(b)). This also holds for the energy
components, which show only a slight exchange between $V_{AI}$ and $K_{A}$
(see figs. 14(c) and 14(d)). On the level of energies, the eigenstate seems
quite robust to perturbations by atom-atom interactions, though the atom-atom
separation distribution $\rho_{1}^{AA}$ features significant structural
changes with increasing $g$ leading to increased separation among the atoms
$d_{AA}$ (black dashed curve in the inset of fig. 14(a)).
At $g=0$, there are three pronounced humps in $\rho_{1}^{AA}$ (light blue
dashed line in fig. 14(a)). Comparing this to $\rho_{2}(r,r^{\prime})$ (fig.
15(a)), we can see that these correspond to: (i) atoms being bound to the same
side of the ion ($r_{A}=z_{A}-z^{\prime}_{A}=0$), (ii) atoms being bound to
opposite sides of the ion ($r_{A}\sim 0.4R^{*}$), (iii) and one atom being
bound to the ion whilst the other is unbound ($r_{A}\sim 1.0R^{*}$). With
increasing $g$, we observe a depletion of the central peak at $r_{A}=0$ in
favour of the outer peak, which additionally shifts to larger separations
(dashed curves in fig. 14(a)). The middle peak is essentially unaffected. The
bunching of atoms becomes suppressed and at $g=10$ one ends up with atoms
located on different sides w.r.t. the ion (see fig. 15(b)).
The structure of $\rho_{1}^{AI}$ at $g=0$ has two distinct peaks at $\approx
0.3R^{*}$ and $\approx 0.8R^{*}$, reflecting the different length scales of
the molecular and vibrational orbitals comprising the number state (dashed
curves in fig. 14(b)). $\rho_{1}^{AI}$ is largely unaffected by the
interatomic coupling strength, though there is a peak shift around $\approx
0.8R^{*}$, indicating that the increase in $d_{AI}$ (dashed curve in inset of
fig. 14(b)) arises solely from the unbound atom spreading out as the atom pair
grows increasingly repulsive.
#### IV.4.2 Mobile ion
Several patterns in $\rho_{2}(r,r^{\prime})$ are enhanced for $\beta=1$
(compare rows in fig. 15). When both atoms are in the vicinity of the ion
($r<0.3$, $r^{\prime}<0.3$) the anti-bunching is amplified, while when one
atom is further away, ($r\lessgtr 0.3R^{*}$, $r^{\prime}\gtrless 0.3R^{*}$)
the anti-bunching is suppressed and atoms are most likely to be found on the
same side w.r.t the ion. This causes further depletion of the interparticle
separation distribution at $r_{A}=0$ and suppresses the outer peak at
$r_{A}\sim 1.0R^{*}$ (compare dashed and solid lines in fig. 14(a)). The
relative magnitudes of these effects are different depending on the value of
$g$, resulting in larger $d_{AA}$ at small $g$ and smaller $d_{AA}$ at large
$g$, compared to the static ion case (black curves in the inset of fig.
14(a)). The ion’s mobility creates a positive shift in $d_{AI}$ from the
greater spread of $\rho_{1}^{AI}$. Although the vibrational orbital peak
shifts to greater separations, the molecular orbital peak at $\approx
0.3R^{*}$ is enhanced, leading to a slow decrease of $d_{AI}$.
### IV.5 Fourth excited state
The non-interacting fourth excited state bears similarity to the third excited
state, except that the trap-state atom occupies the next highest-energy
vibrational orbital. Here, the opposing symmetries of the single particle
states suppress the off-diagonal elements ($z_{A}=-z_{A}^{\prime}$) of the
probability density, as was observed already in the first excited state. For
the case of a static ion, the atoms fully-separate to opposite sides of the
ion even at extremely weak atomic interaction strengths and thereafter,
further changes are negligible (see section IV.5.1). For the case of an equal
mass system however, the ion’s mobility reinforces the overlap of the atoms,
which competes against the anti-correlations produced by the repulsive
interatomic interaction. As a result, the total energy of the state grows
rapidly and the on-set of the fully anti-bunched configuration is delayed (see
section IV.5.2).
((a))
((b))
((c))
((d))
((e))
Figure 16: Key observables for the fourth state. (a): interatomic separation
distribution $\rho^{AA}_{1}(z_{A}-z_{A}^{\prime})$ for different atom-atom
coupling strengths $g$. The inset shows the expectation value
$\braket{d_{AA}}$ for the atom-atom separation as a function of $g$ (eq.
(12)). (b): interspecies separation distribution $\rho^{AI}_{1}(z_{A}-z_{I})$
for varying atom-atom coupling strengths $g$. The inset shows the expectation
values $\braket{d_{AI}}$ for the atom-ion separation as a function of $g$ (eq.
(13)). (c)-(e): the evolution of the laboratory frame energy components with
atom-atom coupling strength $g$. Note for (a) and (b): the grey lines indicate
the distance between the minima of the atom-ion interaction potential (1). Due
to the parity symmetry it is sufficient to show only the positive semi-axis.
All subfigures: the solid curves correspond to a mobile ion, whilst dashed
curves correspond to a static ion.
((a))
((b))
((c))
((d))
Figure 17: Snapshots of the atomic probability density
$\rho_{2}^{\text{AA}}=|\psi(r,r^{\prime})|^{2}$ of the fourth excited state
for different interaction strengths $g$ for a static ion ((a) and (b)) and a
mobile ion ((c) and (d)).
#### IV.5.1 Static ion
The fourth excited state for two non-interacting ($g=0$) atoms coupled to a
static ($\beta=0$) ion is given by the number state $\ket{1,0,0,1}$, with one
atom in the even molecular orbital $\phi_{0}$ and the other in the odd
vibrational orbital $\phi_{3}$ (see fig. 1).
Similar to the third excited state, the total energy (given in fig. 2(b)) and
the energy components (see figs. 16(c), 16(d) and 16(e)) are robust to $g$
variation. The small energy gap between the third and fourth eigenstates is
due to a difference in the potential energy $P_{A}$ of the atoms.
Nevertheless, this state also exhibits structural changes in the interatomic
separation distribution $\rho_{1}^{AA}$ (dashed curves in fig. 16(a)).
Contrary to the third excited state, here we observe two humps: a narrow hump
at $r_{A}=z_{A}-z^{\prime}_{A}=0$ and a broad hump at $r_{A}\sim 1.0R^{*}$. As
$g$ increases, the hump at $r_{A}=0$ becomes smaller and broader, whilst the
other becomes higher and narrower. As a result, the distance among the atoms
$d_{AA}$ increases with $g$ (black dashed curve in the inset of fig. 16(a)).
In the two-body density $\rho_{2}(r,r^{\prime})$ at $g=0$ (see fig. 17(a)), we
observe the absence of anti-bunched regions in close proximity to the ion
($r<0.3R^{*},r^{\prime}<0.3R^{*}$). This is in contrast to the third excited
state and explains the absence of the third peak at $r_{A}=0.4R^{*}$.
Additionally, since the occupied orbitals $\phi_{0}$ and $\phi_{3}$ are of
opposite parity symmetry, the off-diagonal at $r=-r^{\prime}$ is zero. At
stronger coupling $g=10$ (see fig. 17(b)) the only eligible configurations are
the ones, where one atom is bound to ion and the other is unbound on the
opposite side w.r.t. the ion.
#### IV.5.2 Mobile ion
For $\beta=1$, the total energy and energy components are not as robust to
variations in $g$. Notably, there is a rapid increase in $V_{AA}$ given by
$g\rho_{1}^{AA}(r=0)$ (solid light blue curve in fig. 16(e)), since the
probability amplitude at $\rho_{1}^{AA}(r=0)$ is unaffected by $g$ (see solid
curves in fig. 16(a)). As a result, the energy gap between the third and
fourth excited state widens with growing $g$ (see fig. 2(b)).
## V Summary and Outlook
In this study, we have analysed the low-energy eigenstates of an atom-ion
hybrid system consisting of a pair of bosons interacting with a single ion,
where both species are confined in a quasi-1D trapping geometry. The
eigenstates were obtained by means of the Multi-Layer Multi-Configuration
Time-Dependent Hartree method for Bosons (ML-MCTDHB), an ab initio method for
simulating entangled mixtures with significant intra-component correlations.
We described the eigenenergies’ dependence on the atom-atom coupling strength
$g$ for the case of an infinitely-heavy ion ($\beta=0$) and contrasted this to
the case of an equal mass system ($\beta=1$). The former we termed the static
ion system and the latter the mobile ion system. Each eigenstate has been
characterised in terms of its interatomic and interspecies separation
distributions, average separations among the particles, and energy allocation
in different Hamiltonian components.
In general, the repulsive interaction between the atoms increases their
average separation, accompanied by a broadening of the interatomic separation
distribution and an energy exchange between atomic kinetic and atom-ion
interaction energies. The average distance to the ion does not necessarily
change in the process, however. When it does change, the atoms separate,
whilst simultaneously remaining on the same side w.r.t the ion (first and
second excited states). On the other hand, a constant atom-ion distance
indicates that the atoms have transitioned to a configuration in which the ion
lies in-between them. In this case, both atoms are either bound (ground state)
or one atom moves freely in the harmonic trap (third and fourth excited
states). Contrary to the repulsive interatomic interaction, the mobility of
the ion works to increase the average separation between all particles,
irrespective of the species.
We explained the apparent ion mobility-induced bunching effect observed in the
ground state through use of an effective potential, which predicts that the
ion mobility morphs the standard double-well-like potential produced by the
static ion into a pseudo-harmonic potential, such that the atoms cluster
together at the trap centre. Likewise, the effective potential for the ion
predicts that for strong interatomic interactions, the anti-bunched atom pair
acts like a pincer, which confines the ion at the centre of the trap. These
predictions agree with the trend in the ion’s energy components obtained via
exact numerical methods.
Regarding possible experimental realisation of our hybrid model, it is
important to note that we have neglected all three-body recombination
processes. However, considering the low particle density in few-body systems
such as ours, the loss rate is expected to be insignificant. For larger
particle numbers in one spatial dimension, such decay channels may be
suppressed by strong repulsion among the atoms [59]. We have additionally
neglected charge transfer and radiative loss resulting from reactions between
the ion and the atoms. For certain heteronuclear atom-ion pairings, these
inelastic processes happen to be of low probability [60], and for other
pairings the chemical reactivity can be controlled through the use of a
magnetic field [35]. We have further assumed our system is at temperatures low
enough for pure s-wave scattering, which has long been the goal of atom-ion
experiments. Recent experiments have attained temperatures at the threshold of
this regime via buffer gas cooling of a single ion in a Paul trap for species
pairings with a large mass-imbalance [36], corresponding to a static ion
system. As for the mobile ion system, the recent advent of optical traps for
ions [30] provides a promising platform that could be used for experiments
with atoms and ions of the same element.
This work has mapped out the landscape of stationary states for an exemplary
few-body mixture characterised by long-range interspecies interactions. It
lays the foundation for the route to more exotic and complex systems, such as
solid-state emulation in Coulomb crystals and dipolar quantum gases.
Additionally, exploring how the properties of the present atom-ion hybrid
system evolve with greater particle numbers, alternative species pairings, and
differing trapping frequencies ($\eta\neq 1$) would be an interesting avenue
for future study. This work also focused solely on a time-independent problem,
therefore a natural extension would be to consider many-body dynamics, e.g.:
ion immersion in an atomic gas and time-resolved monitoring of atom capture,
leading to the formation of mesoscopic molecules. Other theoretical
investigations into atom-ion hybrid systems have employed a variety of
techniques, including: exact diagonalisation [61], quantum Monte Carlo [62,
63], density matrix renormalisation group (DMRG) [64], variational methods
[65], semi-classical methods [66] and hyperspherical coordinates [67], for
which our work may serve as a useful numerical benchmark.
## Acknowledgements
D. J. B. thanks Kevin Keiler and Fabian Köhler for helpful discussions
regarding the numerical implementation. M. P. gratefully acknowledges a
scholarship of the Studienstiftung des deutschen Volkes. This work is funded
by the Cluster of Excellence: ’Advanced Imaging of Matter’ of the Deutsche
Forschungsgemeinschaft (DFG) – EXC 2056 – project ID 390715994.
## References
* Smith _et al._ [2005] W. W. Smith, O. P. Makarov, and J. Lin, J. Mod. Op. 52, 2253 (2005).
* Grier _et al._ [2009] A. T. Grier, M. Cetina, F. Oručević, and V. Vuletić, Phys. Rev. Lett. 102, 223201 (2009).
* Schmid _et al._ [2010] S. Schmid, A. Härter, and J. H. Denschlag, Phys. Rev. Lett. 105, 133202 (2010).
* Zipkes _et al._ [2010] C. Zipkes, S. Palzer, C. Sias, and M. Köhl, Nature 464, 388 (2010).
* Härter and Hecker Denschlag [2014] A. Härter and J. Hecker Denschlag, Contemp. Phys. 55, 33 (2014).
* Tomza _et al._ [2019] M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek, and P. S. Julienne, Rev. Mod. Phys. 91, 035001 (2019).
* Ratschbacher _et al._ [2012] L. Ratschbacher, C. Zipkes, C. Sias, and M. Köhl, Nat. Phys. 8, 649 (2012).
* Hall and Willitsch [2012] F. H. Hall and S. Willitsch, Phys. Rev. Lett. 109, 233202 (2012).
* Härter _et al._ [2012] A. Härter, A. Krükow, A. Brunner, W. Schnitzler, S. Schmid, and J. H. Denschlag, Phys. Rev. Lett. 109, 123201 (2012).
* Meir _et al._ [2016] Z. Meir, T. Sikorsky, R. Ben-Shlomi, N. Akerman, Y. Dallal, and R. Ozeri, Phys. Rev. Lett. 117, 243401 (2016).
* Pérez-Ríos [2020] J. Pérez-Ríos, (2020), arXiv:2011.03963 .
* Krükow _et al._ [2016a] A. Krükow, A. Mohammadi, A. Härter, and J. H. Denschlag, Phys. Rev. A 94, 030701 (2016a).
* Krükow _et al._ [2016b] A. Krükow, A. Mohammadi, A. Härter, J. H. Denschlag, J. Pérez-Ríos, and C. H. Greene, Phys. Rev. Lett. 116, 193201 (2016b).
* Zipkes _et al._ [2011] C. Zipkes, L. Ratschbacher, C. Sias, and M. Köhl, New J. Phys. 13, 053020 (2011).
* Ravi _et al._ [2012] K. Ravi, S. Lee, A. Sharma, G. Werth, and S. Rangwala, Nat. Commun. 3, 1 (2012).
* Côté [2000] R. Côté, Phys. Rev. Lett. 85, 5316 (2000).
* Dieterle _et al._ [2021] T. Dieterle, M. Berngruber, C. Hölzl, R. Löw, K. Jachymski, T. Pfau, and F. Meinert, Phys. Rev. Lett. 126, 033401 (2021).
* Bissbort _et al._ [2013] U. Bissbort, D. Cocks, A. Negretti, Z. Idziaszek, T. Calarco, F. Schmidt-Kaler, W. Hofstetter, and R. Gerritsma, Phys. Rev. Lett. 111, 080501 (2013).
* Casteels _et al._ [2011] W. Casteels, J. Tempere, and J. Devreese, J. Low Temp. Phys. 162, 266 (2011).
* Jachymski and Negretti [2020] K. Jachymski and A. Negretti, Physical Review Research 2 (2020).
* Gerritsma _et al._ [2012] R. Gerritsma, A. Negretti, H. Doerk, Z. Idziaszek, T. Calarco, and F. Schmidt-Kaler, Phys. Rev. Lett. 109, 080402 (2012).
* Schurer _et al._ [2016] J. Schurer, R. Gerritsma, P. Schmelcher, and A. Negretti, Phys. Rev. A 93, 063602 (2016).
* Ratschbacher _et al._ [2013] L. Ratschbacher, C. Sias, L. Carcagni, J. Silver, C. Zipkes, and M. Köhl, Phys. Rev. Lett. 110, 160402 (2013).
* Veit _et al._ [2020] C. Veit, N. Zuber, O. A. Herrera-Sancho, V. S. V. Anasuri, T. Schmid, F. Meinert, R. Löw, and T. Pfau, (2020), arXiv:2008.08512 .
* Cetina _et al._ [2012] M. Cetina, A. T. Grier, and V. Vuletić, Phys. Rev. Lett. 109, 253201 (2012).
* Krych and Idziaszek [2015] M. Krych and Z. Idziaszek, Phys. Rev. A 91, 023430 (2015).
* Wessels _et al._ [2018] P. Wessels, B. Ruff, T. Kroker, A. K. Kazansky, N. M. Kabachnik, K. Sengstock, M. Drescher, and J. Simonet, Commun. Phys. 1, 1 (2018).
* Schneider _et al._ [2010] C. Schneider, M. Enderlein, T. Huber, and T. Schätz, Nat. Phot. 4, 772 (2010).
* Enderlein _et al._ [2012] M. Enderlein, T. Huber, C. Schneider, and T. Schaetz, Phys. Rev. Lett. 109, 233004 (2012).
* Schaetz [2017] T. Schaetz, J. Phys. B 50, 102001 (2017).
* Lambrecht _et al._ [2017] A. Lambrecht, J. Schmidt, P. Weckesser, M. Debatin, L. Karpa, and T. Schaetz, Nat. Phot. 11, 704 (2017).
* Schmidt _et al._ [2020] J. Schmidt, P. Weckesser, F. Thielemann, T. Schaetz, and L. Karpa, Phys. Rev. Lett. 124, 053402 (2020).
* Kleinbach _et al._ [2018] K. S. Kleinbach, F. Engel, T. Dieterle, R. Löw, T. Pfau, and F. Meinert, Phys. Rev. Lett. 120, 193401 (2018).
* Schmid _et al._ [2018] T. Schmid, C. Veit, N. Zuber, R. Löw, T. Pfau, M. Tarana, and M. Tomza, Phys. Rev. Lett. 120, 153401 (2018).
* Tomza _et al._ [2015] M. Tomza, C. P. Koch, and R. Moszynski, Phys. Rev. A 91, 042706 (2015).
* Feldker _et al._ [2020] T. Feldker, H. Fürst, H. Hirzler, N. Ewald, M. Mazzanti, D. Wiater, M. Tomza, and R. Gerritsma, Nat. Phys. , 1 (2020).
* Idziaszek _et al._ [2009] Z. Idziaszek, T. Calarco, P. S. Julienne, and A. Simoni, Phys. Rev. A 79, 010702 (2009).
* Côté _et al._ [2002] R. Côté, V. Kharchenko, and M. Lukin, Phys. Rev. Lett. 89, 093001 (2002).
* Schurer _et al._ [2017] J. Schurer, A. Negretti, and P. Schmelcher, Phys. Rev. Lett. 119, 063001 (2017).
* Massignan _et al._ [2005] P. Massignan, C. J. Pethick, and H. Smith, Phys. Rev. A 71, 023606 (2005).
* Schurer _et al._ [2015] J. Schurer, A. Negretti, and P. Schmelcher, New J. Phys. 17, 083024 (2015).
* Goold _et al._ [2010] J. Goold, H. Doerk, Z. Idziaszek, T. Calarco, and T. Busch, Phys. Rev. A 81, 041601 (2010).
* Schurer _et al._ [2014] J. Schurer, P. Schmelcher, and A. Negretti, Phys. Rev. A 90, 033601 (2014).
* Krönke _et al._ [2013] S. Krönke, L. Cao, O. Vendrell, and P. Schmelcher, New J. Phys. 15, 63018 (2013).
* Cao _et al._ [2013] L. Cao, S. Krönke, O. Vendrell, and P. Schmelcher, J. Chem. Phys. 139, 134103 (2013).
* Keiler _et al._ [2020] K. Keiler, S. I. Mistakidis, and P. Schmelcher, New J. Phys. 22, 083003 (2020).
* Theel _et al._ [2020] F. Theel, K. Keiler, S. I. Mistakidis, and P. Schmelcher, New J. Phys. 22, 023027 (2020).
* Cao _et al._ [2017] L. Cao, V. Bolsinger, S. Mistakidis, G. Koutentakis, S. Krönke, J. Schurer, and P. Schmelcher, J. Chem. Phys. 147, 044106 (2017).
* Chen _et al._ [2018] J. Chen, J. M. Schurer, and P. Schmelcher, Phys. Rev. Lett. 121, 043401 (2018).
* Kwasniok _et al._ [2020] J. Kwasniok, S. I. Mistakidis, and P. Schmelcher, Phys. Rev. A 101, 053619 (2020).
* Meyer _et al._ [1990] H.-D. Meyer, U. Manthe, and L. Cederbaum, Chem. Phys. Lett. 165, 73 (1990).
* Beck _et al._ [2000a] M. Beck, A. Jäckle, G. Worth, and H.-D. Meyer, Phys. Rep. 324, 1 (2000a).
* Alon _et al._ [2008] O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Rev. A 77, 033613 (2008).
* Kato and Kono [2004] T. Kato and H. Kono, Chem. Phys. Lett. 392, 533 (2004).
* Alon _et al._ [2007] O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Rev. A 76, 062501 (2007).
* Light _et al._ [1985] J. C. Light, I. P. Hamilton, and J. V. Lill, J Chem. Phys. 82, 1978 (1985).
* Beck _et al._ [2000b] M. H. Beck, A. Jäckle, G. A. Worth, and H. D. Meyer, Phys. Rep. 324, 1 (2000b).
* Girardeau and Minguzzi [2006] M. D. Girardeau and A. Minguzzi, Phys. Rev. Lett. 96 (2006).
* Gangardt and Shlyapnikov [2003] D. M. Gangardt and G. V. Shlyapnikov, Phys. Rev. Lett. 90, 4 (2003).
* Tomza [2015] M. Tomza, Phys. Rev. A 92, 62701 (2015).
* García-March _et al._ [2014] M. A. García-March, B. Juliá-Díaz, G. E. Astrakharchik, J. Boronat, and A. Polls, Phys. Rev. A 90, 063605 (2014).
* Garcia-March _et al._ [2013] M. A. Garcia-March, B. Juliá-Díaz, G. E. Astrakharchik, T. Busch, J. Boronat, and A. Polls, Phys. Rev. A 88, 063604 (2013).
* Astrakharchik _et al._ [2020] G. E. Astrakharchik, L. A. P. Ardila, R. Schmidt, K. Jachymski, and A. Negretti, (2020), arXiv:2005.12033 .
* Michelsen _et al._ [2019] A. B. Michelsen, M. Valiente, N. T. Zinner, and A. Negretti, Phys. Rev. B 100, 205427 (2019).
* Rath and Schmidt [2013] S. P. Rath and R. Schmidt, Phys. Rev. A 88, 053632 (2013).
* Melezhik _et al._ [2019] V. S. Melezhik, Z. Idziaszek, and A. Negretti, Phys. Rev. A 100, 063406 (2019).
* Pérez-Ríos and Greene [2018] J. Pérez-Ríos and C. H. Greene, Phys. Rev. A 98, 062707 (2018).
|
11institutetext: University of Warwick, UK,
<EMAIL_ADDRESS>University of Warwick, UK,
<EMAIL_ADDRESS>33institutetext: University of Oxford, UK,
<EMAIL_ADDRESS>44institutetext: CNRS, Université de Bordeaux,
France
# Leafy automata for higher-order concurrency
Alex Dixon (🖂) 11 Ranko Lazić 22 Andrzej S. Murawski 33 Igor Walukiewicz 44
###### Abstract
Finitary Idealized Concurrent Algol ($\mathsf{FICA}$) is a prototypical
programming language combining functional, imperative, and concurrent
computation. There exists a fully abstract game model of $\mathsf{FICA}$,
which in principle can be used to prove equivalence and safety of
$\mathsf{FICA}$ programs. Unfortunately, the problems are undecidable for the
whole language, and only very rudimentary decidable sub-languages are known.
We propose leafy automata as a dedicated automata-theoretic formalism for
representing the game semantics of $\mathsf{FICA}$. The automata use an
infinite alphabet with a tree structure. We show that the game semantics of
any $\mathsf{FICA}$ term can be represented by traces of a leafy automaton.
Conversely, the traces of any leafy automaton can be represented by a
$\mathsf{FICA}$ term. Because of the close match with $\mathsf{FICA}$, we view
leafy automata as a promising starting point for finding decidable subclasses
of the language and, more generally, to provide a new perspective on models of
higher-order concurrent computation.
Moreover, we identify a fragment of $\mathsf{FICA}$ that is amenable to
verification by translation into a particular class of leafy automata. Using a
locality property of the latter class, where communication between levels is
restricted and every other level is bounded, we show that their emptiness
problem is decidable by reduction to Petri nets reachability.
###### Keywords:
Finitary Idealized Concurrent Algol, Higher-Order Concurrency, Automata over
Infinite Alphabets, Game Semantics
## 1 Introduction
Game semantics is a versatile paradigm for giving semantics to a wide spectrum
of programming languages [4, 37]. It is well-suited for studying the
observational equivalence of programs and, more generally, the behaviour of a
program in an arbitrary context. About 20 years ago, it was discovered that
the game semantics of a program can sometimes be expressed by a finite
automaton or another simple computational model [21]. This led to algorithmic
uses of game semantics for program analysis and verification [1, 17, 22, 6,
28, 27, 29, 36, 18, 19]. Thus far, these advances concerned mostly languages
without concurrency.
In this work, we consider Finitary Idealized Concurrent Algol
($\mathsf{FICA}$) and its fully abstract game semantics [23]. It is a call-by-
name language with higher-order features, side-effects, and concurrency
implemented by a parallel composition operator and semaphores. It is finitary
since, as it is common in this context, base types are restricted to finite
domains. Quite surprisingly, the game semantics of this language is arguably
simpler than that for the language without concurrency. The challenge comes
from algorithmic considerations.
Following the successful approach from the sequential case [21, 39, 35, 38,
13], the first step is to find an automaton model abstracting the phenomena
appearing in the semantics. The second step is to obtain program fragments
from structural restrictions on the automaton model. In this paper we take
both steps.
We propose _leafy automata_ : an automaton model working on nested data. Data
are used to represent pointers in plays, while the nesting of data reflects
structural dependencies in the use of pointers. Interestingly, the structural
dependencies in plays boil down to imposing a tree structure on the data. We
show a close correspondence between the automaton model and the game semantics
of $\mathsf{FICA}$. For every program, there is a leafy automaton whose traces
(data words) represent precisely the plays in the semantics of the program
(Theorem 6.1). Conversely, for every leafy automaton, there is a program whose
semantics consists of plays representing the traces of the automaton (Theorem
8.1). (The latter result holds modulo a saturation condition we explain
later.) This equivalence shows that leafy automata are a suitable model for
studying decidability questions for $\mathsf{FICA}$.
Not surprisingly, due to their close connection to $\mathsf{FICA}$, leafy
automata turn out to have an undecidable emptiness problem. We use the
undecidability argument to identify the source, namely communication across
several unbounded levels, i.e., levels in which nodes can produce an unbounded
number of children during the lifetime of the automaton. To eliminate the
problem, we introduce a restricted variant of leafy automata, called _local_ ,
in which every other level is bounded and communication is allowed to cross
only one unbounded node. Emptiness for such automata can be decided via
reduction to a number of instances of Petri net reachability problem.
We also identify a fragment of $\mathsf{FICA}$, dubbed _local_ $\mathsf{FICA}$
($\mathsf{LFICA}$), which maps onto local leafy automata. It is based on
restricting the distance between semaphore and variable declarations and their
uses inside the term. This is a first non-rudimentary fragment of
$\mathsf{FICA}$ for which some verification tasks are decidable. Overall, this
makes it possible to use local leafy automata to analyse $\mathsf{LFICA}$
terms and decide associated verification tasks.
#### Related work
Concurrency, even with only first-order recursion, leads to undecidability
[41]. Intuitively, one can encode the intersection of languages of two
pushdown automata. From the automata side, much research on decidable cases
has concentrated on bounding interactions between stacks representing
different threads of the program [40, 31, 5]. From the game semantics side,
the only known decidable fragment of $\mathsf{FICA}$ is Syntactic Control of
Concurrency (SCC) [24], which imposes bounds on the number of threads in which
arguments can be used. This restriction makes it possible to represent the
game semantics of programs by finite automata. In our work, we propose
automata models that correspond to unbounded interactions with arbitrary
$\mathsf{FICA}$ contexts, and importantly that remains true also when we
restrict the terms to $\mathsf{LFICA}$. Leafy automata are a model of
computation over an infinite alphabet. This area has been explored
extensively, partly motivated by applications to database theory, notably XML
[43]. In this context, nested data first appeared in [8], where the authors
considered shuffle expressions as the defining formalism. Later on, data
automata [10] and class memory automata [9] have been adapted to nested data
in [16, 14]. They are similar to leafy automata in that the automaton is
allowed to access states related to previous uses of data values at various
depths. What distinguishes leafy automata is that the lifetime of a data value
is precisely defined and follows a question and answer discipline in
correspondence with game semantics. Leafy automata also feature run-time
“zero-tests”, activated when reading answers.
For most models over nested data, the emptiness problem is undecidable. To
achieve decidability, the authors in [16, 14] relax the acceptance conditions
so that the emptiness problem can eventually be recast as a coverability
problem for a well-structured transition system. In [12], this result was used
to show decidability of equivalence for a first-order (sequential) fragment of
Reduced ML. On the other hand, in [8] the authors relax the order of letters
in words, which leads to an analysis based on semi-linear sets. Both of these
restrictions are too strong to permit the semantics of $\mathsf{FICA}$,
because of the game-semantic $\mathsf{WAIT}$ condition, which corresponds to
waiting until all sub-processes terminate.
Another orthogonal strand of work on concurrent higher-order programs is based
on higher-order recursion schemes [25, 30]. Unlike $\mathsf{FICA}$, they
feature recursion but the computation is purely functional over a single
atomic type $o$.
#### Structure of the paper:
In the next two sections we recall $\mathsf{FICA}$ and its game semantics from
[23]. The following sections introduce leafy automata ($\mathsf{LA}$) and
their local variant ($\mathsf{LLA}$), where we also analyse the associated
decision problems and, in particular, show that the non-emptiness problem for
$\mathsf{LLA}$ is decidable. Subsequently, we give a translation from
$\mathsf{FICA}$ to $\mathsf{LA}$ (and back) and define a fragment
$\mathsf{LFICA}$ of $\mathsf{FICA}$ which can be translated into
$\mathsf{LLA}$.
## 2 Finitary Idealized Concurrent Algol ($\mathsf{FICA}$)
Idealized Concurrent Algol [23] is a paradigmatic language combining higher-
order with imperative computation in the style of Reynolds [42], extended to
concurrency with parallel composition ($||$) and binary semaphores. We
consider its finitary variant $\mathsf{FICA}$ over the finite datatype
$\\{0,\ldots,\mathit{max}\\}$ ($\mathit{max}\geq 0$) with loops but no
recursion. Its types $\theta$ are generated by the grammar
$\theta::=\beta\mid\theta\rightarrow\theta\qquad\qquad\beta::={\bf
com}\mid{\bf exp}\mid{\bf var}\mid{\bf sem}$
where ${\bf com}$ is the type of commands; ${\bf exp}$ that of
$\\{0,\ldots,\mathit{max}\\}$-valued expressions; ${\bf var}$ that of
assignable variables; and ${\bf sem}$ that of semaphores. The typing judgments
are displayed in Figure 1. ${\bf skip}$ and ${\bf div}_{\theta}$ are constants
representing termination and divergence respectively, $i$ ranges over $\\{0,$
$\cdots,$ $\mathit{max}\\}$, and $\mathbf{op}$ represents unary arithmetic
operations, such as successor or predecessor (since we work over a finite
datatype, operations of bigger arity can be defined using conditionals).
Variables and semaphores can be declared locally via $\mathbf{newvar}$ and
$\mathbf{newsem}$. Variables are dereferenced using $!M$, and semaphores are
manipulated using two (blocking) primitives, ${\bf grab}(s)$ and ${\bf
release}(s)$, which grab and release the semaphore respectively.
$\Gamma\vdash{\bf skip}:{\bf com}$ $\Gamma\vdash{\bf div}_{\theta}:\theta$
$\Gamma\vdash i:{\bf exp}$ ${\Gamma}\vdash{M:{\bf exp}}$
${\Gamma}\vdash{\mathbf{op}(M):{\bf exp}}$
$\Gamma\vdash M:{\bf com}$ $\Gamma\vdash N:\beta$ $\Gamma\vdash M;N:\beta$
$\Gamma\vdash M:{\bf com}$ $\Gamma\vdash N:{\bf com}$ $\Gamma\vdash M||N:{\bf
com}$
$\Gamma\vdash M:{\bf exp}$ $\Gamma\vdash N_{1},N_{2}:\beta$ $\Gamma\vdash{\bf
if}\,M\,{\bf then}\,N_{1}\,{\bf else}\,N_{2}:\beta$ $\Gamma\vdash M:{\bf exp}$
$\Gamma\vdash N:{\bf com}$ $\Gamma\vdash{\bf while}\,M\,{\bf do}\,N:{\bf com}$
$\Gamma,x:\theta\vdash x:\theta$ $\Gamma,x:\theta\vdash M:\theta^{\prime}$
$\Gamma\vdash\lambda x.M:\theta\rightarrow\theta^{\prime}$ $\Gamma\vdash
M:\theta\rightarrow\theta^{\prime}$ $\Gamma\vdash N:\theta$ $\Gamma\vdash
MN:\theta^{\prime}$
$\Gamma\vdash M:{\bf var}$ $\Gamma\vdash N:{\bf exp}$ $\Gamma\vdash
M\,\raisebox{0.27986pt}{:}{=}\,N:{\bf com}$ $\Gamma\vdash M:{\bf var}$
$\Gamma\vdash!M:{\bf exp}$
$\Gamma\vdash M:{\bf sem}$ $\Gamma\vdash{\bf release}(M):{\bf com}$
$\Gamma\vdash M:{\bf sem}$ $\Gamma\vdash{\bf grab}(M):{\bf com}$
$\Gamma,x:{\bf var}\vdash M:{\bf com},{\bf exp}$ $\Gamma\vdash{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M:{\bf com},{\bf exp}$
$\Gamma,x:{\bf sem}\vdash M:{\bf com},{\bf exp}$ $\Gamma\vdash{\bf
newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M:{\bf com},{\bf exp}$
Figure 1: $\mathsf{FICA}$ typing rules
The small-step operational semantics of $\mathsf{FICA}$ is reproduced in
Appendix 0.A. In what follows, we shall write ${\bf div}$ for ${\bf div}_{\bf
com}$.
We are interested in _contextual equivalence_ of terms. Two terms are
contextually equivalent if there is no context that can distinguish them with
respect to may-termination. More formally, a term ${}\vdash{M:{\bf com}}$ is
said to terminate, written $M\\!\Downarrow$, if there exists a terminating
evaluation sequence from $M$ to ${\bf skip}$. Then _contextual
(may-)equivalence_ ($\Gamma\vdash M_{1}\cong M_{2}$) is defined by: for all
contexts $\mathcal{C}$ such that ${}\vdash{\mathcal{C}[M]:{\bf com}}$,
$\mathcal{C}[M_{1}]\\!\Downarrow$ if and only if
$\mathcal{C}[M_{2}]\\!\Downarrow$. The force of this notion is quantification
over all contexts.
Since contextual equivalence becomes undecidable for $\mathsf{FICA}$ very
quickly [24], we will look at the special case of testing equivalence with
terms that always diverge, e.g. given $\Gamma\vdash M:\theta$, is it the case
that ${\Gamma}\vdash{M\cong{\bf div}_{\theta}}$? Intuitively, equivalence with
an always-divergent term means that $\mathcal{C}[M]$ will never converge (must
diverge) if $\mathcal{C}$ uses $M$. At the level of automata, this will turn
out to correspond to the emptiness problem.
In verification tasks, with the above equivalence test, we can check whether
uses of $M$ can ever lead to undesirable states. For example, for a given term
${x:{\bf var}}\vdash{M:\theta}$, the term
${f:\theta\rightarrow{\bf com}}\vdash{{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,(f(M)\,||\,{\bf
if}\,!x=13\,{\bf then}\,\,{\bf skip}\,\,{\bf else}\,\,{\bf div})}$
will be equivalent to ${\bf div}$ only when $x$ is never set to $13$ during a
terminating execution. Note that, because of quantification over all contexts,
$f$ may use $M$ an arbitrary number of times, also concurrently or in nested
fashion, which is a very expressive form of quantification.
## 3 Game semantics
Game semantics for programming languages involves two players, called Opponent
(O) and Proponent (P), and the sequences of moves made by them can be viewed
as interactions between a program (P) and a surrounding context (O). In this
section, we briefly present the fully abstract game model for $\mathsf{FICA}$
from [23], which we rely on in the paper. The games are defined using an
auxiliary concept of an arena.
###### Definition 1
An _arena_ $A$ is a triple $\langle{M_{A},\lambda_{A},\vdash_{A}}\rangle$
where:
* •
$M_{A}$ is a set of _moves_ ;
* •
$\lambda_{A}:M_{A}\rightarrow\\{O,P\\}\times\\{Q,A\\}$ is a function
determining for each $m\in M_{A}$ whether it is an _Opponent_ or a _Proponent
move_ , and a _question_ or an _answer_ ; we write
$\lambda_{A}^{OP},\lambda_{A}^{QA}$ for the composite of $\lambda_{A}$ with
respectively the first and second projections;
* •
$\vdash_{A}$ is a binary relation on $M_{A}$, called _enabling_ , satisfying:
if $m\vdash_{A}n$ for no $m$ then $\lambda_{A}(n)=(O,Q)$, if $m\vdash_{A}n$
then $\lambda_{A}^{OP}(m)\neq\lambda_{A}^{OP}(n)$, and if $m\vdash_{A}n$ then
$\lambda_{A}^{QA}(m)=Q$.
We shall write $I_{A}$ for the set of all moves of $A$ which have no enabler;
such moves are called _initial_. Note that an initial move must be an Opponent
question. In arenas used to interpret base types all questions are initial and
P-moves answering them are detailed in the table below, where
$i\in\\{0,\cdots,\mathit{max}\\}$.
$\begin{array}[]{c|c|c||c|c|c}~{}\textrm{Arena}{}&~{}\textrm{O-question}{}&~{}\textrm{P-answers}{}&~{}\textrm{Arena}{}&~{}\textrm{O-question}{}&~{}\textrm{P-answers}{}\\\
\hline\cr{\llbracket}{{\bf
com}}{\rrbracket}&\mathsf{run}&\mathsf{done}&{\llbracket}{{\bf
exp}}{\rrbracket}&\mathsf{q}&i\\\\[4.30554pt] \hline\cr{\llbracket}{{\bf
var}}{\rrbracket}&\mathsf{read}&i&{\llbracket}{{\bf
sem}}{\rrbracket}&\mathsf{grb}&\mathsf{ok}\\\
&\mathsf{write}(i)&\mathsf{ok}&&\mathsf{rls}&\mathsf{ok}\end{array}$
More complicated types are interpreted inductively using the _product_
($A\times B$) and _arrow_ ($A\Rightarrow B$) constructions, given below.
$\begin{array}[]{rcl}M_{A\times B}&=&M_{A}+M_{B}\\\ \lambda_{A\times
B}&=&[\lambda_{A},\lambda_{B}]\\\ \vdash_{A\times
B}&=&\vdash_{A}+\vdash_{B}\\\
\end{array}\qquad\begin{array}[]{rcl}M_{A\Rightarrow B}&=&M_{A}+M_{B}\\\
\lambda_{A\Rightarrow
B}&=&[\langle\lambda_{A}^{PO},\lambda_{A}^{QA}\rangle,\lambda_{B}]\\\
\vdash_{A\Rightarrow B}&=&\vdash_{A}+\vdash_{B}+\\{\,(b,a)\mid b\in
I_{B}\textrm{ and }a\in I_{A}\\}\\\ \end{array}$
where $\lambda_{A}^{PO}(m)=O$ iff $\lambda_{A}^{OP}(m)=P$. We write
${\llbracket}{\theta}{\rrbracket}$ for the arena corresponding to type
$\theta$. Below we draw (the enabling relations of) $A_{1}={\llbracket}{{\bf
com}\rightarrow{\bf com}\rightarrow{\bf com}}{\rrbracket}$ and
$A_{2}={\llbracket}{({\bf var}\rightarrow{\bf com})\rightarrow{\bf
com}}{\rrbracket}$ respectively, using superscripts to distinguish copies of
the same move (the use of superscripts is consistent with our future use of
tags in Definition 9).
---
$\textstyle{O}$$\textstyle{\mathsf{run}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$$\textstyle{\mathsf{run}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{run}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{done}}$$\textstyle{O}$$\textstyle{\mathsf{done}^{2}}$$\textstyle{\mathsf{done}^{1}}$
---
$\textstyle{O}$$\textstyle{\mathsf{run}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$$\textstyle{\mathsf{run}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{done}}$$\textstyle{O}$$\textstyle{\mathsf{read}^{11}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{write}(i)^{11}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{done}^{1}}$$\textstyle{P}$$\textstyle{i^{11}}$$\textstyle{\mathsf{ok}^{11}}$
Given an arena $A$, we specify next what it means to be a legal play in $A$.
For a start, the moves that players exchange will have to form a _justified
sequence_ , which is a finite sequence of moves of $A$ equipped with pointers.
Its first move is always initial and has no pointer, but each subsequent move
$n$ must have a unique pointer to an earlier occurrence of a move $m$ such
that $m\vdash_{A}n$. We say that $n$ is (explicitly) justified by $m$ or, when
$n$ is an answer, that $n$ answers $m$. If a question does not have an answer
in a justified sequence, we say that it is _pending_ in that sequence. Below
we give two justified sequences from $A_{1}$ and $A_{2}$ respectively.
$\rnode{A}{\mathsf{run}}\,\,\rnode{B}{\mathsf{run}}^{1}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}\,\,\rnode{C}{\mathsf{run}}^{2}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{C}{A}}\,\,\rnode{D}{\mathsf{done}^{1}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=140.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{D}{B}}\,\,\rnode{E}{\mathsf{done}^{2}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=140.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{E}{C}}\,\,\rnode{F}{\mathsf{done}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=155.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{F}{A}}\qquad\rnode{A}{\mathsf{run}}\,\,\rnode{B}{\mathsf{run}^{1}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=160.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}\,\,\rnode{C}{\mathsf{read}^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{C}{B}}\,\,\rnode{D}{0^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{D}{C}}\,\,\rnode{E}{\mathsf{write}(1)^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{E}{B}}\,\,\rnode{F}{\mathsf{ok}^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{F}{E}}\,\,\rnode{G}{\mathsf{read}^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=160.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{G}{B}}\,\,\rnode{H}{1^{11}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{H}{G}}$
Not all justified sequences are valid. In order to constitute a legal play, a
justified sequence must satisfy a well-formedness condition that reflects the
“static” style of concurrency of our programming language: any started sub-
processes must end before the parent process terminates. This is formalised as
follows, where the letters $q$ and $a$ to refer to question- and answer-moves
respectively, while $m$ denotes arbitrary moves.
###### Definition 2
The set $P_{A}$ of _plays over $A$_ consists of the justified sequences $s$
over $A$ that satisfy the two conditions below.
FORK
: In any prefix
$s^{\prime}=\cdots\rnode{A}{q}\cdots\rnode{B}{m}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$
of $s$, the question $q$ must be pending when $m$ is played.
WAIT
: In any prefix
$s^{\prime}=\cdots\rnode{A}{q}\cdots\rnode{B}{a}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$
of $s$, all questions justified by $q$ must be answered.
It is easy to check that the justified sequences given above are plays. A
subset $\sigma$ of $P_{A}$ is _O-complete_ if $s\in\sigma$ and $so\in P_{A}$
imply $so\in\sigma$, when $o$ is an O-move.
###### Definition 3
A _strategy_ on $A$, written $\sigma:A$, is a prefix-closed O-complete subset
of $P_{A}$.
Suppose $\Gamma=\\{x_{1}:\theta_{1},\cdots,x_{l}:\theta_{l}\\}$ and
${\Gamma}\vdash{M:\theta}$ is a $\mathsf{FICA}$-term. Let us write
${\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}$ for the arena
${\llbracket}{\theta_{1}}{\rrbracket}\times\cdots\times{\llbracket}{\theta_{l}}{\rrbracket}\Rightarrow{\llbracket}{\theta}{\rrbracket}$.
In [23] it is shown how to assign a strategy on
${\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}$ to any $\mathsf{FICA}$-term
${\Gamma}\vdash{M:\theta}$. We write
${\llbracket}{{\Gamma}\vdash{M}}{\rrbracket}$ to refer to that strategy. For
example, ${\llbracket}{{\Gamma}\vdash{{\bf
div}}}{\rrbracket}=\\{\epsilon,\mathsf{run}\\}$ and
${\llbracket}{{\Gamma}\vdash{{\bf
skip}}}{\rrbracket}=\\{\epsilon,\mathsf{run},\rnode{A}{\mathsf{run}}\,\rnode{B}{\mathsf{done}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}\\}$.
Given a strategy $\sigma$, we denote by $\textsf{comp}(\sigma)$ the set of
non-empty _complete_ plays of $\sigma$, i.e. those in which all questions have
been answered. The game-semantic interpretation
${\llbracket}{\cdots}{\rrbracket}$ turns out to provide a fully abstract model
in the following sense.
###### Theorem 3.1 ([23])
$\Gamma\vdash M_{1}\cong M_{2}$ iff $\textsf{comp}({\llbracket}{\Gamma\vdash
M_{1}}{\rrbracket})=\textsf{comp}({\llbracket}{\Gamma\vdash
M_{2}}{\rrbracket})$.
In particular, since we have $\textsf{comp}({\llbracket}{{\Gamma}\vdash{{\bf
div}_{\theta}}}{\rrbracket})=\emptyset$, ${\Gamma}\vdash{M:\theta}$ is
equivalent to ${\bf div}_{\theta}$ iff
$\textsf{comp}({\llbracket}{{\Gamma}\vdash{M}}{\rrbracket})=\emptyset$.
## 4 Leafy automata
We would like to be able to represent the game semantics of $\mathsf{FICA}$
using automata. To that end, we introduce _leafy automata_ ($\mathsf{LA}$).
They are a variant of automata over nested data, i.e. a type of automata that
read finite sequences of letters of the form $(t,d_{0}d_{1}\cdots d_{j})$
($j\in\mathbb{N}$), where $t$ is a _tag_ from a finite set $\Sigma$ and each
$d_{i}$ ($0\leq i\leq j$) is a _data value_ from an infinite set
$\mathcal{D}$.
In our case, $\mathcal{D}$ will have the structure of a countably infinite
forest and the sequences $d_{0}\cdots d_{j}$ will correspond to branches of a
tree. Thus, instead of $d_{0}\cdots d_{j}$, we can simply write $d_{j}$,
because $d_{j}$ uniquely determines its ancestors: $d_{0},\dots,d_{j-1}$. The
following definition captures the technical assumptions on $\mathcal{D}$.
###### Definition 4
$\mathcal{D}$ is a countably infinite set equipped with a function
$\mathit{pred}:\mathcal{D}\rightarrow\mathcal{D}\cup\\{\bot\\}$ (the _parent_
function) such that the following conditions hold.
* •
Infinite branching: $\mathit{pred}^{-1}(\\{d_{\bot}\\})$ is infinite for any
$d_{\bot}\in\mathcal{D}\cup\\{\bot\\}$.
* •
Well-foundedness: for any $d\in\mathcal{D}$, there exists $i\in\mathbb{N}$,
called the _level of $d$_, such that $\mathit{pred}^{i+1}(d)=\bot$. Level-$0$
data values will be called _roots_.
In order to define configurations of leafy automata, we will rely on finite
subtrees of $\mathcal{D}$, whose nodes will be labelled with states. We say
that $T\subseteq\mathcal{D}$ is a subtree of $\mathcal{D}$ iff $T$ is closed
($\forall x\in T\colon\mathit{pred}(x)\in T\cup\\{\bot\\}$) and rooted
($\exists!x\in T\colon\mathit{pred}(x)=\bot$).
Next we give the formal definition of a level-$k$ leafy automaton. Its set of
states $Q$ will be divided into layers, written $Q^{(i)}$ ($0\leq i\leq k$),
which will be used to label level-$i$ nodes. We will write
$Q^{(i_{1},\cdots,i_{k})}$ to abbreviate $Q^{(i_{1})}\times\cdots\times
Q^{(i_{k})}$, excluding any components $Q^{(i_{j})}$ where $i_{j}<0$. We
distinguish $Q^{(0,-1)}=\\{\dagger\\}$.
###### Definition 5
A level-$k$ leafy automaton ($k$-$\mathsf{LA}$) is a tuple
$\mathcal{A}=\langle\Sigma,k,Q,\delta\rangle$, where
* •
$\Sigma=\Sigma_{\mathsf{Q}}+\Sigma_{\mathsf{A}}$ is a finite alphabet,
partitioned into questions and answers;
* •
$k\geq 0$ is the level parameter;
* •
$Q=\sum_{i=0}^{k}Q^{(i)}$ is a finite set of states, partitioned into sets
$Q^{(i)}$ of level-$i$ states;
* •
$\delta=\delta_{\mathsf{Q}}+\delta_{\mathsf{A}}$ is a finite transition
function, partitioned into question- and answer-related transitions;
* •
$\delta_{\mathsf{Q}}=\sum_{i=0}^{k}\delta^{(i)}_{\mathsf{Q}}$, where
$\delta^{(i)}_{\mathsf{Q}}\subseteq
Q^{(0,1,\cdots,i-1)}\times\Sigma_{\mathsf{Q}}\times Q^{(0,1,\cdots,i)}$ for
$0\leq i\leq k$;
* •
$\delta_{\mathsf{A}}=\sum_{i=0}^{k}\delta^{(i)}_{\mathsf{A}}$, where
$\delta^{(i)}_{\mathsf{A}}\subseteq
Q^{(0,1,\cdots,i)}\times\Sigma_{\mathsf{A}}\times Q^{(0,1,\cdots,i-1)}$ for
$0\leq i\leq k$.
Configurations of $\mathsf{LA}$ are of the form $(D,E,f)$, where $D$ is a
finite subset of $\mathcal{D}$ (consisting of data values that have been
encountered so far), $E$ is a finite subtree of $\mathcal{D}$, and
$f:E\rightarrow Q$ is a level-preserving function, i.e. if $d$ is a level-$i$
data value then $f(d)\in Q^{(i)}$. A leafy automaton starts from the empty
configuration $\kappa_{0}=(\emptyset,\emptyset,\emptyset)$ and proceeds
according to $\delta$, making two kinds of transitions. Each kind manipulates
a single leaf: for questions one new leaf is added, for answers one leaf is
removed. Let the current configuration be $\kappa=(D,E,f)$.
* •
On reading a letter $(t,d)$ with $t\in\Sigma_{\mathsf{Q}}$ and $d\not\in D$ a
fresh level-$i$ data, the automaton adds a new leaf $d$ in a configuration and
updates the states on the branch to $d$. So it changes its configuration to
$\kappa^{\prime}=(D\cup\\{d\\},E\cup\\{d\\},f^{\prime})$ provided that
$\mathit{pred}(d)\in E$ and $f^{\prime}$ satisfies:
$(f(\mathit{pred}^{i}(d)),\cdots,f(\mathit{pred}(d)),t,f^{\prime}(\mathit{pred}^{i}(d)),\cdots,f^{\prime}(\mathit{pred}(d)),f^{\prime}(d))\in\delta^{(i)}_{\mathsf{Q}},$
$\mathsf{dom}(f^{\prime})=\mathsf{dom}(f)\cup\\{d\\}$, and
$f^{\prime}(x)=f(x)$ for all
$x\not\in\\{\mathit{pred}(d),\cdots,\mathit{pred}^{i}(d)\\}$.
* •
On reading a letter $(t,d)$ with $t\in\Sigma_{\mathsf{A}}$ and $d\in E$ a
level-$i$ data which is a leaf, the automaton deletes $d$ and updates the
states on the branch to $d$. So it changes its configuration to
$\kappa^{\prime}=(D,E\setminus\\{d\\},f^{\prime})$ where $f^{\prime}$
satisfies:
$(f(\mathit{pred}^{i}(d)),\cdots,f(\mathit{pred}(d)),f(d),t,f^{\prime}(\mathit{pred}^{i}(d)),\cdots,f^{\prime}(\mathit{pred}(d)))\in\delta^{(i)}_{\mathsf{A}},$
$\mathsf{dom}(f^{\prime})=\mathsf{dom}(f)\setminus\\{d\\}$ and
$f^{\prime}(x)=f(x)$ for all
$x\not\in\\{\mathit{pred}(d),\cdots,\mathit{pred}^{i}(d)\\}$.
* •
Initially $D$,$E$, and $f$ are empty; we proceed to
$\kappa^{\prime}=(\\{d\\},\\{d\\},\\{d\mapsto q^{(0)}\\})$ if $(t,d)$ is read
where $\dagger{\xlongrightarrow{t}}q^{(0)}\in\delta^{(0)}_{\mathsf{Q}}$. The
last move is treated symmetrically.
In all cases, we write $\kappa{\xlongrightarrow{(t,d)}}\kappa^{\prime}$. Note
that a single transition can only change states on the branch ending in $d$.
Other parts of the tree remain unchanged.
###### Example 1
Below we illustrate the effect of $\mathsf{LA}$ transitions. Let
$D_{1}=\\{d_{0},d_{1},d_{1}^{\prime}\\}$ and $d_{2}\not\in D_{1}$. Let
$\kappa_{1}=(D_{1},E_{1},f_{1})$,
$\kappa_{2}=(D_{1}\cup\\{d_{2}\\},E_{2},f_{2})$,
$\kappa_{3}=(D_{1}\cup\\{d_{2}\\},E_{1},f_{1})$, where the trees $E_{1},E_{2}$
are displayed below and node annotations of the form $(q)$ correspond to
values of $f_{1},f_{2}$, e.g. $f_{1}(d_{0})=q^{(0)}$.
$\textstyle{d_{0}(q^{(0)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1},f_{1}:}$$\textstyle{d_{1}^{\prime}(q)}$$\textstyle{d_{1}(q^{(1)})}$
$\textstyle{d_{0}(r^{(0)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2},f_{2}:}$$\textstyle{d_{1}^{\prime}(q)}$$\textstyle{d_{1}(r^{(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{2}(r^{(2)})}$
For $\kappa_{1}$ to evolve into $\kappa_{2}$ (on $(t,d_{2})$), we need
$(q^{(0)},q^{(1)},t,r^{(0)},r^{(1)},r^{(2)})\in\delta^{(2)}_{\mathsf{Q}}$. On
the other hand, to go from $\kappa_{2}$ to $\kappa_{3}$ (on $(t,d_{2})$), we
want $(r^{(0)},$ $r^{(1)},$ $r^{(2)},$ $t,$ $q^{(0)},$
$q^{(1)})\in\delta^{(2)}_{\mathsf{A}}$.
###### Definition 6
A _trace_ of a leafy automaton $\mathcal{A}$ is a sequence $w=l_{1}\cdots
l_{h}\in(\Sigma\times\mathcal{D})^{\ast}$ such that
$\kappa_{0}{\xlongrightarrow{l_{1}}}\kappa_{1}\dots\kappa_{h-1}{\xlongrightarrow{l_{h}}}\kappa_{h}$
where $\kappa_{0}=(\emptyset,\emptyset,\emptyset)$. A configuration
$\kappa=(D,E,f)$ is _accepting_ if $E$ and $f$ are empty. A trace $w$ is
accepted by $\mathcal{A}$ if there is a non-empty sequence of transitions as
above with $\kappa_{h}$ accepting. The set of traces (resp. accepted traces)
of $\mathcal{A}$ is denoted by $\mathit{Tr}(\mathcal{A})$ (resp.
$\mathit{L}(\mathcal{A})$).
###### Remark 1
When writing states, we will often use superscripts $(i)$ to indicate the
intended level. So,
$(q^{(0)},\cdots,q^{(i-1)}){\xlongrightarrow{t}}(r^{(0)},\cdots,r^{(i)})$
refers to $(q^{(0)},\cdots,q^{(i-1)},t,$ $r^{(0)},$
$\cdots,r^{(i)})\in\delta^{(i)}_{\mathsf{Q}}$; similarly for
$\delta^{(i)}_{\mathsf{A}}$ transitions. For $i=0$, this degenerates to
$\dagger{\xlongrightarrow{t}}r^{(0)}$ and
$r^{(0)}{\xlongrightarrow{t}}\dagger$.
###### Example 2
Consider the $1$-$\mathsf{LA}$ over
$\Sigma_{\mathsf{Q}}=\\{\mathsf{start},\mathsf{inc}\\},\Sigma_{\mathsf{A}}=\\{\mathsf{dec},\mathsf{end}\\}$.
Let $Q^{(0)}=\\{0\\}$, $Q^{(1)}=\\{0\\}$ and define $\delta$ by:
$\dagger{\xlongrightarrow{\mathsf{start}}}0$,
$0{\xlongrightarrow{\mathsf{inc}}}(0,0)$,
$(0,0){\xlongrightarrow{\mathsf{dec}}}0$,
$0{\xlongrightarrow{\mathsf{end}}}\dagger$. The accepted traces of this
$1$-$\mathsf{LA}$ have the form
$(\mathsf{start},d_{0})\,\,(||_{i=0}^{n}(\mathsf{inc},d_{1}^{i})$
$(\mathsf{dec},d_{1}^{i}))\,\,(\mathsf{end},d_{0})$, i.e. they are valid
histories of a single non-negative counter (histories such that the counter
starts and ends at 0). In this case, all traces are simply prefixes of such
words.
###### Remark 2
Note that, whenever a leafy automaton reads $(t,d)$
($t\in\Sigma_{\mathsf{Q}}$) and the level of $d$ is greater than $0$, then it
must have read a unique question $(t^{\prime},\mathit{pred}(d))$ earlier.
Also, observe that an $\mathsf{LA}$ trace contains at most two occurrences of
the same data value, such that the first is paired with a question and the
second is paired with an answer. Because the question and the answer share the
same data value, we can think of the answer as answering the question, like in
game semantics. Indeed, justification pointers from answers to questions will
be represented in this way in Theorem 6.1. Finally, we note that $\mathsf{LA}$
traces are invariant under tree automorphisms of $\mathcal{D}$.
###### Lemma 1
The emptiness problem for $2$-$\mathsf{LA}$ is undecidable. For
$1$-$\mathsf{LA}$, it is reducible to the reachability problem for VASS in
polynomial time and there is a reverse reduction in exponential time, so it is
decidable in Ackermannian time [33] but not elementary [15].
###### Proof
For $2$-$\mathsf{LA}$ we reduce from the halting problem on two-counter-
machines. Two counters can be simulated using configurations of the form
$\textstyle{q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\star}$$\textstyle{\star}$$\textstyle{\star}$$\textstyle{\star}$$\textstyle{\star}$$\textstyle{\star}$$\textstyle{\star}$
where there are two level-$1$ nodes, one for each counter. The number of
children at level $2$ encodes the counter value. Zero tests can be implemented
by removing the corresponding level-$1$ node and creating a new one. This is
possible only when the node is a leaf, i.e., it does not have children at
level $2$. The state of the 2-counter machine can be maintained at level $0$,
the states at level $1$ indicate the name of the counter, and the level-$2$
states are irrelevant.
The translation from $1$-$\mathsf{LA}$ to VASS is straightforward and based on
representing $1$-$\mathsf{LA}$ configurations by the state at level $0$ and,
for each state at level $1$, the count of its occurrences. The reverse
translation is based on the same idea and extends the encoding of a non-
negative counter in Example 2, where the exponential blow up is simply due to
the fact that vector updates in VASS are given in binary whereas
$1$-$\mathsf{LA}$ transitions operate on single branches.∎
###### Lemma 2
$1$-$\mathsf{LA}$ equivalence is undecidable.
###### Proof
We provide a direct reduction from the halting problem for 2-counter machines,
where both counters are required to be zero initially as well as finally. The
main obstacle is that implementing zero tests as in the proof of the first
part of Lemma 1 is not available because we are restricted to leafy automata
with levels $0$ and $1$ only. To overcome it, we exploit the power of the
equivalence problem where one of the $1$-$\mathsf{LA}$ will have the task not
of correctly simulating zero tests but recognising zero tests that are
incorrect. The full argument can be found in Appendix 0.B.∎
## 5 Local leafy automata ($\mathsf{LLA}$)
Here we identify a restricted variant of $\mathsf{LA}$ for which the emptiness
problem is decidable. We start with a technical definition.
###### Definition 7
A $k$-$\mathsf{LA}$ is _bounded_ at level $i$ ($0\leq i\leq k$) if there is a
bound $b$ such that each node at level $i$ can create at most $b$ children
during a run. We refer to $b$ as the _branching bound_.
Note that we are defining a “global” bound on the number of children that a
node at level $i$ may create across a whole run, rather than a “local” bound
on the number of children a node may have in a given configuration.
To motivate the design of $\mathsf{LLA}$, we observe that the undecidability
argument (for the emptiness problem) for $2$-$\mathsf{LA}$ used two
consecutive levels ($0$ and $1$) that are not bounded. For the node at level
$0$, this corresponded to the number of zero tests, while an unbounded counter
is simulated at level $1$. In the following we will eliminate consecutive
unbounded levels by introducing an alternating pattern of bounded and
unbounded levels. Even-numbered layers ($i=0,2,...$) will be bounded, while
odd-numbered layers will be unbounded. Observe in particular that the root
(layer $0$) is bounded. As we will see later, this alternation reflects the
term/context distinction in game semantics: the levels corresponding to terms
are bounded, and the levels coresponding to contexts are unbounded.
With this restriction alone, it is possible to reconstruct the undecidability
argument for $4$-$\mathsf{LA}$, as two unbounded levels may still communicate.
Thus we introduce a restriction on how many levels a transition can read and
modify.
* •
when adding or removing a leaf at an odd level $2i+1$, the automaton will be
able to access levels $2i$, $2i-1$ and $2i-2$; while
* •
when adding or removing a leaf at an even level $2i$, the automaton will be
able to access levels $2i-1$ and $2i-2$.
In particular, when an odd level produces a leaf, it will not be able to see
the previous odd level. The above constraints mean that the transition
functions $\delta^{(i)}_{\mathsf{Q}},\delta^{(i)}_{\mathsf{Q}}$ can be
presented in a more concise form, given below.
$\delta^{(i)}_{\mathsf{Q}}\subseteq\begin{cases}Q^{(i-2,i-1)}\times\Sigma_{\mathsf{Q}}\times
Q^{(i-2,i-1,i)}&\text{if $i$ is even}\\\
Q^{(i-3,i-2,i-1)}\times\Sigma_{\mathsf{Q}}\times Q^{(i-3,i-2,i-1,i)}&\text{if
$i$ is odd}\end{cases}$
$\delta^{(i)}_{\mathsf{A}}\subseteq\begin{cases}Q^{(i-2,i-1,i)}\times\Sigma_{\mathsf{A}}\times
Q^{(i-2,i-1)}&\text{if $i$ is even}\\\
Q^{(i-3,i-2,i-1,i)}\times\Sigma_{\mathsf{A}}\times Q^{(i-3,i-2,i-1)}&\text{if
$i$ is odd}\end{cases}$
In terms of the previous notation developed for $\mathsf{LA}$,
$(q^{(i-2)},q^{(i-1)},x,r^{(i-2)},r^{(i-1)},$
$r^{(i)})\in\delta^{(i)}_{\mathsf{Q}}$ represents all tuples of the form
$(\vec{q},q^{(i-2)},q^{(i-1)},x,\vec{q},r^{(i-2)},r^{(i-1)},r^{(i)})$, where
$\vec{q}$ ranges over $Q^{(0,\cdots,i-3)}$.
###### Definition 8
A level-$k$ _local leafy automaton_ ($k$-$\mathsf{LLA}$) is a
$k$-$\mathsf{LA}$ whose transition function admits the above-mentioned
presentation and which is bounded at all even levels.
###### Theorem 5.1
The emptiness problem for $\mathsf{LLA}$ is decidable.
###### Proof (Sketch)
Let $b$ be a bound on the number of children created by each even node during
a run.
The critical observation is that, once a node $d$ at even level $2i$ has been
created, all subsequent actions of descendants of $d$ access (read and/or
write) the states at levels $2i-1$ and $2i-2$ at most $2b$ times. The shape of
the transition function dictates that this can happen only when child nodes at
level $2i+1$ are added or removed. In addition, the locality property ensures
that the automaton will never access levels $<2i-2$ at the same time as node
$d$ or its descendants.
We will make use of these facts to construct _summaries_ for nodes on even
levels which completely describe such a node’s lifetime, from its creation as
a leaf until its removal, and in between performing at most $2b$ reads-writes
of the parent and grandparent states. A summary is a sequence quadruples of
states: two pairs of states of levels $2i-2$ and $2i-1$. The first pair are
the states we expect to find on these levels, while the second are the states
to which we update these levels. Hence a summary at level $2i$ is a complete
record of a valid sequence of read-writes and stateful changes during the
lifetime of a node on level $2i$.
We proceed by induction and show how to calculate the complete set of
summaries at level $2i$ given the complete set of summaries at level $2i+2$.
We construct a program for deciding whether a given sequence is a summary at
level $2i$. This program can be evaluated via Vector Addition Systems with
States (VASS). Since we can finitely enumerate all candidate summaries at
level $2i$, this gives us a way to compute summaries at level $2i$. Proceeding
this way, we finally calculate summaries at level $2$. At this stage, we can
reduce the emptiness problem for the given $\mathsf{LLA}$ to a reachability
test on a VASS.
The complete argument is given in Appendix 0.C. ∎
Let us remark also that the problem becomes undecidable if we remove either
boundedness restriction, or allow transitions to look one level further.
## 6 From FICA to LA
Recall from Section 3 that, to interpret base types, game semantics uses moves
from the set
$\begin{array}[]{rcl}{\mathcal{M}}&=&M_{{\llbracket}{{\bf
com}}{\rrbracket}}\cup M_{{\llbracket}{{\bf exp}}{\rrbracket}}\cup
M_{{\llbracket}{{\bf var}}{\rrbracket}}\cup M_{{\llbracket}{{\bf
sem}}{\rrbracket}}\\\
&=&\\{\,\mathsf{run},\,\mathsf{done},\,\mathsf{q},\,\mathsf{read},\,\mathsf{grb},\,\mathsf{rls},\,\mathsf{ok}\,\\}\cup\\{\,i,\,\mathsf{write}(i){}\,|\,0\leq
i\leq\max\,\\}.\end{array}$
The game semantic interpretation of a term-in-context
${\Gamma}\vdash{M:\theta}$ is a strategy over the arena
${\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}$, which is obtained through
product and arrow constructions, starting from arenas corresponding to base
types. As both constructions rely on the disjoint sum, the moves from
${\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}$ are derived from the base
types present in types inside $\Gamma$ and $\theta$. To indicate the exact
occurrence of a base type from which each move originates, we will annotate
elements of ${\mathcal{M}}$ with a specially crafted scheme of superscripts.
Suppose $\Gamma=\\{x_{1}:\theta_{1},\cdots,x_{l}:\theta_{l}\\}$. The
superscripts will have one of the two forms, where
$\vec{i}\in\mathbb{N}^{\ast}$ and $\rho\in\mathbb{N}$:
* •
$(\vec{i},\rho)$ will be used to represent moves from $\theta$;
* •
$(x_{v}\vec{i},\rho)$ will be used to represent moves from $\theta_{v}$
($1\leq v\leq l$).
The annotated moves will be written as $m^{(\vec{i},\rho)}$ or
$m^{(x_{v}\vec{i},\rho)}$, where $m\in{\mathcal{M}}$. We will sometimes omit
$\rho$ on the understanding that this represents $\rho=0$. Similarly, when
$\vec{i}$ is omitted, the intended value is $\epsilon$. Thus, $m$ stands for
$m^{(\epsilon,0)}$.
The next definition explains how the $\vec{i}$ superscripts are linked to
moves from ${\llbracket}{\theta}{\rrbracket}$. Given
$X\subseteq\\{m^{(\vec{i},\rho)}\,|\,\vec{i}\in\mathbb{N}^{\ast},\,\rho\in\mathbb{N}\\}$
and $y\in\mathbb{N}\cup\\{x_{1},\cdots,x_{l}\\}$, we let
$yX=\\{m^{(y\vec{i},\rho)}\,|\,m^{(\vec{i},\rho)}\in X\\}$.
###### Definition 9
Given a type $\theta$, the corresponding alphabet $\mathcal{T}_{\theta}$ is
defined as follows
$\begin{array}[]{rcl}\mathcal{T}_{\beta}&=&\\{\,m^{(\epsilon,\rho)}\,|\,m\in
M_{{\llbracket}{\beta}{\rrbracket}},\,\rho\in\mathbb{N}\,\\}\qquad\beta={\bf
com},{\bf exp},{\bf var},{\bf sem}\\\
\mathcal{T}_{\theta_{h}\rightarrow\ldots\rightarrow\theta_{1}\rightarrow\beta}&=&\bigcup_{u=1}^{h}(u\mathcal{T}_{\theta_{u}})\cup\mathcal{T}_{\beta}\end{array}$
For $\Gamma=\\{x_{1}:\theta_{1},\cdots,x_{l}:\theta_{l}\\}$, the alphabet
$\mathcal{T}_{{\Gamma}\vdash{\theta}}$ is defined to be
$\mathcal{T}_{{\Gamma}\vdash{\theta}}=\bigcup_{v=1}^{l}(x_{v}\mathcal{T}_{\theta_{v}})\cup\mathcal{T}_{\theta}$.
###### Example 3
The alphabet $\mathcal{T}_{{f:{\bf com}\rightarrow{\bf com},x:{\bf
com}}\vdash{{\bf com}}}$ is
$\\{\mathsf{run}^{(f1,\rho)},\mathsf{done}^{(f1,\rho)},\mathsf{run}^{(f,\rho)},\mathsf{done}^{(f,\rho)},\mathsf{run}^{(x,\rho)},\mathsf{done}^{(x,\rho)},\mathsf{run}^{(\epsilon,\rho)},\mathsf{done}^{(\epsilon,\rho)}\,|\,\rho\in\mathbb{N}\\}.$
To represent the game semantics of terms-in-context, of the form
${\Gamma}\vdash{M:\theta}$, we are going to use _finite subsets_ of
$\mathcal{T}_{{\Gamma}\vdash{\theta}}$ as alphabets in leafy automata. The
subsets will be finite, because $\rho$ will be bounded. Note that
$\mathcal{T}_{\theta}$ admits a natural partitioning into questions and
answers, depending on whether the underlying move is a question or answer.
We will represent plays using data words in which the underpinning sequence of
tags will come from an alphabet as defined above. Superscripts and data are
used to represent justification pointers. Intuitively, we represent
occurrences of questions with data values. Pointers from answers to questions
just refer to these values. Pointers from questions use bounded indexing with
the help of $\rho$.
Initial question-moves do not have a pointer and to represent such questions
we simply use $\rho=0$. For non-initial questions, we rely on the tree
structure of $\mathcal{D}$ and use $\rho$ to indicate the ancestor of the
currently read data value that we mean to point at. Consider a trace
$w(t_{i},d_{i})$ ending in a non-initial question, where $d_{i}$ is a
level-$i$ data value and $i>0$. In our case, we will have
$t_{i}\in\mathcal{T}_{{\Gamma}\vdash{\theta}}$, i.e.
$t_{i}=m^{(\cdots,\rho)}$. By Remark 2, trace $w$ contains unique occurrences
of questions $(t_{0},d_{0}),\cdots,(t_{i-1},d_{i-1})$ such that
$\mathit{pred}(d_{j})=d_{j-1}$ for $j=1,\cdots,i$. The pointer from
$(t_{i},d_{i})$ goes to one of these questions, and we use $\rho$ to represent
the scenario in which the pointer goes to $(t_{i-(1+\rho)},d_{i-(1+\rho)})$.
Pointers from answer-moves to question-moves are represented simply by using
the same data value in both moves (in this case we use $\rho=0$).
We will also use $\epsilon$-tags $\epsilon_{\mathsf{Q}}$ (question) and
$\epsilon_{\mathsf{A}}$ (answer), which do not contribute moves to the
represented play. Each $\epsilon_{\mathsf{Q}}$ will always be answered with
$\epsilon_{\mathsf{A}}$. Note that the use of
$\rho,\epsilon_{\mathsf{Q}},\epsilon_{\mathsf{A}}$ means that several data
words may represent the same play (see Examples 4, 6).
###### Example 4
Suppose that
$d_{0}=\mathit{pred}(d_{1}),d_{1}=\mathit{pred}(d_{2})=\mathit{pred}(d_{2}^{\prime}),d_{2}=\mathit{pred}(d_{3})$,
$d_{2}^{\prime}=\mathit{pred}(d_{3}^{\prime})$. Then the data word
$(\mathsf{run},d_{0})$ $(\mathsf{run}^{f},d_{1})$ $(\mathsf{run}^{f1},d_{2})$
$(\mathsf{run}^{f1},d_{2}^{\prime})$ $(\mathsf{run}^{(x,2)},d_{3})$
$(\mathsf{run}^{(x,2)},d_{3}^{\prime})$ $(\mathsf{done}^{x},d_{3})$, which is
short for $(\mathsf{run}^{(\epsilon,0)},d_{0})$ $(\mathsf{run}^{(f,0)},d_{1})$
$(\mathsf{run}^{(f1,0)},d_{2})$ $(\mathsf{run}^{(f1,0)},d_{2}^{\prime})$
$(\mathsf{run}^{(x,2)},d_{3})$ $(\mathsf{run}^{(x,2)},d_{3}^{\prime})$
$(\mathsf{done}^{(x,0)},d_{3})$, represents the play
$\begin{array}[]{ccccccc}\rnode{Z}{\mathsf{run}}&\rnode{A}{\mathsf{run}^{f}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{A}{Z}}&\rnode{B}{\mathsf{run}^{f1}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}&\rnode{C}{\mathsf{run}^{f1}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=140.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{C}{A}}&\rnode{D}{\mathsf{run}^{x}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=150.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{D}{Z}}&\rnode{E}{\mathsf{run}^{x}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=150.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{E}{Z}}&\rnode{F}{\mathsf{done}^{x}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=140.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{F}{D}}\\\
O&P&O&O&P&P&O.\end{array}$
###### Example 5
Consider the $\mathsf{LA}$ $\mathcal{A}=\langle Q,3,\Sigma,\delta\rangle$,
where $Q^{(0)}=\\{0,1,2\\}$, $Q^{(1)}=\\{0\\}$, $Q^{(2)}=\\{0,1,2\\}$,
$Q^{(3)}=\\{0\\}$,
$\Sigma_{\mathsf{Q}}=\\{\mathsf{run},\mathsf{run}^{f},\mathsf{run}^{f1},\mathsf{run}^{(x,2)}\\}$,
$\Sigma_{\mathsf{A}}=\\{\mathsf{done},\mathsf{done}^{f},$
$\mathsf{done}^{f1},\mathsf{done}^{x}\\}$, and $\delta$ is given by
$\begin{array}[]{c}\dagger{\xlongrightarrow{\mathsf{run}}}0\qquad
0{\xlongrightarrow{\mathsf{run}^{f}}}{(1,0)}\qquad(1,0){\xlongrightarrow{\mathsf{done}^{f}}}2\qquad
2{\xlongrightarrow{\mathsf{done}}}\dagger\qquad(1,0){\xlongrightarrow{\mathsf{run}^{f1}}}(1,0,0)\\\
(1,0,0){\xlongrightarrow{\mathsf{run}^{(x,2)}}}(1,0,1,0)\qquad(1,0,1,0){\xlongrightarrow{\mathsf{done}^{(x,0)}}}(1,0,2)\qquad(1,0,2){\xlongrightarrow{\mathsf{done}^{f1}}}(1,0)\end{array}$
Then traces from $\mathit{Tr}(\mathcal{A})$ represent all plays from
$\sigma=\llbracket f:{\bf com}\rightarrow{\bf com},\,x:{\bf
com}\,\vdash\,fx\rrbracket$, including the play from Example 4, and
$\mathit{L}(\mathcal{A})$ represents $\textsf{comp}(\sigma)$.
###### Example 6
One might wish to represent plays of $\sigma$ from the previous Example using
data values
$d_{0},d_{1},d_{1}^{\prime},d_{1}^{\prime\prime},d_{2},d_{2}^{\prime}$ such
that
$d_{0}=\mathit{pred}(d_{1})=\mathit{pred}(d_{1}^{\prime})=\mathit{pred}(d_{1}^{\prime\prime})$,
$d_{1}=\mathit{pred}(d_{2})=\mathit{pred}(d_{2}^{\prime})$, so that the play
from Example 4 is represented by $(\mathsf{run}^{(\epsilon,0)},d_{0})$
$(\mathsf{run}^{(f,0)},d_{1})$ $(\mathsf{run}^{(f1,0)},d_{2})$
$(\mathsf{run}^{(f1,0)},d_{2}^{\prime})$
$(\mathsf{run}^{(x,0)},d_{1}^{\prime})$
$(\mathsf{run}^{(x,0)},d_{1}^{\prime\prime})$
$(\mathsf{done}^{(x,0)},d_{1}^{\prime})$. Unfortunately, it is impossible to
construct a $2$-$\mathsf{LA}$ that would accept all representations of such
plays. To achieve this, the automaton would have to make sure that the number
of $\mathsf{run}^{f1}$s is the same as that of $\mathsf{run}^{x}$s. Because
the former are labelled with level-$2$ values and the latter with incomparable
level-$1$ values, the only point of communication (that could be used for
comparison) is the root. However, the root cannot accommodate unbounded
information, while plays of $\sigma$ can feature an unbounded number of
$\mathsf{run}^{f1}$s, which could well be consecutive.
Before we state the main result linking $\mathsf{FICA}$ with leafy automata,
we note some structural properties of the automata. Questions will create a
leaf, and answers will remove a leaf. P-moves add leaves at odd levels
(questions) and remove leaves at even levels (answers), while O-moves have the
opposite effect at each level. Finally, when removing nodes at even levels we
will not need to check if a node is a leaf. We call the last property _even-
readiness_.
Even-readiness is a consequence of the WAIT condition in the game semantics.
The condition captures well-nestedness of concurrent interactions – a term can
terminate only after subterms terminate. In the leafy automata setting, this
is captured by the requirement that only leaf nodes can be removed, i.e. a
node can be removed only if all of its children have been removed beforehand.
It turns out that, for _P-answers_ only, this property will come for free.
Formally, whenever the automaton arrives at a configuration $\kappa=(D,E,f)$,
where $d\in E$ and there is a transition
$(f(\mathit{pred}^{(2i)}(d)),\cdots,f(\mathit{pred}(d)),f(d),t,f^{\prime}(\mathit{pred}^{(2i)}(d)),\cdots,f^{\prime}(\mathit{pred}(d)))\in\delta^{(2i)}_{\mathsf{A}},$
then $d$ is a leaf. In contrast, our automata will not satisfy the same
property for O-answers (the environment) and for such transitions it is
crucial that the automaton actually checks that only leaves can be removed.
###### Theorem 6.1
For any $\mathsf{FICA}$-term ${\Gamma}\vdash{M:\theta}$, there exists an even-
ready leafy automaton $\mathcal{A}_{M}$ over a finite subset of
$\mathcal{T}_{{\Gamma}\vdash{\theta}}+\\{\epsilon_{\mathsf{Q}},\epsilon_{\mathsf{A}}\\}$
such that the set of plays represented by data words from
$\mathit{Tr}(\mathcal{A}_{M})$ is exactly
${\llbracket}{{\Gamma}\vdash{M:\theta}}{\rrbracket}$. Moreover,
$\mathit{L}(\mathcal{A}_{M})$ represents
$\textsf{comp}({\llbracket}{{\Gamma}\vdash{M:\theta}}{\rrbracket})$ in the
same sense.
###### Proof (Sketch)
Because every $\mathsf{FICA}$-term can be converted to $\beta\eta$-normal
form, we use induction on the structure of such normal forms. The base cases
are: ${\Gamma}\vdash{{\bf skip}:{\bf com}}$ ($Q^{(0)}=\\{0\\}$;
$\dagger{\xlongrightarrow{\mathsf{run}}}0$,
$0{\xlongrightarrow{\mathsf{done}}}\dagger$), ${\Gamma}\vdash{{\bf div}:{\bf
com}}$ ($Q^{(0)}=\\{0\\}$; $\dagger{\xlongrightarrow{\mathsf{run}}}0$), and
${\Gamma}\vdash{i:{\bf exp}}$ ($Q^{(0)}=\\{0\\}$;
$\dagger{\xlongrightarrow{\mathsf{q}}}0$, $0{\xlongrightarrow{i}}\dagger$).
The remaining cases are inductive. When referring to the inductive hypothesis
for a subterm $M_{i}$, we shall use subscripts $i$ to refer to the automata
components, e.g. $Q_{i}^{(j)}$, ${\xlongrightarrow{\mathsf{m}}}_{i}$ etc. In
contrast, $Q^{(j)}$, ${\xlongrightarrow{\mathsf{m}}}$ will refer to the
automaton that is being constructed. Inference lines $\frac{\qquad}{\qquad}$
will indicate that the transitions listed under the line should be added to
the new automaton provided the transitions listed above the line are present
in the automaton obtained via induction hypothesis. We discuss a selection of
technical cases below.
#### ${\Gamma}\vdash{M_{1}||M_{2}}$
In this case we need to run the automata for $M_{1}$ and $M_{2}$ concurrently.
To this end, their level-$0$ states will be combined
($Q^{(0)}=Q_{1}^{(0)}\times Q_{2}^{(0)}$), but not deeper states
($Q^{(j)}=Q_{1}^{(j)}+Q_{2}^{(j)},1\leq j\leq k$). The first group of
transitions activate and terminate the two components respectively:
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1}q_{1}^{(0)}\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}(q_{1}^{(0)},q_{2}^{(0)})}$,
$\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger\qquad
q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger}{(q_{1}^{(0)},q_{2}^{(0)}){\xlongrightarrow{\mathsf{done}}}\dagger}$.
The remaining transitions advance each component:
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
q_{2}^{(0)}\in
Q_{2}^{(0)}}{((q_{1}^{(0)},q_{2}^{(0)}),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},q_{2}^{(0)}),\cdots,r_{1}^{(j^{\prime})})}$,
$\frac{q_{1}^{(0)}\in
Q_{1}^{(0)}\qquad(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}{((q_{1}^{(0)},q_{2}^{(0)}),\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}((q_{1}^{(0)},r_{2}^{(0)}),\cdots,r_{2}^{(j^{\prime})})}$,
where $\mathsf{m}\neq\mathsf{run},\mathsf{done}$.
#### ${\Gamma}\vdash{{\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M_{1}}$
By [23], the semantics of this term is obtained from the semantics of
${\llbracket}{{\Gamma,x}\vdash{M_{1}}}{\rrbracket}$ by
1. 1.
restricting to plays in which the moves $\mathsf{read}^{x}$,
$\mathsf{write}(n)^{x}$ are followed immediately by answers,
2. 2.
selecting those plays in which each answer to a $\mathsf{read}^{x}$-move is
consistent with the preceding $\mathsf{write}(n)^{x}$-move (or equal to $i$,
if no $\mathsf{write}(n)^{x}$ was made),
3. 3.
erasing all moves related to $x$, e.g. those of the form $m^{(x,\rho)}$.
To implement 1., we will lock the automaton after each $\mathsf{read}^{x}$\-
or $\mathsf{write}(n)^{x}$-move, so that only an answer to that move can be
played next. Technically, this will be done by adding an extra bit (lock) to
the level-$0$ state. To deal with 2., we keep track of the current value of
$x$, also at level $0$. This makes it possible to ensure that answers to
$\mathsf{read}^{x}$ are consistent with the stored value and that
$\mathsf{write}(n)^{x}$ transitions cause the right change. Erasing from
condition 3 is implemented by replacing all moves with the $x$ subscript with
$\epsilon_{\mathsf{Q}},\epsilon_{\mathsf{A}}$-tags.
Accordingly, we have
$Q^{(0)}=(Q_{1}^{(0)}+(Q_{1}^{(0)}\times\\{\mathit{lock}\\}))\times\\{0,\cdots,\mathit{max}\\}$
and $Q^{(j)}=Q_{1}^{(j)}$ ($1\leq j\leq k$). As an example of a transition, we
give the transition related to writing:
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{write}(z)^{(x,\rho)}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
0\leq
n,z\leq\mathit{max}}{((q_{1}^{(0)},n),\cdots,q_{1}^{(j)}){\xlongrightarrow{\epsilon_{\mathsf{Q}}}}((r_{1}^{(0)},\mathit{lock},z),\cdots,r_{1}^{(j^{\prime})})}$.
#### ${\Gamma}\vdash{fM_{h}\cdots M_{1}:{\bf com}}$ with
$(f:\theta_{h}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow{\bf com})$
Here we will need $Q^{(0)}=\\{0,1,2\\}$, $Q^{(1)}=\\{0\\}$,
$Q^{(j+2)}=\sum_{u=1}^{h}Q_{u}^{(j)}$ ($0\leq j\leq k$). The first group of
transitions corresponding to calling and returning from $f$:
$\dagger{\xlongrightarrow{\mathsf{run}}}0$,
$0{\xlongrightarrow{\mathsf{run}^{f}}}(1,0)$,
$(1,0){\xlongrightarrow{\mathsf{done}^{f}}}2$,
$2{\xlongrightarrow{\mathsf{done}}}\dagger$. Additionally, in state $(1,0)$ we
want to enable the environment to spawn an unbounded number of copies of each
of ${\Gamma}\vdash{M_{u}:\theta_{u}}$ ($1\leq u\leq h$). This is done through
rules that embed the actions of the automata for $M_{u}$ while (possibly)
relabelling the moves in line with our convention for representing moves from
game semantics. Such transitions have the general form
$\frac{(q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(t,\rho)}}}_{u}(q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}{(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(t^{\prime},\rho^{\prime})}}}(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}$.
Note that this case also covers $f:{\bf com}$ ($h=0$).
More details and the remaining cases are covered in Appendix 0.D. In Appendix
0.D.2 we give an example of a term and the corresponding $\mathsf{LA}$. ∎
## 7 Local $\mathsf{FICA}$
In this section we identify a family of $\mathsf{FICA}$ terms that can be
translated into $\mathsf{LLA}$ rather than $\mathsf{LA}$. To achieve
boundedness at even levels, we remove $\mathsf{while}$111The automaton for
${\bf while}\,M\,{\bf do}\,N$ may repeatedly visit the automata for $M$ and
$N$, generating an unbounded number of children at level $0$ in the process..
To achieve restricted communication, we will constrain the distance between a
variable declaration and its use. Note that in the translation, the
application of function-type variables increases $\mathsf{LA}$ depth. So in
$\mathsf{LFICA}$ we will allow the link between the binder
$\mathbf{newvar}/\mathbf{newsem}\,x$ and each use of $x$ to “cross” at most
one occurrence of a free variable. For example, the following terms
* •
${\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf
in}\,x\,\raisebox{0.27986pt}{:}{=}\,1\,||\,f(x\,\raisebox{0.27986pt}{:}{=}\,2)$,
* •
${\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,f({\bf
newvar}\,y\,{\bf in}\,f(y\,\raisebox{0.27986pt}{:}{=}\,1)\,||\,x:=!y)$
will be allowed, but not ${\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf
in}\,f(f(x\,\raisebox{0.27986pt}{:}{=}\,1))$.
To define the fragment formally, given a term $Q$ in $\beta\eta$-normal form,
we use a notion of the _applicative depth of a variable $x:\beta$ ($\beta={\bf
var},{\bf sem}$) inside $Q$_, written $\mathit{ad}_{x}(Q)$ and defined
inductively by the table below. The applicative depth is increased whenever a
functional identifier is applied to a term containing $x$.
$\begin{array}[]{lcl}\textrm{shape of $Q$}&&\mathit{ad}_{x}(Q)\\\ \hline\cr
x&&1\\\ y\,(y\neq x),\,{\bf skip},\,{\bf div},\,i&&0\\\
\mathbf{op}(M),\,!M,\,{\bf release}(M),\,{\bf grab}(M)&&\mathit{ad}_{x}(M)\\\
M;N,\,M||N,\,M\,\raisebox{0.27986pt}{:}{=}\,N,\,{\bf while}\,M\,{\bf
do}\,N&&\max(\mathit{ad}_{x}(M),\mathit{ad}_{x}(N))\\\ {{\bf if}\,M\,{\bf
then}\,N_{1}\,{\bf
else}\,N_{2}}&&\max(\mathit{ad}_{x}(M),\mathit{ad}_{x}(N_{1}),\mathit{ad}_{x}(N_{2}))\\\
{\lambda y.M},{\bf
newvar}\,\textbf{/newsem}\,y\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M&&\mathit{ad}_{x}(M[z/y]),\textrm{where $z$ is fresh}\\\ {fM_{1}\cdots
M_{k}}&&1+\max(\mathit{ad}_{x}(M_{1}),\cdots,\mathit{ad}_{x}(M_{k}))\end{array}$
Note that in our examples above, in the first two cases the applicative depth
of $x$ is $2$; and in the third case it is $3$.
###### Definition 10 (Local $\mathsf{FICA}$)
A $\mathsf{FICA}$-term ${\Gamma}\vdash{M:\theta}$ is _local_ if its
$\beta\eta$-normal form does not contain any occurrences of $\mathbf{while}$
and, for every subterm of the normal form of the shape ${\bf
newvar}\,/\mathbf{newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,N$, we
have $\mathit{ad}_{x}(N)\leq 2$. We write $\mathsf{LFICA}$ for the set of
local $\mathsf{FICA}$ terms.
###### Theorem 7.1
For any $\mathsf{LFICA}$-term ${\Gamma}\vdash{M:\theta}$, the automaton
$\mathcal{A}_{M}$ obtained from the translation in Theorem 6.1 can be
presented as a $\mathsf{LLA}$.
###### Proof (Sketch)
We argue by induction that the constructions from Theorem 6.1 preserve
presentability as a $\mathsf{LLA}$.
The case of parallel composition involves running copies of $M_{1}$ and
$M_{2}$ in parallel without communication, with their root states stored as a
pair at level $0$. Note, though, that each of the automata transitions
independently of the state of the other automaton. In consequence, if the
automata $M_{1}$ and $M_{2}$ are $\mathsf{LLA}$, so will be the automaton for
$M_{1}||M_{2}$. The branching bound after the construction is the sum of the
two bounds for $M_{1}$ and $M_{2}$.
For ${\Gamma}\vdash{{\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M}$, because the term is in $\mathsf{LFICA}$, so is ${\Gamma,x:{\bf
var}}\vdash{M}$ and we have $\mathit{ad}_{x}(M)\leq 2$. Then we observe that
in the translation of Theorem 6.1 (${\Gamma,x:{\bf var}}\vdash{M:\theta}$) the
questions related to $x$, (namely $\mathsf{write}(i)^{(x,\rho)}$ and
$\mathsf{read}^{(x,\rho)}$) correspond to creating leaves at levels $1$ or
$3$, while the corresponding answers ($\mathsf{ok}^{(x,\rho)}$ and
$i^{(x,\rho)}$ respectively) correspond to removing such leaves. In the
construction for ${\Gamma}\vdash{{\bf newvar}\,x\,{\bf in}\,M}$, such
transitions need access to the root (to read/update the current state) and the
root is indeed within the allowable range: in an $\mathsf{LLA}$ transitions
creating/destroying leaves at level $3$ can read/write at level $0$. All other
transitions (not labelled by $x$) proceed as in $M$ and need not consult the
root for additional information about the current state, as it is propagated.
Consequently, if $M$ is represented by a $\mathsf{LLA}$ then the
interpretation of ${\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M$ is also a $\mathsf{LLA}$. The construction does not affect the
branching bound, because the resultant runs can be viewed as a subset of runs
of the automaton for $M$, i.e. those in which reads and writes are related.
For $fM_{h}\cdots M_{1}$, we observe that the construction first creates two
nodes at levels $0$ and $1$, and the node at level $1$ is used to run an
unbounded number of copies of (the automaton for) $M_{i}$. The copies do not
need access to the states stored at levels $0$ and $1$, because they are never
modified when the copies are running. Consequently, if each $M_{i}$ can be
translated into a $\mathsf{LLA}$, the outcome of the construction in Theorem
6.1 is also a $\mathsf{LLA}$. The new branching bound is the maximum over
bounds from $M_{1},\cdots,M_{h}$, because at even levels children are produced
as in $M_{i}$ and level $0$ produces only $1$ child. ∎
###### Corollary 1
For any $\mathsf{LFICA}$-term ${\Gamma}\vdash{M:\theta}$, the problem of
determining whether
$\textsf{comp}({\llbracket}{{\Gamma}\vdash{M}}{\rrbracket})$ is empty is
decidable.
Theorems 3.1 and 5.1 imply the above. Thanks to Theorem 3.1, it is decidable
if a $\mathsf{LFICA}$ term is equivalent to a term that always diverges (cf.
example on page 2). In case of inequivalence, our results could also be
applied to extract the distinguishing context, first by extracting the
witnessing trace from the argument underpinning Theorem 5.1 and then feeding
it to the Definability Theorem (Theorem 41 [23]). This is a valuable property
given that in the concurrent setting bugs are difficult to replicate.
## 8 From LA to FICA
In this section, we show how to represent leafy automata in $\mathsf{FICA}$.
Let $\mathcal{A}=\langle\Sigma,k,Q,\delta\rangle$ be a leafy automaton. We
shall assume that $\Sigma,Q\subseteq\\{0,\cdots,\mathit{max}\\}$ so that we
can encode the alphabet and states using type ${\bf exp}$. We will represent a
trace $w$ generated by $\mathcal{A}$ by a play $\mathsf{play}(w)$, which
simulates each transition with two moves, by $O$ and $P$ respectively. The
child-parent links in $\mathcal{D}$ will be represented by justification
pointers. We refer the reader to Appendix 0.F for details. Below we just state
the lemma that identifies the types that correspond to our encoding, where we
write $\theta^{\mathit{max}+1}\rightarrow\beta$ for
$\underbrace{\theta\rightarrow\cdots\rightarrow\theta}_{\mathit{max}+1}\rightarrow\beta$.
###### Lemma 3
Let $\mathcal{A}$ be a $k$-$\mathsf{LA}$ and $w\in\mathit{Tr}(\mathcal{A})$.
Then $\mathsf{play}(w)$ is a play in ${\llbracket}{\theta_{k}}{\rrbracket}$,
where $\theta_{0}={\bf com}^{\mathit{max}+1}\rightarrow{\bf exp}$ and
$\theta_{i+1}=(\theta_{i}\rightarrow{\bf com})^{\mathit{max}+1}\rightarrow{\bf
exp}$ ($i\geq 0$).
Before we state the main result, we recall from [23] that strategies
corresponding to $\mathsf{FICA}$ terms satisfy a closure condition known as
_saturation_ : swapping two adjacent moves in a play belonging to such a
strategy yields another play from the same strategy, as long as the swap
yields a play and it is not the case that the first move is by O and the
second one by P. Thus, saturated strategies express causal dependencies of
P-moves on O-moves. Consequently, one cannot expect to find a
$\mathsf{FICA}$-term such that the corresponding strategy is the smallest
strategy containing
$\\{\,\mathsf{play}(w)\,|\,w\in\mathit{Tr}(\mathcal{A})\,\\}$. Instead, the
best one can aim for is the following result.
###### Theorem 8.1
Given a $k$-$\mathsf{LA}$ $\mathcal{A}$, there exists a $\mathsf{FICA}$ term
${}\vdash{M_{\mathcal{A}}:\theta_{k}}$ such that
${\llbracket}{{}\vdash{M_{\mathcal{A}}:\theta_{k}}}{\rrbracket}$ is the
smallest saturated strategy containing
$\\{\,\mathsf{play}(w)\,|\,w\in\mathit{Tr}(\mathcal{A})\,\\}$.
###### Proof (Sketch)
Our assumption $Q\subseteq\\{0,\cdots,\mathit{max}\\}$ allows us to maintain
$\mathcal{A}$-states in the memory of $\mathsf{FICA}$-terms. To achieve
$k$-fold nesting, we rely on the higher-order structure of the term: $\lambda
f^{(0)}.f^{(0)}(\lambda f^{(1)}.f^{(1)}(\lambda f^{(2)}.f^{(2)}(\cdots\lambda
f^{(k)}.f^{(k)})))$. In fact, instead of the single variables $f^{(i)}$, we
shall use sequences $f^{(i)}_{0}\cdots f^{(i)}_{\mathit{max}}$, so that a
question $t_{\mathsf{Q}}^{(i)}$ read by $\mathcal{A}$ at level $i$ can be
simulated by using variable $f^{(i)}_{t_{\mathsf{Q}}^{(i)}}$ (using our
assumption $\Sigma\subseteq\\{0,\cdots,\mathit{max}\\}$). Additionally, the
term contains state-manipulating code that enables moves only if they are
consistent with the transition function of $\mathcal{A}$.∎
## 9 Conclusion and further work
We have introduced leafy automata, $\mathsf{LA}$, and shown that they
correspond to the game semantics of Finitary Idealized Concurrent Algol
($\mathsf{FICA}$). The automata formulation makes combinatorial challenges
posed by the equivalence problem explicit. This is exemplified by a very
transparent undecidability proof of the emptiness problem for $\mathsf{LA}$.
Our hope is that $\mathsf{LA}$ will allow to discover interesting fragments of
$\mathsf{FICA}$ for which some variant of the equivalence problem is
decidable. We have identified one such instance, namely local leafy automata
($\mathsf{LLA}$), and a fragment of $\mathsf{FICA}$ that can be translated to
them. The decidability of the emptiness problem for $\mathsf{LLA}$ implies
decidability of a simple instance of the equivalence problem. This in turn
allows to decide some verification questions as in the example on page 2.
Since these types of questions involve quantification over all contexts, the
use of a fully-abstract semantics appears essential to solve them.
The obvious line of future work is to find some other subclasses of
$\mathsf{LA}$ with decidable emptiness problem. Another interesting target is
to find an automaton model for the call-by-value setting, where answers enable
questions [2, 26]. It would also be worth comparing our results with abstract
machines [20], the Geometry of Interaction [32], and the $\pi$-calculus [7].
## References
* [1] Abramsky, S., Ghica, D.R., Murawski, A.S., Ong, C.H.L.: Applying game semantics to compositional software modelling and verification. In: Proceedings of TACAS, Lecture Notes in Computer Science, vol. 2988, pp. 421–435. Springer-Verlag (2004)
* [2] Abramsky, S., McCusker, G.: Call-by-value games. In: Proceedings of CSL. Lecture Notes in Computer Science, vol. 1414, pp. 1–17. Springer-Verlag (1997)
* [3] Abramsky, S., McCusker, G.: Linearity, sharing and state: a fully abstract game semantics for Idealized Algol with active expressions. In: O’Hearn, P.W., Tennent, R.D. (eds.) Algol-like languages, pp. 297–329. Birkhaüser (1997)
* [4] Abramsky, S., McCusker, G.: Game semantics. In: Schwichtenberg, H., Berger, U. (eds.) Logic and Computation. Springer-Verlag (1998), proceedings of the NATO Advanced Study Institute, Marktoberdorf
* [5] Aiswarya, C., Gastin, P., Kumar, K.N.: Verifying communicating multi-pushdown systems via split-width. In: Automated Technology for Verification and Analysis - 12th International Symposium, ATVA 2014. Lecture Notes in Computer Science, vol. 8837, pp. 1–17. Springer (2014)
* [6] Bakewell, A., Ghica, D.R.: On-the-fly techniques for games-based software model checking. In: Proceedings of TACAS, Lecture Notes in Computer Science, vol. 4963, pp. 78–92. Springer (2008)
* [7] Berger, M., Honda, K., Yoshida, N.: Sequentiality and the pi-calculus. In: Proceedings of TLCA, Lecture Notes in Computer Science, vol. 2044, pp. 29–45. Springer-Verlag (2001)
* [8] Björklund, H., Bojańczyk, M.: Shuffle expressions and words with nested data. In: Proceedings of MFCS. Lecture Notes in Computer Science, vol. 4708, pp. 750–761 (2007)
* [9] Björklund, H., Schwentick, T.: On notions of regularity for data languages. Theor. Comput. Sci. 411(4-5), 702–715 (2010)
* [10] Bojańczyk, M., David, C., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data words. ACM Trans. Comput. Log. 12(4), 27:1–27:26 (2011)
* [11] Castellan, S., Clairambault, P., Rideau, S., Winskel, G.: Games and strategies as event structures. Logical Methods in Computer Science 13(3) (2017)
* [12] Cotton-Barratt, C., Hopkins, D., Murawski, A.S., Ong, C.L.: Fragments of ML decidable by nested data class memory automata. In: Proceedings of FOSSACS. Lecture Notes in Computer Science, vol. 9034, pp. 249–263. Springer (2015)
* [13] Cotton-Barratt, C., Murawski, A.S., Ong, C.L.: ML, visibly pushdown class memory automata, and extended branching vector addition systems with states. ACM Trans. Program. Lang. Syst. 41(2), 11:1–11:38 (2019)
* [14] Cotton-Barratt, C., Murawski, A.S., Ong, C.L.: Weak and nested class memory automata. In: Proceedings of LATA. LNCS, vol. 8977, pp. 188–199. Springer (2015)
* [15] Czerwiński, W., Lasota, S., Lazic, R., Leroux, J., Mazowiecki, F.: The reachability problem for Petri nets is not elementary. In: Proceedings of STOC. pp. 24–33. ACM (2019)
* [16] Decker, N., Habermehl, P., Leucker, M., Thoma, D.: Ordered navigation on multi-attributed data words. In: Proceedings of CONCUR. LNCS, vol. 8704, pp. 497–511. Springer (2014)
* [17] Dimovski, A., Ghica, D.R., Lazic, R.: A counterexample-guided refinement tool for open procedural programs. In: Proceedings of SPIN. Lecture Notes in Computer Science, vol. 3925, pp. 288–292. Springer-Verlag (2006)
* [18] Dimovski, A.S.: Symbolic game semantics for model checking program families. In: Proceedings of SPIN. Lecture Notes in Computer Science, vol. 9641, pp. 19–37. Springer (2016)
* [19] Dimovski, A.S.: Probabilistic analysis based on symbolic game semantics and model counting. In: Proceedings of GandALF. EPTCS, vol. 256, pp. 1–15 (2017)
* [20] Fredriksson, O., Ghica, D.R.: Abstract machines for game semantics, revisited. In: Proceedings of LICS. pp. 560–569 (2013)
* [21] Ghica, D.R., McCusker, G.: Reasoning about Idealized Algol using regular expressions. In: Proceedings of ICALP, Lecture Notes in Computer Science, vol. 1853, pp. 103–115. Springer-Verlag (2000)
* [22] Ghica, D.R., Murawski, A.S.: Compositional model extraction for higher-order concurrent programs. In: Proceedings of TACAS, Lecture Notes in Computer Science, vol. 3920, pp. 303–317. Springer (2006)
* [23] Ghica, D.R., Murawski, A.S.: Angelic semantics of fine-grained concurrency. Annals of Pure and Applied Logic 151(2-3), 89–114 (2008)
* [24] Ghica, D.R., Murawski, A.S., Ong, C.H.L.: Syntactic control of concurrency. Theoretical Computer Science pp. 234–251 (2006)
* [25] Hague, M.: Saturation of concurrent collapsible pushdown systems. In: Proceedings of FSTTCS. LIPIcs, vol. 24, pp. 313–325. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)
* [26] Honda, K., Yoshida, N.: Game-theoretic analysis of call-by-value computation. Theoretical Computer Science 221(1–2), 393–456 (1999)
* [27] Hopkins, D., Murawski, A.S., Ong, C.H.L.: Hector: An Equivalence Checker for a Higher-Order Fragment of ML. In: Proceedings of CAV, Lecture Notes in Computer Science, vol. 7358, pp. 774–780. Springer (2012)
* [28] Hopkins, D., Ong, C.H.L.: Homer: A Higher-order Observational equivalence Model checkER. In: Proceedings of CAV, Lecture Notes in Computer Science, vol. 5643, pp. 654–660. Springer (2009)
* [29] Kiefer, S., Murawski, A.S., Ouaknine, J., Wachter, B., Worrell, J.: APEX: An Analyzer for Open Probabilistic Programs. In: Proceedings of CAV, Lecture Notes in Computer Science, vol. 7358, pp. 693–698. Springer (2012)
* [30] Kobayashi, N., Igarashi, A.: Model-checking higher-order programs with recursive types. In: Proceedings of ESOP. Lecture Notes in Computer Science, vol. 7792, pp. 431–450. Springer (2013)
* [31] La Torre, S., Madhusudan, P., Parlato, G.: Reducing context-bounded concurrent reachability to sequential reachability. In: Proceedings of CAV. Lecture Notes in Computer Science, vol. 5643, pp. 477–492. Springer (2009)
* [32] Lago, U.D., Tanaka, R., Yoshimizu, A.: The geometry of concurrent interaction: handling multiple ports by way of multiple tokens. In: Proceedings of LICS. pp. 1–12 (2017)
* [33] Leroux, J., Schmitz, S.: Reachability in vector addition systems is primitive-recursive in fixed dimension. In: Proceedings of LICS. pp. 1–13. IEEE (2019)
* [34] Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall (1967)
* [35] Murawski, A.S.: Games for complexity of second-order call-by-name programs. Theoretical Computer Science 343(1/2), 207–236 (2005)
* [36] Murawski, A.S., Ramsay, S.J., Tzevelekos, N.: Game semantic analysis of equivalence in IMJ. In: Proceedings of ATVA. Lecture Notes in Computer Science, vol. 9364, pp. 411–428. Springer (2015)
* [37] Murawski, A.S., Tzevelekos, N.: An invitation to game semantics. SIGLOG News 3(2), 56–67 (2016)
* [38] Murawski, A.S., Walukiewicz, I.: Third-order Idealized Algol with iteration is decidable. Theoretical Computer Science 390(2-3), 214–229 (2008)
* [39] Ong, C.H.L.: Observational equivalence of 3rd-order Idealized Algol is decidable. In: Proceedings of IEEE Symposium on Logic in Computer Science. pp. 245–256. Computer Society Press (2002)
* [40] Qadeer, S., Rehof, J.: Context-bounded model checking of concurrent software. In: Proceedings of TACAS. Lecture Notes in Computer Science, vol. 3440, pp. 93–107. Springer (2005)
* [41] Ramalingam, G.: Context-sensitive synchronization-sensitive analysis is undecidable. ACM Trans. Program. Lang. Syst. 22(2), 416–430 (2000). https://doi.org/10.1145/349214.349241, https://doi.org/10.1145/349214.349241
* [42] Reynolds, J.C.: The essence of Algol. In: de Bakker, J.W., van Vliet, J. (eds.) Algorithmic Languages, pp. 345–372. North Holland (1978)
* [43] Schwentick, T.: Automata for XML - A survey. J. Comput. Syst. Sci. 73(3), 289–315 (2007)
## Appendix 0.A Additional material for Section 2
### 0.A.1 Operational semantics of $\mathsf{FICA}$
The operational semantics is defined using a (small-step) transition relation
$\mathcal{V}\vdash M,\,s\longrightarrow M^{\prime},\,s^{\prime}$, where
$\mathcal{V}$ is a set of variable names denoting active _memory cells_ and
_semaphore locks_. $s,s^{\prime}$ are states, i.e. functions
$s,s^{\prime}:\mathcal{V}\rightarrow\\{0,\cdots,\mathit{max}\\}$, and
$M,M^{\prime}$ are terms. We write $s\otimes(v\mapsto i)$ for the state
obtained by augmenting $s$ with $(v\mapsto i)$, assuming
$v\not\in\mathsf{dom}(s)$. The basic reduction rules are given in Figure 2,
where $c$ stands for any language constant ($i$ or ${\bf skip}$) and
$\widehat{\mathbf{op}}:\\{0,\cdots,\mathit{max}\\}\rightarrow\\{0,\cdots,\mathit{max}\\}$
is the function corresponding to $\mathbf{op}$. In-context reduction is given
by the schemata:
$\mathcal{V},v\vdash M[v/x],s\otimes(v\mapsto i)\longrightarrow
M^{\prime},s^{\prime}\otimes(v\mapsto i^{\prime})$ $M\neq c$
$\mathcal{V}\vdash{\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M,s\longrightarrow{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i^{\prime}\,{\bf
in}\,M^{\prime}[x/v],s^{\prime}$
$\mathcal{V},v\vdash M[v/x],s\otimes(v\mapsto i)\longrightarrow
M^{\prime},s^{\prime}\otimes(v\mapsto i^{\prime})$ $M\neq c$
$\mathcal{V}\vdash{\bf newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M,s\longrightarrow{\bf
newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i^{\prime}\,{\bf
in}\,M^{\prime}[x/v],s^{\prime}$
$\mathcal{V}\vdash M,\,s\longrightarrow M^{\prime},\,s^{\prime}$
$\mathcal{V}\vdash\mathcal{E}[M],\,s\longrightarrow\mathcal{E}[M^{\prime}],\,s^{\prime}$
where reduction contexts $\mathcal{E}[-]$ are produced by the grammar:
$\begin{array}[]{rcl}\mathcal{E}[-]&::=&[-]\mid\mathcal{E};N\mid(\mathcal{E}\,||\,N)\mid(M\,||\,\mathcal{E})\mid{\mathcal{E}}N\mid\mathbf{op}(\mathcal{E})\mid{\bf
if}\,\mathcal{E}\,{\bf then}\,N_{1}\,{\bf else}\,N_{2}\\\
&&\mid{!}\mathcal{E}\mid\mathcal{E}\,\raisebox{0.27986pt}{:}{=}\,m\mid
M\,\raisebox{0.27986pt}{:}{=}\,\mathcal{E}\mid{\bf grab}(\mathcal{E})\mid{\bf
release}(\mathcal{E}).\end{array}$
$\begin{array}[]{rclcrcl}\mathcal{V}\vdash{\bf skip}||{\bf
skip},\,s&\longrightarrow&{\bf skip},\,s&&\mathcal{V}\vdash{\bf if}\,i\,{\bf
then}\,N_{1}\,{\bf else}\,N_{2},\,s&\longrightarrow&N_{1},\,s,\quad i\neq 0\\\
\mathcal{V}\vdash{\bf skip};c,\,s&\longrightarrow&c,\,s&&\mathcal{V}\vdash{\bf
if}\,0\,{\bf then}\,N_{1}\,{\bf else}\,N_{2},\,s&\longrightarrow&N_{2},\,s\\\
\mathcal{V}\vdash\mathbf{op}(i),\,s&\longrightarrow&\widehat{\mathbf{op}}(i),\,s&&\mathcal{V}\vdash(\lambda
x.M)N,\,s&\longrightarrow&M[N/x],\,s\\\ \mathcal{V}\vdash{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,c,\,s&\longrightarrow&c,\,s&&\mathcal{V}\vdash{!}v,\,s\otimes(v\mapsto
i)&\longrightarrow&i,\,s\otimes(v\mapsto i)\\\ \mathcal{V}\vdash{\bf
newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,c,\,s&\longrightarrow&c,\,s&&\mathcal{V}\vdash
v\,\raisebox{0.27986pt}{:}{=}\,i^{\prime},\,s\otimes(v\mapsto
i)&\longrightarrow&{\bf skip},\,s\otimes(v\mapsto i^{\prime})\end{array}$
$\begin{array}[]{rcl}\mathcal{V}\vdash{\bf grab}(v),\,s\otimes(v\mapsto
0)&\longrightarrow&{\bf skip},\,s\otimes(v\mapsto 1)\\\ \mathcal{V}\vdash{\bf
release}(v),\,s\otimes(v\mapsto i)&\longrightarrow&{\bf
skip},\,s\otimes(v\mapsto 0),\quad i\neq 0\\\ \mathcal{V}\vdash{\bf
while}\,M\,{\bf do}\,N,\,s&\longrightarrow&{\bf if}\,M\,{\bf then}\,(N;{\bf
while}\,M\,{\bf do}\,N)\,{\bf else}\,{\bf skip},\,s\end{array}$
Figure 2: Reduction rules for $\mathsf{FICA}$
${}\vdash{M:{\bf com}}$ is said to terminate, written $M\Downarrow$, if
$\emptyset\vdash\emptyset,\,M\longrightarrow^{\ast}\emptyset,{\bf skip}$.
Idealized Concurrent Algol [23] also features variable and semaphore
constructors, called mkvar and mksem respectively, which play a technical role
in the full abstraction argument, similarly to [3]. We omit them in the main
body of the paper, because they do not present technical challenges, but they
are covered in the Appendix for the sake of completeness.
#### Typing rules
$\Gamma\vdash M:{\bf exp}\rightarrow{\bf com}$ $\Gamma\vdash N:{\bf exp}$
$\Gamma\vdash{\bf mkvar}(M,N):{\bf var}$ $\Gamma\vdash M:{\bf com}$
$\Gamma\vdash N:{\bf com}$ $\Gamma\vdash{\bf mksem}(M,N):{\bf sem}$
#### Reduction rules
$\displaystyle\mathcal{V}\vdash({\bf
mkvar}(M,N))\,\raisebox{0.27986pt}{:}{=}\,M^{\prime},\,s$
$\displaystyle\longrightarrow MM^{\prime},\,s$
$\displaystyle\mathcal{V}\vdash{!}({\bf mkvar}(M,N),\,s$
$\displaystyle\longrightarrow N,\,s$ $\displaystyle\mathcal{V}\vdash{\bf
grab}(\mathbf{mksem}\,MN),\,s$ $\displaystyle\longrightarrow M,\,s$
$\displaystyle\mathcal{V}\vdash{\bf release}(\mathbf{mksem}\,MN),\,s$
$\displaystyle\longrightarrow N,\,s$
#### $\eta$ rules for ${\bf var},{\bf sem}$
$\begin{array}[]{rcl}M&\longrightarrow&{\bf mkvar}((\lambda x^{\bf
exp}.M\,\raisebox{0.27986pt}{:}{=}\,x),!M)\\\ M&\longrightarrow&{\bf
mksem}({\bf grab}(M),{\bf release}(M))\end{array}$
Using $\mathbf{mkvar}$ and $\mathbf{mksem}$, one can define ${\bf
div}_{\theta}$ as syntactic sugar using ${\bf div}={\bf div}_{\bf com}$ only.
${\bf div}_{\theta}=\left\\{\begin{array}[]{lcl}{\bf div}&&\theta={\bf com}\\\
{\bf div};0&&\theta={\bf exp}\\\ {\bf mkvar}(\lambda x^{\bf exp}.{\bf
div},{\bf div}_{\bf exp})&&\theta={\bf var}\\\ {\bf mksem}({\bf div},{\bf
div})&&\theta={\bf sem}\\\ \lambda x^{\theta_{1}}.{\bf
div}_{\theta_{2}}&&\theta=\theta_{1}\rightarrow\theta_{2}\\\
\end{array}\right.$
## Appendix 0.B Additional material for Section 4
### 0.B.1 Proof of Lemma 2
We proceed by reducing from the halting problem for deterministic two-counter
machines [34, pp. 255–258].
The input to the halting problem is a deterministic two-counter machine
$\mathcal{C}=(Q_{\mathcal{C}},q_{0},q_{F},T)$, where $Q_{\mathcal{C}}$ is the
set of states, $q_{0},q_{F}\in Q_{\mathcal{C}}$ are the initial and final
states respectively, and
$T:Q_{\mathcal{C}}~{}\setminus~{}\\{q_{F}\\}\rightarrow(\mathsf{INC}\cup\mathsf{JZDEC})$
is the step function. Steps in $\mathsf{INC}$ are of the form
$(i,q^{\prime})\in\\{1,2\\}\times Q_{\mathcal{C}}$ (increment counter $i$ and
go to state $q^{\prime}$). Steps in $\mathsf{JZDEC}$ are of the form
$(i,q^{\prime},q^{\prime\prime})\in\\{1,2\\}\times Q_{\mathcal{C}}\times
Q_{\mathcal{C}}$ (if counter $i$ is zero then go to state $q^{\prime}$; else
decrement counter $i$ and go to state $q^{\prime\prime}$). The question is
whether, starting from $q_{0}$ with both counters zero, $\mathcal{C}$
eventually reaches $q_{F}$ with both counters zero.
We first construct a $1$-$\mathsf{LA}$ that recognises the language of all
data words such that:
* •
the underlying word (i.e., the projection onto the finite alphabet) encodes a
path through the transition relation of $\mathcal{C}$ from the initial state
to the final state, in other words a pseudo-run where the non-negativity of
counters and the correctness of zero tests are ignored;
* •
the occurrences of the letters that encode increments and decrements of
$\mathcal{C}$ form pairs that are labelled by the same level-$1$ data values,
where each increment is earlier than the corresponding decrement, which
assuming that both counters are zero initially ensures their non-negativity
throughout the pseudo-run and their being zero finally.
The second $1$-$\mathsf{LA}$ is slightly more complex. It accepts data words
that have the same properties as those accepted by the first
$1$-$\mathsf{LA}$, and in addition:
* •
there exists some increment followed by a zero test of the same counter before
a decrement with the same data value has occurred, in other words there is at
least one incorrect zero test in the pseudo-run.
The two sets of accepted traces will be equal if and only if all pseudo-runs
that satisfy the initial, non-negativity and final conditions necessarily
contain some incorrect zero test, i.e. if and only if $\mathcal{C}$ does not
halt as required. We give the formal construction below.
The two LAs we compute are
$\mathcal{A}_{1}(\mathcal{C})=\langle\Sigma,1,Q,\delta_{1}\rangle$ and
$\mathcal{A}_{2}(\mathcal{C})=\langle\Sigma,1,Q,\delta_{2}\rangle$.
The alphabet, $\Sigma=\Sigma_{\mathsf{Q}}\cup\Sigma_{\mathsf{A}}$, is defined
as follows:
$\Sigma_{\mathsf{Q}}=\\{\mathsf{start},\mathsf{inc_{1}},\mathsf{inc_{2}},\mathsf{zero_{1}},\mathsf{zero_{2}}\\}\qquad\Sigma_{\mathsf{A}}=\\{\mathsf{end},\mathsf{dec_{1}},\mathsf{dec_{2}},\mathsf{zero^{\prime}_{1}},\mathsf{zero^{\prime}_{2}}\\}$
Traces of $\mathcal{A}_{1}(\mathcal{C})$ and $\mathcal{A}_{2}(\mathcal{C})$
represent pseudo-runs of $\mathcal{C}$, i.e. sequences of steps of the
machine. Aside from $\mathsf{start}$ and $\mathsf{end}$, each letter in the
trace corresponds to the machine performing either an $\mathsf{INC}$ step
($\mathsf{inc}$), the “then” of a $\mathsf{JZDEC}$ step ($\mathsf{zero}$), or
the “else” of a $\mathsf{JZDEC}$ step ($\mathsf{dec}$). The
$\mathsf{zero^{\prime}}$ transition is a necessity which allows us to erase
leaves added by $\mathsf{zero}$. Each of $\mathsf{inc}$, $\mathsf{dec}$,
$\mathsf{zero}$, $\mathsf{zero^{\prime}}$ has two variants which encode $i$,
the counter number in the corresponding step. We will say that two letters
_match_ if they have the same data value.
By construction $\mathcal{A}_{1}(\mathcal{C})$ will accept exactly the traces
with the following properties, which correspond to the high-level description
of our first $1$-$\mathsf{LA}$:
* •
The first letter in the trace is $\mathsf{start}$ and the last is a matching
$\mathsf{end}$.
* •
For each occurrence of $\mathsf{inc_{i}}$, there is a matching
$\mathsf{dec_{i}}$ later in the trace.
* •
For each occurrence of $\mathsf{zero_{i}}$, there is a matching
$\mathsf{zero^{\prime}_{i}}$ later in the trace.
* •
The letters in the trace (excluding $\mathsf{start}$ and $\mathsf{end}$) form
a sequence $(a_{0},\ldots,a_{n-1})$; there exists some sequence of states
$(s_{0},\ldots,s_{n})\in Q_{\mathcal{C}}^{n+1}$ such that for all
$i\in(0,\ldots,n-1)$, $s_{i+1}$ appears as the second or third component of
$T(s_{i})$, and $a_{i}$ is a step which may be performed at state $s_{i}$
(irrespective of counter values).
The state space of the root,
$Q^{(0)}=Q_{\mathcal{C}}\times\\{\circ,\star,\mathbf{1},\mathbf{2}\\}$,
comprises pairs where the first component corresponds to a state of
$\mathcal{C}$ and the second tracks an observation of some invalid sequence.
The second component is only used in $\mathcal{A}_{2}(\mathcal{C})$. We denote
the pair at the root by square brackets. The states of the leaves at level 1
are
$Q^{(1)}=\bigcup\big{\\{}~{}\\{i,0_{i},i\star\\}~{}\big{|}~{}i\in\\{1,2\\}~{}\big{\\}}$,
where $0_{i}$ denotes a temporary leaf generated by $\mathsf{zero_{i}}$, $i$
denotes a counter, and $i\star$ denotes a counter being observed in
$\mathcal{A}_{2}(\mathcal{C})$.
The transition function $\delta_{1}$ of $\mathcal{A}_{1}(\mathcal{C})$ is
defined as follows.
$\dagger{\xlongrightarrow{\mathsf{start}}}_{1}[q_{0},\circ]\qquad[q_{F},\circ]{\xlongrightarrow{\mathsf{end}}}_{1}\dagger\qquad\frac{q{\xlongrightarrow{\mathsf{INC}}}(i,q^{\prime})~{}\in
T}{[q,\circ]{\xlongrightarrow{\mathsf{inc_{i}}}}_{1}([q^{\prime},\circ],i)}$
$\frac{q{\xlongrightarrow{\mathsf{JZDEC}}}(i,q^{\prime},q^{\prime\prime})~{}\in
T}{([q,\circ],i){\xlongrightarrow{\mathsf{dec_{i}}}}_{1}[q^{\prime\prime},\circ]\qquad[q,\circ]{\xlongrightarrow{\mathsf{zero_{i}}}}_{1}([q^{\prime},\circ],0_{i})}\qquad\frac{q\in
Q_{\mathcal{C}}}{([q,\circ],0_{i}){\xlongrightarrow{\mathsf{zero^{\prime}_{i}}}}_{1}[q,\circ]}$
By construction $\mathcal{A}_{2}(\mathcal{C})$ accepts exactly those traces of
$\mathcal{A}_{1}(\mathcal{C})$ where at least one $\mathsf{zero_{i}}$ letter
occurs in between an $\mathsf{inc_{i}}$ letter and the matching letter
$\mathsf{dec_{i}}$. In other words, the “then” of a $\mathsf{JZDEC}$ step has
been taken while the counter was nonzero. This is not a legal step, and so
such a trace does not represent a computation of $\mathcal{C}$. This
implements the high-level description of our second $1$-$\mathsf{LA}$.
In order to accept a word, $\mathcal{A}_{2}(\mathcal{C})$ must change the
second component of the root’s state from $\star$ to $\circ$. It does this by
nondeterministically choosing to observe some $\mathsf{inc}$ transition. From
here, it proceeds as in $\mathcal{A}_{1}(\mathcal{C})$ until either it meets
the matching $\mathsf{dec}$, in which case the automaton rejects, or it meets
an $\mathsf{ifz}$ transition on the same counter, at which point it marks the
second component with $\circ$ and proceeds as in
$\mathcal{A}_{1}(\mathcal{C})$.
The transition function $\delta_{2}$ of $\mathcal{A}_{2}(\mathcal{C})$ is
defined as follows:
$\dagger{\xlongrightarrow{\mathsf{start}}}_{2}[q_{0},\star]\qquad[q_{F},\circ]{\xlongrightarrow{\mathsf{end}}}_{2}\dagger\qquad\frac{q{\xlongrightarrow{\mathsf{INC}}}(i,q^{\prime})~{}\in
T\qquad
x\in\\{\circ,\star,\mathbf{1},\mathbf{2}\\}}{[q,x]{\xlongrightarrow{\mathsf{inc_{i}}}}_{2}([q^{\prime},x],i)\qquad[q,\star]{\xlongrightarrow{\mathsf{inc_{i}}}}_{2}([q,\mathbf{i}],i\star)}$
$\frac{q{\xlongrightarrow{\mathsf{JZDEC}}}(i,q^{\prime},q^{\prime\prime})~{}\in
T\qquad
x\in\\{\circ,\star,\mathbf{1},\mathbf{2}\\}}{[q,x]{\xlongrightarrow{\mathsf{zero_{i}}}}_{2}([q^{\prime},x],0_{i})\qquad[q,\mathbf{i}]{\xlongrightarrow{\mathsf{zero_{i}}}}_{2}([q^{\prime},\circ],0_{i})}\qquad\frac{q\in
Q_{\mathcal{C}}\qquad
x\in\\{\circ,\star,\mathbf{1},\mathbf{2}\\}}{([q,x],0_{i}){\xlongrightarrow{\mathsf{zero^{\prime}_{i}}}}_{2}[q,x]}$
$\frac{q{\xlongrightarrow{\mathsf{JZDEC}}}(i,q^{\prime},q^{\prime\prime})~{}\in
T\qquad
x\in\\{\circ,\star,\mathbf{1},\mathbf{2}\\}}{([q,x],i){\xlongrightarrow{\mathsf{dec_{i}}}}_{2}[q^{\prime\prime},x]\qquad([q,\circ],i\star){\xlongrightarrow{\mathsf{dec_{i}}}}_{2}[q^{\prime\prime},\circ]}$
$\mathcal{A}_{1}(\mathcal{C})$ captures every correctness condition for
halting computations of $\mathcal{C}$ except the legality of $\mathsf{zero}$
steps. Hence, $\mathcal{A}_{2}(\mathcal{C})$ accepts exactly those accepted
traces of $\mathcal{A}_{1}(\mathcal{C})$ which are _not_ halting computations
of $\mathcal{C}$, and so $\mathcal{C}$ performs a halting computation if and
only if $\mathcal{A}_{1}(\mathcal{C})\neq\mathcal{A}_{2}(\mathcal{C})$.
## Appendix 0.C Additional material for Section 5
### 0.C.1 Proof of Theorem 5.1
We present a proof of decidability of the emptiness problem for
$\mathsf{LLA}$, Theorem 5.1. There are two main steps in the proof. The first
step uses a notion of summary for some even layer $2i$. This allows to
restrict an automaton to first $2i$ layers. The second step is a method for
computing a summary for layer $2i$ from a summary for layer $2i+2$.
### Summaries
The structure of transitions of $\mathsf{LLA}$ provides a notation of a domain
for data values. The _domain_ of a data value $d\in\mathcal{D}$ is the set of
data values whose associated state may be modified by a transition that adds
or removes $d$, i.e., when reading a letter annotated by $d$.
$\mathsf{dom}(d)=\begin{cases}\\{\mathit{pred}^{2}(d),\mathit{pred}(d),d\\}&\text{if
d is at an even level}\\\
\\{\mathit{pred}^{3}(d),\mathit{pred}^{2}(d),\mathit{pred}(d),d\\}&\text{if d
is at an odd level}\end{cases}$
Domains give us a notion of independence: Two letters $(t_{1},d_{1})$,
$(t_{2},d_{2})$ are _independent_ if the domains of $d_{1}$ and $d_{2}$ are
disjoint. We remark that if $w$ is a trace of some $\mathsf{LLA}$ then every
sequence obtained by permuting adjacent independent letters of $w$ is also a
trace of the same $\mathsf{LLA}$ ending in the same configuration.
Let us fix an $k$-$\mathsf{LLA}$ automaton
$\mathcal{A}=\langle\Sigma_{\mathcal{A}},k_{\mathcal{A}},Q_{\mathcal{A}},\delta_{\mathcal{A}}\rangle$,
and let $b$ be its even-layer bound.
Suppose, on an accepting trace on $\mathcal{A}$, we encounter some data value
$d$ at even layer $2i$. On an accepting trace value $d$ occurs twice: the
first occurrence corresponds to adding $d$, the second to deleting $d$. Let
$w$ be the part of the trace in between, and including, these two occurrences
of $d$.
We can classify letters $(t^{\prime},d^{\prime})$ in $w$ into one of three
categories:
1. 1.
_$d$ -internal_, when $\mathsf{dom}(d^{\prime})$ is included in the subtree
rooted at $d$;
2. 2.
_$d$ -external_, when $\mathsf{dom}(d^{\prime})$ is disjoint from the subtree
rooted at $d$;
3. 3.
_$d$ -frontier_, when $\mathsf{dom}(d^{\prime})$ contains $d$ and its parent.
Note that these three categories partition the set of all letters in $w$. The
frontier letters are the ones with data value $d$, as well as those with
children of $d$. The later are from layer $2i+1$. Letters with data values
from bigger layers are either $d$-internal or $d$-external.
At this point we use branching bound $b$ of the automaton. The number of
children of $d$ is bounded by $b$, and every child of $d$ appears twice in
$w$. Hence, the number of $d$-frontier letters in $w$ is at most $b+2$,
counting the letters with $d$.
The $d$-frontier letters divide $w$ into subwords, giving us a sequence of
transitions:
$\kappa_{1}{\xlongrightarrow{m_{1}}}\kappa^{\prime}_{1}{\xlongrightarrow{w_{1}}}\kappa_{2}{\xlongrightarrow{m_{2}}}\kappa^{\prime}_{2}{\xlongrightarrow{w_{2}}}\dots\kappa_{l}{\xlongrightarrow{m_{l}}}\kappa^{\prime}_{l}{\xlongrightarrow{w_{l}}}\kappa_{l+1}{\xlongrightarrow{m_{l+1}}}\kappa^{\prime}_{l+1}$
(1)
where $m_{1},\dots,m_{l}$ are $d$-frontier letters; $m_{1}$ adds node $d$
while $m_{l+1}$ deletes $d$.
Configuration $\kappa^{\prime}_{1}$ is the first in which $d$ appears in the
tree, so $d$ is a leaf node in $\kappa^{\prime}_{1}$. Likewise, $\kappa_{l}$
is the last configuration in which $d$ appears, as it is removed by $m_{l+1}$,
so $d$ is a leaf node in $\kappa_{l+1}$.
We now use independence properties. Every word $w_{j}$ contains only
$d$-internal and $d$-external letters. Due to independence, $w_{j}$ is
equivalent to some $u_{j}v_{j}$, with $u_{j}$ containing only $d$-internal
letters of $w_{j}$ and $v_{j}$ containing only the $d$-external letters of
$w_{j}$. (Actually $u_{1}$ and $u_{l}$ are empty but we do not need to make a
case distinction in the rest of the argument)
From here, we can see that the $d$-internal parts $u_{1},\cdots,u_{l}$ of $w$
only interact with the $d$-external parts at a bounded number of positions,
and those positions exactly correspond to the frontier transitions
$m_{2},\cdots,m_{l}$. Hence, if we could characterize the interactions that
can occur at level $2i$, then we could replace the sequences of transitions on
every $u_{j}$ by a single short-cut transition. This would eliminate the need
for levels $\geq 2i$ in the automaton.
We introduce a notion of a summary to implement the idea of short-cut
transitions. A _summary_ for level $2i$ is a function
$f\colon\\{1,\dots,2(l+1)\\}\to Q^{2i-2}\times Q^{2i-1}$; for some $l\leq
b+1$. Intuitively, from some trace $w$ expanded as in Equation 1, we can
extract $f$ such that $f(2j-1)$ is a pair of states labelling
$\mathit{pred}^{2}(d)$ and $\mathit{pred}(d)$ in $\kappa_{j}$, while $f(2j)$
is a pair of states labelling these nodes in $\kappa^{\prime}_{j}$. This is
only the intuition because we do not have runs of $\mathcal{A}$ at hand to
compute $f$.
To formalise the idea of summaries for a given automaton, we will introduce
the notion of a _cut automaton_. Intuitively, the behaviour of a cut automaton
$\mathcal{A}^{\downarrow}(2i,f)$ will represent the behaviours of
$\mathcal{A}$ contained within some subtree rooted in a data value at layer
$2i$.
The states and transitions of $\mathcal{A}^{\downarrow}(2i,f)$ are those of
$\mathcal{A}$ but lifted up so that level $2i$ becomes the root level:
$\mathsf{Q}^{\downarrow(l-2i)}=\mathsf{Q}^{(l)}\qquad\delta_{\mathsf{Q}}^{\downarrow(l-2i)}=\delta_{\mathsf{Q}}^{(l)}\qquad\delta_{\mathsf{A}}^{\downarrow(l-2i)}=\delta_{\mathsf{A}}^{(l)}\qquad\text{for
$l\geq 2i+2$}$
The two to layers, $0$ and $1$, are special as just lifting transitions would
make them stick above the root. Here is also the place where we use the
summary $f$.
$Q^{\downarrow(0)}=Q^{(2i)}\times\mathsf{dom}(f)\qquad
Q^{\downarrow(1)}=Q^{(2i+1)}$
The extra component at layer $0$ will be used for layer $1$ transitions.
Before defining transitions we introduce some notation. For a summary $f$ we
write $\max(\mathsf{dom}(f))$ for the maximal element in the domain of $f$. We
use an abbreviated notation for transitions. If
$f(j)=(q^{(2i-2)},q^{(2i-1)})$, and
$f(j+1)=(q^{\prime(2i-2)},q^{\prime(2i-1)})$ then we write
$f(j){\xlongrightarrow{a}}(f(j+1),q^{\prime(2i)})\ \text{instead of}\
(q^{(2i-2)},q^{(2i-1)}){\xlongrightarrow{a}}(q^{\prime(2i-2)},q^{\prime(2i-1)},q^{\prime(2i)})\
.$
Transitions at levels $0$ and $1$ are adaptations of those of levels $2i$ and
$2i+1$ in the original automaton. A node that was at level $2i$ is now the
root so it has no predecessors anymore. The initial and final moves of
$\mathcal{A}^{\downarrow}(2i,f)$ create and destroy the root. They use $f$ to
predict what are states of predecessors in a corresponding move of
$\mathcal{A}$.
$\displaystyle\delta_{\mathsf{Q}}^{\downarrow(0)}\text{ contains }$
$\displaystyle\ \dagger{\xlongrightarrow{a}}(q^{\prime(2i)},1)$ if there is a
transition $f(1){\xlongrightarrow{a}}(f(2),q^{\prime(2i)})$ in
$\delta_{\mathsf{Q}}^{(2i)}$
$\displaystyle\delta_{\mathsf{A}}^{\downarrow(0)}\text{ contains }$
$\displaystyle\ (q,r){\xlongrightarrow{a}}\dagger$ if
$r=\max(\mathsf{dom}(f))-1$ and there is $(f(r),q){\xlongrightarrow{a}}f(r+1)$
in $\delta_{\mathsf{A}}^{(2i)}$
Finally, we have transitions that add and delete nodes on level $1$:
$\displaystyle\text{in }\delta_{\mathsf{Q}}^{\downarrow(1)}\text{ we have }$
$\displaystyle(q^{(2i)},r){\xlongrightarrow{a}}((q^{\prime(2i)},r+2),q^{\prime(2i+1)})$
$\displaystyle\text{\qquad if
}(f(r),q^{(2i)}){\xlongrightarrow{a}}(f(r+1),q^{\prime(2i)},q^{\prime(2i+1)})\in\delta_{\mathsf{Q}}^{(2i+1)}$
$\displaystyle\text{in }\delta_{\mathsf{A}}^{\downarrow(1)}\text{ we have }$
$\displaystyle((q^{(2i)},r),q^{(2i+1)}){\xlongrightarrow{a}}((q^{\prime(2i)},r+2))$
$\displaystyle\text{\qquad if
}(f(r),q^{(2i)},q^{(2i+1)}){\xlongrightarrow{a}}(f(r+1),q^{\prime(2i)})\in\delta_{\mathsf{A}}^{(2i+1)}$
We can now formally define the set of summaries for an even layer $2i$:
$\mathit{Summary}(\mathcal{A},2i)=\\{f\colon\mathcal{A}^{\downarrow}(2i,f)\text{
accepts some trace}\\}$
The next step is to define an automaton that uses such a set of summaries. The
idea is that when a node of layer $2i$ is created it is assigned a summary
from the set of summaries. Then all moves below this node are simulated by
consulting this summary. So we will never need layers below $2i$.
Let $\mathcal{S}$ be a set of summaries at level $2i$. We will now define
$\mathcal{A}^{\uparrow}(2i,\mathcal{S})$. It will be $(2i+1)$-$\mathsf{LLA}$
automaton. The states and transitions of
$\mathcal{A}^{\uparrow}(2i,\mathcal{S})$ are exactly the states and
transitions of $\mathcal{A}$ for levels $0$ to $2i-1$. The set of states at
level $2i$ is
$Q^{(2i)}=\\{(f,r)\colon f\in\mathcal{S},r\in\mathsf{dom}(f)\\}\ .$
So a state at layer $2i$ is a summary function and a _use counter_ indicating
the part of the summary that has been used.
For technical reasons we will also need one state at layer $2i+1$. We set
$Q^{(2i+1)}=\\{\bullet\\}$.
The transitions $\delta_{\mathsf{Q}}^{\uparrow(2i)}$ and
$\delta_{\mathsf{A}}^{\uparrow(2i)}$ are defined as follows.
$\displaystyle\text{in }\delta_{\mathsf{Q}}^{\uparrow(2i)}\text{ we have }$
$\displaystyle f(1){\xlongrightarrow{a}}(f(2),(f,3))$
$\displaystyle\quad\text{if }f\in\mathcal{S}$ $\displaystyle\text{in
}\delta_{\mathsf{A}}^{\uparrow(2i)}\text{ we have }$
$\displaystyle(f(r),(f,r)){\xlongrightarrow{a}}f(r+1)$
$\displaystyle\quad\text{if }r=\max(\mathsf{dom}(f))-1$
These transitions imply that for every node created at level $2i$, the
automaton guesses a summary and sets the summary’s use counter to $3$. It is
$3$ and not $1$ because the first two values of $f$ are used for the creation
of the node. The node can be deleted once this bounded counter value is
maximal.
Finally, we define the transitions in $\delta_{\mathsf{Q}}^{\uparrow(2i+1)}$
and $\delta_{\mathsf{A}}^{\uparrow(2i+1)}$:
$\displaystyle\text{In }\delta_{\mathsf{Q}}^{\uparrow(2i+1)}\text{ we have }$
$\displaystyle(f(r),(f,r)){\xlongrightarrow{a}}(f(r+1),(f,r+2),\bullet)$
$\displaystyle\qquad\text{if }r<\max(\mathsf{dom}(f))-1$
$\displaystyle\text{In }\delta_{\mathsf{A}}^{\uparrow(2i+1)}\text{ we have }$
$\displaystyle(f(r),(f,r),\bullet){\xlongrightarrow{a}}(f(r),(f,r))$
$\displaystyle\text{if }r=\max(\mathsf{dom}(f))-1$
So the automaton creates a child node whenever it uses a summary. The use
counter is increased by $2$ at such a transition. Once the use counter cannot
be increased anymore, $\delta_{\mathsf{A}}^{\uparrow(2i+1)}$ provides
transitions for deleting children at layer $2i+1$. No other transitions are
applicable at this point. Once there are no children, the root can be removed
by a $\delta_{\mathsf{A}}^{\uparrow(2i)}$ transition.
The next lemma states formally the relation between the two automata we have
introduced and the original one. Recall that $\mathcal{A}^{\downarrow}$ is
used to define a set of summaries. The lemma is proved by stitching runs of
$\mathcal{A}^{\uparrow}$ and $\mathcal{A}^{\downarrow}$.
###### Lemma 4
For every $k$-level automaton $\mathcal{A}$ and level $2i<k$, $\mathcal{A}$
accepts a trace iff
$\mathcal{A}^{\uparrow}(2i,\mathit{Summary}(\mathcal{A},2i))$ accepts a trace.
The next lemma shows how to use summaries of level $2i+1$ to compute summaries
at level $2i$.
###### Lemma 5
Take a summary $f$ of some level $2i$, and consider
$\mathcal{B}=\mathcal{A}^{\downarrow}(2i,f)$. Then $\mathcal{B}$ accepts some
trace iff $\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$
accepts some trace.
###### Proof
Follows from
$\mathit{Summary}(\mathcal{A},2i+2)=\mathit{Summary}(\mathcal{B},2)$ and the
previous lemma.
The lemma reduces the task of computing summaries to checking emptiness of
automata with $3$ layers. In the next subsection we show how to reduce the
later problem to the reachability problem in VASS. With this lemma we can
compute $\mathit{Summary}(\mathcal{A},2i)$ inductively. Once we compute
$\mathit{Summary}(\mathcal{A},2)$, we can reduce testing emptiness of
$\mathcal{A}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2))$ to VASS
reachability. This turns out to be degenerate case of computing summaries, so
the same technique as for computing summaries applies.
### Computing summaries
We compute $\mathit{Summary}(\mathcal{A},2i)$ assuming that we know
$\mathit{Summary}(\mathcal{A},2i+2)$. For this we use Lemma 5. We reduce
testing emptiness of
$\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$ from that lemma
to VASS reachability. Since presenting a VASS directly would be quite
unreadable, we present a nondeterministic program that will use variables
ranging over bounded domains and some fixed set of non-negative counters. By
construction, every counter will be tested for $0$ only at the end of the
computation. This structure allows us to emulate our nondeterministic program
in a VASS, such that acceptance by the program is equivalent to reachability
of a particular configuration in the VASS.
We fix a summary $\widehat{f}$ of level $2i$. Observe that the number of
summaries at level $2i$ is bounded, and so it is sufficient to check whether a
given candidate summary $\widehat{f}$ is a valid summary.
The variables of the program are as follows:
$\displaystyle\widehat{r}\in~{}$ $\displaystyle\mathsf{dom}(\widehat{f})$
$\displaystyle\mathit{state}\in~{}$ $\displaystyle Q_{2i}\cup\\{\bot\\}$
$\displaystyle\mathit{state}[j]\in~{}$ $\displaystyle
Q_{2i+1}\cup\\{\bot,\top\\}$ $\displaystyle j\in\\{1,\dots,b\\}$
$\displaystyle\mathit{children}[j,f,r]\in~{}$ $\displaystyle\mathbb{N}$ $f$
summary at level $(2i+2)$, $r\in\mathsf{dom}(f)$
Intuitively, $\mathit{state}$ and $\widehat{r}$ represent a state from
$Q^{(0)}$ of $\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$.
The initial configuration is empty so $\mathit{state}=\bot$. Variable
$\mathit{state}[j]$, represents the state of $j$-th child of the root. By
boundedness, the root can have at most $b$ children. Value
$\mathit{state}[j]=\bot$ means that the child has not yet been yet created,
and $\mathit{state}[j]=\top$ that the child has been deleted. Counter
$\mathit{children}[j,f,r]$ indicates the number of children of the $j$-th
child of the root with a particular summary $f$ of level $2i+2$ and usage
counter $r$.
Following these intuitions the initial values of the variables are
$\widehat{r}=1$, $\mathit{state}=\bot$, $\mathit{state}[j]=\bot$ for every
$j$, and $\mathit{children}[j,f,r]=0$ for every $j$, $f$ and $r$.
The program $\mathtt{TEST}(\widehat{f})$ we are going to write is a set of
rules that are executed nondeterministically. Either the program will
eventually accept, or it will block with no further rules that can be applied.
We later show that the program has an accepting run for $\widehat{f}$ iff
$\widehat{f}\in\mathit{Summary}(\mathcal{A},2i)$. The rules of the program
refer to transitions of $\mathcal{A}$ and simulate the definition of
$\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$ from Lemma 5.
They are defined as follows.
#### Initializing the root
We have a rule
if $\displaystyle\mathit{state}=\bot$ then $\displaystyle\begin{aligned}
&\mathit{state}=q^{\prime(2i)}\\\ &\widehat{r}=3\end{aligned}$
for every transition $f(1){\xlongrightarrow{a}}(f(2),q^{\prime(2i)})$ in
$\delta_{\mathsf{Q}}^{(2i)}$.
#### Removing the root and accepting.
The program is able to accept when it has completed all of its interaction
with the outside world. Observe that this is the only time that the counters
are tested for zero. Since this occurs at the end of the program, it can be
easily checked by VASS reachability.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\
&\widehat{r}=\max(\mathsf{dom}(\widehat{f}))-1\\\ &\forall
j\colon\mathit{state}[j]=\top\\\
&\forall(j,f,r)\colon\mathit{children}[j,f,r]=0\end{aligned}$ then accept
for every $(f(\widehat{r}),q^{(2i)}){\xlongrightarrow{a}}f(\widehat{r}+1)$ in
$\delta_{\mathsf{A}}^{(2i)}$.
#### Adding a node at level $2i+1$.
We ensure that we are in the correct state and ensure that the summary we are
testing aligns with some transition from the automaton.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\
&\widehat{f}(\widehat{r})=(q^{(2i-2)},q^{(2i-1)})\\\
&\widehat{f}(\widehat{r}+1)=(q^{\prime(2i-2)},q^{\prime(2i-1)})\\\
&\widehat{r}+2<\max(\mathsf{dom}(\widehat{f}))\\\ &\exists
j\colon\mathit{state}[j]=\bot\end{aligned}$ then $\displaystyle\begin{aligned}
&\mathit{state}:=q^{\prime(2i)}\\\ &\mathit{state}[j]:=q^{\prime(2i+1)}\\\
&\widehat{r}=\widehat{r}+2\end{aligned}$
for every transition
$(q^{(2i-2)},q^{(2i-1)},q^{(2i)})\xrightarrow{t}(q^{\prime(2i-2)},q^{\prime(2i-1)},q^{\prime(2i)},q^{\prime(2i+1)})\in\delta_{\mathsf{Q}}^{(2i+1)}$
#### Removing a node at level $2i+1$.
We delete a child according to some transition from
$\delta_{\mathsf{Q}}^{(2i+1)}$. While the zero test (ensuring $j$ is a leaf)
is not performed here directly, no further operations will be made on children
counters of this child and hence the zero test performed at the end of the
simulation does the job.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\
&\widehat{f}(\widehat{r})=(q^{(2i-2)},q^{(2i-1)})\\\
&\widehat{f}(\widehat{r}+1)=(q^{\prime(2i-2)},q^{\prime(2i-1)})\\\
&\widehat{r}+2<\max(\mathsf{dom}(\widehat{f}))\\\ &\exists
j\colon\mathit{state}[j]=q^{(2i+1)}\end{aligned}$ then
$\displaystyle\begin{aligned} &\mathit{state}:=q^{\prime(2i)}\\\
&\mathit{state}[j]:=\top\\\ &\widehat{r}=\widehat{r}+2\end{aligned}$
for every transition
$(q^{(2i-2)},q^{(2i-1)},q^{(2i)},q^{(2i+1)})\xrightarrow{t}(q^{\prime(2i-2)},q^{\prime(2i-1)},q^{\prime(2i)})\in\delta_{\mathsf{A}}^{(2i+1)}$
#### Adding a node at level $2i+2$.
Firstly we ensure that there is some child $j$ where such a node can be
appended. We simulate creation of a child by nondeterministically choosing a
summary and increasing the corresponding unbounded counter. Index $3$ in
$\mathit{children}[j,f,3]$ means that this child is after the first
interaction with its ancestors at levels $2i$ and $2i+1$, that happened at its
creation.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\ &\exists
j\colon\mathit{state}[j]=q^{(2i+1)}\end{aligned}$ then
$\displaystyle\begin{aligned} &\mathit{state}=q^{\prime(2i)}\\\
&\mathit{state}[j]=q^{\prime(2i+1)}\\\ &\mathit{children}[j,f,3]\text{ +=
}1\\\ &\text{ for some $f\in\mathit{Summary}(2i+2)$ s.t.}\\\
&\text{\qquad\qquad\qquad$f(1)=(q^{(2i)},q^{(2i+1)})$ and
$f(2)=(q^{\prime(2i)},q^{\prime(2i+1)})$}\end{aligned}$
#### Progressing a child at level $2i+2$.
We identify an appropriate child $j$ which itself has a child in state
$(f,r)$. We use the test $r+2<\max(\mathsf{dom}(f))$ to ensure that the last
interaction of the node is reserved for deletion of our root node.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\
&\exists(j,f,r)\colon\mathit{state}[j]=q^{(2i+1)}\\\ &\hskip 42.67912pt\text{
and }f(r)=(q^{(2i)},q^{(2i+1)})\\\ &\hskip 42.67912pt\text{ and
}f(r+1)=(q^{\prime(2i)},q^{\prime(2i+1)})\\\ &\hskip 42.67912pt\text{ and
}(r+2)<\max(\mathsf{dom}(f))\\\ &\hskip 42.67912pt\text{ and
}\mathit{children}[j,f,r]\geq 1\end{aligned}$ then
$\displaystyle\begin{aligned} &\mathit{state}:=q^{\prime(2i)}\\\
&\mathit{state}[j]:=q^{\prime(2i+1)}\\\ &\mathit{children}[j,f,r+2]\text{ +=
}1\\\ &\mathit{children}[j,f,r]\text{ -= }1\end{aligned}$
Observe that the test $\mathit{children}[j,f,r]\geq 1$ can be simulated by a
VASS because we have $\mathit{children}[j,f,r]\text{ -= }1$ in the statement
that follows.
#### Removing a node at level $2i+2$.
We find a child which has completed its summary to the point that it can now
be removed. We use the last values in $f$ to determine how to remove the node.
if $\displaystyle\begin{aligned} &\mathit{state}=q^{(2i)}\\\
&\exists(j,f,r)\colon\mathit{state}[j]=q^{(2i+1)}\\\ &\hskip 42.67912pt\text{
and }f(r)=(q^{(2i)},q^{(2i+1)})\\\ &\hskip 42.67912pt\text{ and
}f(r+1)=(q^{\prime(2i)},q^{\prime(2i+1)})\\\ &\hskip 42.67912pt\text{ and
}(r+1)=\max(\mathsf{dom}(f))\\\ &\hskip 42.67912pt\text{ and
}\mathit{children}[j,f,r]\geq 1\end{aligned}$ then
$\displaystyle\begin{aligned} &\mathit{state}:=q^{\prime(2i)}\\\
&\mathit{state}[b]:=q^{\prime(2i+1)}\\\ &\mathit{children}[j,f,r]\text{ -=
}1\end{aligned}$
###### Lemma 6
Program $\mathtt{TEST}(\widehat{f})$ accepts iff
$\widehat{f}\in\mathit{Summary}(\mathcal{A},2i)$.
###### Proof
By definition, $\widehat{f}\in\mathit{Summary}(\mathcal{A},2i)$ if automaton
$\mathcal{B}=\mathcal{A}^{\downarrow}(2i,\widehat{f})$ accepts a trace. By
Lemma 5 this is equivalent to
$\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$ accepting some
trace. It can be checked that the instructions of $\mathtt{TEST}(\widehat{f})$
correspond one-to-one to transitions of
$\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$. So an
accepting run of $\mathtt{TEST}(\widehat{f})$ can be obtained from a trace
accepted by $\mathcal{B}^{\uparrow}(2,\mathit{Summary}(\mathcal{A},2i+2))$,
and vice versa.
## Appendix 0.D Additional material for Section 6
### 0.D.1 Proof of Theorem 6.1
Because every $\mathsf{FICA}$-term can be converted to $\beta\eta$-normal
form, we use induction on the structure of such normal forms. The base cases
are:
* •
${\Gamma}\vdash{{\bf skip}:{\bf com}}$: $Q^{(0)}=\\{0\\}$,
$\dagger{\xlongrightarrow{\mathsf{run}}}0$,
$0{\xlongrightarrow{\mathsf{done}}}\dagger$;
* •
${\Gamma}\vdash{{\bf div}_{\bf com}:{\bf com}}$: $Q^{(0)}=\\{0\\}$,
$\dagger{\xlongrightarrow{\mathsf{run}}}0$;
* •
${\Gamma}\vdash{{\bf div}_{\theta}:\theta}$: $Q^{(0)}=\\{0\\}$,
$\dagger{\xlongrightarrow{m}}0$, assuming
$\theta=\theta_{l}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow\beta$ and
$m$ ranges over question-moves from $M_{{\llbracket}{\beta}{\rrbracket}}$;
* •
${\Gamma}\vdash{i:{\bf exp}}$: $Q^{(0)}=\\{0\\}$,
$\dagger{\xlongrightarrow{\mathsf{q}}}0$, $0{\xlongrightarrow{i}}\dagger$.
Observe that they are clearly even-ready, because only one node is ever
created.
The remaining cases are inductive. Note that we will use $\mathsf{m}$ to range
over
$\mathcal{T}_{{\Gamma}\vdash{\theta}}+\\{\epsilon_{\mathsf{Q}},\epsilon_{\mathsf{A}}\\}$,
i.e. not only $M_{{\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}}$, and
recall our convention that $m\in
M_{{\llbracket}{{\Gamma}\vdash{\theta}}{\rrbracket}}$ stands for
$m^{(\epsilon,0)}$.
When referring to the inductive hypothesis, i.e. the automaton constructed for
some subterm $M_{i}$, we will use the subscript $i$ to refer to its
components, e.g. $Q_{i}^{(j)}$, ${\xlongrightarrow{\mathsf{m}}}_{i}$ etc. In
contrast, we shall use $Q^{(j)}$, ${\xlongrightarrow{\mathsf{m}}}$ to refer to
the automaton that is being constructed. The construction will often use
inference lines $\frac{\qquad}{\qquad}$ to indicate that the transitions
listed under the line should be added to the new automaton as long as the
transitions listed above the line are present in an automaton given by the
inductive hypothesis. Sometimes we will invoke the inductive hypothesis for
several terms, which can provide several automata of different depths. Without
loss of generality, we will then assume that they all have the same depth $k$,
because an automaton of lower depth can be viewed as one of higher depth.
* •
${\Gamma}\vdash{\mathbf{op}(M_{1}):{\bf exp}}$: $Q^{(j)}=Q_{1}^{(j)}$ ($0\leq
j\leq k$). In order to interpret unary operators it suffices to modify
transitions carrying the final answer in the automaton for $M_{1}$. Formally,
this is done as follows.
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq
i}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\widehat{\mathbf{op}}(i)}}\dagger}$
Above, $j$ ranges over $\\{-1,0,\cdots,k\\}$, so that
$(q_{1}^{(0)},\cdots,q_{1}^{(j)})$ can also stand for $\dagger$. Even-
readiness is preserved by the construction, because the configuration graph of
the original automaton is preserved.
* •
${\Gamma}\vdash{M_{1}||M_{2}}:{\bf com}$: $Q^{(0)}=Q_{1}^{(0)}\times
Q_{2}^{(0)}$, $Q^{(j)}=Q_{1}^{(j)}+Q_{2}^{(j)}$ $(1\leq j\leq k)$. The first
group of transitions activate and terminate the two components respectively:
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1}q_{1}^{(0)}\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}(q_{1}^{(0)},q_{2}^{(0)})}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger\qquad
q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger}{(q_{1}^{(0)},q_{2}^{(0)}){\xlongrightarrow{\mathsf{done}}}\dagger}.$
The remaining transitions allow each component to progress.
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
q_{2}^{(0)}\in
Q_{2}^{(0)}\qquad\mathsf{m}\neq\mathsf{run},\mathsf{done}}{((q_{1}^{(0)},q_{2}^{(0)}),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},q_{2}^{(0)}),\cdots,r_{1}^{(j^{\prime})})}\qquad$
$\frac{q_{1}^{(0)}\in
Q_{1}^{(0)}\qquad(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run},\mathsf{done}}{((q_{1}^{(0)},q_{2}^{(0)}),\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}((q_{1}^{(0)},r_{2}^{(0)}),\cdots,r_{2}^{(j^{\prime})})}$
Even-readiness at even levels different from $0$ follows from even-readiness
of the automata obtained in IH, because the construction simply runs them
concurrently without interaction at these levels. For level $0$, we observe
that, whenever the root reaches state $(q_{1}^{(0)},q_{2}^{(0)})$, even-
readiness of the two automata implies that each of them has removed all nodes
below the root, i.e. the root will be a leaf.
* •
${\Gamma}\vdash{M_{1};M_{2}}:{\bf com}$: $Q^{(i)}=Q^{(i)}_{1}+Q^{(i)}_{2}$
($0\leq i\leq k$). We let the automaton for $M_{1}$ run first (except for the
final step $\mathsf{done}$):
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}q_{1}^{(0)}}\qquad\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{done}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}.$
Whenever the automaton $M_{1}$ can terminate, we pass control to the automaton
for $M_{2}$ via
$\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}\qquad
q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run}}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
and allow it to continue
$\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run}}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}.$
Note that the construction relies crucially on even-readiness of the automaton
for $M_{1}$, because we move to the automaton for $M_{2}$ as soon as the
automaton $M_{1}$ arrives at a configuration with level-$0$ state
$q_{1}^{(0)}$ such that
$q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger$. Thanks to even-
readiness, we can conclude that the root will be the only node in the
configuration then and the transition can indeed fire, i.e. $M_{1}$ is really
finished.
Even-readiness of the new automaton follows from the fact that the original
automata were even-ready, because we are re-using their transitions (and when
the automaton for $M_{2}$ is active, that for $M_{1}$ has not left any nodes).
* •
${\Gamma}\vdash{M_{1};M_{2}:\beta}$
The general case is nearly the same as the ${\bf com}$ case presented above
except that we need to keep track of what initial move has been played in
order to perform the transition to $M_{2}$ correctly. This is especially
important for $\beta={\bf var},{\bf sem}$, where there are multiple initial
moves. This extra information will be stored at level $0$, while the automaton
corresponding to $M_{1}$ is active. Below we present a general construction
parameterized by the set $I$ of initial moves. The set $I$ is defined as
follows.
* –
$\beta={\bf com}$: $I=\\{\mathsf{run}\\}$
* –
$\beta={\bf exp}$: $I=\\{\mathsf{q}\\}$
* –
$\beta={\bf var}$:
$I=\\{\mathsf{read},\mathsf{write}(0),\cdots,\mathsf{write}(\mathit{max})\\}$
* –
$\beta={\bf sem}$: $I=\\{\mathsf{grb},\mathsf{rls}\\}$
States
$\begin{array}[]{rcl}Q^{(0)}&=&(Q^{(0)}_{1}\times I)+Q^{(0)}_{2}\\\
Q^{(i)}&=&Q^{(i)}_{1}+Q^{(i)}_{2}\qquad(0<i\leq k)\end{array}$
Transitions
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1}q_{1}^{(0)}\qquad x\in
I}{\dagger{\xlongrightarrow{x}}(q_{1}^{(0)},x)}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{done}\qquad
x\in
I}{((q_{1}^{(0)},x),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},x),\cdots,r_{1}^{(j^{\prime})})}.$
$\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger\qquad\dagger{\xlongrightarrow{x}}_{2}q_{2}^{(0)}\qquad
q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad
x\in I\qquad\mathsf{m}\not\in
I}{(q_{1}^{(0)},x){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
$\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\not\in
I}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
None of the $M_{1};M_{2}$ cases requires an adjustment of pointers, because
the inherited indices are accurate.
* •
${\Gamma}\vdash{{\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M_{1}}:\beta$. By [23], ${\llbracket}{{\Gamma}\vdash{{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M_{1}}}{\rrbracket}$ can
be obtained by
* –
first restricting ${\llbracket}{{\Gamma,x}\vdash{M_{1}}}{\rrbracket}$ to plays
in which the moves $\mathsf{read}^{x}$, $\mathsf{write}(n)^{x}$ are followed
immediately by answers,
* –
selecting only those plays in which each answer to a $\mathsf{read}^{x}$-move
is consistent with the preceding $\mathsf{write}(n)^{x}$-move (or equal to
$i$, if no preceding $\mathsf{write}(n)^{x}$ was made),
* –
erasing all moves related to $x$, e.g. those of the form $m^{(x,\rho)}$.
To implement the above recipe, we will lock the automaton after each
$\mathsf{read}^{x}$\- or $\mathsf{write}(n)^{x}$-move, so that only an answer
to that move can be played next. Technically, this will be done by annotating
the level-$0$ state with a $\mathit{lock}$-tag. Moreover, at level $0$, we
will also keep track of the current value of $x$. This will help us ensure
that answers to $\mathsf{read}^{x}$ are consistent with the stored value and
that $\mathsf{write}(n)^{x}$ transitions cause the right change. Eventually,
all moves with the $x$ subscript will be replaced with
$\epsilon_{\mathsf{Q}},\epsilon_{\mathsf{A}}$ to model hiding.
Accordingly, we take
$Q^{(0)}=(Q_{1}^{(0)}+(Q_{1}^{(0)}\times\\{\mathit{lock}\\}))\times\\{0,\cdots,\mathit{max}\\}$
and $Q^{(j)}=Q_{1}^{(j)}$ ($1\leq j\leq k$). First, we make sure that the
state component is initialised to $i$ and that it can be arbitrary at the very
end:
$\frac{\dagger{\xlongrightarrow{\mathsf{m}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{m}}}(q_{1}^{(0)},i)}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{1}\dagger\qquad
0\leq
n\leq\mathit{max}}{(q_{1}^{(0)},n){\xlongrightarrow{\mathsf{m}}}\dagger}.$
Transitions involving moves different from $\mathsf{write}(z)^{x}$,
$\mathsf{ok}^{x}$, $\mathsf{read}^{x}$, $z^{x}$ (and the moves handled above)
progress unaffected while preserving $n$ (the current value of $x$ recorded at
level $0$):
$\frac{\begin{array}[]{lcl}(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})&&\mathsf{m}\neq\mathsf{read}^{x},z^{x},\mathsf{write}(z)^{x},\mathsf{ok}^{x}\\\
&&0\leq j,j^{\prime}\qquad 0\leq
n\leq\mathit{max}\end{array}}{((q^{(0)}_{1},n),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},n),\cdots,r_{1}^{(j^{\prime})})}.$
Transitions using $\mathsf{read}^{x}$, $\mathsf{write}(z)^{x}$ add a lock at
level $0$. The lock can be lifted only if a corresponding answer is played
(because of the lock, a unique $\mathsf{write}(z)^{x}$ or $\mathsf{read}^{x}$
will be pending). Its value must be consistent with the value of $x$ recorded
at level $0$.
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{write}(z)^{(x,\rho)}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
0\leq
n,z\leq\mathit{max}}{((q_{1}^{(0)},n),\cdots,q_{1}^{(j)}){\xlongrightarrow{\epsilon_{\mathsf{Q}}}}((r_{1}^{(0)},\mathit{lock},z),\cdots,r_{1}^{(j^{\prime})})}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{read}^{(x,\rho)}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
0\leq
n\leq\mathit{max}}{((q_{1}^{(0)},n),\cdots,q_{1}^{(j)}){\xlongrightarrow{\epsilon_{\mathsf{Q}}}}((r_{1}^{(0)},\mathit{lock},n),\cdots,r_{1}^{(j^{\prime})}))}$
$\frac{(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{\mathsf{ok}^{x}}}_{1}(t_{1}^{(0)},\cdots,t_{1}^{(j)})\qquad
0\leq
n\leq\mathit{max}}{((r_{1}^{(0)},\mathit{lock},n),\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{\epsilon_{\mathsf{A}}}}((t_{1}^{(0)},n),\cdots,t_{1}^{(j)})}$
$\frac{(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{n^{x}}}_{1}(t_{1}^{(0)},\cdots,t_{1}^{(j)})\qquad
0\leq
n\leq\mathit{max}}{((r_{1}^{(0)},\mathit{lock},n),\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{\epsilon_{\mathsf{A}}}}((t_{1}^{(0)},n),\cdots,t_{1}^{(j)})}$
As the construction involves running the original automaton and transitions
corresponding to P-answers are not modified, even-readiness follows directly
from IH. For the same reason, the indices corresponding to justification
pointers need no adjustment.
* •
The case of ${\bf newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M_{1}$
is similar to ${\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M_{1}$. We represent the state of the semaphore using an additional bit
at level $0$, where $0$ means free and $1$ means taken. We let
$Q^{(0)}=(Q_{1}^{(0)}+(Q_{1}^{(0)}\times\\{\mathit{lock}\\}))\times\\{0,1\\}$
and $Q^{(j)}=Q_{1}^{(j)}$ ($1\leq j\leq k$). First, we make sure the bit is
initialised to $i$ and can be arbitrary at the very end.
$\frac{\dagger{\xlongrightarrow{\mathsf{m}}}_{1}q_{1}^{(0)}\qquad
i=0}{\dagger{\xlongrightarrow{\mathsf{m}}}(q_{1}^{(0)},0)}\qquad\frac{\dagger{\xlongrightarrow{\mathsf{m}}}_{1}q_{1}^{(0)}\qquad
i>0}{\dagger{\xlongrightarrow{\mathsf{m}}}(q_{1}^{(0)},1)}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{1}\dagger\qquad
z\in\\{0,1\\}}{(q_{1}^{(0)},z){\xlongrightarrow{\mathsf{m}}}\dagger}$
Transitions involving moves other than $\mathsf{rls}^{(x,\rho)}$,
$\mathsf{grb}^{(x,\rho)}$ and $\mathsf{ok}^{x}$ proceed as before, while
preserving the state of the semaphore.
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad
z\in\\{0,1\\}\qquad\mathsf{m}\neq\mathsf{rls}^{(x,\rho)},\mathsf{grb}^{(x,\rho)},\mathsf{ok}^{x}}{((q_{1}^{(0)},z),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},z),\cdots,r_{1}^{(j^{\prime})})}$
Transitions using $\mathsf{rls}^{(x,\rho)}$, $\mathsf{grb}^{(x,\rho)}$ proceed
only if they are compatible with the current state of the semaphore, as
represented by the extra bit. At the same time, each time
$\mathsf{grb}^{(x,\rho)}$ or $\mathsf{rls}^{(x,\rho)}$ is played, we lock the
automaton so that the corresponding answer can be played next. The moves are
then hidden and replaced with $\epsilon_{\mathsf{Q}}$ and
$\epsilon_{\mathsf{A}}$.
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{grb}^{(x,\rho)}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}{((q_{1}^{(0)},0),\cdots,q_{1}^{(j)}){\xlongrightarrow{\epsilon_{\mathsf{Q}}}}((r_{1}^{(0)},\mathit{lock},1),\cdots,r_{1}^{(j^{\prime})})}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{rls}^{(x,\rho)}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}{((q_{1}^{(0)},1),\cdots,q_{1}^{(j)}){\xlongrightarrow{\epsilon_{\mathsf{Q}}}}((r_{1}^{(0)},\mathit{lock},0),\cdots,r_{1}^{(j^{\prime})})}$
$\frac{(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{\mathsf{ok}^{x}}}_{1}(t_{1}^{(0)},\cdots,t_{1}^{(j)})\qquad
z\in\\{0,1\\}}{((r_{1}^{(0)},\mathit{lock},z),\cdots,r_{1}^{(j^{\prime})}){\xlongrightarrow{\epsilon_{\mathsf{A}}}}((t_{1}^{(0)},z),\cdots,t_{1}^{(j)})}$
* •
${\Gamma}\vdash{fM_{h}\cdots M_{1}:{\bf com}}$ with
$(f:\theta_{h}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow{\bf
com})\in\Gamma$. Note that this also covers the case $f:{\bf com}$.
$Q^{(0)}=\\{0,1,2\\}$, $Q^{(1)}=\\{0\\}$, $Q^{(j+2)}=Q^{(j)}$ ($0\leq j\leq
k$). First we add transitions corresponding to calling and returning from $f$:
$\dagger{\xlongrightarrow{\mathsf{run}}}0$,
$0{\xlongrightarrow{\mathsf{run}^{f}}}(1,0)$,
$(1,0){\xlongrightarrow{\mathsf{done}^{f}}}2$,
$2{\xlongrightarrow{\mathsf{done}}}\dagger$.
In state $(1,0)$ we want to enable the environment to spawn an unbounded
number of copies of each of ${\Gamma}\vdash{M_{u}:\theta_{u}}$ ($1\leq u\leq
h$). This is done through the following rules, which embed the actions of the
automata for $M_{u}$ while relabelling the moves.
* –
Moves from $M_{u}$ corresponding to $\theta_{u}$ obtain an additional
annotation $fu$, as they are now the $u$th argument of
$f:\theta_{h}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow{\bf com}$.
$\frac{(q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(\vec{i},\rho)}}}_{u}(q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}{(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(fu\vec{i},\rho)}}}(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}$
Note that above we mean $j,j^{\prime}$ to range over $\\{-1,0,\cdots,k\\}$, so
that $(q_{u}^{(0)},\cdots,q_{u}^{(j)})$ and
$(q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})$ can also stand for $\dagger$. The
pointer structure is simply inherited in this case, but an additional pointer
needs to be created to $\mathsf{run}^{f}$ from the old initial move for
$M_{u}$, i.e. $m^{(\epsilon,0)}$, which did not have a pointer earlier.
Fortunately, because we also use $\rho=0$ in initial moves to represent the
lack of a pointer, by copying $0$ now we indicate that the move $m^{fu,\rho}$
points one level up, i.e. at the new $\mathsf{run}^{f}$ move, as required.
* –
The moves from $M_{u}$ that originate from $\Gamma$, i.e. moves of the form
$m^{(x_{v}\vec{i},\rho)}$ ($1\leq v\leq l$), where
$(x_{v}\in\theta_{v})\in\Gamma$, need no relabelling except for question moves
that should point at the initial move. These moves correspond to question-tags
of the form $m^{(x_{v},\rho)}$. Leaving $\rho$ unchanged in this case would
mean pointing at $m^{fu,0}$, whereas we need to point at $\mathsf{run}$
instead. To readjust such pointers, we simply add $2$ to $\rho$, and preserve
$\rho$ in other moves.
$\frac{(q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(x_{v},\rho)}}}_{u}(q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})\qquad\textrm{$m$
is a
question}}{(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(x_{v},\rho+2)}}}(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}$
$\frac{(q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(x_{v}\vec{i},\rho)}}}_{u}(q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})\qquad\textrm{$\vec{i}\neq\epsilon$
or ($\vec{i}=\epsilon$ and $m$ is an
answer)}}{(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j)}){\xlongrightarrow{m^{(x_{v}\vec{i},\rho)}}}(1,0,q_{u}^{(0)},\cdots,q_{u}^{(j^{\prime})})}$
The construction clearly preserves even-readiness at level $0$. For other even
levels, this follows directly from IH as we are simply running copies of the
automata from IH.
* •
${\Gamma}\vdash{fM_{h}\cdots M_{1}:{\bf exp}}$. Here we follow the same recipe
as for ${\bf com}$ except that the initial and final transitions need to be
changed from
$\dagger{\xlongrightarrow{\mathsf{run}}}0\qquad
0{\xlongrightarrow{\mathsf{run}^{f}}}(1,0)\qquad(1,0){\xlongrightarrow{\mathsf{done}^{f}}}2\qquad
2{\xlongrightarrow{\mathsf{done}}}\dagger$
to
$\dagger{\xlongrightarrow{\mathsf{q}}}0\qquad
0{\xlongrightarrow{\mathsf{q}^{f}}}(1,0)\qquad(1,0){\xlongrightarrow{i^{f}}}2^{i}\qquad
2^{i}{\xlongrightarrow{i}}\dagger.$
* •
${\Gamma}\vdash{fM_{h}\cdots M_{1}:{\bf var}}$. Here a slightly more
complicated adjustment is needed to account for the two kinds of initial
moves. Consequently, we need to distinguish two copies of $1$, i.e. $1^{r}$
and $1^{w}$.
$\dagger{\xlongrightarrow{\mathsf{read}}}0\qquad
0{\xlongrightarrow{\mathsf{read}^{f}}}(1^{r},0)\qquad(1^{r},0){\xlongrightarrow{i^{f}}}2^{i}\qquad
2^{i}{\xlongrightarrow{i}}\dagger.$
$\dagger{\xlongrightarrow{\mathsf{write}(i)}}0^{i}\qquad
0^{i}{\xlongrightarrow{\mathsf{write}(i)^{f}}}(1^{w},0)\qquad(1^{w},0){\xlongrightarrow{\mathsf{ok}}}2\qquad
2{\xlongrightarrow{\mathsf{ok}}}\dagger.$
All the other rules allowing for transitions between states of the form
$(1,0,\cdots)$ need to be replicated for $(1^{r},0,\cdots)$ and
$(1^{w},0,\cdots)$.
* •
${\Gamma}\vdash{fM_{h}\cdots M_{1}:{\bf sem}}$. This is similar to the
previous case. To account for the two kinds of initial moves, we use states
$1^{g}$ and $1^{r}$.
$\dagger{\xlongrightarrow{\mathsf{grb}}}0^{g}\qquad
0^{g}{\xlongrightarrow{\mathsf{grb}^{f}}}(1^{g},0)\qquad(1^{g},0){\xlongrightarrow{\mathsf{ok}^{f}}}2^{g}\qquad
2^{g}{\xlongrightarrow{\mathsf{ok}}}\dagger$
$\dagger{\xlongrightarrow{\mathsf{rls}}}0^{r}\qquad
0^{r}{\xlongrightarrow{\mathsf{rls}^{f}}}(1^{r},0)\qquad(1^{r},0){\xlongrightarrow{\mathsf{ok}^{f}}}2^{r}\qquad
2^{r}{\xlongrightarrow{\mathsf{ok}}}\dagger$
All the other rules allowing for transitions between states of the form
$(1,0,\cdots)$ need to be replicated for $(1^{r},0,\cdots)$ and
$(1^{g},0,\cdots)$.
* •
${\Gamma}\vdash{\lambda
x.M_{1}:\theta_{h}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow\beta}$:
This is simply dealt with by renaming labels in the automaton for
${\Gamma,x:\theta_{h}}\vdash{M_{1}:\theta_{h-1}\rightarrow\cdots\rightarrow\theta_{1}\rightarrow\beta}$:
tags of the form $m^{(x\vec{i},\rho)}$ must be renamed as
$m^{(h\vec{i},\rho)}$.
* •
${\Gamma}\vdash{{\bf if}\,M_{1}\,{\bf then}\,M_{2}\,{\bf else}\,M_{3}:\beta}$
This case is similar to $M_{1};M_{2}$ except that $M_{1}$ of type ${\bf exp}$,
so the associated move is $\mathsf{q}$ rather than $\mathsf{run}$. Morever,
once $M_{1}$ terminates, the automaton for either $M_{2}$ or $M_{3}$ must be
activated, as appropriate.
States
$\begin{array}[]{rcl}Q^{(0)}&=&(Q^{(0)}_{1}\times
I)+Q^{(0)}_{2}+Q^{(0)}_{3}\\\
Q^{(i)}&=&Q^{(i)}_{1}+Q^{(i)}_{2}+Q^{(i)}_{3}\qquad(0<i\leq k)\end{array}$
Transitions
$\frac{\dagger{\xlongrightarrow{\mathsf{q}}}_{1}q_{1}^{(0)}\qquad x\in
I}{\dagger{\xlongrightarrow{x}}(q_{1}^{(0)},x)}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\not\in\\{0,\cdots,\mathit{max}\\}\qquad
x\in
I}{((q_{1}^{(0)},x),\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}((r_{1}^{(0)},x),\cdots,r_{1}^{(j^{\prime})})}.$
$\frac{q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger\qquad
i>0\qquad\dagger{\xlongrightarrow{x}}_{2}q_{2}^{(0)}\qquad
q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad
x\in I\qquad\mathsf{m}\not\in
I}{(q_{1}^{(0)},x){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
$\frac{q_{1}^{(0)}{\xlongrightarrow{0}}_{1}\dagger\qquad\dagger{\xlongrightarrow{x}}_{3}q_{3}^{(0)}\qquad
q_{3}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{3}(r_{3}^{(0)},\cdots,r_{3}^{(j^{\prime})})\qquad
x\in I\qquad\mathsf{m}\not\in
I}{(q_{1}^{(0)},x){\xlongrightarrow{\mathsf{m}}}(r_{3}^{(0)},\cdots,r_{3}^{(j^{\prime})})}$
$\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\not\in
I}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
$\frac{(q_{3}^{(0)},\cdots,q_{3}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{3}(r_{3}^{(0)},\cdots,r_{3}^{(j^{\prime})})\qquad\mathsf{m}\not\in
I}{(q_{3}^{(0)},\cdots,q_{3}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{3}^{(0)},\cdots,r_{3}^{(j^{\prime})})}$
None of the cases requires an adjustment of pointers, because the inherited
indices are accurate. Even-readiness follows directly from IH.
* •
${\Gamma}\vdash{{\bf while}\,M_{1}\,{\bf do}\,M_{2}:{\bf com}}$:
States
$Q^{(j)}=Q_{1}^{(j)}+Q_{2}^{(j)}\qquad 0\leq j\leq k$
Transitions
$\frac{\dagger{\xlongrightarrow{\mathsf{q}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}q_{1}^{(0)}}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{0}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\not\in\\{\mathsf{q},0,\cdots,\mathit{max}\\}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}$
$\frac{q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger\qquad
i>0\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},r_{2}^{(1)})\qquad\mathsf{m}\neq\mathsf{done}}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},r_{2}^{(1)})}$
$\frac{q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger\qquad
i>0\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{q}}}_{1}r_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{1}(u_{1}^{(0)},u_{1}^{(1)})\qquad\mathsf{m}\not\in\\{0,\cdots,\mathit{max}\\}}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}(u_{1}^{(0)},u_{1}^{(1)})}$
$\frac{q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger\qquad
i>0\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{q}}}_{1}r_{1}^{(0)}{\xlongrightarrow{0}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
—
$\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\not\in\\{\mathsf{run},\mathsf{done}\\}}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
$\frac{q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{q}}}_{1}q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},r_{1}^{(1)})\qquad\mathsf{m}\not\in\\{0,\cdots,\mathit{max}\\}}{q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},r_{1}^{(1)})}$
$\frac{q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{q}}}_{1}q_{1}^{(0)}{\xlongrightarrow{0}}_{1}\dagger}{q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
$\frac{q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger\qquad\dagger{\xlongrightarrow{\mathsf{q}}}_{1}q_{1}^{(0)}{\xlongrightarrow{i}}_{1}\dagger\qquad
i>0\qquad\dagger{\xlongrightarrow{\mathsf{run}}}_{2}r_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{2}(u_{2}^{(0)},u_{2}^{(1)})\qquad\mathsf{m}\neq\mathsf{done}}{q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}(u_{2}^{(0)},u_{2}^{(1)})}$
As before, no pointers need adjustment, even-readiness is inherited.
* •
${\Gamma}\vdash{!M_{1}:{\bf exp}}$
To model dereferencing, it suffices to explore the plays that start with
$\mathsf{read}$ in the automaton for $M_{1}$, the $\mathsf{read}$ gets
relabelled to $\mathsf{q}$.
States
$Q^{(j)}=Q_{1}^{(j)}\qquad(0\leq j\leq k)$
Transitions
$\frac{\dagger{\xlongrightarrow{\mathsf{read}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{q}}}q_{1}^{(0)}}\qquad\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{read},\mathsf{write}(i){},\mathsf{ok}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}\qquad$
Note that the second rule will also handle transitions with the tag $i$. No
pointer readjustment is needed, as the inherited pointers are accurate. Even-
readiness follows from IH.
* •
${\Gamma}\vdash{M_{1}\,\raisebox{0.27986pt}{:}{=}\,M_{2}:{\bf com}}$
For assignment, we first direct the computation into the automaton for $M_{2}$
and, depending on the final move $i$, continue in the automaton for $M_{1}$ as
if $\mathsf{write}(i)$ was played. This is similar to $M_{1};M_{2}$.
States
$\begin{array}[]{rcl}Q^{(i)}&=&Q^{(i)}_{1}+Q^{(i)}_{2}\qquad(0\leq i\leq
k)\end{array}$
Transitions
$\frac{\dagger{\xlongrightarrow{\mathsf{q}}}_{2}q_{2}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}q_{2}^{(0)}}$
$\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\not\in\\{0,\cdots,\mathit{max}\\}}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}$
$\frac{q_{2}^{(0)}{\xlongrightarrow{i}}_{2}\dagger\qquad
i\in\\{0,\cdots,\mathit{max}\\}\qquad\dagger{\xlongrightarrow{\mathsf{write}(i)}}_{1}q_{1}^{(0)}\qquad
q_{1}^{(0)}{\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{ok}}{q_{2}^{(0)}{\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\not\in\\{\mathsf{read},\mathsf{write}(0),\cdots,\mathsf{write}(\mathit{max}),0,\cdots,\mathit{max},\mathsf{ok}\\}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}$
$\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{ok}}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
None of the cases requires an adjustment of pointers, because the inherited
indices are accurate.
* •
${\Gamma}\vdash{{\bf grab}(M_{1}):{\bf com}}$: $Q^{(j)}=Q_{1}^{(j)}$ ($0\leq
j\leq k$). Here we simply need to direct the automaton to perform the same
transitions as $M_{1}$ would, starting from $\mathsf{grb}$. At the same time,
$\mathsf{grb}$ and the corresponding answer $\mathsf{ok}$ have to be
relabelled as $\mathsf{run}$ and $\mathsf{done}$ respectively.
$\frac{\dagger{\xlongrightarrow{\mathsf{grb}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}q_{1}^{(0)}}\qquad\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{grb},\mathsf{rls},\mathsf{ok}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{ok}}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
* •
${\Gamma}\vdash{{\bf release}(M_{1}):{\bf com}}$: $Q^{(j)}=Q_{1}^{(j)}$
($0\leq j\leq k$). Here we simply need to direct the automaton to perform the
same transitions as $M_{1}$ would, starting from $\mathsf{rls}$. At the same
time, $\mathsf{rls}$ and the corresponding answer $\mathsf{ok}$ have to be
relabelled as $\mathsf{run}$ and $\mathsf{done}$ respectively.
$\frac{\dagger{\xlongrightarrow{\mathsf{rls}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{run}}}q_{1}^{(0)}}\qquad\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{grb},\mathsf{rls},\mathsf{ok}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{ok}}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}\dagger}$
* •
${\Gamma}\vdash{{\bf mkvar}(M_{1},M_{2}):{\bf var}}$. Recall that
${\Gamma}\vdash{M_{1}:{\bf exp}\rightarrow{\bf com}}$. Because we are using
terms in normal form $M_{1}=\lambda x^{\bf exp}.M_{1}^{\prime}$. For $0\leq
i\leq\mathit{max}$, consider $N_{i}=M_{1}^{\prime}[i/x]$, which is of smaller
size than $M_{1}$. Let us apply IH to $N_{i}$ and write $Q_{1i}^{(j)}$ and
${\xlongrightarrow{}}_{1i}$ for components of the resultant automaton.
Let $Q^{(j)}=\sum_{i=0}^{\mathit{max}}Q_{1i}^{(j)}+Q_{2}^{(j)}$ ($0<j\leq k$).
In this case, after $\mathsf{write}(i)$ we redirect transitions to the
automaton for $N_{i}$, and after $\mathsf{read}$ \- to $M_{2}$, relabelling
the initial and final moves as appropriate.
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1i}q_{1i}^{(0)}\qquad 0\leq
i\leq\mathit{max}}{\dagger{\xlongrightarrow{\mathsf{write}(i)}}q_{1i}^{(0)}}\qquad\frac{q_{1i}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1i}\dagger\qquad
0\leq i\leq\mathit{max}}{q_{1i}^{(0)}{\xlongrightarrow{\mathsf{ok}}}\dagger}$
$\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1i}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run},\mathsf{done}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}$
$\frac{\dagger{\xlongrightarrow{\mathsf{q}}}_{2}q_{2}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{read}}}q_{2}^{(0)}}\qquad\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{q},i}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}\qquad\frac{q_{2}^{(0)}{\xlongrightarrow{i}}_{2}\dagger}{q_{2}^{(0)}{\xlongrightarrow{i}}\dagger}$
* •
${\Gamma}\vdash{{\bf mksem}(M_{1},M_{2}):{\bf sem}}$.
$Q^{(j)}=Q_{1}^{(j)}+Q_{2}^{(j)}$ ($0\leq j\leq k$). In this case, after
$\mathsf{grb}$ we redirect transitions to the automaton for $M_{1}$, and after
$\mathsf{rls}$ \- to $M_{2}$.
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{1}q_{1}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{grb}}}q_{1}^{(0)}}\qquad\frac{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{1}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run},\mathsf{done}}{(q_{1}^{(0)},\cdots,q_{1}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{1}^{(0)},\cdots,r_{1}^{(j^{\prime})})}\qquad\frac{q_{1}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{1}\dagger}{q_{1}^{(0)}{\xlongrightarrow{\mathsf{ok}}}\dagger}$
$\frac{\dagger{\xlongrightarrow{\mathsf{run}}}_{2}q_{2}^{(0)}}{\dagger{\xlongrightarrow{\mathsf{rls}}}q_{2}^{(0)}}\qquad\frac{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}_{2}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})\qquad\mathsf{m}\neq\mathsf{run},\mathsf{done}}{(q_{2}^{(0)},\cdots,q_{2}^{(j)}){\xlongrightarrow{\mathsf{m}}}(r_{2}^{(0)},\cdots,r_{2}^{(j^{\prime})})}\qquad\frac{q_{2}^{(0)}{\xlongrightarrow{\mathsf{done}}}_{2}\dagger}{q_{2}^{(0)}{\xlongrightarrow{\mathsf{ok}}}\dagger}$
### 0.D.2 Example
Here is a worked example of Theorem 6.1 for the term $t=$
${f:{\bf com}\rightarrow{\bf com}}\vdash{{\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf
in}\,(f(x\,\raisebox{0.27986pt}{:}{=}\,1||x\,\raisebox{0.27986pt}{:}{=}\,13)\,||\,{\bf
if}\,!x=13\,{\bf then}\,\,{\bf skip}\,\,{\bf else}\,\,{\bf div})}$
We will show some simple subterms of this term, and then how to combine them
using $||$ and introduce newvar. We will first construct the sub-automaton
representing the following subterm:
$f(x\,\raisebox{0.27986pt}{:}{=}\,1||x\,\raisebox{0.27986pt}{:}{=}\,13)$
For convenience we will call this subterm $w$ as in “write”. The states for
$\mathcal{A}(w)$ are as follows:
$\displaystyle Q_{w}^{(0)}$ $\displaystyle=\\{0_{w},1_{w},2_{w}\\}$
$\displaystyle\qquad Q_{w}^{(1)}$ $\displaystyle=\\{0_{w}\\}$ $\displaystyle
Q_{w}^{(2)}$
$\displaystyle=\\{0_{1},1_{1},2_{1}\\}\times\\{0_{13},1_{13},2_{13}\\}$
$\displaystyle\qquad Q_{w}^{(3)}$ $\displaystyle=\\{0_{1},0_{13}\\}$
Note: in the standard construction, the subterms will not be annotated with
the subscripts given. We show them here to emphasise that the union operation
performed by combining branches is the _disjoint_ union of the states from
each side.
The transitions for $\mathcal{A}(w)$ are as follows. When we write transitions
here, places where values are symbolic (e.g. $u$ or $v$) represent one
transition for every possible value that may appear in those places.
$\dagger{\xlongrightarrow{\mathsf{run}}}0_{w}\qquad
2_{w}{\xlongrightarrow{\mathsf{done}}}\dagger$
$0_{w}{\xlongrightarrow{\mathsf{run}^{(f,0)}}}(1_{w},0_{w})\qquad(1_{w},0_{w}){\xlongrightarrow{\mathsf{done}^{(f,0)}}}2_{w}$
$(1_{w},0_{w}){\xlongrightarrow{\mathsf{run}^{(f1,0)}}}(1_{w},0_{w},(0_{1},0_{13}))\qquad(1_{w},0_{w},(2_{1},2_{13})){\xlongrightarrow{\mathsf{done}^{(f1,0)}}}(1_{w},0_{w})$
$(1_{w},0_{w},(0_{1},v)){\xlongrightarrow{\mathsf{write}(1)^{(x,2)}}}(1_{w},0_{w},(1_{1},v),0_{1})\qquad(1_{w},0_{w},(1_{1},v),0_{1}){\xlongrightarrow{\mathsf{ok}^{(x,0)}}}(1_{w},0_{w},(2_{1},v))$
$(1_{w},0_{w},(u,0_{13})){\xlongrightarrow{\mathsf{write}(1)^{(x,2)}}}(1_{w},0_{w},(u,1_{13}),0_{13})\qquad(1_{w},0_{w},(u,1_{13}),0_{13}){\xlongrightarrow{\mathsf{ok}^{(x,0)}}}(1_{w},0_{w},(u,2_{13}))$
where $u\in\\{0_{1},1_{1},2_{1}\\}$ and $v\in\\{0_{13},1_{13},2_{13}\\}$.
We now do the same for the following term, $r$ (for “read”):
${\bf if}\,!x=13\,{\bf then}\,\,{\bf skip}\,\,{\bf else}\,\,{\bf div}$
The states for $\mathcal{A}(r)$ are simpler, as this term is shallow.
$Q_{r}^{(0)}=\\{0_{r},1_{r},2_{r}^{0},\cdots,2_{r}^{\mathit{max}}\\}\qquad
Q_{r}^{(1)}=\\{0_{r}\\}$
The transitions for $\mathcal{A}(r)$ are as follows.
$\dagger{\xlongrightarrow{\mathsf{run}}}0_{r}\qquad
2_{r}^{13}{\xlongrightarrow{\mathsf{done}}}\dagger$
$0_{r}{\xlongrightarrow{\mathsf{read}^{(x,0)}}}(1_{r},0_{r})\qquad(1_{r},0_{r}){\xlongrightarrow{z^{(x,0)}}}2_{r}^{z}$
where $z\in\\{0,\cdots,\mathit{max}\\}$. Observe that only reaching state
$2_{r}^{13}$ (hence, reading a value $13$ from $x$) will allow this automaton
to terminate.
Combining these two automata is relatively simple. We will first apply the
procedure for parallel composition ($||$), and then apply the newvar context.
See Theorem 6.1 for the precise workings of these steps. The final automaton
$\mathcal{A}(t)$ for our term $t$ is as follows.
States:
$\displaystyle Q^{(0)}=(Q^{\prime(0)}+Q^{\prime(0)}\times\\{lock\\})\times X$
$\displaystyle\text{ where }Q^{\prime(0)}=Q_{w}^{(0)}\times Q_{r}^{(0)}\text{
and }X=\\{0,\cdots,\mathit{max}\\}$
$Q^{(1)}=Q_{r}^{(1)}+Q_{w}^{(1)}\qquad\qquad Q^{(2)}=Q_{w}^{(2)}\qquad\qquad
Q^{(3)}=Q_{w}^{(3)}$
Transitions:
$\dagger{\xlongrightarrow{\mathsf{run}}}((0_{r},0_{w}),0)\qquad((2_{w},2_{r}^{13}),n){\xlongrightarrow{\mathsf{done}}}\dagger$
$((0_{w},b),n){\xlongrightarrow{\mathsf{run}^{(f,0)}}}(((1_{w},b),n),0_{w})\qquad(((1_{w},b),n),0_{w}){\xlongrightarrow{\mathsf{done}^{(f,0)}}}((2_{w},b),n)$
$\displaystyle(((1_{w},b),n),0_{w})$
$\displaystyle{\xlongrightarrow{\mathsf{run}^{(f1,0)}}}(((1_{w},b),n),0_{w},(0_{1},0_{13}))$
$\displaystyle(((1_{w},b),n),0_{w},(2_{1},2_{13}))$
$\displaystyle{\xlongrightarrow{\mathsf{done}^{(f1,0)}}}(((1_{w},b),n),0_{w})$
$\displaystyle(((1_{w},b),n),0_{w},(0_{1},v))$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{Q}}}}(((1_{w},b),lock,1),0_{w},(1_{1},v),0_{1})$
$\displaystyle(((1_{w},b),lock,n),0_{w},(1_{1},v),0_{1})$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{A}}}}(((1_{w},b),n),0_{w},(2_{1},v))$
$\displaystyle(((1_{w},b),n),0_{w},(u,0_{13}))$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{Q}}}}(((1_{w},b),lock,13),0_{w},(u,1_{13}),0_{13})$
$\displaystyle(((1_{w},b),lock,n),0_{w},(u,1_{13}),0_{13})$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{A}}}}(((1_{w},b),n),0_{w},(u,2_{13}))$
$\displaystyle((a,0_{r}),n)$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{Q}}}}(((a,1_{r}),lock,n),0_{r})$
$\displaystyle(((a,1_{r}),lock,n),0_{r})$
$\displaystyle{\xlongrightarrow{\epsilon_{\mathsf{A}}}}((a,2_{r}^{n}),n)$
where $u\in\\{0_{1},1_{1},2_{1}\\}$, $v\in\\{0_{13},1_{13},2_{13}\\}$,
$a\in\\{0_{w},1_{w},2_{w}\\}$ and $b\in\\{0_{r},1_{r},2_{r}\\}$.
## Appendix 0.E Additional material for Section 7
### 0.E.1 Proof of Theorem 7.1
We start with a technical lemma that identifies the level of moves
corresponding to free variables of type ${\bf var}$ and ${\bf sem}$. Given
$x:{\bf var}$, moves of the form $\mathsf{write}(i)^{(x,\rho)}$ and
$\mathsf{read}^{(x,\rho)}$ (by P) will be referred to as the associated
questions, while $\mathsf{ok}^{(x,\rho)}$ and $i^{(x,\rho)}$ (by O) will be
called the associated answers. We use analogous terminology for $x:{\bf sem}$:
the associated questions are $\mathsf{grb}^{(x,\rho)}$ and
$\mathsf{rls}^{(x,\rho)}$, while the associated answer is
$\mathsf{ok}^{(x,\rho)}$.
###### Lemma 7
Given a $\mathsf{FICA}$-term ${\Gamma}\vdash{M:\theta}$ in $\beta\eta$-normal
form, let $\mathcal{A}_{M}$ be the automaton produced by Theorem 6.1. For any
$x:{\bf var}$ or $x:{\bf sem}$ such that $\mathit{ad}_{x}(M)=i$, the
transitions corresponding to the moves associated with $x$ add/remove leaves
at odd levels $1,3,\cdots,2i-1$.
###### Proof
We reason by induction on $M$, inspecting each construction in turn.
For $M\equiv{\bf skip},{\bf div},i$, the result holds vacuously, because there
are no moves associated with $x$ ($i=0$).
In the following cases, $\mathit{ad}_{x}(M)$ is calculated by taking the
maximum of $\mathit{ad}_{x}(M^{\prime})$ for subterms and the automata
constructions never modify the level of transitions in automata obtained by
IH. Consequently, the lemma can be established by appeal to IH:
$M_{1}||M_{2}$, $M_{1};M_{2}$, ${\bf if}\,M_{1}\,{\bf then}\,M_{2}\,{\bf
else}\,M_{3}$, ${\bf while}\,M_{1}\,{\bf do}\,M_{2}$, $!M_{1}$,
$M_{1}\,\raisebox{0.27986pt}{:}{=}\,M_{2}$, ${\bf grab}(M_{1})$, ${\bf
release}(M_{1})$, ${\bf newvar}\,y\,{\bf in}\,M_{1}$, ${\bf newsem}\,y\,{\bf
in}\,M_{1}$.
The remaining case is $M\equiv fM_{h}\cdots M_{1}$.
* •
Note that this case also covers $f\equiv x$, in which case
$\mathit{ad}_{x}(M)=1$ and transitions associated with $x$ involved leaves at
level $2\cdot 1-1=1$, as required.
* •
If $f\not\equiv x$ then
$\mathit{ad}_{x}(M)=1+\max(\mathit{ad}_{x}(M_{1}),\cdots,\mathit{ad}_{x}(M_{h}))$.
In this case, the automata construction lowers transitions associated with $x$
by exactly two levels, so by IH, they will appear at levels
$1+2,\cdots,(2i-1)+2$. Note that $(2i-1)+2=2(i+1)-1$, i.e. the lemma holds.
Observe that subterms of $\mathsf{LFICA}$ terms are in $\mathsf{LFICA}$, i.e.
we can reason by structural induction.
###### Lemma 8
Suppose ${\Gamma}\vdash{M:\theta}$ is from $\mathsf{LFICA}$. The automaton
$\mathcal{A}_{M}$ obtained from the translation in Theorem 6.1 is presentable
as a $\mathsf{LLA}$.
###### Proof
In many cases, the construction merely relabels the given automaton. Then a
simple appeal to the inductive hypothesis will suffice. The relevant cases
are: $!M_{1},\mathbf{op}(M_{1}),{\bf release}(M_{1}),{\bf grab}(M_{1}),\lambda
x.M_{1}$.
$M\equiv M_{1}||M_{2}$
The case of parallel composition involves running copies of $M_{1}$ and
$M_{2}$ in parallel without communication, with their root states stored as a
pair at level $0$. Note, though, that each of the automata transitions
independently of the state of the other automaton, which means that, if the
automata $M_{1}$ and $M_{2}$ are $\mathsf{LLA}$, so will be the automaton for
$M_{1}||M_{2}$. The branching bound after the construction is the sum of the
two bounds for $M_{1}$ and $M_{2}$.
$M\equiv M_{1};M_{2}$
The construction schedules the automaton for $M_{1}$ first and there is a
transition to (a disjoint copy of) the second one only after the configuration
of the first automaton consists of the root only. Otherwise the automata never
communicate. As the transition from the first to the second automaton happens
at the root, it can be captured as a $\mathsf{LLA}$ transition. Consequently,
if the automata for $M_{1},M_{2}$ are $\mathsf{LLA}$, so is the automaton for
$M$. Here the branching bound is simply the maximum of the bounds for $M_{1}$
and $M_{2}$.
The same argument applies to ${\bf if}\,M_{1}\,{\bf then}\,M_{2}\,{\bf
else}\,M_{3}$, $M_{1}\,\raisebox{0.27986pt}{:}{=}\,M_{2}$.
$M\equiv{\bf newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M_{1}$
Transitions not associated with $x$ are embedded into the automaton for $M$
except that at level $0$, the new automaton keeps track of the current value
stored in $x$. Because these transitions proceed uniformly without ever
depending on the value stored at the root, this is consistent with
$\mathsf{LLA}$ behaviour.
For transitions associated with $x$, we note that, because $M$ is from
$\mathsf{LFICA}$, we have $\mathit{ad}_{x}(M_{1})\leq 2$. By Lemma 7, this
means that the transitions related to $x$ correspond to creating/removing
leaves at either level $1$ or $3$. These transitions need to read/write the
root but, because they concern nodes at level $0$ or $3$, they will be
consistent with the definition of a $\mathsf{LLA}$. All other transitions (not
labelled by $x$) proceed as in $M$ and need not consult the additional
information about the current state stored in the root (the extra information
is simply propagated). Consequently, if $M$ is represented by a $\mathsf{LLA}$
then the interpretation of ${\bf
newvar}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf in}\,M$ is also a
$\mathsf{LLA}$. The construction does not affect the branching bound, because
the resultant runs can be viewed as a subset of runs of the automaton for $M$,
i.e. those in which reads and writes are related.
The case of $M\equiv{\bf newsem}\,x\,\raisebox{0.27986pt}{:}{=}\,i\,{\bf
in}\,M_{1}$ is analogous.
$M\equiv fM_{h}\cdots M_{1}$
For $fM_{h}\cdots M_{1}$, we observe that the construction first creates two
nodes at levels $0$ and $1$, and the node at level $1$ is used to run an
unbounded number of copies of (the automaton for) $M_{i}$. The copies do not
need access to the states stored at levels $0$ and $1$, because they are never
modified when the copies are running. Consequently, if each $M_{i}$ can be
translated into a $\mathsf{LLA}$, the outcome of the construction in Theorem
6.1 is also a $\mathsf{LLA}$. The new branching bound is the maximum over
bounds from $M_{1},\cdots,M_{h}$, because at even levels children are produced
as in $M_{i}$ and level $0$ produces only $1$ child.
## Appendix 0.F Additional material for Section 8
#### Word representation
Let $\mathcal{A}=\langle\Sigma,k,Q,\delta\rangle$ be a leafy automaton. We
shall assume that $\Sigma,Q\subseteq\\{0,\cdots,\mathit{max}\\}$ so that we
can encode the alphabet and states using type ${\bf exp}$. First we discuss
how to assign a play $\mathsf{play}(w)$ to a trace $w$ of $\mathcal{A}$. The
basic idea is to simulate each transition with two moves, by $O$ and $P$
respectively. The child-parent links in $\mathcal{D}$ will be represented by
justification pointers.
* •
Suppose $w=w^{\prime}(t,d)$ with $t\in\Sigma_{\mathsf{Q}}$. We will represent
$(t,d)$ by a segment of the form
$\rnode{A}{{\mathsf{q}}^{{\vec{i}}}}\quad\rnode{B}{\mathsf{run}}^{t\vec{i}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$.
If $w^{\prime}=\epsilon$, we let
$\mathsf{play}(w)=\rnode{A}{\mathsf{q}}\,\,\rnode{B}{\mathsf{run}^{t}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$,
i.e. $\vec{i}=\epsilon$. If $w^{\prime}\neq\epsilon$ then, because $w$ is a
trace, $w^{\prime}$ must contain a unique occurrence of
$(t^{\prime},\mathit{pred}(d))$ for some $t^{\prime}\in\Sigma_{\mathsf{Q}}$.
Then, if $(t^{\prime},\mathit{pred}(d))$ was represented by
$\rnode{A}{\mathsf{q}^{\vec{i^{\prime}}}}\rnode{B}{\mathsf{run}}^{t^{\prime}\vec{i^{\prime}}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=130.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$
in $\mathsf{play}(w^{\prime})$, we let
$\mathsf{play}(w)=\rnode{C}{\mathsf{play}(w^{\prime})}\,\,\,\rnode{K}{\rnode{D}{{\mathsf{q}}}^{1t^{\prime}{\vec{i^{\prime}}}}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=100.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{D}{C}}\,\,\,\rnode{B}{\mathsf{run}}^{t1t^{\prime}\vec{i^{\prime}}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=100.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{K}}$,
where $\mathsf{q}^{1t^{\prime}\vec{i^{\prime}}}$ points at
$\mathsf{run}^{t^{\prime}\vec{i^{\prime}}}$.
* •
Suppose $w=w^{\prime}(t,d)$ with $t\in\Sigma_{\mathsf{A}}$. Because $w$ is a
trace, $w^{\prime}$ must contain a unique occurrence $(t^{\prime},d)$ for some
$t^{\prime}\in\Sigma_{\mathsf{Q}}$. If $(t^{\prime},d)$ is represented by the
segment
$\rnode{A}{\mathsf{q}^{\vec{i}}}\rnode{B}{\mathsf{run}^{t^{\prime}\vec{i}}}{\psset{arrowsize=2.5pt
0.0,nodesep=0.5pt,offsetB=-2pt,linewidth=0.4pt,angleA=110.0,angleB=30.0,linecolor=darkgray}\nccurve{->}{B}{A}}$
in $\mathsf{play}(w^{\prime})$, we set
$\mathsf{play}(w)=\mathsf{play}(w^{\prime})\,\,\mathsf{done}^{t^{\prime}\vec{i}}\,\,t^{\vec{i}}$,
where the two answer-moves are justified by $\mathsf{run}^{t^{\prime}\vec{i}}$
and ${\mathsf{q}^{\vec{i}}}$ respectively. Because $w$ is a trace, we can be
sure that after processing $w^{\prime}$, $\mathcal{A}$ enters a configuration
in which $d$ is a leaf. Thus, the two answers will satisfy the game-semantic
$\mathsf{WAIT}$ condition, and $\mathsf{play}(w)$ will be well-defined.
The $\mathsf{FORK}$ condition is satisfied for $\mathsf{play}(w)$, because
reading an answer removes the corresponding data value from the configuration
and, hence, it cannot be used as a justifier afterwards. In what follows, we
write $\theta^{n}\rightarrow\beta$ for
$\underbrace{\theta\rightarrow\cdots\rightarrow\theta}_{n}\rightarrow\beta$
for $n\in\mathbb{N}$. The lemma below identifies the types that correspond to
our encoding of traces.
###### Lemma 9
Let $N=\mathit{max}+1$. Suppose $\mathcal{A}$ is a $k$-$\mathsf{LA}$ and
$w\in\mathit{Tr}(\mathcal{A})$. Then $\mathsf{play}(w)$ is a play in
${\llbracket}{\theta_{k}}{\rrbracket}$, where $\theta_{0}={\bf
com}^{N}\rightarrow{\bf exp}$ and $\theta_{i+1}=(\theta_{i}\rightarrow{\bf
com})^{N}\rightarrow{\bf exp}$ ($i\geq 0$).
### 0.F.1 Saturation
The game model [23] of $\mathsf{FICA}$ consists of _saturated_ strategies
only: the saturation condition stipulates that all possible (sequential)
observations of (parallel) interactions must be present in a strategy: actions
of the environment (O) can always be observed earlier if possible, actions of
the program (P) can be observed later. To formalize this, for any arena $A$,
we define a preorder $\preceq$ on $P_{A}$, as the least transitive relation
$\preceq$ satisfying $s\,o\,m\,s^{\prime}\preceq s\,m\,o\,s^{\prime}$ and
$s\,m\,p\,s^{\prime}\preceq s\,p\,m\,s^{\prime}$ for all $s,s^{\prime}$, where
$o$ and $p$ are an O- and a P-move respectively (in the above pairs of plays
moves on the left-hand-side of $\preceq$ are assumed to have the same
justifiers as on the right-hand-side).
###### Definition 11
A strategy $\sigma:A$ is _saturated_ iff, for all $s,s^{\prime}\in P_{A}$, if
$s\in\sigma$ and $s^{\prime}\preceq s$ then $s^{\prime}\in\sigma$.
###### Remark 3
Definition 11 states that saturated strategies are stable under certain
rearrangements of moves. Note that $s_{0}\,p\,o\,s_{1}\not\preceq
s_{0}\,o\,p\,s_{1}$, while other move-permutations are allowed. Thus,
saturated strategies express causal dependencies of P-moves on O-moves. This
partial-order aspect is captured explicitly in concurrent games based on event
structures [11].
### 0.F.2 Proof of Theorem 8.1
###### Proof
Our assumption $Q\subseteq\\{0,\cdots,\mathit{max}\\}$ allows us to maintain
$\mathcal{A}$-states in the memory of $\mathsf{FICA}$-terms. A question
$t_{\mathsf{Q}}^{(i)}$ read by $\mathcal{A}$ at level $i$ is represented by
the variable $f^{(i)}_{t_{\mathsf{Q}}^{(i)}}$, the corresponding answers
$t_{\mathsf{A}}^{(i)}$ are represented by constants $t_{\mathsf{A}}^{(i)}$
(using our assumption $\Sigma\subseteq\\{0,\cdots,\mathit{max}\\}$). The level
$i$ of the data tree is encoded by the order of the variable
$f^{(i)}_{t_{\mathsf{Q}}^{(i)}}$. For $0\leq i<k$, the variables $f_{t}^{(i)}$
are meant to have type $\theta_{k-i-1}\rightarrow{\bf com}$ and
$f_{t}^{(k)}:{\bf com}$. This ensures that questions and answers respect the
tree structure on data. To achieve nesting, we rely on a higher-order
structure of the term: $\lambda f^{(0)}.f^{(0)}(\lambda
f^{(1)}.f^{(1)}(\lambda f^{(2)}.f^{(2)}(\cdots\lambda f^{(k)}.f^{(k)})))$.
Recall that the semantics of $fM$ consists of an arbitrary number of
interleavings of $M$. This feature is used to mimic the fact that a leafy
automaton can spawn unboundedly many offspring. Finally, instead of single
variables $f^{(i)}$, we will actually use sequences $f^{(i)}_{0}\cdots
f^{(i)}_{\mathit{max}}$, which will be used to induce the right move
$\mathsf{run}^{t\vec{i}}$ when representing
$t\in\Sigma_{\mathsf{Q}}\subseteq\\{0,\cdots,\mathit{max}\\}$. Additionally,
the term contains state-manipulating code that enables $P$-moves only if they
are consistent with the transition function of $\mathcal{A}$. To achieve this,
every level is equipped with a local variable $X^{(i)}$ of type ${\bf exp}$,
so that states on a single branch are represented by
$\overrightarrow{X^{(i)}}=(X^{(0)},\cdots,X^{(i)})$.
Given $\alpha\in\\{\mathsf{Q},\mathsf{A}\\}$ and $-1\leq j\leq k$, we write
$\overrightarrow{r_{\alpha}^{(j)}}$ for a tuple of values
$(r_{\alpha}^{(0)},\cdots,r_{\alpha}^{(j)})$ on the understanding that
$\overrightarrow{r_{\alpha}^{(-1)}}=\dagger$. A similar convention will apply
to $\overrightarrow{u_{\alpha}^{(j)}}$. Then we use
$\overrightarrow{X^{(i)}[{u_{\alpha}^{(j^{\prime})}}/{r_{\alpha}^{(j)}}]}$,
where $-1\leq j,j^{\prime}\leq i$, as shorthand for $\mathsf{FICA}$ code that
checks componentwise whether the values of $\overrightarrow{X^{(j)}}$ equal
$\overrightarrow{r_{\alpha}^{(j)}}$ and, if so, updates
$\overrightarrow{X^{(j^{\prime})}}$ to
$\overrightarrow{u_{\alpha}^{(j^{\prime})}}$ (if the check fails, the code
should diverge). For $j=-1$ (resp. $j^{\prime}=-1$), there is nothing to check
(resp. update). All occurrences of
$\overrightarrow{X^{(i)}[{u_{\alpha}^{(j^{\prime})}}/{r_{\alpha}^{(j)}}]}$
will be protected by a semaphore to ensure mutual exclusion. Consequently,
they will induce exactly the causal dependencies (cf. Remark 3) consistent
with sequences of $\mathcal{A}$-transitions, i.e. with the shape of
$\mathsf{play}(w)$ for some $w\in\mathit{Tr}(\mathcal{A})$. To select
transitions at each stage, we rely on non-deterministic choice $\bigoplus$,
which can be encoded in $\mathsf{FICA}$222 $M_{1}\oplus M_{2}={\bf
newvar}\,X\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf
in}\,((X\,\raisebox{0.27986pt}{:}{=}\,0\,||\,X\,\raisebox{0.27986pt}{:}{=}\,1);{\bf
if}\,!X\,{\bf then}\,M_{1}\,{\bf else}\,M_{2})$..
Below we define inductively a family of terms ${}\vdash{M_{i}:\theta_{k-i}}$
($0\leq i\leq k$). Term $M_{\mathcal{A}}$ is then obtained by making a simple
change to $M_{0}$. For any $0\leq i\leq k$, let $M_{i}$ be the term
$\begin{array}[]{rl}\lambda f_{0}^{(i)}\cdots f_{\mathit{max}}^{(i)}.&{\bf
newvar}\,X^{(i)}\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\\\
\bigoplus\limits_{(\overrightarrow{r_{\mathsf{Q}}^{(i-1)}},{\displaystyle
t_{\mathsf{Q}}^{(i)}},\overrightarrow{u_{\mathsf{Q}}^{(i)}})\in\delta_{\mathsf{Q}}^{(i)}}&\Big{(}{\bf
grab}(s);\overrightarrow{X^{(i)}[{u_{\mathsf{Q}}^{(i)}}/{r_{\mathsf{Q}}^{(i-1)}}]};{\bf
release}(s);\quad f_{t_{\mathsf{Q}}^{(i)}}^{(i)}\,M_{i+1};\\\\[-11.38109pt]
&\quad\bigoplus_{(\overrightarrow{r_{\mathsf{A}}^{(i)}},{\displaystyle
t_{\mathsf{A}}^{(i)}},\overrightarrow{u_{\mathsf{A}}^{(i-1)}})\in\delta_{\mathsf{A}}^{(i)}}\big{(}{\bf
grab}(s);\overrightarrow{X^{(i)}[{u_{\mathsf{A}}^{(i-1)}}/{r_{\mathsf{A}}^{(i)}}]};{\bf
release}(s);t_{\mathsf{A}}^{(i)}\big{)}\Big{)}.\end{array}$
We write $M_{k+1}$ for empty space (this is for a good reason, because
$f_{t}^{(k)}:{\bf com}$). The above term $M_{i}$ declares a new variable to
store the state, and then makes a non-deterministic choice for question
transitions that create data values at level $i$. The update of the state is
protected by a semaphore. Then the appropriate $f^{(i)}_{t}$ is applied to
term $M_{i+1}$ that simulates moves of the automaton on data in the subtree of
the freshly created node. This is followed by the code making a non-
deterministic choice over all answer transitions. To define $M_{\mathcal{A}}$,
it now suffices to declare the semaphore in $M_{0}$, i.e. given $M_{0}=\lambda
f_{0}^{(0)}\cdots f_{\mathit{max}}^{(0)}.{{\bf
newvar}\,X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,M}$ we let
$M_{\mathcal{A}}$ be
$\lambda f_{0}^{(0)}\cdots f_{\mathit{max}}^{(0)}.{\bf
newsem}\,s\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,{\bf
newvar}\,X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,M.$
###### Example 7
We illustrate the outcome of the construction from Theorem 8.1 for $k=1$.
$\begin{array}[]{lll}\lambda f_{0}^{(0)}\cdots
f_{\mathit{max}}^{(0)}.&\lx@intercol{\bf
newsem}\,s\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\,{\bf
newvar}\,X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}\hfil\lx@intercol\\\
\bigoplus\limits_{(\dagger,{t_{\mathsf{Q}}^{(0)}},{u_{\mathsf{Q}}^{(0)}})\in\delta_{\mathsf{Q}}^{(0)}}&\lx@intercol\Bigg{(}{\bf
grab}(s);\,\overrightarrow{X^{(0)}[{u_{\mathsf{Q}}^{(0)}}/\dagger]};\,{\bf
release}(s);\hfil\lx@intercol\\\ &\lx@intercol\hfil\quad
f_{t_{\mathsf{Q}}^{(0)}}^{(0)}\,\,\bigg{(}\,\,\lambda f_{0}^{(1)}\cdots
f_{\mathit{max}}^{(1)}.\lx@intercol&{{\bf
newvar}\,X^{(1)}\,\raisebox{0.27986pt}{:}{=}\,0\,{\bf in}}\\\
&\lx@intercol\hfil\bigoplus\limits_{({r_{\mathsf{Q}}^{(0)}},{t_{\mathsf{Q}}^{(1)}},\overrightarrow{u_{\mathsf{Q}}^{(1)}})\in\delta_{\mathsf{Q}}^{(1)}}\lx@intercol&{\Big{(}{\bf
grab}(s);\,\overrightarrow{X^{(1)}[{u_{\mathsf{Q}}^{(1)}}/{r_{\mathsf{Q}}^{(0)}}]};\,{\bf
release}(s);\,\,f_{t_{\mathsf{Q}}^{(1)}}^{(1)};}\\\
&&{\quad\bigoplus_{(\overrightarrow{r_{\mathsf{A}}^{(1)}},t_{\mathsf{A}}^{(1)},{u_{\mathsf{A}}^{(0)}})\in\delta_{\mathsf{A}}^{(1)}}}\big{(}{\bf
grab}(s);\,\overrightarrow{X^{(1)}[{u_{\mathsf{A}}^{(0)}}/{r_{\mathsf{A}}^{(1)}}]};\,{\bf
release}(s);\,t_{\mathsf{A}}^{(1)}\big{)}\Big{)}\bigg{)};\\\
&\lx@intercol\quad\bigoplus_{({r_{\mathsf{A}}^{(0)}},t_{\mathsf{A}}^{(0)},\dagger)\in\delta_{\mathsf{A}}^{(0)}}\big{(}{\bf
grab}(s);\,\overrightarrow{X^{(0)}[\dagger/{r_{\mathsf{A}}^{(0)}}]};\,{\bf
release}(s);\,t_{\mathsf{A}}^{(0)})\Bigg{)}\hfil\lx@intercol\\\ \end{array}$
where
$\begin{array}[]{rcl}\overrightarrow{X^{(0)}[{u_{\mathsf{Q}}^{(0)}}/\dagger]}&=&X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,u_{\mathsf{Q}}^{(0)}\\\
\overrightarrow{X^{(1)}[{u_{\mathsf{Q}}^{(1)}}/{r_{\mathsf{Q}}^{(0)}}]}&=&{\bf
if}\,(X^{(0)}=r_{\mathsf{Q}}^{(0)})\,{\bf
then}\,(X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,u_{\mathsf{Q}}^{(0)};X^{(1)}\,\raisebox{0.27986pt}{:}{=}\,u_{\mathsf{Q}}^{(1)})\,{\bf
else}\,\Omega\\\
\overrightarrow{X^{(1)}[{u_{\mathsf{A}}^{(0)}}/{r_{\mathsf{A}}^{(1)}}]}&=&{\bf
if}\,((X^{(0)}=r_{\mathsf{A}}^{(0)})\wedge(X^{(1)}=r_{\mathsf{A}}^{(1)}))\,{\bf
then}\,(X^{(0)}\,\raisebox{0.27986pt}{:}{=}\,u_{\mathsf{A}}^{(0)})\,{\bf
else}\,\Omega\\\
\overrightarrow{X^{(0)}[\dagger/{r_{\mathsf{A}}^{(0)}}]}&=&{\bf
if}\,(X^{(0)}=r_{\mathsf{A}}^{(0)})\,{\bf then}\,{\bf skip}\,{\bf
else}\,\Omega\\\ \Omega&=&{\bf while}\,1\,{\bf do}\,{\bf skip}\end{array}$
|
# High-accuracy longitudinal position measurement using self-accelerating
light
Shashi Prabhakar<EMAIL_ADDRESS>Stephen Plachta Marco Ornigotti
Robert Fickler Photonics Laboratory, Physics Unit, Tampere University,
Tampere, FI-33720, Finland
###### Abstract
Radially self-accelerating light exhibits an intensity pattern that describes
a spiraling trajectory around the optical axis as the beam propagates. In this
article, we show in simulation and experiment how such beams can be used to
perform a high-accuracy distance measurement with respect to a reference using
simple off-axis intensity detection. We demonstrate that generating beams
whose intensity pattern simultaneously spirals with fast and slow rotation
components enables a distance measurement with high accuracy over a broad
range, using the high and low rotation frequency, respectively. In our
experiment, we achieve an accuracy of around 2 $\mu$m over a longitudinal
range of more than 2 mm using a single beam and only two quadrant detectors.
As our method relies on single-beam interference and only requires a static
generation and simple intensity measurements, it is intrinsically stable and
might find applications in high-speed measurements of longitudinal position.
## I Introduction
Structuring the spatial shape of light fields has become a broad research
field spanning areas from the foundations of optics to optical communication,
materials processing, quantum optics, and microscopy, to name a few
rubinsztein2016roadmap . Amongst many interesting features structured light
may have, one in particular has attracted a lot of attention and might even be
seen as the starting point of the field, namely the azimuthal phase structure
connected to the orbital angular momentum (OAM) of light padgett2017orbital .
Light fields carrying such OAM have a transverse phase of the form
$\exp(-i\ell\varphi)$, where $\varphi$ is the azimuthal coordinate and $\ell$
defines the quanta of OAM each photon carries allen1992orbital . These
transverse scalar modes are commonly known as vortex modes, or donut beams as
they have a phase singularity and, thus, an intensity null along the optical
axis. Over the last decades, various techniques to imprint such twisted
structures have been established, e.g., spiral phase plates
beijersbergen1994helical ; holographic generation using spatial light
modulators heckenberg1992generation ; carpentier2008making ;
forbes2016creation ; cylindrical lenses beijersbergen1993astigmatic ; $q$\-
and $J$-plates marrucci2006optical ; larocque2016arbitrary ;
devlin2017arbitrary ; or direct generation of the light field inside the
cavity of a laser forbes2019structured .
One particular family of modes, whose higher orders have an azimuthal phase
ramp, is known as Bessel beams. Bessel beams are propagation invariant light
fields described by Bessel functions gori1987bessel . They have received
significant attention due to being diffraction-less zahid1989directionality ;
vetter2019realization and self-healing chu2012analytical ; mcgloin2005bessel
. While zero-order Bessel beams have an intensity maximum along the optical
axis and do not carry OAM, higher orders have a twisted phase structure
leading to well-defined quanta of OAM per photon. In addition, Bessel beams
are comparatively easy to generate using a ring aperture, with or without an
azimuthally varying phase, at the back focal plane of a lens ($k$-space). The
light fields in the focal plane of the lens, which have undergone an optical
Fourier transformation, then resemble the theoretical Bessel beams; however,
they have a finite beam extent and are therefore called Bessel-Gauss beams. If
not only one ring but multiple rings of different radii and different orders
are put into the back focal plane of the lens, a superposition of multiple
Bessel beams is generated. Interestingly, the obtained superposition
structures show the peculiar feature of spiraling around the optical axis
along the propagation direction if the constituents forming the superposition
have different OAM values chavez1996nondiffracting ; schechner1996wave ;
abramochkin1997generation ; paakkonen1998rotating . This property of complex
superpositions of higher-order Bessel beams has been the focus of various
research efforts. Thorough theoretical and experimental studies of such
spiraling beams have been performed, which have been enabled by the progress
in experimental techniques of generating such beams with high precision and
flexibility tervo2001rotating ; kotlyar2007rotation ; vasilyeu2009generating ;
vetter2015optimization ; schulze2015accelerated .
Contrary to Airy beams, whose self-accelerating character is essentially given
by the fact that an observer in a reference frame solidal with an Airy beam
would experience a tangential fictitious force greenberger1980comment that
results in their characteristic parabolic propagation profile, radially self-
accelerating beams (RSABs) are characterised by a centrifugal fictitious force
linked to their characteristic spiraling motion vetter2014generalized . As
such they are also distinct from another class of self-accelerating beams,
recently investigated in vetter2017real ; webster2017radially .
In this work, we demonstrate a novel application of spiraling beams as a means
to determine the longitudinal distance with high accuracy by only measuring
the intensity using a quadrant detector, i.e., in a limited number of off-axis
locations. First, we briefly introduce the theory behind spiraling beams and
describe how superpositions of three Bessel modes can lead to a complex
rotating pattern having both quickly and slowly rotating parts at the same
time. We then show in simulations and experiments that using such a spiraling
beam enables the accurate determination of distance over a long range when the
intensity using two quadrant detectors (or in minimally three off-axis
positions) is recorded. In the experimental implementation, we are able to
achieve an accuracy of around 2 $\mu$m over a range of 2 mm. The obtained
result is mainly limited by the aperture of our optical system, as well as the
resolution of the generating and detecting devices. Hence, the proposed and
demonstrated method of measuring a longitudinal distance using structured
light might find promising applications, similar to self-accelerating Airy
beams, which for example have been used to resolve depth in microscopy
applications recently jia2014isotropic ; he2019depth . Due to its simplicity
and high accuracy, our method nicely complements other available techniques
using light, e.g., time-of-flights measurements as used in LIDAR systems
jarvis1983laser , interferometric approaches kubota1987interferometer , or
schemes that rely on complex scattering of structured light fields
berg2020microsphere , to name a few.
## II Theoretical background
### II.1 Spiraling light fields
Measuring the longitudinal position, i.e., a certain distance with respect to
a fixed reference, using an intensity structure that changes over propagation
requires a light field with well-defined propagation dynamics. The recently
demonstrated radially self-accelerating, or spiraling, light fields, which
show a constant rotation of the intensity pattern along the propagation
direction, are a very convenient solution to use. While light with more
complex propagation dynamics would require sophisticated evaluation
procedures, rotating structures allow the determination of the longitudinal
position through simple measurements of the rotation angle. Importantly, such
light fields can be easily realized by superimposing (at least) two vortex
beams, each having a different OAM value $\ell$ as well as two different
longitudinal wave vectors $k_{z}$ defining the propagation dynamics
vetter2014generalized . The two-component solution, i.e., the so-called
helicon beams, can then be written as
$u(r,\phi,z)=A_{\ell_{1}}(r)\exp{[i(k_{1z}z+\ell_{1}\varphi)]}+A_{\ell_{2}}(r)\exp{[i(k_{2z}z+\ell_{2}\varphi)]},$
(1)
where the indices $1,2$ label the two constituents and $A_{\ell}(r)$ is a
radially-dependent envelope function. The resulting intensity,
$I=u(r,\varphi,z)u(r,\varphi,z)^{*}$, can then be obtained to be
$I(r,\varphi,z)\propto\cos^{2}{[\Delta kz+\Delta\ell\varphi]},$ (2)
where $\Delta k=(k_{1z}-k_{2z})/2$ is the difference between the wave vectors
of the two beams and $\Delta\ell=(\ell_{1}-\ell_{2})/2$ is the difference
between their OAM values. Notice, moreover, that we have only kept the part of
the intensity that is of importance, i.e., the one including the required
angular and $z$-dependence. For more details we refer the interested reader to
earlier works vetter2014generalized . From (2) we find that the angular
orientation of the intensity profile $\phi(z)$ changes along the beam
propagation according to the relation
$\phi(z)=\frac{z\Delta k}{\Delta\ell},$ (3)
from which the angular velocity can be calculated as
$\omega=\frac{\partial\phi(z)}{\partial z}=\frac{\Delta k}{\Delta\ell}.$ (4)
One such beam propagation is shown in Fig. 1, where the spiraling of the mode
along the propagation axis is depicted.
Figure 1: Normalised three-dimensional representation of the propagation of
the central lobe of a radially self-accelerating beam along the $z$ direction,
as defined in (1), consisting of the superposition of $\ell_{1}=0$ and
$\ell_{2}=1$ Bessel modes, which matches the form of the single-frequency beam
used in the experiment. The transverse directions $\\{k_{0}x,k_{0}y\\}$ are
normalised to the central $k$-vector of the beam, i.e., $k_{0}$, while the
longitudinal direction (i.e., the propagation direction $z$) has been
normalised to the rotation period $\Lambda=2\pi/\omega$, with $\omega$ being
the rotation speed defined by (4). The plot shows propagation up to the first
two periods. The colour scale in the picture represents different iso-
intensity surfaces, with brighter colours indicating regions of higher
intensity. As can be seen, the whole intensity distribution rigidly rotates
around the $z$ axis with rotation speed $2\pi/\Lambda$. This peculiar
propagation pattern is the result of interference between the various Bessel
beam components constituting the radially self-accelerating beam, as described
by (1).
In other words, if we measure the intensity over a certain angular region, the
intensity along the beam propagation follows a periodic $\cos^{2}$-function
and can be used to determine the longitudinal distance unambiguously within
half a period. In principle, a simple measurement of the intensity at an off-
axis transverse position therefore allows the determination of a distance with
arbitrary precision. In practical situations, however, errors induced by the
generation or measurement of the structure result in an uncertainty in the
determination of the intensity’s angular position, which leads to a limitation
of the longitudinal accuracy. One direct way to improve the measurement
accuracy despite these imperfections is to increase the rotation frequency of
the structure with respect to its propagation. The faster the rotation, the
better the accuracy in measuring the longitudinal distance $z$. Hence, one aim
in high-accuracy distance measurement using self-accelerating light fields is
to achieve the largest possible difference in the longitudinal wave vectors
$\Delta k$ allowed by the optical system. We also see that the difference in
the OAM values $\Delta\ell$ should be kept as small as possible, i.e., the
$\ell$-values of the two constituent beams should only differ by 1.
Obviously, improving the longitudinal accuracy by increasing the rotation
frequency comes at the cost of a reduced longitudinal range over which an
unambiguous determination of the angular position is possible by only
examining the intensity pattern. To circumvent this limitation, it is possible
to realize a more complex structure, which shows a rotating intensity pattern
that includes two (or more) well-defined rotation frequencies. Ideally, the
intensity structure should have one very high rotation frequency used to
obtain locally a high-accuracy measurement of the longitudinal position. The
intensity dynamics should further include a rotating structure with very low
frequency from which it is possible to determine the global distance and
discriminate between different fast-varying periods. Both frequencies need to
be adjusted such that each period of the high-accuracy measurement can be
distinguished from any other using the slow rotation pattern. In the
theoretical description, this idea can be implemented by adding a third term
to the equation earlier introduced (1), such that we obtain
$\displaystyle u(r,\phi,z)$ $\displaystyle=$ $\displaystyle
A_{\ell_{1}}(r)\exp{[i(k_{1z}z+\ell_{1}\varphi)]}$ (5)
$\displaystyle+A_{\ell_{2}}(r)\exp{[i(k_{2z}z+\ell_{2}\varphi)]}$
$\displaystyle+A_{\ell_{3}}(r)\exp{[i(k_{3z}z+\ell_{3}\varphi)]}.$
As can be seen, the electric field defined above contains three different
contributions, each characterised by its spatial frequency $k_{i}$ and OAM
$\ell_{i}$. If we now calculate the intensity distribution generated by such a
field, we will have, together with the contributions of the single terms in
(5) – i.e., terms proportional to $|A_{\ell_{k}}|^{2}$ – also all the possible
interference terms between the three beams composing the field above. We can
therefore write, neglecting the $z$\- independent terms, which amount only to
an overall normalisation factor (supplemental material of
vetter2014generalized ),
$\displaystyle I$ $\displaystyle\propto$ $\displaystyle\cos^{2}{[\Delta
k_{1,2}z+\Delta\ell_{1,2}\varphi]}+\cos^{2}{[\Delta
k_{1,3}z+\Delta\ell_{1,3}\varphi]}$ (6) $\displaystyle+\cos^{2}{[\Delta
k_{2,3}z+\Delta\ell_{2,3}\varphi]},$
where we labelled $\Delta k_{i,j}=(k_{iz}-k_{jz})/2$ and
$\Delta\ell_{i,j}=(\ell_{i}-\ell_{j})/2$ as the pairwise differences between
the wave vectors and OAM values of the three fields. By choosing
$\Delta\ell_{2,3}=0$, i.e., $\ell_{2}=\ell_{3}$, the angular dependence of the
propagation dynamics of the last term vanishes. Hence, we obtain the required
light field, whose structure rotates with only two rotation frequencies at the
same time.
### II.2 Experimental implementation
In an experiment, it is convenient to realize these radially accelerating
beams in the framework of Bessel beams, i.e., $A_{\ell}(r)=J_{\ell}(rk_{r})$
ornigotti2018vector ; rop2012measuring . For Bessel beams the longitudinal
wavevector $k_{z}$ can straightforwardly controlled by adjusting its radial
counterpart $k_{r}$. Both quantities are related through
$k_{z}=\sqrt{k^{2}-k_{r}^{2}},$ (7)
where $k=2\pi/\lambda$ labels the wavenumber and $\lambda$ corresponds to the
wavelength of the utilized light field.
As the angular spectrum of a Bessel beam forms a ring in $k$-space, such beams
are relatively simple to generate in the laboratory. By modulating an incoming
light field to have a ring-shaped amplitude at one plane ($k$-space), we can
transform the light into a Bessel beam (real space) by implementing an optical
Fourier transform using a properly placed lens. The lens is placed one focal
distance $f$ away from the initial modulation plane, i.e., in the back focal
plane of the lens, such that at around another focal distance behind the lens
a Fourier transform leads to a Bessel beam with a well-defined longitudinal
wave vector $k_{z}$. In the modulation plane is a phase-only spatial light
modulator (SLM) whose screen displays a holographic pattern, generated by a
MATLAB script, that modulates the amplitude and phase of the beam into the
required ring shape rosales2017shape . Depending on the radius $r$ of the ring
in the modulation plane, the Bessel beam with longitudinal wave vector
$k_{z}=\frac{2\pi}{\lambda}\cos{\left(\frac{r}{f}\right)}$ (8)
will be obtained, which follows from simple geometric arguments
vasilyeu2009generating ; rop2012measuring .
A radially self-accelerating beam, such as the one described above, follows
from simply modulating the light field to have two (or more) rings with
different radii $r_{i}$ and individual OAM values $\ell_{i}$. The spiraling
around the optical axis can thus be tuned by changing the radii of the two
rings, leading to a wave vector difference
$\displaystyle\Delta k_{i,j}$ $\displaystyle=$
$\displaystyle(k_{iz}-k_{jz})/2$ (9) $\displaystyle=$
$\displaystyle\frac{\pi}{\lambda}\left[\cos{\left(\frac{r_{i}}{f}\right)}-\cos{\left(\frac{r_{j}}{f}\right)}\right].$
As discussed earlier, to achieve the best possible longitudinal accuracy, we
aim at generating a structure that contains both a very high and a very low
rotation frequency at the same time. To realise such a rotating structure, the
light field generated by the SLM needs to contain three Bessel beam
components, so that their mutual coupling gives rise to a fast-rotating and a
slow-rotating self-accelerating beam. The former, for example, results from
Bessel beam components 1 and 2, whose difference in radii should be as large
as the optical system allows, while the OAM value differs by only a single
quanta, i.e., $\Delta\ell=1$. As one of the rings, ring 2, necessarily has to
have a large radius $r_{2}$ leading to a large difference in longitudinal wave
vectors $\Delta k_{1,2}$, the interfering light field and, thus, the rotating
structure will be strongly confined at a small transverse region around the
optical axis.
The slow-rotating part, on the other hand, results from Bessel components 1
and 3, so the radii of the two rings should be very similar to obtain a small
difference in wave vectors $\Delta k_{1,3}$. However, the difference should
also be large enough that the resulting rotation frequency allows us to
discriminate between the repeating periods of the fast-oscillating signal. If
these two rings are chosen to be similar in radius but much smaller than ring
2, the rotating light field in the Fourier plane of the lens, i.e., the slowly
rotating structure, will cover not only the area in close proximity of the
optical axis but also the outer region. Note that examples of the ring-shaped
modulation patterns and the resulting propagation dynamics of the spiraling
beams can be found in Figures 3 and 5 in later sections. This difference in
the radial intensity compared to the fast-rotating structure enables us to
discriminate the two differently varying patterns by observing the intensity
in different radial regions. In the simplest case, these regions might be
defined by a single transverse location where the intensity is evaluated.
However, higher experimental accuracy can be obtained using two quadrant
detectors, one for each rotating structure, which evaluate the differences
between opposing quadrants to increase the signal-to-noise ratio. As such
detectors can work with tens of nanoseconds of rise time, the proposed method
might also find applications in high-speed longitudinal position measurements.
Another important aspect is the overall distance over which the spiraling
intensity can be observed. Here, the ultimate limit is given by the width of
the ring in the modulation plane that generates the Bessel beam. The narrower
the ring, the longer will the spiraling beam survive, thus allowing a
measurement for longer distances. On the contrary, the wider the ring width
is, the shorter will be the self-accelerating beam in the focal region, thus
allowing measurements over only a very short distance. Obviously, in
applications the ring width is determined by the resolution of the modulation
device, which is in our case the SLM (see below), as well as the minimum
amount of light required to detect the rotating intensity patterns. Using
high-resolution generation methods, self-accelerating beams over a distance of
70 mm have been demonstrated already vetter2019realization .
We further note that, in principle, one can also tune the rotation frequency
by adjusting the difference in the OAM values $\Delta l$ of the two
constituent beams, as can be seen in formula (3). However, by doing so, one
should keep in mind that large differences in OAM result in beams of more
complex angular structures, which might also require a detection system that
is able to resolve those structures. Moreover, if the two constituents differ
by many OAM quanta, they also show a significant difference in their OAM-
induced divergence of the beam padgett2015divergence . As this difference also
leads to a fast decrease in their spatial overlap, the region over which the
interference and, thus, the rotation can be observed is reduced. Hence,
improving the accuracy as well as the distance over which it can be measured
is preferably done by tuning the rotation frequency through adjustments to the
longitudinal wave vector difference $\Delta k$.
## III Results
### III.1 High-accuracy measurements
As a first task, we investigated the largest possible rotation frequency and,
thus, the highest possible accuracy. Before testing the theory in the
laboratory, we performed the so-called split-step propagation
methodpoon2017engineering to simulate the entire setup. A sketch of the setup
can be seen in Fig. 2.
During the first set of simulations, the main aim was to verify the idea and
find the fastest rotation of the angular modes that was still within the
limitations of our experimental system. These limitations were mainly the
aperture of our optics (1-inch); the pixel resolution of our modulating
device, a phase-only SLM (Holoeye, Pluto-2.1-NIR-011 LCOS, 1920$\times$1080
pixels, 8 $\mu$m pixel size); and the pixels of the detection system, a camera
(ZWO ASI120MM Mini, 1280$\times$960 pixels, 3.75 $\mu$m pixel size).
Figure 2: Sketch of the experimental setup used in simulation and experiment.
A laser beam with a wavelength of 780 nm is enlarged to a wide radius using a
telescope (lenses $f_{1}$ and $f_{2}$). For maximum efficiency, the beam’s
polarization is controlled by a half-wave plate $\lambda/2$ such that it
aligns with the orientation of the diffraction grating on the SLM, which
modulates the diffracted light to have a shape with a central circle and
ring(s) and imparts angular momentum to the ring(s). The beam is filtered
through a 4f optical system (lenses $f_{3}$) that extracts the first
diffraction order, thereby removing the un-modulated light. Beyond the image
plane, the structured beam is focused by another lens $f_{4}$. The resulting
spiraling structure is imaged by a camera on a translation stage (TS).
In order to achieve the best matching between simulation and experiment, we
included in the simulation the spatial resolution of our modulation and
detection plane using the values of our laboratory system. In particular, we
set the pixel size of the simulation to 1.875 $\mu$m, which is half the size
of the physical pixels of our camera used in the experiment.
At first, we started by simulating a beam that only rotates with a single
frequency, i.e., a beam consisting of two Bessel components, as described in
equation (1). To achieve the largest difference in their longitudinal wave
vectors, we generated two Bessel beam components from two ring-shaped patterns
of very different radii in the Fourier domain. The larger ring size was mainly
limited by the screen size of the SLM as well as the optical apertures in our
system. The ring we utilized had an inner radius of $r_{1}$=3.5 mm, with a
ring width of 0.1 mm, and imprinted an OAM value of $\ell=$1 onto the beam. As
the length over which the rotating intensity pattern exists is determined by
the ring width, we chose the smallest possible width allowed by the resolution
of the modulation device, i.e., the pixel size of the SLM. Utilizing an SLM
with a pixel size of 8 $\mu$m, we chose a ring width of 100 $\mu$m in order to
have around 11-13 pixels at any given angular position, which allowed an
efficient generation using holographic methods. We chose the smaller ring to
be a circular area of radius $r_{0}=0.4$ mm with a flat phase, i.e., an OAM
value of $\ell=0$ (see Fig. 3a). The circular area was optimized to have a
similar intensity as the ring, taking the Gaussian shape of our input light
field, with a beam waist of 4.1 mm, into account. This optimization was done
because equal amplitudes of the two components result in a rotating pattern
with improved visibility vetter2015optimization ; ornigotti2018vector , which
in our task improves the accuracy.
Figure 3: Simulation for high-accuracy measurements. a) In the inset, the
modulation pattern is shown to generate the spiraling structure. Its
brightness corresponds to the amplitude of the light, and the color depicts
the phase of the modulation. For clarity only the modulation is shown, not the
holographic pattern required in the experiment. The green lines depict the
region utilized to emulate a quadrant detector. b) The simulated angular
intensity changes over propagation distance. Four exemplary intensity patterns
are shown as insets to visualize the spiraling behaviour. Comparing the
intensities found in 4 quadrants, shown in a), around the optical axis enables
the determination of the longitudinal position. The intensity difference
between quadrants 1,2 and 3,4 (2,3 and 1,4) leads to a sinusoidal curve shown
in black (red) with a period of 311.2 $\mu$m. Using both fast-varying curves
the longitudinal positions can be unambiguously determined over one half
period.
Finally, the ring-shaped intensity patterns were transformed into a spiraling
beam through an optical Fourier transform using a lens of 50 mm focal length.
The focal length was mainly limited by the pixel size of the camera, as
shorter focal lengths lead to beams of smaller extent in the focus. We note
that a magnification system might be used to circumvent this constraint if
shorter focal length lenses are required.
In order to determine the propagation distance from the change in angular
intensity, we simulated the intensity in a 300$\times$300-pixel region
centered on the optical axis during propagation. We then registered the
intensity within a circular region of radius 3.75 $\mu$m around the optical
axis. To emulate a quadrant detector we summed up the registered intensity of
each of the four quadrants and evaluated the difference between the
intensities of quadrants 1,2 and 3,4, i.e., the upper and lower halves (see
Fig. 3a). We obtained a sinusoidal curve with a period of 311.2 $\mu$m, as
shown in Fig. 3b, which matches the value of 311 $\mu$m expected from theory.
In addition, if one evaluates the difference in intensity between quadrants
2,3 and 1,4, i.e., the left and right halves, it is possible to shift the
steepest slope of the sinusoidal curve by a quarter of a period (an effective
phase shift of $\pi/2$). When using both signals, it is possible to achieve
the same longitudinal accuracy at the positions where the shallow slope of one
signal would cause a significant decrease in accuracy (see Fig. 3b).
After having verified the method in simulations, we turned to the experimental
implementation to determine the actual accuracy limits of our system due to
experimental imperfections and errors. In the experiment, sketched in Fig. 2,
we used a fiber-coupled single-frequency Toptica laser (DLpro) at a wavelength
of 780 nm. We enlarged the laser beam using a telescope system to a beam waist
radius of approximately 4.1 mm, such that it illuminated the whole screen of
the SLM. Via reflection off the SLM screen, we modulated the beam to have the
multiple-ring shaped intensity structure with the dimensions described above.
As the SLM is a liquid-crystal phase-only modulation device zhang2014LCOS , we
used holographic modulation techniques to perform a complex amplitude
modulation. This is done by displaying the diffractive holographic pattern
only at the ring-shaped regions where we wanted to obtain a light field
rosales2017shape . Through filtering only the first diffraction order using an
aperture in a Fourier plane of a 4f-system, we not only imprint the required
phase but also carve out the required intensity structure. At the image plane,
we obtained the required ring-shaped amplitude that leads to a spiraling beam
after a Fourier transformation, as described earlier. Analogous to the
simulations, we performed this Fourier transformation using another lens with
a focal distance of 50 mm.
To record the spiraling structure, we placed the camera on a high-accuracy
motorized translation stage, which was scanned over 1 mm in steps of 1 $\mu$m.
The recording and translation of the camera were automated by interfacing the
camera and translation stage with LabVIEW. At each position, we recorded 50
frames, from which we obtained the average and standard deviation of the
angular intensity along the optical axis. Analogous to the simulation, we
emulated a quadrant detector by registering the intensity difference between
different quadrants illuminated by the beam. As the beam waist in the focal
plane is very small, we only used 4 pixels of size 3.75 $\mu$m, each
corresponding to one quadrant.
Figure 4: Experimental high-accuracy distance measurements. a) The scatter
plot of intensity differences between quadrants 1,2 and 3,4 (2,3 and 1,4) are
shown in black (red). The intensities are obtained by averaging 50 images for
each position. The solid lines show the corresponding sinusoidal fits to the
measurements. b) The displacement accuracy analysis is obtained using the
fitted sine functions and the error propagation method from the standard
deviation of intensity fluctuations at each position. We find a best accuracy
of 1-2 $\mu$m at the points of steepest slope. For maintaining the minimum
error, one must hop between the steep slopes of the red and black curves,
which leads to obtaining the accuracy of 2.2$\pm$0.9 $\mu$m over the full
range (larger symbols).
The resulting variations in intensity when comparing quadrants 1,2 and 3,4 are
shown in Fig. 4a, demonstrating a period of about 332.6$\pm$0.3 $\mu$m. This
result matches the expected period from theory and simulation, with the small
discrepancy attributed to experimental imperfections such as finite resolution
and misalignment. Using the standard deviation of the intensity at each
position as the experimentally determined error and taking the known
sinusoidal curve into account, we determined the longitudinal accuracy or
displacement accuracy through error propagation. Hence, we define the
displacement accuracy as the minimal displacement for which the errors of two
data point do not overlap, i.e., the two data points that can be discriminated
with a 1-$\sigma$ confidence interval. We found that at the steep slope of the
curve, two different longitudinal positions that are 1 to 2 $\mu$m apart can
still be resolved with one standard deviation significance (see Fig. 4b). As
expected, the accuracy decreases dramatically around the extremal regions of
the sinusoidal curve. However, as mentioned earlier, evaluating the difference
in intensity between the left and right halves (quadrants 2,3 and 1,4) results
in a periodic signal that is shifted by a quarter of a period relative to the
signal of the upper and lower halves. We can therefore always refer to a fast-
varying signal regardless of position, so the strong reduction of accuracy due
to slow intensity variations at the extremal points of the sinusoidal curve is
circumvented. When switching between the two signals such that the quarter
period with the steepest slope is always used for a given longitudinal region,
we obtain an average accuracy of 2.2$\pm$0.9 $\mu$m over the whole scanning
range.
### III.2 Long range measurements
In order to overcome the ambiguity between multiple fast-varying periods, we
then studied a beam that spirals with two components simultaneously: a slow
one and a fast one. The former can be used to determine the coarse position,
while the latter can be used to obtain the longitudinal distance with high
accuracy. Again, we first investigated the method in simulation before
implementing the scheme in the laboratory.
Figure 5: Simulation for distance measurements over a longer range. a) An
intensity structure spiraling simultaneously with two frequencies is generated
using the modulation pattern shown in the inset. As in Fig. 3, only the
complex modulation is shown and the brightness depicts the amplitude and the
color depicts the phase. The simulated beam close the focal region shows the
small off-axis intensity, that is spiraling fast as before, as well as larger
regions of increased intensity, which is simultaneously spiraling slower
around the optical axis. The regions used to emulate the quadrant detectors
are depicted with green lines. b) The angular intensity variation found in the
inner and out regions around the optical axis over propagation distance.
Comparing the intensity differences found in four quadrants in the inner
region (1,2-3,4 in black and 2,3-1,4 in red) around the optical axis leads to
a fast-varying pattern used as a local high-accuracy measurement analog as
before. Comparing the intensity difference in the outer region (ring-quad,
1,2-3,4), however, leads to a slow-varying function (blue line) with a
periodicity of 12.63 mm, which will give information about the global position
such that the high-accuracy distance measurement can be extended over multiple
periods, i.e., a longer region. Small oscillations in the slow-varying curve
are due to an inevitable cross-talk from the strong fast-oscillating signal.
As discussed in the theory section, such a dual-frequency beam can be
generated by adding an additional ring to the modulation pattern. In our
realization, we used an additional ring that was slightly bigger than the
inner circular area, with an inner ring radius of $r_{1}$ = 0.6 mm and a width
of 100 $\mu$m. The modulation pattern to generate such a beam can be found in
Fig. 5a. As before, the ring width was limited by the pixel size of the
modulating device used in the subsequent experiment. The additional ring
results in a beam having a second, much smaller rotation frequency in the
intensity, as described by equation (5). To prevent a third rotation frequency
from appearing in the intensity pattern, we imprinted the additional ring with
an OAM value of $\ell=1$, such that the pairwise interference only appears
between the circular central area and each of the rings. Again, the width of
the additional ring was chosen such that all three beam components were
similar in amplitude, thus obtaining a high-visibility structure
vetter2015optimization . The rest of the simulation remained the same as
before.
Figure 6: High-accuracy distance measurements over a long range. a) The
connected-scatter plot of intensity differences between the inner quadrants
(red and black) and the ring quadrant (blue) as shown in Fig. 5a. At every
position, 25 images were recorded. b) Displacement accuracy analysis for the
slow-varying curve (blue), again obtained using the fitted function from a),
the standard deviation of the intensity fluctuations at each position, and
error propagation. To distinguish the different steep slopes of the fast-
oscillating signals, an accuracy of around 80 $\mu$m is required (dashed
line), which we achieved over a region of about 2 mm (orange shaded region).
Because the components of the light field that rotate slower are stemming from
the components on the inner parts of the generation hologram, the resulting
interference is distributed over a larger region around the optical axis (see
Fig. 5a and insets in b). In other words, the angular intensity found in
regions of larger radii is strongly determined by the slow-varying pattern.
This spatial separation enables a simultaneous measurement of both variations:
the fast-spiraling part close to the optical axis and the slow-spiraling part
further away from the beam center. As before, we used the small circular
region with a radius of 3.75 $\mu$m around the optical axis to determine the
fast variation. The slow-varying signal was obtained by recording the
intensity in a circular, ring-shaped region with an inner radius of 7.5 $\mu$m
and a width of 22.5 $\mu$m. Upon propagation, the intensity follows a fast as
well as slow-varying sinusoidal curve when the differences between the
quadrants are evaluated (see Fig. 5b). As expected, the fast oscillation
period is again 311.2 $\mu$m, while the slow-varying signal has a periodicity
of around 12.63 mm. While the fast-oscillating period again matches nicely
with the theoretical value of 311 $\mu$m, we find a bigger discrepancy of the
slow-oscillating signal to the theoretically expected periodicity of 10.2 mm.
We attribute the latter to the fact that we only obtained only one full
fringe, which also shows an additional modulation. The additional slow-varying
signal now allows a discrimination between the different fast-oscillating
periods and, thus, a global determination of the longitudinal position. In
other words, after a first gauging of slope of the slow-varying curve with
respect to a coarse distance measure, the fast-oscillating signal can then be
used to determine the distance with high accuracy. However, a closer
inspection of the slow-varying curve shows that it is also slightly modulated
by the fast-oscillating signal, as the two differently spiraling fields are
not complete separable. While this leads to an increase of the slope in some
regions, it also flattens the curve whenever the fast-varying modulation
counteracts the change of the slow-oscillating curve. Obviously, in these
regions the accuracy of the slow-varying signal will be reduced. Thus, an
important experimental question is to determine if the detrimental effect of
this additional modulation is small enough to allow a discrimination between
the different fringes using our quadrant measurement technique: in other
words, over what range is the displacement accuracy of the slow-varying signal
less than $1/4$ of the period of the fast-varying signal?
Apart from changing the modulation pattern at the SLM, the rest of the
experimental setup remained the same as before. However, this time we scanned
over a range of 10 mm with a step size of 10 $\mu$m and recorded 25 image
frames of the spiraling structure at each position. In the data analysis, we
again used the four pixels around the optical axis (2$\times$2 pixel array) as
a quadrant detector to obtain the fast-oscillating signal. Additionally, to
measure the slow-varying signal, we measured the intensity differences between
the upper and lower quadrants (1,2 and 3,4) in a disk-shaped region with a
radius of 30 $\mu$m around the optical axis, excluding the four inner pixels
used to determine the fast-oscillating signal. As can be seen in Fig. 6a, we
find a slow-varying curve with a periodicity of 11.29$\pm$0.03 mm and fast-
oscillating fringes with periods of 312.9$\pm$0.4 $\mu$m, as expected. We also
obtain, analogous to the simulations, the additional high-frequency modulation
of the slow-varying signal. To evaluate the displacement accuracy, we first
determine the function that describes the slow-varying curve, for which we use
a sum of two sinusoidal functions whose amplitude and periodicity we obtain
from fits. Using this function; experimentally obtained errors given by the
standard deviation of the measured intensities; and error propagation, we find
that we obtain the required accuracy of less than than 80 $\mu$m ($\sim 1/4$
of the period of the fast curve) over a range of more than 7 fast-oscillating
fringes, or around 2 mm. Most displacement accuracies for the slow-varying
curve in this region are as low as 10-20 $\mu$m (see Fig. 6b). This accuracy
is enough to distinguish the different fast-oscillating fringes and the
corresponding regions having a steep slope, thereby demonstrating an accuracy
of around 2 $\mu$m over the full region, i.e., three orders of magnitude or 2
mm. We note that in principle our method does not require an initial
calibration, as the longitudinal-varying angular intensities can be
theoretically obtained from the beam dimensions as well as the focusing lens.
However, in experiments the system might be initially characterized once, such
that a camera (or quadrant detector) placed anywhere within the possible
measurement range can be used to determine an absolute position without
rescanning the entire translation region.
## IV Conclusion
We have demonstrated that radially self-accelerating beams can be used to
determine the distance with respect to a reference with an accuracy of 2
$\mu$m over three orders of magnitude. The main benefit of our technique is
its simplicity, as only a single beam having the appropriate structure and two
quadrant detectors are required. Apart from this strong benefit, there is also
an important precaution worth mentioning. The technique requires a very
accurate alignment of the beam’s optical axis with the center of the detector
such that the recorded intensity does not move transversely while the detector
is translated. Even a single pixel of transverse displacement can cause large
discontinuous jumps in the measured intensity differences. However, once
alignment has been ensured, the measurement is stable and only requires
minimal post-processing (after having gauged the system), such that very high
read-out speeds on the order of nano-seconds might be feasible. The obtained
accuracy of the distance measurements can be further improved by using
stronger focusing optics and custom-fabricated optics for beam generation, as
well as an additional imaging system to lift the limitation of the finite
resolution of the detection system. In addition, it might be interesting to
consider more complex spiraling structures, such as an accelerated rotation
schulze2015accelerated , which should further increase the accuracy over small
regions at the cost of the accuracy elsewhere. Finally, we hope to stimulate
further research into applications that benefit from the propagation-dependent
intensity variations of radially self-accelerating beams.
## Acknowledgments
SP, SP, MO, and RF acknowledge the support from Academy of Finland through the
Competitive Funding to Strengthen University Research Profiles (decision
301820) and the Photonics Research and Innovation Flagship (PREIN - decision
320165). RF also acknowledges support from Academy of Finland through the
Academy Research Fellowship (decision 332399).
## Disclosures
The authors declare that there are no conflicts of interest related to this
article.
## References
* (1) H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer _et al._ , Journal of Optics 19, 013001 (2016).
* (2) M. J. Padgett, Optics Express 25, 11265 (2017).
* (3) L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Physical Review A 45, 8185 (1992).
* (4) M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Optics communications 112, 321 (1994).
* (5) N. Heckenberg, R. McDuff, C. Smith, and A. White, Optics Letters 17, 221 (1992).
* (6) A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, American Journal of Physics 76, 916 (2008).
* (7) A. Forbes, A. Dudley, and M. McLaren, Advances in Optics and Photonics 8, 200 (2016).
* (8) M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. Woerdman, Optics Communications 96, 123 (1993).
* (9) L. Marrucci, C. Manzo, and D. Paparo, Physical Review Letters 96, 163905 (2006).
* (10) H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, Journal of Optics 18, 124002 (2016).
* (11) R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, Science 358, 896 (2017).
* (12) A. Forbes, Laser & Photonics Reviews 13, 1900140 (2019).
* (13) F. Gori, G. Guattari, and C. Padovani, Optics Communications 64, 491 (1987).
* (14) M. Zahid and M. Zubairy, Optics Communications 70, 361 (1989).
* (15) C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, Laser & Photonics Reviews 13, 1900103 (2019).
* (16) X. Chu, The European Physical Journal D 66, 259 (2012).
* (17) D. McGloin and K. Dholakia, Contemporary Physics 46, 15 (2005).
* (18) S. Chávez-Cerda, G. McDonald, and G. New, Optics Communications 123, 225 (1996).
* (19) Y. Y. Schechner, R. Piestun, and J. Shamir, Physical Review E 54, R50 (1996).
* (20) E. Abramochkin, N. Losevsky, and V. Volostnikov, Optics Communications 141, 59 (1997).
* (21) P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. Khonina, V. Kotlyar, V. Soifer, and A. Friberg, Journal of Modern Optics 45, 2355 (1998).
* (22) J. Tervo and J. Turunen, Optics Express 9, 9 (2001).
* (23) V. Kotlyar, S. N. Khonina, R. Skidanov, and V. Soifer, Optics Communications 274, 8 (2007).
* (24) R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, Optics Express 17, 23389 (2009).
* (25) C. Vetter, T. Eichelkraut, M. Ornigotti, and A. Szameit, Applied Physics Letters 107, 211104 (2015).
* (26) C. Schulze, F. S. Roux, A. Dudley, R. Rop, M. Duparré, and A. Forbes, Physical Review A 91, 043821 (2015).
* (27) D. M. Greenberger, American Journal of Physics 48, 256 (1980).
* (28) C. Vetter, T. Eichelkraut, M. Ornigotti, and A. Szameit, Physical Review Letters 113, 183901 (2014).
* (29) C. Vetter, A. Dudley, A. Szameit, and A. Forbes, Optics Express 25, 20530 (2017).
* (30) J. Webster, C. Rosales-Guzmán, and A. Forbes, Optics Letters 42, 675 (2017).
* (31) S. Jia, J. C. Vaughan, and X. Zhuang, Nature Photonics 8, 302 (2014).
* (32) H. He, C. Kong, X.-J. Tan, K. Y. Chan, Y.-X. Ren, K. K. Tsia, and K. K. Wong, Optics Letters 44, 5238 (2019).
* (33) R. A. Jarvis, IEEE Transactions on pattern analysis and machine intelligence PAMI-5, 505 (1983).
* (34) T. Kubota, M. Nara, and T. Yoshino, Optics Letters 12, 310 (1987).
* (35) S. Berg-Johansen, M. Neugebauer, A. Aiello, G. Leuchs, P. Banzer, and C. Marquardt, arXiv preprint arXiv:2010.16387 (2020).
* (36) M. Ornigotti and A. Szameit, Journal of Optics 20, 125601 (2018).
* (37) R. Rop, A. Dudley, C. López-Mariscal, and A. Forbes, Journal of Modern Optics 59, 259 (2012).
* (38) C. Rosales-Guzmán and A. Forbes, _How to shape light with spatial light modulators_ (SPIE Press, 2017).
* (39) M. J. Padgett, F. M. Miatto, M. P. Lavery, A. Zeilinger, and R. W. Boyd, New Journal of Physics 17, 023011 (2015).
* (40) T.-C. Poon and T. Kim, _Engineering Optics with MATLAB_ (World Scientific Publishing Company, 2017).
* (41) Z. Zhang, Z. You, and D. Chu, Light: Science & Applications 3, e213 (2014).
|
A final functor between categories $\functor{F} \colon \catname{A} \to
\catname{B}$ is a functor that allows the restriction of diagrams on $\catname{B}$ to $\catname{A}$
without changing their colimits. More precisely, the functor $\functor{F}$ is final if, for
any diagram $\functor{D} \colon \catname{B} \to \catname{B}$, there is a
canonical isomorphism
\[
\colim_{\catname{B}}\functor{D} \cong \colim_{\catname{A}}\functor{D} \circ \functor{F}
\]
where either colimit exists whenever the other one does. There is a classical
criterion for final
functors [§IX.3]maclanesaundersCategoriesWorkingMathematician1971:
a functor $\functor{F} \colon \catname{A} \to \catname{B}$ is final if and only if, for any object $b \in
\catname{B}$, the slice category $b / \functor{F}$ is nonempty and connected.
Such a criterion also exists for $(\infty,
1)$-categories [§4.1]lurieHigherToposTheory2009: an $(\infty,
1)$-functor $\ffunctor{F} \colon \catname{A} \to \catname{B}$ is final (with respect to any
$\infty$-diagram) if and only if for any object $b \in \catname{B}$, the slice
$\infty$-category $b / \functor{F}$ is weakly contractible. One would expect a
similar result for any dimension: an $(n, 1)$-functor $\ffunctor{F} \colon \catname{A} \to
\catname{B}$ is final (with respect to any $n$-diagram) if and only if, for any
object $b \in \catname{B}$, the slice $(n, 1)$-category $b / \functor{F}$ is
nonempty and has trivial homotopy groups $\pi_k$ for $0 \leq k \leq n-1$. Note
that these are not consequences of the known criterion for $1$-functors and
$(\infty, 1)$-functors (see <ref> and <ref>). This paper presents a
combinatorial proof in the case $n = 2$ (<ref>). An application of
this criterion will appear in my Ph.D. thesis maillard.
§ BICATEGORICAL NOTIONS
We will follow the naming conventions of johnsonDimensionalCategories2021
for bicategories. In particular, the terms $2$-category and $2$-functor
denote the strict ones. We will use the term $(2,1)$-category for a
$2$-category with only invertible $2$-morphisms, and the term $(2,1)$-functor
for a $2$-functor between $(2,1)$-categories.
We recall some usual constructions and properties of $2$-categories we will use,
and introduce some notations.
The symbol $\simeq$ denotes an isomorphism between two objects (in the $1$-categorical
The symbol $\cong$ denotes an equivalence between two objects (in the
$2$-categorical sense).
Let $\catname{C}$ be a 2-category. The opposite 2-category of $\catname{C}$, written
$\catop{\catname{C}}$, is the 2-category with
* Objects: the objects of $\catname{C}$
* Hom-categories: $\catop{\catname{C}}(A, B) = \catname{C}(B, A)$
We write $[\catname{A},\catname{B}]$ for the $2$-category of pseudofunctors,
pseudonatural transformations and modifications, between two $2$-categories
$\catname{A}$ and $\catname{B}$.
Let $\catname{I}, \catname{C}$ be $2$-categories and $T$ be an object of
$\catname{C}$. We denote by $\Delta{T}$ the constant functor $\catname{I} \to
\catname{C}$ with value $T$.
Let $\catname{I}, \catname{C}$ be $2$-categories and $\ffunctor{D} \colon
\catname{I} \to \catname{C}$ be a $2$-functor. A (pseudo) bicolimit of
$\ffunctor{D}$ is an object $L$ of $\catname{C}$ and a family of equivalences
\[
\Psi_{T} \colon \catname{C}(L, T) \cong [\catname{I},\catname{C}](\ffunctor{D}, \Delta{T})
\]
pseudonatural in $T$. When it exists, the bicolimit of $\ffunctor{D}$ is unique
up to equivalence and the object $L$ is noted $\ccolim_{\catname{I}}\ffunctor{D}$.
We will use the term $2$-diagram to denote a $2$-functor we introduce with
the intent to take its bicolimit.
We will use the term cone under $\ffunctor{D}$ with vertex $T$ to denote
objects of the category of pseudonatural transformations and modifications $[\catname{I},\catname{C}](\ffunctor{D}, \Delta{T})$.
Let $\catname{C}$ be a $(2,1)$-category. Fix a $(2,1)$-functor $\ffunctor{F}
\colon \catname{I} \to \catname{C}$ and an object $c$ of $\catname{C}$. The
slice $c / \ffunctor{F}$ is the $(2, 1)$-category with:
* Objects: the pairs $(i, f)$ consisting of an object $i$ of
$\catname{I}$ and a morphism $f \colon c \to \ffunctor{F}i$
* Morphisms $(i, f) \to (i', f')$: the pairs $(u, \mu)$ consisting
of a morphism $u \colon i \to i'$ of $\catname{I}$ and a 2-isomorphism $\mu \colon f' \to
\ffunctor{F}(u)f$ of $\catname{C}$:
\[
\begin{tikzcd}
\ffunctor{F}i \arrow[rr, "\ffunctor{F}u"] & & \ffunctor{F}i' \\
& c \arrow[lu, "f", bend left] \arrow[ru, "f'"{name=n}, bend right, swap]
\arrow[from=n, to=1-1, Rightarrow, "\mu", shorten=0.5cm] &
\end{tikzcd}
\]
* 2-Morphisms $(u, \mu) \Rightarrow (v, \nu)$: the 2-morphisms
$\alpha \colon u \Rightarrow v$ satisfying:
\[
\begin{tikzcd}[row sep=2cm]
\ffunctor{F}i
\arrow[rr, "\ffunctor{F}v"{name=nv}, bend left]
\arrow[rr, "\ffunctor{F}u"{name=nu}, bend right, swap] & & \ffunctor{F}i' \\
& c \arrow[lu, "f", bend left] \arrow[ru, "f'"{name=n}, bend right, swap]
\arrow[from=n, to=1-1, Rightarrow, "\mu", shorten=0.5cm, shift left=0.4cm]
\arrow[from=nu, to=nv, Rightarrow, "\ffunctor{F}\alpha", shorten=0.2cm]&
\end{tikzcd} =
\begin{tikzcd}[row sep=2cm]
\ffunctor{F}i \arrow[rr, "\ffunctor{F}v", bend left] & & \ffunctor{F}i' \\
& c \arrow[lu, "f", bend left] \arrow[ru, "f'"{name=n}, bend right, swap]
\arrow[from=n, to=1-1, Rightarrow, "\nu", shorten=0.5cm] &
\end{tikzcd}
\]
* Compositions are induced by the compositions of $\catname{I}$ and $\catname{C}$.
A slice $2$-category $c/\ffunctor{F}$ is endowed with a canonical forgetful $2$-functor:
\[
\left\{
\begin{array}{lll}
c / \ffunctor{F} & \to & \catname{I} \\
(i, f) & \mapsto & i \\
(u, \mu) & \mapsto & u \\
\alpha & \mapsto & \alpha
\end{array}
\right.
\]
§ COMBINATORIAL PATHS AND HOMOTOPIES
A 2-category $\catname{C}$ has an associated CW-complex $|\catname{C}|$, defined using the
Duskin nerve [§5.4]johnsonDimensionalCategories2021, which maps
objects of $C$ to vertices, $1$-morphisms to 1-simplices and $2$-morphisms to
2-simplices. There are thus notions of paths and homotopies of paths in
$\catname{C}$. We give in this section a combinatorial approach to these, for
We fix a $(2,1)$-category $\catname{C}$.
A path (of 1-morphism) in C is a finite sequence of objects $(a_i)_{0 \leq i \leq
n}$ and a family of pairs $(\varepsilon_i, f_i)_{1 \leq i \leq n}$ of a sign
$\varepsilon \in \{ -1, 1 \}$ and a morphism
\[
f_i \colon \left\{
\begin{array}{llll}
a_{i-1} & \to & a_i & \text{if } \varepsilon_i = 1 \\
a_{i} & \to & a_{i-1} & \text{if } \varepsilon_i = -1 \\
\end{array}
\right.
\]
Such a path is said to have source $a_0$ and target $a_n$.
We write $p \colon a_0 \leadsto a_n$ to denote a path with source $a_0$ and
target $a_n$.
A path can be pictured as a zig-zag of morphisms (potentially with consecutive
morphisms in the same direction):
\[
a_0 \xrightarrow{f_1} a_1 \xleftarrow{f_2} a_2 \xleftarrow{f_3} \ldots
\xrightarrow{f_n} a_n
\]
Following the usual conventions, left-to-right arrows represents pairs with
$\varepsilon = 1$ and right-to-left arrows pairs with $\varepsilon = -1$. The
empty path (at an object $a$) should be represented by $a$.
There is an obvious notion of concatenation of paths with compatible target and
source, given by the concatenation of the sequence of morphisms.
We say two path $p, p'$ of $\catname{C}$ are elementary homotopic,
written $p \sim_{\text{elem}} p'$, in any of the following cases:
* $a \xrightarrow{\id} a \: \sim_{\text{elem}} \: a$, for any object $a$
* $a \xleftarrow{\id} a \: \sim_{\text{elem}} \: a$, for any object $a$
* $a_0 \xrightarrow{f_1} a_1 \xrightarrow{f_2} a_2 \: \sim_{\text{elem}} \: a_0
\xrightarrow{f_2f_1} a_2$, for any composable pair $f_1, f_2$ of morphisms
* $a_0 \xleftarrow{f_1} a_1 \xleftarrow{f_2} a_2 \: \sim_{\text{elem}} \: a_0
\xleftarrow{f_1f_2} a_2$, for any composable pair $f_1, f_2$ of morphisms
* $a_0 \xleftarrow{u} a_1 \xrightarrow{v} a_2 \:
\sim_{\text{elem}} \: a_0
\xrightarrow{u'} a_1' \xleftarrow{v'} a_2$, for any 2-isomorphism
\[
\begin{tikzcd}
& a_1' & \\
a_0 \arrow[ru, "u'"] \arrow[rr, Rightarrow, shorten=0.5cm] & {} & a_2
\arrow[lu, "v'", swap] \\
& a_1 \arrow[lu, "u"] \arrow[ru, "v", swap] &
\end{tikzcd}
\]
We then define a homotopy relation $\sim$ on paths as the smallest congruent
(for the concatenation of paths), reflexive, symmetric and transitive relation
encompassing the relation $\sim_{\text{elem}}$.
We should pause to consider two consequences of <ref>:
* a 2-morphism $\begin{tikzcd}[cramped] a_0 \arrow[r, bend left=2cm,
"f_0"{name=n0}] \arrow[r, bend right=2cm, swap, "f_1"{name=n1}] & a_1
\arrow[from=n0, to=n1, Rightarrow, shorten=0.2cm] \end{tikzcd}$ can be
arranged into the following square:
\[
\begin{tikzcd}
& a_1 & \\
a_0 \arrow[ru, "f_0"] \arrow[rr, Rightarrow, shorten=0.5cm] & {} & a_1
\arrow[lu, "\id", swap] \\
& a_0 \arrow[lu, "\id"] \arrow[ru, "f_1", swap] &
\end{tikzcd}
\]
This shows, together with <ref> and <ref>, that $a_0
\xrightarrow{f_0} a_1 \: \sim \: a_0 \xrightarrow{f_1} a_1$, as one would expect.
* a 1-morphism $a_0 \xrightarrow{f} a_1$ can be used to form the square:
\[
\begin{tikzcd}
& a_1 & \\
a_1 \arrow[ru, "\id"] \arrow[rr, Rightarrow, shorten=0.5cm] & {} & a_1
\arrow[lu, "\id", swap] \\
& a_0 \arrow[lu, "f"] \arrow[ru, "f", swap] &
\end{tikzcd}
\]
Once again using <ref> and <ref>, this proves that
$(a_1 \xleftarrow{f} a_0 \xrightarrow{f} a_1) \sim a_1$. A similar argument
(putting $f$ on the upper side of the square) shows that $(a_0
\xrightarrow{f} a_1 \xleftarrow{f} a_0) \sim a_0$. Hence, up to homotopy,
the paths $a_0 \xrightarrow{f} a_1$ and $a_1 \xleftarrow{f} a_0$ are mutual
Note that for two paths to be homotopic, they must have the same source and the
same target.
It is natural to look for a category of paths up-to homotopy:
The (algebraic) fundamental groupoid $\Pi_1(\catname{C})$ of
$\catname{C}$ is the 1-category with:
* Objects: the objects of $\catname{C}$.
* Morphisms: the classes of paths between objects modulo the
homotopy relation.
* Composition is induced by the concatenation of paths.
The (2,1)-category $\catname{C}$ is said to be connected if for any
pair of objects $a, a'$ there is a path with source $a$ and target $a'$.
The (2,1)-category $\catname{C}$ is said to be simply connected if $p
\sim p'$ for any pair of
paths $p, p'$ with same source and same target.
A (2,1)-category $\catname{C}$ is nonempty, connected and simply connected if and
only if its fundamental groupoid $\Pi_1(\catname{C})$ is equivalent to $1$, the
category with exactly one object and one morphism.
Given a (2,1)-category $\catname{C}$ which is nonempty, connected and simply
connected, its nerve $|\catname{C}|$ is not necessarily weakly contractible.
Indeed higher homotopy groups may be nontrivial. For instance, one can realize
the sphere $S^2$ as the nerve of the $(2,1)$-category with two objects, two
parallel 1-morphisms between these objects, and two parallel 2-isomorphisms
between these 1-morphisms.
For any algebraic path $p$ in $\catname{C}$, there is an
associated topological path $|p| \colon I \to |\catname{C}|$. The following
assertions, which should result from simplicial approximation, motivate
the definitions of this section:
The $2$-category $\catname{C}$ is connected (resp. simply connected) if and
only if the CW-complex $|\catname{C}|$ is connected (resp. simply connected).
Two algebraic paths $p, p'$ in $\catname{C}$ are homotopic if and only if the
topological paths $|p|, |p'|$ are homotopic.
The categories $\Pi_1(\catname{C})$ and $\Pi_1(|\catname{C}|)$ are equivalent.
§ A CRITERION FOR 2-FINAL 2-FUNCTORS
A $2$-functor $\ffunctor{F} \colon \catname{A} \to \catname{B}$ between
$(2,1)$-categories is $2$-final if for any 2-diagram $\ffunctor{D} \colon \catname{B} \to
\catname{E}$, the pseudo bicolimits $\ccolim_{\catname{B}} \ffunctor{D}$ and
$\ccolim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}}$ each exists if and only if the other one
exists, and the canonical comparison morphism
\[
\ccolim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}} \to \ccolim_{\catname{B}}{\ffunctor{D}}
\]
is an equivalence.
In the above definition, $\catname{E}$ is only assumed to be a $2$-category.
However, since $\catname{B}$ is a $(2,1)$-category, the pseudo
bicolimits can be equivalently computed in $\catname{E}_g$, the
$(2,1)$-category with the objects of $\catname{E}$, the 1-morphisms
of $\catname{E}$, and the invertible 2-morphisms of $\catname{E}$. Hence we could assume
$\catname{E}$ to be a $(2,1)$-category, without changing the meaning of the definition.
A $1$-final $1$-functor $\ffunctor{F} \colon \catname{A} \to \catname{B}$ between
$1$-categories is a functor such that, for any diagram $\ffunctor{D} \colon \catname{B} \to
\catname{E}$, the colimits $\colim_{\catname{B}} \ffunctor{D}$ and
$\colim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}}$ each exists if and only if the other one
exists, and the canonical comparison morphism
\[
\colim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}} \to \colim_{\catname{B}}{\ffunctor{D}}
\]
is an isomorphism.
A $2$-final $1$-functor $\ffunctor{F} \colon \catname{A} \to \catname{B}$ between
$1$-categories (seen as $2$-categories with only the identities as
$2$-morphisms) is $1$-final, since any diagram is also a $2$-diagram. The
converse is not true, though: there are $1$-final functors which are not
Let $\catname{A}$, $\catname{B}$ be two $(2,1)$-categories. A 2-functor
$\ffunctor{F} \colon \catname{A} \to \catname{B}$ is 2-final
(<ref>) if and only if, for any object $b \in \catname{B}$, the
slice (2,1)-category $b / \ffunctor{F}$ is nonempty, connected and simply
connected (<ref>).
We will first prove the backward implication.
Fix a 2-functor $\ffunctor{D} \colon \catname{B} \to \catname{E}$. We will
construct a pseudoinverse to the canonical comparison morphism
\[
\ccolim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}} \to \ccolim_{\catname{B}}{\ffunctor{D}}
\]
This morphism correspond to a family of functors, pseudonatural in $e$:
\[
\functor{K} \colon [\catname{B}, \catname{E}](\ffunctor{D}, \Delta{e})
\to [\catname{A}, \catname{E}](\ffunctor{D} \circ \ffunctor{F}, \Delta{e}).
\]
We will construct a pseudoinverse $\ffunctor{L}$ to $\ffunctor{K}$. Given a cone
$\phi \colon \ffunctor{D} \circ \ffunctor{F} \Rightarrow \Delta{e}$, we obtain a
cone $\ffunctor{L}(\phi) \colon \ffunctor{D} \Rightarrow \Delta{e}$ as follows:
* objects in the slice categories $b / \ffunctor{F}$ define the 1-morphism components
$\functor{L}(\phi)_b$ (see <ref>),
* paths in $b / \ffunctor{F}$ define natural transformations between the
components (see <ref>),
* homotopies between paths ensure the cohesion of these constructions (see <ref>).
Consider an
arbitrary cone under $\ffunctor{D} \circ \ffunctor{F}$ with vertex $e \in \catname{E}$, that is, a
pseudo natural transformation $\phi \colon \ffunctor{D} \circ \ffunctor{F} \to
\Delta(e)$. We first want to define a cone $\psi$ under $\ffunctor{D}$ with vertex $e$,
using the cone $\phi$.
As a first step, we fix an object $b$ and we want to define the component at $b$
$\psi_b \colon \ffunctor{D}(b) \to e$. Since the slice (2,1)-category $b /
\ffunctor{F}$ is nonempty, we consider the following candidate.
We fix an object in $b / \ffunctor{F}$, that is an object $a(b) \in
\catname{A}$ and a morphism $\alpha(b) \colon b \to \ffunctor{F}(a(b))$. Define
\[
\psi_{(a(b), \alpha(b))} \colon \begin{tikzcd}
\ffunctor{D}(b) \arrow[r, "\ffunctor{D}(\alpha(b))"] &
\ffunctor{D}(\ffunctor{F}(a(b))) \arrow[r, "\phi_{a(b)}"] & e
\end{tikzcd}
\]
We then consider the dependence of $\psi_{(a(b), \alpha(b))}$ on $(a(b),
\alpha(b))$. Fix another object $(a'(b),\alpha'(b)) \in b / \ffunctor{F}$.
Since $b / \ffunctor{F}$ is connected there is a path
\[
p \colon (a_0, \alpha_0) = (a(b), \alpha(b)) \leadsto (a_n, \alpha_n) =(a'(b),
\alpha'(b))
\]
which can be pictured as:
\[
\begin{tikzcd}[row sep=2cm]
\ffunctor{F}(a_0) & \ffunctor{F}(a_1) \arrow[l, "\ffunctor{F}u_1", swap] \arrow[r, "\ffunctor{F}u_2"] &
\ffunctor{F}(a_2) \arrow[r, phantom, "\cdots"] & \ffunctor{F}(a_n) \\
& & b
\arrow[llu, "\alpha_0"{name=n0}, bend left]
\arrow[lu, "\alpha_1"{name=n1}]
\arrow[u, "\alpha_2"{name=n2}, swap]
\arrow[ru, "\alpha_n", bend right, swap]
\arrow[to=1-2, from=n0, "\mu_1", Rightarrow, shorten=0.3cm]
\arrow[to=1-2, from=n2, "\mu_2", Rightarrow, shorten=0.3cm]
\end{tikzcd}
\]
Applying the 2-functor $\ffunctor{D}$ and using the cone $\phi$, we obtain the
pasting diagram:
\begin{equation}\label{DiagDefJ}
\begin{tikzcd}[row sep=1.5cm, column sep=3cm]
& \ffunctor{D}\ffunctor{F}(a_0) \arrow[rd, ""{name=nphia0}, bend left] & \\
\ffunctor{D}(b)
\arrow[ru, ""{name=nalp0}, bend left, swap]
\arrow[r] \arrow[rd, ""{name=nalp2}, bend right]
\arrow[rdd, bend right] &
\ffunctor{D}\ffunctor{F}(a_1)
\arrow[u, "\ffunctor{D}\ffunctor{F}(u_1)"description]
\arrow[d, "\ffunctor{D}\ffunctor{F}(u_2)"description]
\arrow[r, ""{name=nphia1}] & e \\
& \ffunctor{D}\ffunctor{F}(a_2)
\arrow[d, phantom, "\vdots", near start]
\arrow[ru, ""{name=nphia2}, bend right, swap] & \\
& \ffunctor{D}\ffunctor{F}(a_n) \arrow[ruu, bend right] &
\arrow[from=nalp0, to=2-2, "\ffunctor{D}(\mu_1)", Rightarrow, shorten=0.2cm, swap]
\arrow[from=1-2, to=nphia1, "\phi_{u_1}", Rightarrow, shorten=0.6cm, swap]
\arrow[from=2-2, to=nalp2, "\ffunctor{D}(\mu_2)^{-1}", Rightarrow, shorten=0.2cm, swap]
\arrow[from=nphia1, to=3-2, "\phi_{u_2}^{-1}", Rightarrow, shorten=0.6cm, swap]
\end{tikzcd}
\end{equation}
We can thus define:
Any path $p \colon (a, \alpha) \leadsto (a', \alpha')$ in $b / \ffunctor{F}$ defines a
2-isomorphism in $\catname{E}$
\[
j(p) \colon \psi_{(a, \alpha)} \to \psi_{(a', \alpha')}
\]
as given by the above pasting diagram <ref>.
For any paths $p, p' \colon (a, \alpha) \leadsto (a', \alpha')$ in $b /
\ffunctor{F}$ with same source and target,
\[
j(p) = j(p').
\]
We first prove that two elementary homotopic paths $p \sim_{\text{elem}} p'$ induce the same
2-isomorphism $j(p) = j(p')$. The four first cases are immediate consequences
of the pseudonaturality of $\phi$. We can thus assume that
\[
p =
\begin{tikzcd}
\ffunctor{F}(a_0) & \ffunctor{F}(a_1) \arrow[l, "\ffunctor{F}u", swap]
\arrow[r, "\ffunctor{F}v"] & \ffunctor{F}(a_2) \\
& b \arrow[lu, bend left, ""{name=n1}] \arrow[u] \arrow[ru, bend right,
""{name=n2}, swap] &
\arrow[from=n1, to=1-2, "\mu", Rightarrow, shorten=0.2cm]
\arrow[from=n2, to=1-2, "\nu", Rightarrow, shorten=0.2cm, swap]
\end{tikzcd}
\]
\[
p' =
\begin{tikzcd}
\ffunctor{F}(a_0) \arrow[r, "\ffunctor{F}u'"] & \ffunctor{F}(a_1') &
\ffunctor{F}(a_2) \arrow[l, "\ffunctor{F}v'", swap] \\
& b \arrow[lu, bend left] \arrow[u] \arrow[ru, bend right] &
\arrow[from=2-2, to=1-1, "\mu'", Rightarrow, shorten=0.2cm, swap]
\arrow[from=2-2, to=1-3, "\nu'", Rightarrow, shorten=0.2cm]
\end{tikzcd}
\]
and that there is a 2-isomorphism $\zeta \colon u'u \Rightarrow v'v$ such that
\[
\begin{tikzcd}[column sep=0.25cm, baseline=(current bounding box.center)]
& {} \arrow[d, Rightarrow, "\mu'", shorten=0.2cm, swap] & \ffunctor{F}(a_1') & \\
b \arrow[rru, bend left] \arrow[r] \arrow[rrd, bend right] &
\ffunctor{F}(a_0)
\arrow[ru, "\ffunctor{F}u'"]
\arrow[d, Rightarrow,"\mu", shorten=0.2cm, swap]
\arrow[rr, Rightarrow, "\ffunctor{F}\zeta", shorten=0.5cm]
& & \ffunctor{F}(a_0) \arrow[lu, "\ffunctor{F}v'", swap] \\
& {} & \ffunctor{F}(a_1) \arrow[lu, "\ffunctor{F}u"] \arrow[ru,
"\ffunctor{F}v", swap] &
\end{tikzcd} \quad = \quad
\begin{tikzcd}[baseline=(current bounding box.center)]
& & \ffunctor{F}(a_1') \arrow[d, Rightarrow, "\nu'", shorten=0.2cm] & \\
b \arrow[rru, bend left] \arrow[rrr] \arrow[rrd, bend right] & & {}
\arrow[d, Rightarrow, "\nu", shorten=0.2cm]& \ffunctor{F}(a_0)
\arrow[lu, "\ffunctor{F}v'", swap] \\
& & \ffunctor{F}(a_1) \arrow[ru, "\ffunctor{F}v", swap] &
\end{tikzcd}
\]
We can apply the functor $\ffunctor{D}$ and express this relation using
string diagrams (see [§3.7]johnsonDimensionalCategories2021):
\begin{equation} \label{ZetaMorphSlice}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (2, 0) {$\ffunctor{D}\alpha_1'$};
\node[tip] (end0) at (0, -4) {$\ffunctor{D}\alpha_1$};
\node[tip] (end1) at (1, -4) {$\ffunctor{D}\ffunctor{F}v$};
\node[tip] (end2) at (2, -4) {$\ffunctor{D}\ffunctor{F}v'$};
\node[naturaltr] (dmu0) at (1.5, -1) {$\ffunctor{D}\mu'$};
\node[naturaltr] (dmu1) at (0.5, -2) {$\ffunctor{D}\mu$};
\node[naturaltr] (zeta) at (1.5, -3) {$\ffunctor{D}\ffunctor{F}\zeta$};
\draw (sta0) to[out=270, in=90] (dmu0.north east);
\draw (dmu0.south west) to[out=270, in=90] (dmu1.north east);
\draw (dmu0.south east) to[out=270, in=90] (zeta.north east);
\draw (dmu1.south west) to[out=270, in=90] (end0);
\draw (dmu1.south east) to[out=270, in=90] (zeta.north west);
\draw (zeta.south west) to[out=270, in=90] (end1);
\draw (zeta.south east) to[out=270, in=90] (end2);
\end{tikzpicture} \xspacedequal{}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (2, 0) {$\ffunctor{D}\alpha_1'$};
\node[tip] (end0) at (0, -4) {$\ffunctor{D}\alpha_1$};
\node[tip] (end1) at (1, -4) {$\ffunctor{D}\ffunctor{F}v$};
\node[tip] (end2) at (2, -4) {$\ffunctor{D}\ffunctor{F}v'$};
\node[naturaltr] (dnu0) at (1.5, -1) {$\ffunctor{D}\nu'$};
\node[naturaltr] (dnu1) at (0.5, -2) {$\ffunctor{D}\nu$};
\draw (sta0) to[out=270, in=90] (dnu0.north east);
\draw (dnu0.south west) to[out=270, in=90] (dnu1.north east);
\draw (dnu0.south east) to[out=270, in=90] (end2);
\draw (dnu1.south west) to[out=270, in=90] (end0);
\draw (dnu1.south east) to[out=270, in=90] (end1);
\end{tikzpicture}
\end{equation}
Similarly, the pseudonaturality of $\phi$ gives the relation:
\begin{equation} \label{PsNatPhiZeta}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (0, 0) {$\ffunctor{D}\ffunctor{F}u$};
\node[tip] (sta1) at (1, 0) {$\ffunctor{D}\ffunctor{F}u'$};
\node[tip] (sta2) at (2, 0) {$\phi_{a_1'}$};
\node[tip] (end0) at (0, -4) {$\phi_{a_1}$};
\node[naturaltr] (u') at (1.5, -1) {$\phi_{u'}$};
\node[naturaltr] (u) at (0.5, -2) {$\phi_{u}$};
\draw (sta0) to[out=270, in=90] (u.north west);
\draw (sta1) to[out=270, in=90] (u'.north west);
\draw (sta2) to[out=270, in=90] (u'.north east);
\draw (u'.south west) to[out=270, in=90] (u.north east);
\draw (u.south west) to[out=270, in=90] (end0);
\end{tikzpicture} =
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (0, 0) {$\ffunctor{D}\ffunctor{F}u$};
\node[tip] (sta1) at (1, 0) {$\ffunctor{D}\ffunctor{F}u'$};
\node[tip] (sta2) at (2, 0) {$\phi_{a_1'}$};
\node[tip] (end0) at (0, -4) {$\phi_{a_1}$};
\node[naturaltr] (zeta) at (0.5, -1) {$\ffunctor{D}\ffunctor{F}\zeta$};
\node[naturaltr] (v') at (1.5, -2) {$\phi_{v'}$};
\node[naturaltr] (v) at (0.5, -3) {$\phi_{v}$};
\draw (sta0) to[out=270, in=90] (zeta.north west);
\draw (sta1) to[out=270, in=90] (zeta.north east);
\draw (sta2) to[out=270, in=90] (v'.north east);
\draw (zeta.south west) to[out=270, in=90] (v.north west);
\draw (zeta.south east) to[out=270, in=90] (v'.north west);
\draw (v'.south west) to[out=270, in=90] (v.north east);
\draw (v.south west) to[out=270, in=90] (end0);
\end{tikzpicture}
\end{equation}
We can now compute $j(p)$:
\begin{align*}
j(p) &=
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (1, 0) {$\ffunctor{D}\alpha_0$};
\node[tip] (sta1) at (2, 0) {$\phi_{a_0}$};
\node[tip] (end0) at (1, -5) {$\ffunctor{D}\alpha_2$};
\node[tip] (end1) at (2, -5) {$\phi_{a_2}$};
\node[naturaltr] (dmu) at (0.5, -1) {$\ffunctor{D}\mu$};
\node[naturaltr] (u) at (1.5, -2) {$\phi_{u}$};
\node[naturaltr] (v) at (1.5, -3) {$\phi_{v}^{-1}$};
\node[naturaltr] (dnu) at (0.5, -4) {$\ffunctor{D}\nu^{-1}$};
\draw (sta0) to[out=270, in=90] (dmu.north east);
\draw (sta1) to[out=270, in=90] (u.north east);
\draw (dmu.south west) to[out=270, in=90] (dnu.north west);
\draw (dmu.south east) to[out=270, in=90] (u.north west);
\draw (u.south west) to[out=270, in=90] (v.north west);
\draw (u.south east) to[out=270, in=90] (v.north east);
\draw (v.south west) to[out=270, in=90] (dnu.north east);
\draw (v.south east) to[out=270, in=90] (end1);
\draw (dnu.south east) to[out=270, in=90] (end0);
\end{tikzpicture} \xspacedequal{\eqref{PsNatPhiZeta}}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (1, 0) {};
\node[tip] (sta1) at (2, 0) {};
\node[tip] (end0) at (1, -6) {};
\node[tip] (end1) at (2, -6) {};
\node[naturaltr] (dmu) at (0.5, -1) {$\ffunctor{D}\mu$};
\node[naturaltr] (u) at (2.5, -2) {$\phi_{u'}^{-1}$};
\node[naturaltr] (zeta) at (1.5, -3) {$\ffunctor{D}\ffunctor{F}\zeta$};
\node[naturaltr] (v) at (2.5, -4) {$\phi_{v'}$};
\node[naturaltr] (dnu) at (0.5, -5) {$\ffunctor{D}\nu^{-1}$};
\draw (sta0) to[out=270, in=90] (dmu.north east);
\draw (sta1) to[out=270, in=90] (u.north west);
\draw (dmu.south west) to[out=270, in=90] (dnu.north west);
\draw (dmu.south east) to[out=270, in=90] (zeta.north west);
\draw (u.south west) to[out=270, in=90] (zeta.north east);
\draw (u.south east) to[out=270, in=90] (v.north east);
\draw (zeta.south west) to[out=270, in=90] (dnu.north east);
\draw (zeta.south east) to[out=270, in=90] (v.north west);
\draw (v.south west) to[out=270, in=90] (end1);
\draw (dnu.south east) to[out=270, in=90] (end0);
\end{tikzpicture} \\
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (1, 0) {};
\node[tip] (sta1) at (2, 0) {};
\node[tip] (end0) at (1, -6) {};
\node[tip] (end1) at (2, -6) {};
\node[naturaltr] (u) at (2.5, -1) {$\phi_{u'}^{-1}$};
\node[naturaltr] (dmu) at (0.5, -2) {$\ffunctor{D}\mu$};
\node[naturaltr] (zeta) at (1.5, -3) {$\ffunctor{D}\ffunctor{F}\zeta$};
\node[naturaltr] (dnu) at (0.5, -4) {$\ffunctor{D}\nu^{-1}$};
\node[naturaltr] (v) at (2.5, -5) {$\phi_{v'}$};
\draw (sta0) to[out=270, in=90] (dmu.north east);
\draw (sta1) to[out=270, in=90] (u.north west);
\draw (u.south west) to[out=270, in=90] (zeta.north east);
\draw (u.south east) to[out=270, in=90] (v.north east);
\draw (dmu.south west) to[out=270, in=90] (dnu.north west);
\draw (dmu.south east) to[out=270, in=90] (zeta.north west);
\draw (zeta.south west) to[out=270, in=90] (dnu.north east);
\draw (zeta.south east) to[out=270, in=90] (v.north west);
\draw (dnu.south east) to[out=270, in=90] (end0);
\draw (v.south west) to[out=270, in=90] (end1);
\end{tikzpicture} \xspacedequal{\eqref{ZetaMorphSlice}}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[tip] (sta0) at (1, 0) {$\ffunctor{D}\alpha_0$};
\node[tip] (sta1) at (2, 0) {$\phi_{a_0}$};
\node[tip] (end0) at (1, -5) {$\ffunctor{D}\alpha_2$};
\node[tip] (end1) at (2, -5) {$\phi_{a_2}$};
\node[naturaltr] (u) at (2.5, -1) {$\phi_{u'}^{-1}$};
\node[naturaltr] (dmu) at (1.5, -2) {$\ffunctor{D}\mu'^{-1}$};
\node[naturaltr] (dnu) at (1.5, -3) {$\ffunctor{D}\nu'$};
\node[naturaltr] (v) at (2.5, -4) {$\phi_{v'}$};
\draw (sta0) to[out=270, in=90] (dmu.north west);
\draw (sta1) to[out=270, in=90] (u.north west);
\draw (u.south west) to[out=270, in=90] (dmu.north east);
\draw (u.south east) to[out=270, in=90] (v.north east);
\draw (dmu.south east) to[out=270, in=90] (dnu.north east);
\draw (dnu.south west) to[out=270, in=90] (end0);
\draw (dnu.south east) to[out=270, in=90] (v.north west);
\draw (v.south west) to[out=270, in=90] (end1);
\end{tikzpicture} \spacedequal{} j(p')
\end{align*}
We now show that, for two homotopic paths $p \sim p'$, we have $j(p) = j(p')$. It suffices to show
that the relation $\mathcal{R}$ on paths defined by
\[
p \mathcal{R} p' \Longleftrightarrow j(p) = j(p')
\]
is reflexive, symmetric, transitive and congruent, since we have already
proved that it contains $\sim_{\text{elem}}$. The three first
properties are obviously satisfied. The last one is a direct consequence of
the compatibility of $j$ with the concatenation of paths: $j(p \cdot p') =
Since $b / \ffunctor{F}$ is simply connected by hypothesis and $j$ is homotopy invariant, the $2$-isomorphism $j(p)$
only depends on the source and the target of $p$. Hence for any two objects
$(a, \alpha)$ and $(a', \alpha')$ in $b / \ffunctor{F}$, there is a unique
$2$-isomorphism $\psi_{(a,
\alpha)} \Rightarrow \psi_{(a', \alpha')}$ in $\catname{E}$ induced by a path in $b/
\ffunctor{F}$.
Given a morphism $u \colon b \to b'$ in B, there is a
base change functor:
\[
u^* \colon \left\{
\begin{array}{lll}
b' / \ffunctor{F} & \to & b / \ffunctor{F} \\
(x, \chi) & \mapsto & (x, \chi \circ u) \\
(v, \nu) & \mapsto & (v, \nu \cdot u)
\end{array}
\right.
\]
Note that this functor also extends to a function between the respective sets of
The application $j$ maps base change to whiskering:
\[
j(u^*p) = j(p) \cdot \ffunctor{D}u
\]
We can now use the above properties to construct a cone $\psi$ under $\ffunctor{D}$ with
vertex $e$. For any $b \in \catname{B}$, fix an arbitrary object $(a(b),
\alpha(b))$ in $b / \ffunctor{F}$. This defines the components $\psi_b =
\psi_{(a(b), \alpha(b))}$, as stated in <ref>. For a morphism $u \colon b \to b'$, note that
$\psi_{(a(b'), \alpha(b')} \circ \ffunctor{D}u =
\psi_{u^*(a(b'),\alpha(b'))}$; hence we can define $\psi_u$ as the unique
2-isomorphism $j(p)$ induced by any path $p \colon u^*(a(b'), \alpha(b')) \leadsto (a(b),
\alpha(b))$. We must check that $\psi$ is indeed a pseudonatural
transformation. The compatibility of $j$ with the whiskering and the
concatenation of paths implies the required compatibility of $\psi$ with the
composition of morphisms.
It remains to check the compatibility with 2-morphisms. Let $u, u' \colon b
\to b'$ be two parallel 1-morphisms and $\delta \colon u \Rightarrow u'$ be a
2-morphism in B. By unicity of
2-morphisms induced by a path (<ref>), it suffices to check that the pasting
\begin{equation}\label{diagComp2Morph}
\begin{tikzcd}
\ffunctor{D}b' \arrow[rrd, bend left] & & \\
& & e \\
\ffunctor{D}b \arrow[uu, bend left=2cm, "\ffunctor{D}u"{name=n1, description}] \arrow[uu,
bend right=2cm, "\ffunctor{D}u'"{name=n2, description}] \arrow[from=n1, to=n2, Rightarrow,
shorten=0.1cm, "\ffunctor{D}\delta"] \arrow[from=n2, to=2-3, Rightarrow,
shorten=0.1cm, "\psi_{u'}"] \arrow[rru, bend right]
\end{tikzcd}
\end{equation}
is induced by a path. Indeed, fix a path $p \colon (u')^*(a(b'), \alpha(b'))
\leadsto (a(b), \alpha(b))$ and recall that, by definition, $\psi_{u'} =
We consider the path $p'$ of length one:
\[
p' \spacedequal
(a(b'), u'\alpha(b')) \xleftarrow{(\id, \alpha(b')\delta)} (a(b'),
u'\alpha(b')) \spacedequal
\begin{tikzcd}[baseline=(current bounding box.center), sep=0.25cm]
a(b') \arrow[rr, equal] & & a(b') \\
b' \arrow[u] \arrow[rr, equal] & & b' \arrow[u] \\
& b \arrow[lu, bend left, "u"{name=n1}, pos=0.9] \arrow[ru, bend right,
"u'"{name=n2}, swap, pos=0.9]
\arrow[from=n1, to=n2, Rightarrow, "\delta", shorten=0.4cm]
\end{tikzcd}
\]
The above pasting (<ref>) is then induced by the concatenation $p' \cdot p$ of $p'$ and $p$.
Through similar arguments, we can see that any other choice of the objects
$(a(b),\alpha(b))_{b \in \catname{B}}$ leads to an isomorphic cone.
From now on, we assume that the objects $(a(b), \alpha(b))_{b \in \catname{B}}$ are fixed and
we write $\functor{L}(\phi)$ for the cone $\psi$ under $\ffunctor{D}$ induced
from the cone $\phi$ under $\ffunctor{D} \circ \ffunctor{F}$. Since we will not
work with a single fixed cone $\phi$ anymore, we should write $j_{\phi}$ instead
of $j$.
We would like to extend this mapping $\phi \mapsto \functor{L}{\phi}$ to a
\[
\functor{L} \colon [\catname{A},\catname{E}](\ffunctor{D} \circ \ffunctor{F}, \Delta{e}) \to
[\catname{B}, \catname{E}](\ffunctor{D}, \Delta{e}).
\]
We use the proposition:
Let $m \colon \phi \to \phi'$ be a modification between two cones
\[
\phi, \phi' \colon \ffunctor{D} \circ \ffunctor{F} \Rightarrow \Delta{e}
\]
For any path $p \colon (a, \alpha) \leadsto (a', \alpha')$ in $b /
\ffunctor{F}$, we have the following equality:
\begin{equation}\label{eqModifJ}
\begin{tikzcd}[column sep=0.35cm, row sep=1cm]
e \arrow[rr, equal] \arrow[from=d, bend left, "\phi_a", pos=0.45] & &
e \arrow[from=d, bend left, "\phi_{a'}"{name=n1}, pos=0.45]
\arrow[from=d, bend right, "\phi'_{a'}"{name=n2}, pos=0.45, swap]
\arrow[from=n1, to=n2, "m_{a'}", Rightarrow, shorten=0.2cm] \\
\ffunctor{D}\ffunctor{F}(a) \arrow[from=rd] \arrow[rr, Rightarrow,
"j_{\phi}(p)", shorten=0.5cm] & &
\ffunctor{D}\ffunctor{F}(a') \arrow[from=ld] \\
& \ffunctor{D}(b)
\end{tikzcd} \spacedequal
\begin{tikzcd}[column sep=0.35cm, row sep=1cm]
e \arrow[rr, equal] \arrow[from=d, bend left, "\phi_a"{name=n1}, pos=0.45]
\arrow[from=d, bend right, "\phi'_a"{name=n2}, pos=0.45, swap]
\arrow[from=n1, to=n2, "m_a", Rightarrow, shorten=0.2cm] & &
e \arrow[from=d, bend right, "\phi_{a'}", pos=0.45]\\
\ffunctor{D}\ffunctor{F}(a) \arrow[from=rd] \arrow[rr, Rightarrow,
"j_{\phi'}(p)", shorten=0.5cm] & &
\ffunctor{D}\ffunctor{F}(a') \arrow[from=ld] \\
& \ffunctor{D}(b)
\end{tikzcd}
\end{equation}
For an empty path $p$, the proposition reduce to the tautology $m_a = m_a$.
For a path $p = (a, \alpha) \xrightarrow{(u, \mu)} (a', \alpha')$ of length 1,
we can decompose the equation as:
\[
\begin{tikzcd}[column sep=0.35cm, row sep=1cm]
e \arrow[rr, equal] \arrow[from=d, bend left, "\phi_a"{name=n0}, pos=0.45,
swap] & &
e \arrow[from=d, bend left, "\phi_{a'}"{name=n1}, pos=0.45]
\arrow[from=d, bend right, "\phi'_{a'}"{name=n2}, pos=0.45, swap]
\arrow[from=n0, to=n1, "\phi_{u}^{-1}", Rightarrow, shorten=0.2cm]
\arrow[from=n1, to=n2, "m_{a'}", Rightarrow, shorten=0.2cm] \\
\ffunctor{D}\ffunctor{F}(a) \arrow[from=rd, ""{name=n3}]
\arrow[rr, "\ffunctor{D}\ffunctor{F}u"] & &
\ffunctor{D}\ffunctor{F}(a') \arrow[from=ld, ""{name=n4}, swap]
\arrow[from=n3, to=n4, Rightarrow, "\ffunctor{D}\mu^{-1}", shorten=0.2cm] \\
& \ffunctor{D}(b)
\end{tikzcd} \spacedequal
\begin{tikzcd}[column sep=0.35cm, row sep=1cm]
e \arrow[rr, equal] \arrow[from=d, bend left, "\phi_a"{name=n1}, pos=0.45]
\arrow[from=d, bend right, "\phi'_a"{name=n2}, pos=0.45, swap]
\arrow[from=n1, to=n2, "m_a", Rightarrow, shorten=0.2cm] & &
e \arrow[from=d, bend right, "\phi_{a'}"{name=n0}, pos=0.45]
\arrow[from=n2, to=n0, "\phi_u^{'-1}", Rightarrow, shorten=0.3cm] \\
\ffunctor{D}\ffunctor{F}(a) \arrow[from=rd, ""{name=n3}]
\arrow[rr, "\ffunctor{D}\ffunctor{F}u"] & &
\ffunctor{D}\ffunctor{F}(a') \arrow[from=ld, ""{name=n4}, swap]
\arrow[from=n3, to=n4, Rightarrow, "\ffunctor{D}\mu^{-1}", shorten=0.2cm]\\
& \ffunctor{D}(b)
\end{tikzcd}
\]
The lower parts of these diagrams are the same and the upper parts are equal,
by the property of the modification $m$. Hence <ref> holds for a
path $p = ((a, \alpha) \xrightarrow{(u, \mu)} (a', \alpha'))$. A similar
decomposition of the diagrams shows that it also holds for a path $p = ((a,
\alpha) \xleftarrow{(u, \mu)} (a', \alpha'))$ of length one in the reverse direction.
Since $j_\phi$ and $j_{\phi'}$ are compatible with paths concatenation, if
<ref> holds for two composable paths $p$ and $p'$, it also holds for
their concatenation $pp'$. We can thus conclude that it holds for any path $p$, as
the path $p$ is generated by paths of length 1.
This property directly implies that the components
\[
\functor{L}(m)_b \: = \: \begin{tikzcd}[column sep=1.2cm]
\ffunctor{D}(b) \arrow[r] & \ffunctor{D}(a(b))
\arrow[r, bend left, ""{name=n1}, pos=0.4, swap] \arrow[r, bend right,
""{name=n2}, pos=0.4]
& e
\arrow[from=n1, to=n2, "m_{a(b)}", Rightarrow]
\end{tikzcd}
\]
define a 2-morphism $\functor{L}(\phi) \to \functor{L}(\phi')$. The
functoriality of $\functor{L}$ is straightforward.
Now consider the canonical functor
\[
\functor{K} \colon \left\{
\begin{array}{lll}
[\catname{B}, \catname{E}](\ffunctor{D}, \Delta{e}) & \to
& [\catname{A}, \catname{E}](\ffunctor{D} \circ \ffunctor{F}, \Delta{e}) \\
\psi_{\bullet} & \mapsto & \psi_{\ffunctor{F}(\bullet)} \\
m_{\bullet} & \mapsto & m_{\ffunctor{F}(\bullet)}
\end{array}
\right.
\]
sending cones under D with vertex $e$ to cones under $\ffunctor{D}
\circ \ffunctor{F}$ with vertex $e$. We are now ready to show that
$\functor{L}$ and $\functor{K}$ are mutual pseudo-inverses.
There is a natural isomorphism $\eta \colon \id \Rightarrow \functor{K}\functor{L}$.
Let $\phi \in [\catname{A}, \catname{E}](\ffunctor{D} \circ \ffunctor{F},
\Delta{e})$ be a cone under $\ffunctor{D} \circ \ffunctor{F}$ with vertex $e$.
We will write $j = j_{\phi}$.
We want to define the component $\eta_{\phi} \colon \phi \to
\functor{K}\functor{L}\phi$ of $\eta$ at $\phi$. Since $\eta_{\phi}$ must be
a modification, we have to define its components at each object $a_0 \in \catname{A}$:
\[
\eta_{\phi, a_0} \colon \phi_{a_0} \Rightarrow \phi_{a(\ffunctor{F}a_0)} \circ \ffunctor{D}\alpha{\ffunctor{F}a_0}.
\]
Both $(a_0, \id_{\ffunctor{F}a_0})$ and $(a(\ffunctor{F}a_0),
\alpha(\ffunctor{F}a_0))$ are objects of $\ffunctor{F}a_0 / \ffunctor{F}$, which
is connected. Hence, there is a path in $\ffunctor{F}a_0 / \ffunctor{F}$:
\[
p \colon (a_0, \id_{\ffunctor{F}a_0}) \leadsto (a(\ffunctor{F}a_0),\alpha(\ffunctor{F}a_0))
\]
We have to check that $\eta_{\phi}$ is a modification. That is, for any
morphism $f \colon a_0 \to a_1$ in $\catname{A}$, we have to check the
commutativity of:
\[
\begin{tikzcd}[column sep=2cm]
\phi_a
\arrow[r, "\eta_{\phi, a_0}", Rightarrow]
\arrow[d, "\phi_f", Rightarrow]
& \phi_{a(\ffunctor{F}a_0)} \circ \ffunctor{D}\alpha(\ffunctor{F}a_0)
\arrow[d, "\functor{L}(\phi)_{\ffunctor{F}f}", Rightarrow] \\
\phi_{a_1} \circ \ffunctor{D}\ffunctor{F}f
\arrow[r, "\eta_{\phi, a_1} \cdot \ffunctor{D}\ffunctor{F}f", Rightarrow]
& \phi_{a(\ffunctor{F}a_1)} \circ \ffunctor{D}\alpha(\ffunctor{F}a_1)
\circ \ffunctor{D}\ffunctor{F}f
\end{tikzcd}
\]
We first remark that there is a path $p_0 = ((a_0, \id) \xrightarrow{(f, \id)}
(a_1, \ffunctor{F}f))$ in $\ffunctor{F}a_0 / \ffunctor{F}$ and the induced 2-isomorphism is $\phi_f = j(p_0)$.
Moreover, expanding the definitions, we have
\begin{align*}
&\eta_{\phi, a_0} = j(p_1) && \text{for some } p_1 \colon (a_0, \id) \leadsto (a(\ffunctor{F}a_0),\alpha(\ffunctor{F}a_0)) \\
&\eta_{\phi, a_1} = j(p_2) && \text{for some } p_2 \colon (a_1, \id) \leadsto (a(\ffunctor{F}a_1),\alpha(\ffunctor{F}a_1)) \\
& \functor{L}(\phi)_{\ffunctor{F}f} = j(p_3) && \text{for some } p_3 \colon (a(\ffunctor{F}a_0), \alpha(\ffunctor{F}a_0)) \leadsto \ffunctor{F}(f)^*(a(\ffunctor{F}a_1), \alpha(\ffunctor{F}a_1))
\end{align*}
where $p_1$ and $p_3$ are paths in $\ffunctor{F}a_0 / \ffunctor{F}$, and $p_2$
is a path in $\ffunctor{F}a_1 / \ffunctor{F}$.
We can check that $p_1 \cdot p_3$ and $p_0 \cdot \ffunctor{F}(f)^*p_2$
are paths
\[
(a_0, \id) \leadsto \ffunctor{F}(f)^*(a(\ffunctor{F}a_1),\alpha(\ffunctor{F}a_1)).
\]
\begin{align*}
(\eta_{\phi, a_1} \cdot \ffunctor{D}\ffunctor{F}f) \circ \phi_f
&= j(\ffunctor{F}(f)^*p_2) \circ j(p_0) \\
&= j(p_0 \cdot \ffunctor{F}(f)^*p_2) \\
&= j(p_1 \cdot p_3) \\
&= j(p_3) \circ j(p_1) \\
&= \functor{L}(\phi)_{\ffunctor{F}f} \circ \eta_{\phi, a_0} \\
\end{align*}
We also have to check the naturality of $\eta$. For any modification $m \colon
\phi \to \phi'$, we want the commutativity of the square:
\[
\begin{tikzcd}
\phi \arrow[r, "\eta_{\phi}"] \arrow[d, "m"] &
\functor{K}\functor{L}\phi \arrow[d, "\functor{K}\functor{L}m"] \\
\phi' \arrow[r, "\eta_{\phi'}"] &
\functor{K}\functor{L}\phi'
\end{tikzcd}
\]
That is, for any object $a_0 \in \catname{A}$:
\[
\begin{tikzcd}
\phi_{a_0} \arrow[r, "j_{\phi}(p)"] \arrow[d, "m_{a_0}"] &
\phi_{a(\ffunctor{F}a_0)} \circ \ffunctor{D}\alpha(\ffunctor{F}a_0) \arrow[d, "m_{\ffunctor{F}a_0}"] \\
\phi'_{a_0} \arrow[r, "j_{\phi'}(p)"] &
\functor{K}\functor{L}\phi'_{a_0}
\end{tikzcd}
\]
where $p \colon (a_0, \id) \leadsto (a(\ffunctor{F}a_0), \alpha(\ffunctor{F}a_0))$
is a path in $\ffunctor{F}a_0 / \ffunctor{F}$. This last square commutes by <ref>.
In the reverse direction we show:
There is a natural isomorphism $\epsilon : \functor{L}\functor{K} \Rightarrow \id$.
Fix a cone $\psi \colon \ffunctor{D} \Rightarrow \Delta{e}$ under $\ffunctor{D}$. Write $\psi' =
\functor{L}\functor{K}(\psi)$. For any $b \in \catname{B}$, we have:
\begin{align*}
\psi'_b &= \functor{K}(\psi)_{a(b)} \circ \ffunctor{D}(\alpha(b))
= \psi_{\ffunctor{F}(a(b))} \circ \ffunctor{D}(\alpha(b))
\end{align*}
Hence we can define a 2-morphism $\epsilon_{\psi,b} \colon \psi'_b \Rightarrow \psi_b$ in $\catname{E}$ by:
\begin{align*}
\epsilon_{\psi, b} = \psi_{\alpha(b)}
\end{align*}
When $b$ ranges over all objects of $\catname{B}$, these morphisms then form a modification $\epsilon_{\psi} \colon \psi' \to \psi$.
Indeed for any morphism $(u, \mu) \colon (a, \alpha) \to (a', \alpha')$ in $b
/ \ffunctor{F}$, we have:
\[
\begin{tikzcd}[row sep=1.5cm, column sep=1.5cm]
& e \\
\ffunctor{D}\ffunctor{F}(a') \arrow[ru, bend left] \arrow[ru, Rightarrow,
shorten=0.2cm, swap, "\psi_{\ffunctor{F}(u)}"] & \ffunctor{D}\ffunctor{F}(a) \arrow[l] \arrow[u] \\
& \ffunctor{D}(b) \arrow[lu, bend left, ""{name=n2}, swap] \arrow[u] \arrow[uu, bend right=3cm,
\arrow[from=2-2, to=n1, Rightarrow, "\psi_{\alpha}"]
\arrow[from=n2, to=2-2, Rightarrow, "\ffunctor{D}\mu"]
\end{tikzcd} \spacedequal
\begin{tikzcd}[row sep=1.5cm, column sep=1.5cm]
& e \\
\ffunctor{D}\ffunctor{F}(a') \arrow[ru, bend left] & \\
& \ffunctor{D}(b) \arrow[lu, bend left] \arrow[uu, ""{name=n1}]
\arrow[from=2-1, to=n1, Rightarrow, "\psi_{\alpha'}"]
\end{tikzcd}
\]
which implies a similar formula for any path $p \colon (a', \alpha') \leadsto (a, \alpha)$ in $b / \ffunctor{F}$:
\[
\begin{tikzcd}[row sep=1.5cm, column sep=1.5cm]
& e \\
\ffunctor{D}\ffunctor{F}(a') \arrow[ru, bend left] \arrow[r, Rightarrow, "j_{\functor{K}(\psi)}(p)"] & \ffunctor{D}\ffunctor{F}(a) \arrow[u] \\
& \ffunctor{D}(b) \arrow[lu, bend left] \arrow[u] \arrow[uu, bend right=3cm,
\arrow[from=2-2, to=n1, Rightarrow, "\psi_{\alpha}"]
\end{tikzcd} \spacedequal
\begin{tikzcd}[row sep=1.5cm, column sep=1.5cm]
& e \\
\ffunctor{D}\ffunctor{F}(a') \arrow[ru, bend left] & \\
& \ffunctor{D}(b) \arrow[lu, bend left] \arrow[uu, ""{name=n1}]
\arrow[from=2-1, to=n1, Rightarrow, "\psi_{\alpha'}"]
\end{tikzcd}
\]
This in turn implies that $\epsilon_{\psi}$ is a modification. Fix a morphism
$u \colon b \to b'$ and consider a path $p \colon u^*(a(b'),\alpha(b'))
\leadsto (a(b), \alpha(b))$ (hence we have $\psi'_u =
j_{\functor{K}(\psi)}(p)$). We check the modification axiom at $u$:
\begin{align*}
&\begin{tikzcd}[ampersand replacement=\&, row sep=1.2cm]
e \arrow[r, equal] \& e \\
\ffunctor{D}\ffunctor{F}(a(b')) \arrow[u] \arrow[r, Rightarrow, "\psi'_u"] \& \ffunctor{D}\ffunctor{F}(a(b)) \arrow[u] \\
\ffunctor{D}(b') \arrow[u] \& \ffunctor{D}(b) \arrow[l] \arrow[u] \arrow[uu, bend
right=3cm, ""{name=n1}] \arrow[from=2-2, to=n1, Rightarrow,
"\epsilon_{\psi, b}"]
\end{tikzcd} \spacedequal
\begin{tikzcd}[ampersand replacement=\&, row sep=1.2cm]
\& e \\
\ffunctor{D}\ffunctor{F}(a(b')) \arrow[ru, bend left] \arrow[r, Rightarrow, "j_{\functor{K}(\psi)}(p)"] \& \ffunctor{D}\ffunctor{F}(a(b)) \arrow[u] \\
\& \ffunctor{D}(b) \arrow[lu, bend left] \arrow[u] \arrow[uu, bend
right=3cm, ""{name=n1}] \arrow[from=2-2, to=n1, Rightarrow,
\end{tikzcd} \\ &=
\begin{tikzcd}[ampersand replacement=\&]
\& e \\
\ffunctor{D}\ffunctor{F}(a(b')) \arrow[ru, bend left] \& \\
\& \ffunctor{D}(b) \arrow[lu, bend left] \arrow[uu, ""{name=n1}]
\arrow[from=2-1, to=n1, Rightarrow, "\psi_{\alpha(b')u}"]
\end{tikzcd} =
\begin{tikzcd}[ampersand replacement=\&]
\& e \arrow[r, equal] \& e\\
\ffunctor{D}\ffunctor{F}(a(b')) \arrow[ru, bend left] \& \& \\
\& \ffunctor{D}(b') \arrow[lu, bend left] \arrow[uu, ""{name=n1}] \arrow[ruu, Rightarrow, shorten=0.1cm, "\psi_u"]
\& \ffunctor{D}(b) \arrow[l] \arrow[uu]
\arrow[from=2-1, to=n1, Rightarrow, "\epsilon_{\psi, b'}"]
\end{tikzcd}
\end{align*}
Finally we have to check the naturality of $\epsilon \colon \functor{L}\functor{K} \to
\id$, that is, for any modification of cones $m \colon \psi \to \psi'$, the
commutativity of the square:
\[
\begin{tikzcd}
\functor{L}\functor{K}\psi \arrow[r, "\epsilon_{\psi}"] \arrow[d,
"\functor{L}\functor{K}m"] & \psi \arrow[d, "m"] \\
\functor{L}\functor{K}\psi' \arrow[r, "\epsilon_{\psi'}"] & \psi'
\end{tikzcd}
\]
Indeed, for any object $b \in \catname{B}$:
\[
\epsilon_{\psi', b} \circ (\functor{L}\functor{K}m)_b = \psi'_{\alpha(b)}
\circ m_{a(b)}\alpha(b) = m_{b} \circ \psi_{\alpha(b)}.
\]
Putting together <ref> and <ref> we deduce:
The canonical functor
\[
\functor{K} \colon [\catname{B}, \catname{E}](\ffunctor{D}, \Delta{e}) \to [\catname{A},
\catname{E}](\ffunctor{D} \circ \ffunctor{F}, \Delta{e})
\]
is an equivalence.
Since this is true for any object $e$ of $\catname{E}$, clearly
$\ccolim \ffunctor{D}$ exists if and only if $\ccolim \ffunctor{D} \circ
\ffunctor{F}$ exists and, if it is the case, they are canonically equivalent.
We have thus proved one implication of <ref>:
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a $(2,1)$-functor.
If for any object $b \in \catname{B}$, the slice $(2,1)$-category $b /
\ffunctor{F}$ is nonempty, connected and simply connected, then the $(2,
1)$-functor $\ffunctor{F}$ is $2$-final.
The reverse implication is proved by observing the following fact:
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a $(2, 1)$-functor.
\[
\Pi_1(b / \ffunctor{F}) \cong \ccolim_{a \in \catname{A}}\catname{B}(b , \ffunctor{F}a).
\]
The wanted equivalence can be proved by constructing a family of equivalences,
pseudonatural in the category $T$:
\[
C_{T} \colon [\catname{A}, \catcat](\catname{B}(b, \ffunctor{F}-), \Delta{T})
\cong [\Pi_1(b / \ffunctor{F}), T].
\]
Fix $\psi \colon \catname{B}(b, \ffunctor{F}-) \Rightarrow \Delta{T}$ a
pseudonatural transformation. We want to define a functor $C_{T}(\psi) \colon
\Pi_1(b / \ffunctor{F}) \to T$.
For any object $(a, \alpha \colon b \to \ffunctor{F}(a))$ of $\Pi_1(b /
\ffunctor{F})$, set:
\[
C_{T}(\psi)(a, \alpha) = \psi_a(\alpha)
\]
For any morphism $(u, \mu \colon u\alpha \Rightarrow \alpha') \colon (a,
\alpha) \to (a', \alpha')$ of $b /
\ffunctor{F}$, define the composite isomorphism:
\[
C_{T}(\psi)(u, \mu) :
\begin{tikzcd}
\psi_a(\alpha) \arrow[r, "(\psi_u)_{\alpha}"] & \psi_{a'}(u \circ \alpha)
\arrow[r, "\psi_{a'}(\mu)"] & \psi_{a'}(\alpha')
\end{tikzcd}
\]
This can be extended to paths using the relations:
\begin{align*}
C_{T}(\psi)((a', \alpha') \xleftarrow{(u, \mu)} (a, \alpha)) &= C_{T}(\psi)((a, \alpha) \xrightarrow{(u, \mu)} (a', \alpha'))^{-1} \\
C_{T}(\psi)((a, \alpha)) &= \id_{\psi_a(\alpha)} \\
C_{T}(\psi)(p \cdot p') &= C_{T}(\psi)(p') \circ C_{T}(\psi)(p)
\end{align*}
On can check that such a definition is homotopy invariant, and gives a
well-defined functor $C_{T}(\psi) \colon \Pi_1(b / \ffunctor{F}) \to T$.
For a modification $m \colon \psi \to \psi'$, we define a natural transformation
\[
C_{T}(m) \colon C_{T}(\psi) \Rightarrow C_{T}(\psi')
\]
with components:
\begin{equation}\label{defCModif}
C_{T}(m)_{(a, \alpha)} = (m_a)_{\alpha}
\end{equation}
To show that $C_T$ is an equivalence, we show that it is a fully faithful and
essentially surjective functor.
Indeed it is clear that (<ref>) defines a bijection
between modifications $\psi \to \psi'$ and natural transformations $C_{T}(\psi)
\Rightarrow C_{T}(\psi')$. Hence $C_T$ is fully faithful.
Moreover, given any functor $F \colon \Pi_1(b / \ffunctor{F}) \to T$, one can
define a pseudonatural transformation $\psi \colon \catname{B}(b,
\ffunctor{F}-) \to \Delta{T}$ by:
\[
\psi_a : \left\{
\begin{array}{lll}
\catname{B}(b, \ffunctor{F}a) & \to & T \\
\alpha & \mapsto & F(a, \alpha) \\
\nu & \mapsto & F(\id_a, \nu)
\end{array}
\right.
\]
\[
(\psi_u)_{\alpha} \colon
F(a, \alpha) \xrightarrow{F(u, \id)} F(a', u \circ \alpha)
\]
for any object $a$ and morphism $u \colon a \to a'$ of $\catname{A}$. It is
straightforward to check:
\[
F = C_T(\psi)
\]
We can now prove:
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a $2$-final $(2,
1)$-functor. Then, for any object $b$ in $\catname{B}$:
\[
\Pi_1(b / \ffunctor{F}) \cong 1
\]
We have a chain of equivalences:
\[
\Pi_1(b / \ffunctor{F})
\overset{\ref{propPiColimit}}{\cong} \ccolim_{a \in
\catname{A}}\catname{B}(b, \ffunctor{F}a)
\overset{(1)}{\cong} \ccolim_{b' \in
\catname{B}}\catname{B}(b, b')
\overset{(2)}{\cong} 1
\]
The equivalence $(1)$ is an application of the $2$-finality of $\ffunctor{F}$.
The equivalence $(2)$ is a consequence of the Yoneda lemma for $2$-categories.
Indeed we have the chain of equivalences, for any $1$-category $T$, and
pseudonatural in $T$:
\[
[\catname{B}, \catcat](\catname{B}(b, -), \Delta{T}) \cong \Delta{T}(b)
\cong T \cong \catcat(1, T)
\]
By combining <ref> and <ref>, we have:
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a $2$-final $(2,
1)$-functor. Then, for any object $b$ in $\catname{B}$, the $(2, 1)$-category
$b / \ffunctor{F}$ is nonempty, connected and simply connected.
There is a dual notion of 2-initial 2-functor, with a dual criterion,
proven by a duality argument.
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a $2$-functor between
$(2,1)$-categories. The 2-functor is said to be 2-initial if, for any
$2$-diagram $\ffunctor{D} \colon \catname{B} \to \catname{E}$, each of the bilimits
$\llim_{\catname{B}}{\ffunctor{D}}$ and $\llim_{\catname{A}}{\ffunctor{D} \circ
\ffunctor{F}}$ exists whenever the other one exists, and the canonical
comparison $1$-morphism
\[
\llim_{\catname{B}}{\ffunctor{D}} \to \llim_{\catname{A}}{\ffunctor{D} \circ \ffunctor{F}}
\]
is an equivalence.
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a 2-functor between
$(2,1)$-categories. The 2-functor $\ffunctor{F}$ is initial if and only if the
2-functor $\catop{\ffunctor{F}} \colon \catop{\catname{A}} \to
\catop{\catname{B}}$ is final.
Let $\ffunctor{F} \colon \catname{A} \to \catname{B}$ be a 2-functor between
$(2,1)$-categories. The 2-functor $\ffunctor{F}$ is initial if and only if,
for any object $b \in B$, the slice $(2, 1)$-category $\ffunctor{F}/b$ is
nonempty, connected and simply connected.
§ FURTHER DIRECTIONS
There are various direction in which one may try to improve the finality
criterion presented in the previous section.
The most straightforward one is to work in the context of bicategories, where
composition of $1$-morphisms is only associative up to isomorphism. One should note
that the correct notions of $2$-finality for pseudofunctors between bicategories
with invertible $2$-morphisms should be weakened to include any pseudofunctor
as diagram, and not only the strict ones as we do in <ref>. Since
not all pseudofunctors can be strictified (lackBicatNotTriequivalent2007),
the analogous result for pseudofunctors is not a direct corollary of <ref>.
Another natural route is to prove it for higher dimensions $n$. An analogous
combinatorial proof would require a combinatorial presentation of higher
homotopies in an $n$-category, and probably to set up a machinery for working
inductively on the dimension. An alternative, potentially more reasonable
approach may be to adapt Lurie's topological proof to finite dimensions.
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
New York
Oxford University Press
title2 Dimensional Categories
journaltitleTheory and Applications of Categories
shortjournalTheory Appl. Categ.
titleBicat Is Not Triequivalent to Gray
pagesNo. 113
Princeton, N.J
Princeton University Press
annotationOCLC: ocn244702012
numberno. 170
seriesAnnals of Mathematics Studies
titleHigher Topos Theory
Categories (Mathematics),Toposes
family=Mac Lane Saunders,
New York
seriesGraduate Texts in Mathematics
titleCategories for the Working Mathematician
Categories (Mathematics),Catégories (mathématiques),Problèmes aux limites
typePh.D. Thesis (In preparation)
|
# Self-Adaptive Training: Bridging Supervised and Self-Supervised Learning
Lang Huang, Chao Zhang, and Hongyang Zhang The material in this paper was
presented in part at Thirty-fourth Conference on Neural Information Processing
Systems, December 2020 [1]. L. Huang is with the Department of Information &
Communication Engineering, The University of Tokyo, Tokyo, Japan. C. Zhang is
with the Key Laboratory of Machine Perception (MOE), School of Intelligence
Science and Technology, Peking University, Beijing, China. H. Zhang is with
the David R. Cheriton School of Computer Science, University of Waterloo,
Waterloo, Canada.E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>received Month Date, Year; revised
Month Date, Year.
###### Abstract
We propose self-adaptive training—a unified training algorithm that
dynamically calibrates and enhances training processes by model predictions
without incurring an extra computational cost—to advance both supervised and
self-supervised learning of deep neural networks. We analyze the training
dynamics of deep networks on training data that are corrupted by, e.g., random
noise and adversarial examples. Our analysis shows that model predictions are
able to magnify useful underlying information in data and this phenomenon
occurs broadly even in the absence of _any_ label information, highlighting
that model predictions could substantially benefit the training processes:
self-adaptive training improves the generalization of deep networks under
noise and enhances the self-supervised representation learning. The analysis
also sheds light on understanding deep learning, e.g., a potential explanation
of the recently-discovered double-descent phenomenon in empirical risk
minimization and the collapsing issue of the state-of-the-art self-supervised
learning algorithms. Experiments on the CIFAR, STL, and ImageNet datasets
verify the effectiveness of our approach in three applications: classification
with label noise, selective classification, and linear evaluation. To
facilitate future research, the code has been made publicly available at
https://github.com/LayneH/self-adaptive-training.
###### Index Terms:
Deep Learning, Supervised Learning, Self-Supervised Learning, Generalization,
Robust Learning under Noise.
## 1 Introduction
Deep neural networks have received significant attention in machine learning
and computer vision, in part due to their impressive performance achieved by
supervised learning approaches in the ImageNet challenge. With the help of
massive labeled data, deep neural networks advance the state-of-the-art to an
unprecedented level on many fundamental tasks, such as image classification
[2, 3], object detection [4], and semantic segmentation [5]. However, data
acquisition is notoriously costly, error-prone, and even infeasible in certain
cases. Furthermore, deep neural networks suffer significantly from overfitting
in these scenarios. On the other hand, the great success of self-supervised
pre-training in natural language processing (e.g., GPT [6, 7, 8] and BERT [9])
highlights that learning universal representations from unlabeled data can be
even more beneficial for a broad range of downstream tasks.
Regarding this, much effort has been devoted to learning representations
without human supervision for computer vision. Several recent studies show
promising results and largely close the performance gap between supervised and
self-supervised learning. To name a few, the contrastive learning approaches
[10, 11, 12] solve the instance-wise discrimination task [13] as a proxy
objective of representation learning. Extensive studies demonstrate that self-
supervisedly learned representations are generic and even outperform the
supervised pre-trained counterparts when they are fine-tuned on certain
downstream tasks.
Our work advances both supervised learning and self-supervised learning
settings. Instead of designing two distinct algorithms for each learning
paradigm separately, in this paper, we explore the possibility of a unified
algorithm that bridges supervised and self-supervised learning. Our
exploration is based on two observations on the learning of deep neural
networks.
(a) Accuracy curves of the model trained by ERM.
(b) Accuracy curves of the model trained by our method.
Figure 1: Accuracy curves of models on the CIFAR10 dataset with 40% of
corrupted data. All models can only have access to the noisy training set
(corresponding to the red dashed curve) during training and are tested on the
other three sets for illustration. The horizontal dotted line displays the
percentage of clean data in the training sets. It shows that while ERM suffers
from overfitting to noise, our method enjoys improved generalization and
higher validation accuracy (e.g., the improvement is larger than 10% in the
left-most column).
Observation I: Under various _supervised learning_ settings with noisy data,
deep neural networks are able to magnify useful underlying information of data
by their predictions.
Observation II: Observation I extends to the extreme noise where the
supervised signals are completely random, which is equivalent to _self-
supervised learning_.
These two observations indicate that model predictions can magnify useful
information in data, which further indicates that incorporating predictions
into the training processes could significantly benefit model learning. With
this in mind, we propose _self-adaptive training_ —a carefully designed
approach that dynamically uses model predictions as a guiding principle in the
design of training algorithms—that bridges supervised learning and self-
supervised learning in a unified framework. Our approach is conceptually
simple yet calibrates and significantly enhances the learning of deep models
in multiple ways.
### 1.1 Summary of our contributions
Self-adaptive training sheds light on understanding and improving the learning
of deep neural networks.
* •
We analyze the Empirical Risk Minimization (ERM) training processes of deep
models on four kinds of corruptions (see Fig. 1(a)). We describe the failure
scenarios of ERM and observe that useful information from data has been
distilled to model predictions in the first few epochs. We show that this
phenomenon occurs broadly even in the absence of any label information (see
Fig. 2). These insights motivate us to propose self-adaptive training—a
unified training algorithm for both supervised and self-supervised learning—to
improve the learning of deep neural networks by dynamically incorporating
model predictions into training, requiring no modification to existing network
architecture and incurring almost no extra computational cost.
* •
We show that self-adaptive training improves the generalization of deep
networks under both label-wise and instance-wise random noise (see Fig. 1 and
3). Besides, self-adaptive training exhibits a single-descent error-capacity
curve (see Fig. 4). This is in sharp contrast to the recently-discovered
double-descent phenomenon in ERM, which might be a result of overfitting to
noise. Moreover, while adversarial training may easily overfit adversarial
noise, our approach mitigates the overfitting issue and improves the
adversarial accuracy by $\sim$3% over the state-of-the-art (see Fig. 5).
* •
Self-adaptive training questions and alleviates the dependency of recent self-
supervised algorithms on the dominant training mechanism that typically
involves multiple augmented views of the same images at each training step:
self-adaptive training achieves remarkable performance despite requiring only
a single view of each image for training, which significantly reduces the
heavy cost of data pre-processing and model training on extra views.
Self-adaptive training has three applications and advances the state-of-the-
art by a significant gap.
* •
Learning with noisy labels, where the goal is to improve the performance of
deep networks on clean test data in the presence of training label noise. On
the CIFAR datasets, our approach obtains up to 9% absolute classification
accuracy improvement over the state-of-the-art. On the ImageNet dataset, our
approach improves over ERM by 3% under 40% noise rate.
* •
Selective classification, which trades prediction coverage for classification
accuracy. Our approach achieves up to 50% relative improvement over the state-
of-the-art on three datasets under various coverage rates.
* •
Linear evaluation, which evaluates the representations of a self-supervised
pre-trained model using a linear classifier. Our approach performs on par with
or even better than the state-of-the-art on various datasets. In particular,
on the ImageNet dataset, self-adaptive training achieves 72.8% top1 linear
accuracy using only 200 pre-training epochs, surpassing the other methods by
2.2% in absolute under the same setting.
## 2 Self-Adaptive Training
### 2.1 Blessing of model predictions
On corrupted data Recent works [14, 15] cast doubt on the ERM training:
techniques such as uniform convergence might be unable to explain the
generalization of deep neural networks because ERM easily overfits the
training data even though the training data are partially or completely
corrupted by random noise. To take a closer look at this phenomenon, we
conduct the experiments on the CIFAR10 dataset [16], splitting the original
training data into a training set (consists of the first 45,000 data pairs)
and a validation set (consists of the last 5,000 data pairs). We measure four
random noise schemes according to prior work [14], where the data are
_partially_ corrupted with probability $p$: 1) _Corrupted labels_. Labels are
assigned uniformly at random; 2) _Gaussian_. Images are replaced by random
Gaussian samples with the same mean and standard deviation as the original
image distribution; 3) _Random pixels_. Pixels of each image are shuffled
using independent random permutations; 4) _Shuffled pixels_. Pixels of each
image are shuffled using a fixed permutation pattern. We consider the
performance on both the noisy and the clean sets (i.e., the original
uncorrupted data), while the models can only have access to the noisy training
sets.
Fig. 1(a) displays the accuracy curves of ERMs that are trained on the noisy
training sets under four kinds of random corruptions: ERM easily overfits
noisy training data and achieves nearly perfect training accuracy. However,
the four subfigures exhibit very different generalization behaviors which are
indistinguishable if we only look at the accuracy curve on the noisy training
set (the red curve). In Fig. 1(a), the accuracy increases in the early stage,
and the generalization errors grow quickly only after a certain number of
epochs. Intuitively, early-stopping improves the generalization in the
presence of label noise (see the first column in Fig. 1(a)); however, it
remains unclear how to properly identify such an epoch without using
validation data. Moreover, the early-stop mechanism may significantly hurt the
performance on the clean validation sets, as we can see in the last three
columns of Fig. 1(a). Our approach is motivated by the failure scenarios of
ERM and goes beyond ERM. We begin by making the following observations in the
leftmost subfigure of Fig. 1(a): the peak of the accuracy curve on the clean
training set (>80%) is much higher than the percentage of clean data in the
noisy training set ($\sim$60%). This finding was also previously reported by
[17, 18, 19] under label corruption and suggested that model predictions might
be able to magnify useful underlying information in data. We confirm this
finding and show that the pattern occurs under various kinds of corruption
more broadly (see the last three subfigures of Fig. 1(a)).
On unlabelled data We notice that supervised learning with 100% noisy labels
is equivalent to unsupervised learning if we simply discard the meaningless
labels. Therefore, it would be interesting to analyze how deep models behave
in such an extreme case. Here, we conduct experiments on the CIFAR10 [16]
dataset and consider two kinds of random noise as the training (real-valued)
targets for deep learning models: 1) the output features of another model on
the same training images, where the model is randomly initialized and then
frozen; 2) random noise that is drawn i.i.d. from standard Gaussian
distribution and then fixed. The training of the deep model is then formulated
as minimizing the mean square error between $\ell_{2}$-normalized model
predictions and these two kinds of random noise. To monitor the training, we
learn a linear classifier on the top of each model to evaluate its
representation at each training epoch (see Appendix B.4 for the detailed setup
of this classifier).
Figure 2: Linear evaluation accuracy on CIFAR10 of checkpoints of the model
trained by fitting the noise as described in Sec. 2.1 (the red and green
curves) and fitting the accumulated representation of the model (the blue
curve). The horizontal dashed line indicates the accuracy of linear evaluation
directly using a randomly initialized network. All the curves are smoothed for
better demonstration.
Fig. 2 shows the linear evaluation accuracy curves of models trained on these
two kinds of noise, i.e., the green and red curves (the explanation of the
blue curve is deferred to Sec. 6.1). We observe that, perhaps surprisingly,
the model trained by predicting fixed random Gaussian noise (the green curve)
achieves 57% linear evaluation accuracy, which is substantially higher than
the 38% accuracy of a linear classifier trained on top of a randomly
initialized network (the dashed horizontal line). This intriguing observation
shows that deep neural networks are able to distill underlying information
from data to its predictions, even when supervision signals are completely
replaced by random noise. Furthermore, although predicting the output of
another network can also improve the representation learning (the red curve),
its performance is worse than that of the second scheme. We hypothesize that
this might be a result of inconsistent training targets: the output of a
network depends on the input training image, which is randomly transformed by
the data augmentation at each epoch. This suggests that the consistency of
training targets should also be taken into account in the design of the
training algorithm.
Inspiration for our methodology Based on our analysis, we propose a unified
algorithm, _Self-Adaptive Training_ , for both supervised and self-supervised
learning. Self-adaptive training incorporates model predictions to augment the
training processes of deep models in a _dynamic_ and _consistent_ manner. Our
methodology is general and very effective: self-adaptive training
significantly improves (a) the generalization of deep models on the corrupted
data, and (b) the representation learning of deep models without human
supervision.
### 2.2 Meta algorithm: Self-Adaptive Training
Given a set of training images $\\{\bm{x}_{i}\\}_{n}$ and a deep network
$f_{\theta}(\cdot)$ parametrized by $\theta$, our approach records training
targets $\\{\bm{t}_{i}\\}_{n}$ for all data points accordingly. We first
obtain the predictions of the deep network as
$\displaystyle\bm{p}_{i}$ $\displaystyle=\rho(f_{\theta}(\bm{x}_{i})),$ (1)
where $\rho(\cdot)$ is a normalization function. Then, the training targets
track all historical model predictions during training and are updated by
Exponential-Moving-Average (EMA) scheme as
$\displaystyle\bm{t}_{i}$
$\displaystyle\leftarrow\alpha\times\bm{t}_{i}+(1-\alpha)\times\bm{p}_{i}.$
(2)
The EMA scheme in Equation (2) alleviates the instability issue of model
predictions, smooths out $\bm{t}_{i}$ during the training process, and enables
our algorithm to completely change the training labels if necessary. The
momentum term $\alpha$ controls the weight of the model predictions. Finally,
we can update the weights $\theta$ of the deep network $f_{\theta}$ by
Stochastic Gradient Descent (SGD) on the loss function
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})$ at each training iteration.
We summarize the meta algorithm of self-adaptive training in Algorithm 1. The
algorithm is conceptually simple, flexible, and has three components adapting
to different learning settings: 1) the training _targets initialization_ ; 2)
_normalization function_ $\rho(\cdot)$; 3) _loss function_
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})$. In the following sections, we
will elaborate on the instantiations of these components for specific learning
settings.
Algorithm 1 Self-Adaptive Training
0: Data $\\{\bm{x}_{i}\\}_{n}$, deep network $f_{\theta}(\cdot)$ parametrized
by $\theta$, momentum term $\alpha$, normalization function $\rho(\cdot)$
1: Initialize targets $\\{\bm{t}_{i}\\}_{n}$
2: repeat
3: Fetch mini-batch data $\\{(\bm{x}_{i},\bm{t}_{i})\\}_{m}$
4: for $i=1$ to $m$ (in parallel) do
5: $\bm{p}_{i}=\rho(f_{\theta}(\bm{x}_{i}))$
6: $\bm{t}_{i}\leftarrow\alpha\times\bm{t}_{i}+(1-\alpha)\times\bm{p}_{i}$
7: end for
8: Update $f_{\theta}$ by SGD on
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})$
9: until end of training
Convergence analysis To simplify the analysis, we consider a linear regression
problem with data $\\{\bm{x}_{i}\\}_{n}$, training targets $\\{t_{i}\\}_{n}$
and a linear model $f_{\bm{\theta}}(\bm{x})=\bm{x}\bm{\theta}$, where
$\bm{x}_{i}\in\mathbb{R}^{d}$, $t_{i}\in\mathbb{R}$ and
$\theta\in\mathbb{R}^{d}$. Let
$\mathbf{X}=[\bm{x}_{1}|\bm{x}_{2}|\cdots|\bm{x}_{n}]\in\mathbb{R}^{n\times
d}$, $\bm{t}=[t_{1}|t_{2}|\cdots|t_{n}]\in\mathbb{R}^{n}$. Then the
optimization for this regression problem (corresponding to
$\min_{\bm{\theta}}\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})$ in Algorithm
1) can be written as
$\displaystyle\arg\min_{\bm{\theta}}\left\lVert\mathbf{X}\bm{\theta}-\bm{t}\right\rVert_{2}^{2}.$
(3)
Let $\bm{\theta}^{(k)}$ and $\bm{t}^{(k)}$ be the model parameters and
training targets at the $k$-th training step, respectively. Let $\eta$ denote
the learning rate for gradient descent update. The Algorithm 1 alternatively
minimizes the problem (3) over $\bm{\theta}^{(k)}$ and $\bm{t}^{(k)}$ as
$\displaystyle\begin{split}\bm{\theta}^{(k)}&=\bm{\theta}^{(k-1)}-\eta\nabla
f_{\bm{\theta}^{(k-1)}}(\mathbf{X})\\\
&=\bm{\theta}^{(k-1)}-\eta\mathbf{X}^{\intercal}(\mathbf{X}\bm{\theta}^{(k-1)}-\bm{t}^{(k-1)}),\end{split}$
(4)
$\displaystyle\begin{split}\bm{t}^{(k)}&=\alpha\bm{t}^{(k-1)}+(1-\alpha)f_{\bm{\theta}^{(k)}}(\mathbf{X})\\\
&=\alpha\bm{t}^{(k-1)}+(1-\alpha)\mathbf{X}\bm{\theta}^{(k)}.\end{split}$ (5)
###### Proposition 1.
Let $d_{\mathrm{max}}$ be the maximal eigenvalue of the matrix
$\mathbf{X}\mathbf{X}^{\intercal}$, if the learning rate
$\eta<\frac{\alpha+1}{\alpha d_{\mathrm{max}}}$, then
$\displaystyle\lim_{k\rightarrow\infty}\left\lVert\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}\right\rVert_{2}^{2}=0,$
(6)
###### Corollary 1.
Under the same condition as in Proposition 1, we have
$\displaystyle\lim_{k\rightarrow\infty}\frac{\left\lVert\mathbf{X}\bm{\theta}^{(k+1)}-\bm{t}^{(k+1)}\right\rVert_{2}^{2}}{\left\lVert\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}\right\rVert_{2}^{2}}<1,$
(7)
Combining Proposition 1 and Corollary 1, we see that, with a proper learning
rate for gradient descent, the optimization of problem (3) converges at least
$Q$-linearly to 0. The proofs of Proposition 1 and Corollary 1 are presented
in Appendix A.
Limitations We note that one disadvantage of self-adaptive training is the
extra storage cost of the training targets, which might be a problem when
training on an extremely large dataset. However, we find this cost is actually
moderate in most realistic cases. Take the large-scale ImageNet dataset [20]
as an example. The ImageNet consists of about 1.2 million images categorized
into 1,000 classes. The storage of such vectors in the single precision format
for the entire dataset requires $1.2\times 10^{6}\times 1000\times 32$ bit
$\approx 4.47$GB, which is reduced to $\sim$$1.12$GB under the self-supervised
learning setting that records a $256$-d feature for each image. The cost is
acceptable since modern GPUs usually have 20GB or more dedicated memory, e.g.,
NVIDIA Tesla A100 has 40GB of memory. Moreover, the vectors can be (a) sharded
to all GPU devices [21], (b) stored on the CPU memory, or even (c) stored on
the disk and loaded along with the images, to further reduce the storage cost.
Figure 3: Generalization error and clean validation error under four kinds of
random noise (represented by different colors) for ERM (the dashed curves) and
our approach (the solid curves) on the CIFAR10 dataset. We zoom in on the
dashed rectangle region and display it in the third column for a clear
demonstration.
## 3 Improved Generalization of Deep Models
### 3.1 Supervised Self-Adaptive Training
Instantiation We consider a $c$-class classification problem and denote the
images by $\bm{x}_{i}$, labels by
$\bm{y}_{i}\in\\{0,1\\}^{c},\bm{y}_{i}^{\intercal}\mathbf{1}=1$. Given a data
pair $(\bm{x}_{i},\bm{y}_{i})$, our approach instantiates the three components
of meta Algorithm 1 for supervised learning as follows:
1. 1.
_Targets initialization_. Since the labels are provided, the training target
$\bm{t}_{i}$ is directly initialized as $\bm{t}_{i}\leftarrow\bm{y}_{i}$.
2. 2.
_Normalization function_. We use the softmax function
$\mathrm{softmax}(\bm{u})_{j}=\exp^{\bm{u}_{j}}/\sum_{k}\exp^{\bm{u}_{k}}$ to
normalize the model predictions into probability vectors
$\bm{p}_{i}=\mathrm{softmax}(f_{\theta}(\bm{x}_{i}))$, such that
$\bm{p}_{i}\in[0,1]^{c},\bm{p}_{i}^{\intercal}\mathbf{1}=1$.
3. 3.
_Loss function_. Following the common practice in supervised learning, the
loss function is implemented as the cross entropy loss between model
predictions $\bm{p}_{i}$ and training targets $\bm{t}_{i}$, i.e.,
$\sum_{j}\bm{t}_{i,j}~{}\mathrm{log}~{}\bm{p}_{i,j}$, where $\bm{t}_{i,j}$ and
$\bm{p}_{i,j}$ represent the $j$th entry of $\bm{t}_{i}$ and $\bm{p}_{i}$,
respectively.
During the training process, we fix $\bm{t}_{i}$ in the first $\mathrm{E}_{s}$
training epochs and update the training targets $\bm{t}_{i}$ according to
Equation (2) in each following training epoch. The number of initial epochs
$\mathrm{E}_{s}$ allows the model to capture informative signals in the data
set and excludes ambiguous information that is provided by model predictions
in the early stage of training.
Sample re-weighting Based on the scheme presented above, we introduce a simple
yet effective sample re-weighting scheme on each sample. Concretely, given
training target $\bm{t}_{i}$, we set
$w_{i}=\max_{j}~{}\bm{t}_{i,j}.$ (8)
The sample weight $w_{i}\in[\frac{1}{c},1]$ reveals the labeling confidence of
this sample. Intuitively, all samples are treated equally in the first
$\mathrm{E}_{s}$ epochs. As the target $\bm{t}_{i}$ being updated, our
algorithm pays less attention to potentially erroneous data and learns more
from potentially clean data. This scheme also allows the corrupted samples to
re-attain attention if they are confidently corrected.
Putting everything together We use stochastic gradient descent to minimize:
$\mathcal{L}(f)=-\frac{1}{\sum_{i}w_{i}}\sum_{i}w_{i}\sum_{j}\bm{t}_{i,j}~{}\mathrm{log}~{}\bm{p}_{i,j}$
(9)
during the training process. Here, the denominator normalizes per sample
weight and stabilizes the loss scale. We summarize the _Supervised Self-
Adaptive Training_ and display the pseudocode in Algorithm 2. Intuitively, the
optimal choice of hyper-parameter $\mathrm{E}_{s}$ should be related to the
epoch where overfitting occurs, which is around the $60$th epoch according to
Fig. 1(a). For convenience, we directly fix the hyper-parameters
$\mathrm{E}_{s}=60$, $\alpha=0.9$ by default if not specified. Experiments on
the sensitivity of our algorithm to hyper-parameters are deferred to Sec. 4.2.
Our approach requires no modification to existing network architecture and
incurs almost no extra computational cost.
Algorithm 2 Supervised Self-Adaptive Training
0: Data $\\{(\bm{x}_{i},\bm{y}_{i})\\}_{n}$; classifier $f_{\theta}$; initial
epoch $\mathrm{E}_{s}=60$; momentum term $\alpha=0.9$
1: Initialize training targets by
$\\{\bm{t}_{i}\\}_{n}\leftarrow\\{\bm{y}_{i}\\}_{n}$
2: repeat
3: Fetch mini-batch data $\\{(\bm{x}_{i},\bm{t}_{i})\\}_{m}$ at current epoch
$e$
4: for $i=1$ to $m$ (in parallel) do
5: $\bm{p}_{i}=\mathrm{softmax}(f_{\theta}(\bm{x}_{i}))$
6: if $e>\mathrm{E}_{s}$ then
7: $\bm{t}_{i}\leftarrow\alpha\times\bm{t}_{i}+(1-\alpha)\times\bm{p}_{i}$
8: end if
9: $w_{i}=\max_{j}~{}\bm{t}_{i,j}$
10: end for
11:
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})=-\frac{1}{\sum_{i}w_{i}}\sum_{i}w_{i}\sum_{j}\bm{t}_{i,j}~{}\mathrm{log}~{}\bm{p}_{i,j}$
12: Update $f_{\theta}$ by SGD on
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta})$
13: until end of training
Methodology differences with prior work Supervised self-adaptive training
consists of two components: a) label correction; b) sample re-weighting. With
the two components, our algorithm is robust to both instance-wise and label-
wise noise and is ready to combine with various training schemes such as
natural and adversarial training, without incurring multiple rounds of
training. In contrast, the vast majority of works on learning from corrupted
data follow a preprocessing-training scheme with an emphasis on the label-wise
noise only: this line of research either discards samples based on the
disagreement between noisy labels and model predictions [22, 23, 24, 25], or
corrects noisy labels [26, 27]; [28] investigated a more generic approach that
corrects both label-wise and instance-wise noise. However, their approach
inherently suffers from extra computational overheads. Besides, unlike the
general scheme in robust statistics [29] and other re-weighting methods [30,
31] that used an additional optimization step to update the sample weights,
our approach directly obtains the weights based on accumulated model
predictions and thus is much more efficient.
### 3.2 Improved generalization under random noise
We consider the noise scheme (including noise type and noise level) and model
capacity as two factors that affect the generalization of deep networks under
random noise. We analyze self-adaptive training by varying one of the two
factors while fixing the other.
Varying noise schemes We use ResNet-34 [3] and rerun the same experiments as
in Fig. 1(a) by replacing ERM with our approach. In Fig. 1(b), we plot the
accuracy curves of models trained with our approach on four corrupted training
sets and compare them with those in Fig. 1(a). We highlight the following
observations.
* •
Our approach mitigates the overfitting issue in deep networks. The accuracy
curves on noisy training sets (i.e., the red dashed curves in Fig. 1(b))
nearly converge to the percentage of clean data in the training sets and do
not reach perfect accuracy.
* •
The generalization errors of self-adaptive training (the gap between the red
and blue dashed curves in Fig. 1(b)) are much smaller than Fig. 1(a). We
further confirm this observation by displaying the generalization errors of
the models trained on the four noisy training sets under various noise rates
in the leftmost subfigure of Fig. 3. The generalization errors of ERM
consistently grow as we increase the injected noise level. In contrast, our
approach significantly reduces the generalization errors across all noise
levels from 0% (no noise) to 90% (overwhelming noise).
* •
The accuracy on the clean sets (cyan and yellow solid curves in Fig. 1(b)) is
monotonously increasing and converges to higher values than that of the
counterparts in Fig. 1(a). We also show the clean validation errors in the
right two subfigures in Fig. 3. The figures show that the error of self-
adaptive training is consistently much smaller than that of ERM.
Figure 4: Double-descent ERM _vs._ single-descent self-adaptive training on
the error-capacity curve. The model of width 64 corresponds to the standard
ResNet-18. The vertical dashed line represents the interpolation threshold
[32, 33].
Varying model capacity We notice that such analysis is related to a recently
discovered intriguing phenomenon [34, 35, 36, 37, 32, 38, 33] in modern
machine learning models: as the capacity of the model increases, the test
error initially decreases, then increases, and finally shows a second descent.
This phenomenon is termed _double descent_ [32] and has been widely observed
in deep networks [33]. To evaluate the double-descent phenomenon on self-
adaptive training, we follow exactly the same experimental settings as [33]:
we vary the width parameter of ResNet-18 [3] and train the networks on the
CIFAR10 dataset with 15% of training labels being corrupted at random (details
are given in Appendix B.1).
Fig. 4 shows the test error curves. It shows that self-adaptive training
overall achieves a much lower test error than that of ERM except when using
extremely small models that underfit the training data. This suggests that our
approach can improve the generalization of deep networks especially when the
model capacity is reasonably large. Besides, we observe that the curve of ERM
clearly exhibits the double-descent phenomenon, while the curve of our
approach is monotonously decreasing as the model capacity increases. Since the
double-descent phenomenon may vanish when label noise is absent [33], our
experiment indicates that this phenomenon may be a result of overfitting to
noise and we can bypass it by a properly designed training process such as
self-adaptive training.
### 3.3 Improved generalization under adversarial noise
Adversarial noise [39] is different from random noise in that the noise is
model-dependent and imperceptible to humans. We use the state-of-the-art
adversarial training algorithm TRADES [40] as our baseline to evaluate the
performance of self-adaptive training under adversarial noise.
Algorithmically, TRADES minimizes
$\mathbb{E}_{\bm{x},\bm{y}}\Bigg{\\{}\mathrm{CE}(\bm{p}(\bm{x}),\bm{y})+\max_{\|\widetilde{\bm{x}}-\bm{x}\|_{\infty}\leq\epsilon}\mathrm{KL}(\bm{p}(\bm{x}),\bm{p}(\widetilde{\bm{x}}))/\lambda\Bigg{\\}},$
(10)
where $\bm{p}(\cdot)$ is the model prediction, $\epsilon$ is the maximally
allowed perturbation, CE stands for the cross entropy, KL stands for the
Kullback–Leibler divergence, and the hyper-parameter $\lambda$ controls the
trade-off between robustness and accuracy. We replace the CE term in TRADES
loss with our method. The models are evaluated using robust accuracy
$\frac{1}{n}\sum_{i}\mathbbm{1}\\{\operatorname{argmax}~{}\bm{p}(\widetilde{\bm{x}}_{i})=\operatorname{argmax}~{}\bm{y}_{i}\\}$,
where adversarial examples $\widetilde{\bm{x}}$ are generated by white box
$\ell_{\infty}$ AutoAttack [41] with $\epsilon$ = 0.031 (the evaluation of
projected gradient descent attack [42] is given in Fig. 14 of Appendix C). We
set the initial learning rate as 0.1 and decay it by a factor of 0.1 in epochs
75 and 90, respectively. We choose $1/\lambda=6.0$ as suggested by [40] and
use $\mathrm{E}_{s}$ = 70, $\alpha$ = 0.9 for our approach. Experimental
details are given in Appendix B.2.
We display the robust accuracy on the CIFAR10 test set after $\mathrm{E}_{s}$
= 70 epochs in Fig. 5. It shows that the robust accuracy of TRADES reaches its
highest value around the epoch of the first learning rate decay (epoch 75) and
decreases later, which suggests that overfitting might happen if we train the
model without early stopping. On the other hand, our method considerably
mitigates the overfitting issue in the adversarial training and consistently
improves the robust accuracy of TRADES by 1%$\sim$3%, which indicates that
self-adaptive training can improve the generalization in the presence of
adversarial noise.
Figure 5: Robust Accuracy (%) on the CIFAR10 test set under white box $\ell_{\infty}$ AutoAttack [41] ($\epsilon$=0.031). The plot of the first 70 epochs is omitted because the number of initial epochs $\mathrm{E}_{s}$ is set as 70. The vertical lines indicate learning rate decay. It shows that our method consistently improves TRADES. TABLE I: Test Accuracy (%) on the CIFAR datasets with various levels of symmetric noise (SN) or asymmetric noise (AN) injected into the training labels. We compare with previous works under exactly the same experiment settings. It shows that, in all settings, self-adaptive training improves over the state-of-the-art by at least $1$%, and sometimes the improvement is as significant as $9$%. The best entries are bold-faced. Backbone | | SN@CIFAR10 | SN@CIFAR100 | AN@CIFAR10 | AN@CIFAR100
---|---|---|---|---|---
Label Noise Rate | 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.2 | 0.4
ResNet-34 | ERM + Early Stop | 85.57 | 81.82 | 76.43 | 60.99 | 63.70 | 48.60 | 37.86 | 17.28 | 89.06 | 79.33 | 65.37 | 46.22
Label Smooth [43] | 85.64 | 71.59 | 50.51 | 28.19 | 67.44 | 53.84 | 33.01 | 9.74 | - | - | - | -
Forward $\hat{T}$ [44] | 87.99 | 83.25 | 74.96 | 54.64 | 39.19 | 31.05 | 19.12 | 8.99 | 89.09 | 83.55 | 42.46 | 34.44
Mixup [45] | 93.58 | 89.46 | 78.32 | 66.32 | 69.31 | 58.12 | 41.10 | 18.77 | - | - | - | -
Trunc $\mathcal{L}_{q}$ [46] | 89.70 | 87.62 | 82.70 | 67.92 | 67.61 | 62.64 | 54.04 | 29.60 | 89.33 | 76.74 | 66.59 | 47.22
Joint Opt [27] | 92.25 | 90.79 | 86.87 | 69.16 | 58.15 | 54.81 | 47.94 | 17.18 | - | - | - | -
SCE [47] | 90.15 | 86.74 | 80.80 | 46.28 | 71.26 | 66.41 | 57.43 | 26.41 | 90.44 | 82.51 | 72.56 | 69.32
DAC [48] | 92.91 | 90.71 | 86.30 | 74.84 | 73.55 | 66.92 | 57.17 | 32.16 | - | - | - | -
SELF [25] | - | 91.13 | - | 63.59 | - | 66.71 | - | 35.56 | 92.76 | 89.07 | 70.53 | 53.83
ELR [49] | 92.12 | 91.43 | 88.87 | 80.69 | 74.68 | 68.43 | 60.05 | 30.27 | 93.28 | 90.35 | 74.20 | 73.73
| Ours | 94.14 | 92.64 | 89.23 | 78.58 | 75.77 | 71.38 | 62.69 | 38.72 | 94.07 | 90.45 | 77.80 | 74.08
WRN28-10 | ERM + Early Stop | 87.86 | 83.40 | 76.92 | 63.54 | 68.46 | 55.43 | 40.78 | 20.25 | 90.61 | 79.93 | 69.85 | 50.30
MentorNet [30] | 92.0 | 89.0 | - | 49.0 | 73.0 | 68.0 | - | 35.0 | - | - | - | -
DAC [48] | 93.25 | 90.93 | 87.58 | 70.80 | 75.75 | 68.20 | 59.44 | 34.06 | - | - | - | -
SELF [25] | - | 93.34 | - | 67.41 | - | 72.48 | - | 42.06 | - | - | - | -
Ours | 94.84 | 93.23 | 89.42 | 80.13 | 77.71 | 72.60 | 64.87 | 44.17 | 95.11 | 91.07 | 80.59 | 75.30
## 4 Application I: Learning with Noisy Label
Given the improved generalization of self-adaptive training over ERM under
noise, we show the applications of our approach which outperforms the state-
of-the-art with a significant gap.
### 4.1 Problem formulation
Given a set of noisy training data
$\\{(\bm{x}_{i},\widetilde{\bm{y}}_{i})\\}_{n}\in\mathcal{\widetilde{D}}$,
where $\mathcal{\widetilde{D}}$ is the distribution of noisy data and
$\widetilde{\bm{y}}_{i}$ is the noisy label for each uncorrupted sample
$\bm{x}_{i}$, the goal is to be robust to the label noise in the training data
and improve the classification performance on clean test data that are sampled
from the clean distribution $\mathcal{D}$.
### 4.2 Experiments on CIFAR datasets
Setup We consider the cases that, with different noise rates, the labels are
(a) assigned uniformly at random (i.e., the symmetric noise), and (b) flipped
according to the class-conditioned rules [44] (i.e., the asymmetric noise):
for CIFAR10, the label noise is generated by mapping Truck $\rightarrow$
Automobile, Bird $\rightarrow$ Airplane, Deer $\rightarrow$ Horse, Cat
$\leftrightarrow$ Dog; for CIFAR100, the noise is generated by flipping each
class to the next class circularly. Following previous work [46, 48], we
conduct the experiments on the CIFAR10 and CIFAR100 datasets [16] and use the
ResNet-34 [3] / Wide ResNet-28-10 [50] (WRN28-10) as our base classifier. The
networks are implemented on PyTorch [51] and are optimized using SGD with an
initial learning rate of 0.1, a momentum of 0.9, a weight decay of 0.0005, a
batch size of 256, and the total training epochs of 200. The learning rate is
decayed to zero using the cosine annealing schedule [52]. We use the standard
data augmentation with random horizontal flipping and cropping. We report the
average performance over 3 trials.
TABLE II: Influence of the two components of our approach. | CIFAR10 | CIFAR100
---|---|---
Label Noise Rate | 0.4 | 0.8 | 0.4 | 0.8
Ours | 92.64 | 78.58 | 71.38 | 38.72
\- Re-weighting | 92.49 | 78.10 | 69.52 | 36.78
\- Exponential Moving Average | 72.00 | 28.17 | 50.93 | 11.57
Main results We summarize the experiments in Table I. Most of the results are
directly cited from original papers with the same experiment settings; the
results of Label Smoothing [43], Mixup [45], Joint Opt [27], and SCE [47] are
reproduced by rerunning the official open-sourced implementations. From the
table, we can see that our approach outperforms the state-of-the-art methods
in most entries by 1% $\sim$ 5% on both CIFAR10 and CIFAR100 datasets, using
different backbones. We observe that the improvements by our approach are
consistent under both symmetric and asymmetric noise, demonstrating the
robustness of self-adaptive training against various kinds of noise. Notably,
unlike the Joint Opt, DAC, and SELF methods that require multiple iterations
of training, our method enjoys the same computational budget as ERM.
TABLE III: Parameters sensitivity when label noise of 40% is injected into the CIFAR10 training set. $\alpha$ | 0.6 | 0.8 | 0.9 | 0.95 | 0.99
---|---|---|---|---|---
Fix $\mathrm{E}_{s}=60$ | 90.17 | 91.91 | 92.64 | 92.54 | 84.38
$\mathrm{E}_{s}$ | 20 | 40 | 60 | 80 | 100
Fix $\alpha=0.9$ | 89.58 | 91.89 | 92.64 | 92.26 | 88.83
Ablation study and hyper-parameter sensitivity First, we report the
performance of ERM equipped with the simple early stopping scheme in the first
row of Table I. We observe that our approach achieves substantial improvements
over this baseline. This demonstrates that simply early stopping the training
process is a sub-optimal solution. Then, we further report the influences of
the two individual components of our approach: Exponential Moving Average
(EMA) and sample re-weighting scheme. As displayed in Table II, removing any
component considerably hurts the performance under all noise rates, and
removing the EMA scheme leads to a significant performance drop. This suggests
that properly incorporating model predictions is important in our approach.
Finally, we analyze the sensitivity of our approach to the parameters $\alpha$
and $\mathrm{E}_{s}$ in Table III, where we fix one parameter while varying
the other. The performance is stable for various choices of $\alpha$ and
$\mathrm{E}_{s}$, indicating that our approach is not sensitive to
hyperparameter tuning.
### 4.3 Experiments on ImageNet dataset
The work of [53] suggested that the ImageNet dataset [20] contains annotation
errors on its own even after several rounds of cleaning. Therefore, in this
subsection, we use ResNet-50/101 [3] to evaluate self-adaptive training on the
large-scale ImageNet under both the standard setup (i.e., using original
labels) and the case that 40% of the training labels are corrupted. We provide
the experimental details in Appendix B.3 and report model performance on the
ImageNet validation set in terms of top1 accuracy in Table IV. We can see that
self-adaptive training consistently improves the ERM baseline by a
considerable margin under all settings using different models. Specifically,
the improvement can be as large as 3% in absolute for the larger ResNet-101
when 40% of the training labels are corrupted. The results validate the
effectiveness of our approach on the large-scale dataset and larger model.
Figure 6: Confusion matrix of recovered labels w.r.t clean labels on CIFAR10 training set with 40% of label noise. The overall recovery accuracy is 94.65%. TABLE IV: Top1 Accuracy (%) on ImageNet validation set. | ResNet-50 | ResNet-101
---|---|---
Label Noise Rate | 0.0 | 0.4 | 0.0 | 0.4
ERM | 76.8 | 69.5 | 78.2 | 70.2
Ours | 77.2 | 71.5 | 78.7 | 73.5
### 4.4 Label recovery of self-adaptive training
We demonstrate that our approach is able to recover the true labels from noisy
training labels: we obtain the recovered labels by the moving average targets
$\bm{t}_{i}$ and compute the recovered accuracy as
$\frac{1}{n}\sum_{i}\mathbbm{1}\\{\operatorname{argmax}~{}\bm{y}_{i}=\operatorname{argmax}~{}\bm{t}_{i}\\}$,
where $\bm{y}_{i}$ is the clean label of each training sample. When 40% of
labels are corrupted in the CIFAR10 and ImageNet training set, our approach
successfully corrects a huge amount of labels and obtains recovered accuracy
of 94.6% and 81.1%, respectively. We also display the confusion matrix of
recovered labels w.r.t the clean labels on CIFAR10 in Fig. 6, where we can see
that our approach performs well for all classes.
### 4.5 Investigation of sample weights
We further inspect the re-weighting scheme of self-adaptive training.
Following the procedure in Section 4.4, we display the average sample weights
in Fig. 7. In the figure, the $(i,j)$th block contains the average weight of
samples with clean label $i$ and recovered label $j$, the white areas
represent the case that no sample lies in the cell. We see that the weights on
the diagonal blocks are clearly higher than those on non-diagonal blocks. The
figure indicates that, aside from its impressive ability to recover the
correct labels, self-adaptive training could properly down-weight the noisy
examples.
Figure 7: Average sample weights $w_{i}$ under various labels. The white areas represent the case that no sample lies in the cell. TABLE V: Selective classification error rate (%) on CIFAR10, SVHN, and Dogs vs. Cats datasets for various coverage rates (%). Mean and standard deviation are calculated over 3 trials. The best entries and those overlap with them are marked bold. Dataset | Coverage | Ours | Deep Gamblers [54] | SelectiveNet [55] | SR [56] | MC-dropout [56]
---|---|---|---|---|---|---
CIFAR10 | 100 | 6.05$\pm$0.20 | 6.12$\pm$0.09 | 6.79$\pm$0.03 | 6.79$\pm$0.03 | 6.79$\pm$0.03
95 | 3.37$\pm$0.05 | 3.49$\pm$0.15 | 4.16$\pm$0.09 | 4.55$\pm$0.07 | 4.58$\pm$0.05
90 | 1.93$\pm$0.09 | 2.19$\pm$0.12 | 2.43$\pm$0.08 | 2.89$\pm$0.03 | 2.92$\pm$0.01
85 | 1.15$\pm$0.18 | 1.09$\pm$0.15 | 1.43$\pm$0.08 | 1.78$\pm$0.09 | 1.82$\pm$0.09
80 | 0.67$\pm$0.10 | 0.66$\pm$0.11 | 0.86$\pm$0.06 | 1.05$\pm$0.07 | 1.08$\pm$0.05
75 | 0.44$\pm$0.03 | 0.52$\pm$0.03 | 0.48$\pm$0.02 | 0.63$\pm$0.04 | 0.66$\pm$0.05
70 | 0.34$\pm$0.06 | 0.43$\pm$0.07 | 0.32$\pm$0.01 | 0.42$\pm$0.06 | 0.43$\pm$0.05
SVHN | 100 | 2.75$\pm$0.09 | 3.24$\pm$0.09 | 3.21$\pm$0.08 | 3.21$\pm$0.08 | 3.21$\pm$0.08
95 | 0.96$\pm$0.09 | 1.36$\pm$0.02 | 1.40$\pm$0.01 | 1.39$\pm$0.05 | 1.40$\pm$0.05
90 | 0.60$\pm$0.05 | 0.76$\pm$0.05 | 0.82$\pm$0.01 | 0.89$\pm$0.04 | 0.90$\pm$0.04
85 | 0.45$\pm$0.02 | 0.57$\pm$0.07 | 0.60$\pm$0.01 | 0.70$\pm$0.03 | 0.71$\pm$0.03
80 | 0.43$\pm$0.01 | 0.51$\pm$0.05 | 0.53$\pm$0.01 | 0.61$\pm$0.02 | 0.61$\pm$0.01
Dogs vs. Cats | 100 | 3.01$\pm$0.17 | 2.93$\pm$0.17 | 3.58$\pm$0.04 | 3.58$\pm$0.04 | 3.58$\pm$0.04
95 | 1.25$\pm$0.05 | 1.23$\pm$0.12 | 1.62$\pm$0.05 | 1.91$\pm$0.08 | 1.92$\pm$0.06
90 | 0.59$\pm$0.04 | 0.59$\pm$0.13 | 0.93$\pm$0.01 | 1.10$\pm$0.08 | 1.10$\pm$0.05
85 | 0.25$\pm$0.11 | 0.47$\pm$0.10 | 0.56$\pm$0.02 | 0.82$\pm$0.06 | 0.78$\pm$0.06
80 | 0.15$\pm$0.06 | 0.46$\pm$0.08 | 0.35$\pm$0.09 | 0.68$\pm$0.05 | 0.55$\pm$0.02
## 5 Application II: Selective Classification
### 5.1 Problem formulation
Selective classification, a.k.a. classification with rejection, trades
classifier coverage for accuracy [57], where the coverage is defined as the
fraction of classified samples in the dataset; the classifier is allowed to
output “don’t know” for certain samples. The task focuses on the noise-free
setting and allows the classifier to abstain from potential out-of-
distribution samples or samples that lies in the tail of data distribution,
that is, making prediction only on the samples with confidence.
Formally, a selective classifier is a composition of two functions
$(f_{\theta},g)$, where $f_{\theta}$ is the conventional $c$-class classifier
and $g$ is the selection function that reveals the underlying uncertainty of
inputs. Given an input $\bm{x}$, selective classifier outputs
$(f_{\theta},g)(\bm{x})=\begin{cases}\mathrm{Abstain},&g(\bm{x})>\tau;\\\
f_{\theta}(\bm{x}),&\mathrm{otherwise,}\end{cases}$ (11)
for a given threshold $\tau$ that controls the trade-off.
### 5.2 Approach
Inspired by [48, 54], we adapt our presented approach in Algorithm 1 to the
selective classification task. We introduce an extra ($c+1$)th class
(represents _abstention_) during training and replace selection function
$g(\cdot)$ in Equation (11) by $f_{\theta}(\cdot)_{c}$. In this way, we can
train a selective classifier in an end-to-end fashion. Besides, unlike
previous works that provide no explicit signal for learning abstention class,
we use model predictions as a guideline when designing the learning process.
Given a mini-batch of data pairs $\\{(\bm{x}_{i},\bm{y}_{i})\\}_{m}$, the
model prediction $\bm{p}_{i}$ and its exponential moving average $\bm{t}_{i}$
for each sample, we optimize the classifier $f$ by minimizing:
$\mathcal{L}(f_{\theta})=-\frac{1}{m}\sum_{i}[\bm{t}_{i,y_{i}}\log\bm{p}_{i,y_{i}}+(1-\bm{t}_{i,y_{i}})\log\bm{p}_{i,c}],$
(12)
where $y_{i}$ is the index of the non-zero element in the one hot label vector
$\bm{y}_{i}$. The first term measures the cross entropy loss between
prediction and original label $\bm{y}_{i}$, in order to learn a good multi-
class classifier. The second term acts as the selection function and
identifies uncertain samples in datasets. $\bm{t}_{i,y_{i}}$ dynamically
trades-off these two terms: if $\bm{t}_{i,y_{i}}$ is very small, the sample is
deemed as uncertain and the second term enforces the selective classifier to
learn to abstain from this sample; if $\bm{t}_{i,y_{i}}$ is close to 1, the
loss recovers the standard cross entropy minimization and enforces the
selective classifier to make perfect predictions.
### 5.3 Experiments
Setup We conduct the experiments on three datasets: CIFAR10 [16], SVHN [58],
and Dogs vs. Cats [59]. We compare our method with previous state-of-the-art
methods on selective classification, including Deep Gamblers [54],
SelectiveNet [55], Softmax Response (SR), and MC-dropout [56]. The experiments
are based on the official open-sourced
implementation111https://github.com/Z-T-WANG/NIPS2019DeepGamblers of Deep
Gamblers to ensure a fair comparison. We use the VGG-16 network [2] with batch
normalization [60] and dropout [61] as the base classifier in all experiments.
The network is optimized using SGD with an initial learning rate of 0.1, a
momentum of 0.9, a weight decay of 0.0005, a batch size of 128, and a total
training epoch of 300. The learning rate is decayed by 0.5 in every 25 epochs.
For our method, we set the hyper-parameters $\mathrm{E}_{s}=0,\alpha=0.99$.
Main results The results of prior methods are cited from original papers and
are summarized in Table V. We see that our method achieves up to 50% relative
improvements compared with all other methods under various coverage rates, on
all datasets. Notably, Deep Gamblers also introduces an extra abstention class
in their method but without applying model predictions. The improvement of our
method comes from the use of model predictions in the training process.
## 6 Improved Representations Learning
### 6.1 Self-Supervised Self-Adaptive Training
Instantiation We consider training images $\\{\bm{x}_{i}\\}_{n}$ without label
and use a deep network followed by a non-linear projection as encoder
$f_{\theta}$. Then, we instantiate the meta Algorithm 1 for self-supervised
learning as follows:
1. 1.
_Target initialization._ Since the labels are absent, each training target
$\bm{t}_{i}$ is randomly and independently drawn from a standard normal
distribution.
2. 2.
_Normalization function._ We directly normalize each representation by
dividing its $\ell_{2}$ norm.
3. 3.
_Loss function._ The loss function is implemented as the Mean Square Error
(MSE) between the normalized model prediction $\bm{p}_{i}$ and training target
$\bm{t}_{i}$.
The above instantiation of our meta algorithm, i.e., fitting models’ own
accumulated representations, suffices to learn decent representations, as
exhibited by the blue curve in Fig. 2. However, as discussed in Sec. 2.1 that
the consistency of the training target also plays an essential role,
especially in self-supervised representation learning, we introduce two
components that further improve the representation learning with self-adaptive
training.
Momentum encoder and predictor We follow prior works [10, 62] to employ a
momentum encoder $f_{\theta_{m}}$, whose parameters $\theta_{m}$ is also
updated by the EMA scheme as
$\theta_{m}\leftarrow\beta\times\theta_{m}+(1-\beta)\times\theta.$ (13)
With the slowly-evolving $f_{\theta_{m}}$, we can obtain a representation
$\bm{z}_{i}=f_{\theta_{m}}(\bm{x}^{t}_{i}),$ (14)
and construct the target $\bm{t}_{i}$ following the EMA scheme in Equation
(2).
Furthermore, to prevent the model from outputting the same representation for
every image in each iteration (i.e., collapsing), we further use a predictor
$g$ to transform the output of encoder $f_{\theta}$ to prediction
$\bm{p}_{i}=g(f_{\theta}(\bm{x}_{i})),$ (15)
where $g$ has the same number of output units as $f_{\theta}$.
Figure 8: Pipeline of Self-Supervised Self-Adaptive Training.
Putting everything together We $\ell_{2}$-normalize $\bm{p}_{i}$ to
$\bm{\widetilde{p}}_{i}=\bm{p}_{i}/\left\lVert\bm{p}_{i}\right\rVert_{2}$ and
$\bm{t}_{i}$ to
$\bm{\widetilde{t}}_{i}=\bm{t}_{i}/\left\lVert\bm{t}_{i}\right\rVert_{2}$.
Finally, the MSE loss between the normalized predictions and accumulated
representations
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta},g)=\frac{1}{m}\sum_{i}\left\lVert\bm{\widetilde{p}}_{i}-\bm{\widetilde{t}}_{i}\right\rVert^{2}_{2}$
(16)
is minimized to update the encoder $f_{\theta}$ and predictor $g$. We term
this variant _Self-Supervised Self-Adaptive Training_ , and summarize the
pseudo-code in Algorithm 3 and the overall pipeline in Fig. 8. Our approach is
straightforward to implement in practice and requires only single-view
training, which significantly alleviates the heavy computation burden of data
augmentation operations.
Algorithm 3 Self-Supervised Self-Adaptive Training
0: Data $\\{\bm{x}_{i}\\}_{n}$; encoder $f_{\theta}$ and momentum encoder
$f_{\theta_{m}}$; predictor $g$; momentum terms $\alpha=0.7$, $\beta=0.99$
1: Initialize $\theta_{m}\leftarrow\theta$
2: Randomly initialize training target
$\\{\bm{t}_{i}\\}_{n}\sim\mathcal{N}(\bm{0},\mathbf{I})$
3: repeat
4: Fetch augmented mini-batch data $\\{\bm{x}^{t}_{i}\\}_{m}$
5: for $i=1$ to $m$ (in parallel) do
6: $\bm{p}_{i}=g(f_{\theta}(\bm{x}^{t}_{i}))$
7: $\bm{z}_{i}=f_{\theta_{m}}(\bm{x}^{t}_{i})$
8: $\bm{\widetilde{z}}_{i}=\bm{z}_{i}/\left\lVert\bm{z}_{i}\right\rVert_{2}$;
$\bm{\widetilde{t}}_{i}=\bm{t}_{i}/\left\lVert\bm{t}_{i}\right\rVert_{2}$
9:
$\bm{t}_{i}\leftarrow\alpha\times\bm{\widetilde{t}}_{i}+(1-\alpha)\times\bm{\widetilde{z}}_{i}$
10: end for
11:
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta},g)=\frac{1}{m}\sum_{i}\left\lVert\frac{\bm{p}_{i}}{\left\lVert\bm{p}_{i}\right\rVert_{2}}-\frac{\bm{t}_{i}}{\left\lVert\bm{t}_{i}\right\rVert_{2}}\right\rVert^{2}_{2}$
12: Update $f_{\theta}$ and $g$ by SGD on
$\mathcal{L}(\bm{p}_{i},\bm{t}_{i};f_{\theta},g)$
13: Update $\theta_{m}\leftarrow\beta\times\theta_{m}+(1-\beta)\times\theta$
14: until end of training
Methodology differences with prior works BYOL [62] formulated the self-
supervised training of deep models as predicting the representation of one
augmented view of an image from the other augmented view of the same image.
The self-supervised self-adaptive training shares some similarities with BYOL
since both methods do not need to contrast with negative examples to prevent
the collapsing issue. Instead of directly using the outputs of a momentum
encoder as training targets, our approach uses the _accumulated predictions_
as training targets, which contain all historical view information for each
image. As a result, our approach requires only a single view during training,
which is much more efficient as shown in Sec. 6.3 and Fig. 9. Besides, NAT
[63] used an online clustering algorithm to assign noise as the training
target for each image. Unlike NAT that fixes the noise while updating the
noise assignment during training, our approach uses the noise as the _initial_
training target and updates the noise by model predictions in the subsequent
training process. InstDisc [13] used a memory bank to store the representation
of each image in order to construct positive and negative samples for the
contrastive objective. By contrast, our method gets rid of the negative
samples and only matches the prediction with the training target of the same
image (i.e., positive sample).
### 6.2 Bypassing collapsing issues
We note that there exist trivial local minima when we directly optimize the
MSE loss between predictions and training targets due to the absence of
negative pairs: the encoder $f_{\theta}$ can simply output a constant feature
vector for every data point to minimize the training loss, a.k.a. collapsing
issue [62]. Despite the existence of a collapsing solution, self-adaptive
training intrinsically prevents the collapsing. The initial targets
$\\{\bm{t}_{i}\\}_{n}$ are different for different classes (under the
supervised setting) or even for different images (under the self-supervised
setting), enforcing the models to learn different representations for
different classes/images. Our empirical studies in Fig. 1, 2 of the main body
and Fig. 11 of the Appendix strongly support that deep neural networks are
able to learn meaningful information from corrupted data or even random noise,
and bypass the model collapse. Based on this, our approach significantly
improves the supervised learning (see Fig. 1(a)) and self-supervised learning
(see the blue curve in Fig. 2) of deep neural networks.
Figure 9: Training time comparison on the ImageNet dataset in terms of the
data preprocessing/forward pass/backward pass time. The comparison is
performed on a machine with 8 V100 GPUs. It shows the efficiency of our
method.
### 6.3 Is multi-view training indispensable?
The success of state-of-the-art self-supervised learning approaches [10, 62]
largely hinges on the multi-view training scheme: they essentially use strong
data augmentation operations to create multiple views (crops) of the same
image and then match the representation of one view with the other views of
the same image. Despite their promising performance, these methods suffer
heavily from the computational burden of data pre-processing and the training
on extra views. Concretely, as shown in Fig. 9, prior methods BYOL [62] and
MoCo [10] incur doubled training time compared with standard supervised cross
entropy training. In contrast, since our method requires only single-view
training, its training is only slightly slower than the supervised method and
is much faster than MoCo and BYOL. See Appendix B.5 for the detailed setup of
this comparison.
We further conduct experiments to evaluate the performance of the multi-view
training scheme on the CIFAR10 and STL10 datasets, which cast doubt on its
necessity for learning a good representation. Concretely, we first pre-train a
ResNet-18 encoder using different algorithms and then train a linear
classifier on top of the encoder to evaluate the classification performance
(see Sec. 7.1 for details). As shown in Table VI, although the performance of
MoCo and BYOL is nearly halved on both datasets when using the single-view
training scheme, the self-supervised adaptive training achieves comparable
results under both settings. Moreover, our approach with single-view training
even slightly outperforms the MoCo and BYOL with multi-view training. We
attribute this superiority to the guiding principle of our algorithm: by
dynamically incorporating the model predictions, each training target contains
the relevant information about all the historical views of each image and,
therefore, implicitly forces the model learning representations that are
invariant to the historical views.
### 6.4 On the power of momentum encoder and predictor
Momentum encoder and predictor are two important components of our approach
and BYOL. The work of BYOL [62] showed that the algorithm may converge to a
trivial solution if one of them is removed, not to mention removing both of
them. As shown in Table VII, however, our results challenge their conclusion:
1) with the predictor, the linear evaluation accuracy of either BYOL (> 85%)
or our method (> 90%) is non-trivial, regardless of the absence and the
configuration of momentum encoder; 2) without the predictor, the momentum
encoder with a sufficiently large momentum can also improve the performance
and bypass the collapsing issue. The results suggest that although both the
predictor and the momentum encoder are indeed crucial for the performance of
representation learning, either one of them with a proper configuration (i.e.,
the momentum term) suffices to avoid the collapse. We note that the latest
version of [62] also found that the momentum encoder can be removed without
collapse when carefully tuning the learning rate of the predictor. Our
results, however, are obtained using the same training setting.
Moreover, we find that self-supervised self-adaptive training exhibits
impressive resistance to collapsing, despite using only single-view training.
Our approach can learn a decent representation even when the predictor and
momentum encoder are both removed (see the seventh row of Table VII). We
hypothesize that the resistance comes from the consistency of the training
target due to our EMA scheme. This hypothesis is also partly supported by the
observation that the learning of BYOL heavily depends on the slowly-evolving
momentum encoder.
TABLE VI: Necessity of the multi-view training scheme in self-supervised learning. Top1 test Accuracy (%) on CIFAR10 and STL10 datasets using the ResNet-18 backbone. Method | # Views | CIFAR10 | STL10
---|---|---|---
MoCov2 [64] | 1 | 46.36 | 39.65
2 | 91.68 | 88.90
BYOL [62] | 1 | 44.36 | 42.60
2 | 91.78 | 89.60
Ours | 1 | 92.27 | 89.55
2 | 91.56 | 90.05
TABLE VII: Influence of the momentum encoder and predictor. The performance is measured by the top1 test Accuracy (%) on the CIFAR10 dataset using the ResNet-18 backbone. The entries with † indicate that $\alpha=0.9$. Method | Predictor | Momentum $\beta$ | Accuracy
---|---|---|---
BYOL [62] | $\times$ | 0.0 | 25.64
0.99 | 26.87
0.999 | 72.44
$\checkmark$ | 0.0 | 85.22
0.99 | 91.78
0.999 | 90.68
Ours | $\times$ | 0.0 † | 78.36
0.99 † | 79.68
0.999 † | 83.58
$\checkmark$ | 0.0 | 90.18
0.99 | 92.27
0.999 | 90.92
## 7 Application III: Linear Evaluation Protocol in Self-Supervised Learning
### 7.1 Experimental setup
Datasets and data augmentations We conduct experiments on four benchmarks:
CIFAR10/CIFAR100 [16] with 50k images; STL10 [65] with 105k images; ImageNet
[20] with $\sim$1.2m images. The choice of data augmentations follows prior
works [10, 11]: we take a random crop from each image and resize it to a fixed
size (i.e., $32\times 32$ for CIFAR10/CIFAR100, $96\times 96$ for STL10, and
$224\times 224$ for ImageNet); the crop is then randomly transformed by color
jittering, horizontal flip, and grayscale conversion (and Gaussian blurring
for ImageNet).
Network architecture The encoders $f_{\theta}$ and $f_{\theta_{m}}$ consist of
a backbone of ResNet-18 [3]/ResNet-50 [3]/AlexNet [66] and a projector that is
instantiated by a multi-layer perceptron (MLP). We use the output of the last
global average pooling layer of the backbone as the extracted feature vectors.
Following prior works [11], the output vectors of the backbone are transformed
by the projector MLP to a dimension of 256. Besides, the predictor $g$ is also
instantiated by an MLP with the same architecture as the projector in
$f_{\theta}$. In our implementation, all the MLPs have one hidden layer of
size 4,096, followed by a batch normalization [60] layer and the ReLU
activation [67].
TABLE VIII: Linear evaluation protocol on CIFAR10, CIFAR100, and STL10 datasets using different backbones. Backbone | Method | CIFAR10 | CIFAR100 | STL10
---|---|---|---|---
AlexNet | SplitBrain [68] | 67.1 | 39.0 | -
DeepCluster [69] | 77.9 | 41.9 | -
InstDisc [13] | 70.1 | 39.4 | -
AND [70] | 77.6 | 47.9 | -
SeLa [71] | 83.4 | 57.4 | -
CMC [72] | - | - | 83.28
| Ours | 83.55 | 59.80 | 83.75
ResNet-50 | MoCov2 [64] | 93.20 | 69.48 | 91.95
SimCLR[11] | 93.08 | 67.92 | 90.90
BYOL [62] | 93.48 | 68.48 | 92.40
Ours | 94.04 | 70.16 | 92.60
Self-supervised pre-training settings On the CIFAR10, CIFAR100, and STL10
datasets, we optimize the networks using an SGD optimizer with a momentum of
0.9 and a weight decay of 0.00005. We use a batch size of 512 for all methods
in all experiments and train the networks for 800 epochs using 4 NVIDIA GTX
1080Ti GPUs. The hyper-parameters of our method are set to $\alpha=0.7$,
$\beta=0.99$. On the ImageNet dataset, we use a batch size of 1024, a LARS
[73] optimizer with a weight decay of 0.000001, and train the networks for 200
epochs on 8 GPUs. The base learning rate is set to 2.0 and is scaled linearly
with respect to the base batch size 256 following [74]. For our method, we set
the initial values of the momentum terms $\alpha$/$\beta$ to 0.5/0.99, which
are then increased to 0.8/1.0 following the prior work [62]. During training,
the learning rate is warmed up for the first 30 epochs (10 epochs when using
the ImageNet dataset) and is then adjusted according to the cosine annealing
schedule [52].
Linear evaluation protocol Following the common practice [10, 11, 62], we
evaluate the representation learned by self-supervised pre-training using
linear classification protocol. That is, we remove the projector in
$f_{\theta}$ and the predictor $g$, fix the parameters of the backbone of the
encoder $f_{\theta}$, and train a supervised linear classifier on top of the
features extracted from the encoder. On the ImageNet dataset, the linear
classifier is trained for 100 epochs with a weight decay of 0.0, a batch size
of 512, and an initial learning rate of 2.0 that is decayed to 0 according to
the cosine annealing schedule [52]. On the rest datasets, the linear
classifier is trained for 100 epochs with a weight decay of 0.0 and a batch
size of 512. The initial learning rate is set to 0.4 and is decayed by a
factor of 0.1 at the 60th and 80th training epochs. The performance is
measured in terms of the top1 test accuracy of the linear classifier.
### 7.2 Comparison with the state-of-the-art
We firstly conduct experiments on the CIFAR10/100 [16] and STL10 [65] datasets
and compare our self-supervised self-adaptive training with three state-of-
the-art methods, including the contrastive learning methods MoCo [10], SimCLR
[11], and bootstrap method BYOL [62]. For fair comparisons, we use the same
code base and the same experimental settings for all the methods, following
their official open-sourced implementations. We carefully adjust the hyper-
parameters on the CIFAR10 dataset for each method and use the same parameters
on the rest datasets. Besides, we also conduct experiments using AlexNet as
the backbone and compare the performance of our method with the reported
results of prior methods. The results are summarized in Table VIII. We can see
that, despite using only single-view training, our self-supervised self-
adaptive training consistently obtains better performance than all other
methods on all datasets with different backbones.
TABLE IX: Linear evaluation protocol on ImageNet dataset using the ResNet-50 [3] backbone. The entries marked with ${\dagger}$ are cited from [75]; those marked by ${\ddagger}$ are cited from [12]. Method | # Views | Epochs | Linear Acc. (%)
---|---|---|---
InstDisc [13] | 2 | 200 | 56.5
MoCo [10] | 2 | 200 | 60.6
CPCv2 [76] | 2 | 200 | 63.8
SimCLR [11] | 2 | 200 | 66.6
MoCov2 [64] | 2 | 200 | 67.5
BYOL [62]† | 2 | 200 | 70.6
SwAV [12]† | 2 | 200 | 69.1
SimSiam [75]† | 2 | 200 | 70.0
Ours | 1 | 200 | 67.4
Ours | 2 | 200 | 72.8
CMC [72] | 2 | 240 | 60.0
BYOL [62] | 2 | 300 | 72.4
BarlowTwins [77] | 2 | 300 | 71.4
MoCov3 [78] | 2 | 300 | 72.8
SeLa [71] | 2 | 400 | 61.5
SeLav2 [71]‡ | 2 | 400 | 67.2
DeepClusterv2 [69]‡ | 2 | 400 | 70.2
SwAV [12]‡ | 2 | 400 | 70.1
PIRL [79] | 2 | 800 | 63.6
SimCLRv2 [80] | 2 | 800 | 71.7
(a) Sensitivity to data augmentation.
(b) Sensitivity to training batch size.
Figure 10: Sensitivity of our method to data augmentations and training batch
size. The performance is measured by the top1 test Accuracy (%) on the CIFAR10
dataset using the ResNet-18 backbone.
Moreover, we follow the standard setups in self-supervised learning on the
large-scale ImageNet [53] dataset (i.e., using ResNet-50 backbone, 200 pre-
training epochs, and the linear evaluation protocol), and report the
comparisons with the state-of-the-art methods in Table IX. We find that, while
using 50% fewer augmented views during training, the single-view self-adaptive
training achieves 67.4% top1 accuracy, which is comparable with the two-view
SimCLR and MoCov2 and slightly worse than BYOL. We conjecture that, on the
large-scale dataset, the training targets are updated less frequently than
those on a smaller dataset, which results in performance degradation.
Regarding this, to make a fair comparison, we also pre-train a ResNet-50 model
based on the two-view variant of SAT, at a similar training cost as BYOL. From
the results in Table IX. We can observe that the two-view variant of SAT (a)
outperforms all other methods under the same setting by a large margin, and
(b) even performs on par with BYOL and MoCov3 trained by 300 epochs while
outperforming the other methods with $>2\times$ training epochs.
TABLE X: Transfer learning to Pascal-VOC object detection and semantic segmentation with models pre-trained on ImageNet-1K datasets for 200 epochs. All entries are based on the Faster R-CNN [81] architecture with the ResNet-50 C4 backbone [82] for detection, and are based on ResNet-50 FCN [5] for segmentation. ${\dagger}$: results cited from [75]. $*$: reproduction based on the publicly available checkpoint. Method | # Views | VOC 07+12 Det. | VOC 12 Seg.
---|---|---|---
AP | AP50 | AP75 | mIoU
Supervised† | 1 | 53.5 | 81.3 | 58.8 | 67.7∗
SimCLR [11]† | 2 | 55.5 | 81.8 | 61.4 | -
MoCov2 [64] | 2 | 57.0 | 82.4 | 63.6 | 66.7∗
SwAV [12]† | 2 | 55.4 | 81.5 | 61.4 | -
BYOL [62]† | 2 | 55.3 | 81.4 | 61.1 | -
SimSiam [75]† | 2 | 56.4 | 82.0 | 62.8 | -
Ours | 1 | 55.2 | 81.5 | 60.9 | 64.8
Ours | 2 | 56.4 | 82.5 | 63.0 | 68.2
Finally, in addition to the above experiments, we also evaluate the pre-
trained models on two downstream tasks, object detection and semantic
segmentation on the Pascal VOC dataset [83]. We follow exactly the same
settings as in prior works [10, 75] for fair comparisons, where all models are
pre-trained for 200 epochs on ImageNet. We use the Faster R-CNN [81]
architecture with the ResNet-50 C4 backbone [82] for detection and the
ResNet-50 dilated FCN [5] architecture for segmentation. The experiments are
conducted on the opensource codebases $\operatorname{detectron2}$
[82]/$\operatorname{mmsegmentation}$ [84] for the detection/segmentation
tasks. According to the results in Table X, self-adaptive training performs on
par with or even better than state-of-the-art self-supervised learning methods
on both transfer learning tasks. Moreover, we can also observe that SAT
consistently achieves superior performance to the supervised baseline.
TABLE XI: Parameters sensitivity on the CIFAR10 dataset. $\alpha$ | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
---|---|---|---|---|---
Fix $\beta=0.99$ | 91.68 | 91.76 | 92.27 | 92.08 | 91.68
$\beta$ | 0.9 | 0.95 | 0.99 | 0.995 | 0.999
Fix $\alpha=0.7$ | 90.68 | 91.60 | 92.27 | 92.16 | 90.92
### 7.3 Sensitivity of self-supervised self-adaptive training
Sensitivity to hyper-parameters We study how the two momentum parameters
$\alpha$ and $\beta$ affect the performance of our approach and report the
results in Table XI. By varying one of the parameters while fixing the other,
we observe that self-supervised self-adaptive training performs consistently
well, which suggests that our approach is not sensitive to the choice of
hyper-parameters.
Sensitivity to data augmentation Data augmentation is one of the most
essential ingredients of recent self-supervised learning methods: a large body
of these methods, including ours, formulate the training objective as learning
representations that encode the shared information across different views
generated by data augmentation. As a result, prior methods, like MoCo and
BYOL, fail in the single-view training setting. Self-supervised self-adaptive
training, on the other hand, maintains a training target for each image, which
contains all historical view information of this image. Therefore, we
conjecture that our method should be more robust to data augmentations. To
validate our conjecture, we evaluate our method and BYOL under the settings
that some of the augmentation operators are removed. The results are shown in
Fig. 10(a). Removing any augmentation operators hurts the performance of both
methods while our method is less affected. Specifically, when all augmentation
operators except random crop and flip are removed, the performance of BYOL
drops to 86.6% while our method still obtains 88.6% accuracy.
Sensitivity to training batch size Recent contrastive learning methods require
large batch size training (e.g., 1024 or even larger) for optimal performance,
due to the need of comparing with massive negative samples. BYOL does not use
negative samples and suggests that this issue can be mitigated. Here, since
our method also gets rid of the negative samples, we make direct comparisons
with BYOL at different batch sizes to evaluate the sensitivity of our method
to batch size. For each method, the base learning rate is linearly scaled
according to the batch size while the rest settings are kept unchanged. The
results are shown in Fig. 10(b). We can see that our method exhibits a smaller
performance drop than BYOL at various batch sizes. Concretely, the accuracy of
BYOL drops by 2% at batch size 128 while that of ours drops by only 1%.
## 8 Related Works
Generalization of deep networks Previous work [14] systematically analyzed the
capability of deep networks to overfit random noise. Their results show that
traditional wisdom fails to explain the generalization of deep networks.
Another line of works [34, 35, 36, 37, 32, 38, 33] observed an intriguing
double-descent risk curve from the bias-variance trade-off. [32, 33] claimed
that this observation challenges the conventional U-shaped risk curve in the
textbook. Our work shows that this observation may stem from overfitting to
noise; the phenomenon vanishes by a properly designed training process such as
self-adaptive training. To improve the generalization of deep networks, [43,
85] proposed label smoothing regularization that uniformly distributes
$\epsilon$ of labeling weight to all classes and uses this soft label for
training; [45] introduced mixup augmentation that extends the training
distribution by dynamic interpolations between random paired input images and
the associated targets during training. This line of research is similar to
ours as both methods use soft labels in the training. However, self-adaptive
training is able to recover true labels from noisy labels and is more robust
to noise.
Robust learning from corrupted data Aside from the preprocessing-training
approaches that have been discussed in the last paragraph of Section 3.1,
there have also been many other works on learning from noisy data. To name a
few, [86, 19] showed that deep neural networks tend to fit clean samples first
and overfitting to noise occurs in the later stage of training. [19] further
proved that early stopping can mitigate the issues that are caused by label
noise. [87, 88] incorporated model predictions into training by simple
interpolation of labels and model predictions. We demonstrate that our
exponential moving average and sample re-weighting schemes enjoy superior
performance. Other works [46, 47] proposed alternative loss functions to cross
entropy that are robust to label noise. They are orthogonal to ours and are
ready to cooperate with our approach as shown in Appendix C.4. Beyond the
corrupted data setting, recent works [89, 90] propose a self-training scheme
that also uses model predictions as training targets. However, they suffer
from the heavy cost of multiple iterations of training, which is avoided by
our approach. Temporal Ensembling [91] incorporated the ’ensemble’ predictions
as pseudo-labels for training. Different from ours, Temporal Ensembling
focuses on the semi-supervised learning setting and only accumulates
predictions for unlabeled data.
Self-supervised learning Aiming to learn powerful representations, most self-
supervised learning approaches typically first solve a proxy task without
human supervision. For example, prior works proposed recovering input using
auto-encoder [92, 93], generating pixels in the input space [94, 95],
predicting rotation [96] and solving jigsaw [97]. Recently, contrastive
learning methods [13, 98, 72, 10, 11, 99] significantly advanced self-
supervised representation learning. These approaches essentially used strong
data augmentation techniques to create multiple views (crops) of the same
image and discriminated the representation of different views of the same
images (i.e., positive samples) from the views of other images (i.e., negative
samples). Bootstrap methods eliminated the discrimination of positive and
negative data pairs: the works of [69, 71] alternatively performed clustering
on the representations and then used the cluster assignments as classification
targets to update the model; [12] swapped the cluster assignments between the
two views of the same image as training targets; [62] simply predicted the
representation of one view from the other view of the same image; [63]
formulated the self-supervised training objective as predicting a set of
predefined noise. Our work follows the path of bootstrap methods. Going
further than them, self-adaptive training is a general training algorithm that
bridges supervised and self-supervised learning paradigms. Our approach casts
doubt on the necessity of the costly multi-view training and works well with
the single-view training scheme.
## 9 Conclusion
In this paper, we explore the possibility of a unified framework to bridge the
supervised and self-supervised learning of deep neural networks. We first
analyze the training dynamic of deep networks under these two learning
settings and observe that useful information from data is distilled to model
predictions. The observation occurs broadly even in the presence of data
corruptions and the absence of labels, which motivates us to propose Self-
Adaptive Training—a general training algorithm that dynamically incorporates
model predictions into the training process. We demonstrate that our approach
improves the generalization of deep neural networks under various kinds of
training data corruption and enhances the representation learning using
accumulated model predictions. Finally, we present three applications of self-
adaptive training on learning with noisy labels, selective classification, and
linear evaluation protocol in self-supervised learning, where our approach
significantly advances the state-of-the-art.
## Acknowledgments
Lang Huang and Chao Zhang were supported by the National Nature Science
Foundation of China under Grant 62071013 and 61671027, and the National Key
R&D Program of China under Grant 2018AAA0100300. Hongyang Zhang was supported
in part by an NSERC Discovery Grant.
## References
* [1] L. Huang, C. Zhang, and H. Zhang, “Self-adaptive training: beyond empirical risk minimization,” in _Advances in Neural Information Processing Systems_ , vol. 33, 2020.
* [2] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” in _International Conference on Learning Representations_ , 2015.
* [3] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2016, pp. 770–778.
* [4] R. Girshick, J. Donahue, T. Darrell, and J. Malik, “Rich feature hierarchies for accurate object detection and semantic segmentation,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2014, pp. 580–587.
* [5] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2015, pp. 3431–3440.
* [6] A. Radford, K. Narasimhan, T. Salimans, and I. Sutskever, “Improving language understanding by generative pre-training,” 2018.
* [7] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever, “Language models are unsupervised multitask learners,” _OpenAI blog_ , vol. 1, no. 8, p. 9, 2019.
* [8] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell _et al._ , “Language models are few-shot learners,” in _Advances in Neural Information Processing Systems_ , vol. 33, 2020.
* [9] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” in _Proceedings of the Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 2019, pp. 4171–4186.
* [10] K. He, H. Fan, Y. Wu, S. Xie, and R. Girshick, “Momentum contrast for unsupervised visual representation learning,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 9729–9738.
* [11] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton, “A simple framework for contrastive learning of visual representations,” in _International Conference on Machine Learning_ , 2020.
* [12] M. Caron, I. Misra, J. Mairal, P. Goyal, P. Bojanowski, and A. Joulin, “Unsupervised learning of visual features by contrasting cluster assignments,” in _Advances in Neural Information Processing Systems_ , vol. 33, 2020.
* [13] Z. Wu, Y. Xiong, S. X. Yu, and D. Lin, “Unsupervised feature learning via non-parametric instance discrimination,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 3733–3742.
* [14] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals, “Understanding deep learning requires rethinking generalization,” in _International Conference on Learning Representations_ , 2017.
* [15] V. Nagarajan and J. Z. Kolter, “Uniform convergence may be unable to explain generalization in deep learning,” in _Advances in Neural Information Processing Systems_ , 2019, pp. 11 611–11 622.
* [16] A. Krizhevsky and G. E. Hinton, “Learning multiple layers of features from tiny images,” University of Toronto, Tech. Rep., 2009.
* [17] D. Rolnick, A. Veit, S. Belongie, and N. Shavit, “Deep learning is robust to massive label noise,” _arXiv preprint arXiv:1705.10694_ , 2017.
* [18] M. Y. Guan, V. Gulshan, A. M. Dai, and G. E. Hinton, “Who said what: Modeling individual labelers improves classification,” in _Thirty-Second AAAI Conference on Artificial Intelligence_ , 2018.
* [19] M. Li, M. Soltanolkotabi, and S. Oymak, “Gradient descent with early stopping is provably robust to label noise for overparameterized neural networks,” in _International Conference on Artificial Intelligence and Statistics_. PMLR, 2020, pp. 4313–4324.
* [20] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. Ieee, 2009, pp. 248–255.
* [21] M. Baines, S. Bhosale, V. Caggiano, N. Goyal, S. Goyal, M. Ott, B. Lefaudeux, V. Liptchinsky, M. Rabbat, S. Sheiffer, A. Sridhar, and M. Xu, “Fairscale: A general purpose modular pytorch library for high performance and large scale training,” https://github.com/facebookresearch/fairscale, 2021.
* [22] C. E. Brodley, M. A. Friedl _et al._ , “Identifying and eliminating mislabeled training instances,” in _Proceedings of the National Conference on Artificial Intelligence_ , 1996, pp. 799–805.
* [23] C. E. Brodley and M. A. Friedl, “Identifying mislabeled training data,” _Journal of Artificial Intelligence Research_ , vol. 11, pp. 131–167, 1999\.
* [24] X. Zhu, X. Wu, and Q. Chen, “Eliminating class noise in large datasets,” in _Proceedings of the 20th International Conference on Machine Learning (ICML-03)_ , 2003, pp. 920–927.
* [25] D. T. Nguyen, C. K. Mummadi, T. P. N. Ngo, T. H. P. Nguyen, L. Beggel, and T. Brox, “SELF: Learning to filter noisy labels with self-ensembling,” in _International Conference on Learning Representations_ , 2020.
* [26] H. Bagherinezhad, M. Horton, M. Rastegari, and A. Farhadi, “Label refinery: Improving imagenet classification through label progression,” _arXiv preprint arXiv:1805.02641_ , 2018.
* [27] D. Tanaka, D. Ikami, T. Yamasaki, and K. Aizawa, “Joint optimization framework for learning with noisy labels,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 5552–5560.
* [28] C.-M. Teng, “Correcting noisy data.” in _International Conference on Machine Learning_. Citeseer, 1999, pp. 239–248.
* [29] P. J. Rousseeuw and A. M. Leroy, _Robust regression and outlier detection_. John Wiley & Sons, 2005, vol. 589.
* [30] L. Jiang, Z. Zhou, T. Leung, L.-J. Li, and L. Fei-Fei, “Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels,” in _International Conference on Machine Learning_ , 2018, pp. 2304–2313.
* [31] M. Ren, W. Zeng, B. Yang, and R. Urtasun, “Learning to reweight examples for robust deep learning,” in _International Conference on Machine Learning_ , 2018, pp. 4334–4343.
* [32] M. Belkin, D. Hsu, S. Ma, and S. Mandal, “Reconciling modern machine-learning practice and the classical bias–variance trade-off,” _Proceedings of the National Academy of Sciences_ , vol. 116, no. 32, pp. 15 849–15 854, 2019\.
* [33] P. Nakkiran, G. Kaplun, Y. Bansal, T. Yang, B. Barak, and I. Sutskever, “Deep double descent: Where bigger models and more data hurt,” in _International Conference on Learning Representations_ , 2020.
* [34] M. Opper, “Statistical mechanics of learning: Generalization,” _The Handbook of Brain Theory and Neural Networks,_ , pp. 922–925, 1995.
* [35] ——, “Learning to generalize,” _Frontiers of Life_ , vol. 3, no. part 2, pp. 763–775, 2001.
* [36] M. S. Advani, A. M. Saxe, and H. Sompolinsky, “High-dimensional dynamics of generalization error in neural networks,” _Neural Networks_ , vol. 132, pp. 428–446, 2020.
* [37] S. Spigler, M. Geiger, S. d’Ascoli, L. Sagun, G. Biroli, and M. Wyart, “A jamming transition from under-to over-parametrization affects loss landscape and generalization,” _arXiv preprint arXiv:1810.09665_ , 2018.
* [38] M. Geiger, S. Spigler, S. d’Ascoli, L. Sagun, M. Baity-Jesi, G. Biroli, and M. Wyart, “Jamming transition as a paradigm to understand the loss landscape of deep neural networks,” _Physical Review E_ , vol. 100, no. 1, p. 012115, 2019.
* [39] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus, “Intriguing properties of neural networks,” in _International Conference on Learning Representations_ , 2014.
* [40] H. Zhang, Y. Yu, J. Jiao, E. Xing, L. El Ghaoui, and M. Jordan, “Theoretically principled trade-off between robustness and accuracy,” in _International Conference on Machine Learning_ , 2019, pp. 7472–7482.
* [41] F. Croce and M. Hein, “Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,” in _International Conference on Machine Learning_ , 2020.
* [42] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models resistant to adversarial attacks,” in _International Conference on Learning Representations_ , 2018.
* [43] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2016, pp. 2818–2826.
* [44] G. Patrini, A. Rozza, A. Krishna Menon, R. Nock, and L. Qu, “Making deep neural networks robust to label noise: A loss correction approach,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 1944–1952.
* [45] H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz, “mixup: Beyond empirical risk minimization,” in _International Conference on Learning Representations_ , 2018.
* [46] Z. Zhang and M. Sabuncu, “Generalized cross entropy loss for training deep neural networks with noisy labels,” in _Advances in Neural Information Processing Systems_ , 2018, pp. 8778–8788.
* [47] Y. Wang, X. Ma, Z. Chen, Y. Luo, J. Yi, and J. Bailey, “Symmetric cross entropy for robust learning with noisy labels,” in _Proceedings of the IEEE International Conference on Computer Vision_ , 2019, pp. 322–330.
* [48] S. Thulasidasan, T. Bhattacharya, J. Bilmes, G. Chennupati, and J. Mohd-Yusof, “Combating label noise in deep learning using abstention,” in _International Conference on Machine Learning_ , 2019, pp. 6234–6243.
* [49] S. Liu, J. Niles-Weed, N. Razavian, and C. Fernandez-Granda, “Early-learning regularization prevents memorization of noisy labels,” in _Advances in Neural Information Processing Systems_ , 2020.
* [50] S. Zagoruyko and N. Komodakis, “Wide residual networks,” in _Proceedings of the British Machine Vision Conference_ , 2016, pp. 87.1–87.12.
* [51] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga _et al._ , “Pytorch: An imperative style, high-performance deep learning library,” in _Advances in Neural Information Processing Systems_ , 2019, pp. 8024–8035.
* [52] I. Loshchilov and F. Hutter, “SGDR: Stochastic gradient descent with warm restarts,” in _International Conference on Learning Representations_ , 2017\.
* [53] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein _et al._ , “Imagenet large scale visual recognition challenge,” _International Journal of Computer Vision_ , vol. 115, no. 3, pp. 211–252, 2015.
* [54] Z. Liu, Z. Wang, P. P. Liang, R. Salakhutdinov, L.-P. Morency, and M. Ueda, “Deep gamblers: Learning to abstain with portfolio theory,” in _Advances in Neural Information Processing Systems_ , 2019.
* [55] Y. Geifman and R. El-Yaniv, “Selectivenet: A deep neural network with an integrated reject option,” in _International Conference on Machine Learning_ , 2019, pp. 2151–2159.
* [56] ——, “Selective classification for deep neural networks,” in _Advances in Neural Information Processing Systems_ , 2017, pp. 4878–4887.
* [57] R. El-Yaniv and Y. Wiener, “On the foundations of noise-free selective classification,” _Journal of Machine Learning Research_ , vol. 11, no. May, pp. 1605–1641, 2010.
* [58] Y. Netzer, T. Wang, A. Coates, A. Bissacco, B. Wu, and A. Y. Ng, “Reading digits in natural images with unsupervised feature learning,” 2011.
* [59] “Dogs vs. cats dataset,” https://www.kaggle.com/c/dogs-vs-cats.
* [60] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in _International Conference on Machine Learning_ , 2015, pp. 448–456.
* [61] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” _The Journal of Machine Learning Research_ , vol. 15, no. 1, pp. 1929–1958, 2014.
* [62] J.-B. Grill, F. Strub, F. Altché, C. Tallec, P. Richemond, E. Buchatskaya, C. Doersch, B. Avila Pires, Z. Guo, M. Gheshlaghi Azar _et al._ , “Bootstrap your own latent-a new approach to self-supervised learning,” in _Advances in Neural Information Processing Systems_ , vol. 33, 2020.
* [63] P. Bojanowski and A. Joulin, “Unsupervised learning by predicting noise,” in _International Conference on Machine Learning_ , 2017, pp. 517–526.
* [64] X. Chen, H. Fan, R. Girshick, and K. He, “Improved baselines with momentum contrastive learning,” _arXiv preprint arXiv:2003.04297_ , 2020.
* [65] A. Coates, A. Ng, and H. Lee, “An analysis of single-layer networks in unsupervised feature learning,” in _Proceedings of International Conference on Artificial Intelligence and Statistics_ , 2011, pp. 215–223.
* [66] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in _Advances in Neural Information Processing Systems_ , vol. 25, 2012, pp. 1097–1105.
* [67] V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in _International Conference on Machine Learning_ , 2010.
* [68] R. Zhang, P. Isola, and A. A. Efros, “Split-brain autoencoders: Unsupervised learning by cross-channel prediction,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 1058–1067.
* [69] M. Caron, P. Bojanowski, A. Joulin, and M. Douze, “Deep clustering for unsupervised learning of visual features,” in _Proceedings of the European Conference on Computer Vision_ , 2018, pp. 132–149.
* [70] J. Huang, Q. Dong, S. Gong, and X. Zhu, “Unsupervised deep learning by neighbourhood discovery,” in _International Conference on Machine Learning_ , 2019, pp. 2849–2858.
* [71] Y. M. Asano, C. Rupprecht, and A. Vedaldi, “Self-labelling via simultaneous clustering and representation learning,” in _International Conference on Learning Representations_ , 2020.
* [72] Y. Tian, D. Krishnan, and P. Isola, “Contrastive multiview coding,” in _European Conference on Computer Vision_ , 2020.
* [73] Y. You, I. Gitman, and B. Ginsburg, “Large batch training of convolutional networks,” _arXiv preprint arXiv:1708.03888_ , 2017.
* [74] P. Goyal, P. Dollár, R. Girshick, P. Noordhuis, L. Wesolowski, A. Kyrola, A. Tulloch, Y. Jia, and K. He, “Accurate, large minibatch sgd: Training imagenet in 1 hour,” _arXiv preprint arXiv:1706.02677_ , 2017.
* [75] X. Chen and K. He, “Exploring simple siamese representation learning,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 15 750–15 758.
* [76] O. Henaff, “Data-efficient image recognition with contrastive predictive coding,” in _International Conference on Machine Learning_. PMLR, 2020, pp. 4182–4192.
* [77] J. Zbontar, L. Jing, I. Misra, Y. LeCun, and S. Deny, “Barlow twins: Self-supervised learning via redundancy reduction,” in _International Conference on Machine Learning_ , 2021.
* [78] X. Chen, S. Xie, and K. He, “An empirical study of training self-supervised vision transformers,” in _Proceedings of the IEEE International Conference on Computer Vision_ , 2021.
* [79] I. Misra and L. v. d. Maaten, “Self-supervised learning of pretext-invariant representations,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 6707–6717.
* [80] T. Chen, S. Kornblith, K. Swersky, M. Norouzi, and G. E. Hinton, “Big self-supervised models are strong semi-supervised learners,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 22 243–22 255, 2020\.
* [81] S. Ren, K. He, R. Girshick, and J. Sun, “Faster r-cnn: Towards real-time object detection with region proposal networks,” _Advances in Neural Information Processing Systems_ , vol. 28, pp. 91–99, 2015.
* [82] Y. Wu, A. Kirillov, F. Massa, W.-Y. Lo, and R. Girshick, “Detectron2,” https://github.com/facebookresearch/detectron2, 2019.
* [83] M. Everingham, S. M. A. Eslami, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman, “The pascal visual object classes challenge: A retrospective,” _International Journal of Computer Vision_ , vol. 111, no. 1, pp. 98–136, Jan. 2015.
* [84] Contributors, “MMSegmentation: Openmmlab semantic segmentation toolbox and benchmark,” https://github.com/open-mmlab/mmsegmentation, 2020.
* [85] G. Pereyra, G. Tucker, J. Chorowski, Ł. Kaiser, and G. Hinton, “Regularizing neural networks by penalizing confident output distributions,” _arXiv preprint arXiv:1701.06548_ , 2017.
* [86] D. Arpit, S. Jastrzębski, N. Ballas, D. Krueger, E. Bengio, M. S. Kanwal, T. Maharaj, A. Fischer, A. Courville, Y. Bengio _et al._ , “A closer look at memorization in deep networks,” in _International Conference on Machine Learning_. JMLR. org, 2017, pp. 233–242.
* [87] S. Reed, H. Lee, D. Anguelov, C. Szegedy, D. Erhan, and A. Rabinovich, “Training deep neural networks on noisy labels with bootstrapping,” in _Workshop at the International Conference on Learning Representation_ , 2015\.
* [88] B. Dong, J. Hou, Y. Lu, and Z. Zhang, “Distillation $\approx$ early stopping? harvesting dark knowledge utilizing anisotropic information retrieval for overparameterized neural network,” _arXiv preprint arXiv:1910.01255_ , 2019\.
* [89] T. Furlanello, Z. Lipton, M. Tschannen, L. Itti, and A. Anandkumar, “Born again neural networks,” in _International Conference on Machine Learning_ , 2018, pp. 1607–1616.
* [90] Q. Xie, M.-T. Luong, E. Hovy, and Q. V. Le, “Self-training with noisy student improves imagenet classification,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 10 687–10 698.
* [91] S. Laine and T. Aila, “Temporal ensembling for semi-supervised learning,” in _International Conference on Learning Representations_ , 2017.
* [92] P. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol, “Extracting and composing robust features with denoising autoencoders,” in _International Conference on Machine Learning_ , 2008, pp. 1096–1103.
* [93] D. Pathak, P. Krahenbuhl, J. Donahue, T. Darrell, and A. A. Efros, “Context encoders: Feature learning by inpainting,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2016, pp. 2536–2544.
* [94] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” in _International Conference on Learning Representation_ , 2014.
* [95] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” in _Advances in Neural Information Processing Systems_ , 2014, pp. 2672–2680.
* [96] S. Gidaris, P. Singh, and N. Komodakis, “Unsupervised representation learning by predicting image rotations,” in _International Conference on Learning Representation_ , 2018.
* [97] M. Noroozi and P. Favaro, “Unsupervised learning of visual representations by solving jigsaw puzzles,” in _European Conference on Computer Vision_. Springer, 2016, pp. 69–84.
* [98] A. v. d. Oord, Y. Li, and O. Vinyals, “Representation learning with contrastive predictive coding,” _arXiv preprint arXiv:1807.03748_ , 2018\.
* [99] J. Li, P. Zhou, C. Xiong, R. Socher, and S. C. Hoi, “Prototypical contrastive learning of unsupervised representations,” _International Conference on Learning Representations_ , 2021.
* [100] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in _International Conference on Learning Representations_ , 2015.
* [101] D. Hendrycks and T. Dietterich, “Benchmarking neural network robustness to common corruptions and perturbations,” in _International Conference on Learning Representations_ , 2018.
| Lang Huang received the Master’s degree from the Department of Machine
Intelligence, School of Electronics Engineering and Computer Science, Peking
University in 2021. He is currently a Ph.D. student at the Department of
Information & Communication Engineering, The University of Tokyo. His research
interests include self-supervised representation learning, learning from noisy
data, and their applications.
---|---
| Chao Zhang received the Ph.D. degree in Electrical Engineering from Beijing
Jiaotong University, Beijing, China in 1995. He is currently a Research
Professor at the Key Laboratory of Machine Perception (MOE), School of
Intelligence Science and Technology, Peking University. His research interests
include computer vision, image processing, machine learning, and pattern
recognition.
---|---
| Hongyang Zhang received the Ph.D. degree from Machine Learning Department,
Carnegie Mellon University in 2019. He is currently an Assistant Professor at
David R. Cheriton School of Computer Science, University of Waterloo, and a
Faculty affiliated with Vector Institute for AI. Before joining University of
Waterloo, he was a Postdoctoral Research Associate with Toyota Technological
Institute at Chicago. His research interests include machine learning, AI
security, and trustworthy AI.
---|---
(a) Accuracy curves of the model trained using ERM.
(b) Accuracy curves of the model trained using our method.
Figure 11: Accuracy curves of the model trained on the noisy CIFAR10 training
set with a noise rate of 80%. The horizontal dotted line displays the
percentage of clean data in the training sets. It shows that our observations
in Section 3 hold even when extreme label noise is injected.
## Appendix A Proofs
###### Proposition 1.
Let $d_{\mathrm{max}}$ be the maximal eigenvalue of the matrix
$\mathbf{X}^{\intercal}\mathbf{X}$, if the learning rate
$\eta<\frac{\alpha+1}{\alpha d_{\mathrm{max}}}$, then
$\displaystyle\lim_{s\rightarrow\infty}\left\lVert\mathbf{X}\bm{\theta}^{(s)}-\bm{t}^{(s)}\right\rVert_{2}^{2}=0.$
(17)
###### Proof.
Inserting Equation (4) into (5), we have
$\displaystyle\begin{split}\bm{t}^{(k)}&=\alpha\bm{t}^{(k-1)}+(1-\alpha)\mathbf{X}\bm{\theta}^{(k)}\\\
&=\alpha\bm{t}^{(k-1)}\\\
&\quad~{}+(1-\alpha)\mathbf{X}[\bm{\theta}^{(k-1)}-\eta\mathbf{X}^{\intercal}(\mathbf{X}\bm{\theta}^{(k-1)}-\bm{t}^{(k-1)})]\\\
&=[\alpha\mathbf{I}+(1-\alpha)\eta\mathbf{X}\mathbf{X}^{\intercal}]\bm{t}^{(k-1)}\\\
&\quad~{}+(1-\alpha)(\mathbf{I}-\eta\mathbf{X}\mathbf{X}^{\intercal})\mathbf{X}\bm{\theta}^{(k-1)}.\end{split}$
(18)
Note that $\mathbf{X}\mathbf{X}^{\intercal}$ is positive semi-definite and can
be diagonalized as
$\mathbf{X}\mathbf{X}^{\intercal}=\mathbf{V}^{\intercal}\mathbf{D}\mathbf{V}$,
where the diagonal matrix $\mathbf{D}$ contains the eigenvalue of
$\mathbf{X}^{\intercal}\mathbf{X}$ and the matrix $\mathbf{V}$ contains the
corresponding eigenvectors, $\mathbf{V}\mathbf{V}^{\intercal}=\mathbf{I}$. And
let
$\bm{r}^{(k)}=\mathbf{V}\bm{t}^{(k)},\bm{s}^{(k)}=\mathbf{V}\mathbf{X}\bm{\theta}^{(k)}$.
Multiplying both sides of Equation (18) by $\mathbf{V}$, we have
$\displaystyle\begin{split}\mathbf{V}\bm{t}^{(k)}&=\mathbf{V}[\alpha\mathbf{V}^{\intercal}\mathbf{V}+(1-\alpha)\eta\mathbf{V}^{\intercal}\mathbf{D}\mathbf{V}]\bm{t}^{(k-1)}\\\
&\quad~{}+(1-\alpha)\mathbf{V}(\mathbf{V}^{\intercal}\mathbf{V}-\eta\mathbf{V}^{\intercal}\mathbf{D}\mathbf{V})\mathbf{X}\bm{\theta}^{(k-1)}\\\
\mathbf{V}\bm{t}^{(k)}&=[\alpha\mathbf{I}+(1-\alpha)\eta\mathbf{D}]\mathbf{V}\bm{t}^{(k-1)}\\\
&\quad~{}+(1-\alpha)(\mathbf{I}-\eta\mathbf{D})\mathbf{V}\mathbf{X}\bm{\theta}^{(k-1)}\\\
\bm{r}^{(k)}&=[\alpha\mathbf{I}+(1-\alpha)\eta\mathbf{D}]\bm{r}^{(k-1)}\\\
&\quad~{}+(1-\alpha)(\mathbf{I}-\eta\mathbf{D})\bm{s}^{(k-1)}.\end{split}$
(19)
From Equation (4), we have
$\displaystyle\begin{split}\bm{s}^{(k)}&=\mathbf{V}\mathbf{X}\bm{\theta}^{(k)}\\\
&=\mathbf{V}\mathbf{X}[\bm{\theta}^{(k-1)}-\eta\mathbf{X}^{\intercal}(\mathbf{X}\bm{\theta}^{(k-1)}-\bm{t}^{(k-1)})]\\\
&=\bm{s}^{(k-1)}-\eta\mathbf{D}(\bm{s}^{(k-1)}-\bm{r}^{(k-1)})\\\
&=\eta\mathbf{D}\bm{r}^{(k-1)}+(\mathbf{I}-\eta\mathbf{D})\bm{s}^{(k-1)}.\end{split}$
(20)
Subtracting the the both sides of Equation (20) by $\bm{r}^{(k)}$, we obtain
$\displaystyle\begin{split}\bm{s}^{(k)}-\bm{r}^{(k)}&=\alpha(\mathbf{I}-\eta\mathbf{D})(\bm{s}^{(k-1)}-\bm{r}^{(k-1)})\\\
&=[\alpha(\mathbf{I}-\eta\mathbf{D})]^{k}(\bm{s}^{(0)}-\bm{r}^{(0)})\\\
&=\mathbf{A}^{k}\mathbf{V}\bm{b},\end{split}$ (21)
where
$\mathbf{A}=\alpha(\mathbf{I}-\eta\mathbf{D}),\bm{b}=\mathbf{X}\bm{\theta}^{(0)}-\bm{t}^{(0)}$.
Therefore,
$\displaystyle\begin{split}\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}&=\mathbf{V}^{\intercal}\mathbf{A}^{k}\mathbf{V}\bm{b}.\end{split}$
(22)
Because $\mathbf{X}\mathbf{X}^{\intercal}$ is positive semi-definite and
$\alpha\in(0,1)$, all elements in
$\mathbf{A}=\alpha(\mathbf{I}-\eta\mathbf{D})$ is smaller than 1. When
$0<\eta<\frac{\alpha+1}{\alpha d_{\mathrm{max}}}$, each elements of the
diagonal matrix $\mathbf{A}$ is greater than -1, and we have
$\displaystyle\begin{split}\lim_{k\rightarrow\infty}\left\lVert\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}\right\rVert_{2}^{2}&=\bm{b}^{\intercal}\mathbf{V}^{\intercal}\mathbf{A}^{2k}\mathbf{V}\bm{b}\\\
&=0.\end{split}$ (23)
∎
###### Corollary 1.
Under the same condition as in Proposition 1, we have
$\displaystyle\lim_{k\rightarrow\infty}\frac{\left\lVert\mathbf{X}\bm{\theta}^{(k+1)}-\bm{t}^{(k+1)}\right\rVert_{2}^{2}}{\left\lVert\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}\right\rVert_{2}^{2}}=a_{i}^{2}<1,$
(24)
###### Proof.
Let $a_{i}$ be the element of the diagonal matrix $\mathbf{A}$ that has the
maximal absolute value, where $i$ is its index on $\mathbf{A}$, i.e.,
$i=\operatorname{argmax}_{j}|\mathbf{A}_{jj}|,|a_{i}|<1$. Then, we have
$\displaystyle\begin{split}&\lim_{k\rightarrow\infty}\frac{\left\lVert\mathbf{X}\bm{\theta}^{(k+1)}-\bm{t}^{(k+1)}\right\rVert_{2}^{2}}{\left\lVert\mathbf{X}\bm{\theta}^{(k)}-\bm{t}^{(k)}\right\rVert_{2}^{2}}\\\
=&\lim_{k\rightarrow\infty}\frac{\bm{b}^{\intercal}\mathbf{V}^{\intercal}\mathbf{A}^{2k+2}\mathbf{V}\bm{b}}{\bm{b}^{\intercal}\mathbf{V}^{\intercal}\mathbf{A}^{2k}\mathbf{V}\bm{b}}\\\
=&\frac{a_{i}^{2k+2}(\mathbf{V}\bm{b})_{i}^{2}}{a_{i}^{2k}(\mathbf{V}\bm{b})_{i}^{2}}\\\
=&a_{i}^{2}<1.\end{split}$ (25)
∎
## Appendix B Experimental Setups
### B.1 Double descent phenomenon
Following previous work [33], we optimize all models using Adam [100]
optimizer with a fixed learning rate of 0.0001, a batch size of 128, the
common data augmentations, and a weight decay of 0 for 4,000 epochs. For our
approach, we use the hyper-parameters $\mathrm{E}_{s}=40,\alpha=0.9$ for
standard ResNet-18 (width of 64) and dynamically adjust them for other models
according to the relation of model capacity $r=\frac{64}{\mathrm{width}}$ as:
$\mathrm{E}_{s}=40\times r;\quad\alpha=0.9^{\frac{1}{r}}.$ (26)
### B.2 Adversarial training
[39] reported that imperceptible small perturbations around input data (i.e.,
adversarial examples) can cause ERM-trained deep neural networks to make
arbitrary predictions. Since then, a large literature devoted to improving the
adversarial robustness of deep neural networks. Among them, the adversarial
training algorithm TRADES [40] achieves state-of-the-art performance. TRADES
decomposed robust error (w.r.t adversarial examples) to the sum of natural
error and boundary error, and proposed to minimize:
$\mathbb{E}_{\bm{x},\bm{y}}\Bigg{\\{}\mathrm{CE}(\bm{p}(\bm{x}),\bm{y})+\max_{\|\widetilde{\bm{x}}-\bm{x}\|_{\infty}\leq\epsilon}\mathrm{KL}(\bm{p}(\bm{x}),\bm{p}(\widetilde{\bm{x}}))/\lambda\Bigg{\\}},$
(27)
where $\bm{p}(\cdot)$ is the model prediction, $\epsilon$ is the maximal
perturbation, CE stands for cross entropy, and KL stands for Kullback–Leibler
divergence. The first term corresponds to ERM that maximizes the natural
accuracy; the second term pushes the decision boundary away from data points
to improve adversarial robustness; the hyper-parameter $1/\lambda$ controls
the trade-off between natural accuracy and adversarial robustness. We evaluate
self-adaptive training on this task by replacing the first term of Equation
(27) with our approach.
Our experiments are based on the official open-sourced
implementation222https://github.com/yaodongyu/TRADES of TRADES [40].
Concretely, we conduct experiments on the CIFAR10 dataset [16] and use
WRN-34-10 [50] as base classifier. For training, we use an initial learning
rate of 0.1, a batch size of 128, and 100 training epochs. The learning rate
is decayed at the 75th and 90th epoch by a factor of 0.1. The adversarial
example $\widetilde{\bm{x}}_{i}$ is generated dynamically during training by
projected gradient descent (PGD) attack [42] with a maximal $\ell_{\infty}$
perturbation $\epsilon$ of 0.031, perturbation step size of 0.007, number of
perturbation steps of 10. The hyper-parameter $1/\lambda$ of TRADES is set to
6 as suggested by the original paper, $\mathrm{E}_{s},\alpha$ of our approach
is set to 70, 0.9, respectively. For evaluation, we report robust accuracy
$\frac{1}{n}\sum_{i}\mathbbm{1}\\{\operatorname{argmax}~{}p(\widetilde{\bm{x}}_{i})=\operatorname{argmax}~{}\bm{y}_{i}\\}$,
where adversarial example $\widetilde{\bm{x}}$ is generated by two kinds of
white box $\ell_{\infty}$ attacks with $\epsilon$ of 0.031: 1) AutoAttack [41]
(as in Fig. 5); 2) untargeted PGD attack (as in Fig. 14), with a perturbation
step size of 0.007, the number of perturbation steps of 20.
Figure 12: Accuracy curves on different portions of the CIFAR10 training set
(with 40% of label noise) w.r.t. correct labels. We split the training set
into two portions: 1) _Untouched portion_ , i.e., the elements in the
training set which were left untouched; 2) _Corrupted portion_ , i.e., the
elements in the training set which were indeed randomized. It shows that ERM
fits correct labels in the first few epochs and then eventually overfits the
corrupted labels. In contrast, self-adaptive training calibrates the training
process and consistently fits the correct labels. Figure 13: Generalization
error and clean validation error under four kinds of random noise (represented
by different colors) for ERM (the dashed curves) and our approach (the solid
curves) on CIFAR10 when data augmentation is turned off. We zoom in on the
dashed rectangle region and display it in the third column for a clear
demonstration.
### B.3 Supervised learning on ImageNet
We use ResNet-50/ResNet-101 [3] as base classifier. Following the original
papers [3] and [52, 74], we use SGD to optimize the networks with a batch size
of 768, a base learning rate of 0.3, a momentum of 0.9, a weight decay of
0.0005, and a total training epoch of 95. The learning rate is linearly
increased from 0.0003 to 0.3 in the first 5 epochs (i.e., warmup), and then
decayed using the cosine annealing schedule [52] to 0. Following common
practice, we use the random resizing, cropping, and flipping augmentations
during training. The hyper-parameters of our approach are set to
$\mathrm{E}_{s}=50$ and $\alpha=0.99$ under standard setup, and are set to
$\mathrm{E}_{s}=60$ and $\alpha=0.95$ under 40% label noise setting. The
experiments are conducted on PyTorch [51] with distributed training and mixed-
precision training333https://github.com/NVIDIA/apex for acceleration.
### B.4 Linear classifier during self-supervised training
For the horizontal line in Fig. 2, we randomly initialize a network and train
a linear classifier atop this network. The horizontal line corresponds to the
_final_ accuracy of this linear classifier. For the other three curves in Fig.
2, we adopt an online classifier scheme, following the official implementation
of BOYL [62]. Concretely, in the self-supervised pre-training stage, we train
1) a network to fit the output of another randomly initialized network, and 2)
a linear classifier on top of the network whose gradient will not
backpropagate to the network. This scheme bypasses the heavy cost to train a
linear classifier from scratch at each training epoch and allows us to
directly evaluate the linear classifier on the validation set. In practice, we
use a separate learning rate for this classier, i.e., 0.4 in our experiments.
### B.5 Training time comparison
In this comparison, we calculate the running time for all methods using a
server with 8 V100 GPUs, a 40-core CPU, CUDA10.1, and PyTorch1.8. For MoCov2,
we follow the official implementation using a batch size of 256 and ShuffleBN
[10]. For the BYOL, we use SyncBn and a batch size of 512 to fit into a single
machine. For our method, we use SyncBN and a batch size of 1024 so that the
number of augmented views (1024x1) in each batch is identical to that of BYOL
(512x2). For the supervised cross entropy baseline, we use the standard BN and
a batch size of 256 following the common practice. According to Fig. 9, we can
see that, though data preprocessing is not the bottleneck on a modern machine,
our method is still 2x/1.3x faster than BYOL/MoCov2.
## Appendix C Additional Experimental Results
### C.1 ERM may suffer from overfitting of noise
In [14], the authors showed that the model trained by standard ERM can easily
fit randomized data. However, they only analyzed the generalization errors in
the presence of corrupted labels. In this paper, we report the whole training
process and also consider the performance on clean sets (i.e., the original
uncorrupted data). Fig. 1(a) shows the four accuracy curves (on clean and
noisy training, validation set, respectively) for each model that is trained
on one of four corrupted training data. Note that the models can only have
access to the noisy training sets (i.e., the red curve), and the other three
curves are shown only for illustration purposes. We conclude with two
principal observations from the figures: (1) The accuracy on noisy training
and validation sets is close at the beginning and the gap is monotonously
increasing w.r.t. epoch. The generalization errors (i.e., the gap between the
accuracy on noisy training and validation sets) are large at the end of
training. (2) The accuracy on the clean training and validation sets is
consistently higher than the percentage of clean data in the noisy training
set. This occurs around the epochs between underfitting and overfitting.
Figure 14: Robust Accuracy (%) on CIFAR10 test set under white box
$\ell_{\infty}$ PGD-20 attack ($\epsilon$=0.031). The vertical dashed lines
indicate learning rate decay. It shows that self-adaptive training
consistently improves TRADES.
Our first observation poses concerns on the overfitting issue of ERM training
dynamic which has also been reported by [19]. However, the work of [19] only
considered the case of corrupted labels and proposed using the early-stop
mechanism to improve the performance on the clean data. On the other hand, our
analysis of the broader corruption schemes shows that the early stopping might
be sub-optimal and may hurt the performance under other types of corruption
(see the last three columns in Fig. 1(a)).
The second observation implies that model predictions by ERM can capture and
amplify useful signals in the noisy training set, although the training
dataset is heavily corrupted. While this was also reported in [14, 17, 18, 19]
for the case of corrupted labels, we show that a similar phenomenon occurs
under other kinds of corruption more generally. This observation sheds light
on our approach, which incorporates model predictions into the training
procedure.
### C.2 Improved generalization of self-adaptive training on random noise
Training accuracy w.r.t. correct labels on different portions of data For a
more intuitive demonstration, we split the CIFAR10 training set (with 40%
label noise) into two portions: 1) _Untouched portion_ , i.e., the elements
in the training set which were left untouched; 2) _Corrupted portion_ , i.e.,
the elements in the training set which were indeed randomized. The accuracy
curves on these two portions w.r.t correct training labels are shown in Fig.
12. We can observe that the accuracy of ERM on the corrupted portion first
increases in the first few epochs and then eventually decreases to 0. In
contrast, self-adaptive training calibrates the training process and
consistently fits the correct labels.
Figure 15: Self-adaptive training _vs._ ERM on the error-epoch curve. We train the standard ResNet-18 networks (i.e., the width of 64) on the CIFAR10 dataset with 15% randomly-corrupted labels and report the test errors on the clean data. The dashed vertical line represents the initial epoch $\mathrm{E}_{s}$ of our approach. It shows that self-adaptive training has a significantly diminished epoch-wise double-descent phenomenon. TABLE XII: Test Accuracy (%) on CIFAR datasets with various levels of uniform label noise injected into the training set. We show that considerable gains can be obtained when combined with SCE loss. | CIFAR10 | CIFAR100
---|---|---
Label Noise Rate | 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8
SCE [47] | 90.15 | 86.74 | 80.80 | 46.28 | 71.26 | 66.41 | 57.43 | 26.41
Ours | 94.14 | 92.64 | 89.23 | 78.58 | 75.77 | 71.38 | 62.69 | 38.72
Ours + SCE | 94.39 | 93.29 | 89.83 | 79.13 | 76.57 | 72.16 | 64.12 | 39.61
Study on extreme noise We further rerun the same experiments as in Fig. 1 of
the main text by injecting extreme noise (i.e., noise rate of 80%) into the
CIFAR10 dataset. We report the corresponding accuracy curves in Fig. 11, which
shows that our approach significantly improves the generalization over ERM
even when random noise dominates training data. This again justifies our
observations in Section 3.
Effect of data augmentation All our previous studies are performed with common
data augmentation (i.e., random cropping and flipping). Here, we further
report the effect of data augmentation. We adjust the introduced hyper-
parameters as $\mathrm{E}_{s}=25$, $\alpha=0.7$ due to severer overfitting
when data augmentation is absent. Fig. 13 shows the corresponding
generalization errors and clean validation errors. We observe that, for both
ERM and our approach, the errors clearly increase when data augmentation is
absent (compared with those in Fig. 3). However, the gain is limited and the
generalization errors can still be very large, with or without data
augmentation for standard ERM. Directly replacing the standard training
procedure with our approach can bring bigger gains in terms of generalization
regardless of data augmentation. This suggests that data augmentation can help
but is not of the essence to improving the generalization of deep neural
networks, which is consistent with the observation in [14].
### C.3 Epoch-wise double descent phenomenon
[33] reported that, for a sufficiently large model, the test error-training
epoch curve also exhibits a double-descent phenomenon, which they termed
_epoch-wise double descent_. In Fig. 15, we reproduce the epoch-wise double
descent phenomenon on ERM and inspect self-adaptive training. We observe that
our approach (the red curve) exhibits a slight double-descent due to the
overfitting starts before the initial $\mathrm{E}_{s}$ epochs. As the training
targets being updated (i.e., after $\mathrm{E}_{s}$ = 40 training epochs), the
red curve undergoes a monotonous decrease. This observation again indicates
that the double-descent phenomenon may stem from the overfitting of noise and
can be avoided by our algorithm.
### C.4 Cooperation with Symmetric Cross Entropy
[47] showed that Symmetric Cross Entropy (SCE) loss is robust to underlying
label noise in training data. Formally, given training target $\bm{t}_{i}$ and
model prediction $\bm{p}_{i}$, SCE loss is defined as:
$\mathcal{L}_{sce}=-w_{1}\sum_{j}\bm{t}_{i,j}~{}\log~{}\bm{p}_{i,j}-w_{2}\sum_{j}\bm{p}_{i,j}~{}\log~{}\bm{t}_{i,j},$
(28)
where the first term is the standard cross entropy loss and the second term is
the reversed version. In this section, we show that self-adaptive training can
cooperate with this noise-robust loss and enjoy further performance boost
without extra cost.
Setup Most of the experimental settings are kept the same as Section 4.2. For
the introduced hyper-parameters $w_{1},w_{2}$ of SCE loss, we directly set
them to 1, 0.1, respectively, in all our experiments.
Results We summarize the results in Table XII. We can see that, although self-
adaptive training already achieves very strong performance, considerable gains
can be obtained when equipped with SCE loss. Concretely, the improvement is as
large as 1.5% when 60% of label noise is injected into the CIFAR100 training
set. It also indicates that our approach is flexible and can be further
extended.
TABLE XIII: Average Accuracy (%) on the CIFAR10 test set and out-of-distribution dataset CIFAR10-C at various corruption levels. Method | CIFAR10 | Corruption Level@CIFAR10-C
---|---|---
1 | 2 | 3 | 4 | 5
ERM | 95.32 | 88.44 | 83.22 | 77.26 | 70.40 | 58.91
Ours | 95.80 | 89.41 | 84.53 | 78.83 | 71.90 | 60.77
### C.5 Out-of-distribution generalization
In this section, we consider the out-of-distribution (OOD) generalization,
where the models are evaluated on unseen test distributions outside the
training distribution.
Setup To evaluate the OOD generalization performance, we use the CIFAR10-C
benchmark [101] that is constructed by applying 15 types of corruption to the
original CIFAR10 test set at 5 levels of severity. The performance is measured
by average accuracy over 15 types of corruption. We mainly follow the training
details in Section 4.2 and adjust $\alpha=0.95,\mathrm{E}_{s}=80$.
Results We summarize the results in Table XIII. Regardless of the presence of
corruption and corruption levels, our method consistently outperforms ERM by a
considerable margin, which becomes large when the corruption is more severe.
The experiment indicates that self-adaptive training may provide implicit
regularization for OOD generalization.
|
0 11institutetext: MPI-SWS, Kaiserslautern and Saarbrücken, Germany, 11email:
<EMAIL_ADDRESS>22institutetext: Karlsruhe Institute of
Technology, Karlsruhe, Germany, 22email<EMAIL_ADDRESS>33institutetext: Chalmers University of Technology, Göteborg, Sweden, 33email:
<EMAIL_ADDRESS>
# Deductive Verification of Floating-Point
Java Programs in KeY
Rosa Abbasi (🖂) 11 Jonas Schiffl 22 Eva Darulova 11
Mattias Ulbrich 22 Wolfgang Ahrendt 33
###### Abstract
Deductive verification has been successful in verifying interesting properties
of real-world programs. One notable gap is the limited support for floating-
point reasoning. This is unfortunate, as floating-point arithmetic is
particularly unintuitive to reason about due to rounding as well as the
presence of the special values infinity and ‘Not a Number’ (NaN). In this
paper, we present the first floating-point support in a deductive verification
tool for the Java programming language. Our support in the KeY verifier
handles arithmetic via floating-point decision procedures inside SMT solvers
and transcendental functions via axiomatization. We evaluate this integration
on new benchmarks, and show that this approach is powerful enough to prove the
absence of floating-point special values—often a prerequisite for further
reasoning about numerical computations—as well as certain functional
properties for realistic benchmarks.
###### Keywords:
Deductive Verification Floating-point Arithmetic Transcendental Functions.
## 1 Introduction
Deductive verification has been successful in providing functional
verification for programs written in popular programming languages such as
Java [3, 48, 40, 22], Python [28], Rust [5], C [24, 53], and Ada [49, 18].
Deductive verifiers allow a user to annotate methods in a program with pre-
and postconditions, from which they automatically generate verification
conditions (VCs). These are then either proven directly by the verifier
itself, or discharged with external tools such as automated (SMT) solvers or
interactive proof assistants.
While deductive verifiers fully implement many sophisticated data
representations (including heap data structures, objects, and ownership),
support for floating-point numbers remains rather limited – solely Frama-C and
SPARK offer automated support for floating-point arithmetic in C and Ada [31].
This state of affairs is at least partially a result of previous limitations
in floating-point support in SMT solvers. Consequently, deductive verification
has been used for floating-point programs only by experts with considerable
manual effort [31, 14]. This is unfortunate as it makes deductive verification
unavailable for a large number of programs across many domains including
embedded systems, machine learning, and scientific computing. With the
increasing need for parallelization in code, scientific computing specifically
has recently experienced algorithmic challenges for which formal methods may
contribute to a solution [55, 9].
One of the main challenges of floating-point arithmetic is its unintuitive
behavior and the special values that the IEEE 754 standard [38] introduces.
For instance, an overflow or a division by zero results in the special value
(positive or negative) _infinity_ , and not a runtime exception. Similarly,
invalid operations like sqrt(-1.0) result in a _Not a Number_ (NaN) value.
These special values are problematic as seemingly straight-forward identities
do not hold (x == x or x * 0.0 == 0.0). In addition, every operation on
floating-point numbers potentially involves rounding, which compromises
familiar rules like associativity and distributivity. Hence, reasoning support
for writing correct floating-point programs is indispensable.
Abstract interpretation-based tools can prove the absence of runtime errors
and special values [19, 42], and bound roundoff errors due to floating-point’s
finite precision [25, 56, 10, 20, 35]. SMT decision procedures [17] or SAT-
based model-checking [55, 23], on the other hand, can prove intricate
properties requiring bit-precise reasoning. However, these techniques and
tools largely support only purely floating-point programs or program snippets,
or analyze programs only up to a predefined depth of the call stack. General
reasoning about real-world object-oriented programs, however, also requires
support for features such as the (unbounded) heap, necessitating different
analyses which need to be combined with floating-point reasoning.
Handling floating-points in a deductive verifier has unique advantages. First,
the deductive verification approach already comes with the infrastructure for
reasoning about complex control and data structures (like exception handling
and heap). Second, it allows one to flexibly combine the verifier’s symbolic
execution reasoning with external decision procedures. Third, depending on the
theory support, the verifier or external solver may also generate
counterexamples of a property and thus help program debugging – something an
abstract interpretation-based approach fundamentally cannot provide.
We report on adding floating-point support to the KeY deductive verifier,
providing the first automated deductive floating-point support for the Java
programming language. We focus mainly on proving the absence of the special
values infinity and NaN. While these are helpful in certain circumstances, for
most applications they signal an error. Hence, showing their absence is a
prerequisite for further (functional) reasoning. That said, our extension also
allows one to express and discharge arbitrary functional properties
expressible in floating-point arithmetic, including bounds on roundoff errors
for certain programs, and bounds on differences between two similar floating-
point programs
We exploit both KeY’s symbolic execution and external SMT support. On the one
hand, we handle arithmetic operations by relying on a combination of KeY’s
symbolic execution to handle the heap and SMT based decision procedures to
handle the floating-point part of the VCs. On the other hand, we support
transcendental functions via axiomatization in the KeY prover itself.
Transcendental functions such as sine are a common feature in numerical
programs, but are not supported by floating-point decision procedures. We
explore two ways of supporting them soundly but approximately, by encoding
them as axiomatized uninterpreted function symbols once directly in the SMT
queries, and once in additional calculus rules in KeY. Our evaluation shows
that even though such reasoning is approximate, it is nonetheless sufficient
to prove the absence of special values in many interesting programs.
We evaluate KeY’s floating-point support on a number of real-world floating-
point Java programs. Our benchmark set allows us to evaluate recent progress
in SMT floating-point support in Z3 [27], CVC4 [7] and MathSAT [21] on yet
unseen benchmarks. For instance, we observe that quantifiers are challenging
even if they do not affect satisfiability of SMT queries. Our benchmarks are
openly available, and we expect our insights to be useful for further solver
development.
##### Contributions
In summary, we make the following contributions:
* •
we implement and evaluate the first automated deductive verification of
floating-point Java programs by combining the strength of rule based and SMT
based deduction;
* •
we collect a new set of challenging real-world floating-point benchmarks in
Java (available at https://gitlab.mpi-sws.org/AVA/key-float-benchmarks/);
* •
we compare different SMT solvers for discharging floating-point VCs on this
new set of benchmarks;
* •
and we develop novel automated support for reasoning about transcendental
functions in a deductive verifier.
## 2 Background
### 2.1 Introduction to KeY
KeY [3] is a platform for deductive verification of Java programs, working at
a source code level. The input is a Java program annotated in the Java
Modeling Language (JML) [44], encouraging a Design by Contract ([50, 45])
approach to software development. The user specifies the expected behavior of
Java classes with _class invariants_ that the program has to maintain at
critical points. Methods are specified with _method contracts_ , consisting
mainly of pre- and postconditions, with the understanding that if the
precondition holds when the method is called, the postcondition has to hold
after the method returns.
After loading an annotated program, KeY translates it to a formula in Java
Dynamic Logic [3] (JavaDL), an instance of Dynamic Logic [36] which enables
logical reasoning about Java programs. Logical rules are provided for the
translation of programs into first-order logic, and for closing the resulting
goals, or proof obligations. KeY is semi-interactive in that it allows manual
rule application, while also offering powerful built-in automation and macros.
In addition, it is also possible to translate an open goal into SMT-LIB format
[8] and call an external SMT solver. For specific theories, SMT solvers can be
much more efficient than KeY’s own automation. This makes it possible to prove
some goals, which depend on SMT supported theories, by using an SMT solver,
while others are proved internally, using KeY’s own automation.
### 2.2 Floating-Point Arithmetic in Java
In the following, we summarize some central characteristics of Java floating-
point numbers, loosely following [52]. Each _normal_ floating-point number $x$
can be represented as a triplet $(s,m,e)$, such that $x=(-1)^{s}*m*2^{e}$,
where $s\in\\{0,1\\}$ is the _sign_ , $m$ (called _significand_) is a binary
fixed-point number with one digit before the radix point and $p-1$ digits
after the radix point (note that $0\leq m<2$), and $e$ (_exponent_) is an
integer such that $e_{min}\leq e\leq e_{max}$. Java supports two floating-
point formats (both in base $2$): float (‘single’) precision with $p=24$, and
minimal and maximal exponent $e_{min}=-126,e_{max}=127$ and double precision
with $p=53,e_{min}=-1022,e_{max}=1023$.
Whenever the result of a computation cannot be exactly represented with the
given precision, it is rounded. IEEE 754 defines various rounding modes, of
which Java only supports _round to nearest, ties to even_. Rounding is exact,
as if one would first compute the ideal real number, and round afterwards.
The triple representation gives us two zeros, $+0$ and $-0$, represented by
$(0,0,0)$ and $(1,0,0)$, respectively. If the absolute value of the ideal
result of a computation is too small to be representable as a floating-point
number of the given format, the resulting floating point number is $+0$ or
$-0$. In addition, there are three special values, $+\infty$, $-\infty$, and
NaN (Not a Number). If the absolute value of the ideal result of a computation
is too big to be representable as a floating-point number of the given format,
the result is $+\infty$ or $-\infty$. Also, division by zero will give an
infinite result (e.g., $7.13/+0=+\infty$). Computing further with infinity may
give an infinite result (e.g., $+\infty++\infty=+\infty$), but may also result
in the additional ‘error value’ NaN (e.g., $+\infty-+\infty=\text{NaN}$). Due
to the presence of infinities and NaN, floating-point operations do _not_
throw Java exceptions.
By default, the Java virtual machine is allowed to make use of higher-
precision formats provided by the hardware. This can make computation more
accurate, but it also leads to platform dependent behaviour. This can be
avoided by using the strictfp modifier, ensuring that only the single and
double precision types are used. This modifier ensures portability.
## 3 Floating-Point Support in KeY
### 3.1 Arithmetics
In order to be able to specify and verify programs containing floating-point
numbers, we made several extensions to the KeY tool. First, we added the float
and double types to the KeY type system, together with an enum type for the
different rounding modes of the IEEE 754 Standard.
We further introduced functions and predicate symbols to formalize operations
(+, *, …) and comparisons (<, ==, …) on floating-point expressions. The
translation supports both code with and without the strictfp modifier.
However, since the actual precision of non-strictfp operations is not known,
the function symbols remain uninterpreted. We extended KeY’s parser to
correctly handle programs and annotations containing floating-point numbers,
and added logic rules for translating floating-point expressions from Java or
JML to JavaDL.
Listing 1: The Rectangle.scale benchmark
⬇
/*@ public normal_behavior
@ requires \fp_nice(arg0.x) && \fp_nice(arg0.y)
@ && \fp_nice(arg1) && \fp_nice(arg2);
@ ensures !\fp_nan(\result.x) && !\fp_nan(\result.y) &&
@ !\fp_nan(\result.width) && !\fp_nan(\result.height);
@ also
@ public normal_behavior
@ requires -5.53 <= arg0.x && arg0.x <= -3.38 &&
@ -5.53 <= arg0.y && arg0.y <= -3.38 &&
@ 3.1 < arg0.width && arg0.width <= 3.7332 &&
@ 3.0000001 < arg0.height && arg0.height <=4.0004 &&
@ 3.0003001 < arg1 && arg1 <= 4.0024 &&
@ -6.4000003 < arg2 && arg2 <= 3.0001;
@ ensures !\fp_nan(\result.x) && !\fp_nan(\result.y)&&
@ !\fp_nan(\result.width) &&!\fp_nan(\result.height);
@*/
public Rectangle scale(Rectangle arg0, double arg1, double arg2){
Area v1 = new Area(arg0);
AffineTransform v2 = AffineTransform.getScaleInstance(arg1, arg2);
Area v3 = v1.createTransformedArea(v2);
Rectangle v4 = v3.getRectangle2D();
return v4;
}
As an example, 1 shows JML specifications of our Rectangle benchmark that
contains floating-point literals and makes use of the fp_nan and fp_nice
predicates. fp_nan states that a floating-point expression is NaN and fp_nice,
which is shorthand for “not infinity and not NaN”, states that a floating-
point expression is not NaN or infinity. The scale method contains two
contracts that are checked separately, ensuring that the class fields of a
scaled rectangle object are not NaN, considering different preconditions. For
the first contract, the SMT solver produces a counterexample. In the second,
we bound inputs by concrete ranges that we picked arbitrarily and get the
valid result. In practice, such ranges would come from the context, e.g. from
the kind of rectangles that appear in an application, or from known ranges of
sensor values.
Concerning discharging the resulting proof obligations, there were two main
ways to consider. One is to create a floating-point theory within KeY by
adding axioms and deduction rules, so that the desired properties can be
proven in KeY’s sequent calculus. The other way is to translate the proof
obligations from JavaDL to SMT-LIB and call an external SMT solver. While the
KeY approach traditionally favors conducting proofs within KeY, for this work,
we partially deviated from this way in order to harness the greater experience
and efficiency of SMT solvers when it comes to floating-point arithmetic. Our
approach attempts to get the best of both worlds by distinguishing between
basic floating-point arithmetic, i. e., elementary operations and comparisons,
and more complex functions which do not have an SMT-LIB equivalent (e. g., the
transcendental functions), or where the SMT-LIB function is not usefully
implemented by current SMT solvers (see subsubsection 3.2.2).
Elementary operations and comparisons get translated to the corresponding SMT-
LIB functions. In SMT-LIB, all floating-point computations conform to the IEEE
754 Standard. Therefore, only Java programs with the strictfp modifier can be
directly translated to SMT-LIB without loss of correctness.
We developed a translation from KeY’s floating-point theory to SMT-LIB. In
order to integrate it into KeY, we also overhauled the existing translation
from JavaDL to SMT-LIB to create a new, more modular framework, which now
supports all the features of the original translation, e. g., heaps and
integer arithmetic, but also floating-point expressions at the same time.
Floating-point intricacies sometimes require extra caution. For example, there
are two different notions of equality for floats: bitwise equality and IEEE754
equality. Our implementation ensures these are distinguished correctly, and
that the specification language remains intuitive for a developer to use.
Using the translation to SMT-LIB, we can specify and prove two classes of
properties in KeY: The absence of special values is specified using the fp_nan
and fp_infinite predicates (or the fp_nice equivalent). Furthermore, one can
specify _functional_ properties that are expressible in floating-point
arithmetic, e.g. one can compare the result of a computation against the
result of a different program which is known to produce a good result or a
reference value.
### 3.2 Transcendental Functions
Floating-point decision procedures in SMT solvers successfully handle programs
consisting of arithmetic and square root operations. Many numerical real-world
programs, however, include transcendental functions such as sin and cos. In
Java programs, these functions are implemented as static library functions in
the class java.lang.Math.
Unlike arithmetic operations, transcendental functions are much more loosely
specified by the IEEE 754 Standard—only an upper bound on the roundoff error
is given. Libraries are thus free to provide different implementations, and
even tighter error bounds. Exact reasoning in the same spirit as floating-
point arithmetic would thus have to encode a specific implementation. Given
that these implementations are highly optimized, this approach would be
arguably complex. We observe, however, that such exact reasoning about
transcendental functions is often not necessary and a sound approximate
approach is sufficient and efficient.
In this section, we introduce an axiomatic approach for reasoning about
programs containing transcendental functions. We observe that with the
flexibility of deductive verification and KeY itself, we can instantiate it in
two different ways. We encode transcendental functions as uninterpreted
functions and axiomatize them in the SMT queries. Alternatively, we encode
these axioms in KeY as logical inference rules.
#### 3.2.1 (A) Axiomatization in SMT
We encode library functions as uninterpreted functions and include a set of
axioms in the SMT-LIB translation for each method that is called in a
benchmark. That is, we extended KeY such that when a transcendental function
exists in the proof obligation, its definition alongside all the axioms for
that function are added to the translation.
For the axiomatization of transcendentals, we did _not_ add rules that expand
to a definition or allow a repeated approximation of the function value (like
expansion into a Taylor series). Instead, we added a number of lemmata
encoding interesting properties related to special values. For instance, the
following axiom states that if the input to the sin function is not a NaN or
infinity, then the returned value of sin is between $-1.0$ and $1.0$:
⬇
(assert (forall ((a Float64)) (=>
(and (not (fp.isNaN a)) (not (fp.isInfinite a)))
(and (fp.leq (sinDouble a) (fp #b0 #b01111111111 #b0000...000000))
(fp.geq (sinDouble a) (fp #b1 #b01111111111 #b0000...000000))))))
Note that this implies that the result is not a NaN or infinity. The other
axioms are similar in spirit, so we do not list them.
These axioms are expressed as quantified floating-point formulas and capture
high-level properties of library functions complying with the specifications
in the IEEE 754 Standard. Clearly, since we do not have the actual
implementations of these functions, we are not able to prove arbitrary
properties. However, such an axiomatization is often sufficient to check for
the (absence of) special values, i.e. NaN and infinity, as our experiments in
subsection 4.4 show.
#### 3.2.2 (B) Taclets in KeY
Reasoning about quantified formulas in SMT is a long-lasting challenge [33].
We have also observed in our experiments with only arithmetic operations
(subsection 4.3) that SMT solvers struggle with quantifiers in combination
with floating-points. We have therefore implemented an alternative approach
encoding the axioms not in the SMT queries, but instead as deductive inference
rules (so-called taclets) in KeY.
The rules encode the same logical information as the universally quantified
assertions that we add in SMT-LIB (and where we leave the choice of
instantiations entirely to the SMT/SAT solver). With our taclet approach, we
instantiate a quantifier (only) to one’s needs. We note that for proving a
property correct, this results in a correct (under)approximation. However, the
prize for achieving more closed proofs and shorter running times is that for
disproving a property, not considering all possible quantifier instantiations
may lead to spurious counterexamples, i.e., false positives.
A heuristic strategy applies the rules automatically using the occurrences of
transcendentals as instantiation triggers. However, instantiating the axioms
too eagerly, considerably increases the number of open goals, which is why we
assume that the user selects the axioms to apply manually (and did so in the
experiments). After the application the proof obligation can either be closed,
i.e proven, by KeY automatically, or be given to the SMT solver as before for
final solving.
Currently, the set of axioms (in the SMT-LIB translation and as taclets in
KeY) only contains axioms for the transcendental functions occurring in our
benchmarks. So far we have $10$ axioms; however, adding more axioms (also for
further transcendentals like exponentiation or logarithm) is straightforward.
The full set of axioms is included in the Appendix of the technical report.
## 4 Evaluation
### 4.1 Benchmark Programs
Benchmark Details Automode Statistics benchmark # classes # method calls #
arith. ops library functions # goals closed by KeY # goals to be closed
externally # rules applied automode time (s) Complex.add (2) 1 0 2 - 3 / 3 1 /
4 185 / 286 0.7 / 0.2 Complex.divide (2) 1 0 11 - 10 / 8 2 / 8 483 / 625 0.7 /
0.8 Complex.compare 1 0 2 - 3 2 216 0.2 Complex.reciprocal (2) 1 1 6 - 1 / 1 2
/ 2 402 / 406 0.4 / 0.5 Circuit.impedance 2 1 3 - 1 4 360 0.5 Circuit.current
(2) 2 3 14 - 11 / 11 4 / 1 1267 / 1238 4.0 / 4.1 Matrix2.transposedEq 1 3 3 -
3 1 735 0.9 Matrix3.transposedEq 1 4 34 - 3 1 1786 5.1 Matrix3.transposedEqV2
1 4 34 - 3 1 1796 5.4 Rectangle.scale (2) 3 + 1 23 22 - 32 / 32 32 / 16 5990 /
5617 18.4 / 14.5 Rotate.computeError 1 + 1 6 26 - 108 8 3693 74.2
Rotate.computeRelErr 1 + 1 6 28 - 120 8 3898 79.6 FPLoop.fploop 1 0 1 - 2 4 99
0.1 FPLoop.fploop2 1 0 1 - 2 4 99 0.1 FPLoop.fploop3 1 0 1 - 2 4 99 0.1
Cartesian.toPolar 2 + 1 3 6 sqrt, atan 1 4 438 0.5 Cartesian.distanceTo 1 + 1
1 5 sqrt 2 1 191 0.1 Polar.toCartesian 2 + 1 3 4 cos, sin 1 2 364 0.5
Circuit.instantCurrent 2 + 1 14 23 sqrt, atan, cos 17 2 1686 14.1
Circuit.instantVoltage 1 + 1 1 4 cos 0 2 138 0.1
Table 1: Benchmark details and KeY automode statistics, time is measured in
seconds
We collected a set of existing floating-point Java programs representing real-
world applications in order to evaluate the feasibility and performance of
KeY’s floating-point support.
The left half of Table 1 provides an overview of our benchmarks. Each
benchmark consists of one method, which is composed of arithmetic operations
and method calls to potentially other classes. The invocations of methods from
java.lang.Math (e.g. Math.abs) are marked by “+1” in Table 1; these are
resolved by inlining the method implementation. For benchmarks that contain
calls to transcendental functions and square root, the called functions are
listed; these are handled by our axiomatization. We include sqrt in this list,
as we have observed that exact support can be expensive, so it may be
advantageous to handle sqrt axiomatically. Benchmarks Rectangle, Circuit,
Matrix3 and Rotation are partially shown in Listings 1, 2, 3 and 4
respectively.
Each benchmark also includes a JML contract that is to be checked. For some
methods, we specify two contracts (marked by “(2)” in the first column of
Table 1), each serving as an independent benchmark. The contracts for most of
these benchmarks check that the methods do not return a special value i.e
infinity and/or NaN, the preconditions being that the variables are not
themselves special values and possibly are bounded in a given range. For the
Matrix, FPLoop and Rotate benchmarks, we check a _functional_ property (see
subsection 4.3). FPLoop, which has three contracts, additionally shows how to
specify floating-point loop behavior using loop invariants.
Listing 2: The Circuit.instantCurrent benchmark
⬇
public class Circuit {
double maxVoltage, frequency, resistance, inductance;
// ...
/*@ public normal_behavior
@ requires 1.0 < this.maxVoltage && this.maxVoltage < 12.0 &&
@ 1.0 < this.frequency && this.frequency < 100.0 &&
@ 1.0 < this.resistance && this.resistance < 50.0 &&
@ 0.001 < this.inductance && this.inductance < 0.004 &&
@ 0.0 < time && time < 300.0;
@ ensures !\fp_nan(\result) && !\fp_infinite(\result);
@*/
public double instantCurrent(double time) {
Complex current = computeCurrent();
double maxCurrent = Math.sqrt(current.getRealPart() * current.getRealPart() +
current.getImaginaryPart() * current.getImaginaryPart());
double theta = Math.atan(current.getImaginaryPart() / current.getRealPart());
return maxCurrent * Math.cos((2.0 * Math.PI * frequency * time) + theta);
}}
### 4.2 Proof Obligation Generation
To reason about the contract of a selected benchmark, we apply KeY, which
generates proof obligations or ‘goals’. Some of these goals (heap-related) are
closed by KeY automatically. The remaining open goals are closed by either SMT
solvers with floating-point support directly (subsection 3.1 and subsubsection
3.2.1), or with a combination of transcendental KeY taclets and floating-point
SMT solving (subsubsection 3.2.2).
Columns 6 and 7 in Table 1 show the number of proof obligations closed by KeY
directly and to be discharged by external solvers, respectively. The next two
columns show the number of taclet rules that KeY applied in order to close its
goals, and the time this takes. For benchmarks with two contracts we show the
respective values separated by ‘/’.
We run our experiments on a server with 1.5 TB memory and 4x12 CPU cores at 3
GHz. However, KeY runs single-threadedly and does not use more than 8GB of
memory.
For our set of benchmarks, the symbolic execution process is fully automated.
Note that the machinery can deal with loop invariants, if they are provided.
Loop invariant generation is, however, particularly challenging for floating-
points due to roundoff errors [26, 39], and a research topic in itself.
### 4.3 Evaluation of SMT Floating-Point Support
Previous work [31] reported that SMT support for floating-point arithmetic is
rather limited. However, with recent advances [17], we evaluate the situation
again. Most benchmarks used to evaluate SMT solvers’ decision procedures [1]
aim to check (individual) specialized (corner case) properties of floating-
point arithmetic. The proof obligations generated from our set of benchmarks
are complementary in that they are more arithmetic heavy, while nonetheless
relying on accurate reasoning about special values and functional properties.
For each open goal not automatically closed, KeY generates one SMT-LIB file
that is fed to the solvers for validation. We compare the performance of the
three major SMT solvers with floating-point support CVC4 [7] (version 1.8,
with the SymFPU library [17] enabled), Z3 (4.8.9) [27] and MathSAT (5.6.3)
[21]. For this we set a timeout of 300s for each proof obligation. While KeY
is able to discharge proof obligations in parallel, for our experiments, we do
so sequentially to maintain comparability.
KeY’s default translation to SMT includes quantifiers. These quantifications
are not related to floating-point arithmetic, but are used to logically encode
important properties of the Java memory model, like the type hierarchy and the
absence of dangling references on any valid Java heap. If we reason about
floating-point problems in isolation, they are not needed, but if we want to
consider Java verification more holistically with questions combining aspects
of heap and floating point reasoning, they become essential. We manually
inspected that the proof obligations without our axiomatized treatment of
transcendental functions do not depend on these properties and investigate the
quantifier support by including or removing them from the SMT translations. We
do not report results with quantifiers for MathSAT, since it does not support
them.
index experiment quantified axioms # goals CVC4 Z3 MathSAT # goals decided
avg. # goals decided avg. # goals decided avg. 1 valid contracts ✓ 80 79 4.1
25 18.4 - - 2 ✗ 80 79 4.0 52 35.0 80 8.8 3 invalid contracts ✓ 9 0 3.4 0 3.4 -
- 4 ✗ 9 8 36.7 7 27.6 9 3.9 5 axioms in SMT ✓ 10 9 33.2 4 63.4 - - 6 axioms as
taclets ✗ 10 10 33.4 5 74.2 8 0.9 7 fp.sqrt ✗ 7 7 46.2 1 23.5 5 0.4 8
axiomatized sqrt ✗ 7 5 2.4 5 282.8 5 5.7
Table 2: Summary of valid / invalid goals correctly decided and average
running times of each solver for the SMT translations with and without
quantified axioms
Figure 1: Runtimes for valid goals with SMT translations _with_ quantifiers
Figure 2: Runtimes for valid goals with SMT translations _without_ quantifiers
Table 2 summarizes the results of our experiments. Column 4 shows the number
of expected valid or invalid goals for all benchmarks. For each solver we show
the number of goals that each solver can validate or invalidate, together with
the average time (in seconds) needed. The goals resulting in timeout were
excluded from the computation of the average time. Column 3 shows whether the
SMT queries include quantifiers or not.
Rows 1 and 2 of Table 2 show the results for benchmarks with valid contracts.
This experiment thus represents the common behavior of KeY, whose main goal is
to _prove_ contracts correct. Rows 3 and 4 of Table 2 demonstrate the results
for benchmarks with invalid contracts, i.e. for those we expect a
counterexample for at least one of the goals. The Appendix (Appendix 0.A)
contains the detailed results for each experiment separated by benchmark.
subsection 4.3 and Figure 2 show a more detailed view of the solvers’ running
time for the valid benchmarks. The x-axis shows the number of open goals that
are discharged by the SMT solvers, sorted by running time for each solver
individually. The $k$-th point of one graph shows the minimum running time
needed by the solver to close each of the $k$ fastest goals. Note that each
solver may have different goals which are its $k$ fastest. The y-axis shows
the time on a logarithmic scale.
We conclude that in the presence of quantified axioms and floating-point
arithmetic solvers’ performance deteriorate for both valid and invalid goals.
In particular, none of the solvers is able to find counterexamples for any of
the invalid goals. However, when the quantified axioms are removed from the
SMT translations, their performance improves. For valid contracts, CVC4 and
MathSAT perform better than Z3, in terms of both number of goals validated and
the running time per goal. In particular, MathSAT is able to prove all goals.
However, the running time performance of CVC4 is better than MathSAT’s. For
invalid contracts, solvers are able to produce the expected counterexamples at
least partially. Particularly, MathSAT has a better performance than CVC4 and
Z3 in terms of both running time and the number of proof obligations for which
it can produce counterexamples.
We conducted another experiment on our Rectangle.scale benchmark to assess the
solvers’ sensitivity to various changes, applied to the benchmark’s contract
or its implementation. We considered modifications such as reducing the number
of classes while keeping the same functionality, having tighter and larger
bounds for variables, reducing the number of arithmetic operations etc. The
details of this experiment can be found in the Appendix of the technical
report. In summary, solvers’ performance seems to be sensitive to slight
innocuous looking changes such as the number of classes involved and variable
bounds. For example, constraining arg2 in the original benchmark more tightly
allows CVC4 to validate all goals (1 more). This behavior could be potentially
exploited by e.g. relaxing a variable’s bounds.
Listing 3: The Matrix3 benchmark
⬇
public class Matrix3 {
double a, b, c, d, e, f, g, h, i; //The matrix: [[a b c],[d e f],[g h i]]
double det;
// method transpose not shown
double determinant() {
return (a * e * i + b * f * g + c * d * h) -
(c * e * g + b * d * i + a * f * h);
}
double determinantNew() {
return (a * (e * i) + (g * (b * f) + c * (d * h))) -
(e * (c * g) + (i * (b * d) + a * (f * h)));
}
/*@ ensures \fp_normal(\result) ==> (\result == det); @*/
double transposedEq() {
det = determinant();
return transpose().determinant();
}
/*@ ensures \fp_normal(\result) ==> (\result == det); @*/
double transposedEqV2() {
det = determinantNew();
return transpose().determinantNew();
}
}
Listing 4: The Rotation benchmark
⬇
public class Rotation {
final static double cos90 = 6.123233995736766E-17;
final static double sin90 = 1.0;
// rotates a 2D vector by 90 degrees
public static double[] rotate(double[] vec) {
double x = vec[0] * cos90 - vec[1] * sin90;
double y = vec[0] * sin90 + vec[1] * cos90;
return new double[]{x, y};
}
/*@ requires (\forall int i; 0 <= i && i < vec.length;
@ \fp_nice(vec[i]) && vec[i] > 1.0 && vec[i] < 2.0) && vec.length == 2;
@ ensures \result[0] < 1.0E-15 && \result[1] < 1.0E-15;
*/
public static double[] computeError(double[] vec) {
double[] temp = rotate(rotate(rotate(rotate(vec))));
return new double[]{Math.abs(temp[0] - vec[0]), Math.abs(temp[1] - vec[1])};
}
}
##### Proving Functional Properties
Listings 3 and 4 show examples of functional properties that are expressible
in floating-point arithmetic and that KeY can handle. The verification results
are included in rows 1 and 2 of Table 2, for more details see the Appendix of
the technical report.
For Matrix, we check that the determinants of a matrix and its transpose are
equal. Note that this property holds trivially under real arithmetic, but not
necessarily under floating-points. After feeding transposedEq (which uses the
determinant method) and its contract to KeY, increasing the default timeout
sufficiently and discharging the created goal, CVC4 generates a counterexample
in 170.2s seconds and MathSAT in 16.2s. Z3 times out after 30 minutes. By
feeding transposedEqV2 (which uses the determinantNew method) to KeY, CVC4
validates the contract in 1.1s, MathSAT in 3.9s and Z3 times out again. One
thing worth noting is that the way programs are written can greatly influence
the computational complexity needed to reject or verify the contract. This is
evident from the fact that slightly modifying the order of operations (using
determinantNew instead) substantially reduces verification time and changes
the verification result for MathSAT and CVC4.
For Rotate, we check that the difference between an original vector and the
one that is rotated four times by 90 degrees, must not be larger than 1.0E-15.
We also verified the same bound for the relative difference (by exploiting
another method and contract) for this benchmark. The constant cos90 in 4 is
not precisely 0.0 to account for rounding effects in the computation of the
cosine. FPLoop includes three loops, for which the contracts check that the
return value is bigger than a given constant.
Though not always very fast, these examples show that verification of
functional floating-point properties is viable.
### 4.4 Evaluation of Support for Transcendental Functions in KeY
We evaluated the two approaches from Section 3.2.1 on our set of benchmarks;
rows 5 and 6 in Table 2 summarize the results. (The detailed results of these
experiments are included in the Appendix of the technical report.) Note that
both approaches are fully automated.
We conclude that the SMT solvers perform better when the axiomatization is
applied at the KeY level. When axioms for transcendental functions are added
to the SMT-LIB translation directly Z3 validates 4 out of 10 goals. With the
axiomatization at the KeY level, solvers are able to validate more goals (with
quantified formulas removed from the SMT translations), e.g. Z3 is able to
validate 5 goals and CVC4 can validate all. Therefore, it is preferable to
apply them on the KeY side via taclet rules.
All the solvers we have used in this work comply with the IEEE 754 standard
and therefore have bit-precise support for the square root function. They
provide bit-precise reasoning by effectively encoding the behavior of
floating-point circuits over bitvectors (which is naturally expensive),
together with different heuristics and abstractions to speed up solving time.
However, depending on the property, we do not always need bit-precise
reasoning, so we propose handling the square root function with the same
taclet-based axiomatization as introduced in subsubsection 3.2.2.
To this end, we conducted an experiment on the benchmarks containing sqrt,
comparing the approach from subsubsection 3.2.2 (adding the necessary axioms,
resp. taclet rules) to using the square root implemented in SMT solvers
(fp.sqrt). We chose to include only axioms specified in or inferred from the
IEEE 754 standard (e.g. if the argument of the square root function is NaN or
less than zero, then the square root results in NaN). The full set of axioms
that we used is included in the Appendix of the technical report.
Rows 7 and 8 in Table 2 summarize the results for this experiment; the
detailed results are included in the Appendix of the technical report. We
observed that for two out of the three benchmarks, the average running time of
all solvers decreases using the axiomatized square root. Furthermore, Z3 is
able to reason about more proof obligations with the axiomatized version.
However, the success of this approach depends on the axioms added to KeY and
may not always work if we do not have suitable axioms. For example, for the
Circuit.instantCurrent benchmark (2), using the axiomatized square root, CVC4
is not able to validate the contract, but with fp.sqrt the contract is
validated.
In summary, treating sqrt axiomatically can result in shorter solving times
than performing bit-precise reasoning, but the approach may not always succeed
when the axioms are not sufficient to prove a particular property.
### 4.5 Discussion and insights
The experiments show that highly automated floating point program verification
is viable for relevant properties (handling of special values and some
functional properties), up to a certain level of complexity (given by the SMT
solvers). The choices of which parts of a proof obligation are delegated to
SMT, and how they are translated to SMT, are crucial for achieving effective
and efficient program verification. Arithmetic operations proved to be more
efficiently dealt with by delegation to SMT, whereas for transcendental
functions, axiomatization and rule based treatment in the theorem prover,
outside the SMT solver, performs clearly better.
## 5 Related Work
Our implementation uses the floating-point SMT-LIB theory [16], which however
does not handle transcendental functions, as their semantics is (library)
implementation dependent. Some real-valued automated solvers do handle
transcendental functions [32, 4], but to the best of our knowledge, the
combination of floating-points and reals in SMT solvers is still severely
limited.
None of the existing deductive verifiers support floating-point transcendental
functions automatically. The Why3 deductive verification framework [29] has
support for floating-point arithmetic, with front-ends for the C and Ada
programming languages through Frama-C [24] and SPARK [18, 31], respectively.
Why3 has back-end support for different SMT solvers, as well as interactive
proof assistants like Coq. Until recently, Why3 would discharge still many
interesting floating-point problems with help of Coq, relying on significant
user interaction. In later work [31] (in the context with floating-point
verification for Ada programs), Why3 can achieve a higher degree of
automation. Note, however, that the user is still required to add code
assertions as well as ‘ghost code’ to a significant extent.
The Boogie intermediate verification language [46] also supports floating-
point expressions, and targets Z3 for discharging proof obligations. In the
Boogie community, it was observed that writing a specification in Boogie leads
to decreases in SMT solver performance when compared to writing the goal in
SMT-LIB directly, probably due to an inherent mixing of theories when using
Boogie [2]. This matches our own experiences, and separation of theories
should be considered an important task for the further development of
floating-point verification.
Other deductive verifiers for Java have only rudimentary support for floating-
points. Verifast [40] treats floating-point operations as if they were real
values, and OpenJML [22] parses programs with floating-point operations, but
essentially treats float and double as uninterpreted sorts.
The Java category of verification competition SV-COMP [11] contains a number
of benchmarks that make use of floating-point variables. However, the focus of
these benchmarks is usually not on arithmetical properties of expressions, but
on the completeness of the Java language support. Amongst the participants of
SV-COMP 2020, the Symbolic (Java) Pathfinder (SPF) [54] (and various
extensions) and the Java Bounded Model Checker (JBMC) [23] support floating-
point arithmetic. Besides being limited to exploring the state space up to a
bounded depth, their constraint languages do not support quantifiers and
abstracting of method calls—which are features that we have used in this work.
Floating-point arithmetic has also been formalized in several interactive
theorem provers [41, 15, 30]. While one can prove intricate properties about
floating-point programs [13, 14, 37], proofs using interactive provers are to
a large part manual and require significant expertise.
Abstract interpretation based techniques can show the absence of special
values in floating-point code fully automatically, and several abstract
domains which are sound with respect to floating-point arithmetic exist [19,
42]. While the analysis itself is fully automated, applying it successfully to
real-world programs in general requires adaptation to each program analyzed by
end-users, e.g. the selection of suitable abstract domains or widening
thresholds [12].
Besides showing the absence of special values, recent research has developed
static analyses to bound floating-point roundoff errors [34, 25, 56, 47, 51].
These analyses currently work only for small arithmetic kernels and the tools
in particular do not accept programs with objects.
Dynamic analyses generally scale well on real-world programs, but can only
identify bugs (when given failure-triggering input), rather than proving
correctness for _all_ possible inputs. Executing a floating-point program
together with a higher-precision one allows one to find inputs which cause
large roundoff errors [10, 20, 43]. Ariadne [6] uses a combination of symbolic
execution, real-valued SMT solving and testing to find inputs that trigger
floating-point exceptions, including overflow and invalid operations. Our work
subsumes this approach as the SMT solvers that we use can directly generate
counterexamples, but more importantly, KeY is able to prove the absence of
such exceptions.
## 6 Conclusion
By joining the forces of rule-based deduction and SAT-based SMT solving, we
presented the first working floating-point support in a deductive verification
tool for Java and by that close a remaining gap in KeY to now support full
sequential Java. Our evaluation shows that for specifications dealing with
value ranges and absence of NaN and infinity, our approach can verify
realistic programs within a reasonable time frame. We observe that the MathSAT
and CVC4 solver’s floating-point support scales sufficiently for our
benchmarks, as long as the queries do not include any quantifiers, and that
our axiomatized approach for handling transcendental functions is best
realized using calculus rules in KeY’s internal reasoning engine. While our
work is implemented within the KeY verifier, we expect our approach to be
portable to other verifiers.
### Acknowledgements
This research was partially funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) project 387674182. The authors would like to
thank Daniel Eddeland, who together with co-author W. Ahrendt performed
prestudies which impacted the current work.
## References
* [1] QF_FP SMT benchmarks. https://clc-gitlab.cs.uiowa.edu:2443/SMT-LIB-benchmarks/QF_FP (2019)
* [2] Slow verification of programs combining multiple floating point values (Github issue) (2019 (accessed May 11, 2020)), https://github.com/boogie-org/boogie/issues/109
* [3] Ahrendt, W., Beckert, B., Bubel, R., Hähnle, R., Schmitt, P.H., Ulbrich, M. (eds.): Deductive Software Verification - The KeY Book - From Theory to Practice, LNCS, vol. 10001. Springer (2016)
* [4] Akbarpour, B., Paulson, L.C.: MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions. Journal of Automated Reasoning 44(3) (2010)
* [5] Astrauskas, V., Müller, P., Poli, F., Summers, A.J.: Leveraging Rust Types for Modular Specification and Verification. In: Object-Oriented Programming Systems, Languages, and Applications (OOPSLA) (2019)
* [6] Barr, E.T., Vo, T., Le, V., Su, Z.: Automatic Detection of Floating-point Exceptions. In: Principles of Programming Languages (POPL) (2013)
* [7] Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanovi’c, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Computer Aided Verification (CAV) (2011), snowbird, Utah
* [8] Barrett, C., Stump, A., Tinelli, C., et al.: The SMT-LIB Standard: Version 2.0. In: Proceedings of the 8th International Workshop on Satisfiability Modulo Theories (2010)
* [9] Beckert, B., Nestler, B., Kiefer, M., Selzer, M., Ulbrich, M.: Experience Report: Formal Methods in Material Science. CoRR abs/1802.02374 (2018)
* [10] Benz, F., Hildebrandt, A., Hack, S.: A Dynamic Program Analysis to Find Floating-Point Accuracy Problems. In: Programming Language Design and Implementation (PLDI) (2012)
* [11] Beyer, D.: Advances in automatic software verification: Sv-comp 2020. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2020)
* [12] Blanchet, B., Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: A Static Analyzer for Large Safety-Critical Software. In: Programming Language Design and Implementation (PLDI) (2003)
* [13] Boldo, S., Clément, F., Filliâtre, J.C., Mayero, M., Melquiond, G., Weis, P.: Wave Equation Numerical Resolution: A Comprehensive Mechanized Proof of a C Program. Journal of Automated Reasoning 50(4) (2013)
* [14] Boldo, S., Filliâtre, J.C., Melquiond, G.: Combining Coq and Gappa for Certifying Floating-Point Programs. In: Intelligent Computer Mathematics (2009)
* [15] Boldo, S., Melquiond, G.: Flocq: A Unified Library for Proving Floating-Point Algorithms in Coq. In: IEEE Symposium on Computer Arithmetic (ARITH) (2011)
* [16] Brain, M., Tinelli, C., Rümmer, P., Wahl, T.: An Automatable Formal Semantics for IEEE-754 Floating-Point Arithmetic. In: IEEE Symposium on Computer Arithmetic (ARITH) (2015)
* [17] Brain, M., Schanda, F., Sun, Y.: Building Better Bit-Blasting for Floating-Point Problems. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2019)
* [18] Chapman, R., Schanda, F.: Are We There Yet? 20 Years of Industrial Theorem Proving with SPARK. In: Interactive Theorem Proving (ITP) (2014)
* [19] Chen, L., Miné, A., Cousot, P.: A Sound Floating-Point Polyhedra Abstract Domain. In: Asian Symposium on Programming Languages and Systems (APLAS) (2008)
* [20] Chiang, W.F., Gopalakrishnan, G., Rakamaric, Z., Solovyev, A.: Efficient Search for Inputs Causing High Floating-point Errors. In: Principles and Practice of Parallel Programming (PPoPP) (2014)
* [21] Cimatti, A., Griggio, A., Schaafsma, B., Sebastiani, R.: The MathSAT5 SMT Solver. In: Proceedings of Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2013)
* [22] Cok, D.R.: OpenJML: JML for Java 7 by extending OpenJDK. In: NASA Formal Methods (2011)
* [23] Cordeiro, L.C., Kesseli, P., Kroening, D., Schrammel, P., Trtík, M.: JBMC: A Bounded Model Checking Tool for Verifying Java Bytecode. In: Computer Aided Verification (CAV) (2018)
* [24] Cuoq, P., Kirchner, F., Kosmatov, N., Prevosto, V., Signoles, J., Yakobowski, B.: Frama-C. In: Software Engineering and Formal Methods (SEFM) (2012)
* [25] Darulova, E., Izycheva, A., Nasir, F., Ritter, F., Becker, H., Bastian, R.: Daisy - Framework for Analysis and Optimization of Numerical Programs. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2018)
* [26] Darulova, E., Kuncak, V.: Towards a Compiler for Reals. TOPLAS 39(2) (2017)
* [27] De Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2008)
* [28] Eilers, M., Müller, P.: Nagini: A Static Verifier for Python. In: Computer Aided Verification (CAV) (2018)
* [29] Filliâtre, J.C., Paskevich, A.: Why3 — Where Programs Meet Provers. In: European Symposium on Programming (ESOP) (2013)
* [30] Fox, A., Harrison, J., Akbarpour, B.: A Formal Model of IEEE Floating Point Arithmetic. HOL4 Theorem Prover Library (2017), https://github.com/HOL-Theorem-Prover/HOL/tree/master/src/floating-point
* [31] Fumex, C., Marché, C., Moy, Y.: Automating the Verification of Floating-Point Programs. In: Verified Software: Theories, Tools, and Experiments (VSTTE) (2017)
* [32] Gao, S., Kong, S., Clarke, E.M.: dReal: An SMT Solver for Nonlinear Theories over the Reals. In: Automated Deduction – CADE-24 (2013)
* [33] Ge, Y., de Moura, L.: Complete Instantiation for Quantified Formulas in Satisfiabiliby Modulo Theories. In: Computer Aided Verification (CAV) (2009)
* [34] Goubault, E., Putot, S.: Static Analysis of Finite Precision Computations. In: Verification, Model Checking, and Abstract Interpretation (VMCAI) (2011)
* [35] Goubault, E., Putot, S.: Robustness Analysis of Finite Precision Implementations. In: Asian Symposium on Programming Languages and Systems (APLAS) (2013)
* [36] Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. In: Handbook of Philosophical Logic, pp. 99–217. Springer (2001)
* [37] Harrison, J.: Floating Point Verification in HOL Light: The Exponential Function. Formal Methods in System Design 16(3) (2000)
* [38] IEEE, C.S.: IEEE Standard for Floating-Point Arithmetic. IEEE Std 754-2008 (2008)
* [39] Izycheva, A., Darulova, E., Seidl, H.: Counterexample and Simulation-Guided Floating-Point Loop Invariant Synthesis. In: Static Analysis Symposium (SAS) (2020)
* [40] Jacobs, B., Smans, J., Philippaerts, P., Vogels, F., Penninckx, W., Piessens, F.: VeriFast: A Powerful, Sound, Predictable, Fast Verifier for C and Java. In: NASA Formal Methods (NFM) (2011)
* [41] Jacobsen, C., Solovyev, A., Gopalakrishnan, G.: A Parameterized Floating-Point Formalizaton in HOL Light. Electronic Notes in Theoretical Computer Science 317 (2015)
* [42] Jeannet, B., Miné, A.: Apron: A Library of Numerical Abstract Domains for Static Analysis. In: Computer Aided Verification (CAV) (2009)
* [43] Lam, M.O., Hollingsworth, J.K., Stewart, G.W.: Dynamic Floating-point Cancellation Detection. Parallel Comput. 39(3) (2013)
* [44] Leavens, G.T., Baker, A.L., Ruby, C.: Preliminary design of JML: A behavioral interface specification language for Java. ACM SIGSOFT Software Engineering Notes 31(3) (2006)
* [45] Leavens, G.T., Cheon, Y.: Design by Contract with JML (2006), http://www.jmlspecs.org/jmldbc.pdf
* [46] Leino, K.R.M.: This is Boogie 2 (June 2008), https://www.microsoft.com/en-us/research/publication/this-is-boogie-2-2/
* [47] Magron, V., Constantinides, G., Donaldson, A.: Certified Roundoff Error Bounds Using Semidefinite Programming. ACM Trans. Math. Softw. 43(4) (2017)
* [48] Marché, C., Paulin-Mohring, C., Urbain, X.: The KRAKATOA tool for certification of Java/JavaCard programs annotated in JML. The Journal of Logic and Algebraic Programming 58(1) (2004)
* [49] McCormick, J.W., Chapin, P.C.: Building High Integrity Applications with SPARK. Cambridge University Press (2015)
* [50] Meyer, B.: Applying “Design by Contract”. Computer 25(10) (1992)
* [51] Moscato, M., Titolo, L., Dutle, A., Muñoz, C.: Automatic Estimation of Verified Floating-Point Round-Off Errors via Static Analysis. In: SAFECOMP (2017)
* [52] Muller, J., Brisebarre, N., de Dinechin, F., Jeannerod, C., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of Floating-Point Arithmetic. Birkhäuser (2010)
* [53] Müller, P., Schwerhoff, M., Summers, A.J.: Viper: A Verification Infrastructure for Permission-Based Reasoning. In: Verification, Model Checking, and Abstract Interpretation (VMCAI) (2016)
* [54] Pasareanu, C.S., Mehlitz, P.C., Bushnell, D.H., Gundy-Burlet, K., Lowry, M.R., Person, S., Pape, M.: Combining unit-level symbolic execution and system-level concrete execution for testing NASA software. In: International Symposium on Software Testing and Analysis (ISSTA) (2008)
* [55] Siegel, S.F., Mironova, A., Avrunin, G.S., Clarke, L.A.: Using Model Checking with Symbolic Execution to Verify Parallel Numerical Programs. In: International Symposium on Software Testing and Analysis (ISSTA) (2006)
* [56] Solovyev, A., Jacobsen, C., Rakamaric, Z., Gopalakrishnan, G.: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions. In: Formal Methods (FM) (2015)
## Appendix 0.A Appendix
### 0.A.1 Axioms for Transcendental Functions in KeY
Here we present the axioms that we implemented to prove properties for
benchmarks with transcendental functions:
* •
If arg is NaN or an infinity, then sin(arg) is NaN.
* •
If arg is zero, then the result of sin(arg) is a zero with the same sign as
arg.
* •
if arg is not NaN or infinity, then the returned value of sin is between
$-1.0$ and $1.0$.
* •
if arg is not NaN or infinity, then the returned value of sin is not NaN.
* •
If arg is NaN or an infinity, then cos(arg) is NaN.
* •
if arg is not NaN or infinity, then the returned value of cos is between
$-1.0$ and $1.0$.
* •
if arg is not NaN or infinity, then the returned value of cos is not NaN.
* •
If arg is NaN or an infinity, then atan(arg) is NaN.
* •
If arg is zero, then the result of atan(arg) is a zero with the same sign as
arg.
* •
if arg is not NaN, then the returned value of atan is between $-\pi/2$ and
$\pi/2$.
In our Evaluation we showed that handling square root axiomatically can
improve performance. Here is the list of axioms we used for this function:
* •
If arg is NaN or less than zero, then sqrt(arg) is NaN.
* •
If arg is positive infinity, then sqrt(arg) is positive infinity.
* •
If arg is positive zero or negative zero, then sqrt(arg) is the same as arg.
* •
If arg is not NaN and greater or equal to zero, then sqrt(arg) is not NaN.
* •
If arg is not infinity and is greater than one then sqrt(arg) $<$ arg.
### 0.A.2 Detailed Evaluation Results
Here we present the tables that did not fit in the main body of the paper and
contain detailed results of our experiments. In each table we show the number
of goals per benchmark that each solver can validate or invalidate, together
with the average and maximum time (in seconds) needed. ‘TO’ in the maximum
column denotes that at least one goal timed out. The goals resulting in
timeout were excluded from the computation of the average time.
Table 3 shows the results for benchmarks with valid contracts with the
quantified formulas included in the SMT translations. We have summarized this
table in row 1 of Table 2.
benchmark | # goals | CVC4 | Z3
---|---|---|---
| # goals
---
proven
avg. | max. | | # goals
---
proven
avg. | max.
Complex.add(1) | 1 | 1 | 0.6 | 0.6 | 1 | 1.6 | 1.6
Complex.divide(1) | 2 | 2 | 1.7 | 1.9 | 2 | 1.8 | 2.3
Complex.divide(2) | 8 | 8 | 3.4 | 11.5 | 4 | 3.2 | TO
Complex.compare | 2 | 2 | 0.5 | 0.6 | 2 | 1.5 | 1.5
Complex.reciprocal(1) | 2 | 2 | 1.0 | 1.7 | 0 | - | TO
Complex.reciprocal(2) | 2 | 2 | 1.8 | 2.4 | 2 | 2.7 | 4.0
Circuit.impedance | 4 | 4 | 0.8 | 0.9 | 3 | 87.5 | TO
Circuit.current(1) | 4 | 4 | 6.3 | 13.2 | 0 | - | TO
Circuit.current(2) | 1 | 1 | 5.5 | 5.5 | 0 | - | TO
Matrix2.transposedEq | 1 | 1 | 0.5 | 0.5 | 0 | - | TO
Matrix3.transposedEqV2 | 1 | 1 | 1.3 | 1.3 | 0 | - | TO
Rectangle.scale(1) | 32 | 31 | 2.2 | TO | 7 | 46.6 | TO
Rotate.computeError | 8 | 8 | 9.8 | 13.9 | 0 | - | TO
Rotate.computeRelErr | 8 | 8 | 25.3 | 45.6 | 0 | - | TO
FPLoop.fploop | 4 | 4 | 0.5 | 1.0 | 4 | 2.3 | 8.1
Table 3: Summary of valid goals proved and running times of each solver for
the SMT translations _with_ quantified axioms
Table 4 demonstrates the results for the same benchmarks when the quantified
axioms are removed form the SMT translations which is summarized in row 2 of
Table 2.
benchmark # goals CVC4 Z3 MathSAT # goals proven avg. max. # goals proven avg.
max. # goals proven avg. max. Complex.add(1) 1 1 0.5 0.5 1 1.3 1.3 1 0.2 0.2
Complex.divide(1) 2 2 1.6 1.9 2 2.1 2.7 2 1.5 1.9 Complex.divide(2) 8 8 3.4
11.4 4 2.5 TO 8 29.0 209.5 Complex.compare 2 2 0.5 0.6 2 1.1 1.3 2 0.2 0.2
Complex.reciprocal(1) 2 2 0.9 1.5 1 198.5 TO 2 2.2 2.4 Complex.reciprocal(2) 2
2 1.7 2.3 2 2.8 3.4 2 1.5 1.9 Circuit.impedance 4 4 0.7 0.7 4 9.5 17.1 4 0.4
0.5 Circuit.current(1) 4 4 6.3 13.0 0 - TO 4 13.3 36.2 Circuit.current(2) 1 1
5.2 5.2 0 - TO 1 26.3 26.3 Matrix2.transposedEq 1 1 0.5 0.5 0 - TO 1 0.6 0.6
Matrix3.transposedEqV2 1 1 1.1 1.1 0 - TO 1 3.9 3.9 Rectangle.scale(1) 32 31
2.1 TO 32 95.1 251.8 32 2.4 4.2 Rotate.computeError 8 8 9.7 13.5 0 - TO 8 22.7
35.7 Rotate.computeRelErr 8 8 25.0 45.0 0 - TO 8 27.5 46.4 FPLoop.fploop 4 4
0.5 1.1 4 2.0 7.2 4 0.4 1.0
Table 4: Summary of valid goals proved and running times of each solver for
the SMT translations _without_ quantified axioms
Table 5 shows the detailed results of the experiments with benchmarks with
invalid contracts, when the quantified formulas are included in and removed
form the SMT translations. This results are summarized in rows 3 and 4 of
Table 2.
benchmark # goals CVC4 Z3 MathSAT # goals avg. max. # goals avg. max. # goals
avg. max. valid invalid valid invalid valid invalid valid invalid with
quantified axioms Matrix3.transposedEq 0 1 0 0 - TO 0 0 - TO - - - -
Rectangle.scale(2) 12 4 12 0 12.2 TO 8 0 4.6 TO - - - - Complex.add(2) 2 2 2 0
0.6 0.7 2 0 1.4 TO - - - - FPLoop.fploop2 3 1 3 0 0.9 1.7 3 0 0.5 TO - - - -
FPLoop.fploop3 3 1 3 0 0.4 1.7 3 0 0.3 TO - - - - without quantified axioms
Matrix3.transposedEq 0 1 0 1 170.2 170.2 0 0 - TO 0 1 16.2 16.2
Rectangle.scale(2) 12 4 12 3 12.2 TO 12 3 108.2 TO 12 4 2.4 9.5 Complex.add(2)
2 2 2 2 0.5 0.5 2 2 0.7 1.0 2 2 0.2 0.2 FPLoop.fploop2 3 1 3 1 0.4 0.6 3 1 0.9
1.7 3 1 0.3 0.5 FPLoop.fploop3 3 1 3 1 0.3 0.6 3 1 0.6 1.7 3 1 0.2 0.4
Table 5: Summary of invalid goals proved and running times of each solver for
the SMT translations with and without quantified axioms
benchmark # goals CVC4 Z3 MathSAT # goals validated avg. max. # goals
validated avg. max. # goals validated avg. max. _fp.sqrt_ Cartesian.toPolar 4
4 6.9 7.5 1 23.5 TO 4 1.2 1.7 Cartesian.distanceTo 1 1 8.2 8.2 0 - TO 1 1.0
1.0 Circuit.instantCurrent 2 2 123.5 127.5 0 - TO 0 - TO _axiomatized sqrt_
Cartesian.toPolar 4 4 2.0 2.9 4 49.81 163.0 4 1.0 1.6 Cartesian.distanceTo 1 1
2.7 2.7 1 233.0 233.0 1 1.0 1.0 Circuit.instantCurrent 2 0 - TO 0 - TO 0 (2
CE) 11.1 13.8
Table 6: Summary statistics for benchmarks containing the square root
function, with quantified formulas removed from the SMT-LIB translation
The first two sections of Table 7 show the results from applying the two
approaches for handling transcendental functions in sections 3.2.1 and 3.2.2,
using the default SMT translation in KeY. The last section of the table
depicts the results of applying the approach in subsubsection 3.2.2, while the
quantified formulas are removed from the SMT translations. This table is
summarized in rows 5 and 6 of Table 2
benchmark # goals CVC4 Z3 MathSAT # goals validated avg. max. # goals
validated avg. max. # goals validated avg. max. _axioms in SMT-LIB
translation_ Cartesian.toPolar 4 4 7.1 9.2 1 16.8 TO - - - Polar.toCartesian 2
2 0.9 0.9 2 69.7 95.7 - - - Circuit.instantCurrent 2 1 123.6 TO 0 - TO - - -
Circuit.instantVoltage 2 2 1.1 1.1 1 103.8 TO - - - _axioms as taclet rules in
KeY with quantified formulas_ Cartesian.toPolar 4 4 6.8 7.7 1 40.0 TO - - -
Polar.toCartesian 2 2 1.4 1.9 1 288.0 TO - - - Circuit.instantCurrent 2 2
123.8 128.3 0 - TO - - - Circuit.instantVoltage 2 2 1.3 1.3 0 - TO - - -
_axioms as taclet rules in KeY without quantified formulas_ Cartesian.toPolar
4 4 6.9 7.5 1 23.5 TO 4 1.2 1.7 Polar.toCartesian 2 2 1.5 2.3 2 52.6 81.2 2
0.6 0.8 Circuit.instantCurrent 2 2 123.5 127.5 0 - TO 0 - TO
Circuit.instantVoltage 2 2 1.5 1.7 2 146.4 160.7 2 0.8 0.8
Table 7: Summary statistics with axioms in SMT-LIB translations and as taclet
rules in KeY
Table 6 shows the detailed results of conducting the experiment on the
benchmarks containing sqrt, comparing the approach from subsubsection 3.2.2
(adding the necessary axioms, resp. taclet rules) to using the square root
implemented in SMT solvers (fp.sqrt), when the quantified formulas are removed
from the SMT translations. We have summarized the results of these experiments
in rows 7 and 8 of Table 2.
#### 0.A.2.1 Sensitivity to Contract Variations
We conducted an experiment on our Rectangle.scale benchmark to assess the
solver’s sensitivity to various changes, applied to the benchmark’s contract
or its implementation. We considered the following modifications:
* •
$v0$: is the original version of the benchmark (1 using the second contract)
and our baseline;
* •
$v1$: reduces the number of classes involved to two, while keeping the same
functionality;
* •
$v2$: reduces the number of classes involved to one, while keeping the same
functionality;
* •
$v3$: modifies $v2$ such that variable bounds in the precondition become more
“complicated” in terms of longer fractional parts (e.g. the bounds for arg2
become [3.0000001, -6.4000000003] instead of [3.0001, -6.4000003]);
* •
$v4$: simplifies the mathematical expression of $v2$ (less arithmetic
operations)
* •
$v5$: modifies $v3$ such that arg2 has a tighter bound, i.e. the interval
width is smaller
* •
$v6$: modifies $v2$ such that arg2 has a larger bound, i.e. the interval width
is larger
* •
$v7$: modifies $v2$ such that only arg2 has a “complicated” bound
* •
$v8$: modifies $v0$ such that arg2 has a tighter bound
Table 8 summarizes the results for this experiment. With the quantified
forulas included in the SMT translation, Both CVC4 and Z3 are able to prove
more goals when the number of classes is reduced, and also when the number of
arithmetic operations is reduced. Z3 further seems to be sensitive to whether
variable bounds are “complicated” or not, whereas CVC4 is not. We obtain a
somewhat surprising result when arg2 has a tighter bound. While Z3’s
performance improves, CVC4 validates two goals less. On the other hand,
increasing the bounds on arg2 does not seem to make a difference.
It seems that arg2 is the bottleneck for this benchmark; when only arg2 has a
“complicated” input interval, CVC4 proves less goals. Finally, constraining
arg2 in the original benchmark more tightly allows CVC4 to validate all goals
but Z3’s performance remains unaffected.
With the quantified formulas removed from the SMT-LIB translation, we can see
that CVC4’s results, in terms of number of goals validated, are the same as
before, and Z3 performs much better than before. MathSAT is able to validate
all goals of all versions.
In summary, solvers’ performance seems to be sensitive to slight innocuous
looking changes such as the number of classes involved and variable bounds.
This behavior could be potentially exploited by e.g. relaxing a variable’s
bounds.
version applied change # goals CVC4 Z3 # goals validated avg. max. # goals
validated avg. max. v0 none (original) 32 31 2.3 TO 7 46.2 TO v1 fewer classes
(2) in v0 32 32 2.5 4.8 9 42.7 TO v2 fewer classes (1) in v0 32 32 2.4 4.2 7
85.2 TO v3 complicated intervals for all vars in v2 32 32 2.5 5.5 6 59.3 TO v4
simpler math in v2 16 16 1.0 1.3 12 35.3 TO v5 shorter interval for arg2 in v3
32 30 2.3 TO 9 95.1 TO v6 longer interval for arg2 in v2 32 32 2.6 7.4 7 14.8
TO v7 complicated interval for arg2 in v2 32 31 2.4 TO 7 23.9 TO v8 shorter
interval for arg2 in v0 32 32 2.5 4.2 7 46.5 TO
Table 8: SMT solvers summary statistics for various versions of the Rectangle
benchmark with quantified axioms in the SMT translations
|
11institutetext: TU Dortmund University, Germany
11email<EMAIL_ADDRESS>22institutetext:
Universität zu Lübeck, Germany
22email<EMAIL_ADDRESS>University of Zurich,
Switzerland
33email<EMAIL_ADDRESS>44institutetext: Ruhr University Bochum,
Germany
44email<EMAIL_ADDRESS>
# Work-sensitive Dynamic Complexity of Formal Languages
Jonas Schmidt 11 Thomas Schwentick 11 Till Tantau 22 Nils Vortmeier 33 Thomas
Zeume 44
###### Abstract
Which amount of parallel resources is needed for updating a query result after
changing an input? In this work we study the amount of work required for
dynamically answering membership and range queries for formal languages in
parallel constant time with polynomially many processors. As a prerequisite,
we propose a framework for specifying dynamic, parallel, constant-time
programs that require small amounts of work. This framework is based on the
dynamic descriptive complexity framework by Patnaik and Immerman.
###### Keywords:
Dynamic complexity work parallel constant time.
## 1 Introduction
Which amount of parallel resources is needed for updating a query result after
changing an input, in particular if we only want to spend constant parallel
time?
In classical, non-dynamic computations, parallel constant time is well
understood. Constant time on CRAMs, a variant of CRCW-PRAMs used by Immerman
[16], corresponds to constant-depth in circuits, so, to the circuit class
$\textsf{AC}^{0}$, as well as to expressibility in first-order logic with
built-in arithmetic (see, for instance, the books of Immerman [16, Theorem
5.2] and Vollmer [28, Theorems 4.69 and 4.73]). Even more, the amount of work,
that is, the overall number of operations of all processors, is connected to
the number of variables required by a first-order formula [16, Theorem 5.10].
However, the work aspect of constant parallel time algorithms is less
understood for scenarios where the input is subject to changes. To the best of
our knowledge, there is only little previous work on constant-time PRAMs in
dynamic scenarios. A notable exception is early work showing that spanning
trees and connected components can be computed in constant time by CRCW-PRAMs
with $O(n^{4})$ and $O(n^{2})$ processors, respectively [26].
In an orthogonal line of research, parallel dynamic constant time has been
studied from a logical perspective in the dynamic complexity framework by
Patnaik and Immerman [23] and Dong, Su, and Topor [7, 6]. In this framework,
the update of query results after a change is expressed by first-order
formulas. The formulas may refer to auxiliary relations, whose updates in turn
are also specified by first-order formulas (see Section 3 for more details).
The queries maintainable in this fashion constitute the dynamic complexity
class DynFO. Such queries can be updated by PRAMs in constant time with a
polynomial number of processors. In this line of work, the main focus in
recent years has been on proving that queries are in DynFO, and thus
emphasised the constant time aspect. It has, for instance, been shown that all
context-free languages [12] and the reachability query [5] are in DynFO.
However, if one tries to make the “DynFO approach” for dynamic problems
relevant for practical considerations, the work that is needed to carry out
the specified updates, hence the _work_ of a parallel algorithm implementing
them, is a crucial factor. The current general polynomial upper bounds are too
coarse. In this paper, we therefore initiate the investigation of more work-
efficient dynamic programs that can be specified by first-order logic and that
can therefore be carried out by PRAMs in constant time. To do so, we propose a
framework for specifying such dynamic, parallel, constant-time programs, which
is based on the DynFO framework, but allows for more precise (and better)
bounds on the necessary work of a program.
###### Goal 1.1
Extend the formal framework of dynamic complexity towards the consideration of
parallel work.
Towards this goal, we link the framework we propose to the CRAM framework in
Section 3. In fact, the new framework also takes a somewhat wider perspective,
since it does not focus exclusively at one query under a set of change
operations, but rather considers dynamic problems that may have several change
and query operations (and could even have operations that combine the two).
Therefore, from now on we speak about dynamic problems and not about (single)
queries.
###### Goal 1.2
Find work-efficient DynFO-programs for dynamic problems that are known to be
in DynFO (but whose dynamic programs111In the field of dynamic complexity the
term “dynamic program” is traditionally used for the programs for updating the
auxiliary data after a change. The term should not be confused with the
“dynamic programming” technique used in algorithm design. are not competitive,
work-wise).
Ideally we aim at showing that dynamic problems can be maintained in DynFO
with sublinear or even polylogarithmic work. One line of attack for this goal
is to study dynamic algorithms and to see whether they can be transformed into
parallel $\mathcal{O}(1)$-time algorithms with small work. There is a plethora
of work that achieves polylogarithmic sequential update time (even though,
sometimes only amortised), see for instance [3, 9, 13, 14]. For many of these
problems, it is known that they can be maintained in constant parallel time
with polynomial work, e.g. as mentioned above, it has been shown that
connectivity and maintenance of regular (and even context-free) languages is
in DynFO.
In this paper, we follow this approach for dynamic string problems, more
specifically, dynamic problems that allow membership and range queries for
regular and context-free languages. Our results can be summarised as follows.
We show in Section 5 that regular languages can be maintained in constant time
with $\mathcal{O}(n^{\epsilon})$ work for all $\epsilon>0$ and that for star-
free languages even work $\mathcal{O}(\log n)$ can be achieved. These results
hold for range and membership queries.
For context-free languages, the situation is not as nice, as we observe in
Section 6. We show that subject to a well-known conjecture, we cannot hope for
maintaining membership in general context-free languages in DynFO with less
than $\mathcal{O}(n^{1.37-\epsilon})$ work. The same statement holds even for
the bound $\mathcal{O}(n^{2-\epsilon})$ and “combinatorial dynamic programs”.
For Dyck languages, that is, sets of well-formed strings of parentheses, we
show that this barrier does not apply. Their membership problem can be
maintained with $\mathcal{O}(n(\log n)^{3})$ work in general, and with
polylogarithmic work if there is only one kind of parentheses. By a different
approach, range queries can be maintained with work
$\mathcal{O}(n^{1+\epsilon})$ in general, and $\mathcal{O}(n^{\epsilon})$ for
one parenthesis type.
_Related work._ A complexity theory of incremental time has been developed in
[22]. We discuss previous work on dynamic complexity of formal languages in
Sections 5 and 6.
## 2 Preliminaries
Since dynamic programs are based on first-order logic, we represent inputs
like graphs and strings as well as “internal” data structures as logical
structures.
A _schema_ $\tau$ consists of a set of relation symbols and function symbols
with a corresponding arity. A constant symbol is a function symbol with arity
$0$. A _structure_ ${\mathcal{D}}$ over schema $\tau$ with finite domain $D$
has, for every $k$-ary relation symbol $R\in\tau$, a relation
$R^{\mathcal{D}}\subseteq D^{k}$, as well as a function $f^{\mathcal{D}}\colon
D^{k}\to D$ for every $k$-ary function symbol $f\in\tau$. We allow partially
defined functions and write $f^{\mathcal{D}}(\bar{a})=\bot$ if
$f^{\mathcal{D}}$ is not defined for $\bar{a}$ in ${\mathcal{D}}$. Formally,
this can be realized using an additional relation that contains the domain of
$f^{\mathcal{D}}$. We occasionally also use functions $f^{\mathcal{D}}\colon
D^{k}\to D^{\ell}$ for some $\ell>1$. Formally, such a function represents
$\ell$ functions $f^{\mathcal{D}}_{1},\ldots,f^{\mathcal{D}}_{\ell}\colon
D^{k}\to D$ with
$f^{\mathcal{D}}(\bar{a})\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;(f^{\mathcal{D}}_{1}(\bar{a}),\ldots,f^{\mathcal{D}}_{\ell}(\bar{a}))$.
Throughout this work, the structures we consider provide a linear order $\leq$
on their domain $D$. As we can thus identify $D$ with an initial sequence of
the natural numbers, we usually just assume that
$D=[n]\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\\{0,\ldots,n-1\\}$
for some natural number $n$.
We assume familiarity with first-order logic FO, and refer to [19] for basics
of Finite Model Theory. In this paper, unless stated otherwise, first-order
formulas _always_ have access to a linear order on the domain, as well as
compatible functions $+$ and $\times$ that express addition and
multiplication, respectively. This holds in particular for formulas in dynamic
programs. We use the following “if-then-else” construct: if $\varphi$ is a
formula, and $t_{1}$ and $t_{2}$ are terms, then
$\@ifmtarg{\varphi}{\textsc{ITE}}{\textsc{ITE}\text{$(\varphi,t_{1},t_{2})$}}$
is a term. Such a term evaluates to the result of $t_{1}$ if $\varphi$ is
satisfied, otherwise to $t_{2}$.
Following [12], we encode words of length (at most) $n$ over an alphabet
$\Sigma$ by _word structures_ , that is, as relational structures $W$ with
universe $\\{0,\ldots,n-1\\}$, one unary relation $R_{\sigma}$ for each symbol
$\sigma\in\Sigma$ and the canonical linear order $\leq$ on
$\\{0,\ldots,n-1\\}$. We only consider structures for which, for every
position $i$, $R_{\sigma}(i)$ holds for at most one $\sigma\in\Sigma$ and
write $W(i)=\sigma$ if $R_{\sigma}(i)$ holds and $W(i)=\epsilon$ if no such
$\sigma$ exists. We write $\text{word}(W)$ for the word represented by $W$,
that is, the concatenation $w=W(0)\circ\ldots\circ W(n-1)$. As an example, the
word structure $W_{0}$ with domain $\\{0,1,2,3\\}$, $W(1)=a$, $W(3)=b$ and
$W(0)=W(2)=\epsilon$ represents the string $ab$. We write
$\text{word}(W)[\ell,r]$ for the word $W(\ell)\circ\ldots\circ W(r)$.
Informally, a _dynamic problem_ can be seen as a data type: it consists of
some underlying structure together with a set $\Delta$ of operations. We
distinguish between _change operations_ that can modify the structure and
_query operations_ that yield information about the structure, but combined
operations could be allowed, as well. Thus, a dynamic problem is characterised
by the schema of its underlying structures and the operations that it
supports.222This view is a bit broader than the traditional setting of Dynamic
Complexity, where there can be various change operations but usually only one
fixed query is supported.
In this paper, we are particularly interested in dynamic language problems,
defined as follows. Words are represented as word structures $W$ with
elementary change operations $\textsc{set}_{\sigma}(i)$ (with the effect that
$W(i)$ becomes $\sigma$ if it was $\epsilon$ before) and $\textsc{reset}(i)$
(with the effect that $W(i)$ becomes $\epsilon$).
For some fixed language $L$ over some alphabet $\Sigma$, the dynamic problem
${\textsc{RangeMember}}(L)$ further supports one query operation
$\textsc{range}(\ell,r)$. It yields the result true, if
$\text{word}(W)[\ell,r]$ is in $L$, and otherwise false.
In the following, we denote a word structure $W$ as a sequence $w_{0}\ldots
w_{n-1}$ of letters with $w_{i}\in\Sigma\cup\\{\epsilon\\}$ in order to have
an easier, less formal notation. Altogether, the dynamic problem
${\textsc{RangeMember}}(L)$ is defined as follows.
Problem: | ${\textsc{RangeMember}}(L)$
---|---
Input: | A sequence $w=w_{0}\ldots w_{n-1}$ of letters with $w_{i}\in\Sigma\cup\\{\epsilon\\}$
Changes: | $\textsc{set}_{\sigma}(i)$ for $\sigma\in\Sigma$: Sets $w_{i}$ to $\sigma$, if $w_{i}=\epsilon$ $\textsc{reset}(i)$: Sets $w_{i}$ to $\epsilon$
Queries: | $\textsc{range}(\ell,r)$: Is $w_{\ell}\circ\cdots\circ w_{r}\in L$?
In this example, the query range maps (binary) pairs of domain elements to a
truth value and thus defines a (binary) relation over the universe of the
input word structure. We call such a query _relational_. We will also consider
_functional_ queries mapping tuples of elements to elements.
Another dynamic problem considered here is ${\textsc{Member}}(L)$ which is
defined similarly as ${\textsc{RangeMember}}(L)$ but instead of range only has
the Boolean query operation member that yields true if $w_{0}\circ\ldots\circ
w_{n-1}\in L$ holds.
## 3 Work-sensitive Dynamic Complexity
Since we are interested in the work that a dynamic program does, our
specification mechanism for dynamic programs is considerably more elaborated
than the one used in previous papers on dynamic complexity. We introduce the
mechanism in this section in two steps. First the general form of dynamic
programs and then a more pseudo-code oriented syntax. Afterwards, we discuss
how these dynamic programs translate into work-efficient constant-time
parallel programs.
### 3.1 The Dynamic Complexity Framework
Our general form of dynamic programs mainly follows [25], but is adapted to
the slightly broader view of a dynamic problem as a data type. For a more
gentle introduction to dynamic complexity, we refer to [24].
The goal of a _dynamic program_ for a dynamic problem $\Pi$ is to support all
its operations $\Delta$. To do so, it stores and updates an auxiliary
structure ${\mathcal{A}}$ over some schema $\tau_{\text{aux}}$, over the same
domain as the input structure ${\mathcal{I}}$ for $\Pi$.
A (first-order) dynamic program ${\mathcal{P}}$ consists of a set of (first-
order) _update rules_ for change operations and _query rules_ for query
operations. More precisely, a program has one query rule over schema
$\tau_{\text{aux}}$ per query operation that specifies how the (relational)
result of that operation is obtained from the auxiliary structure.
Furthermore, for each change operation $\delta\in\Delta$, it has one update
rule per auxiliary relation or function that specifies the updates after a
change based on $\delta$.
A query rule is of the form $\textbf{on query}\ Q(\bar{p})\ \textbf{yield}\
\varphi_{Q}(\bar{p}),$ where $\varphi_{Q}$ is the (first-order) _query
formula_ with free variables from $\bar{p}$.
An update rule for a $k$-ary auxiliary relation $R$ is of the form
$\textbf{on change}\ \delta(\bar{p})\ \textbf{update}\ R\ \textbf{at}\
(t_{1}(\bar{p};\bar{x}),\ldots,t_{k}(\bar{p};\bar{x}))\ \textbf{as}\
\varphi_{\delta}^{R}(\bar{p};\bar{x})\ \textbf{where}\ C(\bar{x}).$
Here, $\varphi^{R}_{\delta}$ is the (first-order) _update formula_ ,
$t_{1},\ldots,t_{k}$ are first-order terms (possibly using the ITE construct)
over $\tau_{\text{aux}}$, and $C(\bar{x})$, called a _constraint_ for the
tuple $\bar{x}=x_{1},\ldots,x_{\ell}$ of variables, is a conjunction of
inequalities $x_{i}\leq f_{i}(n)$ using functions
$f_{i}\colon\mathbb{N}\to\mathbb{N}$, where $n$ is the size of the domain and
$1\leq i\leq\ell$. We demand that all functions $f_{i}$ are first-order
definable from $+$ and $\times$.
The effect of such an update rule after a change operation $\delta(\bar{a})$
is as follows: the new relation $R^{{\mathcal{A}}^{\prime}}$ in the updated
auxiliary structure ${\mathcal{A}}^{\prime}$ contains all tuples from
$R^{\mathcal{A}}$ that are _not_ equal to
$(t_{1}(\bar{a};\bar{b}),\ldots,t_{k}(\bar{a};\bar{b}))$ for any tuple
$\bar{b}$ that satisfies the constraints $C$; and additionally
$R^{{\mathcal{A}}^{\prime}}$ contains all tuples
$(t_{1}(\bar{a};\bar{b}),\ldots,t_{k}(\bar{a};\bar{b}))$ such that $\bar{b}$
satisfies $C$ and ${\mathcal{A}}\models\varphi_{\delta}^{R}(\bar{a};\bar{b})$
holds.
Phrased more operationally, an update is performed by enumerating all tuples
$\bar{b}$ that satisfy $C$, evaluating $\varphi_{\delta}^{R}(\bar{a};\bar{b})$
on the old auxiliary structure ${\mathcal{A}}$, and depending on the result
adding the tuple $(t_{1}(\bar{a};\bar{b}),\ldots,t_{k}(\bar{a};\bar{b}))$ to
$R$ (if it was not already present), or removing that tuple from $R$ (if it
was present).
Update rules for auxiliary functions are similar, but instead of an update
formula that decides whether a tuple of the form
$(t_{1}(\bar{a};\bar{b}),\ldots,t_{k}(\bar{a};\bar{b}))$ is contained in the
updated relation, it features an update term that determines the new function
value for a function argument of the form
$(t_{1}(\bar{a};\bar{b}),\ldots,t_{k}(\bar{a};\bar{b}))$.
We say that ${\mathcal{P}}$ is a dynamic program for a dynamic problem $\Pi$
if it supports all its operations and, in particular, always yields correct
results for query operations. More precisely, if the result of applying a
query operation after a sequence $\alpha$ of change operations on an initial
structure ${\mathcal{I}}_{0}$ yields the same result as the evaluation of the
query rule on the auxiliary structure that is obtained by applying the update
rules corresponding to the change operations in $\alpha$ to an initial
auxiliary structure ${\mathcal{A}}_{0}$. Here, an initial input structure
${\mathcal{I}}_{0}$ over some domain $D$ is _empty_ , that is, it is a
structure with empty relations and with all function values being undefined
($\bot$). The initial auxiliary structure ${\mathcal{A}}_{0}$ is over the same
domain $D$ as ${\mathcal{I}}_{0}$ and is defined from ${\mathcal{I}}_{0}$ by
some FO-definable initialization.
By DynFO, we denote the class of all dynamic problems that have a dynamic
program in the sense we just defined.
### 3.2 A syntax for work-efficient dynamic programs
In this paper we are particularly interested in dynamic programs that require
little work to update the auxiliary structure after every change operation and
to compute the result of a query operation. However, since dynamic programs do
not come with an execution model, there is no direct way to define, say, when
a DynFO-programs has polylogarithmic-work, syntactically. But it is even not
clear how a _semantic_ definition could be obtained: the obvious approach to
require that the program has an implementation as a parallel program that only
needs polylogarithmic-work does not work, since it is not clear how to define
when a parallel program _implements_ a given dynamic program. This difficulty
occurs in particular if one wants to prove that some problem does _not_ have a
work-efficient dynamic program.
Since, we are not interested in lower bounds in this paper, we follow a
pragmatic approach here. We define a pseudo-code-based syntax for _update_ and
_query procedures_ that will be used in place of the update and query
_formulas_ in rules of dynamic programs. This syntax has three important
properties: (1) it is reasonably well readable (as opposed to strict first-
order logic formulas), (2) it allows a straightforward translation of rules
into proper DynFO-programs, and (3) it allows to associate a “work-bounding
function” to each rule and to translate it into a PRAM program with
$\mathcal{O}(1)$ parallel time and work bounded by this function.
The syntax of the pseudo-code has similarities with Abstract State Machines
[4] and the PRAM-syntax of [17]. For simplicity, we describe a minimal set of
syntactic elements that suffice for the dynamic programs in this paper. We
encourage readers to have a look at Section 4 for examples of update rules
with pseudo-code syntax. However, we mention some extensions that could be
considered for more complicated programs.
We only spell out a syntax for _update procedures_ that can be used in place
of the update formula $\varphi_{\delta}^{R}(\bar{p};\bar{x})$ of an update
rule
$\textbf{on change}\ \delta(\bar{p})\ \textbf{update}\ R\ \textbf{at}\
(t_{1}(\bar{p};\bar{x}),\ldots,t_{k}(\bar{p};\bar{x}))\ \textbf{as}\
\varphi_{\delta}^{R}(\bar{p};\bar{x})\ \textbf{where}\ C(\bar{x}).$
Query procedures are defined similarly, but they can not invoke any change
operations for supplementary instances, and their only free variables are from
$\bar{p}$.
We allow some compositionality: a dynamic program on some _main instance_ can
use _supplementary instances_ of other dynamic problems and invoke change or
query operations of other dynamic programs on those instances. These
supplementary instances are declared on a global level of the dynamic program
and each has an associated identifier.
Update procedures $P=P_{1};P_{2}$ consist of two parts. In the _initial
procedure_ $P_{1}$ no reference to the free variables from $\bar{x}$ are
allowed, but change operations for supplementary instances can be invoked. We
require that, for each change operation $\delta$ of the main instance and each
supplementary instance ${\mathcal{S}}$, at most one update rule for $\delta$
invokes change operations for ${\mathcal{S}}$. In general, some more
flexibility could be added, e.g., additional local instances.
In the _main procedure_ $P_{2}$, no change operations for supplementary
instances can be invoked, but references to $\bar{x}$ are allowed.
More precisely, both $P_{1}$ and $P_{2}$ can use (a series of) instructions of
the following forms:
* •
assignments $f(\bar{y})\leftarrow\texttt{term}$ of a function value,
* •
assignments $R(\bar{y})\leftarrow\texttt{condition}$ of a Boolean value,
* •
conditional branches if condition then $P^{\prime}$ else $P^{\prime\prime}$,
and
* •
parallel branches for $z\leq g(n)$ pardo $P^{\prime}$.
Semantically, here and in the following $n$ always refers to the size of the
domain of the main instance. The initial procedure $P_{1}$ can further use
change invocations $\texttt{instance}.\delta(\bar{y})$.
* •
change invocations $\texttt{instance}.\delta(\bar{y})$.
However, they are not allowed in the scope of parallel branches. And we recall
that in $P_{1}$ no variables from $\bar{x}$ can be used.
The main procedure $P_{2}$ can further use return statements return condition
or return term, but not inside parallel branches.
* •
return statements return condition or return term, but not inside parallel
branches.
Of course, initial procedures can only have initial procedures $P^{\prime}$
and $P^{\prime\prime}$ in conditional and parallel branches, and analogously
for main procedures.
Conditions and terms are defined as follows. In all cases, $\bar{y}$ denotes a
tuple of terms and $z$ is a _local variable_ , not occurring in $\bar{p}$ or
$\bar{x}$. In general, a _term_ evaluates to a domain element (or to $\bot$).
It is built from
* •
local variables and variables from $\bar{p}$ and $\bar{x}$,
* •
function symbols from $\tau_{\text{aux}}$ and previous function assignments,
* •
if-then-else terms if condition then term′ else term′′,
* •
functional queries $\texttt{instance}.Q(\bar{y})$, and
* •
expressions getUnique($z\leq g(n)$ $\mid$ condition).
For the latter expression it is required that there is always exactly one
domain element $a\leq g(n)$ satisfying condition.
A _condition_ evaluates to true or false. It may be
* •
an atomic formula with relation symbols from $\tau_{\text{aux}}$ or previous
assignments, with terms as above,
* •
an expression exists($z\leq g(n)$ $\mid$ condition),
* •
a relational query $\texttt{instance}.Q(\bar{y})$ with terms $\bar{y}$, and
* •
a Boolean combination of conditions.
All functions $g\colon\mathbb{N}\to\mathbb{N}$ in these definitions are
required to be FO-definable. For assignments of relations $R$ and functions
$f$ we demand that these symbols do _not_ appear in $\tau_{\text{aux}}$. If an
assignment with a head $f(\bar{y})$ or $R(\bar{y})$ occurs in the scope of a
parallel branch that binds variable $z$, then $z$ has to occur as a term
$y_{i}$ in $\bar{y}$. We further demand that update procedures are well-
formed, in the sense that every execution path ends with a return statement of
appropriate type.
In our pseudo-code algorithms, we display update procedures $P=P_{1};P_{2}$
with initial procedure $P_{1}$ and main procedure $P_{2}$ as
on change $\delta(\bar{p})$ with $P_{1}$
update $R$ at $(t_{1}(\bar{p},\bar{x}),\ldots,t_{k}(\bar{p},\bar{x}))$, for
all $C(\bar{x})$, by: $P_{2}$.
to emphasise that $P_{1}$ only needs to be evaluated once for the update of
$R$, and not once for every different value of $\bar{x}$.
In a nutshell, the semantics of an update rule
$\textbf{on change}\ \delta(\bar{p})\ \textbf{update}\ R\ \textbf{at}\
(t_{1}(\bar{p};\bar{x}),\ldots,t_{k}(\bar{p};\bar{x}))\ \textbf{as}\ P\
\textbf{where}\ C(\bar{x})$
is defined as in Subsection 3.1, but
${\mathcal{A}}\models\varphi_{\delta}^{R}(\bar{a},\bar{b})$ has to be replaced
by the condition that $P$ returns true under the assignment
$(\bar{p}\mapsto\bar{a};\bar{x}\mapsto\bar{b})$.
For update rules for auxiliary functions, $P$ returns the new function value
instead of a Boolean value.
Since $P_{1}$ is independent of $\bar{x}$, in the semantics, it is only
evaluated once. In particular, any change invocations are triggered only once.
We refer to the above class of dynamic update programs as Procedural-DynFO-
programs. Here and later we will introduce abbreviations as syntactic sugar,
for example the sequential loop for $z\leq m$ do $P$, where $m\in\mathbb{N}$
needs to be a fixed natural number.
We show next that update and query procedures can be translated into constant-
time CRAM programs. Since the latter can be translated into FO-formulas [15,
Theorem 5.2], therefore Procedural-DynFO-programs can be translated in DynFO-
programs.
It is not hard to see that Procedural-DynFO-programs can be transformed into
DynFO-programs, as stated in the following proposition.
###### Proposition 1
If a dynamic problem $\Pi$ can be specified by a Procedural-DynFO-program then
$\Pi\in\textsf{DynFO}$.
However, we do not need to prove this proposition, since we show next that
Procedural-DynFO-programs can be translated into constant-time CRAMs and it is
known that such programs can be translated into FO$(\leq,+,\times)$-formulas
[15, Theorem 5.2].
### 3.3 Implementing Procedural-DynFO-programs as PRAMs
We use _Parallel Random Access Machines_ (PRAMs) as the computational model to
measure the work of our dynamic programs. A PRAM consists of a number of
processors that work in parallel and use a shared memory. We only consider
_CRAMs_ , a special case of Concurrent-Read Concurrent-Write model (CRCW
PRAM), i.e. processors are allowed to read and write concurrently from and to
the same memory location, but if multiple processors concurrently write the
same memory location, then all of them need to write the same value. For an
input of size $n$ we denote the _time_ that a PRAM algorithm needs to compute
the solution as $T(n)$. The _work_ $W(n)$ of a PRAM algorithm is the sum of
the number of all computation steps of all processors made during the
computation. For further details we refer to [15, 17].
It is easy to see that Procedural-DynFO programs ${\mathcal{P}}$ can be
translated into $\mathcal{O}(1)$-time CRAM-programs ${\mathcal{C}}$. To be
able to make a statement about (an upper bound of) the work of
${\mathcal{C}}$, we associate a function $w$ with update rules and show that
every update rule $\pi$ can be implemented by a $\mathcal{O}(1)$-time CRAM-
program with work $\mathcal{O}(w)$. Likewise for query rules.
In a nutshell, the work of an update procedure mainly depends on the scopes of
the (nested) parallel branches and the amount of work needed to query and
update the supplementary instances. The work of a whole update rule is then
determined by adding the work of the initial procedure once and adding the
work of the main procedure for each tuple that satisfies the constraint of the
update rule.
The function $w$ is defined as follows. Let the update rule $\pi$ be of the
form
on change $\delta(\bar{p})$
update $R$ at $(t_{1}(\bar{p},\bar{x}),\ldots,t_{k}(\bar{p},\bar{x}))$, for
all $x_{1}\leq g_{1}(n)\land\ldots\land x_{\ell}\leq g_{\ell}(n)$, by: $P$
with $P=P_{1};P_{2}$ consisting of an initial procedure $P_{1}$ and a main
procedure $P_{2}$. For simplicity we require that for each variable $x_{i}$
there is an inequality $x_{i}\leq g_{i}(n)$, but it could be $g_{i}(n)=n$. We
set
$w(\pi)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\max(w(P_{1}),g_{1}(n)\cdot\ldots\cdot
g_{\ell}(n)\cdot w(P_{2}))$, where $w(P)$ is inductively defined as follows.
For terms $t$ and conditions $C$, we define $w(t)$ and $w(C)$, inductively.
$w(t)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\begin{cases}1&\text{if
$t$ is a constant or a variable}\\\ \max(w(t_{1}),\ldots,w(t_{\ell}))&\text{if
$t=f(t_{1},\ldots,t_{\ell})$,}\\\ \max(w(C),w(t_{1}),w(t_{2}))&\text{if
$t=\textbf{if }C\textbf{ then }t_{1}\textbf{ else }t_{2}$}\\\ g(n)\cdot
w(C)&\text{if $t=\textbf{getUnique}(z\leq g(n)\mid C)$.}\end{cases}$
$w(C)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\begin{cases}1&\text{if
$C$ is atomic,}\\\ \max(w(C_{1}),\ldots,w(C_{m}))&\text{if $C$ is a Boolean
combination}\\\ &\quad\text{of conditions }C_{1},\ldots,C_{m},\\\ g(n)\cdot
w(C^{\prime})&\text{if $C=\textbf{exists}(z\leq g(n)\mid C^{\prime})$},\\\
w(\pi^{\prime})&\text{if $C$ is $\texttt{instance}.Q(\bar{y})$ and
$\pi^{\prime}$ is}\\\ &\quad\text{its query rule.}\end{cases}$
Furthermore,
* •
$w(f(\bar{y})\leftarrow
C)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;w(C)$,
* •
$w(R(\bar{y})\leftarrow
t)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;w(t)$,
* •
$w(\textbf{if }C\textbf{ then }P^{\prime}\textbf{ else
}P^{\prime\prime})\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\max(w(C),w(P^{\prime}),w(P^{\prime\prime}))$,
* •
$w(\textbf{for }z\leq g(n)\textbf{ pardo
}P^{\prime})\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;g(n)\cdot
w(P^{\prime})$,
* •
$w(\textbf{return
}\varphi)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;1$,
* •
$w(\texttt{instance}.\delta(\bar{y}))\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\max(w(\pi_{1}),\ldots,w(\pi_{m}))$,
where $\pi_{1},\ldots,\pi_{m}$ are the update rules for change operation
$\delta$.
###### Proposition 2
For every update rule $\pi$ a $\mathcal{O}(1)$-time PRAM-program with work
$\mathcal{O}(w(\pi))$ can be constructed. Likewise for query rules.
A sketch for the straightforward proof can be found in the appendix.
It follows from Proposition 2 that a dynamic program ${\mathcal{P}}$ can be
implemented by a $\mathcal{O}(1)$-time PRAM-program with work
$\mathcal{O}(f)$, if for each update rule $\pi$ of ${\mathcal{P}}$ it holds
$w(\pi)\leq f$.
## 4 A simple work-efficient Dynamic Program
In this section we consider a simple dynamic problem with a fairly work-
efficient dynamic program. It serves as an example for our framework but will
also be used as a subroutine in later sections.
The dynamic problem is to maintain a subset $K$ of an ordered set $D$ of
elements under insertion and removal of elements in $K$, allowing for
navigation from an element of $D$ to the next larger and smaller element in
$K$. That is, we consider the following dynamic problem:
Problem: | NextInK
---|---
Input: | A set $K\subseteq D$ with canonical linear order $\leq$ on $D$
Changes: | $\textsc{ins}(i)$: Inserts $i\in D$ into $K$ $\textsc{del}(i)$: Deletes $i\in D$ from $K$
Queries: | $\textsc{pred}(i)$: Returns predecessor of $i$ in $K$, that is, $\max\\{j\in K\mid i>j\\}$ $\textsc{succ}(i)$: Returns successor of $i$ in $K$, that is, $\min\\{j\in K\mid i<j\\}$
For the smallest (largest) element the result of a pred (succ) query is
undefined, i.e. $\bot$. For simplicity, we assume in the following that $D$ is
always of the form $[n]$, for some $n\in\mathbb{N}$.
Sequentially, the changes and queries of NextInK can be handled in sequential
time $\mathcal{O}(\log\log n)$ [9]. Here we show that the problem also has a
dynamic program with parallel time $\mathcal{O}(1)$ and work $\mathcal{O}(\log
n)$.
###### Lemma 1
There is a DynFO-program for NextInK with $\mathcal{O}(\log n)$ work per
change and query operation.
###### Proof
The dynamic program uses an ordered binary balanced tree $T$ with leave set
$[n]$, and with $0$ as its leftmost leaf. Each inner node $v$ represents the
interval $S(v)$ of numbers labelling the leafs of the subtree of $v$. To
traverse the tree, the dynamic program uses functions 1st and 2nd that map an
inner node to its first or second child, respectively, and a function
$\text{anc}(v,j)$ that returns the $j$-th ancestor of $v$ in the tree.
Formally, the $2|D|$ nodes of $T$ can be represented by pairs $(a,b)$ of
elements from the domain $D$, where $b$ indicates the height of the node in
the tree, and $a$ gives the left-to-right position of the node among all nodes
with the same height. Then, we have to use pairs of binary functions
$\text{1st}^{1},\text{1st}^{2}$ etc. that map encodings of a node to the
components of the encoding of another node. We disregard these technical
issues and use nodes of $T$ just as domain elements, and also identify an
element $i\in[n]$ with the $i$-th leaf of $T$. The technical translation is
straightforward and does not affect our reasoning regarding FO-expressibility
and work of dynamic programs. So, $\text{anc}(v,2)$ returns the parent of the
parent of $v$. If there is no $j$-th ancestor of a node $v$, then
$\text{anc}(v,j)$ is undefined.
The functions 1st, 2nd and anc are static, that is, they are initialized
beforehand and not affected by change operations.
The idea of the dynamic program is to maintain, for each node $v$, the maximal
and minimal element in $K\cap S(v)$ (which is undefined if $K\cap
S(v)=\emptyset$), by maintaining two functions $\min$ and $\max$. It is easy
to see that this information can be updated and queries be answered in
$\mathcal{O}(\log n)$ time as the tree has depth $\mathcal{O}(\log n)$. For
achieving $\mathcal{O}(\log n)$ work and constant time, we need to have a
closer look.
Using $\min$ and $\max$, it is easy to determine the $K$-successor of an
element $i\in D$: if $v$ is the lowest ancestor of $i$ with $\max(v)>i$, then
the $K$-successor of $i$ is $\min(w)$ for the second child
$w\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\text{2nd}(v)$
of $v$. Algorithm 1 shows a query rule for the query operation
$\textsc{succ}(i)$.
1:on query $\textsc{succ}(i)$:
2: if $\max(T.\text{root})\leq i$ then
3: return $\bot$
4: else
5: $k\leftarrow\textbf{getUnique}(1\leq
k\leq\log(n)\mid\max(T.\text{anc}(i,k))>i)$
6: $\land\;\max(T.\text{anc}(i,k-1))\leq i$
7: return $\min(T.\text{2nd}(T.\text{anc}(i,k)))$
Algorithm 1 Querying a successor.
The update of these functions is easy when an element $i$ is inserted into
$K$. This is spelled out for $\min$ in Algorithm 2. The dynamic program only
needs to check if the new element becomes the minimal element in $S(v)$, for
every node $v$ that is an ancestor of the leaf $i$.
1:on change $\textsc{ins}(i)$ update $\min$ at $T.\text{anc}(i,k)$, for all
$k\leq\log n$, by:
2: $v\leftarrow T.\text{anc}(i,k)$
3: if $\min(v)>i$ then
4: return $i$
5: else
6: return $\min(v)$
Algorithm 2 Updating $\min$ after an insertion.
Algorithm 3 shows how $\min$ can be updated if an element $i$ is deleted from
$K$: if $i$ is the minimal element of $K$ in $S(v)$, for some node $v$, then
$\min(v)$ needs to be replaced by its $K$-successor, assuming it is in $S(v)$.
1:on change $\textsc{del}(i)$
2: with
3: $s\leftarrow\textsc{succ}(i)$
4: update $\min$ at $T.\text{anc}(i,k)$, for all $k\leq\log n$, by:
5: $v\leftarrow T.\text{anc}(i,k)$
6: if $\min(v)\neq i$ then
7: return $\min(v)$
8: else if $\max(v)=i$ then
9: return $\bot$
10: else
11: return $s$
Algorithm 3 Updating $\min$ after a deletion.
It is easy to verify the claimed work upper bounds for ${\mathcal{P}}$.
Querying a successor or predecessor via Algorithm 1 needs $\mathcal{O}(\log
n)$ work, since Line 6 requires $\mathcal{O}(\log n)$ and all others require
$\mathcal{O}(1)$ work. For maintaining the function $\min$ the programs in
Algorithms 2 and 3 update the value of $\log n$ tuples, but the work per tuple
is constant. In the case of a deletion, Line 3 requires $\mathcal{O}(\log n)$
work but is executed only once. The remaining part consists of
$\mathcal{O}(\log n)$ parallel executions of statements, each with
$\mathcal{O}(1)$ work.
The handling of $\max$ and its work analysis is analogous. ∎
## 5 Regular Languages
In this section, we show that the range problem can be maintained with $o(n)$
work for all regular languages and with polylogarithmic work for star-free
languages. For the former we show how to reduce the work of a known DynFO-
program. For the latter we translate the idea of [9] for maintaining the range
problem for star-free languages in $\mathcal{O}(\log\log n)$ sequential time
into a dynamic program with $\mathcal{O}(1)$ parallel time.
### 5.1 DynFO-programs with sublinear work for regular languages
###### Theorem 5.1
Let $L$ be a regular language. Then ${\textsc{RangeMember}}(L)$ can be
maintained in DynFO with work $\mathcal{O}(n^{\epsilon})$ per query and change
operation, for every $\epsilon>0$.
The proof of this theorem makes use of the algebraic view of regular
languages. For readers not familiar with this view, the basic idea is as
follows: for a fixed DFA ${\mathcal{A}}=(Q,\Sigma,\delta,q_{0},F)$, we first
associate with each string $w$ a function $f_{w}$ on $Q$ that is induced by
the behaviour of ${\mathcal{A}}$ on $w$ via
$f_{w}(q)\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\delta^{*}(q,w)$,
where $\delta^{*}$ is the extension of the transition function $\delta$ to
strings. The set of all functions $f\colon Q\to Q$ with composition as binary
operation is a _monoid_ , that is, a structure with an associative binary
operation $\circ$ and a neutral element, the identity function. Thus,
composing the effect of ${\mathcal{A}}$ on subsequent substrings of a string
corresponds to multiplication of the monoid elements associated with these
substrings. The _syntactic monoid_ $M(L)$ of a regular language $L$ is
basically the monoid associated with its minimal automaton.
It is thus clear that, for the dynamic problem ${\textsc{RangeMember}}(L)$
where $L$ is regular, a dynamic program can be easily obtained from a dynamic
program for the dynamic problem ${\textsc{RangeEval}}(M(L))$, where
${\textsc{RangeEval}}(M)$, for finite monoids $M$, is defined as follows.333We
note that, unlike for words, each position always carries a monoid element.
However, the empty string of the word case corresponds to the neutral element
in the monoid case. In particular, the initial “empty” sequence consists of
$n$ copies of the neutral element.
Problem: | ${\textsc{RangeEval}}(M)$
---|---
Input: | A sequence $m_{0}\ldots m_{n-1}$ of monoid elements $m_{i}\in M$
Changes: | $\textsc{set}_{m}(i)$ for $m\in M$: Replaces $m_{i}$ by $m$
Queries: | $\textsc{range}(\ell,r)$: $m_{\ell}\circ\cdots\circ m_{r}$
For the proof of Theorem 5.1 we do not need any insights into monoid theory.
However, when studying languages definable by first-order formulas in Theorem
5.2 below, we will make use of a known decomposition result.
From the discussion above it is now clear that in order to prove Theorem 5.1,
it suffices to prove the following result.
###### Proposition 3
Let $M$ be a finite monoid. For every $\epsilon>0$, ${\textsc{RangeEval}}(M)$
can be maintained in DynFO with work $\mathcal{O}(n^{\epsilon})$ per query and
change operation.
###### Proof (Proof sketch)
In [12], it was (implicitly) shown that ${\textsc{RangeMember}}(L)$ is in
DynProp (that is, quantifier-free DynFO), for regular languages $L$. The idea
was to maintain the effect of a DFA for $L$ on $w[\ell,r]$, for each interval
$(\ell,r)$ of positions. This approach can be easily used for
${\textsc{RangeEval}}(M)$ as well, but it requires a quadratic number of
updates after a change operation, in the worst case.
We adapt this approach and only store the effect of the DFA for
$\mathcal{O}(n^{\epsilon})$ intervals, by considering a hierarchy of intervals
of bounded depth.
The first level in the hierarchy of intervals is obtained by decomposing the
input sequence into intervals of length $t$, for a carefully chosen $t$. We
call these intervals _base intervals_ of height $1$ and their subintervals
_special intervals_ of height $1$. The latter are _special_ in the sense that
they are exactly the intervals for which the dynamic program maintaines the
product of monoid elements. In particular, each base interval of height $1$
gives rise to $\mathcal{O}(t^{2})$ special intervals of height $1$. The second
level of the hierarchy is obtained by decomposing the sequence of base
intervals of height $1$ into sequences of length $t$. Each such sequence of
length $t$ is combined to one base interval of height $2$; and each contiguous
subsequence of such a sequence is combined to one special interval of height
$2$. Again, each base interval of height $2$ gives rise to
$\mathcal{O}(t^{2})$ special intervals of height $2$. This process is
continued recursively for the higher levels of the hierarchy, until only one
base interval of height $h$ remains. We refer to Figure 1 for an illustration
of this construction.
The splitting factor $t$ is chosen in dependence of $n$ and $\epsilon$ such
that the height of this hierarchy of special intervals only depends on
$\epsilon$ and is thus constant for all $n$. More precisely, we fix
$\lambda\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\frac{\epsilon}{2}$
and
$t\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;n^{\lambda}$,
rounded up. For simplicity of exposition, we assume from now on that
$n=t^{\frac{1}{\lambda}}$. Therefore, $h=\log_{t}(n)=\frac{1}{\lambda}$.
The idea for the dynamic program is to store the product of monoid elements
for each special interval. The two crucial observations are then, that (1) the
product of each (not necessary special) interval can be computed with the help
of a constant number of special intervals, and (2) that each change operation
affects at most $t^{2}$ special intervals per level of the hierarchy and thus
at most $ht^{2}\in\mathcal{O}(n^{\epsilon})$ special intervals in total. We
refer to the appendix for more details.
$m_{0}$$m_{1}$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$m_{13}$$m_{14}$$m_{15}$$m_{16}$$m_{17}$$m_{18}$$m_{19}$$m_{20}$$m_{21}$$m_{22}$$m_{23}$$m_{24}$$m_{25}$$m_{26}$level
$1$level $2$level $3$
Figure 1: Illustration of special intervals, for $t=3$. The special intervals
of level $3$ are $[0,9),[9,18),[18,27),[0,18)$ and $[9,27)$ with base interval
$[0,27)$. The result of a query $\textsc{range}(2,22)$ can be computed as
$\prod_{i=2}^{22}m_{i}=\big{(}m[2,3)\circ m[3,9)\big{)}\circ
m[9,18)\circ\big{(}m[18,21)\circ m[21,23)\big{)}$, illustrated above in blue.
The affected base intervals for a change at position $23$ are marked in red.
E.g., the new product $m^{\prime}[18,27)$ can be computed by
$m^{\prime}[18,27)=m[18,21)\circ m^{\prime}[21,24)\circ m[24,27)$. As the
products are recomputed bottom up, $m^{\prime}[21,24)$ is already updated.
∎
### 5.2 DynFO-programs with polylogarithmic work for star-free languages
Although the work bound of Theorem 5.1 for regular languages is strongly
sublinear, one might aim for an even more work-efficient dynamic program,
especially, since ${\textsc{RangeMember}}(L)$ can be maintained _sequentially_
with logarithmic update time for regular languages [9]. We leave it as an open
problem whether for every regular language $L$ there is a DynFO-program for
${\textsc{RangeMember}}(L)$ with a polylogarithmic work bound. However, we
show next that such programs exist for star-free regular languages, in fact
they even have a logarithmic work bound. The star-free languages are those
that can be expressed by regular expressions that do not use the Kleene star
operator but can use complementation.
###### Theorem 5.2
Let $L$ be a star-free regular language. Then ${\textsc{RangeMember}}(L)$ can
be maintained in DynFO with work $\mathcal{O}(\log n)$ per query and change
operation.
It is well-known that star-free regular languages are just the regular
languages that can be defined in first-order logic (without arithmetic!) [20].
Readers might ask why we consider dynamic first-order maintainability of a
problem that can actually be _expressed_ in first-order logic. The key point
is the parallel work here: even though the membership problem for star-free
languages can be solved by a parallel algorithm in time $\mathcal{O}(1)$, it
inherently requires parallel work $\Omega(n)$.
###### Proof (Proof sketch)
The proof uses the well-known connection between star-free languages and
group-free monoids (see, e.g., [27, Chapter V.3] and [27, Theorem V.3.2]). It
thus follows the approach of [9].
In a nutshell, our dynamic program simply implements the algorithms of the
proof of Theorem 2.4.2 in [9]. Those algorithms consist of a constantly
bounded number of simple operations and a constantly bounded number of
searches for a next neighbour in a set. Since the latter can be done in DynFO
with work $\mathcal{O}(\log n)$ thanks to Lemma 1, we get the desired result
for group-free monoids and then for star-free languages. We refer to the
appendix for more details. ∎
## 6 Context Free Languages
As we have seen in Section 5, range queries to regular languages can be
maintained in DynFO with strongly sublinear work. An immediate question is
whether context-free languages are equally well-behaved. Already the initial
paper by Patnaik and Immerman showed that DynFO can maintain the membership
problem for _Dyck languages_ $D_{k}$, for $k\geq 1$, that is, the languages of
well-balanced parentheses expressions with $k$ types of parentheses [23]. It
was shown afterwards in [12, Theorem 4.1] that DynFO actually captures the
membership problem for all context-free languages and that Dyck languages even
do not require quantifiers in formulas (but functions in the auxiliary
structure) [12, Proposition 4.4]. These results can easily be seen to apply to
range queries as well. However, the dynamic program of [12, Theorem 4.1] uses
4-ary relations and three nested existential quantifiers, yielding work in the
order of $n^{7}$.
In the following, we show that the membership problem for context-free
languages is likely _not_ solvable in DynFO with sublinear work, but that the
Dyck language $D_{1}$ with one bracket type can be handled with
polylogarithmic work for the membership problem and work
$\mathcal{O}(n^{\epsilon})$ for the range problem, and that for other Dyck
languages these bounds hold with an additional linear factor $n$.
### 6.1 A conditional lower bound for context-free languages
Our conditional lower bound for context-free languages is based on a result
from Abboud et al. [2] and the simple observation that the word problem for a
language $L$ can be solved, given a dynamic program for its membership
problem.
###### Lemma 2
Let $L$ be a language. If ${\textsc{Member}}(L)$ can be maintained in DynFO
with work $f(n)$, then the word problem for $L$ can be decided sequentially in
time $\mathcal{O}(n\cdot f(n))$.
The announced lower bound is relative to the following conjecture [1].
###### Conjecture 1 ($k$-Clique conjecture)
For any $\epsilon>0$, and $k\geq 3$, $k$-Clique has no algorithm with time
bound $\mathcal{O}(n^{(1-\epsilon)\frac{\omega}{3}k})$.
Here, $\omega$ is the matrix multiplication exponent [11, 29], which is known
to be smaller than $2.373$ and believed to be exactly two [11, 29].
In [2], the word problem for context-free languages was linked to the
$k$-Clique problem as follows.
###### Theorem 6.1 ([2, Theorem 1.1])
There is a context free grammar $G$ such that, if the word problem for $L(G)$
can be solved in time $T(n)$, $k$-Clique can be solved on $n$ node graphs in
$\mathcal{O}(T(n^{\frac{k}{3}+1}))$ time, for any $k\geq 3$.
Putting Lemma 2 and Theorem 6.1 together, we get the following result.
###### Theorem 6.2
There is a context free grammar $G$ such that, if the membership problem for
$L(G)$ can be solved by a DynFO-program with work
$\mathcal{O}(n^{\omega-1-\epsilon})$, for some $\epsilon>0$, then the
$k$-Clique conjecture fails.
The simple proofs of Lemma 2 and Theorem 6.2 are presented in the appendix.
Thus, we can not reasonably expect any DynFO-programs for general context-free
languages with considerable less work than $\mathcal{O}(n^{1.37})$ barring any
breakthroughs for matrix multiplication. In fact, for “combinatorial DynFO-
programs”, an analogous reasoning yields a work lower bound of
$\mathcal{O}(n^{2-\epsilon})$.
### 6.2 On work-efficient dynamic programs for Dyck languages
We next turn to Dyck languages. Clearly, all Dyck languages are deterministic
context-free, their word problem can therefore be solved in linear time, and
thus the lower bound approach of the previous subsection does not work for
them. In [8] it was shown that $D_{1}$ is maintainable by a sequential dynamic
algorithm in time $\mathcal{O}(\log n)$, per change and query operation, and
$D_{k}$ in time $\mathcal{O}((\log n)^{3}\cdot\log^{\ast}n)$, for every $k>1$.
We already mentioned that their membership problem can be maintained in DynFO.
The challenge is to find dynamic programs (with parallel time
$\mathcal{O}(1)$) with little work.
We show that it is actually possibly by suitably adapting the idea from [8] to
get a DynFO-program for $D_{1}$ with polylogarithmic work, but for $D_{k}$,
for $k>1$, the approach so far only yields DynFO-programs with an additional
linear factor in the work bound. We first present the DynFO-program with
polylogarithmic work for the membership problem of $D_{1}$. It basically
mimics the sequential algorithm from [8] that maintains $D_{1}$.
###### Theorem 6.3
${\textsc{Member}}(D_{1})$ can be maintained in DynFO with $\mathcal{O}((\log
n)^{3})$ work.
###### Proof (Proof sketch)
Let $\Sigma_{1}=\\{\langle,\rangle\\}$ be the alphabet underlying $D_{1}$. The
dynamic program uses an ordered binary tree $T$ such that each leaf
corresponds to one position from left-to right. A parent node corresponds to
the set of positions of its children. We assume for simplicity that the domain
is $[n]$, for some number $n$ that is a power of 2. If the input string is $w$
then each node $x$ of $T$ represents a substring $\textnormal{str}(x)$ of $w$
via the leaves of the induced subtree at $x$ in a natural fashion. The main
idea from [8] is to maintain for each node $x$ of $T$ information about the
unmatched parentheses of $\textnormal{str}(x)$. The input word is in the Dyck
language if there are no unmatched parentheses in the root of $T$.
In a nutshell, the dynamic program for ${\textsc{Member}}(D_{1})$ maintains
for each node $x$ of $T$ the numbers $\ell(x)$ and $r(x)$ that represent the
number of unmatched closing and unmatched opening brackets of the string
$\textnormal{str}(x)$. E.g., if that string is
$\rangle\langle\rangle\rangle\langle\langle\rangle$ for $x$, then $\ell(x)=2$
and $r(x)=1$. The overall string $w$ is in $D_{1}$ exactly if
$r(\textnormal{root})=\ell(\textnormal{root})=0$.
In the algorithm of [8], the functions $\ell$ and $r$ are updated in a bottom-
up fashion. However, we will observe that they do not need to be updated
sequentially in that fashion, but can be updated in parallel constant time. In
the following, we describe how ${\mathcal{P}}$ can update $\ell(x)$ and $r(x)$
for all ancestor nodes $x$ of a position $p$, after a closing parenthesis
$\rangle$ was inserted at $p$. Maintaining $\ell$ and $r$ for the other change
operations is analogous.
There are two types of effects that an insertion of a closing parenthesis
could have on $x$: either $\ell(x)$ is increased by one and $r(x)$ remains
unchanged, or $r(x)$ is decreased by one and $\ell(x)$ remains unchanged. We
denote these effects by the pairs $(+1,0)$ and $(0,-1)$, respectively.
Table 1 shows how the effect of a change at a position $p$ below a node $x$
with children $y_{1}$ and $y_{2}$ relates to the effect at the affected child.
This depends on whether $r(y_{1})\leq\ell(y_{2})$ and whether the affected
child is $y_{1}$ or $y_{2}$.
| $p$ is in $\textnormal{str}(y_{1})$ | $p$ is in $\textnormal{str}(y_{2})$
---|---|---
$r(y_{1})\leq\ell(y_{2})$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(+1,0)\end{array}$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(0,-1)\end{array}$
$r(y_{1})>\ell(y_{2})$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(0,-1)\end{array}$ | $\begin{array}[]{c}(+1,0)\to(0,-1)\\\ (0,-1)\to(0,-1)\end{array}$
Table 1: The effect on $x$ after a closing parenthesis was inserted at
position $p$. The effects depend on the effect on the children $y_{1}$ and
$y_{2}$ of $x$: for example, an entry ’$(0,-1)\to(+1,0)$’ in the column ’$p$
is in $\textnormal{str}(y_{1})$’ means that if the change operation has effect
$(0,-1)$ on $y_{1}$ then the change operation has effect $(+1,0)$ on $x$.
A closer inspection of Table 1 reveals a crucial observation: in the upper
left and the lower right field of the table, the effect on $x$ is
_independent_ of the effect on the child (being it $y_{1}$ or $y_{2}$). That
is, these cases induce an effect on $x$ independent of the children. We thus
call these cases _effect-inducing_. In the other two fields, the effect on $x$
depends on the effect at the child, but in the simplest possible way: they are
just the same. That is the effect at the child is just adopted by $x$. We call
these cases _effect-preserving_. To determine the effect at $x$ it is thus
sufficient to identify the highest affected descendant node $z$ of $x$, where
an effect-inducing case applies, such that for all intermediate nodes between
$x$ and $z$ only effect-preserving cases apply.
Our dynamic program implements this idea. First it determines, for each
ancestor $x$ of the change position $p$, whether it is effect-inducing and
which effect is induced. Then it identifies, for each $x$, the node $z$
(represented by its height $i$ above $p$) as the unique effect-inducing node
that has no effect-inducing node on its path to $x$. The node $z$ can be
identified with work $\mathcal{O}((\log n)^{2})$, as $z$ is one of at most
$\log n$ many nodes on the path from $x$ to the leaf of $p$, and one needs to
check that all nodes between $x$ and $z$ are effect-preserving. As the
auxiliary relations need to be updated for $\log n$ many nodes, the overall
work of ${\mathcal{P}}$ is $\mathcal{O}((\log n)^{3})$. We refer to the
appendix for more details. ∎
#### A work-efficient dynamic program for range queries for $D_{1}$ and
$D_{k}$
Unfortunately, the program of Theorem 6.3 does not support range queries,
since it seems that one would need to combine the unmatched parentheses of
$\log n$ many nodes of the binary tree in the worst case. However, its idea
can be combined with the idea of Proposition 3, yielding a program that
maintains $\ell$ and $r$ for $\mathcal{O}(n^{\epsilon})$ special intervals on
a constant number of levels.
In fact, this approach even works for $D_{k}$ for $k>1$. Indeed, with the help
of $\ell$ and $r$, it is possible to identify for each position of an opening
parenthesis the position of the corresponding closing parenthesis in
$\mathcal{O}(1)$ parallel time with work $n^{\epsilon}$, and then one only
needs to check that they match everywhere. The latter contributes an extra
factor $\mathcal{O}(n)$ to the work, for $k>1$, but can be skipped for $k=1$.
###### Theorem 6.4
For all $\epsilon>0$, $k>1$,
1. a)
${\textsc{RangeMember}}(D_{1})$ can be maintained in DynFO with
$\mathcal{O}(n^{\epsilon})$ work, and
2. b)
${\textsc{RangeMember}}(D_{k})$ can be maintained in DynFO with
$\mathcal{O}(n^{\epsilon})$ work per change operation and
$\mathcal{O}(n^{1+\epsilon})$ work per query operation.
###### Proof (Proof sketch)
In the following we reuse the definition of _special intervals_ from the proof
of Proposition 3 as well as the definition of $\ell$ and $r$ from the proof of
Proposition 6.3. We first describe a dynamic program ${\mathcal{P}}$ for
${\textsc{RangeMember}}(D_{1})$. It maintains $\ell$ and $r$ for all special
intervals, which is clearly doable with $\mathcal{O}(n^{\epsilon})$ work per
change operation. Similar to the proof of Proposition 3, the two crucial
observations (justified in the appendix) are that (1) a range query can be
answered with the help of a constant number of special intervals, and (2) the
change operation affects only a bounded number of special intervals per level.
As stated before, the program for ${\textsc{RangeMember}}(D_{k})$ also
maintains $\ell$ and $r$, but it should be emphasised that also in the case of
several parenthesis types, the definition of these functions ignores the
bracket type. With that information it computes, for each opening bracket the
position of its matching closing bracket, with the help of $\ell$ and $r$, and
checks that they match. This can be done in parallel and with work
$\mathcal{O}(n^{\epsilon})$ per position. We refer to the appendix for more
details. ∎
#### Moderately work-efficient dynamic programs for $D_{k}$
We now turn to the membership query for $D_{k}$ with $k>1$. Again, our program
basically mimics the sequential algorithm from [8] which heavily depends on
the dynamic problem StringEquality that asks whether two given strings are
equal.
Problem: | StringEquality
---|---
Input: | Two Sequences $u=u_{0}\ldots u_{n-1}$ and $v=v_{0}\ldots v_{n-1}$ of letters with $u_{i},v_{i}\in\Sigma\cup\\{\epsilon\\}$
Changes: | $\textsc{set}_{x,\sigma}(i)$ for $\sigma\in\Sigma,x\in\\{u,v\\}$: Sets $x_{i}$ to $\sigma$, if $x_{i}=\epsilon$ $\textsc{reset}_{x}(i)$ for $x\in\\{u,v\\}$: Sets $x_{i}$ to $\epsilon$
Queries: | equals: Is $u_{0}\circ\ldots\circ u_{n-1}=v_{0}\circ\ldots\circ v_{n-1}$?
It is easy to show that a linear amount of work is sufficient to maintain
StringEquality.
###### Lemma 3
StringEquality is in DynFO with work $\mathcal{O}(n)$.
Because of the linear work bound for StringEquality our dynamic program for
${\textsc{Member}}(D_{k})$ also has a linear factor in the work bound.
###### Theorem 6.5
${\textsc{Member}}(D_{k})$ is maintainable in DynFO with $\mathcal{O}(n\log
n+(\log n)^{3})$ work for every fixed $k\in\mathbb{N}$.
###### Proof (Proof sketch)
The program can be seen as an extension of the one for
${\textsc{Member}}(D_{1})$. As unmatched parentheses are no longer well-
defined if we have more than one type of parenthesis the idea of [8] is to
maintain the parentheses to the left and right that remain if we reduce the
string by matching opening and closing parentheses regardless of their type.
To be able to answer ${\textsc{Member}}(D_{k})$, the dynamic program maintains
the unmatched parentheses for every node $x$ of a tree spanning the input
word, and a bit $M(x)$ that indicates whether the types of the parentheses
match properly.
How the unmatched parentheses can be maintained for a node $x$ after a change
operation depends on the “segment” of $\textnormal{str}(x)$ in which the
change happened and in some cases reduces to finding a node $z$ with a local
property on the path from $x$ to the leaf that corresponds to the changed
position.
To update $M(x)$ for a node $x$ with children $y_{1}$ and $y_{2}$ the dynamic
program compares the unmatched parentheses to the right of $y_{1}$ with the
ones to the left of $y_{2}$ using StringEquality. We refer to the appendix for
more details. ∎
Maintaining string equality and membership in $D_{k}$ for $k>1$ is even closer
related which is stated in the following lemma.
###### Lemma 4
1. a)
If StringEquality can be maintained in DynFO with work $W(n)$ then
${\textsc{Member}}(D_{k})$ can be maintained in DynFO with work
$\mathcal{O}(W(n)\cdot\log n+(\log n)^{3})$, for each $k\geq 1$.
2. b)
If ${\textsc{Member}}(D_{k})$ can be maintained in DynFO with work $W(n)$ for
all $k$, then StringEquality can be maintained in DynFO with work
$\mathcal{O}(W(n))$.
## 7 Conclusion
In this paper we proposed a framework for studying the aspect of work for the
dynamic, parallel complexity class DynFO. We established that all regular
languages can be maintained in DynFO with $\mathcal{O}(n^{\epsilon})$ work for
all $\epsilon>0$, and even with $\mathcal{O}(\log n)$ work for star-free
regular languages. For context-free languages we argued that it will be hard
to achieve work bounds lower than $\mathcal{O}(n^{\omega-1-\epsilon})$ in
general, where $\omega$ is the matrix multiplication exponent. For the special
case of Dyck languages $D_{k}$ we showed that $\mathcal{O}(n\cdot(\log
n)^{3})$ work suffices, which can be further reduced to
$\mathcal{O}(\log^{3}n)$ work for $D_{1}$. For range queries, dynamic programs
with work $\mathcal{O}(n^{1+\epsilon})$ and $\mathcal{O}(n^{\epsilon})$ exist,
respectively.
We highlight some research directions. One direction is to improve the upper
bounds on work obtained here. For instance, it would be interesting to know
whether all regular languages can be maintained with polylog or even
$\mathcal{O}(\log n)$ work and how close the lower bounds for context-free
languages can be matched. Finding important subclasses of context-free
languages for which polylogarithmic work suffices is another interesting
question. Apart from string problems, many DynFO results concern problems on
dynamic graphs, especially the reachability query [5]. How large is the work
of the proposed dynamic programs, and are more work-efficient dynamic programs
possible?
The latter question also leads to another research direction: to establish
further lower bounds. The lower bounds obtained here are relative to strong
conjectures. Absolute lower bounds are an interesting goal which seems in
closer reach than lower bounds for DynFO without bounds on the work.
## References
* [1] Abboud, A., Backurs, A., Bringmann, K., Künnemann, M.: Fine-grained complexity of analyzing compressed data: Quantifying improvements over decompress-and-solve. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017. pp. 192–203. IEEE Computer Society (2017). https://doi.org/10.1109/FOCS.2017.26
* [2] Abboud, A., Backurs, A., Williams, V.V.: If the current clique algorithms are optimal, so is Valiant’s parser. SIAM J. Comput. 47(6), 2527–2555 (2018)
* [3] Alstrup, S., Husfeldt, T., Rauhe, T.: Dynamic nested brackets. Inf. Comput. 193(2), 75–83 (2004). https://doi.org/10.1016/j.ic.2004.04.006, https://doi.org/10.1016/j.ic.2004.04.006
* [4] Börger, E.: Abstract state machines: a unifying view of models of computation and of system design frameworks. Ann. Pure Appl. Log. 133(1-3), 149–171 (2005). https://doi.org/10.1016/j.apal.2004.10.007
* [5] Datta, S., Kulkarni, R., Mukherjee, A., Schwentick, T., Zeume, T.: Reachability is in DynFO. J. ACM 65(5), 33:1–33:24 (2018). https://doi.org/10.1145/3212685
* [6] Dong, G., Su, J.: First-order incremental evaluation of datalog queries. In: Database Programming Languages (DBPL-4), Proceedings of the Fourth International Workshop on Database Programming Languages - Object Models and Languages, Manhattan, New York City, USA, 30 August - 1 September 1993. pp. 295–308 (1993)
* [7] Dong, G., Topor, R.W.: Incremental evaluation of datalog queries. In: Database Theory - ICDT’92, 4th International Conference, Berlin, Germany, October 14-16, 1992, Proceedings. pp. 282–296 (1992). https://doi.org/10.1007/3-540-56039-4_48
* [8] Frandsen, G.S., Husfeldt, T., Miltersen, P.B., Rauhe, T., Skyum, S.: Dynamic algorithms for the Dyck languages. In: Akl, S.G., Dehne, F.K.H.A., Sack, J., Santoro, N. (eds.) Algorithms and Data Structures, 4th International Workshop, WADS ’95, Kingston, Ontario, Canada, August 16-18, 1995, Proceedings. Lecture Notes in Computer Science, vol. 955, pp. 98–108. Springer (1995). https://doi.org/10.1007/3-540-60220-8_54
* [9] Frandsen, G.S., Miltersen, P.B., Skyum, S.: Dynamic word problems. J. ACM 44(2), 257–271 (1997). https://doi.org/10.1145/256303.256309
* [10] Freydenberger, D.D., Thompson, S.M.: Dynamic complexity of document spanners. In: Lutz, C., Jung, J.C. (eds.) 23rd International Conference on Database Theory, ICDT 2020, March 30-April 2, 2020, Copenhagen, Denmark. LIPIcs, vol. 155, pp. 11:1–11:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.ICDT.2020.11, https://www.dagstuhl.de/dagpub/978-3-95977-139-9
* [11] Gall, F.L.: Powers of tensors and fast matrix multiplication. In: International Symposium on Symbolic and Algebraic Computation, ISSAC ’14, Kobe, Japan, July 23-25, 2014. pp. 296–303 (2014)
* [12] Gelade, W., Marquardt, M., Schwentick, T.: The dynamic complexity of formal languages. ACM Trans. Comput. Log. 13(3), 19 (2012). https://doi.org/10.1145/2287718.2287719
* [13] Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001). https://doi.org/10.1145/502090.502095
* [14] Holm, J., Rotenberg, E.: Fully-dynamic planarity testing in polylogarithmic time. In: Makarychev, K., Makarychev, Y., Tulsiani, M., Kamath, G., Chuzhoy, J. (eds.) Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020. pp. 167–180. ACM (2020). https://doi.org/10.1145/3357713.3384249
* [15] Immerman, N.: Descriptive complexity. Graduate texts in computer science, Springer (1999). https://doi.org/10.1007/978-1-4612-0539-5
* [16] Immerman, N.: Descriptive complexity. Springer Science & Business Media (2012)
* [17] JáJá, J.: An Introduction to Parallel Algorithms. Addison-Wesley (1992)
* [18] Krohn, K., Rhodes, J.: Algebraic theory of machines. i. prime decomposition theorem for finite semigroups and machines. Transactions of the American Mathematical Society 116, 450–464 (1965)
* [19] Libkin, L.: Elements of Finite Model Theory. Springer (2004). https://doi.org/10.1007/978-3-662-07003-1
* [20] McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press (1971)
* [21] Mehlhorn, K., Sundar, R., Uhrig, C.: Maintaining dynamic sequences under equality tests in polylogarithmic time. Algorithmica 17(2), 183–198 (1997). https://doi.org/10.1007/BF02522825
* [22] Miltersen, P.B., Subramanian, S., Vitter, J.S., Tamassia, R.: Complexity models for incremental computation. Theor. Comput. Sci. 130(1), 203–236 (1994). https://doi.org/10.1016/0304-3975(94)90159-7
* [23] Patnaik, S., Immerman, N.: Dyn-FO: A parallel, dynamic complexity class. In: PODS. pp. 210–221 (1994)
* [24] Schwentick, T., Vortmeier, N., Zeume, T.: Sketches of dynamic complexity. SIGMOD Rec. 49(2), 18–29 (2020). https://doi.org/10.1145/3442322.3442325
* [25] Schwentick, T., Zeume, T.: Dynamic Complexity: Recent Updates. SIGLOG News 3(2), 30–52 (2016). https://doi.org/10.1145/2948896.2948899
* [26] Sherlekar, D.D., Pawagi, S., Ramakrishnan, I.V.: O(1) parallel time incremental graph algorithms. In: Maheshwari, S.N. (ed.) Foundations of Software Technology and Theoretical Computer Science, Fifth Conference, New Delhi, India, December 16-18, 1985, Proceedings. Lecture Notes in Computer Science, vol. 206, pp. 477–495. Springer (1985). https://doi.org/10.1007/3-540-16042-6_27
* [27] Straubing, H.: Finite automata, formal logic, and circuit complexity. Birkhauser Verlag (1994)
* [28] Vollmer, H.: Introduction to circuit complexity: a uniform approach. Springer Science & Business Media (2013)
* [29] Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Karloff, H.J., Pitassi, T. (eds.) Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012. pp. 887–898. ACM (2012). https://doi.org/10.1145/2213977.2214056
## Appendix 0.A Appendix for Section 3
###### Proof (Proof sketch)
[for Proposition 2] We briefly sketch, how a PRAM ${\mathcal{P}}$ can evaluate
an update rule $\pi$ of the form
on change $\delta(\bar{p})$
update $R$ at $(t_{1}(\bar{p},\bar{x}),\ldots,t_{k}(\bar{p},\bar{x}))$, for
all $x_{1}\leq g_{1}(n)\land\ldots\land x_{\ell}\leq g_{\ell}(n)$, by: $P$.
with $P=P_{1};P_{2}$ as before, in constant time and with work
$\mathcal{O}(w(\pi))$. The initial procedure $P_{1}$ has to be evaluated only
once and therefore contributes at most $w(P_{1})$. The main procedure $P_{2}$
has to be evaluated once, for every tuple $\bar{x}$ that fulfils $x_{1}\leq
g_{1}(n)\land\ldots\land x_{k}\leq g_{\ell}(n)$. To this end, it can use
$m\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;g_{1}(n)\cdot\ldots\cdot
g_{\ell}(n)$ processors, each taking care of one evaluation, as in the
following pseudo-code.
1:procedure $\text{Evaluate}_{\delta}^{R}$($\bar{p}$)
2: Evaluate $P_{1}$
3: for $x_{1}=1\ \textbf{to}\ g_{1}(n)$ pardo
4: $\ddots$
5: for $x_{\ell}=1\ \textbf{to}\ g_{\ell}(n)$ pardo
6:
$R\big{(}t_{1}(\bar{p},\bar{x}),\ldots,t_{k}(\bar{p},\bar{x})\big{)}\leftarrow\textsc{$\text{Eval}_{P_{2}}$}(\bar{p},\bar{x})$
Algorithm 4 Evaluation of the update rule for $\delta(\bar{p})$.
This amounts to at most $m$ evaluations of $P_{2}$ and it thus suffices to
show that one evaluation of $P_{2}$ only requires work
$\mathcal{O}(w(P_{2}))$.
The proof of this is relatively straightforward, but tedious. Each computation
path consists only of a constant number of steps (unless a query towards a
supplementary instance occurs). Thus, assuming sufficient parallelisation, the
PRAM only needs time $\mathcal{O}(1)$. Along each path, $w$ basically yields
only 1, but any constant number is captured by the big $\mathcal{O}$. This
explains, why we can use $\max$ instead of addition, everywhere. Parallel
branches, exists-expressions and unique-expressions translate into parallel
branching of the PRAM, and that is captured in the definition of $w$ by
respective factors. Likewise, query operations to supplementary instances are
captured, by respective factors in the definition of $w$. ∎
## Appendix 0.B Appendix for Section 5
###### Proof (Proof sketch)
[of Proposition 3, continued] We start by introducing necessary notation. In
the following we fix the length $n$ of a sequence $m_{0},\ldots,m_{n-1}$ of
monoid elements, we chose $t$ as described above, and assume that $t^{h}=n$.
We write $m[\ell,r)$ for the subsequence $m_{\ell},\ldots,m_{r-1}$, and,
depending on the context, for the product $m_{\ell}\circ\ldots\circ m_{r-1}$.
If $r\leq\ell$ then $m[\ell,r)$ is the neutral monoid element.
In order to define special intervals, we consider positions $j\in[n]$ in
$t$-adic representation as $\sum_{i=0}^{h-1}j_{i}t^{i}$, where $j_{i}\in[t]$,
for every $0\leq i<h$. We say that a number $j$ has _height_ $k$ if $k<h$ is
the maximum number such that
$\sum_{i=k}^{h-1}j_{i}t^{i}=\sum_{i=0}^{h-1}j_{i}t^{i}$, that is, if
$j_{0},\ldots,j_{k-1}$ are zero and $j_{k}$ is not (or $j=0$ and $k=h-1$).
We say that an interval $[i,j)$ of positions is _special_ if, for some $k$ and
$p$, the height of $i$ and $j$ is at least $k$ and $pt^{k+1}\leq i\leq
j\leq(p+1)t^{k+1}$. It is easy to see that, for each $k$ and $p$ there are
$\mathcal{O}(t^{2})$ such special intervals.
The dynamic program maintains the product $m[i,j)$ for all special intervals
$[i,j)$.
To show that the two crucial observations (1) and (2) stated above are
correct, we need some more notation. For a number $j\in[n]$ we write $\lfloor
j\rfloor_{k}$ for the largest number $p$ of height at least $k$ with $p\leq j$
and $\lceil j\rceil^{k}$ for the smallest number $q$ of height at least $k$
with $j\leq q$, i.e.
$\lfloor
j\rfloor_{k}\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\displaystyle
j_{k}t^{k}+\sum_{i=k+1}^{h-1}j_{i}t^{i}\quad\text{ and }\quad\lceil
j\rceil^{k}\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\displaystyle(j_{k}+1)t^{k}+\sum_{i=k+1}^{h-1}j_{i}t^{i}.$
Towards observation (1), it is not hard to see that the following equality
holds for all $0\leq\ell,r\leq n-1$ and that the product consists of
$2(h-1)+1=\mathcal{O}(1)$ factors. Suppose that $k<h$ is the smallest number
such that $\lceil\ell\rceil^{k+1}>\lfloor r\rfloor_{k+1}$, then:
$m[\ell,r)=\prod_{j=0}^{k-1}m[\lceil\ell\rceil^{j},\lceil\ell\rceil^{j+1})\;\circ\;m[\lceil\ell\rceil^{k},\lfloor
r\rfloor_{k})\;\circ\;\prod_{j=0}^{k-1}m[\lfloor r\rfloor_{j+1},\lfloor
r\rfloor_{j})$
It is thus straightforward to answer range queries by a dynamic program with
work $\mathcal{O}(1)$.
Furthermore, towards observation (2), if position $p$ is changed, only the
information for the $ht^{2}$ special intervals that include $p$ needs to be
updated. This can be done in $h=\mathcal{O}(1)$ phases. In phase $k$, the
special intervals with height $k$ are updated. The new product
$m^{\prime}[i,j)$ can be obtained via
$m^{\prime}[i,j)=m[i,\lfloor p\rfloor_{k})\circ m^{\prime}[\lfloor
p\rfloor_{k},\lceil p\rceil^{k})\circ m[\lceil p\rceil^{k},j),$
where $m^{\prime}[\lfloor p\rfloor_{k},\lceil p\rceil^{k})$ is the already
computed new value of the special interval $[\lfloor p\rfloor_{k},\lceil
p\rceil^{k})$ of height $k-1$. This completes the proof sketch for Proposition
3. ∎
Towards a proof of Theorem 5.2, we employ the following algebraic property of
syntactic monoids of star-free languages, due to Schützenberger, McNaughton
and Papert (see, e.g., [27, Chapter V.3]). A monoid $M$ is called a _group_ if
for each $m\in M$ there is an inverse $m^{-1}$ with $m\cdot m^{-1}=1$. A
monoid is called _group-free_ if it does not contain a nontrivial group.
###### Lemma 5 ([27, Theorem V.3.2])
A language $L$ is star-free iff $M(L)$ is finite and group-free.
In [9] it was shown that ${\textsc{RangeEval}}(M)$ for a group-free monoid $M$
can be maintained in $\mathcal{O}(\log\log n)$ sequential time on RAMs with
cell size $\mathcal{O}(\log n)$. We adapt their algorithm to show that
${\textsc{RangeEval}}(M)$ can be maintained in DynFO with logarithmic work.
The main ingredient for the algorithm of [9] is the following consequence of a
decomposition theorem by Krohn and Rhodes.
###### Lemma 6 ([18][9])
Let $M$ be a finite, group-free monoid. Then one of the following holds:
1. (a)
$M=\\{1\\}$,
2. (b)
there is a $k$ such that
$M-\\{1\\}=\\{\sigma,\sigma^{2},\ldots,\sigma^{k}=\sigma^{k+1}\\}$,
3. (c)
$\sigma\sigma^{\prime}=\sigma$, for all $\sigma,\sigma^{\prime}\in M-\\{1\\}$,
or
4. (d)
$M=V\cup T$ where $T\neq M$ and $V\neq M$ are submonoids of $M$ and
$T-\\{1\\}$ is a left ideal of $M$, i.e. $\sigma_{M}\sigma_{T}\in T-\\{1\\}$
for all $\sigma_{M}\in M$ and $\sigma_{T}\in T-\\{1\\}$
We illustrate this lemma by the following example. It will be reused to
illustrate the construction in our dynamic program for maintaining
${\textsc{RangeEval}}(M)$.
###### Example 1
Consider the language
$L=L\big{(}c^{\ast}ac^{\ast}ac^{\ast}b(a+b+c)^{\ast}\big{)}$ that contains all
words over the alphabet $\Sigma=\\{a,b,c\\}$ having at least one $b$ and at
least two $a$’s somewhere before the first $b$. It syntactic monoid is
$M=\\{1,A,A^{2},B,D,E\\}$ with the following multiplication table:
${}_{x}\setminus{}^{y}$ | $1$ | $A$ | $A^{2}$ | $B$ | $D$ | $E$
---|---|---|---|---|---|---
$1$ | $1$ | $A$ | $A^{2}$ | $B$ | $D$ | $E$
$A$ | $A$ | $A^{2}$ | $A^{2}$ | $D$ | $E$ | $E$
$A^{2}$ | $A^{2}$ | $A^{2}$ | $A^{2}$ | $E$ | $E$ | $E$
$B$ | $B$ | $B$ | $B$ | $B$ | $B$ | $B$
$D$ | $D$ | $D$ | $D$ | $D$ | $D$ | $D$
$E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
The elements of $M$ (and thus the states of the minimal automaton for $L$)
correspond to the following languages. In particular, $E$ corresponds to $L$.
$1$ | $L\big{(}c^{\ast}\big{)}$
---|---
$A$ | $L\big{(}c^{\ast}ac^{\ast}\big{)}$
$A^{2}$ | $L\big{(}c^{\ast}ac^{\ast}ac^{\ast}\big{)}$
$B$ | $L\big{(}c^{\ast}b(a+b+c)^{\ast}\big{)}$
$D$ | $L\big{(}c^{\ast}ac^{\ast}b(a+b+c)^{\ast}\big{)}$
$E$ | $L\big{(}c^{\ast}ac^{\ast}ac^{\ast}b(a+b+c)^{\ast}\big{)}$
The monoid can be decomposed as $M=V\cup T$ with $T=\\{1,B,D,E\\}$ and
$V=\\{1,A,A^{2}\\}$, which are Type (c) and Type (b) submonoids, respectively,
of $M$ in the sense of Lemma 6.
The characterization of group-free monoids provided by Lemma 6 is the basis
for a recursive approach for maintaining ${\textsc{RangeEval}}(M)$.
###### Theorem 0.B.1
Let $M$ be a group-free monoid. Then ${\textsc{RangeEval}}(M)$ can be
maintained in DynFO with work $\mathcal{O}(\log n)$ for each change and query
operation.
###### Proof
The proof is by induction on the decomposition of $M$. We describe, for each
case of Lemma 6, how a dynamic program ${\mathcal{P}}$ can maintain
${\textsc{RangeEval}}(M)$ with $\mathcal{O}(\log n)$ work.
Let $m=m_{0}\ldots m_{n-1}$ be an input sequence. Case (a) from Lemma 6 is
trivial.
For Case (b), suppose that
$M=\\{1,\sigma,\sigma^{2},\ldots,\sigma^{k}=\sigma^{k+1}\\}$ for some fixed
$k$. As in this case the monoid elements (except for $1$) only differ in their
exponents, the result of a range query $\textsc{range}(\ell,r)$ can be
computed by summing over the exponents of all elements $\neq 1$ between $\ell$
and $r$. A closer look reveals that it suffices to sum over the exponents of
the first up to $k$ elements differing from the identity element $1$, because
$\sigma^{k}=\sigma^{k+1}$. For finding these up to $k$ elements quickly, the
program uses a supplementary instance of NextInK for the set $K=\\{i\mid
w_{i}\neq 1\\}$.
These ideas lead to Algorithm 5 for answering range queries. As only a
constant number of successors are queried from $K$, it requires at most
$\mathcal{O}(\log n)$ work due to Lemma 1. For changes of the input instance,
only the supplementary NextInK instance needs to be updated, which also
requires at most $\mathcal{O}(\log n)$ work by Lemma 1.
1:on query $\textsc{range}(\ell,r)$:
2: $s\leftarrow 0$
3: $i\leftarrow(\ell-1)$
4: for $1\ \textbf{to}\ k$ do
5: $i=K.\textsc{succ}(i)$
6: if $i\leq r$ then
7: $s\leftarrow s+\textnormal{exponent}(i)$
8: if $s>0$ then
9: return $a^{s}$
10: else
11: return $1$
Algorithm 5 Querying the product $m_{\ell}\cdot\ldots\cdot m_{r}$ in Case (b)
of the proof of Theorem 0.B.1.
For Case (c), suppose that $M$ is a monoid with $\sigma\sigma^{\prime}=\sigma$
for all $\sigma,\sigma^{\prime}\in M-\\{1\\}$. In this case, a range query
$\textsc{range}(\ell,r)$ results in the first element in the interval
$[\ell,r]$ which is not the identity element (if such an element does not
exist, then the result is the identity element). In order to find such an
element quickly, the dynamic program maintains a supplementary NextInK
instance for the set $K=\\{i\mid w_{i}\neq 1\\}$.
Now range queries can be answered according to Algorithm 6, which clearly
requires at most $O(\log n)$ work. For changes of the input instance, again
only the supplementary NextInK instance needs to be updated, which also
requires at most $\mathcal{O}(\log n)$ work.
1:on query $\textsc{range}(\ell,r)$:
2: $i\leftarrow K.\textsc{succ}(\ell-1)$
3: if $i\leq r$ then
4: return $m_{i}$
5: else
6: return $1$
Algorithm 6 Querying the product $m_{\ell}\cdot\ldots\cdot m_{r}$ in Case (c)
of the proof of Theorem 0.B.1.
For Case (d), suppose that $M=V\cup T$ where $T\neq M$ and $V\neq M$ are
submonoids of $M$ and $T-\\{1\\}$ is a left ideal of $M$. Because of $T\neq M$
and $V\neq M$, by induction we can assume that ${\textsc{RangeEval}}(T)$ and
${\textsc{RangeEval}}(V)$ can be maintained with work $\mathcal{O}(\log n)$.
Denote $T-\\{1\\}$ by $T_{\neq 1}$. The idea of the dynamic program is to
split the input sequence $m$ into $T_{\neq 1}$-blocks, consisting of a maximal
sequence of consecutive elements of $T_{\neq 1}$ only, and $\overline{T_{\neq
1}}$-blocks consisting of maximal sequence of elements of $\overline{T_{\neq
1}}$ only. The program maintains precomputed products of each
$\overline{T_{\neq 1}}$-block combined with its successive $T_{\neq 1}$-block.
Because $T_{\neq 1}$ is a left ideal, all these precomputed block products
result in elements of $T_{\neq 1}$. A range query $\textsc{range}(\ell,r)$ for
$m$ can then be answered by computing the product of these precomputed block
products between $\ell$ and $r$ and adding the at most two incomplete block
products at the beginning and the end of the queried range.
For encoding the blocks as well as the precomputed products, the program uses
three sequences $t$, $v$ and $u$ with the following intention. The sequences
$v$ and $t$ partition $m$ into elements of $T_{\neq 1}$ and $V-T_{\neq 1}$,
respectively. For the definition of $u$, let $K=\\{i\mid w_{i}\in T_{\neq
1}\land w_{i+1}\notin T_{\neq 1}\\}$ be the set of _switching positions_ ,
i.e. positions of the input sequence where the monoid element changes from
$T_{\neq 1}$ to $V-T_{\neq 1}$. At a switching position $i$, $u_{i}$ stores
the product of elements between the previous switching position and $i$. In
order to find switching positions quickly, the dynamic program maintains a
supplementary NextInK instance for the set $K$.
More precisely, the three sequences are defined as
$\displaystyle t_{i}=$ $\displaystyle\begin{cases}m_{i}&\textnormal{if
$m_{i}\in T_{\neq 1}$}\\\ 1&\textnormal{otherwise}\end{cases},$ $\displaystyle
v_{i}=$ $\displaystyle\begin{cases}m_{i}&\textnormal{if $m_{i}\in V-T_{\neq
1}$}\\\ 1&\textnormal{otherwise}\end{cases},$ $\displaystyle u_{i}=$
$\displaystyle\begin{cases}\prod_{j=k_{i}+1}^{i}m_{j}&\textnormal{if $i\in
K$}\\\ 1_{T}&\textnormal{otherwise}\end{cases},$
where $k_{i}=K.\textsc{pred}(i)$ if $K.\textsc{pred}(i)\neq\bot$ and $k_{i}=0$
otherwise. Note that $m_{i}\in T_{\neq 1}$ for all $i$ with $u_{i}\neq 1$
because $T_{\neq 1}$ is a left interval. We refer to Example 2 for an
illustration of these sequences.
The result of a range query $\textsc{range}(\ell,r)$ can then be computed as
$\prod_{i=\ell}^{r}m_{i}=\prod_{i=\ell}^{k_{1}}v_{i}\prod_{i=\ell}^{k_{1}}t_{i}\prod_{i=k_{1}+1}^{k_{q}}u_{i}\prod_{i=k_{q}+1}^{r}v_{i}\prod_{i=k_{q}+1}^{r}t_{i}$
where $k_{1}<\ldots<k_{q}$ are all positions in $K\cap\\{\ell-1,\ldots,r\\}$.
The resulting dynamic program, see Algorithm 7, clearly requires at most
$O(\log n)$ work.
1:on query $\textsc{range}(\ell,r)$:
2: $k_{1}\leftarrow K.\textsc{succ}(\ell-1)$
3: $k_{q}\leftarrow K.\textsc{pred}(r)$
4: return $v.\textsc{range}(\ell,k_{1})\circ t.\textsc{range}(\ell,k_{1})\circ
u.\textsc{range}(k_{1}+1,k_{q})$
5: $\circ\;v.\textsc{range}(k_{q}+1,r)\circ t.\textsc{range}(k_{q}+1,r)$
Algorithm 7 Querying the product $m_{\ell}\cdot\ldots\cdot m_{r}$ in Case (d)
of the proof of Theorem 0.B.1.
For changes of the input instance, the supplementary NextInK instance for the
switching positions needs to be updated, which requires at most
$\mathcal{O}(\log n)$ work. Also the sequences $t$, $v$ and $u$ need to be
updated. For $v$ and $t$ this is easy, but updating $u$ requires some effort.
For maintaining $u$ after a change of the input instance at position $p$, we
distinguish the cases as summarized in Table 2. In every case, the precomputed
block product stored in the next switching position after $p$ needs to be
recomputed. Additionally, if the insertion created a new block by splitting a
block (see Cases (4) and (5)) or removed a block by merging two blocks (see
Cases (2) and (3)) either $u_{p}$ or $u_{p-1}$ has to be updated as well.
Case condition | Changes to $u$
---|---
(1) Switching positions do not change, i.e. either (a) $m_{p}\in T_{\neq 1}$ iff $m^{\prime}_{p}\in T_{\neq 1}$, (b) $m_{p-1}\notin T_{\neq 1}$ and $m^{\prime}_{p},m_{p+1}\in T_{\neq 1}$, or (c) $m_{p-1},m^{\prime}_{p}\notin T_{\neq 1}$ and $m_{p+1}\in T_{\neq 1}$ | $\displaystyle u_{k}\leftarrow\prod_{i=j+1}^{k}v_{i}\circ\prod_{i=j+1}^{k}t_{i}$
(2) Two $T_{\neq 1}$-blocks are merged, i.e. $m_{p-1},m^{\prime}_{p},m_{p+1}\in T_{\neq 1}$ and $m_{p}\notin T_{\neq 1}$, | $u_{k}$ as in Case (1) and $u_{p-1}\leftarrow 1$
(3) Two $\overline{T_{\neq 1}}$-blocks are merged, i.e. $m_{p-1},m^{\prime}_{p},m_{p+1}\notin T_{\neq 1}$ and $m_{p}\in T_{\neq 1}$, | analogous to (2)
(4) A $T_{\neq 1}$-block is split, i.e. $m_{p-1},m_{p}\in T_{\neq 1}$ and $m^{\prime}_{p}\notin T_{\neq 1}$, | $\displaystyle u_{k}\leftarrow\prod_{i=j+1}^{p-1}v_{i}\circ\prod_{i=j+1}^{p-1}t_{i}$ and $\displaystyle u_{p-1}\leftarrow\prod_{i=p}^{k}v_{i}\circ\prod_{i=p}^{k}t_{i}$,
(5) A $\overline{T_{\neq 1}}$-block is split, i.e. $m^{\prime}_{p},m_{p+1}\notin T_{\neq 1}$ and $m_{p}\in T_{\neq 1}$, | analogous to (4)
Table 2: Summary of the cases for updating $u$ after a change at position
$p$. Suppose that $v,t$ and $K$ are already updated. By $m_{p}$ and
$m^{\prime}_{p}$ we denote the elements at position $p$ before and after the
change operation, respectively. In all cases $k=K.\textsc{succ}(p)$ is defined
as the next switching position after $p$. For Case (1), $j$ is defined as
$K.\textsc{pred}(p)$; for Case (3) as $K.\textsc{pred}(p-1)$.
In all cases, $u$ is updated with a constant number of range queries to $v$
and $t$ which results overall in $\mathcal{O}(\log n)$ work. ∎
Combining Lemma 5 and Theorem 0.B.1 we get the desired upper work bound for FO
definable languages.
###### Example 2
We revisit the monoid $M$ with decomposition into $T$ and $V$ introduced in
Example 1. Consider the sequence $m$ with its sequences $t,v$ and $u$ as
defined in the proof of Theorem 0.B.1:
1 2 3 01234567890123456789012345678901 $\displaystyle m=$
AA1AABBA11A1AABAA1B11A1B1A1A11BB $\displaystyle t=$
11111BB1111111B111B1111B111111BB $\displaystyle v=$
AA1AA11A11A1AA1AA1111A111A1A1111 $\displaystyle u=$
111111E1111111E111E1111D1111111E
The sequences $t$ and $u$ consist of $T_{\neq 1}$-elements and $v$ of
$V-T_{\neq 1}$-elements only. For a query $\textsc{range}(1,26)$ on $m$ the
set of switching positions is $\\{6,14,18,23\\}$, so the result of the query
can be computed as follows:
$\displaystyle\prod_{i=1}^{26}m_{i}$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\prod_{i=1}^{6}v_{i}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\prod_{i=1}^{6}t_{i}}{\color[rgb]{0.0,0.42,0.24}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.42,0.24}\prod_{i=7}^{23}u_{i}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\prod_{i=24}^{26}v_{i}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\prod_{i=24}^{26}t_{i}}$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\texttt{A}^{2}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\texttt{B}}{\color[rgb]{0.0,0.42,0.24}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.42,0.24}\texttt{E}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\texttt{A}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\texttt{1}}$
$\displaystyle=\texttt{E}$
Note, that the results of the five subproducts are determined by subqueries to
$t,v$ and $u$ and can be evaluated as described in Case (b) for $v$ and Case
(c) for $t$ and $u$.
## Appendix 0.C Appendix for Section 6
###### Proof (Proof sketch)
[of Lemma 2] Let ${\mathcal{P}}$ be a dynamic program that maintains
${\textsc{Member}}(L)$ with work $f(n)$. Whether $w\in L$ holds for a given
input string $w=\sigma_{0}\sigma_{1}\ldots\sigma_{n-1}$ can be decided by
simulating ${\mathcal{P}}$ on the sequence of change operations that inserts
$\sigma_{1}$ to $\sigma_{n}$ one after the other. By simulating the PRAM
sequentially, the work $\mathcal{O}(f(n))$ for one change step translates into
sequential time $\mathcal{O}(f(n))$, thus yielding an overall sequential time
$\mathcal{O}(n\cdot f(n))$. ∎
###### Proof (Proof of Theorem 6.1)
Let $G$ be the grammar from Theorem 6.1 and let us assume, towards a
contradiction, that there is a dynamic program for $L(G)$ with work
$\mathcal{O}(n^{\omega-1-\epsilon})$. By Lemma 2 this would yield an algorithm
for the word problem for $L(G)$ with time $\mathcal{O}(n^{\omega-\epsilon})$.
Theorem 6.1 would then give an algorithm for $k$-Clique with time bound
$\mathcal{O}(n^{(\frac{k}{3}+1)(\omega-\epsilon)})$.
However, for $k\geq\frac{18}{\epsilon}$, this yields a contradiction as
follows.
We have
$(\frac{k}{3}+1)(\omega-\epsilon)=\frac{\omega}{3}k+\omega-(\frac{\epsilon}{3}k+\epsilon)\leq\frac{\omega}{3}k-\frac{\epsilon}{6}k$,
since
$\frac{\epsilon}{3}k+\epsilon=\frac{\epsilon}{6}k+\frac{\epsilon}{6}k+\epsilon\geq
3+\frac{\epsilon}{6}k+\epsilon$. Thus, $k$-Clique could be solved in time
$\mathcal{O}(n^{(1-\frac{\epsilon}{6})k})$. ∎
###### Proof (Proof of Theorem 6.3)
We give a complete proof here, repeating parts of the proof sketch in the
body, for convenience.
With a string $w$ over $\Sigma_{1}$ we associate the _reduced string_ $\mu(w)$
that basically consists of the unmatched closing parentheses of $w$ followed
by the unmatched opening parentheses of $w$.
Formally, $\mu(w)$ can be defined for strings $w\in\Sigma_{1}^{\ast}$ as
$\mu(\epsilon)=\epsilon$ and
$\mu(z\sigma)=\begin{cases}z^{\prime}&\textnormal{if
$\mu(z)=z^{\prime}\langle$ and $\sigma=\rangle$}\\\
\mu(z)\sigma&\textnormal{else}\end{cases}$
for every $\sigma\in\Sigma_{1}$ and $z\in\Sigma_{1}^{\ast}$. As indicated
before, a reduced string can be split into two strings $u$ and $v$ of the
forms $\rangle^{\ast}$ and $\langle^{\ast}$, respectively. We call $u$ the
unmatched (closing) parentheses of $w$ to the left and $v$ the unmatched
(opening) parentheses of $w$ to the right.
The algorithm of [8] stores, for each node $x$ of $T$, the reduced string
$\mu(\textnormal{str}(x))$ of its associated string, and updates these strings
bottom up after each change, starting from the changed position, resulting in
logarithmic (sequential) update-time. We show that the new reduced strings can
be computed in constant parallel time. To this end, we show that for each
affected node $x$ of $T$ its new reduced string can be determined by a
parallel computation that does not need to know the new reduced string of its
affected child.
We describe a dynamic program ${\mathcal{P}}$ that maintains
${\textsc{Member}}(D_{1})$, using a binary tree $T$ as sketched above, which
is constructed by the first-order initialisation as in the proof of Lemma 1.
For each node $x$ of the tree, ${\mathcal{P}}$ represents its reduced string
by the number of unmatched closing parentheses of $\textnormal{str}(x)$ to the
left and to the right as auxiliary functions $\ell(x)$ and $r(x)$,
respectively. In particular, the current word is well-balanced if
$r(\textnormal{root})=\ell(\textnormal{root})=0$.
In the following, we describe how ${\mathcal{P}}$ can update $\ell(x)$ and
$r(x)$ for a node $x$ after a closing parenthesis $\rangle$ was inserted at
some position $p$. Maintaining $\ell$ and $r$ for the other change operations
is analogous. Note that $\ell$ and $r$ only need to be updated for nodes whose
substring contains the changed position $p$, that is, for ancestors of the
leaf at position $p$.
There are two types of effects that an insertion of a closing parenthesis
could have on $x$: Either $\ell(x)$ is increased by one and $r(x)$ remains
unchanged, or $r(x)$ is decreased by one and $\ell(x)$ remains unchanged. We
denote these effects by the pairs $(+1,0)$ and $(0,-1)$, respectively. We next
analyse how the effect that applies to $x$ depends on the effect of its
affected child. We will see then that the effect on the child needs not to be
known to update $x$, but that it can be determined by a direct parallel
exploration of the nodes on the path from $x$ to $p$.
To this end, let $y_{1}$ be the left child of $x$ and $y_{2}$ its right child.
The effect on $x$ depends on whether the changed position $p$ is in
$\textnormal{str}(y_{1})$ or in $\textnormal{str}(y_{2})$, and on the relation
between $r(y_{1})$ and $\ell(y_{2})$ before the change. Table 3 lists all
possible combinations for (1) whether the changed position $p$ is in
$\textnormal{str}(y_{1})$ or in $\textnormal{str}(y_{2})$, (2) which effect
occurs there, and (3) the relation between $r(y_{1})$ and $\ell(y_{2})$, and
which effect is applied to $x$ in each case. For example, if $p$ is in the
subtree of $y_{1}$, the effect on $y_{1}$ is $(0,-1)$, and
$r(y_{1})\leq\ell(y_{2})$ holds. Then, the inserted closing parenthesis
matches (and is to the right of) a former unmatched opening parenthesis in
$\textnormal{str}(y_{1})$. As $r(y_{1})\leq\ell(y_{2})$, all unmatched opening
parentheses of $\textnormal{str}(y_{1})$ could previously be matched by
unmatched closing parentheses of $\textnormal{str}(y_{2})$. Since after the
change there is one unmatched opening parenthesis less in
$\textnormal{str}(y_{1})$, the unmatched opening parentheses of $y_{1}$ can
still be matched but, on the other hand, one more closing parenthesis of
$y_{2}$ cannot be matched in $\textnormal{str}(x)$, so the effect on $x$ is
$(+1,0)$.
| $p$ is in $\textnormal{str}(y_{1})$ | $p$ is in $\textnormal{str}(y_{2})$
---|---|---
$r(y_{1})\leq\ell(y_{2})$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(+1,0)\end{array}$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(0,-1)\end{array}$
$r(y_{1})>\ell(y_{2})$ | $\begin{array}[]{c}(+1,0)\to(+1,0)\\\ (0,-1)\to(0,-1)\end{array}$ | $\begin{array}[]{c}(+1,0)\to(0,-1)\\\ (0,-1)\to(0,-1)\end{array}$
Table 3: The effect on $x$ after a closing parenthesis was inserted at
position $p$. The effects depend on the effect on the children $y_{1}$ and
$y_{2}$ of $x$: for example, an entry ’$(0,-1)\to(+1,0)$’ in the column ’$p$
is in $\textnormal{str}(y_{1})$’ means that if the change operation has effect
$(0,-1)$ on $y_{1}$ then the change operation has effect $(+1,0)$ on $x$.
A closer inspection of Table 3 reveals a crucial observation: in the upper
left and the lower right field of the table, the effect on $x$ is
_independent_ of the effect on the child (being it $y_{1}$ or $y_{2}$). That
is, these cases induce an effect on $x$ independent of the children. We thus
call these cases _effect-inducing_. In the other two fields, the effect on $x$
depends on the effect at the child, but in the simplest possible way: they are
just the same. That is the effect at the child is just adopted by $x$. We call
these cases _effect-preserving_. To determine the effect at $x$ it is thus
sufficient to identify the highest affected descendant node $z$ of $x$, where
an effect-inducing case applies, such that for all intermediate nodes between
$x$ and $z$ only effect-preserving cases apply.
Algorithm 8 implements this idea. First it determines, for each ancestor $x$
of the change position $p$, whether it is effect-inducing (Ind) and which
effect is induced (Ind$+$ for $(+1,0)$ and Ind$-$ for $(0,-1)$). Then it
identifies, for each $x$, the node $z$ (represented by its height $i$ above
$p$) as the unique effect-inducing node that has no effect-inducing node on
its path to $x$.
1:on change $\textsc{ins}_{\rangle}(p)$
2: with
3: for $1\leq i\leq\log n$ pardo
4: $x(i)\leftarrow T.\text{anc}(p,i)$
5: $c(i)\leftarrow T.\text{anc}(p,i-1)$
6: $\text{Ind+}(i)\leftarrow\text{1st}(x(i))=c(i)\land
r(\text{1st}(x(i)))\leq\ell(\text{2nd}(x(i)))$
7: $\text{Ind-}(i)\leftarrow\text{2nd}(x(i))=c(i)\land
r(\text{1st}(x(i)))>\ell(\text{2nd}(x(i)))$
8: $\text{Ind}(i)\leftarrow\text{Ind+}(i)\lor\text{Ind-}(i)$
9: update $(\ell,r)$ at $T.\text{anc}(p,k)$, for all $k\leq\log n$, by:
10: if $k=0$ then
11: return $(1,0)$
12: else
13: $x\leftarrow T.\text{anc}(p,k)$
14: if exists($1\leq i\leq k$ $\mid$ $\text{Ind}(i)$) then
15: $i\leftarrow\textbf{getUnique}(1\leq i\leq
k\mid\text{Ind}(i)\land\neg\textbf{exists}(i<j\leq k\mid\text{Ind}(j)))$
16: if $\text{Ind+}(i)$ then
17: return $(\ell(x)+1,r(x))$
18: else
19: return $(\ell(x),r(x)-1)$
20: else
21: return $(\ell(x)+1,r(x))$
Algorithm 8 Updating $(\ell,r)$ after the insertion of a closing parenthesis
in the proof of Theorem 6.3.
The node $z$ can be identified with work $\mathcal{O}((\log n)^{2})$, as $z$
is one of at most $\log n$ many nodes on the path from $x$ to $v$, and one
needs to check that all nodes between $x$ and $z$ are effect-preserving. As
the auxiliary relations need to be updated for $\log n$ many nodes, the
overall work of ${\mathcal{P}}$ is $\mathcal{O}((\log n)^{3})$.
∎
###### Proof (Proof sketch)
[of Theorem 6.4] In the following we reuse the definition of _special
intervals_ from the proof of Proposition 3 as well as the definition of $\ell$
and $r$ from the proof of Proposition 6.3. We first describe a dynamic program
${\mathcal{P}}$ for ${\textsc{RangeMember}}(D_{1})$. It maintains $\ell$ and
$r$ for all special intervals. Similar to the proof of Proposition 3, the two
crucial observations are that (1) a range query can be answered with the help
of a constant number of special intervals, and (2) the change operation
affects only a bounded number of special intervals per level.
Towards observation (1), recall that each (not necessarily special) interval
can be split into a constant number of special intervals. To answer a query
$\textsc{range}(p,q)$ the program can verify whether the _balance factor_ ,
that is the sum of the differences $\ell-r$ over all special intervals
splitting $[p,q]$, is zero, and that the balance factor is not negative for
any prefix of the sequence of the special intervals.
Towards observation (2), note that a change operation only affects the special
intervals that contain the changed position and that $\ell$ and $r$ can be
updated bottom up in the same fashion as in the proof of Proposition 3.
We turn now to ${\textsc{RangeMember}}(D_{k})$. Here, ${\mathcal{P}}$
maintains again $\ell$ and $r$ for all special intervals but ignoring the type
of parenthesis. If the query asks whether the string from position $p$ to
position $q$ is in $D_{k}$, then the program checks that inside $[p,q]$ all
opening parentheses have consistent closing parentheses and that the “level”
of $p$ is the same as of $q$, that is, that the number of opening parentheses
is the same as the number of closing parentheses. This can be done by
inspecting the special intervals that lead from $p$ to $q$ in the canonical
way.
As there are a bounded number of levels in the hierarchy the work is
$\mathcal{O}(n^{\epsilon})$ per position and therefore
$\mathcal{O}(n^{1+\epsilon})$ in total. ∎
###### Proof (Proof sketch)
[of Lemma 3] The proof is not entirely trivial, as we allow strings to have
positions labelled by $\epsilon$, so we cannot simply compare the positions of
the input strings one by one in parallel. To deal with positions labelled by
$\epsilon$, the dynamic program maintain a bijection between the
non-$\epsilon$ positions of the two given strings $s_{1}$ and $s_{2}$ such
that the $i$-th non-$\epsilon$ position of $s_{1}$ is mapped to the $i$-th
non-$\epsilon$ position in $s_{2}$, and vice versa. The strings are equal if
all mapped positions have the same label, which can be checked with work
$\mathcal{O}(n)$ after each change.
Such a bijection can be defined using the functions $\textnormal{rank}_{j}(i)$
that map a non-$\epsilon$ position $i$ in the string $s_{j}$ to the number
$m$, if $i$ is the $m$-th non-$\epsilon$ position $i$ in $s_{j}$, and the
inverse $\textnormal{rank}_{j}^{-1}(i)$ of this function. The function
$\textnormal{rank}_{j}$ for a string $s_{j}$ can be maintained as follows. If
a (non-$\epsilon$) symbol is inserted at position $p$, the rank of all
non-$\epsilon$ positions $i>p$ is increased by $1$ and the changed position
$p$ gets $\textnormal{rank}_{j}(i^{\prime})+1$ where $i^{\prime}$ is the next
non-$\epsilon$ position to the left of $p$. Finding this position can be done
in DynFO with $\mathcal{O}(\log n)$ work thanks to Lemma 1. If a symbol is
deleted from a position $p$, the rank of all non-$\epsilon$ positions $i>p$ is
decreased by $1$ and the $\textnormal{rank}_{j}(p)$ is no longer defined. The
inverse can be maintained in a very similar way. The necessary work for
updating the functions $\textnormal{rank}_{j}$ and
$\textnormal{rank}_{j}^{-1}$ at the changed position $p$ is bounded by
$\mathcal{O}(\log n)$, for all other positions it is bounded by a constant.
So, the overall work is $\mathcal{O}(n)$. ∎
Unfortunately, if we generalize the definition of a reduced string $\mu(w)$ to
arbitrary many types of parentheses, the result is in general not a string of
closing parentheses followed by a string of opening ones. The idea of [8] is
now to maintain the parentheses to the left and right that remain if we reduce
the string by matching opening and closing parenthesis regardless of their
type.
For a string $w$ over the alphabet
$\Sigma_{k}=\Sigma_{k}^{\langle}\cup\Sigma_{k}^{\rangle}=\\{\langle_{1},\ldots,\langle_{k}\\}\cup\\{\rangle_{1},\ldots,\rangle_{k}\\}$
we define a _type-aware reduced string_ $\mu_{a}(w)$ and a _type-unaware
reduced string_ $\mu_{u}(w)$ as $\mu_{a}(\epsilon)=\mu_{u}(\epsilon)=\epsilon$
and
$\mu_{a}(z\sigma)=\begin{cases}z^{\prime}&\textnormal{if
$\mu_{a}(z)=z^{\prime}\langle_{i}$ and $\sigma=\rangle_{i}$, for some $i\leq
k$}\\\ \mu_{a}(z)\sigma&\textnormal{else}\end{cases}$
$\mu_{u}(z\sigma)=\begin{cases}z^{\prime}&\textnormal{if
$\mu_{u}(z)=z^{\prime}\langle_{i}$ and $\sigma=\rangle_{j}$, for some $i,j\leq
k$}\\\ \mu_{u}(z)\sigma&\textnormal{else.}\end{cases}$
Note that, as before, the type-unaware reduced string can always be split into
a first and second part that consist only of closing and opening parentheses,
respectively. For a string $w\in\Sigma_{k}^{\ast}$, we therefore use the type-
unaware reduced string to define the unmatched parentheses to the left
$\mu_{u}^{\ell}(w)$ and to the right $\mu_{u}^{r}(w)$. We slightly abuse the
notation by writing $\mu_{u}^{\ell}(x)$ for a node $x$ instead of
$\mu_{u}^{\ell}(\textnormal{str}(x))$ (as well as $\mu_{u}^{r}(x)$,
$\mu_{a}(x)$ and $\mu_{u}(x)$).
To be able to answer ${\textsc{Member}}(D_{k})$, the dynamic program we
construct maintains the unmatched parentheses for every node $x$ of a tree
spanning the input word, and a bit $M(x)$ that indicates whether
$\mu_{a}(x)=\mu_{u}(x)$ holds.
In [8] the unmatched parentheses and the bit $M(x)$ are updated bottom up
after a change operation. For a node $x$ with children $y_{1}$ and $y_{2}$ the
bit $M(x)$ is set to $1$ if both $M(y_{1})$ and $M(y_{2})$ are set to $1$ and
if the unmatched opening parentheses of $\textnormal{str}(y_{1})$ match
correctly with the unmatched closing parentheses of $\textnormal{str}(y_{2})$
to the left. The latter condition can be tested by testing whether two strings
are equal, namely the string $\mu_{u}^{r}(y_{1})$ and the string that results
from $\mu_{u}^{\ell}(y_{2})$ by reversing the string and exchanging a closing
parenthesis by an opening parenthesis of the same type (assuming that both
strings have equal length, otherwise surplus symbols in the longer string are
ignored).
In [8], StringEquality is maintained in polylogarithmic sequential time using
[21], in [10] it is shown that maintaining whether two substrings are equal is
in DynFO with a polynomial amount of work. Lemma 3 tells us that a linear
amount of work is sufficient if one is only interested in equality of entire
strings.
###### Proof (Proof sketch)
[of Lemma 6.5] We describe a dynamic program ${\mathcal{P}}$ that maintains
${\textsc{Member}}(D_{k})$, using a precomputed binary tree $T$ spanning the
input word. For each node $x$ of $T$ the program maintains $\mu_{u}^{\ell}(x)$
and $\mu_{u}^{r}(x)$, represented by the subset of positions that constitute
these strings, the bit $M(x)$, as well as the string
$\big{(}\overline{\mu_{u}^{\ell}(x)}\big{)}^{R}$ that results from
$\mu_{u}^{\ell}(x)$ by first exchanging each closing parenthesis by an opening
parenthesis of the same type, and then reversing the string. Further auxiliary
information is introduced later on. The input word is in $D_{k}$ if the root
of the tree satisfies $M(\textnormal{root})=1$ and
$\mu_{u}^{\ell}(\textnormal{root})=\mu_{u}^{r}(\textnormal{root})=\epsilon$.
We first describe how $\mu_{u}^{\ell}(x)$ and $\mu_{u}^{r}(x)$ can be
maintained with work $\mathcal{O}((\log n)^{3})$ for a node $x$ after a
closing parenthesis is inserted at a position $p$ corresponding to a leaf
below $x$. Similar to $D_{1}$, after such an insertion either an opening
parenthesis has to be removed from $\mu_{u}^{r}(x)$ or a closing one has to be
inserted into $\mu_{u}^{\ell}(x)$. In contrast to $D_{1}$ it is not sufficient
to maintain only the number of unmatched parentheses, as we have to check
whether only parenthesis of the same type are matched, so the unmatched
parentheses will be maintained explicitly. But similar to $D_{1}$, one can
infer how the unmatched parentheses need to be updated for a tree node $x$
either directly from the existing auxiliary information of $x$ and its
children, or from the auxiliary information of some easy-to-identify node that
lies on the path from $x$ to the leaf corresponding to the changed position
$p$. Which case applies depends on in which “segment” of $\textnormal{str}(x)$
the change happened.
Suppose that $y_{1}$ and $y_{2}$ are the children of $x$. We call $x$ _right-
heavy_ if
$\lvert\mu_{u}^{r}(y_{1})\rvert\leq\lvert\mu_{u}^{\ell}(y_{2})\rvert$ and
_left-heavy_ otherwise. Denote by $p_{r}(x)$ the position of the first
unmatched opening parenthesis in $\mu_{u}^{r}(x)$ (if it exists), let
$p_{m}(x)$ be the position of the first unmatched opening parenthesis in
$\mu_{u}^{r}(y_{1})$ (if it exists), and let $p_{m}^{\prime}(x)$ be the
position of the parenthesis that is matched with the parenthesis at position
$p_{m}$ (so, the parenthesis at position $\lvert\mu_{u}^{r}(y_{1})\rvert-1$ in
$\mu_{u}^{\ell}(y_{2})$).
We first assume that $x$ is right-heavy. In this case we have that
$p_{m}(x)<p^{\prime}_{m}(x)<p_{r}(x)$ (if these positions exist) and consider
the following four cases:
1. (1)
$p_{r}(x)$ is defined and $p>p_{r}(x)$,
2. (2)
$p_{m}(x)$ is defined and $p_{m}(x)<p<p_{m}^{\prime}(x)$,
3. (3)
Cases (1) and (2) do not apply and
1. (3a)
there is a node $z$ on the path from $x$ to the leaf for $p$ such that Case
(2) applies for $z$, or
2. (3b)
there is no such node $z$.
In Case (1), that is, if $p_{r}(x)$ is defined and $p>p_{r}(x)$, the next
opening parenthesis to the left of $p$ in $\mu_{u}^{r}(x)$ has to be removed
from $\mu_{u}^{r}(x)$ since it is now matched by the new parenthesis. The
problem of finding this position is an instance of NextInK for the set of
opening parentheses, which can be solved with work $\mathcal{O}(\log n)$
thanks to Lemma 1.
In all other cases, a closing parenthesis has to be inserted into
$\mu_{u}^{\ell}(x)$.
In Case (2), that is, if $p_{m}(x)$ is defined and
$p_{m}(x)<p<p_{m}^{\prime}(x)$, the closing parenthesis at position
$p_{m}^{\prime}(x)$ is no longer matched by the opening parenthesis at
position $p_{m}(x)$. It is also not matched by any other opening parenthesis,
as no such unmatched parentheses exist, so it needs to be inserted into
$\mu_{u}^{\ell}(x)$.
In Case (3) we distinguish two subcases. For Case (3a), suppose that $z$ is
the first right-heavy node on the path from $x$ to the leaf of $p$ such that
$p_{m}(z)<p<p_{m}^{\prime}(z)$ holds. The same closing parenthesis that has to
be inserted into $\mu_{u}^{\ell}(z)$ has to be inserted into
$\mu_{u}^{\ell}(x)$, that is, the parenthesis at position $p_{m}^{\prime}(z)$.
For Case (3b), that is, if there is no such node $z$, the newly inserted
parenthesis itself has to be inserted into $\mu_{u}^{\ell}(x)$.
We now argue the correctness of Case (3). As Cases (1) and (2) do not apply,
the position $p$ lays in a substring $v$ of $\textnormal{str}(x)$ that either
(i) begins at the first position of $\textnormal{str}(x)$ and ends at
$p_{m}(x)-1$, or (ii) begins at $p_{m}^{\prime}(x)+1$ and ends at $p_{r}(x)$.
The string $v$ may be balanced, or it may have some unmatched closing
parentheses, but there are no unmatched opening parentheses. So, $v$ can be
split into the unmatched closing parentheses and the balanced substrings
$v_{1},\ldots,v_{j}$ between them, that is,
$v=v_{1}\sigma_{1}v_{2}\sigma_{2}\ldots\sigma_{j-1}v_{j}$ with
$\sigma_{i}\in\Sigma_{k_{i}}^{\rangle}$ and $\mu_{u}(v_{i})=\epsilon$, for all
$i$.
We first consider the case that $p$ is directly before or after an unmatched
parenthesis $\sigma_{i}$. In this case, there is no opening parenthesis before
position $p$ that the newly inserted parenthesis can match with. Therefore,
this parenthesis has to be inserted into $\mu_{u}^{\ell}(x)$. Observe that
there is no node $z$ with $p_{m}(z)<p<p_{m}^{\prime}(z)$ on the path from $x$
to the leaf of $p$, as that would imply that the inserted parenthesis could be
matched. So, in this case, the reasoning above is correct.
Now we consider the case that $p$ lies in some $v_{i}$. The balanced substring
$v_{i}$ can be split again into minimal balanced substrings. Let $p_{\langle}$
and $p_{\rangle}$ be the first and the last position of the minimal balanced
substring that contains $p$. Because a closing parenthesis is inserted between
$p_{\langle}$ and $p_{\rangle}$, the parenthesis at position $p_{\langle}$
gets matched by another parenthesis than before. Therefore, the parenthesis at
position $p_{\rangle}$ remains unmatched and has to be inserted into
$\mu_{u}^{\ell}(x)$.
To see that $p_{\rangle}$ is in fact $p_{m}^{\prime}(z)$ for the first right-
heavy node $z$ with $p_{m}(z)<p<p_{m}^{\prime}(z)$ on the path from $x$ to the
leaf of $p$, let $z^{\prime}$ be the node for which $p_{\langle}$ is in the
left child $z^{\prime}_{1}$ and $p_{\rangle}$ is in the right child
$z^{\prime}_{2}$ of $z^{\prime}$. Since $p_{\langle}$ and $p_{\rangle}$
matched each other before the insertion, they were part of
$\mu_{u}^{r}(z^{\prime}_{1})$ and $\mu_{u}^{\ell}(z^{\prime}_{2})$,
respectively. So, $p$ is strictly between $p_{m}(z^{\prime})$ and
$p_{m}^{\prime}(z^{\prime})$, and $z^{\prime}$ must be right-heavy, since the
string from $p_{\langle}$ to $p_{\rangle}$ is well-balanced, and any prefix of
$\textnormal{str}(z^{\prime})$ that consists of positions smaller than
$p_{\langle}$ cannot contain unmatched opening parentheses. Additionally, if
there was another right-heavy node $z^{\prime\prime}$ between $z^{\prime}$ and
$x$ for which $p$ is also between $p_{m}(z^{\prime\prime})$ and
$p_{m}^{\prime}(z^{\prime\prime})$, this would be a contradiction to the
definition of $p_{\langle}$ and $p_{\rangle}$ as being the first and last
position of a minimal balanced substring of $v_{i}$ that contains $p$: the
existence of such a node $z^{\prime\prime}$ would imply that there are
unmatched opening parentheses before position $p_{\langle}$. It follows that
$z^{\prime}=z$ and therefore $p_{\rangle}=p_{m}^{\prime}(z)$ for the first
right-heavy node $z$ with $p_{m}(z)<p<p_{m}^{\prime}(z)$ on the path from $x$
to the leaf of $p$.
Now, let $x$ be a left-heavy node. As $p_{r}(x)=p_{r}(y_{1})=p_{m}(x)$ (and as
$p_{m}^{\prime}(x)$ is undefined), we have one case less. The reasoning for
the other cases is exactly as for right-heavy nodes.
The strings $\mu_{u}^{\ell}(x)$ and $\mu_{u}^{r}(x)$ need to be updated for
all $\log n$ nodes $x$ on the path from the root to the leaf of $p$. The work
for a single node is $\mathcal{O}((\log n)^{2})$, the dominating factor is the
case where the dynamic program has to find some specific node $z$, for which
it examines all $\mathcal{O}((\log n)^{2})$ pairs of nodes between $x$ and the
leaf of position $p$. So, overall the work to update $\mu_{u}^{\ell}(x)$ and
$\mu_{u}^{r}(x)$ is $\mathcal{O}((\log n)^{3})$.
The updates after an insertion of an opening parenthesis are completely dual.
For deletions of opening or closing parentheses, observe that we can simulate
the deletion of, e.g., an opening parenthesis at position $p$ by inserting a
closing one at position $p+1$ and deleting both afterwards. To ensure that
position $p+1$ is always free ${\mathcal{P}}$ maintains the auxiliary
structure actually on a string $w^{\prime}$ of length $2\cdot\lvert w\rvert$.
If an opening parenthesis is inserted at position $p$ it is inserted at
position $2\cdot p-1$ (i.e. at the $p$-th odd position) in $w^{\prime}$. A
closing parenthesis is inserted at the $p$-th even position. Removing both
parentheses afterwards is easy. Because they match each other, both
parentheses just have to be removed from $\mu_{u}^{\ell}(x)$ and
$\mu_{u}^{r}(x)$ for every node $x$.
Because on each change operation just one parenthesis is inserted to or
removed from $\mu_{u}(x)$, maintaining the positions $p_{r}(x),p_{m}(x)$ and
$p_{m}^{\prime}(x)$ is easy.
As mentioned above, $M(x)=1$ if $M(y_{1})=M(y_{2})=1$ and the first $t$
positions of $\mu_{u}^{r}(y_{1})$ and the last $t$ positions of
$\big{(}\overline{\mu_{u}^{\ell}(y_{2})}\big{)}^{R}$ coincide, where $t$ is
the length of the shorter of the two strings. So, $M(x)=1$ is one if for all
nodes $y$ on the path from $x$ to the leaf of $p$ it holds that
$M(y^{\prime})=1$ for the child $y^{\prime}$ of $y$ that does not lie on this
path, and that the partial string equality condition holds for the other
child. Given the latter information, the necessary work to update $M$ is
bounded by $\mathcal{O}((\log n)^{2})$.
As maintaining string equality needs a linear amount of work, using Lemma 3
the overall work for maintaining the partial string equality for all the $\log
n$ many nodes whose strings $\mu_{u}^{\ell}(x)$ and $\mu_{u}^{r}(x)$ are
updated is $\mathcal{O}(n\cdot\log n)$. In sum, ${\mathcal{P}}$ maintains
$D_{k}$ with work $\mathcal{O}(n\cdot\log n+(\log n)^{3})$. ∎
###### Proof (Proof sketch)
[of Lemma 4] Part a) follows directly from the proof of Theorem 6.5 as the
linear factor in the work bound is due to maintaining string equality for
$\log n$ many nodes. For b), observe that two strings
$s_{1}=\sigma_{1}^{1}\ldots\sigma_{n}^{1}$ and
$s_{2}=\sigma_{1}^{2}\ldots\sigma_{n}^{2}$ over an alphabet of size $k$ are
equal if and only if the word
$w\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle{\text{def}}}}{{=}}}}\;\langle_{\sigma_{1}^{1}}\ldots\langle_{\sigma_{n}^{1}}\rangle_{\sigma_{n}^{2}}\ldots\rangle_{\sigma_{1}^{2}}$
is in $D_{k}$. ∎
|
# Model-based Policy Search
for Partially Measurable Systems
Fabio Amadio
<EMAIL_ADDRESS>
&Alberto Dalla Libera††footnotemark:
<EMAIL_ADDRESS>
&Ruggero Carli††footnotemark:
<EMAIL_ADDRESS>
&Daniel Nikovski
<EMAIL_ADDRESS>
&Diego Romeres††footnotemark:
<EMAIL_ADDRESS>
Deptartment of Information Engineering, University of Padova, Via Gradenigo
6/B, 35131 Padova, ItalyMitsubishi Electric Research Laboratories (MERL),
Cambridge, MA 02139
###### Abstract
In this paper, we propose a Model-Based Reinforcement Learning (MBRL)
algorithm for Partially Measurable Systems (PMS), i.e., systems where the
state can not be directly measured, but must be estimated through proper state
observers. The proposed algorithm, named Monte Carlo Probabilistic Inference
for Learning COntrol for Partially Measurable Systems (MC-PILCO4PMS), relies
on Gaussian Processes (GPs) to model the system dynamics, and on a Monte Carlo
approach to update the policy parameters. W.r.t. previous GP-based MBRL
algorithms, MC-PILCO4PMS models explicitly the presence of state observers
during policy optimization, allowing to deal PMS. The effectiveness of the
proposed algorithm has been tested both in simulation and in two real systems.
## 1 Introduction
Reinforcement Learning (RL) [1] has achieved outstanding results in many
different environments. MBRL algorithms seem a promising solution to apply RL
to real systems, due to their data-efficiency w.r.t. model-free RL algorithms.
In particular, remarkable results have been obtained relying on Gaussian
Processes (GPs) [2] to model the systems dynamics, see for instance [3, 4, 5,
6, 7]. In this paper, we cosider the application of MBRL algorithms to PMS,
i.e., systems where only a subset of the state components can be directly
measured, and the remaining components can be estimated through proper state
observer. PMS are particularly relevant in real world applications, think for
example to mechanical systems, where, typically, only positions are measured,
while velocities are estimated thorough numerical differentiation or more
complex filters. The proposed algorithm, named MC-PILCO4PMS, relies on
Gaussian Processes (GPs) to model the system dynamics, and on a Monte Carlo
approach [8] to optimize the policy parameters. W.r.t. previous GP-based MBRL
algorithms, such as [3, 4, 5, 9], MC-PILCO4PMS models explicitly the presence
of two different state observers during the two phases of model learning and
of policy optimization. This improves the characterization of the PMS in the
two phases and so the control performance. In the following we provide a
description of the proposed algorithm, assuming that it is applied to
mechanical systems where only positions measurement are available. However,
the algorithm generalizes to any PMS.
## 2 Problem Setting
Consider a mechanical system with $d_{\boldsymbol{q}}$ degrees of freedom, and
denote with
$\boldsymbol{x}_{t}=[\boldsymbol{q}_{t}^{T},\boldsymbol{\dot{q}}_{t}^{T}]^{T}$
its state, where $\boldsymbol{q}_{t}\in\mathbb{R}^{d_{\boldsymbol{q}}}$ and
$\boldsymbol{\dot{q}}_{t}\in\mathbb{R}^{d_{\boldsymbol{q}}}$ are,
respectively, the vector of the generalized coordinates and its derivative
w.r.t. time. Assume that joint positions can be directly measured, while
$\boldsymbol{\dot{q}}_{t}$ must be estimated from the history of
$\boldsymbol{q}_{t}$ measurements. Moreover, let the system be Markovian, and
describe its discrete-time dynamics as
$\boldsymbol{x}_{t+1}=f(\boldsymbol{x}_{t},\boldsymbol{u}_{t})+\boldsymbol{w}_{t}$,
where $f(\cdot)$ is an unknown transition function,
$\boldsymbol{u}_{t}\in\mathbb{R}^{d_{\boldsymbol{u}}}$ represents the system
input, while $\boldsymbol{w}_{t}\sim\mathcal{N}(0,\Sigma_{\boldsymbol{w}})$
models uncertainty. The objective of RL algorithms is learning to accomplish a
given task based on interaction data. The task is encoded in a cost function
$c(\boldsymbol{x}_{t})$, defined to characterize the immediate penalty for
being in state $\boldsymbol{x}_{t}$. The system inputs are chosen in
accordance with a policy
$\pi_{\boldsymbol{\theta}}:\boldsymbol{x}\mapsto\boldsymbol{u}$ that depends
on the parameter vector $\boldsymbol{\theta}$. Then, the objective is to find
the policy that minimizes the expected cumulative cost over a finite number of
time steps $T$, with initial state distribution $p(\boldsymbol{x}_{0})$, i.e.,
$J(\boldsymbol{\theta})=\sum_{t=0}^{T}\mathbb{E}_{\boldsymbol{x}_{t}}\left[c(\boldsymbol{x}_{t})\right]$.
## 3 Method
MC-PILCO4PMS consists of the iteration of three phases: (i) model learning,
(ii) policy optimization, and (iii) policy execution. In the first phase, MC-
PILCO4PMS relies on GPR to estimate the one-step-ahead system dynamics, while
for the optimization of the policy parameters, MC-PILCO4PMS implements a
gradient-based strategy. In the following, we briefly discuss the two phases.
### 3.1 Model Learning
Dynamics model. The proposed one-step-ahead GP model exploits the intrinsic
correlation between the position and velocity. In our algorithm a distinct GP
model is learned to predict the velocity change, while positions are obtained
by integration. This approach is different from previous GP-based MBRL
algorithms, such as [3, 4, 5], that learn one independent model for each state
component.
Let us indicate the components of $\boldsymbol{q}_{t}$ and
$\boldsymbol{\dot{q}}_{t}$ with $q_{t}^{(i)}$ and $\dot{q}_{t}^{(i)}$,
respectively, where $i\in\\{1,\ldots,d_{\boldsymbol{q}}\\}$. Then, let
$\Delta^{(i)}_{t}=\dot{q}^{(i)}_{t+1}-\dot{q}^{(i)}_{t}$ be the difference
between the value of the i-th velocity at time $t+1$ and $t$, and
$y^{(i)}_{t}$ the noisy measurement of $\Delta^{(i)}_{t}$. For each velocity
component $i$, we model $\Delta^{(i)}_{t}$ with a distinct GP with zero mean
and kernel function $k(\cdot,\cdot)$, which takes as input
$\tilde{\boldsymbol{x}}_{t}=[\boldsymbol{x}_{t},\boldsymbol{u}_{t}]$. Details
on the kernel choice can be found in Appendix 6.1. In GPR the posterior
distribution of $\Delta^{(i)}_{t}$ given the data is Gaussian, with mean and
covariance available in closed form, see [2]. Then, given the GP input
$\tilde{\boldsymbol{x}}_{t}=[\boldsymbol{x}_{t},\boldsymbol{u}_{t}]$, a
prediction of the velocity changes $\hat{\Delta}_{t}^{(i)}$ can be sampled
from the aforementioned posterior distribution. When considering a
sufficiently small sampling time $T_{s}$, it is reasonable to assume constant
accelerations between two consecutive time-steps, and the predicted positions
and velocities are obtained with the following equations,
$\hat{q}_{t+1}^{(i)}=q_{t}^{(i)}+T_{s}\dot{q}_{t}^{(i)}+\frac{T_{s}}{2}\hat{\Delta}_{t}^{(i)}$
and $\hat{\dot{q}}_{t+1}=\dot{q}_{t}^{(i)}+\hat{\Delta}_{t}^{(i)}$ for
$i\in\\{1,\dots,d_{\boldsymbol{q}}\\}$.
Training data computation. As described before, velocities are not accessible
and have to be estimated from measurements of positions. Notice that the
velocity estimates used to train the GP models can be computed offline,
exploiting the (past and future) history of measurements to improve accuracy.
Well filtered data, that resemble the real states of the system, improve
significantly the adherence between the learnt model and the real system. In
our experiments, we computed offline the velocities used to train the GPs,
using for example, the central difference formula, i.e.,
$\dot{\boldsymbol{q}}_{t}=(\boldsymbol{q}_{t+1}-\boldsymbol{q}_{t-1})/(2T_{s})$,
which is an acausal filter. We would like to underline that these state
estimates are different from the ones computed real-time and provided to the
control policy during system interaction. Typically, due to real-time
constraints, online estimates are less accurate and it is fundamental to keep
this in consideration during policy optimization as we can see in the
following.
### 3.2 Policy optimization
MC-PILCO4PMS optimizes the policy parameters with a gradient-based strategy.
At each optimization step the algorithm performs the following operations: (i)
approximation of the cumulative cost relying on a Monte Carlo approximation;
(ii) computation of the gradient and update of $\boldsymbol{\theta}$. More
precisely, the algorithm samples $M$ particles from the initial state
distribution $p(\boldsymbol{x}_{0})$, and simulates their evolution for $T$
steps. At each simulation step the inputs are selected in accordance with the
current policy, and the next state is predicted with the GP models previously
described. This procedure models the propagation of the model uncertainty for
long-term predictions. Then, the Monte Carlo estimate of the cumulative cost
is
$\hat{J}(\boldsymbol{\theta})=\sum_{t=0}^{T}\left(\frac{1}{M}\sum_{m=1}^{M}c\left(\boldsymbol{x}_{t}^{(m)}\right)\right)$,
where $\boldsymbol{x}_{t}^{(m)}$ denotes the state of the m-th particle at
time $t$. The gradient is computed by backpropagation on the computational
graph of $\hat{J}(\boldsymbol{\theta})$, exploiting the reparametrization
trick [10] to propagate the gradient through the stochastic operations, i.e.,
the sampling from the GP posteriors distribution. Advantages of MC based long-
term predictions w.r.t to e.g., moment matching [3] are that no assumptions on
the state distribution and on the kernel function in the GP models have to be
made. The policy parameters are updated using the Adam solver [11]. In the
remainder of this section we describe the particles simulation, which is the
main novelty introduced to deal with PMS.
Particles simulation with PMS. In order to deal with PMS we not only simulate
the evolution of the system state, but also the evolution of the observed
states, modeling the measurement system and the online state observers
implemented in the real system. A block scheme of the particles generation is
reported in Fig.1. Let
$\boldsymbol{x}^{(m)}_{t}=[\boldsymbol{q}^{(m)}_{t},\dot{\boldsymbol{q}}^{(m)}_{t}]$
be the state of the m-th particle at the simulation step $t$ predicted by the
GP models. In order to transform the prediction of the system state to the
observed state, firstly, we simulate the measurement system by corrupting
positions with a zero mean Gaussian i.i.d noise $\boldsymbol{e}^{(m)}_{t}$:
$\bar{\boldsymbol{q}}^{(m)}_{t}=\boldsymbol{q}^{(m)}_{t}+\boldsymbol{e}^{(m)}_{t}$.
Secondly, the measured states are used to compute an estimate of the observed
states:
$\boldsymbol{z}^{(m)}_{t}=f_{z}(\bar{\boldsymbol{q}}^{(m)}_{t}\dots\bar{\boldsymbol{q}}^{(m)}_{t-m_{q}},\boldsymbol{z}^{(m)}_{t-1}\dots\boldsymbol{z}^{(m)}_{t-1-m_{z}})$,
where $f_{z}$ is the online state observer implemented in the real system,
with memory $m_{q}$ and $m_{z}$. Finally, the control inputs for each particle
are computed as $\pi(\boldsymbol{z}^{(m)}_{t})$, the next particles states are
sampled from the GP dynamics, and the procedure is iterated. The procedure
aims at obtaining robustness w.r.t. delays and distortions introduced by
measurement noise and online observers. Notice that selecting the inputs as
$\pi(\boldsymbol{x}^{(m)}_{t})$, as done in several previous MBRL algorithms,
is equivalent to assume full access to the system state,
Figure 1: Block schemes illustrating particles generation in MC-PILCO4PMS.
which is often an unrealistic assumption when dealing with real systems, since
the difference between the system state and the observed state might be
significant. This is a key differentiation of our method. Let us denote, MC-
PILCO, the version of the proposed algorithm which assumes fully access to the
system state during policy optimization. A numerical comparison between MC-
PILCO and two state-of-the-art GP-based MBRL algorithms is reported in the
Appendix 6.3. The results obtained show that MC-PILCO overperforms both the
algorithms in terms of data-efficiency and accuracy.
## 4 Experiments
MC-PILCO4PMS has been tested both in simulation and in real systems. First, we
validate in simulation the impact of taking into consideration the measurement
system and the online filter during particle simulation. Second, MC-PILCO4PMS
has been successfully applied to two real systems: a Furuta pendulum and a
Ball-and-Plate system. Further details about the implementation of the
algorithm on the presented systems can be found in Appendices 6.2, 6.4, 6.5.
Simulation as proof of concepts. Here, we test the relevance of modeling the
presence of online observers on a simulated cart-pole system. The objective is
to learn a policy able to swing-up the pole and stabilize it in the upwards
equilibrium, while keeping the cart stationary. We assumed to be able to
measure only the cart position and the pole angle. The online estimates of the
velocities were computed by means of causal numerical differentiation followed
by a first order low-pass filter. The velocities used to train the GPs were
derived with the central difference formula. Two policy functions have been
trained: the first has been derived with MC-PILCO, assuming direct access to
the full state predicted by the model; the second policy has been derived
using MC-PILCO4PMS. In Figure 4, we report the results of a Monte Carlo study
with 400 runs. Even though the two policies perform similarly when applied to
the learned models, the results obtained with the cart-pole system are
significantly different. MC-PILCO4PMS solves the task in all 400 attempts. In
contrast, in several attempts, the MC-PILCO policy does not solve the task,
due to delays and discrepancies introduced by the online filter and not
considered during policy optimization.
Figure 2: Comparison of 400 simulated particles rollout (left) and the
trajectories performed applying repetitively the policy 400 times in the
system (right) with the simulated cart-pole system. MC-PILCO results are on
the top plots, while MC-PILCO4PMS are on the bottom.
Figure 3: Trajectories for the pendulum angle (left) and arm angle (right)
obtained at each trial. For all the kernels, the angles are plotted up to the
trial that solved the task.
Figure 4: Ten different ball trajectories obtained on the Ball-and-Plate under
the final policy learned by MC-PILCO4PMS. Steady-state positions are marked
with black crosses. The dashed circle has the same diameter of the used ball.
Furuta Pendulum. The Furuta pendulum [12] is a popular under-actuated
benchmark system that consists of a driven arm, revolving in the horizontal
plane, with a pendulum attached to its end, which rotates in the vertical
plane. Let $\theta^{h}$ be the horizontal angle of the arm, and
$\theta^{v}_{t}$ the vertical angle of the pendulum. The objective is to learn
a controller able to swing-up the pendulum and stabilize it in the upwards
equilibrium ($\theta_{t}^{v}=\pm\pi$ [rad]) with $\theta_{t}^{h}=0$ [rad].
Offline estimates of velocities for the GP model have been computed by means
of central differences. Causal numerical differentiation were used for the
online estimation. MC-PILCO4PMS managed to solve the task using the three
different choices of kernel functions presented in Appendix 6.1. In Figure 4,
we show the resulting trajectories for each trial. These experiments show the
effectivness of MC-PILCO4PMS and confirm the higher data efficiency of more
structured kernels, which is one of the advantage that MC-PILCO4PMS offers by
allowing any kernel function while in methods like PILCO the kernel choice is
limited. For best of our knowledge, with 9 [s] of training data this algorithm
is the most data-efficient to solve a FP.
Ball-and-Plate. The ball-and-plate system is composed of a square plate that
can tilt in two orthogonal directions by means of two motors. On top of it,
there is a camera to track the ball and measure its position on the plate. The
objective of the experiment is to learn how to control the motor angles in
order to stabilize the ball around the center of the plate. Measurements
provided by the camera are very noisy, and cannot be used directly to estimate
velocities from positions. We used a Kalman smoother [13] for the offline
filtering of ball positions and associated velocities. Instead, in real-time
we used a Kalman filter [14] to estimate online the ball state from noisy
measures of positions. MC-PILCO4PMS learnt a policy to stabilize the ball
around the center starting from any initial position after the third trial,
11.33 [s] of interaction with the system. We tested the learned policy
starting from ten different points, see Figure 4. The mean steady-state error,
i.e. the average distance of the final ball position from the center observed
in the ten trials, was 0.0099 [m], while the maximum measured error was 0.0149
[m], which is lower than the ball radius of 0.016 [m].
## 5 Conclusions
We have presented a MBRL algorithm called, MC-PILCO4PMS, which does not assume
that all the components of the state can be measured and we successfully
applied it to robotic systems. The algorithm employs GPs to derive a
probabilistic model of the system dynamics. Policy parameters are updated
through a Monte Carlo gradient-based strategy: expected cumulative cost is
estimated by averaging over hundreds of simulated rollouts, and policy
gradient is computed by backpropagation on the resulting computational graph.
We showed the importance of manipulating the measurements to both provide
accurate state estimates to the model learning algorithm and to reproduce the
measurement system together with the online state observer during policy
optimization.
## References
* [1] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.
* [2] Christopher KI Williams and Carl Edward Rasmussen. Gaussian processes for machine learning. MIT press Cambridge, MA, 2006.
* [3] Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data-efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pages 465–472, 2011.
* [4] Paavo Parmas, Carl Edward Rasmussen, Jan Peters, and Kenji Doya. Pipps: Flexible model-based policy search robust to the curse of chaos. In International Conference on Machine Learning, pages 4065–4074, 2018.
* [5] Konstantinos Chatzilygeroudis, Roberto Rama, Rituraj Kaushik, Dorian Goepp, Vassilis Vassiliades, and Jean-Baptiste Mouret. Black-box data-efficient policy search for robotics. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 51–58. IEEE, 2017.
* [6] Diego Romeres, Devesh K Jha, Alberto DallaLibera, Bill Yerazunis, and Daniel Nikovski. Semiparametrical gaussian processes learning of forward dynamical models for navigating in a circular maze. In 2019 International Conference on Robotics and Automation (ICRA), pages 3195–3202. IEEE, 2019.
* [7] Diego Romeres, Mattia Zorzi, Raffaello Camoriano, Silvio Traversaro, and Alessandro Chiuso. Derivative-free online learning of inverse dynamics models. IEEE Transactions on Control Systems Technology, 28(3):816–830, 2019.
* [8] Russel E Caflisch et al. Monte carlo and quasi-monte carlo methods. Acta numerica, 1998:1–49, 1998.
* [9] Alberto Dalla Libera, Diego Romeres, Devesh K Jha, Bill Yerazunis, and Daniel Nikovski. Model-based reinforcement learning for physical systems without velocity and acceleration measurements. IEEE Robotics and Automation Letters, 5(2):3548–3555, 2020.
* [10] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
* [11] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
* [12] Benjamin Seth Cazzolato and Zebb Prime. On the dynamics of the furuta pendulum. Journal of Control Science and Engineering, 2011, 2011.
* [13] Garry A Einicke. Optimal and robust noncausal filter formulations. IEEE Transactions on Signal Processing, 54(3):1069–1077, 2006.
* [14] R. E. Kalman. A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82(1):35–45, 03 1960.
* [15] Alberto Dalla Libera, Ruggero Carli, and Gianluigi Pillonetto. A novel multiplicative polynomial kernel for volterra series identification. arXiv preprint arXiv:1905.07960, 2019.
* [16] A. D. Libera and R. Carli. A data-efficient geometrically inspired polynomial kernel for robot inverse dynamic. IEEE Robotics and Automation Letters, 5(1):24–31, 2020.
* [17] D. Nguyen-Tuong and J. Peters. Using model knowledge for learning inverse dynamics. In 2010 IEEE International Conference on Robotics and Automation, pages 2677–2682, 2010.
## 6 Appendix
### 6.1 Kernel functions
One of the advantages of the particle-based policy optimization method is the
possibility of choosing any kernel functions without restrictions. Hence, we
considered different kernel functions as examples to model the evolution of
physical systems. But the reader can consider a custom kernel function
appropriate for his application.
Squared exponential (SE). The SE kernel represents the standard choice adopted
in many different works. It is defined as
$k_{SE}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}}):=\lambda^{2}e^{-||\tilde{\boldsymbol{x}}_{t_{j}}-\tilde{\boldsymbol{x}}_{t_{k}}||^{2}_{\Lambda^{-1}}}\text{,}$
(1)
where the scaling factor $\lambda$ and the matrix $\Lambda$ are kernel
hyperparameters which can be estimated by marginal likelihood maximization.
Typically, $\Lambda$ is assumed to be diagonal, with the diagonal elements
named lengthscales.
SE + Polynomial (SE+$\text{P}^{(d)}$). Recalling that the sum of kernels is
still a kernel [2], we considered also a kernel given by the sum of a SE and a
polynomial kernel. In particular, we used the Multiplicative Polynomial (MP)
kernel, which is a refinement of the standard polynomial kernel, introduced in
[15]. The MP kernel of degree $d$ is defined as the product of $d$ linear
kernels, namely,
$k_{P}^{(d)}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}}):=\prod_{r=1}^{d}\left(\sigma^{2}_{P_{r}}+\tilde{\boldsymbol{x}}_{t_{j}}^{T}\Sigma_{P_{r}}\tilde{\boldsymbol{x}}_{t_{k}}\right)\text{.}$
where the $\Sigma_{P_{r}}>0$ matrices are distinct diagonal matrices. The
diagonal elements of the $\Sigma_{P_{r}}$, together with the
$\sigma^{2}_{P_{r}}$ elements are the kernel hyperparameters. The resulting
kernel is
$k_{SE+P^{(d)}}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})=k_{SE}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})+k_{P}^{(d)}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})\text{.}$
(2)
The idea motivating this choice is the following: the MP kernel allows
capturing possible modes of the system that are polynomial functions in
$\tilde{\boldsymbol{x}}$, which are typical in mechanical systems [16], while
the SE kernel models more complex behaviors not captured by the polynomial
kernel.
Semi-Parametrical (SP). When prior knowledge about the system dynamics is
available, for example given by physics first principles, the so called
physically inspired (PI) kernel can be derived. The PI kernel is a linear
kernel defined on suitable basis functions $\phi(\tilde{\boldsymbol{x}})$, see
for instance [6]. More precisely,
$\boldsymbol{\phi}(\tilde{\boldsymbol{x}})\in\mathbb{R}^{d_{\phi}}$ is a,
possibly nonlinear, transformation of the GP input $\tilde{\boldsymbol{x}}$
determined by the physical model. Then we have
$k_{PI}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})=\boldsymbol{\phi}^{T}(\tilde{\boldsymbol{x}}_{t_{j}})\Sigma_{PI}\boldsymbol{\phi}(\tilde{\boldsymbol{x}}_{t_{k}})\text{,}$
where $\Sigma_{PI}$ is a $d_{\phi}\times d_{\phi}$ positive-definite matrix,
whose elements are the $k_{PI}$ hyperparameters; to limit the number of
hyperparameters, a standard choice consists in considering $\Sigma_{PI}$ to be
diagonal. To compensate possible inaccuracies of the physical model, it is
common to combine $k_{PI}$ with an SE kernel, obtaining so called semi-
parametric kernels [17, 6], expressed as
$k_{SP}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})=k_{PI}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})+k_{SE}(\tilde{\boldsymbol{x}}_{t_{j}},\tilde{\boldsymbol{x}}_{t_{k}})\text{.}$
The rationale behind this kernel is the following: $k_{PI}$ encodes the prior
information given by the physics, and $k_{SE}$ compensates for the dynamical
components unmodeled in $k_{PI}$.
### 6.2 Simulated Cart-pole
The physical properties of the cart-pole system considered are the following:
the masses of both cart and pole are 0.5 [kg], the length of the pole is
$L=0.5$ [m], and the coefficient of friction between cart and ground is 0.1.
The state at each time step $t$ is defined as
$\boldsymbol{x}_{t}=[p_{t},\dot{p}_{t},\theta_{t},\dot{\theta}_{t}]$, where
$p_{t}$ represents the position of the cart and $\theta_{t}$ the angle of the
pole. The target states corresponding to the swing-up of the pendulum is given
by $p^{des}=0$ [m] and $|\theta^{des}|=\pi$ [rad]. The downward stable
equilibrium point is defined at $\theta_{t}=0$ [rad]. As done in [3], in order
to avoid singularities due to the angles, $\boldsymbol{x}_{t}$ is replaced
with the state representation
$\bar{\boldsymbol{x}}_{t}=[p_{t},\dot{p}_{t},\dot{\theta}_{t},sin(\theta_{t}),cos(\theta_{t})]$
inside GP inputs. For the GP models SE kernels have been chosen (1). The
control action is the force that pushes the cart horizontally. We considered
white measurement noise with standard deviation of $3\cdot 10^{-3}$, and as
initial state distribution
$\mathcal{N}([0,0,0,0],\text{diag}([10^{-4},10^{-4},10^{-4},10^{-4}]))$. In
order to obtain reliable estimates of the velocities, samples were collected
at 30 [Hz]. The number of particles has been set to $M=400$ in all the tests.
The cost function optimized in MC-PILCO is the following,
$c(\boldsymbol{x}_{t})=1-\text{exp}\left(-\left(\frac{|\theta_{t}|-\pi}{l_{\theta}}\right)^{2}-\left(\frac{p_{t}}{l_{p}}\right)^{2}\right),$
(3)
where $l_{\theta}$ and $l_{p}$ are named lengthscales. Notice that the
lengthscales define the shape of $c(\cdot)$, the cost function goes to its
maximum value more rapidly with small lengthscales. Therefore, higher cost is
associated to the same distance from the target state with lower $l_{\theta}$
and $l_{p}$. The lower the lengthscale the more selective the cost function.
The absolute value on $\theta_{t}$ is needed to allow different swing-up
solutions to both the equivalent target angles of the pole $\pi$ and $-\pi$.
The selected lengthscales were $l_{\theta}=3$ and $l_{p}=1$.
The policy adopted is an RBF network policy with outputs limited by an
hyperbolic tangent function, properly scaled. We call this function squashed-
RBF-network and it is defined as
$\pi_{\boldsymbol{\theta}}(\bar{\boldsymbol{x}}_{t})=u_{max}\;\text{tanh}\left(\frac{1}{u_{max}}\sum_{i=1}^{n_{b}}w_{i}e^{||\boldsymbol{a}_{i}-\bar{\boldsymbol{x}}_{t}||_{\Sigma_{\pi}}^{2}}\right)\text{.}$
(4)
parameters are
$\boldsymbol{\theta}=\left\\{\boldsymbol{w},A,\Sigma_{\pi}\right\\}$, where
$\boldsymbol{w}=[w_{1}\dots w_{n_{b}}]$ and
$A=\left\\{\boldsymbol{a}_{1}\dots\boldsymbol{a}_{n_{b}}\right\\}$ are,
respectively, the weights and the centers of the $n_{b}$ Gaussian basis
functions, while ${\Sigma_{\pi}}$ determines their shapes; in all experiments
we assumed ${\Sigma_{\pi}}$ to be diagonal. $u_{max}$ is the maximum control
action applicable. For this experiment we choose $n_{b}=200$ basis functions
and $u_{max}=10$ [N]. The exploration trajectory is obtained by applying at
each time step a random control action sampled from
$\mathcal{U}(-u_{max},u_{max})$.
### 6.3 Comparison with state of the art algorithms
| Trial 1 | Trial 2 | Trial 3 | Trial 4 | Trial 5
---|---|---|---|---|---
PILCO | 2% | 4% | 20% | 36% | 42%
Black-DROPS | 0% | 4% | 30% | 68% | 86%
MC-PILCO | 0% | 14% | 78% | 94% | 100%
Table 1: Success rate per trial obtained with PILCO, Black-DROPS and MC-PILCO.
We tested PILCO[3], Black-DROPS[5] and MC-PILCO on the simulated cart-pole
system.
Figure 5: Median and confidence interval of the cumulative cost
$c^{pilco}(\cdot)$ per trial obtained with PILCO, Black-DROPS and MC-PILCO.
The setup is equal to the one described in Appendix 6.2, with the only two
difference that here the samples are collected at 20 [Hz] and the noise
standard deviation is $10^{-2}$. In PILCO and Black-DROPS, we considered their
original cost function,
$c^{pilco}(\boldsymbol{x}_{t})=1-\text{exp}\left(-\frac{1}{2}\left(\frac{d_{t}}{0.25}\right)^{2}\right),$
(5)
where $d_{t}^{2}=p_{t}^{2}+2p_{t}Lsin(\theta_{t})+2L^{2}(1+cos(\theta_{t}))$
is the squared distance between the tip of the pole and its position at the
unstable equilibrium point with $p_{t}=0$ [m]. This last cost is also adopted
as a common metric to compare the results obtained by the three algorithms.
Results of the cumulative cost are reported in Figure 5, observed success
rates are shown in Table 1. MC-PILCO achieved the best performance both in
transitory and at convergence, by trial 5, it learned how to swing up the
cart-pole with a success rate of 100%. In each and every trial, MC-PILCO
obtained cumulative costs with lower median and less variability. On the other
hand, the policy in PILCO showed poor convergence properties with only 42% of
success rate after all the 5 trials. Black-DROPS outperforms PILCO, but it
obtained worse results than MC-PILCO in each and every trial, with a success
rate of only 86% at trial 5.
### 6.4 Furuta Pendulum
The Furuta pendulum (FP) [12] is a popular benchmark system used in nonlinear
control and reinforcement learning. The system is composed of two revolute
joints and three links.
Figure 6: Furuta pendulum used in the experiment while being controlled in the
upward equilibrium point by the learned policy.
The first link, called the base link, is fixed and perpendicular to the
ground. The second link, called arm, rotates parallel to the ground, while the
rotation axis of the last link, the pendulum, is parallel to the principal
axis of the second link, see Figure 6. The FP is an under-actuated system as
only the first joint is actuated. In particular, in the FP considered the
horizontal joint is actuated by a DC servomotor, and the two angles are
measured by optical encoders with 4096 [ppr]. The control variable is the
motor voltage. Let the state at time step $t$ be
$\boldsymbol{x}_{t}=[\theta^{h}_{t},\dot{\theta}^{h}_{t},\theta^{v}_{t},\dot{\theta}^{v}_{t}]^{T}$,
where $\theta^{h}_{t}$ is the angle of the horizontal joint and
$\theta^{v}_{t}$ the angle of the vertical joint attached to the pendulum. The
objective is to learn a controller able to swing-up the pendulum and stabilize
it in the upwards equilibrium ($\theta_{t}^{v}=\pm\pi$ [rad]) with
$\theta_{t}^{h}=0$ [rad]. The trial length is 3 [s] with a sampling frequency
of 30 [Hz]. The cost function is defined as
$c(\boldsymbol{x}_{t})=1-\text{exp}\left(-\left(\frac{\theta_{t}^{h}}{2}\right)^{2}-\left(\frac{|\theta_{t}^{v}|-\pi}{2}\right)^{2}\right)+c_{b}(\boldsymbol{x}_{t}),$
(6)
with
$\displaystyle c_{b}(\boldsymbol{x}_{t})=$
$\displaystyle\frac{1}{1+\text{exp}\left(-10\left(-\frac{3}{4}\pi-\theta^{h}_{t}\right)\right)}$
$\displaystyle+\frac{1}{1+\text{exp}\left(-10\left(\theta^{h}_{t}-\frac{3}{4}\pi\right)\right)}\text{.}$
The first part of the function in (6) aims at driving the two angles towards
$\theta_{t}^{h}=0$ and $\theta_{t}^{v}=\pm\pi$, while
$c_{b}(\boldsymbol{x}_{t})$ penalizes solutions where
$\theta_{t}^{h}\leq-\frac{3}{4}\pi$ or $\theta_{t}^{h}\geq\frac{3}{4}\pi$. We
set those boundaries to avoid the risk of damaging the system if the
horizontal joint rotates too much. Offline estimates of velocities for the GP
model have been computed by means of central differences. For the online
estimation, we used causal numerical differentiation:
$\dot{\boldsymbol{q}}_{t}=(\boldsymbol{q}_{t}-\boldsymbol{q}_{t-1})/T_{s}$,
where $T_{s}$ is the sampling time. Instead of $\boldsymbol{x}_{t}$, we
considered the extended state
$\bar{\boldsymbol{x}}_{t}=[\dot{\theta}^{h}_{t},\dot{\theta}^{v}_{t},sin(\theta^{h}_{t}),cos(\theta^{h}_{t}),sin(\theta^{v}_{t}),cos(\theta^{v}_{t})]^{T}$
in GP input. The policy is a squashed-RBF-network with $n_{b}=200$ basis
functions that receives as input
$\bar{\boldsymbol{z}}_{t}=[(\theta^{h}_{t}-\theta^{h}_{t-1})/{T_{s}},(\theta^{v}_{t}-\theta^{v}_{t-1})/T_{s},sin(\theta^{h}_{t}),cos(\theta^{h}_{t}),sin(\theta^{v}_{t}),cos(\theta^{v}_{t})]^{T}$.
We used 400 particles to estimate the policy gradient from model predictions.
The exploration trajectory has been obtained using as input a sum of ten sine
waves of random frequencies and same amplitudes. The initial state
distribution is assumed to be $\mathcal{N}([0,0,0,0]^{T},\text{diag}([5\cdot
10^{-3},5\cdot 10^{-3},5\cdot 10^{-3},5\cdot 10^{-3}])$. $M=400$ particles
were used for gradient estimation.
### 6.5 Ball-and-Plate
The ball-and-plate system is composed of a square plate that can be tilted in
two orthogonal directions by means of two motors. On top of it, there is a
camera to track the ball and measure its position on the plate. Let
$(b^{x}_{t},b^{y}_{t})$ be the position of the center of the ball along X-axis
and Y-axis, while $\theta^{(1)}_{t}$ and $\theta^{(2)}_{t}$ are the angles of
the two motors tilting the plate, at time $t$. So, the state of the system is
defined as
$\boldsymbol{x}_{t}=[b^{x}_{t},b^{y}_{t},\dot{b}^{x}_{t},\dot{b}^{y}_{t},\theta^{(1)}_{t},\theta^{(2)}_{t},\dot{\theta}^{(1)}_{t},\dot{\theta}^{(2)}_{t}]^{T}$.
The drivers of the motors allow only position control, and do not provide
feedback about the motors angles. To keep track of the motor angles, we
defined the control actions as the difference between two consecutive
reference values sent to the motor controllers, and we limited the maximum
input to a sufficiently small value, such that the motor controllers are able
to reach the target angle within the sampling time. Then, in first
approximation, the reference angles and the motor angles coincide, and we have
$u_{t}^{(1)}=\theta^{(1)}_{t+1}-\theta^{(1)}_{t}$ and
$u_{t}^{(2)}=\theta^{(2)}_{t+1}-\theta^{(2)}_{t}$. The objective of the
experiment is to learn how to control the motor angles in order to stabilize
the ball around the center of the plate. Notice that the control task, with
the given definition of inputs, is particularly difficult because the policy
must learn to act in advance, and not only react to changes in the ball
position.
The cost function is defined as
$c(\boldsymbol{x}_{t})=1-\text{exp}\left(-g_{t}(\boldsymbol{x}_{t})\right),\qquad\text{with}$
$g_{t}(\boldsymbol{x}_{t})=\left(\frac{b^{x}_{t}}{0.15}\right)^{2}+\left(\frac{b^{y}_{t}}{0.15}\right)^{2}+\left(\theta_{t}^{(1)}\right)^{2}+\left(\theta_{t}^{(2)}\right)^{2}.$
The trial length is 3 [s], with a sampling frequency of 30 [Hz]. Measurements
provided by the camera are very noisy, and cannot be used directly to estimate
velocities from positions. We used a Kalman smoother for the offline filtering
of ball positions $b^{x}_{t},b^{y}_{t}$ and associated velocities
$\dot{b}^{x}_{t},\dot{b}^{y}_{t}$. In the control loop, instead, we used a
Kalman filter [14] to estimate online the ball state from noisy measures of
positions. Concerning the model, we need to learn only two GPs predicting the
evolution of the ball velocity because we directly control motor angles,
hence, their evolution is assumed deterministic. GP inputs,
$\tilde{\boldsymbol{x}}_{t}=[\bar{\boldsymbol{x}}_{t},u_{t}]$, include an
extended version of the state,
$\bar{\boldsymbol{x}}_{t}=[b^{x}_{t},b^{y}_{t},\dot{b}^{x}_{t},\dot{b}^{y}_{t},sin(\theta^{(1)}_{t}),cos(\theta^{(1)}_{t}),sin(\theta^{(2)}_{t}),cos(\theta^{(2)}_{t}),(\theta^{(1)}_{t}-\theta^{(1)}_{t-1})/T_{s},(\theta^{(2)}_{t}-\theta^{(2)}_{t-1})/T_{s}]^{T}$
where angles have been replaced by their sines and cosines, and motor angular
velocities have been estimated with causal numerical differentiation ($T_{s}$
is the sampling time).
Figure 7: Ball-and-plate system used in the experiment.
The SE+$\text{P}^{(1)}$ kernel (2) is used, where the linear kernel acts only
on a subset of the model inputs,
$\tilde{\boldsymbol{x}}^{lin}_{t}=[sin(\theta^{(1)}_{t}),sin(\theta^{(2)}_{t}),cos(\theta^{(1)}_{t}),cos(\theta^{(2)}_{t}),u_{t}]$.
We considered $M=400$ particles for policy gradient estimation. The policy is
a multi-output squashed-RBF-network, with $n_{b}=400$ basis functions, that
receives as inputs the estimates of
$(b^{x}_{t},b^{y}_{t},\dot{b}^{x}_{t},\dot{b}^{y}_{t},\theta^{(1)}_{t},\theta^{(1)}_{t-1},\theta^{(2)}_{t},\theta^{(2)}_{t-1})$
computed with the Kalman filter; maximum angle displacement is $u_{max}=4$
[deg] for both motors. Initial exploration is given by two different trials,
in which the control signals are two triangular waves perturbed by white
noise. Mostly during exploration and initial trials, the ball might touch the
borders of the plate. In those cases, we kept data up to the collision
instant. A peculiarity of this experiment in comparison to the others seen
before is a wide range of initial conditions. In fact, the ball could be
positioned anywhere on the plate’s surface, and the policy must control it to
the center. The initial distribution of $b^{x}_{0}$ and $b^{y}_{0}$ is a
uniform $\mathcal{U}(-0.15,0.15)$, which covers almost the entire surface (the
plate is a square with sides of about 0.20 [m]). For the other state
components, $\theta^{(1)}_{t}$ and $\theta^{(2)}_{t}$, we assumed tighter
initial distributions $\mathcal{U}(-10^{-6},10^{-6})$.
|
# Renewable Generation Data for European Energy System Analysis
Alexander Kies1, Bruno U. Schyska2, Mariia Bilousova1,3, Omar El Sayed1,3,
Jakub Jurasz4,5,6, Horst Stoecker1,3,7 1Frankfurt Institute for Advanced
Studies, Goethe-University Frankfurt, Ruth-Moufang-Str. 1, 60438 Frankfurt,
Germany
2German Aerospace Center (DLR), Institute of Networked Energy Systems, Carl-
von-Ossietzky-Straße 15, 26129 Oldenburg, Germany
3Goethe-University, Frankfurt, Germany
4Faculty of Environmental Engineering, Wroclaw University of Science and
Technology, 50-370 Wroclaw, Poland
5Faculty of Management, AGH University, 30-059 Cracow, Poland
6School of Business, Society and Engineering, Mälardalens Högskola, 72113
Västerås, Sweden
7GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291
Darmstadt, Germany
###### Abstract
In the process of decarbonization, the global energy mix is shifting from
fossil fuels to renewables. To study decarbonization pathways, large-scale
energy system models are utilized. These models require accurate data on
renewable generation to develop their full potential. Using different data can
lead to conflicting results and policy advice. In this work, we compare
several datasets that are commonly used to study the transition towards highly
renewable European power system. We find significant differences between these
datasets and cost-difference of about 10% result in the different energy mix.
We conclude that much more attention must be paid to the large uncertainties
of the input data.
###### keywords:
Energy System Analysis , Renewable Generation Data , Energy Meteorology ,
MERRA-2 , ERA5, renewables.ninja, EMHires
††journal: Journal Name
## 1 Introduction
Sustainable energy sources are a major solution for the imminent thread of
climate change [1, 2]. Increasing the number of installations of renewable
generation capacities and the electrification of energy sectors like heating
and transportation fosters decarbonization. As part of the Paris Agreement,
many countries around the world commited to reducing their greenhouse gas
emissions. In its turn, the EU set an aim to reduce emissions by at least 50%
by 2030, as compared to 1990 levels [3]. In this context, the sensitivity of
energy systems to weather and climate rises [4, 5]. Energy system models play
an important role in understanding and investigating energy systems of various
scales and scopes. For the renewable future, the use of adequate
meteorological data in the field of energy system analysis and modelling is
essential.
Common large-scale energy models for Europe cover the EU [6] and aggregate
quantities such as generation from renewable sources to country levels. There
are multiple datasets that provide data on hourly generation potentials from
renewable sources such as wind and solar PV. They are based on reanalysis
data, which is an assimilation of historical measurements and numerical models
into a consistent estimate of a state of the atmosphere. These datasets
provide relevant variables such as wind speed, irradiation or temperature used
to compute potential generation from renewables.
A comparison of two of these datasets was performed by Moraes et al. [7]. They
found that datasets diverge from each other even if they are based on the same
meteorological source. This is likely based on differences in technological
assumptions that are made to convert meteorological data to generation. These
include, for instance, assumptions on the wind turbines used, which can take
in historical data or projections into the future such as increasing hub
heights [8], as well as their placement.
In this paper, we compare seven datasets which provide time-resolved
aggregated generation data on the country level. Only five of them cover the
same time period (2003-2012), therefore we focus on them in particular.
Section 2 discusses these datasets and shows the major steps in converting
meteorological data to energy system model input. Section 3 analyses the
datasets with respect to different means such as annual capacity factors, ramp
rates and optimised mixes. Section 4 discusses the results and implications
and Section 5 concludes this paper.
This paper contains novelties relevant for the research community. It provides
the broadest comparison of renewable generation datasets to date, introduces
energy system-related measures to compare them and derives important
conclusions for research on the energy transition.
## 2 Data
Data on renewable generation potentials is essential for the modelling and
analysis of energy systems with significant shares of generation from
renewable sources. Among the most used datasets in the study of weather and
climate are reanalyses. A meteorological reanalysis is a method to create
long-term weather data using numerical weather prediction models and
assimilating historical data. Reanalyses are used to study climate variability
[9] as well as are commonly employed to study energy systems [10]. Wind speeds
for the datasets investigated in this work are mostly based on two reanalyses:
ERA5 and MERRA-2. MERRA [11] is provided by NASA. The official data production
was launched in 2008 with the use of the up-to-date GEOS-5 (Goddard Earth
Observing System Data Assimilation System Version 5) produced at NASA GMAO
(Global Modeling and Assimilation Office). ERA5 [12] is the newest global
reanalysis by the European Centre for Medium-Range Weather Forecasts (ECMWF).
Wind speed data are provided for heights of 10 m as well as 100 m. The RE-
Europe is the only considered dataset that uses ECMWF forecasts (RE-Europe)
and the regional COSMO-REA6 reanalysis for the modelling of wind power [13].
For solar data, CM SAF SARAH is commonly used alongside the reanalyses [14].
It is a satellite-based climate data record of irradiance data and other
variables.
MERRA-2 and ERA5 were compared by various researchers. Olauson [15] compared
ERA5 and MERRA-2 for the modelling of wind power on a country-level as well as
for individual turbines. His findings indicate that ERA5 performs considerably
better than MERRA-2 on both levels. Camargo et al. [16] used ERA5 data to
model multi-annual time series of solar PV generation. They performed a
validation with hourly data of PV plants in Chile and found a slightly
superior performance of ERA5 as compared to modelling based on MERRA-2. Gruber
et al. [17] compared MERRA-2 and ERA5 for wind power simulation bias-corrected
with the global wind atlas for the US, Brazil, New Zealand and South Africa
and found ERA5 to outperform MERRA-2. Urraca et al. [18] evaluated global
horizontal irradiance estimates from ERA5 and COSMO-REA6, a regional
reanalysis from the German Meteorological Service (DWD), and concluded that
both reanalyses reduce the quality gap between reanalysis and satellite data.
Jourdier [19] investigated ERA5, MERRA-2 and other datasets to examine wind
power production in France, and found that ERA5 is skilled, however, it
underestimates wind speeds, especially in mountainous areas. Piasecki et al.
[20] compared ERA5 data with measurements in several locations in Poland and
concluded that they are in good agreement for solar PV, with hourly
correlation coefficients above 0.9, while wind comparison showed a large
variability with differences in capacity factors of up to 15 percentage
points.
Dataset years wind PV capacity layout temporal resolution ref.
renewables.ninja 1980-2019 MERRA Sarah yes 1h [21] EMHires 1986-2015 MERRA
Sarah yes 1h [22] Restore 2003-2012 MERRA Sarah no 1h [23] UReading-E
1979-2018 ERA5 ERA5 yes 3h [24] UReading-M 1979-2018 MERRA MERRA yes 1h [25]
PyPSA-Eur 2013 ERA5 Sarah yes 1h [26] RE-Europe 2012-2014 diff. diff. no 1h
[27]
Table 1: Datasets providing data on renewable generation that were analysed
and compared in this work. Abbreviations used in this work are Ninja,
URead-E/M and RE-Eur-E/C.
Figure 1: Country coverage of the datasets. Purple-colored countries include
both solar PV and wind time series, yellow-colored ones include only wind.
Both UReading datasets exclude Estonia. The Restore dataset excludes Norway
and for countries Ireland, Denmark, Finland and Estonia provides only wind
time series.
Table 1 shows key features of each dataset. Figure 1 shows the countries
covered. To use a sufficiently long time period for every dataset, we set the
time frame of the following analyses to the period 2003-2012 and compare 5
datasets covering it.
For the sake of readability, we only briefly summarize the major steps of
modelling renewable energy generation for the datasets investigated in this
section, except for the Restore dataset, where we describe them in more
detail. In general, similar steps are taken for all datasets (e.g. Fig. 2). A
meteo model converts meteorological data to the needed input for a power
model, which yields space- and time-dependent power per installed capacity.
The output of this model is aggregated by a capacity model, which represents
how much capacity is installed at what location. The aggregated sum of this
model yields the generation time series for the renewable generation.
Figure 2: Schematic flowchart for PV/wind generation modelling for the RESTORE
dataset. Figure from Kies et al. [23].
### 2.1 Restore
The Restore dataset [23] was produced in the context of the RESTORE 2050
project. Besides wind and solar PV generation time series, it also contains
time series of other generation technologies, such as CSP, wave and hydro
inflow. To calculate wind power per grid cell in the Restore dataset, wind
speeds are interpolated to the grid of the COSMO-EU model [28], and
extrapolated to the turbine hub height using the wind speed log profile
$\displaystyle\frac{s(z_{1})}{s(z_{2})}$
$\displaystyle=\frac{\log(z_{1}/z_{0})}{\log(z_{2}/z_{0})},$ (1)
where $z_{1},z_{2}$ are the heights at which wind speeds are given and
desired, respectively and $z_{0}$ is the surface roughness length provided by
the dataset. Wind speeds are corrected using a linear regression according to
COSMO-EU wind speeds. The power curve of an Enercon E-126 turbine is then used
to convert wind speeds to wind power at 100m hub height. Additionally, the
dataset provides capacity factors generated with the same power curve for the
hub heights of 140m and 180m. In the following comparison, we use the dataset
based on 100m wind speeds. Any wind turbine power curve mapping wind speed $s$
to power output $g$ is approximately given by [29, 30]
$\displaystyle g=\begin{cases}0\text{ for }s<s_{0}\vee s>s_{\text{max}}\\\
\propto\left(s-s_{0}\right)^{3}\text{ for }s_{0}<s<s_{\text{nom}}\\\ =G\text{
for }s_{\text{nom}}<s<s_{\text{max}}.\end{cases}$ (2)
For low wind speeds, torque caused by the wind on blades of a turbine is too
low to generate angular momentum. Above a certain speed, referred to as cut-in
speed $s_{0}$, turbines start to rotate and produce power. The power curve now
resembles the characteristic $v^{3}$-dependency of the kinetic energy of wind
passing through a certain area. At the rated wind speed $s_{\text{nom}}$, the
rated power of the turbine is reached. This power is kept constant by the
turbine beyond this speed value, commonly by the adjustment of blade angles.
If speeds increase further, they can reach a critical level, referred to as
cut-out speed $s_{\text{max}}$, at which the rotor blades are turned out of
the wind to prevent a structural damage to the turbine.
For PV Power, global horizontal irradiance is converted to irradiation on
inclined surfaces based on the Klucher model [31]. For tilt angles, optimized
values per country are used.
The efficiency of the PV modules in dependency of incoming irradiation $I$ and
temperature $T$ is modelled via a parametric model for a standard temperature
$\displaystyle\eta(I,25^{\circ}C)=a1+a2\times I+a3\ln I,$ (3)
where $a1,a2,a3$ are device-specific parameters and the temperature dependency
of the efficiency is given via
$\displaystyle\eta(I,T)=\eta(I,25^{\circ}C)(1-0.004\Delta T).$ (4)
PV power output is then directly computable via
$\begin{split}g_{t}&=\eta_{t}I_{t}A.\end{split}$ (5)
Unlike the other datasets investigated in this work, the Restore dataset does
not take the locations of existing generation capacities into account.
Instead, country-wise capacities are distributed proportionally to the
underlying resource [23].
### 2.2 UReading
The UReading datasets [24, 25] were produced by researchers from the
University of Reading, UK. They contain hourly values of aggregated power
generation from wind and solar based on a representative distribution of wind
and solar farms as well as ERA5 (Reading-E) or MERRA-2 (Reading-M) reanalyses
data for 28 European countries. In addition, a daily time series of
electricity demand is provided. It is used in this paper to optimise
generation mixes.
### 2.3 Renewables.ninja
Renewables.ninja [21] is a web tool that provides potential generation of wind
and solar PV for single locations or countries globally. Pfenninger and
Staffell [32, 33] also found that significant correction factors are necessary
to model renewable feed-in from reanalyses in Europe. To model PV power, they
use the model by Huld et al. [34] with irradiation based on SARAH for
different azimuth/tilt combinations. For wind power, they use a virtual wind
farm model [35] and bias-correct it.
### 2.4 EMHires
European Meteorological derived HIgh resolution RES generation time series for
present and future scenarios (EMHires) is a dataset produced by the joint
research centre (JRC) with an aim to allow users to assess the impact of
meteorological and climate variability on renewable generation in Europe [22,
36, 37]. For wind power, this dataset combines MERRA-2 reanalysis data with
wind farms data from thewindpower.net [38]. Wind power time series are further
normalised to the reported ENTSO-E annual production statistics. Wind speeds
are statistically downscaled and interpolated to the desired hub height using
a wind profile power law. Wind speed data are then converted to wind power
using a specific power curve assigned to each wind farm considering the
characteristics of the wind farm, such as manufacturer. For PV generation, the
PVGIS model [39, 40, 41] is used to provide generation in dependency of
irradiation and solar module parameters. Together with assumptions such as
inclinations, PVGIS output is then aggregated to country levels.
### 2.5 PyPSA-Eur
PyPSA-EUR is a dataset for European generation and transmission expansion
planning studies from freely available data [26]. Besides time series of
renewable generation, it contains additional data to model a renewable
European energy system, such as alternating/direct current transmission lines,
substations, data on conventional generators and demand as well as renewable
capacity installation potentials. For renewable generation data, it uses data
from the ERA5 reanalysis with a logarithmic wind power profile to extrapolate
wind speeds to the desired hub height for wind power modelling. For PV
conversion, it also uses the PVGIS model [40].
### 2.6 RE-Europe
RE-Europe [27] is a dataset produced at the Technical University of Denmark
for the modelling of a highly renewable European power system. Analogously to
PyPSA-EUR, alongside renewable generation data it contains other data, for
instance data on transmission lines. To provide meteorological variables for
both, wind and solar PV, meteorological data from the ECMWF forecasts as well
as the COSMO-REA6 reanalysis are used. For the final conversion step, it uses
a smoothed power curve of a specific wind turbine (Siemens SWT 107) as well as
a specific solar panel (Scheuten P6–54 215 Multisol Integra Gold) to convert
meteorological variables to a potential generation. Generation capacities are
assigned to sub-national regions using two heuristic capacity layouts.
## 3 Results
In this section, we study different properties of the provided time series of
generation per unit $g_{n,s,t}^{i}$. Here $i$ denotes the dataset, $n$ is the
country, $s$ is the technology and $t$ is the timestep.
### 3.1 Annual Capacity Factors
The capacity factor of a generator is the ratio of the potential energy output
over a given period of time to the maximum possible energy output as given by
the rated capacity over that period [42],
cf $\displaystyle=\sum_{t}\frac{g^{+}_{n,s,t}}{G_{n,s}}.$ (6)
For renewable generators the capacity factor heavily depends on the resource
availability as well as on the technical parameters. Occasionally, it is
referred to as full load hours. Besides reductions due to resource
inavailability, one can, for instance, factor net congestions into effective
capacity factors [43].
Figure 3: Annual capacity factors for wind and solar PV for the chosen
countries. RE-Europe does not cover the UK. Solar PV capacity factors are the
highest for the RE-Europe dataset. Year-to-year changes are well caught by all
datasets for wind, implying the difference resulting from the conversion
process from meteorological data to generation. For solar PV, capacities in
some countries such as the UK and Italy were very small (well below 1 GW) at
the beginning of the investigated period. Strong changes in year-to-year
reported capacity factors are potentially just caused by rounding or reporting
errors. Also taking the large year-to-year changes in reported capacity
factors into account, it seems questionable, how valueable these values are in
many cases for a comparison with modelled data.
Figure 4: Average capacity factors for wind and solar PV in the years
2003-2012. The greyscale bar indicates the average capacity factor. The
colorbar plots per country show the average deviation in percentage points
from the mean of the ensembles (Restore, EMHires, URead-E, URead-M and Ninja).
Colors are: EMHires (red), URead-E (blue), URead-M (green), Restore (purple),
Ninja (orange) and reported (yellow). For solar PV, trends seem to be
consistent over all for Europe. For wind, there is more variation. For
instance, Restore produces comparably small capacity factors in South-East
Europe and Norway, while for other North Sea countries Restore capacity
factors are above average.
Figure 3 shows annual capacity factors of the different datasets. Annual
reported capacity factors were calculated by dividing reported generation by
reported installed capacities obtained from IRENA [44]. For wind, overall
capacity factors differ significantly between datasets, while relative year-
to-year changes are quite similar. Reading-M in general shows the highest
capacity factor in almost all cases for wind and solar PV, while EMHires shows
the lowest. Absolute capacity factor differences go up to around 20% (UK
wind), and relative capacity factors for some cases differ by a factor of
almost 3 (Italy wind). The UReading-M and UReading-E dataset have different
meteorological reanalyses as input but are based on the same assumptions. They
show a relatively good agreement for wind and are closest to each other in all
cases, except for the UK. For solar PV, UReading-M has relatively low
agreement with the other datasets. This potentially supports the observation
by Camargo et al. [16] that modelled PV power based on ERA5 has better
agreement with measurements than MERRA-2. For wind, differences when switching
between MERRA-2 and ERA5 seem to have a smaller effect than when choosing a
different methodological approach, as demonstrated by the proximity of
capacity factors for UReading-M and -E.
Another important point is the difference of the modelled values to the
reported capacity factors. In some cases, such as German wind, reported values
are well captured by Ninja and the EMHires dataset, while results for PV seem
to not meet the modelled values, in both absolute terms and year-to-year
variations. However, one should keep in mind that not all datasets have the
goal to reproduce historical values of capacity factors. Another likely
interpretation of the fact that year-to-year changes of reported values differ
from the modelled dataset is that modelled values are a subject to
meteorological year-to-year changes only, while reported values are also
influenced by changing capacity layouts or technological parameters. The
discrepancy between modelled and realized capacity factors with the latter
being considerably lower has already led to public debates [45]. There is also
a discrepancy between wind capacity factor estimates and realized values based
on oversimplifcations [46]. For PyPSA-EUR, the year 2013 and for RE-Europe the
years 2012-2014 were considered. Capacity factors in both datasets look
unremarkable with the exception of solar PV for RE-Europe; its capacity
factors are substantially higher in all considered countries based on both,
ECMWF forecasts as well as COSMO-REA6.
Figure 4 shows averaged capacity factors per country for five datasets and
reported values. Reported values for solar PV are for most countries are
significantly below the modelled data mean, except for Sweden and Finland.
U-Read-M reports highest solar PV capacity factors for almost all countries.
For solar PV, it can be concluded, that the methodology and the chosen
datasets are largely independent of the location. For wind, the picture looks
different. Restore, for instance, has mostly lower capacity factors in
Southern Europe, while in North- and Northwestern Europe they are close to or
above the ensemble mean. For EMHires, the picture looks the opposite, with
capacity factors below average in the North and Northwest and close or above
average in the South-East.
### 3.2 Correlation
Figure 5: Correlation of locally normalised three-hourly values from
2003-2012. Wind time series were normalised by computing the average and
standard deviation over a moving window of 30 days. For solar PV, this was
done for each hour of the day separately and a window size of 20 days.
The Pearson correlation coefficient measures linear correlation between two
variables. In the context of renewable generation, the Pearson correlation
coefficient is defined as
$\displaystyle\rho_{g^{\alpha},g^{\beta}}$
$\displaystyle=\frac{\text{cov}\left(g^{\alpha},g^{\beta}\right)}{\sigma_{g^{\alpha}}\sigma_{g^{\beta}}}.$
(7)
The Pearson correlation coefficient is commonly used to study whether
different renewable generation sources or renewable generation sources at
different locations are connected via transmission complement each other [47,
48, 49]. Here, we are interested in the correlations of the fast (in the order
of hours) variations of the wind and solar PV time series. The slow variations
caused by large-scale meteorological phenomena and the seasonal cycle of the
sun should be similar in all datasets. Therefore, we apply a local
normalisation as described in Schäfer and Guhr [50]: Wind time series are
normalised by computing the average and standard deviation over a moving
window of 30 days. Solar PV time series are normalised separately for each
hour of the day and with a window size of 20 days. This approach was used to
reduce the dependency of the time series on the diurnal cycle in case of solar
PV and large-scale synoptic conditions for wind. Figure 5 shows pairwise
correlation coefficients over the ten years from 2003-2012 for exemplary
countries. In general, correlations are very high between datasets despite
emphasising the differences by using the moving window approach, except for
solar Data from UReading-M, which has comparably low correlations in many
cases. However, generally high correlation factors indicate that aggregated
temporal patterns are well captured by every dataset. These patterns are
important to capture infrastructure requirements that couples different time
steps together, such as flexible backup power plant infrastructure or energy
storage.
### 3.3 Low Generation Events
Low generation events are times, when resources for both wind and solar PV are
not available. In Germany, these events are referred to as ”Dunkelflaute”,
which can be translated as dark doldrums. Low-wind-power events for Germany
were studied by Ohlendorf and Schill [51]. They found that long low-wind-power
events are rare. An average wind capacity factor below 10% for around five
consecutive days occurs on yearly basis and for a period of around eight days
every ten years. We characterise these events by the aggregated generation
from wind and solar PV of all countries together. First, normalised time
series are multiplied with installed capacities per country and aggregated
$\displaystyle G_{t}=\sum_{n,s}G_{n,s}g_{n,s,t}.$ (8)
ctry | wind | solar | ctry | wind | solar | ctry | wind | solar
---|---|---|---|---|---|---|---|---
AT | 2.1 | 0.8 | DE | 44.9 | 38.2 | PL | 3.8 | 0.0
BE | 2.0 | 3.1 | GR | 2.0 | 2.6 | PT | 4.9 | 0.4
BG | 0.7 | 1.0 | HU | 0.3 | 0.0 | RO | 3.0 | 1.3
CR | 0.3 | 0.0 | IE | 2.3 | 0.0 | SK | 0.0 | 0.6
CY | 0.2 | 0.0 | IT | 8.7 | 18.4 | SI | 0.0 | 0.3
CZ | 0.3 | 2.1 | LV | 0.0 | 0.0 | ES | 23.0 | 4.8
DK | 4.8 | 0.6 | LT | 0.3 | 0.1 | SE | 5.4 | 0.1
EE | 0.3 | 0.0 | LU | 0.1 | 0.1 | UK | 12.4 | 5.2
FI | 0.6 | 0.0 | MT | 0 | 0.1 | | |
FR | 9.3 | 5.6 | NL | 2.8 | 1.1 | | |
Table 2: Capacities per country in [GW] [52, 53] used to produce Fig. 6 and
Fig. 7.
Generation capacities by country from wind and solar PV for 2014 are given in
Table 2.
The resulting time series $\\{G_{1},G_{2},...G_{|t|}\\}$ are then split into
disjoint subsets $\\{G_{i},...,G_{j}\\}$ with
$G_{k}<\alpha\sum_{n,s}G_{n,s}\forall k\in[i,...,j]$ and
$G_{i-1}>=\alpha\sum_{n,s}G_{n,s}\text{ or }i=0$ and
$G_{j+1}>=\alpha\sum_{n,s}G_{n,s}\text{ or }j=|t|$. The number of events is
then given as the cardinality of the set of subsets
$\left|\\{\\{G_{i},...,G_{j}\\},...\\}\right|$ and the average length by the
average number of elements of this set. $\alpha$ is the threshold, below which
an event is considered to be a low generation event.
Figure 6: Number of EU28-wide low generation events (top) and their average
length (bottom). EMHires shows the highest number of events and length. It
should be noted that UReading-E provides only three-hourly data, which makes
the direct comparison difficult. Besides, the other ERA5-based datasets show
similar slopes for growing thresholds.
Figure 6 shows the statistics on low generation events for the datasets
investigated. Average observed lengths are a few hours and the numbers seems
to grow relatively linearly to the growing threshold until a 10% threshold of
total renewable generation. While all datasets show a similar shape, the
overall number of events differs quite a lot. It should be noted that for the
UReading-E dataset the number of events is influenced by its three-hourly
resolution. It is also interesting that the Ninja-dataset demonstrates a
stronger growth rate, while most datasets show a similar growth of the number
of events with increasing threshold. Really extreme low generation events are
very scarce in Ninja, however the number grows rapidly with the growing
threshold overtaking the UReading-M dataset for $\alpha=0.1$.
### 3.4 Ramp Rates
Another measure of importance for a power system are ramp rates. Ramp rates
describe changes in power output that can impact power system operation
disproportionately. Batteries are considered to be important components that
could deal with ramp rates of renewable generators [54, 55]. Ramp rates are
often considered in energy resource assessment studies [56, 57]. Ramp rates of
a power plant are calculated by taking the difference of the output over
passed time:
$\displaystyle RR_{t,\delta_{t}}=g_{t+\delta_{t}}-g_{t}.$ (9)
Figure 7 shows three-hourly europe-wide ramp rates of the different datasets
for wind and PV generation. While the distribution for wind is monotonous and
symmetric, the solar PV ramp rate distributions show minor peaks at around
$\pm 5$% of generation capacities. These are likely caused by the
deterministic diurnal pattern of PV generation resulting from the rotation of
the Earth. UReading-M ramp rates differ significantly from ramp rates
calculated based on other datasets if wind and solar PV are considered
separately. However, if wind and solar PV are added together, these effects
cancel out and the distribution becomes very similar, with the exception of
the relatively prominent second peaks at values $>\pm 0.1$.
Figure 7: EU28-wide three-hourly ramp rates relative to installed renewable
capacities. The small-shoulder peaks for solar PV are likely caused by the
deterministic diurnal pattern of the sun. Note that the URead-M data show
considerably higher ramp rates for wind than the other datasets. From the
system point of view, high ramp rates lead to additional need for balancing.
### 3.5 Optimised Generation Mixes
Generation data are commonly used to study highly renewable power systems.
These include a variety of technologies, such as conventional and renewable
generation technologies, storage, transmission technologies, etc. The purpose
of power system optimisation is to find the cost-optimal combination of these
technologies and their operation under given constraints, for instance carbon
dioxide reduction targets. Moreover, models often do not portray the
electricity sector alone, but also take heating and transportation into
account. This is mostly driven by the eminent need to decarbonize these
sectors and use the flexibility they offer to integrate renewables via
coupling of the sectors [58, 59].
For renewable generators, the capacity factor of a generator is a crucial
variable. One can compute levelized cost of electricity neglecting system
measures to balance variable renewable generation as the ratio of cost over
generation ($\propto$ capacity factor) [60, 61, 62]. To check the influence of
the different datasets on a cost-optimal power system, we present the results
of a simplified optimisation in this section. We optimise the mix of wind,
solar PV and open cycle gas turbines (OCGT) as flexible backup power to cover
the demand of the EU-28+CH+NO countries. The choice of gas was made because
gas is often considered to be a bridge technology towards clean energy that
buys time to foster the energy transition [63, 64].
The optimisation objective reads
$\displaystyle\min_{g,G,F}$ $\displaystyle\left(\sum_{n,s}c_{n,s}\cdot
G_{s}+\sum_{n,s,t}o_{s}\cdot g_{n,s,t}\right)\text{.}$ (10)
It consists of capital costs $c_{n,s}$ for installed capacity $G_{n,s}$ of a
carrier (wind/solar PV) $s$ at node $n$ and marginal costs of generation
$o_{s}$ for energy generation $g_{n,s,t}$ of carrier $s$ at node $n$ and time
$t$. It is furthermore subject to the following constraints.
$\displaystyle\sum_{s}g_{n,s,t}-d_{n,t}=0\quad\forall\quad n\text{,}t\text{.}$
(11)
Demand in space and time needs to be met by dispatched generation from the
various generating technologies and
$\displaystyle 0\cdot G_{n,s}\leq g_{n,s,t}\leq{g}^{+}_{n,s,t}\cdot
G_{n,s}\quad\forall\quad n\text{,}t\text{.},$ (12)
dispatched generation is limited by generation capacity times a weather-
dependent availability ${g}^{+}_{n,s,t}$ for renewables. This availability is
the hourly renewable capacity factor studied in the beginning of the Results
section.
Technology | Capital Cost | Marginal Cost | Emissions
---|---|---|---
| [USD/MW/a] | [USD/MWh] | CO2 [ton/MWh]
OCGT | 47,235 | 58.385 | 0.635
Wind | 136,428 | 0.005 | 0
Solar PV | 76,486 | 0.003 | 0
Table 3: Annualised cost assumptions for generation and storage technologies
derived from different sources [65, 66].
Cost assumptions for generation capacities $c_{n,s}$ and marginal costs
$o_{s}$ as well as emission assumptions are given in Table 3.
Figure 8: Shares of installed renewable capacities and generation. Capacity
curves are less smooth (i.e. monotoneous) than generation curves. Besides,
there is a shift in capacity between wind and solar for datasets Restore and
Ninja at low CO2 prices indicating a flat optimisation maximum with respect to
capacity. The remaining shares comprise gas power plants.
Figure 8 shows shares of the generation and capacities for renewables in
dependency of the price for emitting carbon dioxide. This price is multiplied
with emissions per generation unit and added to the marginal cost of
generation. A growing CO2 price supports the cost-competitiveness of
renewables and increases their shares
Figure 9: Levelized cost of electricity at CO2 prices of 0 USD/ton (top) and
100 USD/ton (bottom) consisting of capital expenditures (CAPEX) and
operational expenditures (OPEX), which are neglible for renewables. Note that
overall differences are not significantly larger at a high CO2 price. Costs
are calculated without the imposed carbon emission price. At a low CO2 price,
some datasets (URead-E and URead-M) already render wind generation relatively
cost-competitive, while the other datasets do not.
Figure 10: Country-wise LCOE in dependency of the meteorological dataset used
for comparison in relation to the ensemble mean. Colors are: EMHires (red),
URead-E (blue), URead-M (green), Restore (purple), Ninja (orange) and reported
(yellow). These LCOE reflects the cost of autark energy supply per country
from renewables plus gas as a backup transition technology. LCOE strongly
depends on the datasets at high LCOE prices. Datasets that provide relatively
low capacity factors, like EMHires in case of Denmark, lead to a significantly
higher LCOE of more than 25%.
Figure 9 shows levelized cost of electricity and their composition for CO2
prices of 0 USD/ton and 100 USD/ton without the cost-contribution from
emitting carbon dioxide. Significant differences can be observed. Based on
EMHires, Restore or Ninja datasets, renewables are not cost-competitive at a
vanishing carbon price, where they contribute only 4-7%. For the two UReading
datasets, the story is different: even at a very low CO2 price, wind
contributes more than 10% to LCOE indicating relatively large shares of
installed capacities. At 100 USD per ton of carbon dioxide emissions,
renewables gain significant ground with respect to their cost-competitiveness.
Now, for all datasets, wind contributes around 30% to LCOE and solar PV around
10%.
Figure 10 shows levelized cost of electricity on the country level for the
different datasets and carbon emission prices of 0 and 100 USD/ton. At 0
USD/ton, most countries rely entirely on gas, because renewables are not cost-
competitive. However, for some countries, such as UK, Ireland, or Denmark,
where renewables successfully compete with gas, differences between the
datasets are significant. Using datasets such as URead-E and Uread-M that
provide relatively high capacity factors, significantly lowers LCOE, by about
10%. The difference in LCOE for countries that entirely rely on gas is based
on differences in the demand patterns, especially between winter and summer.
The effect of the difference in LCOE between datasets is more remarkable at
the CO2 price of 100 USD/ton. Cost differences are drastical in some countries
such as Denmark, where the results of different datasets for LCOE can differ
by more than 50%. The country-resolved plot shows that differences between
countries can be significant, if different meteorological datasets for power
system optimisation are considered. In case the whole continent is considered,
they level out to a large degree.
## 4 Discussion
Data on generation from renewable sources is commonly used in energy system
models to study their behaviour, transition pathways or create policy advice.
Different data potentially leads to conflicting policy advice and
misallocation of large amounts of money.
The integration of renewable energy is a challenging task due to their weather
dependency. To cope with non-dispatchable renewable energy sources, various
concepts can be applied, such as optimizing the mix of different generation
technologies, energy storage [67, 68], transmission, demand-side management
[69, 70] or sector-coupling [58, 59]. When studying multiple datasets that are
commonly used as input into large-scale energy system models, we observed
significant differences between the models. As the differences are also
observed in datasets based on the same meteorological database, they can
likely be attributed to different assumptions in the conversion process to
country-aggregated time series.
Low generation events are defined as periods of continent-wide low generation
from both, wind and solar PV. As these occur continent-wide, a transmission
grid does not help to tackle these. Instead, storage is a viable solution.
With an average length of a few hours, daily and synoptic storage options seem
to be a suitable choice to cope with these low generation events. Commonly,
lithium-ion batteries are proposed as a daily storage [71] and hydrogen
storage or pumped hydro storage (PHS) is a viable storage for the synoptic
scale [72, 59]. Energy system models suffer from uncertainties not only in
meteorological data, but also in assumptions made. Various approaches have
been studied to tackle the problem of uncertainty in energy system models [73,
74, 75, 76, 77].
Besides uncertainty from the generation data source itself and the assumptions
made in creating it, additional uncertainty arises from what period to choose.
Renewable generation resources tend to vary on decadal scales and models
predict that climate change has also profound effects on renewable generation:
Schlott et al. [78] investigated the effects of climate change on a future
European energy system and found an increasing competitiveness of solar PV due
to changing correlation patterns. Wohland et al. [79] found an increasing need
of backup energy due to climate change, Weber et al. [80] concluded the same
and found besides the increasing need for backup energy an increasing need for
storage due to climate change. Kozarcanin et al. [81] studied various metrics
such as variability of renewable generation or short term dispatchable
capacity under climate change but concluded that there is no discernible
effect on these measures. Bloomfield et al. [82] saw significant uncertainty
in power system design due to climate change and pledged for better
understanding of this climate uncertainty. Besides uncertainty arising from
meteorological data, other uncertainties affect power systems, such as
uncertainty in cost assumptions as well as technological developments [83,
84]. The relevance of these effects should be compared to the relevance of
uncertainty arising from meteorological data.
Considering LCOE that were studied in subsection 3.5 it is interesting to note
that differences in the results did not appear to get larger as renewable
shares grew. This seems contradictory to the naive estimate that the choice of
the renewable generation database should increase the dependency on the choice
of meteorological data. However, in a fully detailed energy system model,
complex interdependencies might exist that render smaller or larger effect of
different datasets. One should also note that capacity factors are not the
sole factor in determining cost-efficiency of renewables, because non-
dispatchable renewables might start cannibalising themselves at high
penetration levels reducing their market value [85, 86]. Brown and Reichenberg
[87] have argued that this is the result of policy. A possible way to deal
with reductions in market value are system-friendly renewable generator
designs [88, 89] that aim at producing more at times of higher prices, for
instance PV modules with different azimuths/tilts [90].
### 4.1 Critical Appraisal
A number of systematic differences increase the difficulty to perform the
analysis of the scenarios compared in this paper: datasets are given with
different temporal resolution. While one-hourly data are considered sufficient
to model renewable country-spanning energy systems with sufficient robustness
[91], for three-hourly values this is less evident. However, due to the fact
that computational limitations and linear optimisation problems are in the
complexity class P-complete, we had to find a sweet spot between model
details, temporal and spatial resolution [92, 93] and computational
feasibility. At last, we decided to focus on onshore wind. Offshore wind
energy was treated differently in the different datasets. However, this effect
is rather small, as the installed offshore capacities contribute only a small
percentage of the overall wind capacities as of 2014. Nevertheless, its shares
are rising quickly, due to technical advances, cost reductions and limited
onshore wind potentials [94, 95, 96, 97], although installation potentials
vary significantly in the literature and some researchers suggests far higher
numbers [98].
## 5 Summary and Conclusions
This paper, compares several distinct datasets, which provide renewable
generation time series for both sources, wind and solar PV, on the country
level. Different measures are used, such as ramp rates, correlation
coefficients, annual capacity factors and optimized mixes of generation, from
wind/solar/gas. From the presented results, the following conclusions can be
drawn:
* 1.
Differences between model statistics are significant, even if they are based
on the same meteorological database. These differences are likely due to
different assumptions about the conversion from weather to energy. There is a
significant need for more research on the effects and interactions of choices
made in the modelling chain weather-to-energy at every step, to achieve better
understanding.
* 2.
These significant differences have severe consequences for the optimisation
and further studies of power systems. Differences in capacity factors directly
affect both, CAPEX and OPEX, of renewable energy generation, transport and
storage technologies. Hence their optimized shares in power system expansion
models are quite sensible to these uncertainties.
* 3.
Emphasis in renewable energy systems research must be put on the use of
adequate generation data and discussion of its properties, before we can offer
reliable research results and robust policy advice.
Future research must focus on the study of these critical issues using a
reliable power system model, which allows to capture complex dependencies
between both, spatial and temporal effects as well as different types of
technologies. The methods proposed by Schyska and Kies [75], Nacken et al.
[73] and Neumann and Brown [74] shall be considered to improve the study of
the effects of the differences in the input weather data on the resulting
output of the large-scale energy system models.
## Acknowledgements
Research is funded by the Federal Ministry of Economic Affairs and Energy
(BMWi) under grant nr. FKZ03EI1028A (EnergiesysAI) and the Federal Ministry of
Research and Education (BMBF) under grant nr. FKZ 03EK3055C (CoNDyNet II). HS
acknowledges the Judah Eisenberg Professor Laureatus of the Fachbereich Physik
and the Walter Greiner Gesellschaft.
## Data Availability
The datasets analysed in this work are available in a harmonised form under
https://github.com/alexfias/compare_met_data/.
## References
* Chu and Majumdar [2012] S. Chu, A. Majumdar, Opportunities and challenges for a sustainable energy future, nature 488 (2012) 294–303.
* Change et al. [2014] I. C. Change, et al., Mitigation of climate change, Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change 1454 (2014).
* Commision [2019] E. Commision, The european green deal. communication from the commission to the european parliment (2019).
* Bloomfield et al. [2020] H. Bloomfield, P. Gonzalez, J. Lundquist, L. Stoop, J. Browell, R. Dargaville, M. De Felice, K. Gruber, A. Hilbers, A. Kies, et al., The importance of weather and climate to energy systems: A workshop on next generation challenges in energy-climate modelling, Bulletin of the American Meteorological Society (2020).
* Staffell and Pfenninger [2018] I. Staffell, S. Pfenninger, The increasing impact of weather on electricity supply and demand, Energy 145 (2018) 65–78.
* Müller et al. [2018] B. Müller, F. Gardumi, L. Hülk, Comprehensive representation of models for energy system analyses: Insights from the energy modelling platform for europe (emp-e) 2017, Energy strategy reviews 21 (2018) 82–87.
* Moraes Jr et al. [2018] L. Moraes Jr, C. Bussar, P. Stoecker, K. Jacqué, M. Chang, D. Sauer, Comparison of long-term wind and photovoltaic power capacity factor datasets with open-license, Applied Energy 225 (2018) 209–220.
* Lantz et al. [2019] E. J. Lantz, J. O. Roberts, J. Nunemaker, E. DeMeo, K. L. Dykes, G. N. Scott, Increasing wind turbine tower heights: Opportunities and challenges, Technical Report, National Renewable Energy Lab.(NREL), Golden, CO (United States), 2019.
* Kravtsov et al. [2014] S. Kravtsov, M. G. Wyatt, J. A. Curry, A. A. Tsonis, Two contrasting views of multidecadal climate variability in the twentieth century, Geophysical Research Letters 41 (2014) 6881–6888.
* Jurasz et al. [2020] J. Jurasz, F. Canales, A. Kies, M. Guezgouz, A. Beluco, A review on the complementarity of renewable energy sources: Concept, metrics, application and future research directions, Solar Energy 195 (2020) 703–724.
* Rienecker et al. [2011] M. M. Rienecker, M. J. Suarez, R. Gelaro, R. Todling, J. Bacmeister, E. Liu, M. G. Bosilovich, S. D. Schubert, L. Takacs, G.-K. Kim, et al., Merra: Nasa’s modern-era retrospective analysis for research and applications, Journal of climate 24 (2011) 3624–3648.
* Hersbach et al. [2020] H. Hersbach, B. Bell, P. Berrisford, S. Hirahara, A. Horányi, J. Muñoz-Sabater, J. Nicolas, C. Peubey, R. Radu, D. Schepers, et al., The era5 global reanalysis, Quarterly Journal of the Royal Meteorological Society 146 (2020) 1999–2049.
* Bollmeyer et al. [2015] C. Bollmeyer, J. Keller, C. Ohlwein, S. Wahl, S. Crewell, P. Friederichs, A. Hense, J. Keune, S. Kneifel, I. Pscheidt, et al., Towards a high-resolution regional reanalysis for the european cordex domain, Quarterly Journal of the Royal Meteorological Society 141 (2015) 1–15.
* Schulz et al. [2009] J. Schulz, P. Albert, H.-D. Behr, D. Caprion, H. Deneke, S. Dewitte, B. Durr, P. Fuchs, A. Gratzki, P. Hechler, et al., Operational climate monitoring from space: the eumetsat satellite application facility on climate monitoring (cm-saf), Atmospheric Chemistry and Physics 9 (2009) 1687–1709.
* Olauson [2018] J. Olauson, Era5: The new champion of wind power modelling?, Renewable energy 126 (2018) 322–331.
* Camargo and Schmidt [2020] L. R. Camargo, J. Schmidt, Simulation of long-term time series of solar photovoltaic power: is the era5-land reanalysis the next big step?, arXiv preprint arXiv:2003.04131 (2020).
* Gruber et al. [2020] K. Gruber, P. Regner, S. Wehrle, M. Zeyringer, J. Schmidt, Towards a global dynamic wind atlas: A multi-country validation of wind power simulation from merra-2 and era-5 reanalyses bias-corrected with the global wind atlas, arXiv preprint arXiv:2012.05648 (2020).
* Urraca et al. [2018] R. Urraca, T. Huld, A. Gracia-Amillo, F. J. Martinez-de Pison, F. Kaspar, A. Sanz-Garcia, Evaluation of global horizontal irradiance estimates from era5 and cosmo-rea6 reanalyses using ground and satellite-based data, Solar Energy 164 (2018) 339–354.
* Jourdier [2020] B. Jourdier, Evaluation of era5, merra-2, cosmo-rea6, newa and arome to simulate wind power production over france, Advances in Science and Research 17 (2020) 63–77.
* Piasecki et al. [2019] A. Piasecki, J. Jurasz, A. Kies, Measurements and reanalysis data on wind speed and solar irradiation from energy generation perspectives at several locations in poland, SN Applied Sciences 1 (2019) 865.
* Pfenninger and Staffell [2016] S. Pfenninger, I. Staffell, Renewables. ninja, URL https://www. renewables. ninja (2016).
* Gonzalez Aparicio et al. [2016] I. Gonzalez Aparicio, A. Zucker, F. Careri, F. Monforti, T. Huld, J. Badger, Emhires dataset, Part I: Wind power generation European Meteorological derived HIgh resolution RES generation time series for present and future scenarios (2016).
* Kies et al. [2016] A. Kies, K. Chattopadhyay, L. von Bremen, E. Lorenz, D. Heinemann, Restore 2050: Simulation of renewable feed-in for power system studies, Technical Report, Tech. Rep, 2016.
* Bloomfield et al. [2020a] H. Bloomfield, D. Brayshaw, A. Charlton-Perez, Era5 derived time series of european country-aggregate electricity demand, wind power generation and solar power generation: hourly data from 1979-2019 (2020a).
* Bloomfield et al. [2020b] H. Bloomfield, D. Brayshaw, A. Charlton-Perez, Merra2 derived time series of european country-aggregate electricity demand, wind power generation and solar power generation (2020b).
* Hörsch et al. [2018] J. Hörsch, F. Hofmann, D. Schlachtberger, T. Brown, Pypsa-eur: An open optimisation model of the european transmission system, Energy Strategy Reviews 22 (2018) 207–215.
* Jensen and Pinson [2017] T. V. Jensen, P. Pinson, Re-europe, a large-scale dataset for modeling a highly renewable european electricity system, Scientific data 4 (2017) 170175.
* Schulz et al. [2014] J.-P. Schulz, U. Schättler, D. Wetterdienst, Kurze beschreibung des lokal-modells europa COSMO-EU (LME) und seiner datenbanken auf dem datenserver des DWD, DWD, 2014.
* Lydia et al. [2014] M. Lydia, S. S. Kumar, A. I. Selvakumar, G. E. P. Kumar, A comprehensive review on wind turbine power curve modeling techniques, Renewable and Sustainable Energy Reviews 30 (2014) 452–460.
* Taslimi-Renani et al. [2016] E. Taslimi-Renani, M. Modiri-Delshad, M. F. M. Elias, N. A. Rahim, Development of an enhanced parametric model for wind turbine power curve, Applied energy 177 (2016) 544–552.
* Klucher [1979] T. M. Klucher, Evaluation of models to predict insolation on tilted surfaces, Solar energy 23 (1979) 111–114.
* Pfenninger and Staffell [2016] S. Pfenninger, I. Staffell, Long-term patterns of european pv output using 30 years of validated hourly reanalysis and satellite data, Energy 114 (2016) 1251–1265.
* Staffell and Pfenninger [2016] I. Staffell, S. Pfenninger, Using bias-corrected reanalysis to simulate current and future wind power output, Energy 114 (2016) 1224–1239.
* Huld et al. [2010] T. Huld, R. Gottschalg, H. G. Beyer, M. Topič, Mapping the performance of pv modules, effects of module type and data averaging, Solar Energy 84 (2010) 324–338.
* Staffell and Green [2014] I. Staffell, R. Green, How does wind farm performance decline with age?, Renewable energy 66 (2014) 775–786.
* Gonzalez-Aparicio et al. [2017] I. Gonzalez-Aparicio, F. Monforti, P. Volker, A. Zucker, F. Careri, T. Huld, J. Badger, Simulating european wind power generation applying statistical downscaling to reanalysis data, Applied Energy 199 (2017) 155–168.
* Gonzalez Aparicio et al. [2017] I. Gonzalez Aparicio, T. Huld, F. Careri, F. Monforti, A. Zucker, Emhires dataset part ii: Solar power generation, European Meteorological Derived HIgh Resolution RES Generation Time Series for Present and Future Scenarios https://doi. org/10.2760/044693 (2017).
* win [????] The wind power, ???? URL: http://www.thewindpower.net/.
* Huld and Amillo [2015] T. Huld, A. M. G. Amillo, Estimating pv module performance over large geographical regions: The role of irradiance, air temperature, wind speed and solar spectrum, Energies 8 (2015) 5159–5181.
* Huld et al. [2012] T. Huld, R. Müller, A. Gambardella, A new solar radiation database for estimating pv performance in europe and africa, Solar Energy 86 (2012) 1803–1815.
* Šúri et al. [2007] M. Šúri, T. A. Huld, E. D. Dunlop, H. A. Ossenbrink, Potential of solar electricity generation in the european union member states and candidate countries, Solar energy 81 (2007) 1295–1305.
* Abed and El-Mallah [1997] K. Abed, A. El-Mallah, Capacity factor of wind turbines, Energy 22 (1997) 487–491.
* Kies et al. [2016] A. Kies, B. U. Schyska, L. Von Bremen, Curtailment in a highly renewable power system and its effect on capacity factors, Energies 9 (2016) 510.
* Agency [2020] I. R. E. Agency, Trends in renewable energy, Available online at https://www.irena.org/Statistics/ (2020).
* Boccard [2009] N. Boccard, Capacity factor of wind power realized values vs. estimates, energy policy 37 (2009) 2679–2688.
* Miller and Keith [2018] L. M. Miller, D. W. Keith, Observation-based solar and wind power capacity factors and power densities, Environmental Research Letters 13 (2018) 104008.
* Giebel [2001] G. Giebel, On the benefits of distributed generation of wind energy in Europe, VDI-Verlag, 2001\.
* Jurasz et al. [2018] J. Jurasz, A. Beluco, F. A. Canales, The impact of complementarity on power supply reliability of small scale hybrid energy systems, Energy 161 (2018) 737–743.
* Francois et al. [2016] B. Francois, M. Borga, J.-D. Creutin, B. Hingray, D. Raynaud, J.-F. Sauterleute, Complementarity between solar and hydro power: Sensitivity study to climate characteristics in northern-italy, Renewable energy 86 (2016) 543–553.
* Schäfer and Guhr [2010] R. Schäfer, T. Guhr, Local normalization: Uncovering correlations in non-stationary financial time series, Physica A: Statistical Mechanics and its Applications 389 (2010) 3856–3865.
* Ohlendorf and Schill [2020] N. Ohlendorf, W.-P. Schill, Frequency and duration of low-wind-power events in germany, Environmental Research Letters (2020).
* Association et al. [2015] E. W. E. Association, et al., Wind in power 2014 european statistics.[online] available at: http://www. ewea. org/fileadmin/files/library/publications/statistics, EWEA-Annual-Statistics-2014. pdf [Accessed 21 April 2015] (2015).
* EurObserv’ER [2015] EurObserv’ER, Photovoltaic barometer 2015 (2015).
* Frate et al. [2019] G. Frate, P. Cherubini, C. Tacconelli, A. Micangeli, L. Ferrari, U. Desideri, Ramp rate abatement for wind power plants: A techno-economic analysis, Applied Energy 254 (2019) 113600.
* Lappalainen et al. [2020] K. Lappalainen, G. C. Wang, J. Kleissl, Estimation of the largest expected photovoltaic power ramp rates, Applied Energy 278 (2020) 115636.
* Zhang et al. [2018] H. Zhang, Y. Cao, Y. Zhang, V. Terzija, Quantitative synergy assessment of regional wind-solar energy resources based on merra reanalysis data, Applied energy 216 (2018) 172–182.
* Widén [2011] J. Widén, Correlations between large-scale solar and wind power in a future scenario for sweden, IEEE transactions on sustainable energy 2 (2011) 177–184.
* Schaber et al. [2013] K. Schaber, F. Steinke, T. Hamacher, Managing temporary oversupply from renewables efficiently: Electricity storage versus energy sector coupling in germany, in: International Energy Workshop, Paris, 2013\.
* Brown et al. [2018] T. Brown, D. Schlachtberger, A. Kies, S. Schramm, M. Greiner, Synergies of sector coupling and transmission reinforcement in a cost-optimised, highly renewable european energy system, Energy 160 (2018) 720–739.
* Lai and McCulloch [2017] C. S. Lai, M. D. McCulloch, Levelized cost of electricity for solar photovoltaic and electrical energy storage, Applied energy 190 (2017) 191–203.
* Talavera et al. [2015] D. Talavera, P. Pérez-Higueras, J. Ruíz-Arias, E. Fernández, Levelised cost of electricity in high concentrated photovoltaic grid connected systems: spatial analysis of spain, Applied Energy 151 (2015) 49–59.
* Bosch et al. [2019] J. Bosch, I. Staffell, A. D. Hawkes, Global levelised cost of electricity from offshore wind, Energy 189 (2019) 116357.
* Brown et al. [2009] S. P. Brown, A. Krupnick, M. A. Walls, Natural gas: a bridge to a low-carbon future, Issue brief (2009) 09–11.
* Gonzalez-Salazar et al. [2018] M. A. Gonzalez-Salazar, T. Kirsten, L. Prchlik, Review of the operational flexibility and emissions of gas-and coal-fired power plants in a future with growing renewables, Renewable and Sustainable Energy Reviews 82 (2018) 1497–1513.
* Carlsson et al. [2014] J. Carlsson, M. d. M. P. Fortes, G. de Marco, J. Giuntoli, M. Jakubcionis, A. Jäger-Waldau, R. Lacal-Arantegui, S. Lazarou, D. Magagna, C. Moles, et al., Etri 2014-energy technology reference indicator projections for 2010-2050, European Commission, Joint Research Centre, Institute for Energy and Transport, Luxembourg: Publications Office of the European Union (2014).
* Schroeder et al. [2013] A. Schroeder, F. Kunz, R. Mendelevitsch, C. von Hirschhausen, Current And Prospective Costs Of Electricity Generation, in: Energy Economics of Phasing out Carbon and Uranium, 13th IAEE European Conference, August 18-21, 2013, International Association for Energy Economics, (2013).
* Zafirakis et al. [2013] D. Zafirakis, K. J. Chalvatzis, G. Baiocchi, G. Daskalakis, Modeling of financial incentives for investments in energy storage systems that promote the large-scale integration of wind energy, Applied Energy 105 (2013) 138–154.
* Zhao et al. [2015] H. Zhao, Q. Wu, S. Hu, H. Xu, C. N. Rasmussen, Review of energy storage system for wind power integration support, Applied energy 137 (2015) 545–553.
* Nebel et al. [2020] A. Nebel, C. Krüger, T. Janßen, M. Saurat, S. Kiefer, K. Arnold, Comparison of the effects of industrial demand side management and other flexibilities on the performance of the energy system, Energies 13 (2020) 4448.
* Kies et al. [2016] A. Kies, B. U. Schyska, L. Von Bremen, The demand side management potential to balance a highly renewable european power system, Energies 9 (2016) 955.
* Jannesar et al. [2018] M. R. Jannesar, A. Sedighi, M. Savaghebi, J. M. Guerrero, Optimal placement, sizing, and daily charge/discharge of battery energy storage in low voltage distribution network with high photovoltaic penetration, Applied energy 226 (2018) 957–966.
* Kroniger and Madlener [2014] D. Kroniger, R. Madlener, Hydrogen storage for wind parks: A real options evaluation for an optimal investment in more flexibility, Applied energy 136 (2014) 931–946.
* Nacken et al. [2019] L. Nacken, F. Krebs, T. Fischer, C. Hoffmanna, Integrated renewable energy systems for germany–a model-based exploration of the decision space, in: 2019 16th International Conference on the European Energy Market (EEM), IEEE, 2019, pp. 1–8.
* Neumann and Brown [2019] F. Neumann, T. Brown, The near-optimal feasible space of a renewable power system model, arXiv preprint arXiv:1910.01891 (2019).
* Schyska et al. [2020] B. U. Schyska, A. Kies, M. Schlott, L. von Bremen, W. Medjroubi, The sensitivity of power system expansion models, arXiv preprint arXiv:2007.09956 (2020).
* Pedersen et al. [2020] T. T. Pedersen, M. Victoria, M. G. Rasmussen, G. B. Andresen, Modeling all alternative solutions for highly renewable energy systems, arXiv preprint arXiv:2010.00836 (2020).
* Hilbers et al. [2020] A. P. Hilbers, D. J. Brayshaw, A. Gandy, Importance subsampling for power system planning under multi-year demand and weather uncertainty, in: 2020 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), IEEE, 2020, pp. 1–6.
* Schlott et al. [2018] M. Schlott, A. Kies, T. Brown, S. Schramm, M. Greiner, The impact of climate change on a cost-optimal highly renewable european electricity network, Applied energy 230 (2018) 1645–1659.
* Wohland et al. [2017] J. Wohland, M. Reyers, J. Weber, D. Witthaut, More homogeneous wind conditions under strong climate change decrease the potential for inter-state balancing of electricity in europe, Earth System Dynamics 8 (2017) 1047.
* Weber et al. [2018] J. Weber, J. Wohland, M. Reyers, J. Moemken, C. Hoppe, J. G. Pinto, D. Witthaut, Impact of climate change on backup energy and storage needs in wind-dominated power systems in europe, PloS one 13 (2018) e0201457.
* Kozarcanin et al. [2019] S. Kozarcanin, H. Liu, G. B. Andresen, 21st century climate change impacts on key properties of a large-scale renewable-based electricity system, Joule 3 (2019) 992–1005.
* Bloomfield et al. [2020] H. Bloomfield, D. Brayshaw, A. Troccoli, C. Goodess, M. De Felice, L. Dubus, P. Bett, Y.-M. Saint-Drenan, Quantifying the sensitivity of european power systems to energy scenarios and climate change projections, Renewable Energy 164 (2020) 1062–1075.
* Schyska and Kies [2020] B. U. Schyska, A. Kies, How regional differences in cost of capital influence the optimal design of power systems, Applied Energy 262 (2020) 114523.
* Monforti and Gonzalez-Aparicio [2017] F. Monforti, I. Gonzalez-Aparicio, Comparing the impact of uncertainties on technical and meteorological parameters in wind power time series modelling in the european union, Applied Energy 206 (2017) 439–450.
* Hirth [2013] L. Hirth, The market value of variable renewables: The effect of solar wind power variability on their relative price, Energy economics 38 (2013) 218–236.
* Kies et al. [2019] A. Kies, J. Jurasz, P. B. Dabek, Market value of pv battery systems for autonomous rural energy supply, Energy Procedia 158 (2019) 1188–1193.
* Brown and Reichenberg [2020] T. Brown, L. Reichenberg, Decreasing market value of variable renewables is a result of policy, not variability, arXiv preprint arXiv:2002.05209 (2020).
* Hirth and Müller [2016] L. Hirth, S. Müller, System-friendly wind power: How advanced wind turbine design can increase the economic value of electricity generated through wind power, Energy Economics 56 (2016) 51–63.
* Tafarte et al. [2020] P. Tafarte, A. Kanngießer, M. Dotzauer, B. Meyer, A. Grevé, M. Millinger, Interaction of electrical energy storage, flexible bioenergy plants and system-friendly renewables in wind-or solar pv-dominated regions, Energies 13 (2020) 1133.
* Chattopadhyay et al. [2017] K. Chattopadhyay, A. Kies, E. Lorenz, L. von Bremen, D. Heinemann, The impact of different pv module configurations on storage and additional balancing needs for a fully renewable european power system, Renewable energy 113 (2017) 176–189.
* Brown et al. [2018] T. W. Brown, T. Bischof-Niemz, K. Blok, C. Breyer, H. Lund, B. V. Mathiesen, Response to ‘burden of proof: A comprehensive review of the feasibility of 100% renewable-electricity systems’, Renewable and sustainable energy reviews 92 (2018) 834–847.
* Raventós and Bartels [2020] O. Raventós, J. Bartels, Evaluation of temporal complexity reduction techniques applied to storage expansion planning in power system models, Energies 13 (2020) 988.
* Siala and Mahfouz [2019] K. Siala, M. Y. Mahfouz, Impact of the choice of regions on energy system models, Energy Strategy Reviews 25 (2019) 75–85.
* Bosch et al. [2017] J. Bosch, I. Staffell, A. D. Hawkes, Temporally-explicit and spatially-resolved global onshore wind energy potentials, Energy 131 (2017) 207–217.
* Czisch [2006] G. Czisch, Szenarien zur zukünftigen Stromversorgung, kostenoptimierte Variationen zur Versorgung Europas und seiner Nachbarn mit Strom aus erneuerbaren Energien, Ph.D. thesis, 2006\.
* Permien and Enevoldsen [2019] F.-H. Permien, P. Enevoldsen, Socio-technical constraints in german wind power planning: An example of the failed interdisciplinary challenge for academia, Energy Research & Social Science 55 (2019) 122–133.
* McKenna et al. [2013] R. McKenna, S. Gantenbein, W. Fichtner, Determination of cost–potential-curves for wind energy in the german federal state of baden-württemberg, Energy Policy 57 (2013) 194–203.
* Ryberg et al. [2019] D. S. Ryberg, D. G. Caglayan, S. Schmitt, J. Linßen, D. Stolten, M. Robinius, The future of european onshore wind energy potential: Detailed distribution and simulation of advanced turbine designs, Energy 182 (2019) 1222–1238.
|
# Soft Genetic Programming Binary Classifiers
Ivan Gridin
<EMAIL_ADDRESS>
###### Abstract
The study of the classifier’s design and it’s usage is one of the most
important machine learning areas. With the development of automatic machine
learning methods, various approaches are used to build a robust classifier
model. Due to some difficult implementation and customization complexity,
genetic programming (GP) methods are not often used to construct classifiers.
GP classifiers have several limitations and disadvantages. However, the
concept of "soft" genetic programming (SGP) has been developed, which allows
the logical operator tree to be more flexible and find dependencies in
datasets, which gives promising results in most cases. This article discusses
a method for constructing binary classifiers using the SGP technique. The test
results are presented. Source code -
https://github.com/survexman/sgp_classifier
## 1 Introduction
Genetic Programming (GP) is a promising machine learning technique based on
the principles of Darwinian evolution to automatically evolve computer
programs to solve problems. GP is especially suitable for building a
classifier of tree representation. GP is a soft computing search technique,
which is used to evolve a tree-structured program toward minimizing the
fitness value of it. The distinctive features of GP make it very convenient
for classification, and the benefit of it is the flexibility, which allows the
algorithm to be adapted to the needs of each particular problem.
A special case of GP studies the logical tree’s development as a solution to a
classification problem [1]. Logical trees are composed of boolean, comparison,
and arithmetic operators, and they output a boolean value ( true or false). A
solution presented as a logical tree is a very convenient way to analyze a
dataset and interpret a solution. Figure 1 shows an example of logical tree.
Figure 1: Logical tree example
This logical tree can be rewritten as the following system of inequalities:
$x^{2}\geq 7\;\lor\;\systeme{y\geq 3,y\leq 8,z<x}$
Evolving logical trees using GP has the benefit of being able to handle both
numerical and categorical data fairly simply. Significant advantage of
evolving logical trees is the fact that the trees are highly interpretable to
a researcher [4]. Logical trees are portable and can be easily implemented by
various tools and programming languages. Another significant advantage is an
ability of feature extraction [6].
However, one of the most critical problems in logical trees is a crucial logic
change when a tree operator changes [9][11]. Such operator changes occur as a
result of crossover and mutation operations [2].
Figure 2: Operator change in Logical Tree
Figure 2 shows a simple change of addition operator to power operator in
logical tree:
$[x>3\;\oplus\;sin(x(y+15)))>0.5]\;\;\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr\scalebox{1.2}{.}\cr$\longrightarrow$\cr}
}}\;\;[x>3\;\oplus\;sin(xy^{15})))>0.5]$
And below at Figure 3 we can see the effect of such modification:
Figure 3: Operator change effect
We see that switching the operator +(y,15) to pow(y,15) changes the logic of
the classifier drastically. The more complex the logical tree is, the more
this feature is noticeable. This sensitivity of logical trees to tiny changes
makes it very difficult to find the appropriate logical tree in a smooth way.
## 2 Study Roadmap
SGP classifier design is based on classical GP classifier design. First, we
will define the architecture of GP classifier. Next, we will introduce the
Soft Genetic Programming approach, its design, and evolution operations. Soft
Genetic Programming’s main goal is to smoothen evolution improvement and
increase the probability of reaching local maxima. After, we will compare the
behavior of classical GP Classifier and SGP Classifier. And finally, we will
present empirical evidence of the applicability of SGP approach.
## 3 Genetic Programming Classifier Design
Genetic programming (GP) is a flexible and powerful evolutionary technique
with some special features that are suitable for building a classifier of tree
representation. [2].
### 3.1 Operators
We will use the following operators for genetic programming trees [3]:
Boolean:
$\displaystyle\textrm{OR}:\ \\{{0,1}\\}^{2}\rightarrow\\{{0,1}\\}$
$\displaystyle\textrm{AND}:\ \\{{0,1}\\}^{2}\rightarrow\\{{0,1}\\}$
$\displaystyle\textrm{NOT}:\ \\{{0,1}\\}\rightarrow\\{{0,1}\\}$
Comparison:
$\displaystyle\textrm{>}:\ \mathbb{R}^{2}\rightarrow\\{{0,1}\\}$
$\displaystyle\textrm{<}:\ \mathbb{R}^{2}\rightarrow\\{{0,1}\\}$
Mathematical:
$\displaystyle\textrm{+}:\ \mathbb{R}^{2}\rightarrow\mathbb{R}$
$\displaystyle\times:\ \mathbb{R}^{2}\rightarrow\mathbb{R}$ $\displaystyle-:\
\mathbb{R}\rightarrow\mathbb{R}$
Terms:
$Symbolic:\ \rightarrow X_{i}\;\;\;i\in
1,n,\;where\;n\;-\;number\;of\;features$ $Constant:\ \rightarrow\mathbb{R}$
### 3.2 Random Tree Generation
Random tree generation is limited by maximum and minimum operator type
subchain [8]:
| min | max
---|---|---
Boolean | 1 | 3
Comparison | 1 | 1
Mathematical | 1 | 4
Terms | 1 | 1
### 3.3 Fitness Function
For fitness function we use Balanced Accuracy:
$Balanced\;Accuracy=\frac{1}{2}(\frac{TP}{TP+FN}+\frac{TN}{TN+FP})$
### 3.4 Crossover
The crossover operator creates two new offspring which are formed by taking
parts (genetic material) from two parents. The operator selects two parents
from the population based on a selection method. A crossover point is then
randomly selected in both trees, say point $p_{1}$ and $p_{2}$ , from tree
$t_{1}$ and $t_{2}$ respectively. The crossover then happens as follows: the
subtree rooted at $p_{1}$ is removed from $t_{1}$ and inserted into the
position $p_{2}$ in $t_{2}$. The same logic applies to the point $p_{2}$ ; the
subtree root at the point is removed from $t_{2}$ and inserted into the place
of $p_{1}$ in $t_{1}$ [4]. Figure 4 illustrates the crossover operation.
Figure 4: Crossover
### 3.5 Operator Mutation
An Operator Mutation works the following way: an operator is randomly selected
and replaced with a random tree. A new randomly generated subtree is inserted
instead of selected operator [2]. Figure 5 illustrates the operator mutation
operation.
Figure 5: Mutation
### 3.6 Term Mutation
Term mutation never replaces a term by a random tree. The term mutation is
defined as follows:
$if\;X\;is\;Symbolic,\;then\;X\;is\;replaced\;by\;random\;X_{i}\;i\in[1,n],\;where\;n\;is\;number\;of\;features\;in\;dataset$
$if\;X\;is\;Constant,\;then\;X\;is\;replaced\;by\;X+r\;where\;r\;is\;normal\;random\;variable\;(0;\;1)$
### 3.7 Mutation Probabilities
During mutation each operator type is being randomly chosen with following
probability table:
Operator Type | Probability
---|---
Boolean | 0.1
Comparison | 0.2
Mathematical | 0.3
Terms | 0.5
### 3.8 Selection
Rank selection with elite size 1 used as the selection operation [2].
### 3.9 Evolution Algorithm
Evolution is implemented via canonical genetic algorithm method [5]:
Result: Best individual
1 max`_`generation = 100;
2 population`_`size = 100;
3 cx`_`prob = 0.5;
4 mut`_`prob = 0.5;
5 population = random`_`population(population`_`size);
6 best`_`ind = select`_`best(population);
7 generation = 0;
8 while _best_ind.fitness < 1 or generation < max_generation_ do
9 selected_population = selection(population) ;
10 crossed_population = crossover(selected_population, cx_prob) ;
11 mutated_population = mutation(crossed_population, mut_prob) ;
12 population = mutated_population best_ind = select_best(population);
13 generation++;
14
15 end while
return best`_`ind
Algorithm 1 GP Classifier Evolution Algorithm
## 4 Soft Genetic Programming Classifier Design
The main idea of SGP is the usage of weighted continuous functions instead of
discontinuous boolean and comparison functions. Weight setting will allow
calibrating each operator’s effect in a pseudo logical tree, leading to a more
adaptive classifier design.
### 4.1 Soft Operators
We change boolean and comparison operators to weighted continuous analogs [10]
(we will call those function sets pseudo boolean and pseudo comparison
operators):
Pseudo Boolean:
$OR$:$[0,1]\times[0,1]^{2}$$(w;x,y)$$\in$$[0,1]$$w\cdot max(x,y)$$\in$
$AND$:$[0,1]\times[0,1]^{2}$$(w;x,y)$$\in$$[0,1]$$w\cdot min(x,y)$$\in$
$NOT$:$[0,1]\times[0,1]$$(w;x)$$\in$$[0,1]$$w(1-x)$$\in$
Also we will add additional operators:
$AND3$:$[0,1]\times[0,1]^{3}$$(w;x,y,z)$$\in$$[0,1]$$w\cdot min(x,y,z)$$\in$
$OR3$:$[0,1]\times[0,1]^{3}$$(w;x,y,z)$$\in$$[0,1]$$w\cdot max(x,y,z)$$\in$
Pseudo Comparison:
$>$:$[0,1]\times\mathbb{R}^{2}$$(w;x,y)$$\in$$[0,1]$$w(\frac{x-y}{|x-y|})$$\in$
$<$:$[0,1]\times\mathbb{R}^{2}$$(w;x,y)$$\in$$[0,1]$$w(\frac{y-x}{|x-y|})$$\in$
Mathematical:
$+$:$\mathbb{R}^{2}$$(x,y)$$\in$$\mathbb{R}$$x+y$$\in$
$\mathbin{\leavevmode\hbox to6.89pt{\vbox
to6.89pt{\pgfpicture\makeatletter\hbox{\hskip 0.43056pt\lower-0.43056pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}\pgfsys@setlinewidth{0.86111pt}\pgfsys@invoke{ }{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.02776pt}{6.02776pt}\pgfsys@moveto{0.0pt}{6.02776pt}\pgfsys@lineto{6.02776pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$:$\mathbb{R}^{2}$$(x,y)$$\in$$\mathbb{R}$$xy$$\in$
$-$:$\mathbb{R}$$x$$\in$$\mathbb{R}$$-x$$\in$
Also we will add additional nonlinear operators:
$sigm$:$\mathbb{R}$$x$$\in$$\mathbb{R}$$\frac{\mathrm{1}}{\mathrm{1}+e^{(}-x)}$$\in$
$lin2$:$\mathbb{R}^{2}\times\mathbb{R}^{2}$$(a,b;x,y)$$\in$$\mathbb{R}$$ax+by$$\in$
$lin3$:$\mathbb{R}^{3}\times\mathbb{R}^{3}$$(a_{i};x_{i})$$\in$$\mathbb{R}$$\sum{a_{i}x_{i}}$$\in$
Terms:
$Symbolic:\ \rightarrow X_{i}\;\;\;i\in
1,n,\;where\;n\;-\;number\;of\;features$ $Constant:\ \rightarrow\mathbb{R}$
Weights:
$Weight:\ \rightarrow
w\;,\;where\;w\;is\;random\;variable\;uniformly\;distributed\;on\;[0,1]$
### 4.2 Soft Tree Representation
SGP tree is a tree where each pseudo boolean and comparison operator has its
weight parameter, as is shown on Figure 6:
Figure 6: SGP Tree
### 4.3 Random Tree Generation
See 3.2
### 4.4 Fitness Function
See 3.3
### 4.5 Weight Adjustment
In SGP we introduce weight adustment operation:
1procedure weight_adjustment($individual,max\\_tries$);
2 for _i in max_tries_ do
3 candidate = copy(individual);
4 coordinate, w = individual.get_random_weight();
5 new_w = w + random_shift();
6 candidate.set_weight(coordinate, new_w);
7 if _candidate.fitness > individual.fitness_ then
8 return candidate
9 end if
10
11 end for
return individual
Algorithm 2 Weight Adjustment
The main point of weight adjustment operation is to find any positive
improvement using weight calibration.
### 4.6 Fitness Driven Genetic Operations
The main problem of classifier trees is that they have very fragile
structures, and the probability of degradation after canonical crossover and
mutation operations is to high. In the SGP evolution algorithm, we will use
only positive improvement evolution operations.
Positive Crossover doesn’t accept an offsrping which is worse then its
parents:
1procedure positive_crossover($ind1,ind2$);
2 child1, child2 = crossover(ind1, ind2);
3 candidate1, candidate2 = max([ind1,ind2,child1,child2], by=fitness, 2);
return [candidate1, candidate2]
Algorithm 3 Positive Crossover
Positive Mutation accepts only those mutations which improves individual’s
genome:
1procedure positive_mutation($ind1,max\\_tries$);
2 for _i in max_tries_ do
3 mutant = mutate(ind);
4 if _mutant.fitness > ind.fitness_ then
5 return mutant
6 end if
7
8 end for
return ind
Algorithm 4 Positive Mutation
### 4.7 Extension Mutation
There is a handy technique for avoiding stucking an improvement of the
population in some specific random subspace. A good way of an individual
improvement is adding OR operator as tree root with random subtree, as it is
shown on the Figure 7.
Figure 7: Extension Mutation
The Extension Mutation operation increases the probability of positive genome
improvement.
### 4.8 Multiple Population
The usage of fitness driven crossover and mutation operation provokes the
problem of lacking gene variation. This problem is solved using the multiple
population technique. On each Nth generation the best individual of each
population is "thrown" to the sequent population [13].
Figure 8: Multiple Population
### 4.9 Evolution Algorithm
SGP evolution algorithm:
Result: Best individual
1 max`_`generation = 100;
2 population`_`size = 100;
3 population_num = 4;
4 cx`_`prob = 0.5;
5 mut`_`prob = 0.5;
6
7populations = [] ;
8 for _i in population_num_ do
9 populations[i] = random_population(population_size);
10
11 end for
12
13best`_`inds = [] ;
14 for _i in population_num_ do
15 best_inds[i] = select_best(populations[i]);
16
17 end for
18
19best`_`ind`_`ever = max(best`_`inds, key = ’fitness’)
20generation = 0;
21 while _best_ind_ever.fitness < 1 or generation < max_generation_ do
22 for _i in population_num_ do
23 selected_population = selection(populations[i]) ;
24 crossed_population = crossover(selected_population, cx_prob) ;
25 mutated_population = mutation(crossed_population, mut_prob) ;
26 weighted_population = weight_adjustment(mutated_population) ;
27 extended_population = weight_adjustment(weighted_population) ;
28 populations[i] = extended_population ;
29 best_inds[i] = select_best(populations[i]);
30
31 end for
32 if _generation mod 5 == 0_ then
33 for _i in population_num_ do
34 populations[i+1].append(best_inds[i]);
35
36 end for
37
38 end if
39 best_ind_ever = max(best_inds, key = ’fitness’);
40 generation++;
41
42 end while
return best`_`ind`_`ever
Algorithm 5 SGP Classifier Evolution Algorithm
## 5 Visualization
Let’s compare the behavior of GP and SGP Classifier on generated 2D datasets
[7].
Figure 9: Linearly separable dataset
Figure 10: Large circle containing a smaller circle dataset
Figure 11: Two interleaving half circles dataset
We can see that both GP and SGP classifiers show confident behavior. As it
could be expected, the SGP Classifier tends to use nonlinear dependencies. SGP
classifier highly likely have strict borders(i.e. $\mu(\\{SGP(x,y)\in]0,1[\
\\})\sim 0$), as it is common behavior for Decision Trees.
## 6 Experimental Results
We have tested SGP and GP classifiers using Large Benchmark Suite [12] for
binary classification problem. As a classification quality score we used
balanced accuracy.
Test results were gathered by following testing algorithm:
1 for _dataset in datasets_ do
2 for _i in [1,20]_ do
3 shuffledDataset = shuffle(dataset) ;
4 train, test = split(shuffledDataset, .7) ;
5 for _cls in classifiers_ do
6 cls.fit(train) score = balancedAccuracy(cls, test)
7 end for
8
9 end for
10
11 end for
Algorithm 6 Testing algorithm
Figure 12: Test Results. GP Classifier - pink, SGP Classifer - blue
Below we provide a heatmap table view with mean balanced accuracy results:
| prnn crabs | heart h | crx | haberman | breast | flare
---|---|---|---|---|---|---
SGP Classifier | 0.978 | 0.7752 | 0.7752 | 0.6792 | 0.9559 | 0.7023
GP Classifier | 0.9724 | 0.7597 | 0.7597 | 0.6522 | 0.9464 | 0.6856
ADA | 0.9095 | 0.7594 | 0.7594 | 0.5752 | 0.9358 | 0.5613
Decision Tree | 0.8815 | 0.7312 | 0.7312 | 0.5585 | 0.9328 | 0.5654
Gausian Process | 0.994 | 0.7925 | 0.7925 | 0.5981 | 0.953 | 0.534
Gausian NB | 0.632 | 0.748 | 0.745 | 0.5748 | 0.9605 | 0.6305
KNeighbors | 0.9079 | 0.7654 | 0.7654 | 0.5857 | 0.96 | 0.5727
Neural Network | 0.9734 | 0.6429 | 0.6429 | 0.5 | 0.9517 | 0.5229
Random Forest | 0.8249 | 0.769 | 0.769 | 0.5702 | 0.958 | 0.5196
| pima | german | heart c | credit g | buddyCrx | prnn synth
---|---|---|---|---|---|---
SGP Classifier | 0.7181 | 0.6791 | 0.7929 | 0.674 | 0.8559 | 0.8642
GP Classifier | 0.7176 | 0.6778 | 0.7938 | 0.6668 | 0.8537 | 0.8543
ADA | 0.703 | 0.6667 | 0.782 | 0.6536 | 0.8427 | 0.8342
Decision Tree | 0.7009 | 0.6394 | 0.7661 | 0.6151 | 0.8309 | 0.8368
Gausian Process | 0.7288 | 0.6255 | 0.8304 | 0.6101 | 0.8614 | 0.8886
Gausian NB | 0.7218 | 0.6633 | 0.8127 | 0.6637 | 0.7866 | 0.844
KNeighbors | 0.6886 | 0.588 | 0.8191 | 0.5941 | 0.8517 | 0.8551
Neural Network | 0.7168 | 0.6255 | 0.8206 | 0.6041 | 0.8598 | 0.8629
Random Forest | 0.6714 | 0.5297 | 0.8146 | 0.5197 | 0.828 | 0.846
SGP provides very positive average results, and especially nice at haberman
and flare datasets. All results can be regathered running script
https://github.com/survexman/sgp_classifier/blob/main/soft/gp_classification.py
## 7 Conclusion
This survey introduces the concept of Soft Genetic Programming. The robustness
of the SGP Classifier is presented. This research aimed to deliver a new
genetic programming design with pseudo boolean and pseudo comparison operators
and show its quality.
Never the less SGP has several drawbacks:
* •
Performance issue. SGP training stage takes much more time than classical
classifers do.
* •
An incapability to use SGP as a feature selection tool. Due to the weighted
operator design, each branch and operator can have a very low weight. Thus the
term that belongs to the weighted operator has low significance and cannot be
used as a significant feature. In classical GP Classifier, each symbolic
variable highly likely has a high significance degree and can thus be selected
as meaningful feature.
Anyway, the improvement of the SGP technique for a particular task can provide
excellent practical results.
## References
* [1] Chan-Sheng Kuo and T. Hong and Chuen-Lung Chen Applying genetic programming technique in classification trees, 2007 Soft Computing, vol. 11, pages 1165-1172
* [2] Koza, John R. Genetic Programming: On the Programming of Computers by Means of Natural Selection 1992 0-262-11170-5 MIT Press Cambridge, MA, USA
* [3] Bhowan Urvesh, Zhang Mengjie and Johnston Mark Genetic Programming for Classification with Unbalanced Data 2010 Springer Berlin Heidelberg pages 1-13 ISBN: 978-3-642-12148-7
* [4] Emmanuel. Dufourq Data classification using genetic programming, 2015
* [5] D. E. Goldberg Genetic Algorithms in Search, Optimization and Machine Learning Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc., 1st ed., 1989.
* [6] Suarez Ranyart, Valencia-Ramírez, José and Graff Mario. Genetic programming as a feature selection algorithm. 2014 IEEE International Autumn Meeting on Power, Electronics and Computing, ROPEC 2014. 1-5. 10.1109/ROPEC.2014.7036345. (2014)
* [7] scikit-learn.org Classifier comparison https://scikit-learn.org/stable/auto_examples/classification/plot_classifier_comparison.html
* [8] Silva, Sara and Almeida, Jonas, Dynamic Maximum Tree Depth, Genetic and Evolutionary Computation — GECCO 2003, 1776–1787 Springer Berlin Heidelberg ISSN 978-3-540-45110-5
* [9] Hamed Hatami. Decision trees and influences of variables over product probability spaces, 2006; arXiv:math/0612405.
* [10] O’Donnell Ryan Analysis of Boolean Functions, 2014 Cambridge University Press DOI: 10.1017/CBO9781139814782
* [11] Moulinath Banerjee and Ian W. McKeague. Confidence sets for split points in decision trees, 2007, Annals of Statistics 2007, Vol. 35, No. 2, 543-574; arXiv:0708.1820. DOI: 10.1214/009053606000001415.
* [12] Randal S. Olson, William La Cava, Patryk Orzechowski, Ryan J. Urbanowicz and Jason H. Moore. PMLB: A Large Benchmark Suite for Machine Learning Evaluation and Comparison, 2017; arXiv:1703.00512.
* [13] Søren B. Vilsen, Torben Tvedebrink and Poul Svante Eriksen. DNA mixture deconvolution using an evolutionary algorithm with multiple populations, hill-climbing, and guided mutation, 2020; arXiv:2012.00513.
|
11institutetext: School of Computer Science and Engineering,
The Hebrew University, Jerusalem, Israel.
11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>22institutetext: Technische Universität München, Munich, Germany.
22email<EMAIL_ADDRESS>
# Certifying Inexpressibility††thanks: This is the full version of an article
with the same title that appears in the FoSSaCS 2021 conference proceedings.
Orna Kupferman is supported in part by the Israel Science Foundation, grant
No. 2357/19. Salomon Sickert is supported in part by the Deutsche
Forschungsgemeinschaft (DFG) under project numbers 436811179 and 317422601
(“Verified Model Checkers”), and in part funded by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement No. 787367 (PaVeS).
Orna Kupferman 11 Salomon Sickert (🖂) 1122
###### Abstract
Different classes of automata on infinite words have different expressive
power. Deciding whether a given language $L\subseteq\Sigma^{\omega}$ can be
expressed by an automaton of a desired class can be reduced to deciding a game
between Prover and Refuter: in each turn of the game, Refuter provides a
letter in $\Sigma$, and Prover responds with an annotation of the current
state of the run (for example, in the case of Büchi automata, whether the
state is accepting or rejecting, and in the case of parity automata, what the
color of the state is). Prover wins if the sequence of annotations she
generates is correct: it is an accepting run iff the word generated by Refuter
is in $L$. We show how a winning strategy for Refuter can serve as a simple
and easy-to-understand certificate to inexpressibility, and how it induces
additional forms of certificates. Our framework handles all classes of
deterministic automata, including ones with structural restrictions like weak
automata. In addition, it can be used for refuting separation of two languages
by an automaton of the desired class, and for finding automata that
approximate $L$ and belong to the desired class.
###### Keywords:
Automata on infinite words Expressive power Games.
## 1 Introduction
Finite automata on infinite objects were first introduced in the 60’s, and
were the key to the solution of several fundamental decision problems in
mathematics and logic [8, 32, 41]. Today, automata on infinite objects are
used for specification, verification, and synthesis of nonterminating systems.
The automata-theoretic approach reduces questions about systems and their
specifications to questions about automata [27, 49], and is at the heart of
many algorithms and tools. Industrial-strength property-specification
languages such as the IEEE 1850 Standard for Property Specification Language
(PSL) [14] include regular expressions and/or automata, making specification
and verification tools that are based on automata even more essential and
popular.
A run $r$ of an automaton on infinite words is an infinite sequence of states,
and acceptance is determined with respect to the set of states that $r$ visits
infinitely often. For example, in Büchi automata, some of the states are
designated as accepting states, denoted by $\alpha$, and a run is accepting
iff it visits states from the accepting set $\alpha$ infinitely often [8].
Dually, in co-Büchi automata, a run is accepting if it visits the set $\alpha$
only finitely often. Then, in parity automata, the acceptance condition maps
each state to a color in some set $C=\\{j,\ldots,k\\}$, for $j\in\\{0,1\\}$
and some index $k\geq 0$, and a run is accepting if the maximal color it
visits infinitely often is odd.
The different classes of automata have different expressive power. For
example, while deterministic parity automata can recognize all
$\omega$-regular languages, deterministic Büchi automata cannot [28]. We use
DBW, DCW, and DPW to denote a deterministic Büchi, co-Büchi, and parity word
automaton, respectively, or (this would be clear from the context) the set of
languages recognizable by the automata in the corresponding class. There has
been extensive research on expressiveness of automata on infinite words [48,
20]. In particular, researchers have studied two natural expressiveness
hierarchies induced by different classes of deterministic automata. The first
hierarchy is the Mostowski Hierarchy, induced by the index of parity automata
[34, 50]. Formally, let DPW[$0,k$] denote a DPW with $C=\\{0,\ldots,k\\}$, and
similarly for DPW[$1,k$] and $C=\\{1,\ldots,k\\}$. Clearly, DPW[$0,k$]
$\subseteq$ DPW[$0,k+1$], and similarly DPW[$1,k$] $\subseteq$ DPW[$1,k+1$].
The hierarchy is infinite and strict. Moreover, DPW[$0,k$] complements
DPW[$1,k+1$], and for every $k\geq 0$, there are languages $L_{k}$ and
$L^{\prime}_{k}$ such that $L_{k}\in$ DPW[$0,k$] $\setminus$ DPW[$1,k+1$] and
$L^{\prime}_{k}\in$ DPW[$1,k+1$] $\setminus$ DPW[$0,k$]. At the bottom of this
hierarchy, we have DBW and DCW. Indeed, DBW=DPW[$0,1$] and DCW=DPW[$1,2$].
While the Mostowski Hierarchy refines DPWs, the second hierarchy, which we
term the depth hierarchy, refines deterministic weak automata (DWWs). Weak
automata can be viewed as a special case of Büchi or co-Büchi automata in
which every strongly connected component in the graph induced by the structure
of the automaton is either contained in $\alpha$ or is disjoint from $\alpha$,
where $\alpha$ is depending on the acceptance condition the set of accepting
or rejecting states. The structure of weak automata captures the alternation
between greatest and least fixed points in many temporal logics, and they were
introduced in this context in [35]. DWWs have been used to represent vectors
of real numbers [6], and they have many appealing theoretical and practical
properties [31, 21]. In terms of expressive power, DWW = DCW $\cap$ DBW.
The depth hierarchy is induced by the depth of alternation between accepting
and rejecting components in DWWs. For this, we view a DWW as a DPW in which
the colors visited along a run can only increase. Accordingly, each run
eventually gets trapped in a single color, and is accepting iff this color is
odd. We use DWW[$0,k$] and DWW[$1,k$] to denote weak-DPW[$0,k$] and weak-
DPW[$1,k$], respectively. The picture obtained for the depth hierarchy is
identical to that of the Mostowski hierarchy, with DWW[$j,k$] replacing
DPW[$j,k$] [50]. At the bottom of the depth hierarchy we have co-safety and
safety languages [2]. Indeed, co-safety languages are DWW[$0,1$] and safety
are DWW[$1,2$].
Beyond the theoretical interest in expressiveness hierarchies, their study is
motivated by the fact many algorithms, like synthesis and probabilistic model
checking, need to operate on deterministic automata [5, 3]. The lower the
automata are in the expressiveness hierarchy, the simpler are algorithms for
reasoning about them. Simplicity goes beyond complexity, which typically
depends on the parity index [16], and involves important practical
considerations like minimization and canonicity (exists only for DWWs [31]),
circumvention of Safra’s determinization [26], and symbolic implementations
[47]. Of special interest is the characterization of DBWs. For example, it is
shown in [25] that given a linear temporal logic formula $\psi$, there is an
alternation-free $\mu$-calculus formula equivalent to $\forall\psi$ iff $\psi$
can be recognized by a DBW. Further research studies typeness for
deterministic automata, examining the ability to define a weaker acceptance
condition on top of a given automaton [19, 21].
Our goal in this paper is to provide a simple and easy-to-understand
explanation to inexpressibility results. The need to accompany results of
decision procedures by an explanation (often termed “certificate”) is not new,
and includes certification of a “correct” decision of a model checker [24,
44], reachability certificates in complex multi-agent systems [1], and
explainable reactive synthesis [4]. To the best of our knowledge, our work is
the first to provide certification to inexpressibility results.
The underlying idea is simple: Consider a language $L$ and a class $\gamma$ of
deterministic automata. We consider a turn-based two-player game in which one
player (Refuter) provides letters in $\Sigma$, and the second player (Prover)
responds with letters from a set $A$ of annotations that describe states in a
deterministic automaton. For example, when we consider a DBW, then
$A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$, and when we consider a DPW[$0,k$],
then $A=\\{0,\ldots,k\\}$. Thus, during the interaction, Refuter generates a
word $x\in\Sigma^{\omega}$ and Prover responds with a word $y\in A^{\omega}$.
Prover wins if for all words $x\in\Sigma^{\omega}$, we have that $x\in L$ iff
$y$ is accepting according to $\gamma$. Clearly, if there is a deterministic
$\gamma$ automaton for $L$, then Prover can win by following its run on $x$.
Dually, a finite-state winning strategy for Prover induces a deterministic
$\gamma$ automaton for $L$. The game-based approach is not new, and has been
used for deciding the membership of given $\omega$-regular languages in
different classes of deterministic automata [26]. Further, the game-based
formulation is used in descriptive set theory to classify sets into
hierarchies, see for example [39, Chapters 4 and 5] for an introduction that
focuses on $\omega$-regular languages. Our contribution is a study of
strategies for Refuter. Indeed, since the above described game is determined
[9] and the strategies are finite-state, Refuter has a winning strategy iff no
deterministic $\gamma$ automaton for $L$ exists, and this winning strategy can
serve as a certificate for inexpressibility.
Figure 1: A refuter for DBW-recognizability of “only finitely many $a$’s”.
###### Example 1
Consider the language $L_{\neg\infty a}\subseteq\\{a,b\\}^{\omega}$ of all
words with only finitely many $a$’s. It is well known that $L$ cannot be
recognized by a DBW [28]. In Figure 1 we describe what we believe to be the
neatest proof of this fact. The figure describes a transducer ${\cal R}$ with
inputs in $\\{\mbox{\sc acc,rej}\\}$ and outputs in $\\{a,b\\}$ – the winning
strategy of Refuter in the above described game. The way to interpret ${\cal
R}$ is as follows. In each round of the game, Prover tells Refuter whether the
run of her DBW for $L_{\neg\infty a}$ is in an accepting or a rejecting state,
and Refuter uses ${\cal R}$ in order to respond with the next letter in the
input word. For example, if Prover starts with acc, namely declaring that the
initial state of her DBW is accepting, then Refuter responds with $a$, and if
Prover continues with rej, namely declaring that the state reachable with $a$
is rejecting, then Refuter responds with $b$. If Prover continues with rej
forever, then Prover continues with $b$ forever. Thus, together Prover and
Refuter generate two words: $y\in\\{\mbox{\sc acc,rej}\\}^{\omega}$ and
$x\in\\{a,b\\}^{\omega}$. Prover wins whenever $x\in L_{\neg\infty a}$ iff $y$
contains infinitely many acc’s. If Prover indeed has a DBW for $L_{\neg\infty
a}$, then she can follow its transition function and win the game. By
following the refuter ${\cal R}$, however, Refuter can always fool Prover and
generate a word $x$ such that $x\in L_{\neg\infty a}$ iff $y$ contains only
finitely many acc’s. $\blacksquare$
We first define refuters for DBW-recognizability, and study their construction
and size for languages given by deterministic or nondeterministic automata.
Our refuters serve as a first inexpressibility certificate. We continue and
argue that each DBW-refuter for a language $L$ induces three words
$x\in\Sigma^{*}$ and $x_{1},x_{2}\in\Sigma^{*}$, such that
$x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\subseteq L$ and
$x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}\cap L=\emptyset$. The triple $\langle
x,x_{1},x_{2}\rangle$ is an additional certificate for $L$ not being in DBW.
Indeed, we show that a language $L$ is not in DBW iff it has a certificate as
above. For example, the language $L_{\neg\infty a}$ has a certificate
$\langle\epsilon,b,a\rangle$. In fact, we show that Landweber’s proof for
$L_{\neg\infty a}$ can be used as is for all languages not in DBW, with
$x_{1}$ replacing $b$, $x_{2}$ replacing $a$, and adding $x$ as a prefix.
We then generalize our results on DBW-refutation and certification in two
orthogonal directions. The first is an extension to richer classes of
deterministic automata, in particular all classes in the two hierarchies
discussed above, as well as all deterministic Emerson-Lei automata (DELWs)
[17]. For the depth hierarchy, we add to the winning condition of the game a
structural restriction. For example, in a weak automaton, Prover loses if the
sequence $y\in A^{\omega}$ of annotations she generates includes infinitely
many alternations between acc and rej. We show how structural restrictions can
be easily expressed in our framework.
The second direction is an extension of the recognizability question to the
questions of separation and approximation: We say that a language
$L\subseteq\Sigma^{\omega}$ is a separator for two languages
$L_{1},L_{2}\subseteq\Sigma^{\omega}$ if $L_{1}\subseteq L$ and $L\cap
L_{2}=\emptyset$. Studies of separation include a search for regular
separators of general languages [11], as well as separation of regular
languages by weaker classes of languages, e.g., FO-definable languages [40] or
piecewise testable languages [12]. In the context of $\omega$-regular
languages, [2] presents an algorithm computing the smallest safety language
containing a given language $L_{1}$, thus finding a safety separator for
$L_{1}$ and $L_{2}$. As far as we know, besides this result there has been no
systematic study of separation of $\omega$-regular languages by deterministic
automata.
In addition to the interest in separators, we use them in the context of
recognizability in two ways. First, a third type of certificate that we
suggest for DBW-refutation of a language $L$ are “simple” languages $L_{1}$
and $L_{2}$ such that $L_{1}\subseteq L$, $L\cap L_{2}=\emptyset$, and
$\langle L_{1},L_{2}\rangle$ are not DBW-separable. Second, we use
separability in order to approximate languages that are not in DBW. Consider
such a language $L\subseteq\Sigma^{\omega}$. A user may be willing to
approximate $L$ in order to obtain DBW-recognizability. Specifically, we
assume that there are languages $I_{\downarrow}\subseteq L$ and
$I_{\uparrow}\subseteq\Sigma^{\omega}\setminus L$ of words that the user is
willing to under- and over-approximate $L$ with. Thus, the user searches for a
language that is a separator for $L\setminus I_{\downarrow}$ and
$\Sigma^{\omega}\setminus(L\cup I_{\uparrow})$. We study DBW-separability and
DBW-approximation, namely separability and approximation by languages in DBW.
In particular, we are interested in finding “small” approximating languages
$I_{\downarrow}$ and $I_{\uparrow}$ with which $L$ has a DBW-approximation,
and we show how certificates that refute DBW-separation can direct the search
to for successful $I_{\downarrow}$ and $I_{\uparrow}$. Essentially, as in
counterexample guided abstraction-refinement (CEGAR) for model checking [10],
we use certificates for non-DBW-separability in order to suggest interesting
radius languages. While in CEGAR the refined system excludes the
counterexample, in our setting the approximation of $L$ excludes the
certificate. As has been the case with recognizability, we extend our results
to all classes of deterministic automata.
## 2 Preliminaries
### 2.1 Transducers and Realizability
Consider two finite alphabets $\Sigma$ and $A$. It is convenient to think
about $\Sigma$ as the “main” alphabet, and about $A$ as an alphabet of
annotations. For two words $x=x_{0}\cdot x_{1}\cdot
x_{2}\cdots\in\Sigma^{\omega}$ and $y=y_{0}\cdot y_{1}\cdot y_{2}\cdots\in
A^{\omega}$, we define $x\oplus y$ as the word in $(\Sigma\times A)^{\omega}$
obtained by merging $x$ and $y$. Thus, $x\oplus
y=(x_{0},y_{0})\cdot(x_{1},y_{1})\cdot(x_{2},y_{2})\cdots$.
A $(\Sigma/A)$-transducer models a finite-state system that responds with
letters in $A$ while interacting with an environment that generates letters in
$\Sigma$. Formally, a $(\Sigma/A)$-transducer is ${\cal
T}=\langle\Sigma,A,\iota,S,s_{0},\rho,\tau\rangle$, where $\iota\in\\{{\it
sys,env}\\}$ indicates who initiates the interaction – the system or the
environment, $S$ is a set of states, $s_{0}\in S$ is an initial state,
$\rho:S\times\Sigma\rightarrow S$ is a transition function, and
$\tau:S\rightarrow A$ is a labelling function on the states. Consider an input
word $x=x_{0}\cdot x_{1}\cdot x_{2}\cdots\in\Sigma^{\omega}$. The run of
${\cal T}$ on $x$ is the sequence $s_{0},s_{1},s_{2}\ldots$ such that for all
$j\geq 0$, we have that $s_{j+1}=\rho(s_{j},x_{j})$. The annotation of $x$ by
${\cal T}$, denoted ${\cal T}(x)$, depends on $\iota$. If $\iota={\it sys}$,
then ${\cal T}(x)=\tau(s_{0})\cdot\tau(s_{1})\cdot\tau(s_{2})\cdots\in
A^{\omega}$. Note that the first letter in $A$ is the output of ${\cal T}$ in
$s_{0}$. This reflects the fact that the system initiates the interaction. If
$\iota={\it env}$, then ${\cal
T}(x)=\tau(s_{1})\cdot\tau(s_{2})\cdot\tau(s_{3})\cdots\in A^{\omega}$. Note
that now, the output in $s_{0}$ is ignored, reflecting the fact that the
environment initiates the interaction.
Consider a language $L\subseteq(\Sigma\times A)^{\omega}$. Let ${\it comp}(L)$
denote the complement of $L$. Thus, ${\it comp}(L)=(\Sigma\times
A)^{\omega}\setminus L$. We say that a language $L\subseteq(\Sigma\times
A)^{\omega}$ is $(\Sigma/A)$-realizable by the system if there is a
$(\Sigma/A)$-transducer ${\cal T}$ with $\iota={\it sys}$ such that for every
word $x\in\Sigma^{\omega}$, we have that $x\oplus{\cal T}(x)\in L$. Then, $L$
is $(A/\Sigma)$-realizable by the environment if there is an
$(A/\Sigma)$-transducer ${\cal T}$ with $i={\it env}$ such that for every word
$y\in A^{\omega}$, we have that ${\cal T}(y)\oplus y\in L$. When the language
$L$ is regular, realizability reduces to deciding a game with a regular
winning condition. Then, by determinacy of games and due to the existence of
finite-memory winning strategies [9], we have the following.
###### Proposition 1
For every $\omega$-regular language $L\subseteq(\Sigma\times A)^{\omega}$,
exactly one of the following holds.
1. 1.
$L$ is $(\Sigma/A)$-realizable by the system.
2. 2.
${\it comp}(L)$ is $(A/\Sigma)$-realizable by the environment.
### 2.2 Automata
A deterministic word automaton over a finite alphabet $\Sigma$ is ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$, where $Q$ is a set of states,
$q_{0}\in Q$ is an initial state, $\delta:Q\times\Sigma\rightarrow Q$ is a
transition function, and $\alpha$ is an acceptance condition. We extend
$\delta$ to words in $\Sigma^{*}$ in the expected way, thus for $q\in Q$,
$w\in\Sigma^{*}$, and letter $\sigma\in\Sigma$, we have that
$\delta(q,\epsilon)=q$ and $\delta(q,w\sigma)=\delta(\delta(q,w),\sigma)$. A
run of ${\cal A}$ on an infinite word
$\sigma_{0},\sigma_{1},\dots\in\Sigma^{\omega}$ is the sequence of states
$r=q_{0},q_{1},\dots$, where for every position $i\geq 0$, we have that
$q_{i+1}=\delta(q_{i},\sigma_{i})$. We use ${\it inf}(r)$ to denote the set of
states that $r$ visits infinitely often. Thus, ${\it inf}(r)=\\{q:\text{
}q_{i}=q\text{ for infinitely many }i\geq 0\\}$.
The acceptance condition $\alpha$ refers to ${\it inf}(r)$ and determines
whether the run $r$ is accepting. For example, in the Büchi, acceptance
condition, we have that $\alpha\subseteq Q$, and a run is accepting iff it
visits states in $\alpha$ infinitely often; that is, $\alpha\cap{\it
inf}(r)\neq\emptyset$. Dually, in co-Büchi, $\alpha\subseteq Q$, and a run is
accepting iff it visits states in $\alpha$ only finitely often; that is,
$\alpha\cap{\it inf}(r)=\emptyset$. The language of ${\cal A}$, denoted
$L({\cal A})$, is then the set of words $w$ such that the run of ${\cal A}$ on
$w$ is accepting.
A parity condition is $\alpha:Q\rightarrow\\{0,\ldots,k\\}$, for $k\geq 0$,
termed the index of $\alpha$. A run $r$ satisfies $\alpha$ iff the maximal
color $i\in\\{0,\ldots,k\\}$ such that $\alpha^{-1}(i)\cap{\it
inf}(r)\neq\emptyset$ is odd. That is, $r$ is accepting iff the maximal color
that $r$ visits infinitely often is odd. Then, a Rabin condition is
$\alpha=\\{\langle G_{1},B_{1}\rangle,\ldots,\langle G_{k},B_{k}\rangle\\}$,
with $G_{i},B_{i}\subseteq Q$, for all $0\leq i\leq k$. A run $r$ satisfies
$\alpha$ iff there is $1\leq i\leq k$ such that ${\it inf}(r)\cap
G_{i}\neq\emptyset$ and ${\it inf}(r)\cap B_{i}=\emptyset$. Thus, there is a
pair $\langle G_{i},B_{i}\rangle$ such that $r$ visits states in $G_{i}$
infinitely often and visits states in $B_{i}$ only finitely often.
All the acceptance conditions above can be viewed as special cases of the
Emerson-Lei acceptance condition (EL-condition, for short) [17], which we
define below. Let $\mathbb{M}$ be a finite set of marks. Given an infinite
sequence $\pi=M_{0}\cdot M_{1}\cdots\in(2^{\mathbb{M}})^{\omega}$ of subsets
of marks, let ${\it inf}(\pi)$ be the set of marks that appear infinitely
often in sets in $\pi$. Thus, ${\it inf}(\pi)=\\{m\in\mathbb{M}$ : there exist
infinitely many $i\geq 0$ such that $m\in M_{i}\\}$. An EL-condition is a
Boolean assertion over atoms in $\mathbb{M}$. For simplicity, we consider
assertions in positive normal form, where negation is applied only to atoms.
Intuitively, marks that appear positively should repeat infinitely often and
marks that appear negatively should repeat only finitely often. Formally, a
deterministic EL-automaton is ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\mathbb{M},\tau,\theta\rangle$, where
$\tau\colon Q\to 2^{\mathbb{M}}$ maps each state to a set of marks, and
$\theta$ is an EL-condition over $\mathbb{M}$. A run $r$ of a ${\cal A}$ is
accepting if ${\it inf}(\tau(r))$ satisfies $\theta$.
For example, a Büchi condition $\alpha\subseteq Q$ can be viewed as an EL-
condition with $\mathbb{M}=\\{\mbox{\sc acc}\\}$ and $\tau(q)=\\{\mbox{\sc
acc}\\}$ for $q\in\alpha$ and $\tau(q)=\emptyset$ for $q\not\in\alpha$. Then,
the assertion $\theta=\mbox{\sc acc}$ is satisfied by sequences $\pi$ induced
by runs $r$ with ${\it inf}(r)\cap\alpha\neq\emptyset$. Dually, the assertion
$\theta=\neg\mbox{\sc rej}$ with $\mathbb{M}=\\{\mbox{\sc rej}\\}$ is
satisfied by sequences $\pi$ induced by runs $r$ with ${\it
inf}(r)\cap\alpha=\emptyset$, and thus corresponds to a co-Büchi condition. In
the case of a parity condition $\alpha:Q\rightarrow\\{0,\ldots,k\\}$, it is
not hard to see that $\alpha$ is equivalent to an EL-condition in which
$\mathbb{M}=\\{0,1,\ldots,k\\}$, for every state $q\in Q$, we have that
$\tau(q)=\\{\alpha(q)\\}$, and $\theta=\theta_{k}$ expresses the parity
condition, where $\theta_{k}$ is inductively defined as:
$\theta_{k}=\begin{cases}\neg 0&\text{if }k=0,\\\ \neg
k\wedge\theta_{k-1}&\text{if }k\text{ is even,}\\\
\phantom{\neg}k\vee\theta_{k-1}&\text{If $k>0$ and $k$ is odd.}\end{cases}$
Lastly, a Rabin condition $\alpha=\\{\langle G_{1},B_{1}\rangle,\ldots,\langle
G_{k},B_{k}\rangle\\}$ is equivalent to an EL-condition with
$\mathbb{M}=\\{G_{1},B_{1},\dots,G_{k},B_{k}\\}$ and
$\tau(q)=\\{m\in\mathbb{M}:q\in m\\}$. Note that now, the mapping $\tau$ is
not to singletons, and each state is marked by all sets in $\alpha$ in which
it is a member. Then, $\theta=\bigvee_{1\leq i\leq k}(G_{i}\wedge\neg B_{i})$.
We use DBW, DCW, DPW, DRW, DELW to denote deterministic Büchi, co-Büchi,
parity, Rabin, and EL word automata, respectively. For parity automata, we
also use DPW[$0,k$] and DPW[$1,k$], for $k\geq 0$, to denote DPWs in which the
colours are in $\\{0,\ldots,k\\}$ and $\\{1,\ldots,k\\}$, respectively. For
Rabin automata, we use DRW[$k$], for $k\geq 0$, to denote DRWs that have at
most $k$ elements in $\alpha$. Finally, we use DELW[$\theta$], to denote DELWs
with EL-condition $\theta$. We sometimes use the above acronyms in order to
refer to the set of languages that are recognizable by the corresponding class
of automata. For example, we say that a language $L$ is in DBW if $L$ is _DBW-
recognizable_ , thus there is a DBW ${\cal A}$ such that $L=L({\cal A})$. Note
that DBW = DPW[$0,1$], DCW = DPW[$1,2$], and DRW[$1$] = DPW[$0,2$]. In fact,
in terms of expressiveness, DRW[$k$] = DPW[$0,2k$] [43, 30].
Consider a directed graph $G=\langle V,E\rangle$. A strongly connected set of
$G$ (SCS) is a set $C\subseteq V$ of vertices such that for every two vertices
$v,v^{\prime}\in C$, there is a path from $v$ to $v^{\prime}$. An SCS $C$ is
maximal if it cannot be extended to a larger SCS. Formally, for every nonempty
$C^{\prime}\subseteq V\setminus C$, we have that $C\cup C^{\prime}$ is not an
SCS. The maximal strongly connected sets are also termed strongly connected
components (SCC). An automaton ${\cal
A}=\langle\Sigma,Q,Q_{0},\delta,\alpha\rangle$ induces a directed graph
$G_{\cal A}=\langle Q,E\rangle$ in which $\langle q,q^{\prime}\rangle\in E$
iff there is a letter $\sigma$ such that $q^{\prime}\in\delta(q,\sigma)$. When
we talk about the SCSs and SCCs of ${\cal A}$, we refer to those of $G_{\cal
A}$. Consider a run $r$ of an automaton ${\cal A}$. It is not hard to see that
the set ${\it inf}(r)$ is an SCS. Indeed, since every two states $q$ and
$q^{\prime}$ in ${\it inf}(r)$ are visited infinitely often, the state
$q^{\prime}$ must be reachable from $q$.
A DBW ${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$ is _weak_ (DWW) if
every SCC $C$ of ${\cal A}$ is accepting, namely $C\subseteq\alpha$, or
rejecting, namely $C\cap\alpha=\emptyset$. Thus, each run of ${\cal A}$
eventually visits either states in $\alpha$ or only states not in $\alpha$. It
is easy to see that every DWW can be viewed as a DBW and as a DCW. In order to
refer to the depth of the SCCs in ${\cal A}$, we also refer to ${\cal A}$ also
as a DPW. Indeed, a DPW ${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$
is _weak_ if for every transition $q^{\prime}=\delta(q,\sigma)$ we have
$\alpha(q^{\prime})\geq\alpha(q)$, i.e., $\alpha$ is monotonically increasing
along a run. We use DWW[$0,k$] and DWW[$1,k$] to denote weak DPW[$0,k$] and
weak DPW[$1,k$], respectively. Finally, note that for each safety
$\omega$-regular language $L$, there exists a DWW[$1,2$] that recognises $L$
and all DWW[$1,2$] recognise a safety language. Dually, co-safety languages
correspond to DWW[$0,1$].
## 3 Refuting DBW-Recognizability
Let $A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$. We use $\infty\mbox{\sc acc}$ to
denote the subset $\\{a_{0}\cdot a_{1}\cdot a_{2}\cdots\in A^{\omega}:\mbox{
there are infinitely many $j\geq 0$ with }a_{j}=\mbox{\sc acc}\\}$ and
$\neg\infty\mbox{\sc acc}={\it comp}(\infty\mbox{\sc acc})=\\{a_{0}\cdot
a_{1}\cdot a_{2}\cdots\in A^{\omega}:\mbox{ there are only finitely many
$j\geq 0$ with }a_{j}=\mbox{\sc acc}\\}$.
A DBW ${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$ can be viewed as a
$(\Sigma/A)$-transducer ${\cal T}_{\cal A}=\langle\Sigma,A,{\it sys}$, $Q$,
$q_{0},\delta,\tau\rangle$, where for every state $q\in Q$, we have that
$\tau(q)=\mbox{\sc acc}$ if $q\in\alpha$, and $\tau(q)=\mbox{\sc rej}$
otherwise. Then, for every word $x\in\Sigma^{\omega}$, we have that $x\in
L({\cal A})$ iff ${\cal T}_{\cal A}(x)\in\infty{\mbox{\sc acc}}$.
For a language $L\subseteq\Sigma^{\omega}$, we define the language
$\operatorname{DBW}(L)\subseteq(\Sigma\times A)^{\omega}$ of words with
correct annotations. Thus,
$\operatorname{DBW}(L)=\\{x\oplus y:x\in L\mbox{ iff }y\in\infty{\mbox{\sc
acc}}\\}.$
Note that ${\it comp}(\operatorname{DBW}(L))$ is the language
$\operatorname{NoDBW}(L)=\\{x\oplus y:(x\in L\mbox{ and
}y\not\in\infty{\mbox{\sc acc}})\mbox{ or }(x\not\in L\mbox{ and
}y\in\infty{\mbox{\sc acc}})\\}.$
A DBW-refuter for $L$ is an $(A/\Sigma)$-transducer with $\iota={\it env}$
realizing $\operatorname{NoDBW}(L)$.
###### Example 2
For every language $R\subseteq\Sigma^{*}$ of finite words, the language
$R^{\omega}\subseteq\Sigma^{\omega}$ consists of infinite concatenations of
words in $R$. It was recently shown that $R^{\omega}$ may not be in DBW [29].
The language used in [29] is $R=\$+(0\cdot\\{0,1,\$\\}^{*}\cdot 1)$. In Figure
2 below we describe a DBW-refuter for $R^{\omega}$.
Figure 2: A DBW-refuter for $(\$+(0\cdot\\{0,1,\$\\}^{*}\cdot 1))^{\omega}$.
Following ${\cal R}$, Refuter starts by generating a prefix $0\cdot 1$ and
then responds to acc with $1$ and responds with $\$$ to rej. Accordingly, if
Prover generates a rejecting run, Prover generates a word in $0\cdot
1\cdot(1+\$)^{*}\cdot\$^{\omega}$, which is in $R^{\omega}$. Also, if Prover
generates an accepting run, Prover generates a word in $0\cdot
1\cdot(1^{+}\cdot\$^{*})^{\omega}$, which has a single $0$ and infinitely many
$1$’s, and is therefore not in $R^{\omega}$. $\blacksquare$
By Proposition 1, we have the following.
###### Proposition 2
Consider a language $L\subseteq\Sigma^{\omega}$. Let $A=\\{\mbox{\sc
acc},\mbox{\sc rej}\\}$. Exactly one of the following holds:
* •
$L$ is in DBW, in which case the language $\operatorname{DBW}(L)$ is
$(\Sigma/A)$-realizable by the system, and a finite-memory winning strategy
for the system induces a DBW for $L$.
* •
$L$ is not in DBW, in which case the language $\operatorname{NoDBW}(L)$ is
$(A/\Sigma)$-realizable by the environment, and a finite-memory winning
strategy for the environment induces a DBW-refuter for $L$.
### 3.1 Complexity
In this section we analyze the size of refuters. We start with the case where
the language $L$ is given by a DPW.
###### Theorem 3.1
Consider a DPW ${\cal A}$ with $n$ states. Let $L=L({\cal A})$. One of the
following holds.
1. 1.
There is a DBW for $L$ with $n$ states.
2. 2.
There is a DBW-refuter for $L$ with $2n$ states.
###### Proof
If $L$ is in DBW, then, as DPWs are Büchi type [19], a DBW for $L$ can be
defined on top of the structure of ${\cal A}$, and so it has $n$ states. If
$L$ is not in DBW, then by Proposition 2, there is a DBW-refuter for $L$,
namely a $(\\{\mbox{\sc acc},\mbox{\sc rej}\\}/\Sigma)$-transducer that
realizes $\operatorname{NoDBW}(L)$. We show we can define a DRW ${\cal U}$
with $2n$ states for $\operatorname{NoDBW}(L)$. The result then follows from
the fact a realizable DRW is realized by a transducer of the same size as the
DRW [15].
We construct ${\cal U}$ by taking the union of the acceptance conditions of a
DRW ${\cal U}_{1}$ for $\\{x\oplus y:x\in L\mbox{ and
}y\not\in\infty{\mbox{\sc acc}}\\}$ and a DRW ${\cal U}_{2}$ for $\\{x\oplus
y:x\not\in L\mbox{ and }y\in\infty{\mbox{\sc acc}}\\}$. We obtain both DRWs by
taking the product of ${\cal A}$, extended to the alphabet
$\Sigma\times\\{\mbox{\sc acc},\mbox{\sc rej}\\}$, with a $2$-state automaton
for $\infty{\mbox{\sc acc}}$, again extended to the alphabet
$\Sigma\times\\{\mbox{\sc acc},\mbox{\sc rej}\\}$.
We describe the construction in detail. Let ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$. Then, the state space of
${\cal U}_{1}$ is $Q\times\\{\mbox{\sc acc},\mbox{\sc rej}\\}$ and its
transition on a letter $\langle\sigma,a\rangle$ follows $\delta$ when it reads
$\sigma$, with $a$ determining whether ${\cal U}_{1}$ moves to the acc or rej
copy. Let $\alpha_{1}$ be the Rabin condition equivalent to $\alpha$. We
obtain the acceptance condition of ${\cal U}_{1}$ by replacing each pair
$\langle G,B\rangle$ in $\alpha_{1}$ by $\langle G\times\\{\mbox{\sc
rej}\\},B\times\\{\mbox{\sc rej}\\}\cup Q\times\\{\mbox{\sc acc}\\}\rangle$.
It is not hard to see that a run of ${\cal U}_{1}$ satisfies the latter pair
iff its projection on $Q$ satisfies the pair $\langle G,B\rangle$ and its
projection on $\\{\mbox{\sc acc},\mbox{\sc rej}\\}$ has only finitely many
acc. The construction of ${\cal U}_{2}$ is similar, with $\alpha_{2}$ being a
Rabin condition that complements $\alpha$, and then replacing each pair
$\langle G,B\rangle$ in $\alpha_{2}$ by $\langle G\times\\{\mbox{\sc
acc}\\},B\times\\{\mbox{\sc acc},\mbox{\sc rej}\\})\rangle$. Since ${\cal
U}_{1}$ and ${\cal U}_{2}$ have the same state space, and we only have to take
the union of the pairs in their acceptance conditions, the $2n$ bound follows.
∎
Now, when $L$ is given by an NBW, an exponential bound follows from the
exponential blow up in determinization [42]. If we are also given an NBW for
${\it comp}(L)$, the complexity can be tightened. Formally, we have the
following.
###### Theorem 3.2
Given NBWs with $n$ and $m$ states, for $L$ and ${\it comp}(L)$, respectively,
one of the following holds.
1. 1.
There is a DBW for $L$ with $\min\\{(1.65n)^{n},3^{m}\\}$ states.
2. 2.
There is a DBW-refuter for $L$ with
$\min\\{2\cdot(1.65n)^{n},2\cdot(1.65m)^{m}\\}$ states.
###### Proof
If $L$ is in DBW, then a DBW for $L$ can be defined on top of a DPW for $L$,
which has at most $(1.65n)^{n}$ states [45], or by dualizing a DCW for ${\it
comp}(L)$. Since the translation of an NBW with $m$ states to a DCW, when it
exists, results in a DCW with $3^{m}$ states [7], we are done. If $L$ is not
in DBW, then we proceed as in the proof of Theorem 3.1, defining ${\cal U}$ on
the top of a DPW for either $L$ or ${\it comp}(L)$. ∎
### 3.2 Certifying DBW-Refutation
Consider a DBW-refuter ${\cal R}=\langle\\{\mbox{\sc acc},\mbox{\sc
rej}\\},\Sigma,{\it env},S,s_{0},\rho,\tau\rangle$. We say that a path
$s_{0},\ldots,s_{m}$ in ${\cal R}$ is an $\mbox{\sc rej}^{+}$-path if it
contains at least one transition and all the transitions along it are labeled
by rej; thus, for all $0\leq j<m$, we have that $s_{j+1}=\rho(s_{j},\mbox{\sc
rej})$. Then, a path $s_{0},\ldots,s_{m}$ in ${\cal R}$ is an acc-path if it
contains at least one transition and its first transition is labeled by acc.
Thus, $s_{1}=\rho(s_{0},\mbox{\sc acc})$.
###### Lemma 1
Consider a DBW-refuter ${\cal R}=\langle\\{\mbox{\sc acc},\mbox{\sc
rej}\\},\Sigma,{\it env},S,s_{0},\rho,\tau\rangle$. Then there exists a state
$s\in S$, a (possibly empty) path $p=s_{0},s_{1},\dots s_{m}$, a $\mbox{\sc
rej}^{+}$-cycle $p_{1}=s^{1}_{0},s^{1}_{1}\dots s^{1}_{m_{1}}$, and an acc-
cycle $p_{2}=s^{2}_{0},s^{2}_{1}\dots s^{2}_{m_{2}}$, such that
$s_{m}=s^{1}_{0}=s^{1}_{m_{1}}=s^{2}_{0}=s^{2}_{m_{2}}=s$.
###### Proof
Let $s_{i}\in S$ be a reachable state that belongs to an ergodic component in
the graph of ${\cal R}$ (that is, $s_{i}\in C$, for a set $C$ of strongly
connected states that can reach only states in $C$). Since ${\cal R}$ is
responsive, in the sense it can read in each round both acc and rej, we can
read from $s_{i}$ the input sequence $\mbox{\sc rej}^{\omega}$. Hence, ${\cal
R}$ has a $\mbox{\sc rej}^{+}$-path $s_{i},\ldots,s_{l},\ldots,s_{k}$ with
$s_{l}=s_{k}$, for $l<k$. It is easy to see that the claim holds with
$s=s_{l}$. In particular, since ${\cal R}$ is responsive and $C$ is strongly
connected, there exists an acc-cycle from $s_{l}$ to itself. ∎
Figure 3: The structure from Lemma 1 that exists in every DBW-refuter.
###### Theorem 3.3
An $\omega$-regular language $L$ is not in DBW iff there exist three finite
words $x\in\Sigma^{*}$ and $x_{1},x_{2}\in\Sigma^{+}$, such that
$x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\subseteq L\quad\text{ and }\quad
x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}\cap L=\emptyset.$
###### Proof
Assume first that $L$ is not in DBW. Then, by Theorem 3.2, there exists a DBW-
refuter ${\cal R}$ for it. Let $p=s_{0},s_{1},\dots s_{m}$,
$p_{1}=s^{1}_{0},s^{1}_{1},\dots,s^{1}_{m_{1}}$, and
$p_{2}=s^{2}_{0},s^{2}_{1},\dots,s^{2}_{m_{2}}$, be the path, $\mbox{\sc
rej}^{+}$-cycle, and acc-cycle that are guaranteed to exist by Lemma 1. Let
$x,x_{1}$, and $x_{2}$ be the outputs that ${\cal R}$ generates along them.
Formally, $x=\tau(s_{1})\cdot\tau(s_{2})\cdots\tau(s_{m})$,
$x_{1}=\tau(s^{1}_{1})\cdot\tau(s^{1}_{2})\cdots\tau(s^{1}_{m_{1}})$, and
$x_{2}=\tau(s^{2}_{1})\cdot\tau(s^{2}_{1})\cdots\tau(s^{2}_{m_{2}})$. Note
that as the environment initiates the interaction, the first letter in the
words $x$, $x_{1}$, and $x_{2}$, are the outputs in the second states in $p$,
$p_{1}$, and $p_{2}$. We prove that $x,x_{1}$, and $x_{2}$ satisfy the two
conditions in the theorem.
Let $y\in\\{\mbox{\sc acc},\mbox{\sc rej}\\}^{*}$, and
$y_{1},y_{2}\in\\{\mbox{\sc acc},\mbox{\sc rej}\\}^{+}$ be the input sequences
read along $p,p_{1}$, and $p_{2}$, respectively. Thus,
$y=a_{0},a_{1},\ldots,a_{m-1}$ is such that for all $0\leq j<m$, we have that
$s_{j+1}=\rho(s_{j},a_{j})$, and similarly for $y_{1}$ and $y_{2}$ with
$p_{1}$ and $p_{2}$.
Consider a word $w\in x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}$. Let $a\in
y\cdot(y_{1}+y_{2})^{*}\cdot y_{1}^{\omega}$ be such that ${\cal R}(a)=w$.
Note we can obtain $a$ from $w$ by replacing each subword $x$ by $y$, $x_{1}$
by $y_{1}$, and $x_{2}$ by $y_{2}$. Since $p_{1}$ is a $\mbox{\sc
rej}^{+}$-cycle, we have that $a\in(\mbox{\sc acc}+\mbox{\sc
rej})^{*}\cdot\mbox{\sc rej}^{\omega}$, and so $a\in\neg\infty\mbox{\sc acc}$.
Since ${\cal R}$ is a refuter for $L$, it follows that ${\cal R}(a)\in L$.
Hence, $x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\subseteq L$.
For this direction it remains to show that $x\cdot(x_{1}^{*}\cdot
x_{2})^{\omega}\cap L=\emptyset$. Consider a word $w\in x\cdot(x_{1}^{*}\cdot
x_{2})^{\omega}$, and let $a\in y\cdot(y_{1}^{*}\cdot y_{2})^{\omega}$ be such
that ${\cal R}(a)=w$. Since $p_{1}$ is an acc-cycle, we have that
$a\in(\mbox{\sc rej}^{*}\mbox{\sc acc})^{\omega}$, and so $a\in\infty\mbox{\sc
acc}$. Since ${\cal R}$ is a refuter for $L$, it follows that ${\cal
R}(a)\notin L$. Hence, $x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}\cap
L=\emptyset$, and we are done.
For the other direction, we adjust Landweber’s proof [28] for the non-DBW-
recognizability of $\neg\infty a$ to $L$. Essentially, $\neg\infty a$ can be
viewed as a special case of $x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}$,
with $x=\epsilon$, $x_{1}=b$, and $x_{2}=a$. Assume by way of contradiction
that there is a DBW ${\cal A}$ with $L({\cal A})=L$. Let ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$. Consider the infinite word
$w_{0}=x\cdot x_{1}^{\omega}$. Since $w_{0}\in x\cdot(x_{1}+x_{2})^{*}\cdot
x_{1}^{\omega}$, and so $w\in L$, the run of ${\cal A}$ on $w_{0}$ is
accepting. Thus, there is $i_{1}\geq 0$ such that ${\cal A}$ visits $\alpha$
when it reads the $x_{1}$ suffix of $x\cdot x_{1}^{i_{1}}$. Consider now the
infinite word $w_{1}=x\cdot x_{1}^{i_{1}}\cdot x_{2}\cdot x_{1}^{\omega}$.
Since $w_{1}$ is also in $L$, the run of ${\cal A}$ on $w_{1}$ is accepting.
Thus, there is $i_{2}\geq 0$ such that ${\cal A}$ visits $\alpha$ when it
reads the $x_{1}$ suffix of $x\cdot x_{1}^{i_{1}}\cdot x_{2}\cdot
x_{1}^{i_{2}}$. In a similar fashion we can continue to find indices
$i_{1},i_{2},\ldots$ such for all $j\geq 1$, we have that ${\cal A}$ visits
$\alpha$ when it reads the $x_{1}$ suffix of $x\cdot x_{1}^{i_{1}}\cdot
x_{2}\cdot x_{1}^{i_{2}}\cdot x_{2}\cdots x_{2}\cdot x_{1}^{i_{j}}$. Since $Q$
is finite, there are iterations $j$ and $k$, such that $1\leq j<k\leq|Q|+1$
and there is a state $q$ such that $q=\delta(q_{0},x\cdot x_{1}^{i_{1}}\cdot
x_{2}\cdot x_{1}^{i_{2}}\cdot x_{2}\cdots x_{2}\cdot
x_{1}^{i_{j}})=\delta(q_{0},x\cdot x_{1}^{i_{1}}\cdot x_{2}\cdot
x_{1}^{i_{2}}\cdot x_{2}\cdots x_{2}\cdot x_{1}^{i_{k}})$. Since $j<k$, the
extension $x_{2}\cdot x_{1}^{i_{j+1}}\cdots x_{1}^{i_{k-1}}\cdot x_{2}\cdot
x_{1}^{i_{k}}$ is not empty and at least one state in $\alpha$ is visited when
${\cal A}$ loops in $q$ while reading it. It follows that the run of ${\cal
A}$ on the word
$w=x\cdot x_{1}^{i_{1}}\cdot x_{2}\cdot x_{1}^{i_{2}}\cdot x_{2}\cdots
x_{2}\cdot x_{1}^{i_{j}}\cdot(x_{2}\cdot x_{1}^{i_{j+1}}\cdots
x_{1}^{i_{k-1}}\cdot x_{2}\cdot x_{1}^{i_{k}})^{\omega}$
is accepting. But $w\in x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}$, so it is not
in $L$, and we have reached a contradiction. ∎
###### Remark 1
Theorem 3.3, as well as the yet to be presented Theorems 6.2 and 6.3 are
special cases of [50, Lemma 14]. However, our alternative proof relies on
Proposition 1 and the analysis of the resulting refuter, while the proof of
[50] examines the structure of a deterministic Muller automaton. Due to the
game-based setting we can easily extend our approach to refuting separability
of languages (Section 4), which requires substantial modifications of the
approach from [50].
We refer to a triple $\langle x,x_{1},x_{2}\rangle$ of words that satisfy the
conditions in Theorem 3.3 as a certificate to the non-DBW-recognizability of
$L$.
###### Example 3
In Example 2, we described a DBW-refuter for
$L=(\$+(0\cdot\\{0,1,\$\\}^{*}\cdot 1))^{\omega}$. A certificate to its non-
DBW-recognizability is $\langle x,x_{1},x_{2}\rangle$, with $x=01$,
$x_{1}=\$$, and $x_{2}=1$. Indeed, $01\cdot(\$+1)^{*}\cdot\$^{\omega}\subseteq
L$ and $01\cdot(\$^{*}\cdot 1)^{\omega}\cap L=\emptyset$. $\blacksquare$
Note that obtaining certificates according to the proof of Theorem 3.3 may not
give us the shortest certificate. For example, for $L$ in Example 3, the proof
would give us $x=01\$$, $x_{1}=\$$, and $x_{2}=1\$,$ with
$01\$\cdot(\$+1\$)^{*}\cdot\$^{\omega}\subseteq L$ and $01\$\cdot(\$^{*}\cdot
1\$)^{\omega}\cap L=\emptyset$. The problem of generating smallest
certificates is related to the problem of finding smallest witnesses to DBW
non-emptiness [22] and is harder. Formally, defining the length of a
certificate $\langle x,x_{1},x_{2}\rangle$ as $|x|+|x_{1}|+|x_{2}|$, we have
the following:
###### Theorem 3.4
Consider a DPW ${\cal A}$ and a threshold $l\geq 1$. The problem of deciding
whether there is a certificate of length at most $l$ for non-DBW-
recognizability of $L({\cal A})$ is NP-complete, for $l$ given in unary or
binary.
###### Proof
We start with membership in NP. Let $n$ be the number of states in ${\cal A}$.
By Theorem 3.1 and the construction in Theorem 3.3 we can bound the length of
a certificate to be at most $6n$, since these are constructed from simple
paths. Given a witness certificate $\langle x,x_{1},x_{2}\rangle$ of length at
most $l$ (the latter can be checked in polynomial time, regardless of how $l$
is given), checking the conditions in Theorem 3.3 involves checking
$x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\subseteq L({\cal A})$, namely
containment of a DCW of size linear in the certificate in the language of a
DPW, which can be done in polynomial time, and checking $x\cdot(x_{1}^{*}\cdot
x_{2})^{\omega}\cap L({\cal A})=\emptyset$, namely emptiness of the
intersection with a DBW, which again can be done in polynomial time.
For the NP-hardness, we describe a reduction from the Hamiltonian-cycle
problem on directed graphs. Formally, given a directed graph $G=\langle
V,E\rangle$, we describe a DPW that is not in DBW and which has a certificate
of length $|V|+1$ iff $G$ has a Hamiltonian cycle, namely a cycle that visits
each vertex in $V$ exactly once. The proof elaborates on the NP-hardness proof
of the problem of finding a shortest witness to DBW non-emptiness [22].
Let $V=\\{1,\ldots,n\\}$, and assume that $n\geq 2$ and $E$ is not empty. We
define a DPW ${\cal A}=\langle E,(V\times V)\cup\\{\langle
1,1\rangle_{\text{err}}\\},\\{\langle 1,1\rangle\\},\delta,\alpha\rangle$,
where $\alpha(\langle n,n\rangle)=1$, $\alpha(\langle
1,1\rangle_{\text{err}})=2$, $\alpha(q)=0$ for all other states $q$, and
$\displaystyle\delta(\langle i,j\rangle,(k,h))$
$\displaystyle=\begin{cases}\langle h,(j\text{ mod }n)+1\rangle&\text{if
}i=k=j,\\\ \langle h,j\rangle&\text{if }i=k\neq j,\\\ \langle
1,1\rangle_{\text{err}}&\text{otherwise.}\end{cases}$
$\displaystyle\delta(\langle 1,1\rangle_{err},(k,h))$
$\displaystyle=\begin{cases}\langle h,2\rangle\hskip 57.50008pt&\text{if
}k=1,\\\ \langle 1,1\rangle_{\text{err}}&\text{otherwise.}\end{cases}$
Intuitively, ${\cal A}$ interprets a word $w\in E^{\omega}$, as an infinite
path starting in vertex $1$, and it verifies that the path is valid on $G$.
Whenever ${\cal A}$ encounters an edge that does not match the current state,
which is tracked in the first component of the state space, it resets and
moves to $\langle 1,1\rangle_{\text{err}}$. The second component of a state
$\langle i,j\rangle$ is the vertex the path owes a visit in order to visit all
vertices infinitely often. It is easy to see that $w\in L({\cal A})$ iff there
is a suffix $w^{\prime}$ of $w$ that describes a valid path in $G$ that visits
every vertex infinitely often. Notice that $L({\cal A})$ is not DBW-
recognizable and that ${\cal A}$ is polynomial in the size of $G$.
Clearly, the reduction is polynomial, we now prove its correctness. Assume
first that $G$ has a Hamiltonian cycle $c$. Then, from the word $w$ read along
$c$ from vertex 1, we construct the certificate
$\langle\epsilon,w,(2,1)\rangle$ showing non-DBW-recognizabilty. Indeed, the
certificate is correct, since $(w+(2,1))^{*}\cdot w^{\omega}\subseteq L({\cal
A})$ and $(w^{*}\cdot(2,1))^{\omega}\cap L({\cal A})=\emptyset$. This
certificate has size $n+1$.
For the other direction, assume that $\langle x,x_{1},x_{2}\rangle$ is a
certificate of size (at most) $n+1$. Then, $x\cdot x_{1}^{\omega}\in L({\cal
A})$ and as $x_{2}$ is not empty, it must be that $|x|+|x_{1}|\leq n$. Let $r$
be the corresponding accepting run and thus $r$ visits $\langle n,n\rangle$
infinitely often. By the definition of $\delta$, the run $r$ also visits the
states $\langle i,i\rangle$, for all $1\leq i\leq n$. Since the transitions to
each of these states are labelled differently, $x_{1}$ must contain at least
$n$ different letters. Hence, $|x_{1}|$ must be $n$ and thus $G$ has a
Hamiltonian cycle.
###### Remark 2
[Relation with existing characterizations] By [28], the language of a DPW
${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$ is in DBW iff for every
accepting SCS $C\subseteq Q$ and SCS $C^{\prime}\supseteq C$, we have that
$C^{\prime}$ is accepting. The proof of Landweber relies on a complicated
analysis of the structural properties of ${\cal A}$. As we elaborate below,
Theorem 3.3, which relies instead on determinacy of games, suggests an
alternative proof. Similarly, [50] examines the structure of a deterministic
Muller automaton, and Theorem 3.3 can be viewed as a special case of Lemma 14
there, with a proof based on the game setting.
We use certificates in order to prove that a DPW ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$ is in DBW iff for every
accepting SCS $C\subseteq Q$ and SCS $C^{\prime}\supseteq C$, we have that
$C^{\prime}$ is accepting. First, an accepting SCS $C\subseteq Q$ and a
rejecting SCS $C^{\prime}\supseteq C$ induce a certificate $\langle
x,x_{1},x_{2}\rangle$. Indeed, taking a state $s\in C$, we can define $x$ to
be a word that leads from $q_{0}$ to $s$, $x_{1}$ to be a word that traverses
$C$, and $x_{2}$ a word that traverses $C^{\prime}$. Then, the set of states
traversed infinitely often in a run on a word in $x\cdot(x_{1}+x_{2})^{*}\cdot
x_{1}^{\omega}$ is $C$, and the set of states traversed infinitely often in a
run on a word in $x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}$ is $C^{\prime}$. For
the other direction, a certificate $\langle x,x_{1},x_{2}\rangle$ induces an
accepting SCS $C\subseteq Q$ and a rejecting SCS $C^{\prime}\supseteq C$ as
follows. Consider a graph $G=\langle Q,E\rangle$, where $E(s,s^{\prime})$ iff
$\delta(s,x_{1})=s^{\prime}$ or $\delta(s,x_{2})=s^{\prime}$. We consider an
ergodic SCC that is reachable from $\delta(q_{0},x)$ in $G$. In this ergodic
SCC, we can traverse both words in $x\cdot(x_{1}+x_{2})^{*}\cdot
x_{1}^{\omega}$ along an accepting cycle $C$, and words in
$x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}$ along a rejecting cycle, whose union
with $C$ can serve as $C^{\prime}$. $\blacksquare$
Being an $(A/\Sigma)$-transducer, every DBW-refuter ${\cal R}$ is responsive
and may generate many different words in $\Sigma^{\omega}$. Below we show that
we can leave ${\cal R}$ responsive and yet let it generate only words induced
by a certificate. Formally, we have the following.
###### Lemma 2
Given a certificate $\langle x,x_{1},x_{2}\rangle$ to non-DBW-recognizability
of a language $L\subseteq\Sigma^{\omega}$, we can define a refuter ${\cal R}$
for $L$ such that for every $y\in A^{\omega}$, if $y\models\infty\mbox{\sc
acc}$, then ${\cal R}(y)\in x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}$, and if
$y\models\neg\infty\mbox{\sc acc}$, then ${\cal R}(y)\in
x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}$.
###### Proof
Intuitively, ${\cal R}$ first ignores the inputs and outputs $x$. It then
repeatedly outputs either $x_{1}$ or $x_{2}$, according to the following
policy: in the first iteration, ${\cal R}$ outputs $x_{1}$. If during the
output of $x_{1}$ all inputs are rej, then ${\cal R}$ outputs $x_{1}$ also in
the next iteration. If an input acc has been detected, thus the prover tries
to accept the constructed word, the refuter outputs $x_{2}$ in the next
iteration, again keeping track of an acc input. If no acc has been input,
${\cal R}$ switches back to outputting $x_{1}$.
Formally, let $\langle x,x_{1},x_{2}\rangle$ be a certificate with
$x=x^{1}\cdots x^{n}$, $x_{1}=x^{1}_{1}\cdots x^{n_{1}}_{1}$, and
$x_{2}=x^{1}_{2}\cdots x_{2}^{n_{2}}$. We define ${\cal R}=\langle\\{\mbox{\sc
acc},\mbox{\sc rej}\\},\Sigma,{\it env},S,s_{0},\rho,\tau\rangle$ with the
components $S$, $\rho$, and $\tau$ defined as follows:
* •
$S=\\{s_{0},s_{1},\dots,s_{n},(s^{1}_{1},a),\dots,(s^{n_{1}}_{1},a),(s^{1}_{2},a),\dots,(s^{n_{2}}_{2},a):a\in\\{\mbox{\sc
acc},\mbox{\sc rej}\\}\\}$
* •
$\rho(s,a)=\begin{cases}s_{1}&\text{if }s=s_{0}\text{ and }n>0,\\\
s_{i+1}&\text{if }s=s_{i}\text{ and }n>i>0,\\\ (s^{1}_{1},\mbox{\sc
rej})&\text{if }s=s_{n},\\\ (s^{1}_{1},\mbox{\sc rej})&\text{if
}s\in\\{(s^{n_{1}}_{1},\mbox{\sc rej}),(s^{n_{2}}_{2},\mbox{\sc rej})\\}\text{
and }a=\mbox{\sc rej},\\\ (s^{1}_{2},\mbox{\sc rej})&\text{if
}s\in\\{(s^{n_{1}}_{1},\mbox{\sc rej}),(s^{n_{2}}_{2},\mbox{\sc rej})\\}\text{
and }a=\mbox{\sc acc},\\\ (s^{1}_{2},\mbox{\sc rej})&\text{if
}s\in\\{(s^{n_{1}}_{1},\mbox{\sc acc}),(s^{n_{2}}_{2},\mbox{\sc acc})\\}\\\
(s^{i+1}_{1},\mbox{\sc rej})&\text{if }s=(s^{i}_{1},\mbox{\sc rej})\text{ and
}n_{1}>i>0\text{ and }a=\mbox{\sc rej},\\\ (s^{i+1}_{1},\mbox{\sc
acc})&\text{if }s=(s^{i}_{1},\mbox{\sc rej})\text{ and }n_{1}>i>0\text{ and
}a=\mbox{\sc acc},\\\ (s^{i+1}_{1},\mbox{\sc acc})&\text{if
}s=(s^{i}_{1},\mbox{\sc acc})\text{ and }n_{1}>i>0,\\\ (s^{i+1}_{2},\mbox{\sc
rej})&\text{if }s=(s^{i}_{2},\mbox{\sc rej})\text{ and }n_{2}>i>0\text{ and
}a=\mbox{\sc rej},\\\ (s^{i+1}_{2},\mbox{\sc acc})&\text{if
}s=(s^{i}_{2},\mbox{\sc rej})\text{ and }n_{2}>i>0\text{ and }a=\mbox{\sc
acc},\\\ (s^{i+1}_{2},\mbox{\sc acc})&\text{if }s=(s^{i}_{2},\mbox{\sc
acc})\text{ and }n_{2}>i>0.\end{cases}$
* •
$\tau(s_{i})=x^{i}$ and $\tau((s^{i}_{j},a))=x^{i}_{j}$. ∎
By Theorem 3.3, every language not in DBW has a certificate $\langle
x,x_{1},x_{2}\rangle$. As we argue below, these certificates are linear in the
number of states of the refuters.
###### Lemma 3
Let ${\cal R}$ be a DBW-refuter for $L\subseteq\Sigma^{\omega}$ with $n$
states. Then, $L$ has a certificate of the form $\langle x,x_{1},x_{2}\rangle$
such that $|x|+|x_{1}|+|x_{2}|\leq 2\cdot n$.
###### Proof
The paths $p$, $p_{1}$, and $p_{2}$ that induce $x$, $x_{1}$ and $x_{2}$ in
the proof of Theorem 3.3 are simple, and so they are all of length at most
$n$. Also, while these paths may share edges, we can define them so that each
edge appears in at most two paths. Indeed, if an edge appears in all three
path, we can shorten $p$. Hence, $|x|+|x_{1}|+|x_{2}|\leq 2\cdot n$, and we
are done. ∎
###### Theorem 3.5
Consider a language $L\subseteq\Sigma^{\omega}$ not in DBW. The length of a
certificate for the non-DBW-recognizability of $L$ is linear in a DPW for $L$
and is exponential in an NBW for $L$. These bounds are tight.
###### Proof
The upper bounds follow from Theorem 3.1 and Lemma 3, and the exponential
determinization of NBWs. The lower bound in the NBW case follows from the
exponential lower bound on the size of shortest non-universality witnesses for
non-deterministic finite word automata (NFW) [33]. We sketch the reduction:
Let $L_{n}\subseteq\\{0,1\\}^{*}$ be a language such that the shortest witness
for non-universality of $L_{n}$ is exponential in $n$, but $L_{n}$ has a
polynomial sized NFW. We then define
$L^{\prime}_{n}=(L_{n}\cdot\$\cdot(0^{*}\cdot
1)^{\omega})+((0+1)^{*}\cdot\$\cdot(0+1)^{*}\cdot 0^{\omega})$. It is clear
that $L^{\prime}_{n}$ has a NBW polynomial in $n$ and is not DBW-recognizable.
Note that for every word $w\in L_{n}$, we have
$w\cdot\$\cdot(0+1)^{\omega}\subseteq L_{n}^{\prime}$. Thus, in order to
satisfy Theorem 3.3, every certificate $\langle x,x_{1},x_{2}\rangle$ needs to
have $w\cdot\$$ as prefix of $x$, for some $w\notin L_{n}$. Hence, it is
exponential in the size of the NBW. ∎
###### Remark 3
[LTL] When the language $L$ is given by an LTL formula $\varphi$, then
$\operatorname{DBW}(\varphi)=\varphi\leftrightarrow\textbf{GF}\mbox{\sc acc}$
and thus an off-the-shelf LTL synthesis tool can be used to extract a DBW-
refuter, if one exists. As for complexity, a doubly-exponential upper bound on
the size of a DPW for $\operatorname{NoDBW}(L)$, and then also on the size of
DBW-refuters and certificates, follows from the double-exponential translation
of LTL formulas to DPWs [49, 42]. The length of certificates, however, and
then, by Lemma 2, also the size of a minimal refuter, is related to the
diameter of the DPW for $\operatorname{NoDBW}(L)$, and we leave its tight
bound open. $\blacksquare$
## 4 Separability and Approximations
Consider three languages $L_{1},L_{2},L\subseteq\Sigma^{\omega}$. We say that
$L$ is a separator for $\langle L_{1},L_{2}\rangle$ if $L_{1}\subseteq L$ and
$L_{2}\cap L=\emptyset$. We say that a pair of languages $\langle
L_{1},L_{2}\rangle$ is DBW-separable iff there exists a language $L$ in DBW
such that $L$ is a separator for $\langle L_{1},L_{2}\rangle$.
###### Example 4
Let $\Sigma=\\{a,b\\}$, $L_{1}=(a+b)^{*}\cdot b^{\omega}$, and
$L_{2}=(a+b)^{*}\cdot a^{\omega}$. By [28], $L_{1}$ and $L_{2}$ are not in
DBW. They are, however, DBW-separable. A witness for this is $L=(a^{*}\cdot
b)^{\omega}$. Indeed, $L_{1}\subseteq L$, $L\cap L_{2}=\emptyset$, and $L$ is
DBW-recognizable. $\blacksquare$
Consider a language $L\subseteq\Sigma^{\omega}$, and suppose we know that $L$
is not in DBW. A user may be willing to approximate $L$ in order to obtain
DBW-recognizability. Specifically, we assume that there is a language
$I\subseteq\Sigma^{\omega}$ of words that the user is indifferent about.
Formally, the user is satisfied with a language in DBW that agrees with $L$ on
all words that are not in $I$. Formally, we say that a language $L^{\prime}$
approximates $L$ with radius $I$ if $L\setminus I\subseteq L^{\prime}\subseteq
L\cup I$. It is easy to see that, equivalently, $L^{\prime}$ is a separator
for $\langle L\setminus I,{\it comp}(L\cup I)\rangle$. Note that the above
formulation embodies the case where the user has in mind different over- and
under-approximation radiuses, thus separating $\langle L\setminus
I_{\downarrow},{\it comp}(L\cup I_{\uparrow})\rangle$ for possibly different
$I_{\downarrow}$ and $I_{\uparrow}$. Indeed, by defining
$I=(I_{\downarrow}\cap L)\cup(I_{\uparrow}\setminus L)$, we get $\langle
L\setminus I,{\it comp}(L\cup I)\rangle=\langle L\setminus I_{\downarrow},{\it
comp}(L)\setminus I_{\uparrow})\rangle$.
It follows that by studying DBW-separability, we also study DBW-approximation,
namely approximation by a language that is in DBW, possibly with different
over- and under-approximation radiuses.
Figure 4: Reduction of approximation to separability.
###### Remark 4
[From recognizability to separation] It is easy to see that DBW-separability
generalizes DBW-recognizability, as $L$ is in DBW iff $\langle L,{\it
comp}(L)\rangle$ is DBW-separable. Given $L\subseteq\Sigma^{\omega}$, we say
that a pair of languages $\langle L_{1},L_{2}\rangle$ is a no-DBW-witness for
$L$ if $L$ is a separator for $\langle L_{1},L_{2}\rangle$ and $\langle
L_{1},L_{2}\rangle$ is not DBW-separable. Note that the latter indeed implies
that $L$ is not in DBW.
A simple no-DBW witness for $L$ can be obtained as follows. Let ${\cal R}$ be
a DBW refuter for $L$. Then, we define $L_{1}=\\{{\cal
R}(y):y\in\neg\infty\mbox{\sc acc}\\}$ and $L_{2}=\\{{\cal
R}(y):y\in\infty\mbox{\sc acc}\\}$. By the definition of DBW-refuters, we have
$L_{1}\subseteq L$ and $L_{2}\cap L=\emptyset$, and so $\langle
L_{1},L_{2}\rangle$ is a no-DBW witness for $L$. It is simple, in the sense
that when we describe $L_{1}$ and $L_{2}$ by a tree obtained by pruning the
$\Sigma^{*}$-tree, then each node has at most two children – these that
correspond to the responses of ${\cal R}$ to acc and rej. $\blacksquare$
### 4.1 Refuting Separability
For a pair of languages $\langle L_{1},L_{2}\rangle$, we define the language
$\operatorname{SepDBW}(L)\subseteq(\Sigma\times A)^{\omega}$ of words with
correct annotations for separation. Thus,
$\operatorname{SepDBW}(L_{1},L_{2})=\\{x\oplus y:(x\in L_{1}\rightarrow
y\in\infty{\mbox{\sc acc}})\wedge(x\in L_{2}\rightarrow
y\not\in\infty{\mbox{\sc acc}})\\}.$
Note that ${\it comp}(\operatorname{SepDBW}(L_{1},L_{2}))$ is then the
language
$\operatorname{NoSepDBW}(L_{1},L_{2})=\\{x\oplus y:(x\in L_{1}\wedge
y\not\in\infty{\mbox{\sc acc}})\vee(x\in L_{2}\wedge y\in\infty{\mbox{\sc
acc}})\\}.$
A DBW-sep-refuter for $\langle L_{1},L_{2}\rangle$ is an
$(A/\Sigma)$-transducer with $\iota={\it env}$ that realizes
$\operatorname{NoSepDBW}(L_{1},L_{2})$.
###### Example 5
Consider the language $L_{\neg\infty a}=(a+b)^{*}\cdot b^{\omega}$, which is
not DBW. Let $I=a^{*}\cdot b^{\omega}+b^{*}\cdot a^{\omega}$, thus we are
indifferent about words with only one alternation between $a$ and $b$. In
Figure 5 we describe a DBW-sep refuter for $\langle L_{\neg\infty a}\setminus
I,{\it comp}(L_{\neg\infty a}\cup I)\rangle$. Note that the refuter generates
only words in $a\cdot b\cdot a\cdot(a+b)^{\omega}$, whose intersection with
$I$ is empty. Consequently, the refutation is similar to the DBW-refutation of
$L_{\neg\infty a}$. $\blacksquare$
Figure 5: A DBW-sep refuter for $\langle L_{\neg\infty a}\setminus I,{\it
comp}(L_{\neg\infty a}\cup I)\rangle$.
By Proposition 1, we have the following extension of Proposition 2.
###### Proposition 3
Consider two languages $L_{1},L_{2}\subseteq\Sigma^{\omega}$. Let
$A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$. Exactly one of the following holds:
* •
$\langle L_{1},L_{2}\rangle$ is DBW-separable, in which case the language
$\operatorname{SepDBW}(L_{1},L_{2})$ is $(\Sigma/A)$-realizable by the system,
and a finite-memory winning strategy for the system induces a DBW for a
language $L$ that separates $L_{1}$ and $L_{2}$.
* •
$\langle L_{1},L_{2}\rangle$ is not DBW-separable, in which case the language
$\operatorname{NoSepDBW}(L)$ is $(A/\Sigma)$-realizable by the environment,
and a finite-memory winning strategy for the environment induces a DBW-sep-
refuter for $\langle L_{1},L_{2}\rangle$.
As for complexity, the construction of the game for
$\operatorname{SepDBW}(L_{1},L_{2})$ is similar to the one described in
Theorem 3.1. Here, however, the input to the problem includes two DPWs. Also,
the positive case, namely the construction of the separator does not follow
from known results.
###### Theorem 4.1
Consider DPWs ${\cal A}_{1}$ and ${\cal A}_{2}$ with $n_{1}$ and $n_{2}$
states, respectively. Let $L_{1}=L({\cal A}_{1})$ and $L_{2}=L({\cal A}_{2})$.
One of the following holds.
1. 1.
There is a DBW ${\cal A}$ with $2\cdot n_{1}\cdot n_{2}$ states such that
$L({\cal A})$ DBW-separates $\langle L_{1},L_{2}\rangle$.
2. 2.
There is a DBW-sep-refuter for $\langle L_{1},L_{2}\rangle$ with $2\cdot
n_{1}\cdot n_{2}$ states.
###### Proof
We show that $\operatorname{SepDBW}(L_{1},L_{2})$ and
$\operatorname{NoSepDBW}(L_{1},L_{2})$ can be recognised by DRWs with at most
$2\cdot n_{1}\cdot n_{2}$ states. Then, by [15], we can construct a DBW or a
DBW-sep-refuter with at most $2\cdot n_{1}\cdot n_{2}$ states. The
construction is similar to the one described in the proof of Theorem 3.1. The
only technical challenge is the fact $\operatorname{SepDBW}(L_{1},L_{2})$ is
defined as the intersection, rather than union, of two languages. For this, we
observe that we can define $\operatorname{SepDBW}(L_{1},L_{2})$ also as
$\\{x\oplus y:(y\in\infty\mbox{\sc acc}\text{ and }x\notin L_{2})\text{ or
}(y\notin\infty\mbox{\sc acc}\text{ and }x\notin L_{1})\\}$. With this
formulation we then can reuse the union construction as seen in Theorem 3.1 to
obtain DRWs with at most $2\cdot n_{1}\cdot n_{2}$ states. ∎
As has been the case with DBW-recognizability, one can generate certificates
from a DBW-sep-refuter. The proof is similar to that of Theorem 3.3, with
membership in $L_{1}$ replacing membership in $L$ and membership in $L_{2}$
replacing being disjoint from $L$. Formally, we have the following.
###### Theorem 4.2
Two $\omega$-regular languages $L_{1},L_{2}\subseteq\Sigma^{\omega}$ are not
DBW-separable iff there exist three finite words $x\in\Sigma^{*}$ and
$x_{1},x_{2}\in\Sigma^{+}$, such that
$x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\subseteq L_{1}\quad\text{ and
}\quad x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}\subseteq L_{2}.$
We refer to a triple $\langle x,x_{1},x_{2}\rangle$ of words that satisfy the
conditions in Theorem 4.2 as a certificate to the non-DBW-separability of
$\langle L_{1},L_{2}\rangle$. Observe that the same way we generated a no-DBW
witness in Remark 4, we can extract, given a DBW-sep-refuter ${\cal R}$ for
$\langle L_{1},L_{2}\rangle$, languages $L^{\prime}_{1}\subseteq L_{1}$ and
$L^{\prime}_{2}\subseteq L_{2}$ that tighten $\langle L_{1},L_{2}\rangle$ and
are still not DBW-separable.
### 4.2 Certificate-Guided Approximation
In this section we describe a method for finding small approximating languages
$I_{\downarrow}$ and $I_{\uparrow}$ such that $\langle L\setminus
I_{\downarrow},{\it comp}(L)\setminus I_{\uparrow}\rangle$ is DBW-separable.
If this method terminates we obtain an approximation for $L$ that is DBW-
recognizable. As in counterexample guided abstraction-refinement (CEGAR) for
model checking [10], we use certificates for non-DBW-separability in order to
suggest interesting approximating languages. Intuitively, while in CEGAR the
refined system excludes the counterexample, here the approximation of $L$
excludes the certificate.
Consider a certificate $\langle x,x_{1},x_{2}\rangle$ for the non-DBW-
separability of $\langle L_{1},L_{2}\rangle$. We suggest the following five
approximations:
$\begin{array}[]{lcl}C_{0}=x\cdot(x_{1}+x_{2})^{\omega}&\rightsquigarrow&\langle
L_{1}\setminus C_{0},L_{2}\setminus C_{0}\rangle\\\
C_{1}=x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}=L_{1}\cap
C_{0}&\rightsquigarrow&\langle L_{1}\setminus C_{1},L_{2}\rangle\\\
C_{2}=x\cdot(x_{2}^{*}\cdot x_{1})^{\omega}\supset
C_{1}&\rightsquigarrow&\langle L_{1},L_{2}\setminus C_{2}\rangle\\\
C_{3}=x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}=L_{2}\cap
C_{0}&\rightsquigarrow&\langle L_{1},L_{2}\setminus C_{3}\rangle\\\
C_{4}=x\cdot(x_{1}+x_{2})^{*}\cdot x_{2}^{\omega}\subset
C_{3}&\rightsquigarrow&\langle L_{1},L_{2}\setminus C_{4}\rangle\end{array}$
First, it is easy to verify that $\langle x,x_{1},x_{2}\rangle$ is indeed not
a certificate for the non-DBW-separability of the obtained candidate pairs
$\langle L_{1}^{\prime},L_{2}^{\prime}\rangle$. If $\langle
L_{1}^{\prime},L_{2}^{\prime}\rangle$ is DBW-separable, we are done (yet may
try to tighten the approximation). Otherwise, we can repeat the process with a
certificate for the non-DBW-separability of $\langle
L_{1}^{\prime},L_{2}^{\prime}\rangle$. As in CEGAR, some suggestions may be
more interesting than others, in some cases the process terminates, in some it
does not, and the user takes part directing the search.
###### Example 6
Consider again the language $L=(a+b)^{*}\cdot b^{\omega}$ and the certificate
$\langle x,x_{1},x_{2}\rangle=\langle\epsilon,b,a\rangle$. Trying to
approximate $L$ by a language in DBW, we start with the pair $\langle L,{\it
comp}(L)\rangle$. Our five suggestions are then as follows.
$\begin{array}[]{lcl}C_{0}=\Sigma^{\omega}&\rightsquigarrow&\langle L\setminus
C_{0},{\it comp}(L)\setminus C_{0}\rangle=\langle\emptyset,\emptyset\rangle\\\
C_{1}=(b+a)^{*}\cdot b^{\omega}&\rightsquigarrow&\langle L\setminus C_{1},{\it
comp}(L)\rangle=\langle\emptyset,{\it comp}(L)\rangle\\\ C_{2}=(a^{*}\cdot
b)^{\omega}&\rightsquigarrow&\langle L,{\it comp}(L)\setminus
C_{2}\rangle=\langle L,(a+b)^{*}\cdot a^{\omega}\rangle\\\ C_{3}=(b^{*}\cdot
a)^{\omega}&\rightsquigarrow&\langle L,{\it comp}(L)\setminus
C_{3}\rangle=\langle L,\emptyset\rangle\\\ C_{4}=(b+a)^{*}\cdot
a^{\omega}&\rightsquigarrow&\langle L,{\it comp}(L)\setminus
C_{4}\rangle=\langle L,(a+b)^{*}\cdot(a\cdot a^{*}\cdot b\cdot
b^{*})^{\omega}\rangle\end{array}$
Candidates $C_{0}$, $C_{1}$, and $C_{3}$ induce trivial approximations. Then,
$C_{2}$ suggests to over-approximate $L$ by setting $I_{\uparrow}$ to
$(a^{*}\cdot b)^{\omega}$, which we view as a nice solution, approximating
“eventually always $b$” by “infinitely often $b$”. Then, the pair derived from
$C_{4}$ is not DBW-separable. We can try to approximate it. Note, however,
that repeated approximations in the spirit of $C_{4}$ are going to only extend
the prefix of $x$ in the certificates, and the process does not terminate.
Let us now consider the slightly different certificate $\langle
x,x_{1},x_{2}\rangle=\langle a,b,a\rangle$ and the derived candidates:
$\begin{array}[]{lcl}C_{0}=a\cdot\Sigma^{\omega}&\rightsquigarrow&\langle
L\setminus C_{0},{\it comp}(L)\setminus C_{0}\rangle=\langle b\cdot
L,b\cdot{\it comp}(L)\rangle\\\ C_{1}=a\cdot(b+a)^{*}\cdot
b^{\omega}&\rightsquigarrow&\langle L\setminus C_{1},{\it
comp}(L)\rangle=\langle b\cdot L,{\it comp}(L)\rangle\\\
C_{2}=a\cdot(a^{*}\cdot b)^{\omega}&\rightsquigarrow&\langle L,{\it
comp}(L)\setminus C_{2}\rangle\\\ &&=\langle L,b\cdot{\it
comp}(L)+a\cdot(a+b)^{*}\cdot a^{\omega}\rangle\\\ C_{3}=a\cdot(b^{*}\cdot
a)^{\omega}&\rightsquigarrow&\langle L,{\it comp}(L)\setminus
C_{3}\rangle=\langle L,b\cdot{\it comp}(L)\rangle\\\
C_{4}=a\cdot(b+a)^{*}\cdot a^{\omega}&\rightsquigarrow&\langle L,{\it
comp}(L)\setminus C_{4}\rangle\\\ &&=\langle L,b\cdot{\it
comp}(L)+a\cdot(a+b)^{*}\cdot(a\cdot a^{*}\cdot b\cdot
b^{*})^{\omega}\rangle\end{array}$
One can easily verify that $\langle x,x_{1},x_{2}\rangle=\langle b\cdot
a,b,a\rangle$ is a certificate showing that none of the suggested pairs are
DBW-separable. In fact $\langle x,x_{1},x_{2}\rangle=\langle b^{i}\cdot
a,b,a\rangle$, for $i=0,1,2,\dots$, describes an infinite sequence such that
no refinement obtained after a finite number of steps is DBW-separable.
$\blacksquare$
## 5 Other Classes of Deterministic Automata
In this section we generalise the idea of DBW-refuters to other classes of
deterministic automata. For this we take again the view that a deterministic
automaton is a $\langle\Sigma,A\rangle$-transducer over a suitable annotation
alphabet $A$. We then characterize each class of deterministic automata by two
languages over $A$:
* •
The language $L_{\textnormal{acc}}\subseteq A^{\omega}$, describing when a run
is accepting. For example, for DBWs, we have $A=\\{\mbox{\sc acc},\mbox{\sc
rej}\\}$ and $L_{\textnormal{acc}}=\infty\mbox{\sc acc}$.
* •
The language $L_{\textnormal{struct}}\subseteq A^{\omega}$, describing
structural conditions on the run. For example, recall that a DWW is a DBW in
which the states of each SCS are either all accepting or all rejecting, and so
each run eventually get trapped in an accepting or rejecting SCS. Accordingly,
the language of runs that satisfy the structural condition is
$L_{\textnormal{struct}}=A^{*}\cdot(\mbox{\sc acc}^{\omega}+\mbox{\sc
rej}^{\omega})$.
We now formalize this intuition. Let $A$ be a finite set of annotations and
let $\gamma=\langle L_{\textnormal{acc}},L_{\textnormal{struct}}\rangle$, for
$L_{\textnormal{acc}},L_{\textnormal{struct}}\subseteq A^{\omega}$. A
deterministic automaton ${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$
is a deterministic $\gamma$ automaton (D$\gamma$W, for short) if there is a
function $\tau\colon Q\to A$ that maps each state to an annotation such that a
run $r$ of ${\cal A}$ satisfies $\alpha$ iff $\tau(r)\in
L_{\textnormal{acc}}$, and all runs $r$ satisfy the structural condition, thus
$\tau(r)\in L_{\textnormal{struct}}$. We then say that a language $L$ is
$\gamma$-recognizable if there a D$\gamma$W ${\cal A}$ such that $L=L({\cal
A})$.
Before we continue to study $\gamma$-recognizability, let us demonstrate the
$\gamma$-characterization of common deterministic automata. We first start
with classes $\gamma$ for which $L_{\textnormal{struct}}$ is trivial; i.e.,
$L_{\textnormal{struct}}=A^{\omega}$.
* •
DBW: $A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$ and
$L_{\textnormal{acc}}=\infty\mbox{\sc acc}$.
* •
DCW: $A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$ and
$L_{\textnormal{acc}}=\neg\infty\mbox{\sc acc}$.
* •
DPW[$i,k$]: $A=\\{i,\dots,k\\}$ and $L_{\textnormal{acc}}=\\{y\in
A^{\omega}:\max({\it inf}(y))\text{ is odd}\\}$.
* •
DELW[$\theta$]: $A=2^{\mathbb{M}}$ and $L_{\textnormal{acc}}=\\{y\in
A^{\omega}:y\models\theta\\}$.
Note that the characterizations for Büchi, co-Büchi, and parity are special
cases of the characterization for DELW. In a similar way, we could define a
language $L_{\textnormal{acc}}$ for DRW[$k$] and other common special cases of
DELWs. We continue to classes in the depth hierarchy, where $\gamma$ includes
also a structural restriction:
* •
DWW: The set $A$ and the language $L_{\textnormal{acc}}$ are as for DBW or
DCW. In addition, $L_{\textnormal{struct}}=A^{*}\cdot(\mbox{\sc
acc}^{\omega}+\mbox{\sc rej}^{\omega})$.
* •
DWW[$j,k$], for $j\in\\{0,1\\}$: The set $A$ and the language
$L_{\textnormal{acc}}$ are as for DPW[$j,k$]. In addition,
$L_{\textnormal{struct}}=\\{y_{0}\cdot y_{1}\cdots\in A^{\omega}:$ for all
$i\geq 0$, we have that $y_{i}\leq y_{i+1}\\}$.
* •
Bounded Languages: A language $L$ is bounded if it is both safety and co-
safety. Thus, every word $w\in\Sigma^{\omega}$ has a prefix $v\in\Sigma^{*}$
such that either for all $u\in\Sigma^{\omega}$ we have $v\cdot u\in L$, or for
all $u\in\Sigma^{\omega}$ we have $v\cdot u\not\in L$ [23]. To capture this,
we use $A=\\{\mbox{\sc acc},\mbox{\sc rej},?\\}$, where $``?"$ is used for
annotating states with both accepting and rejecting continuations. Then,
$L_{\textnormal{acc}}=A^{*}\cdot\mbox{\sc acc}^{\omega}$, and
$L_{\textnormal{struct}}=?^{*}\cdot(\mbox{\sc acc}^{\omega}+\mbox{\sc
rej}^{\omega})$.
* •
Deterministic $(m,n)$-Superparity Automata [39]: $A=\\{(i,j):0\leq i\leq
m,0\leq j\leq n\\}$, $L_{\textnormal{acc}}=\\{y_{m}\oplus y_{n}\in
A^{\omega}:\max({\it inf}(y_{m}))+\max(y_{n})\text{ is odd}\\}$, and
$L_{\textnormal{struct}}=\\{y_{m}\oplus(y_{0}\cdot y_{1}\cdots)\in
A^{\omega}:y_{i}\leq y_{i+1},\mbox{ for all }i\geq 0\\}$.
Let $\Sigma$ be an alphabet, let $A$ be an annotation alphabet, and let
$\gamma=\langle L_{\textnormal{acc}},$ $L_{\textnormal{struct}}\rangle$, for
$L_{\textnormal{acc}},L_{\textnormal{struct}}\subseteq A^{\omega}$. We define
the language $\mathrm{Real}(L,\gamma)\subseteq(\Sigma\times A)^{\omega}$ of
words with correct annotations.
$\mathrm{Real}(L,\gamma)=\\{x\oplus y:y\in L_{\textnormal{struct}}\text{ and
}(x\in L\text{ iff }y\in L_{\textnormal{acc}})\\}.$
Note that the language $\operatorname{DBW}(L)$ can be viewed as a special case
of our general framework. In particular, in cases
$L_{\textnormal{struct}}=A^{\omega}$, we can remove the $y\in
L_{\textnormal{struct}}$ conjunct from $\mathrm{Real}(L,\gamma)$. Note that
${\it comp}(\mathrm{Real}(L,\gamma))$ is the language
$\mathrm{NoReal}(L,\gamma)=\\{x\oplus y:y\not\in L_{\textnormal{struct}}\mbox{
or }(x\in L\mbox{ iff }y\not\in L_{\textnormal{acc}})\\}.$
A $\gamma$-refuter for $L$ is then an $(A/\Sigma)$-transducer with $\iota={\it
env}$ that realizes $\mathrm{NoReal}(L,\gamma)$. We can now state the
“D$\gamma$W-generalization” of Proposition 2.
###### Proposition 4
Consider an $\omega$-regular language $L\subseteq\Sigma^{\omega}$, and a pair
$\gamma=\langle L_{\textnormal{acc}},L_{\textnormal{struct}}\rangle$, for
$\omega$-regular languages
$L_{\textnormal{acc}},L_{\textnormal{struct}}\subseteq A^{\omega}$. Exactly
one of the following holds:
1. 1.
$L$ is in D$\gamma$W, in which case the language $\mathrm{Real}(L,\gamma)$ is
$(\Sigma/A)$-realizable by the system, and a finite-memory winning strategy
for the system induces a D$\gamma$W for $L$.
2. 2.
$L$ is not in D$\gamma$W, in which case the language
$\mathrm{NoReal}(L,\gamma)$ is $(A/\Sigma)$-realizable by the environment, and
a finite-memory winning strategy for the environment induces a
$\gamma$-refuter for $L$.
Note that every DELW can be complemented by dualization, thus by changing its
acceptance condition from $\theta$ to $\neg\theta$. In particular, DBW and DCW
dualize each other. As we argue below, dualization is carried over to
refutation. For example, the $(\\{\mbox{\sc acc},\mbox{\sc
rej}\\}/\Sigma)$-transducer ${\cal R}$ from Figure 1 is both a DBW-refuter for
$\neg\infty a$ and a DCW-refuter for $\infty a$. Formally, we have the
following.
###### Theorem 5.1
Consider an EL-condition $\theta$ over $\mathbb{M}$. Let $A=2^{\mathbb{M}}$.
For every $(A/\Sigma)$-transducer ${\cal R}$ and language $L$, we have that
${\cal R}$ is a DELW$[\theta]$-refuter for $L$ iff ${\cal R}$ is a
DELW$[\neg\theta]$-refuter for ${\it comp}(L)$. In particular, for every
language $L$ and $(\\{\mbox{\sc acc},\mbox{\sc rej}\\}/\Sigma)$-transducer
${\cal R}$, we have that ${\cal R}$ is a DBW-refuter for $L$ iff ${\cal R}$ is
a DCW-refuter for ${\it comp}(L)$.
###### Proof
For DELW[$\theta$]-recognizability of $L$, the language of correct annotations
is $\\{x\oplus y:(x\in L\text{ iff }y\models\theta)\\}$, which is equal to
$\\{x\oplus y:(x\in{\it comp}(L)\text{ iff }y\models\neg\theta)\\}$, which is
the language of correct annotations for DELW[$\neg\theta$]-recognizability of
${\it comp}(L)$. ∎
While dualization is nicely carried over to refutation, this is not the case
for all expressiveness results. For example, while DWW=DBW$\cap$DCW, and in
fact DBW and DCW are weak type (that is, when the language of a DBW is in DWW,
an equivalent DWW can be defined on top of its structure, and similarly for
DCW [21]), we describe below a DWW-refuter that is neither a DBW- nor a DCW-
refuter. Intuitively, this is possible as in DWW refutation, Prover loses when
the input is not in $A^{*}\cdot(\mbox{\sc acc}^{\omega}+\mbox{\sc
rej}^{\omega})$, whereas in DBW and DCW refutation, Refuter has to respond
correctly also for these inputs.
###### Example 7
Let $\Sigma=\\{a,b,c,d\\}$, and $A=\\{\mbox{\sc acc},\mbox{\sc rej}\\}$.
Consider the language $L=(a^{+}\cdot b\cdot c^{*}\cdot d)^{*}\cdot
a^{\omega}+(a\cdot b\cdot d)^{\omega}$. Note that $L$ is in DCW, but not in
DBW, and hence also not in DWW. The $(A/\Sigma)$-transducer ${\cal R}$ in
Figure 6 is a DWW-refuter for $L$. To see this, recall that for DWWs, we have
that $L_{\textnormal{struct}}=A^{*}\cdot(\mbox{\sc acc}^{\omega}+\mbox{\sc
rej}^{\omega})$, and so all input sequences $y\in A^{\omega}$ that satisfy
$L_{\textnormal{struct}}$ eventually gets trapped in the $a^{\omega}$ loop,
generating a rejecting run on a word in the language, or gets trapped in the
$c^{\omega}$ loop, generating an accepting run on a word not in the language.
On the other hand, while $L$ is not in DBW, the transducer ${\cal R}$ is not a
DBW-refuter for $L$. To see this, observe that the DBW ${\cal A}$ in the
figure suggests a winning strategy for Prover in the game corresponding to
DBW. Indeed, when Prover generates $(\mbox{\sc rej}\cdot\mbox{\sc
acc}\cdot\mbox{\sc rej})^{\omega}$, which is accepting, then by following
${\cal R}$, Refuter responds with $(a\cdot b\cdot d)^{\omega}$, which is in
$L$, and so Prover wins. Note that, unsurprisingly, the input generated by
Prover does not satisfy $L_{\textnormal{struct}}$. $\blacksquare$
Figure 6: The DWW-refuter ${\cal R}$ looses as a DBW-refuter when it plays
against ${\cal A}$.
On the other hand, as every DWW is also a DBW and a DCW, every DBW-refuter or
DCW-refuter is also a DWW-refuter.
#### Separability and Approximation.
Consider a characterization $\gamma=\langle
L_{\textnormal{acc}},L_{\textnormal{struct}}\rangle$. Two languages
$L_{1},L_{2}\subseteq\Sigma^{\omega}$ are _$\gamma$ -separable_ if there
exists a D$\gamma$W${\cal A}$ such that $L_{1}\subseteq L({\cal A})$ and
$L_{2}\cap L({\cal A})=\emptyset$. We define the corresponding languages of
correct and incorrect annotations as follows.
* •
$\mathrm{Sep}(L_{1},L_{2},L_{\textnormal{acc}},L_{\textnormal{struct}})=\\\
\\{x\oplus y:y\in L_{\textnormal{struct}}\text{ and }((x\in L_{1}\text{ and
}y\in L_{\textnormal{acc}})\text{ or }(x\in L_{2}\text{ and }y\notin
L_{\textnormal{acc}}))\\}$.
* •
$\mathrm{NoSep}(L_{1},L_{2},L_{\textnormal{acc}},L_{\textnormal{struct}})={\it
comp}(\mathrm{Sep}(L_{1},L_{2},L_{\textnormal{acc}},L_{\textnormal{struct}}))=\\\
\\{x\oplus y:y\notin L_{\textnormal{struct}}\text{ or }((x\in L_{1}\text{ and
}y\notin L_{\textnormal{acc}})\text{ or }(x\in L_{2}\text{ and }y\in
L_{\textnormal{acc}}))\\}$.
Note that the language $\operatorname{SepDBW}(L_{1},L_{2})$ can be viewed as a
special case of our general framework and as before in cases
$L_{\textnormal{struct}}=A^{\omega}$, we can remove the $y\in
L_{\textnormal{struct}}$ conjunct from $\mathrm{Sep}$. A $\gamma$-sep-refuter
for $L$ is an $(A/\Sigma)$-transducer with $\iota={\it env}$ that realizes
$\mathrm{NoSep}(L_{1},L_{2},L_{\textnormal{acc}},L_{\textnormal{struct}})$. By
Proposition 1, exactly one of the following holds:
###### Proposition 5
Consider $\omega$-regular languages $L_{1},L_{2}\subseteq\Sigma^{\omega}$, and
a characterization $\gamma=\langle
L_{\textnormal{acc}},L_{\textnormal{struct}}\rangle$, for $\omega$-regular
languages $L_{\textnormal{acc}},L_{\textnormal{struct}}\subseteq A^{\omega}$.
Exactly one of the following holds:
1. 1.
$\langle L_{1},L_{2}\rangle$ are $\gamma$-separable, in which case the
language $\mathrm{Sep}(L_{1},L_{2},\gamma)$ is $(\Sigma/A)$-realizable by the
system, and a finite-memory winning strategy for the system induces a
D$\gamma$W for some $L$ such that $L_{1}\subseteq L$ and $L\cap
L_{2}=\emptyset$.
2. 2.
$\langle L_{1},L_{2}\rangle$ are not $\gamma$-separable, in which case the
language $\mathrm{NoSep}(L_{1},L_{2},\gamma)$ is $(A/\Sigma)$-realizable by
the environment, and a finite-memory winning strategy for the environment
induces a $\gamma$-sep-refuter for $\langle L_{1},L_{2}\rangle$.
## 6 Certifying D$\gamma$W-Refutation
In this section we extend the three-word certificates for non-DBW-
recognizability to richer classes of deterministic automata. The idea is
similar (and in fact a little tedious): each D$\gamma$W-refuter embodies a
structure (analogous to the one in Lemma 1) from which we can extract finite
words that constitute the corresponding certificate (analogous to the one in
Theorem 3.3). We describe here the details for classes in the Mostowski
hierarchy and well as for classes of the depth-hierarchy. We also restrict
ourselves to word-certificates for non-recognizability and do not show the
word-certificates for non-separability which have an identical structure.
### 6.1 Mostowski Hierarchy
First, by Theorem 5.1, certificates for a class and its dual class are
related. For example, dualizing Theorem 3.3, we obtain certificates for non-
DCW-recognizability as follows.
###### Theorem 6.1
An $\omega$-regular language $L$ is not in DCW iff there exist three finite
words $x\in\Sigma^{*}$ and $x_{1},x_{2}\in\Sigma^{+}$, such that
$x\cdot(x_{1}+x_{2})^{*}\cdot x_{1}^{\omega}\cap L=\emptyset\qquad\text{ and
}\qquad x\cdot(x_{1}^{*}\cdot x_{2})^{\omega}\subseteq L.$
Handling DPWs, we first define the analogue of a $\mbox{\sc rej}^{+}$-path,
and then point to the desired structure and the certificate it induces.
Consider a DPW[$i,k$]-refuter ${\cal R}=\langle\\{i,\dots,k\\},\Sigma,{\it
env},S,s_{0},\rho,\tau\rangle$ with $i\in\\{0,1\\}$ and $i\leq k$. Let
$\ell\in\\{i,\dots,k\\}$. We say that a path $s_{1},\ldots,s_{m}$ in ${\cal
R}$ is an $\ell^{+}_{\leq}$-path if its first transition is labelled $\ell$
and all its other transitions are labeled by colors in $\\{i,\dots\ell\\}$.
Thus, $s_{2}=\rho(s_{1},\ell)$ and, for all $1\leq j<m$, we have that
$s_{j+1}=\rho(s_{j},\ell^{\prime})$, for some $\ell^{\prime}\leq\ell$.
###### Lemma 4
Consider a DPW$[i,k]$-refuter ${\cal R}=\langle\\{i,\dots,k\\},\Sigma,{\it
env},S,s_{0},\rho,\tau\rangle$ with $i\in\\{0,1\\}$ and $i\leq k$. There
exists a state $s\in S$, a (possibly empty) path $p=s_{0},s_{1},\dots s_{m}$,
and for each $\ell\in\\{i,\dots,k\\}$, a $\ell^{+}_{\leq}$-cycle
$p_{\ell}=s^{\ell}_{1}\dots s^{\ell}_{m_{\ell}}$, such that
$s_{m}=s^{\ell}_{1}=s^{\ell}_{m_{\ell}}=s$.
###### Proof
Let ${\cal R}_{\leq j}$ denote the transducer that we obtain from ${\cal R}$
when we restrict $\delta$ to transitions labelled by at most $j$. Note that
${\cal R}$ is ${\cal R}_{\leq k}$. We proceed by induction on $j$ with $i\leq
j\leq k$ and show that in the transducer ${\cal R}_{\leq j}$ for every state
$s\in S$ there exists a state $s^{\prime}\in S$, a (possibly empty) path
$p=s_{1},\dots s_{m}$ with $s=s_{1}$, and that for each
$\ell\in\\{i,\dots,j\\}$ there exists a $\ell^{+}_{\leq}$-cycle
$p_{\ell}=s^{\ell}_{1},s^{\ell}_{2}\dots s^{\ell}_{m_{\ell}}$, such that
$s_{m}=s^{\ell}_{1}=s^{\ell}_{m_{\ell}}=s^{\prime}$. The base case for $j=i$
follows immediately from the fact that ${\cal R}_{\leq i}$ is responsive on
$\\{i\\}$ and by reading $i^{\omega}$ we obtain a lasso with the required
properties.
Let $j>i$ and let $s\in S$ be an arbitrary state. Further, let $s_{j}\in S$ be
a reachable state from $s$ that belongs to an ergodic component in the graph
of ${\cal R}_{\leq j}$ (that is, $s_{j}\in C$, for a set $C$ of strongly
connected states that can reach only states in $C$). By induction hypothesis
there exists $s^{\prime}\in S$, a (possibly empty) path $p=s_{j},s_{j+1},\dots
s_{m}$, and for each $\ell\in\\{i,\dots,j-1\\}$ there exists a
$\ell^{+}_{\leq}$-cycle $p_{\ell}=s^{\ell}_{1},s^{\ell}_{2}\dots
s^{\ell}_{m_{\ell}}$, such that
$s_{m}=s^{\ell}_{1}=s^{\ell}_{m_{\ell}}=s^{\prime}$ for every
$\ell\in\\{i,\dots,j-1\\}$. Since ${\cal R}_{\leq j}$ is responsive on
$\\{i,\dots,j\\}$ we can take from $s^{\prime}$ a transition labelled $\ell$
and since $C$ is ergodic we can find a path back to $s^{\prime}$. Thus we
obtain the missing $j^{+}_{\leq}$-cycle and by concatenating the path from $s$
to $s_{j}$ and the path $p$, we show that $s^{\prime}$ can be reached from
$s$. ∎
###### Theorem 6.2
Let $i\in\\{0,1\\}$ and $i\leq k$. An $\omega$-regular language $L$ is not in
DPW$[i,k]$ iff there exist finite words $x\in\Sigma^{*}$ and
$x_{i},\dots,x_{k}\in\Sigma^{+}$, such that for every even $i\leq\ell\leq k$,
we have
$x\cdot(x_{i}+\dots+x_{k})^{*}\cdot((x_{i}+x_{i+1}+\dots+x_{\ell-1})^{*}\cdot
x_{\ell})^{\omega}\subseteq L,$
and for every odd $i\leq\ell\leq k$, we have
$x\cdot(x_{i}+\dots+x_{k})^{*}\cdot((x_{i}+x_{i+1}+\dots+x_{\ell-1})^{*}\cdot
x_{\ell})^{\omega}\cap L=\emptyset.$
###### Proof
Assume first that $L$ is not in DPW[$i,k$]. Then, by Proposition 4, there
exists a DPW[$i,k$]-refuter ${\cal R}$ for it. From this refuter we can
extract via Lemma 4 a path $p$ and $\ell^{+}_{\leq}$-cycles. We then construct
the postulated finite words in the exact same way as in the proof of Theorem
3.3.
For the other direction, we first simplify the presentation by assuming $i=0$.
The proof for $i=1$ is analogous. Assume by way of contradiction that there is
a DPW$[0,k]$ ${\cal A}$ with $L({\cal A})=L$. Let ${\cal
A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$. Let $n=|Q|$ and consider the
following sequence of words $w_{0}=x_{0}^{n}$, $w_{1}=(w_{0}\cdot x_{1})^{n}$,
…, $w_{k}=(w_{k-1}\cdot x_{k})^{n}$. Let $q=\delta(q_{0},w)$ be a state that
is reached after reading $w\in x\cdot(x_{i}+x_{i+1}+\dots x_{k})^{*}$. Since
$w\cdot w_{0}^{\omega}\in L$, there must be a state $p_{0}$ that is visited
infinitely often and $\alpha(p_{0})$ is odd. Since $|w_{0}|\geq|Q|$, this
state must have been visited while reading $w_{0}$. Now, consider $w\cdot
w_{1}^{\omega}$. This word is rejected and by the same reasoning as before
there must be some $p_{1}$ such that $\alpha(p_{1})$ is even, it is visited
while reading $w_{1}$, and for every $p_{0}$ that belongs to a $w_{0}$
subsequences we have $\alpha(p_{1})>\alpha(p_{0})$. We continue and obtain a
sequence $\alpha(p_{k})>\dots>\alpha(p_{0})$ with $k$ strict inequalities.
Since $\alpha(p_{0})$ is odd, we have $\alpha(p_{0})>0$ and thus
$\alpha(p_{k})>k$, which contradicts the fact that ${\cal A}$ is a DPW$[0,k]$.
∎
Note that, by [38], the “flower”-structure that induces the certificate exists
also in DPWs for $L$. Specifically, while Lemma 4 shows that every
DPW$[i,k]$-refuter contains a “flower” with $k-i+1$ petals, it is shown in
[38] that for every $\omega$-language $L$ not in DPW[$1,k+1$], there exists a
DPW for $L$ that contains a flower with $k+1$ petals and this flower occurs in
some accepting run.
#### Rabin and Streett acceptance.
Recall that for all $k\geq 0$, we have that DRW[$k$] = DPW[$0,2k$] . Hence,
the certificates obtained through Theorem 6.2 carry over to the Rabin case.
Further, in a deterministic generalized Rabin automaton (DGRW), the acceptance
condition is of the form
$\alpha=\\{\langle B_{1},G_{1,1},\dots,G_{1,n_{1}}\rangle,\dots\langle
B_{k},G_{k,1},\dots,G_{n,k_{n}}\rangle\\},$
and a run $r$ is accepting if there is $j\in\\{1,\dots,k\\}$, such that ${\it
inf}(r)\cap B_{j}=\emptyset$ and ${\it inf}(r)\cap G_{j,\ell}\neq\emptyset$
for every $1\leq\ell\leq n_{j}$. Since degeneralization does not increase the
number of Rabin pairs, we have that DGRW[$k$] = DRW[$k$] = DPW[$0,2k$], and so
again the certificates obtained through Theorem 6.2 are applicable.
Nevertheless, a refuter for the DRW[$k$] may be more succinct than a
DPW$[0,2k]$-refuter.
Finally, the Streett and generalized acceptance conditions are dual to Rabin
and generalized Rabin, and certificates for them can be obtained dually.
### 6.2 Depth-Hierarchy
We continue to certificates for non-DWW[$i,k$]-recognizability. Consider a
DWW[$i,k$]-refuter ${\cal R}=\langle\\{i,\dots,k\\},\Sigma,{\it
env},S,s_{0},\rho,\tau\rangle$, with $i\in\\{0,1\\}$ and $i\leq k$. Let
$\ell\in\\{i,\dots,k\\}$. We say that a path $s_{1},\ldots,s_{m}$ in ${\cal
R}$ is an $\ell^{+}$-path if all transitions are labelled by $\ell$. Thus, for
all $1\leq j<m$, we have that $s_{j+1}=\rho(s_{j},\ell)$.
###### Lemma 5
Consider a DWW$[i,k]$-refuter ${\cal R}=\langle\\{i,\dots,k\\},\Sigma,{\it
env},S,s_{0},\rho,\tau\rangle$ with $i\in\\{0,1\\}$ and $i\leq k$. Let
$s^{i-1}$ be an alias for $s_{0}$. Then there exists a sequence of states
$s^{i},s^{i+1},\dots s^{k}\in S$, such that for every $j\in\\{i,\dots,k\\}$
there exists a (possibly empty) $j^{+}$-path $p^{j}=s^{j}_{1},s^{j}_{2},\dots
s^{j}_{m_{j}}$, and a $j^{+}$-cycle
$c^{j}=s^{j}_{m_{j}+1},s^{j}_{m_{j}+2}\dots s^{j}_{m_{j}+m^{\prime}_{j}}$ such
that $s^{j}_{m_{j}}=s^{j}_{m_{j}+1}=s^{j}_{m_{j}+m_{j}^{\prime}}=s^{j}$ and
$s^{j}_{1}=s^{j-1}$.
###### Proof
Such a structure can be found by constructing a sequence of lassos. Start by
reading $i^{\omega}$ from $s_{0}$ to construct an $i^{+}$-path $p^{i}$ and an
$i^{+}$-cycle $c^{i}$. $s^{i}$ is then the last state of $c^{i}$,
respectively. Then, continue by reading $(i+1)^{\omega}$ from $s^{i}$ to find
the next lasso and continue until all lassos are found. ∎
###### Theorem 6.3
Let $i\in\\{0,1\\}$ and $i\leq k$. An $\omega$-regular language $L$ is not in
DWW$[i,k]$ iff there exist finite words
$\hat{x}_{i},\hat{x}_{i+1},\dots,\hat{x}_{k}\in\Sigma^{*}$ and
$x_{i},x_{i+1},\dots,x_{k}\in\Sigma^{+}$, such that for every even
$i\leq\ell\leq k$, we have
$\hat{x}_{i}\cdot x_{i}^{*}\cdot\hat{x}_{i+1}\cdot
x_{i+1}^{*}\cdots\hat{x}_{\ell}\cdot x_{\ell}^{\omega}\subseteq L,$
and for every odd $i\leq\ell\leq k$, we have
$\hat{x}_{i}\cdot x_{i}^{*}\cdot\hat{x}_{i+1}\cdot
x_{i+1}^{*}\cdots\hat{x}_{\ell}\cdot x_{\ell}^{\omega}\cap L=\emptyset.$
###### Proof
Assume first that $L$ is not in DWW[$i,k$]. Then, by Proposition 4, there
exists a DWW[$i,k$]-refuter ${\cal R}$ for it. From this refuter we can
extract via Lemma 5 a sequence of states with the corresponding paths and
cycles. We then obtain words in the same manner as in the proof of Theorem
3.3.
For the remaining direction assume by way of contradiction that there is a
DWW[$i,k$] ${\cal A}=\langle\Sigma,Q,q_{0},\delta,\alpha\rangle$ with $L({\cal
A})=L$. We simplify the presentation by assuming $i=0$. The proof for $i=1$ is
analogous. Let $n=|Q|$ and consider the following sequence of words
$w_{0}=\hat{x}_{0}\cdot x_{0}^{n}$, $w_{1}=w_{0}\cdot\hat{x}_{1}\cdot
x_{1}^{n}$, …, $w_{k}=w_{k-1}\cdot\hat{x}_{k}\cdot x_{k}^{n}$. Since
$w_{0}\cdot x_{0}^{\omega}\in L$ and $w_{0}$ has more letters than ${\cal A}$
has states, we have $\alpha(\delta(q_{0},w_{0}))$ is odd. By the same argument
we have due to $w_{1}\cdot x_{1}^{\omega}\notin L$ that
$\alpha(\delta(q_{0},w_{1}))$ is even and since $w_{0}$ is a prefix of $w_{1}$
we also have $\alpha(\delta(q_{0},w_{1}))>\alpha(\delta(q_{0},w_{0}))$.
Continuing in this manner we obtain a chain of length
$\alpha(\delta(q_{0},w_{k}))>\alpha(\delta(q_{0},w_{k-1}))>\dots>\alpha(\delta(q_{0},w_{0}))$
with $k$ strict inequalities. Since the smallest element is odd, we have
$\alpha(\delta(q_{0},w_{0}))>0$ and thus $\alpha(\delta(q_{0},w_{k}))>k$ which
contradicts ${\cal A}$ being a DWW$[0,k]$. ∎
We continue with general DWWs.
###### Lemma 6
Consider a DWW-refuter ${\cal R}=\langle\\{\mbox{\sc acc},\mbox{\sc
rej}\\},\Sigma,{\it env},S,s_{0},\rho,\tau\rangle$. There exist two states
$s^{1},s^{2}\in S$, (possibly empty) paths $p_{0}=s_{0},s_{1},\dots
s_{m_{0}}$, $p_{1}=s_{m_{0}+1},\dots,s_{m_{0}+m_{1}}$, and
$p_{2}=s_{m_{0}+m_{1}+1},\dots,s_{m_{0}+m_{1}+m_{2}}$, a $\mbox{\sc
rej}^{+}$-cycle $c^{1}=s^{1}_{1},s^{1}_{2}\dots s^{1}_{l_{1}}$, and a
$\mbox{\sc acc}^{+}$-cycle $c^{2}=s^{2}_{1},s^{2}_{2}\dots s^{2}_{l_{2}}$,
such that
$s_{m_{0}}=s_{m_{0}+1}=s_{m_{0}+m_{1}+m_{2}}=s^{1}_{1}=s^{1}_{l_{1}}$ and
$s_{m_{0}+m_{1}}=s_{m_{0}+m_{1}+1}=s^{2}_{1}=s^{2}_{l_{2}}$.
###### Proof
Let $s\in S$ be state in an ergodic SCC of the graph of ${\cal R}$. Then the
$\mbox{\sc acc}^{+}$\- and $\mbox{\sc rej}^{+}$-cycle are obtained from the
lassos formed by reading from $s$ the words $\mbox{\sc acc}^{\omega}$ and
$\mbox{\sc rej}^{\omega}$, respectively. Since $s$ belongs to an ergodic SCC,
there exist paths connecting the first states of these cycles. ∎
We now obtain in the same way as before from Proposition 4 and Lemma 6, the
desired certificate:
###### Theorem 6.4
An $\omega$-regular language $L$ is not in DWW iff there exist five finite
words $x,x_{2},x_{4}\in\Sigma^{*}$ and $x_{1},x_{3}\in\Sigma^{+}$, such that
$x\cdot(x_{1}+x_{2}\cdot x_{3}^{*}\cdot x_{4})^{*}\cdot
x_{1}^{\omega}\subseteq L\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\
x\cdot(x_{1}+x_{2}\cdot x_{3}^{*}\cdot x_{4})^{*}\cdot x_{2}\cdot
x_{3}^{\omega}\cap L=\emptyset.$
Recall that DWW=DBW$\cap$DCW, so one would define a DWW certificate by
disjuncting the certificates for DBW and DCW in Theorems 3.3 and 6.1. Theorem
6.4, however, suggests a different certificate, and it is interesting to
relate it to the ones for DBW and DCW. Also note that while the DBW, DCW, and
DPW certificates are covered by [50, Lemma 14], this is not the case for the
DWW certificate in Theorem 6.4.
Recall that at the bottom of the depth hierarchy we have safety and co-safety
languages, whose intersection is the set of bounded languages.
###### Theorem 6.5
An $\omega$-regular language $L$ is not a bounded language iff there exist six
finite words $\hat{x}_{0},\hat{x}_{1},\hat{x}_{2}\in\Sigma^{*}$ and
$x_{0},x_{1},x_{2}\in\Sigma^{+}$, such that
$\hat{x}_{0}\cdot x_{0}^{*}\cdot\hat{x}_{1}\cdot x_{1}^{\omega}\subseteq
L\qquad\text{ and }\qquad\hat{x}_{0}\cdot x_{0}^{*}\cdot\hat{x}_{2}\cdot
x_{2}^{\omega}\cap L=\emptyset.$
###### Proof
Assume first that $L$ is not bounded. Then, by Proposition 4, there exists a
$\langle
L_{\textnormal{acc}}^{\text{bounded}},L_{\textnormal{struct}}^{\text{bounded}}\rangle$-refuter
${\cal R}$ for it. From this refuter we can extract three lassos: a
$?$-labeled lasso from which we obtain $\hat{x}_{0}$ and $x_{0}$; a rej-
labeled lasso starting at the entry-point of the first lasso from which we
obtain $\hat{x}_{1}$ and $x_{1}$; and a acc-labeled lasso starting at the
entry-point of the first lasso from which we obtain $\hat{x}_{2}$ and $x_{2}$.
For the other direction assume by way of contradiction that there is a
deterministic $\langle
L_{\textnormal{acc}}^{\text{bounded}},L_{\textnormal{struct}}^{\text{bounded}}\rangle$-automaton
${\cal A}=\langle\Sigma,Q,q_{0},\delta,\tau,\gamma\rangle$ with $L({\cal
A})=L$. Assume that $\hat{x}_{0}\cdot x_{0}^{\omega}\in L$. Thus after reading
$|Q|$ letters one state has been repeated and by the constraint it must be
accepting. Thus $\hat{x}_{0}\cdot x_{0}^{|Q|}\cdot\hat{x}_{2}\cdot
x_{2}^{\omega}\in L$ which is a contradiction. The other case is analogous. ∎
## 7 Discussion and Directions for Future Research
The automation of decision procedures makes certification essential. We
suggest to use the winning strategy of the refuter in expressiveness games as
a certificate to inexpressibility. We show that beyond this state-based
certificate, the strategy induces a word-based certificate, generated from
words traversed along a “flower structure” the strategy contains, as well as a
language-based certificate, consisting of languages that under- and over-
approximate the language in question and that are not separable by automata in
the desired class.
While our work considers expressive power, one can use similar ideas in order
to question the size of automata needed to recognize a given language. For
example, in the case of a regular language $L$ of finite words, the Myhill-
Nerode characterization [36, 37] suggests to refute the existence of
deterministic finite word automata (DFW) with $n$ states for $L$ by providing
$n+1$ prefixes that are not right-congruent. Using our approach, one can
alternatively consider the winning strategy of Refuter in a game in which the
set of annotations includes also the state space, and
$L_{\textnormal{struct}}$ ensures consistency of the transition relation. Even
more interesting is refutation of size in the setting of automata on infinite
words. Indeed, there, minimization is NP-complete [46], and there are
interesting connections between polynomial certificates and possible
membership in co-NP, as well as connections between size of certificates and
succinctness of the different classes of automata.
Finally, while the approximation scheme we studied is based on suggested over-
and under-approximating languages, it is interesting to study approximations
that are based on more flexible distance measures [13, 18].
## References
* [1] Almagor, S., Lahijanian, M.: Explainable multi agent path finding. In: Proc. 19th International Conference on Autonomous Agents and Multiagent Systems. pp. 34–42 (2020)
* [2] Alpern, B., Schneider, F.: Recognizing safety and liveness. Distributed computing 2, 117–126 (1987)
* [3] Baier, C., de Alfaro, L., Forejt, V., Kwiatkowska, M.: Model checking probabilistic systems. In: Handbook of Model Checking., pp. 963–999. Springer (2018)
* [4] Baumeister, T., Finkbeiner, B., Torfah, H.: Explainable reactive synthesis. In: 18th Int. Symp. on Automated Technology for Verification and Analysis (2020). https://doi.org/10.1007/978-3-030-59152-6_23
* [5] Bloem, R., Chatterjee, K., Jobstmann, B.: Graph games and reactive synthesis. In: Handbook of Model Checking., pp. 921–962. Springer (2018)
* [6] Boigelot, B., Jodogne, S., Wolper, P.: On the use of weak automata for deciding linear arithmetic with integer and real variables. In: Proc. Int. Joint Conf. on Automated Reasoning. Lecture Notes in Computer Science, vol. 2083, pp. 611–625. Springer (2001)
* [7] Boker, U., Kupferman, O.: Co-ing Büchi made tight and useful. In: Proc. 24th IEEE Symp. on Logic in Computer Science. pp. 245–254 (2009)
* [8] Büchi, J.: On a decision method in restricted second order arithmetic. In: Proc. Int. Congress on Logic, Method, and Philosophy of Science. 1960. pp. 1–12. Stanford University Press (1962)
* [9] Büchi, J., Landweber, L.: Solving sequential conditions by finite-state strategies. Trans. AMS 138, 295–311 (1969)
* [10] Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. Journal of the ACM 50(5), 752–794 (2003)
* [11] Czerwinski, W., Lasota, S., Meyer, R., Muskalla, S., Kumar, K., Saivasan, P.: Regular separability of well-structured transition systems. In: Proc. 29th Int. Conf. on Concurrency Theory. LIPIcs, vol. 118, pp. 35:1–35:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
* [12] Czerwinski, W., Martens, W., Masopust, T.: Efficient separability of regular languages by subsequences and suffixes. In: Proc. 40th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7966, pp. 150–161. Springer (2013)
* [13] Dimitrova, R., Finkbeiner, B., Torfah, H.: Approximate automata for omega-regular languages. In: 17th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 11781, pp. 334–349. Springer (2019)
* [14] Eisner, C., Fisman, D.: A Practical Introduction to PSL. Springer (2006)
* [15] Emerson, E., Jutla, C.: The complexity of tree automata and logics of programs. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 328–337 (1988)
* [16] Emerson, E., Jutla, C.: Tree automata, $\mu$-calculus and determinacy. In: Proc. 32nd IEEE Symp. on Foundations of Computer Science. pp. 368–377 (1991)
* [17] Emerson, E., Lei, C.L.: Modalities for model checking: Branching time logic strikes back. Science of Computer Programming 8, 275–306 (1987)
* [18] Gange, G., Ganty, P., Stuckey, P.: Fixing the state budget: Approximation of regular languages with small dfas. In: 15th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 10482, pp. 67–83. Springer (2017)
* [19] Krishnan, S., Puri, A., Brayton, R.: Deterministic $\omega$-automata vis-a-vis deterministic Büchi automata. In: Algorithms and Computations. Lecture Notes in Computer Science, vol. 834, pp. 378–386. Springer (1994)
* [20] Kupferman, O.: Automata theory and model checking. In: Handbook of Model Checking, pp. 107–151. Springer (2018)
* [21] Kupferman, O., Morgenstern, G., Murano, A.: Typeness for $\omega$-regular automata. International Journal on the Foundations of Computer Science 17(4), 869–884 (2006)
* [22] Kupferman, O., Sheinvald-Faragy, S.: Finding shortest witnesses to the nonemptiness of automata on infinite words. In: Proc. 17th Int. Conf. on Concurrency Theory. Lecture Notes in Computer Science, vol. 4137, pp. 492–508. Springer (2006)
* [23] Kupferman, O., Vardi, M.: On bounded specifications. In: Proc. 8th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning. Lecture Notes in Computer Science, vol. 2250, pp. 24–38. Springer (2001)
* [24] Kupferman, O., Vardi, M.: From complementation to certification. Theoretical Computer Science 305, 591–606 (2005)
* [25] Kupferman, O., Vardi, M.: From linear time to branching time. ACM Transactions on Computational Logic 6(2), 273–294 (2005)
* [26] Kupferman, O., Vardi, M.: Safraless decision procedures. In: Proc. 46th IEEE Symp. on Foundations of Computer Science. pp. 531–540 (2005)
* [27] Kurshan, R.: Computer Aided Verification of Coordinating Processes. Princeton Univ. Press (1994)
* [28] Landweber, L.: Decision problems for $\omega$–automata. Mathematical Systems Theory 3, 376–384 (1969)
* [29] Leshkowitz, O., Kupferman, O.: On repetition languages. In: 45th Int. Symp. on Mathematical Foundations of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs) (2020)
* [30] Löding, C.: Methods for the transformation of automata: Complexity and connection to second order logic (1999), M.Sc. Thesis, Christian-Albrechts-University of Kiel
* [31] Löding, C.: Efficient minimization of deterministic weak $\omega$-automata. Information Processing Letters 79(3), 105–109 (2001)
* [32] McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9, 521–530 (1966)
* [33] Meyer, A., Stockmeyer, L.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proc. 13th IEEE Symp. on Switching and Automata Theory. pp. 125–129 (1972)
* [34] Mostowski, A.: Regular expressions for infinite trees and a standard form of automata. In: Computation Theory. Lecture Notes in Computer Science, vol. 208, pp. 157–168. Springer (1984)
* [35] Muller, D., Saoudi, A., Schupp, P.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Proc. 13th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 226, pp. 275 – 283. Springer (1986)
* [36] Myhill, J.: Finite automata and the representation of events. Tech. Rep. WADD TR-57-624, pages 112–137, Wright Patterson AFB, Ohio (1957)
* [37] Nerode, A.: Linear automaton transformations. Proceedings of the American Mathematical Society 9(4), 541–544 (1958)
* [38] Niwinski, D., Walukiewicz, I.: Relating hierarchies of word and tree automata. In: Proc. 15th Symp. on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1373. Springer (1998)
* [39] Perrin, D., Pin, J.E.: Infinite words - automata, semigroups, logic and games, Pure and applied mathematics series, vol. 141. Elsevier Morgan Kaufmann (2004)
* [40] Place, T., Zeitoun, M.: Separating regular languages with first-order logic. Log. Methods Comput. Sci. 12(1) (2016)
* [41] Rabin, M.: Decidability of second order theories and automata on infinite trees. Transaction of the AMS 141, 1–35 (1969)
* [42] Safra, S.: On the complexity of $\omega$-automata. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 319–327 (1988)
* [43] Safra, S.: Exponential determinization for $\omega$-automata with strong-fairness acceptance condition. In: Proc. 24th ACM Symp. on Theory of Computing (1992)
* [44] S.Almagor, Chistikov, D., Ouaknine, J., Worrell, J.: O-minimal invariants for linear loops. In: Proc. 45th Int. Colloq. on Automata, Languages, and Programming. LIPIcs, vol. 107, pp. 114:1–114:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
* [45] Schewe, S.: Büchi complementation made tight. In: Proc. 26th Symp. on Theoretical Aspects of Computer Science. LIPIcs, vol. 3, pp. 661–672. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)
* [46] Schewe, S.: Beyond Hyper-Minimisation—Minimising DBAs and DPAs is NP-Complete. In: Proc. 30th Conf. on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), vol. 8, pp. 400–411 (2010)
* [47] di Stasio, A., Murano, A., Vardi, M.: Solving parity games: Explicit vs symbolic. In: 23rd International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science, vol. 10977, pp. 159–172. Springer (2018)
* [48] Thomas, W.: Automata on infinite objects. Handbook of Theoretical Computer Science pp. 133–191 (1990)
* [49] Vardi, M., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)
* [50] Wagner, K.: On $\omega$-regular sets. Information and Control 43, 123–177 (1979)
Open Access This chapter is licensed under the terms of the Creative
CommonsAttribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide
a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the
chapter’s Creative Commons license, unless indicated otherwise in a credit
line to the material. If material is not included in the chapter’s Creative
Commons license and your intendeduse is not permitted by statutory regulation
or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.
|
11institutetext: Max-Planck-Institut für extraterrestrische Physik,
Giessenbachstrasse 1, 85748 Garching, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy,
University of Maryland, College Park, MD 20742, USA 33institutetext: Space
Telescope Science Institute, Baltimore, MD 21218, USA 44institutetext:
Department of Astronomy and the Oskar Klein Centre, Stockholm University,
AlbaNova, SE 10691 Stockholm, Sweden 55institutetext: Institute for Astronomy,
University of Hawaii at Manoa, 2680 Woodlawn Dr., Honolulu, HI 96822
66institutetext: The School of Physics and Astronomy, Tel Aviv University, Tel
Aviv 69978, Israel 77institutetext: CIFAR Azrieli Global Scholars program,
CIFAR, Toronto, Canada 88institutetext: Núcleo de Astronomía de la Facultad de
Ingeniería, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago,
Chile 99institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China 1010institutetext: Center for Cosmology and
Particle Physics, New York University, NY 10003, USA 1111institutetext: Leiden
Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
1212institutetext: The Oskar Klein Centre, Department of Physics, AlbaNova,
Stockholm University, SE 10691 Stockholm, Sweden 1313institutetext:
International Centre for Radio Astronomy Research - Curtin University, GPO Box
U1987, Perth, WA 6845, Australia 1414institutetext: INAF - Osservatorio
Astronomico di Trieste, Via G.B. Tiepolo, 11, I-34143 Trieste, Italy
1515institutetext: Department of Physics, University of California, Santa
Barbara, CA 93106-9530, USA 1616institutetext: Las Cumbres Observatory, 6740
Cortona Dr, Suite 102, Goleta, CA 93117-5575, USA 1717institutetext: Graduate
School of Science and Engineering, Saitama Univ., 255 Shimo-Okubo, Sakura-ku,
Saitama City, Saitama 338-8570, Japan 1818institutetext: DIRAC Institute,
Department of Astronomy, University of Washington, 3910 15th Avenue NE,
Seattle, WA 98195, USA 1919institutetext: Division of Physics, Mathematics,
and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
2020institutetext: IPAC, California Institute of Technology, 1200 E.
California Boulevard, Pasadena, CA 91125, USA 2121institutetext: Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
2222institutetext: Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 76100, Israel 2323institutetext: California
Institute of Technology, Pasadena, CA 91125, USA
# AT 2019avd: A novel addition to the diverse population of nuclear transients
A. Malyali 11 A. Rau 11 A. Merloni 11 K. Nandra 11 J. Buchner 11 Z. Liu 11 S.
Gezari 2233 J. Sollerman 44 B. Shappee 55 B. Trakhtenbrot 66 I. Arcavi 6677 C.
Ricci 8899 S. van Velzen 1010221111 A. Goobar 1212 S. Frederick 22 A. Kawka
1313 L. Tartaglia 441414 J. Burke 15151616 D. Hiramatsu 15151616 M. Schramm
1717 D. van der Boom 66 G. Anderson 1313 J. C. A. Miller-Jones 1313 E. Bellm
1818 A. Drake 1919 D. Duev 1919 C. Fremling 1919 M. Graham 1919 F. Masci 2020
B. Rusholme 2020 M. Soumagnac 21212222 R. Walters 2323
(Received XXX; accepted YYY)
We report on SRG/eROSITA, ZTF, ASAS-SN, Las Cumbres, NEOWISE-R, and Swift
XRT/UVOT observations of the unique ongoing event AT 2019avd, located in the
nucleus of a previously inactive galaxy at $z=0.029$. eROSITA first observed
AT 2019avd on 2020-04-28 during its first all sky survey, when it was detected
as an ultra-soft X-ray source ($kT\sim 85$ eV) that was $\gtrsim 90$ times
brighter in the $0.2-2$ keV band than a previous 3$\sigma$ upper flux
detection limit (with no archival X-ray detection at this position). The ZTF
optical light curve in the $\sim 450$ days preceding the eROSITA detection is
double peaked, and the eROSITA detection coincides with the rise of the second
peak. Follow-up optical spectroscopy shows the emergence of a Bowen
fluorescence feature and high-ionisation coronal lines ([Fe X] 6375 Å, [Fe
XIV] 5303 Å), along with persistent broad Balmer emission lines (FWHM$\sim
1400$ km s-1). Whilst the X-ray properties make AT 2019avd a promising tidal
disruption event (TDE) candidate, the optical properties are atypical for
optically selected TDEs. We discuss potential alternative origins that could
explain the observed properties of AT 2019avd, such as a stellar binary TDE
candidate, or a TDE involving a super massive black hole binary.
###### Key Words.:
keyword 1 – keyword 2 – keyword 3
## 1 Introduction
Actively accreting supermassive black holes (SMBHs) have long been known to
exhibit large amplitude flaring behaviour (e.g. Tohline & Osterbrock 1976;
Antonucci & Cohen 1983; Penston & Pérez 1984; Shappee et al. 2014; Storchi-
Bergmann et al. 2017; Frederick et al. 2019), whereby multi-epoch observations
of galaxy nuclei, over year-long timescales, have revealed drastic changes in
their luminosity. The physical mechanisms responsible for producing extreme
accretion rate changes are still unclear, although various models have been
suggested, such as state transitions in the inner disc (Noda & Done, 2018;
Ross et al., 2018), radiation pressure instabilities in the disc (Śniegowska &
Czerny, 2019), or tidal disruption events (TDEs; Merloni et al. 2015; Chan et
al. 2019).
Whilst the sample of ignition events in galactic nuclei was previously limited
to only a few objects, the advance of wide-field, high-cadence surveys over
the last decade has facilitated the discovery of an increasing number of
extreme state changes. This has resulted in tighter constraints on the
timescales of flaring events for these systems. For example, Trakhtenbrot et
al. (2019b) recently reported a new class of SMBH accretion event that sees a
large amplitude rise in the optical/UV luminosity over timescales of months.
In addition to triggering drastic changes in the accretion rate in AGNs, TDEs
can also cause quiescent black holes to transition into short-lived active
phases. In a TDE, a star that passes too close to a BH is torn apart by strong
tidal forces, with a fraction of the bound stellar debris then being accreted
onto the BH (Hills, 1975; Young et al., 1977; Gurzadian & Ozernoi, 1981; Lacy
et al., 1982; Rees, 1988; Phinney, 1989). Early TDE candidates were first
identified through detection of large-amplitude (at least a factor of 20),
ultra-soft X-ray flares (black-body temperatures between 40 and 100 eV) from
quiescent galaxies during the ROSAT survey (Bade et al., 1996; Komossa & Bade,
1999; Komossa & Greiner, 1999; Grupe & Leighly, 1999; Greiner et al., 2000).
Since then, the vast majority of TDE candidates have been optically selected,
such as through the Sloan Digital Sky Survey (SDSS; e.g. van Velzen et al.
2011; Merloni et al. 2015), the Panoramic Survey Telescope and Rapid Response
System (Pan-STARRS; e.g. Gezari et al. 2012; Holoien et al. 2019a), the
Palomar Transient Factory (PTF; e.g. Arcavi et al. 2014), the Intermediate
Palomar Transient Factory (iPTF; e.g. Blagorodnova et al. 2017; Hung et al.
2017), the All Sky Automated Survey for SuperNovae (ASAS-SN; e.g. Holoien et
al. 2014, 2016; Wevers et al. 2019; Holoien et al. 2019b), and the Zwicky
Transient Facility (ZTF; e.g. van Velzen et al. 2019, 2020). Optically
selected TDEs are characterised as blue nuclear transients with light curves
showing longer/ shorter rise and decay timescales relative to supernovae
(SNe)/ AGN111For large, well-defined AGN flares similar to those seen in
Frederick et al. (2019), as opposed to stochastic AGN variability., and a
relatively smooth power-law decline. Optical spectroscopic follow-up of these
events post-peak reveals blue continua (blackbody temperatures $\sim 10^{4}$K)
with various broad emission lines (full width at half maximum, FWHM $\lesssim
10^{4}$ km s-1); a recent characterisation of the different TDE spectroscopic
classes was presented by van Velzen et al. (2020). Although a number of TDE
candidates have also been found through UV selection (GALEX, Gezari et al.
2008, 2009), and X-ray selection (XMM-Newton Slew, Esquej et al. 2007, 2008;
Saxton et al. 2012, 2017), most of our understanding of TDEs is currently
biased towards this set of observed properties of optically-selected TDEs.
Whilst most previous TDE searches focused on identifying TDEs in quiescent
galaxies, an increasing number of candidates for TDEs in AGNs are being
proposed in the literature (Merloni et al., 2015; Blanchard et al., 2017;
Trakhtenbrot et al., 2019a; Liu et al., 2020; Ricci et al., 2020). In certain
cases, the distinction between TDE and non-TDE-induced SMBH accretion state
changes is becoming increasingly blurred (see also Neustadt et al. 2020).
Variants of TDEs have also been proposed to explain more exotic phenomena,
such as the recently observed quasi-periodic eruptions (QPEs) in a few
galactic nuclei (Miniutti et al., 2019; Giustini et al., 2020; King, 2020),
and periodic flaring seen in an AGN (Payne et al., 2020). Other origins for
extreme nuclear transients involve SNe in the AGN accretion disc (Rozyczka et
al., 1995), or interaction of SMBH binaries (SMBHB) with an accretion disc
(Kim et al., 2018). It is clear that such different physical origins may
result in a diverse range of observed variability behaviours.
In this paper, we report on the ongoing extreme event AT 2019avd, which is a
novel addition to the already diverse population of nuclear transients. AT
2019avd is associated to the previously inactive galaxy 2MASX
J08233674+0423027 at $z=0.029$ (see Fig. 1), and was first reported as
ZTF19aaiqmgl at the Transient Name Server (TNS222https://wis-
tns.weizmann.ac.il/) following its discovery by ZTF on 2019-02-09 UT333all
dates in this paper will be reported in UT format. (Nordin et al., 2019). The
transient was independently detected more than a year later on 2020-04-28 as a
new ultra-soft nuclear X-ray source (Malyali et al., 2020) during the first
all-sky survey of the eROSITA instrument (Predehl et al., in press) on-board
the Russian/German Spectrum-Roentgen-Gamma (SRG) mission.
Figure 1: Pan-STARRS $g$-band image centred on the host galaxy of AT 2019avd.
The dark orange star and red circle mark the ZTF position and eROSITA
localisation respectively, where the radius of the circle is set to the 2′′
uncertainty on the eROSITA source position.
This work presents X-ray (SRG/eROSITA, Swift/XRT), optical/UV/mid-infrared
(MIR) photometric (ZTF, ASAS-SN, NEOWISE-R, Swift/UVOT), and optical
spectroscopic (NOT/ALFOSC, Las Cumbres Floyds, ANU/WiFeS) observations of AT
2019avd. In Section 2, we report our X-ray observations and analysis of AT
2019avd, whilst the photometric evolution and host galaxy properties are
presented in Section 3. We then present details of our optical spectroscopic
follow-up campaign in Section 4, before discussing possible origins for AT
2019avd in Section 5, and conclude in Section 6. We adopt a flat $\Lambda$CDM
cosmology throughout this paper, with
$H_{0}=67.7\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$,
$\Omega_{\mathrm{m}}=0.309$ (Planck Collaboration XIII, 2016); $z=0.029$ thus
corresponds to a luminosity distance of 130 Mpc. All magnitudes will be
reported in the AB system, unless otherwise stated.
## 2 X-ray observations
### 2.1 eROSITA discovery
AT 2019avd was discovered in a dedicated search for candidate TDEs in the
first eROSITA all-sky survey (eRASS1). Here, the eROSITA source catalogue
(version 945 of the source detection pipeline of the eROSITA Science Analysis
Software, eSASS, Brunner et al. in prep.) was systematically examined for new
soft X-ray sources associated with the nuclei of galaxies that showed no prior
indication of being an AGN.
The eROSITA data for AT 2019avd are composed of four consecutive scans with
gaps of 4 hr each and a midtime of 2020-04-28. The total on-source exposure
amounts to 140 s (see Table 1). The source was localised to (RAJ2000,
Dec${}_{\mathrm{J2000}})$=(08h23m37s, 04∘23′03′′), with a 1$\sigma$ positional
uncertainty of 2′′, which is consistent with the nucleus of the galaxy 2MASX
J08233674+0423027.
Photons were extracted using the eSASS task SRCTOOL (version 945) choosing a
circular aperture of radius 36′′ centred on the above position (84 counts were
detected within this region). Background counts were selected from a circular
annulus of inner and outer radii 72′′ and 144′′, respectively. Using the best-
fit spectral model (see Section 2.3), we derived a $0.2-2$ keV flux of
(1.4$\pm$0.2)$\times 10^{-12}$ erg cm-2s-1 (1$\sigma$).
No X-ray source has previously been detected at the location of AT 2019avd.
Using both the Upper Limit
Server444http://xmmuls.esac.esa.int/upperlimitserver/ and
webPIMMS555https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl, and
assuming an absorbed black-body spectral model with $kT=80$ eV, and Galactic
neutral hydrogen column density (see also Section 2.3),
$N_{\mathrm{H}}=2.42\times 10^{20}$ cm-2, we infer an $0.2-2$ keV 3$\sigma$
upper limit of 1.7$\times 10^{-14}$ erg cm-2s-1 for a serendipitous 7 ks XMM-
Newton pointed observation obtained on 2015-04-08666XMM-Newton
OBSID=0741580501. Earlier constraints can be derived from ROSAT observations
obtained on 1990-10-14, 1996-11-13, and 1997-04-11 with 3$\sigma$ upper limits
of 4.2$\times 10^{-13}$, 4.0$\times 10^{-13}$ , and 1.2$\times 10^{-13}$ erg
cm-2s-1, respectively.
eROSITA thus first observed AT 2019avd in a state where it had brightened by
at least a factor of 90 in the $0.2-2$ keV band relative to the deepest
archival X-ray observation (luminosity history presented in Fig. 2).
Figure 2: Long-term X-ray light curve in the 0.2–2 keV energy band of AT
2019avd up until the first eROSITA observation. Triangles denote 3$\sigma$
upper limits for ROSAT/PSPC and XMM-Newton/EPIC-pn, whilst the black circle
marks the SRG/eROSITA discovery, where AT 2019avd is at least 90 times
brighter than the XMM-Newton 3$\sigma$ upper limit. The error bar on the
eROSITA marker encloses the 68% credible region on the observed luminosity.
### 2.2 Swift follow-up
Triggered by the eROSITA detection, a series of follow-up observations were
performed with the Neil Gehrels Swift Observatory (P.I.s: A. Malyali & B.
Trakhtenbrot). Observations were obtained roughly every 7 days, until the
source was no longer visible due to Sun angle constraints; a further Swift
observation was then obtained $\sim 3$ months later. A log of the observations
can be found in Table 1. The XRT observations were performed in photon
counting mode. The data were reduced using the xrtpipeline task included in
version 6.25 of the heasoft package. Spectra for each of the five epochs were
extracted using the xrtproducts task. Source counts were extracted from a
circular aperture of radius 47′′ and background counts extracted from a
circular annulus of inner and outer radii 70′′ and 250′′,
respectively777eROSITA and XRT have different PSFs and instrument backgrounds,
thus the radii of the extraction regions were chosen based on each instrument
and differ here..
Table 1: Log of SRG/eROSITA and Swift/XRT observations of AT 2019avd until 2020-09-16. For eROSITA, the mid-date of the coverage in eRASS1 is given. Date | MJD | Telescope | ObsID | Exp. [s]
---|---|---|---|---
2020-04-28 | 58967.7 | SRG/eROSITA | - | 140
2020-05-13 | 58982.4 | Swift/XRT | 00013495001 | 1617
2020-05-19 | 58988.3 | Swift/XRT | 00013495002 | 1966
2020-05-25 | 58994.0 | Swift/XRT | 00013495003 | 1982
2020-06-03 | 59003.3 | Swift/XRT | 00013495004 | 494
2020-06-10 | 59010.6 | Swift/XRT | 00013495005 | 1739
2020-09-16 | 59108.4 | Swift/XRT | 00013495006 | 2967
Observations with the Ultraviolet and Optical Telescope (UVOT; Roming et al.
2005) were obtained simultaneously with the XRT observations. Imaging was
performed at three epochs (00013495001, ..004, ..005) using the UVW1 filter
with exposures of 1.36, 1.95, and 1.93 ks, respectively. The remaining three
observations utilised all six UVOT filters (UVW2, UVM2, UVW1, U, B, V) with
accordingly shorter exposure times.
The UVOT flux was extracted with the uvotsource task using a
$9^{\prime\prime}$ radius aperture centred on the optical position of AT
2019avd, whilst a nearby circular region with $15^{\prime\prime}$ radius was
used for background subtraction. The photometry was extracted from each unique
Swift observation ID, and is presented in Table 2 (we note that this
photometry includes both AGN and host galaxy emission in order to be
consistent with the SED fitting in Section 3.3). Relative to UV photometry
obtained prior to the initial optical outburst (see Section 3.3 and Fig. 7),
AT 2019avd has brightened by $\sim 1$ mag in the UVW1, UVM2, and UVW2 bands,
and brightens only by $\sim 0.1-0.2$ mag over Swift observations between
2020-05-13 and 2020-09-16.
Table 2: Swift UV photometry (corrected for Galactic extinction using the UVOT correction factors in Table 5 of Kataoka et al. 2008). The model magnitudes (for the host galaxy) were obtained by convolving the best-fit SED model (Section 3.3) with the UVOT transmission curves. A hyphen denotes that the given filter was not used on that observation date. Date | UVW1 | UVM2 | UVW2
---|---|---|---
Model | 18.88 | 19.16 | 19.26
2020-05-13 | $18.01\pm 0.04$ | - | -
2020-05-19 | $18.23\pm 0.15$ | $18.28\pm 0.11$ | $18.27\pm 0.10$
2020-05-25 | $17.85\pm 0.07$ | $18.30\pm 0.07$ | $18.31\pm 0.06$
2020-06-03 | $17.89\pm 0.04$ | - | -
2020-06-10 | $17.80\pm 0.04$ | - | -
2020-09-16 | $17.78\pm 0.05$ | $18.17\pm 0.06$ | $18.23\pm 0.05$
### 2.3 X-ray spectral fitting
X-ray spectra were analysed using the Bayesian X-ray Analysis software (BXA,
Buchner et al. 2014), which connects the nested sampling algorithm MultiNest
(Feroz & Hobson, 2008) with the fitting environment CIAO/Sherpa (Freeman et
al., 2001) and XSPEC (Arnaud, 1996). The spectra were fitted unbinned using
the C-statistic (Cash, 1976), and the eROSITA and XRT backgrounds were both
modelled using the principal component analysis (PCA) technique described in
Simmonds et al. (2018). For each set of eROSITA and XRT spectra, a joint fit
on both the source and background spectra was run. Two different models for
the source spectra were used: (i) an absorbed black body (tbabs*blackbody),
and (ii) an absorbed power law (tbabs*powerlaw). The equivalent Galactic
neutral hydrogen column density, $N_{\mathrm{H}}$, was allowed to vary by 20%
from its tabulated value in the HI4PI survey of $2.42\times 10^{20}$ cm-2
(HI4PI Collaboration et al., 2016) during fitting. The complete set of priors
adopted under each model is listed in Table 3, whilst an example of the BXA
fit to the eROSITA spectrum is shown in Fig. 3, and spectral fit results are
presented in Table 4.
Table 3: Summary of priors adopted in the BXA analysis of the eROSITA and XRT spectra. For each fit, a log-uniform prior on $N_{\mathrm{H}}$ between $(0.8N_{\mathrm{H}},1.2N_{\mathrm{H}})$ was defined, where $N_{\mathrm{H}}=2.42\times 10^{20}$ cm-2 (see Section 2.3). $\Gamma$ denotes the slope of a power law, $kT$ the black-body temperature, $A$ the normalisation. The prior over $A$ is in units 1.05$\times 10^{-6}\mathrm{erg}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$. Model | Priors
---|---
tbabs*bbody | $\log[kT/\mathrm{keV}]\sim\mathcal{U}(-2,1)$, $\log[A]\sim\mathcal{U}(-10,10)$
tbabs*powerlaw | $\Gamma\sim\mathcal{U}(0,8)$, $\log[A]\sim\mathcal{U}(-10,10)$
Figure 3: BXA fit to the eROSITA eRASS1 spectrum. Black markers are the binned observed data, whilst the red represents the fitted convolved model for tbabs*blackbody (darker and light red bands enclose the 68 % and 95 % posterior uncertainty on the model at each energy). Both the black-body and power-law fits to the (low count) eRASS1 spectrum suggest that the source is ultra-soft (see Table 4). Table 4: X-ray spectral fit results from applying BXA to the extracted eROSITA and XRT spectra, with uncertainties enclosing 68% of the posterior for each parameter. $F_{\rm 0.2-2keV}$ is the inferred observed (unabsorbed) flux under each model. OBSID | tbabs*blackbody | tbabs*powerlaw |
---|---|---|---
| NH | kT | $F_{\rm 0.2-2keV}$ | NH | $\Gamma$ | $F_{\rm 0.2-2keV}$ |
| [$\times 10^{20}$cm-2] | [eV] | [$\times 10^{-12}$ erg cm-2 s-1] | [$\times 10^{20}$cm-2] | | [$\times 10^{-12}$ erg cm-2 s-1] |
eRASS1 | $2.3_{\rm-0.3}^{\rm+0.3}$ | $85_{\rm-5}^{\rm+6}$ | $1.4_{\rm-0.2}^{\rm+0.2}$ | $2.5_{\rm-0.3}^{\rm+0.3}$ | $4.2_{\rm-0.3}^{\rm+0.3}$ | $1.6_{\rm-0.2}^{\rm+0.2}$ |
00013495001 | $2.4_{\rm-0.3}^{\rm+0.4}$ | $72_{\rm-8}^{\rm+8}$ | $1.4_{\rm-0.2}^{\rm+0.2}$ | $2.4_{\rm-0.3}^{\rm+0.3}$ | $5.3_{\rm-0.4}^{\rm+0.4}$ | $2.5_{\rm-0.5}^{\rm+0.5}$ |
00013495002 | $2.4_{\rm-0.3}^{\rm+0.3}$ | $83_{\rm-11}^{\rm+12}$ | $1.4_{\rm-0.4}^{\rm+0.4}$ | $2.4_{\rm-0.3}^{\rm+0.3}$ | $5.2_{\rm-0.6}^{\rm+0.7}$ | $2.6_{\rm-0.8}^{\rm+0.8}$ |
00013495003 | $2.4_{\rm-0.3}^{\rm+0.3}$ | $132_{\rm-10}^{\rm+10}$ | $1.0_{\rm-0.1}^{\rm+0.1}$ | $2.5_{\rm-0.3}^{\rm+0.3}$ | $3.7_{\rm-0.3}^{\rm+0.2}$ | $1.4_{\rm-0.2}^{\rm+0.2}$ |
00013495004 | $2.4_{\rm-0.3}^{\rm+0.3}$ | $107_{\rm-10}^{\rm+10}$ | $1.0_{\rm-0.2}^{\rm+0.2}$ | $2.4_{\rm-0.3}^{\rm+0.3}$ | $4.2_{\rm-0.3}^{\rm+0.3}$ | $1.6_{\rm-0.3}^{\rm+0.3}$ |
00013495005 | $2.4_{\rm-0.3}^{\rm+0.3}$ | $91_{\rm-6}^{\rm+6}$ | $1.5_{\rm-0.2}^{\rm+0.2}$ | $2.5_{\rm-0.3}^{\rm+0.3}$ | $4.9_{\rm-0.3}^{\rm+0.3}$ | $2.6_{\rm-0.4}^{\rm+0.4}$ |
00013495006 | $2.4_{\rm-0.3}^{\rm+0.3}$ | $115_{\rm-3}^{\rm+3}$ | $9.7_{\rm-0.4}^{\rm+0.4}$ | $2.8_{\rm-0.1}^{\rm+0.1}$ | $4.1_{\rm-0.1}^{\rm+0.1}$ | $14.0_{\rm-0.7}^{\rm+0.7}$ |
Over the course of the six weeks following the initial eROSITA detection,
there was no major variability in the $0.2-2$ keV flux between the eROSITA and
XRT observations (Table 4 and Fig. 4). However, the $0.2-2$ keV flux in the
last Swift epoch increased by a factor of about six relative to the previous
observation.
AT 2019avd remained in an ultra-soft state during the Swift monitoring
campaign, although there is variability in the inferred black-body
temperatures ($kT$ ranges between minimum and maximum values of 72$\pm 8$ eV
and 132$\pm 10$ eV, respectively). The inferred black-body temperatures are
similar to those measured in the X-ray emission of previously observed thermal
TDEs ($45\lesssim kT\lesssim 130$ eV, e.g. van Velzen et al. 2020), and are
also consistent with the temperatures of the soft excess shown in AGN (e.g.
Table A1 in Gliozzi & Williams 2020).
Figure 4: X-ray evolution of AT 2019avd. The empty and filled black markers
represent the eROSITA and XRT observations respectively; error bars enclose
95% of the posterior.
## 3 Photometric evolution and host galaxy properties
### 3.1 Optical evolution
The region around the position of AT 2019avd has been monitored by ZTF (Bellm
et al., 2019; Graham et al., 2019) in the $r$ and $g$ bands from 2019-01-12
until the time of writing. On 2019-02-09 (over a year before the eROSITA
detection), ZTF first detected the transient ZTF19aaiqmgl with an inferred
separation from the galaxy centre of 0$\aas@@fstack{\prime\prime}$04
888https://lasair.roe.ac.uk/object/ZTF19aaiqmgl/, and $r$-band magnitude
$17.64\pm 0.07$ (reference subtracted, Fig. 1).
For MJD$<$58855 (2020-01-07), we obtained a forced photometry ZTF light curve
for AT 2019avd (Masci et al., 2019). For MJD$>$58855, we downloaded the ZTF
light curve of AT 2019avd using the Lasair alert broker (Smith et al., 2019),
which processes and reports to the community on transients detected within the
large ZTF data streams. Both of these light curves are constructed from PSF-
fit photometry measurements run on ZTF difference images. We also obtained
additional photometric observations with the Spectral Energy Distribution
Machine (SEDM; Blagorodnova et al. 2018) on the Palomar 60-inch telescope. The
SEDM photometry was host-subtracted using SDSS reference images, as described
in Fremling et al. (2016). These two light curves, and the host-subtracted
SEDM photometry, were then combined for subsequent analysis, and are shown in
Fig. 5.
After the initial detection on 2019-02-09, AT 2019avd continued to brighten
until reaching its maximum observed brightness of $r\sim 16.8$ mag on
2019-02-20. Between 2019-02-24 and 2020-01-01, the $g$-band magnitude of the
host nucleus decayed nearly monotonically from 17.13$\pm$0.09 mag to
20.08$\pm$0.20 mag, followed by a re-brightening to 18.58$\pm$0.13 mag on
2020-05-03. The late time SEDM photometry around 2020-09-19 revealed a further
brightening to $r$ and $g$-band magnitudes of $\sim 17.6$ mag and $\sim 18.4$
mag respectively. The first eROSITA observation occurred during the rise of
the second major peak of the ZTF light curve (Fig. 5).
The location of AT 2019avd has also been monitored in the $V$-band by ASAS-SN
(Shappee et al., 2014; Kochanek et al., 2017) from February 2012 to November
2018, and in the $g$-band from October 2017 to September 2020 (the time of
writing). No major optical outbursts were seen in the ASAS-SN light curve
prior to the ZTF detection (Fig. 15); given the joint ASAS-SN and ZTF light
curves, it is likely that the system ‘ignited’ around MJD $=58510$
(2019-01-27).
Figure 5: NEOWISE-R (non-host subtracted, top) and ZTF/ SEDM (middle) light
curves of AT 2019avd, with the immediate $0.2-2$ keV X-ray history shown in
the bottom panel. The eROSITA eRASS1 detection and the Swift observation from
2020-09-16 are the empty and filled black markers, respectively. The solid
grey vertical line marks the MJD of the eRASS1 observation, whilst grey dashed
lines mark the times of the NOT and the first FLOYDS spectrum (Table 5). No
significant variability before the initial 2019 outburst is observed in the
host nucleus of AT 2019avd with archival NEOWISE-R and ASAS-SN observations
(Fig. 15). The NEOWISE-R observations pre-outburst are observed with mean W1,
W2 marked out in the top panel by the cream and orange dashed lines
respectively. For plotting clarity, we omit the high-cadence ZTF Partnership
observations obtained between MJD 58820 and 58860, and we rebin the $\sim 3$
SEDM observations in each filter into a single data point.
#### 3.1.1 Rise and decay timescales in the light curve
In the following, we fit the light-curve model presented in equation 1 of van
Velzen et al. (2019), which models the rise with a half-Gaussian function, and
an exponential function for the decay, to the first and second peaks of the
ZTF light curve, using
UltraNest999https://github.com/JohannesBuchner/UltraNest (Buchner, 2016, 2019)
as our sampler. Whilst such a model is not physically motivated, it enables a
comparison of the timescales involved in the light curve of AT 2019avd with
those of the population of ZTF nuclear transients presented in van Velzen et
al. (2019).
While fitting the first peak, we first filter out observations outside of the
MJD period between 58450 and 58650, and we then run a joint fit of the $g$ and
$r$ band observations in flux space. Our model has seven free parameters,
defined following the notation of van Velzen et al. (2019): $\sigma_{r}$ and
$\sigma_{g}$, the rise timescale of the light curve in the $r$ and $g$ bands
respectively; $\tau_{r}$ and $\tau_{g}$, the decay timescale of the light
curve in $r$ and $g$ bands; $F_{\mathrm{peak},r}$ and $F_{\mathrm{peak},g}$,
the peak flux in $r$ and $g$ bands; $t_{\mathrm{peak}}$, the time of the peak
of the light curve (to enable a comparison with van Velzen et al. 2019, we
assume that the light-curve model peaks at the same time in both of these
bands). For the second peak, we filter out observations outside of the MJD
period 58840 and 59115 (the late-time SEDM datapoints are used in the
fitting), and because we do not sample the decay of this peak, we only model
the rise here. The model for the second peak has five free parameters, with
$\tau_{r}$ and $\tau_{g}$ now being omitted. We list our priors in Table 9,
and present the fits in Fig. 6.
From the posterior means, we infer $\sigma_{r}=7.9\pm 0.3$, $\sigma_{g}=7.2\pm
0.2$, $\tau_{r}=58.2\pm 0.5$ and $\tau_{g}=39.8\pm 0.4$ days for the first
optical peak (68% credible intervals). Whilst the rise timescales in each
filter are consistent with each other to within 2$\sigma$, the decay
timescales in each filter significantly differ. With $\tau_{r}>\tau_{g}$, the
first peak shows a potential cooling signature during its decay phase,
although we are unable to constrain the temperature evolution during this
because of a lack of contemporaneous observations in other wavelength bands.
Relative to the population of nuclear transients in van Velzen et al. (2019),
one sees that these are short rise and decay timescales relative to those of
AGN flares, and are thus more similar to those in the van Velzen et al. (2019)
sample of TDEs and SNe (Fig. 6). As expected from Fig. 5, the inferred rise
times for the second peak are longer and more AGN-like, with $\tau_{r}\sim 88$
days and $\tau_{g}\sim 93$ days.
Figure 6: AT 2019avd variability compared with previously classified ZTF
nuclear transients (non-AT 2019avd data presented originally in van Velzen et
al. 2020), with red and green stars computed from the fitted model components
for each respective filter. The red and green vertical lines mark the
e-folding rise time of the second optical peak in the $r$ and $g$ bands,
respectively. We also plot the rise and decay e-fold timescales inferred from
the ASAS-SN V-band light curve of the nuclear transient AT 2017bgt
(Trakhtenbrot et al. 2019b; see also Section 5.1) with a black marker. Not
only is the double-peaked light curve of AT 2019avd clearly distinct from the
other light curves of sources in the AT 2017bgt nuclear transient class, but
the first peak of AT 2019avd decays much faster than the AT 2017bgt flare,
whilst the second peak rises much slower than the AT 2017bgt flare.
### 3.2 Mid-infrared variability
The location of AT 2019avd was observed in the $W1$ (3.4 $\mu$m) and $W2$ (4.6
$\mu$m) bands by the Wide-Field Infrared Survey Explorer mission (WISE, Wright
et al. 2010) in 2010, Near-Earth Object WISE (NEOWISE; Mainzer et al. 2011) in
late 2010 and 2011, and from December 2013 until now, twice per year as part
of the NEOWISE reactivation mission (NEOWISE-R; Mainzer et al. 2014). The
NEOWISE-R light curve was obtained from the NASA/IPAC Infrared Science
Archive101010https://irsa.ipac.caltech.edu/frontpage/ by compiling all source
detections within 5′′ of the ZTF transient position. Individual flux
measurements were rebinned to one data point per NEOWISE-R all-sky scan (using
a weighted mean) and converted into magnitudes. The resulting light curve is
shown in Fig 5.
The MIR light curve was observed to be flat prior to the initial ZTF outburst,
but showed significant brightening in the first NEOWISE-R epoch obtained
thereafter. Observations obtained $\leavevmode\nobreak\ \sim$ 6 months later
found the source to still be in the bright state despite having faded by $\sim
3$ mag in the optical. The MIR brightening was also accompanied by a
significant reddening, evolving from $W1-W2\sim 0.08$ mag in AllWISE, to a
more AGN-like $W1-W2\sim 0.6$ mag during flaring. The $W1-W2$ colour before
the outburst is much lower than the suggested cuts ($W1-W2\gtrsim 0.7$ mag)
for identifying AGNs in previous MIR classification schemes (Stern et al.,
2012; Assef et al., 2013, 2018), further supporting the hypothesis that there
was no strong recent AGN activity in AT 2019avd at that time (although the use
of WISE colours for selecting AGNs is less effective at lower AGN
luminosities; see discussion in Padovani et al. 2017).
### 3.3 Host-galaxy properties
The spectral energy distribution (SED) of the host galaxy of AT 2019avd was
compiled from archival111111‘Archival’ is defined here by photometry taken
prior to the initial ZTF optical outburst. UV to MIR photometry from GALEX
(FUV, NUV), SDSS DR12 ($g$, $r$, $i$, $z$), UKIDSS ($y$, $J$, $H$, $K$), and
AllWISE (W1, W2). The SED was modelled using CIGALE (Burgarella et al., 2005;
Boquien et al., 2019), which allows the estimation of the physical parameters
of a galaxy by fitting composite stellar populations combined with recipes
describing the star formation history and attenuation. The best-fitting model
(see Fig. 7) is that of a galaxy with a stellar mass of $(1.6\pm 0.8)\times
10^{10}M_{\odot}$, a star formation rate (SFR) of $0.17\pm 0.05$
$M_{\odot}$yr-1, and little attenuation, $\mathrm{E(B-V)}=0.03\pm 0.02$ mag,
which experienced a burst of star formation $3.7\pm 0.2$ Gyr ago. The inferred
stellar mass and SFR place the host galaxy of AT 2019avd in the ‘green valley’
between the star-forming main sequence and quenched elliptical galaxies
(adopting the green valley definition presented in Law-Smith et al. 2017).
The SED fit suggests that the host galaxy did not show strong signs of nuclear
activity prior to the detection of AT 2019avd. This is further supported by
the absence of a radio counterpart in the FIRST catalogue (Becker et al.,
1995) within 30′′ of AT 2019avd, with a catalogue upper detection limit at
this position of 0.96 mJy/beam121212http://sundog.stsci.edu/cgi-
bin/searchfirst..
Figure 7: Spectral energy distribution of the host galaxy of AT 2019avd
compiled from archival GALEX, SDSS, UKIDSS, and ALLWISE photometry, with the
best-fit model shown as a red solid line. The three epochs of Swift UVOT
photometry where all filters were used are also plotted. AT 2019avd shows a
$\sim 1$ mag rise in the UVW1, UVM2, and UVW1 bands relative to the best fit
model to the archival photometry.
## 4 Optical spectral analysis
### 4.1 Spectroscopic observations
On 2019-03-15, $\sim$33 days after the first observed peak in the ZTF light
curve, an optical spectrum of AT 2019avd was obtained by Gezari et al. (2020)
with the Alhambra Faint Object Spectrograph and Camera
(ALFOSC)131313http://www.not.iac.es/instruments/alfosc on the 2.56 m Nordic
Optical Telescope (NOT). The spectrum was obtained with a
1$\aas@@fstack{\prime\prime}$0 wide slit, grism #4 (covering the wavelength
region from 3650-9200 Å), and the slit was positioned along the parallactic
angle at the beginning of the 1800s exposure. Reductions were performed in a
standard way using mainly iraf based software, including bias corrections,
flat fielding, wavelength calibration using HeNe arc lamps imaged immediately
after the target and flux calibrations using observations of a
spectrophotometric standard star.
No further spectra were taken until after eROSITA had detected the large-
amplitude soft-X-ray flare from AT 2019avd in late April 2020, which triggered
a further five epochs of spectroscopy (dates listed in Table 5) using the
FLOYDS spectrographs (Brown et al., 2013) mounted on the Las Cumbres
Observatory 2m telescopes at Haleakala, Hawaii, and Siding Spring, Australia.
Each spectrum was taken with a 3.6ks exposure, using the ‘red/blu’ grism and a
slit width of 2′′. The spectra were reduced using PyRAF tasks as described in
Valenti et al. (2014). FLOYDS covers the entire 3500-10000 Å range in a single
exposure by capturing two spectral orders (one red and one blue)
simultaneously, yielding $R\sim 400$. The different orders are usually merged
into a single spectrum using the region between 4900 and 5700 Å, which is
present in both the red and blue orders. However, in this case, in order to
avoid erroneous wavelength shifts at the blue edge of the red order (where
there are fewer arclines), all FLOYDS spectra were merged using a reduced
stitching region of 5400 to 5500 Å141414The most extreme arcline used to
calibrate each order is at $\sim$5460 Å.. This stitching was done manually in
Python, by replacing fluxes in that wavelength range with an average of the
linear interpolations of the two orders.
In addition, a higher resolution spectrum (R$\sim$3000) was obtained on
2020-05-29 with the Wide Field Spectrograph (WiFeS; Dopita et al., 2007, 2010)
mounted on the 2.3m ANU telescope at Siding Spring Observatory. We employed
the R3000 and B3000 gratings, and obtained an arc lamp exposure after each
target exposure. The total spectral range from the two gratings is 3500 to
7000 Å. The data were reduced using the PyWiFeS reduction pipeline (Childress
et al., 2014), which produces three-dimensional data (data cubes). These
spectra are bias subtracted, flat-fielded, wavelength and flux calibrated, and
corrected for telluric absorption. We then extracted the spectra from the
slitlets that captured AT 2019avd using the IRAF (Tody, 1986) task apall which
allowed for background subtraction.
A comparison of the NOT and WiFeS spectra is presented in Fig. 8, and the
spectral evolution in the FLOYDS spectra is shown in Fig. 9. A log of the
spectroscopic observations of AT 2019avd is presented in Table 5. We note that
we have not found any archival optical spectra of the host galaxy that were
obtained prior to the initial 2019 outburst discovered by ZTF.
Table 5: Spectroscopic observations of AT 2019avd. UT Date | Tel. | Instrument | Exp. [ks] | Airmass
---|---|---|---|---
2019-03-15 | NOT | ALFOSC | 1.8 | 1.5
2020-05-10 | FTS | FLOYDS-S | 3.6 | 1.4
2020-05-12 | FTS | FLOYDS-S | 3.6 | 1.6
2020-05-18 | FTN | FLOYDS-N | 3.6 | 1.6
2020-05-29 | ANU | WiFeS | 1.8 | 1.5
2020-05-31 | FTS | FLOYDS-S | 3.6 | 1.7
2020-06-06 | FTS | FLOYDS-S | 3.6 | 1.9
Figure 8: Comparison of NOT and WiFeS spectra (black and blue respectively).
The top panel shows the wavelength range 3800-5450 Å, while the bottom panel
shows the 5600-6800 Å range. The most notable changes are (a) the emergence of
the broad emission feature around rest-frame wavelength 4686 Å and (b) an
increase in intensity of the high-ionisation coronal Fe lines ($\sim$5300 and
6370 Å). The WiFeS spectrum is of much higher resolution relative to the NOT
spectrum, and therefore is able to better resolve narrow emission lines, such
as the [S II] doublet at 6716 and 6731 Å. Neither are shown corrected for
Galactic extinction. The NOT spectrum was normalised by its continuum flux in
the 5100-5200 Å range (rest frame), whilst the blue and red arms of the WiFeS
spectra were normalised in the 5100-5200 Å and 6400-6450 Å ranges respectively
(rest frame).
Figure 9: Evolution of the Bowen+H$\beta$ (top) and H$\alpha$ (bottom) Balmer
emission lines observed through the five epochs of FLOYDS spectroscopy. Grey
dashed lines match those in Fig. 8. Epochs 2020-05-31 and 2020-06-06 were of
low S/N in the blue wavelength range, and thus are omitted from the plot here.
The minor evolution of the H$\alpha$ peak position over the FLOYDS spectra was
deemed to be most likely due to aperture-related effects during observations.
### 4.2 Summary of the main observed features of the optical spectra
The NOT spectrum from 2019-03-15 appears similar to broad line AGN spectra,
showing a relatively flat continuum (in terms of $F_{\lambda}$) and broad
Balmer emission lines (H$\alpha$, H$\beta$, H$\gamma$, H$\delta$; Fig. 8).
However, the strong Fe II complex that is frequently seen in some AGNs is not
present. The H$\alpha$ profile is asymmetric due to the blending of unresolved
H$\alpha$ and narrow [N II] 6549, 6583 Å lines, whilst the asymmetry of the
H$\gamma$ line is likely due to blending of H$\gamma$ and [O III] 4363 Å
emission. The other notable features are the [S II] doublet at 6717 and 6731 Å
(again blended, but later resolved in the WiFeS spectrum), and the weak He I
emission at 5876 Å. As no archival spectrum of the host galaxy is available,
we are unable to judge whether or not the main observed emission features
appeared at the onset of the extreme optical variability. The WiFeS spectrum
from 2020-05-29 (Fig. 8) shows the same emission features as the NOT spectrum,
with the addition of a broad emission feature around 4680 Å and an apparent
increase in intensity of a set of high-ionisation coronal lines ([Fe XIV] 5303
Å and [Fe X] 6375 Å, with ionisation potentials of 392 and 262eV
respectively). We assume that the [Fe X] is not blended with the [O I] 6364 Å
emission feature, because the latter is expected to be a third of the
intensity of the [O I] 6300 Å emission (e.g. Pelat et al. 1987), which is not
detected.
The FLOYDS spectra (Fig. 9) show no major evolution in the Balmer emission
line profiles, and show the broad emission feature around 4680 Å from
2020-05-10 (for epochs with sufficiently high S/N ratios in the blue
wavelength range), which was reported to the TNS (and first identified) in
Trakhtenbrot et al. (2020).
### 4.3 Optical spectrum modelling
For the two higher resolution spectra (NOT and WiFeS), the region around the
main observed emission lines is fitted separately
(H$\gamma,4240{\AA}<\lambda<4440$Å; He II, $4500{\AA}<\lambda<4800{\AA}$;
H$\beta,4700{\AA}<\lambda<5000{\AA}$; H$\alpha,6364$Å $<\lambda<6764{\AA}$; [S
II] doublet, $6650{\AA}<\lambda<6800{\AA}$; and $\pm 100$Å of the line centre
for [O III] 5007 Å, [Fe X] 6375 Å). Each emission line complex is modelled
with multiple Gaussians (an overview of these is presented in Table 10), and
each complex is fitted independently of the others. For all spectral fits, we
assume a flat continuum component during the fitting process, and run our
model fitting using the region slice sampler option within UltraNest. Spectral
fits for the NOT and WiFeS spectra are shown in Figs. 10 and 11, whilst the
spectral fit results are listed in Tables 6, 7, and 8.
Figure 10: Zoomed-in plots of the main emission lines observed in both the NOT and WiFeS spectra (top and bottom panels respectively). The black line is the observed flux density, and the grey error bars are the associated uncertainties. We plot our fitted spectral model to the data for each region in red (including background component), whilst the blue and orange lines along the bottom represent the contribution of each source component to the fit (further described in Table 10). The lower plots in each panel show the residuals in the spectral fitting, where $\delta F_{\lambda}$ is the difference between the observed $F_{\lambda}$ and the model $F_{\lambda}$, normalised by the model $F_{\lambda}$. We note that the double peaked appearance of the He II emission line in the WiFeS spectrum is most likely non-physical and due to the noisy optical spectrum, as no other broad lines show such similar line profiles. Figure 11: Best-fit single Gaussians (red) to the transient [Fe XIV] 5303 Å (top) and [Fe X] 6375 Å (bottom) coronal lines observed in the WiFeS spectrum. The lower ionisation line of the pair, [Fe X] 6375 Å, is more asymmetric, its broad base appears slightly blueshifted, and can also be fitted by a pair of Gaussians of FWHMs $330\pm 40$ km s-1 and $900\pm 100$ km s-1 (blue line), with $F([{Fe\textsc{X}}]\leavevmode\nobreak\ 6375)/F([{O\textsc{III}}]\leavevmode\nobreak\ 5007)\sim 2.6$. Table 6: Emission line ratios relative to [O III] 5007Å, where the inferred [O III] 5007Å flux in each spectrum is $1.34\pm 0.09\,\times 10^{-15}\,\mathrm{erg}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$ and $4.8\pm 0.7\times 10^{-16}\,\mathrm{erg}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$. The two spectra were obtained with different slit widths and orientations, and have not been calibrated with independent photometric measurements, hence the line ratios relative to [O III] 5007Å reported here. A dashed entry indicates that a given emission line was not clearly detected in the optical spectral fitting. Date | N III $4640$ | He II $4686$ | H$\rm{\beta}$ | H$\rm{\alpha}$ | [N II] $(6549+6583$) | [S II] $(6716+6731$)
---|---|---|---|---|---|---
2019-03-15 | - | - | $10_{\rm-3}^{\rm+3}$ | $38_{\rm-3}^{\rm+3}$ | $7_{\rm-1}^{\rm+1}$ | $1.7_{\rm-0.1}^{\rm+0.1}$
2020-05-29 | $4_{\rm-3}^{\rm+5}$ | $6_{\rm-5}^{\rm+7}$ | $11_{\rm-2}^{\rm+2}$ | $41_{\rm-6}^{\rm+6}$ | $5_{\rm-1}^{\rm+1}$ | $1.7_{\rm-0.2}^{\rm+0.2}$
Table 7: Emission line ratios from the WiFeS spectrum, where the narrow components were resolved. The superscript ‘b’ and ‘n’ denote the broad and narrow components, respectively. Line 1, Line 2 | F(Line 1)/ F(Line 2)
---|---
H$\alpha^{\mathrm{n}}$, H$\beta^{\mathrm{n}}$ | $5.8\pm 0.8$
H$\alpha^{\mathrm{b}}$, H$\beta^{\mathrm{b}}$ | $3.4\pm 0.1$
He II 4686, H$\beta^{\mathrm{b}}$ | $0.6\pm 0.1$
N III 4640, H$\beta^{\mathrm{b}}$ | $0.4\pm 0.1$
[Fe X], [O III] 5007 | $2.4\pm 0.3$
[Fe XIV], [O III] 5007 | $3.0\pm 0.5$
Table 8: Line widths inferred from the WiFeS spectrum. Line | FWHM [km s-1]
---|---
N III 4640 | $2813\pm{648}$
He II 4686 | $1959\pm{172}$
H$\beta^{\mathrm{n}}$ | $173\pm{20}$
H$\beta^{\mathrm{b}}$ | $1422\pm{11}$
[O III] 5007 | $384\pm{80}$
[Fe XIV] 5303 | $1558\pm{144}$
[Fe X] 6375 | $768\pm{35}$
H$\alpha^{\mathrm{n}}$ | $182\pm{3}$
H$\alpha^{\mathrm{b}}$ | $1252\pm{9}$
[N II] 6549 | $319\pm{12}$
### 4.4 Emission line diagnostics
#### 4.4.1 Balmer emission
From the best-fitting spectral models, we infer a broad Balmer decrement,
$F(\mathrm{H}\alpha^{\mathrm{b}})/F(\mathrm{H}\beta^{\mathrm{b}})$, of 3.4 in
the WiFeS spectrum (we use superscripts ‘b’ and ‘n’ to refer to the broad and
narrow components of a given emission line when such are clearly detected).
Such a decrement is consistent with what is observed in AGNs (e.g. Dong et al.
2005, 2007; Baron et al. 2016), and is slightly higher than the predicted
value of around 2.74-2.86151515The predicted value is dependent on the assumed
gas density and temperature. for case B recombination (Baker & Menzel, 1938)
and thus a photoionisation origin. Whilst it was originally thought that the
observed distribution in the Balmer decrements above 2.86 may have been due to
a mix of collisional excitation and dust reddening in the centre of the host
galaxy, several papers have suggested that the fundamental driver for this
variance is the reddening (e.g. Dong et al. 2007; Baron et al. 2016; Gaskell
2017). Dong et al. (2007) find that after accounting for reddening, the
intrinsic distribution of Balmer decrements in AGNs is well described by a log
Gaussian of mean 3.06, with a 0.03 dex standard deviation, whilst a recent
work by Gaskell (2017) find the intrinsic distribution is 2.72$\pm$0.04, and
thus consistent with case B recombination.
Using these results, and by working on the assumption that the intrinsic
Balmer decrement is set by Case B recombination to 2.86, we infer an
$E(B-V)\sim 0.17$ and $0.65$ mag from the broad and narrow Balmer emission
lines respectively (using the Calzetti et al. 2000 extinction
law)161616Alternatively, the inferred $E(B-V)$ values are 0.10 and 0.59 if we
assume that the intrinsic Balmer decrement is 3.06 as in Dong et al. (2007)..
We note that the E(B-V) inferred from the Balmer decrement is larger than that
inferred from SED fitting, which was performed on photometry that included
light emitted from a larger region in the host galaxy than that probed by the
Balmer decrement analysis.
#### 4.4.2 Bowen feature around 4680 Å
Both the FLOYDS and the WiFeS spectra show the emergence of a broad emission
feature around 4680 Å, which is likely a blend of He II 4686 Å and N III 4640
Å. Although this feature overlaps with the 4400-4700 Å region, which can often
show prominent Fe II emission in AGNs, we disfavour an Fe II origin here on
the basis of no strong Fe II bump being observed from the strongest Fe II
transitions in the 4500-4600 Å or $\sim$5150-5350 Å ranges (e.g. Kovačević et
al. 2010). When comparing the WiFeS AT 2019avd spectrum to the composite SDSS
quasar spectrum presented in Fig. 2 of Trakhtenbrot et al. (2019b), which was
constructed from about $1000$ SDSS quasars with broad Balmer lines of FWHM
$\sim 2000\,\mathrm{km}\,\mathrm{s}^{-1}$, the He II emission in AT 2019avd is
much stronger relative to the Balmer emission in the AGN composite.
The N III 4640 Å emission suggests the presence of Bowen fluorescence (Bowen,
1928). He II Ly$\alpha$ photons at 303.783 Å are produced after recombination
of He++ 171717The He II ionisation potential is 54.4 eV., and can then either
escape, ionise neutral H or He, or, because of the wavelength coincidence of O
III 303.799 Å and 303.693 Å, be absorbed by O III. If the latter happens, then
the later decay of the excited O III can produce a cascade of emission lines
escaping the region (e.g. 3047, 3133, 3312, 3341, 3444, and 3760
Å181818Unfortunately, our spectra do not cover the 3000-4000 Å range to detect
the other O III Bowen lines.), and eventually a FUV O III 374.436 Å photon.
The 374.436 Å can then be absorbed by ground-state N III, which further
triggers a cascade of emission lines (N III 4100, 4640 Å). Bowen fluorescence
typically requires a high flux of FUV/ soft-X-ray photons in order to produce
the He II Ly$\alpha$ photons.
We measure relative line intensities of
$F(\mathrm{{He\textsc{II}}})/F(\mathrm{H\,\beta}^{\mathrm{b}})\sim 0.57$,
$F(\mathrm{{N\textsc{III}}\,4640})/F(\mathrm{{He\textsc{II}}})\sim 0.65$ and
$F(\mathrm{{N\textsc{III}}\,4640})/F(\mathrm{H\,\beta}^{\mathrm{b}})\sim
0.37$. Netzer et al. (1985) predicted the relative Bowen line intensities in
AGNs under a range of different metal gas densities and abundances, where they
found that to produce the high
$F(\mathrm{{He\textsc{II}}})/F(\mathrm{H\,\beta}^{b})$ ratios seen in AT
2019avd as well as the high observed
$F(\mathrm{{N\textsc{III}}\,4640})/F(\mathrm{H\,\beta}^{b})$ ratio, the gas
producing the Bowen fluorescence must have very high density
($n_{\mathrm{H}}>10^{9.5}$cm-3) and high N and O abundances relative to cosmic
abundances.
#### 4.4.3 Coronal lines
From the line fitting seen on the WiFeS spectrum in Fig. 11, we infer the
luminosities of the [Fe X] 6375 Å and [Fe XIV] 5303 Å emission lines to be
$\sim 2\times 10^{39}$ and $\sim 3\times 10^{39}$ $\mathrm{erg\,s^{-1}}$. We
also infer relative intensities of
$F(\mathrm{[{Fe\textsc{X}}]\,6375})/F(\mathrm{[{O\textsc{III}}]\,5007})\sim
2.4$ and
$F(\mathrm{[{Fe\textsc{XIV}}]\,6375})/F(\mathrm{[{O\textsc{III}}]\,5007})\sim
3$. Based on the coronal line ratio definitions proposed in Wang et al.
(2012), AT 2019avd is classified as an extreme coronal line emitter (ECLE),
where extreme is defined relative to the line ratios seen in coronal line AGNs
(e.g. Nagao et al. 2000 report a maximum line ratio for
$F(\mathrm{[Fe\,X]\,6375})/F(\mathrm{[{O\textsc{III}}]\,5007})$ of 0.24 over a
sample of 124 Seyferts). Also, given the non-detected set of [Fe VII] emission
lines in AT 2019avd which are seen in some ECLEs, and relatively weak [O III]
5007 Å emission, AT 2019avd belongs to the subset of ECLEs that were
designated as TDEs in Wang et al. (2012).
The Fe coronal lines are narrower relative to the He II and N III 4640 Å
emission lines (Table 8), with FWHM for the [Fe XIV] 5303 Å and [Fe X] 6375 Å
of $1560\pm 140$ and $770\pm 40$ km s-1 respectively. Under the assumption
that the line widths are set by the virial motion of the gas, this suggests
that the coronal lines are produced further away from the BH than the Bowen
lines, and also with the higher ionisation coronal lines being produced closer
to the BH than the lower ionisation lines. The width of [Fe XIV] 5303 Å is
comparable to the observed Balmer emission. We also note the differing line
profiles of the [Fe XIV] 5303 Å and [Fe X] 6375 Å emission, with the latter
showing a stronger blue asymmetry (Fig. 11).
As discussed in Wang et al. (2012), the weakness of [Fe VII] emission relative
to [Fe X] and [Fe XIV] may be explained through the coronal line gas either
being overionised under a high X-ray flux, or due to collisional de-excitation
of [Fe VII], because it has a lower critical density ($\sim
10^{7}\,\mathrm{cm}^{-3}$) compared with the higher ionisation lines ($\sim
10^{10}\,\mathrm{cm}^{-3}$, Korista & Ferland 1989).
#### 4.4.4 Black hole mass estimate
We assume that the gas that produces the broad H$\beta$ emission is virialised
around the SMBH at the centre of the galaxy, and use the ‘single epoch’ mass-
estimation technique (e.g. Vestergaard & Peterson 2006) to infer the black
hole mass using the following scaling relation from Assef et al. (2011):
$\log\left(\frac{M_{\mathrm{BH}}}{M_{\odot}}\right)=A+B\log\left(\frac{\lambda
L_{\mathrm{\lambda}}}{10^{44}\mathrm{\,erg\,s^{-1}}}\right)+C\log\left(\frac{\mathrm{FWHM}}{\mathrm{km\,s}^{-1}}\right),$
(1)
with $A=0.895$, $B=0.52$ and $C=2$. From the measured FWHM of the broad
$H\beta$ component 1420 km s-1 and $L_{5100}=\lambda L_{\lambda}(5000\AA)\sim
2\times 10^{42}$ erg s-1 from the WiFeS spectrum191919$L_{\lambda}(5100\AA)$
is computed from the mean of $L_{\lambda}$ between 5095 and 5105 Å, we then
infer $\log[M_{\mathrm{BH}}/M_{\odot}]\sim 6.3$, albeit with a large
uncertainty of $\sim$0.3 dex (Assef et al., 2011). We note that using this
technique requires the correlations between continuum luminosity and radius of
the broad line region (BLR; e.g. Kaspi et al. 2005) obtained in previous AGN
reverberation mapping experiments to also hold for the BLR around the SMBH in
AT 2019avd.
#### 4.4.5 Baldwin, Phillips, and Terlevich line diagnostic
From the fitting of the WiFeS spectrum, we infer line flux ratios of log[[N
II] 6583/H$\alpha^{\mathrm{n}}]=-0.099^{+0.015}_{-0.016}$ and log[[O III]
5007/H$\beta^{\mathrm{n}}]=0.09^{+0.08}_{-0.10}$. According to a Baldwin,
Phillips, and Terlevich (BPT) line diagnostic test (Baldwin et al., 1981),
such line ratios suggest that a blend of star formation and AGN activity is
responsible for producing the narrow line emission in the host galaxy of AT
2019avd (Kauffmann et al., 2003; Kewley et al., 2006). Without an archival
spectrum though, it is unclear whether the [O III] 5007 Å and [N II] 6583 Å
lines have increased in intensity since the initial ZTF outburst, or an AGN-
like ionising source has always been present.
### 4.5 Mapping out the BLR
Assuming that each observed emission line is broadened due to its virial
motion around the central BH, we can use the measured FWHMs to obtain rough
estimates of the distances from the central ionising source at which each line
is produced (Fig. 12). Similar to previous work (e.g. Korista et al. 1995;
Kollatschny 2003; Bentz et al. 2010), we also find evidence for a stratified
BLR, whereby the higher ionisation lines are produced in regions closer to the
BH.
Figure 12: Estimated radii from the BH where different observed optical
emission lines are produced in AT 2019avd, compared with various key physical
length scales predicted in the literature (assuming
$\log[M_{\mathrm{BH}}/M_{\odot}]=6.3$). The pericentre and circularisation
radii are computed assuming a Sun-like star incident on this BH with its
closest approach at the tidal radius. Similarly to Kollatschny et al. (2014),
we see evidence for a stratified BLR. The coloured lines represent length
scales that were obtained based on observations of AT 2019avd, whilst the grey
dashed lines are based on various scaling relations in the literature (BLR
radius based on Kaspi et al. 2005, whilst the inner torus radius was computed
using equation 1 of Nenkova et al. 2008, assuming a dust sublimation
temperature of 1500K).
## 5 Discussion
Based purely on its X-ray luminosity evolution, AT 2019avd most likely
involves an accreting SMBH at the centre of a galaxy. Whilst the large
amplitude X-ray flaring (factor of $\gtrsim$600), soft X-ray spectrum, lack of
previous strong (and sustained) AGN activity, and the implied unabsorbed X-ray
peak luminosity in the $0.2-2$ keV energy range of $2\times 10^{43}$ erg s-1
(using spectroscopic $z=0.029$, see section 4.1) initially made the source a
strong TDE candidate, this is clearly discordant with the double-peaked
optical variability seen in the ZTF observations (it does not look like a
prototypical, single-event TDE as observed elsewhere). In the following
section, we discuss potential origins of the rich phenomenology seen in AT
2019avd.
### 5.1 AT 2019avd as non-TDE-induced AGN variability
If AT 2019avd is related to AGN activity that was not induced by a TDE (herein
referred to simply as AGN ‘activity’ or ‘variability’202020As a TDE may
transform a quiescent BH into an AGN, the variability in BHs induced by TDEs
is also just a subset of AGN variability.), then the combination of its X-ray
and optical light curves make it one of the most extreme cases of AGN
variability observed to date.
It is clear that the X-ray spectrum of AT 2019avd (section 2.3) is far softer
than what is commonly seen in Seyfert 1s; for example, the power-law slope for
Swift OBSID 00013495001 was $5.3^{+0.4}_{-0.4}$, whilst Nandra & Pounds (1994)
model the observed power-law slope distribution with a Gaussian distribution
of mean 1.95 and standard deviation 0.15. However, based on the measured FWHMs
of the broad Balmer emission lines in the optical spectrum, it would be
classified as a NLSy1, and softer spectral indices have also been observed in
the NLSy1 population; a systematic ROSAT study of this by Boller et al. (1996)
found power-law slopes of up to $\sim 5$. NLSy1s are also known to exhibit
rapid, large-amplitude X-ray variability (e.g. Boller et al. 1996). As the
X-ray variability of NLSy1s over longer timescales has not been extensively
monitored before, how common AT 2019avd-like X-ray flares are within this
population is currently unclear. For this reason, the X-ray properties alone
cannot be used to state that the observed variability in AT 2019avd was
induced by a TDE.
However, AT 2019avd shows a number of features in its optical spectrum that
are infrequently seen in NLSy1s. First, NLSy1s commonly show strong Fe II
emission (e.g. Rakshit et al. 2017), whereas this is not seen in the WiFeS
spectrum, and only a weak Fe II complex is seen in the NOT spectrum in AT
2019avd. Instead, the most prominent Fe emission we observe are the transient,
ECLE-like higher ionisation coronal lines of [Fe XIV] 5303 Å and [Fe X] 6375 Å
in the WiFeS spectrum. During our spectroscopic follow-up campaign, we also
observe the appearance of He II 4686 Å and N III 4640 Å emission lines
(attributed to Bowen fluorescence). The optical spectrum at late times appears
similar to the recently identified new class of flaring transients by
Trakhtenbrot et al. (2019b), and we present a comparison of AT 2019avd with
this class in Fig. 13. Whilst AT 2019avd shares the broad emission feature
around 4680Å with the AT 2017bgt flare class, the optical spectrum of AT
2019avd is distinguishable from the other members based on its much weaker [O
III] 5007Å emission line. A likely reason for this is that the host galaxies
of the other flares had persistent, higher luminosity AGNs in them prior to
the optical outburst, relative to AT 2019avd. In addition, AT 2019avd’s large
amplitude, ultra-soft X-ray flare, and its optical light-curve evolution make
it unique amongst the AT 2017bgt flare class.
Figure 13: Comparison of the optical spectrum of AT 2019avd with those of the
three nuclear transients recently identified as a new class of flares from
accreting SMBHs in Trakhtenbrot et al. (2019b). The two dashed grey lines mark
the positions of N III 4640 Å and He II 4686 Å. All objects share high $F$(He
II 4686 Å)/$F$(H$\beta$), and at least one Bowen emission line (N III 4640Å).
Finally, we stress that the double-peaked optical variability shown by AT
2019avd is unprecedented for a NLSy1, which when combined with its X-ray
properties, make AT 2019avd clearly unique relative to all previous examples
of AGN variability. Further examples of NLSy1 variability seen during the ZTF
survey will be presented in a separate publication (Frederick et al., 2020).
### 5.2 An origin related to tidal disruption?
#### 5.2.1 Canonical tidal disruption event
As AT 2019avd shows a very-large-amplitude, soft-X-ray flare from the nucleus
of a galaxy that shows no strong signs of prior AGN activity, it appears
similar to the predicted observational signatures for TDEs (e.g. Rees 1988)
and most of the previous X-ray-selected thermal TDE candidates (Bade et al.,
1996; Komossa & Bade, 1999; Komossa & Greiner, 1999; Grupe & Leighly, 1999;
Greiner et al., 2000; Saxton et al., 2019). On the other hand, its optical
spectrum shows a far weaker blue continuum component relative to that seen in
optically selected TDEs, as well as narrower Balmer emission lines (for TDEs
where these are detected); based on these two pieces of evidence, it would be
straightforward to declare that AT 2019avd is not a TDE candidate, according
to criteria for optical TDE selection in van Velzen et al. (2020).
The observed broad Balmer emission lines in AT 2019avd instead appear more
like those commonly seen in the broad emission lines of Seyfert 1s. With such
similarity, a mechanism analogous to the broad line emission in AGNs is likely
operating in AT 2019avd, whereby the line widths of hydrogen recombination
lines are set by the gas kinematics (whereas some TDEs may have line widths
set by repeated non-coherent electron scattering; e.g. Roth & Kasen 2018), and
the high densities in the BLR result in the line intensity responding
effectively instantaneously to changes in the continuum flux. In the limit of
a weak TDE-like reprocessing layer212121And likely a lack of optically-
selected observed TDE features., the optical spectrum of a TDE may appear
similar to that of an AGN, as has been previously suggested (e.g. Gaskell &
Rojas Lobos 2014). The timescales for the evolution of the spectral features
in such systems may be different from those observed in optically selected
TDEs, as they originate from a region further away from the BH than the
reprocessing layer.
The optical emission mechanism in TDEs is currently not well understood,
although it is thought to arise either from shocks produced from stellar
debris stream self-intersections (Shiokawa et al., 2015; Piran et al., 2015),
or from debris reprocessing the emission from an accretion disc (e.g. Loeb &
Ulmer 1997; Ulmer et al. 1998; Roth et al. 2016; Roth & Kasen 2018). However,
it is unclear how luminous the shocks are from stream self-intersections,
whilst for the reprocessing scenario we still do not understand where the
reprocessor is situated, where it forms, how large its covering angle would be
from the BH, how efficiently it converts disc emission into the optical
wavebands, or how all of these aspects are affected by the properties of the
BH and those of the disrupted star. There is currently not a large enough
sample of TDEs selected through both X-ray and optical surveys to test these
various models of optical emission, and to properly assess the various complex
underlying selection effects likely present in the existing TDE candidate
population. A key example of these effects is the fact that only a small
fraction of optically selected TDEs show transient X-ray emission ($\sim 25$%
of optically selected TDEs in van Velzen et al. 2020 were X-ray bright); Dai
et al. (2018) suggested that the observed properties of a TDE may be dependent
upon the viewing angle to the newly formed disc.
Given the above, and that there are also no X-ray selected, non-relativistic
TDEs in the literature that have high-cadence optical photometric light curves
available222222Although the 4 X-ray bright TDEs in van Velzen et al. (2020)
were monitored at a high cadence with ZTF and Swift UVOT, these were
optically-selected TDEs., we cannot rule out a TDE-related origin for AT
2019avd simply on the basis of a lack of optically selected TDE features in
the optical spectrum. However, we do disfavour the canonical TDE
interpretation (seen in optically selected TDEs) for this flare on the basis
of the double-peaked optical light curve, which has not been observed in any
of the TDEs identified by ZTF so far. Secondary maxima have previously been
seen in the light curves of some TDE candidates (a compilation is presented in
Fig. 8 of Wevers et al. 2019), though not at optical wavelengths and of far
smaller amplitude increase compared with AT 2019avd (with the exception of the
TDE in an AGN candidate in Merloni et al. 2015).
#### 5.2.2 A more exotic variant of a tidal disruption event?
A large fraction of stars may exist in binary systems (e.g. Lada 2006). Mandel
& Levin (2015) studied the various outcomes of a binary star passing close to
a SMBH from a nearly radial orbit. In $\sim 20$% of such approaches, a double
tidal disruption event (DTDE) is produced, whereby both stars in the binary
are disrupted in succession. These latter authors estimated that $\sim$10% of
all stellar tidal disruptions could be associated with DTDEs, with such events
expected to produce double-peaked light curves.
We can use the inferred rise-to-peak timescales from the ZTF light curves to
test the feasibility of whether AT 2019avd may have been triggered by a DTDE,
specifically for the case where each peak is associated with the rise to peak
mass fallback of each successive disruption. Guillochon & Ramirez-Ruiz (2013)
present the time taken for a single TDE to reach peak mass fallback rate (in
their equation A2):
$t_{\mathrm{peak}}=B_{\gamma}\left(\frac{M_{\mathrm{BH}}}{10^{6}M_{\odot}}\right)^{1/2}\left(\frac{M_{\star}}{M_{\odot}}\right)^{-1}\left(\frac{R_{\star}}{R_{\odot}}\right)^{3/2}\,\mathrm{years},$
(2)
where $B_{\gamma}$ is a function of $\beta$, the ratio of the tidal radius of
the BH to the pericentre of the orbit of the star, $\gamma$ is the polytropic
index of the star232323We use $\gamma=4/3$ for
$0.3M_{\odot}<M_{\star}<22M_{\odot}$, and $\gamma=5/3$ for $M_{\star}$ outside
this range, as in Mockler et al. 2019., $M_{\mathrm{BH}}$ is the black hole
mass, and $M_{\star}$ and $R_{\star}$ are the mass and radius of the star
being disrupted.
Similarly to Merloni et al. (2015), we then generate a grid of $M_{\star}$ and
$\beta$, log-uniformly between (0.1$M_{\odot}$, 100$M_{\odot}$) and (0.5, 4),
respectively, and compute $R_{\star}$ for each $M_{\star}$ using the
mass–radius relationship for zero-age main sequence stars presented in Tout et
al. (1996). For each possible combination of $M_{\star}$ and $\beta$, and for
a black hole with $\log[M_{\mathrm{BH}}/M_{\odot}]\sim 6.3$, we check whether
it can produce $t_{\mathrm{peak}}$ (using equation 2) within 20% of the
observed peak timescales in the ZTF light curves ($\sim 24$ days and $\sim
260$ days for the first and second peak respectively). We also enforce the
constraint that its tidal radius lies outside of the Schwarzschild radius for
the system, so that it can produce a TDE with the star being swallowed whole
by the black hole.
Figure 14: Constraints on the $M_{\star}$, $\beta$ parameter space, obtained
for explaining the origin of AT 2019avd as a DTDE on SMBH. Red markers
represent a permitted $M_{\star}$, $\beta$ configuration, whilst a region that
contains grey hashing represents a configuration that is not able to reproduce
the observed timescales for the given peak. Results were obtained for a black
hole with $\log[M_{\mathrm{BH}}/M_{\odot}]\sim 6.3$. Since there are no red
markers on the second optical peak plot, there is no permitted $M_{\star}$,
$\beta$ pairing that can reproduce the observed peak timescale for the second
optical peak. The black dashed lines bound
$0.3M_{\odot}<M_{\star}<22M_{\odot}$, where we adopt $\gamma=4/3$.
We plot the permitted regions of the $M_{\star}$, $\beta$ parameter space in
red in Fig. 14, where we see that no main sequence binary star configuration
can reproduce the observed rise times for both the first and second peaks. It
would also be possible to obtain further constraints on the feasibility of
this scenario based on the observed peak luminosities (similar to Merloni et
al. 2015) and their ratio, as well as from the inferred properties of the
binary itself, such as from the time between the two observed peaks (which
could be used to constrain the semi-major axis) and the inferred mass ratio.
However, the constraints provided from $t_{\mathrm{peak}}$ are perhaps the
simplest to implement and are sufficient to highlight the caveats of a simple
DTDE interpretation.
Bonnerot & Rossi (2019) recently suggested that following the disruption of a
stellar binary, the two separate debris streams may collide prior to their
fallback onto the black hole. These collisions then shock-heat the gas, and
were predicted to produce an optical flare prior to the main flare of the
disruption event. Such a model for a binary TDE could potentially explain the
observed double-peak light curve, and the observed emergence of the Bowen
feature after the second peak (the soft X-rays can only be emitted once the
accretion disc has formed). However, a caveat to this interpretation is that
both a strong ionising flux and high gas densities are required for Bowen
fluorescence to be produced, and we cannot confidently state here that the
reason for not observing Bowen lines in the NOT spectrum is the absence of an
X-ray-emitting accretion disc during that observation, because the absence of
Bowen lines may also be due to insufficiently high gas densities (not all TDEs
that are X-ray bright have displayed Bowen emission lines). We do not rule out
this more complex DTDE scenario for AT 2019avd here, but do not perform a
detailed comparison between the simulations in Bonnerot & Rossi (2019) and AT
2019avd in the present paper. Another alternative could be that AT 2019avd
involved some type of TDE about a SMBH binary (e.g. Liu et al. 2009; Coughlin
et al. 2017), where in such systems, the presence of the secondary BH can
perturb the accretion flow onto the primary, leading to intermittent light
curves.
### 5.3 Could AT 2019avd be supernova-related?
The spectra of Type IIn SNe can appear similar to those of AGNs (e.g.
Filippenko 1989), as they can show broad and narrow emission lines, an absence
of P-Cygni profiles, and higher luminosities and slower decay timescales
relative to normal Type II SNe (Nyholm et al., 2020). Type IIn SNe typically
also show the highest X-ray luminosities amongst all SNe. However, AT 2019avd
has a $L_{\mathrm{0.2-2keV}}$ that is about an order of magnitude higher than
what is seen in most X-ray-luminous Type IIn SNe, when considering the sample
of IIn shown in Fig. 3 of Dwarkadas & Gruszko (2012). Furthermore, the X-ray
emission from Type IIn SNe is predicted to be hard (e.g. Ofek et al. 2013),
whilst that of AT 2019avd is ultra-soft. Based on the X-ray emission alone, we
disfavour the idea that both optical peaks in AT 2019avd are related to a
single Type IIn supernova.
Given the observed peak and decay timescales (Fig. 6), the peak absolute
magnitude of the optical light curve ($\sim-18.5$), the small amount of
reddening seen in the ZTF light curve during the decay phase, and the NOT
spectrum, the first optical peak may have been associated with a Type IIn SN.
The second optical peak would then be associated with a ‘turn on’ event in the
SMBH that sees a vast increase in the accretion rate and the luminosity of the
BH. This scenario would then explain why the He II, Bowen, and coronal lines
are not seen in the NOT spectrum, and only in the spectra taken after the
second peak. However, the probability of observing both a Type IIn SN and an
AGN ‘turn on’ event within just over a year of each other is extremely small
given the apparent rarity of extreme ‘turn-on’ events in AGNs (especially
those showing an AT 2019avd-like X-ray outburst) and the expected detection
rates for Type IIn SNe (e.g. Feindt et al. 2019), and we therefore disfavour a
scenario where AT 2019avd is the chance coincidence of a Type IIn SN and
extreme AGN ignition event within roughly one year of each other.
## 6 Conclusions
This paper presents an overview of a set of multi-wavelength observations of
an exceptional nuclear transient, AT 2019avd, whose main observed features are
as follows:
1. 1.
eROSITA detected an ultra-soft ($kT\sim 85$ eV) X-ray brightening ($\gtrsim
90$ times brighter than a previous 3$\sigma$ upper flux limit) from a
previously X-ray-inactive galaxy (Section 2).
2. 2.
AT 2019avd was initially observed on a weekly basis with Swift XRT/UVOT for 6
weeks following the eROSITA detection. The host had brightened in all UVOT
bands by $\sim 1$ mag relative to archival GALEX observations, and was
observed with $0.2-2$ keV X-ray flux consistent with the eROSITA detection
(Section 2). A further Swift observation $\sim$5 months after the initial
eROSITA detection revealed a brightening by a factor of approximately six in
the $0.2-2$ keV band relative to the eROSITA detection. AT 2019avd therefore
shows a net brightening in the $0.2-2$ keV band by a factor of at least 600
relative to the 3$\sigma$ upper detection limit derived from an XMM-Newton
pointing in 2015.
3. 3.
In the 450 days prior to the eROSITA detection, ZTF observed a double-peaked
light curve (Section 3). The first optical peak shows rise and decay
timescales akin to TDEs and SNe, whilst the rise time of the second peak is
more similar to those seen in AGNs. No optical outbursts were detected during
ASAS-SN observations over the seven years preceding the initial outburst seen
by ZTF.
4. 4.
Optical spectroscopic follow-up finds transient He II emission, Bowen
fluorescence lines, and high-ionisation coronal lines ([Fe X] 6375 Å, [Fe XIV]
5303 Å) in the spectra taken after the second optical peak, but not in the
spectrum taken 30 days after the first peak. The presence of such a set of
lines requires an intense source of soft X-ray emission and extremely high
densities. Broad Balmer emission lines were detected in spectra 30 days after
the first peak in the ZTF light curve, as well as in all spectra taken in the
weeks after the eROSITA detection with FWHM $\sim 1400$km s-1 (Section 4).
AT 2019avd thus shows a set of observed features which have never been
observed together in the same nuclear transient before, and further
complicates the non-trivial task of distinguishing the physical origin of
large-amplitude variability seen in galactic nuclei. Whilst a discussion on
the potential origins of this transient is presented in Section 5, it is still
unclear what has triggered such exotic behaviour. Detailed simulations would
be welcome to distinguish between the various possible scenarios. These will
be well complimented with future planned observations (Swift, NICER, XMM-
Newton) monitoring the late-time evolution of AT 2019avd. Finally, we note
that during its eight successive all-sky surveys in the following years,
eROSITA will systematically monitor the X-ray variability of AGNs and map out
the population of nuclear transients. With this information, we will be able
to better understand the extent of the X-ray variability shown by AT 2019avd,
and make a more informed judgement on the origin of this transient.
###### Acknowledgements.
We thank the anonymous referee, and the journal editor, Sergio Campana, for
constructive comments which helped improve this paper. AM thanks the Yukawa
Institute for Theoretical Physics at Kyoto University, where discussions held
during the YITP workshop YITP-T-19-07 on International Molecule-type Workshop
”Tidal Disruption Events: General Relativistic Transients” were useful to
complete this work. AM thanks Mariuz Gromadzki, Giorgos Leloudas and Clive
Tadhunter for sharing optical spectra. A.M. acknowledges support from and
participation in the International Max-Planck Research School (IMPRS) on
Astrophysics at the Ludwig-Maximilians University of Munich (LMU). BJS is
supported by NSF grant AST-1907570. BJS is also supported by NASA grant
80NSSC19K1717 and NSF grants AST-1920392 and AST-1911074. BT acknowledges
support from the Israel Science Foundation (grant number 1849/19) IA is a
CIFAR Azrieli Global Scholar in the Gravity and the Extreme Universe Program
and acknowledges support from that program, from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(grant agreement number 852097), from the Israel Science Foundation (grant
numbers 2108/18 and 2752/19), from the United States - Israel Binational
Science Foundation (BSF), and from the Israeli Council for Higher Education
Alon Fellowship. L.T. acknowledges support from MIUR (PRIN 2017 grant
20179ZF5KS). GEA is the recipient of an Australian Research Council Discovery
Early Career Researcher Award (project number DE180100346), funded by the
Australian Government. This research was partially supported by the Australian
Government through the Australian Research Council’s Discovery Projects
funding scheme (project DP200102471). This work is based on data from eROSITA,
the primary instrument aboard SRG, a joint Russian-German science mission
supported by the Russian Space Agency (Roskosmos), in the interests of the
Russian Academy of Sciences represented by its Space Research Institute (IKI),
and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft
was built by Lavochkin Association (NPOL) and its subcontractors, and is
operated by NPOL with support from the Max Planck Institute for
Extraterrestrial Physics (MPE). The development and construction of the
eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl
Remeis Observatory Bamberg, the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. This work was based on observations obtained
with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the
Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is
supported by the National Science Foundation under Grant No. AST-1440341 and a
collaboration including Caltech, IPAC, the Weizmann Institute for Science, the
Oskar Klein Center at Stockholm University, the University of Maryland, the
University of Washington, Deutsches Elektronen-Synchrotron and Humboldt
University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee, and Lawrence Berkeley National
Laboratories. Operations are conducted by COO, IPAC, and UW. SED Machine is
based upon work supported by the National Science Foundation under Grant No.
1106171. This publication makes use of data products from the Near-Earth
Object Wide-field Infrared Survey Explorer (NEOWISE), which is a joint project
of the Jet Propulsion Laboratory/California Institute of Technology and the
University of Arizona. NEOWISE is funded by the National Aeronautics and Space
Administration. We thank the Las Cumbres Observatory and its staff for its
continuing support of the ASAS-SN project. ASAS-SN is supported by the Gordon
and Betty Moore Foundation through grant GBMF5490 to the Ohio State
University, and NSF grants AST-1515927 and AST-1908570. Development of ASAS-SN
has been supported by NSF grant AST-0908816, the Mt. Cuba Astronomical
Foundation, the Center for Cosmology and AstroParticle Physics at the Ohio
State University, the Chinese Academy of Sciences South America Center for
Astronomy (CAS- SACA), and the Villum Foundation. The Pan-STARRS1 Surveys
(PS1) have been made possible through contributions of the Institute for
Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-
Planck Society and its participating institutes, the Max Planck Institute for
Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial
Physics, Garching, The Johns Hopkins University, Durham University, the
University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian
Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation under Grant No.
AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE).
This work was partially based on observations made with the Nordic Optical
Telescope, operated at the Observatorio del Roque de los Muchachos, La Palma,
Spain, of the Instituto de Astrofisica de Canarias. Some of the data presented
here were obtained with ALFOSC, which is provided by the Instituto de
Astrofisica de Andalucia (IAA) under a joint agreement with the University of
Copenhagen and NOTSA.
## References
* Alard (2000) Alard, C. 2000, Astronomy and Astrophysics Supplement Series, 144, 363
* Alard & Lupton (1998) Alard, C. & Lupton, R. H. 1998, The Astrophysical Journal, 503, 325
* Antonucci & Cohen (1983) Antonucci, R. R. J. & Cohen, R. D. 1983, The Astrophysical Journal, 271, 564
* Arcavi et al. (2014) Arcavi, I., Gal-Yam, A., Sullivan, M., et al. 2014, The Astrophysical Journal, 793, 38
* Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
* Assef et al. (2011) Assef, R. J., Denney, K. D., Kochanek, C. S., et al. 2011, The Astrophysical Journal, 742, 93
* Assef et al. (2013) Assef, R. J., Stern, D., Kochanek, C. S., et al. 2013, The Astrophysical Journal, 772, 26
* Assef et al. (2018) Assef, R. J., Stern, D., Noirot, G., et al. 2018, The Astrophysical Journal Supplement Series, 234, 23
* Bade et al. (1996) Bade, N., Komossa, S., & Dahlem, M. 1996, Astronomy & Astrophysics, 309
* Baker & Menzel (1938) Baker, J. G. & Menzel, D. H. 1938, The Astrophysical Journal, 88, 52
* Baldwin et al. (1981) Baldwin, A., Phillips, M. M., & Terlevich, R. 1981, Publications of the Astronomical Society of the Pacific, 93, 817
* Baron et al. (2016) Baron, D., Stern, J., Poznanski, D., & Netzer, H. 2016, The Astrophysical Journal, 832, 8
* Becker et al. (1995) Becker, R. H., White, R. L., & Helfand, D. J. 1995, The Astrophysical Journal, 450, 559
* Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 018002
* Bentz et al. (2010) Bentz, M. C., Walsh, J. L., Barth, A. J., et al. 2010, The Astrophysical Journal, 716, 993
* Blagorodnova et al. (2017) Blagorodnova, N., Gezari, S., Hung, T., et al. 2017, The Astrophysical Journal, 844, 46
* Blagorodnova et al. (2018) Blagorodnova, N., Neill, J. D., Walters, R., et al. 2018, Publications of the Astronomical Society of the Pacific, 130, 035003
* Blanchard et al. (2017) Blanchard, P. K., Nicholl, M., Berger, E., et al. 2017, The Astrophysical Journal, 843, 106
* Boller et al. (1996) Boller, T., Brandt, W. N., & Fink, H. 1996, Astronomy & Astrophysics, 305, 53
* Bonnerot & Rossi (2019) Bonnerot, C. & Rossi, E. M. 2019, Monthly Notices of the Royal Astronomical Society, 484, 1301
* Boquien et al. (2019) Boquien, M., Burgarella, D., Roehlly, Y., et al. 2019, Astronomy & Astrophysics, 622, A103
* Bowen (1928) Bowen, I. S. 1928, The Astrophysical Journal, 67, 1
* Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, Publications of the Astronomical Society of the Pacific, 125, 1031
* Buchner (2016) Buchner, J. 2016, Statistics and Computing, 26, 383
* Buchner (2019) Buchner, J. 2019, Publications of the Astronomical Society of the Pacific, 131, 108005
* Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, Astronomy & Astrophysics, 564, A125
* Burgarella et al. (2005) Burgarella, D., Buat, V., & Iglesias-Páramo, J. 2005, Monthly Notices of the Royal Astronomical Society, 360, 1413
* Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, The Astrophysical Journal, 533, 682
* Cash (1976) Cash, W. 1976, Astronomy & Astrophysics, 52, 307
* Chan et al. (2019) Chan, C.-H., Piran, T., Krolik, J. H., & Saban, D. 2019, The Astrophysical Journal, 881, 113
* Childress et al. (2014) Childress, M. J., Vogt, F. P. A., Nielsen, J., & Sharp, R. G. 2014, Astrophysics and Space Science, 349, 617
* Coughlin et al. (2017) Coughlin, E. R., Armitage, P. J., Nixon, C., & Begelman, M. C. 2017, Monthly Notices of the Royal Astronomical Society, 465, 3840
* Dai et al. (2018) Dai, L., McKinney, J. C., Roth, N., Ramirez-Ruiz, E., & Miller, M. C. 2018, The Astrophysical Journal, 859, L20
* Dong et al. (2007) Dong, X., Wang, T., Wang, J., et al. 2007, Monthly Notices of the Royal Astronomical Society, 383, 581
* Dong et al. (2005) Dong, X.-B., Zhou, H., Wang, T., et al. 2005, The Astrophysical Journal, 620, 629
* Dopita et al. (2007) Dopita, M., Hart, J., McGregor, P., et al. 2007, Astrophysics and Space Science, 310, 255
* Dopita et al. (2010) Dopita, M., Rhee, J., Farage, C., et al. 2010, Astrophysics and Space Science, 327, 245
* Dwarkadas & Gruszko (2012) Dwarkadas, V. V. & Gruszko, J. 2012, Monthly Notices of the Royal Astronomical Society, 419, 1515
* Esquej et al. (2007) Esquej, P., Saxton, R. D., Freyberg, M. J., et al. 2007, Astronomy and Astrophysics, 462
* Esquej et al. (2008) Esquej, P., Saxton, R. D., Komossa, S., et al. 2008, Astronomy and Astrophysics, 489, 543
* Feindt et al. (2019) Feindt, U., Nordin, J., Rigault, M., et al. 2019, Journal of Cosmology and Astroparticle Physics, 2019, 005
* Feroz & Hobson (2008) Feroz, F. & Hobson, M. P. 2008, Monthly Notices of the Royal Astronomical Society, 384, 449
* Filippenko (1989) Filippenko, A. V. 1989, The Astronomical Journal, 97, 726
* Frederick et al. (2019) Frederick, S., Gezari, S., Graham, M. J., et al. 2019, The Astrophysical Journal, 883, 31
* Frederick et al. (2020) Frederick, S., Gezari, S., Graham, M. J., et al. 2020, arXiv preprint arXiv:2010.08554
* Freeman et al. (2001) Freeman, P., Doe, S., & Siemiginowska, A. 2001, in Astronomical Data Analysis, ed. J.-L. Starck & F. D. Murtagh, Vol. 4477, International Society for Optics and Photonics (SPIE), 76–87
* Fremling et al. (2016) Fremling, C., Sollerman, J., Taddia, F., et al. 2016, Astronomy & Astrophysics, 593, A68
* Gaskell (2017) Gaskell, C. M. 2017, Monthly Notices of the Royal Astronomical Society, 467, stx094
* Gaskell & Rojas Lobos (2014) Gaskell, C. M. & Rojas Lobos, P. A. 2014, Monthly Notices of the Royal Astronomical Society: Letters, 438, L36
* Gezari et al. (2008) Gezari, S., Basa, S., Martin, D. C., et al. 2008, The Astrophysical Journal, 676, 944
* Gezari et al. (2012) Gezari, S., Chornock, R., Rest, A., et al. 2012, Nature, 485, 217
* Gezari et al. (2020) Gezari, S., Frederick, S., van Velzen, S., et al. 2020, The Astronomer’s Telegram, 13717, 1
* Gezari et al. (2009) Gezari, S., Heckman, T., Cenko, S. B., et al. 2009, The Astrophysical Journal, 698, 1367
* Giustini et al. (2020) Giustini, M., Miniutti, G., & Saxton, R. D. 2020, Astronomy & Astrophysics, 636, L2
* Gliozzi & Williams (2020) Gliozzi, M. & Williams, J. K. 2020, Monthly Notices of the Royal Astronomical Society, 491, 532
* Graham et al. (2019) Graham, M. J., Kulkarni, S. R., Bellm, E. C., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 078001
* Greiner et al. (2000) Greiner, J., Schwarz, R., Zharikov, S., & Orio, M. 2000, Astronomy and Astrophysics, 362
* Grupe & Leighly (1999) Grupe, D. & Leighly, K. 1999, Astronomy and Astrophysics, 350, L31
* Guillochon & Ramirez-Ruiz (2013) Guillochon, J. & Ramirez-Ruiz, E. 2013, The Astrophysical Journal, 767, 25
* Gurzadian & Ozernoi (1981) Gurzadian, V. G. & Ozernoi, L. M. 1981, 95, 39
* Henden et al. (2015) Henden, A. A., Levine, S., Terrell, D., & Welch, D. L. 2015, in American Astronomical Society Meeting Abstracts, Vol. 225, American Astronomical Society Meeting Abstracts #225, 336.16
* HI4PI Collaboration et al. (2016) HI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, Astronomy & Astrophysics, 594, A116
* Hills (1975) Hills, J. G. 1975, Nature, 254, 295
* Holoien et al. (2019a) Holoien, T. W.-S., Huber, M. E., Shappee, B. J., et al. 2019a, The Astrophysical Journal, 880, 120
* Holoien et al. (2016) Holoien, T. W.-S., Kochanek, C. S., Prieto, J. L., et al. 2016, Monthly Notices of the Royal Astronomical Society, 463, 3813
* Holoien et al. (2014) Holoien, T. W.-S., Prieto, J. L., Bersier, D., et al. 2014, Monthly Notices of the Royal Astronomical Society, 445, 3263
* Holoien et al. (2019b) Holoien, T. W.-S., Vallely, P. J., Auchettl, K., et al. 2019b, The Astrophysical Journal, 883, 111
* Hung et al. (2017) Hung, T., Gezari, S., Cenko, S. B., et al. 2017, The Astrophysical Journal Supplement Series, 238, 15
* Kaspi et al. (2005) Kaspi, S., Maoz, D., Netzer, H., et al. 2005, The Astrophysical Journal, 629, 61
* Kataoka et al. (2008) Kataoka, J., Madejski, G., Sikora, M., et al. 2008, The Astrophysical Journal, 672, 787
* Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., Tremonti, C., et al. 2003, Monthly Notices of the Royal Astronomical Society, 346, 1055
* Kewley et al. (2006) Kewley, L. J., Groves, B., Kauffmann, G., & Heckman, T. 2006, Monthly Notices of the Royal Astronomical Society, 372, 961
* Kim et al. (2018) Kim, D.-C., Yoon, I., & Evans, A. S. 2018, The Astrophysical Journal, 861, 51
* King (2020) King, A. 2020, Monthly Notices of the Royal Astronomical Society: Letters, 493, L120
* Kochanek et al. (2017) Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, Publications of the Astronomical Society of the Pacific, 129, 104502
* Kollatschny (2003) Kollatschny, W. 2003, Astronomy & Astrophysics, 407, 461
* Kollatschny et al. (2014) Kollatschny, W., Ulbrich, K., Zetzl, M., Kaspi, S., & Haas, M. 2014, Astronomy & Astrophysics, 566, A106
* Komossa & Bade (1999) Komossa, S. & Bade, N. 1999, Astronomy and Astrophysics, 343, 775
* Komossa & Greiner (1999) Komossa, S. & Greiner, J. 1999, Astronomy and Astrophysics, 349, L45
* Korista et al. (1995) Korista, K. T., Alloin, D., Barr, P., et al. 1995, The Astrophysical Journal Supplement Series, 97, 285
* Korista & Ferland (1989) Korista, K. T. & Ferland, G. J. 1989, The Astrophysical Journal, 343, 678
* Kovačević et al. (2010) Kovačević, J., Popović, L., & Dimitrijević, M. S. 2010, The Astrophysical Journal Supplement Series, 189, 15
* Lacy et al. (1982) Lacy, J. H., Townes, C. H., & Hollenbach, D. J. 1982, The Astrophysical Journal, 262, 120
* Lada (2006) Lada, C. J. 2006, The Astrophysical Journal, 640, L63
* Law-Smith et al. (2017) Law-Smith, J., Ramirez-Ruiz, E., Ellison, S. L., & Foley, R. J. 2017, The Astrophysical Journal, 850, 22
* Liu et al. (2009) Liu, F. K., Li, S., & Chen, X. 2009, The Astrophysical Journal, 706, L133
* Liu et al. (2020) Liu, Z., Li, D., Liu, H.-Y., et al. 2020, The Astrophysical Journal, 894, 93
* Loeb & Ulmer (1997) Loeb, A. & Ulmer, A. 1997, The Astrophysical Journal, 489, 573
* Mainzer et al. (2014) Mainzer, A., Bauer, J., Cutri, R. M., et al. 2014, The Astrophysical Journal, 792, 30
* Mainzer et al. (2011) Mainzer, A., Bauer, J., Grav, T., et al. 2011, The Astrophysical Journal, 731, 53
* Malyali et al. (2020) Malyali, A., Rau, A., Arcodia, R., et al. 2020, The Astronomer’s Telegram, 13712, 1
* Mandel & Levin (2015) Mandel, I. & Levin, Y. 2015, The Astrophysical Journal, 805, L4
* Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 018003
* Merloni et al. (2015) Merloni, A., Dwelly, T., Salvato, M., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 69
* Miniutti et al. (2019) Miniutti, G., Saxton, R. D., Giustini, M., et al. 2019, Nature, 573, 381
* Mockler et al. (2019) Mockler, B., Guillochon, J., & Ramirez-Ruiz, E. 2019, The Astrophysical Journal, 872, 151
* Nagao et al. (2000) Nagao, T., Taniguchi, Y., & Murayama, T. 2000, The Astronomical Journal, 119, 2605
* Nandra & Pounds (1994) Nandra, K. & Pounds, K. A. 1994, Monthly Notices of the Royal Astronomical Society, 268, 405
* Nenkova et al. (2008) Nenkova, M., Sirocky, M. M., Nikutta, R., Ivezić, v., & Elitzur, M. 2008, The Astrophysical Journal, 685, 160
* Netzer et al. (1985) Netzer, H., Elitzur, M., & Ferland, G. J. 1985, The Astrophysical Journal, 299, 752
* Neustadt et al. (2020) Neustadt, J. M. M., Holoien, T. W.-S., Kochanek, C. S., et al. 2020, Monthly Notices of the Royal Astronomical Society, 494, 2538
* Noda & Done (2018) Noda, H. & Done, C. 2018, Monthly Notices of the Royal Astronomical Society, 480, 3898
* Nordin et al. (2019) Nordin, J., Brinnel, V., Giomi, M., et al. 2019, Transient Name Server Discovery Report, 2019-236, 1
* Nyholm et al. (2020) Nyholm, A., Sollerman, J., Tartaglia, L., et al. 2020, Astronomy & Astrophysics, 637, A73
* Ofek et al. (2013) Ofek, E. O., Fox, D., Cenko, S. B., et al. 2013, The Astrophysical Journal, 763, 42
* Padovani et al. (2017) Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The Astronomy and Astrophysics Review, 25, 2
* Payne et al. (2020) Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2020
* Pelat et al. (1987) Pelat, D., Alloin, D., & Bica, E. 1987, Astronomy and astrophysics (Berlin. Print), 182, 9
* Penston & Pérez (1984) Penston, M. V. & Pérez, E. 1984, Monthly Notices of the Royal Astronomical Society, 211, 33P
* Phinney (1989) Phinney, E. 1989, Symposium - International Astronomical Union, 136, 543
* Piran et al. (2015) Piran, T., Svirski, G., Krolik, J., Cheng, R. M., & Shiokawa, H. 2015, The Astrophysical Journal, 806, 164
* Planck Collaboration XIII (2016) Planck Collaboration XIII. 2016, Astronomy & Astrophysics, 594, A13
* Rakshit et al. (2017) Rakshit, S., Stalin, C. S., Chand, H., & Zhang, X.-G. 2017, The Astrophysical Journal Supplement Series, 229, 39
* Rees (1988) Rees, M. J. 1988, Nature, 333, 523
* Ricci et al. (2020) Ricci, C., Kara, E., Loewenstein, M., et al. 2020, The Astrophysical Journal, 898, L1
* Roming et al. (2005) Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Science Reviews, 120, 95
* Ross et al. (2018) Ross, N. P., Ford, K. E. S., Graham, M., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 4468
* Roth & Kasen (2018) Roth, N. & Kasen, D. 2018, The Astrophysical Journal, 855, 54
* Roth et al. (2016) Roth, N., Kasen, D., Guillochon, J., & Ramirez-Ruiz, E. 2016, The Astrophysical Journal, 827, 3
* Rozyczka et al. (1995) Rozyczka, M., Bodenheimer, P., & Lin, D. N. C. 1995, Monthly Notices of the Royal Astronomical Society, 276, 597
* Saxton et al. (2019) Saxton, R., Motch, C., Komossa, S., et al. 2019
* Saxton et al. (2012) Saxton, R. D., Read, A. M., Esquej, P., et al. 2012, Astronomy and Astrophysics, 541, 1
* Saxton et al. (2017) Saxton, R. D., Read, A. M., Komossa, S., et al. 2017, Astronomy & Astrophysics, 598, A29
* Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, The Astrophysical Journal, 788, 48
* Shiokawa et al. (2015) Shiokawa, H., Krolik, J. H., Cheng, R. M., Piran, T., & Noble, S. C. 2015, The Astrophysical Journal, 804, 85
* Simmonds et al. (2018) Simmonds, C., Buchner, J., Salvato, M., Hsu, L.-T., & Bauer, F. E. 2018, Astronomy & Astrophysics, 618, A66
* Smith et al. (2019) Smith, K. W., Williams, R. D., Young, D. R., et al. 2019, Research Notes of the AAS, 3, 26
* Śniegowska & Czerny (2019) Śniegowska, M. & Czerny, B. 2019
* Stern et al. (2012) Stern, D., Assef, R. J., Benford, D. J., et al. 2012, The Astrophysical Journal, 753, 30
* Storchi-Bergmann et al. (2017) Storchi-Bergmann, T., Schimoia, J. S., Peterson, B. M., et al. 2017, The Astrophysical Journal, 835, 236
* Tody (1986) Tody, D. 1986, in Instrumentation in Astronomy VI, ed. D. L. Crawford, Vol. 0627, 733
* Tohline & Osterbrock (1976) Tohline, J. E. & Osterbrock, D. E. 1976, The Astrophysical Journal, 210, L117
* Tout et al. (1996) Tout, C. A., Pols, O. R., Eggleton, P. P., & Han, Z. 1996, Monthly Notices of the Royal Astronomical Society, 281, 257
* Trakhtenbrot et al. (2019a) Trakhtenbrot, B., Arcavi, I., MacLeod, C. L., et al. 2019a, The Astrophysical Journal, 883, 94
* Trakhtenbrot et al. (2020) Trakhtenbrot, B., Arcavi, I., Ricci, C., & Burke, J. 2020, Transient Name Server Classification Report, 2020-1391, 1
* Trakhtenbrot et al. (2019b) Trakhtenbrot, B., Arcavi, I., Ricci, C., et al. 2019b, Nature Astronomy, 3, 242
* Ulmer et al. (1998) Ulmer, A., Paczynski, B., & Goodman, J. 1998, Astronomy & Astrophysics, 333, 379
* Valenti et al. (2014) Valenti, S., Sand, D., Pastorello, A., et al. 2014, Monthly Notices of the Royal Astronomical Society: Letters, 438, L101
* van Velzen et al. (2011) van Velzen, S., Farrar, G. R., Gezari, S., et al. 2011, The Astrophysical Journal, 741, 73
* van Velzen et al. (2019) van Velzen, S., Gezari, S., Cenko, S. B., et al. 2019, The Astrophysical Journal, 872, 198
* van Velzen et al. (2020) van Velzen, S., Gezari, S., Hammerstein, E., et al. 2020, arXiv preprint arXiv:2001.01409
* Vestergaard & Peterson (2006) Vestergaard, M. & Peterson, B. M. 2006, The Astrophysical Journal, 641, 689
* Wang et al. (2012) Wang, T.-G., Zhou, H.-Y., Komossa, S., et al. 2012, The Astrophysical Journal, 749, 115
* Wevers et al. (2019) Wevers, T., Pasham, D. R., van Velzen, S., et al. 2019, Monthly Notices of the Royal Astronomical Society, 488, 4816
* Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, The Astronomical Journal, 140, 1868
* Young et al. (1977) Young, P. J., Shields, G. A., & Wheeler, J. C. 1977, The Astrophysical Journal, 212, 367
## Appendix A Optical spectrum and light-curve fitting
In Table 9, we list the priors adopted in the fitting of the ZTF/ SEDM light
curves, whilst in Table 10, we list the priors used in our fitting of the NOT
and WiFeS optical spectra.
Table 9: Priors adopted in the fitting of the ZTF light curves. The rise and decay timescales are in units of days, whilst $t_{\mathrm{peak}}$ is in MJD. $F_{\mathrm{max}}$ refers to the maximum observed flux within the given peak. | Priors
---|---
Peak 1 | $\log[\tau_{r,g}]\sim\mathcal{U}(0,\log[300])$, $\log[\sigma_{r,g}]\sim\mathcal{U}(0,\log[300])$ $\log[F_{\mathrm{peak},r}]\sim\mathcal{U}(\log[0.9F_{\mathrm{max},r}],\log[10F_{\mathrm{max},r}])$ $\log[F_{\mathrm{peak},g}]\sim\mathcal{U}(\log[0.9F_{\mathrm{max},g}],\log[10F_{\mathrm{max},g}])$ $t_{\mathrm{peak}}\sim\mathcal{U}(58450,58650)$
Peak 2 | $\log[\tau_{r,g}]\sim\mathcal{U}(0,\log[300])$ $\log[F_{\mathrm{peak},r}]\sim\mathcal{U}(\log[0.9F_{\mathrm{max},r}],\log[10F_{\mathrm{max},r}])$ $\log[F_{\mathrm{peak},g}]\sim\mathcal{U}(\log[0.9F_{\mathrm{max},g}],\log[10F_{\mathrm{max},g}])$ $t_{\mathrm{peak}}\sim\mathcal{U}(59000,59300)$
Table 10: Overview of the varying set of Gaussians used for modelling the emission lines in the NOT and WiFeS spectra. Region | Components
---|---
H$\gamma$ | Single Gaussian for each of H$\gamma$ and [O III] 4363 Å.
He II | Single Gaussian component for each of He II 4686 Å, and [N III] 4640 Å.
H$\beta$ | Broad and narrow Gaussian component.
H$\alpha$ | Broad and narrow Gaussian component for H$\alpha$, single Gaussian for each of [N II] 6549 and 6583 Å.
[S II] doublet | Single Gaussian for each of [S II] 6716 and 6731 Å.
[O III] 5007 Å, [Fe XIV] 5303 Å, [Fe X] 6375 Å | Single Gaussian for each.
## Appendix B Long-term light curve of AT 2019avd
In Fig. 15, we plot the long-term light curve of AT 2019avd, including the
ASAS-SN data. ASAS-SN (Shappee et al. 2014) observed the location of AT
2019avd in $V$-band from February 2012 to November 2018 and in $g$-band from
October 2017 to September 2020 (the time of writing). The $V$\- and $g$-band
observations were reduced using a fully automated pipeline detailed in
Kochanek et al. (2017) based on the ISIS image subtraction package (Alard &
Lupton 1998; Alard 2000). During each visit, ASAS-SN observed three 90-second
dithered images that are then subtracted from a reference image. For the
$g$-band we modified the standard pipeline and rebuilt the reference image
without any images with $\rm{JD}\geq 2\,458\,518$ to prevent any flux
contamination from the outbursts.
All subtractions were inspected manually to remove data with clouds, cirrus,
or other issues. We note, however, that the ASAS-SN light curve was negatively
affected by two factors. First, there is a bright nearby star that is not
resolved from the host galaxy in ASAS-SN data and added noise to the
subtractions. Second, the location of AT 2019avd is right on the edge of two
ASAS-SN fields. To help alleviate these issues and increase the ASAS-SN
limiting magnitude we stacked the subtractions within a maximum of 10 days. We
then used the IRAF package apphot to perform aperture photometry with a two-
pixel, or approximately $16.\\!\\!^{\prime\prime}0$, radius aperture on each
subtracted image, generating a differential light curve. The photometry was
calibrated using the AAVSO Photometric All-Sky Survey (Henden et al. 2015).
Figure 15: Long-term NEOWISE-R, ASAS-SN, and ZTF light curves of AT 2019avd.
The early and late black dashed lines mark the 2015 XMM-Newton pointed and the
2020 eROSITA eRASS1 observations respectively. The early and late grey dashed
lines mark the MJD that the NOT and first FLOYDS spectra were taken.
|
# Supersymmetries in non-equilibrium Langevin dynamics
Bastien Marguet Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS,
Université de Lyon 69622 Villeurbanne, France Université Grenoble Alpes,
CNRS, LIPhy, 38000 Grenoble, France Elisabeth Agoritsas Institute of
Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland Léonie Canet Université Grenoble Alpes, CNRS, LPMMC, 38000
Grenoble, France Institut Universitaire de France, 1 rue Descartes, 75005
Paris, France Vivien Lecomte<EMAIL_ADDRESS>Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
(August 27, 2024)
###### Abstract
Stochastic phenomena are often described by Langevin equations, which serve as
a mesoscopic model for microscopic dynamics. It is known since the work of
Parisi and Sourlas that _reversible_ (or equilibrium) dynamics present
supersymmetries (SUSYs). These are revealed when the path-integral action is
written as a function not only of the physical fields, but also of Grassmann
fields representing a Jacobian arising from the noise distribution. SUSYs
leave the action invariant upon a transformation of the fields that mixes the
physical and the Grassmann ones. We show that, contrarily to the common
belief, it is possible to extend the known reversible construction to the case
of arbitrary _irreversible_ dynamics, for overdamped Langevin equations with
additive white noise – provided their steady state is known. The construction
is based on the fact that the Grassmann representation of the functional
determinant is not unique, and can be chosen so as to present a generalization
of the Parisi–Sourlas SUSY. We show how such SUSYs are related to time-
reversal symmetries and allow one to derive modified fluctuation-dissipation
relations valid in non-equilibrium. We give as a concrete example the results
for the Kardar–Parisi–Zhang equation.
The dynamics of a large number of elementary constituents can often be
described by mesoscopic stochastic equations of motion, where the effects of
interactions at small scales are accounted for by friction and noise. Such an
effective Langevin [1] description applies to various examples ranging from
particles in a fluid to chemical or economical processes [2, 3] or
cosmological inflation [4, 5].
Field theory then allows one to write the probability of trajectories followed
by the system using a path-integral representation that encompasses both
classical and quantum problems [6]. The weight of a trajectory takes the form
of the exponential of (minus) an action. It is convenient to make the action
depend not only on the physical fields, but also on non-commuting auxiliary
ones – known as Grassmann fields – representing a Jacobian arising from the
noise distribution. This action possesses a generic ‘supersymmetry’ (SUSY),
known as the BRST (Becchi–Rouet–Stora–Tyutin) symmetry [7, 8, 9, 10]. It
encodes the conservation of probability. Also, when the dynamics is
_reversible_ (_i.e._ forces derive from a potential), a second SUSY was
uncovered by Parisi–Sourlas [11] and by Feigel’man–Tsvelik [12] (after a
similar SUSY was found for the partition function of equilibrium problems
[13]).
Such SUSYs, that mix physical and Grassmann fields, look surprising in a
statistical mechanical context; yet, as other symmetries in Physics, they
turned out to be a powerful tool to study a variety of problems. These range
from the dynamics of spin glasses [14, 15], disordered spin models [16] or
heteropolymers [17], to finite-size effects in critical dynamics [18],
localization [19], renormalization of the random-field Ising model [20, 21,
22], symmetries of Hamiltonian dynamics [23, 24], and metastability in
overdamped [25] and inertial [26] Langevin dynamics, with Witten’s SUSY
version of Morse theory [27]. SUSYs have methodological implications for
renormalization [28] and the derivation of variational principles [29] or of
the Parisi–Wu stochastic quantization [30, 31, 32]. The Parisi–Sourlas SUSY
implies Ward identities yielding the equilibrium fluctuation-dissipation
relation (FDR) [33, 34]. When the dynamics is irreversible, the BRST symmetry
remains valid, but the Parisi–Sourlas one is broken _e.g._ by a driving field
[35, 36] or a colored noise [37]. It has been argued indeed that
microreversibility is at the origin of SUSY [34].
In this paper, we prove the contrary, by extending the previously known
results to the case of arbitrary non-equilibrium Langevin dynamics (in the
overdamped limit and for additive Gaussian white noise). We assume that the
stationary distribution exists and our construction depends explicitly on it.
The key observation is that there are several inequivalent ways to represent
the same Jacobian through Grassmann fields, and we identify one that presents
an extended SUSY generalizing the Parisi–Sourlas one. We show that the
associated Ward identities yield modified FDRs, recovering some known cases
[38, 39, 40]. Then, we explain how this SUSY is directly related to a time-
reversal symmetry between the original Langevin dynamics and its ‘adjoint’. We
identify the mathematical structure at the origin of the extended SUSY. The
construction can be carried out both in the Martin–Siggia–Rose–Janssen–de
Dominicis (MSRJD) framework [41, 42, 43, 44, 45], and in the Onsager–Machlup
one [46, 47], where it takes a particularly simple form. We finally discuss
the cases of spatially correlated noise, continuum space, and the example of
the Kardar–Parisi–Zhang (KPZ) equation [48].
## I BRST SUSY
Consider a set of scalar fields $h_{i}(t)$ evolving in time according to a
Langevin equation
$\partial_{t}h_{i}=f_{i}[h]+\eta_{i}$ (1)
where $f_{i}[h]$ is a deterministic force function of the fields $h=(h_{i})$
at time $t$, and $\eta_{i}(t)$ a centered Gaussian white noise with
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2T\delta_{ij}\delta(t^{\prime}-t)$
[the generalization to anisotropic correlated noise is detailed below]. For
instance $h_{i}(t)$ represents the spatial coordinate of a particle tagged by
a discrete index $i$, or the value of the height of an interface on a lattice
site $i$ (as in the KPZ equation). Eq. (1) is equivalent to a Fokker–Planck
evolution $\partial_{t}P[h,t]=\mathbb{W}P[h,t]$ for the distribution $P[h,t]$
of $h$, with
$\mathbb{W}\,\cdot=-\partial_{i}\big{[}f_{i}[h]\kern-0.75pt\cdot-T\partial_{i}\kern-0.75pt\cdot\kern-0.75pt\big{]}\>.$
(2)
We denote $\partial_{i}\equiv\frac{\partial}{\partial h_{i}}$ and use implicit
summation over repeated indices (including in squares such as $X_{i}^{2}$). We
assume that the dynamics possesses a stationary distribution
${P}_{{\text{st}}}[h]$ such that $\mathbb{W}P_{{\text{st}}}=0$, and define a
functional $\mathcal{H}[h]$ by
${P}_{{\text{st}}}[h]\propto\text{e}^{-\frac{1}{T}\mathcal{H}[h]}$. This is
the so-called quasi-potential, which exists under generic conditions [49].
Then, following Graham [50] and Eyink _et al._ [51], we decompose the total
force as the sum of a conservative force $-\partial_{i}\mathcal{H}[h]$ and a
driving force $g_{i}[h]$ as
$f_{i}[h]=-\partial_{i}\mathcal{H}[h]+g_{i}[h]\,.$ (3)
The case of reversible dynamics is recovered for $g_{i}[h]\equiv 0$. This
decomposition is generic when the quasi-potential exists. From (2), the
stationary condition $\mathbb{W}P_{{\text{st}}}=0$ is equivalent to an
identity that will be used thoroughly:
$\partial_{i}g_{i}[h]=\frac{1}{T}g_{i}[h]\,\partial_{i}\mathcal{H}[h]\,.$ (4)
We consider the distribution of fields on a finite time window
$[0,t_{\text{f}}]$ and denote $\int_{t}=\int_{0}^{t_{\text{f}}}\\!\text{d}t$
(but this time window can also be $\mathbb{R}$). The path-integral
representation [6] of the trajectory probability follows from a mere change of
variable from the Gaussian noise distribution
$\text{Prob}[\eta]\propto\text{e}^{-\int_{t}\eta_{i}^{2}/(4T)}$ to that of the
field $h$, seen from the Langevin equation (1) as a functional of the noise:
$\displaystyle P[h]=\Big{|}\frac{\delta\eta}{\delta
h}\Big{|}\,\text{e}^{-\frac{1}{4T}\int_{t}\eta_{i}[h]^{2}},\quad\eta_{i}[h]\equiv\partial_{t}h_{i}-f_{i}[h]\,.$
(5)
Here $\eta_{i}[h]$ is the expression of the noise as a function of $h$ in the
Langevin equation (1), and $\big{|}\\!\frac{\delta\eta}{\delta
h}\\!\big{|}=\big{|}\\!\det\frac{\delta\eta_{i}[h(t)]}{\delta
h_{j}\kern-0.75pt(t^{\prime})}\big{|}$ is the functional Jacobian of the
change of variables from $\eta$ to $h$. We emphasize that, even if the
Langevin equation (1) is additive and does not depend on its time
discretization, the expressions of the Jacobian and of the path-integral
action do depend on the discretization chosen to write them [52, 53, 54]. We
adopt the Stratonovich convention, that allows one to use the rules of
calculus in the path integral [55], and to reverse time without changing the
discretization [56, 57]. Following Janssen [41], one then linearizes the
square in the exponent of (5) using a ‘response field’
$\smash{\hat{h}_{i}(t)}$ to obtain the MSRJD action. Introducing anticommuting
Grassmann fields $\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}(t)$ and $\Psi_{\kern-0.75pti}(t)$ [58] to represent
$\big{|}\\!\frac{\delta\eta}{\delta h}\\!\big{|}$, we get
$\displaystyle P[h]$
$\displaystyle=\int\mathcal{D}\hat{h}\mathcal{D}\Psi\mathcal{D}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\>\text{e}^{-S_{\text{\tiny{SUSY}}}}$ (6) $\displaystyle
S_{\text{\tiny{SUSY}}}$ $\displaystyle=\int_{t}\Big{\\{}\hat{h}_{i}\kern
0.75pt\eta_{i}[h]-T\hat{h}_{i}^{2}-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi\Big{\\}}.$ (7)
The response field $\smash{\hat{h}_{i}}$ is integrated on the imaginary axis,
and $\eta_{i}\kern-0.75pt\text{{'}}[h]$ is the Fréchet derivative of
$\eta_{i}[h]$ which is a linear operator acting on the vector $\Psi$ as
$\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi=\partial_{j}\eta_{i}[h]\Psi_{\kern-0.75ptj}$
111 It is more rigorously defined as
$\varphi_{i}[h+h^{1}]=\varphi_{i}[h]+\varphi_{i}[h]\text{{'}}h^{1}+o(h^{1})$,
implying that $\partial_{t}(\varphi[h])=\varphi\text{{'}}\partial_{t}h$, and
$\varphi_{i}[h+\varepsilon\Psi]=\varphi_{i}[h]+\varepsilon\varphi_{i}[h]\text{{'}}\Psi$.
One has for instance for $\eta_{i}[h]$ defined in Eq. (5):
$\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi=\big{(}\delta_{ij}\partial_{t}-\partial_{j}f_{i}[h]\big{)}\Psi_{\kern-0.75ptj}$.
. The BRST SUSY, which originates in the conservation of probability, is a
Grassmann symmetry: it depends on a Grassmann parameter $\varepsilon$ that
allows one to mix the anticommuting Grassmann and the commuting physical
fields as $h\mapsto h+\delta h$, $\hat{h}\mapsto h+\delta\hat{h}$, etc., with
$\text{BRST:}\ \delta
h_{i}=\varepsilon\Psi_{\kern-0.75pti}\quad\>\delta\hat{h}_{i}=0\quad\>\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}=\varepsilon\hat{h}_{i}\quad\>\delta\Psi_{\kern-0.75pti}=0\,.$
(8)
$S_{\text{\tiny{SUSY}}}$ is invariant under (8), since
$\delta(\eta_{i}[h])=\eta_{i}\kern-0.75pt\text{{'}}[h]\delta
h=\varepsilon\kern 0.75pt\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi$. (We denote
$\delta(X)=X[h+\delta h,...]-X[h,...]$).
## II Extended Parisi–Sourlas SUSY
When forces derive from a potential ($g_{i}[h]\equiv 0$), another SUSY was
found by Parisi–Sourlas [11] and by Feigel’man–Tsvelik [12], in relation with
the former work of Nicolai [60, 61, 62] (see [63]). It yields the equilibrium
FDR [33, 34] (as discussed below). We now extend these results to the generic
Langevin dynamics (1). The key observation is that one can identify a
Grassmann action, different from (7), but that still fully represents the
Langevin equation (1) and possesses a SUSY:
$\displaystyle S_{\text{\tiny{SUSY}}}^{\dagger}$
$\displaystyle=\\!\int_{t}\\!\Big{\\{}\hat{h}_{i}\kern
0.75pt\eta_{i}-T\hat{h}_{i}^{2}+\frac{1}{T}g_{i}\kern
0.75pt\partial_{i}\mathcal{H}-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\tilde{\eta}_{i}\kern-0.75pt\text{{'}}^{\dagger}\Psi\Big{\\}}$
(9) $\displaystyle\tilde{\eta}_{i}[h]$
$\displaystyle=\partial_{t}h_{i}+\partial_{i}\mathcal{H}[h]+g_{i}[h]$ (10)
(for compacity we drop some dependencies in $h$). For an operator $A$, we set
$(A^{\dagger})_{ij}=A_{ji}$ 222Hence, explicitly: $\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\tilde{\eta}_{i}\kern-0.75pt\text{{'}}^{\dagger}\Psi=\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}(\delta_{ij}\partial_{t}+\partial_{ij}\mathcal{H}+\partial_{i}g_{j})\Psi_{\kern-0.75ptj}$
. The Grassmann part of $S_{\text{\tiny{SUSY}}}^{\dagger}$ is involving
$\tilde{\eta}_{i}[h]$, whose signification as the noise of an ‘adjoint’
dynamics becomes clear below when relating SUSYs to time reversal.
For a reversible dynamics ($g_{i}[h]\equiv 0$), one sees that
$S_{\text{\tiny{SUSY}}}=S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$: the
actions (7) and (9) are identical. For an arbitrary irreversible dynamics
($g_{i}[h]\neq 0$), one has $S_{\text{\tiny{SUSY}}}\neq
S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$, and yet, as we now show in
detail, _the actions_ (7) _and_ (9) _represent the same Langevin equation_ (1)
(_and thus the same Fokker–Planck operator_ (2)). This is due to the fact that
when integrating over $\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu,\Psi$, the extra term
$\frac{1}{T}g_{i}\kern 0.75pt\partial_{i}\mathcal{H}$ in (9) ensures that the
Jacobian $\big{|}\\!\frac{\delta\eta}{\delta h}\\!\big{|}$ is correctly
represented. To show this, we first recall that in Stratonovich discretization
[65, 66, 42, 67, 28, 68, 69, 70, 71]:
$\Big{|}\dfrac{\delta\eta[h]}{\delta
h}\Big{|}=\exp\Big{\\{}-\frac{1}{2}\int_{t}\operatorname{tr}f_{i}\text{{'}}[h]\Big{\\}}$
(11)
which can be obtained by time discretization 333Indeed, discretizing with a
time step $\Delta t$, one has $\eta_{i}[h]_{t}=\frac{h_{i,t+\Delta
t}-h_{i,t}}{\Delta t}-f_{i}[h]\big{|}_{h=\frac{1}{2}(h_{i,t+\Delta
t}+h_{i,t})}$ where the time is in index (and discretization is Stratonovich).
Hence the matrix of coordinates $(i,t;j,t^{\prime})$ in the definition of the
Jacobian after Eq. (5) is upper triangular in the time direction (this is
causality), so that only its equal-time components matter. Importantly, since
the time-discrete Langevin equation is read as $h_{t+\Delta t}$ function of
$h_{t}$ and $\eta_{t}$, one must pay attention that the change of variables is
between $h_{t+\Delta t}$ and $\eta_{t}$. Its Jacobian is thus
$\partial\eta_{i}[h]_{t}/\partial h_{j,t+\Delta t}=\frac{1}{\Delta
t}\delta_{ij}-\frac{1}{2}\partial_{j}f_{i}[h]$. Factorizing by
$\frac{1}{\Delta t}$ [which yields a field-independent normalization factor of
the Jacobian], using the formula $\log\det=\operatorname{tr}\log$, one thus
obtains $\log\big{|}\\!\frac{\delta\eta}{\delta
h}\\!\big{|}=\sum_{t}\operatorname{tr}\log(\mathbf{1}-\frac{1}{2}\Delta
tf_{i}\text{{'}}[h])$. Expanding at small $\Delta t$, one recovers Eq. (11). ,
and where the trace is
$\operatorname{tr}f_{i}\text{{'}}[h]=\partial_{i}f_{i}[h]\,$. The time
discretization of $\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu,\Psi$ in (7) is crucial to correctly represent the Jacobian (11)
444Denoting by $X^{\text{\tiny{s}}}_{t}=\frac{X_{t+\Delta t}+X_{t}}{2}$ the
Stratonovich discretization, $\int_{t}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi=\int_{t}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}(\delta_{ij}\partial_{t}-\partial_{j}\kern-0.75ptf_{i}[h])\Psi_{\kern-0.75ptj}$
must be discretized as $\sum_{t}\\!\Delta t\kern 0.75pt\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti,t+\Delta t}\big{(}\frac{\Psi_{\kern-0.75pti,t+\Delta
t}-\Psi_{\kern-0.75pti,t}}{\Delta
t}-\partial_{j}\kern-0.75ptf_{i}[h^{\text{\tiny{s}}}_{t}]\Psi^{\text{\tiny{s}}}_{\kern-0.75ptj,t}\big{)}=\sum_{tt^{\prime}}\\!\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti,t^{\prime}}M_{i,t^{\prime};j,t}\Psi_{\kern-0.75ptj,t}$
with the matrix elements given by
$M_{i,t^{\prime}\\!;j,t}=\delta_{ij}(\delta_{t^{\prime}\\!,t}-\delta_{t^{\prime}\\!,t+\Delta
t})-\\!\Delta t\kern
0.75pt\partial_{j}\kern-0.75ptf_{i}[h_{t^{\prime}}]\frac{\delta_{t^{\prime}\\!,t}+\delta_{t^{\prime}\\!,t+\Delta
t}}{2}$. As the Grassmann integral yields the determinant of $M$, and as $M$
is triangular in the time coordinate, only the diagonal $t^{\prime}=t$ matters
and $\det M=\sum_{t}\det(\delta_{ij}-\\!\Delta t\kern
0.75pt\partial_{j}f_{i}[h])$. One thus recovers the Jacobian [72]. .
Integrating over the fields $\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu,\Psi$ in
$\text{e}^{-S_{\text{\tiny{SUSY}}}^{\dagger}}$ yields
$|\kern-0.75pt\frac{\delta\tilde{\eta}}{\delta
h}\kern-0.75pt|=\text{e}^{\frac{1}{2}\\!\int_{t}\\!\operatorname{tr}\kern
0.75pt(\partial_{i}\mathcal{H}+g_{i})\text{{'}}[h]}$, see Eq. (11). Thus, the
contribution of the last two terms in (9) is
$\text{e}^{-\int_{t}\\{\frac{1}{T}g_{i}\partial_{i}\kern-0.75pt\mathcal{H}-\frac{1}{2}\operatorname{tr}(\partial_{i}\mathcal{H}+g_{i})\text{{'}}\\}}$
but this expression is in fact equal to the Jacobian (11), because the
stationary condition (4) implies $\frac{1}{T}g_{i}\kern
0.75pt\partial_{i}\kern-0.75pt\mathcal{H}=\operatorname{tr}g_{i}\kern-0.75pt\text{{'}}$.
We thus have shown $\int\mathcal{D}\hat{h}\mathcal{D}\Psi\mathcal{D}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\>\text{e}^{-S_{\text{\tiny{SUSY}}}}=\int\mathcal{D}\hat{h}\mathcal{D}\Psi\mathcal{D}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\>\text{e}^{-S_{\text{\tiny{SUSY}}}^{\dagger}}$.
Hence, despite being different in general, the actions
$S_{\text{\tiny{SUSY}}}$ and $S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ both
correctly represent the trajectory probability of the Langevin equation (1)
(and we denote by $\langle\kern 0.75pt\cdot\kern 0.75pt\rangle$ and
$\langle\kern 0.75pt\cdot\kern 0.75pt\rangle^{\kern-0.75pt\smash{{\dagger}}}$
the corresponding averages). Physically, this means that observables depending
only on $h$ and $\smash{\hat{h}}$ have the same average:
$\langle\mathcal{O}[h,\hat{h}]\rangle=\langle\mathcal{O}[h,\hat{h}]\rangle^{\kern-0.75pt{\dagger}}$.
This is of course not the case if $\mathcal{O}\kern-0.75pt$ depends on $\Psi$
or $\smash{\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu}$.
This freedom of representation originates in the fact that the Jacobian
depends only on the diagonal components of the operator
$\eta_{i}\kern-0.75pt\text{{'}}[h]$ through the trace
$\operatorname{tr}f_{i}\kern-0.75pt\text{{'}}[h]=\partial_{i}f_{i}[h]$, and
not on all its components
$(\eta_{i}\kern-0.75pt\text{{'}}[h])_{j}=\delta_{ij}\partial_{t}-\partial_{j}f_{i}[h]$
555This explains why one cannot transform $S_{\text{\tiny{SUSY}}}$ into
$\smash{S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}}$: these actions contain
the same information after integrating on the Grassmann fields, but not
before. .
Then, one checks by direct computation that
$\text{PS}_{1}\\!\kern-0.75pt:\left\\{\begin{aligned} &\delta
h_{i}=\varepsilon T\kern 0.75pt\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\qquad\delta\hat{h}_{i}=\varepsilon\big{(}\delta_{ij}\partial_{t}-\partial_{j}g_{i}[h]\big{)}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75ptj}\\\
&\delta\Psi_{\kern-0.75pti}=\varepsilon\big{(}\partial_{t}h_{i}-g_{i}[h]-T\hat{h}_{i}\big{)}\qquad\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}=0\end{aligned}\right.\\!\\!$ (12)
leaves $S^{\smash{{\dagger}}}_{\text{\tiny{SUSY}}}$ invariant, up to time
boundary terms. This SUSY generalizes the Parisi–Sourlas one to arbitrary
irreversible dynamics (1) since, for reversible dynamics ($g_{i}[h]\equiv 0$)
we have $S_{\text{\tiny{SUSY}}}^{\dagger}=S_{\text{\tiny{SUSY}}}$, and (12)
yields the known SUSY [11]. An important difference with the reversible case
$g_{i}[h]\equiv 0$ is that this transformation is now non-linear in general,
because of the terms $\propto g_{i}[h]$ in (12). We also uncover a dual SUSY
$\text{PS}_{2}\\!:\left\\{\begin{aligned} &\delta h_{i}=\varepsilon T\kern
0.75pt\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\qquad\delta\hat{h}_{i}=\varepsilon\partial_{ij}\mathcal{H}[h]\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75ptj}\\\
&\delta\Psi_{\kern-0.75pti}=-\varepsilon\big{(}{\partial_{i}\mathcal{H}[h]}-T\hat{h}_{i}\big{)}\qquad\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}=0\end{aligned}\right.$ (13)
which seems to have been unnoticed even for $g_{i}[h]\equiv 0$ (perhaps
because it is non-linear, even in this case).
We emphasize that this construction can also be formulated using the
superfield, with explicit expressions for the generators of PS1,2 [75]. One
can also transpose it to the Onsager–Machlup formalism straightforwardly:
indeed the passage from the MSRJD to the Onsager–Machlup action is done by
integrating over the response field, which amounts to replacing $\hat{h}$ by
its optimal value $\hat{h}^{{\text{opt}}}=\frac{1}{2T}\eta[h]$ [75]. The
corresponding SUSY transformation is obtained likewise, as made explicit
below.
The non-equilibrium SUSY we derived is more intricate than in equilibrium,
since it involves two actions ($S_{\text{\tiny{SUSY}}}$ invariant only under
BRST, and $S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ only under PS1,2), and
depends explicitly on the steady state. However, it allows one to derive
physical consequences, as shown now.
## III Modified FDRs
Symmetries of the action imply Ward identities for correlation functions:
denoting $h_{1}=h_{i_{1}\\!}(t_{1})$ (and similarly for other indices,
functions or operators), the BRST symmetry (8) implies in particular $\langle
h_{1}\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{2}\rangle=\langle(h_{1}+\delta h_{1})(\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{2}+\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{2})\rangle$, hence
$\langle h_{1}\delta\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{2}\rangle+\langle\delta h_{1}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{2}\rangle=0$ which
means:
$\langle h_{1}\hat{h}_{2}\rangle=-\langle\Psi_{1}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{2}\rangle$ (14)
and we find that the 2-point correlator of the Grassmann fields is a response
function. In particular, these correlators are 0 for $t_{1}<t_{2}$. From the
invariance of $\langle h_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ under the
SUSYs PS1,2 we similarly infer:
$\displaystyle\big{\langle}h_{1}\big{(}\partial_{t}h_{2}-g_{2}[h_{2}]\big{)}\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle-T\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ (15)
$\displaystyle\big{\langle}h_{1}\kern
0.75pt\partial_{2}\kern-0.75pt\mathcal{H}[h_{2}]\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle+T\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ (16)
where we used that for observables independent of $\Psi,\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$, the actions
$S_{\text{\tiny{SUSY}}}$ and $S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$
yield the same averages. The causal structure of the Grassmann contribution to
$S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ shows that $\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}=0$ for $t_{1}>t_{2}$ 666With
the notations of [73], we have that $\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{i,t^{\prime}}\Psi_{j,t}\rangle^{\\!{\dagger}}={(M^{T})^{-1}}_{i,t^{\prime};j,t}$,
but $M$ is lower triangular in the time direction, so that $(M^{T})^{-1}$ is
upper triangular. This implies the causality $\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{i,t^{\prime}}\Psi_{j,t}\rangle^{\\!{\dagger}}=0$ for $t^{\prime}>t$.
(which can also be inferred from the interpretation of $\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ as a response function in
the adjoint dynamics, see below). We thus obtain two modified FDRs:
$\displaystyle\big{\langle}h_{1}\big{(}\partial_{t}h_{2}-g_{2}[h_{2}]\big{)}\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle\qquad\text{if }t_{1}>t_{2}$
(17) $\displaystyle\big{\langle}h_{1}\kern
0.75pt\partial_{2}\kern-0.75pt\mathcal{H}[h_{2}]\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle\qquad\text{if }t_{1}>t_{2}\,.$
(18)
Note that adding (15) and (16), or (17) and (18), one obtains $\langle
h_{1}(\eta_{2}[h_{2}]-2T\hat{h}_{2})\rangle=0$ which is always valid, as can
be checked using
$\delta\kern-0.75pt(\text{e}^{-S_{\text{\tiny{SUSY}}}})/\delta\hat{h}_{2}=(\eta_{2}[h_{2}]-2T\hat{h}_{2})\kern
0.75pt\text{e}^{-S_{\text{\tiny{SUSY}}}}$ and a functional integration by
part.
Since the r.h.s. of the relation (17) is the response function $\langle
h_{1}\hat{h}_{2}\rangle=\langle\delta
h_{1}/\delta\mathfrak{f}_{2}\rangle_{\mathfrak{f}=0}$ to a perturbation
$f\mapsto f+\mathfrak{f}$ of the total force, this relation entails a modified
FDR, valid in non-equilibrium (the equilibrium one, $\langle
h_{1}\partial_{t}h_{2}\rangle=T\langle h_{1}\hat{h}_{2}\rangle$, is recovered
for $g[h]\equiv 0$, and can be derived from the Parisi–Sourlas SUSY [33, 34]).
A relation similar in spirit was derived in [50, 51], but in a particular
setting where the perturbation is acting only on the conservative part of the
force, so that the l.h.s. of (17) has no contribution from $g[h]$. One checks
that (17)-(18) are equivalent to the Agarwal FDR [38] and its equivalent
formulations (e.g. [77, 49, 39, 78, 79, 80, 40, 81, 81, 82, 83]). Also, Eqs.
(17)-(18) and other Ward identities can be read as providing information on
the quasi-potential, when it is not known.
## IV Structure of the extended SUSY
Noting that $\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}(\partial_{i}\mathcal{H}+g_{i})\text{{'}}[h]^{\kern-0.75pt{\dagger}}\Psi=-\Psi_{\kern-0.75pti}(\partial_{i}\mathcal{H}+g_{i})\text{{'}}[h]\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$, and that
integrating by parts $\int_{t}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pti}\partial_{t}\Psi_{\kern-0.75pti}=\int_{t}\Psi_{\kern-0.75pti}\partial_{t}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti}$, we
define a new action
$\mathbb{S}_{\text{\tiny{SUSY}}}^{\dagger}\kern-0.75pt=\\!S_{\text{\tiny{SUSY}}}^{\dagger}-\frac{1}{T}\\!\big{[}\mathcal{H}[h]\kern
0.75pt\big{]}_{0}^{t_{\text{f}}}\kern-0.75pt\kern-0.75pt$ which writes
$\displaystyle\mathbb{S}_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$
$\displaystyle=\\!\\!\int_{t}\\!\Big{\\{}\kern-0.75pt\hat{h}_{i}\kern
0.75pt\eta_{i}\\!-\\!T\hat{h}_{i}^{2}\\!-\\!\frac{1}{T}\big{(}\partial_{t}h_{i}\\!-\\!g_{i}\big{)}\partial_{i}\mathcal{H}\\!+\\!\Psi_{\kern-0.75pti}\bar{\eta}_{i}\kern-0.75pt\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\Big{\\}}$ (19)
$\displaystyle\bar{\eta}_{i}[h]$
$\displaystyle=-\partial_{t}h_{i}+\partial_{i}\mathcal{H}[h]+g_{i}[h]\>.$ (20)
With $\frac{\eta_{i}+\bar{\eta}_{i}}{2}=\partial_{i}\mathcal{H}$ and
$\frac{\eta_{i}-\bar{\eta}_{i}}{2}=\partial_{t}h_{i}-g_{i}$, one obtains
$\displaystyle\mathbb{S}_{\text{\tiny{SUSY}}}^{\dagger}=\\!\\!\int_{t}\\!\Big{\\{}\kern-0.75pt\\!-\\!\frac{1}{T}\Big{(}\kern-0.75pt\frac{\eta_{i}-\bar{\eta}_{i}}{2}\\!-\\!T\hat{h}_{i}\\!\Big{)}\\!\Big{(}\kern-0.75pt\frac{\eta_{i}+\bar{\eta}_{i}}{2}\\!-\\!T\hat{h}_{i}\\!\Big{)}\kern-0.75pt\\!+\\!\Psi_{\kern-0.75pti}\bar{\eta}_{i}\kern-0.75pt\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\kern-0.75pt\Big{\\}}\\!\\!\\!\\!\\!$ (21)
Such a rewriting renders manifest that
$\mathbb{S}_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ is invariant under the
SUSYs PS${}_{1,2}\\!$ (without generating any boundary term). Indeed from
(12):
$\text{PS}_{1}\Rightarrow\left\\{\begin{aligned}
&\delta\kern-0.75pt\big{(}\tfrac{\eta_{i}-\bar{\eta}_{i}}{2}-T\hat{h}_{i}\big{)}=0,\quad\delta\Psi_{\kern-0.75pti}=\varepsilon\delta\kern-0.75pt\big{(}\tfrac{\eta_{i}-\bar{\eta}_{i}}{2}-T\hat{h}_{i}\big{)}\\\
&\delta\kern-0.75pt\big{(}\tfrac{\eta_{i}+\bar{\eta}_{i}}{2}-T\hat{h}_{i}\big{)}=\varepsilon
T\bar{\eta}_{i}\kern-0.75pt\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\end{aligned}\right.$
(22)
so that the variations of the two products in (21) cancel each other very
simply. PS2 presents a similar structure with the roles of
$\eta_{i}+\bar{\eta}_{i}$ and $\eta_{i}-\bar{\eta}_{i}$ exchanged. The
identified structure explains how $\partial_{i}\mathcal{H}$ and the ‘covariant
derivative’ $\partial_{t}h-g[h]$ (see [84] for KPZ) play a dual role in the
SUSYs PS1,2 and in the modified FDRs (15)-(16).
The actions $S_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ and
$\mathbb{S}_{\text{\tiny{SUSY}}}^{\smash{{\dagger}}}$ have an equivalent
physical content as they are equal up to time-boundary terms. A careful
treatment of these shows that the averages in (15)-(16) on a finite time
window are those sampled by the steady state $P_{\text{st}}[h]$ at initial
time [75].
In the Onsager–Machlup formalism, the corresponding actions are particularly
simple: $S_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\eta^{2}}{4T}-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\eta\text{{'}}\Psi\big{\\}}$ and
$\mathbb{S}^{\dagger}_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\bar{\eta}^{2}}{4T}+\Psi\bar{\eta}\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\big{\\}}$, with
$S_{\text{\tiny{OM}}}$ verifying the BRST SUSY (8), and
$\mathbb{S}^{\dagger}_{\text{\tiny{OM}}}$ being invariant by the PS SUSY
$\delta h=\varepsilon T\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$,
$\delta\Psi=-\frac{\varepsilon}{2}\bar{\eta}$, $\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu=0$ corresponding to
PS1,2.
## V Time reversal without Grassmann
One can also represent $P[h]$ as a path integral on the response field only,
$P[h]=\int\mathcal{D}\hat{h}\>\text{e}^{-S_{\text{\tiny{MSR}}}}$, with the
Jacobian contribution (11) included in the action:
$\displaystyle S_{\text{\tiny{MSR}}}[h,\hat{h}]$
$\displaystyle=\int_{t}\Big{\\{}\hat{h}_{i}\kern
0.75pt\eta_{i}[h]-T\hat{h}_{i}^{2}+\tfrac{1}{2}\partial_{i}f_{i}[h]\Big{\\}}\kern
0.75pt.$ (23)
Consider a time reversal of the field
$h_{i}(t)=h^{\smash{\text{S}}}_{i}\kern-0.75pt(t_{\text{R}})$ (with
$t_{\text{R}}=t_{\text{f}}-t$) combined with either one of these two response-
field transformations (denoting $\dot{\varphi}=\partial_{t}\varphi$)
$\displaystyle\text{TR}_{1}\\!:\ \hat{h}_{i}(t)$
$\displaystyle=\hat{h}_{i}^{\text{S}}\kern-0.75pt(t_{\text{R}})-\tfrac{1}{T}\big{(}\dot{h}_{i}^{\text{S}}\kern-0.75pt(t_{\text{R}})+g_{i}[h^{\text{S}}]\big{)}$
(24) $\displaystyle\text{TR}_{2}\\!:\ \hat{h}_{i}(t)$
$\displaystyle=-\hat{h}_{i}^{\text{S}}\kern-0.75pt(t_{\text{R}})+\tfrac{1}{T}\partial_{i}\mathcal{H}[h^{\text{S}}]\>.$
(25)
The adjoint process [78, 85] of (1) is the one with a force
$\tilde{f}_{i}[h]=-\partial_{i}\mathcal{H}[h]-g_{i}[h]$ instead of $f_{i}[h]$.
It is the process followed by time-reversed trajectories [86]. The action
$\tilde{S}_{\text{\tiny{MSR}}}$ of the adjoint process present a mapping with
$S_{\text{\tiny{MSR}}}$:
$S_{\text{\tiny{MSR}}}[h,\hat{h}]=\tilde{S}_{\text{\tiny{MSR}}}[h^{\text{S}},\hat{h}^{\text{S}}]-\tfrac{1}{T}\big{[}\mathcal{H}[h^{\text{S}}]\big{]}_{0}^{t_{\text{f}}}\,.$
(26)
To derive it, one uses the stationary condition (4), and we stress that the
Jacobian and non-Jacobian contributions to the action (23) interfere. TR1,2
imply respectively
$\displaystyle\big{\langle}h_{1}\big{(}\partial_{t}h_{2}-g_{2}[h_{2}]\big{)}\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle-T\langle
h^{\text{R}}_{1}\hat{h}^{\text{R}}_{2}\tilde{\rangle}$ (27)
$\displaystyle\big{\langle}h_{1}\kern
0.75pt\partial_{2}\kern-0.75pt\mathcal{H}[h_{2}]\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle+T\langle
h^{\text{R}}_{1}\hat{h}^{\text{R}}_{2}\tilde{\rangle}$ (28)
where the superscript $\cdot\kern 0.75pt\vphantom{|}^{\smash{\text{R}}}$
indicates that the field is time-reversed and $\langle\kern 0.75pt\cdot\kern
0.75pt\smash{\tilde{\rangle}}$ is the average for the adjoint process. These
relations imply the modified FDRs (17)-(18), because $\langle
h^{\text{R}}_{1}\hat{h}^{\text{R}}_{2}\tilde{\rangle}=0$ for $t_{1}>t_{2}$ (as
this response function is causal). Note that these modified FDRs were derived
above from PS1,2, which are infinitesimal Grassmann SUSYs, in contrast to
TR1,2 which are discrete symmetries. The mapping (26) also allows one to
recover that $\text{e}^{-\mathcal{H}/T}$ is the steady state [75]. Comparing
(27)-(28) to (15)-(16), we also identify the Grassmann correlator
$\langle\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ for
$\mathbb{S}_{\text{\tiny{SUSY}}}^{\dagger}$ as being equal to the time-
reversed response function $\langle
h^{\text{R}}_{1}\hat{h}^{\text{R}}_{2}\tilde{\rangle}$ in the adjoint
dynamics. This allows one to relate such Grassmann correlators to physical
correlation and response functions.
As we now show, this can be inferred from a BRST SUSY. One can check by direct
computation that either of the time-reversal transformations TR1,2 yields:
$\mathbb{S}^{\dagger}[h,\hat{h},\Psi,\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu]=\tilde{S}_{\text{\tiny{SUSY}}}[h^{\text{S}},\hat{h}^{\text{S}},-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu^{\text{R}},\Psi^{\text{R}}]$ (29)
(note the exchange of $\Psi$ and $\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$) where
$\tilde{S}_{\text{\tiny{SUSY}}}$ is the original SUSY action (7) but for the
adjoint process. It possesses a BRST symmetry of the type (8) from which we
infer that $\langle\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}\smash{\stackrel{{\scriptstyle(\ref{eq:SSSUSYTRtilde})}}{{=}}}-\langle\Psi_{1}^{\text{R}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{2}^{\text{R}}\tilde{\rangle}\stackrel{{\scriptstyle(\ref{eq:wardPsiPsiresp})}}{{=}}\langle
h_{1}^{\text{R}}\hat{h}_{2}^{\text{R}}\tilde{\rangle}$. Hence $\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ is a (time-reversed)
response function for the adjoint dynamics, as noted above. Eq. (29) also
implies identities for higher-order correlations of $\Psi$ and $\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$.
## VI Correlated noise
For noises correlated as
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2TD_{ij}\delta(t^{\prime}-t)$
with a symmetric invertible matrix $D$, the previous results can be
generalized as follows. Keeping the same definition for the quasi-potential
$\mathcal{H}$, the force is now decomposed as
$f_{i}=D_{ij}(-\partial_{j}\mathcal{H}+g_{j})$ instead of (3) and the
stationary condition (4) becomes
$\frac{1}{T}g_{i}D_{ij}\partial_{j}\mathcal{H}=D_{ij}\partial_{i}g_{j}$. The
action $S_{\text{\tiny{SUSY}}}$ is the same with $\hat{h}_{i}^{2}$ replaced by
$\smash{\hat{h}_{i}D_{ij}\hat{h}_{j}}$, and it verifies the BRST (8). Taking
matrix notations and defining now
$\tilde{\eta}=\partial_{t}h+D(\nabla\mathcal{H}+g)$ and
$\bar{\eta}=-\partial_{t}h+D(\nabla\mathcal{H}+g)$, the actions
$\displaystyle S_{\text{\tiny{SUSY}}}^{\dagger}$
$\displaystyle=\int_{t}\Big{\\{}\hat{h}\big{(}\eta-
TD\hat{h}\big{)}+\frac{1}{T}gD\nabla\mathcal{H}-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\kern
0.75pt\tilde{\eta}\text{{'}}^{\dagger}\kern-0.75pt\Psi\Big{\\}}$
$\displaystyle\mathbb{S}_{\text{\tiny{SUSY}}}^{\dagger}$
$\displaystyle=\\!\\!\int_{t}\\!\Big{\\{}\kern-0.75pt\hat{h}\big{(}\eta-
TD\hat{h}\big{)}\\!-\\!\frac{1}{T}\big{(}\partial_{t}h-Dg\big{)}\nabla\mathcal{H}+\Psi\bar{\eta}\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\Big{\\}}$
generalize (9) and (19), and a factorized form similar to (21) can be
identified [75]. SUSYs PS1,2 become 777The Onsager–Machlup actions take a
simple form: $S_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\eta
D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\eta}{4T}-\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\eta\text{{'}}\Psi\big{\\}}$ and
$\mathbb{S}^{\dagger}_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\bar{\eta}D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\bar{\eta}}{4T}+\Psi\bar{\eta}\text{{'}}\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\big{\\}}$, with
$S_{\text{\tiny{OM}}}$ verifying the BRST SUSY (8), and
$\mathbb{S}^{\dagger}_{\text{\tiny{OM}}}$ being invariant by the PS SUSY
$\delta h=\varepsilon T\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$,
$\delta\Psi=-\frac{\varepsilon}{2}D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\bar{\eta}$,
$\delta\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu=0$
corresponding to PS1,2.
$\displaystyle\text{PS}_{1}\\!\kern-0.75pt:\left\\{\begin{aligned} &\delta
h=\varepsilon T\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\qquad\delta\hat{h}=\varepsilon
D^{-1}\kern-0.75pt(\partial_{t}h-Dg[h])\kern-0.75pt\text{{'}}\kern
0.75pt\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\\\
&\delta\Psi=\varepsilon
D^{-1}\kern-0.75pt(\partial_{t}h-Dg[h]-TD\hat{h})\qquad\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu=0\end{aligned}\right.\\!\\!$
$\displaystyle\text{PS}_{2}\\!:\left\\{\begin{aligned} &\delta h=\varepsilon
T\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu\qquad\delta\hat{h}=\varepsilon(\nabla\mathcal{H}[h])\text{{'}}\kern
0.75pt\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\\\
&\delta\Psi=-\varepsilon
D^{-1}\kern-0.75pt(D\nabla\mathcal{H}[h]-TD\hat{h})\qquad\delta\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu=0\end{aligned}\right.$
and they imply the following modified FDR:
$\displaystyle\big{\langle}h_{1}\big{(}\partial_{t}h_{2}-Dg_{2}[h_{2}]\big{)}\big{\rangle}$
$\displaystyle=T\langle h_{1}D\hat{h}_{2}\rangle-T\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pt1}D\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}$ (30)
$\displaystyle\big{\langle}h_{1}\kern
0.75pt\nabla\kern-0.75pt\mathcal{H}[h_{2}]\big{\rangle}$
$\displaystyle=T\langle h_{1}\hat{h}_{2}\rangle+T\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{\kern-0.75pt1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}.$ (31)
One has $\langle\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}=\langle\mkern
1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern
1.6mu_{1}D\Psi_{2}\rangle^{\kern-0.75pt{\dagger}}=0$ if $t_{1}>t_{2}$.
## VII KPZ equation and continuous space
Choosing $\mathcal{H}[h]=\frac{\nu}{2}\sum_{i}(\nabla_{i}h)^{2}$ [with
$\nabla_{i}h=h_{i+1}-h_{i}$] and
$g_{i}[h]=\frac{\lambda}{6}\big{[}(\nabla_{i}h)^{2}+\nabla_{i}h\nabla_{i-1}h+(\nabla_{i-1}h)^{2}\big{]}$,
the Langevin equation (1) is a discretized version of the continuum KPZ
equation
$\partial_{t}h=\nu\partial_{x}^{2}h+\frac{\lambda}{2}(\partial_{x}h)^{2}+\eta$.
It possesses the SUSYs we have derived together with the modified FDRs, since
the chosen discretizations of $\mathcal{H}$ and of the non-linear term
$g_{i}[h]$ ensure that both sides of the stationary condition (4) is 0. Such a
situation with an orthogonal decomposition of the force
($g_{i}\partial_{i}\mathcal{H}=0$) and a zero-divergence
($\partial_{i}g_{i}=0$) could be generic [51].
If $i$ is a lattice index, the continuous-space limit of our results is
obtained directly. For KPZ one has for instance $\langle
h_{1}(\partial_{t}h-\frac{\lambda}{2}(\partial_{x}h)^{2})_{2}\rangle=T\langle
h_{1}\hat{h}_{2}\rangle$ and $\langle
h_{1}\partial^{2}_{x}h_{2}\rangle=T\langle h_{1}\hat{h}_{2}\rangle$ if
$t_{1}>t_{2}$. The second relation was derived in [84]. Note that _not all
spatial discretizations of the non-linear term satisfy_ (4): hence, in general
the discretization of gradients must be specified when it comes to SUSY, FDR
and time reversal, because $\partial_{i}g_{i}$ is ambiguous in the continuum
if $g[h]$ depends on gradients – as also seen in singularities of the
functional Fokker–Planck equation [75].
## VIII Discussion and Outlook
We have identified SUSYs related to arbitrary Langevin equations with Gaussian
additive white noise, generalizing long-known results that were restricted to
reversible settings [11, 12]. They can be expressed both in the MSRJD
formalism and in the Onsager–Machlup one. The price to pay is an explicit
dependency on the stationary state, and a more complex structure: two actions
both representing the same physical process and each presenting different
SUSYs. The important outcome is that they entail modified non-equilibrium FDRs
[38] (that provide information on the steady-state when it is not known). As
illustrated for the KPZ equation, the case of spatially continuous models is
obtained directly from the results we presented, but the spatial
discretization of gradients has to be specified (to make sense of
$\partial_{i}g_{i}$ in the continuum).
The construction we presented is reminiscent of the derivation of the
Jarzynski relation by Mallick _et al._ [88], and it would be interesting to
find a unified framework. Our results apply to non-equilibrium models with
known steady state, such as the zero-range process [89, 90, 91] or mass
transport models [92], and other cases [93, 94, 95]. In the small-noise limit,
the adjoint dynamics is often known in Macroscopic Fluctuation Theory [96],
and thus the SUSYs PS1,2 should be applicable. We note in general that, in the
small-noise asymptotics of Langevin process [97], the quasi-potential
$\mathcal{H}[h]$ can become a singular (non-differentiable) function of its
argument [98, 99, 100], even though $\mathcal{H}[h]$ is regular as long as $T$
is finite. This implies that the $T\to 0$ limit has to be taken in a careful
way. The case of non-Gaussian noise could be investigated [28]. The extensions
to inertial Langevin equations, or singular ($D$ not invertible) or colored
noise, or multiplicative noise deserve further investigations.
The SUSYs we have unveiled are defined for path integrals, but the reversible
SUSY also has an operator version, with the Fokker–Planck operator completed
by fermionic operators representing the Grassmann variables; it was used by
Kurchan _et al._ to study metastability in overdamped [25] and inertial [26]
Langevin dynamics, see also [101]. It would be interesting to translate our
results in these settings. It is a non-trivial task already in the overdamped
case, since in the reversible case the equality of the actions (7) and (9)
corresponds to the fact that the extended (fermionic) Fokker–Planck operator
can be made Hermitian (which is an essential aspect of Kurchan _et al._ ’s
construction), while the same property does not hold in the generic
irreversible case that we are considering. Last, it could be instructive to
identify the relation between our results and the slave process of Refs. [102,
101], and more generally with cohomology [103, 104].
###### Acknowledgements.
The authors thank Matthieu Tissier for very useful discussions. E.A.
acknowledges support from the Swiss National Science Foundation by the SNSF
Ambizione Grant PZ00P2_173962. L.C. acknowledges support from the
ANR-18-CE92-0019 Grant NeqFluids and support from Institut Universitaire de
France. V.L. acknowledges financial support from the ERC Starting Grant No.
680275 MALIG, the ANR-18-CE30-0028-01 Grant LABS and the ANR-15-CE40-0020-03
Grant LSD.
## References
* [1] P. Langevin. Sur la théorie du mouvement brownien. C. R. Acad. Sci. (Paris) 146, 530 (1908).
* [2] N. G. van Kampen. Stochastic processes in physics and chemistry. North-Holland personal library. Elsevier Amsterdam ; Boston 3rd edition 2007.
* [3] C. W. Gardiner. Handbook of stochastic methods for physics, chemistry, and the natural sciences. Number 13 in Springer series in synergetics. Springer-Verlag Berlin ; New York 2nd edition 1994.
* [4] A. A. Starobinsky. Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations. Physics Letters B 117, 175 (1982).
* [5] L. Pinol, S. Renaux-Petel, and Y. Tada. A manifestly covariant theory of multifield stochastic inflation in phase space. arXiv:2008.07497 [astro-ph.CO] 2020.
* [6] J. Zinn-Justin. Quantum field theory and critical phenomena. Number 113 in International series of monographs on physics. Clarendon Press ; Oxford University Press Oxford : New York 4th edition 2002.
* [7] C. Becchi, A. Rouet, and R. Stora. The abelian Higgs Kibble model, unitarity of the S-operator. Physics Letters B 52, 344 (1974).
* [8] I. V. Tyutin. Gauge invariance in field theory and statistical physics in operator formalism. Lebedev Institute preprint, arXiv:0812.0580 [hep-th] 1975.
* [9] C. Becchi, A. Rouet, and R. Stora. Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975).
* [10] C. Becchi, A. Rouet, and R. Stora. Renormalization of gauge theories. Annals of Physics 98, 287 (1976).
* [11] G. Parisi and N. Sourlas. Supersymmetric field theories and stochastic differential equations. Nuclear Physics B 206, 321 (1982).
* [12] M. V. Feigel’man and A. M. Tsvelik. Hidden supersymmetry of stochastic dissipative dynamics. Sov. Phys. JETP 56, 823 (1982).
* [13] G. Parisi and N. Sourlas. Random magnetic fields, supersymmetry, and negative dimensions. Phys. Rev. Lett. 43, 744 (1979).
* [14] J. Kurchan. Replica trick to calculate means of absolute values: applications to stochastic equations. J. Phys. A: Math. Gen. 24, 4969 (1991).
* [15] J. Kurchan. Supersymmetry in spin glass dynamics. J. Phys. I France 2, 1333 (1992).
* [16] G. Semerjian, L. F. Cugliandolo, and A. Montanari. On the stochastic dynamics of disordered spin models. J. Stat. Phys. 115, 493 (2004).
* [17] A. I. Olemskoi. Supersymmetric field theory of a nonequilibrium stochastic system as applied to disordered heteropolymers. Physics-Uspekhi 44, 479 (2001).
* [18] J. C. Niel and J. Zinn-Justin. Finite size effects in critical dynamics. Nuclear Physics B 280, 355 (1987).
* [19] A. Schwarz and O. Zaboronsky. Supersymmetry and localization. Commun. Math. Phys. 183, 463 (1997).
* [20] M. Tissier and G. Tarjus. Supersymmetry and its spontaneous breaking in the Random Field Ising Model. Phys. Rev. Lett. 107, 041601 (2011).
* [21] G. Tarjus and M. Tissier. Random-field Ising and O(N) models: theoretical description through the functional renormalization group. Eur. Phys. J. B 93, 50 (2020).
* [22] A. Kaviraj, S. Rychkov, and E. Trevisani. Random Field Ising Model and Parisi-Sourlas supersymmetry II. Renormalization group. arXiv:2009.10087 [cond-mat, physics:hep-th] 2020.
* [23] E. Gozzi. Hidden BRS invariance in classical mechanics. Physics Letters B 201, 525 (1988).
* [24] E. Gozzi, M. Reuter, and W. D. Thacker. Hidden BRS invariance in classical mechanics. II. Phys. Rev. D 40, 3363 (1989).
* [25] S. Tănase-Nicola and J. Kurchan. Metastable states, transitions, basins and borders at finite temperatures. J. Stat. Phys. 116, 1201 (2004).
* [26] J. Tailleur, S. Tănase-Nicola, and J. Kurchan. Kramers equation and supersymmetry. J. Stat. Phys. 122, 557 (2006).
* [27] E. Witten. Supersymmetry and Morse theory. J. Differential Geom. 17, 661 (1982).
* [28] J. Zinn-Justin. Renormalization and stochastic quantization. Nuclear Physics B 275, 135 (1986).
* [29] R. Balian and M. Vénéroni. Static and dynamic variational principles for expectation values of observables. Annals of Physics 187, 29 (1988).
* [30] G. Parisi and Y. S. Wu. Perturbation theory without gauge fixing. Scientia Sinica 24, 483 (1981).
* [31] E. Gozzi. Functional-integral approach to Parisi-Wu stochastic quantization: scalar theory. Phys. Rev. D 28, 1922 (1983).
* [32] P. H. Damgaard and H. Hüffel. Stochastic quantization. Physics Reports 152, 227 (1987).
* [33] S. Chaturvedi, A. K. Kapoor, and V. Srinivasan. Ward Takahashi identities and fluctuation-dissipation theorem in a superspace formulation of the Langevin equation. Z. Physik B Condensed Matter 57, 249 (1984).
* [34] E. Gozzi. Onsager principle of microscopic reversibility and supersymmetry. Phys. Rev. D 30, 1218 (1984).
* [35] M. F. Zimmer. Fluctuations in nonequilibrium systems and broken supersymmetry. J. Stat. Phys. 73, 751 (1993).
* [36] K. Gawedzki and A. Kupiainen. Critical behaviour in a model of stationary flow and supersymmetry breaking. Nuclear Physics B 269, 45 (1986).
* [37] S. Trimper. Supersymmetry breaking for dynamical systems. J. Phys. A: Math. Gen. 23, L169 (1990).
* [38] G. S. Agarwal. Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications. Z Physik A Hadrons and nuclei 252, 25 (1972).
* [39] T. Speck and U. Seifert. Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. EPL (Europhysics Letters) 74, 391 (2006).
* [40] J. Prost, J.-F. Joanny, and J. M. R. Parrondo. Generalized fluctuation-dissipation theorem for steady-state systems. Phys. Rev. Lett. 103, 090601 (2009).
* [41] H.-K. Janssen. On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties. Z. Physik B Condensed Matter 23, 377 (1976).
* [42] H.-K. Janssen. Field-theoretic method applied to critical dynamics. In Charles P. Enz, editor, Dynamical Critical Phenomena and Related Topics Lecture Notes in Physics pages 25 Berlin, Heidelberg 1979\. Springer.
* [43] C. De Dominicis. Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques. J. Phys. Colloques 37, 247 (1976).
* [44] C. De Dominicis and L. Peliti. Field-theory renormalization and critical dynamics above ${T}_{c}$: Helium, antiferromagnets, and liquid-gas systems. Phys. Rev. B 18, 353 (1978).
* [45] P. C. Martin, E. D. Siggia, and H. A. Rose. Statistical dynamics of classical systems. Phys. Rev. A 8, 423 (1973).
* [46] L. Onsager and S. Machlup. Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953).
* [47] S. Machlup and L. Onsager. Fluctuations and irreversible process. II. Systems with kinetic energy. Phys. Rev. 91, 1512 (1953).
* [48] M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889 (1986).
* [49] H. Risken. The Fokker-Planck equation: methods of solution and applications. Number v. 18 in Springer series in synergetics. Springer-Verlag New York 2nd ed edition 1996.
* [50] R. Graham. Covariant formulation of non-equilibrium statistical thermodynamics. Z. Physik B Condensed Matter 26, 397 (1977).
* [51] G. L. Eyink, J. L. Lebowitz, and H. Spohn. Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems. J. Stat. Phys. 83, 385 (1996).
* [52] F. Langouche, D. Roekaerts, and E. Tirapegui. General Langevin equations and functional integration. In Field Theory, Quantization and Statistical Physics: in Memory of Bernard Jouvet. Springer Dordrecht 1981.
* [53] M. Itami and S-i. Sasa. Universal form of stochastic evolution for slow variables in equilibrium systems. J. Stat. Phys. 167, 46 (2017).
* [54] L. F. Cugliandolo and V. Lecomte. Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach. J. Phys. A: Math. Theor. 50, 345001 (2017).
* [55] L. F. Cugliandolo, V. Lecomte, and F. van Wijland. Building a path-integral calculus: a covariant discretization approach. J. Phys. A: Math. Theor. 52, 50LT01 (2019).
* [56] Z. G. Arenas and D. G. Barci. Supersymmetric formulation of multiplicative white-noise stochastic processes. Phys. Rev. E 85, 041122 (2012).
* [57] Z. G. Arenas and D. G. Barci. Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes. J. Stat. Mech. 2012, P12005 (2012).
* [58] F. A. Berezin. The method of second quantization. Academic Press 1966.
* [59] It is more rigorously defined as $\varphi_{i}[h+h^{1}]=\varphi_{i}[h]+\varphi_{i}[h]\text{{'}}h^{1}+o(h^{1})$, implying that $\partial_{t}(\varphi[h])=\varphi\text{{'}}\partial_{t}h$, and $\varphi_{i}[h+\varepsilon\Psi]=\varphi_{i}[h]+\varepsilon\varphi_{i}[h]\text{{'}}\Psi$. One has for instance for $\eta_{i}[h]$ defined in Eq. (5): $\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi=\big{(}\delta_{ij}\partial_{t}-\partial_{j}f_{i}[h]\big{)}\Psi_{\kern-0.75ptj}$.
* [60] H. Nicolai. Supersymmetry and functional integration measures. Nuclear Physics B 176, 419 (1980).
* [61] H. Nicolai. On a new characterization of scalar supersymmetric theories. Physics Letters B 89, 341 (1980).
* [62] H. Nicolai. On the functional integration measure of supersymmetric Yang-Mills theories. Physics Letters B 117, 408 (1982).
* [63] S. Cecotti and L. Girardello. Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry. Annals of Physics 145, 81 (1983).
* [64] Hence, explicitly: $\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti}\tilde{\eta}_{i}\kern-0.75pt\text{{'}}^{\dagger}\Psi=\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti}(\delta_{ij}\partial_{t}+\partial_{ij}\mathcal{H}+\partial_{i}g_{j})\Psi_{\kern-0.75ptj}$.
* [65] R. Graham. Statistical theory of instabilities in stationary nonequilibrium systems with applications to lasers and nonlinear optics. In G. Höhler, editor, Springer Tracts in Modern Physics: Ergebnisse der exakten Naturwissenschaftenc; Volume 66 pages 1. Springer Berlin, Heidelberg 1973.
* [66] F. Langouche, D. Roekaerts, and E. Tirapegui. Functional integrals and the Fokker-Planck equation. Il Nuovo Cimento B 53, 135 (1979).
* [67] F. Langouche, D. Roekaerts, and E. Tirapegui. Functional integration and semiclassical expansions. Kluwer Academic Publishers Dordrecht 1982.
* [68] H. K. Janssen. On the renormalized field theory of nonlinear critical relaxation. In From Phase Transitions to Chaos pages 68. World Scientific 1992.
* [69] A. W. C. Lau and T. C. Lubensky. State-dependent diffusion: thermodynamic consistency and its path integral formulation. Phys. Rev. E 76, 011123 (2007).
* [70] Z. G. Arenas and D. G. Barci. Functional integral approach for multiplicative stochastic processes. Phys. Rev. E 81, 051113 (2010).
* [71] C. Aron, D. G. Barci, L. F. Cugliandolo, Z. G. Arenas, and G. S. Lozano. Dynamical symmetries of Markov processes with multiplicative white noise. J. Stat. Mech. 2016, 053207 (2016).
* [72] Indeed, discretizing with a time step $\Delta t$, one has $\eta_{i}[h]_{t}=\frac{h_{i,t+\Delta t}-h_{i,t}}{\Delta t}-f_{i}[h]\big{|}_{h=\frac{1}{2}(h_{i,t+\Delta t}+h_{i,t})}$ where the time is in index (and discretization is Stratonovich). Hence the matrix of coordinates $(i,t;j,t^{\prime})$ in the definition of the Jacobian after Eq. (5) is upper triangular in the time direction (this is causality), so that only its equal-time components matter. Importantly, since the time-discrete Langevin equation is read as $h_{t+\Delta t}$ function of $h_{t}$ and $\eta_{t}$, one must pay attention that the change of variables is between $h_{t+\Delta t}$ and $\eta_{t}$. Its Jacobian is thus $\partial\eta_{i}[h]_{t}/\partial h_{j,t+\Delta t}=\frac{1}{\Delta t}\delta_{ij}-\frac{1}{2}\partial_{j}f_{i}[h]$. Factorizing by $\frac{1}{\Delta t}$ [which yields a field-independent normalization factor of the Jacobian], using the formula $\mathop{log}\nolimits\mathop{det}\displaylimits=\operatorname{tr}\mathop{log}\nolimits$, one thus obtains $\log\big{|}\\!\frac{\delta\eta}{\delta h}\\!\big{|}=\sum_{t}\operatorname{tr}\log(\mathbf{1}-\frac{1}{2}\Delta tf_{i}\text{{'}}[h])$. Expanding at small $\Delta t$, one recovers Eq. (11).
* [73] Denoting by $X^{\text{{s}}}_{t}=\frac{X_{t+\Delta t}+X_{t}}{2}$ the Stratonovich discretization, $\int_{t}\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti}\eta_{i}\kern-0.75pt\text{{'}}[h]\Psi=\int_{t}\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti}(\delta_{ij}\partial_{t}-\partial_{j}\kern-0.75ptf_{i}[h])\Psi_{\kern-0.75ptj}$ must be discretized as $\sum_{t}\\!\Delta t\kern 0.75pt\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti,t+\Delta t}\big{(}\frac{\Psi_{\kern-0.75pti,t+\Delta t}-\Psi_{\kern-0.75pti,t}}{\Delta t}-\partial_{j}\kern-0.75ptf_{i}[h^{\text{\tiny{s}}}_{t}]\Psi^{\text{\tiny{s}}}_{\kern-0.75ptj,t}\big{)}=\sum_{tt^{\prime}}\\!\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{\kern-0.75pti,t^{\prime}}M_{i,t^{\prime};j,t}\Psi_{\kern-0.75ptj,t}$ with the matrix elements given by $M_{i,t^{\prime}\\!;j,t}=\delta_{ij}(\delta_{t^{\prime}\\!,t}-\delta_{t^{\prime}\\!,t+\Delta t})-\\!\Delta t\kern 0.75pt\partial_{j}\kern-0.75ptf_{i}[h_{t^{\prime}}]\frac{\delta_{t^{\prime}\\!,t}+\delta_{t^{\prime}\\!,t+\Delta t}}{2}$. As the Grassmann integral yields the determinant of $M$, and as $M$ is triangular in the time coordinate, only the diagonal $t^{\prime}=t$ matters and $\det M=\sum_{t}\det(\delta_{ij}-\\!\Delta t\kern 0.75pt\partial_{j}f_{i}[h])$. One thus recovers the Jacobian [72].
* [74] This explains why one cannot transform $S_{\text{{SUSY}}}$ into $\smash{S_{\text{{SUSY}}}^{\smash{}{\dagger}}}$: these actions contain the same information after integrating on the Grassmann fields, but not before.
* [75] B. Marguet, E. Agoritsas, L. Canet, and V. Lecomte. (In preparation) 2021.
* [76] With the notations of [73], we have that $\langle\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{i,t^{\prime}}\Psi_{j,t}\rangle^{\\!{\dagger}}={(M^{T})^{-1}}_{i,t^{\prime};j,t}$, but $M$ is lower triangular in the time direction, so that $(M^{T})^{-1}$ is upper triangular. This implies the causality $\langle\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu_{i,t^{\prime}}\Psi_{j,t}\rangle^{\\!{\dagger}}=0$ for $t^{\prime}>t$.
* [77] Massimo Falcioni, Stefano Isola, and Angelo Vulpiani. Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics. Physics Letters A 144, 341 (1990).
* [78] G. E. Crooks. Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E 61, 2361 (2000).
* [79] R. Chetrite, G. Falkovich, and K. Gawedzki. Fluctuation relations in simple examples of non-equilibrium steady states. J. Stat. Mech. 2008, P08005 (2008).
* [80] M. Baiesi, C. Maes, and B. Wynants. Fluctuations and response of nonequilibrium states. Phys. Rev. Lett. 103, 010602 (2009).
* [81] U. Seifert and T. Speck. Fluctuation-dissipation theorem in nonequilibrium steady states. EPL 89, 10007 (2010).
* [82] G. Verley, K. Mallick, and D. Lacoste. Modified fluctuation-dissipation theorem for non-equilibrium steady states and applications to molecular motors. EPL 93, 10002 (2011).
* [83] Sara Dal Cengio, Demian Levis, and Ignacio Pagonabarraga. Fluctuation–dissipation relations in the absence of detailed balance: formalism and applications to active matter. Journal of Statistical Mechanics: Theory and Experiment 2021, 043201 (2021).
* [84] L. Canet, H. Chaté, B. Delamotte, and N. Wschebor. Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation: general framework and first applications. Phys. Rev. E 84, 061128 (2011).
* [85] V. Y. Chernyak, M. Chertkov, and C. Jarzynski. Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech. 2006, P08001 (2006).
* [86] Brian Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications 12, 313 (1982).
* [87] The Onsager–Machlup actions take a simple form: $S_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\eta D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\eta}{4T}-\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\eta\text{{'}}\Psi\big{\\}}$ and $\mathbb{S}^{\dagger}_{\text{\tiny{OM}}}=\int_{t}\\!\big{\\{}\frac{\bar{\eta}D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\bar{\eta}}{4T}+\Psi\bar{\eta}\text{{'}}\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu\big{\\}}$, with $S_{\text{{OM}}}$ verifying the BRST SUSY (8), and $\mathbb{S}^{\dagger}_{\text{{OM}}}$ being invariant by the PS SUSY $\delta h=\varepsilon T\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu$, $\delta\Psi=-\frac{\varepsilon}{2}D^{\kern-0.75pt-\kern-0.75pt1\kern-0.75pt}\bar{\eta}$, $\delta\mkern 1.6mu\overline{\mkern-1.6mu\Psi\mkern-1.6mu}\mkern 1.6mu=0$ corresponding to PS1,2.
* [88] K. Mallick, M. Moshe, and H. Orland. A field-theoretic approach to non-equilibrium work identities. J. Phys. A: Math. Theor. 44, 095002 (2011).
* [89] F. Spitzer. Interaction of Markov processes. Advances in Mathematics 5, 246 (1970).
* [90] E. Levine, D. Mukamel, and G. M. Schütz. Zero-range process with open boundaries. J. Stat. Phys. 120, 759 (2005).
* [91] M. R. Evans and T. Hanney. Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A: Math. Gen. 38, R195 (2005).
* [92] J. Guioth and E. Bertin. A mass transport model with a simple non-factorized steady-state distribution. J. Stat. Mech. 2017, 063201 (2017).
* [93] G. M. Schütz. Exactly Solvable Models for Many-Body Systems Far from Equilibrium. In C. Domb and J. L. Lebowitz, editor, Phase Transitions and Critical Phenomena volume 19 pages 1. Academic Press 2001.
* [94] Cristian Giardinà, Jorge Kurchan, Frank Redig, and Kiamars Vafayi. Duality and Hidden Symmetries in Interacting Particle Systems. Journal of Statistical Physics 135, 25 (2009).
* [95] Rouven Frassek, Cristian Giardinà, and Jorge Kurchan. Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes. Journal of Statistical Physics 180, 135 (2020).
* [96] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim. Macroscopic fluctuation theory. Rev. Mod. Phys. 87, 593 (2015).
* [97] Mark I. Freidlin and Alexander D. Wentzell. Random Perturbations of Dynamical Systems. Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg 3 edition 2012.
* [98] R. Graham and T. Tél. On the weak-noise limit of Fokker-Planck models. Journal of Statistical Physics 35, 729 (1984).
* [99] R. Graham and T. Tél. Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems. Physical Review A 31, 1109 (1985).
* [100] Alex Kamenev. Field theory of non-equilibrium systems. Cambridge University Press Cambridge ; New York 2011.
* [101] I. T. Drummond and R. R. Horgan. Stochastic processes, slaves and supersymmetry. J. Phys. A: Math. Theor. 45, 095005 (2012).
* [102] D. S. Dean, I. T. Drummond, R. R. Horgan, and S. N. Majumdar. Equilibrium statistics of a slave estimator in Langevin processes. Phys. Rev. E 71, 031103 (2005).
* [103] I. V. Ovchinnikov. Introduction to supersymmetric theory of stochastics. Entropy 18, 108 (2016).
* [104] F. Wegner. Supermathematics and its applications in statistical physics volume 920 of Lecture Notes in Physics. Springer Berlin Heidelberg 2016.
|
# Intersecting geodesics on the modular surface
Junehyuk Jung and Naser Talebizadeh Sardari Department of Mathematics, Brown
University, Providence, RI 02912 USA<EMAIL_ADDRESS>Penn State
department of Mathematics, McAllister Building, Pollock Rd, State College, PA
16802 USA<EMAIL_ADDRESS>
###### Abstract.
We introduce the modular intersection kernel, and we use it to study how
geodesics intersect on the full modular surface
$\mathbb{X}=PSL_{2}\left(\mathbb{Z}\right)\backslash\mathbb{H}$. Let $C_{d}$
be the union of closed geodesics with discriminant $d$ and let
$\beta\subset\mathbb{X}$ be a compact geodesic segment. As an application of
Duke’s theorem to the modular intersection kernel, we prove that
$\\{\left(p,\theta_{p}\right)~{}:~{}p\in\beta\cap C_{d}\\}$ becomes
equidistributed with respect to $\sin\theta dsd\theta$ on $\beta\times[0,\pi]$
with a power saving rate as $d\to+\infty$. Here $\theta_{p}$ is the angle of
intersection between $\beta$ and $C_{d}$ at $p$. This settles the main
conjectures introduced by Rickards [Ric19].
We prove a similar result for the distribution of angles of intersections
between $C_{d_{1}}$ and $C_{d_{2}}$ with a power-saving rate in $d_{1}$ and
$d_{2}$ as $d_{1}+d_{2}\to\infty$. Previous works on the corresponding problem
for compact surfaces do not apply to $\mathbb{X}$, because of the singular
behavior of the modular intersection kernel near the cusp. We analyze the
singular behavior of the modular intersection kernel by approximating it by
general (not necessarily spherical) point-pair invariants on
$PSL_{2}\left(\mathbb{Z}\right)\backslash PSL_{2}\left(\mathbb{R}\right)$ and
then by studying their full spectral expansion.
We thank D. Jakobson, V. Blomer, D. Milicevic, C. Pagano, M. Lee, M.
Lipnowski, Y. Kim, and J. Yim for valuable comments. J.J. thanks A. Reid for
the discussion that led to this project. J.J. was supported by NSF grant
DMS-1900993, and by Sloan Research Fellowship. N.T.S. was supported partially
by the National Science Foundation under Grant No. DMS-2015305, and is
grateful to Max Planck Institute for Mathematics in Bonn for its hospitality
and financial support.
## 1\. Introduction
Let $Y$ be a negatively curved surface of finite area. The prime geodesic
theorem [Sar80] states that the number of primitive closed geodesics having
length less than $L$, which we denote by $\pi\left(L\right)$, satisfies
$\pi\left(L\right)\sim\frac{e^{L}}{L},$
as $L\to\infty$. A natural problem is to understand how primitive closed
geodesics of length less than $L$ are positioned or distributed in $Y$ as
$L\to\infty$. In particular, one may ask
1. (1)
how the number of transversal intersections
$I\left(\alpha_{1},\alpha_{2}\right)$ between two primitive closed geodesics
$\alpha_{1}$ and $\alpha_{2}$ is distributed, or
2. (2)
how the set of angles of intersections between $\alpha_{1}$ and $\alpha_{2}$
is distributed,
as one varies $\alpha_{1}$, or both $\alpha_{1}$ and $\alpha_{2}$. Bonahon
[Bon86] defined the intersection form
$i:\mathcal{C}\times\mathcal{C}\to\mathbb{R}^{+}$ on the space of currents
$\mathcal{C}$ such that when $\mu_{i}$ ($i=1,2$) is the unique invariant
measure corresponding to $\alpha_{i}$, then
$i\left(\mu_{1},\mu_{2}\right)=I\left(\alpha_{1},\alpha_{2}\right)$. When $Y$
is compact, Pollicott and Sharp [PS06] used an extension of the intersection
form to understand the distribution of angles of self-intersections of closed
geodesic $\alpha$ having length less than $L$, as $L\to\infty$. When $Y$ is a
compact hyperbolic surface, using the intersection form, Herrera [HJ15] proved
that the distribution of
$I\left(\alpha_{1},\alpha_{2}\right)/\left(l\left(\alpha_{1}\right)l\left(\alpha_{2}\right)\right)$
for closed geodesics $\alpha_{1},\alpha_{2}$ of length $<L$, is concentrated
near
$1/\left(2\pi^{2}\left(g-1\right)\right)=2/\left(\pi\mathrm{vol}\left(Y\right)\right)$
with exponentially decaying tail, as $L\to\infty$. Here $l\left(\cdot\right)$
is the length function, and $g$ is the genus of $Y$.
In this article, we study a refined problem:
* (3)
how are the locations and angles of intersections between $\alpha_{1}$ and
$\alpha_{2}$ jointly distributed relative to $\alpha_{2}$, as one varies
$\alpha_{1}$, or both $\alpha_{1}$ and $\alpha_{2}$?
To state our main theorem, we let $\mathbb{X}$ be the full modular surface
$PSL_{2}\left(\mathbb{Z}\right)\backslash\mathbb{H}$. On $\mathbb{X}$,
primitive oriented closed geodesics are in one-to-one correspondence with
conjugacy classes of primitive hyperbolic elements of
$PSL_{2}\left(\mathbb{Z}\right)$. Moreover there is a bijection between the
primitive hyperbolic conjugacy classes and the $SL_{2}\left(\mathbb{Z}\right)$
equivalence classes of primitive integral binary quadratic forms of non-square
positive discriminant [LRS09, Sar82]. So by the discriminant of a primitive
closed geodesic, we mean the discriminant of the corresponding binary
quadratic form. In particular, if the hyperbolic class $\gamma$ is associated
to the binary quadratic form $Q$ then $\gamma^{-1}$ is associated to $-Q$.
Each oriented primitive closed geodesics of discriminant $d$ has a unique lift
to a closed geodesic of length $2\log\varepsilon_{d}$ in the unit tangent
bundle $S\mathbb{X}$. Let $h\left(d\right)$ be the number of inequivalent
primitive integral binary quadratic forms of discriminant $d$. We denote the
disjoint union of these $h\left(d\right)$ closed geodesics by
$\mathscr{C}_{d}\subset S\mathbb{X},$ which has total length
$2h\left(d\right)\log\varepsilon_{d}$.
Note that the closed geodesic on $\mathbb{X}$ has length $\log\varepsilon_{d}$
or $2\log\varepsilon_{d}$ according as $Q$ is or is not equivalent to $-Q$
[Duk88, p.75]. We now let $C_{d}$ be the union of primitive (unoriented)
closed geodesics of discriminant $d$ on $\mathbb{X}$, and note that
$l\left(C_{d}\right)=h\left(d\right)\log\varepsilon_{d}$ is the total length
of $C_{d}$.
###### Theorem 1.1.
Fix $T>100$, and let $\beta$ be a compact oriented geodesic segment of length
$<1$ in the region determined by $y<T$ on $\mathbb{X}$. For
$0<\theta_{1}<\theta_{2}<\pi$, let
$I_{\theta_{1},\theta_{2}}\left(\beta,C_{d}\right)$ be the number of
intersections between $\beta$ and $C_{d}$ with the angle between $\theta_{1}$
and $\theta_{2}$. (Here the angle between $\beta$ and $C_{d}$ at
$p\in\beta\cap C_{d}$ is measured counterclockwise from the tangent to $\beta$
at $p$ to the tangent to $C_{d}$ at $p$.)
Then we have
$\frac{I_{\theta_{1},\theta_{2}}\left(\beta,C_{d}\right)}{l\left(\beta\right)l\left(C_{d}\right)}=\frac{3}{\pi^{2}}\int_{\theta_{1}}^{\theta_{2}}\sin\theta
d\theta+O_{\epsilon}\left(d^{-\frac{25}{3584}+\epsilon}\right),$
uniformly in $\beta$, $\theta_{1}$, and $\theta_{2}$, under the assumption
that
$\theta_{2}-\theta_{1}\gg d^{-\frac{25}{7168}},$
and that
$l\left(\beta\right)\gg d^{-\frac{25}{7168}}.$
###### Remark 1.2.
This statement is false if $C_{d}$ is replaced by individual geodesics. For
instance, the set of intersections between $\beta$ and a closed geodesic
$\alpha$ does not necessarily become equidistributed as
$l\left(\alpha\right)\to\infty$. To see this, take a finite sheeted covering
$S$ of $\mathbb{X}$ whose genus is $\geq 2$. Then according to Rivin’s work
[Riv01] there are arbitrarily long simple closed geodesics on $S$. Note that
these simple closed geodesics must be contained in a compact part of $S$
[JR21]. This implies that there is a compact set $C\subset\mathbb{X}$ which
contains arbitrarily long primitive closed geodesics. Take a geodesic segment
$\beta$ in $\mathbb{X}-C$. Then there are infinitely many closed geodesics
which do not intersect $\beta$.
###### Remark 1.3.
The exponent $-\frac{25}{3584}$ can be improved slightly by refining our
argument, but in order to keep the exposition simple, we do not discuss the
optimal rate in the current article.
As an immediate consequence, we deduce that the intersection points and
corresponding angles become equidistributed, resolving the main conjectures
introduced by Rickards [Ric19].
###### Corollary 1.4.
Fix a closed geodesic $\alpha$. Then for any fixed segment
$\beta\subset\alpha$, and any fixed $0<\theta_{1}<\theta_{2}<\pi$, we have
$\lim_{d\to\infty}\frac{I_{\theta_{1},\theta_{2}}\left(\beta,C_{d}\right)}{I\left(\alpha,C_{d}\right)}=\frac{l\left(\beta\right)}{l\left(\alpha\right)}\int_{\theta_{1}}^{\theta_{2}}\frac{\sin\theta}{2}d\theta.$
###### Remark 1.5.
Rickards’s work is motivated by the work of Darmon and Vonk [DV21] on the
arithmetic ($p$-arithmetic) intersection between pairs of oriented closed
geodesics on the modular surfaces (Shimura curves). The arithmetic
intersection between oriented closed geodesics $\alpha_{1}$ and $\alpha_{2}$
of discriminants $D_{1}$ and $D_{2}$ only depends on $D_{1}$ and $D_{2}$ and
the angles of intersections between $\alpha_{1}$ and $\alpha_{2}$. Darmon and
Vonk conjectured that [DV21, Conjecture 2] that the $p$-arithmetic
intersection is algebraic and belongs to the composition of the Hilbert class
field of real quadratic fields of discriminants $D_{1}$ and $D_{2}$.
To prove our main results, we introduce the modular intersection kernel. For
$\delta>0$ and $\theta_{1},\theta_{2}\in\left(0,\pi\right)$, let
$k_{\delta}^{\theta_{1},\theta_{2}}:S\mathbb{H}\times
S\mathbb{H}\to\mathbb{R}$ be the integral kernel defined by
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(x_{1},\xi_{1}\right),\left(x_{2},\xi_{2}\right)\right)=1,$
if the geodesic segments of length $\delta$ from $x_{i}$ with the initial
vector $\xi_{i}$ intersect at an angle
$\in\left(\theta_{1},\theta_{2}\right)$, and $0$ otherwise. Under the
identification $S\mathbb{H}\cong PSL_{2}\left(\mathbb{R}\right)$, for a given
discrete subgroup $\Gamma\subset PSL_{2}\left(\mathbb{R}\right)$, we define
the modular intersection kernel
$K_{\delta}^{\theta_{1},\theta_{2}}:\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\times\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\to\mathbb{R}$ by taking the average of
$k_{\delta}^{\theta_{1},\theta_{2}}$ over $\Gamma$:
$K_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}k_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},\gamma
g_{2}\right).$
The basic idea of the proof of Theorem 1.1 then is as follows. Heuristically,
$I_{\theta_{1},\theta_{2}}\left(\beta,C_{d}\right)$
should be well approximated by
$\frac{1}{2\delta^{2}}\int_{\mathscr{C}_{d}}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2},$
(1.1)
where $\widetilde{\beta}\subset S\mathbb{X}$ is a lift of $\beta$ with either
of orientations of $\beta$
$\widetilde{\beta}\left(t\right)=\left(\beta\left(t\right),\beta^{\prime}\left(t\right)\right),$
under assuming that $\beta\left(t\right)$ is parameterized by the arc length.
As noted from [LRS09], Duke’s theorem [Duk88] can be extended to the
equidistribution of $\mathscr{C}_{d}$ in
$PSL_{2}\left(\mathbb{Z}\right)\backslash PSL_{2}\left(\mathbb{R}\right)$ as
$d\to\infty$. Observing that
$\frac{1}{2\delta^{2}}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},g\right)ds_{1}$
(1.2)
is a compactly supported function in $g$ for compact $\beta$, (1.1) is
$\sim\frac{l\left(\mathscr{C}_{d}\right)}{2\delta^{2}}\int_{g}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},g\right)ds_{1}d\mu_{g},$
which is asymptotically
$\frac{3}{\pi^{2}}l\left(C_{d}\right)l\left(\beta\right)\int_{\theta_{1}}^{\theta_{2}}\sin\alpha
d\alpha$ as $\delta\to 0$, by an explicit computation.
Note that (1.2) is a discontinuous function. Therefore, in order to obtain the
rate of convergence, we need a smooth approximation of (1.2), and a quantified
version of Duke’s theorem with explicit dependency on the test functions. To
this end, we follow the argument sketched in [LRS09] to prove:
###### Theorem 1.6.
Assume that $f\in C_{0}^{\infty}\left(PSL_{2}\left(\mathbb{Z}\right)\backslash
PSL_{2}\left(\mathbb{R}\right)\right)$ has support in the region determined by
$y<T$. Then we have
$\frac{1}{l\left(\mathscr{C}_{d}\right)}\int_{\mathscr{C}_{d}}f\left(s\right)ds=\frac{3}{\pi^{2}}\int_{PSL_{2}\left(\mathbb{Z}\right)\backslash
PSL_{2}\left(\mathbb{R}\right)}f\left(g\right)d\mu_{g}+O_{\epsilon}\left(\log
Td^{-\frac{25}{512}+\epsilon}\|f\|_{W^{6,\infty}}\right).$
Here $\|\cdot\|_{W^{k,p}}$ is the Sobolev norm:
$\|f\|_{W^{k,p}}=\max_{|\alpha|\leq
k}\|\partial_{\theta}^{\alpha_{1}}\left(y\partial_{x}\right)^{\alpha_{2}}\left(y\partial_{y}\right)^{\alpha_{3}}f\|_{L^{p}}.$
###### Remark 1.7.
The proof of Theorem 1.1 is based on the equidistribution of the lifts of
$C_{d}$ in the unit tangent bundle. For this reason, one may generalize
Theorem 1.1 to any surfaces and any sequence of closed geodesics whose lifts
become equidistributed on the unit tangent bundle.
### 1.1. Intersecting two closed geodesics
We now consider the number of intersections between two closed geodesics when
both vary.
###### Theorem 1.8.
The following estimate holds uniformly in $d_{1},d_{2}>0$, and
$0<\theta_{1}<\theta_{2}<\pi$ such that
$\theta_{2}-\theta_{1}\gg\left(d_{1}d_{2}\right)^{-\frac{25}{3072}}$
$\frac{I_{\theta_{1},\theta_{2}}\left(C_{d_{1}},C_{d_{2}}\right)}{l\left(C_{d_{1}}\right)l\left(C_{d_{2}}\right)}=\frac{3}{\pi^{2}}\int_{\theta_{1}}^{\theta_{2}}\sin\theta
d\theta+O_{\epsilon}\left(\left(d_{1}d_{2}\right)^{-\frac{25}{6144}+\epsilon}\right).$
Note that if $\Gamma$ is co-compact, then the modular intersection kernel
coincides with the intersection kernel from [Lal14] when $\theta=\pi$ and
$\delta>0$ is sufficiently small. However, when $\Gamma\backslash\mathbb{H}$
is non-compact, then they are never the same; for instance, we have
$K_{\delta}^{\alpha_{1},\alpha_{2}}\left(g,g\right)=\Omega\left(y\right)$ as
$y\to\infty$ (Proposition 2.2). In particular,
$K_{\delta}^{\alpha_{1},\alpha_{2}}$ is not a Hilbert—Schmit kernel, so the
usual spectral theory does not apply. This is the main technical difficulty of
dealing with the modular intersection kernel for non-compact quotients of
$\mathbb{H}$. As it will be shown in the subsequent chapters, when both
$\alpha_{1}$ and $\alpha_{2}$ are closed geodesics,
$I_{0,\theta}\left(\alpha_{1},\alpha_{2}\right)/\left(l\left(\alpha_{1}\right)l\left(\alpha_{2}\right)\right)$
is the integral of
$\delta^{-2}K_{\delta}^{\theta_{1},\theta_{2}}/\left(l\left(\alpha_{1}\right)l\left(\alpha_{2}\right)\right)$
over $\alpha_{1}\times\alpha_{2}$. When $\alpha_{1}$ and $\alpha_{2}$ vary
over closed geodesics of length $<L$, as $L\to\infty$, we expect that the
integral converges to the integral of
$\delta^{-2}K_{\delta}^{\theta_{1},\theta_{2}}$ over $\Gamma\backslash
S\mathbb{H}\times\Gamma\backslash S\mathbb{H}$, since
$\alpha_{1}\times\alpha_{2}$ becomes equidistributed in $\Gamma\backslash
S\mathbb{H}\times\Gamma\backslash S\mathbb{H}$, as $L\to\infty$. However,
unboundedness of the modular intersection kernel $K$ causes issues of
interchanging the limit and the integral. In particular, the argument of
[PS06] using intersection form does not apply in this case. Hence, in order to
prove Theorem 1.8, we study the full spectral expansion of
$K_{\delta}^{\alpha_{1},\alpha_{2}}\left(g_{1},g_{2}\right)$. This is similar
to the existing work on the weight-$m$ Selberg’s trace formula [Hej76], except
that we have to deal with all weights simultaneously, and that the modular
intersection kernel is not diagonalizable in general. We go over this
carefully in Section 5. Once the spectral expansion is obtained, the integral
of $\delta^{-2}K_{\delta}^{\theta_{1},\theta_{2}}$ over
$\alpha_{1}\times\alpha_{2}$ becomes a linear combination of the period
integrals of the form
$\int_{\alpha_{1}}\phi_{1}ds\times\int_{\alpha_{2}}\phi_{2}ds.$
We may now use the same estimates that we use in order to prove the effective
Duke’s theorem to bound these, which leads to Theorem 1.8, generalizing [PS06]
to a non-compact hyperbolic surface.
## 2\. The Modular Intersection Kernel
### 2.1. Parametrization
Recall that $PSL_{2}\left(\mathbb{R}\right)$ acts transitively on $\mathbb{H}$
and on $S\mathbb{H}$ with the fractional transformations. For $g\in
PSL_{2}\left(\mathbb{R}\right),$ $z\in\mathbb{H}$ and $u\in S\mathbb{H}$ we
write these actions by $gz$ and $gu$. We parameterize the points of
$\mathbb{H}$ and $S\mathbb{H}$ with $x+iy$ and
$\left(x+iy,\exp\left(i\theta\right)\right)$. Let
$\Pi\left(\left(x+iy,\exp\left(i\theta\right)\right)\right):=x+iy,$
be the projection map from $S\mathbb{H}$ to $\mathbb{H}$.
Fix $z_{0}=i$ and $u_{0}=\left(i,\exp\left(i\frac{\pi}{2}\right)\right)$. Let
$g=naR_{\theta}\in PSL_{2}\left(\mathbb{R}\right)$ be the Iwasawa
decomposition where
$n=n\left(x\right)=\begin{pmatrix}1&x\\\
0&1\end{pmatrix},~{}a=a\left(y\right)=\begin{pmatrix}y^{\frac{1}{2}}&0\\\
0&y^{-\frac{1}{2}}\end{pmatrix},~{}\text{ and
}~{}R_{\theta}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}.$
Then we have
$gz_{0}=x+iy$
and
$gu_{0}=\left(x+iy,\exp\left(i\left(\frac{\pi}{2}+2\theta\right)\right)\right).$
For the rest of the paper, we identify $S\mathbb{H}$ with
$PSL_{2}\left(\mathbb{R}\right)$ by sending $g\in
PSL_{2}\left(\mathbb{R}\right)$ to $gu_{0}$. We often use the following fact
in our computation without mentioning.
###### Proposition 2.1.
The image under $\gamma\in SL_{2}\left(\mathbb{R}\right)$ of the geodesic
segment of length $\delta$ corresponding to $g=\left(x,\xi\right)$ is the
geodesic segment of length $\delta$ corresponding to $\gamma g$.
We use the volume form given by $dV=\frac{dxdyd\theta}{y^{2}}$. The volume of
$S\mathbb{X}$ is then $\frac{\pi^{2}}{3}$.
### 2.2. Preliminary estimates
We first recall here the definition of the modular intersection kernel
described in the introduction. For $\delta>0$ and
$\theta_{1},\theta_{2}\in\left(0,\pi\right)$, we define the integral kernel
$k_{\delta}^{\theta_{1},\theta_{2}}:S\mathbb{H}\times
S\mathbb{H}\to\mathbb{R}$
by
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(x_{1},\xi_{1}\right),\left(x_{2},\xi_{2}\right)\right)=1,$
if the geodesic segment of length $\delta$ on $\mathbb{H}$ from $x_{1}$ with
the initial vector $\xi_{1}$ and the segment from $x_{2}$ with the initial
vector $\xi_{2}$ intersect at an angle
$\in\left(\theta_{1},\theta_{2}\right)$, and $0$ otherwise. Here the angle of
the intersection of geodesic segments $l_{1}$ and $l_{2}$ at $p\in l_{1}\cap
l_{2}$ is measured counterclockwise from $l_{1}$ to $l_{2}$. Under the
identification $S\mathbb{H}\cong PSL_{2}\left(\mathbb{R}\right)$ from §2.1, we
note here that
$k_{\delta}^{\theta_{1},\theta_{2}}\left(gg_{1},gg_{2}\right)=k_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},g_{2}\right)$
for any $g,g_{1},g_{2}\in PSL_{2}\left(\mathbb{R}\right)$.
Now for a given discrete subgroup $\Gamma\subset
PSL_{2}\left(\mathbb{R}\right)$, we define the modular intersection kernel
$K_{\delta}^{\theta_{1},\theta_{2}}:\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\times\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\to\mathbb{R}$ by taking the average of
$k_{\delta}^{\theta_{1},\theta_{2}}$ over $\Gamma$:
$K_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}k_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},\gamma
g_{2}\right).$
Note that when $\Gamma$ is co-compact, and $\delta>0$ is less than a half of
the injectivity radius of $\Gamma\backslash\mathbb{H}$, we have
$K_{\delta}^{\theta_{1},\theta_{2}}\leq 1$. However, when
$\Gamma\backslash\mathbb{H}$ is non-compact,
$K_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},g_{2}\right)$ becomes
arbitrarily large near the diagonal $g_{1}=g_{2}$ as $y_{1},y_{2}\to\infty$.
This is illustrated in the following proposition when
$\Gamma=PSL_{2}(\mathbb{Z})$.
###### Proposition 2.2.
Fix $0<\theta<\pi$. Then for any $1>\delta>0$, we have
$K_{\delta}^{0,\theta}\left(g,g\right)=\Omega_{\theta}\left(\delta y\right).$
###### Proof.
Consider
$g=\left(Re^{i\left(\frac{\pi}{2}+\alpha\left(\delta\right)\right)},e^{i\alpha\left(\delta\right)}\right)\in
S\mathbb{H},$
where $\alpha\left(\delta\right)$ is chosen such that the geodesic segment
$\beta_{g}:=\\{Re^{i\theta}~{}:~{}|\theta-\frac{\pi}{2}|<\alpha\left(\delta\right)\\}\subset\mathbb{H}$
has length $\delta$. Note that the length of the segment does not depend on
$R$ and that $\alpha\left(\delta\right)\sim\delta$ as $\delta\to 0$. From
this, we infer that $\beta_{g}$ and $\beta_{g}+n$ with $0<n\ll R\delta$
intersect.
The angle of intersection is explicitly given by $2\arcsin\frac{n}{R}$. So for
all sufficiently small $0<\delta<\theta$, we have
$k_{\delta}^{\theta_{1},\theta_{2}}\left(g,\begin{pmatrix}1&n\\\
0&1\end{pmatrix}g\right)=1,$
for $0<n\ll R\delta$. This implies that
$K_{\delta}^{\theta_{1},\theta_{2}}\left(g,g\right)\gg\delta R\gg\delta
y.\qed$
In view of Proposition 2.2, the following proposition provides a nice upper
bound of the modular intersection kernel.
###### Proposition 2.3.
Let $\Gamma=PSL_{2}\left(\mathbb{Z}\right)$ and let $1>\delta>0$. Let $h$ be a
compactly supported function on $S\mathbb{H}$, where we assume that
$h\left(\left(\cdot,\xi\right)\right)$ is supported in
$B_{\delta}\left(i\right)$ for any $\xi\in S^{1}$. Define $H:\Gamma\backslash
S\mathbb{H}\times\Gamma\backslash S\mathbb{H}$ by
$H\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}h\left(g_{1}^{-1}\gamma
g_{2}\right)$
for $g_{1},g_{2}\in\Gamma\backslash PSL_{2}\left(\mathbb{R}\right)$. Then for
$g_{i}=\left(z_{i},\xi_{i}\right)$ with
$\textrm{dist}_{\Gamma\backslash\mathbb{H}}\left(z_{1},z_{2}\right)>2\delta$,
we have
$H\left(g_{1},g_{2}\right)=0.$
When $y_{1}>0$ and $y_{2}>0$ are sufficiently large, we have
$H\left(g_{1},g_{2}\right)\ll\delta\sqrt{y_{1}y_{2}}\|h\|_{L^{\infty}}.$
###### Proof.
If $H>0$, then there exists $\gamma\in\Gamma$ such that
$h\left(g_{1}^{-1}\gamma g_{2}\right)>0.$
This implies that the balls of radius $\delta$ centered at $z_{1}$ and $\gamma
z_{2}$ intersect, hence $\mathrm{dist}_{\mathbb{H}}\left(z_{1},\gamma
z_{2}\right)<2\delta$, which contradicts the assumption.
Now to prove the second estimate, we first note that when $y_{2}$ is
sufficiently large, we have $y\left(\gamma g_{2}\right)<1$ unless
$\gamma=\begin{pmatrix}1&n\\\ 0&1\end{pmatrix}$. Therefore
$h\left(g_{1}^{-1}\gamma g_{2}\right)>0$ only if $\gamma=\begin{pmatrix}1&n\\\
0&1\end{pmatrix}$. Note that $h\left(g_{1}^{-1}\gamma g_{2}\right)=1$ holds
only if $\textrm{dist}_{\mathbb{H}}\left(z_{1},n+z_{2}\right)<2\delta$. This
is equivalent to
$\mathrm{arccosh}\left(1+\frac{\left(n+x_{2}-x_{1}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}{y_{1}y_{2}}\right)<2\delta,$
and so
$\left(n+x_{2}-x_{1}\right)^{2}<y_{1}y_{2}\left(\cosh\left(2\delta\right)-1\right)-\left(y_{1}-y_{2}\right)^{2}\leq
y_{1}y_{2}\left(\cosh\left(2\delta\right)-1\right),$
from which we infer that there are at most $\ll\delta\sqrt{y_{1}y_{2}}$
choices of $\gamma$ which makes $h\left(g_{1},\gamma g_{2}\right)>0$. ∎
Now we analyze the modular intersection kernel when one variable is assumed to
be contained in a compact set. We first note that if $\delta$ is less than
half of the injectivity radius of $g_{0}$ in $\Gamma\backslash S\mathbb{H}$,
then for each $g\in S\mathbb{H}$, there is at most one $\gamma\in\Gamma$ such
that
$k_{\delta}^{\theta_{1},\theta_{2}}\left(g_{0},\gamma g\right)\neq 0.$
Therefore $K_{\delta}^{\theta_{1},\theta_{2}}\left(g_{0},\cdot\right)$
coincides with $k_{\delta}^{\theta_{1},\theta_{2}}\left(g_{0},\cdot\right)$ in
the $2\delta$-neighborhood of $g_{0}$, which is a translation of
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),\cdot\right)$ around
$\left(i,i\right)$.
###### Lemma 2.4.
For $0<\theta_{1}<\theta_{2}<\pi$, we have
$\int_{\mathbb{H}}k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),g\right)dV=\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2}.$
Assume that $0<\delta<1$. Then for any $\varepsilon=o\left(\delta\right)$ and
$\varepsilon=o\left(\theta_{2}-\theta_{1}\right)$ there exist a smooth
majorant $M_{\delta}^{\theta_{1},\theta_{2}}$ and a smooth minorant
$m_{\delta}^{\theta_{1},\theta_{2}}$, i.e.,
$0\leq m_{\delta}^{\theta_{1},\theta_{2}}\leq
k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),\cdot\right)\leq
M_{\delta}^{\theta_{1},\theta_{2}},$
such that
$\int m_{\delta}^{\theta_{1},\theta_{2}}dV~{}\text{ and }~{}\int
M_{\delta}^{\theta_{1},\theta_{2}}dV$
are both
$\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2}\left(1+O\left(\varepsilon\right)\right),$
and that
$\|m_{\delta}^{\theta_{1},\theta_{2}}\|_{W^{k,\infty}}+\|M_{\delta}^{\theta_{1},\theta_{2}}\|_{W^{k,\infty}}=O_{k}\left(\varepsilon^{-k}\right).$
###### Proof.
Note that the action of the geodesic flow of time $t$ on
$S\mathbb{H}=PSL_{2}\left(\mathbb{R}\right)$ is the multiplication from the
right by $a\left(e^{t}\right)$. For given
$\varphi\in\left(\theta_{1},\theta_{2}\right)$, we describe the collection of
$g\in PSL_{2}\left(\mathbb{R}\right)$ for which the corresponding geodesic
segment of length $\delta$ intersects $\\{iy~{}:~{}e^{\delta}>y>1\\}$
transversally at angle $\varphi$. Note that this happens only when
$ga\left(e^{\frac{t_{2}}{2}}\right)=\begin{cases}a\left(e^{\frac{t_{1}}{2}}\right)R_{\frac{\varphi}{2}},\\\
a\left(e^{\frac{t_{1}}{2}}\right)R_{\frac{\varphi+\pi}{2}}.\end{cases}$
for some $0<t_{1},t_{2}<\delta$. Hence
$g=\begin{cases}a\left(e^{\frac{t_{1}}{2}}\right)R_{\frac{\varphi}{2}}a\left(e^{-\frac{t_{2}}{2}}\right),\\\
a\left(e^{\frac{t_{1}}{2}}\right)R_{\frac{\varphi+\pi}{2}}a\left(e^{-\frac{t_{2}}{2}}\right).\end{cases}$
Consider $\Psi:AKA\to PSL_{2}\left(\mathbb{R}\right)$ given by
$\left(t_{1},\varphi,t_{2}\right)\mapsto
a\left(e^{\frac{t_{1}}{2}}\right)R_{\frac{\varphi}{2}}a\left(e^{-\frac{t_{2}}{2}}\right)$
The determinant of the Jacobian of $\Psi$ is a nonzero multiple of
$|\sin\varphi|$ (we refer the readers to Appendix A for the computation), and
so this defines a local diffeomorphism away from $\varphi=0$ and $\pi$.
Observe that $\Psi$ is injective away from $\varphi=0$ and $\pi$. From this we
infer that the support of
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),g\right)$ is the
image of the open box
$\\{\left(t_{1},\varphi,t_{2}\right)~{}:~{}0<t_{1},t_{2}<\delta,~{}\theta_{1}<\varphi<\theta_{2}~{}\text{
or }~{}\theta_{1}+\pi<\varphi<\theta_{2}+\pi\\}$
under $\Psi$, and
$\int_{\mathbb{H}}k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),g\right)dV=\frac{1}{2}\int_{0}^{\delta}\int_{0}^{\delta}\int_{\theta_{1}}^{\theta_{2}}|\sin\left(\varphi\right)|d\varphi
dt_{1}dt_{2}+\frac{1}{2}\int_{0}^{\delta}\int_{0}^{\delta}\int_{\theta_{1}+\pi}^{\theta_{2}+\pi}|\sin\left(\varphi\right)|d\varphi
dt_{1}dt_{2}\\\ =\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2},$
where we used $dV=\frac{1}{2}|\sin\varphi|d\varphi dt_{1}dt_{2}$ ((A.1)).
Note that the support of
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(i,i\right),\cdot\right)$ is an
open set which has a piecewise smooth boundary. Therefore, under the
assumption that $\varepsilon=o\left(\delta\right)$ and
$\varepsilon=o\left(\theta_{2}-\theta_{1}\right)$, there exist smooth majorant
and minorant whose $L^{1}$ norms are
$\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2}\left(1+O\left(\varepsilon\right)\right)$,
and whose $k$-th derivatives are $O_{k}\left(\varepsilon^{-k}\right)$. ∎
As an immediate application, we have the following corollary.
###### Corollary 2.5.
Fix a compact subset $C\subset\Gamma\backslash S\mathbb{H}$, and assume that
$\delta$ is less than the half of the infimum of injectivity radius of $g\in
C$ in $\Gamma\backslash S\mathbb{H}$. Then for any given compact geodesic
segment $\beta\subset C$, and for any given $\varepsilon>0$ which is
$o\left(\delta\right)$ and $o\left(\theta_{2}-\theta_{1}\right)$,
$\int_{\beta}K_{\delta}^{\theta_{1},\theta_{2}}\left(s,\cdot\right)ds$
admits a smooth majorant $M_{\beta,\delta}^{\theta_{1},\theta_{2}}$ and a
smooth minorant $m_{\beta,\delta}^{\theta_{1},\theta_{2}}$ such that
$\|m_{\beta,\delta}^{\theta_{1},\theta_{2}}\|_{L^{1}},\|M_{\beta,\delta}^{\theta_{1},\theta_{2}}\|_{L^{1}}=l\left(\beta\right)\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2}\left(1+O\left(\varepsilon\right)\right),$
and that
$\|m_{\beta,\delta}^{\theta_{1},\theta_{2}}\|_{W^{k,\infty}}+\|M_{\beta,\delta}^{\theta_{1},\theta_{2}i}\|_{W^{k,\infty}}=O_{k}\left(l\left(\beta\right)\varepsilon^{-k}\right).$
### 2.3. Intersection numbers
In this section, we prove formulas relating the number of intersections
between two geodesics to the integral of the modular intersection kernel over
the two geodesics.
###### Lemma 2.6.
Let
$\alpha_{i}=\\{\alpha_{i}\left(t\right)~{}:~{}t\in[0,l\left(\alpha_{i}\right))\\}$
be closed geodesics in $\Gamma\backslash\mathbb{H}$ parameterized by the arc
length, and let
$\widetilde{\alpha}_{i}=\\{\left(\alpha_{i}\left(t\right),\alpha_{i}^{\prime}\left(t\right)\right)~{}:~{}t\in[0,l\left(\alpha_{i}\right))\\}\subset
S\mathbb{H}$ be the lifts of $\alpha_{i}$ for $i=1,2$. Then for any
$\delta>0$,
$I_{\theta_{1},\theta_{2}}\left(\alpha_{1},\alpha_{2}\right)=\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\alpha}_{1}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}.$
###### Remark 2.7.
For each $\alpha_{i}$, there are two choices of parameterization by the arc
length, namely $\alpha_{i}\left(t\right)$ and $\alpha_{i}\left(-t\right)$, but
the integral does not depend on the choice of the parameterizations.
###### Proof.
By abuse of notations, we think of each $\alpha_{i}$ with
$t\in[0,l\left(\alpha_{i}\right))$ a geodesic segment in $\mathbb{H}$ and
accordingly $\widetilde{\alpha}_{i}$ a corresponding curve in $S\mathbb{H}$.
For a geodesic segment $\alpha\subset\mathbb{H}$ parameterized by $t\in[a,b]$,
let $[\alpha]\subset\mathbb{H}$ be the bi-infinite geodesic
$\\{\alpha\left(t\right)~{}:~{}t\in\mathbb{R}\\}$ that contains $\alpha$. Then
we express the integral as follows:
$\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\alpha}_{1}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\sum_{\gamma\in\Gamma}\frac{1}{\delta^{2}}\int_{\gamma\widetilde{\alpha}_{2}}\int_{\widetilde{\alpha}_{1}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\\\
=\sum_{\gamma\in\Gamma/\Gamma_{[\alpha_{2}]}}\frac{1}{\delta^{2}}\int_{\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{\alpha}_{1}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\\\
=\sum_{\gamma\in\Gamma_{[\alpha_{1}]}\backslash\Gamma/\Gamma_{[\alpha_{2}]}}\sum_{\gamma^{\prime}\in\Gamma_{[\alpha_{1}]}}\frac{1}{\delta^{2}}\int_{\gamma^{\prime}\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{\alpha}_{1}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\\\
=\sum_{\gamma\in\Gamma_{[\alpha_{1}]}\backslash\Gamma/\Gamma_{[\alpha_{2}]}}\frac{1}{\delta^{2}}\int_{\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{[\alpha_{1}]}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}.$
Here $\Gamma_{[\alpha_{i}]}$ is the stabilizer subgroup of $\Gamma$ with
respect to $[\alpha_{i}]$.
Now because two geodesics in $\mathbb{H}$ may intersect at most once, for each
intersection point $p\in\alpha_{1}\cap\alpha_{2}$ on
$\Gamma\backslash\mathbb{H}$, there exists a unique
$\gamma\in\Gamma/\Gamma_{[\alpha_{2}]}$ such that $\alpha_{1}$ and
$\gamma[\alpha_{2}]$ intersect at a lift of $p$. Also, because $[\alpha_{1}]$
is a disjoint union of $\gamma^{\prime}\alpha_{1}$ with
$\gamma^{\prime}\in\Gamma_{[\alpha_{1}]}$, each
$\\{\gamma^{\prime}\gamma~{}:~{}\gamma^{\prime}\in\Gamma_{[\alpha_{1}]}\\}$
contains at most one $\gamma^{\prime}\gamma$ such that
$\gamma^{\prime}\gamma[\alpha_{2}]$ intersects $\alpha_{1}$.
Therefore the intersections of $\alpha_{1}$ and $\alpha_{2}$ are in one-to-one
correspondence with
$\gamma\in\Gamma_{[\alpha_{1}]}\backslash\Gamma/\Gamma_{[\alpha_{2}]}$ such
that $\gamma[\alpha_{2}]$ intersects $[\alpha_{1}]$. We complete the proof by
observing that
$\int_{\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{[\alpha_{1}]}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=1,$
if $[\alpha_{1}]$ and $\gamma[\alpha_{2}]$ intersect at an angle
$\in\left(\theta_{1},\theta_{2}\right)$, and $=0$ otherwise. ∎
Now let $\beta=\\{\beta\left(t\right)~{}:~{}t\in[0,l\left(\beta\right))\\}$ be
a compact geodesic segment in $\Gamma\backslash\mathbb{H}$, and let
$\alpha_{2}$ be a closed geodesic as before. Then
$\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}$
does not always give $I\left(\beta,\alpha_{2}\right)$. Instead, it is a
weighted sum over the intersections of
$\beta_{0}:=\\{\beta\left(t\right)~{}:~{}t\in[0,l\left(\beta\right)+\delta)\\}$
and $\alpha_{2}$. We prove the following.
###### Lemma 2.8.
With the same notations as above, assume that $0<\delta<l\left(\beta\right)$
and that $\beta_{0}$ has no self intersection. For
$0<\theta_{1}<\theta_{2}<\pi$, let
$S\left(\beta_{0},\alpha_{2}\right)_{\theta_{1},\theta_{2}}$ be the set of
intersections between $\beta_{0}$ and $\alpha_{2}$ where the intersection
angle is $\in\left(\theta_{1},\theta_{2}\right)$. Then we have
$\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\sum_{p\in
S\left(\beta_{0},\alpha_{2}\right)_{\theta_{1},\theta_{2}}}\min\left\\{\frac{\beta^{-1}\left(p\right)}{\delta},1,\frac{l\left(\beta\right)+\delta-\beta^{-1}\left(p\right)}{\delta}\right\\}.$
###### Proof.
As in the proof of Lemma 2.6, we first have
$\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\beta}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\sum_{\gamma\in\Gamma}\frac{1}{\delta^{2}}\int_{\gamma\widetilde{\alpha}_{2}}\int_{\widetilde{\beta}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\\\
=\sum_{\gamma\in\Gamma/\Gamma_{[\alpha_{2}]}}\frac{1}{\delta^{2}}\int_{\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{\beta}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}.$
Note that because we assumed that $\beta_{0}$ has no self-intersection, $p\in
S\left(\beta_{0},\alpha_{2}\right)_{\theta_{1},\theta_{2}}$ is in one-to-one
correspondence with $\gamma\in\Gamma/\Gamma_{[\alpha_{2}]}$ such that
$\beta_{0}$ and $\gamma\widetilde{[\alpha_{2}]}$ intersect at $p$ at an angle
$\in\left(\theta_{1},\theta_{2}\right)$. We denote by $\gamma_{p}$ the
$\gamma$ corresponding to $p$. Observe that
$\int_{\gamma\widetilde{[\alpha_{2}]}}\int_{\widetilde{\beta}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=0,$
if $\gamma\widetilde{[\alpha_{2}]}\cap\beta_{0}=\emptyset$. So it is
sufficient to prove that
$\frac{1}{\delta^{2}}\int_{\gamma_{p}\widetilde{[\alpha_{2}]}}\int_{\widetilde{\beta}}k_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\min\left\\{\frac{\beta^{-1}\left(p\right)}{\delta},1,\frac{l\left(\beta\right)+\delta-\beta^{-1}\left(p\right)}{\delta}\right\\}.$
This follows by observing that
$k_{\delta}^{\theta_{1},\theta_{2}}\left(\left(\beta\left(t_{1}\right),\beta^{\prime}\left(t_{1}\right)\right),\left(\gamma_{p}\alpha_{2}\left(t_{2}\right),\left(\gamma_{p}\alpha_{2}\right)^{\prime}\left(t_{2}\right)\right)\right)=1$
for
$\left(t_{1},t_{2}\right)\in\left(\beta^{-1}\left(p\right)-\delta,\beta^{-1}\left(p\right)\right)\times\left(\alpha_{2}^{-1}\left(p\right)-\delta,\alpha_{2}^{-1}\left(p\right)\right),$
and $0$ otherwise, whereas the integral over $\widetilde{\beta}$ is over the
range $t_{1}\in(0,l(\beta))$. ∎
## 3\. Spectral theory
### 3.1. Spectral expansion
We first go over the spectral decomposition of
$L^{2}\left(S\mathbb{X}\right)$. Readers may find more details on the subject
from [Kub73] and [Lan85]. On $G=PSL_{2}\left(\mathbb{R}\right)$, there is a
differential operator of order $2$ that commutes with $G$ action:
$\Omega=y^{2}\partial_{x}^{2}+y^{2}\partial_{y}^{2}+y\partial_{x}\partial_{\theta},$
which is called the Casimir operator. An equivariant eigenfunctions of
$\Omega$ is a function $f\in C^{\infty}(S\mathbb{X})$ that satisfies
$\Omega f=\lambda f$
for some $\lambda\in\mathbb{R}$, and
$f\left(gR_{\theta}\right)=e^{-im\theta}f\left(g\right)$ (3.1)
for some $m\in 2\mathbb{Z}$. We say that a function has weight $m$ if it
satisfies (3.1).
Each irreducible (cuspidal) sub-representation of the right regular
representation
$\rho_{g}:f\left(h\right)\mapsto f\left(hg\right)$
on $L^{2}\left(S\mathbb{X}\right)$ is generated by an equivariant
eigenfunction of $\Omega$.
We let $\mathbf{E}^{+}$ and $\mathbf{E}^{-}$ to be the raising and lowering
operator acting on equivariant functions on $L^{2}\left(S\mathbb{X}\right)$,
which are given by [Jak94]
$\begin{split}\mathbf{E}^{+}&=e^{-2i\theta}\left(2iy\partial_{x}+2y\partial_{y}+i\partial_{\theta}\right),\text{
and}\\\
\mathbf{E}^{-}&=e^{2i\theta}\left(2iy\partial_{x}-2y\partial_{y}+i\partial_{\theta}\right).\end{split}$
(3.2)
Note that $\mathbf{E}^{+}$ (resp. $\mathbf{E}^{-}$) maps a weight $m$
eigenfunction of $\Omega$ to a weight $m+2$ (resp. $m-2$) eigenfunction of
$\Omega$.
For an even integer $m$ let
$\psi_{s,m}\left(g\right)=y^{s}e^{-im\theta}.$
Note that $\psi_{s,m}$ is invariant under the action of the unipotent upper
triangular matrices. The weight-$m$ Eisenstein series is then given by
$E_{m}\left(g,s\right)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\psi_{s,m}\left(\gamma
g\right),$
where $\Gamma_{\infty}=\left\\{\begin{pmatrix}1&n\\\
&1\end{pmatrix}~{}:~{}n\in\mathbb{Z}\right\\}$ is the stabilizer subgroup of
$\Gamma$ with respect to the cusp $i\infty$. Although the right-hand side of
the equation is absolutely convergent only for $\mathrm{Re}(s)>1$, the
weight-$m$ Eisenstein series has a meromorphic continuation to the entire
complex plane.
Let $\Theta$ be the closure of
$\left\\{\int_{-\infty}^{\infty}h\left(t\right)E_{m}\left(g,\frac{1}{2}+it\right)dt~{}:~{}h\left(t\right)\in
C_{0}^{\infty}\left(\mathbb{R}\right),~{}m\in 2\mathbb{Z}\right\\}$
in $L^{2}\left(S\mathbb{X}\right)$, and let
$L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)=\\{f\in
L^{2}\left(S\mathbb{X}\right)~{}:~{}\int_{0}^{1}f\left(n\left(x\right)g\right)dx=0\text{
for almost every }g\in S\mathbb{X}\\}$
be the space of cusp forms. Then we have the decomposition
$L^{2}\left(S\mathbb{X}\right)=\langle\\{1\\}\rangle\oplus\Theta\oplus
L_{\text{cusp}}^{2}\left(S\mathbb{X}\right),$
where $\langle\\{1\\}\rangle$ is the subspace spanned by a constant function.
We express the cuspidal subspace as a direct sum of subspaces generated by
Maass forms and modular forms as in [LRS09, (1.10)],
$L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)=\sum_{j=1}^{\infty}W_{\pi_{j}^{0}}\bigoplus\sum_{m\geq
12}\sum_{j=1}^{d_{m}}\left(W_{\pi_{j}^{m}}\oplus W_{\pi_{j}^{-m}}\right),$
where each $W_{\pi_{j}^{m}}$ corresponds to a $G$ and Hecke irreducible
subspace of a right regular representation on $L_{\text{cusp}}^{2}$. Here
$d_{m}$ is the dimension of the space of holomorphic cusp forms of weight $m$
for $PSL_{2}\left(\mathbb{Z}\right)$. Each $\pi_{j}^{0}$ corresponds to a
Maass—Hecke cusp form which we denote by $\phi_{j}^{0}$. For $m>0$,
$\pi_{j}^{m}$ corresponds to a holomorphic Hecke cusp form $\phi_{j}^{m}$. We
identify a weight $m$ function on $\Gamma\backslash\mathbb{H}$
$f\left(\gamma z\right)=\left(cz+d\right)^{m}f\left(z\right)~{}\text{ for
}~{}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\gamma\in\Gamma$
with a weight $m$ $\Gamma$-invariant function $F$ on
$PSL_{2}\left(\mathbb{R}\right)$ via
$F\left(g\right)=y^{\frac{m}{2}}f\left(z\right)e^{-im\theta}.$ (3.3)
When $m\geq 0$, viewing $\phi_{j}^{m}$ as a function on $S\mathbb{X}$, each
$W_{\pi_{j}^{m}}$ is spanned by
$\ldots,~{}\left(\mathbf{E}^{-}\right)^{3}\phi_{j}^{m},~{}\left(\mathbf{E}^{-}\right)^{2}\phi_{j}^{m},~{}\mathbf{E}^{-}\phi_{j}^{m},~{}\phi_{j}^{m},~{}\mathbf{E}^{+}\phi_{j}^{m},~{}\left(\mathbf{E}^{+}\right)^{2}\phi_{j}^{m},~{}\left(\mathbf{E}^{+}\right)^{3}\phi_{j}^{m},\ldots$
Note that when $m>0$, $\mathbf{E}^{-}\phi_{j}^{m}=0$.
For $m<0$, we set
$W_{\pi_{j}^{-m}}=\overline{W_{\pi_{j}^{m}}}=\\{\bar{f}~{}:~{}f\in
W_{\pi_{j}^{m}}\\}.$
Now let
$U_{\pi_{j}^{0}}=W_{\pi_{j}^{0}},\text{ and
}U_{\pi_{j}^{m}}=W_{\pi_{j}^{m}}\oplus W_{\pi_{j}^{-m}},$
when $m>0$. We specify an orthonormal basis of each $U_{\pi_{j}^{m}}$ as
follows.
The Maass cusp form case $m=0$. Let $-\left(\frac{1}{4}+t_{j}^{2}\right)$ be
the Laplacian eigenvalue of $\phi_{j}^{0}$222Formally, it is the eigenvalue of
the Laplace—Beltrami operator on $\mathbb{X}$ that corresponds to
$\phi_{j}^{0}$., for some real $t_{j}$. We set $\phi_{j,0}^{0}=\phi_{j}^{0}$,
and define $\phi_{j,l}^{0}$ for $l\in 2\mathbb{Z}$ inductively by
$\begin{split}\mathbf{E}^{-}\phi_{j,l}^{0}&=\left(l+1-2it_{j}\right)\phi_{j,l-2}^{0},\text{
and}\\\
\mathbf{E}^{+}\phi_{j,l}^{0}&=\left(l+1+2it_{j}\right)\phi_{j,l+2}^{0}.\end{split}$
(3.4)
The holomorphic Hecke cusp form case $m>0$. We set
$\phi_{j,m}^{m}=\phi_{j}^{m}$ and $\phi_{j,-m}^{m}=\overline{\phi_{j}^{m}}$,
and define $\phi_{j,l}^{m}$ for $l\in 2\mathbb{Z}$ inductively by
$\begin{split}\mathbf{E}^{-}\phi_{j,l}^{m}&=\left(l-m\right)\phi_{j,l-2}^{m},\text{
and}\\\
\mathbf{E}^{+}\phi_{j,l}^{m}&=\left(l+m\right)\phi_{j,l+2}^{m}.\end{split}$
(3.5)
Finally, note that we have the following relation among the weight $m$
Eisenstein series.
$\displaystyle\mathbf{E}^{-}E_{m}\left(g,\frac{1}{2}+it\right)$
$\displaystyle=\left(m+1-2it\right)E_{m-2}\left(g,\frac{1}{2}+it\right),\text{
and}$ $\displaystyle\mathbf{E}^{+}E_{m}\left(g,\frac{1}{2}+it\right)$
$\displaystyle=\left(m+1+2it\right)E_{m+2}\left(g,\frac{1}{2}+it\right).$
With these notations, we have
###### Proposition 3.1.
Let $f\in L^{2}\left(S\mathbb{X}\right)$. Then we have
$f\left(g\right)=\frac{3}{\pi^{2}}\int_{S\mathbb{X}}f\left(g_{1}\right)dg_{1}+\sum_{\begin{subarray}{c}m\geq
0\\\ 2|m\end{subarray}}\sum_{j=1}^{d_{m}}\sum_{\begin{subarray}{c}l\in
2\mathbb{Z}\\\ |l|\geq m\end{subarray}}\left\langle
f,\phi_{j,l}^{m}\right\rangle_{S\mathbb{X}}\phi_{j,l}^{m}\left(g\right)\\\
+\sum_{m\in 2\mathbb{Z}}\frac{1}{4\pi}\int_{-\infty}^{\infty}\left\langle
f,E_{m}\left(\cdot,\frac{1}{2}+it\right)\right\rangle_{S\mathbb{X}}E_{m}\left(g,\frac{1}{2}+it\right)dt,$
where we set $d_{0}=+\infty$.
## 4\. Effective Equidistribution
### 4.1. Invariant linear form
Define $\mu_{d}$ to be the integral over discriminant $d$ oriented closed
geodesics on $S\mathbb{X}$,
$\mu_{d}\left(F\right):=\int_{\mathscr{C}_{d}}F\left(s\right)ds=\sum_{\text{disc}\left(q\right)=d}\int_{C\left(q\right)}F\left(s\right)ds.$
where $C\left(q\right)\subset S\mathbb{X}$ is the oriented closed geodesic
associated to the binary quadratic form $q$ [LRS09, 2.3]. Then for any $F\in
U_{\pi_{j}^{m}}$, we have
$\mu_{d}\left(F\right)=\mu_{d}\left(\phi_{j}^{m}\right)\eta_{j}^{m}\left(F\right)$
for some linear form $\eta_{j}^{m}$ on $U_{\pi_{j}^{m}}$ invariant under the
diagonal action [LRS09, §3.7.1], which we describe below following [LRS09,
§3.2]. (Note that the parameter $s$ in [LRS09] is replaced by $2it$ in this
article for consistency.)
The Maass cusp form case $m=0$. Let $\phi_{j,l}^{0}$ be the Maass form defined
by (3.4). When $4|l$ and $l\geq 4$, we have
$\eta_{j}^{0}\left(\phi_{j,l}^{0}\right)=\eta_{j}^{0}\left(\phi_{j,-l}^{0}\right)=\frac{\left(1-2it_{j}\right)\left(5-2it_{j}\right)\cdots\left(l-3-2it_{j}\right)}{\left(3+2it_{j}\right)\left(7+2it_{j}\right)\cdots\left(l-1+2it_{j}\right)},$
(4.1)
and $\eta_{j}^{0}\left(\phi_{j,l}^{0}\right)$ is identically $0$ if $l\equiv
2\pmod{4}$. Note that $\\{\phi_{j,l}^{0}\\}_{l\in 2\mathbb{Z}}$ is an
orthogonal basis of $U_{\pi_{j}^{0}}$, and normalized so that,
$\|\phi_{j,l}^{0}\|_{L^{2}}=\|\phi_{j}^{0}\|_{L^{2}}.$
The holomorphic Hecke cusp form case $m>0$. Let $\phi_{j,l}^{m}$ be the
holomorphic Hecke cusp form defined by (3.5). When $m\equiv 2\pmod{4}$,
$\eta_{j}^{m}$ is identically $0$.
When $m\equiv 0\pmod{4}$, for $l\geq 4$ with $4|l$,
$\eta_{j}^{m}\left(\phi_{j,m+l}^{m}\right)=\eta_{j}^{m}\left(\phi_{j,-m-l}^{m}\right)=\frac{1\cdot
3\cdot
5\cdots\left(\frac{l}{2}-1\right)}{\left(m+1\right)\left(m+3\right)\cdots\left(m+\frac{l}{2}-1\right)},$
(4.2)
and $\eta_{j}^{m}\left(\phi_{j,m+l}^{m}\right)$ vanishes for $l\equiv
2\pmod{4}$.
Note that $\\{\phi_{j,l}^{m}\\}_{l\in 2\mathbb{Z},~{}|l|\geq m}$ is an
orthogonal basis of $U_{\pi_{j}^{m}}$, and normalized so that
$\|\phi_{j,l}^{m}\|_{L^{2}}=\|\phi_{j}^{m}\|_{L^{2}}.$
for $l\in 2\mathbb{Z},~{}|l|\geq m$.
Eisenstein series case. By the above identities and following [LRS09, Section
3], we have
$\mu_{d}\left(E_{m}\left(g,\frac{1}{2}+it\right)\right)=\eta\left(m,t\right)\mu_{d}\left(E_{0}\left(g,\frac{1}{2}+it\right)\right),$
where for $m\geq 4$ such that $4|m$,
$\eta\left(m,t\right)=\eta\left(-m,t\right)=\frac{\left(1-2it\right)\left(5-2it\right)\cdots\left(2m-3-2it\right)}{\left(3+2it\right)\left(7+2it\right)\cdots\left(2m-1+2it\right)},$
(4.3)
and $\eta\left(m,t\right)$ is identically $0$ if $m\equiv 2\pmod{4}$.
### 4.2. Period integrals
#### 4.2.1. Holomorphic cusp forms
In this section, we give an upper bound on the period integrals of holomorphic
forms. We first use the results of Shintani to relate the period integrals of
holomorphic cusp forms to the Fourier coefficients of half integral
holomorphic forms. We then apply the result of Kohnen and Zagier [KZ81] which
gives an explicit version of the Waldspurger’s formula for the Fourier
coefficients of half integral holomorphic forms. An upper bound on these
period integrals is deduced by using the subconvexity bounds on the central
value of the $L$-functions and the Ramanujan bound on the Fourier coefficients
of holomorphic modular forms.
Note that $c(d)$ is identically zero when $m\equiv 2\pmod{4}$, and so we
assume that $4|m$. Let $\hat{\phi}_{j}^{m}$ be a normalization of the Hecke
holomorphic cusp form $\phi_{j}^{m}$ of weight $m$ such that $a_{1}=1$. Let
$c\left(d\right):=\sum_{\text{disc}\left(q\right)=d}\int_{C\left(q\right)}\hat{\phi}_{j}^{m}\left(z\right)q\left(z,1\right)^{\frac{m}{2}-1}dz,$
where $\hat{\phi}_{j}^{m}\left(z\right)$ is the associated holomorphic modular
form defined on the upper half plane and the integration is on the upper half
plane (3.3). By [LRS09, equation (2.4) p.14], we have
$|c\left(d\right)|=|d|^{\frac{m}{4}-\frac{1}{2}}|\mu_{d}\left(\hat{\phi}_{j}^{m}\right)|.$
(4.4)
Let
$\theta\left(z,\hat{\phi}_{j}^{m}\right):=\sum_{d\geq
1}c\left(d\right)e\left(dz\right).$
By [Shi75, Theorem 2], $\theta\left(z,\phi_{j}^{m}\right)$ is a Hecke
holomorphic cusp form of weight $\frac{m+1}{2}$ and level
$\Gamma_{0}\left(4\right)$. By [LRS09, (6.2), p.37], we have the following
explicit version of Rallis inner product formula
$\left\langle\theta\left(\hat{\phi}_{j}^{m}\right),\theta\left(\hat{\phi}_{j}^{m}\right)\right\rangle=\frac{\left(\frac{m}{2}-1\right)!}{2^{m}\pi^{\frac{m}{2}}}L\left(\frac{1}{2},\phi_{j}^{m}\right)\left\langle\hat{\phi}_{j}^{m},\hat{\phi}_{j}^{m}\right\rangle.$
Suppose that $d=Db^{2}$ with $D$ a fundamental discriminant. By [KZ81, Theorem
1], for $D$ a fundamental discriminant with $D>0$ and $4|m$, we have
$\frac{c\left(D\right)^{2}}{\left\langle\theta\left(\hat{\phi}_{j}^{m}\right),\theta\left(\hat{\phi}_{j}^{m}\right)\right\rangle}=\frac{\left(\frac{m}{2}-1\right)!}{\pi^{\frac{m}{2}}}D^{\frac{m-1}{2}}\frac{L\left(\frac{1}{2},\phi_{j}^{m}\otimes\chi_{D}\right)}{\left\langle\hat{\phi}_{j}^{m},\hat{\phi}_{j}^{m}\right\rangle},$
which implies that
$|c\left(D\right)|=D^{\frac{m-1}{4}}\frac{\left(\frac{m}{2}-1\right)!}{2^{\frac{m}{2}}\pi^{\frac{m}{2}}}\left(L\left(\frac{1}{2},\phi_{j}^{m}\right)L\left(\frac{1}{2},\phi_{j}^{m}\otimes\chi_{D}\right)\right)^{\frac{1}{2}}.$
By using the Ramanujan bound on the Fourier coefficients of integral weight
cusp forms and the above, we have
$|c\left(d\right)|\ll_{\epsilon}b^{\frac{m-1}{2}+\epsilon}|c\left(D\right)|\ll_{\epsilon}d^{\frac{m-1}{4}+\epsilon}\frac{\left(\frac{m}{2}-1\right)!}{2^{\frac{m}{2}}\pi^{\frac{m}{2}}}\left(L\left(\frac{1}{2},\phi_{j}^{m}\right)L\left(\frac{1}{2},\phi_{j}^{m}\otimes\chi_{D}\right)\right)^{\frac{1}{2}},$
and so
$|\mu_{d}\left(\hat{\phi}_{j}^{m}\right)|\ll_{\epsilon}|d|^{\frac{1}{4}+\epsilon}\frac{\left(\frac{m}{2}-1\right)!}{2^{\frac{m}{2}}\pi^{\frac{m}{2}}}\left(L\left(\frac{1}{2},\phi_{j}^{m}\right)L\left(\frac{1}{2},\phi_{j}^{m}\otimes\chi_{D}\right)\right)^{\frac{1}{2}},$
by (4.4).
We now use the convexity bound
$L\left(\frac{1}{2},\phi_{j}^{m}\right)\ll_{\epsilon}m^{\frac{1}{2}+\epsilon},$
and the subconvexity bound [BHM07, Theorem 1]
$L\left(\frac{1}{2},\phi_{j}^{m}\otimes\chi_{D}\right)\ll_{\epsilon}m^{\frac{75+12\theta}{16}}D^{\frac{1}{2}-\frac{1}{8}\left(1-2\theta\right)+\epsilon},$
where $\theta=\frac{7}{64}$ is the best exponent toward Ramanujan conjecture
for Maass forms, to see that
$|\mu_{d}\left(\hat{\phi}_{j}^{m}\right)|\ll_{\epsilon}d^{\frac{1}{4}+\epsilon}\frac{\left(\frac{m}{2}-1\right)!}{2^{\frac{m}{2}}\pi^{\frac{m}{2}}}m^{2.64}D^{\frac{1}{4}-\frac{25}{512}}.$
It is well-known that
$\left\langle\hat{\phi}_{j}^{m},\hat{\phi}_{j}^{m}\right\rangle=\frac{\Gamma\left(m\right)}{\left(4\pi\right)^{m}}L\left(1,\text{sym}^{2}\phi_{j}^{m}\right)$
up to a constant. Hence, by Stirling’s approximation
$|\mu_{d}\left(\phi_{j}^{m}\right)|\ll_{\epsilon}d^{\frac{1}{4}+\epsilon}m^{2.9}D^{\frac{1}{4}-\frac{25}{512}}\ll
d^{\frac{1}{2}-\frac{25}{512}+\epsilon}m^{2.9}.$ (4.5)
#### 4.2.2. Maass forms
In this section, we give an upper bound on the period integrals of Maass
forms. We first recall some results of Katok and Sarnak [KS93] that generalize
the work of Shintani [Shi75] to Maass forms and related the period integrals
to the Fourier coefficients of half integral Maass forms. Then we use an
explicit version of the Waldspurger formula [BM10] and give a non-trivial
bound on these period integrals by using the subconvexity bound on the central
value of the $L$-functions and the best bound toward Ramanujan conjecture for
Maass forms.
Let $\phi_{j}^{0}$ be a Hecke—Maass form with
$\left\langle\phi_{j}^{0},\phi_{j}^{0}\right\rangle=1$ and with the Laplacian
eigenvalue $-\left(\frac{1}{4}+t_{j}^{2}\right)$. For $d>0$, let
$\rho\left(d\right):=\frac{1}{\sqrt{8}\pi^{\frac{1}{4}}d^{\frac{3}{4}}}\sum_{disc\left(q\right)=d}\int_{C\left(q\right)}\phi_{j}^{0}ds$
be the associated period integral, and let
$\theta\left(\left(u+iv\right),\phi_{j}^{0}\right):=\sum_{d\neq
0}\rho\left(d\right)W_{\frac{\text{sgn}\left(d\right)}{4},\frac{it_{j}}{2}}\left(4\pi|d|v\right)e\left(du\right),$
where $W_{\frac{\text{sgn}\left(d\right)}{4},\frac{it_{j}}{2}}$ is the usual
Whittaker function. Here $\rho\left(d\right)$ for $d<0$ is the sum of
$\phi_{j}^{0}$ over the CM points with the discriminant $d$ appropriately
normalized (see [KS93, p.197] or [TS20, section 3.3] for a detailed
discussion.)
Note from [KS93] that $\theta\left(\left(u+iv\right),\phi_{j}^{0}\right)$ is a
weight-$\frac{1}{2}$ Hecke—Maass form with the Laplacian eigenvalue
$-\left(\frac{1}{4}+\frac{t_{j}^{2}}{4}\right)$. By [KS93, (5.6), p.224] or
[LRS09, (6.4), p.38], we have the following version of the Rallis inner
product formula
$\left\langle\theta\left(\phi_{j}^{0}\right),\theta\left(\phi_{j}^{0}\right)\right\rangle=\frac{3}{2}\Lambda\left(\frac{1}{2},\phi_{j}^{0}\right),$
where
$\Lambda\left(s,\phi_{j}^{0}\right)=\pi^{-s}\Gamma\left(\frac{s+it_{j}}{2}\right)\Gamma\left(\frac{s-it_{j}}{2}\right)L\left(s,\phi_{j}^{0}\right)$
is the completed $L$-function.
By an explicit form of Waldspurger formula [BM10, Theorem 1.4], and the best
exponent toward the Ramanujan conjecture [LRl20, Corollary 6.1], we have
$\frac{\rho\left(d\right)}{\left\langle\theta\left(\phi_{j}^{0}\right),\theta\left(\phi_{j}^{0}\right)\right\rangle^{\frac{1}{2}}}\ll_{\epsilon}\frac{1}{\sqrt{|d|}}\left(\frac{L\left(\frac{1}{2},\phi_{j}^{0}\otimes\chi_{D}\right)}{L\left(1,\text{sym}^{2}\phi_{j}^{0}\right)}\right)^{\frac{1}{2}}b^{\frac{7}{64}+\epsilon}|t_{j}|^{-\frac{\text{sgn}\left(d\right)}{4}}e^{\frac{\pi|t_{j}|}{4}},$
where $d=Db^{2}$ with $D$ a fundamental discriminant. Note from Stirling’s
formula that
$\Gamma\left(\frac{\frac{1}{2}+it_{j}}{2}\right)\Gamma\left(\frac{\frac{1}{2}-it_{j}}{2}\right)\ll|t_{j}|^{-\frac{1}{2}}e^{-\frac{\pi|t_{j}|}{2}},$
from which we infer that
$\displaystyle\mu_{d}\left(\phi_{j}^{0}\right)$ $\displaystyle\ll
d^{\frac{3}{4}}|\rho\left(d\right)|$
$\displaystyle\ll_{\epsilon}d^{\frac{1}{4}}\left(\Lambda\left(\frac{1}{2},\phi_{j}^{0}\right)\right)^{\frac{1}{2}}\left(\frac{L\left(\frac{1}{2},\phi_{j}^{0}\otimes\chi_{D}\right)}{L\left(1,\text{sym}^{2}\phi_{j}^{0}\right)}\right)^{\frac{1}{2}}b^{\frac{7}{64}+\epsilon}|t_{j}|^{-\frac{\text{sgn}\left(d\right)}{4}}e^{\frac{\pi|t_{j}|}{4}}$
$\displaystyle\ll_{\epsilon}d^{\frac{1}{4}}\left(L\left(\frac{1}{2},\phi_{j}^{0}\right)L\left(\frac{1}{2},\phi_{j}^{0}\otimes\chi_{D}\right)\right)^{\frac{1}{2}}b^{\frac{7}{64}+\epsilon}|t_{j}|^{-\left(\frac{\text{sgn}\left(d\right)+1}{4}\right)+\epsilon}.$
We now use the convexity bound,
$L\left(\frac{1}{2},\phi_{j}^{0}\right)\ll_{\epsilon}|t_{j}|^{\frac{1}{2}+\epsilon},$
and the subconvexity bound [BHM07, Theorem 1],
$L\left(\frac{1}{2},\phi_{j}^{0}\otimes\chi_{D}\right)\ll_{\epsilon}|t_{j}|^{\frac{31+4\theta+\epsilon}{16}}D^{\frac{1}{2}-\frac{1}{8}\left(1-2\theta\right)+\epsilon},$
to conclude that
$\mu_{d}\left(\phi_{j}^{0}\right)\ll_{\epsilon}d^{\frac{1}{4}+\epsilon}|t_{j}|^{\frac{3}{4}}b^{\frac{7}{64}+\epsilon}D^{\frac{1}{4}-\frac{25}{512}}\ll
d^{\frac{1}{2}-\frac{25}{512}+\epsilon}|t_{j}|^{\frac{3}{4}}.$ (4.6)
#### 4.2.3. Eisenstein series
For a non-square integer $d\equiv 0,1\pmod{4}$, let $d=Db^{2}$ where $D$ is a
fundamental discriminant. Then we have the following explicit formula for the
period integral of the Eisenstein series [Zag81, p.282]222When $b=1$, this is
a classical result due to Hecke [Sie65, p.88].:
$\mu_{d}\left(E_{0}\left(\cdot,s\right)\right)=\frac{\Gamma(\frac{s}{2})^{2}d^{\frac{s}{2}}L(s,d)}{\Gamma(s)\zeta(2s)},$
(4.7)
where
$L(s,d)=L(s,\chi_{D})\left(\sum_{a|b}\mu(a)\left(\frac{D}{a}\right)a^{-s}\sigma_{1-2s}\left(\frac{b}{a}\right)\right).$
(4.8)
Here $L(s,\chi_{D})$ is the Dirichlet $L$-function attached to the quadratic
Dirichlet character $\chi_{D}(\cdot)=\left(\frac{D}{\cdot}\right)$,
$\mu(\cdot)$ is the Möbius function, and
$\sigma_{v}(\cdot)=\sum_{a|\cdot}a^{v}$ is the divisor function.
Now assume that $s=\frac{1}{2}+it$ for some $t\in\mathbb{R}$. By Stirling’s
formula, we have
$\frac{\Gamma(\frac{s}{2})^{2}}{\Gamma(s)}\ll|t|^{-\frac{1}{2}}.$
By the zero free region of $\zeta(2s)$ around $2s=1+2it$, we have
$|\zeta(2s)|\gg_{\epsilon}t^{-\epsilon}.$
We also have the convexity bound
$\zeta\left(s\right)\ll|t|^{\frac{1}{4}},$
and we know from [HB80] that
$L\left(\frac{1}{2}+it,\chi_{D}\right)\ll_{\epsilon}\left(\left(|t|+1\right)D\right)^{\frac{3}{16}+\epsilon}.$
Finally, observe that we have
$\sum_{a|b}\mu(a)\left(\frac{D}{a}\right)a^{-s}\sigma_{1-2s}\left(\frac{b}{a}\right)\ll_{\epsilon}d^{\epsilon}.$
Combining all these estimates, we deduce the following estimate from (4.7) for
$s=\frac{1}{2}+it$:
$\mu_{d}\left(E_{0}\left(\cdot,s\right)\right)\ll_{\epsilon}d^{\frac{1}{2}-\frac{1}{16}+\epsilon}.$
(4.9)
### 4.3. Proof of Theorem 1.6
For any compactly supported smooth function $F\in
C_{0}^{\infty}\left(S\mathbb{X}\right)$, recall from Proposition 3.1 that we
have
$F\left(g\right)=\frac{3}{\pi^{2}}\int_{S\mathbb{X}}F\left(g_{1}\right)dg_{1}+\sum_{\begin{subarray}{c}m\geq
0\\\ 2|m\end{subarray}}\sum_{j=1}^{d_{m}}\sum_{\begin{subarray}{c}l\in
2\mathbb{Z}\\\ |l|\geq m\end{subarray}}\left\langle
F,\phi_{j,l}^{m}\right\rangle_{S\mathbb{X}}\phi_{j,l}^{m}\left(g\right)\\\
+\sum_{m\in 2\mathbb{Z}}\frac{1}{4\pi}\int_{-\infty}^{\infty}\left\langle
F,E_{m}\left(\cdot,\frac{1}{2}+it\right)\right\rangle_{S\mathbb{X}}E_{m}\left(g,\frac{1}{2}+it\right)dt,$
and so from the discussion of Section 4.1, we have
$\mu_{d}\left(F\right)=\mu_{d}\left(\frac{3}{\pi^{2}}\right)\int_{S\mathbb{X}}F\left(g\right)dg+\sum_{\begin{subarray}{c}m\geq
0\\\
4|m\end{subarray}}\sum_{j=1}^{d_{m}}\mu_{d}\left(\phi_{j}^{m}\right)\sum_{\begin{subarray}{c}l\in
4\mathbb{Z}\\\ |l|\geq m\end{subarray}}\left\langle
F,\phi_{j,l}^{m}\right\rangle_{S\mathbb{X}}\eta_{j}^{m}\left(\phi_{j,l}^{m}\right)\\\
+\sum_{m\in 4\mathbb{Z}}\frac{1}{4\pi}\int_{-\infty}^{\infty}\left\langle
F,E_{m}\left(\cdot,\frac{1}{2}+it\right)\right\rangle_{S\mathbb{X}}\eta\left(m,\frac{1}{2}+it\right)\mu_{d}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)dt.$
Firstly, we have from (4.1), (4.2), and (4.3) that
$\eta_{j}^{m}\left(\phi_{j,l}^{m}\right)$ and
$\eta\left(m,\frac{1}{2}+it\right)$ are both $O\left(1\right)$. Note by
successive integration by parts and Cauchy—Schwarz inequality, we have for all
$N\geq 1$,
$\langle
F,\phi_{j,l}^{m}\rangle\ll_{N}\left(|l|^{2}+1\right)^{-N}\|F\|_{W^{2N,2}\left(S\mathbb{X}\right)},$
when $m>0$, and
$\langle
F,\phi_{j,l}^{0}\rangle\ll_{N}\left(|l|^{2}+|t_{j}|^{2}+1\right)^{-N}\|F\|_{W^{2N,2}\left(S\mathbb{X}\right)}.$
Likewise, assuming that the support of $F$ is contained in $y<T$, we have
$\left\langle
F,E_{m}\left(\cdot,\frac{1}{2}+it\right)\right\rangle_{S\mathbb{X}}\ll_{N}\left(|m|^{2}+t^{2}+1\right)^{-N}\|F\|_{W^{2N,2}\left(S\mathbb{X}\right)}\log
T,$
where we used [Kub73, (6.1.6)] and [Jak94, (1.6),(1.7)].
Now for $m>0$, we take $N=3$ and apply (4.5) to see that
$\sum_{\begin{subarray}{c}m>0\\\
4|m\end{subarray}}\sum_{j=1}^{d_{m}}\mu_{d}\left(\phi_{j}^{m}\right)\sum_{\begin{subarray}{c}l\in
4\mathbb{Z}\\\ |l|\geq m\end{subarray}}\left\langle
F,\phi_{j,l}^{m}\right\rangle_{S\mathbb{X}}\eta_{j}^{m}\left(\phi_{j,l}^{m}\right)\ll_{\epsilon}d^{\frac{1}{2}-\frac{25}{512}+\epsilon}\|F\|_{W^{6,2}\left(S\mathbb{X}\right)},$
and for $m=0$, we take $N=2$ and apply (4.6) to deduce
$\sum_{j=1}^{\infty}\mu_{d}\left(\phi_{j}^{0}\right)\sum_{l\in
4\mathbb{Z}}\left\langle
F,\phi_{j,l}^{0}\right\rangle_{S\mathbb{X}}\eta_{j}^{0}\left(\phi_{j,l}^{0}\right)\ll_{\epsilon}d^{\frac{1}{2}-\frac{25}{512}+\epsilon}\|F\|_{W^{4,2}\left(S\mathbb{X}\right)}.$
For the Eisenstein series contribution, we take $N=2$ and apply (4.9) to see
$\sum_{m\in 4\mathbb{Z}}\frac{1}{4\pi}\int_{-\infty}^{\infty}\left\langle
F,E_{m}\left(\cdot,\frac{1}{2}+it\right)\right\rangle_{S\mathbb{X}}\eta\left(m,\frac{1}{2}+it\right)\mu_{d}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)dt\ll_{\epsilon}\log
Td^{\frac{7}{16}+\epsilon}\|F\|_{W^{4,2}\left(S\mathbb{X}\right)}.$
Therefore Theorem 1.6 will follow once we establish the following lower bound
for the total length of $\mathscr{C}_{d}$:
$l\left(\mathscr{C}_{d}\right)=2h(d)\log\epsilon_{d}\gg_{\epsilon}d^{\frac{1}{2}-\epsilon}.$
(4.10)
To see this, let $d=Db^{2}$ where $D$ is a fundamental discriminant. Then by
Dirichlet class number formula [Dav67, p.50] for binary quadratic forms
discriminant $d$ (or by letting $s\to 1$ in (4.7)), we have
$h(d)\log(\epsilon_{d})=d^{\frac{1}{2}}L(1,d)$
with the same $L(\cdot,d)$ given in (4.8), i.e.,
$L(1,d)=L(1,\chi_{D})\left(\sum_{a|b}\mu(a)\left(\frac{D}{a}\right)a^{-1}\sigma_{-1}\left(\frac{b}{a}\right)\right).$
Note that
$\displaystyle\sum_{a|b}\mu(a)\left(\frac{D}{a}\right)a^{-1}\sigma_{-1}\left(\frac{b}{a}\right)$
$\displaystyle=\sum_{ca|b}\mu(a)\left(\frac{D}{a}\right)\frac{c}{b}$
$\displaystyle=\frac{1}{b}\sum_{e|b}e\prod_{p|e}\left(1-\left(\frac{D}{p}\right)p^{-1}\right),$
where $e=ac$, and that
$\frac{1}{b}\sum_{e|b}e\prod_{p|e}\left(1-\left(\frac{D}{p}\right)p^{-1}\right)\gg
b^{-\epsilon}.$
Now (4.10) follows by using Siegel’s lower bound [Sie35]
$L(1,\chi_{D})\gg_{\epsilon}D^{-\epsilon},$
and this completes the proof of Theorem 1.6.
### 4.4. Proof of Theorem 1.1
We are now ready to prove Theorem 1.1. Assume that
$\beta:[0,l\left(\beta\right)]\to\mathbb{X}$ is a sufficiently short compact
geodesic segment in the region determined by $y<T$ such that
$\beta\left([-l\left(\beta\right),2l\left(\beta\right)]\right)$ has no self
intersection. (We fix $T$ for simplicity, but it is possible to vary $T$ with
$d$.) For $\delta=d^{-a}$ with $a>0$ to be chosen later, such that
$l\left(\beta\right)\gg\delta$, let
$\beta_{1}:=\\{\beta\left(t\right)~{}:~{}t\in[0,l\left(\beta\right)-\delta]\\}$
and
$\beta_{2}:=\\{\beta\left(t\right)~{}:~{}t\in[-\delta,l\left(\beta\right)]\\}.$
Then from Lemma 2.8, we have
$\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\beta_{1}}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\leq
I^{\theta_{1},\theta_{2}}\left(\beta,\alpha_{2}\right)\leq\frac{1}{\delta^{2}}\int_{\widetilde{\alpha}_{2}}\int_{\widetilde{\beta_{1}}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}$
for any closed geodesic $\alpha_{2}$. Now define $f_{1},f_{2}\in
C_{0}^{\infty}\left(S\mathbb{X}\right)$ using Lemma 2.4 by
$f_{1}\left(g\right)=\frac{1}{\delta^{2}}\int_{\widetilde{\beta_{1}}}m_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1}^{-1}g\right)ds_{1}$
and
$f_{2}\left(g\right)=\frac{1}{\delta^{2}}\int_{\widetilde{\beta_{2}}}M_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1}^{-1}g\right)ds_{1},$
with $\varepsilon=d^{-2a}$, where we assume that $\theta_{2}-\theta_{1}\gg
d^{-a}$. Note that $m\left(g_{1}^{-1}g_{2}\right)$ and
$M\left(g_{1}^{-1}g_{2}\right)$ are minorant and majorant of
$K_{\delta}^{\theta_{1},\theta_{2}}\left(g_{1},g_{2}\right)$ for
$g_{1}\in\beta_{i}$, $g_{2}\in S\mathbb{X}$ for all sufficiently large $d$.
Hence, for all sufficiently large $d$ (independent of $\alpha_{2}$), we have
$\int_{\widetilde{\alpha}_{2}}f_{1}\left(s\right)ds\leq
I^{\theta_{1},\theta_{2}}\left(\beta,\alpha_{2}\right)\leq\int_{\widetilde{\alpha}_{2}}f_{2}\left(s\right)ds,$
and so
$\int_{\mathscr{C}_{d}}f_{1}\left(s\right)ds\leq
2I^{\theta_{1},\theta_{2}}\left(\beta,C_{d}\right)\leq\int_{\mathscr{C}_{d}}f_{2}\left(s\right)ds,$
(4.11)
where the factor $2$ amounts to the fact that $\mathscr{C}_{d}$ is a double
cover of $C_{d}$.
We now apply Theorem 1.6 to see that
$\frac{1}{l\left(\mathscr{C}_{d}\right)}\int_{\mathscr{C}_{d}}f_{i}\left(s\right)ds=\frac{3}{\pi^{2}}\int_{S\mathbb{X}}f_{i}\left(g\right)d\mu_{g}+O_{\epsilon}\left(d^{-\frac{25}{512}+\epsilon}\|f_{i}\|_{W^{6,\infty}}\right).$
Because of the choice of $f_{1}$ and $f_{2}$, we have
$\|f_{i}\|_{W^{6,\infty}}\ll\varepsilon^{-6}l\left(\beta\right)\ll
d^{12a}l\left(\beta\right),$
and
$\int_{S\mathbb{X}}f_{i}\left(g\right)d\mu_{g}=\left(\cos\theta_{1}-\cos\theta_{2}\right)\left(l\left(\beta\right)+O\left(\delta\right)\right)\left(1+O\left(\varepsilon\right)\right)=\left(\cos\theta_{1}-\cos\theta_{2}\right)l\left(\beta\right)\left(1+O\left(d^{-2a}\right)\right)$
by Lemma 2.4. Now we complete the proof of Theorem 1.1 for sufficiently short
geodesic segments by choosing $a=\frac{25}{7168}$ and applying these estimates
to (4.11). This then implies Theorem 1.1 for any geodesic segment of length
$<1$ by dividing the segment into finitely many sufficiently short geodesic
segments, and then applying Theorem 1.1 to each of them.
## 5\. Selberg’s pre-Trace Formula for $PSL_{2}\left(\mathbb{R}\right)$
Let $k\in C_{0}^{\infty}\left(PSL_{2}\left(\mathbb{R}\right)\right)$, and let
$K$ be the integral kernel on $S\mathbb{X}$ defined by
$K\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}k\left(g_{1},\gamma
g_{2}\right),$
where $k\left(g_{1},g_{2}\right)=k\left(g_{1}^{-1}g_{2}\right)$. The
corresponding integral operator $T_{K}$ acts on $f\in
L^{2}\left(S\mathbb{X}\right)$ by
$T_{K}\left(f\right):=\int_{S\mathbb{X}}K\left(g_{1},g_{2}\right)f\left(g_{2}\right)dg_{2}=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(g_{1}^{-1}g_{2}\right)f\left(g_{2}\right)dg_{2}.$
It follows that $T_{K}(f)\in L^{2}\left(S\mathbb{X}\right)$. In this section,
we study the spectral expansion of $K$ in terms of the equivariant
eigenfunctions of the Casimir operator, which are explicitly described in
§3.1. In other words, we derive Selberg’s pre-trace formula for
$PSL_{2}\left(\mathbb{Z}\right)\backslash PSL_{2}\left(\mathbb{R}\right)$.
### 5.1. Cuspidal spectrum
In this section, we describe explicitly the spectrum of $T_{K}$ acting on the
cuspidal subspace $L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)$. Let
$R_{g}\left(f\right)\left(x\right)=f\left(xg\right)$ be the right regular
action of $PSL_{2}\left(\mathbb{R}\right)$ on
$L_{\text{cusp}}^{2}\left(\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\right)=L_{\text{cusp}}^{2}\left(S\mathbb{X}\right).$
###### Lemma 5.1.
Let $\pi$ be an irreducible unitary representation of
$PSL_{2}\left(\mathbb{R}\right)$. Then for any $f\in W_{\pi}\subset
L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)$, we have
$T_{K}(f)\in W_{\pi}.$
###### Proof.
Observe that
$T_{K}(f)\left(g_{1}\right)=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(g_{1},g_{2}\right)f\left(g_{2}\right)dg_{2}=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(g_{1}^{-1}g_{2}\right)f\left(g_{2}\right)dg_{2}=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)f\left(g_{1}u\right)du,$
(5.1)
where $u=g_{1}^{-1}g_{2}$. Hence, we have
$T_{K}(f)=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)R_{u}\left(f\right)du,$
and because $R_{u}\left(f\right)\in W_{\pi}$ for every $u$, we conclude that
$T_{K}(f)\in W_{\pi}$. ∎
From (5.1), for an abstract irreducible unitary representation $\pi$ of
$PSL_{2}\left(\mathbb{R}\right)$ and $f\in W_{\pi}$, we define the action of
$k$ on $f$ by
$k*f=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)\pi\left(u\right)\left(f\right)du,$
which agrees with $T_{K}(f)$ when $W_{\pi}$ is a subspace of
$L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)$.
Let $\psi:W_{\pi}\to W_{\pi^{\prime}}$ be an isomorphism of representations
$\pi$ and $\pi^{\prime}$. Note that for $f\in W_{\pi}$ and $f^{\prime}\in
W_{\pi^{\prime}}$ with $\psi\left(f\right)=f^{\prime}$, we have
$\psi\left(k*f\right)=k*f^{\prime}$. We denote by $\phi_{m}\in W_{\pi}$ the
unique (up to a unit scalar) vector of norm $1$ and weight $m$. We fix the
unit scalar except for the spherical or the lowest weight vector, by using the
normalized lowering and raising operator that we introduced in (3.4) and
(3.5).
Now let
$h\left(k,m,n,\pi\right):=\left\langle k*\phi_{m},\phi_{n}\right\rangle,$
(5.2)
and let
$M_{\pi}\left(m,n\right)\left(g\right)=\left\langle\pi\left(g\right)\phi_{m},\phi_{n}\right\rangle$
be the matrix coefficient of $\pi$. We note that $h\left(k,m,n,\pi\right)$ and
$M_{\pi}\left(m,n\right)\left(g\right)$ do not depend on the choice of the
unit scalar of the spherical or the lowest weight vector.
We recall some properties of $M_{\pi}\left(m,n\right)\left(g\right)$ in the
following lemma.
###### Lemma 5.2.
We have for every $g\in PSL_{2}\left(\mathbb{R}\right)$,
$|M_{\pi}\left(m,n\right)\left(g\right)|\leq 1,$
and
$M_{\pi}\left(m,n\right)\left(R_{\theta^{\prime}}gR_{\theta}\right)=e^{-im\theta}e^{-in\theta^{\prime}}M_{\pi}\left(m,n\right)\left(g\right).$
###### Proof.
We have
$1=|\pi\left(g\right)\phi_{m}|^{2}=\sum_{n}\left\langle\pi\left(g\right)\phi_{m},\phi_{n}\right\rangle^{2},$
from which it is immediate that $|M_{\pi}\left(m,n\right)\left(g\right)|\leq
1$. For the second identity, we have
$M_{\pi}\left(m,n\right)\left(R_{\theta^{\prime}}gR_{\theta}\right)=\left\langle\pi\left(g\right)\pi\left(R_{\theta}\right)\phi_{m},\pi\left(R_{-\theta^{\prime}}\right)\phi_{n}\right\rangle=e^{-im\theta}e^{-in\theta^{\prime}}M_{\pi}\left(m,n\right)\left(g\right).\qed$
Define $k_{m,n}\in C_{0}^{\infty}\left(PSL_{2}\left(\mathbb{R}\right)\right)$
by
$k_{m,n}\left(g\right):=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}k\left(R_{\theta^{\prime}}gR_{\theta}\right)e^{-in\theta^{\prime}-im\theta}d\theta^{\prime}d\theta.$
(5.3)
Note that
$k_{m,n}\left(R_{\theta_{1}}gR_{\theta_{2}}\right)=e^{in\theta_{1}}k_{m,n}\left(g\right)e^{im\theta_{2}}.$
(5.4)
The following lemma holds for every unitary irreducible representation of
$PSL_{2}\left(\mathbb{R}\right)$.
###### Lemma 5.3.
We have
$h\left(k,m,n,\pi\right)=\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(u\right)M_{\pi}\left(m,n\right)\left(u\right)du,$
and for all non-negative integers $N_{1},N_{2}$, we have the following
estimate
$h\left(k,m,n,\pi\right)\ll_{N=N_{1}+N_{2}}\left(1+|m|\right)^{-N_{1}}\left(1+|n|\right)^{-N_{2}}\|k\|_{W^{N,1}}.$
###### Proof.
Recall from the definition that
$h\left(k,m,n,\pi\right)=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)\left\langle\pi\left(u\right)\phi_{m},\phi_{n}\right\rangle
du=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)M_{\pi}\left(m,n\right)\left(u\right)du,$
and so
$\displaystyle h\left(k,m,n,\pi\right)$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)M_{\pi}\left(m,n\right)\left(u\right)du.$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{PSL_{2}\left(\mathbb{R}\right)}\int_{\theta}\int_{\theta^{\prime}}k\left(R_{\theta^{\prime}}uR_{\theta}\right)M_{\pi}\left(m,n\right)\left(R_{\theta^{\prime}}uR_{\theta}\right)d\theta
d\theta^{\prime}du$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{PSL_{2}\left(\mathbb{R}\right)}M_{\pi}\left(m,n\right)\left(u\right)\int_{\theta}\int_{\theta^{\prime}}k\left(R_{\theta^{\prime}}uR_{\theta}\right)e^{-im\theta}e^{-in\theta^{\prime}}d\theta
d\theta^{\prime}du$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(u\right)M_{\pi}\left(m,n\right)\left(u\right)du.$
Therefore, by integration by parts, we have
$\displaystyle
h\left(k,m,n,\pi\right)\leq\int_{PSL_{2}\left(\mathbb{R}\right)}|k_{m,n}\left(u\right)|du$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}\left|\frac{1}{4\pi^{2}}\int_{\theta}\int_{\theta^{\prime}}k\left(R_{\theta^{\prime}}uR_{\theta}\right)e^{-im\theta}e^{-in\theta^{\prime}}d\theta
d\theta^{\prime}\right|du$
$\displaystyle\ll_{N}\left(1+|m|\right)^{-N_{1}}\left(1+|n|\right)^{-N_{2}}\|k\|_{W^{N,1}},$
where we used $|M_{\pi}\left(m,n\right)\left(u\right)|\leq 1$ from Lemma 5.2.
This completes the proof of our lemma. ∎
#### 5.1.1. Principal series representation of
$SL_{2}\left(\mathbb{R}\right)$
For our application in the subsequent chapters, we need a refined estimate for
$h\left(k,m,n,\pi\right)$ when $\pi$ is a unitary principal series
representation. We first give an explicit representation of
$h\left(k,m,n,\pi\right)$.
###### Lemma 5.4.
Let $W_{\pi}$ be a unitary principal series representation of
$SL_{2}\left(\mathbb{R}\right)$ with the parameter $it$ [Kna01, Cahpter VII].
Let
$h\left(k,m,n,t\right):=\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(g\right)y^{\frac{1}{2}+it}e^{-im\theta}dg,$
(5.5)
where $g=na\left(y\right)R_{\theta}$. Then we have
$h\left(k,m,n,\pi\right)=h\left(k,m,n,t\right).$
###### Proof.
We note that principal series representations are induced from the unitary
characters of the upper triangular matrices to
$PSL_{2}\left(\mathbb{R}\right)$ [Kna01, Cahpter VII]. In this model, a dense
subspace of a representation is given by
$\\{f:PSL_{2}\left(\mathbb{R}\right)\to\mathbb{C}\text{ continuous
}~{}:~{}f\left(xan\right)=e^{\left(it+\frac{1}{2}\right)\log\left(a\right)}f\left(x\right)\\}$
with the norm
$|f|^{2}=\frac{1}{2\pi}\int_{\theta}|f\left(R_{\theta}\right)|^{2}d\theta,$
and the $PSL_{2}\left(\mathbb{R}\right)$ action is given by
$\pi\left(g\right)f\left(x\right)=f\left(g^{-1}x\right).$
The weight-$m$ unit vectors are explicitly given by
$\phi_{m}\left(R_{\theta}a\left(y\right)n\right)=e^{im\theta}y^{-\left(\frac{1}{2}+it\right)}.$
Note that the orthonormal basis $\\{\phi_{m}\\}$ is normalized as our
convention in (3.4), i.e.,
$\displaystyle\mathbf{E}^{-}\phi_{m}$
$\displaystyle=\left(m+1-2it\right)\phi_{m-2},\text{ and}$
$\displaystyle\mathbf{E}^{+}\phi_{m}$
$\displaystyle=\left(m+1+2it\right)\phi_{m+2}.$
With these, we first see that
$\displaystyle k*\phi_{m}\left(R_{\theta^{\prime}}\right)$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(u\right)y\left(u^{-1}R_{\theta^{\prime}}\right)^{-\left(\frac{1}{2}+it\right)}e^{im\theta\left(u^{-1}R_{\theta^{\prime}}\right)}du$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(R_{\theta^{\prime}}v^{-1}\right)y\left(v\right)^{-\left(\frac{1}{2}+it\right)}e^{im\theta\left(v\right)}dv,$
where $v=u^{-1}R_{\theta^{\prime}}$ and
$v=R_{\theta\left(v\right)}a\left(y\left(v\right)\right)n\left(v\right)$. We
therefore have
$\displaystyle h\left(k,m,n,\pi\right)=\left\langle
k*f_{m},f_{n}\right\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{\theta^{\prime}}k*f_{m}\left(R_{\theta^{\prime}}\right)\bar{f}_{n}\left(R_{\theta^{\prime}}\right)d\theta^{\prime}$
$\displaystyle=\frac{1}{2\pi}\int_{\theta^{\prime}}e^{-in\theta^{\prime}}\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(R_{\theta^{\prime}}v^{-1}\right)y\left(v\right)^{-\left(\frac{1}{2}+it\right)}e^{im\theta\left(v\right)}dvd\theta^{\prime}$
$\displaystyle=\frac{1}{2\pi}\int_{PSL_{2}\left(\mathbb{R}\right)}y^{\frac{1}{2}+it}\int_{\theta^{\prime}}e^{-in\theta^{\prime}}e^{-im\theta}k\left(R_{\theta^{\prime}}w\right)d\theta^{\prime}dw$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(w\right)y^{\frac{1}{2}+it}e^{-im\theta}dw,$
where $w=v^{-1}$ and $w=na\left(y\right)R_{\theta}$. Note that
$y=y\left(v\right)^{-1}$ and $\theta=-\theta\left(v\right)$. ∎
We now prove that $h\left(k,m,n,t\right)$ decays fast in all parameters
uniformly.
###### Lemma 5.5.
Suppose that $k$ is supported inside the compact subset $C\subset
SL_{2}\left(\mathbb{R}\right)$. Then we have
$\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(g\right)y^{\frac{1}{2}+it}e^{-im\theta}dg\ll_{N,C}\left(1+|m|\right)^{-N_{1}}\left(1+|n|\right)^{-N_{2}}\left(1+|t|\right)^{-N_{3}}\|k\|_{W^{N,\infty}}$
for any $N_{1},N_{2},N_{3}\geq 0$, where $N=N_{1}+N_{2}+N_{3}$.
###### Proof.
From the definition, we have
$\int_{PSL_{2}\left(\mathbb{R}\right)}k_{m,n}\left(g\right)y^{\frac{1}{2}+it}e^{-im\theta}dg=\frac{1}{4\pi}\int_{\mathbb{H}}\int_{0}^{2\pi}\int_{0}^{2\pi}k\left(R_{\theta_{1}^{\prime}}n\left(x\right)a\left(y\right)R_{\theta_{2}^{\prime}}\right)y^{\frac{1}{2}+it}e^{-in\theta_{1}^{\prime}-im\theta_{2}^{\prime}}d\theta_{1}^{\prime}d\theta_{2}^{\prime}\frac{dxdy}{y^{2}},$
and so the statement follows from integration by parts. ∎
### 5.2. Continuous spectrum
For $k_{m,n}$ given by (5.3), let
$K_{m,n}\left(g_{1},g_{2}\right):=\sum_{\gamma\in\Gamma}k_{m,n}\left(g_{1}^{-1}\gamma
g_{2}\right).$ (5.6)
Then we infer from (5.4) that
$K_{m,n}\left(g_{1}R_{\theta_{1}},g_{2}R_{\theta_{2}}\right)=e^{-in\theta_{1}}K_{m,n}\left(g_{1},g_{2}\right)e^{im\theta_{2}},$
and so it defines an integral operator that maps weight $m$-forms to weight
$n$-forms. Denote by $S^{m}\subset L^{2}\left(\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\right)$ the space of weight $m$ forms and by
$S_{\text{cusp}}^{m}$ the space of weight $m$ forms in
$L_{\text{cusp}}^{2}\left(S\mathbb{X}\right)$. We first recall the following
result regarding the decomposition of $K_{m,m}$.
###### Theorem 5.6 ([Hej76]).
The integral kernel
$K_{m,m}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}h\left(k,m,m,t\right)E_{m}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt$
defines a compact operator $S_{\text{cusp}}^{m}\to S_{\text{cusp}}^{m}$ that
acts trivially on $\Theta$. (Here $h\left(k,m,m,t\right)$ is given by (5.5).)
We define $\mathbf{E}^{a}$ to be $\left(\mathbf{E}^{+}\right)^{a}$ if $a>0$,
and $\left(\mathbf{E}^{-}\right)^{|a|}$ if $a<0$. We have
$\overline{\mathbf{E}^{a}}=\left(-\mathbf{E}\right)^{-a},$
which follows directly from (3.2). Let $c_{m,n}$ be given by
$\mathbf{E}^{n-m}E_{m}\left(g,s\right)=c_{m,n}\left(s\right)E_{n}\left(g,s\right).$
Observe that
$\mathbf{E}^{n-m}y^{s}e^{-im\theta}=c_{m,n}\left(s\right)y^{s}e^{-in\theta},$
and that
$\overline{c_{m,n}\left(\frac{1}{2}+it\right)}=c_{n,m}\left(\frac{1}{2}+it\right)$
(5.7)
for $t\in\mathbb{R}$.
###### Theorem 5.7.
For $m,n\in 2\mathbb{Z}$,
$K_{m,n}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt$
defines a compact operator $S_{\text{cusp}}^{m}\to S_{\text{cusp}}^{n}$ that
acts trivially on $\Theta$.
###### Proof.
Note that
$\int\mathbf{E}^{m-n}_{g_{2}}\left(K\left(g_{1},g_{2}\right)f\left(g_{2}\right)\right)dg_{2}=0$
for every $g_{1}$, $m\neq n$, and $f\in C^{\infty}_{0}\left(\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\right)$. Hence
$T_{K}\mathbf{E}^{m-n}:C^{\infty}_{0}\left(\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\right)\to C^{\infty}_{0}\left(\Gamma\backslash
PSL_{2}\left(\mathbb{R}\right)\right)$
is an integral operator with the integral kernel
$K^{\prime}\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}k^{\prime}\left(g_{1}^{-1}\gamma
g_{2}\right),$
where
$k^{\prime}\left(g\right)=\left(-\mathbf{E}\right)^{m-n}k\left(g\right)=\overline{\mathbf{E}^{n-m}}k\left(g\right).$
Then by Theorem 5.6, we see that
$K^{\prime\prime}\left(g_{1},g_{2}\right)=K^{\prime}_{n,n}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}h\left(k^{\prime},n,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{n}\left(g_{2},\frac{1}{2}+it\right)}dt$
defines a compact operator $T_{K^{\prime\prime}}:S_{\text{cusp}}^{n}\to
S_{\text{cusp}}^{n}$ that acts trivially on $\Theta$. Note that
$\int_{-\infty}^{\infty}h\left(k^{\prime},n,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{n}\left(g_{2},\frac{1}{2}+it\right)}dt\\\
=\int_{-\infty}^{\infty}\frac{h\left(k^{\prime},n,n,t\right)}{\overline{c_{m,n}\left(\frac{1}{2}+it\right)}}E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{\mathbf{E}^{n-m}E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt.$
Let
$K^{\prime\prime\prime}\left(g_{1},g_{2}\right):=K_{m,n}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{h\left(k^{\prime},n,n,t\right)}{\overline{c_{m,n}\left(\frac{1}{2}+it\right)}}E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt.$
Note that
$T_{K^{\prime\prime}}=T_{K^{\prime\prime\prime}}\circ\mathbf{E}^{m-n}.$
Firstly, since $\mathbf{E}^{m-n}$ does not annihilate the Eisenstein series,
$T_{K^{\prime\prime\prime}}$ acts trivially on $\Theta$.
If $m>n\geq 0$ or $m<n\leq 0$, then as a map $S_{cusp}^{n}\to S_{cusp}^{m}$,
$ker\left(\mathbf{E}^{m-n}\right)$ is empty, and we may decompose
$S_{cusp}^{m}$ as
$S_{cusp}^{m}=Im\left(\mathbf{E}^{m-n}\right)\oplus R,$
where $R$ is a finite dimensional subspace of $S_{cusp}^{m}$ spanned by
modular forms of weight $>n$ and their images under raising operators in
$S_{cusp}^{m}$. Note that
$\left(\mathbf{E}^{m-n}\right)^{-1}:Im\left(\mathbf{E}^{m-n}\right)\to
S_{cusp}^{n}$
is a bounded operator, hence
$T_{K^{\prime\prime\prime}}|_{Im\left(\mathbf{E}^{m-n}\right)}=T_{K^{\prime\prime}}\circ\left(\mathbf{E}^{m-n}\right)^{-1}$
is a compact operator. This implies that $T_{K^{\prime\prime\prime}}$ is a
direct sum of a compact operator and finite dimensional linear operator, which
is a compact operator.
If $n>m\geq 0$ or $n<m\leq 0$, then $\mathbf{E}^{m-n}:S_{\text{cusp}}^{n}\to
S_{\text{cusp}}^{m}$ is surjective, and so we may define a bounded operator
$\left(\mathbf{E}^{m-n}\right)^{-1}:S_{m}\to\left(ker\left(\mathbf{E}^{m-n}\right)\right)^{\perp}$
from which it follows that
$T_{K^{\prime\prime\prime}}=T_{K^{\prime\prime}}\circ\left(\mathbf{E}^{m-n}\right)^{-1}$
is a compact operator.
If $n>0>m$ or $m>0>n$, then we further decompose $T_{K^{\prime\prime}}$ to
$S_{\text{cusp}}^{n}\xrightarrow{\mathbf{E}^{-n}}S_{\text{cusp}}^{0}\xrightarrow{\mathbf{E}^{m}}S_{\text{cusp}}^{m}\xrightarrow{T_{K^{\prime\prime\prime}}}S_{\text{cusp}}^{n},$
and then combine the above arguments to see that $T_{K^{\prime\prime}}$ is a
compact operator.
Finally, observe that
$\displaystyle h\left(k^{\prime},n,n,t\right)$
$\displaystyle=\int_{PSL_{2}\left(\mathbb{R}\right)}\left(\overline{\mathbf{E}^{n-m}}k\left(g\right)\right)y^{\frac{1}{2}+it}e^{in\theta}dg$
$\displaystyle=c_{n,m}\left(\frac{1}{2}+it\right)\int_{PSL_{2}\left(\mathbb{R}\right)}k\left(g\right)y^{\frac{1}{2}+it}e^{im\theta}dg,$
and we complete the proof using (5.7). ∎
### 5.3. General case
We are now ready to describe Selberg’s pre-trace formula for
$PSL_{2}\left(\mathbb{R}\right)$.
###### Theorem 5.8.
For $k\in C_{0}^{\infty}\left(PSL_{2}\left(\mathbb{R}\right)\right)$, let $K$
be the integral kernel on $S\mathbb{X}$ defined by
$K\left(g_{1},g_{2}\right)=\sum_{\gamma\in\Gamma}k\left(g_{1},\gamma
g_{2}\right).$
Then we have
$K\left(g_{1},g_{2}\right)=\frac{9}{\pi^{4}}\iint
K\left(g_{1},g_{2}\right)dg_{1}dg_{2}+\sum_{\begin{subarray}{c}e\geq 0\\\
2|e\end{subarray}}\sum_{j=1}^{d_{e}}\sum_{\begin{subarray}{c}m,n\in
2\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}h\left(k,m,n,\pi_{j}^{e}\right)\phi_{j,n}^{e}\left(g_{1}\right)\overline{\phi_{j,m}^{e}\left(g_{2}\right)}\\\
+\frac{1}{4\pi}\sum_{m,n\in
2\mathbb{Z}}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt,$
where $\pi_{j}^{e}$ is the irreducible unitary representation of
$PSL_{2}\left(\mathbb{R}\right)$ associated to $\phi_{j}^{e}$.
###### Proof.
We first note from (5.3) and (5.6) that
$\displaystyle K_{m,n}\left(g_{1},g_{2}\right)$
$\displaystyle=\sum_{\gamma\in\Gamma}\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}k\left(R_{\theta_{1}^{\prime}}g_{1}^{-1}\gamma
g_{2}R_{\theta_{2}^{\prime}}\right)e^{-in\theta_{1}^{\prime}-im\theta_{2}^{\prime}}d\theta_{1}^{\prime}d\theta_{2}^{\prime}$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\sum_{\gamma\in\Gamma}k\left(R_{-\theta_{1}^{\prime}}g_{1}^{-1}\gamma
g_{2}R_{\theta_{2}^{\prime}}\right)e^{in\theta_{1}^{\prime}-im\theta_{2}^{\prime}}d\theta_{1}^{\prime}d\theta_{2}^{\prime}$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}K\left(g_{1}R_{\theta_{1}^{\prime}},g_{2}R_{\theta_{2}^{\prime}}\right)e^{in\theta_{1}^{\prime}-im\theta_{2}^{\prime}}d\theta_{1}^{\prime}d\theta_{2}^{\prime}$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}K\left(\left(x_{1},y_{1},\theta_{1}^{\prime}\right),\left(x_{2},y_{2},\theta_{2}^{\prime}\right)\right)e^{in\theta_{1}^{\prime}-im\theta_{2}^{\prime}}d\theta_{1}^{\prime}d\theta_{2}^{\prime}e^{-in\theta_{1}+im\theta_{2}}.$
Therefore, we have the Fourier expansion of $K$,
$K\left(g_{1},g_{2}\right)=\sum_{n,m\in
2\mathbb{Z}}K_{m,n}\left(g_{1},g_{2}\right),$
where the summation is uniform for $g_{1}$ and $g_{2}$ in compacta.
We infer from Theorem 5.7 that
$K_{m,n}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt$
defines a compact operator acting on $L_{\text{cusp}}$ that acts trivially on
$\Theta$. Because it only acts non-trivially on weight $m$ forms, we see that
$K_{m,n}\left(g_{1},g_{2}\right)-\frac{1}{4\pi}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt\\\
=\frac{9}{\pi^{4}}\iint
K_{m,n}\left(g_{1},g_{2}\right)dg_{1}dg_{2}+\sum_{\begin{subarray}{c}e\geq
0\\\
2|e\end{subarray}}^{\min\\{|m|,|n|\\}}\sum_{j=1}^{d_{e}}h\left(k,m,n,\pi_{j}^{e}\right)\phi_{j,n}^{e}\left(g_{1}\right)\overline{\phi_{j,m}^{e}\left(g_{2}\right)},$
where we used (5.2), and the fact that
$\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt$
acts trivially on $L_{\text{cusp}}^{2}$. Note that the integral on the right-
hand side of the equation vanishes unless $m=n=0$, in which case it is
identical to
$\frac{9}{\pi^{4}}\iint K\left(g_{1},g_{2}\right)dg_{1}dg_{2}.\qed$
### 5.4. Proof of Theorem 1.8
We now present a proof of Theorem 1.8. Recall from Theorem 5.8 that we have
$K\left(g_{1},g_{2}\right)=\frac{9}{\pi^{4}}\iint
K\left(g_{1},g_{2}\right)dg_{1}dg_{2}+\sum_{\begin{subarray}{c}e\geq 0\\\
2|e\end{subarray}}\sum_{j=1}^{d_{e}}\sum_{\begin{subarray}{c}m,n\in
2\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}h\left(k,m,n,\pi_{j}^{e}\right)\phi_{j,n}^{e}\left(g_{1}\right)\overline{\phi_{j,m}^{e}\left(g_{2}\right)}\\\
+\frac{1}{4\pi}\sum_{m,n\in
2\mathbb{Z}}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)E_{n}\left(g_{1},\frac{1}{2}+it\right)\overline{E_{m}\left(g_{2},\frac{1}{2}+it\right)}dt.$
Therefore we have
$\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K\left(s_{1},s_{2}\right)ds_{1}ds_{2}=M+D+\frac{1}{4\pi}E,$
where
$M=\frac{9}{\pi^{4}}\iint K\left(g_{1},g_{2}\right)dg_{1}dg_{2},$
$D=\sum_{\begin{subarray}{c}e\geq 0\\\
2|e\end{subarray}}\sum_{j=1}^{d_{e}}\sum_{\begin{subarray}{c}m,n\in
2\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}h\left(k,m,n,\pi_{j}^{e}\right)\frac{\mu_{d_{1}}\left(\phi_{j,n}^{e}\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(\phi_{j,m}^{e}\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\\\
=\sum_{\begin{subarray}{c}e\geq 0\\\
4|e\end{subarray}}\sum_{j=1}^{d_{e}}\frac{\mu_{d_{1}}\left(\phi_{j}^{e}\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(\phi_{j}^{e}\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\sum_{\begin{subarray}{c}m,n\in
4\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}h\left(k,m,n,\pi_{j}^{e}\right)\eta_{j}^{e}\left(\phi_{j,n}^{e}\right)\overline{\eta_{j}^{e}\left(\phi_{j,m}^{e}\right)},$
and
$E=\sum_{m,n\in
2\mathbb{Z}}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)\frac{\mu_{d_{1}}\left(E_{n}\left(\cdot,\frac{1}{2}+it\right)\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(E_{m}\left(\cdot,\frac{1}{2}+it\right)\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}dt\\\
=\sum_{m,n\in
4\mathbb{Z}}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)\frac{\mu_{d_{1}}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\eta\left(n,\frac{1}{2}+it\right)\overline{\eta\left(m,\frac{1}{2}+it\right)}dt.$
For $D$ with $e>0$, we use (4.2), (4.5), Lemma 5.3 with $N_{1}=N_{2}=5$, and
(4.10) to see that
$\displaystyle\sum_{\begin{subarray}{c}e>0\\\
4|e\end{subarray}}\sum_{j=1}^{d_{e}}\frac{\mu_{d_{1}}\left(\phi_{j}^{e}\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(\phi_{j}^{e}\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\sum_{\begin{subarray}{c}m,n\in
4\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}h\left(k,m,n,\pi_{j}^{e}\right)\eta_{j}^{e}\left(\phi_{j,n}^{e}\right)\overline{\eta_{j}^{e}\left(\phi_{j,m}^{e}\right)}$
$\displaystyle\ll_{\epsilon}$ $\displaystyle\sum_{\begin{subarray}{c}e>0\\\
4|e\end{subarray}}e^{6.8}\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\sum_{\begin{subarray}{c}m,n\in
4\mathbb{Z}\\\ |m|,|n|\geq
e\end{subarray}}|m|^{-5}|n|^{-5}\|k\|_{W^{10,\infty}}$ $\displaystyle\ll$
$\displaystyle\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\|k\|_{W^{10,\infty}}.$
For $D$ with $e=0$, we use (4.1), (4.6), Lemma 5.5 with $N_{1}=N_{2}=2$ and
$N_{3}=4$, and (4.10) to see that
$\displaystyle\sum_{j=1}^{\infty}\frac{\mu_{d_{1}}\left(\phi_{j}^{0}\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(\phi_{j}^{0}\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\sum_{\begin{subarray}{c}m,n\in
4\mathbb{Z}\end{subarray}}h\left(k,m,n,\pi_{j}^{0}\right)\eta_{j}^{0}\left(\phi_{j,n}^{0}\right)\overline{\eta_{j}^{0}\left(\phi_{j,m}^{0}\right)}$
$\displaystyle\ll_{\epsilon}$
$\displaystyle\sum_{j=1}^{\infty}\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}|t_{j}|^{\frac{3}{2}}\sum_{\begin{subarray}{c}m,n\in
4\mathbb{Z}\end{subarray}}\left(1+|m|\right)^{-2}\left(1+|n|\right)^{-2}\left(1+|t_{j}|\right)^{-4}\|k\|_{W^{8,\infty}}$
$\displaystyle\ll$
$\displaystyle\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\|k\|_{W^{8,\infty}}.$
For $E$, we use (4.3), (4.9), Lemma 5.5 with $N_{1}=N_{2}=2$ and $N_{3}=3$,
and (4.10) to see that
$\displaystyle\sum_{m,n\in
4\mathbb{Z}}\int_{-\infty}^{\infty}h\left(k,m,n,t\right)\frac{\mu_{d_{1}}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)}{l\left(\mathscr{C}_{d_{1}}\right)}\frac{\overline{\mu_{d_{2}}\left(E_{0}\left(\cdot,\frac{1}{2}+it\right)\right)}}{l\left(\mathscr{C}_{d_{2}}\right)}\eta\left(n,\frac{1}{2}+it\right)\overline{\eta\left(m,\frac{1}{2}+it\right)}dt$
$\displaystyle\ll_{\epsilon}$ $\displaystyle\sum_{m,n\in
4\mathbb{Z}}\int_{-\infty}^{\infty}\left(d_{1}d_{2}\right)^{-\frac{1}{16}+\epsilon}\left(1+|m|\right)^{-2}\left(1+|n|\right)^{-2}\left(|t|+1\right)^{-2}\|k\|_{W^{7,\infty}}dt$
$\displaystyle\ll$
$\displaystyle\left(d_{1}d_{2}\right)^{-\frac{1}{16}+\epsilon}\|k\|_{W^{7,\infty}}.$
Now observe that
$\iint
K\left(g_{1},g_{2}\right)dg_{1}dg_{2}=\int_{S\mathbb{X}}\int_{S\mathbb{H}}k\left(g_{1}^{-1}g_{2}\right)dg_{2}dg_{1}=\frac{\pi^{2}}{3}\int_{S\mathbb{H}}k\left(g\right)dg,$
and so
$M=\frac{3}{\pi^{2}}\int_{S\mathbb{H}}k\left(g\right)dg.$
So far, we proved the following:
###### Theorem 5.9.
For any $k\in C_{0}^{\infty}\left(S\mathbb{H}\right)$, we have
$\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\frac{3}{\pi^{2}}\int_{S\mathbb{H}}k\left(g\right)dg+O_{\epsilon}\left(\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\|k\|_{W^{10,\infty}}\right).$
###### Remark 5.10.
Note that this is not the same as equidistribution of
$\mathscr{C}_{d_{1}}\times\mathscr{C}_{d_{2}}$ in $S\mathbb{X}\times
S\mathbb{X}$. For instance, if we replace $K$ with any compactly supported
smooth function in $S\mathbb{X}\times S\mathbb{X}$, then the equality may not
hold when $d_{1}$ is fixed and $d_{2}$ tends to $\infty$.
In order to prove Theorem 1.8, we make specific choices of $k$ in Theorem 5.9.
We let $K_{1}$ and $K_{2}$ to be the kernel corresponding to
$k=m_{\delta}^{\theta_{1},\theta_{2}}$ and
$k=M_{\delta}^{\theta_{1},\theta_{2}}$ defined in Lemma 2.4, respectively.
Then by Lemma 2.4, we have
$\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K_{1}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\leq\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}\\\
\leq\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K_{2}\left(s_{1},s_{2}\right)ds_{1}ds_{2},$
while we know from Lemma 2.6 that
$\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K_{\delta}^{\theta_{1},\theta_{2}}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=4\delta^{2}I_{\theta_{1},\theta_{2}}\left(C_{d_{1}},C_{d_{2}}\right).$
We now apply Theorem 5.9 and Lemma 2.4 to see that
$\frac{1}{l\left(\mathscr{C}_{d_{1}}\right)l\left(\mathscr{C}_{d_{2}}\right)}\int_{\mathscr{C}_{d_{2}}}\int_{\mathscr{C}_{d_{1}}}K_{i}\left(s_{1},s_{2}\right)ds_{1}ds_{2}=\frac{3}{\pi^{2}}\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta^{2}\left(1+O\left(\varepsilon\right)\right)+O_{\epsilon}\left(\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\varepsilon^{-10}\right).$
Therefore, we have
$\frac{I_{\theta_{1},\theta_{2}}\left(C_{d_{1}},C_{d_{2}}\right)}{l\left(C_{d_{1}}\right)l\left(C_{d_{2}}\right)}=\frac{3}{\pi^{2}}\left(\cos\theta_{1}-\cos\theta_{2}\right)\left(1+O\left(\delta^{2}\right)\right)\left(1+O\left(\varepsilon\right)\right)+O_{\epsilon}\left(\left(d_{1}d_{2}\right)^{-\frac{25}{512}+\epsilon}\varepsilon^{-10}\delta^{-2}\right),$
and by choosing
$\delta^{2}=\varepsilon=\left(d_{1}d_{2}\right)^{-\frac{25}{6144}}$, we
complete the proof of Theorem 1.8.
## References
* [BHM07] V. Blomer, G. Harcos, and P. Michel. A Burgess-like subconvex bound for twisted $L$-functions. Forum Math., 19(1):61–105, 2007. Appendix 2 by Z. Mao.
* [BM10] Ehud Moshe Baruch and Zhengyu Mao. A generalized Kohnen-Zagier formula for Maass forms. J. Lond. Math. Soc. (2), 82(1):1–16, 2010.
* [Bon86] Francis Bonahon. Bouts des variétés hyperboliques de dimension $3$. Ann. of Math. (2), 124(1):71–158, 1986.
* [Dav67] Harold Davenport. Multiplicative number theory, volume 1966 of Lectures given at the University of Michigan, Winter Term. Markham Publishing Co., Chicago, Ill., 1967.
* [Duk88] W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math., 92(1):73–90, 1988.
* [DV21] Henri Darmon and Jan Vonk. Arithmetic intersections of modular geodesics. Journal of Number Theory, 2021.
* [HB80] D. R. Heath-Brown. Hybrid bounds for Dirichlet $L$-functions. II. Quart. J. Math. Oxford Ser. (2), 31(122):157–167, 1980.
* [Hej76] Dennis A. Hejhal. The Selberg trace formula for ${\rm PSL}(2,R)$. Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.
* [HJ15] Yoe Alexander Herrera Jaramillo. Intersection numbers of geodesic arcs. Rev. Colombiana Mat., 49(2):307–319, 2015.
* [Jak94] Dmitry Jakobson. Quantum unique ergodicity for Eisenstein series on ${\rm PSL}_{2}({\bf Z})\backslash{\rm PSL}_{2}({\bf R})$. Ann. Inst. Fourier (Grenoble), 44(5):1477–1504, 1994.
* [JR21] Junehyuk Jung and Alan W. Reid. Embedding closed totally geodesic surfaces in Bianchi orbifolds. to appear Math. Res. Lett., 2021.
* [Kna01] Anthony W. Knapp. Representation theory of semisimple groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001. An overview based on examples, Reprint of the 1986 original.
* [KS93] Svetlana Katok and Peter Sarnak. Heegner points, cycles and Maass forms. Israel J. Math., 84(1-2):193–227, 1993.
* [Kub73] Tomio Kubota. Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York-London-Sydney, 1973.
* [KZ81] W. Kohnen and D. Zagier. Values of $L$-series of modular forms at the center of the critical strip. Invent. Math., 64(2):175–198, 1981.
* [Lal14] Steven P. Lalley. Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces. Duke Math. J., 163(6):1191–1261, 2014.
* [Lan85] Serge Lang. ${\rm SL}_{2}({\bf R})$, volume 105 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1985. Reprint of the 1975 edition.
* [LRl20] Stephen Lester and Maksym Radziwił ł. Quantum unique ergodicity for half-integral weight automorphic forms. Duke Math. J., 169(2):279–351, 2020.
* [LRS09] Wenzhi Luo, Zeév Rudnick, and Peter Sarnak. The variance of arithmetic measures associated to closed geodesics on the modular surface. J. Mod. Dyn., 3(2):271–309, 2009.
* [PS06] Mark Pollicott and Richard Sharp. Angular self-intersections for closed geodesics on surfaces. Proc. Amer. Math. Soc., 134(2):419–426, 2006.
* [Ric19] James Rickards. Intersections of closed geodesics on the modular curve. arXiv:1909.04103, 2019.
* [Riv01] Igor Rivin. Simple curves on surfaces. Geom. Dedicata, 87(1-3):345–360, 2001.
* [Sar80] Peter Clive Sarnak. PRIME GEODESIC THEOREMS. ProQuest LLC, Ann Arbor, MI, 1980. Thesis (Ph.D.)–Stanford University.
* [Sar82] Peter Sarnak. Class numbers of indefinite binary quadratic forms. J. Number Theory, 15(2):229–247, 1982.
* [Shi75] Takuro Shintani. On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J., 58:83–126, 1975.
* [Sie35] C. L. Siegel. Über die Classenzahl quadratischer Zahlkörper. Acta Arith., 1:83–86, 1935.
* [Sie65] Carl Ludwig Siegel. Lectures on advanced analytic number theory. Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23. Tata Institute of Fundamental Research, Bombay, 1965.
* [TS20] Naser Talebizadeh Sardari. The least prime number represented by a binary quadratic form. J. Eur. Math. Soc., 2020.
* [Zag81] D. Zagier. Eisenstein series and the Riemann zeta function. In Automorphic forms, representation theory and arithmetic (Bombay, 1979), volume 10 of Tata Inst. Fund. Res. Studies in Math., pages 275–301. Tata Inst. Fundamental Res., Bombay, 1981.
## Appendix A Jacobian computation
Recall that $\Psi:AKA\to SL_{2}\left(\mathbb{R}\right)$ is given by
$\left(t_{1},\varphi,t_{2}\right)\mapsto\begin{pmatrix}e^{\frac{t_{1}}{2}}&0\\\
0&e^{-\frac{t_{1}}{2}}\end{pmatrix}R_{\frac{\varphi}{2}}\begin{pmatrix}e^{-\frac{t_{2}}{2}}&0\\\
0&e^{\frac{t_{2}}{2}}\end{pmatrix}=\begin{pmatrix}e^{\frac{t_{1}-t_{2}}{2}}\cos\frac{\varphi}{2}&-e^{\frac{t_{1}+t_{2}}{2}}\sin\frac{\varphi}{2}\\\
e^{\frac{-t_{1}-t_{2}}{2}}\sin\frac{\varphi}{2}&e^{\frac{t_{2}-t_{1}}{2}}\cos\frac{\varphi}{2}\end{pmatrix}.$
In this section, we compute the pullback of $dV=\frac{dxdyd\theta}{y^{2}}$
under $\Psi$. We start with the identity
$\begin{pmatrix}e^{\frac{t_{1}-t_{2}}{2}}\cos\frac{\varphi}{2}&-e^{\frac{t_{1}+t_{2}}{2}}\sin\frac{\varphi}{2}\\\
e^{\frac{-t_{1}-t_{2}}{2}}\sin\frac{\varphi}{2}&e^{\frac{t_{2}-t_{1}}{2}}\cos\frac{\varphi}{2}\end{pmatrix}=n\left(x\right)a\left(y\right)R_{\theta}=\begin{pmatrix}*&*\\\
\frac{\sin\theta}{\sqrt{y}}&\frac{\cos\theta}{\sqrt{y}}\end{pmatrix}.$
By comparing the image of $i\in\mathbb{H}$, we have
$x+iy=\frac{e^{\frac{t_{1}-t_{2}}{2}}\cos\frac{\varphi}{2}i-e^{\frac{t_{1}+t_{2}}{2}}\sin\frac{\varphi}{2}}{e^{\frac{-t_{1}-t_{2}}{2}}\sin\frac{\varphi}{2}i+e^{\frac{t_{2}-t_{1}}{2}}\cos\frac{\varphi}{2}},$
and for simplicity, we write this as $\frac{A}{B}$. By comparing the second
row of each matrix, we have
$\frac{e^{i\theta}}{\sqrt{y}}=B.$
From a quick computation, we see that
$A_{t_{1}}=\frac{A}{2},~{}B_{t_{1}}=-\frac{B}{2},~{}A_{t_{2}}=\frac{\overline{A}}{2},~{}B_{t_{2}}=\frac{\overline{B}}{2},~{}A_{\varphi}=-\frac{e^{t_{1}}}{2}B,~{}B_{\varphi}=\frac{e^{-t_{1}}}{2}A,~{}\text{Im}A\overline{B}=1,~{}y=\frac{1}{|B|^{2}}.$
We use these to express the Jacobian matrix in terms of $A$ and $B$ as follows
$\frac{\partial\left(x,y,\theta\right)}{\partial\left(t_{1},t_{2},\varphi\right)}=\begin{pmatrix}\text{Re}\frac{A}{B}&\text{Im}\frac{1}{B^{2}}&\text{Re}\left(-\frac{e^{t_{1}}}{2}-\frac{e^{-t_{1}}}{2}\frac{A^{2}}{B^{2}}\right)\\\
\text{Im}\frac{A}{B}&-\text{Re}\frac{1}{B^{2}}&\text{Im}\left(-\frac{e^{t_{1}}}{2}-\frac{e^{-t_{1}}}{2}\frac{A^{2}}{B^{2}}\right)\\\
0&\frac{1}{2}\text{Im}\frac{\overline{B}}{B}&\frac{e^{-t_{1}}}{2|B|^{2}}\end{pmatrix}.$
From this, we have
$\displaystyle\frac{1}{y^{2}}\left|\frac{\partial\left(x,y,\theta\right)}{\partial\left(t_{1},t_{2},\varphi\right)}\right|$
$\displaystyle=|B|^{4}\left|\frac{\partial\left(x,y,\theta\right)}{\partial\left(t_{1},t_{2},\varphi\right)}\right|$
$\displaystyle=\left|-\frac{1}{2}e^{-t_{1}}\text{Re}\left(\frac{\overline{A}}{B}\right)+\frac{1}{4}\text{Im}\left(\overline{B}^{2}\right)\text{Im}\left(\overline{A}B\left(e^{t_{1}}+e^{-t_{1}}\frac{A^{2}}{B^{2}}\right)\right)\right|$
$\displaystyle=\left|\frac{e^{t_{1}}}{2}\text{Im}\left(B^{2}\right)+\frac{e^{-t_{1}}}{4|B|^{2}}\left(-2\text{Re}\left(AB\right)-|A|^{2}\text{Im}\left(B^{2}\right)\right)\right|.$
Now we use the definition of $A$ and $B$ to compute each term explicitly as
follows
$\displaystyle 2\text{Re}\left(AB\right)$
$\displaystyle=-\left(e^{t_{2}}+e^{-t_{1}}\right)\sin\varphi$ $\displaystyle
e^{t_{1}}\text{Im}\left(B^{2}\right)$ $\displaystyle=\sin\varphi$
$\displaystyle e^{-t_{1}}|A|^{2}$
$\displaystyle=e^{t_{2}}\sin^{2}\frac{\varphi}{2}+e^{-t_{2}}\cos^{2}\frac{\varphi}{2}$
$\displaystyle e^{t_{1}}|B|^{2}$
$\displaystyle=e^{-t_{2}}\sin^{2}\frac{\varphi}{2}+e^{t_{2}}\cos^{2}\frac{\varphi}{2},$
and so
$\frac{1}{y^{2}}\left|\frac{\partial\left(x,y,\theta\right)}{\partial\left(t_{1},t_{2},\varphi\right)}\right|=\frac{1}{2}|\sin\varphi|.$
Therefore, we conclude that
$dV=\frac{1}{2}|\sin\varphi|dt_{1}dt_{2}d\varphi.$ (A.1)
|
In this paper we study gapless fermionic and bosonic systems in
$d$-dimensional continuum space with $U(1)$ particle-number conservation and
$\mathbb{R}^{d}$ translation symmetry. We write down low energy effective
field theories for several gapless phases where $U(1)\times\mathbb{R}^{d}$ is
viewed as internal symmetry. The $U(1)\times\mathbb{R}^{d}$ symmetry, when
viewed as an internal symmetry, has a mixed anomaly, and the different
effective field theories for different phases must have the same mixed
anomaly. Such a mixed anomaly is proportional to the particle number density,
and can be measured from the distribution of the total momentum
$\boldsymbol{k}_{\text{tot}}$ for low energy many-body states (i.e. how such a
distribution is shifted by $U(1)$ symmetry twist $\boldsymbol{a}$), as well as
some other low energy universal properties of the systems. In particular, we
write down low energy effective field theory for Fermi liquid with infinite
number of fields, in the presence of both real space magnetic field and
$\boldsymbol{k}$-space “magnetic” field. The effective field theory also
captures the mixed anomaly, which constraints the low energy dynamics, such as
determine the volume of Fermi surface (which is another formulation of
Luttinger-Ward-Oshikawa theorem).
# Low energy effective field theories of fermion liquids and mixed
$U(1)\times\mathbb{R}^{d}$ anomaly
Xiao-Gang Wen Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
###### Contents
1. I Introduction
2. II 1d boson liquid with $U(1)\times\mathbb{R}$ symmetry
3. III 1d fermion liquid with $U(1)\times\mathbb{R}$ symmetry
4. IV Low energy effective theory of $d$-dimensional Fermi liquid and the mixed $U(1)\times\mathbb{R}^{d}$ anomaly
5. V Fermion-pair liquid and the mixed $U(1)\times\mathbb{R}^{d}$ anomaly
6. A Dynamical variational approach and low energy effective theory
## I Introduction
A gapless Fermi liquid for spinless fermions in $d$-dimensional space is
described by the following integrated Boltzmann equation (if we ignore the
collision term that is irrelevant under the renormalization group scaling) Kim
_et al._ (1995)
$\displaystyle\partial_{t}u(\boldsymbol{x},\boldsymbol{k}_{F},t)+\partial_{\boldsymbol{x}}\cdot\big{(}\boldsymbol{v}_{F}(\boldsymbol{k}_{F})u(\boldsymbol{x},\boldsymbol{k}_{F},t)\big{)}$
$\displaystyle+\partial_{\boldsymbol{k}_{F}}\cdot\big{(}\boldsymbol{f}(\boldsymbol{x})u(\boldsymbol{x},\boldsymbol{k}_{F},t)\big{)}=0,$
(1)
where $\boldsymbol{k}_{F}$ parametrizes the Fermi surface,
$\boldsymbol{v}_{F}$ is the Fermi velocity,
$u(\boldsymbol{x},\boldsymbol{k}_{F})$ describes the Fermi surface
displacement at Fermi momentum $\boldsymbol{k}_{F}$ and spacial location
$\boldsymbol{x}$, and $\boldsymbol{f}$ is the force acting on a fermion. We
can view the integrated Boltzmann equation as the equation of motion for the
bosonized Fermi liquid.Luther (1979); Haldane (1992); Houghton and Marston
(1993); Castro Neto and Fradkin (1994) Together with the total energy
(assuming $\boldsymbol{f}=0$)
$\displaystyle E=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}u^{2}(\boldsymbol{x},\boldsymbol{k}_{F}),$
(2)
we obtain a phase-space low energy effective Lagrangian for Fermi liquid (see
eqn. (IV.1)). Note that the low energy effective field theory contain infinite
number of fields labeled by $\boldsymbol{k}_{F}$. Or alternatively, the low
energy effective field theory can be viewed as having a single scaler field in
$(d+d-1)$-dimensional space, but the interaction in the $d-1$ dimensions
(parametrized by $\boldsymbol{k}_{F}$) is allowed to be non-local.
However, such a low energy effective theory fails to capture one of the most
important properties of Fermi liquid: the volume enclosed by the Fermi surface
is $(2\pi)^{d}\bar{\rho}$ where $\bar{\rho}$ is the density of the fermion in
the ground state,Luttinger and Ward (1960); Oshikawa (2000) since the fermion
density $\bar{\rho}$ does not even appear in the above formulation.
In recent years, it was realized that the Lieb-Schultz-Mattis (LSM)
theoremLieb _et al._ (1961) and its higher-dimensional generalizations by
OshikawaOshikawa (2000) and HastingsHastings (2004) can be understood in term
of a mixed anomaly between translation symmetry and an internal
symmetry.Furuya and Oshikawa (2017); Cheng _et al._ (2016); Po _et al._
(2017); Lu _et al._ (2020); Lu (2017); Cheng (2019); Jiang _et al._ (2019)
For a 1-dimensional system with $U(1)$ symmetry and translation symmetry,
there is a similar LSM theorem when the $U(1)$ charge per site is not an
integer.Chen _et al._ (2011) This suggests that such a system also has a
mixed anomaly when the $U(1)$ charge per site is not an integer. Similarly,
for continuum systems with $U(1)$ and translation $\mathbb{R}^{d}$ symmetries,
there should also be a mixed anomaly, whenever the $U(1)$ charge density is
non-zero.
The low energy effective field theory for systems with
$U(1)\times\mathbb{R}^{d}$ symmetry should capture this mixed
$U(1)\times\mathbb{R}^{d}$ anomaly. Here we like to remark that the
$U(1)\times\mathbb{R}^{d}$ symmetry is an exact symmetry. In the low energy
effective field theory, $U(1)\times\mathbb{R}^{d}$ can be viewed as an
internal symmetry. But when viewed as an internal symmetry,
$U(1)\times\mathbb{R}^{d}$ may have a mixed anomaly, and this is the so called
mixed anomaly discussed in this paper and in Ref. Furuya and Oshikawa, 2017;
Cheng _et al._ , 2016; Po _et al._ , 2017; Lu _et al._ , 2020; Lu, 2017;
Cheng, 2019; Jiang _et al._ , 2019
The Fermi liquid at low energies also has many emergent symmetries. In
particular, the $U(1)$ fermion-number-conservation symmetry is enlarged to
$U^{\infty}(1)$ emergent symmetry.Luther (1979); Haldane (1992); Houghton and
Marston (1993); Castro Neto and Fradkin (1994) Recently, it was pointed out
that such an emergent $U^{\infty}(1)$ symmetry also has an anomaly in the
presence of $U(1)$ flux.Else _et al._ (2020) Using such an $U^{\infty}(1)$
anomaly, one can also derive the relation between the volume enclosed by the
Fermi surface and the density of the fermion.
In this paper, we will carefully write down the low energy effective field
theories for some gapless phases of bosons and fermions. The low energy
effective field theories contain a proper topological term that captures the
mixed $U(1)\times\mathbb{R}^{d}$ anomaly. Such a mixed
$U(1)\times\mathbb{R}^{d}$ anomaly ensures that the system must be gapless.
In particular, the low energy effective field theory (IV.1) for Fermi liquid
is obtained, that contains the proper mixed $U(1)\times\mathbb{R}^{d}$
anomaly. Such a mixed $U(1)\times\mathbb{R}^{d}$ anomaly determines the volume
enclosed by the Fermi surface, within the low energy effective field theory.
We also write down the low energy effective field theory (85) for Fermi liquid
with real space magnetic field and $\boldsymbol{k}$-space “magnetic”
field.Thouless _et al._ (1982); Sundaram and Niu (1999); Xiao _et al._
(2010) Those are the main results of this paper.
We will also discuss the universal low energy properties of the gapless phases
for systems with $U(1)\times\mathbb{R}^{d}$ symmetry. Some of the features in
the universal low energy properties are determined by the mixed
$U(1)\times\mathbb{R}^{d}$ anomaly, and we identify those features.
In section II, we will first discuss low energy effective field theory of 1d
weakly interacting bosons. Then in section III, we will consider 1d weakly
interacting fermions. In section IV, we will obtain a low energy effective
field theory for Fermi liquid in a general dimension. Section V discusses
another gapless phase of fermions – a fermion-pair liquid, and its low energy
effective field theory. All those effective field theories capture the mixed
$U(1)\times\mathbb{R}^{d}$ anomaly.
In this paper, we will use the natural unit where $\hbar=e=c=1$.
## II 1d boson liquid with $U(1)\times\mathbb{R}$ symmetry
### II.1 Gapless phase of weakly interacting bosons
In this section, we are going to consider 1d gapless systems in continuum
space with $U(1)$ particle-number-conservation symmetry and $\mathbb{R}$
translation symmetry. The systems are formed by bosons with weak interaction,
which gives rise to a gapless state: a “superfluid” state for bosons. We
assume the system to have a size $L$ with a periodic boundary condition. We
will compute distribution of the total momentum for many-body low energy
excitations, and how such a distribution depends on the $U(1)$ symmetry twist
described by a constant $U(1)$ background vector potential $a$. We will see
that such a dependence directly measure a mixed anomaly in
$U(1)\times\mathbb{R}$ symmetry, if we view $U(1)\times\mathbb{R}$ as an
internal symmetry in the effctive field theory.
Using the results from a careful calculation in Appendix A, we find the
following low energy effective field theory for the gapless phase of bosons
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}x\;$
$\displaystyle\big{(}\bar{\rho}\dot{\phi}(x,t)+\delta\rho(x,t)\dot{\phi}(x,t)$
(3) $\displaystyle\
-\frac{\bar{\rho}}{2M_{b}}|\partial\phi|^{2}-\frac{g}{2}\delta\rho^{2}+\
\cdots\big{)},$ $\displaystyle L_{\text{co}}=\int\hskip 1.0pt\mathrm{d}x\;$
$\displaystyle\big{(}\bar{\rho}\dot{\phi}(x,t)-\frac{\bar{\rho}}{2M_{b}}|\partial\phi|^{2}+\frac{1}{2g}(\dot{\phi})^{2}+\
\cdots\big{)},$
where $\delta\rho$ is the boson density fluctuation, $\bar{\rho}=\bar{N}/L$
the boson density in the ground state, and $\phi$ an angular field
$\phi(x,t)\sim\phi(x,t)+2\pi$. $L_{\text{ph}}$ is the phase-space Lagrangian
and $L_{\text{co}}$ is the coordinate-space Lagrangian. They both describe the
system at low energies. People usually drop the total derivative term (also
called topological term) $\bar{\rho}\dot{\phi}(x,t)$, since it does not affect
the classical equation of motion of the fields. We will see that the
topological term affects the dynamics in quantized theory and should not be
dropped.
Figure 1: (a) The distribution of the total energies and total momenta for low
energy states of 1d weakly interacting boson liquid. (b) The distribution of
the total momenta $k_{\text{tot}}$ for low energy many-body states, and its
dependence on the $U(1)$ symmetry twist $\theta$.
From the low energy effective field theory, we find that the low energy
excitations are labeled by
$(N\in\mathbb{N},m\in\mathbb{Z},n_{k}\in\mathbb{N})$. The total energy and
total momentum of those excitations are given by (in the presendence a
constant $U(1)$ connection $a$ describing the $U(1)$ symmetry twist)
$\displaystyle E$ $\displaystyle=\frac{N}{2M_{b}}(\frac{2\pi
m}{L}+a)^{2}+\frac{g}{2}(N-\bar{N})^{2}+\sum_{k\neq
0}(n_{k}+\frac{1}{2})v|k|,$ $\displaystyle k_{\text{tot}}$
$\displaystyle=\int\hskip
1.0pt\mathrm{d}x\,\rho(\partial_{x}+a)\phi=N(\frac{2\pi m}{L}+a)+\sum_{k\neq
0}n_{k}k,$ (4)
where $N=\bar{N}+\delta N$ is the total number of bosons in the excited state.
When $N=\bar{N}$ and $a=0$, the possible values of $(E,k_{\text{tot}})$ are
plotted in Fig. 1a.
After quantization, we have the following operator algebra
$\displaystyle[\phi(x),\delta\rho(y)]=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\delta(x-y),$ (5)
where $\delta\rho(x)$ is the boson density operator. Since
$\displaystyle[\delta\rho(y),\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\phi(x)}]=\delta(x-y)\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\phi(x)},$ (6)
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\phi(x)}$ is the
boson creation operator.
Let $\varphi(x)=2\pi\int^{x}\hskip
1.0pt\mathrm{d}x^{\prime}\delta\rho(x^{\prime})$. We find that
$\displaystyle[\varphi(x),\phi(y)]$ $\displaystyle=-2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Theta(x-y),$ $\displaystyle\Theta(x)$
$\displaystyle=\begin{cases}1&\ \ \text{ for }x>0,\\\ 0&\ \ \text{ for
}x<0,\\\ \end{cases}$ (7)
or
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha\varphi(x)}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\beta\phi(y)}$ $\displaystyle=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\beta\phi(y)}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\alpha\varphi(x)}\hskip
1.0pt\mathrm{e}^{\alpha\beta[\phi(y),\varphi(x)]}$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\beta\phi(y)}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\alpha\varphi(x)}\hskip
1.0pt\mathrm{e}^{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha\beta\Theta(x-y)}$ (8)
which can also be rewritten as
$\displaystyle\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha\varphi(x)}\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\beta\phi(y)}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha\varphi(x)}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\beta\phi(y)}$ $\displaystyle=\hskip 1.0pt\mathrm{e}^{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\alpha\beta\Theta(x-y)},$ $\displaystyle\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\beta\phi(y)}\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\alpha\varphi(x)}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\beta\phi(y)}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\alpha\varphi(x)}$
$\displaystyle=\hskip 1.0pt\mathrm{e}^{-2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha\beta\Theta(x-y)}.$ (9)
The above expression tells us that that the operator $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\alpha\varphi(x)}$ cause a
$\hskip 1.0pt\mathrm{e}^{-2\pi\hskip 1.0pt\mathrm{i}\hskip 1.0pt\alpha}$ phase
shift for the operator $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\phi(y)}$ for $y<x$, and keep $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\phi(y)}$ unchanged for $y>x$. So the operator
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\varphi(x)}$
increases $m$ by 1, i.e. increase the total momentum by $2\pi\bar{\rho}$.
Similarly, the operator $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\beta\phi(x)}$ cause a $\hskip 1.0pt\mathrm{e}^{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\beta}$ phase shift for the operator $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\varphi(y)}$ for $y<x$, and
keep $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\varphi(y)}$
unchanged for $y>x$. So the operator $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\phi(x)}$ increases $N=\frac{1}{2\pi}\int\hskip
1.0pt\mathrm{d}x\,\partial_{x}\varphi=\int\hskip 1.0pt\mathrm{d}x\,\delta\rho$
by 1.
From the above results, we see that under the $U(1)$ transformation $\theta$
$\displaystyle\phi(x)\to\phi(x)+\theta,\ \ \ \ \varphi(x)\to\varphi(x).$ (10)
Under the $\mathbb{R}$ translation transformation $\delta x$
$\displaystyle\phi(x)\to\phi(x+\delta x),\ \ \ \ \varphi(x)\to\varphi(x+\delta
x)+2\pi\bar{\rho}\delta x.$ (11)
However, at low energies, for smooth fields and small $\delta x$, we have
$\phi(x)\approx\phi(x+\delta x)$ and $\varphi(x)\approx\varphi(x+\delta x)$.
Thus under the $\mathbb{R}$ transformation $\delta x$
$\displaystyle\phi(x)\to\phi(x),\ \ \ \
\varphi(x)\to\varphi(x)+2\pi\bar{\rho}\delta x.$ (12)
This way, the $\mathbb{R}$ symmetry becomes an internal symmetry in the low
energy effective field theory (3). We will see that $U(1)\times\mathbb{R}$,
when viewed as an internal symmetry, has a mixed anomaly.
We note that $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\varphi(x)}$ is a local operator and $\int\hskip
1.0pt\mathrm{d}x\,\delta\rho=\frac{1}{2\pi}\int\hskip
1.0pt\mathrm{d}x\,\partial_{x}\varphi$ is an integer. Both imply that
$\varphi$ is also an angular variable $\varphi(x,t)\sim\varphi(x,t)+2\pi$.
Using the two angular fields $\phi_{1}=:\phi$ and $\phi_{2}=:\varphi$, the low
energy effective theory can be written as
$\displaystyle L_{\text{ph}}=\int\hskip
1.0pt\mathrm{d}x\;\big{(}\bar{\rho}\partial_{t}\phi_{1}+\frac{K_{IJ}}{4\pi}\partial_{x}\phi_{I}\partial_{t}\phi_{J}-\frac{V_{IJ}}{2}\partial_{x}\phi_{I}\partial_{x}\phi_{J}\big{)}$
(13)
where
$\displaystyle K=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}.$ (14)
and $V$ is a positive definite symmetric matrix.
### II.2 The universal properties of the gapless phase
From eqn. (II.1), we see that the total momentum $k_{\text{tot}}$ depends on
the $U(1)$ symmetry twist
$\displaystyle k_{\text{tot}}=\rho\theta,\ \ \ \text{ where }\theta=\int\hskip
1.0pt\mathrm{d}x\;a=aL,\ \ \rho=\frac{N}{L}.$ (15)
However, since the state is gapless, there are many low energy states with
different momenta. So we do not know $k_{\text{tot}}$ to be the momentum of
which low energy states. To make our statement meaningful, we consider two low
energy states $|\Psi_{N}\rangle$ of $N$ bosons. Let $k_{\text{tot}}$ be the
total momenta of $|\Psi_{N}\rangle$. Since there are many different low energy
states $|\Psi_{N}\rangle$’s, we have many different values of
$k_{\text{tot}}$. In other words, we have a distribution of
$k_{\text{tot}}$’s. Such a distribution is plotted in Fig. 1b.
From the distribution pattern in Fig. 1, we see two universal properties: the
period in the distribution and the $\theta=aL$ dependence of the distribution
$\displaystyle k_{0}=2\pi\bar{\rho},\ \ \ \ \ \frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}=\bar{\rho},$ (16)
which do not depend on the small changes in the interactions and the
dispersion of the bosons, unless those changes cause a phase transition. Thus
we say they are universal properties that characterize the gapless phase. The
two universal properties are closely related $(2\pi)^{-1}k_{0}=\frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}=\bar{\rho}$. We
call $\bar{\rho}$ an index for the gapless phase. Physically $\bar{\rho}$ is
nothing but the density of the $U(1)$ charges in the ground state.
Let us give a argument why $\frac{\hskip 1.0pt\mathrm{d}k_{\text{tot}}}{\hskip
1.0pt\mathrm{d}\theta}$ is universal. Let us assume the $U(1)$ symmetry twist
is described by a boundary condition on single-particle wave function at
$x_{0}$: $\psi(x_{0}+0^{+})=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta}\psi(x_{0}-0^{+})$. A usual translation
$x\to x+\Delta x$ will shift the symmetry twist from $x_{0}$ to $x_{0}+\Delta
x$. So the symmetry twist breaks the translation symmetry. But we can redefine
the translation operator to be the usual translation plus a $U(1)$
transformation $\psi(x)\to\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta}\psi(x)$ for $x\in[x_{0},x_{0}+\Delta x]$. The new translation
operator generates the translation symmetry in the presence of the $U(1)$
symmetry twist. Due to the $U(1)$ transformation $\psi(x)\to\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta}\psi(x)$ for
$x\in[x_{0},x_{0}+\Delta x]$, the eigenvalue of the new translation operator
has a $\theta$ dependence given by $(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta})^{\bar{\rho}\Delta x}$, where
$\bar{\rho}\Delta x$ is the total $U(1)$ charges in the interval
$[x_{0},x_{0}+\Delta x]$. In other words the total momentum has a $\theta$
dependence given by $\theta\bar{\rho}$. This is the reason why $\frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}=\bar{\rho}$.
Since $\theta=0$ and $\theta=2\pi$ are equivalent, therefore $\frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}=\bar{\rho}$
implies the periodicy in Fig. 1, with the period $k_{0}=2\pi\bar{\rho}$. The
above discussion does not depend on interactions and boson dispersion. So the
results (16) are universal properties.
### II.3 Mixed anomaly for $U(1)\times\mathbb{R}$ symmetry
From the above argument, we also see that the shift of the low energy momentum
distribution by the $U(1)$ symmetry twist, $\frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}=\bar{\rho}$, is
an invariant not only against small perturbations that preserve the
$U(1)\times\mathbb{R}$ symmetry, but is also an invariant against large
symmetry-preserving perturbations that can drive through a phase transition.
The invariant for large perturbations is actually an anomaly.’t Hooft (1980)
This is because, under a new point of view,Wen (2013); Kong and Wen (2014) an
anomaly corresponds to an SPT or topological order in one higher dimension.Wen
(2013); Kong and Wen (2014) Any large perturbations and phase transitions
cannot change the SPT or topological order in one higher dimension.
In our case, we can view $\frac{\hskip 1.0pt\mathrm{d}k_{\text{tot}}}{\hskip
1.0pt\mathrm{d}\theta}=\bar{\rho}$ as an anomaly in the low energy effective
field theories (3) and (13). We see that the topological term
$\bar{\rho}(x,t)\dot{\phi}(x,t)$ in the low energy effective field theory
determine the anomaly. In fact, such an anomaly is a mixed anomaly between
$U(1)$ symmetry and the translation $\mathbb{R}$ symmetry, which describe how
an $U(1)$ symmetry twist can change the total momentum (i.e. $\frac{\hskip
1.0pt\mathrm{d}k_{\text{tot}}}{\hskip 1.0pt\mathrm{d}\theta}\neq 0$).
The presence of the anomaly implies that the ground state of the system must
be either gapless or have a non-trivial topological order. Since there is no
non-trivial topological order in 1d, the ground state must be gapless. In
other words,
the field theories in eqn. (3) and eqn. (13) with $\bar{\rho}\neq 0$ must be
gapless regardless the interaction term described by $\cdots$, as long as the
$U(1)\times\mathbb{R}$ symmetry is preserved. On the other hand, when
$\bar{\rho}=0$, the field theories in eqn. (3) and eqn. (13) allow a gapped
phase with $U(1)\times\mathbb{R}$ symmetry.
The mixed anomaly between the $U(1)$ symmetry and the $\mathbb{R}$ symmetry
can also be detected via the patch symmetry transformations studied in Ref. Ji
and Wen, 2020. The $U(1)$ patch symmetry transformations are given by
$\displaystyle W_{U(1)}(x,y)=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi\alpha\int_{x}^{y}\hskip
1.0pt\mathrm{d}x\rho}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\alpha(\varphi(y)-\varphi(x))}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi\alpha\bar{\rho}(y-x)},$ (17)
which perform the $U(1)$ transformation, $\phi\to\phi+\alpha$, on the segment
$[x,y]$. The $\mathbb{R}$ patch symmetry transformations are given by
$\displaystyle W_{\mathbb{R}}(x,y)=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\rho}\Delta x(\phi(y)-\phi(x))}.$ (18)
which perform the $\mathbb{R}$ transformation (the translation $\Delta x$) on
the segment $[x,y]$. In the low energy limit, $k\to 0$. So for a finite
$\Delta x$ the translation is trivial for the phonon modes. The translation
$\Delta x$ has a non-trivial actions only on sector labeled by different $N$’s
and $m$’s. For a translation $\Delta x$ that acts on a segment $[x,y]$, its
effect is to transfer $U(1)$ charge-$\bar{\rho}\Delta x$ from $x$ to $y$. This
is why the $\mathbb{R}$ patch symmetry transformations are given by eqn. (18).
In the low energy effective theories (3) and (13), the $U(1)$ transformation
is given by $\phi\to\phi+\theta$. The term $\bar{\rho}\dot{\phi}$ implies
$\bar{\rho}$ is the background $U(1)$ charge density. Therefore, the patch
translation transformation has a form eqn. (18).
Assume $x_{2}>x_{1}$. We have shown that $W_{U(1)}(x_{1},x_{2})$ shifts
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\beta\phi(y)}$ by a
phase $\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\alpha\beta}$, if $x_{1}<y<x_{2}$. Therefore
$\displaystyle\ \ \ \ W_{U(1)}(x_{1},x_{2})W_{\mathbb{R}}(y_{1},y_{2})$
$\displaystyle=W_{\mathbb{R}}(y_{1},y_{2})W_{U(1)}(x_{1},x_{2})\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt2\pi\alpha\bar{\rho}\Delta
x},$ (19)
for $x_{1}<y_{1}<x_{2}<y_{2}$. The extra phase factor $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt2\pi\alpha\bar{\rho}\Delta
x}$ indicates the appearance of the mixed $U(1)\times\mathbb{R}$ anomaly.
Using the terminology of Ref. Ji and Wen, 2020, we say, the $U(1)$ symmetry
and the $\mathbb{R}$ symmetry have a “mutual statistics” between them, as a
consequence of the mixed anomaly. So, according to Ref. Ji and Wen, 2020, the
$U(1)$ symmetry and the $\mathbb{R}$ are not independent, and we may denote
the combined symmetry as $U(1)\vee\mathbb{R}$ to stress the mixed anomaly.
We like to remark that $\phi$ and $\varphi$ fields are cannonial conjugate to
each other. We see that the symmetries that shift $\phi$ and $\varphi$ have a
mixed anomaly, as captured by non-trivial commutation relation between the
patch operators for the symmetry transformation. This is a general mechanism
of the appearance of anomaly.
## III 1d fermion liquid with $U(1)\times\mathbb{R}$ symmetry
### III.1 Weakly interacting 1d gapless fermionic systems
1d gapless fermionic systems with $U(1)\times\mathbb{R}$ symmetry and weak
repulsive interaction are also in a gapless phase – a Tomonaga-Luttinger
liquid for fermions. The low energy effective theory also has a form
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}x\;$
$\displaystyle\big{(}\bar{\rho}\dot{\phi}(x,t)+\delta\rho(x,t)\dot{\phi}(x,t)$
$\displaystyle-\frac{\bar{\rho}}{2M_{f}}|\partial\phi|^{2}+\frac{1}{2g}(\dot{\phi})^{2}+\
\cdots\big{)}.$ (20)
Considering non-interacting fermions in a system with periodic boundary
condition on a ring of size $L$, we find that the low energy excitations are
also labeled by $(N\in\mathbb{N},m\in\mathbb{Z},n_{k}\in\mathbb{N})$. However,
the total energies and total momenta of those excitations are given by, for
$N=$ odd,
$\displaystyle E=\frac{N}{2M_{f}}(2\pi\frac{m}{L}+a)^{2}+\frac{g}{2}\delta
N^{2}+\sum_{k\neq 0}(n_{k}+\frac{1}{2})v|k|,$ $\displaystyle
k_{\text{tot}}=N(2\pi\frac{m}{L}+a)+\sum_{k\neq 0}n_{k}k,$ (21)
and for $N=$ even,
$\displaystyle
E=\frac{N}{2M_{f}}(2\pi\frac{m+\frac{1}{2}}{L}+a)^{2}+\frac{g}{2}\delta
N^{2}+\sum_{k\neq 0}(n_{k}+\frac{1}{2})v|k|,$ $\displaystyle
k_{\text{tot}}=N(2\pi\frac{m+\frac{1}{2}}{L}+a)+\sum_{k\neq 0}n_{k}k,$ (22)
where $N(2\pi\frac{m}{L}+a)$ or $N(2\pi\frac{m+\frac{1}{2}}{L}+a)$ are the
momentum of center of mass.
Figure 2: The distribution of the total momenta $\Delta k_{\text{tot}}$ and
its dependence on the $U(1)$ symmetry twist, for a weakly interacting 1d
fermion liquid.
Again, we consider low energy states $|\Psi_{N}\rangle$ of $N$ fermions. Let
$k_{\text{tot}}$ be the total momenta of $|\Psi_{N}\rangle$. Since there are
many different low energy states $|\Psi_{N}\rangle$’s, we have a distribution
of $k_{\text{tot}}$’s. Such a distribution is plotted in Fig. 2a for $N_{=}$
odd case, and in Fig. 2b for $N=$ even case. We see that $N=$ odd case and
$N=$ even case have different distributions for $k_{\text{tot}}$’s.
The shift of the distribution of $k_{\text{tot}}$ by the $U(1)$ symmetry twist
(see Fig. 2) can be interpreted as $U(1)$ symmetry twist producing or pumping
momentum. This directly measures the mixed $U(1)\times\mathbb{R}$ anomaly.
We also see that a local operator that create a fermion (i.e. change $N$ by 1)
must also change $m$ by $\frac{1}{2}$. Those operators have a form $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(\phi\pm\frac{1}{2}\varphi)}$, where
$\partial_{x}\varphi=2\pi\delta\rho$. Since the allowed operators are
generated by $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(\phi\pm\frac{1}{2}\varphi)}$, the following fields
$\displaystyle\phi_{1}=\phi+\frac{1}{2}\varphi,\ \ \ \ \
\phi_{2}=\phi-\frac{1}{2}\varphi$ (23)
are angular fields: $\phi_{i}\sim\phi_{i}+2\pi$. Using
$\displaystyle\phi=\frac{1}{2}(\phi_{1}+\phi_{2}),\ \ \ \ \
\varphi=\phi_{1}-\phi_{2},$ (24)
we find the low energy effective theory to be
$\displaystyle L$ $\displaystyle=\int\hskip
1.0pt\mathrm{d}x\;\big{(}\frac{\bar{\rho}}{2}\partial_{t}(\phi_{1}+\phi_{2})$
$\displaystyle\ \ \ \ \ \ \ \
+\frac{1}{4\pi}\partial_{x}(\phi_{1}-\phi_{2})\partial_{t}(\phi_{1}+\phi_{2})-\frac{V_{IJ}}{2}\partial_{x}\phi_{I}\partial_{x}\phi_{J}\big{)}$
$\displaystyle=\int\hskip
1.0pt\mathrm{d}x\;\big{(}\frac{\bar{\rho}}{2}\partial_{t}(\phi_{1}+\phi_{2})+\frac{1}{4\pi}(\partial_{x}\phi_{1}\partial_{t}\phi_{2}-\partial_{x}\phi_{2}\partial_{t}\phi_{1})$
$\displaystyle\ \ \ \ \ \ \ \
+\frac{K_{IJ}}{4\pi}\partial_{x}\phi_{I}\partial_{t}\phi_{J}-\frac{V_{IJ}}{2}\partial_{x}\phi_{I}\partial_{x}\phi_{J}\big{)},$
$\displaystyle K$ $\displaystyle=\begin{pmatrix}1&0\\\ 0&-1\\\ \end{pmatrix},\
\ \ \ V=\text{positive definite}.$ (25)
Here, we have been careful to keep the total derivative terms
$\frac{\bar{\rho}}{2}\partial_{t}(\phi_{1}+\phi_{2})+\frac{1}{4\pi}(\partial_{x}\phi_{1}\partial_{t}\phi_{2}-\partial_{x}\phi_{2}\partial_{t}\phi_{1})$.
Those are topological terms that do not affect the classical equation of
motion, but have effects in quantum theory.
Effective theory similar to the above form has been obtained before for edge
state of fractional quantum Hall states.Wen (1992, 1995) But here we have to
be more careful in keeping the topological term
$\bar{\rho}\partial_{t}\phi_{1}$, which describes the mixed anomaly of
$U(1)\times\mathbb{R}$ symmetry for the fermionic system.
We like to mention that the patch symmetry transformations are determined from
the low energy effective theories (III.1) or (III.1), and are still given by
eqn. (17) and eqn. (18). So the mixed anomaly can still be detected via
commutation relation of the patch symmetry transformations (II.3).
In fact, eqn. (III.1) is the low energy effective theory for a fermion system
with Fermi momentum $k_{F}=\pi\bar{\rho}$.
$\frac{1}{2\pi}\partial_{x}\phi_{1}$ describes the density of right-moving
fermions and $\frac{1}{2\pi}\partial_{x}\phi_{2}$ describes the density of
left-moving fermions. For example, the low energy effective theory for right-
moving fermions is given by
$\displaystyle L=\int\hskip
1.0pt\mathrm{d}x\;\big{(}\frac{k_{F}}{2\pi}\partial_{t}\phi_{1}+\frac{1}{4\pi}\partial_{x}\phi_{1}\partial_{t}\phi_{1}-\frac{V_{11}}{2}\partial_{x}\phi_{1}\partial_{x}\phi_{1}\big{)},$
$\displaystyle=\int\hskip
1.0pt\mathrm{d}x\;\big{(}\bar{\rho}_{1}\partial_{t}\phi_{1}+\frac{1}{4\pi}\partial_{x}\phi_{1}\partial_{t}\phi_{1}-\frac{V_{11}}{2}\partial_{x}\phi_{1}\partial_{x}\phi_{1}\big{)}.$
(26)
In the above expression, we stress the direction connection between the
topological term and the Fermi momentum $k_{F}$, as well as the direction
connection between the topological term and the density of the right-moving
fermions: $\bar{\rho}_{1}=\frac{k_{F}}{2\pi}=\frac{1}{2}\bar{\rho}$.
We see that the mixed anomaly of $U(1)\times\mathbb{R}$ symmetry is nothing
but a non-zero Fermi momentum $k_{F}$. When $k_{F}=0$, i.e. when mixed anomaly
vanishes, the fermion system can have a gapped ground state that does not
break the $U(1)\times\mathbb{R}$ symmetry. But when $k_{F}\neq 0$, i.e. in the
presence of mixed anomaly, the fermion system cannot have a gapped ground
state that does not break the $U(1)\times\mathbb{R}$ symmetry. This is a well
known result, but restated in terms of mixed anomaly of $U(1)\times\mathbb{R}$
symmetry.
## IV Low energy effective theory of $d$-dimensional Fermi liquid and the
mixed $U(1)\times\mathbb{R}^{d}$ anomaly
### IV.1 Effective theory for the Fermi surface dynamics
In the last section, we discussed the low energy effective theory of 1d Fermi
liquid, which contain a proper topological term that reflects the mixed
$U(1)\times\mathbb{R}$ anomaly. In this section, we are going to generalize
this result to higher dimensions. The generalization is possible since the
higher dimensional Fermi liquid can be viewed as a collection of 1d Fermi
liquids.
Let us use $\boldsymbol{k}_{F}$ to parametrize the Fermi surface. We introduce
$u(\boldsymbol{x},\boldsymbol{k}_{F})$ to describe the shift of the Fermi
surface. Thus the total fermion number is given by
$\displaystyle N=\bar{N}+\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,u(\boldsymbol{x},\boldsymbol{k}_{F}).$
(27)
The total energy is
$\displaystyle E=\bar{E}+\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}u^{2}(\boldsymbol{x},\boldsymbol{k}_{F}),$
(28)
where
$\displaystyle\boldsymbol{v}_{F}=:\partial_{\boldsymbol{k}}H(\boldsymbol{k}),$
(29)
is the Fermi velocity and $H(\boldsymbol{k})$ is the single fermion energy.
The equation of motion for the field $u(\boldsymbol{x},\boldsymbol{k}_{F})$ is
given by
$\displaystyle(\partial_{t}+\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}})u(\boldsymbol{x},\boldsymbol{k}_{F})=0$
(30)
Let us introduce a field $\phi(\boldsymbol{x},\boldsymbol{k}_{F})$ via
$\displaystyle-\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})=u(\boldsymbol{x},\boldsymbol{k}_{F}),$
(31)
where
$\displaystyle\boldsymbol{n}_{F}=:\frac{\boldsymbol{v}_{F}}{|\boldsymbol{v}_{F}|}.$
(32)
The equation of motion for $\phi$ becomes
$\displaystyle(\partial_{t}+\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}})\big{(}\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})\big{)}=0$
(33)
and the total energy becomes
$\displaystyle E=\bar{E}+\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})]^{2}.$
(34)
The phase-space Lagrangian that produces the above equation of motion and
total energy is given by
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}$
$\displaystyle\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\Big{(}-\frac{1}{2}\partial_{t}\phi(\boldsymbol{x},\boldsymbol{k}_{F})\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle-\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}[\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})]^{2}\Big{)}$
(35)
Repeating a calculation similar to 1d chiral Luttinger liquid,Wen (1992, 1995)
we find that after quantization, the operator
$\phi(\boldsymbol{x},\boldsymbol{k}_{F})$ has the following commutation
relation
$\displaystyle\ \ \ \
[\phi(\boldsymbol{x}^{\prime},\boldsymbol{k}_{F}^{\prime}),\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})]$
$\displaystyle=-[\phi(\boldsymbol{x}^{\prime},\boldsymbol{k}_{F}^{\prime}),u(\boldsymbol{x},\boldsymbol{k}_{F})]$
$\displaystyle=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt(2\pi)^{d}\delta^{d}(\boldsymbol{x}-\boldsymbol{x}^{\prime})\delta^{d-1}(\boldsymbol{k}_{F}-\boldsymbol{k}_{F}^{\prime})$
(36)
which reproduce the equation of motion
$\displaystyle\partial_{t}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)=\hskip
1.0pt\mathrm{i}\hskip
1.0pt[H,\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)]=-\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t),$
$\displaystyle H=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}[\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})]^{2}$
(37)
We also see that
$\displaystyle[N,\phi(\boldsymbol{x},\boldsymbol{k}_{F})]=-\hskip
1.0pt\mathrm{i}\hskip 1.0pt,\ \ \ [N,\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\phi(\boldsymbol{x},\boldsymbol{k}_{F})}]=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\phi(\boldsymbol{x},\boldsymbol{k}_{F})}.$ (38)
Thus $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)}$ is the operator that
increases $N$ by 1, and the $U(1)$ symmetry transformation is given by
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta
N}\phi(\boldsymbol{x},\boldsymbol{k}_{F})\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta
N}=\phi(\boldsymbol{x},\boldsymbol{k}_{F})+\theta.$ (39)
We see that $\phi(\boldsymbol{x},\boldsymbol{k}_{F})$ is an angular field
$\phi(\boldsymbol{x},\boldsymbol{k}_{F})\sim\phi(\boldsymbol{x},\boldsymbol{k}_{F})+2\pi$.
However, in eqn. (IV.1) we only have terms that
$\phi(\boldsymbol{x},\boldsymbol{k}_{F})$ couples to $\delta N$. In a compete
Lagrangian, $\phi(\boldsymbol{x},\boldsymbol{k}_{F})$ must also couple to
$\bar{\rho}$ – the density of the $U(1)$ charge in the ground state. The
complete phase-space Lagrangian is given by
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}$
$\displaystyle\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\Big{(}\frac{\bar{\rho}}{A_{F}}\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle-\frac{1}{2}\dot{\phi}(\boldsymbol{x},-\boldsymbol{k}_{F})\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle-\frac{1}{2}\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle-\frac{|\boldsymbol{v}_{F}(\boldsymbol{k}_{F})|}{2}[\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})]^{2}\Big{)}$
(40)
where
$\displaystyle A_{F}=:\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}.$ (41)
and we have assumed a central reflection symmetry
$\boldsymbol{k}_{F}\to-\boldsymbol{k}_{F}$, i.e.
$\boldsymbol{v}_{F}(\boldsymbol{k}_{F})=-\boldsymbol{v}_{F}(-\boldsymbol{k}_{F})$
and
$\boldsymbol{n}_{F}(\boldsymbol{k}_{F})=-\boldsymbol{n}_{F}(-\boldsymbol{k}_{F})$.
The two terms,
$\frac{\bar{\rho}}{A_{F}}\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})$ and
$\displaystyle\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\frac{1}{2}\dot{\phi}(\boldsymbol{x},-\boldsymbol{k}_{F})\boldsymbol{n}_{F}(\boldsymbol{k}_{F})\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F}),$
(42)
are total derivative topological terms.
We know that the volume enclosed by the Fermi surface is directly related to
the fermion density $\bar{\rho}$.Luttinger and Ward (1960); Oshikawa (2000)
Naively, in our effective theory (IV.1), the parameter $\bar{\rho}$ and Fermi
surface $\boldsymbol{k}_{F}$ are not related. In the following, we like to
show that in fact $\bar{\rho}$ and the Fermi surface $\boldsymbol{k}_{F}$ are
related, from within the effective field theory (IV.1).
Consider a field configuration
$\displaystyle\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle=\boldsymbol{a}\cdot\boldsymbol{x}$ $\displaystyle\text{or }\ \
\boldsymbol{n}_{F}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle=\boldsymbol{a}\cdot\boldsymbol{n}_{F}=u(\boldsymbol{x},\boldsymbol{k}_{F}).$
(43)
There are two ways to compute the momentum for such a field configuration.
In the first way, the total momentum is computed via the deformation
$u(\boldsymbol{x},\boldsymbol{k}_{F})$ of the Fermi surface (assuming the
total momentum of the ground state to be zero)
$\displaystyle\boldsymbol{k}_{\text{tot}}$ $\displaystyle=\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\boldsymbol{k}_{F}u(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\boldsymbol{k}_{F}(\boldsymbol{a}\cdot\boldsymbol{n}_{F}).$
(44)
Note that $\boldsymbol{n}_{F}$ is the normal direction of the Fermi surface.
Therefore
$\displaystyle\boldsymbol{k}_{\text{tot}}=\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\boldsymbol{k}_{F}(\boldsymbol{a}\cdot\boldsymbol{n}_{F})$
(45) $\displaystyle=\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\int_{\boldsymbol{k}\in\boldsymbol{k}_{F}+\boldsymbol{a}}\frac{\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\boldsymbol{k}-\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\int_{\boldsymbol{k}\in\boldsymbol{k}_{F}}\frac{\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\boldsymbol{k},$
where $\int_{\boldsymbol{k}\in\boldsymbol{k}_{F}}\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{k}_{F}$ means integration over $\boldsymbol{k}$
inside the Fermi surface, and
$\int_{\boldsymbol{k}\in\boldsymbol{k}_{F}+\boldsymbol{a}}\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{k}_{F}$ means integration over $\boldsymbol{k}$
inside the shifted Fermi surface (shifted by $\boldsymbol{a}$). Let
$\displaystyle
V_{\boldsymbol{k}_{F}}=:\int_{\boldsymbol{k}\in\boldsymbol{k}_{F}}\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{k}_{F}$ (46)
be the volume enclosed by the Fermi surface, we see that
$\displaystyle\boldsymbol{k}_{\text{tot}}=V\frac{V_{\boldsymbol{k}_{F}}}{(2\pi)^{d}}\boldsymbol{a}$
(47)
where $V$ is the volume of the system.
There is a second way to compute the total momentum
$\boldsymbol{k}_{\text{tot}}$. We consider a time dependent translation of the
above configuration
$\displaystyle\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
$\displaystyle=\boldsymbol{a}\cdot(\boldsymbol{x}+\boldsymbol{x}_{0}(t)).$
(48)
The effective phase-space Lagrangian for $\boldsymbol{x}_{0}(t)$ is given by
$\displaystyle
L_{\text{ph}}=V\bar{\rho}\boldsymbol{a}\cdot\dot{\boldsymbol{x}}_{0}(t).$ (49)
We see that $V\bar{\rho}\boldsymbol{a}$ is the canonical momentum of the
translation $\boldsymbol{x}_{0}$. Thus, the total momentum of the
configuration is
$\displaystyle\boldsymbol{k}_{tot}=V\bar{\rho}\boldsymbol{a}.$ (50)
Compare eqn. (47) and eqn. (50), we see that the volume included by the Fermi
surface and the fermion density is related
$\displaystyle\frac{V_{\boldsymbol{k}_{F}}}{(2\pi)^{d}}=\bar{\rho}.$ (51)
This is the Luttinger theorem.
### IV.2 The mixed anomaly in $U(1)\times\mathbb{R}^{d}$ symmetry
As we have pointed out that the topological term $\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\big{(}\frac{\bar{\rho}}{A_{F}}\partial_{t}\phi(\boldsymbol{x},\boldsymbol{k}_{F})$
represents a mixed anomaly of $U(1)\times\mathbb{R}^{d}$ symmetry. To see this
point, we note that $\boldsymbol{a}$ in eqn. (IV.1) can be viewed as the
$U(1)$ symmetry twist. The fact that the $U(1)$ symmetry twist can induce the
$\mathbb{R}^{d}$ quantum number (i.e. the momentum) reflects the presence of
the mixed anomaly of $U(1)\times\mathbb{R}^{d}$ symmetry. Eqn. (51) indicates
that the mixed anomaly can constraint the low energy dynamics, in this case
determines the volume enclosed by the Fermi surface.
Figure 3: The total momentum distributions for low energy many-body states
for a 2d Fermi liquid, with 1-particle excitations (left, red channal),
2-particle 1-hole excitations (left, green channal), 1-particle 1-hole
excitations (right, red channal), and 2-particle 2-hole excitations (right,
green channal). The horizental axis is $k_{x}$ and the vertical axis is
$k_{y}$.
In the above, we discussed how the $U(1)$ symmetry twist shifts the total
momentum of a particular low energy many-body state. However, in practice, we
cannot pick a particular low energy many-body state, and see how its momentum
is shifted by the $U(1)$ symmetry twist. What can be done is to examine all
the low energy many-body low energy states, and their total momentum
distribution. The shift of the total momentum distribution by the $U(1)$
symmetry twist measure the mixed $U(1)\times\mathbb{R}^{d}$ anomaly. In Fig.
3, we plot the total momentum distributions for low energy many-body states,
with 1-particle excitations, 2-particle 1-hole excitations, 1-particle 1-hole
excitations, and 2-particle 2-hole excitations.
The mixed anomaly not only appears in Fermi liquid phases of fermions, it also
appears in any other phases of fermions. Thus the mixed anomaly constrain the
low energy dynamics in any of those phases. In next section, we consider a
phase of fermion, where fermions pair-up to form a boson liquid in
$d$-dimensional space.
### IV.3 Effective theory of a Fermi liquid in most general setting
In section IV.1, we considered Fermi liquid in free space. In this section, we
like to include electromagnetic field in real space, as well as “magnetic
field” in $\boldsymbol{k}$ space. We like to find the low energy effective
theory of Fermi liquid for this more general situation.
First we consider the dynamics of a single particle in a very general setting.
The classical state of the particle is described by a point in phase-space
parametrized by $\xi^{I}$. The single particle dynamics is described by a
single-particle phase-space Lagrangian:
$\displaystyle L(\dot{\xi}^{I},\xi^{I})=\int\hskip 1.0pt\mathrm{d}t\
\big{[}a_{I}(\xi^{I})\dot{\xi}^{I}-H(\xi^{I})\big{]},$ (52)
which gives rise to the following single-particle equation of motion
$\displaystyle b_{IJ}\dot{\xi}^{J}$ $\displaystyle=\frac{\partial
H}{\partial\xi^{I}},\ \ \ \ \ \
b_{IJ}=\partial_{\xi^{I}}a_{J}-\partial_{\xi^{J}}a_{I}.$ (53)
Here $H(\xi^{I})$ is the single-particle energy for the state $\xi^{I}$, and
$a_{I}(\xi^{I})$ is a phase-space vector potential that describes the phase
“magnetic” field $b_{IJ}(\xi^{I})$. The phase space “magnetic” field includes
both the real space magnetic field and $\boldsymbol{k}$-space “magnetic”
field.Thouless _et al._ (1982); Sundaram and Niu (1999); Xiao _et al._
(2010)
For a particle in a $d$-dimensional free space described by coordinate-
momentum pair $(\boldsymbol{x},\boldsymbol{k})=(x^{i},k_{i})$, $i=1,\cdots,d$,
the phase-space magnetic field $b_{IJ}(\xi^{I})$ is a constant (i.e.
independent of $\xi^{I}$), since
$a_{I}(\xi^{I})\dot{\xi}^{I}=\boldsymbol{k}\cdot\dot{\boldsymbol{x}}$ (i.e.
$a_{x^{i}}=k_{i},\ a_{k_{i}}=0$). On the other hand, if there is a non-uniform
real space and/or $\boldsymbol{k}$-space magnetic fields, phase-space
“magnetic” field $b_{IJ}(\xi^{I})$ will not be uniform.
As a example, let us consider a particle in 3-dimensional space. The phase
space is 6-dimensional and is parametrized by
$(\xi^{I})=(\boldsymbol{x},\boldsymbol{k})$. The phase-space Lagrangian is
given by
$\displaystyle L$
$\displaystyle=[\boldsymbol{k}\cdot\dot{\boldsymbol{x}}+\boldsymbol{A}(\boldsymbol{x})\cdot\dot{\boldsymbol{x}}+\boldsymbol{\widetilde{A}}(\boldsymbol{k})\cdot\dot{\boldsymbol{k}}]-H(\boldsymbol{k})-V(\boldsymbol{x}).$
(54)
Here $\boldsymbol{A}(\boldsymbol{x})$ is the real space vector potential for
electromagnetic field that only depends on $\boldsymbol{x}$.
$\boldsymbol{\widetilde{A}}(\boldsymbol{k})$ is the $\boldsymbol{k}$-space
vector potential that is assumed to depend only on $\boldsymbol{k}$. Such a
$\boldsymbol{k}$-space vector potential can appear for an electron in a
crystal with spin orbital couplings. The corresponding equation of motion is
given by
$\displaystyle\dot{k}_{i}$ $\displaystyle=-\frac{\partial V}{\partial
x^{i}}+B_{ij}\dot{x}^{j},$ $\displaystyle\dot{x}^{i}$
$\displaystyle=\frac{\partial H}{\partial
k_{i}}-\widetilde{B}^{ij}\dot{k}_{j},$ $\displaystyle\text{or }\ \
\dot{\boldsymbol{k}}$ $\displaystyle=-\frac{\partial
V}{\partial\boldsymbol{x}}+\dot{\boldsymbol{x}}\times\boldsymbol{B},$
$\displaystyle\dot{\boldsymbol{x}}$ $\displaystyle=\frac{\partial
H}{\partial\boldsymbol{k}}-\dot{\boldsymbol{k}}\times\widetilde{\boldsymbol{B}}.$
(55)
where
$\displaystyle B_{ij}$
$\displaystyle=\partial_{x^{i}}A_{j}-\partial_{x^{j}}A_{i},$
$\displaystyle\widetilde{B}^{ij}$
$\displaystyle=\partial_{k_{i}}\widetilde{A}^{j}-\partial_{k_{j}}\widetilde{A}^{i},$
$\displaystyle\text{or }\ \ \boldsymbol{B}$
$\displaystyle=\partial_{\boldsymbol{x}}\times\boldsymbol{A},$
$\displaystyle\widetilde{\boldsymbol{B}}$
$\displaystyle=\partial_{\boldsymbol{k}}\times\widetilde{\boldsymbol{A}}.$
(56)
Now consider a many-fermion system which is described by a particle-number
distribution $g(\xi^{I})$. The meaning of the distribution $g(\xi^{I})$ is
given by
$\displaystyle\hskip
1.0pt\mathrm{d}N=g(\xi^{I})\text{Pf}[b(\xi^{I})]\frac{\hskip
1.0pt\mathrm{d}^{2d}\xi^{I}}{(2\pi)^{d}},$ (57)
where $\hskip 1.0pt\mathrm{d}N$ is the number of fermions in the phase-space
volume $\hskip 1.0pt\mathrm{d}^{2d}\xi^{I}$, and $\text{Pf}[b(\xi^{I})]$ is
the Pfaffian of the $2d\times 2d$ anti-symmetric matrix
$\displaystyle[b(\xi^{I})]_{IJ}=b_{IJ}(\xi^{I}).$ (58)
In fact $g(\xi^{I})$ have a meaning as the occupation number per orbital,
since the number of orbitals (i.e. the single particle quantum states) in the
phase-space volume $\hskip 1.0pt\mathrm{d}^{2d}\xi^{I}$ is given by
$\text{Pf}[b(\xi^{I})]\frac{\hskip 1.0pt\mathrm{d}^{2d}\xi^{I}}{(2\pi)^{d}}$.
The above interpretation is correct since under the time evolution (53), the
scaled phase-space volume $\text{Pf}[b(\xi^{I})]\frac{\hskip
1.0pt\mathrm{d}^{2d}\xi^{I}}{(2\pi)^{d}}$ is time independent, which
corresponds to the unitary time evolution in quantum theory. To show such a
result, we first choose a phase space coordinate such that $b_{IJ}$ is uniform
in the phase space. In this case, the time evolution $\dot{\xi}^{I}$ is
described by a divergent-less vector field, $\frac{\partial
H}{\partial\xi^{I}}$, in the phase space, and the phase-space volume $\hskip
1.0pt\mathrm{d}^{2d}\xi^{I}$ is time independent. We note that the phase-space
volume given by the combination $\text{Pf}[b(\xi^{I})]\frac{\hskip
1.0pt\mathrm{d}^{2d}\xi^{I}}{(2\pi)^{b}}$ invariant under the coordinate
transformation. Such an invariant combination is invariant under the time
evolution (53) for a general coordinate, since the equation of motion is
covariant under the coordinate transformation. We see that the phase space has
a simpletic geometry.
For our example (54), the 6-by-6 matrix $b_{IJ}$ is given by
$\displaystyle(b_{IJ})=\begin{pmatrix}B_{ij}&\delta_{ij}\\\
-\delta_{ij}&\widetilde{B}^{ij}\\\ \end{pmatrix}$ (59)
We find that
$\displaystyle\text{Pf}(b)$
$\displaystyle=\text{Pf}\begin{pmatrix}B_{ij}&\delta_{ij}\\\
-\delta_{ij}&\widetilde{B}^{ij}\\\ \end{pmatrix}$
$\displaystyle=\text{Pf}(B,\widetilde{B})=1+B_{ij}\widetilde{B}^{ji}+O(B_{ik}\widetilde{B}^{kj})^{2}.$
(60)
The effective theory for the Fermi liquid in such a general setting is simply
a hydrodynamical theory for an incompressible fluild in the phase space. In
the following, we will write down such a theory for small fluctuations near
the ground state. First, the ground state of the Fermi liquid is described by
the following distribution (or phase-space density)
$\displaystyle\bar{g}(\xi^{I})=\begin{cases}1,&\text{ for }H(\xi^{I})<0,\\\
0,&\text{ for }H(\xi^{I})>0.\\\ \end{cases}$ (61)
The generalized Fermi surface is the $(2d-1)$-dimensional sub-manifold in the
phase space where $\bar{g}(\xi^{I})$ has a jump. A many-body collective
excitation is described by another incompressible distribution
$g(\xi^{I})=0,1$. For low energy collective excitations near the ground state,
we may describe such an incompressible distribution via the displacement of
the generalized Fermi surface
$\displaystyle u(\xi^{I}_{F})=\sqrt{\sum_{I}(\Delta\xi^{I})^{2}},$ (62)
where $\xi^{I}_{F}$ parametrize the $2d-1$-dimensional generalized Fermi
surface, and $\Delta\xi^{I}$ describe the shift of the generalized Fermi
surface in the normal direction.
Let us introduce an integration over the generalized Fermi surface
$\displaystyle\int\text{Pf}[b(\xi^{I})]\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}=:\int\text{Pf}[b(\xi^{I})]\frac{\hskip
1.0pt\mathrm{d}^{2d}\xi}{(2\pi)^{d}}\,|\partial_{\xi^{I}}\bar{g}|$ (63)
The number of fermions in the collective excited state described by
$u(\xi^{I})$ is given by
$\displaystyle N$ $\displaystyle=\int\text{Pf}(b)\frac{\hskip
1.0pt\mathrm{d}^{2d}\xi}{(2\pi)^{d}}\,g(\xi^{I})$
$\displaystyle=\bar{N}+\int\text{Pf}(b)\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,u(\xi^{I}_{F})$ (64)
The energy of the collective excited state is given by
$\displaystyle E$ $\displaystyle=\bar{E}+\int\text{Pf}(b)\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,\frac{1}{2}|h_{.}|u^{2}(\xi^{I}_{F})$
(65)
where
$\displaystyle h_{I}=:\partial_{\xi^{I}}H,\ \ \ \ \
|h_{.}|=:\sqrt{\sum_{I}h_{I}^{2}}.$ (66)
The equation of the motion of $u(\xi^{I}_{F},t)$ can be obtained in two ways.
First, we note that $|h_{I}|u$ is the single particle energy, which is
invariant under the single particle time evolution $\xi^{I}_{F}(t)$ that
satisfies the single-particle equation of motion (53). Thus
$\displaystyle\frac{\hskip 1.0pt\mathrm{d}}{\hskip
1.0pt\mathrm{d}t}|h_{.}|u(\xi^{I}_{F}(t),t)=0,$ (67)
This allows us to obtain the equation of motion for $u(\xi^{I}_{F},t)$ field
using the single-particle equation of motion (53)
$\displaystyle\Big{(}\partial_{t}+h^{I}(\xi^{I}_{F})\partial_{\xi^{I}}\Big{)}|h_{.}|(\xi^{I}_{F})u(\xi^{I}_{F},t)=0,$
(68)
where
$\displaystyle h^{I}=b^{IJ}h_{J}$ (69)
and the repeated index $J$ is summed. Here $b^{IJ}$ is the matrix inversion of
of $b_{IJ}$:
$\displaystyle b_{IJ}b^{JK}=\delta_{IK}.$ (70)
Second, we note that $\text{Pf}(b)u$ is the density of fermions on the
generalized Fermi surface (see eqn. (IV.3)). The corresponding current density
is given by $h^{I}\text{Pf}(b)u$, since $\dot{\xi}^{I}=h^{I}$ (see eqn. (53)).
The fermion conservation gives us another equation of motion for
$u(\xi^{I}_{F},t)$:
$\displaystyle\partial_{t}\text{Pf}\big{(}b(\xi^{I}_{F})\big{)}u(\xi^{I}_{F},t)+\partial_{\xi^{I}}\Big{(}h^{I}(\xi^{I}_{F})\text{Pf}\big{(}b(\xi^{I}_{F})\big{)}u(\xi^{I}_{F},t)\Big{)}=0.$
(71)
Since the single-particle dynamics leads to the two equations, so they they
must be consistent. This requires that
$\displaystyle|h.|(\partial_{t}+\partial_{\xi^{I}}h^{I})\text{Pf}(b)u=\text{Pf}(b)(\partial_{t}+h^{I}\partial_{\xi^{I}})|h.|u.$
(72)
In other words, $h^{I}$, $|h.|$, and $\text{Pf}(b)$ are related, and they
satisfy
$\displaystyle\frac{|h.|}{\text{Pf}(b)}(\partial_{t}+\partial_{\xi^{I}}h^{I})\frac{\text{Pf}(b)}{|h_{.}|}=\partial_{t}+h^{I}\partial_{\xi^{I}}.$
(73)
Let us introduce a scalar field $\phi(\xi^{I}_{F},t)$ via
$\displaystyle-|h_{.}|^{-1}h^{I}\partial_{\xi^{I}_{F}}\phi(\xi^{I}_{F},t)=u(\xi^{I}_{F},t),$
(74)
The equation of motion for $\phi$ is given by
$\displaystyle(\partial_{t}+h^{I}\partial_{\xi^{I}})h^{J}\partial_{\xi^{I}_{F}}\phi=0,$
(75)
which can be simplified further as
$\displaystyle(\partial_{t}+h^{I}\partial_{\xi^{I}})\phi=0.$ (76)
since $h^{I}(\xi^{I}_{F})$ does not depend on time.
The above equation of motion and the expression of total energy (65) allow us
to determine the phase-space Lagrangian
$\displaystyle L_{\text{ph}}=-\int\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,\frac{\text{Pf}(b)}{2|h_{.}|}\Big{(}\dot{\phi}h^{I}\partial_{\xi^{I}_{F}}\phi+[h^{I}\partial_{\xi^{I}_{F}}\phi]^{2}\Big{)},$
(77)
up to total derivative topological terms.
To include topological terms, we assume a symmetry described by a map in phase
space
$\displaystyle\xi^{I}\to\bar{\xi}^{I},\ \ \ \
h^{I}(\xi^{I})=-h^{I}(\bar{\xi}^{I}),$ (78)
which generalize the $\boldsymbol{k}_{F}\to-\boldsymbol{k}_{F}$ symmetry used
before. The phase-space Lagrangian can now be written as
$\displaystyle L_{\text{ph}}$ $\displaystyle=\int\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,\Big{[}\frac{\bar{N}}{VA_{F}}\dot{\phi}(\xi^{I},t)$
$\displaystyle\ \ \ \ \ \ \
-\frac{\text{Pf}\big{(}b(\xi^{I})\big{)}}{2|h_{.}(\xi^{I})|}\Big{(}\dot{\phi}(\bar{\xi}^{I},t)h^{I}(\xi^{I})\partial_{\xi^{I}_{F}}\phi(\xi^{I},t)$
(79) $\displaystyle+$
$\displaystyle\dot{\phi}(\xi^{I},t)h^{I}(\xi^{I})\partial_{\xi^{I}_{F}}\phi(\xi^{I},t)+\big{[}h^{I}(\xi^{I})\partial_{\xi^{I}_{F}}\phi(\xi^{I},t)\big{]}^{2}\Big{)}\Big{]},$
where $\bar{N}$ is the number of fermnions in the ground state, and
$\displaystyle VA_{F}=\int\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,1$ (80)
is the total volume of generalized Fermi surface.
Fro the first three terms in eqn. (IV.3), we see that $\dot{\phi}(\xi^{I},t)$
directly couples to the total density of fermions
$\displaystyle N=\int\frac{\hskip
1.0pt\mathrm{d}^{2d-1}\xi_{F}}{(2\pi)^{d}}\,\Big{[}\frac{\bar{N}}{VA_{F}}$
$\displaystyle-\frac{\text{Pf}\big{(}b(\xi^{I})\big{)}}{2|h_{.}(\xi^{I})|}\Big{(}h^{I}(\xi^{I})\partial_{\xi^{I}_{F}}\phi(\xi^{I},t)$
$\displaystyle-h^{I}(\xi^{I})\partial_{\bar{\xi}^{I}_{F}}\phi(\bar{\xi}^{I},t)\Big{)}\Big{]}.$
(81)
In particular, the uniform part of $\dot{\phi}(\xi^{I},t)$ couple to total
number of fermions
$\displaystyle L_{\text{ph}}=N\dot{\phi}_{\text{uniform}}(t)+\cdots.$ (82)
This indicates that $\phi\sim\phi+2\pi$ is an angular field, and the $U(1)$
transformation is given by
$\displaystyle\phi(\xi^{I},t)\to\phi(\xi^{I},t)+\theta.$ (83)
### IV.4 Effective theory for a Fermi liquid with real space and
$\boldsymbol{k}$-space magnetic fields
Now, let us apply the above formalism to develop the low energy effective
theory of Fermi liquid, for 3-dimensional fermions with real space magnetic
field $\boldsymbol{A}(\boldsymbol{x})$ and $\boldsymbol{k}$-space “magnetic”
field $\widetilde{\boldsymbol{A}}(\boldsymbol{k})$. The dynamics of a single
fermion is described by eqn. (54) (with $V=0$). We have
$\displaystyle(h_{I})$ $\displaystyle=(0,{\boldsymbol{v}}_{F}),\ \ \ \ \
|h_{.}|=|\boldsymbol{v}_{F}|,\ \
{\boldsymbol{v}}_{F}=:\partial_{\boldsymbol{k}}H,$ $\displaystyle(h^{I})$
$\displaystyle=(\widetilde{\boldsymbol{v}}_{F},\boldsymbol{f}),\ \ \ \ \
\text{Pf}(b)=1-2\boldsymbol{B}\cdot\widetilde{\boldsymbol{B}}+\cdots,$ (84)
$\displaystyle\widetilde{\boldsymbol{v}}_{F}$
$\displaystyle={\boldsymbol{v}}_{F}-({\boldsymbol{v}}_{F}\times\boldsymbol{B})\times\widetilde{\boldsymbol{B}}+\cdots,\
\ \boldsymbol{f}=:{\boldsymbol{v}}_{F}\times\boldsymbol{B}+\cdots.$
Here $\boldsymbol{f}=\dot{\boldsymbol{k}}$ has a physical meaning as the force
acting on each fermion. $\widetilde{\boldsymbol{v}}_{F}=\dot{\boldsymbol{x}}$
has a physical meaning as the velocity of each fermion. When
$\widetilde{\boldsymbol{A}}\neq 0$, the velocity of a fermion at the Fermi
surface is not given by $\boldsymbol{v}_{F}=\partial_{\boldsymbol{k}}H$.
$\widetilde{\boldsymbol{v}}_{F}$ is also called the anomalous
velocity.Sundaram and Niu (1999)
Substitute the above into eqn. (IV.3), we obtain
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}$
$\displaystyle\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\Big{(}\frac{\bar{\rho}}{A_{F}}\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})-\frac{\text{Pf}(b)\dot{\phi}(\boldsymbol{x},-\boldsymbol{k}_{F})}{2|\boldsymbol{v}_{F}|}(\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)$
(85)
$\displaystyle-\frac{\text{Pf}(b)\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})}{2|\boldsymbol{v}_{F}|}(\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)-\frac{\text{Pf}(b)}{2|\boldsymbol{v}_{F}|}\big{[}(\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)\big{]}^{2}\Big{)}$
$\displaystyle\hskip-43.36243pt-\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\,\frac{\text{Pf}(b)\text{Pf}(b^{\prime})V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})}{2|\boldsymbol{v}_{F}||\boldsymbol{v}_{F}^{\prime}|}\Big{(}(\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)\Big{)}\Big{(}(\widetilde{\boldsymbol{v}}_{F}^{\prime}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}^{\prime}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)\Big{)}.$
Up to first order in $\boldsymbol{B}$ and $\widetilde{\boldsymbol{B}}$, the
above can be simplified
$\displaystyle L_{\text{ph}}=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}$
$\displaystyle\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\Big{(}\frac{\bar{\rho}}{A_{F}}\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})-\frac{\dot{\phi}(\boldsymbol{x},-\boldsymbol{k}_{F})}{2|\boldsymbol{v}_{F}|}(\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)$
(86)
$\displaystyle-\frac{\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})}{2|\boldsymbol{v}_{F}|}(\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)-\frac{1}{2|\boldsymbol{v}_{F}|}\big{[}(\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)\big{]}^{2}\Big{)}$
$\displaystyle\hskip-43.36243pt-\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\,\frac{V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})}{2|\boldsymbol{v}_{F}||\boldsymbol{v}_{F}^{\prime}|}\Big{(}(\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)\Big{)}\Big{(}(\boldsymbol{v}_{F}^{\prime}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}^{\prime}\cdot\partial_{\boldsymbol{k}_{F}})\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)\Big{)}.$
Here, we have assumed a central reflection symmetry
$\boldsymbol{k}\to-\boldsymbol{k}$, and the mapping $\xi^{I}\to\bar{\xi}^{I}$
is given by
$(\boldsymbol{x},\boldsymbol{k})\to(\boldsymbol{x},-\boldsymbol{k})$. We also
included the interaction term for the Fermi surface fluctuations, the
$V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})$ term, where
$\boldsymbol{v}_{F},\boldsymbol{v}_{F}^{\prime}$ are the Fermi velocities at
$\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime}$.
Note that the fermion density at the Fermi surface $\boldsymbol{k}_{F}$ is
given by (see eqn. (74))
$\displaystyle
u(\boldsymbol{x},\boldsymbol{k}_{F})=-\frac{\boldsymbol{v}_{F}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)-\frac{\boldsymbol{f}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{k}_{F}}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t),$
(87)
and thus the total fermion number density is given by
$\displaystyle\rho=\bar{\rho}-\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}}{(2\pi)^{d}}\,\Big{(}\frac{\boldsymbol{v}_{F}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{x}}\phi+\frac{\boldsymbol{f}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{k}_{F}}\phi\Big{)}.$
(88)
This expression helps us to understand why the interaction term for the Fermi
surface fluctuations has a form given in eqn. (86).
The above expression also allows us to see that
$\dot{\phi}(\boldsymbol{x},\boldsymbol{k}_{F})$ couples to the total fermion
density (see the first three terms in eqn. (86)). Thus
$\phi(\boldsymbol{x},\boldsymbol{k}_{F})\sim\phi(\boldsymbol{x},\boldsymbol{k}_{F})+2\pi$
is an angular field and the $U(1)$ symmetry transformation is given by
$\displaystyle\phi(\boldsymbol{x},\boldsymbol{k}_{F})\to\phi(\boldsymbol{x},\boldsymbol{k}_{F})+\theta.$
(89)
Eqn. (73) now becomes (to the first order in $\boldsymbol{B}$ and
$\widetilde{\boldsymbol{B}}$)
$\displaystyle\partial_{\boldsymbol{x}}\cdot\boldsymbol{v}_{F}+\partial_{\boldsymbol{k}_{F}}\cdot\boldsymbol{f}=|\boldsymbol{v}_{F}|^{-1}\big{(}\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}|\boldsymbol{v}_{F}|$
(90)
This will help us to compute the equation of motion for the $\phi$ field. The
resulting equation of motion is given by (written in terms of
$u(\boldsymbol{x},\boldsymbol{k}_{F},t)$)
$\displaystyle\big{(}\partial_{t}+\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}|\boldsymbol{v}_{F}|u(\boldsymbol{x},\boldsymbol{k}_{F},t)+\big{(}\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})u(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)=0.$
(91)
The above is the Boltzmann equation which can be use to compute the transport
properties after adding the collision terms. In terms of
$\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)$, we have
$\displaystyle\big{(}\partial_{t}+\boldsymbol{v}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)+\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\frac{V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})}{|\boldsymbol{v}_{F}^{\prime}|}\big{(}\boldsymbol{v}_{F}^{\prime}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)+\boldsymbol{f}^{\prime}\cdot\partial_{\boldsymbol{k}_{F}}\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)\big{)}=0.$
(92)
The above equations of motion are valid only to the first order in
$\boldsymbol{B}$ and $\widetilde{\boldsymbol{B}}$. The exact equations of
motion, in several different forms, are given by
$\displaystyle\big{(}\partial_{t}+\partial_{\boldsymbol{x}}\cdot\widetilde{\boldsymbol{v}}_{F}+\partial_{\boldsymbol{k}_{F}}\cdot\boldsymbol{f}\big{)}\text{Pf}(b)u(\boldsymbol{x},\boldsymbol{k}_{F},t)+\text{Pf}(b)\big{(}\frac{\widetilde{\boldsymbol{v}}_{F}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{x}}+\frac{\boldsymbol{f}}{|\boldsymbol{v}_{F}|}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\text{Pf}(b^{\prime})V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})u(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)=0,$
$\displaystyle\big{(}\partial_{t}+\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}|\boldsymbol{v}_{F}|u(\boldsymbol{x},\boldsymbol{k}_{F},t)+\big{(}\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\text{Pf}(b^{\prime})V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})u(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)=0,$
(93)
$\displaystyle\big{(}\partial_{t}+\widetilde{\boldsymbol{v}}_{F}\cdot\partial_{\boldsymbol{x}}+\boldsymbol{f}\cdot\partial_{\boldsymbol{k}_{F}}\big{)}\phi(\boldsymbol{x},\boldsymbol{k}_{F},t)+\int\frac{\hskip
1.0pt\mathrm{d}^{d-1}\boldsymbol{k}_{F}^{\prime}}{(2\pi)^{d}}\,\frac{\text{Pf}(b^{\prime})}{|\boldsymbol{v}_{F}^{\prime}|}V(\boldsymbol{k}_{F},\boldsymbol{k}_{F}^{\prime})\big{(}\boldsymbol{v}_{F}^{\prime}\cdot\partial_{\boldsymbol{x}}\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)+\boldsymbol{f}^{\prime}\cdot\partial_{\boldsymbol{k}_{F}}\phi(\boldsymbol{x},\boldsymbol{k}_{F}^{\prime},t)\big{)}=0.$
### IV.5 Emergent $U^{\infty}(1)$ symmetry
From the effective theory (85), we see when there is no real space magnetic
field $\boldsymbol{B}=0$, we have $\boldsymbol{f}=0$ and the effective theory
has a $U^{\infty}(1)$ symmetry generated by
$\displaystyle\phi(\boldsymbol{x},\boldsymbol{k}_{F})\to\phi(\boldsymbol{x},\boldsymbol{k}_{F})+\theta(\boldsymbol{k}_{F}),$
(94)
where $\theta(\boldsymbol{k}_{F})$ can be any function of
$\boldsymbol{k}_{F}$. This is the so called emergent $U^{\infty}(1)$ symmetry,
which a key character of Fermi liquid.Luther (1979); Haldane (1992); Houghton
and Marston (1993); Castro Neto and Fradkin (1994); Else _et al._ (2020) We
also see that the above transformation is no longer a symmetry in the presence
of real space magnetic field $\boldsymbol{B}\neq 0$. This may be related to
the anomaly in the emergent $U^{\infty}(1)$ symmetry discussed in Ref. Else
_et al._ , 2020
## V Fermion-pair liquid and the mixed $U(1)\times\mathbb{R}^{d}$ anomaly
In this section, we are going to consider a fermion system in $d$-dimensional
continuous space, with $U(1)$ particle-number-conservation symmetry and
$\mathbb{R}^{d}$ translation symmetry. We assume the space to have a size
$L_{1}\times L_{2}\times\cdots\times L_{d}$, and has a periodic boundary
condition. We will compute distribution of the total momentum for many-body
low energy excitations $\boldsymbol{k}_{\text{tot}}$, and how such a
distribution depends on the $U(1)$ symmetry twist described by a constant
vector potential $\boldsymbol{a}$.
Using the results from the Appendix A, we find that the low energy effective
theory is described by the following phase-space Lagrangian (see eqn. (A.2))
$\displaystyle L=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}\;$
$\displaystyle\big{(}\bar{\rho}_{p}\dot{\phi}_{p}(\boldsymbol{x},t)+\delta\rho_{p}(\boldsymbol{x},t)\dot{\phi}_{p}(\boldsymbol{x},t)$
$\displaystyle\
-\frac{\bar{\rho}}{2M_{p}}|\boldsymbol{\partial}\phi_{p}|^{2}-\frac{g}{2}\delta\rho_{p}^{2}+\
\cdots\big{)},$ (95)
where $\bar{\rho}_{p}=\frac{1}{2}\bar{\rho}$ is the fermion-pair density in
the ground state, and $\phi_{p}\sim\phi_{p}+2\pi$ is the angular field for the
fermion-pair. The total energy and total crystal momentum of those excitations
are given by
$\displaystyle E$
$\displaystyle=\frac{\bar{N}_{p}}{2M_{p}}\sum_{\mu}(\frac{m_{\mu}}{L_{\mu}}+2a_{\mu})^{2}+\sum_{\boldsymbol{k}\neq
0}(n_{\boldsymbol{k}}+\frac{1}{2})v|\boldsymbol{k}|,$
$\displaystyle\boldsymbol{k}_{\text{tot}}$
$\displaystyle=N_{p}\sum_{\mu}(\frac{m_{\mu}}{L_{\mu}}+2a_{\mu})\hat{\boldsymbol{x}}_{\mu}+\sum_{\boldsymbol{k}\neq
0}n_{\boldsymbol{k}}\boldsymbol{k},$ (96)
We see that the $U(1)$ symmetry twist $\boldsymbol{a}$ induces a change in the
total momentum
$\displaystyle\boldsymbol{k}_{\text{tot}}=2N_{p}\boldsymbol{a}=2\bar{\rho}_{p}\boldsymbol{a}V=\bar{\rho}\boldsymbol{a}V.$
(97)
Such momentum dependence of the $U(1)$ symmetry twist $\boldsymbol{a}$
reflects the mixed $U(1)\times\mathbb{R}^{d}$ anomaly. Eqn. (97) and eqn. (50)
are identical, implies the identical mixed anomaly, which is captured by the
topological term $\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\,\big{(}\bar{\rho}_{p}\dot{\phi}_{p}(\boldsymbol{x},t)$.
The mixed anomaly can also be measured by the periodicy $k_{0\mu}$ in the
distribution of $\boldsymbol{k}_{\text{tot}}$, and the periodicy $\Delta N=2$
in the distribution of $N$, for the low energy excitations. The period in
$\mu$-direction times $\Delta N$ is
$\displaystyle\Delta Nk_{0\mu}=2\frac{N_{p}}{L_{\mu}}=\frac{N}{L_{\mu}}.$ (98)
The periodices $k_{0\mu}$ and $\Delta N$ are universal low energy properties
of the fermion-pair gapless phase. Their product $\Delta Nk_{0\mu}$ is even
more robust, since it is invariant even across any phase transitions, and thus
correspond to an anomaly.
For the gapless state formed by four-fermion bound states, the periodicies
will be
$\displaystyle k_{0\mu}=\frac{1}{4}\frac{N}{L_{\mu}},\ \ \ \ \ \ \ \ \Delta
N=4.$ (99)
Their product is still $\Delta Nk_{0\mu}=\frac{N}{L_{\mu}}$.
I would like to thank Maissam Barkeshli, Dominic Else, and Senthil Todadri for
discussions and comments. This research is partially supported by NSF
DMR-2022428 and by the Simons Collaboration on Ultra-Quantum Matter, which is
a grant from the Simons Foundation (651440).
## Appendix A Dynamical variational approach and low energy effective theory
### A.1 Coherent state approach
A quantum state is described by a complex vector
$|\psi\rangle=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\ \vdots\\\ \end{pmatrix}$
in a Hilbert state, with inner product
$\displaystyle\langle\phi|\psi\rangle=\sum_{n}\phi^{*}_{n}\psi_{n}.$ (100)
The motion of a quantum state is described by time dependent vector:
$|\psi(t)\rangle$, which satisfy an equation of motion (called Shrödinger
equation) with only first order time derivative:
$\displaystyle\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\hskip
1.0pt\mathrm{d}}{\hskip
1.0pt\mathrm{d}t}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle,$ (101)
where the hermitian operator $\hat{H}$ is the Hamiltonian.
The Shrödinger equation (101) also has a phase-space Lagrangian description.
If we choose the Lagrangian $L$ to be
$\displaystyle L(\frac{\hskip 1.0pt\mathrm{d}}{\hskip
1.0pt\mathrm{d}t}|\psi\rangle,|\psi\rangle)$
$\displaystyle=\langle\psi(t)|\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\hskip
1.0pt\mathrm{d}}{\hskip 1.0pt\mathrm{d}t}-\hat{H}|\psi(t)\rangle$
$\displaystyle=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\psi_{n}^{*}(t)\dot{\psi}_{n}(t)-\psi_{m}^{*}(t)H_{mn}\psi_{n}(t),$ (102)
then the action $S=\int\hskip 1.0pt\mathrm{d}t\,L(\frac{\hskip
1.0pt\mathrm{d}}{\hskip 1.0pt\mathrm{d}t}|\psi\rangle,|\psi\rangle)$ will be a
functional for the paths $|\psi(t)\rangle$ in the Hilbert space. The
stationary paths $|\psi_{\text{sta}}(t)\rangle$ of the action will correspond
to the solutions of the Shrödinger equation. Since the Shrödinger equation can
be derived from the Lagrangian, we can say that the Lagrangian $L(\frac{\hskip
1.0pt\mathrm{d}}{\hskip 1.0pt\mathrm{d}t}|\psi\rangle,|\psi\rangle)$ provides
a complete description of a quantum system.
In the variational approach to the ground state, we consider a variational
state $|\psi_{\xi^{i}}\rangle$ that depends on variational parameters
$\xi^{i}$. We then found an approximation of the ground state
$|\psi_{\bar{\xi}^{i}}\rangle$ by choosing $\bar{\xi}^{i}$ that minimize the
average energy
$\displaystyle\bar{H}(\xi^{i})=\langle\psi_{\xi^{i}}|\hat{H}|\psi_{\xi^{i}}\rangle.$
(103)
If we choose the variational parameters $\xi^{i}$ properly, the low energy
excitations are also described by the fluctuations of variational parameters.
In other words, the dynamics of the variational parameters $\xi^{i}$ described
the low energy excitations. This leads to a dynamical variational approach (or
coherent state approach) that gives us a description of both ground state and
low energy excitations.
The dynamics of the full quantum system is described by phase-space Lagrangian
$L=\langle\psi|\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\hskip
1.0pt\mathrm{d}}{\hskip 1.0pt\mathrm{d}t}-\hat{H}|\psi\rangle$. The dynamics
of the variational parameters is described by the evolution of the quantum
states in a submanifold of the total Hilbert space, given by the variational
states $|\psi_{\xi^{i}}\rangle$. Here we want to obtain the dynamics of the
quantum states, restricted to the submanifold parametrized by $\xi^{i}$. Such
a dynamics is described by the same phase-space Lagrangian restricted in the
submanifold:
$L(\dot{\xi}^{i},\xi^{i})=\langle\psi_{\xi^{i}(t)}|\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{\hskip 1.0pt\mathrm{d}}{\hskip
1.0pt\mathrm{d}t}-\hat{H}|\psi_{\xi^{i}(t)}\rangle=a_{i}(\xi^{i})\dot{\xi}^{i}-\bar{H}(\xi^{i})$
(104)
where
$a_{i}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\langle\psi_{\xi^{i}}|\frac{\partial}{\partial\xi^{i}}|\psi_{\xi^{i}}\rangle,\
\ \ \ \ \
\bar{H}(\xi^{i})=\langle\psi_{\xi^{i}}|\hat{H}|\psi_{\xi^{i}}\rangle$ (105)
The resulting equation of motion is given by
$b_{ij}\dot{\xi}^{j}=\frac{\partial\bar{H}}{\partial\xi^{i}},\ \ \ \ \ \ \
b_{ij}=\partial_{i}a_{j}-\partial_{j}a_{i}$ (106)
which describes the classical motion of $\xi^{i}$.
The above phase-space Lagrangian actually only described the classical
dynamics of the variables $\xi^{i}$. The obtain the low energy effective
theory for the quantum dynamics of the variables $\xi^{i}$, we need quantize
the phase-space Lagrangian (104) to obtain the low energy effective Hilbart
space ${\cal H}_{\text{eff}}$ and the low energy effective Hamiltonian
$H_{\text{eff}}$ acting with ${\cal H}_{\text{eff}}$. Roughly, the low energy
effective Hilbart space ${\cal H}_{\text{eff}}$ is an representation of the
operators algebra
$\displaystyle\hskip 1.0pt\mathrm{i}\hskip
1.0pt[\hat{\xi}^{i},\hat{\xi}^{j}]=b^{ij}(\hat{\xi}^{i}),$ (107)
where $b^{ij}$ is the inverse of $b_{ij}$: $b_{ij}b^{jk}=\delta_{ik}$. The low
energy effective Hamiltonian is given by
$\displaystyle H_{\text{eff}}=\bar{H}(\hat{\xi}^{i}).$ (108)
Let us use the above approach to describe hardcore boson on a single site. The
total Hilbert space is 2-dimensional, spanned by $|0\rangle$ (no boson) and
$|1\rangle$ (one boson). The coherent state is described by a unit vector
$\boldsymbol{n}=(n_{x},n_{y},n_{z})\in S^{2}$. We may also use $\theta,\phi$
to describe $\boldsymbol{n}$:
$\displaystyle n_{x}(\theta,\phi)$ $\displaystyle=\cos\phi\sin\theta,$
$\displaystyle n_{x}(\theta,\phi)$ $\displaystyle=\sin\phi\sin\theta,$
$\displaystyle n_{z}(\theta,\phi)$ $\displaystyle=\cos\theta.$ (109)
We can choose the coherent state to be
$\displaystyle|\boldsymbol{n}(\theta,\phi)\rangle=\begin{pmatrix}\cos\frac{\theta}{2}\\\
\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\phi}\sin\frac{\theta}{2}\\\ \end{pmatrix}$ (110)
The phase-space Lagrangian to describe the classical dynamics of $\theta,\phi$
is given by
$\displaystyle L$
$\displaystyle=\sin^{2}\frac{\theta}{2}\dot{\phi}-\bar{H}(\theta,\phi)$
$\displaystyle=\rho\dot{\phi}-\bar{H}(\rho,\phi),\ \ \ \ \
\rho\equiv\frac{1-n_{z}}{2}=\sin^{2}\frac{\theta}{2}$ (111)
We will use $(\rho,\phi)$ to parametrize the phase space, where $\rho\in[0,1]$
has a physical meaning being the average number of bosons on the site. Here we
stress that $(\rho,\phi)$ parametrize $S^{2}$. Quantizing the above classical
phase-space Lagrangian, we suppose to obtain a quantum system with Hilbert
space ${\cal H}=\text{span}(|0\rangle,|1\rangle)$.
### A.2 Low energy effective theory of bosonic superfluid phase
Using the above result, we obtain the following low energy effective theory
for interacting bosons in a $d$-dimensional cubic lattice with periodic
boundary condition, whose sites are labeled by $\boldsymbol{i}$:
$\displaystyle
L=\sum_{\boldsymbol{i}}\rho_{\boldsymbol{i}}\dot{\phi}_{\boldsymbol{i}}-\bar{H}(\rho_{\boldsymbol{i}},\phi_{\boldsymbol{i}}),$
(112)
where $\phi_{\boldsymbol{i}}\sim\phi_{\boldsymbol{i}}+2\pi$ is an angular
variable. If we assume $\rho_{\boldsymbol{i}},\phi_{\boldsymbol{i}}$ to have a
smooth dependence on the space coordinate $\boldsymbol{x}\sim\boldsymbol{i}$,
the above can be rewritten as a field theory
$\displaystyle L=\int\hskip 1.0pt\mathrm{d}^{d}\boldsymbol{x}\;$
$\displaystyle\big{(}\bar{\rho}\dot{\phi}(\boldsymbol{x},t)+\delta\rho(\boldsymbol{x},t)\dot{\phi}(\boldsymbol{x},t)$
$\displaystyle\
-\frac{\bar{\rho}}{2m}|\boldsymbol{\partial}\phi|^{2}-\frac{g}{2}\delta\rho^{2}+\
\cdots\big{)},$ (113)
where we have assumed that $\bar{H}(\rho,\phi)$ is minimized as
$\rho=\bar{\rho}$ which correspond to average number of bosons per site.
To quantize the above low-energy-effective theory, we expand
$\displaystyle\delta\rho$ $\displaystyle=\rho_{0}+\sum_{\boldsymbol{k}\neq
0}\rho_{\boldsymbol{k}}\frac{\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\boldsymbol{k}\cdot\boldsymbol{x}}}{L^{d/2}}$
$\displaystyle\phi$
$\displaystyle=\phi_{0}+2\pi\frac{\boldsymbol{m}\cdot\boldsymbol{x}}{L}+\sum_{\boldsymbol{k}\neq
0}\phi_{\boldsymbol{k}}\frac{\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\boldsymbol{k}\cdot\boldsymbol{x}}}{L^{d/2}},$
(114)
where $L$ is the size of the cubic lattice. The $\boldsymbol{k}\neq 0$ modes
give rise to a collection of quantum oscillators after quantization.
$\rho_{0},\phi_{0},\boldsymbol{m}$ describe the $\boldsymbol{k}=0$ mode, where
the integer vector $\boldsymbol{m}=(m_{1},\cdots,m_{d})$ describes the winding
numbers of the phase $\phi$. The effective Lagrangian for the
$\boldsymbol{k}=0$ modes is given by
$\displaystyle
L_{0}=\big{(}L^{d}\bar{\rho}\dot{\phi}_{0}+L^{d}\rho_{0}\dot{\phi}_{0}-\frac{\bar{\rho}L^{d-2}}{2m}(2\pi)^{2}\boldsymbol{m}^{2}-L^{d}\frac{g}{2}\rho_{0}^{2}\big{)}.$
(115)
After quantization, the $\boldsymbol{k}=0$ mode describes a particle on a ring
with $L^{d}\bar{\rho}$ flux through the ring. Let $\hat{L}_{z}\sim
L^{d}(\bar{\rho}+\rho_{0})$ be the angular operator of the quantized particle.
After quantization, the Hamiltonian is given by
$\displaystyle\hat{H}_{0}=\frac{\bar{\rho}L^{d-2}}{2m}(2\pi)^{2}\boldsymbol{m}^{2}+\frac{g}{2L^{d}}(\hat{L}_{z}-L^{d}\bar{\rho})^{2}$
(116)
The many-body low energy excitations are labeled by
$(\boldsymbol{m},N,n_{\boldsymbol{k}\neq 0})$, where integer $N$ is the
eigenvalues of $\hat{L}_{z}$ (the total number of bosons) and integer
$n_{\boldsymbol{k}\neq 0})$ is the number of excited phonons for
$\boldsymbol{k}$ mode. The total energy and the total crystal momentum are
given by
$\displaystyle
E=\bar{N}\frac{(2\pi)^{2}\boldsymbol{m}^{2}/L^{2}}{2m}+\frac{g}{2}\frac{(N-\bar{N})^{2}}{L^{d}}+\sum_{\boldsymbol{k}\neq
0}(n_{\boldsymbol{k}}+\frac{1}{2})v|\boldsymbol{k}|^{2},$
$\displaystyle\boldsymbol{k}_{\text{tot}}=\int\hskip
1.0pt\mathrm{d}^{d}\boldsymbol{x}\;(\bar{\rho}+\delta\rho)\boldsymbol{\partial}\phi=N\frac{2\pi\boldsymbol{m}}{L}+\sum_{\boldsymbol{k}\neq
0}n_{\boldsymbol{k}}\boldsymbol{k}$ (117)
where the phonon velocity $v=\sqrt{\frac{g\bar{\rho}}{m}}$, and $\bar{N}\equiv
L^{d}\bar{\rho}_{0}$.
The above results are very standard, except that we carefully keep the
topological term $\bar{\rho}(\boldsymbol{x},t)\dot{\phi}(\boldsymbol{x},t)$ in
the Lagrangian. Our quantum system has $U(1)$ particle number conservation
symmetry and $\mathbb{Z}^{d}$ lattice translation symmetry. In the continuum
field theory (A.2), we take the limit of zero lattice spacing. In this case,
both the $U(1)$ and $\mathbb{Z}^{d}$ symmetries are internal symmetries of the
field theory. It turns out that the $U(1)\times\mathbb{Z}^{d}$ symmetry has a
mixed ’t Hooft anomaly when $\bar{\rho}$ is not an integer, which constrains
the low energy dynamics of the interacting bosons.
## References
* Kim _et al._ (1995) Y. B. Kim, P. A. Lee, and X.-G. Wen, Phys. Rev. B 52, 17275 (1995), arXiv:cond-mat/9504063 .
* Luther (1979) A. Luther, Phys. Rev. B 19, 320 (1979).
* Haldane (1992) F. D. M. Haldane, Helv. Phys. Acta. 65, 152 (1992), arXiv:cond-mat/0505529 .
* Houghton and Marston (1993) A. Houghton and J. B. Marston, Phys. Rev. B 48, 7790 (1993).
* Castro Neto and Fradkin (1994) A. H. Castro Neto and E. Fradkin, Phys. Rev. Lett. 72, 1393 (1994).
* Luttinger and Ward (1960) J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960).
* Oshikawa (2000) M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000), arXiv:cond-mat/0002392 .
* Lieb _et al._ (1961) E. H. Lieb, T. D. Schultz, and D. C. Mattis, Ann. Phys. (N.Y) 16, 407 (1961).
* Hastings (2004) M. B. Hastings, Phys. Rev. B 69, 104431 (2004), arXiv:cond-mat/0305505 .
* Furuya and Oshikawa (2017) S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 118, 021601 (2017), arXiv:1503.07292 .
* Cheng _et al._ (2016) M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, Physical Review X 6, 041068 (2016), arXiv:1511.02263 .
* Po _et al._ (2017) H. C. Po, H. Watanabe, C.-M. Jian, and M. P. Zaletel, Phys. Rev. Lett. 119, 127202 (2017), arXiv:1703.06882 .
* Lu _et al._ (2020) Y.-M. Lu, Y. Ran, and M. Oshikawa, Annals of Physics 413, 168060 (2020), arXiv:1705.09298 .
* Lu (2017) Y.-M. Lu, (2017), arXiv:1705.04691 .
* Cheng (2019) M. Cheng, Phys. Rev. B 99, 075143 (2019), arXiv:1804.10122 .
* Jiang _et al._ (2019) S. Jiang, M. Cheng, Y. Qi, and Y.-M. Lu, (2019), arXiv:1907.08596 .
* Chen _et al._ (2011) X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011), arXiv:1008.3745 .
* Else _et al._ (2020) D. V. Else, R. Thorngren, and T. Senthil, , arXiv:2007.07896 (2020), arXiv:2007.07896 .
* Thouless _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
* Sundaram and Niu (1999) G. Sundaram and Q. Niu, Phys. Rev. 59, 14915 (1999), arXiv:cond-mat/9908003 .
* Xiao _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010), arXiv:0907.2021 .
* ’t Hooft (1980) G. ’t Hooft, in _Recent Developments in Gauge Theories. NATO Advanced Study Institutes Series (Series B. Physics)_, Vol. 59, edited by G. ’t Hooft et al. (Springer, Boston, MA., 1980) pp. 135–157.
* Wen (2013) X.-G. Wen, Phys. Rev. D 88, 045013 (2013), arXiv:1303.1803 .
* Kong and Wen (2014) L. Kong and X.-G. Wen, (2014), arXiv:1405.5858 .
* Ji and Wen (2020) W. Ji and X.-G. Wen, Phys. Rev. Research 2, 033417 (2020), arXiv:1912.13492 .
* Wen (1992) X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992).
* Wen (1995) X.-G. Wen, Adv. Phys. 44, 405 (1995), arXiv:cond-mat/9506066 .
|
We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.
§ INTRODUCTION
Quandles are non-associative algebraic structures that are modeled on the three Reidemester moves in classical knot theory. Thus, they are appropriate algebraic structures for constructing invariants of knots and links in $3$-space and knotted surfaces in $4$-space. Quandles were introduced independently by Joyce <cit.> and Matveev <cit.> in the 1980s. Quandles have been investigated in many areas of mathematics such as quasigroups and Moufang loops <cit.>, the Yang-Baxter equation <cit.>, representation theory <cit.>, and ring theory <cit.>. For more information on quandles, the reader is advised to consult the book <cit.>. Knot theory has been extended in several directions, for example, singular knot theory. In <cit.>, connections between Jones type invariants defined in <cit.> and Vassiliev invariants of singular knots defined in <cit.> were established. In <cit.>, a variation of the Hecke algebra was used to construct a Jones-type invariant for singular knots. Combinatorial singular knot theory has the so-called generalized Reidemeister moves <cit.>. These generalized moves were used in <cit.> to extend the concept of quandle to an algebraic structure called singquandle to provide invariants for singular knots. In <cit.>, generating sets of the generalized Reidemeister moves for oriented singular links were introduced and used to distinguish singular knots and links. In <cit.>, the quandle cocycle invariant <cit.> was extended to oriented singular knots and used to construct a state sum invariant for singular links. Furthermore in <cit.>, the quandle polynomial invariant was extended to the case of singquandles. A singular link invariant was constructed from the singquandle polynomial and it was shown to generalize the singquandle counting invariant in <cit.>.
The article is organized as follows. In Section <ref>, the basics of quandle theory are reviewed, and some examples are given. Section <ref> provides a review of the basic constructions of oriented singquandles as well as the singquandle counting invariant. In Section <ref>, we discuss the singquandle polynomial and the subsingquandle polynomial, which we introduced in a previous paper <cit.> and used to define a singular link invariant. Section <ref> defines singquandle shadows which is used to generalize the shadow colorings of knot diagrams by quandles previously defined in CN. Furthermore, the shadow singquandle polynomial and the singquandle shadow polynomial invariant for a singular link $L$ is defined. In Section <ref>, the shadow counting invariant is used to define an enhanced shadow link invariant by combining the shadow singquandle counting invariant and the shadow singquandle polynomial. Lastly, Section <ref> examines the strength of the singquandle shadow polynomial invariant. Two examples are provided to illustrate that the singquandle shadow polynomial is sensitive to differences in singular links not detected by the singquandle coloring invariant and singquandle polynomial invariant.
§ BASICS OF QUANDLES
In this paper we will consider only finite quandles and singquandles.
We'll review the basics of quandles; more details on the topic can be found in EN, Joyce, Matveev.
A set $X$ with binary operation $\ast$ is called a quandle if the following three identities are satisfied.
\begin{eqnarray*}
& &\mbox{\rm (i) \ } \mbox{\rm For all $x \in X$,
$x* x =x$.} \label{axiom1} \\
& & \mbox{\rm (ii) \ }\mbox{\rm For all $y,z \in X$, there is a unique $x \in X$ such that
$ x*y=z$.} \label{axiom2} \\
& &\mbox{\rm (iii) \ }
\mbox{\rm For all $x,y,z \in X$, we have
$ (x*y)*z=(x*z)*(y*z). $} \label{axiom3}
\end{eqnarray*}
From Axiom (ii) of Definition <ref> we can write the element $x$ as $z \bar{*} y = x$. Notice that this operation $\bar{*}$ defines a quandle structure on $X$. The axioms of a quandle correspond respectively to the three Reidemeister moves of types I, II and III (see <cit.> for examples). In fact, one of the motivations of defining quandles came from knot diagrammatic.
A quandle homomorphism between two quandles $(X,*)$ and $(Y,\triangleright)$ is a map $f: X \rightarrow Y$ such that $f(x *y)=f(x) \triangleright f(y) $, where
$*$ and $\triangleright$
denote respectively the quandle operations of $X$ and $Y$. Furthermore, if $f$ is a bijection, then it is called a
quandle isomorphism between $X$ and $Y$.
Some typical examples of quandles:
* Any non-empty set $X$ with the operation $x*y=x,$ for all $x,y \in X,$ is
a quandle called a trivial quandle.
* Any group $X=G$ with conjugation $x*y=y^{-1} xy$ is a quandle.
* Let $G$ be an abelian group.
For elements
$x,y \in G$,
$x*y \equiv 2y-x$.
Then $\ast$ defines a quandle
structure on $G$ called Takasaki quandle. In case $G=\mathbb{Z}_n$ (integers mod $n$) the quandle is called dihedral quandle.
This quandle can be identified with the
set of reflections of a regular $n$-gon
with conjugation
as the quandle operation.
* Any $\Lambda = (\mathbb{Z }[T^{\pm 1}])$-module $M$
is a quandle with
$x*y=Tx+(1-T)y$, $x,y \in M$, called an Alexander quandle.
§ REVIEW OF THE QUANDLE POLYNOMIAL
In this section we recall the definition of the quandle polynomial, subquandle polynomial, and the link invariants obtained from the subquandle polynomial. For a detail construction of these polynomials, see EN,N.
Let $(Q,*)$ be a finite quandle. For any element $x \in Q$, let
\[ C(x) = \lbrace y \in Q \, : \, y *x = y \rbrace \quad \text{and} \quad R(x) = \lbrace y \in Q \, : \, x *y = x \rbrace \]
and set $r(x) = \vert R(x) \vert$ and $c(x) = \vert C(x) \vert$. Then the quandle polynomial of Q, $qp_Q(s,t)$, is
\[ qp_Q(s,t) = \sum_{x \in Q} s^{r(x)}t^{c(x)}.\]
In <cit.>, the quandle polynomial was shown to be an effective invariant of finite quandles. In addition to being an invariant of finite quandles, the quandle polynomial was generalized to give information about how a subquandle is embedded in a larger quandle.
Let $S \subset Q$ be a subquandle of $Q$. The subquandle polynomial of S, $qp_{S \subset Q}(s,t)$, is
\[qp_{S \subset Q}(s,t) = \sum_{x \in S} s^{r(x)}t^{c(x)}\]
where $r(x)$ and $c(x)$ are defined above.
Note that for any knot or link $K$, there is an associated fundamental quandle, $Q(K)$, and for any given finite quandle $T$ the set of quandle homomorphism, denoted by $\textup{Hom}(Q(K),T)$, has been used to define computable link invariants, for example, the cardinality of the set is known as the quandle counting invariant. In <cit.>, the subquandle polynomial of the image of each homomorphism was used to enhance the counting invariant.
Let $K$ be a link and $T$ a finite quandle. Then for every $f \in \textup{Hom}(Q(K),T)$, the image of $f$ is a subquandle of $T$. The subquandle polynomial invariant $\Phi_{qp}(K,T)$ be the set with multiplicities
\[ \Phi_{qp}(K,T) = \lbrace qp_{\textup{Im}(f) \subset T} (s,t) \, \vert \, f \in \textup{Hom}(Q(K),T) \rbrace,\]
or the multiset can be represented in polynomial form
\[ \phi_{qp}(K,T) = \sum_{f \in \textup{Hom}(Q(K),T)} u^{qp_{\textup{Im}(f) \subset T} (s,t)}.\]
§ ORIENTED SINGQUANDLES AND THE COUNTING INVARIANT
We will provide a basic overview of an oriented singquandle as well as the singquandle counting invariant. For a detailed construction of oriented singquandle and the singquandle counting invariant, see BEHY, CEHN. We will be adopting
the following conventions at classical and singular crossings.
[use Hobby shortcut, scale=1.5, add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=1,>=stealth]>]
consider self intersections=true,
clip width=5,
ignore endpoint intersections=false,
(1,1) ..(-1,-1.1)[add arrow];
(-1,1) ..(1,-1.1)[add arrow];
(2,1) ..(4,-1.1)[add arrow];
(4,1) ..(2,-1.1)[add arrow];
(5,1) ..(7,-1.1)[add arrow];
(7,1) ..(5,-1.1)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (6,-.05) ;
[above] at (-1,1) $x$;
[above] at (1,1) $y$;
[below] at (1,-1.1) $x*y$;
[below] at (-1,-1.1) $y$;
[above] at (2,1) $y$;
[above] at (4,1) $x$;
[below] at (2,-1.1) $x\bar{*}y$;
[below] at (4,-1.1) $y$;
[above] at (7,1) $y$;
[above] at (5,1) $x$;
[below] at (5,-1.1) $R_1(x,y)$;
[below] at (7,-1.1) $R_2(x,y)$;
[below] at (0,-1.5) $(a)$;
[below] at (3,-1.5) $(b)$;
[below] at (6,-1.5) $(c)$;
(a) Coloring of arcs at a positive crossing, (b) coloring of arcs at a negative crossing, (c) colorings of semi-arcs at a singular crossing.
Generating sets of oriented singular Reidemeister moves were studied and were used to define oriented singquandles in <cit.>. We will follow the naming convention for oriented singular Reidemeister moves used in <cit.>. Note that if we let $(S, *)$ be a quandle, we only need to consider the colorings from singular Reidemeister moves in Figures <ref>, <ref>, and <ref>.
[use Hobby shortcut,, scale=1, add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=1,>=stealth]>]
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
flip crossing/.list=2,3
(-1,2) .. (2,-2)[add arrow];
(2,2) .. (-1,-2)[add arrow];
(.7,-2.4)..(0,-1.5).. (-1,0)..(0,1.5)..(.7,2.4)[add arrow];
[left] at (-1,2) $a$;
[right] at (0,.8)$a\bar{*}b$;
[left] at (-1,-2) $R_1(a\bar{*}b,c)*b$;
[right] at (2,2) $c$;
[left] at (.5,-2) $b$;
[right] at (2,-2) $R_2(a \bar{*}b,c)$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (.5,0) ;
[very thick, <->] (3,0) – (4,0);
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
flip crossing/.list=2,3
(8,2) .. (5,-2)[add arrow];
(5,2).. (8,-2)[add arrow];
(6.3,-2.4).. (7,-1.5).. (8,0).. (7,1.5)..(6.3,2.4)[add arrow];
[left] at (5,2) $a$;
[left] at (8,2) $c$;
[left] at (6.3,-2) $b$;
[left] at (7,.8) $c*b$;
[left] at (5.5,-1.4) $R_1(a,c*b)$;
[right] at (8,-2) $R_2(a,c*b)\bar{*}b$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (6.5,0) ;
The Reidemeister move $\Omega 4a$ with colors.
[use Hobby shortcut,, scale=1, add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=1,>=stealth]>]
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
(-1,2) .. (2,-2)[add arrow];
(2,2) .. (-1,-2)[add arrow];
(.7,-2.4)..(0,-1.5).. (-1,0)..(0,1.5)..(.7,2.4)[add arrow];
[left] at (-1,2) $a$;
[left] at (-1,0)$b \bar{*} R_1(a,c)$;
[left] at (-1,-2) $R_1(a,c)$;
[above] at (.7,2.4) $(b \bar{*} R_1(a,c))*a$;
[right] at (2,2) $c$;
[left] at (.5,-2) $b$;
[right] at (2,-2) $R_2(a,c)$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (.5,0) ;
[very thick, <->] (3,0) – (4,0);
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
(8,2) .. (5,-2)[add arrow];
(5,2).. (8,-2)[add arrow];
(6.3,-2.4).. (7,-1.5).. (8,0).. (7,1.5)..(6.3,2.4)[add arrow];
[left] at (5,2) $a$;
[left] at (8,2) $c$;
[left] at (6.3,-2) $b$;
[right] at (8,0) $b*R_2(a,c)$;
[left] at (5.5,-1.4) $R_1(a,c)$;
[right] at (8,-2) $R_2(a,c)$;
[above] at (6.3,2.4) $(b*R_2(a,c))\bar{*}c$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (6.5,0) ;
The Reidemeister move $\Omega 4e$ with colors.
[use Hobby shortcut,, scale=1, add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=1,>=stealth]>]
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
flip crossing/.list=2
(-6,2) .. (-6,1.8).. (-7,.2) .. (-7,0)..(-7,-.2) .. (-6,-1.8).. (-6,-2.2)[add arrow];
(-7,2).. (-7,1.8).. (-6,.2).. (-6,0)..(-6,-.2).. (-7,-1.8)..(-7,-2.2)[add arrow] ;
[left] at (-7,2) $a$;
[left] at (-7,0) $R_1(a,b)$;
[left] at (-7,-2) $R_2(a,b)$;
[right] at (-6,2) $b$;
[right] at (-6,0) $R_2(a,b)$;
[right] at (-6,-2) $R_1(a,b)*R_2(a,b)$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (-6.5,1) ;
[very thick, <->] (-3,0) – (-1,0);
consider self intersections=true,
clip width=4,
ignore endpoint intersections=false,
flip crossing/.list=1
(2,2) .. (2,1.8).. (3,.2) .. (3,0)..(3,-.2) .. (2,-1.8).. (2,-2.3)[add arrow];
(3,2).. (3,1.8).. (2,.2).. (2,0)..(2,-.2).. (3,-1.8)..(3,-2.3)[add arrow];
[left] at (2,2) $a$;
[left] at (2,0) $b$;
[left] at (2,-2) $R_1(b,a*b)$;
[right] at (3,2) $b$;
[right] at (3,0) $a * b$;
[right] at (3,-2) $R_2(b,a*b)$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (2.5,-1.1) ;
The Reidemeister move $\Omega 5a$ with colors.
Let $(X, *)$ be a quandle. Let $R_1$ and $R_2$ be two maps from $X \times X$ to $X$. The triple $(X, *, R_1, R_2)$ is called an oriented singquandle if the following axioms are satisfied for all $a,b,c \in X$:
\begin{eqnarray}
R_1(a\bar{*}b,c)*b&=&R_1(a,c*b) \label{eq1}\\
R_2(a\bar{*}b, c) & =& R_2(a,c*b)\bar{*}b \label{eq2}\\
(b\bar{*}R_1(a,c))*a &=& (b*R_2(a,c))\bar{*}c \label{eq3}\\
R_2(a,b)&=&R_1(b,a*b) \label{eq4}\\
R_1(a,b)*R_2(a,b)&=&R_2(b,a*b). \label{eq5}
\end{eqnarray}
We will only consider oriented singquandles in this paper. Therefore, we will simply refer to oriented singquandle as singquandles. For a description of unoriented singquandles, see <cit.>.
In <cit.>, the following family of singquandles was introduced.
Let $n$ be a positive integer, let $a$ be an invertible element in $\mathbb{Z}_n$ and let $b,c \in \mathbb{Z}_n$, then the binary operations $x*y = ax+(1-a)y$, $ R_1(x,y) = bx + cy$ and $R_2(x,y)= acx + [b+ c(1 - a)]y $ make the quadruple $(\mathbb{Z}_n,*, R_1,R_2)$ into an oriented singquandle.
The singquandles used in this paper are obtained from Example <ref> with specific values of $a,b,$ and $c$.
Let $(X, *, R_1, R_2)$ be a singquandle. A subset $M \subset X$ is called a subsingquandle if $(M, *, R_1, R_2)$ is itself a singquandle. In particular, $M$ is closed under the operations $*, R_1$ and $R_2$.
We can define the notion of a homomorphism and isomorphism of oriented singquandles.
A map $f: X \rightarrow Y$ is called a homomorphism of oriented singquandles $(X, *, R_1, R_2)$ and $(Y, \triangleright, R'_1, R'_2)$ if the following conditions are satisfied for all $x,y \in X$
\begin{eqnarray}
f(x*y)&=&f(x) \triangleright f(y)\label{3.6}\\
\end{eqnarray}
An oriented singquandle isomorphism is a bijective oriented singquandle homomorphism. We say two oriented singquandles are isomorphic if there exists an oriented singquandle isomorphism between them.
The authors of this paper introduced the idea of a fundamental singquandle associated to a singular link $L$, denoted by $\mathcal{SQ}(L)$, in <cit.>. Therefore, for any oriented singular link $L$ and an oriented singquandle $(S,*,R_1', R_2')$, the set of singquandle homomorphism from $(\mathcal{SQ}(L),\triangleright ,R_1,R_2)$ to $(S, *, R_1', R_2')$ is defined by:
\begin{equation*}
\begin{split}
\textup{Hom}(\mathcal{SQ}(L),S) = \lbrace f \, : \, &\mathcal{SQ}(L) \rightarrow S \, \vert \, f( x\triangleright y) = f(x) * f(y),\\ &f(R_1(x,y))= R_1'(f(x),f(y)),f(R_2(x,y))= R_2'(f(x),f(y)) \rbrace.
\end{split}
\end{equation*}
The set defined above was shown to be an invariant of oriented singular links in <cit.>. Furthermore, this set can be used to define computable invariants of oriented singular links. For example, by taking the cardinality of $\textup{Hom}(\mathcal{SQ}(L),S)$, we obtain the following invariant of oriented singular links.
Let $L$ be an oriented singular link and $(S,*,R_1,R_2)$ be an oriented singquandles. The singquandle counting invariant is
\[ \#\textup{Col}_S (L)=\vert \textup{Hom}(\mathcal{SQ}(L),S) \vert. \]
We also note that the image, $\textup{Im}(f)$, for each $f \in\textup{Hom}(\mathcal{SQ}(L),S)$ is a subsingquandle of $S$ as shown in <cit.>.
§ REVIEW OF THE SINGQUANDLE POLYNOMIAL
The quandle polynomial was introduced in <cit.> and generalized in <cit.>. In <cit.>, the authors of this paper defined the singquandle polynomial, the subsingquandle polynomial and a polynomial invariant of singular links. In this section, we will give an overview of the construction of the singquandle polynomial, the subsingquandle polynomial, and the polynomial invariant of singular links. We will follow the construction and notation introduced in <cit.>.
Let $(X,*,R_1,R_2)$ be a finite singquandle. For every $x \in X$, define
\[ C^1(x) = \lbrace y \in X \, \vert \, y * x = y \rbrace \quad \text{and} \quad R^1(x) = \lbrace y \in X \, \vert \, x * y = x \rbrace, \]
\[ C^2(x) = \lbrace y \in X \, \vert \, R_1(y , x) = y \rbrace \quad \text{and} \quad R^2(x) = \lbrace y \in X \, \vert \, R_1(x , y) = x \rbrace, \]
\[ C^3(x) = \lbrace y \in X \, \vert \, R_2(y , x) = y \rbrace \quad \text{and} \quad R^3(x) = \lbrace y \in X \, \vert \, R_2(x , y) = x \rbrace. \]
Let $c^i(x) = \vert C^i(x)\vert$ and $r^i(x) = \vert R^i(x)\vert$ for $i=1,2,3$. Then the singquandle polynomial of X is
\[ sqp(X) = \sum_{x\in X} s_1^{r^1(x)}t_1^{c^1(x)}s_2^{r^2(x)}t_2^{c^2(x)}s_3^{r^3(x)}t_3^{c^3(x)}. \]
We note that the value $r^i(x)$ is the number of elements in $X$ that act trivially on $x$, while $c^i(x)$ is the number of elements of $X$ on which $x$ acts trivially via $*, R_1$ and $R_2.$ Furthermore, if $Y \subset X$ is a subsingquandle we can define the following singquandle polynomial for $Y$ as a subsignquandle of $X$
Let $(X,*, R_1,R_2)$ be a finite singquandle and $S \subset X$ a subsingquandle. Then the subsingquandle polynomial is
\[ Ssqp(S \subset X ) = \sum_{x \in S} s_1^{r^1(x)}t_1^{c^1(x)}s_2^{r^2(x)}t_2^{c^2(x)}s_3^{r^3(x)}t_3^{c^3(x)}. \]
The subsingquandle polynomials can be thought of as the contributions to the singquandle polynomial coming from the subsingquandles we are considering. Using the subsingquandles polynomial we can now define the following polynomial invariant of singular links.
Let $L$ be a singular link, $(X,*,R_1,R_2)$ a finite singquandle. Then the multiset
\[ \Phi_{Ssqp}(L,X) = \lbrace Ssqp(Im(f) \subset X) \, \vert \, f \in \text{Hom}(\mathcal{SQ}(L), X\rbrace \]
is the subsingquandle polynomial invariant of $L$ with respect to $X$. We can also represent this invariant in the following polynomial-style form by converting the multiset elements to exponents of a formal variable $u$ and converting their multiplicities to coefficients:
\[ \phi_{Ssqp}(L,X) = \sum_{f \in \textup{Hom}(\mathcal{SQ}(L),X)} u^{Ssqp(Im(f) \subset X)}.\]
Consider the 2-bouquet graphs of type $L$ listed as $1^l_1$ in <cit.>. Let $(S,*,R_1,R_2)$ be the singquandle with $S=\mathbb{Z}_4$ and operations $x*y = 3x-2y = x\bar{*}y$, $R_1(x,y)=2 x + 3 y$, and $R_2(x,y)=x$.
[use Hobby shortcut]
clip width=4
[decoration=markings,mark=at position .25 with
[scale=1,>=stealth]<,postaction=decorate](-1,0) circle[radius=2cm];
[decoration=markings,mark=at position .25 with
[scale=1,>=stealth]>,postaction=decorate] (1,0) circle[radius=2cm];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (0,1.7) ;
[left] at (-1,0) $z$;
[left] at (-2,2) $x$;
[right] at (2,2) $y$;
Diagram of $1_1^l$.
We can identify each coloring of $1_1^l$ by $S$ with the triple $(f(x),f(y),f(z))$. Using the fact that $z = R_1(x,y), x = R_2(x,y)$ and $z * x =y,$ by a straightforward computation we obtain the following colorings:
\begin{equation*}
\begin{split}
\textup{Hom}(\mathcal{SQ}(1_1^l),S)=\{(1, 1, 1), (1, 2, 0), (1, 3, 3), (1, 0, 2), (2, 1, 3), (2, 2, 2), (2, 3, 1), (2, 0, 0),\\ (3, 1, 1), (3, 2, 0), (3, 3, 3), (3, 0, 2), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 0, 0)\}.
\end{split}
\end{equation*}
Therefore, $\#\textup{Col}_S (1_1^l)=16$. We compute $r^i$ and $c^i$ for $i=1,2,3$:
\[
\begin{tabular}{ r| c c }
$x$ & $r^1(x)$ & $c^1(x)$\\
\hline
1& 2 & 2 \\
2& 2 & 2 \\
3& 2 & 2 \\
0& 2 & 2 \\
\end{tabular}
\qquad
\begin{tabular}{ r| c c }
$x$ & $r^2(x)$ & $c^2(x)$\\
\hline
1& 1 & 1 \\
2& 1 & 1 \\
3& 1 & 1 \\
0& 1 & 1 \\
\end{tabular}
\qquad
\begin{tabular}{ r| c c }
$x$ & $r^3(x)$ & $c^3(x)$\\
\hline
1& 4& 4\\
2& 4 & 4 \\
3& 4 & 4 \\
0& 4 & 4 \\
\end{tabular}.
\]
In order to compute the subsingquandle polynomial invariant of $1_1^l$ we consider the corresponding subsingquandle for each coloring. Therefore, we obtain
$$\phi_{Ssqp}(1_1^l,S) = 4 u^{s_1^2 t_1^2 s_2 t_2 s_3^4 t_3^4}+4 u^{2 s_1^2 t_1^2 s_2 t_2 s_3^4 t_3^4}+8
u^{4 s_1^2 t_1^2 s_2 t_2 s_3^4 t_3^4}.$$
§ SINGQUANDLE SHADOWS
In this section, we define singquandle shadows which can be used to generalize the shadow colorings of knot diagrams by quandles previously defined in CN.
Let $(S,*,R_1,R_2)$ be a singquandle. An S-set is a set $X$ and a map $\cdot : X \times S \rightarrow X$ satisfying the following conditions:
* For all $s \in S$, $\cdot s : X \rightarrow X$ mapping $x$ to $x \cdot s$ is a bijection.
* For all $s_1, s_2 \in S$ and $x \in X$,
\begin{eqnarray}
(x \cdot s_1)\cdot s_2 &=& (x \cdot s_2)\cdot ( s_1 * s_2)\\
(x \cdot s_1) \cdot s_2 &=& (x \cdot R_1(s_1,s_2)) \cdot R_2(s_1,s_2).
\end{eqnarray}
The meaning of these two equations will become clear from Figure <ref>.
A singquandle shadow or $S$-shadow is the pair of an oriented sinquandle $(S,*,R_1,R_2)$ and a $S$-set $(X,\cdot)$, denoted by $(S,X,*,R_1,R_2,\cdot)$ or simply by $(S,X)$. Let $S^{\prime}$ be a subsingquandle of $S$. A subset $Y$ of $X$ closed under the action of $S^{\prime}$ is an subshadow of $(S,X)$, which we will denote by $(S^\prime, Y) \subset (S,X)$.
The following definition will allow us to present a shadow operation in an alternate form that will be useful in later sections.
When $(X,S)$ is a singquandle shadow with $X$ and $S$ finite, the shadow matrix of the singquandle shadow $(X = \lbrace x_1, \dots, x_m \rbrace$, $S = \lbrace s_1,\dots, s_n \rbrace)$ is the $m \times n$ matrix whose $(i,j)$ entry is $k$ where $x_k = x_i \cdot s_j$.
Let $(S,*,R_1,R_2)$ and $(S',\triangleright, R'_1,R'_2)$ be singquandles. Furthermore, let $(X, \cdot)$ be an $S$-set and $(X', \bullet)$ be an $S'$-set. A map $f: S \rightarrow S'$ makes $X'$ inherit a natural action of $S$ via the map $f$ by $x' \cdot s :=x' \bullet f(s)$.
A homomorphism of sinquandle shadows between $(S,X,*,R_1,R_2,\cdot)$ and
$(S',Y,\triangleright,R'_1,R'_2,\bullet)$ is a pair of maps $\phi:(X, \cdot) \rightarrow (Y,\bullet)$ and $f:(S,*,R_1,R_2) \rightarrow (S',\triangleright,R'_1,R'_2)$, such that $f$ is a singquandle homomorphism, that is the identities (<ref>), (<ref>) and (<ref>) are satisfied and for all $x \in X$ and $s \in S$, we have
\begin{eqnarray}\label{action}
\phi(x \cdot s)=\phi(x) \bullet f(s).
\end{eqnarray}
Furthermore, if $\phi$ and $f$ are bijections then we have a singquandle shadow isomorphism.
From this definition it is straightforward to obtain the following lemma.
$(Im(f),Im(\phi),\triangleright,R'_1,R'_2,\bullet)$ is a subshadow of $(S',Y,\triangleright,R'_1,R'_2,\bullet)$.
Let $(S,*,R_1,R_2)$ be an oriented singquandle with $S=\mathbb{Z}_8=\{ 1,2,3,4,5,6,7,0 \}$, $x*y = 5x-4y = x\bar{*}y$, $R_1(x,y) =3x+4y $, and $R_2(x,y)= 4x+3y$. Then the four element set $X= \mathbb{Z}_4=\{ 1,2,3,0 \}$ with map $\cdot s : \mathbb{Z}_4 \rightarrow \mathbb{Z}_4$ for each $s \in S$ defined by $x\cdot s = x + 2 s + s^2$ is a singquandle shadow. Note that $\cdot$ has the following operation table
\[ \begin{array}{r|cccccccc}
\cdot & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 0\\
\hline
1& 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
2& 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 \\
3& 2 & 3 & 2 & 3 & 2 & 3 & 2 & 3 \\
0& 3 & 0 & 3 & 0 & 3 & 0 & 3 & 0 \\
\end{array}. \]
Furthermore, by Definition <ref> the shadow operation $\cdot$ can be presented by the shadow matrix,
\[
\left[
\begin{array}{cccccccc}
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 \\
2 & 3 & 2 & 3 & 2 & 3 & 2 & 3 \\
3 & 0 & 3 & 0 & 3 & 0 & 3 & 0 \\
\end{array}
\right].
\]
Let $(S,*,R_1,R_2)$ be any oriented singquandle and let $X = S$. Then $X$ is a singquandle shadow under the shadow operation $x \cdot s = x * s$ for all $x,y \in S$, since we have
\begin{eqnarray}
(x \cdot s_1) \cdot s_2 &=& (x * s_1)*s_2 \nonumber\\
&=& (x * s_2) * (s_1 * s_2) \nonumber\\
&=& (x\cdot s_2) \cdot (s_1 \cdot s_2), \label{Eq1}
\end{eqnarray}
\begin{eqnarray}
(x \cdot s_1) \cdot s_2 &=& (x * s_1)*s_2 \nonumber\\
&=& (x * R_1 (s_1,s_2)) * R_2(s_1,s_2) \nonumber\\
&=& (x \cdot R_1 (s_1,s_2)) \cdot R_2(s_1,s_2).\label{Eq2}
\end{eqnarray}
Equation (<ref>) is satisfied by the self-distributitive property of $*$. On the other hand, equation (<ref>) is satisfied by applying a combination of singquandle properties. Consider equation (<ref>) in the definition of a singquandle, let $b = c$. Therefore, we obtain
\begin{eqnarray*}
(c \bar{*} R_1(a,c))* a &=& (c * R_2(a,c)) \bar{*} c,
\end{eqnarray*}
which can be written as
\begin{eqnarray*}
((c \bar{*} R_1(a,c))* a)*c &=& c * R_2(a,c).
\end{eqnarray*}
Now, we let $w = c \bar{*} R_1(a,c) \iff w * R_1(a,c) = c$. Next, we make the appropriate substitution to obtain
\[ (w *a) * c = (w * R_1(a,c)) * R_2(a,c).\]
Lastly, let $w=x$, $a = s_1$, and $b = s_2$ to obtain
\[ (x * s_1)*s_2 = (x * R_1 (s_1,s_2)) * R_2(s_1,s_2). \]
Let $D$ be a diagram of an oriented singular link $L$. We will denoted the set of arcs of $D$ by $\mathcal{A}(D)$ and the connected regions of $\mathbb{R}^2\setminus D$ by $\mathcal{R}(D)$. Using the notion of a singquandle homomorphism given in Definition <ref>, we have the following notion of colorings by singquandles.
Let $(S,*, R_1, R_2)$ be an oriented singquandle. An $S$-coloring of $D$ is a map $f: \mathcal{A}(D) \rightarrow S$ such that at a crossing with $u_1, u_2, o_1 \in \mathcal{A}(D)$ and at a singular crossing with $a_1,a_2,a_3,a_4 \in \mathcal{A}(D)$ the following conditions are satisfied,
\begin{eqnarray}
f(u_2) = f(u_1) * f(o_1), \\
f(a_3) = R_1(f(a_1),f(a_2)), \\
f(a_4) = R_2(f(a_1),f(a_2)).
\end{eqnarray}
The conditions above are illustrated in Figure <ref>.
Let $(S,X,*, R_1, R_2, \cdot)$ be a shadow singquandle. An $(S,X)$-coloring of $D$ is a map $f \times \phi: \mathcal{A}(D) \times \mathcal{R}(D) \rightarrow S \times X$ satisfying the following conditions,
* $f$ is an $S$-coloring of $D$.
* $\phi(\mathcal{R}(D)) \subset X$.
* For $a \in \mathcal{A}(D)$ and $x_1, x_2 \in \mathcal{R}(D)$ the following
\begin{equation}
\phi(x_1) \cdot f(a) = \phi(x_2).
\end{equation}
The condition above is illustrated in Figure <ref>.
When there is no confusion we will refer to an $(S,X)$-coloring by a shadow coloring of $D$.
[use Hobby shortcut,scale=1.2]
consider self intersections=true,
clip width=5,
ignore endpoint intersections=false,
flip crossing/.list=3,4,5,6,11,13
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (0,1) ..(0,-1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (-1,0) ..(1,0);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (4,1) ..(4,-1);
(1,-2)..(3,-4.1)[decoration=markings,mark=at position 1 with
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (3,-2)..(1,-4.1);
[above] at (-1,0) $s_1=f(u_1)$;
[above] at (0,1) $s_2=f(o_1)$;
[above] at (1,0) $s_1 * s_2 =f(u_2)$;
(3,0) node [draw] $\phi(x_1)$;
(5.5,0) node [draw] $\phi(x_2) = \phi(x_1)\cdot f(a)$;
[above] at (4,1) $f(a)$;
[above] at (1,-2) $f(a_1)$;
[above] at (3,-2) $f(a_2)$;
[right] at (3,-4.1) $f(a_4)=R_2 (f(a_1),f(a_2))$;
[left] at (1,-4.1) $f(a_3)=R_1(f(a_1),f(a_2))$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (2,-3.05) ;
Arcs and regions of diagram $D$.
We will denote a region coloring by a box around the shadow element and we will denote an arc coloring by an element of a singquandle without a box. Note that the conditions required for the set $X$ to be an $S$-set for some oriented sinquandle, are the conditions needed to guarantee that shadow colorings are well defined at crossings, see Figure <ref>.
[use Hobby shortcut,scale=1.5]
consider self intersections=true,
clip width=5,
ignore endpoint intersections=false,
flip crossing/.list=3,4,5,6,11,13
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (1,1) ..(-1,-1.1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (-1,1) ..(1,-1.1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (3,1) ..(5,-1.1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (5,1) ..(3,-1.1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (1,-2) ..(3,-4.1);
[decoration=markings,mark=at position 1 with
[scale=1,>=stealth]>,postaction=decorate] (3,-2) ..(1,-4.1);
[above] at (-1,1) $s_1$;
[above] at (1,1) $s_2$;
[below] at (-1,-1.1) $s_2$;
[below] at (1,-1.1) $s_1 *s_2$;
(-.5,0) node [draw] $x$;
(0,.6) node [draw] $x \cdot s_1$;
(1.25,.3) node [draw] $(x \cdot s_1) \cdot s_2$;
(0,-.6) node [draw] $x \cdot s_2$;
(1.5,-.3) node [draw] $(x \cdot s_2) \cdot (s_1 * s_2)$;
at (1,0) $=$;
[below] at (3,-1.1) $s_1$;
[below] at (5,-1.1) $s_2$;
[above] at (3,1) $s_2$;
[above] at (5,1) $s_1 * s_2$;
(3.5,0) node [draw] $x$;
(4,.6) node [draw] $x \cdot s_2$;
(5.5,.3) node [draw] $(x \cdot s_2) \cdot (s_1 * s_2)$;
(4,-.6) node [draw] $x \cdot s_1$;
(5.2,-.3) node [draw] $(x \cdot s_1) \cdot s_2$;
at (5,0) $=$;
[above] at (1,-2) $s_1$;
[above] at (3,-2) $s_2$;
[below] at (1,-4.1) $R_1(s_1,s_2)$;
[below] at (3,-4.1) $R_2(s_1,s_2)$;
(1.5,-3) node [draw] $x$;
(2,-2.4) node [draw] $x \cdot s_1$;
(3.3,-2.7) node [draw] $(x \cdot s_1) \cdot s_2$;
(2,-3.97) node [draw] $x \cdot R_1(s_1,s_2)$;
(4,-3.3) node [draw] $(x \cdot R_1(s_1,s_2)) \cdot R_2(s_1,s_2) $;
at (3,-3) $=$;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (2,-3.05) ;
Shadow coloring at positive, negative, and singular crossings.
Let $L$ be a singular link diagram and $(S,X,*,R_1,R_2,\cdot)$ be a singquandle shadow. Then for each singquandle coloring of $L$ by $S$ and each element of $X$ there is exactly one shadow coloring of $L$.
Consider a singquandle coloring of $L$ and choose a region of $L$. Any element of $X$ can be assigned to the chosen region, and any choice determines a unique shadow color of each region by following the rule in Figure <ref>.
Let $L$ be singular link diagram and $(S,X)$ is a singquandle shadow. The shadow counting invariant, $\#\textup{Col}_{(S,X)}(L)$, is the number of shadow colorings of $L$ by $(S,X)$.
Let $(S,*,R_1,R_2)$ be the singquandle with $S=\mathbb{Z}_{10}$ and operations defined by $x*y = 3x-2y$, $x \bar{*} y = 7 x - 6 y$, $R_1(x,y) = 4x+6y$, and $R_2(x,y) = 8x+2y$. We can define a shadow structure by $X=\mathbb{Z}_4$ with the map $\cdot s: \mathbb{Z}_4 \rightarrow \mathbb{Z}_4$ for each $s \in S$ defined by $x\cdot s= 2+x + 2x^2$. We also represent this shadow operation by the shadow matrix
\[
\left[
\begin{array}{cccccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
\end{array}
\right].
\]
We will compute a shadow coloring for the following 2 bouquet graph of type $K$ listed as $3_1^k$ in <cit.>.
[scale=.6,use Hobby shortcut,add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=2,>=stealth]<]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $s_2$;
[above] at (0,1.5) $s_4$;
[right] at (3,-3.5) $s_3$;
[left] at (-3,-3.5) $s_1$;
(0,-2) node [draw] $x_i$;
(-2,-3) node [draw] $x_i \cdot s_4$;
(2,-3) node [draw] $x_i \cdot s_1$;
(0,0) node [draw] $x_i \cdot s_2$;
(-5,-2.8) node [draw] $(x_i \cdot s_4) \cdot s_1$;
(5,-2.8) node [draw] $(x_i \cdot s_1) \cdot s_3$;
(0,2.5) node [draw] $(x_i \cdot s_2) \cdot s_4$;
Shadow coloring of $3_1^k$.
This singular knot has one coloring by $S$ given by
\[\textup{Hom}(\mathcal{SQ}(3_1^k),S) =\lbrace (s_1\to 0,s_2 \to 0, s_3 \to 0, s_4 \to 0)\rbrace. \]
For this coloring we also obtain one shadow coloring for each element of $X$. Therefore, we have the following four shadow colorings by the above shadow singquandle.
[scale=.4,use Hobby shortcut,add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=2,>=stealth]<]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $0$;
[above] at (0,1.5) $0$;
[right] at (3,-3.5) $0$;
[left] at (-3,-3.5) $0$;
(0,-2) node [draw] $1$;
(-2,-3) node [draw] $1$;
(2,-3) node [draw] $1$;
(0,.5) node [draw] $1$;
(-4.5,-3) node [draw] $1$;
[scale=.4,use Hobby shortcut,add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=2,>=stealth]<]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $0$;
[above] at (0,1.5) $0$;
[right] at (3,-3.5) $0$;
[left] at (-3,-3.5) $0$;
(0,-2) node [draw] $2$;
(-2,-3) node [draw] $0$;
(2,-3) node [draw] $0$;
(0,.5) node [draw] $0$;
(-4.5,-3) node [draw] $2$;
[scale=.4,use Hobby shortcut,add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=2,>=stealth]<]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $0$;
[above] at (0,1.5) $0$;
[right] at (3,-3.5) $0$;
[left] at (-3,-3.5) $0$;
(0,-2) node [draw] $3$;
(-2,-3) node [draw] $3$;
(2,-3) node [draw] $3$;
(0,.5) node [draw] $3$;
(-4.5,-3) node [draw] $3$;
[scale=.4,use Hobby shortcut,add arrow/.style=postaction=decorate, decoration=
mark=at position 1 with [scale=2,>=stealth]<]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $0$;
[above] at (0,1.5) $0$;
[right] at (3,-3.5) $0$;
[left] at (-3,-3.5) $0$;
(0,-2) node [draw] $0$;
(-2,-3) node [draw] $2$;
(2,-3) node [draw] $2$;
(0,.5) node [draw] $2$;
(-4.5,-3) node [draw] $0$;
Shadow colorings of $3_1^k$ by singquandle shadow $X$.
Therefore, $\#\textup{Col}_{(S,X)}(3_1^k) = 4.$
The following corollary implies that the shadow counting invariant does not contain any information not contained by the singquandle counting invariant. We obtain the following result by noticing that for each coloring of an oriented singular link by an oriented singquandle we have a different shadow coloring for each element in the $S$-set.
The shadow counting invariant of a singular link $L$ by the $S$-shadow $(S,X)$ is given by
\[\# \textup{Col}_{(S,X)}(L) = \vert X \vert \, \# \textup{Col}_S(L) ,\]
where $\#\textup{Col}_S(L)$ is the singquandle counting invariant.
We can define the following polynomial for a sinquandle shadow to obtain a singquandle shadow invariant.
The shadow singquandle polynomial, denoted by $\textup{sp}(S,X)$, of the shadow singquandle $(S,X,*,R_1,R_2,\cdot)$ is the sum
\[ \textup{sp}(S,X) = \sum_{x \in X} t^{r(x)},\]
where $r(x) = \vert \lbrace s \in S \, ; \, x \cdot s = x \rbrace\vert$. Furthermore, If $(S^{\prime},Y)$ is a subshadow of $(S,X)$, then the subshadow singquandle polynomial of $(S^{\prime},Y)$ is
\[ \textup{Subsp} ((S',Y) \subset (S,X)) = \sum_{x \in Y} t^{r(x)}, \]
where $r(x) =\vert \lbrace s' \in S' \, ; \, x \cdot s' = x \rbrace \vert$.
Let $S= \mathbb{Z}_6 =\{ 1,2,3,4,5,0\}$ with singquandle operations defined by $x* y = 5x-4y = x\bar{*}y$, $R_1(x,y)=2x+y$ and $R_2(x,y)= 5x+4y$. Consider the shadow structure defined by $X = \mathbb{Z}_2 = \{1,0\}$ with shadow matrix
\[ \left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right].\]
Note that we can compute $r(x)$ for each $x \in X$ by going through the row of the shadow matrix and counting the occurrences of the row number. Therefore, $r(1)=6$ and $r(0) = 6$, and the shadow singquandle polynomial of $(S,X)$ is
\[ \textup{sp}(S,X) = 2t^6. \]
We will consider two types of subshadows. We will first consider a subset of $X$ closed under the action of $S$. When we consider the subshadow $(S, Y) \subset (S,X)$, where $Y = \{1\}$, note that we can check from the shadow matrix that $Y$ is closed under the action of $S$. The subshadow $(S,Y)$ has the following subshadow singquandle polynomial
\[ \textup{Subsp}(S,Y) = t^6.\]
Next, we consider the subshadow $(S',Y) \subset (S,X)$, where $S'$ is the subsingquandle consisting of $\{2,4,0\}$ and $Y = \{1 \}$. A straightforward computation shows that $S^\prime$ is closed under the singquandle operations. Furthermore, we can check from the shadow matrix that $Y$ is closed under the action of $S^\prime$. The subshadow $(S',Y)$ has the following subshadow singquandle polynomial,
\[ \textup{Subsp}(S^\prime,Y)= t^3. \]
We now prove that the shadow singquandle polynomial is an invariant of shadow singquandles.
Let $(S,X)$ and $(S',Y)$ be two singquandle shadows.
If $(S,X)$ and $(S',Y)$ are isomorphic, then they have equal shadow polynomials, $\textup{sp}(S,X) = \textup{sp}(S',Y)$.
Suppose the pair $\phi: X \rightarrow Y$ and $f: S \rightarrow S'$ is a shadow singquandle isomorphism. Then $r(\phi(x)) = r(x)$ and the contribution to $\textup{sp}(S,X)$ from $x \in X$ is the same as the contribution of $\phi(x) \in Y$ to $\textup{sp}(S',Y)$. Since $\phi$ and $f$ are bijective maps satisfying equations (<ref>), (<ref>), (<ref>) and (<ref>), the result follows.
The shadow polynomial can be used to distinguish and classify shadow singqundles. In the following example, we distinguish two shadow singquandles.
Let $(S,*,R_1,R_2)$ be a singquandle with $S= \mathbb{Z}_8$, $x*y= 5x-4y = x \bar{*} y$, $R_1(x,y) = 3 x + 4 y$, and $R_2(x,y)= 4x+3y$. Let $X = \mathbb{Z}_4$ with shadow matrix
\[ \left[
\begin{array}{cccccccc}
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 \\
2 & 3 & 2 & 3 & 2 & 3 & 2 & 3 \\
3 & 0 & 3 & 0 & 3 & 0 & 3 & 0 \\
\end{array}
\right]. \]
The $(S,X, *, R_1,R_2,\cdot)$ is a singquandle shadow with shadow polynomial
\[ \textup{sp}(S,X) = 4t^4. \]
Let $W = \mathbb{Z}_4$ with shadow matrix
\[ \left[
\begin{array}{cccccccc}
3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right]. \]
The $(S,W, *, R_1,R_2,\bullet )$ is a singquandle shadow with shadow polynomial
\[ \textup{sp}(S,W) = 2+2t^8. \]
We see that the shadow singquandle polynomial is an effective invariant of singquandle shadows.
In this section, we see that by simply computing the shadow counting invariant of an oriented singular link we do not obtain any more information than that obtained from the singquandle counting invariant. Therefore, in the following section we will enhance the shadow counting invariant in order to obtain a stronger invariant.
§ ENHANCED SHADOW COUNTING INVARIANT
In this section, we will jazz up the shadow counting invariant from the previous section. We will combine the $S$-shadow counting invariant and the shadow polynomial in order to define an enhanced shadow singquandle invariant for singular link.
Let $f \times \phi$ be a shadow coloring of an oriented singular link diagram $D$. The closure of the set of shadow colors under the action of the image subsingquandle $\textup{Im}(f) \subset S$ of $f \times \phi$ is a subshadow called the shadow image of $f \times \phi$, which we denote by $\textup{om}(f \times \phi)$[The choice of $\textup{om}(f \times \phi)$ to denote the shadow image of $f \times \phi$ was derived from the french word ombre for
Let $(S,X)$ be an $S$-shadow and let $L$ be an oriented singular link with diagram $D$. The singquandle shadow polynomial invariant of $L$ with respect $(S,X)$ is
\[ SP(L) = \sum_{f \times \phi \in \textup{shadow coloring}} u^{\textup{Subsp} (\textup{om}(f \times \phi) \subset (S,X))}.\]
§ EXAMPLES
In this section, we present two examples in which we show that the shadow sinquandle polynomial is an enhancement of the singquandle counting invariant. In the first example, we include a pair of singular knots with the same singquandle counting invariant and the same singquandle polynomial but are distinguished by the singquandle shadow polynomial invariant. The computations were performed by Mathematica and python independently and checked by hand.
Let $(S,X,*,R_1,R_2,\cdot)$ be the shadow singquandle with $S=\mathbb{Z}_8$, $X=\mathbb{Z}_6$ and operations $x*y = 3x-2y = x \bar{*} y$, $R_1(x,y) = 7x+6y$, $R_2(x,y) = 2x+3y$, and shadow matrix
\[ \left[
\begin{array}{cccccccc}
4 & 1 & 4 & 1 & 4 & 1 & 4 & 1 \\
5 & 2 & 5 & 2 & 5 & 2 & 5 & 2 \\
0 & 3 & 0 & 3 & 0 & 3 & 0 & 3 \\
1 & 4 & 1 & 4 & 1 & 4 & 1 & 4 \\
2 & 5 & 2 & 5 & 2 & 5 & 2 & 5 \\
3 & 0 & 3 & 0 & 3 & 0 & 3 & 0 \\
\end{array}
\right]. \]
The following two 2 bouquet graph of type $K$ listed as $4_1^k$ and $5_4^k$ in <cit.>. We obtain the following coloring equations from the singular knot $4_1^k$,
\begin{eqnarray*}
s_1 &=& s_6 \bar{*} s_2 =-2 s_2 + 3 s_6 \\
s_2 &=& s_5 \bar{*} s_6 = 3 s_5 - 2 s_6 \\
s_3 &=& R_1(s_1,s_2) = 7 s_1 + 6 s_2 \\
s_4 &=& R_2(s_1,s_2) =2 s_1 + 3 s_2 \\
s_5 &=& s_4 \bar{*}s_3 =-2 s_3 + 3 s_4 \\
s_6 &=& s_3 \bar{*} s_5 = 3 s_3 - 2 s_5.
\end{eqnarray*}
From these equations we obtain the colorings listed below. In the following list we identify each coloring $f \in \textup{Hom}(\mathcal{SQ}(4_1^k),S)$ with the $6$-tuple $(f(s_1),f(s_2),f(s_3),f(s_4),f(s_5),f(s_6))$.
\begin{equation*}
\begin{split}
\textup{Hom}(\mathcal{SQ}(4_1^k),S)=
\{(1, 7, 1, 7, 3, 5), (1, 3, 1, 3, 7, 5), (2, 2, 2, 2, 2, 2), (2, 6, 2, 6, 6, 2), \\
(3, 5, 3, 5, 1, 7), (3, 1, 3, 1, 5, 7), (4, 0, 4, 0, 0, 4),(4, 4, 4, 4, 4, 4), \\
(5, 3, 5, 3, 7, 1), (5, 7, 5, 7, 3, 1), (6, 6, 6, 6, 6, 6), (6, 2, 6, 2, 2, 6), \\
(7, 1, 7, 1, 5, 3), (7, 5, 7, 5, 1,3), (0, 4, 0, 4, 4, 0), (0, 0, 0, 0, 0, 0)\}.
\end{split}
\end{equation*}
We obtain the following coloring equations from the singular knot $5_4^k$,
\begin{eqnarray*}
s_1 &=& s_7 \bar{*} s_5 = -2 s_5 + 3 s_7\\
s_2 &=& s_4 * s_6 = 3 s_4 - 2 s_6\\
s_3 &=& R_1(s_1,s_2) = 7 s_1 + 6 s_2 \\
s_4 &=& R_2(s_1,s_2) =2 s_1 + 3 s_2 \\
s_5 &=& s_3 \bar{*}s_7 = 3 s_3 - 2 s_7 \\
s_6 &=& s_5 * s_2 = 2 s_2 + 3 s_5\\
s_7 &=& s_6 \bar{*}s_3 =-2 s_3 + 3 s_6 .
\end{eqnarray*}
From these equations we obtain the colorings listed below. In the following list we identify each coloring $f \in \textup{Hom}(\mathcal{SQ}(5_4^k),S)$ with the $6$-tuple $(f(s_1),f(s_2),f(s_3),f(s_4),f(s_5),f(s_6))$.
\begin{equation*}
\begin{split}
\textup{Hom}(\mathcal{SQ}(5_4^k),S)=\{
(2, 2, 2, 2, 2, 2, 2), (2, 4, 6, 0, 6, 2, 2), (2, 6, 2, 6, 2, 2, 2), (2, 0, 6, 4, 6, 2, 2), \\
(4, 2, 0, 6, 0, 4, 4), (4, 4, 4, 4, 4, 4, 4), (4, 6, 0, 2, 0, 4, 4), (4, 0, 4, 0, 4, 4, 4),\\
(6, 2, 6, 2, 6, 6, 6), (6, 4, 2, 0, 2, 6, 6), (6, 6, 6, 6, 6, 6, 6), (6, 0, 2, 4, 2, 6, 6), \\
(0, 2, 4, 6, 4, 0, 0), (0, 4, 0, 4, 0, 0, 0), (0, 6, 4, 2, 4, 0, 0),
(0, 0, 0, 0, 0, 0, 0) \}.
\end{split}
\end{equation*}
Therefore, both singular knots have the same singquandle counting invariant $\#\textup{Col}_S(4_1^k) = 16 = \#\textup{Col}_S(5_4^k)$. Therefore, by Theorem <ref> we obtain that the two singular knots have the same shadow counting invariant $\#\textup{Col}_{(S,X)}(4_1^k) = 96 = \#\textup{Col}_{(S,X)}(5_4^k)$. Furthermore, the two singular knots have the same singquandle polynomial $\phi_{Ssqp}(4_1^k)=4 u^{s_1^2 t_1^2 s_2^2 t_2^2 s_3 t_3}+4 u^{2 s_1^2 t_1^2 s_2^2 t_2^2 s_3
t_3}+8 u^{4 s_1^2 t_1^2 s_2^2 t_2^2 s_3 t_3} =\phi_{Ssqp}(5_4^k) $. However, the singquandle shadow polynomial invariant distinguishes the two singular knots:
[use Hobby shortcut,scale=1,add arrow/.style=postaction=decorate, decoration=
mark=at position .5 with [scale=1.5,>=stealth]<]
consider self intersections=true,
clip width=3,
flip crossing/.list=2,4,6
([closed]90:2) foreach įn 1,...,5 .. (90-360/5+*̨720/5:1.5) .. (90+*̨720/5:2) (90:2)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (1,1.35) ;
[above] at (0,1) $s_1$;
[above] at (0,2) $s_2$;
[left] at (1.5,.5) $s_3$;
[right] at (2,.5) $s_4$;
[left] at (-1.5,-1.5) $s_5$;
[left] at (-2,.5) $s_6$;
(0,0) node [draw] $x$;
[below] at (0,-2.5)$SP(4_1^k)= 24 u^{t^2}+24 u^t+48 u^2$;
[use Hobby shortcut,scale=1,add arrow/.style=postaction=decorate, decoration=
mark=at position .5 with [scale=1.5,>=stealth]>]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=7,4,9,
clip width=4,
only when rendering/.style=
([closed]0,2)..(-.5,1.75)..(0,0)..(.5,-.2)..(2,-2)..(-.5,-1)..(-2,1.5)..(0,1.5)..(2,1.5)..(.5,-1)..(-2,-2)..(-.5,-.2)..(0,0)..(.5,1.75)..(0,2)[add arrow];
[circle,draw=black, fill=black, inner sep=0pt,minimum size=6pt] (a) at (0,0) ;
[right] at (-1,0) $s_1$;
[right] at (-1,.5) $s_2$;
[right] at (.5,0) $s_3$;
[right] at (.5,.5) $s_4$;
[left] at (-2,.5) $s_5$;
[right] at (2,.5) $s_6$;
[left] at (-2,-1) $s_7$;
(0,-.5) node [draw] $x$;
[below] at (0,-2.5)$SP(5_4^k)=48 u^{t^4}+24 u^{t^2}+24 u^t $;
Singular knots $4_1^k$ and $5_4^k$ and corresponding $SP$ invariant.
Let $(S,X,*,R_1,R_2,\cdot)$ with $S=\mathbb{Z}_{12}$, $X=\mathbb{Z}_8$, $x*y = 5x-4y = x \bar{*} y$, $R_1(x,y) = 5x+10y$, $R_2(x,y) = 2x+y$, and shadow matrix
\[ \left[
\begin{array}{cccccccccccc}
3 & 7 & 7 & 7 & 3 & 7 & 7 & 7 & 3 & 7 & 7 & 7 \\
2 & 6 & 6 & 6 & 2 & 6 & 6 & 6 & 2 & 6 & 6 & 6 \\
1 & 5 & 5 & 5 & 1 & 5 & 5 & 5 & 1 & 5 & 5 & 5 \\
0 & 4 & 4 & 4 & 0 & 4 & 4 & 4 & 0 & 4 & 4 & 4 \\
7 & 3 & 3 & 3 & 7 & 3 & 3 & 3 & 7 & 3 & 3 & 3 \\
6 & 2 & 2 & 2 & 6 & 2 & 2 & 2 & 6 & 2 & 2 & 2 \\
5 & 1 & 1 & 1 & 5 & 1 & 1 & 1 & 5 & 1 & 1 & 1 \\
4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\
\end{array}
\right]. \]
We compute the shadow polynomial for the following singular knots derived from the classical trefoil knot
[scale=.55,use Hobby shortcut]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)[decoration=markings,mark=at position .5 with
[scale=3,>=stealth]<,postaction=decorate]..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5);
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $s_1$;
[above] at (0,1.5) $s_2$;
[right] at (3,-3.5) $s_3$;
[left] at (-3,-3.5) $s_4$;
(0,-2) node [draw] $x$;
(-2,-3) node [draw] $x\cdot s_2$;
(2,-3) node [draw] $x \cdot s_4$;
(0,.5) node [draw] $x \cdot s_1$;
(-4.5,0) node [draw] $(x \cdot s_1)\cdot s_2$;
at (0,-5) $K_1$;
[scale=.55,use Hobby shortcut]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)[decoration=markings,mark=at position .5 with
[scale=3,>=stealth]<,postaction=decorate]..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5);
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (-1.24,-1.35) ;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[above] at (0,-1) $s_1$;
[above] at (0,1.5) $s_2$;
[right] at (3,-3.5) $s_3$;
[left] at (-3,-3.5) $s_4$;
[right] at (-.55,-2.9) $s_5$;
(0,-2) node [draw] $x$;
(-2,-3) node [draw] $x\cdot s_5$;
(2,-3) node [draw] $x \cdot s_4$;
(0,.5) node [draw] $x \cdot s_1$;
(-4.5,0) node [draw] $(x \cdot s_1)\cdot s_2$;
at (0,-5) $K_2$;
[scale=.55,use Hobby shortcut]
consider self intersections=true,
ignore endpoint intersections=false,
flip crossing/.list=4,
clip width=5,
only when rendering/.style=
([closed]0,1.5)[decoration=markings,mark=at position .5 with
[scale=3,>=stealth]<,postaction=decorate]..(-1.2,-1.5).. (3,-3.5) ..(0,-1.2) ..(-3,-3.5) ..(1.2,-1.5)..(0,1.5);
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (-1.24,-1.35) ;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (1.24,-1.35) ;
[circle,draw=black, fill=black, inner sep=0pt,minimum size=8pt] (a) at (0,-3.6) ;
[above] at (0,-1) $s_1$;
[above] at (0,1.5) $s_2$;
[right] at (3,-3.5) $s_3$;
[left] at (-3,-3.5) $s_4$;
[right] at (-.55,-2.9) $s_5$;
[right] at (1,-2.1) $s_6$;
(0,-2) node [draw] $x$;
(-2,-3) node [draw] $x\cdot s_5$;
(2,-3) node [draw] $x \cdot s_6$;
(0,.5) node [draw] $x \cdot s_1$;
(-4.5,0) node [draw] $(x \cdot s_1)\cdot s_2$;
at (0,-5) $K_3$;
Colorings of the singular knots derived from the trefoil.
$\#\textup{Col}_S$ $\#\textup{Col}_{(S,X)}$ $SP$ Singular knot
4 32 $4 u^{t^2}+4 u^t+24 u^2$ $K_1$, $K_3$
$4 u^t+8 u^{2 t}+8 u^3+12 u^2$ $K_2$
In this example we have a collection of singular knots all with the same singquandle counting invariant with respect to $X$. If we then compute the shadow singquandle polynomial invariant we can distinguish $K_2$ from $K_1$ and $K_3$.
§ ACKNOWLEDGMENTS
The authors of this paper would like to thank Sam Nelson for fruitful discussion. The authors would also like to thank Hamza Elhamdadi for providing python code to verify the examples in Section 7.
author=Bataineh, Khaled,
author=Elhamdadi, Mohamed,
author=Hajij, Mustafa,
author=Youmans, William,
title=Generating sets of Reidemeister moves of oriented singular links
and quandles,
journal=J. Knot Theory Ramifications,
pages=1850064, 15,
author=Birman, Joan S.,
author=Lin, Xiao-Song,
title=Knot polynomials and Vassiliev's invariants,
journal=Invent. Math.,
author=Carter, J. Scott,
author=Elhamdadi, Mohamed,
author=Saito, Masahico,
title=Homology theory for the set-theoretic Yang-Baxter equation and
knot invariants from generalizations of quandles,
journal=Fund. Math.,
author=Carter, J. Scott,
author=Jelsovsky, Daniel,
author=Kamada, Seiichi,
author=Langford, Laurel,
author=Saito, Masahico,
title=Quandle cohomology and state-sum invariants of knotted curves and
journal=Trans. Amer. Math. Soc.,
author=Ceniceros, Jose,
author=Churchill, Indu R. U.,
author=Elhamdadi, Mohamed,
author=Hajij, Mustafa,
title=Cocycle Invariants and Oriented Singular Knots,
journal= arXiv:2009.07950 (2020), 15 pp.
author=Chang, Wesley,
author=Nelson, Sam,
title=Rack shadows and their invariants,
journal=J. Knot Theory Ramifications,
author=Ceniceros, Jose,
author=Churchill, Indu R. U.,
author=Elhamdadi, Mohamed,
title=Polynomial Invariants of Singular Knots and Links,
journal=to appear in J. Knot Theory Ram.; arXiv:2010.13295.
author=Ceniceros, Jose,
author=Nelson, Sam,
title=Virtual Yang-Baxter cocycle invariants,
journal=Trans. Amer. Math. Soc.,
author=Churchill, Indu R. U.,
author=Elhamdadi, Mohamed,
author=Hajij, Mustafa,
author=Nelson, Sam,
title=Singular Knots and Involutive Quandles,
journal= J. Knot Theory Ramifications 26 (2017), no. 14, 1750099, 14 pp.
review=1362791 (96i:57015),
author=Elhamdadi, Mohamed,
title=Distributivity in quandles and quasigroups,
title=Algebra, geometry and mathematical physics,
series=Springer Proc. Math. Stat.,
publisher=Springer, Heidelberg,
author=Elhamdadi, Mohamed,
author=Fernando, Neranga,
author=Tsvelikhovskiy, Boris,
title=Ring theoretic aspects of quandles,
journal=J. Algebra,
author=Elhamdadi, Mohamed,
author=Moutuou, El-kaïoum M.,
title=Finitely stable racks and rack representations,
journal=Comm. Algebra,
author=Elhamdadi, Mohamed,
author=Nelson, Sam,
title=Quandles—an introduction to the algebra of knots,
series=Student Mathematical Library,
publisher=American Mathematical Society, Providence, RI,
author=Jones, V. F. R.,
title=Hecke algebra representations of braid groups and link
journal=Ann. of Math. (2),
author=Joyce, David,
title=A classifying invariant of knots, the knot quandle,
journal=J. Pure Appl. Algebra,
title=An invariant for singular knots,
author=Juyumaya, Jesús,
author=Lambropoulou, Sofia,
journal=Journal of Knot Theory and Its Ramifications,
publisher=World Scientific
author=Matveev, S. V.,
title=Distributive groupoids in knot theory,
journal=Mat. Sb. (N.S.),
pages=78–88, 160,
author=Nelson, Sam,
title=Generalized quandle polynomials,
journal=Canad. Math. Bull.,
author=Nelson, Sam,
title=A polynomial invariant of finite quandles,
journal=J. Algebra Appl.,
author=Oyamaguchi, Natsumi,
title=Enumeration of spatial 2-bouquet graphs up to flat vertex isotopy,
journal=Topology Appl.,
number=part B,
part=part B,
author=Vassiliev, V. A.,
title=Cohomology of knot spaces,
title=Theory of singularities and its applications,
series=Adv. Soviet Math.,
publisher=Amer. Math. Soc., Providence, RI,
|
# AI Choreographer: Music Conditioned 3D Dance Generation with AIST++
Ruilong Li∗1 Shan Yang∗2 David A. Ross2 Angjoo Kanazawa2,3
1University of Southern California 2Google Research 3University of California,
Berkeley
###### Abstract
We present AIST++, a new multi-modal dataset of 3D dance motion and music,
along with FACT, a Full-Attention Cross-modal Transformer network for
generating 3D dance motion conditioned on music. The proposed AIST++ dataset
contains 5.2 hours of 3D dance motion in 1408 sequences, covering 10 dance
genres with multi-view videos with known camera poses—the largest dataset of
this kind to our knowledge. We show that naively applying sequence models such
as transformers to this dataset for the task of music conditioned 3D motion
generation does not produce satisfactory 3D motion that is well correlated
with the input music. We overcome these shortcomings by introducing key
changes in its architecture design and supervision: FACT model involves a deep
cross-modal transformer block with full-attention that is trained to predict
$N$ future motions. We empirically show that these changes are key factors in
generating long sequences of realistic dance motion that are well-attuned to
the input music. We conduct extensive experiments on AIST++ with user studies,
where our method outperforms recent state-of-the-art methods both
qualitatively and quantitatively. The code and the dataset can be found at:
https://google.github.io/aichoreographer. †† ∗ equal contribution. Work
performed while Ruilong was an intern at Google.
Figure 1: AI Choreographer. We present a new 3D dance dataset, AIST++, which
contains $5.2$ hours of 3D motion reconstructed from real dancers paired with
music (left) and a novel Full-Attention Cross-modal Transformer (FACT) network
that can generate realistic 3D dance motion with global translation
conditioned on music (right). We output our 3D motion in representations that
allow for instant motion retargeting to a novel character. Here we use a
character from Mixamo [1]
## 1 Introduction
Figure 2: Cross-Modal Music Conditioned 3D Motion Generation Overview. Our
proposed a Full-Attention Cross-modal Transformer (FACT) network (details in
Figure 3) takes in a music piece and a $2$-second sequence of seed motion,
then auto-regressively generates long-range future motions that correlates
with the input music.
The ability to dance by composing movement patterns that align to musical
beats is a fundamental aspect of human behavior. Dancing is an universal
language found in all cultures [50], and today, many people express themselves
through dance on contemporary online media platforms. The most watched videos
on YouTube are dance-centric music videos such as “Baby Shark Dance”, and
“Gangnam Style” [75], making dance a more and more powerful tool to spread
messages across the internet. However, dancing is a form of art that requires
practice—even for humans, professional training is required to equip a dancer
with a rich repertoire of dance motions to create an expressive choreography.
Computationally, this is even more challenging as the task requires the
ability to generate a continuous motion with high kinematic complexity that
captures the non-linear relationship with the accompanying music.
In this work, we address these challenges by presenting a novel Full Attention
Cross-modal Transformer (FACT) network, which can robustly generate realistic
3D dance motion from music, along with a large-scale multi-modal 3D dance
motion dataset, AIST++, to train such a model. Specifically, given a piece of
music and a short (2 seconds) seed motion, our model is able to generate a
long sequence of realistic 3D dance motions. Our model effectively learns the
music-motion correlation and can generate dance sequences that varies for
different input music. We represent dance as a 3D motion sequence that
consists of joint rotation and global translation, which enables easy transfer
of our output for applications such as motion retargeting as shown in Figure
1.
In order to generate 3D dance motion from music, we propose a novel Full
Attention Cross-modal Transformer (FACT) model, which employs an audio
transformer and seed motion transformer to encode the inputs, which are then
fused by a cross-modal transformer that models the distribution between audio
and motion. This model is trained to predict $N$ future motion sequences and
at test time is applied in an auto-regressive manner to generate continuous
motion. The success of our model relies on three key design choices: 1) the
use of full-attention in an auto-regressive model, 2) future-N supervision,
and 3) early fusion of two modalities. The combination of these choices is
critical for training a model that can generate a long realistic dance motion
that is attuned to the music. Although prior work has explored using
transformers for motion generation [3], we find that naively applying
transformers to the 3D dance generation problem without these key choices does
not lead to a very effective model.
In particular, we notice that because the context window in the motion domain
is significantly smaller than that of language models, it is possible to apply
full-attention transformers in an auto-regressive manner, which leads to a
more powerful model. It is also critical that the full-attention transformer
is trained to predict $N$ possible future motions instead of one. These two
design choices are key for preventing 3D motion from freezing or drifting
after several auto-regressive steps as reported in prior works on 3D motion
generation [4, 3]. Our model is trained to predict 20 future frames, but it is
able to produce realistic 3D dance motion for over 1200 frames at test time.
We also show that fusing the two modalities early, resulting in a deep cross-
modal transformer, is important for training a model that generates different
dance sequences for different music.
In order to train the proposed model, we also address the problem of data.
While there are a few motion capture datasets of dancers dancing to music,
collecting mocap data requires heavily instrumented environments making these
datasets severely limited in the number of available dance sequences, dancer
and music diversity. In this work, we propose a new dataset called AIST++,
which we build from the existing multi-view dance video database called AIST
[82]. We use the multi-view videos to recover reliable 3D motion from this
data. We will release code and this dataset for research purposes, where
AIST++ can be a new benchmark for the task of 3D dance generation conditioned
on music.
In summary, our contributions are as follows:
* •
We propose Full Attention Cross-Modal Transformer model, FACT, which can
generate a long sequence of realistic 3D dance motion that is well correlated
with the input music.
* •
We introduce AIST++ dataset containing 5.2 hours of 3D dance motions
accompanied with music and multi-view images, which to our knowledge is the
largest dataset of such kind.
* •
We provide extensive evaluations validating our design choices and show that
they are critical for high quality, multi-modal, long motion sequence
generation.
## 2 Related Work
##### 3D Human Motion Synthesis
The problem of generating realistic and controllable 3D human motion sequences
has long been studied. Earlier works employ statistical models such as kernel-
based probability distribution [64, 10, 25, 11] to synthesize motion, but
abstract away motion details. Motion graphs [53, 7, 47] address this problem
by generating motions in a non-parametric manner. Motion graph is a directed
graph constructed on a corpus of motion capture data, where each node is a
pose and the edges represent the transition between poses. Motion is generated
by a random walk on this graph. A challenge in motion graph is in generating
plausible transition that some approaches address via parameterizing the
transition [30]. With the development in deep learning, many approaches
explore the applicability of neural networks to generate 3D motion by training
on a large-scale motion capture dataset, where network architectures such as
CNNs [35, 34], GANs [31], RBMs [80], RNNs [24, 4, 40, 27, 16, 18, 88, 12, 87]
and Transformers [3, 9] have been explored. Auto-regressive models like RNNs
and vanilla Transformers are capable of generating unbounded motion in theory,
but in practice suffer from regression to the mean where motion “freezes”
after several iterations, or drift to unnatural motions [4, 3]. Some works [8,
56, 49] propose to ease this problem by periodically using the network’s own
outputs as inputs during training. Phase-functioned neural networks and it’s
variations [94, 33, 73, 74] address this issue via conditioning the network
weights on phase, however, they do not scale well to represent a wide variety
of motion.
##### Audio To Human Motion Generation
Audio to motion generation has been studied in 2D pose context either in
optimization based approach [81], or learning based approaches [52, 72, 51,
67, 68, 21] where 2D pose skeletons are generated from a conditioning audio.
Training data for 2D pose and audio is abundant thanks to the high reliability
of 2D pose detectors [14]. However, predicting motion in 2D is limited in its
expressiveness and potential for downstream applications. For 3D dance
generation, earlier approaches explore matching existing 3D motion to music
[71] using motion graph based approach [20]. More recent approach employ LSTMs
[5, 79, 90, 97, 42], GANs [51, 78, 28], transformer encoder with RNN decoder
[36] or convolutional [2, 92] sequence-to-sequence models. Concurrent to our
work, Chen _et al_. [15] proposed a method that is based on motion graphs
with learned embedding space. Many prior works [72, 68, 42, 28, 92] solve this
problem by predicting future motion deterministically from audio without seed
motion. When the same audio has multiple corresponding motions, which often
occurs in dance data, these methods collapse to predicting a mean pose. In
contrast, we formulate the problem with seed motion as in [55, 96], which
allows generation of multiple motion from the same audio even with a
deterministic model.
Closest to our work is that of Li _et al_. [55], which also employ
transformer based architecture but only on audio and motion. Furthermore,
their approach discretize the output joint space in order to account for
multi-modality, which generates unrealistic motion. In this work we introduce
a novel full-attention based cross-modal transformer (FACT model) for audio
and motion, which can not only preserve the correlation between music and 3D
motion better, but also generate more realistic long 3D human motion with
global translation. One of the biggest bottleneck in 3D dance generation
approaches is that of data. Recent work of Li _et al_. [55] reconstruct 3D
motion from dance videos on the Internet, however the data is not public.
Further, using 3D motion reconstructed from monocular videos may not be
reliable and lack accurate global 3D translation information. In this work we
also reconstruct the 3D motion from 2D dance video, but from multi-view video
sequences, which addresses these issues. While there are many large scale 3D
motion capture datasets [39, 59, 1, 37], mocap dataset of 3D dance is quite
limited as it requires heavy instrumentation and expert dancers for capture.
As such, many of these previous works operate on either small-scale or private
motion capture datasets [79, 5, 96]. We compare our proposed dataset with
these public datasets in Table 1.
##### Cross-Modal Sequence-to-Sequence Generation
Beyond of the scope of human motion generation, our work is closely related to
the research of using neural network on cross-modal sequence to sequence
generation task. In natural language processing and computer vision, tasks
like text to speech (TTS) [69, 41, 43, 83] and speech to gesture [22, 28, 23],
image/video captioning (pixels to text) [13, 44, 58, 48] involve solving the
cross-modal sequence to sequence generation problem. Initially, combination of
CNNs and RNNs [86, 85, 91, 93] were prominent in approaching this problem.
More recently, with the development of attention mechanism [84], transformer
based networks achieve top performance for visual-text [95, 77, 19, 54, 38,
76, 76], visual-audio [26, 89] cross-modal sequence to sequence generation
task. Our work explores audio to 3D motion in a transformer based
architecture. While all cross-modal problems induce its own challenges, the
problem of music to 3D dance is uniquely challenging in that there are many
ways to dance to the same music and that the same dance choreography may be
used for multiple music. We hope the proposed AIST++ dataset advances research
in this relatively under-explored problem.
## 3 AIST++ Dataset
Dataset | Music | 3D $\text{Joint}_{\text{pos}}$ | 3D $\text{Joint}_{\text{rot}}$ | 2D Kpt | Views | Images | Genres | Subjects | Sequences | Seconds
---|---|---|---|---|---|---|---|---|---|---
AMASS[59] | ✗ | ✓ | ✓ | ✗ | 0 | 0 | 0 | 344 | 11265 | 145251
Human3.6M[39] | ✗ | ✓ | ✓ | ✓ | 4 | 3.6M | 0 | 11 | 210 | 71561
Dance with Melody[79] | ✓ | ✓ | ✗ | ✗ | 0 | 0 | 4 | - | 61 | 5640
GrooveNet [5] | ✓ | ✓ | ✗ | ✗ | 0 | 0 | 1 | 1 | 2 | 1380
DanceNet [96] | ✓ | ✓ | ✗ | ✗ | 0 | 0 | 2 | 2 | 2 | 3472
EA-MUD [78] | ✓ | ✓ | ✗ | ✗ | 0 | 0 | 4 | - | 17 | 1254
AIST++ | ✓ | ✓ | ✓ | ✓ | 9 | 10.1M | 10 | 30 | 1408 | 18694
Table 1: 3D Dance Datasets Comparisons. The proposed AIST++ dataset is the
largest dataset with 3D dance motion paired with music. We also have the
largest variety of subjects and genres. Furthermore, our dataset is the only
one that comes with image frames, as other dance datasets only contain motion
capture dataset. We include popular 3D motion dataset without any music in the
first two rows for reference.
Data Collection We generate the proposed 3D motion dataset from an existing
database called AIST Dance Database [82]. AIST is only a collection of videos
without any 3D information. Although it contains multi-view videos of dancers,
these cameras are not calibrated, making 3D reconstruction of dancers a non-
trivial effort. We recover the camera calibration parameters and the 3D human
motion in terms of SMPL parameters. Please find the details of this algorithm
in the Appendix. Although we adopt the best practices in reconstructing this
data, no code base exist for this particular problem setup and running this
pipeline on a large-scale video dataset requires non-trivial amount of compute
and effort. We will make the 3D data and camera parameters publicly available,
which allows the community to benchmark on this dataset on an equal footing.
##### Dataset Description
Resulting AIST++ is a large-scale 3D human dance motion dataset that contains
a wide variety of 3D motion paired with music. It has the following extra
annotations for each frame:
* •
$9$ views of camera intrinsic and extrinsic parameters;
* •
$17$ COCO-format[70] human joint locations in both 2D and 3D;
* •
$24$ SMPL [57] pose parameters along with the global scaling and translation.
Besides the above properties, AIST++ dataset also contains multi-view
synchronized image data unlike prior 3D dance dataset, making it useful for
other research directions such as 2D/3D pose estimation. To our knowledge,
AIST++ is the largest 3D human dance dataset with $\mathbf{1408}$ sequences,
$\mathbf{30}$ subjects and $\mathbf{10}$ dance genres with basic and advanced
choreographies. See Table. 1 for comparison with other 3D motion and dance
datasets. AIST++ is a complementary dataset to existing 3D motion dataset such
as AMASS [59], which contains only $17.8$ minutes of dance motions with no
accompanying music.
Owing to the richness of AIST, AIST++ contains 10 dance genres: Old School
(Break, Pop, Lock and Waack) and New School (Middle Hip-hop, LA-style Hip-hop,
House, Krump, Street Jazz and Ballet Jazz). Please see the Appendix for more
details and statistics. The motions are equally distributed among all dance
genres, covering wide variety of music tempos denoted as beat per minute
(BPM)[61]. Each genre of dance motions contains $85\%$ of basic choreographies
and $15\%$ of advanced choreographies, in which the former ones are those
basic short dancing movements while the latter ones are longer movements
freely designed by the dancers. However, note that AIST is an instructional
database and records multiple dancers dancing the same choreography for
different music with varying BPM, a common practice in dance. This posits a
unique challenge in cross-modal sequence-to-sequence generation. We carefully
construct non-overlapping train and val subsets on AIST++ to make sure neither
choreography nor music is shared across the subsets.
## 4 Music Conditioned 3D Dance Generation
Here we describe our approach towards the problem of music conditioned 3D
dance generation. Specifically, given a $2$-second seed sample of motion
represented as $\mathbf{X}=(x_{1},\dots,x_{T})$ and a longer conditioning
music sequence represented as $\mathbf{Y}=({y}_{1},\dots,{y}_{T^{\prime}})$,
the problem is to generate a sequence of future motion
${\mathbf{X^{\prime}}}=({x}_{T+1},\dots,{x}_{T^{\prime}})$ from time step
$T+1$ to $T^{\prime}$, where $T^{\prime}\gg T$.
Figure 3: FACT Model Details. (a) The structure of the audio/motion/cross-
modal transformer with $N$ attention layers. (b) Attention and supervision
mechanism as a simplified two-layer model. Models like GPT [66] and the motion
generator of [55] use causal attention (left) to predict the immediate next
output for each input nodes. We employ full-attention and predict $n$ future
from the last input timestamp $m$ (right). The dots on the bottom row are the
input tensors, which are computed into context tensors through causal (left)
and full (right) attention transformer layer. The output (predictions) are
shown on the top. We empirically show that these design choices are critical
in generating non-freezing, more realistic motion sequences.
##### Preliminaries
Transformer [84] is an attention based network widely applied in natural
language processing. A basic transformer building block (shown in of Figure 3
(a)) has multiple layers with each layer composed of a multi-head attention-
layer (Attn) followed by a feed forward layer (FF). The multi-head attention-
layer embeds input sequence $\mathbf{X}$ into an internal representation often
referred to as the context vector $\mathbf{C}$. Specifically, the output of
the attention layer, the context vector $\mathbf{C}$ is computed using the
query vector ${\mathbf{Q}}$ and the key ${\mathbf{K}}$ value ${\mathbf{V}}$
pair from input with or without a mask ${\mathbf{M}}$ via,
$\displaystyle{\mathbf{C}}$
$\displaystyle=\text{FF}(\text{Attn}(\mathbf{Q,K,V,M}))$
$\displaystyle=\text{FF}(\text{softmax}\Bigg{(}\frac{\mathbf{QK}^{T}+\mathbf{M}}{\sqrt{D}}\Bigg{)}\mathbf{V}),$
$\displaystyle\mathbf{Q}$
$\displaystyle=\mathbf{X}\mathbf{W}^{Q},\mathbf{K}=\mathbf{X}\mathbf{W}^{K},\mathbf{V}=\mathbf{X}\mathbf{W}^{V}$
(1)
where $D$ is the number of channels in the attention layer and $\mathbf{W}$
are trainable weights. The design of the mask function is a key parameter in a
transformer. In natural language generation, causal models such as GPT [66]
uses an upper triangular look-ahead mask $\mathbf{M}$ to enable causal
attention where each token can only look at past inputs. This allows efficient
inference at test time, since intermediate context vectors do not need to be
recomputed, especially given the large context window in these models (2048).
On the other hand, models like BERT [17] employ full-attention for feature
learning, but rarely are these models employed in an auto-regressive manner,
due to its inefficiency at test time.
### 4.1 Full Attention Cross-Modal Transformer
We propose Full Attention Cross-Modal Transformer (FACT) model for the task of
3D dance motion generation. Given the seed motion ${\mathbf{X}}$ and audio
features $\mathbf{Y}$, FACT first encodes these inputs using a motion
transformer $f_{\text{mot}}$ and audio transformer $f_{\text{audio}}$ into
motion and audio embeddings ${\mathbf{h}^{x}}_{1:T}$ and
${\mathbf{h}^{y}}_{1:T^{\prime}}$ respectively. These are then concatenated
and sent to a cross-modal transformer $f_{\text{cross}}$, which learns the
correspondence between both modalities and generates $N$ future motion
sequences $\mathbf{X^{\prime}}$, which is used to train the model in a self-
supervised manner. All three transformers are jointly learned in an end-to-end
manner. This process is illustrated in Figure 2. At test time, we apply this
model in an auto-regressive framework, where we take the first predicted
motion as the input of the next generation step and shift all conditioning by
one.
FACT involves three key design choices that are critical for producing
realistic 3D dance motion from music. First, all of the transformers use full-
attention mask. We can still apply this model efficiently in an auto-
regressive framework at test time, since our context window is not
prohibitively large (240). The full-attention model is more expressive than
the causal model because internal tokens have access to all inputs. Due to
this full-attention design, we train our model to only predict the unseen
future after the context window. In particular, we train our model to predict
$N$ futures beyond the current input instead of just $1$ future motion. This
encourages the network to pay more attention to the temporal context, and we
experimentally validate that this is a key factor training a model that does
not suffer from motion freezing or diverging after a few generation steps.
This attention design is in contrast to prior work that employ transformers
for the task of 3D motion [3] or dance generation [55], which applies GPT [66]
style causal transformer trained to predict the immediate next future token.
We illustrate this difference in Figure 3 (b).
Lastly, we fuse the two embeddings early and employ a deep 12-layer cross-
modal transformer module. This is in contrast to prior work that used a single
MLP to combine the audio and motion embeddings [55], and we find that deep
cross-modal module is essential for training a model that actually pays
attention to the input music. This is particularly important as in dance,
similar choreography can be used for multiple music. This also happens in AIST
dataset, and we find that without a deep cross-modal module, the network is
prone to ignoring the conditioning music. We experimentally validate this in
Section 5.2.3.
## 5 Experiments
| Motion Quality | Motion Diversity | Motion-Music Corr | User Study
---|---|---|---|---
| | $\text{FID}_{k}\downarrow$
---
| $\text{FID}_{g}\downarrow$
---
| $\text{Dist}_{k}\uparrow$
---
| $\text{Dist}_{g}\uparrow$
---
| $\text{BeatAlign}\uparrow$
---
| FACT WinRate$\downarrow$
---
AIST++ | – | – | 9.057 | 7.556 | 0.292 | –
AIST++ (random) | – | – | – | – | 0.213 | 25.4%
Li _et al_. [55] | 86.43 | 20.58 | 6.85* | 4.93 | 0.232 | 80.6%
Dancenet [96] | 69.18 | 17.76 | 2.86 | 2.72 | 0.232 | 71.1%
DanceRevolution [36] | 73.42 | 31.01 | 3.52 | 2.46 | 0.220 | 77.0%
FACT (ours) | 35.35 | 12.40 | 5.94 | 5.30 | 0.241 | –
Table 2: Conditional Motion Generation Evaluation on AIST++ dataset. Comparing
to the three recent state-of-the-art methods, our model generates motions that
are more realistic, better correlated with input music and more diversified
when conditioned on different music. *Note Li _et al_. [55]’s generated
motions are discontinuous making its average kinetic feature distance
($\text{FID}_{k}$) abnormally high.
### 5.1 AIST++ Motion Quality Validation
We first carefully validate the quality of our 3D motion reconstruction.
Possible error sources that may affect the quality of our 3D reconstruction
include inaccurate 2D keypoints detection and the estimated camera parameters.
As there is no 3D ground-truth for AIST dataset, our validation here is based-
on the observation that the re-projected 2D keypoints should be consistent
with the predicted 2D keypoints which have high prediction confidence in each
image. We use the 2D mean per joint position error MPJPE-2D, commonly used for
3D reconstruction quality measurement [46, 39, 65]) to evaluate the
consistency between the predicted 2D keypoints and the reconstructed 3D
keypoints along with the estimated camera parameters. Note we only consider 2D
keypoints with prediction confidence over 0.5 to avoid noise. The MPJPE-2D of
our entire dataset is $6.2$ pixels on the $1920\times 1080$ image resolution,
and over $86\%$ of those has less than $10$ pixels of error. Besides, we also
calculate the PCKh metric introduced in [6] on our AIST++. The<EMAIL_ADDRESS>on the
whole set is $98.7\%$, meaning the reconstructed 3D keypoints are highly
consistent with the predicted 2D keypoints. Please refer to the Appendix for
detailed analysis of MPJPE-2D and PCKh on AIST++.
### 5.2 Music Conditioned 3D Motion Generation
#### 5.2.1 Experimental Setup
##### Dataset Split
All the experiments in this paper are conducted on our AIST++ dataset, which
to our knowledge is the largest dataset of this kind. We split AIST++ into
_train_ and _test_ set, and report the performance on the _test_ set only. We
carefully split the dataset to make sure that the music and dance motion in
the _test_ set does not overlap with that in the _train_ set. To build the
_test_ set, we first select one music piece from each of the 10 genres. Then
for each music piece, we randomly select two dancers, each with two different
choreographies paired with that music, resulting in total $40$ unique
choreographies in the _test_ set. The _train_ set is built by excluding all
test musics and test choreographies from AIST++, resulting in total $329$
unique choreographies in the _train_ set. Note that in the test set we
intentionally pick music pieces with different BPMs so that it covers all
kinds of BPMs ranging from $80$ to $135$ in AIST++.
##### Implementation Details
In our main experiment, the input of the model contains a seed motion sequence
with $120$ frames (2 seconds) and a music sequence with $240$ frames (4
seconds), where the two sequences are aligned on the first frame. The output
of the model is the future motion sequence with $N=20$ frames supervised by
$L2$ loss. During inference we continually generate future motions in a auto-
regressive manner at $60$ FPS, where only the first predicted motion is kept
in every step. We use the publicly available audio processing toolbox Librosa
[60] to extract the music features including: 1-dim _envelope_ , 20-dim _MFCC_
, 12-dim _chroma_ , 1-dim _one-hot peaks_ and 1-dim _one-hot beats_ ,
resulting in a 35-dim music feature. We combine the 9-dim rotation matrix
representation for all $24$ joints, along with a 3-dim global translation
vector, resulting in a 219-dim motion feature. Both these raw audio and motion
features are first embedded into $800$-dim hidden representations with linear
layers, then added with learnable positional encoding, before they were input
into the transformer layers. All the three (audio, motion, cross-modal)
transformers have $10$ attention heads with $800$ hidden size. The number of
attention layers in each transformer varies based on the experiments, as
described in Sec. 5.2.3. We disregard the last linear layer in the
audio/motion transformer and the positional encoding in the cross-modal
transformer, as they are not necessary in the FACT model. All our experiments
are trained with $16$ batch size using Adam [45] optimizer. The learning rate
starts from $1\mathrm{e}{-4}$ and drops to {$1\mathrm{e}{-5}$,
$1\mathrm{e}{-6}$} after {$60k$, $100k$} steps. The training finishes after
$300k$, which takes $3$ days on $4$ TPUs. For baselines, we compare with the
latest work on 3D dance generation that take music and seed motion as input,
including Dancenet [96] and Li _et al_. [55]. For a more comprehensive
evaluation we also compare with the recent state-of-the-art 2D dance
generation method DanceRevolution [36]. We adapt this work to output 3D joint
locations which can be directly compared with our results quantitatively,
though joint locations do not allow immediate re-targeting. We train and test
these baselines on the same dataset with ours using the _official_ code
provided by the authors.
#### 5.2.2 Quantitative Evaluation
In this section, we evaluate our proposed model FACT on the following aspects:
(1) motion quality, (2) generation diversity and (3) motion-music correlation.
Experiments results (shown in Table 2) show that our model out-performs state-
of-the-art methods [55, 36, 96], on those criteria.
##### Motion Quality
Similar to prior works [55, 36], we evaluate the generated motion quality by
calculating the distribution distance between the generated and the ground-
truth motions using Frechet Inception Distance (FID) [32] on the extracted
motion features. As prior work used motion-encoders that are not public, we
measure FID with two well-designed motion feature extractors [62, 63]
implemented in _fairmotion_ [29]: (1) a geometric feature extractor that
produces a boolean vector $\mathbf{z}_{g}\in\mathbb{R}^{33}$ expressing
geometric relations between certain body points in the motion sequence
$X\in\mathbb{R}^{T\times N\times 3}$, (2) a kinetic feature extractor [63]
that maps a motion sequence $X$ to $\mathbf{z}_{k}\in\mathbb{R}^{72}$, which
represents the kinetic aspects of the motion such as velocity and
accelerations. We denote the FID based on these geometric and kinetic features
as $\text{FID}_{g}$and $\text{FID}_{k}$, respectively. The metrics are
calculated between the real dance motion sequences in AIST++ test set and $40$
generated motion sequences each with $T=1200$ frames (20 secs). As shown in
Table 2, our generated motion sequences have a much closer distribution to
ground-truth motions compared with the three baselines. We also visualize the
generated sequences from the baselines in our supplemental video.
Figure 4: Diverse Generation Results. Here we visualize 4 different dance
motions generated using different music but the same seed motion. On the left
we illustrate the 2 second seed motion and on the right we show the generated
3D dance sequences subsampled by 2 seconds. For rows top to bottom, the genres
of the conditioning music are: Break, Ballet Jazz, Krump and Middle Hip-hop.
Note that the seed motion come from hip-hop dance.Our model is able to adapt
the dance style when given a more modern dance music (second row: Ballet
Jazz). Please see more results in the supplementary video.
##### Generation Diversity
We also evaluate our model’s ability to generate diverse dance motions when
given various input music compared with the baseline methods. Similar to the
prior work [36], we calculate the average Euclidean distance in the feature
space across $40$ generated motions on the AIST++ _test_ set to measure the
diversity. The motion diversity in the geometric feature space and in the
kinetic feature space are noted as $\text{Dist}_{m}$and $\text{Dist}_{k}$,
respectively. Table 2 shows that our method generates more diverse dance
motions comparing to the baselines except Li _et al_. [55], which discretizes
the motion, leading to discontinuous outputs that results in high
$\text{Dist}_{k}$. Our generated diverse motions are visualized in Figure 4.
##### Motion-Music Correlation
Further, we evaluate how much the generated 3D motion correlates to the input
music. As there is no well-designed metric to measure this property, we
propose a novel metric, Beat Alignment Score (BeatAlign), to evaluate the
motion-music correlation in terms of the similarity between the kinematic
beats and music beats. The music beats are extracted using _librosa_ [60] and
the kinematic beats are computed as the local minima of the kinetic velocity,
as shown in Figure 5. The Beat Alignment Score is then defined as the average
distance between every kinematic beat and its nearest music beat.
Specifically, our Beat Alignment Score is defined as:
$\text{BeatAlign}=\frac{1}{m}\sum_{i=1}^{m}{\exp(-\frac{\min_{\forall
t_{j}^{y}\in B^{y}}||t_{i}^{x}-t_{j}^{y}||^{2}}{2\sigma^{2}})}$ (2)
where $B^{x}=\\{t_{i}^{x}\\}$ is the kinematic beats, $B^{y}=\\{t_{j}^{y}\\}$
is the music beats and $\sigma$ is a parameter to normalize sequences with
different FPS. We set $\sigma=3$ in all our experiments as the FPS of all our
experiments sequences is 60. A similar metric Beat Hit Rate was introduced in
[51, 36], but this metric requires a dataset dependent handcrafted threshold
to decide the alignment (“hit”) while ours directly measure the distances.
This metric is explicitly designed to be uni-directional as dance motion does
not necessarily _have to_ match with every music beat. On the other hand,
every kinetic beat is expected to have a corresponding music beat. To
calibrate the results, we compute the correlation metrics on the entire AIST++
dataset (upper bound) and on the random-paired data (lower bound). As shown in
Table 2, our generated motion is better correlated with the input music
compared to the baselines. We also show one example in Figure 5 that the
kinematic beats of our generated motion align well with the music beats.
However, when comparing to the real data, all four methods including ours have
a large space for improvement. This reflects that music-motion correlation is
still a challenging problem.
Figure 5: Beats Alignment between Music and Generated Dance. Here we visualize
the kinetic velocity (blue curve) and kinematic beats (green dotted line) of
our generated dance motion, as well as the music beats (orange dotted line).
The kinematic beats are extracted by finding local minima from the kinetic
velocity curve.
#### 5.2.3 Ablation Study
We conduct the following ablation experiments to study the effectiveness of
our key design choices: Full-Attention Future-N supervision, and early cross-
modal fusion. Please refer to our supplemental video for qualitative
comparison. The effectiveness of different model architectures is measured
quantitatively using the motion quality ($\text{FID}_{k}$, $\text{FID}_{g}$)
and the music-motion correlation (BeatAlign) metrics, as shown in Table 4 and
Table 3.
##### Full-Attention Future-N Supervision
Here we dive deep into the attention mechanism and our future-N supervision
scheme. We set up four different settings: causal-attention shift-by-1
supervision, and full-attention with future-$\\{1,10,20\\}$ supervision.
Qualitatively, we find that the motion generated by the causal-attention with
shift-by-1 supervision (as done in [55, 66, 3]) starts to freeze after several
seconds (please see the supplemental video). Similar problem was reported in
the results of [3]. Quantitatively (shown in the Table 3), when using causal-
attention shift-by-1 supervision, the FIDs are large meaning that the
difference between generated and ground-truth motion sequences is substantial.
For the full-attention with future-1 supervision setting, the results rapidly
drift during long-range generation. However, when the model is supervised with
$10$ or $20$ future frames, it pays more attention to the temporal context.
Thus, it learns to generate good quality (non-freezing, non-drifting) long-
range motion.
Attn-Supervision | $\text{FID}_{k}\downarrow$ | $\text{FID}_{g}\downarrow$ | $\text{BeatAlign}\uparrow$
---|---|---|---
Causal-Attn-Shift-by-1 | 111.69 | 21.43 | 0.217
Full-Attn-F1 (FACT-1) | 207.74 | 19.35 | 0.233
Full-Attn-F10 (FACT-10) | 35.10 | 15.17 | 0.239
Full-Attn-F20 (FACT-20) | 35.35 | 12.39 | 0.241
Table 3: Ablation Study on Attention and Supervision Mechanism. Causal-
attention shift-by-1 supervision tends to generate freezing motions in the
long-term. While Full-attention supervised more future frames boost the
ability of generating more realistic dance motions. Figure 6: Attention
Weights Visualization. We compare the attention weights from the last layer of
the (a) 12-layer cross-modal transformer and (b) 1-layer cross-modal
transformer. Deeper cross-modal transformer pays equal attention to motion and
music, while a shallower one pays more attention to motion.
##### Early Cross-Modal Fusion
Here we investigate when to fuse the two input modalities. We conduct
experiments in three settings, (1) _No-Fusion_ : 14-layer motion transformer
only; (2) _Late-Fusion_ : 13-layer motion/audio transformer with 1-layer
cross-modal transformer; (3) _Early-Fusion_ : 2-layer motion/audio transformer
with 12-layer cross-modal transformer. For fair comparison, we change the
number of attention layers in the motion/audio transformer and the cross-modal
transformer but keep the total number of the attention layers fixed. Table 4
shows that the early fusion between two input modalities is critical to
generate motions that are well correlated with input music. Also we show in
Figure 6 that Early-Fusion allows the cross-model transformer pays more
attention to the music, while Late-Fusion tend to ignore the conditioning
music. This also aligns with our intuition that the two modalities need to be
fully fused for better cross-modal learning, as contrast to prior work that
uses a single MLP to combine the audio and motion [55].
#### 5.2.4 User Study
Finally, we perceptually evaluate the motion-music correlation with a user
study to compare our method with the three baseline methods and the “random”
baseline, which randomly combines AIST++ motion-music. (Refer to the Appendix
for user study details.) In this study, each user is asked to watch 10 videos
showing one of our results and one random counterpart, and answer the question
_“which person is dancing more to the music? LEFT or RIGHT”_ for each video.
For user study on each of the four baselines, we invite 30 participants,
ranging from professional dancers to people who rarely dance. We analyze the
feedback and the results are: (1) $81\%$ of our generated dance motion is
better than Li _et al_. [55]; (2) $71\%$ of our generated dance motion is
better than Dancenet [96]; (3) $77\%$ of our generated dance motion is better
than DanceRevolution [36]; (4) $75\%$ of the unpaired AIST++ dance motion is
better than ours. Clearly we surpass the baselines in the user study. But
because the “random” baseline consists of real advanced dance motions that are
extremely expressive, participants are biased to prefer it over ours. However,
quantitative metrics show that our generated dance is more aligned with music.
Cross-Modal Fusion | $\text{FID}_{k}\downarrow$ | $\text{FID}_{g}\downarrow$ | $\text{BeatAlign}\uparrow$
---|---|---|---
No-Fusion | 45.66 | 13.27 | 0.228*
Late-Fusion | 45.76 | 14.30 | 0.234
Early-Fusion | 35.35 | 12.39 | 0.241
Table 4: Ablation Study on Cross-modal Fusion. Early fusion of the two
modalities allows the model to generate motion sequences align better with the
conditioning music. *Note this number is calculated using the music paired
with the input motion.
## 6 Conclusion and Discussion
In this paper, we present a cross-modal transformer-based neural network
architecture that can not only learn the audio-motion correspondence but also
can generate non-freezing high quality 3D motion sequences conditioned on
music. We also construct the largest 3D human dance dataset: AIST++. This
proposed, multi-view, multi-genre, cross-modal 3D motion dataset can not only
help research in the conditional 3D motion generation research but also human
understanding research in general. While our results shows a promising
direction in this problem of music conditioned 3D motion generation, there are
more to be explored. First, our approach is kinematic based and we do not
reason about physical interactions between the dancer and the floor. Therefore
the global translation can lead to artifacts such as foot sliding and
floating. Second, our model is currently deterministic. Exploring how to
generate multiple realistic dance per music is an exciting direction.
## 7 Acknowledgement
We thank Chen Sun, Austin Myers, Bryan Seybold and Abhijit Kundu for helpful
discussions. We thank Emre Aksan and Jiaman Li for sharing their code. We also
thank Kevin Murphy for the early attempts on this direction, as well as Peggy
Chi and Pan Chen for the help on user study experiments.
## References
* [1] Mixamo. https://www.mixamo.com/.
* [2] Hyemin Ahn, Jaehun Kim, Kihyun Kim, and Songhwai Oh. Generative autoregressive networks for 3d dancing move synthesis from music. IEEE Robotics and Automation Letters, 5(2):3500–3507, 2020.
* [3] Emre Aksan, Peng Cao, Manuel Kaufmann, and Otmar Hilliges. Attention, please: A spatio-temporal transformer for 3d human motion prediction. arXiv preprint arXiv:2004.08692, 2020.
* [4] Emre Aksan, Manuel Kaufmann, and Otmar Hilliges. Structured prediction helps 3d human motion modelling. In Proceedings of the IEEE International Conference on Computer Vision, pages 7144–7153, 2019.
* [5] Omid Alemi, Jules Françoise, and Philippe Pasquier. Groovenet: Real-time music-driven dance movement generation using artificial neural networks. networks, 8(17):26, 2017.
* [6] Mykhaylo Andriluka, Leonid Pishchulin, Peter Gehler, and Bernt Schiele. 2d human pose estimation: New benchmark and state of the art analysis. In Proceedings of the IEEE Conference on computer Vision and Pattern Recognition, pages 3686–3693, 2014.
* [7] Okan Arikan and David A Forsyth. Interactive motion generation from examples. ACM Transactions on Graphics (TOG), 21(3):483–490, 2002.
* [8] Samy Bengio, Oriol Vinyals, Navdeep Jaitly, and Noam Shazeer. Scheduled sampling for sequence prediction with recurrent neural networks. In Advances in neural information processing systems, 2015.
* [9] Uttaran Bhattacharya, Nicholas Rewkowski, Abhishek Banerjee, Pooja Guhan, Aniket Bera, and Dinesh Manocha. Text2gestures: A transformer-based network for generating emotive body gestures for virtual agents. arXiv preprint arXiv:2101.11101, 2021.
* [10] Richard Bowden. Learning statistical models of human motion. In IEEE Workshop on Human Modeling, Analysis and Synthesis, CVPR, volume 2000, 2000.
* [11] Matthew Brand and Aaron Hertzmann. Style machines. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 183–192, 2000.
* [12] Judith Bütepage, Michael J Black, Danica Kragic, and Hedvig Kjellström. Deep representation learning for human motion prediction and classification. In CVPR, page 2017, 2017.
* [13] Slides by Saheel. Baby talk: Understanding and generating image descriptions.
* [14] Zhe Cao, Gines Hidalgo, Tomas Simon, Shih-En Wei, and Yaser Sheikh. OpenPose: realtime multi-person 2D pose estimation using Part Affinity Fields. In arXiv preprint arXiv:1812.08008, 2018.
* [15] Kang Chen, Zhipeng Tan, Jin Lei, Song-Hai Zhang, Yuan-Chen Guo, Weidong Zhang, and Shi-Min Hu. Choreomaster: choreography-oriented music-driven dance synthesis. ACM Transactions on Graphics (TOG), 40(4):1–13, 2021.
* [16] Hsu-kuang Chiu, Ehsan Adeli, Borui Wang, De-An Huang, and Juan Carlos Niebles. Action-agnostic human pose forecasting. In 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), pages 1423–1432. IEEE, 2019.
* [17] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.
* [18] Xiaoxiao Du, Ram Vasudevan, and Matthew Johnson-Roberson. Bio-lstm: A biomechanically inspired recurrent neural network for 3-d pedestrian pose and gait prediction. IEEE Robotics and Automation Letters, 4(2):1501–1508, 2019.
* [19] Xuguang Duan, Wenbing Huang, Chuang Gan, Jingdong Wang, Wenwu Zhu, and Junzhou Huang. Weakly supervised dense event captioning in videos. In Advances in Neural Information Processing Systems, pages 3059–3069, 2018.
* [20] Rukun Fan, Songhua Xu, and Weidong Geng. Example-based automatic music-driven conventional dance motion synthesis. IEEE transactions on visualization and computer graphics, 18(3):501–515, 2011.
* [21] Joao P Ferreira, Thiago M Coutinho, Thiago L Gomes, José F Neto, Rafael Azevedo, Renato Martins, and Erickson R Nascimento. Learning to dance: A graph convolutional adversarial network to generate realistic dance motions from audio. Computers & Graphics, 94:11–21.
* [22] Ylva Ferstl and Rachel McDonnell. Investigating the use of recurrent motion modelling for speech gesture generation. In Proceedings of the 18th International Conference on Intelligent Virtual Agents, pages 93–98, 2018.
* [23] Ylva Ferstl, Michael Neff, and Rachel McDonnell. Adversarial gesture generation with realistic gesture phasing. Computers & Graphics, 2020.
* [24] Katerina Fragkiadaki, Sergey Levine, Panna Felsen, and Jitendra Malik. Recurrent network models for human dynamics. In Proceedings of the IEEE International Conference on Computer Vision, pages 4346–4354, 2015.
* [25] Aphrodite Galata, Neil Johnson, and David Hogg. Learning variable-length markov models of behavior. Computer Vision and Image Understanding, 81(3):398–413, 2001.
* [26] Chuang Gan, Deng Huang, Peihao Chen, Joshua B Tenenbaum, and Antonio Torralba. Foley music: Learning to generate music from videos. arXiv preprint arXiv:2007.10984, 2020.
* [27] Partha Ghosh, Jie Song, Emre Aksan, and Otmar Hilliges. Learning human motion models for long-term predictions. In 2017 International Conference on 3D Vision (3DV), pages 458–466. IEEE, 2017.
* [28] Shiry Ginosar, Amir Bar, Gefen Kohavi, Caroline Chan, Andrew Owens, and Jitendra Malik. Learning individual styles of conversational gesture. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3497–3506, 2019.
* [29] Deepak Gopinath and Jungdam Won. fairmotion - tools to load, process and visualize motion capture data. Github, 2020.
* [30] Rachel Heck and Michael Gleicher. Parametric motion graphs. In Proceedings of the 2007 symposium on Interactive 3D graphics and games, pages 129–136, 2007.
* [31] Alejandro Hernandez, Jurgen Gall, and Francesc Moreno-Noguer. Human motion prediction via spatio-temporal inpainting. In Proceedings of the IEEE International Conference on Computer Vision, pages 7134–7143, 2019.
* [32] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in neural information processing systems, pages 6626–6637, 2017.
* [33] Daniel Holden, Taku Komura, and Jun Saito. Phase-functioned neural networks for character control. ACM Transactions on Graphics (TOG), 36(4):1–13, 2017.
* [34] Daniel Holden, Jun Saito, and Taku Komura. A deep learning framework for character motion synthesis and editing. ACM Transactions on Graphics (TOG), 35(4):1–11, 2016.
* [35] Daniel Holden, Jun Saito, Taku Komura, and Thomas Joyce. Learning motion manifolds with convolutional autoencoders. In SIGGRAPH Asia 2015 Technical Briefs, pages 1–4. 2015.
* [36] Ruozi Huang, Huang Hu, Wei Wu, Kei Sawada, Mi Zhang, and Daxin Jiang. Dance revolution: Long-term dance generation with music via curriculum learning. In International Conference on Learning Representations, 2021.
* [37] Yinghao Huang, Manuel Kaufmann, Emre Aksan, Michael J. Black, Otmar Hilliges, and Gerard Pons-Moll. Deep inertial poser learning to reconstruct human pose from sparseinertial measurements in real time. ACM Transactions on Graphics, (Proc. SIGGRAPH Asia), 37(6):185:1–185:15, Nov. 2018.
* [38] Vladimir Iashin and Esa Rahtu. Multi-modal dense video captioning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 958–959, 2020.
* [39] Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3.6m: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(7):1325–1339, jul 2014.
* [40] Ashesh Jain, Amir R Zamir, Silvio Savarese, and Ashutosh Saxena. Structural-rnn: Deep learning on spatio-temporal graphs. In Proceedings of the ieee conference on computer vision and pattern recognition, pages 5308–5317, 2016.
* [41] Ye Jia, Melvin Johnson, Wolfgang Macherey, Ron J Weiss, Yuan Cao, Chung-Cheng Chiu, Naveen Ari, Stella Laurenzo, and Yonghui Wu. Leveraging weakly supervised data to improve end-to-end speech-to-text translation. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7180–7184. IEEE, 2019.
* [42] Hsuan-Kai Kao and Li Su. Temporally guided music-to-body-movement generation. In Proceedings of the 28th ACM International Conference on Multimedia, pages 147–155, 2020.
* [43] Shigeki Karita, Nanxin Chen, Tomoki Hayashi, Takaaki Hori, Hirofumi Inaguma, Ziyan Jiang, Masao Someki, Nelson Enrique Yalta Soplin, Ryuichi Yamamoto, Xiaofei Wang, et al. A comparative study on transformer vs rnn in speech applications. In 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 449–456. IEEE, 2019.
* [44] Andrej Karpathy and Li Fei-Fei. Deep visual-semantic alignments for generating image descriptions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3128–3137, 2015.
* [45] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
* [46] Muhammed Kocabas, Nikos Athanasiou, and Michael J. Black. Vibe: Video inference for human body pose and shape estimation, 2019.
* [47] Lucas Kovar, Michael Gleicher, and Frédéric Pighin. Motion graphs. In ACM SIGGRAPH 2008 classes, pages 1–10. 2008.
* [48] Ranjay Krishna, Kenji Hata, Frederic Ren, Li Fei-Fei, and Juan Carlos Niebles. Dense-captioning events in videos. In Proceedings of the IEEE international conference on computer vision, pages 706–715, 2017.
* [49] Jogendra Nath Kundu, Himanshu Buckchash, Priyanka Mandikal, Anirudh Jamkhandi, Venkatesh Babu RADHAKRISHNAN, et al. Cross-conditioned recurrent networks for long-term synthesis of inter-person human motion interactions. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 2724–2733, 2020.
* [50] Kimerer LaMothe. The dancing species: how moving together in time helps make us human. Aeon, June 2019.
* [51] Hsin-Ying Lee, Xiaodong Yang, Ming-Yu Liu, Ting-Chun Wang, Yu-Ding Lu, Ming-Hsuan Yang, and Jan Kautz. Dancing to music, 2019.
* [52] Juheon Lee, Seohyun Kim, and Kyogu Lee. Listen to dance: Music-driven choreography generation using autoregressive encoder-decoder network. arXiv preprint arXiv:1811.00818, 2018.
* [53] Jehee Lee and Sung Yong Shin. A hierarchical approach to interactive motion editing for human-like figures. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pages 39–48, 1999.
* [54] Guang Li, Linchao Zhu, Ping Liu, and Yi Yang. Entangled transformer for image captioning. In Proceedings of the IEEE International Conference on Computer Vision, pages 8928–8937, 2019.
* [55] Jiaman Li, Yihang Yin, Hang Chu, Yi Zhou, Tingwu Wang, Sanja Fidler, and Hao Li. Learning to generate diverse dance motions with transformer. arXiv preprint arXiv:2008.08171, 2020.
* [56] Zimo Li, Yi Zhou, Shuangjiu Xiao, Chong He, Zeng Huang, and Hao Li. Auto-conditioned recurrent networks for extended complex human motion synthesis. ICLR, 2018.
* [57] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multi-person linear model. SIGGRAPH Asia, 2015.
* [58] Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Neural baby talk. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7219–7228, 2018.
* [59] Naureen Mahmood, Nima Ghorbani, Nikolaus F Troje, Gerard Pons-Moll, and Michael J Black. Amass: Archive of motion capture as surface shapes. In Proceedings of the IEEE International Conference on Computer Vision, pages 5442–5451, 2019.
* [60] Brian McFee, Colin Raffel, Dawen Liang, Daniel PW Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto. librosa: Audio and music signal analysis in python. In Proceedings of the 14th python in science conference, volume 8, 2015.
* [61] Dirk Moelants. Dance music, movement and tempo preferences. In Proceedings of the 5th Triennial ESCOM Conference, pages 649–652. Hanover University of Music and Drama, 2003.
* [62] Meinard Müller, Tido Röder, and Michael Clausen. Efficient content-based retrieval of motion capture data. In ACM SIGGRAPH 2005 Papers, pages 677–685. 2005.
* [63] Kensuke Onuma, Christos Faloutsos, and Jessica K Hodgins. Fmdistance: A fast and effective distance function for motion capture data. In Eurographics (Short Papers), pages 83–86, 2008.
* [64] Katherine Pullen and Christoph Bregler. Animating by multi-level sampling. In Proceedings Computer Animation 2000, pages 36–42. IEEE, 2000\.
* [65] Haibo Qiu, Chunyu Wang, Jingdong Wang, Naiyan Wang, and Wenjun Zeng. Cross view fusion for 3d human pose estimation. In International Conference on Computer Vision (ICCV), 2019.
* [66] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training, 2018.
* [67] Xuanchi Ren, Haoran Li, Zijian Huang, and Qifeng Chen. Music-oriented dance video synthesis with pose perceptual loss. arXiv preprint arXiv:1912.06606, 2019.
* [68] Xuanchi Ren, Haoran Li, Zijian Huang, and Qifeng Chen. Self-supervised dance video synthesis conditioned on music. In Proceedings of the 28th ACM International Conference on Multimedia, pages 46–54, 2020.
* [69] Yi Ren, Yangjun Ruan, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, and Tie-Yan Liu. Fastspeech: Fast, robust and controllable text to speech. In Advances in Neural Information Processing Systems, pages 3171–3180, 2019.
* [70] Matteo Ruggero Ronchi and Pietro Perona. Benchmarking and error diagnosis in multi-instance pose estimation. In The IEEE International Conference on Computer Vision (ICCV), Oct 2017.
* [71] Takaaki Shiratori, Atsushi Nakazawa, and Katsushi Ikeuchi. Dancing-to-music character animation. In Computer Graphics Forum, volume 25, pages 449–458. Wiley Online Library, 2006.
* [72] Eli Shlizerman, Lucio Dery, Hayden Schoen, and Ira Kemelmacher-Shlizerman. Audio to body dynamics. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7574–7583, 2018.
* [73] Sebastian Starke, He Zhang, Taku Komura, and Jun Saito. Neural state machine for character-scene interactions. ACM Trans. Graph., 38(6):209–1, 2019.
* [74] Sebastian Starke, Yiwei Zhao, Taku Komura, and Kazi Zaman. Local motion phases for learning multi-contact character movements. ACM Transactions on Graphics (TOG), 39(4):54–1, 2020.
* [75] Statista. https://www.statista.com/statistics/249396/top-youtube-videos-views/, 2020\. Accessed: 2020-11-09.
* [76] Chen Sun, Fabien Baradel, Kevin Murphy, and Cordelia Schmid. Learning video representations using contrastive bidirectional transformer. arXiv preprint arXiv:1906.05743, 2019.
* [77] Chen Sun, Austin Myers, Carl Vondrick, Kevin Murphy, and Cordelia Schmid. Videobert: A joint model for video and language representation learning. In Proceedings of the IEEE International Conference on Computer Vision, pages 7464–7473, 2019.
* [78] Guofei Sun, Yongkang Wong, Zhiyong Cheng, Mohan S Kankanhalli, Weidong Geng, and Xiangdong Li. Deepdance: Music-to-dance motion choreography with adversarial learning. IEEE Transactions on Multimedia, 2020.
* [79] Taoran Tang, Jia Jia, and Hanyang Mao. Dance with melody: An lstm-autoencoder approach to music-oriented dance synthesis. In Proceedings of the 26th ACM international conference on Multimedia, pages 1598–1606, 2018.
* [80] Graham W Taylor and Geoffrey E Hinton. Factored conditional restricted boltzmann machines for modeling motion style. In Proceedings of the 26th annual international conference on machine learning, pages 1025–1032, 2009.
* [81] Purva Tendulkar, Abhishek Das, Aniruddha Kembhavi, and Devi Parikh. Feel the music: Automatically generating a dance for an input song. arXiv preprint arXiv:2006.11905, 2020.
* [82] Shuhei Tsuchida, Satoru Fukayama, Masahiro Hamasaki, and Masataka Goto. Aist dance video database: Multi-genre, multi-dancer, and multi-camera database for dance information processing. In Proceedings of the 20th International Society for Music Information Retrieval Conference, ISMIR 2019, pages 501–510, Delft, Netherlands, Nov. 2019.
* [83] Jean-Marc Valin and Jan Skoglund. Lpcnet: Improving neural speech synthesis through linear prediction. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5891–5895. IEEE, 2019.
* [84] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.
* [85] Subhashini Venugopalan, Marcus Rohrbach, Jeffrey Donahue, Raymond Mooney, Trevor Darrell, and Kate Saenko. Sequence to sequence-video to text. In Proceedings of the IEEE international conference on computer vision, pages 4534–4542, 2015.
* [86] Subhashini Venugopalan, Huijuan Xu, Jeff Donahue, Marcus Rohrbach, Raymond Mooney, and Kate Saenko. Translating videos to natural language using deep recurrent neural networks. arXiv preprint arXiv:1412.4729, 2014.
* [87] Ruben Villegas, Jimei Yang, Duygu Ceylan, and Honglak Lee. Neural kinematic networks for unsupervised motion retargetting. In CVPR, 2018.
* [88] Borui Wang, Ehsan Adeli, Hsu-kuang Chiu, De-An Huang, and Juan Carlos Niebles. Imitation learning for human pose prediction. In Proceedings of the IEEE International Conference on Computer Vision, pages 7124–7133, 2019.
* [89] Haoming Xu, Runhao Zeng, Qingyao Wu, Mingkui Tan, and Chuang Gan. Cross-modal relation-aware networks for audio-visual event localization. In Proceedings of the 28th ACM International Conference on Multimedia, pages 3893–3901, 2020.
* [90] Nelson Yalta, Shinji Watanabe, Kazuhiro Nakadai, and Tetsuya Ogata. Weakly-supervised deep recurrent neural networks for basic dance step generation. In 2019 International Joint Conference on Neural Networks (IJCNN), pages 1–8. IEEE, 2019.
* [91] Li Yao, Atousa Torabi, Kyunghyun Cho, Nicolas Ballas, Christopher Pal, Hugo Larochelle, and Aaron Courville. Describing videos by exploiting temporal structure. In Proceedings of the IEEE international conference on computer vision, pages 4507–4515, 2015.
* [92] Zijie Ye, Haozhe Wu, Jia Jia, Yaohua Bu, Wei Chen, Fanbo Meng, and Yanfeng Wang. Choreonet: Towards music to dance synthesis with choreographic action unit. In Proceedings of the 28th ACM International Conference on Multimedia, pages 744–752, 2020.
* [93] Haonan Yu, Jiang Wang, Zhiheng Huang, Yi Yang, and Wei Xu. Video paragraph captioning using hierarchical recurrent neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4584–4593, 2016.
* [94] He Zhang, Sebastian Starke, Taku Komura, and Jun Saito. Mode-adaptive neural networks for quadruped motion control. ACM Transactions on Graphics (TOG), 37(4):1–11, 2018.
* [95] Luowei Zhou, Yingbo Zhou, Jason J Corso, Richard Socher, and Caiming Xiong. End-to-end dense video captioning with masked transformer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 8739–8748, 2018.
* [96] Wenlin Zhuang, Congyi Wang, Siyu Xia, Jinxiang Chai, and Yangang Wang. Music2dance: Music-driven dance generation using wavenet. arXiv preprint arXiv:2002.03761, 2020.
* [97] Wenlin Zhuang, Yangang Wang, Joseph Robinson, Congyi Wang, Ming Shao, Yun Fu, and Siyu Xia. Towards 3d dance motion synthesis and control. arXiv preprint arXiv:2006.05743, 2020.
## Appendix
## Appendix A AIST++ Dataset Details
##### 3D Reconstruction
Here we describe how we reconstruct 3D motion from the AIST dataset. Although
the AIST dataset contains multi-view videos, they are not calibrated meaning
their camera intrinsic and extrinsic parameters are not available. Without
camera parameters, it is not trivial to automatically and accurately
reconstruct the 3D human motion. We start with 2D human pose detection
[papandreou2017towards] and manually initialized the camera parameters. On
this we apply bundle adjustment [triggs1999bundle] to refine the camera
parameters. With the improved camera parameters, the 3D joint locations
$\hat{J}\in\mathbb{R}^{M\times 3}$($M=17$) are then triangulated from the
multi-view 2D human pose keypoints locations. During the triangulation phase,
we introduce temporal smoothness and bone length constraints to improve the
quality of the reconstructed 3D joint locations. We further fit SMPL human
body model [loper2015smpl] to the triangulated joint locations $\hat{J}$ by
minimizing an objective with respect to $\Theta=\\{\theta_{i}\\}_{i}^{M}$,
global scale parameter $\alpha$ and global transformation $\gamma$ for each
frame:
$\min_{\Theta,\gamma,\alpha}\sum_{i=1}^{M}\|\hat{J}-J(\theta_{i},\beta,\gamma,\alpha)\|_{2}$.
We fix $\beta$ to the average shape as the problem is under-constrained from
3D joint locations alone.
##### Statistics
We show the detailed statistics of our AIST++ dataset in Table 5. Thanks to
the AIST Dance Video Database [82], our dataset contains in total $5.2$-hour
($1.1$M frame, $1408$ sequences) of 3D dance motion accompanied with music.
The dataset covers 10 dance genre (shown in Figure 8) and $60$ pieces of
music. For each genre, there are $6$ different pieces of music, ranging from
$29$ seconds to $54$ seconds long, and from $80$ BPM to $130$ BPM (except for
House genre which is $110$ BPM to $135$ BPM). Among those motion sequences for
each genre, $120$ ($85$%) of them are _basic_ choreographies and $21$ ($15$%)
of them are _advanced_. Advanced choreographies are longer and more
complicated dances improvised by the dancers. Note for the _basic_ dance
motion, dancers are asked to perform the same choreography on all the $6$
pieces of music with different speed to follow different music BPMs. So the
total _unique_ choreographies in for each genre is $120/6+21=41$. In our
experiments we split the AIST++ dataset such that there is no overlap between
_train_ and _test_ for both music and choreographies (see Sec. 5.2.1 in the
paper).
Genres | Musics | Music Tempo | Motions | Choreographs | Motion Duration (sec.) | Total Seconds
---|---|---|---|---|---|---
ballet jazz | 6 | 80 - 130 | 141 | 85% basic + 15% advanced | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1910.8
street jazz | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 14.9 - 48.0 adv. | 1875.3
krump | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1904.3
house | 6 | 110 - 135 | 141 | 7.1 - 8.7 basic / 28.4 - 34.9 adv. | 1607.6
LA-style hip-hop | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1935.8
middle hip-hop | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1934.0
waack | 6 | 80 - 130 | 140 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1897.1
lock | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1898.5
pop | 6 | 80 - 130 | 140 | 7.4 - 12.0 basic / 29.5 - 48.0 adv. | 1872.9
break | 6 | 80 - 130 | 141 | 7.4 - 12.0 basic / 23.8 - 48.0 adv. | 1858.3
total | 60 | | 1408 | | | 18694.6
Table 5: AIST++ Dataset Statistics. AIST++ is built upon a subset of AIST
database [82] that contains single-person dance. Figure 7: PCKh Metric on
AIST++. We analyze the PCKh (percentage of correct keypoints) metric between
re-projected 2D keypoints and detected 2D keypoints on AIST++. Averaged
<EMAIL_ADDRESS>is $98.4\%$ on all joints shows that our reconstructed 3D keypoints
are highly consistent with the predicted 2D keypoints. Figure 8: AIST++ Motion
Diversity Visualization. Here we show the 10 types of 3D human dance motion in
our dataset. Figure 9: MPJPE-2D Distribution on AIST++. We analyze the
distribution of MPJPE-2D among all video sequences on 1920x1080 resolution.
MPJPE-2D is calculated between the re-projected 2D keypoints and the detected
2D keypoints. Over $86\%$ of the videos have less than average $10$ pixels of
error. Figure 10: Participant Demography of the _Comparison_ User Study.
Figure 11: User study interface. The interface of our User study. We ask each
participant to watch 10 videos and answer the question _”which person is
dancing more to the music? LEFT or RIGHT”_.
##### Validation
As described in Sec. 5.1 in the paper, we validate the quality of our
reconstructed 3D motion by calculating the overall MPJPE-2D (in pixel) between
the re-projected 2D keypoints and the detected 2D keypoints with high
confidence ($>0.5$). We provide here the distribution of MPJPE-2D among all
video sequences (Figure 9). Moreover, we also analyze the PCKh metric with
various thresholds on the AIST++, which measures the consistency between the
re-projected and detected 2D keypoints. Averaged<EMAIL_ADDRESS>is 98.4% on all
joints shows that our reconstructed 3D keypoints are highly consistent with
the detected 2D keypoints.
## Appendix B User Study Details
### B.1 Comparison User Study
As mentioned in Sec. 5.2.5 in the main paper, we qualitatively compare our
generated results with several baselines in a user study. Here we describe the
details of this user study. Figure 11 shows the interface that we developed
for this user study. We visualize the dance motion using stick-man and conduct
side-by-side comparison between our generated results and the baseline
methods. The left-right order is randomly shuffled for each video to make sure
that the participants have absolutely no idea which is ours. Each video is
$10$-second long, accompanied with the music. The question we ask each
participant is _“which person is dancing more to the music? LEFT or RIGHT”_ ,
and the answers are collected through a Google Form. At the end of this user
study, we also have an exit survey to ask for the dance experience of the
participants. There are two questions: _“How many years have you been
dancing?”_ , and _“How often do you watch dance videos?”_. Figure 10 shows
that our participants ranges from professional dancers to people rarely dance,
with majority with at least 1 year of dance experience.
|
# NGTS and HST insights into the long period modulation in GW Librae
P. Chote1, B. T. Gänsicke1, J. McCormac1, A. Aungwerojwit2, D. Bayliss1, M. R.
Burleigh3, S. L. Casewell3, Ph. Eigmüller4, S. Gill1, M. R. Goad3, J. J.
Hermes5, J. S. Jenkins6,7, A. S. Mukadam8, S. Poshyachinda9, L. Raynard3, D.
E. Reichart10, P. Szkody8, O. Toloza1, R. G. West1, P. J. Wheatley1
1Department of Physics, University of Warwick, Coventry CV4 7AL, United
Kingdom
2Department of Physics, Faculty of Science, Naresuan University, Phitsanulok
65000, Thailand
3Department of Physics and Astronomy, University of Leicester, Leicester, LE1
7RH, UK
4Institute of Planetary Research, German Aerospace Center, Rutherfordstrasse
2, 12489 Berlin, Germany
5Department of Astronomy, Boston University, Boston, MA 02215, USA
6Departamento de Astronomía, Universidad de Chile, Camino El Observatorio
1515, Las Condes, Santiago, Chile
7Centro de Astrofísica y Tecnologías Afines (CATA), Casilla 36-D, Santiago,
Chile
8Department of Astronomy, University of Washington, Box 351580, Seattle, WA
98195, USA
9National Astronomical Research Institute of Thailand (Public Organization),
Chiangmai, 50180, Thailand
10Department of Physics and Astronomy, University of North Carolina at Chapel
Hill, Chapel Hill, NC 27599, USA
E-mail<EMAIL_ADDRESS>
###### Abstract
Light curves of the accreting white dwarf pulsator GW Librae spanning a 7.5
month period in 2017 were obtained as part of the Next Generation Transit
Survey. This data set comprises 787 hours of photometry from 148 clear nights,
allowing the behaviour of the long (hours) and short period (20 min)
modulation signals to be tracked from night to night over a much longer
observing baseline than has been previously achieved. The long period
modulations intermittently detected in previous observations of GW Lib are
found to be a persistent feature, evolving between states with periods
$\simeq$ 83 min and 2 – 4 h on time-scales of several days. The 20 min signal
is found to have a broadly stable amplitude and frequency for the duration of
the campaign, but the previously noted phase instability is confirmed.
Ultraviolet observations obtained with the Cosmic Origin Spectrograph onboard
the Hubble Space Telescope constrain the ultraviolet-to-optical flux ratio to
$\simeq 5$ for the 4 h modulation, and $\lesssim 1$ for the 20 minute period,
with caveats introduced by non-simultaneous observations. These results add
further observational evidence that these enigmatic signals must originate
from the white dwarf, highlighting our continued gap in theoretical
understanding of the mechanisms that drive them.
###### keywords:
stars: individual: GW Librae, stars: variables: general, stars: dwarf novae,
white dwarfs
††pagerange: NGTS and HST insights into the long period modulation in GW
Librae–References
## 1 Introduction
It has now been over 20 years since coherent short-term variability was
discovered in the dwarf nova GW Librae (Warner & van Zyl, 1998), and
attributed to non-radial pulsations of the central white dwarf. This revealed
a new class of accreting white dwarf pulsators, of which more than a dozen are
now known: all residing in short-period, low accretion rate, cataclysmic
variables (CVs) (e.g. Warner & van Zyl, 1998; Woudt & Warner, 2004; Warner &
Woudt, 2004; Araujo-Betancor et al., 2005; Vanlandingham et al., 2005;
Patterson et al., 2005; Gänsicke et al., 2006; Nilsson et al., 2006; Mukadam
et al., 2007; Patterson et al., 2008; Pavlenko, 2009; Woudt & Warner, 2011;
Uthas et al., 2012; Mukadam et al., 2017). Their physical and
asteroseismological properties differ noticeably from those of the well-
studied single white dwarf pulsators with hydrogen-envelopes (DAV, ZZ Ceti
stars, Mukadam et al. 2004; Gianninas et al. 2006; Van Grootel et al. 2012)
and helium-envelopes (DBV, V777 Her stars, Beauchamp et al. 1999): the
envelopes of accreting white dwarf pulsators are hydrogen-dominated, but
enriched by non-negligible amounts of helium and trace metals from their
companion star. The H/He ratio affects the driving of pulsations, and has
hence to be considered as a third dimension, effectively establishing an
instability volume rather than the two-dimensional strips in
$T_{\mathrm{eff}}$ and $\log g$. Consequently, an additional HeII partial
ionisation zone may form that can drive pulsations at higher temperatures
compared to the ZZ Ceti stars (Townsley et al., 2004; Arras et al., 2006; Van
Grootel et al., 2015). In fact, the known accreting white dwarf pulsators span
a wide range of effective temperatures from 10500 K to above 15000 K (Szkody
et al., 2010). In addition, white dwarfs in CVs can be spun up to very short
periods (e.g. King et al., 1991; Cheng et al., 1997); $\sim$100 s for GW Lib
(van Spaandonk et al., 2010), compared with $\sim$ tens of hours for typical
isolated white dwarfs (e.g. Hermes et al., 2017), which further complicates
interpretation of the observed pulsation signals.
Observations of GW Lib obtained between 1997 and 2005 determined a binary
period of 76.78 minutes (Thorstensen et al., 2002) from H$\alpha$ radial
velocity measurements, and established the presence of three pulsation modes
near 650, 380, and 230 s visible in both optical and ultraviolet light curves
(van Zyl et al., 2000; Szkody et al., 2002; van Zyl et al., 2004; Copperwheat
et al., 2009). While these three pulsation signals were consistently detected,
they did not have the frequency stability seen in many ZZ Ceti or V777 Her
white dwarf pulsators, and the spread of periods observed between runs and the
apparent splitting of frequencies within individual long runs was interpreted
as signs of true underlying structure (van Zyl et al., 2004). In addition to
these short-period pulsation modes, Woudt & Warner (2002) discovered a quasi-
periodic 2.1 h modulation, whose origin remained unexplained.
The CVs hosting accreting white dwarf pulsators undergo dwarf novae
‘superoutbursts’ with recurrence times of tens of years, during which the
accretion disc reaches a critical density, becomes unstable, and rapidly dumps
its contents onto the surface of the white dwarf. This material compresses and
heats the envelope (Godon & Sion, 2003; Piro et al., 2005) so that its
temperature increases to a value outside the instability strip, halting
pulsations. Monitoring the white dwarf as it returns to its quiescent
temperature provides an opportunity to study the onset and evolution of
pulsations over human time-scales – a few years, as compared with $\sim
10^{8}\,$years for evolutionary cooling.
GW Lib was discovered on 1983 August 10, when it underwent a superoutburst
(Maza & Gonzalez, 1983), i.e. the white dwarf pulsations were identified after
the system had been 14 years back into quiescence. The next superoutburst was
detected on 2007 April 11 (Templeton et al., 2007), and observations during
the outburst only revealed superhumps from the accretion disc (Vican et al.
2011, hereafter VI11). The first short-period modulations were detected in
2008, though not at the periods seen prior to the outburst, but at $\simeq 19$
min and $\simeq 290$ s (Copperwheat et al. 2009, hereafter CO09), in line with
the expectations that heating of the envelope during the superoutburst affects
the driving of the pulsations. GW Lib was intensively monitored at optical and
ultraviolet wavelengths over the next decade, and presented overall a
consistent behaviour, exhibiting brightness modulations at $\simeq 20$ min and
$\simeq 300$ sec (Schwieterman et al. 2010, hereafter SC10; VI11; Bullock et
al. 2011, hereafter BU11; Szkody et al. 2012, hereafter SZ12; Chote & Sullivan
2016, hereafter CH16; Toloza et al. 2016, hereafter TO16; Szkody et al. 2016,
hereafter SZ16; Chote et al. 2017, hereafter CH17). Two puzzling facts emerged
from this vast array of data: (1) After an initial brief phase of cooling
(SZ12), the white dwarf has remained at a constant temperature since 2011, and
is still $\simeq 3000$ K hotter than in quiescence (TO16, SZ16) – contrasting
observations of similar large amplitude outburst dwarf novae which showed the
white dwarfs monotonously cooling back to their pre-outburst temperature on a
time-scale of $\simeq 5$ yr, e.g. WZ Sge (Slevinsky et al., 1999; Godon et
al., 2006) and AL Com (Szkody et al., 2003). (2) The long-period modulation
seen before the outburst was initially detected at 2.1 h in 2008 in both the
optical (CO09) and UV (BU11), but now often shows up at $\simeq 4$ h, which
led VI11 to suggest that this is the fundamental of the $\simeq 2.1$ h signal
and that this modulation originates in the accretion disc. However, TO16
demonstrated that the $\simeq 4$ h brightening event captured by ultraviolet
spectroscopy was clearly related to the heating and cooling of the white dwarf
in GW Lib. A more detailed assessment of the long-period modulations remains
particularly difficult as their time-scale is similar to that of a typical
observing run.
GW Lib was observed in the survey footprint of the Next Generation Transit
Survey (NGTS, Wheatley et al. 2018) between January and September 2017. Here
we report on the analysis of 787 h of time-series photometry obtained over 148
clear nights spanning 7.5 months. The observations and data reduction are
described in §2, before analysing the long- and short-timescale variability in
§3 and §4. Finally, §5 attempts to put these results into context against
earlier observations of GW Lib and other systems before closing in §6 with our
conclusions.
## 2 Observations and Data Reduction
### 2.1 NGTS
The NGTS facility (Wheatley et al., 2018) consists of 12 robotic 20 cm
telescopes that are housed in a common enclosure at the European Southern
Observatory (ESO) Cerro Paranal site in Chile. Each telescope observes a
$2.8\times 2.8$ degree field of view through a broad red (roughly $R+I$)
filter using a $2\text{k}\times 2\text{k}$ back-illuminated deep-depleted CCD
camera, producing a $5$ arcsec per pixel plate scale.
The NGTS survey strategy, as it operated in 2017, observed pre-selected fields
continuously with the same telescope every night while $\gtrsim 30^{\circ}$
elevation. Closed-loop auto-guiding (McCormac et al., 2013) ensured that stars
were observed using the same CCD pixels for the entire observing season. The
standard survey exposure was 10 s and it took 3 s to read out the CCD, giving
a survey cadence of 13 s.
The survey field NG1518-2518 containing GW Lib was scheduled for observations
between 2017 January 25 and 2017 September 21, but we chose to truncate the
data set on 2017 September 15 when the observability dropped below 1 h each
night. This provided a data set containing 218045 exposures from 148 nights,
corresponding to a total of 606 h integration time from 787 h time on target.
Due to issues with the shutter, the camera was unmounted twice during the
season. When the camera is remounted there is a small shift in the field
rotation and hence a new auto-guiding reference image is acquired for each
subset of nights.
GW Lib is too faint to be included in the standard NGTS analysis pipeline, so
we developed a stand-alone reduction procedure that operated on the raw
images. This allowed us to automatically extract light curves at the end of
each night and follow the behaviour of GW Lib in near-real time. The final
reduction procedure operates as follows:
Figure 1: The full 7.5 month NGTS light curve of GW Lib, smoothed with a 40
min boxcar filter to emphasize the long-period behaviour. Each fortnight is
offset vertically by 0.5 units, and the observation duty cycle for each
fortnight is listed in parentheses below the start date. The regular gaps in
the light curve are due to the day/night cycle; the gaps on February 1 – 2 and
August 11 – 15 were due to technical issues; the remaining gaps are nights
when NGTS was not operating due to cloud. Several nights are marked with
indicative periods to highlight the detection of the previously observed long
period modulations. The time windows of the HST observations are indicated
with shaded rectangles.
1. (1)
A deep-stack reference frame is manually created for each auto-guider
reference by averaging 100 frames from the same night’s data.
2. (2)
Accurate pixel positions are determined in each reference frame for GW Lib and
five nearby comparison stars. Comparison stars were selected based on their
brightness and lack of blending with nearby stars and confirmed to be non-
variable by inspection of their light curves.
3. (3)
A SExtractor (Bertin & Arnouts, 1996) segmentation map is generated using the
SEP Python library (Barbary, 2016). The segmentation map is converted to a
pixel mask that identifies pixels that are suitable for calculating the
background sky level.
4. (4)
Master bias and sky flat frames are created for each night using the afternoon
and morning calibration frames. On nights where calibration frames were not
taken or when the sky flats were affected by cloud (verified by manual
inspection) they are substituted with frames from the closest good night.
5. (5)
Science frames are prepared by (optionally) applying the calibration frames
and then subtracting a background sky map that is calculated using the
standard SEP/SExtractor routine over the pixels defined by the reference
background mask.
6. (6)
Aperture positions for the comparison stars are calculated by adding the auto-
guider offset (specified in the frame metadata) to the reference positions.
The apertures are then recentred to correct for the small drifts that occur
each night as the field distortion changes with airmass. The aperture for GW
Lib is placed using a blind offset from the mean comparison star position.
7. (7)
Aperture photometry is done using one of two ways:
1. (a)
Photometry for the long period analysis in Section 3 is extracted using the
SEP circular aperture routines with a 2.5 pixel aperture radius. This method
minimises systematic effects at the expense of increased noise in a given
measurement. This essentially white noise bins down as we combine many
measurements using a running mean.
2. (b)
A second set of photometry is extracted omitting the bias and flat field
corrections and using a smaller 1.5 pixel aperture radius for GW Lib while
keeping the 2.5 pixel radius for the comparison stars. This extraction is used
in Section 4 where it improves the sensitivity to the short time-scale
variability at a trade-off of introducing long ($\simeq$ hours) time-scale
systematic trends.
8. (8)
The extracted flux measurements and UTC timestamps are converted to tabular
files containing UTC and BJD${}_{\text{TDB}}$ time, relative flux, and
relative flux error using the tsreduce (Chote et al., 2014) software. These
files, along with the other data described below, are made available as
supplementary data in the online journal.
As GW Lib is bluer than most of the nearby stars, differential extinction
leads to reduced flux ratios at larger airmasses. This effect is reduced in
the NGTS data due to the redder bandpass and limited airmass range of
observations, but it can still be measured and removed following the technique
from CH16. The target / comparison flux ratio is fitted with a linear
coefficient as a function of airmass simultaneously for the entire data set.
The phase and amplitude of the long period modulations average out over many
nights, allowing the airmass effect to be extracted much more robustly than
the standard technique of fitting a low-order polynomial to each run.
### 2.2 APO & Prompt
Additional observations were obtained on 2017 March 4 using the Agile
photometer on the 3.5 m telescope at Apache Point Observatory, and on 2017 May
5 and 2017 May 6 using the 60 cm robotic Prompt8 telescope at CTIO. The APO
observations consisted of 515 30 s integrations obtained through a BG39
filter, and the Prompt observations consisted of 253 and 300 30 s integrations
obtained through a Baader Clear (C) filter.
The APO and Prompt data were reduced following our standard procedures (see
e.g. Chote et al., 2014, SZ16) to extract aperture photometry, with one
exception: the polynomial fit to remove long-period trends was omitted as
there was insufficient data to separate the effect of differential extinction
from the long-period variability intrinsic to GW Lib.
### 2.3 Hubble Space Telescope
Quasi-simultaneous Hubble Space Telescope (HST) far-ultraviolet (UV)
spectroscopy of GW Lib was obtained using the Cosmic Origins Spectrograph
(COS, Green et al. 2012) on the nights of 2017 August 31, September 6, and
September 13, near the end of the NGTS observation window. The first two
observations occurred during bad weather at Paranal, so simultaneous
observations were not possible. The weather was clear during the final run,
but GW Lib set past the NGTS elevation limit less than 2 min before the HST
observations started.
The time-tag COS data were reduced following the procedure from TO16 to create
ultraviolet light curves. Data were binned into 5 s sub-spectra, which were
then background-subtracted and the Ly$\alpha$ and OI airglow lines at 1216
and 1302 Å were masked. Each spectrum was then binned in the spectral
direction to create individual photometric points with a 5 s cadence. We
present limited results from this data set where they provide important
context to the NGTS observations, but defer the full analysis to a future
publication (Toloza et al, in prep).
## 3 Long Period Variability
Photometry demonstrating the long-period behaviour was extracted using a 2.5
pixel aperture and all calibrations. Points more than 5 $\sigma$ from the
nightly mean (42 in total) were rejected, and then the data were smoothed with
a running 40 min mean (roughly 190 points) to reduce the scatter introduced by
noise and the 20 min signal. The resulting 34-week long light curve is
presented in Fig. 1.
The familiar signals near 83 minutes and between 2 – 4 h from earlier
observations (illustrated in CH17) are clearly visible, and we can see for the
first time these signals switching on and off, with each state lasting for 1 –
2 weeks. The larger amplitude modulations appear to grow rather haphazardly
out of the quieter states.
Figure 2: The 83 min periodicity has been seen frequently over the last
several years with a ragged multi-peaked appearance. The top panels show
examples observed with larger telescopes (from SZ12, CH16, CH17), and the
bottom panels show examples from the NGTS data set. Vertical black lines
denote the common 83 min period, and the blue curve shows the running 40 min
mean.
Fig. 2 compares the 83 min signals seen on several nights of NGTS observations
with earlier observations from larger aperture telescopes. The high-cadence
archival observations demonstrate the ragged nature of this signal, which is
smoothed out by the running mean. Similar average profiles are visible in the
NGTS light curves, and so we infer that this behaviour was present but not
resolved due to the low signal-to-noise ratio in the raw NGTS photometry. A
few nights in the smoothed NGTS data show amplitudes in excess of 10%, which
implies potentially much larger changes in the unresolved variability, more
than has been seen in earlier observations.
Figure 3: Near-simultaneous observations from NGTS and HST on the night of
2017 September 13 captured a cycle of the $\simeq 4$ h modulation, confirming
that these modulations are visible in the UV. The disparity in UV/Optical mode
amplitudes for the 20 min and 300 s periods is also visible. Note that
different scales have been used for the NGTS vs HST flux.
Fig. 3 presents the near-simultaneous optical and UV light curves on 2017
September 13. The NGTS light curve captures the rise on one of the long period
modulations, and the HST light curve captures the subsequent fall.
The full NGTS light curve shows that the brightness in successive minima of
the long period modulation is relatively constant (at the 10–20% level), so we
can estimate that the UV to optical amplitude ratio,
$A_{\text{UV}}/A_{\text{Optical}}\simeq 1.15/0.23\simeq 5$ assuming that the
same holds true in the UV, and that the modulation in the UV and optical are
in phase. This is comparable to the UV/Optical ratios measured for the
pulsation periods (e.g. Szkody et al., 2002, SZ16), reinforcing the suggestion
from TO16 that this long-period variability arises from the white dwarf rather
than the accretion disc.
## 4 Short-period signals
To improve the sensitivity to the short-period pulsations we extracted
photometry omitting the bias and flat field corrections, and using a smaller
(1.5 pixel) extraction radius for GW Lib. Reducing the aperture size removes
the sky and read noise contributed by the pixels in the wings of the PSF where
the signal is negligible. Omitting the calibration frames avoids their noise
contribution, taking advantage of the precise sub-pixel telescope guiding to
know that any per-pixel effects will remain consistent between exposures. The
stars do move on the CCD due to field distortion changes with airmass, but the
systematic trends this introduces are on a $\sim$hours time-scale, much longer
than the periods of interest (200 – 2000 seconds).
The time series data is converted to the frequency domain by normalising light
curves relative to their mean level and calculating the Discrete Fourier
Transform (DFT). Results are presented as amplitude spectra where the vertical
scale corresponds to the amplitude of an equivalent sinusoid in the time
domain data. The window function for each night is computed by taking the DFT
of a noise-free sinusoid sampled at the same times as the observations, and
shows the pattern of aliases that are associated with each ‘real’ sinusoidal
signal in the time domain data.
Fig. 4 presents an illustrative NGTS light curve alongside the APO and Prompt
light curves obtained during the NGTS campaign. The 20 min period is clearly
visible as the dominant feature in the light curves, and the NGTS DFTs compare
surprisingly well to the larger aperture telescopes in a large part due to the
much longer run lengths (up to 8.8 h per night).
Figure 4: Light curves and DFTs of the APO and Prompt8 runs, with a segment of
one NGTS run for comparison. The left column presents raw photometric
measurements as grey dots, with a blue line showing a running 5 min mean. The
$\simeq 4$ h modulation is clearly visible in the 2017 May 6 run. The right
column presents the DFTs and window functions for each night, with the 20 min
period and its harmonics marked with vertical red dashes and the 0.1% FAP
significance thresholds shown as horizontal grey lines. The vertical grey dash
on 2017 May 5 indicates a potential but non-significant detection of a 310 s
pulsation period. Figure 5: Trailed amplitude spectra computed from the NGTS
light curves. Each row represents the colour-coded amplitude of the DFT. The
Window column shows the associated window function for each night using the
same frequency scale. The FAP column indicates the amplitude at which a
feature in the DFT is deemed to be statistically significant. The vertical red
dashes indicate the location of the 20 min period and its harmonics. Periods
longer than 2000 s are omitted as they are contaminated by systematic effects
introduced by the reduction procedure.
We define an amplitude threshold for statistical significance by applying a
False Alarm Probability (FAP; e.g. Sullivan et al., 2008) test. The 20 min
signal and its first harmonic are subtracted from the run, and any residual
coherent signals are destroyed by randomly shuffling the times associated with
each measurement. The DFT is calculated covering 200 – 2000 s, and the
amplitude of the largest peak is recorded. This is repeated 1000 times to
simulate different realizations with the same white noise characteristics. The
largest peak then defines the amplitude where there is a 1 in 1000 (0.1%)
probability that a peak of the same amplitude can be produced by white noise.
Fig. 5 presents the nightly runs as a single trailed DFT. Each horizontal
slice shows the DFT, window function, and FAP threshold of the corresponding
night with amplitude represented using colour. Several partial nights that
were interrupted by bad weather (ten in total) and have a very poor window
function were excluded to aid readability.
The 20 min period is clearly visible at $\simeq$1200 s with a $\simeq 3\,\%$
amplitude throughout the entire 7.5 month observing campaign. Traces of the
first and second harmonics can be seen by eye near 590 and 390 s. The $\simeq
300\,$s periods that have been repeatedly seen in the far-ultraviolet (SZ12,
SZ16) and occasionally in the optical (CO09, SZ12, SZ16, TO16, CH16) are not
detected in the NGTS data. The historical amplitude of this signal was
typically $\simeq 0.5-2$% in the optical, and so they would only be apparent
above the $\simeq 0.7$% mean noise floor if they remained coherent over
several nights – previous observations (e.g. SZ16, CH17) have shown this is
usually not the case.
Figure 6: Amplitude and frequency/period fits to the 20 min signal in the NGTS
run (blue) and APO, Prompt, and HST (see Section 3) runs (red). The histograms
on the right show that the majority of measurements are within $3\sigma$ of
the mean 1182 s period and 2.9% amplitude.
The period and amplitude of the 20 min signal on each night were fitted with a
sinusoid in the time domain. $1\sigma$ uncertainties were bootstrapped by
resampling the light curves with replacement (Efron, 1979) to generate 5000
alternative realizations of each light curve, which were each fitted to
produce a distribution of frequency and amplitude values. The majority of
these showed the expected normal distribution, which were fitted with a
Gaussian to measure $\sigma$. Nights with $\sigma>20\,\mu$Hz (five in total)
or non-Gaussian profiles (a further nine) were excluded as they do not
robustly measure a unique fit.
Fig. 6 presents the fitted frequency/period and amplitudes, which show that
the 20 min signal remains broadly stable in both frequency and amplitude
across the full run. The phase and frequency instability shown in CH16
remains, however, as DFTs calculated over multiple nights were found to smear
power in the frequency domain in a similar way.
Fourier transforms of the HST/COS observations obtained on 2017 August 31 and
2017 September 6 are presented in Fig. 7. Both observations were dominated by
the 300 s period, which had an amplitude in the range of 2 – 4%. Previous
observations (SZ12, SZ16) have shown $A_{\text{UV}}/A_{\text{Optical}}\approx
5$, which implies an optical amplitude below 1% – consistent with the NGTS
non-detection. A more detailed analysis on the behaviour of this pulsation
mode will be presented in a future publication.
The key result from the perspective of the NGTS observations is that the 20
min signal is only weakly detected in the UV, with an amplitude < 2%
constraining $A_{\text{UV}}/A_{\text{Optical}}\lesssim 1$. These observations
were obtained during a gap in NGTS observations, so we cannot rule out that
the modulation was intrinsically weaker at the time of the HST observations,
but this seems unlikely considering that the six months of optical
measurements before the observation and the few nights after show no evidence
of significant amplitude variability.
Figure 7: DFTs of the 2017-08-31 and 2017-09-06 HST COS observations show that
the $\sim$20 min signal is only weakly detected at periods consistent with the
optical observations. The $\sim 300\,$s pulsation period dominates the
observed variability, but demonstrates poor coherancy in the second visit with
power being smeared over a range of frequencies in the DFT.
## 5 Discussion
The 83 min signal was originally thought to be associated with a late
superhump period (BU11). As it has now been regularly observed to come and go
for nearly 10 years, it is likely related to the ‘quiescent superhump’ that
has been seen to come and go in EQ Lyn (Mukadam et al., 2013) and V455 And
(Araujo-Betancor et al., 2005). In EQ Lyn, the absence of any period in the UV
while near-simultaneous optical data showed superhumps suggested that the
periodicity may come from the outer accretion disc.
The UV flux from GW Lib is dominated by the white dwarf, with the cooler
accretion disc contributing just a few percent (see e.g. SZ12) to the total.
The accretion disc is more significant at optical wavelengths (VI11 suggest up
to 40% of the total flux), diluting the amplitude of any signals from the
white dwarf. Our near-simultaneous measurements showing the amplitude of the
$\simeq 4\,$h signal as $\simeq 5$ times larger in the UV than the optical
therefore further supports the suggestion from TO16 that this signal
originates from the white dwarf. The GALEX data from BU11 showed that this
signal had a larger amplitude in FUV than NUV, pointing to the inner disc or
white dwarf, but the optical amplitude was also comparable. Unfortunate timing
of the inter-orbit gaps in the 2015 observations (SZ16) mean that we can
neither definitively detect nor rule out the presence of the 2 h signal in the
UV at that time.
The NGTS observations appear to show that the behaviour switches betweenthese
2 – 4 h and 83 min states on a time-scale of roughly a week, which suggests
that they may share a common source.
The 20 min signal is similarly enigmatic. It was not recognized as a
persistent feature in the pre-2007 outburst photometry, despite van Zyl et al.
(2004) noting a detection in their Table 2. There have been at least three
extended periods in 2008, 2012, and 2015 – 2017 where this signal has been the
dominant source of variability. It was initially suggested (BU11, VI11) that
this signal was a disc-related feature due to its poor phase coherency between
nights and its prior non-detections. In the years since, it has become clear
that it must originate on the WD, as there are no known mechanisms that could
generate signals that maintain the level of observed stability over such a
long time-scale.
Hermes et al. (2017) showed that the stable pulsations in isolated DAVs also
lose phase coherency at periods greater than 800 s, so the lack of coherency
of the 20 min signal does not exclude the pulsation mode hypothesis.
The UV/optical amplitude ratio $A_{\text{UV}}/A_{\text{Optical}}<1$ is more
problematic, as the current understanding of g-mode pulsations predict that
the pulsation amplitude should increase significantly in the UV (Robinson et
al., 1995) due to increased limb darkening reducing the geometric cancellation
effect. It is not unprecedented, however: Szkody et al. (2010) report similar
ratios for two other accreting pulsators (the 1285 s period in PQ And and the
582 and 655 s periods in REJ 1255+266); Kotak et al. (2004) shows
$A_{\text{UV}}/A_{\text{Optical}}\simeq 1$ for the 272 and 304 s modes in the
prototypical ‘stable’ DAV G117-B15A; and Kepler et al. (2000) likewise for the
141 s mode in G185-32. So, while we do not yet understand the origin of these
signals, the amplitude ratios alone are not sufficient to rule out an origin
on the white dwarf.
Saio (2019) presents a case for the variability in these accreting systems not
being g-mode pulsations at all, but rather r-mode oscillations trapped in the
H-rich layers of the WD. While they can model the shorter period modulations
in GW Lib, they are not able to explain the $\sim$4 hour modulations and do
not attempt to model the 20 minute signal.
## 6 Conclusions
Observations of GW Librae between 2017 January 25 and 2017 September 21
provide the longest continuous monitoring campaign yet obtained on this
accreting white dwarf pulsator.
These observations demonstrate that the long-period modulations in the system
appear to change between states on a time-scale of days/weeks. While the
individual modes are not strictly periodic, they do appear to have distinctive
and repeating characteristics that can be identified in archival observations
going back nearly ten years. Near-simultaneous NGTS and HST observations
confirm that the 4 h modulation seen by (TO16) in the UV has the same origin
as in the optical, and provide an estimate of the UV/Optical flux ratio of
$\simeq 5$.
The steady presence of the 20 min signal over the full observation baseline
strongly suggests that this signal originates on the WD, and is not a
transient disc phenomenon. The weak detection of this signal in UV is
difficult to explain, but has been seen before in both accreting and non-
accreting WD pulsators.
These NGTS observations add significantly to the body of observational data on
GW Lib, which document several phenomenological behaviours that so far remain
unexplained. Future advancements in explaining these observations will require
theoretical developments towards the pulsation (or other potential) mechanisms
that can drive near-coherent variability in these systems.
## Acknowledgments
Based on data collected under the NGTS project at the ESO La Silla Paranal
Observatory. The NGTS facility is operated by the consortium institutes with
support from the UK Science and Technology Facilities Council (STFC) under
projects ST/M001962/1 and ST/S002642/1. The research leading to these results
has received funding from the European Research Council under the European
Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.
320964 (WDTracer). BG, OT and PJW received support from the UK STFC
consolidated grants ST/L000733/1 and ST/P000495/1. PS and AM acknowledge
support from HST GO-13807 and GO-14912.002-A, as well as NSF AST-1514737. AA
received support from the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation grant B05F630110. JSJ
acknowledges support by FONDECYT grant 1201371 and partial support from
CONICYT project Basal AFB-170002. This work is based on observations made with
the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are
associated with program #14912. This work has made use of observations
obtained with the Apache Point Observatory (APO) 3.5-meter telescope, which is
owned and operated by the Astrophysical Research Consortium (ARC). This work
has made use of data obtained by the PROMPT-8 telescope, owned by National
Astronomical Research Institute of Thailand, and operated by the Skynet
Robotic Telescope Network.
## Data availability
The data underlying this article are available in the article and in its
online supplementary material.
## References
* Araujo-Betancor et al. (2005) Araujo-Betancor S., et al., 2005, A&A, 430, 629
* Arras et al. (2006) Arras P., Townsley D. M., Bildsten L., 2006, ApJ, 643, L119
* Barbary (2016) Barbary K., 2016, Journal of Open Source Software, 1, 58
* Beauchamp et al. (1999) Beauchamp A., Wesemael F., Bergeron P., Fontaine G., Saffer R. A., Liebert J., Brassard P., 1999, ApJ, 516, 887
* Bertin & Arnouts (1996) Bertin E., Arnouts S., 1996, A&AS, 117, 393
* Bullock et al. (2011) Bullock E., et al., 2011, AJ, 141, 84
* Cheng et al. (1997) Cheng F. H., Sion E. M., Szkody P., Huang M., 1997, ApJ, 484, L149
* Chote & Sullivan (2016) Chote P., Sullivan D. J., 2016, MNRAS, 458, 1393
* Chote et al. (2014) Chote P., Sullivan D. J., Brown R., Harrold S. T., Winget D. E., Chandler D. W., 2014, MNRAS, 440, 1490
* Chote et al. (2017) Chote P., et al., 2017, in Tremblay P. E., Gaensicke B., Marsh T., eds, Astronomical Society of the Pacific Conference Series Vol. 509, 20th European White Dwarf Workshop. p. 335
* Copperwheat et al. (2009) Copperwheat C. M., et al., 2009, MNRAS, 393, 157
* Efron (1979) Efron B., 1979, The Annals of Statistics, 7, 1
* Gänsicke et al. (2006) Gänsicke B. T., et al., 2006, MNRAS, 365, 969
* Gianninas et al. (2006) Gianninas A., Bergeron P., Fontaine G., 2006, AJ, 132, 831
* Godon & Sion (2003) Godon P., Sion E. M., 2003, ApJ, 586, 427
* Godon et al. (2006) Godon P., Sion E. M., Cheng F., Long K. S., Gänsicke B. T., Szkody P., 2006, ApJ, 642, 1018
* Green et al. (2012) Green J. C., et al., 2012, ApJ, 744, 60
* Hermes et al. (2017) Hermes J. J., et al., 2017, ApJS, 232, 23
* Kepler et al. (2000) Kepler S. O., Robinson E. L., Koester D., Clemens J. C., Nather R. E., Jiang X. J., 2000, ApJ, 539, 379
* King et al. (1991) King A. R., Regev O., Wynn G. A., 1991, MNRAS, 251, 30P
* Kotak et al. (2004) Kotak R., van Kerkwijk M. H., Clemens J. C., 2004, A&A, 413, 301
* Maza & Gonzalez (1983) Maza J., Gonzalez L. E., 1983, IAU Circ., 3854, 2
* McCormac et al. (2013) McCormac J., Pollacco D., Skillen I., Faedi F., Todd I., Watson C. A., 2013, PASP, 125, 548
* Mukadam et al. (2004) Mukadam A. S., Winget D. E., von Hippel T., Montgomery M. H., Kepler S. O., Costa A. F. M., 2004, ApJ, 612, 1052
* Mukadam et al. (2007) Mukadam A. S., Gänsicke B. T., Szkody P., Aungwerojwit A., Howell S. B., Fraser O. J., Silvestri N. M., 2007, ApJ, 667, 433
* Mukadam et al. (2013) Mukadam A. S., et al., 2013, AJ, 146, 54
* Mukadam et al. (2017) Mukadam A. S., Szkody P., Gänsicke B. T., Pala A., 2017, in Tremblay P. E., Gaensicke B., Marsh T., eds, Astronomical Society of the Pacific Conference Series Vol. 509, 20th European White Dwarf Workshop. p. 341
* Nilsson et al. (2006) Nilsson R., Uthas H., Ytre-Eide M., Solheim J.-E., Warner B., 2006, MNRAS, 370, L56
* Patterson et al. (2005) Patterson J., Thorstensen J. R., Kemp J., 2005, PASP, 117, 427
* Patterson et al. (2008) Patterson J., Thorstensen J. R., Knigge C., 2008, PASP, 120, 510
* Pavlenko (2009) Pavlenko E., 2009, Journal of Physics Conference Series, 172, 012071
* Piro et al. (2005) Piro A. L., Arras P., Bildsten L., 2005, ApJ, 628, 401
* Robinson et al. (1995) Robinson E. L., et al., 1995, ApJ, 438, 908
* Saio (2019) Saio H., 2019, MNRAS, 487, 2177
* Schwieterman et al. (2010) Schwieterman E. W., et al., 2010, Journal of the Southeastern Association for Research in Astronomy, 3, 6
* Slevinsky et al. (1999) Slevinsky R. J., Stys D., West S., Sion E. M., Cheng F. H., 1999, PASP, 111, 1292
* Sullivan et al. (2008) Sullivan D. J., et al., 2008, MNRAS, 387, 137
* Szkody et al. (2002) Szkody P., Gänsicke B. T., Howell S. B., Sion E. M., 2002, ApJ, 575, L79
* Szkody et al. (2003) Szkody P., Gänsicke B. T., Sion E. M., Howell S. B., Cheng F. H., 2003, AJ, 126, 1451
* Szkody et al. (2010) Szkody P., et al., 2010, ApJ, 710, 64
* Szkody et al. (2012) Szkody P., et al., 2012, ApJ, 753, 158
* Szkody et al. (2016) Szkody P., et al., 2016, AJ, 152, 48
* Templeton et al. (2007) Templeton M., Stubbings R., Waagen E. O., Schmeer P., Pearce A., Nelson P., 2007, Central Bureau Electronic Telegrams, 922, 1
* Thorstensen et al. (2002) Thorstensen J. R., Patterson J., Kemp J., Vennes S., 2002, PASP, 114, 1108
* Toloza et al. (2016) Toloza O., et al., 2016, MNRAS, 459, 3929
* Townsley et al. (2004) Townsley D. M., Arras P., Bildsten L., 2004, ApJ, 608, L105
* Uthas et al. (2012) Uthas H., et al., 2012, MNRAS, 420, 379
* Van Grootel et al. (2012) Van Grootel V., Dupret M. A., Fontaine G., Brassard P., Grigahcène A., Quirion P. O., 2012, A&A, 539, A87
* Van Grootel et al. (2015) Van Grootel V., Fontaine G., Brassard P., Dupret M. A., 2015, A&A, 575, A125
* Vanlandingham et al. (2005) Vanlandingham K. M., Schwarz G. J., Howell S. B., 2005, PASP, 117, 928
* Vican et al. (2011) Vican L., et al., 2011, PASP, 123, 1156
* Warner & Woudt (2004) Warner B., Woudt P. A., 2004, in Kurtz D. W., Pollard K. R., eds, Astronomical Society of the Pacific Conference Series Vol. 310, IAU Colloq. 193: Variable Stars in the Local Group. p. 382 (arXiv:astro-ph/0310072)
* Warner & van Zyl (1998) Warner B., van Zyl L., 1998, in Deubner F.-L., Christensen-Dalsgaard J., Kurtz D., eds, IAU Symposium Vol. 185, New Eyes to See Inside the Sun and Stars. p. 321 (arXiv:cond-mat/9701105)
* Wheatley et al. (2018) Wheatley P. J., et al., 2018, MNRAS, 475, 4476
* Woudt & Warner (2002) Woudt P. A., Warner B., 2002, Ap&SS, 282, 433
* Woudt & Warner (2004) Woudt P. A., Warner B., 2004, MNRAS, 348, 599
* Woudt & Warner (2011) Woudt P. A., Warner B., 2011, Ap&SS, 333, 119
* van Spaandonk et al. (2010) van Spaandonk L., Steeghs D., Marsh T. R., Parsons S. G., 2010, ApJ, 715, L109
* van Zyl et al. (2000) van Zyl L., Warner B., O’Donoghue D., Sullivan D., Pritchard J., Kemp J., 2000, Baltic Astronomy, 9, 231
* van Zyl et al. (2004) van Zyl L., et al., 2004, MNRAS, 350, 307
|
MPP-2021-6
# Combining cosmological and local bounds on bimetric theory
Angelo Caravano Marvin Lüben 0000-0002-1259-7356 Jochen Weller
0000-0002-8282-2010
###### Abstract
Ghost-free bimetric theory describes two nonlinearly interacting spin-2
fields, one massive and one massless, thus extending general relativity. We
confront bimetric theory with observations of Supernovae type 1a, Baryon
Acoustic Oscillations and the Cosmic Microwave Background in a statistical
analysis, utilising the recently proposed physical parametrisation. This
directly constrains the physical parameters of the theory, such as the mass of
the spin-2 field and its coupling to matter. We find that all models under
consideration are in agreement with the data. Next, we compare these results
to bounds from local tests of gravity. Our analysis reveals that all two- and
three parameter models are observationally consistent with both cosmological
and local tests of gravity. The minimal bimetric model (only $\beta_{1}$) is
ruled out by our combined analysis.
## 1 Introduction
In our current era of precision cosmology, cosmological tests of gravity with
remarkable accuracy become available. On the one hand, the direct detection of
gravitational waves by the LIGO/Virgo collaboration [1] has confirmed some of
the most fundamental predictions of general relativity – gravitational waves
[2]. Moreover, the joint detection of gravitational waves with their
electromagnetic counterpart [3] show that gravitational waves indeed travel at
the speed of light to large precision [4]. On the other hand, measurements of
the cosmic microwave background [5, 6] and local observables [7, 8, 9, 10]
challenge the standard model of cosmology. In fact, the discrepancy between
the inferred value for the Hubble parameter today ($H_{0}$) from early and
late-time measurements represents a substantial tension [11]. From a
theoretical perspective, the cosmological constant problems remain unsolved
[12, 13]. In addition, concerns about the quantum consistency of a positive
cosmological constant have been raised, in the context of quantum breaking
[14, 15, 16, 17] and the swampland111For a critical perspective on the de
Sitter conjectures and the cosmological implications, see e.g. [18]. [19, 20,
21, 22].
Therefore, it is worth investigating theories beyond the standard model of
cosmology, which modify the gravitational theory or replace the cosmological
constant by some dark energy [23]. To modify gravity, one has to break one of
the axioms of Lovelock’s theorem [24]. A particularly simple possibility is to
add new degrees of freedom to the gravitational sector. The resulting scalar-
tensor [25, 26] and vector-tensor [27, 28] theories are already tightly
constrained by gravitational wave observations and forced into their simplest
forms [29, 30, 31, 32]. Also massive gravity [33, 34] is tightly constrained
observationally [35]. Contrarily, bimetric theory [36] remains mostly
unconstrained by the determination of the propagation speed of gravitational
waves [37, 38]. In the present paper, we will further investigate the
observational viability of the latter theory.
Historically, the question of whether the graviton can have a finite mass goes
back to Fierz and Pauli, who presented a linear theory of a massive graviton
in [39]. However, this theory failed even basic solar-system tests of gravity
due to the van Dam-Veltman-Zakharov (vDVZ) discontinuity [40, 41]. Shortly
after, Vainshtein argued that nonlinear terms might cure the discontinuity and
render a nonlinear theory of a massive graviton observationally viable [42].
However, Boulware and Deser argued that a fully nonlinear theory of a massive
graviton will inevitably introduce an additional degree of freedom (d.o.f.)
with negative norm222This additional degree of freedom is commonly referred to
as Boulware-Deser ghost, which represents a Laplacian instability [43, 44,
45]. Such an instability spoils unitarity and renders the theory ill-defined..
Rather recently, de Rham, Gabadadze and Tolley identified a loop-hole in the
argument, which lead to the construction of a fully nonlinear and ghost-free
theory of massive gravity [46, 33, 34]333Since then, the ghost-freedom of
massive gravity and its extensions was confirmed in a series of papers
utilising various methods, see e.g. [47, 48, 49, 50, 51]. For a pedagogical
paper on how to consistently count the number of propagating degrees of
freedom in first-order field theories, see [52].. The construction of a
nonlinear theory for a massive graviton requires the introduction of a
reference metric, which is kept fixed. A natural extension is to promote this
fixed reference metric to be dynamical, which leads to ghost-free bimetric
theory [36].
While massive gravity describes a single massive graviton, bimetric theory
contains a massless and a massive spin-2 field [36]. This yields distinct
phenomenological features. The mass $m_{\mathrm{FP}}$ of a single massive
graviton is tightly constrained. While gravitational wave observations provide
the upper bound of $m_{\mathrm{FP}}\lesssim 10^{-22}\,{\rm eV}$ [32], the most
stringent constraints come from the solar system with $m_{\mathrm{FP}}\lesssim
10^{-33}\,{\rm eV}$ [35]. Moreover, massive gravity does not give rise to
viable homogenous and isotropic cosmological solutions [53, 54, 55, 56].
Reviews on theoretical and phenomenological aspects of massive gravity can be
found in [57, 58, 35].
Contrarily, bimetric theory has a viable and rich phenomenology. In the
context of cosmology, the theory allows for homogeneous and isotropic
cosmological solutions with late-time de Sitter attractor. The
(self-)interactions of the massive and massless spin-2 fields give rise to
dynamical dark energy, even in the absence of vacuum energy, i.e. so-called
self-accelerating solutions [59, 60, 61, 62, 63, 64, 65, 66, 67]. Bimetric
theory can alleviate the $H_{0}$-tension if the spin-2 mass is of the order of
the Hubble constant [68, 69] Cosmological perturbations and cosmic structure
formation remain challenging due to the gradient instability at early times
[70, 71, 72, 73, 74, 75, 65, 76, 77, 78, 79, 80], which might be cured by
nonlinear terms in analogy to the Vainshtein mechanism [81, 82, 69]. The
theory contains the massive spin-2 field as dark-matter candidate [83, 84, 85,
86].
Spherically symmetric solutions [87] have been confronted with observations on
galaxy cluster [88], galactic [89, 90, 88], and solar system scales [90, 91,
92]. The nonlinear Vainshtein mechanism suppresses modifications in the
gravitational sector on small scales [93, 90]. Going further, gravitational
waves were studied in [94, 78, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104],
with an application to binary mergers [37, 38]. Summarising, bimetric theory
is in agreement with the current corresponding observational bounds and even
leads to possibly detectable effects, but it is not clear, whether the theory
is consistent with all theoretical and observational constraints
simultaneously.
To consistently combine the various observational and theoretical constraints
on the parameter space, a new parametrisation of bimetric solutions was
developed in [105]. The idea is to formulate solutions in terms of the
following physical parameters: mass of the spin-2 field, its coupling constant
to ordinary matter and the effective asymptotic cosmological constant. The
procedure automatically yields theoretical constraints that ensure a viable
cosmic expansion history, i.e. real-valued, non-singular and devoid of the
Higuchi ghost.
In this paper, we extend the analysis of [105] to test bimetric cosmology
against data from Supernovae type 1a (SN1a), Baryon Acoustic Oscillations
(BAOs) and the Cosmic Microwave Background (CMB). While bimetric theory has
been tested against these data sets beforehand [60, 64, 67, 106, 68, 107],
here we directly constrain the physical parameters. We restrict our analysis
to all theoretically viable bimetric models with up to three free interaction
parameters444To be more precise, we consider all models with up to three free
parameters $\beta_{n}$. Strictly speaking, the parameters $\beta_{1,2,3}$
parametrise interactions, while the parameters $\beta_{0,4}$ parametrise
vacuum energy.. In addition, we allow for non-zero spatial curvature in the
statistical analysis.
Next, we compare the cosmological bounds to bounds from local tests of
gravity. We use tests of the Newtonian gravitational potential from laboratory
to planetary scale as summarised in [108, 109, 110, 111]. The Yukawa
parametrisation, where the gravitational potential is written as a
superposition of the usual $1/r$-term and a Yukawa-term, is appropriate for
our purposes. We use the existing bounds to constrain the physical parameters
of bimetric theory. We add further bounds from tests of the scalar curvature,
i.e. measurements of the deflection and time delay of light [112, 113, 91].
The aforementioned frameworks are appropriate only in regimes without
Vainshtein screening. In screened spacetime regions these bounds are not
trustworthy, because deviations from general relativity (GR) are suppressed.
However, only a subregion of the full bimetric parameter space supports
Vainshtein screening. In this paper, we relate the bounds of [93, 90], which
ensure a working Vainshtein mechanism, to the physical parameters and identify
the region of the physical parameter space that supports screening. We will
see that the Vainshtein mechanism does not affect the observational bounds
from local tests of gravity, such that these are directly applicable.
Summarising, we consistently combine for the first time these observational
and theoretical constraints from cosmology and local tests of gravity with
each other.
This paper is organised as follows. We start with a brief summary of bimetric
theory, the cosmological solutions and the physical parameters in section 2.
We proceed with the statistical analysis using cosmological data in section 3.
In section 4, we discuss the bounds from local tests of gravity and the
Vainshtein screening, which we confront with the cosmological constraints. We
summarise our analysis and conclude in section 5.
## 2 Review of bimetric theory
In this section we briefly discuss bimetric theory, its mass spectrum,
cosmological solutions and the physical parametrisation. For a thorough
introduction to the field we refer to [114].
The construction of a nonlinear mass term involves two metric tensors, which
we call $g_{\mu\nu}$ and $f_{\mu\nu}$. The action of ghost-free bimetric
theory for these metrics reads [46, 33, 115, 34, 36]
$\displaystyle S=m_{\rm g}^{2}\int\mathrm{d}^{4}x\Bigg{[}\sqrt{-g}R^{\rm
g}+\alpha^{2}\sqrt{-f}R^{\rm
f}-2\mathcal{U}\left(\sqrt{g^{-1}f}\right)\Bigg{]}+\int\mathrm{d}^{4}x\sqrt{-g}\mathcal{L}_{\rm
m}\left(g,\Phi\right)$ (2.1)
where $R^{\rm g}$ and $R^{\rm f}$ are the Ricci scalars of $g_{\mu\nu}$ and
$f_{\mu\nu}$, resp. The parameter $m_{\rm g}$ is the Planck mass of
$g_{\mu\nu}$ and $\alpha$ is the Planck mass of $f_{\mu\nu}$ normalised to
$m_{\rm g}$. The ghost-free bimetric potential $\mathcal{U}$ is defined in
terms of the square-root matrix $S=\sqrt{g^{-1}f}$ and given by555Note that
bimetric theory is well-defined only if both metrics have a common time
direction [116].
$\displaystyle\mathcal{U}\left(\sqrt{g^{-1}f}\right)=\sum_{n=0}^{4}\beta_{n}e_{n}\left(\sqrt{g^{-1}f}\right)\,.$
(2.2)
Here, $\beta_{n}$ are real parameters with mass dimension two in our
parametrisation. The functions $e_{n}$ are the elementary symmetric
polynomials [115]. Matter fields are collectively denoted by $\Phi$, which
minimally couple666This setup is referred to singly-coupled bimetric theory.
More general ghost-free matter couplings exist in bimetric theory [117, 118,
119, 120, 121, 122]. to the metric $g_{\mu\nu}$ via the matter Lagrangian
$\mathcal{L}_{\rm m}$. Hence, $g_{\mu\nu}$ is the physical metric that defines
the geometry in which the matter fields propagate.
Varying the action with respect to $g^{\mu\nu}$ and $f^{\mu\nu}$ yields the
modified Einstein equations
$\displaystyle G^{\rm g}_{\mu\nu}+\mathcal{U}^{\rm g}_{\mu\nu}=\frac{1}{m_{\rm
g}^{2}}T_{\mu\nu}\,,\qquad G^{\rm f}_{\mu\nu}+\alpha^{-2}\mathcal{U}^{\rm
f}_{\mu\nu}=0\,,$ (2.3)
where $G^{\rm g,f}_{\mu\nu}$ are the Einstein tensors of $g_{\mu\nu}$ and
$f_{\mu\nu}$, resp. The terms coming from the variation of the bimetric
potential are given by
$\displaystyle\mathcal{U}^{\rm
g}_{\mu\nu}=\sum_{n=0}^{3}(-1)^{n}\beta_{n}g_{\mu\lambda}Y^{\lambda}_{(n)\nu}(S)\,,\qquad\mathcal{U}^{\rm
f}_{\mu\nu}=\sum_{n=0}^{3}(-1)^{n}\beta_{4-n}f_{\mu\lambda}Y^{\lambda}_{(n)\nu}(S^{-1})$
(2.4)
where the matrices $Y^{\lambda}_{(n)\nu}$ are given in, e.g. [115]. Since
matter couples to the physical metric $g_{\mu\nu}$, the matter stress-energy
tensor is given by
$\displaystyle
T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta\sqrt{-g}\mathcal{L}_{\rm
m}}{\delta g^{\mu\nu}}\,.$ (2.5)
If the matter sector is invariant under diffeomorphisms, the stress-energy
tensor of matter is conserved,
$\displaystyle\nabla^{\mu}\,T_{\mu\nu}=0\,,$ (2.6)
where $\nabla_{\mu}$ is the covariant derivative compatible with $g_{\mu\nu}$.
The Bianchi identity, $\nabla^{\mu}G^{\rm g}_{\mu\nu}=0$, then yields the so-
called Bianchi constraint,
$\displaystyle\nabla^{\mu}\mathcal{U}^{\rm g}_{\mu\nu}=0\,.$ (2.7)
The Bianchi identity in the $f_{\mu\nu}$-sector yields another Bianchi
constraint, which however coincides with eq. (2.7) by diffeomorphism
invariance.
### 2.1 Vacuum solutions and mass eigenstates
Before moving to cosmology, we briefly review an important class of solutions
in bimetric theory [36, 123]. Let both metric tensors be proportional as
$f_{\mu\nu}=c^{2}g_{\mu\nu}$ with conformal factor $c$. This ansatz is a
solution to the modified Einstein equations only in vacuum, i.e. for
$T_{\mu\nu}=0$. The Bianchi constraint (2.7) implies that $c$ is a constant.
Upon this ansatz, the modified Einstein equations (2.3) simplify to
$\displaystyle G^{\rm g}+\Lambda_{\rm g}\,g_{\mu\nu}=0\,,\qquad G^{\rm
f}_{\mu\nu}+c^{-2}\Lambda_{\rm f}\,f_{\mu\nu}=0\,,$ (2.8)
where the effective cosmological constants are given in terms of the bimetric
parameters as
$\displaystyle\Lambda_{\rm
g}=\beta_{0}+3\beta_{1}c+3\beta_{2}c^{2}+\beta_{3}c^{3}\,,\qquad\Lambda_{\rm
f}=\frac{1}{\alpha^{2}c^{2}}\left(\beta_{1}c+3\beta_{2}c^{2}+\beta_{3}c^{3}+\beta_{4}c^{4}\right)\,.$
(2.9)
Subtracting both equations in (2.8) and using that both metrics are
conformally related yields $\Lambda_{\rm g}=\Lambda_{\rm f}\equiv\Lambda$, or
in terms of the bimetric parameters
$\displaystyle\alpha^{2}\beta_{3}c^{4}+(3\alpha^{2}\beta_{2}-\beta_{4})c^{3}+3(\alpha^{2}\beta_{1}-\beta_{3})c^{2}+(\alpha^{2}\beta_{0}-\beta_{3})c-\beta_{1}=0\,.$
(2.10)
This is a polynomial in $c$ of degree 4. Hence, there are up to four real-
valued solutions for $c$ in terms of the bimetric parameters $\alpha$ and
$\beta_{n}$, each corresponds to a vacuum solution.
Bimetric theory has a well-defined mass spectrum around proportional
solutions. Let us perturb the metrics around the background $\bar{g}_{\mu\nu}$
as
$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\frac{1}{m_{\rm g}}\delta
g_{\mu\nu}\,,\qquad f_{\mu\nu}=c^{2}\bar{g}_{\mu\nu}+\frac{c}{\alpha m_{\rm
g}}\delta f_{\mu\nu}\,,$ (2.11)
where $\delta g_{\mu\nu}\ll m_{\rm g}$ and $\delta f_{\mu\nu}\ll\alpha m_{\rm
g}$ are small fluctuations around the proportional background
$\bar{f}_{\mu\nu}=c^{2}\bar{g}_{\mu\nu}$. The mass eigenstates are linear
superpositions of the metric fluctuations. The spectrum contains a massless
mode, $\delta G_{\mu\nu}$, and a massive mode, $\delta M_{\mu\nu}$, with
Fierz-Pauli mass777The mass term of $\delta M_{\mu\nu}$ has the Fierz-Pauli
structure [39].
$\displaystyle
m_{\mathrm{FP}}^{2}=\left(1+\frac{1}{\alpha^{2}c^{2}}\right)c(\beta_{1}+2\beta_{2}c+\beta_{3}c^{2})\,.$
(2.12)
We can express the metric fluctuations in terms of the mass eigenstates as
$\displaystyle\delta
g_{\mu\nu}=\frac{1}{\sqrt{1+\bar{\alpha}^{2}}}\left(\delta
G_{\mu\nu}-\bar{\alpha}\,\delta M_{\mu\nu}\right)\,,\qquad\delta
f_{\mu\nu}=\frac{1}{\sqrt{1+\bar{\alpha}^{2}}}\left(\delta
M_{\mu\nu}+\bar{\alpha}\,\delta G_{\mu\nu}\right)\,.$ (2.13)
Here, we defined $\bar{\alpha}=\alpha c$ that parametrises the mixing of the
massless and the massive mode in the metric fluctuations. From here we can
already identify the GR and massive gravity limit of bimetric theory [80, 92].
The GR-limit is $\bar{\alpha}\rightarrow 0$ because then the fluctuation of
$g_{\mu\nu}$ is aligned with the massless mode while the fluctuation of
$f_{\mu\nu}$ is aligned with the massive mode. In the opposite limit
$\bar{\alpha}\rightarrow\infty$ bimetric theory approaches massive gravity as
then $\delta g_{\mu\nu}$ is aligned with the massive mode while $\delta
f_{\mu\nu}$ coincides with the massless mode.
### 2.2 FLRW solutions
Assuming homogeneity and isotropy according to the cosmological principle,
both metric can be cast into the Friedmann-Lemaître-Robertson-Walker (FLRW)
form [59, 60, 61],
$\mathrm{d}s_{\rm
g}^{2}=-\mathrm{d}t^{2}+a^{2}\left(\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}\Omega^{2}\right)\,,\qquad\mathrm{d}s_{\rm
f}^{2}=-X^{2}\mathrm{d}t^{2}+b^{2}\left(\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}\Omega^{2}\right)\,,$
(2.14)
where $a$ and $b$ are the scale factors of $g_{\mu\nu}$ and $f_{\mu\nu}$,
respectively, while $X$ is the lapse of $f_{\mu\nu}$. These metric functions
depend on timte $t$ only. Further, $k$ is the spatial curvature, which must be
common to both metrics [124]. In this coordinate system, $k<0$, $k=0$, $k>0$
describes a closed, flat, or open universe, resp. Let us define the Hubble
rates and the scale factor ratio as
$\displaystyle H=\frac{\dot{a}}{a}\,,\ \ H_{\rm f}=\frac{\dot{b}}{Xb}\,,\ \
y=\frac{b}{a}\,,$ (2.15)
where dot denotes derivative with respect to cosmic time $t$.
On the bidiagonal FLRW ansatz (2.14), the Bianchi constraint (2.7) simplifies
to
$\displaystyle(\dot{b}-X\dot{a})(\beta_{1}+2\beta_{2}y+\beta_{3}y^{2})=0\,.$
(2.16)
This equation has two branches of solution. We use the so-called dynamical
branch solution with $X=\dot{b}/\dot{a}$ for the remainder of this paper888On
the other branch of solutions, referred to as algebraic, the scale factor
ratio is forced to be constant, $y={\rm const}$. Since this solution is found
to be pathological [70, 125], we focus on the dynamical branch solution.. Then
the Hubble rates are related as $H=yH_{\rm f}$.
We take the matter source to be a perfect fluid with stress-energy tensor
$\displaystyle T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+p\,g^{\mu\nu}$ (2.17)
with energy density $\rho$, pressure $p$ and $u^{\mu}$ the $4$-velocity of the
fluid. We split energy density and pressure into a non-relativistic and a
relativistic part as $\rho=\rho_{\rm m}+\rho_{\rm r}$ and $p=p_{\rm m}+p_{\rm
r}$, resp. The subscript $\rm m$ refers to non-relativistic matter, while the
subscript $\rm r$ refers to radiation. The conservation of energy-momentum
(2.6) leads to the continuity equation
$\displaystyle\dot{\rho}_{i}=-3H(1+w_{i})\rho_{i}\,,\quad i={\rm m,r}\,,$
(2.18)
with $w_{i}=p_{i}/\rho_{i}$ the equation of state. Therefore, the energy
density of matter and radiation evolve with the scale factor $a$ of the
physical metric as in standard cosmology. To be explicit, the continuity
equation is solved by
$\displaystyle\rho_{i}=\rho_{i,0}a^{-3(1+w_{i})}\,,$ (2.19)
unless $w_{i}=-1$, with the integration constants chosen such that
$\rho_{i,0}=\rho_{i}(1)$, where $a=1$ corresponds to the present time.
Radiation has an equation of state of $w_{\rm r}=1/3$ while for matter $w_{\rm
m}=0$.
The time-time component of the two modified Einstein equations (2.3) become
$3H^{2}+3\frac{k}{a^{2}}=\frac{1}{m_{\rm g}^{2}}\left(\rho_{\rm de}+\rho_{\rm
m}+\rho_{\rm r}\right)\,,\qquad 3H^{2}+3\frac{k}{a^{2}}=\frac{1}{m_{\rm
g}^{2}}\rho_{\rm pot}\,.$ (2.20)
Here we defined the energy density of the dynamical dark energy induced by the
bimetric potential as
$\frac{\rho_{\rm de}}{m_{\rm
g}^{2}}=\beta_{0}+3\beta_{1}y+3\beta_{2}y^{2}+\beta_{3}y^{3}$ (2.21)
with the time-dependent function $y$. The Friedmann equation of $f_{\mu\nu}$
is sourced by potential energy only with energy density
$\displaystyle\frac{\rho_{\rm pot}}{m_{\rm
g}^{2}}=\frac{1}{\alpha^{2}}\left(\frac{\beta_{1}}{y}+3\beta_{2}+3\beta_{3}y+\beta_{4}y^{2}\right)\,.$
(2.22)
Interpreting the effect of spatial curvature as an energy source, we can
define the energy density
$\displaystyle\rho_{\rm k}=-3m_{\rm g}^{2}ka^{-2}\,.$ (2.23)
Then, the Friedmann eq. 2.20 of $g_{\mu\nu}$ can be written as
$\displaystyle 3H^{2}=\frac{1}{m_{\rm g}^{2}}\left(\rho_{\rm de}+\rho_{\rm
k}+\rho_{\rm m}+\rho_{\rm r}\right)\,.$ (2.24)
It remains to determine the time-evolution of $\rho_{\rm de}$, which in
general cannot be written as a polynomial in $1/a$. Subtracting the the
Friedmann equations in (2.20) yields
$\displaystyle\alpha^{2}\beta_{3}y^{4}+(3\alpha^{2}\beta_{2}-\beta_{4})y^{3}+3(\alpha^{2}\beta_{1}-\beta_{3})y^{2}+\left(\alpha^{2}\beta_{0}-3\beta_{2}+\alpha^{2}\frac{\rho_{\rm
m}+\rho_{\rm r}}{m_{\rm g}^{2}}\right)y-\beta_{1}=0\,.$ (2.25)
This equation determines the time evolution of $y$ in terms of the time
evolution of the matter energy densities. It is a quartic polynomial in $y$,
hence there are up to four real-valued solutions with different time
evolutions of $y$. Following [60, 67], we can distinguish three classes of
solutions by inspecting the early-time behavior, for which the energy
densities classically diverge:
* •
Finite branch. The first possible early-time behavior is that $y$ approaches
zero so as to compensate the diverging value of the energy density in the
linear piece of the polynomial (2.25). Hence, during early times the scale
factor ratio can be approximated as $y\sim m_{\rm
g}^{2}\beta_{1}/(\alpha^{2}\rho)$. Existence of the finite branch requires
$\beta_{1}>0$. The scale factor ratio monotonically increases in time. In the
asymptotic future, the energy densities vanish and $y$ approaches a constant
value $c$, which corresponds to the lowest-lying, strictly positive root of
eq. 2.10 [105]. For positive matter and radiation energy densities, this
branch of solution satisfies the cosmological stability bound, a
generalization of the Higuchi bound to FLRW spacetime [126, 79].
* •
Infinite branch. The alternative is that $y$ is large as the energy densities
are large. Then, at early times, the evolution can be approximated as
$y^{3}\sim-\rho/(m_{\rm g}^{2}\beta_{3})$. This reveals that the infinite
branch exists only if $\beta_{3}<0$. The scale factor ratio monotonically
decreases in time. In the asymptotic future, when the energy densities vanish,
$y$ approaches the highest lying, strictly positive root of eq. 2.10 [105].
This branch, however, inevitably violates the cosmological stability bound and
is therefore not a physical solution [126, 79].
Due to the high degree of the polynomial (2.25), there are further exotic
branches of solutions, which however do not give rise to a viable expansion
history [79]. Therefore, we pick out the finite branch solution in section 3.
The scale factor ratio then satisfies $0<y<c$ [79, 105].
Let us pause for a moment and summarise. The bimetric potential contributes a
dynamical dark energy to the Friedmann equation in the form of $\rho_{\rm
de}$, which depends on the scale factor ratio $y$. Since $y$ approaches a
constant value $c$ in the asymptotic future, $\rho_{\rm de}$ approximates the
effect of a cosmological constant at late times. We have identified two
possible time evolutions for $y$ yielding a consistent expansion history. Out
of these, only the finite branch solution satisfies the cosmological stability
bound. Therefore, $y$ decreases when looking back in time, implying that also
$\rho_{\rm de}$ decreases. In other words, the (self-)interactions of the
massive and massless spin-2 field are dynamically increasing as time proceeds.
Hence, the (self-)interaction energy is necessarily phantom or negative [79,
69].
### 2.3 Physical parametrisation
The solutions presented thus far are parametrised in terms of the Planck mass
ratio $\alpha$ and the interaction parameters $\beta_{n}$. Vacuum solutions
are labeled by $c$ that are the roots of eq. 2.10. The bimetric action and
hence the equations of motion are invariant under a rescaling of these
parameters implying that one of them is redundant.
In [105] a parametrisation that is invariant under this rescaling is proposed
and worked out. It relies on the following physical parameters: the mass
$m_{\mathrm{FP}}$ of the spin-2 field, its coupling to matter $\bar{\alpha}$,
and the effective cosmological constant $\Lambda$. The hence baptised physical
parametrisation allows to unambiguously combine various theoretical and
observational bounds on the parameter space of bimetric theory.
Let us give more detail on the physical parametrisation and on the already
inferred theoretical consistency bounds. Since there are up to four real-
valued roots $c$ of eq. 2.10, the relation between the physical and
interaction parameters is not unique. However, a vacuum solution and the
cosmic expansion history are well-defined only under the following conditions:
* •
Consistent de Sitter vacuum. Without loss of generality we restrict ourselves
to roots with $c>0$999Solutions with $c<0$ can be mapped onto solutions with
$c>0$ because the equations of motion are invariant under the combined
rescaling $c\rightarrow-c$, $\beta_{n}\rightarrow(-1)^{n}\beta_{n}$.. We
impose that the mass of the spin-2 field is positive, $m_{\mathrm{FP}}>0$, and
we restrict ourselves to de Sitter vacua, $\Lambda>0$. For a massive spin-$2$
field propagating in de Sitter space, unitarity forbids the mass to be
arbitrarily small. Instead, the Higuchi bound [127]
$\displaystyle 3m_{\mathrm{FP}}^{2}>2\Lambda$ (2.26)
has to be satisfied. If the bound is violated, the helicity-$0$ mode of the
massive spin-$2$ field has a negative norm (Higuchi ghost).
* •
Consistent asymptotic future. The cosmic expansion has to approach a
consistent de Sitter vacuum in the asymptotic future. Since only the finite
branch solution to the Friedmann equations is physical, the scale factor ratio
satisfies $0<y<c$. Hence, the lowest-lying strictly positive root of eq. 2.10
as the unique physical vacuum solution. Identifying the asymptotic future of
the cosmic expansion history with the physical vacuum solution implies
further, model-dependent conditions on the bimetric parameters.
* •
Consistent cosmic expansion history. The consistency of the early universe for
a cosmic expansion history on the finite branch requires [67]
$\displaystyle\beta_{1}>0\,,$ (2.27)
as can be seen from eq. 2.20. Otherwise $H^{2}\leq 0$ during early times.
These restrictions imply a unique relation between the physical and
interaction parameters. The details for all bimetric models with up to three
non-vanishing interaction parameters are worked out in Ref. [105]. For
completeness, we summarise the explicit expressions in table 4. Using the
physical parametrisation, in section 3 we compute bounds from cosmological
observations.
## 3 Bounds from cosmological observations
We now compare bimetric theory to cosmological observables. We restrict
ourselves to background observables that constrain the Hubble rate at
different redshifts. In particular, we use Supernovae type 1a, Baryon Acoustic
Oscillations and the first peak of the oscillations in the Cosmic Microwave
Background, which we discuss in greater detail in section 3.2.
Measurements of perturbative quantities are not included because of the
aforementioned gradient instability in the scalar sector [70, 71, 72, 73, 74,
75, 65, 76, 77, 78, 79, 80]. The scale at which the gradient instability kicks
in is set by the spin-2 mass [69]. Taking nonlinearities into account possibly
stabilises the perturbations at early times [82, 81, 128]. However, a
framework to study cosmological phenomena on the perturbative level within
bimetric theory is still pending.
### 3.1 Parametrisation used for model fitting
We have already explained how to get the time-evolution of the scale factor
ratio $y$, the time evolution of $\rho_{\rm de}$ and hence the time evolution
of $H$. To ease these computations, it is convenient to introduce the energy
density parameters
$\displaystyle\Omega_{i}=\frac{\rho_{i}}{3m_{\rm g}^{2}H^{2}}\,,\quad
i=\\{{\rm de,k,m,b,r},\gamma\\}\,.$ (3.1)
The Friedmann equation (2.24) can then be written in the compact form
$\Omega_{\rm de}+\Omega_{\rm k}+\Omega_{\rm m}+\Omega_{\rm r}=1\,.$ (3.2)
Evaluating this equation at present times yields the following relation
between the parameters as
$\Omega_{\rm de0}+\Omega_{\rm k0}+\Omega_{\rm m0}+\Omega_{\rm r0}=1\,.$ (3.3)
We choose to express the parameter $\Omega_{\rm m0}$ in terms of the other
parameters in our statistical analysis. In order to compute $\Omega_{\rm de}$,
we introduce the following constant parameters:
$B_{n}=\frac{\alpha^{-n}\beta_{n}}{3H_{0}^{2}}\,,\ \Omega_{\rm
FP}=\frac{m_{\mathrm{FP}}^{2}}{3H_{0}^{2}}\,,\
\Omega_{\Lambda}=\frac{\Lambda}{3H_{0}^{2}}\,.$ (3.4)
The relations between $B_{n}$ and the physical parameters $\bar{\alpha}$,
$\Omega_{\mathrm{FP}}$, and $\Omega_{\Lambda}$ are presented in appendix A as
derived in [105]. Note that $B_{n}$ is rescaling-invariant and hence
observable. Further, in analogy to the definition of $\bar{\alpha}$, we define
the rescaled scale factor ratio as $\bar{y}=\alpha y$. To be explicit, the
energy density parameter of the induced dynamical dark energy is given as
$E^{2}\Omega_{\rm
de}=B_{0}+3B_{1}\bar{y}+3B_{2}\bar{y}^{2}+B_{3}\bar{y}^{3}\,,$ (3.5)
where $E=H/H_{0}$ is the normalised Hubble rate, as follows from eq. 2.21. The
scale factor ratio $\bar{y}$ is determined by the quartic polynomial (2.25),
which in terms of the energy density parameters reads
$B_{3}\bar{y}^{4}+(3B_{2}-B_{4})\bar{y}^{3}+3(B_{1}-B_{3})\bar{y}^{2}+(B_{0}-3B_{2}+E^{2}\Omega_{\rm
m}+E^{2}\Omega_{\rm r})\bar{y}-B_{1}=0\,.$ (3.6)
As last ingredient, we need to determine $\Omega_{\rm de0}$ as initial
condition. In order to do so, we determine the value of the scale factor ratio
at present time $\bar{y}_{0}$ using the $f_{\mu\nu}$-Friedmann equation (2.20)
evaluated today:
$0=B_{1}+(3B_{2}+\Omega_{\rm
k0}-1)\bar{y}_{0}+3B_{3}\bar{y}_{0}^{2}+B_{4}\bar{y}_{0}^{3}$ (3.7)
This polynomial has up to three real-valued roots, out of which we pick the
one that satisfies $0<\bar{y}_{0}<\bar{\alpha}$ in order to ensure that we are
on the finite branch. Plugging the result into eq. 3.5 evaluated today,
determines $\Omega_{\rm de0}$, which in turn determines $\Omega_{\rm m0}$ via
eq. 3.3.
### 3.2 Cosmological data
We now present the cosmological observables that are used in our statistical
analysis. We use three different types of observations: Supernovae type 1a
(SN1a), Baryon Acoustic Oscillations (BAOs) and Cosmic Microwave Background
(CMB). Before discussing them in detail, let us first introduce some useful
quantities to enlighten the notation.
We start with the definition of the following integral:
$\displaystyle\mathcal{I}_{k}(z)=\begin{cases}\frac{1}{\sqrt{|\Omega_{\rm
k0}|}}\sinh\left(\sqrt{|\Omega_{\rm
k0}|}\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime})}\right)&,\ k<0\\\
\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime})}&,\ k=0\\\
\frac{1}{\sqrt{|\Omega_{\rm k0}|}}\sin\left(\sqrt{|\Omega_{\rm
k0}|}\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime})}\right)&,\
k>0\end{cases},$ (3.8)
where $E(z)=H(z)/H_{0}$. We use this quantity to write the various distance
indicators, such as the luminosity distance $d_{\rm L}$ and the angular
diameter distance $d_{\rm A}$:
$\displaystyle d_{\rm L}(z)=\frac{c}{H_{0}}(1+z)\mathcal{I}_{k}(z)\,,\qquad
d_{\rm A}(z)=\frac{c}{H_{0}}\frac{1}{1+z}\mathcal{I}_{k}(z)\,.$ (3.9)
Another important quantity is the comoving sound horizon, defined as
$\displaystyle r_{\rm
s}(z)=\frac{c}{\sqrt{3}}\int_{z}^{\infty}\frac{dz^{\prime}}{H(z^{\prime})\sqrt{1+(3\Omega_{\rm
b0}/4\Omega_{\rm\gamma 0})/(1+z^{\prime})}}\,.$ (3.10)
Here $\Omega_{\rm b0}$ and $\Omega_{\rm\gamma 0}$ are the baryons and photons
density parameters. $\Omega_{\rm\gamma 0}$ is fixed by the temperature $T_{\rm
CMB}$ of the CMB [129]:
$\displaystyle\frac{3}{4\Omega_{\rm\gamma 0}h^{2}}=31500(T_{\rm
CMB}/2.7\text{K})^{-4},\qquad T_{\rm CMB}=2.7255\text{K}\,.$ (3.11)
Here, $h=H_{0}/100\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ is the normalised
Hubble rate today. The photons density parameter is related to the total
radiation density parameter via
$\displaystyle\Omega_{\rm r0}=\Omega_{\rm\gamma 0}(1+0.2271N_{\text{eff}})\,.$
(3.12)
$N_{\text{eff}}$ is the effective number of neutrino species, that we fix to
the standard value $N_{\text{eff}}=3.046$ for our analysis. We now proceed
introducing separately the different observables.
#### 3.2.1 Supernovae type 1a
Supernovae of type 1a serve as standard candles whose luminosities can be
inferred independently of their redshift. This allows to build up the local
distance ladder. We use the catalogue of the Joint Light-curve Analysis (JLA)
that contains 740 SN1a events as reported in Ref. [130].
The observed quantity in the case of Supernovae is the apparent magnitude $m$
$\displaystyle m$ $\displaystyle=M+25+5\log_{10}d_{\rm
L}=\mathcal{M}+5\log_{10}D_{\rm L}\,,$ (3.13)
where $M$ is the absolute magnitude, $D_{\rm L}=H_{0}d_{\rm L}$ and
$\mathcal{M}=M-5\log_{10}H_{0}+25$. In this formula $\mathcal{M}$ is just an
additive constant and appears as a nuisance parameter. This implies that the
absolute magnitude $M$ and $H_{0}$ are degenerate parameters.
In order to compare the luminosity distance to the observed apparent magnitude
$m_{\rm obs}$, we have to callibrate the supernovae as [130]
$\displaystyle m=m_{\rm obs}-\Delta_{\rm M}+\alpha X_{1}-\beta C\,,$ (3.14)
where $m$ is related to the luminosity distance as in Eq. (3.13). Here,
$X_{1}$ and $C$ are the light-curve parameters of the supernova. For the
nuisance parameters $\alpha$, $\beta$ and $\Delta_{\rm M}$ we use the best-fit
values $\alpha=0.140\pm 0.006$ and $\beta=3.139\pm 0.072$ as obtained in Ref.
[130]. The parameter $\Delta_{\rm M}$ is a correction to the absolute
magnitude $M$ of the supernova that depends on the stellar mass $M_{\ast}$ of
the host galaxy of the supernova as
$\displaystyle\Delta_{\rm M}=\begin{cases}-0.060\pm 0.012&,\ {\rm
if}\,M_{\ast}<10^{10}M_{\odot}\\\ 0&,\ {\rm otherwise.}\end{cases}$ (3.15)
Marginalising over $\mathcal{M}$, the log-likelihood for the SN-data is given
by
$-2\log\mathcal{L}_{\rm SN}(\theta)=S_{2}-\frac{S_{1}^{2}}{S_{0}}\,,$ (3.16)
where we have defined
$S_{0}=\sum_{ij}(C^{-1}_{\rm SN})_{ij}\,,\qquad S_{1}=\sum_{ij}(C^{-1}_{\rm
SN})_{ij}y_{i}\,,\qquad S_{2}=\sum_{ij}(C^{-1}_{\rm SN})_{ij}y_{i}y_{j}\,,$
(3.17)
in terms of $y=m-5\log_{10}D_{\rm L}$. The covariance matrix $C_{\rm SN}$ is
given in Ref. [130]. The errors on $\alpha$, $\beta$ and $\Delta_{M}$ are
added in quadrature.
#### 3.2.2 CMB
CMB observations constrain background cosmology through the measurement of two
different distance ratios [131]. The first is the ratio between the angular
diameter distance $d_{\rm A}(z_{\ast})$ and the sound horizon $r_{\rm
s}(z_{\ast})$ at decoupling epoch $z_{\ast}$. This ratio is measured through
the location of the first peak of the CMB power spectrum:
$\displaystyle l_{\rm A}=(1+z_{\ast})\frac{\pi d_{\rm A}(z_{\ast})}{r_{\rm
s}(z_{\ast})}\,.$ (3.18)
The second is the angular distance divided by the Hubble horizon at the
decoupling epoch, $d_{\rm A}(z_{\ast})H(z_{\ast})/c$. This last information is
usually implemented in the shift parameter, defined as:
$\displaystyle R(z_{\ast})=H_{0}\sqrt{\Omega_{\rm m0}}(1+z_{\ast})d_{\rm
A}(z_{\ast})/c\,,$ (3.19)
which differs from $d_{\rm A}(z_{\ast})H(z_{\ast})/c$ by a factor of
$\sqrt{1+z_{\ast}}$. This parameter is obtained assuming
$\displaystyle H^{2}(z_{\ast})=H^{2}_{0}\Omega_{\rm m0}(1+z_{\ast})^{3},$
(3.20)
which correspond to neglect all contributions (other than $\Omega_{\rm m0}$)
to the evolution history at the time of decoupling $H(z_{\ast})$. We use that
the early-universe is not altered in bimetric theory compared to general
relativity as our working assumption. This is justified because on the finite
branch the energy density arising from the bimetric potential does not
contribute to the Hubble rate at large redshifts.
For our statistical analysis we follow the standard procedure [132],
implementing CMB data in the three distance priors $(l_{A},\text{ }R,\text{
}\Omega_{\rm b0}h^{2})$. We use the latest Planck values [129]
$\displaystyle l_{\rm A}(z_{\ast})=301.471^{+0.089}_{-0.090}\,,\qquad
R(z_{\ast})=1.7502\pm 0.0046\,,\qquad\Omega_{\rm b0}h^{2}=0.02236\pm 0.00015$
(3.21)
In our analysis $R(z_{\ast})$ and $l_{A}(z_{\ast})$ are computed from eq. 3.19
and eq. 3.18, where the decoupling epoch $z_{\ast}$ is obtained via the
fitting function [133]:
$\begin{split}z_{\ast}&=1048(1+0.00124\,(\Omega_{\rm
b0}h^{2})^{-0.738})(1+g_{1}(\Omega_{\rm m0}h^{2})^{g_{2}})\,,\\\
g_{1}&=\frac{0.0783\,(\Omega_{\rm b0}h^{2})^{-0.238}}{1+39.5\,(\Omega_{\rm
b0}h^{2})^{0.763}}\,,\qquad g_{2}=\frac{0.560}{1+21.1\,(\Omega_{\rm
b0}h^{2})^{1.81}}.\end{split}$ (3.22)
Defining the CMB data vector as
$\displaystyle X_{\rm CMB}^{\rm T}=\left(\begin{matrix}l_{\rm
A}-301.471,\text{ }R(z_{\ast})-1.7502,\text{ }\Omega_{\rm
b0}h^{2}-0.02236\end{matrix}\right)$ (3.23)
the likelihood for the CMB is written as:
$\displaystyle-2\log\mathcal{L}_{\text{CMB}}=X_{\rm CMB}^{\rm T}C_{\rm
CMB}^{-1}X_{\rm CMB}\,.$ (3.24)
where the covariance matrix $C_{\rm CMB}$ can be found in [129].
#### 3.2.3 BAO
Barionic Acoustic Oscillations (BAOs) are another important tool to constrain
background cosmology. We follow a procedure similar to [107], using
measurements at 10 different redshifts $z=0.106,\text{ }0.15,\text{
}0.38,\text{ }0.51,\text{ }0.61,\text{ }0.72,\text{ }0.978,\text{ }1.23,\text{
}1.526,\text{ }1.944$. The data that we consider are taken from the surveys
6dFGS [134], SDSS MGS [135], BOSS DR12 [136], BOSS DR14 [137], eBOSS QSO [138]
and are summarised, e.g. in table 2 of [107].
The relevant length scale in the case of BAOs is the sound horizon at the drag
epoch $r_{\rm s}(z_{\rm d})$. The drag epoch can be derived with the following
fitting formula [133]:
$\displaystyle\begin{split}&z_{\rm d}=\frac{1291(\Omega_{\rm
m0}h^{2})^{0.251}}{1+0.659(\Omega_{\rm m0}h^{2})^{0.828}}[1+b_{1}(\Omega_{\rm
b0}h^{2})^{b_{2}}]\,,\\\ &b_{1}=0.313(\Omega_{\rm
m0}h^{2})^{-0.419}[1+0.607(\Omega_{\rm m0}h^{2})^{0.674}]\,,\qquad
b_{2}=0.238(\Omega_{\rm m0}h^{2})^{0.223}\end{split}$ (3.25)
The other relevant quantities in our study are the effective distance measure
$d_{\rm V}$ and the redshift-weighted comoving distance $d_{\rm M}$:
$d_{\rm V}(z)=\Biggl{[}(1+z)^{2}d_{\rm
A}^{2}(z)\frac{cz}{H(z)}\Biggr{]}^{1/3}\,,\qquad d_{\rm M}(z)=(1+z)\,d_{\rm
A}(z).$ (3.26)
The likelihood for the statistical analysis is derived using the covariance
matrices taken from the original references [134, 135, 136, 137, 138].
### 3.3 Statistical analysis and results
Using the aforementioned data sets, we perform a statistical analysis via MCMC
sampling for all models with up to three non-vanishing interaction and vacuum
energy parameters $\beta_{n}$. The $\Lambda$CDM-model corresponds to the
$\beta_{0}$-model in this sense. We classify models according to their number
of free parameters $\beta_{n}$: models with one, two or three free parameters
$\beta_{n}$ are referred to as one-, two- or three-parameter models,
respectively. In terms of the physical parameters, that means that one, two or
all three of the parameters $\bar{\alpha}$, $\Omega_{\rm FP}$ and
$\Omega_{\Lambda}$ are independent. The best-fit values and the 1-dimensional
($1\sigma$) error intervals for all free and derived parameters are reported
in tables 1 and 2.
In order to estimate the goodness-of-fit for each model, we compute the
Bayesian Information Criterion (BIC) [139],
${\rm BIC}=\chi^{2}+k\ln N$ (3.27)
where $\chi^{2}=-2\ln\mathcal{L}$ is evaluated at maximum likelihood, $k$ is
the number of model parameters and $N$ is the number of data points. For our
combined data set SN+CMB+BAO we have $N=740+3+10=753$. The number of model
parameters is given by $k=3+\hat{k}$ with $\hat{k}$ the number of independent
physical parameters as will become clear from the next paragraph. We will use
the BIC of the $\Lambda\rm CDM$-model as reference. The difference $\Delta\rm
BIC$ allows to estimate the preference of one model over another as [140,
107]: strong support ($\Delta\rm BIC<-12$), favorable ($\Delta\rm BIC<-6$),
inconclusive ($\Delta\rm BIC<6$), disfavored ($\Delta\rm BIC<12$), strongly
disfavored ($\Delta\rm BIC\geq 12$). The corresponding $\Delta\rm BIC$ are
reported in tables 1 and 2 as well.
Independent parameters and priors. We use the theoretical bounds of [105] that
ensure a consistent cosmology, which we summarise in table 4, as flat priors.
For the parameter $\Omega_{\Lambda}$ we use $[0,1]$ as flat prior. For the
other physical parameters $\bar{\alpha}$ and $\Omega_{\mathrm{FP}}$ we use a
log-scale to resolve large regions of the parameter space. We again use flat
priors with $\log_{10}(\bar{\alpha})\in[-100,100]$ and $\log_{10}(\Omega_{\rm
FP})\in[-2,122]$ if applicable101010The $\Lambda$CDM-model does not contain
these parameters. For the two-parameter models, $\bar{\alpha}$ and
$\Omega_{\mathrm{FP}}$ are not independent. For the $\beta_{0}\beta_{1}$-model
we scan over the parameter $\log_{10}(\bar{\alpha})$, while we scan over
$\log_{10}(\Omega_{\mathrm{FP}})$ for the $\beta_{1}\beta_{2,3,4}$-models..
The lower bound on $\Omega_{\mathrm{FP}}$ is chosen because of the Higuchi
bound, while the upper bound corresponds to the Planck-scale. In addition to
the physical parameters, the models further depend on the cosmological
parameters. We choose to scan over $H_{0}$, $\Omega_{\rm k0}$ and $\Omega_{\rm
b0}$ as free parameters with flat priors
$[45{\rm\frac{km/s}{Mpc}},85{\rm\frac{km/s}{Mpc}}]$, $[0,1]$ and $[0,1]$,
respectively. Note that we analytically marginalise over $H_{0}$ in the case
of supernovae. The other energy density parameters $\Omega_{\rm de0}$ and
$\Omega_{\rm m0}$ are computed from eqs. 3.5 and 3.3. We use $[-1,1]$ and
$[0,1]$ as flat priors on these parameters, respectively.
Model | $\beta_{0}$ | $\beta_{1}$ | $\beta_{0}\beta_{1}$ | $\beta_{1}\beta_{n}$
---|---|---|---|---
$\bar{\alpha}$ | | $3^{-1/2}$ | $<0.2$ | $<0.016$
$m_{\mathrm{FP}}$ [eV] | | $(1.82\pm 0.02)\times 10^{-32}$ | $(1.33\pm 0.02)\times 10^{-32}$ | $>4.24\times 10^{-31}$
$\Lambda$ [$10^{-64}{\rm eV}^{2}$] | $1.76\pm 0.05$ | $2.5\pm 0.1$ | $1.8\pm 0.1$ | $1.76^{+0.1}_{-0.09}$
$H_{0}$ [$\frac{{\rm km}/{\rm s}}{{\rm Mpc}}$] | $68.8^{+1.4}_{-1.2}$ | $73.1^{+1.6}_{-1.4}$ | $68.8\pm 0.2$ | $69.0\pm 1.0$
$\Omega_{\Lambda}$ | $0.69\pm 0.01$ | $0.86\pm 0.01$ | $0.69^{+0.03}_{-0.02}$ | $0.69\pm 0.01$
$\Omega_{\rm m0}$ | $0.31\pm 0.01$ | $0.27\pm 0.01$ | $0.31\pm 0.02$ | $0.31\pm 0.01$
$\Omega_{\rm de0}$ | $0.69\pm 0.01$ | $0.73\pm 0.01$ | $0.69\pm 0.02$ | $0.69\pm 0.01$
$\Omega_{\rm k0}$ | $0.002^{+0.004}_{-0.003}$ | $-0.007^{+0.004}_{-0.003}$ | $0.002^{+0.005}_{-0.001}$ | $0.002\pm 0.004$
$\Omega_{\rm b0}$ | $0.0224\pm 0.0001$ | $0.0223\pm 0.0003$ | $0.0224\pm 0.0004$ | $0.0224\pm 0.0003$
$\chi^{2}$ | $694.7$ | $726.0$ | $694.7$ | $694.7$
$\Delta{\rm BIC}$ | $721$ | $+31$ | $+6.6$ | $+6.6$
Table 1: The best-fit values and errors at 68% c.l. from SN+CMB+BAO for the
one- and two-parameter models are shown. To compute the values of
$m_{\mathrm{FP}}$ and $\Lambda$ from $\Omega_{\mathrm{FP}}$ and
$\Omega_{\Lambda}$, we use the obtained value of $H_{0}$ for each model. The
$\beta_{0}$-model is the $\Lambda$CDM-model and hence does not contain the
parameters $\bar{\alpha}$ and $m_{\mathrm{FP}}$. The last row contains the
parameter values for the $\beta_{1}\beta_{2}$-, $\beta_{1}\beta_{3}$\- and
$\beta_{1}\beta_{4}$-model as indicated by the subscript $n$, for which the
statistical analysis yielded the exact same results as expected.
Model | $\beta_{0}\beta_{1}\beta_{4}$ | $\beta_{0}\beta_{1}\beta_{3}$ | $\beta_{0}\beta_{1}\beta_{2}$
---|---|---|---
$\bar{\alpha}$ | $<0.14$ | $<0.03$ | $<0.06$
$m_{\mathrm{FP}}$ [eV] | $>1.01\times 10^{-32}$ | $>1.17\times 10^{-32}$ | $>1.08\times 10^{-32}$
$\Lambda\,[10^{-64}{\rm eV}^{2}]$ | $1.77^{+0.10}_{-0.09}$ | $1.77\pm 0.09$ | $1.77^{+0.10}_{-0.09}$
$H_{0}$ [$\frac{{\rm km}/{\rm s}}{{\rm Mpc}}$] | $68.8^{+1.5}_{-1.3}$ | $68.9^{+1.4}_{-1.3}$ | $68.9^{+1.4}_{-1.3}$
$\Omega_{\Lambda}$ | $0.69^{+0.02}_{-0.01}$ | $0.69^{+0.02}_{-0.01}$ | $0.69^{+0.03}_{-0.01}$
$\Omega_{\rm m0}$ | $0.31\pm 0.01$ | $0.30\pm 0.01$ | $0.31\pm 0.01$
$\Omega_{\rm de0}$ | $0.69\pm 0.01$ | $0.69\pm 0.01$ | $0.69\pm 0.01$
$\Omega_{\rm k0}$ | $0.002\pm 0.004$ | $0.002^{+0.004}_{-0.003}$ | $0.002^{+0.006}_{-0.001}$
$\Omega_{\rm b0}$ | $0.0224\pm 0.0003$ | $0.0224\pm 0.0003$ | $0.0224\pm 0.0003$
$\chi^{2}$ | $694.7$ | $694.7$ | $694.7$
$\Delta{\rm BIC}$ | $+11.6$ | $+11.6$ | $+11.6$
Model | $\beta_{1}\beta_{2}\beta_{3}$ | $\beta_{1}\beta_{2}\beta_{4}$ | $\beta_{1}\beta_{3}\beta_{4}$
---|---|---|---
$\bar{\alpha}$ | $<0.004$ | $<0.01$ | $<0.002$
$m_{\mathrm{FP}}$ [eV] | $>1.10\times 10^{-32}$ | $>1.15\times 10^{-32}$ | $>1.10\times 10^{-32}$
$\Lambda\,[10^{-64}{\rm eV}^{2}]$ | $1.78\pm^{+0.09}_{-0.1}$ | $1.77\pm 0.09$ | $1.77^{+0.10}_{-0.09}$
$H_{0}$ [$\frac{{\rm km}/{\rm s}}{{\rm Mpc}}$] | $68.9^{+1.4}_{-1.3}$ | $69.0^{+1.3}_{-1.2}$ | $68.9^{+1.4}_{-1.3}$
$\Omega_{\Lambda}$ | $0.69\pm 0.01$ | $0.69^{\pm}0.01$ | $0.69\pm 0.01$
$\Omega_{\rm m0}$ | $0.31\pm 0.01$ | $0.30\pm 0.01$ | $0.31\pm 0.01$
$\Omega_{\rm de0}$ | $0.69\pm 0.01$ | $0.69\pm 0.01$ | $0.69\pm 0.01$
$\Omega_{\rm k0}$ | $0.002\pm 0.004$ | $0.0023^{+0.0003}_{-0.004}$ | $0.002\pm 0.003$
$\Omega_{\rm b0}$ | $0.0224\pm 0.0003$ | $0.0224\pm 0.0003$ | $0.0224\pm 0.0003$
$\chi^{2}$ | $694.7$ | $694.7$ | $694.7$
$\Delta{\rm BIC}$ | $+11.6$ | $+11.6$ | $+11.6$
Table 2: The best-fit values and errors at 68% c.l. from SN+CMB+BAO for the
three-parameter models are shown. To compute the values of $m_{\mathrm{FP}}$
and $\Lambda$ from $\Omega_{\mathrm{FP}}$ and $\Omega_{\Lambda}$, we use the
obtained value of $H_{0}$ for each model.
Reference model and consistency of data sets. Let us start with a discussion
of the results for the $\Lambda$CDM-model as reference. The 1- and
2-dimensional posterior distributions for the parameters $H_{0}$, $\Omega_{\rm
m0}$ and $\Omega_{\Lambda}=\Omega_{\rm de0}$ are presented in the left panel
of fig. 1. While supernovae do not constrain $H_{0}$ due to the degeneracy
with the absolute magnitude, also CMB and BAOs do not yield robust constraints
individually. However, this geometric degeneracy is broken when combining all
three sets of data. We then obtain the value $H_{0}=(68.8\pm
1.2)\rm{km/s/Mpc}$, which is in good agreement with current constraints [130,
141, 5]. Our value is slightly but not significantly larger, which can be
understood as an artifact of using the distance priors [129]. Combining the
data sets stabilises spatial curvature to the value $\Omega_{\rm k0}=0.002\pm
0.003$ such that the universe is spatially flat within $68\%$ c.l. Also the
other parameters as reported in table 1 are in good agreement with current
constraints. We conclude that our analysis is robust. At the best-fit point we
have $\chi^{2}=694.7$ leading to ${\rm BIC}=721.2$.
#### 3.3.1 $\beta_{1}$-model
Figure 1: Marginalised posterior distribution of the cosmological parameters
in the $\Lambda\rm CDM$-model (left) and the $\beta_{1}$-model (right). The
contour lines correspond to $68\%$ and $95\%$ c.l. The results reflect a
tension between the early- and late-time data sets for the $\beta_{1}$-model.
As discussed before, the $\beta_{1}$-model is the only bimetric one-parameter
model that can possibly give rise to a viable expansion history. The posterior
distributions are presented in the right panel of fig. 1. Also this model is
subject to the geometric degeneracy such that both $H_{0}$ and $\Omega_{\rm
k0}$ are unconstrained by CMB and BAO individually. Combining all three data
sets stabilises these parameters. Data then favors a slightly spatially curved
universe with $\Omega_{\rm k0}=-0.007^{+0.004}_{-0.003}$. The Hubble rate
today is constrained to the remarkable high value of
$H_{0}=73.1^{+1.6}_{-1.4}{\rm km/s/Mpc}$, which is in perfect agreement with
local determinations of $H_{0}$ [142]. We emphasise that we obtained this
result without local prior on $H_{0}$. The value of the spin-2 mass is
constrained to be small with $m_{\mathrm{FP}}=(1.82\pm 0.02)\times
10^{-32}\,{\rm eV}$, which is close to the Higuchi bound.
However, the cosmological early- and late-time measurements are incompatible
within this model, as can be seen already from the right panel of fig. 1.
Quantitatively, $\Delta{\rm BIC}=+31$ indicates that this model is indeed
strongly disfavored by cosmological background data. Therefore, this model
does not solve the $H_{0}$-tension, despite its large favored value. We
conclude that the bimetric $\beta_{1}$-model is statistically ruled out.
Nonetheless, we summarise the obtained best-fit values for the other
parameters in table 1.
#### 3.3.2 Two-parameter models
As already discussed, consistency of the early universe on the finite branch
requires $\beta_{1}>0$. This means that there are four possibilities for the
two-parameter models: the $\beta_{0}\beta_{1}$ model and the
$\beta_{1}\beta_{n}$ models with $n=2,3,4$. From our analysis it emerges that
the results for all the $\beta_{1}\beta_{n}$ models are very similar and
statistically compatible. For this reason, we explicitly show only the results
for the $\beta_{1}\beta_{3}$ model and we refer to it as $\beta_{1}\beta_{n}$
for the rest of this section. In fig. 2 we plot the 1- and 2- dimensional
marginalised posterior distribution for these models, where we can see that
the data sets are compatible. This is true also for SN data, that are not
shown explicitly in this figure. We conclude that the data sets can be
combined in the statistical analysis.
In table 1 we summarise the results of the combined analysis. The main
difference between the $\beta_{0}\beta_{1}$ model and the $\beta_{0}\beta_{n}$
models are the values of $\bar{\alpha}$ and $m_{\mathrm{FP}}$. In both cases,
$\bar{\alpha}$ is constrained only from above with a maximum value of $0.2$
for the $\beta_{0}\beta_{1}$ model and $0.016$ for the $\beta_{1}\beta_{n}$
models. In the $\beta_{0}\beta_{1}$ model the value of $m_{\mathrm{FP}}$ is
constrained to be $(1.33\pm 0.02)\times 10^{-32}\,{\rm eV}$, which lies close
to the Higuchi bound. In the $\beta_{1}\beta_{n}$ models, however, the value
of the Fierz-Pauli mass is constrained only weakly from below with
$m_{\mathrm{FP}}>4.24\times 10^{-31}\,{\rm eV}$. We will discuss the
constrains on $\bar{\alpha}$ and $m_{\mathrm{FP}}$ in more detail in section
4.3, where we will compare the results of the statistical analysis with local
tests of gravity.
Despite these differences, the other parameters for the $\beta_{0}\beta_{1}$
and $\beta_{1}\beta_{n}$ models are very similar with each other and
compatible with $\Lambda\rm CDM$. The only exception is spatial curvature,
that seems to be slightly positive for the $\beta_{0}\beta_{1}$ model.
However, this model still prefers an almost flat universe with $\Omega_{\rm
k0}=0.002^{+0.005}_{-0.001}$, which remains very close to $0$ at $1\sigma$ and
compatible with the $\Lambda\rm CDM$ value. Moreover, the value of $\chi^{2}$
in both models coincides with $\Lambda\rm CDM$. This makes the
$\beta_{0}\beta_{1}$ and $\beta_{1}\beta_{n}$ models slightly disfavored with
a $\Delta{\rm BIC}=+6.6$, since these models have more free parameters with
respect to $\Lambda\rm CDM$.
Figure 2: Marginalised posterior distribution of the cosmological parameters
in the $\beta_{0}\beta_{1}$-model (left) and the $\beta_{1}\beta_{3}$-model
(right). The contour lines correspond to $68\%$ and $95\%$ c.l.
#### 3.3.3 Three-parameter models
We finish by discussing models with three free parameters. We will see that
the parameter constraints are similar across these models. Data sets are
compatible for all the three parameter models, as we can see in figs. 3, 4 and
5. Therefore, combining the data sets for these models is
trustworthy111111This is true also for SN data, even if they are not shown
explicitly in figs. 3, 4 and 5..
Figure 3: Marginalised posterior distribution of the cosmological parameters
in the $\beta_{0}\beta_{1}\beta_{2}$-model (left) and the
$\beta_{0}\beta_{1}\beta_{3}$-model (right). The contour lines correspond to
99% and 95% c.l.
$\bullet$ Models with $\beta_{4}=0$. We first discuss the three-parameter
models without vacuum energy in the $f$-sector. We present the 1- and 2-
dimensional marginalised posterior distribution in the left and right panel of
fig. 3 and the left panel of fig. 4 for the $\beta_{0}\beta_{1}\beta_{2}$-,
$\beta_{0}\beta_{1}\beta_{3}$\- and $\beta_{0}\beta_{1}\beta_{4}$-model,
respectively. The parameter constraints are reported in table 2 and agree with
the $\Lambda\rm CDM$ values within $1\sigma$ and are all quite similar across
these models. The only exception is that the
$\beta_{0}\beta_{1}\beta_{2}$-model seems to prefer a slightly curved universe
at $1\sigma$. However, the value of the curvature is very small and compatible
with $\Lambda\rm CDM$ at $1\sigma$ and therefore we do not consider it as a
statistically relevant difference. The bimetric parameters $\bar{\alpha}$ and
$m_{\mathrm{FP}}$ are constrained only weakly. We will discuss more about the
constraints on these parameters in the $\bar{\alpha}-m_{\rm FP}$ plane when
comparing with local tests in section 4.3. We find that these models are
consistent with the current data because $\chi^{2}=694.7$ as in the
$\Lambda$CDM-model. However, the BIC is higher due to the two additional free
parameters indicating that these models are statistically disfavored.
Figure 4: Marginalised posterior distribution of the cosmological parameters
in the $\beta_{0}\beta_{1}\beta_{4}$-model (left) and the
$\beta_{1}\beta_{2}\beta_{3}$-model (right)
Figure 5: Marginalised posterior distribution of the cosmological parameters
in the $\beta_{1}\beta_{2}\beta_{4}$-model (left) and the
$\beta_{1}\beta_{3}\beta_{4}$-model (right)
$\bullet$ Models with $\beta_{0}=0$. The result for this models is very
similar to models with $\beta_{4}=0$. The only exception is that
$\bar{\alpha}$ is more constrained of about one order of magnitude, as it can
be seen in table 2. We show the 1- and 2- dimensional marginalised posterior
distribution in the right panel of fig. 4 for the
$\beta_{1}\beta_{3}\beta_{4}$-model and in fig. 5 for the
$\beta_{1}\beta_{2}\beta_{4}$\- and $\beta_{1}\beta_{3}\beta_{4}$-model.
Again, these models have the same $\chi^{2}$ of the $\Lambda$CDM-model, which
makes them statistically disfavored with a higher BIC.
## 4 Bounds from local tests of gravity
Having identified those regions of the parameter space that are in agreement
with current observations in background cosmology, we now turn to local tests
of gravity. In section 4.1 we give a brief review on local tests of gravity
and present the existing constraints on the parameters $\bar{\alpha}$ and
$m_{\mathrm{FP}}$. However, the existing bounds were derived without taking
into account the Vainshtein screening mechanism. In regions where Vainshtein
screening is active, the bounds on $\bar{\alpha}$ and $m_{\mathrm{FP}}$ are
not trustworthy. In section 4.2 we identify the regions of the parameter space
that give rise to Vainshtein screening in spherically symmetric systems. We
finally confront cosmological with local bounds in section 4.3 in parameter
regions without Vainshtein screening.
### 4.1 Local tests of gravity
In this section we briefly summarise local tests of gravity. For details and
reviews we refer to Ref. [108, 109, 110, 113, 111]. Extra degrees of freedom
(such as the massive spin-2 field of bimetric theory) modify the Newtonian
gravitational potential as compared to GR. This is usually attributed to a
fifth force. If the force mediator is massive, it contributes a Yukawa
potential to the gravitational potential felt by a massive test body.
Following standard notation in the literature, the gravitational potential for
a static and spherically-symmetric configuration around a massive source can
be parametrised as
$V(r)=-\frac{1}{m_{\rm
P}}\left(\frac{1}{r}+\xi\frac{e^{-r/\lambda}}{r}\right)\,,$ (4.1)
with $m_{\rm P}$ the Planck mass. This is referred to as Yukawa
parametrisation. The first term corresponds to the usual potential arising
from massless gravitons. The second term arises from a massive force mediator
with Compton wavelength $\lambda$ and coupling to matter $\xi$. While in the
limit $\xi\ll 1$ the fifth force is suppressed, it is most prominent on length
scales $r\sim\lambda$. As we will see in section 4.2, these parameters are
related to the bimetric parameters as
$\xi=\frac{4\bar{\alpha}^{2}}{3}\,,\qquad\lambda=m_{\mathrm{FP}}^{-1}\,.$
(4.2)
In fig. 6 we present the constraints from local tests of gravity on the
bimetric parameters $\bar{\alpha}$ and $m_{\mathrm{FP}}$ as indicated by the
black lines. The gray-shaded region is excluded by 95% c.l. To compute the
constraints, we follow the procedure as described in [108, 109, 110]. The
laboratory constraints correspond to the Irvine [143]121212Note that the bound
from [144] is not included, yet., Eöt-Wash [145] and Stanford [146]
experiments. The geophysical constraints correspond to the lake [147, 148] and
tower [149, 150, 151] experiments. The constraints from measurements with the
LAGEOS satellite are reported in [152, 153, 154]. The constraint from Lunar-
Laser-Ranging (LLR) results from lunar precession [109]. The planetary
constraints are reported in [155].
The aforementioned tests of the Newtonian potential provide only weak
constraints on modifications if the mass of the fifth force mediator is small.
In this region of the parameter space, tests of the scalar curvature, i.e. the
deflection and delay of light induced by matter, provide powerful constraints.
We use the gravitational slip parameter $\gamma$, which is unity in GR,
$\gamma=1$. Within the context of bimetric theory, the parameter has been
calculated to be [91]
$\gamma=\frac{3+2\bar{\alpha}^{2}e^{-m_{\mathrm{FP}}r}}{3+4\bar{\alpha}^{2}e^{-m_{\mathrm{FP}}r}}\,.$
(4.3)
The most precise value has been obtained within the solar system. The
measurement of the time delay of radar signals sent between the Earth and the
Cassini spacecraft on its way to Saturn yields $\gamma-1=(2.1\pm 2.3)\times
10^{-5}$ [112]. The radio signals were passing by the sun at a distance of
$1.6$ solar radii, i.e. $r\simeq 7.44\times 10^{-3}\,{\rm AU}\simeq 1.31\times
10^{-15}\,{\rm eV}^{-1}$. Using these values defines the red line in fig. 6.
The region above that line is excluded by 95% c.l. For small spin-2 masses,
the coupling to matter is constraint to be $\bar{\alpha}\lesssim 1.7\times
10^{-3}$. A collection of bounds on $\gamma$ from time delay and deflection of
light form galaxy halo to solar system scales can be found in [113], which
however are less constraining than the bound used here.
Summarising, the most stringent constraint comes from LLR with
$\bar{\alpha}<8.9\times 10^{-6}$ at $95\%$ c.l. for $m_{\mathrm{FP}}\simeq
6.5\times 10^{-15}\,{\rm eV}$. On the other hand, for $m_{\mathrm{FP}}\gtrsim
10^{-2}\,{\rm eV}$ the bounds on $\bar{\alpha}$ are weak. This, however, is
not the end of the story in the framework of bimetric theory due to the
Vainshtein screening mechanism as we discuss in the next section.
Figure 6: Bounds on the bimetric parameters $\bar{\alpha}$ and
$m_{\mathrm{FP}}$ from local tests of gravity. The black lines result from
tests of the Newtonian gravitational potential as reported in [155, 108, 109,
145, 146, 110]. Tests of the scalar curvature yield the red line [112, 91].
The gray-shaded region is excluded by $95\%$ c.l. Note that the Vainshtein
mechanism is not taken into account in the derivation of these bounds.
### 4.2 Vainshtein screening
Local tests of gravity put tight constraints on modifications of the
gravitational potential. Historically, the linear theory of Fierz and Pauli
[39] was discarded because it did not pass basic solar system tests. This is
due to the so-called vDVZ discontinuity [41, 40]: the helicity-0 mode of the
massive graviton couples to the trace of the energy-momentum tensor even in
the limit $m_{\mathrm{FP}}\rightarrow 0$. Vainshtein argued that including
nonlinear interaction terms for the massive spin-2 field restores GR in that
limit thus curing the problem [42]. This is indeed the case as explicitly
demonstrated in [156, 157, 158]. For a review on the Vainshtein mechanism we
refer to [159].
Bimetric theory incorporates the Vainshtein mechanism as explicitly
demonstrated for static and spherically symmetric systems [160, 93, 89, 90].
We leave the technical details to appendix B, where we repeat and extend the
analysis by relaxing the assumption of asymptotic flatness to allow for a non-
vanishing cosmological constant. Here we limit ourselves to discussing the
general features. The radius below which nonlinearities have to be taken into
account is referred to as Vainshtein radius $r_{\rm V}=(r_{\rm
S}/m_{\mathrm{FP}}^{2})^{1/3}$ with $r_{\rm S}$ the Schwarzschild radius of
the matter source. For larger radii, $r\gg r_{\rm V}$, the linear
approximation is valid. The resulting gravitational potential for small and
large radii can be written as [93, 89]
$\displaystyle V(r)=\begin{cases}-\frac{1}{m_{\rm g}^{2}}\frac{1}{r}&r\ll
r_{V}\\\ -\frac{1}{m_{\rm
P}^{2}}\left(\frac{1}{r}+\frac{4\bar{\alpha}^{2}}{3}\frac{e^{-m_{\mathrm{FP}}r}}{r}\right)&r\gg
r_{V}\end{cases}\,,$ (4.4)
where $m_{\rm P}^{2}=(1+\bar{\alpha}^{2})m_{\rm g}^{2}$ is the unscreened
Planck mass. This justifies our identification (4.2). On the other hand, since
$r_{\rm V}$ depends on the spin-2 mass and the mass of the central source, the
constraints on $\bar{\alpha}$ and $m_{\mathrm{FP}}$ summarised in fig. 6 are
not trustworthy.
However, Vainshtein screening exists only in a subregion of the parameter
space [93, 90], as we review in appendix B. In this section, we will identify
this subregion in terms of $\bar{\alpha}$ and $m_{\mathrm{FP}}$. The
Vainshtein mechanism relies on the following three necessary conditions:
* •
Consistent screening regime. The existence of a nonlinear solution that
restores GR in spacetime regions close to the matter source requires
$\frac{c^{3}\beta_{3}}{c\beta_{1}+2c^{2}\beta_{2}+c^{3}\beta_{3}}>1\,.$ (4.5)
Since the denominator is positive if $m_{\mathrm{FP}}^{2}>0$, this bound
implies that $\beta_{3}$ must be strictly positive131313Recall that the
infinite branch solution in the context of background cosmology is well-
defined only for $\beta_{3}<0$. Hence, requiring of a working Vainshtein
mechanism rules out the infinite branch., $\beta_{3}>0$. That means, any
bimetric model with $\beta_{3}=0$ does not have a working Vainshtein mechanism
(for some related caveats, see [93]).
* •
Consistent asymptotics. The nonlinear equations must give rise to a solution
that matches the linearised solution for $r\gg r_{\rm V}$. This requires
$\frac{c^{2}\beta_{2}+c^{3}\beta_{3}}{c\beta_{1}+2c^{2}\beta_{2}+c^{3}\beta_{3}}<\sqrt{\frac{c^{3}\beta_{3}}{c\beta_{1}+2c^{2}\beta_{2}+c^{3}\beta_{3}}}$
(4.6)
to be satisfied. Combined with eq. 4.5 it follows that $\beta_{2}$ must be
strictly negative, $\beta_{2}<0$. Bimetric models with $\beta_{2}=0$ do not
give rise to Vainshtein screening.
* •
Consistent Vainshtein-Yukawa solution. The nonlinear solution realising
Vainshtein screening has to be smoothly connected to the linear solution
realising the Yukawa-type fifth force without branch cuts. The existence of
such a well-defined Vainshtein-Yukawa solution imposes another restriction on
the parameters. The explicit expression is lengthy so that we shift its
presentation to eq. B.28.
Following [105], we express these conditions in terms of the physical
parameters $\bar{\alpha}$, $m_{\mathrm{FP}}$, and $\Lambda$ in this section.
The relations between the interaction and physical parameters are collected in
appendix A. It suffices to study models with all interaction parameters
$\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ being free as only those models can
possibly give rise to a viable cosmic expansion history while incorporating
the local Vainshtein mechanism.
Figure 7: The physical parameter space of the $\beta_{1}\beta_{2}\beta_{3}$
model is shown. The model gives rise to a consistent cosmology only outside
the blue-shaded region, which indicates where the vacuum point is not well-
defined, and the red-shaded region, in which $\beta_{1}$ is non-positive. The
Vainshtein mechanism works only outside the purple-shaded region, which does
not give rise to a consistent screening regime, the green-shaded region, which
indicates where the Vainshtein solution does not have the right asymptotics,
and the yellow-shaded region, in which there is no Vainshtein-Yukawa solution.
Finally, the Higuchi bound is violated in the gray-shaded region. Summarising,
this bimetric model gives rise to a viable expansion history while
incorporating a working Vainshtein mechanism in the region that is left white.
Cosmologically viable one- and two-parameter models are not able to
incorporate Vainshtein screening. The $\beta_{1}\beta_{2}\beta_{3}$-model is
the only three-parameter model with a possibly viable background cosmology and
a working Vainshtein mechanism. In fig. 7 we collect all the aforementioned
theoretical bounds on the physical parameter space of that model. The
expressions are too lengthy to display them here explicitly. The blue- and
red-shaded regions are excluded as these do not give rise to a viable
expansion history. The purple-shaded region violates the bound in eq. 4.5. In
the green-shaded region the bound in eq. 4.6 is violated. The Vainshtein-
Yukawa solution does not exist in the yellow-shaded region, which represents
the most stringent bound. To summarise, we expand the most-stringent bound on
the physical parameters for $m_{\mathrm{FP}}^{2}\gg\Lambda$ to find
$\displaystyle 16\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}\lesssim\Lambda\,.$ (4.7)
If this approximate bound is satisfied, the
$\beta_{1}\beta_{2}\beta_{3}$-model incorporates the local Vainshtein
mechanism. As can be seen from fig. 7, the parameter region giving rise to a
viable cosmic expansion history is only slightly larger.
Before moving on, let us point out the following caveats. The equivalence
principle is violated in two-body systems due to nonlinearities [161]. The
assumption of spherical symmetry entering the derivation of the gravitational
potential might not be appropriate to describe, e.g. the earth-moon system.
Further, Vainshtein screening is weaker or even completely absent in systems
with cylindrical or planar symmetry, respectively, as demonstrated in the
context of galileons [162]. This has to be taken into account for gravity
tests without spherical symmetry.
### 4.3 Local vs. cosmological bounds
We are finally in the position to compare the constraints from local tests of
gravity, shown in fig. 6, to the parameter regions that give rise to
Vainshtein screening. Further, we compare the local constraints to the
constraints that we inferred from background cosmology in section 3.3. Since
only the $\beta_{1}\beta_{2}\beta_{3}$-model incorporates Vainshtein
screening, the local bounds are directly applicable to all the other models
considered in this paper. The discussed observational bounds in this section
always refer to 95% confidence level.
Figure 8: Cosmological constraints compared to local tests for the
$\beta_{0}\beta_{1}$-model (left) and for the $\beta_{1}\beta_{3}$-model
(right). The blue lines delimit the $2\sigma$ regions from the statistical
analysis. The gray region is excluded by local tests at $2\sigma$.
Figure 9: The combined bounds from local and cosmological tests for the
$\beta_{0}\beta_{1}\beta_{2}$\- (left) and $\beta_{0}\beta_{1}\beta_{3}$-model
(right). The gray-shaded region is excluded by local tests of gravity at
$95\%$ c.l. The red-shaded region is excluded by theoretical constraints. The
region above the blue line is excluded by our bounds from cosmology at $95\%$
c.l.
Figure 10: The combined bounds from local and cosmological tests for the
$\beta_{0}\beta_{1}\beta_{4}$\- and $\beta_{1}\beta_{2}\beta_{3}$-model. The
gray-shaded region is excluded by local tests of gravity at $95\%$ c.l. The
red-shaded region is excluded by theoretical constraints. The region above the
blue line is excluded by our bounds from cosmology at $95\%$ c.l. The hatched
region in the right panel represents where the Vainshtein mechanism is
working.
Figure 11: The combined bounds from local and cosmological tests for the
$\beta_{1}\beta_{2}\beta_{4}$\- and $\beta_{1}\beta_{3}\beta_{4}$-model. The
region above the gray line is excluded by local tests of gravity at $95\%$
c.l. The red-shaded region is excluded by theoretical constraints. The region
above the blue line is excluded by our bounds from cosmology at $95\%$ c.l.
Let us start discussing the $\beta_{1}$-model. As mentioned before, the
cosmological data sets are not compatible, strongly disfavoring this model. If
we nonetheless combine the data sets, the spin-2 mass is constrained to
$m_{\mathrm{FP}}=(1.82\pm 0.02)\times 10^{-32}\,{\rm eV}$. The coupling to
matter is fixed to the value $\bar{\alpha}=1/\sqrt{3}\simeq 0.58$, which is
incompatible with the observational bound from Cassini $\bar{\alpha}\lesssim
1.7\times 10^{-3}$ in that mass regime. Since this model does not give rise to
Vainshtein screening, we conclude that the $\beta_{1}$-model is completely
ruled out by the combined analysis of cosmological and local data.
Next let us discuss the two-parameter models, which do not give rise to
Vainshtein screening as well. This implies that the bounds from local tests of
gravity are indeed trustworthy. In fig. 8 we compare the bounds from local and
cosmological tests for the two parameter models $\beta_{0}\beta_{1}$ and
$\beta_{1}\beta_{3}$. The blue line defines the boundary of the region that is
consistent with cosmological data. The gray-shaded region is excluded by local
tests of gravity. For the $\beta_{0}\beta_{1}$-model, cosmological data
constraints the spin-2 mass to $m_{\mathrm{FP}}=(1.33\pm 0.02)\times
10^{-32}\,{\rm eV}$ while Cassini constrains the coupling to matter to
$\bar{\alpha}\lesssim 1.7\times 10^{-3}$, which is slightly more stringent
than the cosmological bound. For the $\beta_{1}\beta_{n}$-model, the physical
parameters are correlated as can be seen in the right panel of fig. 8. Indeed,
they are approximately related as
$\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}\simeq\frac{12}{n}\Lambda$ for
$n=\\{2,3,4\\}$ (see section A.1) [105]. As before, Cassini provides the most
stringent constraint on $\bar{\alpha}$. Due to the parameter correlation, this
implies a more stringent constraint on the spin-2 mass than inferred from
cosmology of $m_{\mathrm{FP}}\gtrsim 1.85\times 10^{-30}\,{\rm eV}$ at 95%
c.l. Therefore, all these two-parameter models are in agreement with
observational constraints from cosmology and local tests of gravity, but
driven to their GR-limits.
In figs. 9, 10 and 11 we combine theoretical, cosmological and local bounds in
single plots for all the three parameter models. In all these plots, the
regions above the blue lines are excluded at 95% c.l. by our cosmological
bounds inferred in section 3.2. The red-shaded regions in the plots are
excluded by theoretical constraints that ensure a consistent cosmic expansion
history, which we used as hard priors in the cosmological analysis. The region
excluded by local bounds at 95% c.l. (if the Vainshtein mechanism is not taken
into account) is shown as a gray-shaded region. For large spin-2 masses, the
bounds from cosmology are the most stringent ones. For smaller spin-2 masses,
the most stringent bound is provided by tests of the scalar curvature with the
Cassini spacecraft. Summarising, without taking Vainshtein screening into
account, all three-parameter models are consistent with both local tests of
gravity and background cosmology. However, since the coupling of the massive
spin-2 field to matter is forced to be small, i.e. $\bar{\alpha}\lesssim
1.7\times 10^{-3}$, all these models are essentially forced into their GR-
limits. Although local tests of gravity allow for large couplings to matter if
the spin-2 mass is large, $m_{\mathrm{FP}}\gtrsim 10^{-2}\,{\rm eV}$,
cosmological data excludes this parameter region. Therefore, deviations from
GR are substantially suppressed for these models on all scales.
Let us discuss further about the cosmological bounds on $\bar{\alpha}$ and
$m_{\mathrm{FP}}$. From inspecting figs. 9, 10 and 11 we notice that the 95%
contour line falls into two regimes. The first is the small spin-2 mass
regime. Here, below some critical value that depends on the model, the
observational constraints becomes independent of $m_{\mathrm{FP}}$. This is
particularly evident for the models with $\beta_{0}\neq 0$. In this regime,
the coupling to matter $\bar{\alpha}$ is bounded from above as reported in
tables 1 and 2. The second regime is above the critical value for
$m_{\mathrm{FP}}$, where the contour line can be approximated as:
$\bar{\alpha}^{c_{1}}\cdot{m_{\rm FP}}\leq 10^{-c_{2}}\text{ }\text{eV}.$
(4.8)
We give the explicit value of the coefficients $c_{1}$ and $c_{2}$ in table 3,
as the result of a polynomial fit. This behavior can be seen in figs. 9, 10
and 11, that however show only a portion of the $\bar{\alpha}-m_{\rm FP}$
plane scanned by the cosmological analysis141414The wiggles in the contour
lines of figs. 9, 10 and 11 are a consequence of zooming in on a portion of
the full scanned space, and are interpreted as artifacts of the Markov chains
having finite size.. Indeed, the approximation (4.8) is valid up to the Planck
scale, i.e. beyond the zoomed-in regions in the plots151515The limits on
$\bar{\alpha}$ and $m_{\rm FP}$ used in our statistical analysis can be found
in section 3.3..
Model | $c_{1}$ | $c_{2}$
---|---|---
$\beta_{0}\beta_{1}\beta_{4}$ | $0.97\pm 0.01$ | $23.6\pm 0.4$
$\beta_{0}\beta_{1}\beta_{3}$ | $1.117\pm 0.006$ | $32.9\pm 0.2$
$\beta_{0}\beta_{1}\beta_{2}$ | $0.964\pm 0.006$ | $24.6\pm 0.2$
$\beta_{1}\beta_{2}\beta_{3}$ | $1.047\pm 0.007$ | $35.5\pm 0.2$
$\beta_{1}\beta_{2}\beta_{4}$ | $1.021\pm 0.006$ | $32.7\pm 0.2$
$\beta_{1}\beta_{3}\beta_{4}$ | $0.959\pm 0.008$ | $32.7\pm 0.3$
Table 3: The value of numerical coefficients for the fitting formula (4.8),
obtained with a polynomial fit. We also show $1\sigma$ errors in the
determination of the coefficients.
For the $\beta_{1}\beta_{2}\beta_{3}$-model, we need to take into account the
Vainshtein mechanism when including local tests of gravity. The parameter
region, which supports Vainshtein screening, is depicted as a hatched area in
the right panel of fig. 10. As we have seen before, the region incorporating
Vainshtein screening is almost exactly complementary to the region that is
excluded by our theoretical bounds. In particular, the entire parameter region
that is consistent with cosmological data gives rise to Vainshtein screening.
From the right panel of fig. 10 we see that there is a small overlap between
the region excluded by local tests and the region where the Vainshtein
mechanism is active. In this interesting region, which is characterized by a
potentially large value of $\bar{\alpha}$, local tests are not trustworthy.
Unfortunately this small region seems to be excluded by cosmological data at
95% c.l. in our statistical analysis. However, this is probably caused by
prior domination. Indeed, this is a very small region of the full parameter
space explored by the statistical analysis and it is close to the hard prior
given by the theoretical constraints. To further investigate this issue, one
should perform another statistical analysis focusing on this small region of
parameter space.
For the models with $\beta_{0}=0$, the $95\%$ contour lies very close to the
theoretical priors except for a few wiggles. We interpret the wiggles as an
artifact of the finite size of the Markov chains and conclude that the used
data sets are not constraining these parameters. Instead, the statistically
favored regions almost exactly coincide with the parameter regions allowed by
our theoretical priors. To stabilise the value of $\bar{\alpha}$ and
$m_{\mathrm{FP}}$, we would need to include other data sets, which however is
beyond the scope of the present analysis.
On the other hand, models with $\beta_{4}=0$ are subject to rather weak
theoretical priors. Nonetheless, the $95\%$ lines show the same tendency as
for the other models. Hence, the data sets constrain these parameters
substantially beyond the prior knowledge, but is not able to stabilise their
values. To understand why the upper-right region in the
$\bar{\alpha}-m_{\mathrm{FP}}$-plane is excluded also for these models, we
note the following. Expanding the expression for $\beta_{0}$ in the limit
$m_{\mathrm{FP}}^{2}\gg\Lambda$, we find that
$\beta_{0}\simeq-\frac{12}{n}\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}$ for the
$\beta_{0}\beta_{1}\beta_{n}$-model with $n=\\{2,3,4\\}$. This allows to
conclude that data disfavors large negative values of $\beta_{0}$. This means
that, in order to be cosmologically viable, $\rho_{\rm de}$ is allowed to
change its sign only at sufficiently large redshifts.
## 5 Conclusions and outlook
Massive spin-2 fields might play a crucial role in our universe. Their
(self-)interaction energy can drive the observed cosmic accelerated expansion
at late times. Furthermore, a massive spin-2 field serves as ideal dark matter
candidate as it interacts with ordinary matter only gravitationally [84, 85,
86]. Recently, another intriguing aspect has been explored within a completely
different context. As argued in [163], massive spin-2 states could be
essential in solving the black hole information paradoxon.
In this paper, we have further studied the observational viability of bimetric
theory. We confronted all models with up to three free interaction parameters
$\beta_{n}$ with measurements of SN1a, BAOs and CMB. We find that all two and
three parameter models are in perfect agreement with these data sets. Compared
to the $\Lambda$CDM model, bimetric models are statistically disfavored due to
the larger number of free parameters. However, the self-accelerating solutions
with $\beta_{0}=0$ appear more appealing from a theoretical perspective as
these models do not suffer the cosmological constant problems [12, 13].
The best-fit values for the cosmological parameters do not significantly
differ from the best-fit values of the $\Lambda$CDM model. We have also
studied spatial curvature within bimetric theory, and our results show that a
spatially flat universe is always preferred.
The only exception from the previous summary is the $\beta_{1}$-model, which
represents the simplest bimetric model that gives rise to self-accelerating
solutions on the finite branch. Our statistical analysis shows that the
aforementioned data sets are inconsistent leading to a substantially higher
$\chi^{2}$ at the best fit point. We therefore conclude that the
$\beta_{1}$-model is statistically disfavored. This conclusion is further
strengthened when simultaneously confronting this model with local tests of
gravity. Cosmological data selects a spin-2 mass of $m_{\mathrm{FP}}=(1.82\pm
0.02)\times 10^{-32}\,{\rm eV}$, for which local tests of gravity demand a
coupling to matter of $\bar{\alpha}\lesssim 1.7\times 10^{-3}$ at 95% c.l.
Within the $\beta_{1}$-model, this parameter is fixed to be
$\bar{\alpha}=3^{-1/2}$. Therefore, this model is in severe conflict with
observations.
We also analysed the other bimetric models with up to three free parameters,
for which we compared the bounds from background cosmology to the ones
obtained from local tests of gravity. All models are consistent with both
types of observations even if the Vainshtein mechanism is not active. However,
the coupling of the massive spin-2 field to ordinary matter is forced to be
small, $\bar{\alpha}\lesssim 10^{-3}$ (see section 4.1 and tables 1 and 2 for
the precise values). For large spin-2 masses, the cosmological observations
force the coupling to be even smaller. This suppresses the deviations from GR
on all scales for these models.
For the $\beta_{1}\beta_{2}\beta_{3}$-model, the Vainshtein mechanism removes
the constraint on $\bar{\alpha}$ and $m_{\mathrm{FP}}$ implied by local tests
for sufficiently small values of the spin-2 mass. In this region, where
$\bar{\alpha}$ can be large, the theory is not in its GR limit. As we
mentioned in section 4.3, the fact that this small region is excluded by our
cosmological analysis might be a numerical effect, and it would be interesting
to further investigate this issue in another analysis. Further, more general
bimetric models, i.e. those with four or all five parameters $\beta_{n}$ being
free, might not be forced into their GR-limits. In any case, even if the
restricted models studied in this paper turn out to be forced into their GR-
limits, their theoretical motivation would persist: self-accelerating models
with $\beta_{0}=0$ have an effective cosmological constant that is technical
natural in the sense of t’Hooft [164].
The data sets used in this paper are only weekly constraining the physical
parameter space of $\bar{\alpha}$ and $m_{\mathrm{FP}}$. To stabilise their
values and effectively constrain bimetric theories, a possibility would be
taking into account further measurements. Cosmological perturbations around
the FLRW background solution are expected to have constraining power as these
probe different redshifts and scales. However, in bimetric theory linear
scalar perturbations are plagued by gradient instabilities at early times [70,
74, 77, 78, 65, 79]. The instability sets in when the Hubble rate is of the
order of the spin-2 mass, $H\simeq m_{\mathrm{FP}}$ [69]. Nonlinear terms
become as important as the linear term rendering perturbation theory invalid.
Indeed, taking nonlinearities into account might stabilise the solution in
analogy to the Vainshtein mechanism [82, 81, 128, 69]. Structure formation and
nonlinear cosmological perturbations are therefore main open issues within
bimetric theory, which should be addressed in the future.
## Acknowledgements
We thank Marcus Högås and Edvard Mörtsell for interesting discussions and
useful comments on the manuscript. We are grateful to Thomas Hahn, Martin
Kerscher and Kerstin Paech for their kind help regarding the statistical
analysis and numerical implementation. We further acknowledge the usage of the
Getdist package, which we we extensively used for the statistical analysis
[165]. The work of ML is supported by a grant from the Max Planck Society. The
work of AC and JW is supported by the Excellence Cluster ORIGINS which is
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.
## Appendix A Details on the physical parametrisation
In order to provide an overview over the relation between the physical and
interaction parameters, we summarise the results of Ref. [105] in this
appendix. We immediately work in terms of the cosmological parameters that we
use for parameter fitting.
In the $\beta_{1}$-model, there is only one free physical parameter, in terms
of which the interaction parameter is given as
$B_{1}=\frac{1}{\sqrt{3}}\Omega_{\Lambda}=\frac{\sqrt{3}}{4}\Omega_{\mathrm{FP}}\,,$
(A.1)
while the mixing angle is fixed to be $\bar{\alpha}=1/\sqrt{3}$.
### A.1 Two-parameter models
Next, we turn to the two-parameter models with $\beta_{1}\neq 0$. We choose to
express the interaction parameters on terms of $\Omega_{\mathrm{FP}}$ and
$\Omega_{\Lambda}$, while the mixing angle $\bar{\alpha}$ is a derived
parameter. However, one can equivalently eliminate one of the physical
parameters by $\bar{\alpha}$. For the $\beta_{0}\beta_{1}$-model, the relation
between the parameters is
$B_{0}=-3\Omega_{\mathrm{FP}}+4\Omega_{\Lambda}\,,\
B_{1}=\sqrt{(\Omega_{\mathrm{FP}}-\Omega_{\Lambda})\Omega_{\Lambda}}\,,$ (A.2)
in terms of which the mixing angle is determined to be
$\bar{\alpha}=\sqrt{\Omega_{\mathrm{FP}}/\Omega_{\Lambda}-1}$.
For the $\beta_{1}\beta_{2}$-model, the relation is given by
$\displaystyle\begin{split}B_{1}&=\frac{1}{2}\sqrt{\frac{3\Omega_{\mathrm{FP}}-2\Omega_{\Lambda}-\Delta_{12}}{3\Omega_{\Lambda}}}\left(3\Omega_{\mathrm{FP}}-3\Omega_{\Lambda}+\Delta_{12}\right)\,,\\\
B_{2}&=-\frac{1}{6}\left(3\Omega_{\mathrm{FP}}-5\Omega_{\Lambda}+\Delta_{12}\right)\end{split}$
(A.3)
where we defined
$\Delta_{12}=\sqrt{9\Omega_{\mathrm{FP}}^{2}-12\Omega_{\mathrm{FP}}\Omega_{\Lambda}+\Omega_{\Lambda}^{2}}$.
The mixing angle is given by
$\bar{\alpha}^{2}=(3\Omega_{\mathrm{FP}}-2\Omega_{\Lambda}-\Delta_{12})/(3\Omega_{\Lambda})$.
To avoid numerical instabilities, it is convenient to expand the expressions
for $\Omega_{\mathrm{FP}}\gg\Omega_{\Lambda}$. In this limit, the parameter
relation reads
$\displaystyle\bar{\alpha}\simeq\sqrt{\frac{\Omega_{\Lambda}}{6\Omega_{\mathrm{FP}}}}\,,\qquad
B_{1}\simeq\sqrt{\frac{3\Omega_{\mathrm{FP}}\Omega_{\Lambda}}{2}}\,,\qquad
B_{2}\simeq-\Omega_{\mathrm{FP}}\,.$ (A.4)
Next, we turn to the $\beta_{1}\beta_{3}$-model, where the relations read
$\displaystyle B_{1}$
$\displaystyle=\frac{1}{4}\sqrt{\frac{2\Omega_{\mathrm{FP}}-\Omega_{\Lambda}-2\Delta_{13}}{\Omega_{\Lambda}}}\sqrt{3\Omega_{\mathrm{FP}}-2\Omega_{\Lambda}+3\Delta_{13}}\,,$
(A.5) $\displaystyle B_{3}$
$\displaystyle=-\sqrt{\frac{2\Omega_{\mathrm{FP}}-\Omega_{\Lambda}-2\Delta_{13}}{\Omega_{\Lambda}}}\left(4\Omega_{\mathrm{FP}}^{2}-7\Omega_{\mathrm{FP}}\Omega_{\Lambda}+2\Omega_{\Lambda}^{2}+\Delta_{13}(4\Omega_{\mathrm{FP}}-5\Omega_{\Lambda})\right)\,.$
(A.6)
with
$\Delta_{13}=\sqrt{(\Omega_{\mathrm{FP}}-\Omega_{\Lambda})\Omega_{\mathrm{FP}}}$.
The mixing angle is given by
$\bar{\alpha}^{2}=(2\Omega_{\mathrm{FP}}-\Omega_{\Lambda}-2\Delta_{13})/(\Omega_{\Lambda})$.
Again, to avoid numerical instabilities, we expand the expressions for
$\Omega_{\mathrm{FP}}\gg\Omega_{\Lambda}$ yielding
$\displaystyle\bar{\alpha}\simeq\frac{1}{2}\sqrt{\frac{\Omega_{\Lambda}}{\Omega_{\mathrm{FP}}}}\,,\qquad
B_{1}\simeq\frac{3}{4}\sqrt{\Omega_{\mathrm{FP}}\Omega_{\Lambda}}\,,\qquad
B_{3}\simeq-\sqrt{\frac{\Omega_{\mathrm{FP}}^{3}}{\Omega_{\Lambda}}}\,.$ (A.7)
Finally, in the $\beta_{1}\beta_{4}$-model the parameter relations read
$\displaystyle B_{1}$
$\displaystyle=\frac{1}{3}\sqrt{(3\Omega_{\mathrm{FP}}-\Omega_{\Lambda})\Omega_{\Lambda}}\,,$
(A.8) $\displaystyle B_{4}$
$\displaystyle=-\frac{9\Omega_{\mathrm{FP}}^{2}-15\Omega_{\mathrm{FP}}\Omega_{\Lambda}+4\Omega_{\Lambda}^{2}}{3\Omega_{\Lambda}}$
(A.9)
while the mixing angle reads
$\bar{\alpha}^{2}=\Omega_{\Lambda}/(3\Omega_{\mathrm{FP}}-\Omega_{\Lambda})$.
Model | Viable cosmology | Vainshtein screening
---|---|---
$\beta_{0}\beta_{1}$ | $\Omega_{\mathrm{FP}}>\Omega_{\Lambda}$ | –
$\beta_{1}\beta_{2}$ | $3\Omega_{\mathrm{FP}}>(2+\sqrt{3})\Omega_{\Lambda}$ | –
$\beta_{1}\beta_{3}$ | $\Omega_{\mathrm{FP}}>\Omega_{\Lambda}$ | –
$\beta_{0}\beta_{1}\beta_{2}$ | $3\Omega_{\mathrm{FP}}>2(1+\bar{\alpha}^{2})\Omega_{\Lambda}$ | –
$\beta_{0}\beta_{1}\beta_{3}$ | $4\Omega_{\mathrm{FP}}>(1+\bar{\alpha}^{2})^{2}\Omega_{\Lambda}$ | –
$\beta_{1}\beta_{2}\beta_{3}$ | $4\bar{\alpha}^{4}\Omega_{\mathrm{FP}}<(1+\bar{\alpha}^{2})^{2}\Omega_{\Lambda}$ | $16\bar{\alpha}^{2}\Omega_{\mathrm{FP}}\lesssim\Omega_{\Lambda}$
| $6\bar{\alpha}^{2}\Omega_{\mathrm{FP}}<(3+4\bar{\alpha}^{2}+\bar{\alpha}^{4})\Omega_{\Lambda}$ |
$\beta_{1}\beta_{2}\beta_{4}$ | $3\bar{\alpha}^{2}\Omega_{\mathrm{FP}}<2(1+\bar{\alpha}^{2})\Omega_{\Lambda}$ | –
| $3\bar{\alpha}^{4}\Omega_{\mathrm{FP}}<(1+\bar{\alpha}^{2})^{2}\Omega_{\Lambda}$ |
$\beta_{1}\beta_{3}\beta_{4}$ | $\bar{\alpha}^{2}\Omega_{\mathrm{FP}}<(1+\bar{\alpha}^{2})\Omega_{\Lambda}$ | –
Table 4: This table summarises the theoretical constraints on all bimetric
models with up to three non-vanishing interaction parameters. The cosmology
constraints were derived in Ref. [105] and follow from requiring a consistent
cosmic expansion history. We only state the most stringent constraints other
than the Higuchi bound, $3\Omega_{\mathrm{FP}}>2\Omega_{\Lambda}$. Only the
$\beta_{1}\beta_{2}\beta_{3}$-model has a working Vainshtein mechanism. We
report the approximation of the most stringent bound.
### A.2 Three-parameter models
We continue by stating the parameter relations for the three-parameter models
with $\beta_{1}\neq 0$. For the $\beta_{0}\beta_{1}\beta_{2}$-model, the
relations read
$\displaystyle\begin{split}B_{0}&=-\frac{6\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+4\bar{\alpha}^{2}+3\bar{\alpha}^{4})\Omega_{\Lambda}}{1+\bar{\alpha}^{2}}\,,\\\
B_{1}&=\bar{\alpha}\left(\frac{3\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}-2\Omega_{\Lambda}\right)\,\\\
B_{2}&=-\frac{\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}+\Omega_{\Lambda}\,.\end{split}$
(A.10)
In the $\beta_{0}\beta_{1}\beta_{3}$-model, the physical and interaction
parameters are related as
$\displaystyle\begin{split}B_{0}&=-\frac{4\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+2\bar{\alpha}^{2}+\bar{\alpha}^{4})\Omega_{\Lambda}}{1+\bar{\alpha}^{2}}\,,\\\
B_{1}&=\frac{\bar{\alpha}}{2}\left(\frac{3\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}-\Omega_{\Lambda}\right)\,,\\\
B_{3}&=-\frac{1}{2\bar{\alpha}}\left(\frac{\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}-\Omega_{\Lambda}\right)\,.\end{split}$
(A.11)
Next, the $\beta_{0}\beta_{1}\beta_{4}$-model gives rise the the following
relations,
$B_{0}=-\frac{3\bar{\alpha}^{2}\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}+\Omega_{\Lambda}\,,\qquad
B_{1}=\frac{\bar{\alpha}\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}\,,\qquad
B_{4}=-\frac{1}{\bar{\alpha}^{2}}\left(\frac{\Omega_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}-\Omega_{\Lambda}\right)\,.$
(A.12)
Turning to models with $\beta_{0}=0$, the parameter relations for the
$\beta_{1}\beta_{2}\beta_{3}$-model read
$\displaystyle\begin{split}B_{1}&=\frac{-6\bar{\alpha}^{2}\Omega_{\mathrm{FP}}+(3+4\bar{\alpha}^{2}+\bar{\alpha}^{4})\Omega_{\Lambda}}{4\bar{\alpha}(1+\bar{\alpha}^{2})}\,,\\\
B_{2}&=\frac{4\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+2\bar{\alpha}^{2}+\bar{\alpha}^{4})\Omega_{\Lambda}}{2\bar{\alpha}^{2}(1+\bar{\alpha}^{2})}\,,\\\
B_{3}&=-\frac{6\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+4\bar{\alpha}^{2}+3\bar{\alpha}^{4})\Omega_{\Lambda}}{4\bar{\alpha}^{3}(1+\bar{\alpha}^{2})}\,.\end{split}$
(A.13)
Next, the $\beta_{1}\beta_{2}\beta_{4}$-model has the following parameter
relations,
$\displaystyle\begin{split}B_{1}&=\frac{-3\bar{\alpha}^{2}\Omega_{\mathrm{FP}}+2(1+\bar{\alpha}^{2})\Omega_{\Lambda}}{3\bar{\alpha}(1+\bar{\alpha}^{2})}\,,\\\
B_{2}&=\frac{3\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+\bar{\alpha}^{2})\Omega_{\Lambda}}{3\bar{\alpha}^{2}(1+\bar{\alpha}^{2})}\,,\\\
B_{4}&=-\frac{6\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+4\bar{\alpha}^{2}+3\bar{\alpha}^{4})\Omega_{\Lambda}}{3\bar{\alpha}^{4}(1+\bar{\alpha}^{2})}\,.\end{split}$
(A.14)
Finally, in the $\beta_{1}\beta_{3}\beta_{4}$-model the interaction and
physical parameters are related as
$\displaystyle\begin{split}B_{1}&=\frac{-\bar{\alpha}^{2}\Omega_{\mathrm{FP}}+(1+\bar{\alpha}^{2})\Omega_{\Lambda}}{2\bar{\alpha}(1+\bar{\alpha}^{2})}\,,\\\
B_{3}&=\frac{3\bar{\alpha}^{2}\Omega_{\mathrm{FP}}-(1+\bar{\alpha}^{2})\Omega_{\Lambda}}{2\bar{\alpha}^{3}(1+\bar{\alpha}^{2})}\,,\\\
B_{4}&=\frac{-4\bar{\alpha}^{2}\Omega_{\mathrm{FP}}+(1+2\bar{\alpha}^{2}+\bar{\alpha}^{4})\Omega_{\Lambda}}{\bar{\alpha}^{4}(1+\bar{\alpha}^{2})}\,.\end{split}$
(A.15)
## Appendix B Vainshtein screening in static, spherically symmetric systems
In this appendix, we will solve the bimetric equations of motion for the
static, spherically symmetric, and bidiagonal ansatz. This allows to study the
Vainshtein mechanism in these systems, which restores General Relativity on
small length scales. However, a solution that incorporates the Vainshtein
mechanism does not exist for every set of bimetric parameters. Instead,
demanding existence of such a solution implies conditions on the bimetric
parameters. The situation was studied in detail in Refs. [93, 90] however with
flat asymptotics. We aim at generalizing the analysis allowing for a non-zero
cosmological constant [166] and compute the conditions for the Vainshtein
mechanism to work in this more general setup. As we will see, our results
reduce to the ones of Ref. [93, 90] in the limit of vanishing cosmological
constant.
Assuming staticity and spherical symmetry, the metrics $g_{\mu\nu}$ and
$f_{\mu\nu}$ can be written in the bidiagonal case as [93]
$\displaystyle\begin{split}\mathrm{d}s_{\rm g}^{2}&=-e^{-\nu_{\rm
g}}\mathrm{d}t^{2}+e^{\lambda_{\rm
g}}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}\,,\\\ \mathrm{d}s_{\rm
f}^{2}&=-e^{-\nu_{\rm f}}\mathrm{d}t^{2}+e^{\lambda_{\rm f}}(r+r\mu)^{\prime
2}\mathrm{d}r^{2}+(r+r\mu)^{2}\mathrm{d}\Omega^{2}\,.\end{split}$ (B.1)
The metric functions $\nu_{\rm g,f}$, $\lambda_{\rm g,f}$, and $\mu$ depend on
radius $r$ only. The field $\mu$ can be thought of as a Stückelberg field that
restores diffeomorphisms in the $r$-dimension.
First, let us introduce the following short-hand notation,
$\displaystyle\beta=\frac{1+\bar{\alpha}^{2}}{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}c^{2}(\beta_{2}+\beta_{3}c)\,,\qquad\gamma=\frac{1+\bar{\alpha}^{2}}{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}c^{3}\beta_{3}\,.$
(B.2)
To simplify the equation of motion, we assume that the gravitational fields
and their derivatives are small, $\\{\lambda_{\rm g,f},\nu_{\rm g,f}\\}\ll 1$
and $\\{r\lambda^{\prime}_{\rm g,f},r\nu^{\prime}_{\rm g,f}\\}\ll 1$, but we
keep all nonlinearities in $\mu$. This results in the following set of
Einstein equation for $g_{\mu\nu}$,
$\displaystyle\begin{split}\frac{(r\lambda_{\rm
g})^{\prime}}{r^{2}}&=\Lambda+\frac{\rho(r)}{m_{\rm
g}^{2}}+\frac{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\left(\frac{1}{2}(\lambda_{\rm
f}-\lambda_{\rm
g})+\frac{1}{r^{2}}\Big{[}r^{3}\left(\mu+\beta\mu^{2}+\frac{\gamma}{3}\mu^{3}\right)\Big{]}^{\prime}\right)\,\\\
\frac{\lambda_{\rm g}}{r^{2}}-\frac{\nu^{\prime}_{\rm
g}}{r}&=\Lambda+\frac{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\left(\frac{1}{2}(\nu_{\rm
f}-\nu_{\rm g})+2\mu+\beta\mu^{2}\right)\,,\\\ -\frac{\lambda^{\prime}_{\rm
g}}{2r}+\frac{(r\nu_{\rm
g}^{\prime})^{\prime}}{2r}&=\Lambda+\frac{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\left(\frac{1}{2}(\lambda_{\rm
f}-\lambda_{\rm g}+\nu_{\rm f}-\nu_{\rm
g})+\frac{1}{r}\Big{[}r^{2}\Big{(}\mu+\frac{\beta}{2}\mu^{2}\Big{)}\Big{]}^{\prime}\right)\,.\end{split}$
(B.3)
With the same approximations, the Einstein equations of $f_{\mu\nu}$ can be
arranged to
$\displaystyle\begin{split}\frac{\left((r+r\mu)\lambda_{\rm
f}\right)^{\prime}}{r^{2}}&=\Lambda-\frac{m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\Bigg{(}\frac{1}{2}(\lambda_{\rm
f}-\lambda_{\rm
g})+\frac{1}{r^{2}}\Big{[}r^{3}\Big{(}\mu+(1+\beta)\mu^{2}+\frac{1+\beta+\gamma}{3}\mu^{3}\Big{)}\Big{]}^{\prime}\Bigg{)}\,,\\\
(r+r\mu)^{\prime}\frac{\lambda_{\rm f}}{r^{2}}-(1+\mu)\frac{\nu^{\prime}_{\rm
f}}{r}&=\Lambda-\frac{m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\Big{(}\frac{1}{2}(\nu_{\rm
f}-\nu_{\rm g})+2\mu+(1+\beta)\mu^{2}\Big{)}(r+r\mu)^{\prime}\,,\\\
\frac{\lambda_{\rm f}}{2r}-\frac{1}{2r}\left(\frac{(r+r\mu)\nu_{\rm
f}^{\prime}}{(r+r\mu)^{\prime}}\right)^{\prime}&=\Lambda-\frac{m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\Bigg{(}\frac{1}{2}(\lambda_{\rm
f}-\lambda_{\rm g}-\nu_{\rm f}-\nu_{\rm
g})++\frac{1}{r}\Big{[}r^{2}\left(\mu+\frac{1+\beta}{2}\mu^{2}\right)\Big{]}^{\prime}\Bigg{)}\end{split}$
(B.4)
Note that we do not present the $\phi\phi$-components of the Einstein
equations as these coincide with the $\theta\theta$-components. The last
ingredient is the Bianchi constraint, that simplifies to
$\displaystyle\frac{(r+r\mu)^{\prime}}{r}(1+\beta\mu)(\lambda_{\rm
f}-\lambda_{\rm g})-\frac{1}{2}(1+2\beta\mu+\gamma\mu^{2})(\nu^{\prime}_{\rm
f}-(r+r\mu)^{\prime}\nu^{\prime}_{\rm g})=0\,.$ (B.5)
Even in this simplified approximation, the equations are not solvable
analytically. To find analytic solutions, we will go to large radii where all
metric functions and their derivatives have to be small because we assume
spacetime to be asymptotically de Sitter while remaining far inside the de
Sitter horizon $3/\sqrt{\Lambda}$. The other limit is for small radii compared
to the Compton wavelength of the massive spin-2 field.
### B.1 Approximate solutions
#### B.1.1 Linear regime
First, we will solve the equations of motion for $r\gg r_{\rm S}$ with $r_{\rm
S}$ the Schwarzschild radius of a localised source to be defined later. For
large radii smaller than the de Sitter horizon, all metric functions,
including $\mu$, are assumed to be small such that both metrics approach de
Sitter solution asymptotically. We expand the Einstein equations and the
Bianchi constraint further in $\mu\ll 1$ and $r\mu^{\prime}\ll 1$. It is
convenient to introduce the auxiliary functions,
$\displaystyle\lambda_{\pm}=\lambda_{f}\pm\lambda_{g}\,,\
\nu_{\pm}=\nu_{f}\pm\nu_{g}$ (B.6)
In terms of these functions, the linearised Bianchi constraint (B.5) reads,
$\displaystyle 2\lambda_{-}=r\nu^{\prime}_{-}\,,$ (B.7)
which is independent of the $+$-fields. We can find linear combinations of the
Einstein equations for $g_{\mu\nu}$ and $f_{\mu\nu}$ such that the $+$-fields
drop out of the expression,
$\displaystyle\begin{split}0&=2r\lambda^{\prime}_{-}+\left(2+m_{\mathrm{FP}}^{2}r^{2}\right)\lambda_{-}+2m_{\mathrm{FP}}^{2}r^{2}(3\mu+r\mu^{\prime})\,,\\\
0&=-2\lambda_{-}+m_{\mathrm{FP}}^{2}r^{2}(\nu_{-}+4\mu)\,,\\\
0&=r\lambda^{\prime}_{-}-m_{\mathrm{FP}}^{2}r^{2}(\lambda_{-}+\nu_{-}+4\mu+2r\mu^{\prime})\,,\end{split}$
(B.8)
where we have used the Bianchi constrained already to eliminate
$\nu^{\prime}_{-}$ and $\nu^{\prime\prime}_{-}$ in terms of $\lambda_{-}$ and
$\lambda^{\prime}_{-}$. The equations reduce to two coupled differential
equations for $\lambda_{-}$ and $\mu$,
$\displaystyle\lambda^{\prime}_{-}=-2m_{\mathrm{FP}}^{2}\,r\mu\,,\qquad\mu^{\prime}=-\frac{(2+m_{\mathrm{FP}}^{2}\,r^{2})\lambda_{-}+2m_{\mathrm{FP}}^{2}\,r^{2}\mu}{2m_{\mathrm{FP}}^{2}\,r^{3}}\,,$
(B.9)
while $\nu_{-}$ is algebraically determined by the other two fields. These
equations are solved by
$\displaystyle\begin{split}\nu_{-}&=\frac{C_{1}e^{-m_{\mathrm{FP}}\,r}}{r}\,,\\\
\lambda_{-}&=-\frac{C_{1}(1+m_{\mathrm{FP}}\,r)e^{-m_{\mathrm{FP}}\,r}}{2r}\,,\\\
\mu&=-\frac{C_{1}\left(1+m_{\mathrm{FP}}^{2}\,r+m_{\mathrm{FP}}^{2}r^{2}\right)e^{-m_{\mathrm{FP}}\,r}}{4m_{\mathrm{FP}}^{2}r^{3}}\,,\end{split}$
(B.10)
where $C_{1}$ is a constant of integration to be determined later. We already
fixed the other constant of integration by requiring that the fields vanish in
the limit $r\rightarrow\infty$ in order to ensure that the metrics are
asymptotically bidiagonal and de Sitter. Next, we turn to the other set of
linearly independent equations that contain also the $+$-fields. These can be
arranged to
$\displaystyle\begin{split}0&=(\bar{\alpha}^{2}-1)(\lambda_{-}+r\lambda^{\prime}_{-})+(1+\bar{\alpha}^{2})(\lambda_{+}+r\lambda^{\prime}_{+})-2(1+\bar{\alpha}^{2})\Lambda
r^{2}\,,\\\
0&=(\bar{\alpha}^{2}-1)\lambda_{-}-(1+\bar{\alpha}^{2})(\lambda_{+}-r\nu^{\prime}_{+})+2(1+\bar{\alpha}^{2})\Lambda
r^{2}\,,\\\
0&=(\bar{\alpha}^{2}-1)\lambda^{\prime}_{-}-(1+\bar{\alpha}^{2})(\lambda^{\prime}_{+}-\nu^{\prime}_{+}-r\nu^{\prime\prime}_{+}-4\Lambda
r)\end{split}$ (B.11)
Upon using eq. B.10, these equations are solved by
$\displaystyle\begin{split}\nu_{+}&=-\frac{2\Lambda
r^{2}}{3}-\frac{2C_{2}}{r}-\frac{C_{1}e^{-m_{\mathrm{FP}}\,r}}{(1+\bar{\alpha}^{2})r}\,,\\\
\lambda_{+}&=\frac{2\Lambda
r^{2}}{3}+\frac{2C_{2}}{r}+\frac{C_{1}(1+m_{\mathrm{FP}}\,r)e^{-m_{\mathrm{FP}}\,r}}{2(1+\bar{\alpha}^{2})r}\,,\end{split}$
(B.12)
where $C_{2}$ is another constant of integration. Again, we have already used
that the functions have to be bidiagonal and de Sitter in the limit
$r\rightarrow\infty$ to fix one of the constants of integration. Having solved
the differential equations for all the fields, we can present the final
solutions for the metric fields in the linearised limit. They are given by
$\displaystyle\begin{split}\mu&=-\frac{C_{1}(1+m_{\mathrm{FP}}\,r+m_{\mathrm{FP}}^{2}r^{2})e^{-m_{\mathrm{FP}}\,r}}{4m_{\mathrm{FP}}^{2}r^{3}}\,,\\\
\nu_{\rm g}&=-\frac{\Lambda
r^{2}}{3}-\frac{C_{2}}{r}-\frac{C_{1}\bar{\alpha}^{2}e^{-m_{\mathrm{FP}}\,r}}{(1+\bar{\alpha}^{2})r}\,,\\\
\nu_{\rm f}&=-\frac{\Lambda
r^{2}}{3}-\frac{C_{2}}{r}+\frac{C_{1}e^{-m_{\mathrm{FP}}\,r}}{(1+\bar{\alpha}^{2})r}\,,\\\
\lambda_{\rm g}&=\frac{\Lambda
r^{2}}{3}+\frac{C_{2}}{r}+\frac{C_{1}\bar{\alpha}^{2}(1+m_{\mathrm{FP}}\,r)e^{-m_{\mathrm{FP}}\,r}}{2(1+\bar{\alpha}^{2})r}\,,\\\
\lambda_{\rm f}&=\frac{\Lambda
r^{2}}{3}+\frac{C_{2}}{r}-\frac{C_{1}(1+m_{\mathrm{FP}}\,r)e^{-m_{\mathrm{FP}}\,r}}{2(1+\bar{\alpha}^{2})r}\,.\end{split}$
(B.13)
In the limit of a vanishing cosmological constant, $\Lambda=0$, our linearised
solutions reduce the the results obtained in Refs. [93, 90].
Let us comment on the region of validity of these solutions. As we will see
later, $C_{1}\sim r_{\rm S}$. Hence, all metric fields are small on scales
$r\gg r_{\rm S}$ and inside the de Sitter horizon, $r\ll\Lambda^{-1/2}$. This
is not true however for the Stückelberg field $\mu$, which is small only on
scales $r\gg(r_{\rm S}/m_{\mathrm{FP}}^{2})^{1/3}$. In general, this scale is
larger than the Schwarzschild radius of the source. These solutions are hence
not appropriate to describe the scales $r_{\rm S}\ll r\ll(r_{\rm
S}/m_{\mathrm{FP}}^{2})^{1/3}$. In the next section, we seek solutions that
are valid on these scales.
#### B.1.2 Compton regime
Next, we want to solve the equations on scales much smaller than the Compton
wavelength of the massive spin-2 field, i.e. for $r\ll m_{\mathrm{FP}}^{-1}$.
Technically that means that we omit the terms
$m_{\mathrm{FP}}^{2}r^{2}\cdot\\{\lambda_{\rm g,f},\nu_{\rm g,f}\\}\ll 1$ in
the modified Einstein equations.
Under this approximation we can integrate the $tt$-component of eq. B.3 to
$\displaystyle\lambda_{\rm g}=\frac{r_{\rm
S}}{r}+\frac{\bar{\alpha}^{2}m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}r^{2}\left(\mu+\beta\mu^{2}+\frac{\gamma}{3}\mu^{3}\right)\,,$
(B.14)
where we defined the Schwarzschild radius
$\displaystyle r_{\rm S}=\frac{1}{4\pi m_{\rm g}^{2}}\int_{0}^{R_{*}}\rho
r^{2}\mathrm{d}r$ (B.15)
of a source of mass $M$ and radius $R_{*}$. The solution is valid outside the
compact object and we fixed the constants of integration by demanding
regularity at the surface of the compact object and at the origin $r=0$. Upon
using the solution for $\lambda_{\rm g}$, the $rr$-component of eq. B.3
simplifies to
$\displaystyle r\nu^{\prime}_{\rm g}=\frac{r_{\rm
S}}{r}-\frac{\bar{\alpha}^{2}m_{\mathrm{FP}}}{1+\bar{\alpha}^{2}}r^{2}\left(\mu-\frac{\gamma}{3}\mu^{3}\right)\,.$
(B.16)
The $tt$-component of eq. B.4 can easily be integrated to
$\displaystyle\lambda_{\rm
f}=-\frac{m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\frac{r^{2}}{1+\mu}\left(\mu+(1+\beta)\mu^{2}+\frac{1+\beta+\gamma}{3}\mu^{3}\right)\,.$
(B.17)
Here, we fixed the constant of integration by demanding regularity at the
origin $r=0$. With this result, the $rr$-component of eq. B.3 simplifies to
$\displaystyle r\nu^{\prime}_{\rm
f}=\frac{m_{\mathrm{FP}}^{2}}{1+\bar{\alpha}^{2}}\frac{r^{2}(r+r\mu)^{\prime}}{(1+\mu)^{2}}\left(\mu+2\mu^{2}+\frac{2+2\beta-\gamma}{3}\mu^{3}\right)\,.$
(B.18)
We have obtained expression for all the fields that show up in the Bianchi
constraint in terms of $\mu$. Plugging these results into the Bianchi
constraint yields the following algebraic polynomial for $\mu$:
$\displaystyle\begin{split}-3(3+3\bar{\alpha}^{2})\mu\ \,&\\\
-2\left(9(1+\bar{\alpha}^{2})(1+\beta)\right)\mu^{2}&\\\
-\left(10+9\bar{\alpha}^{2}+2(17+18\bar{\alpha}^{2})\beta+6(1+\bar{\alpha}^{2})\beta^{2}+4(1+\bar{\alpha}^{2})\gamma\right)\mu^{3}&\\\
-2\left(1+(7+9\bar{\alpha}^{2})\beta+6(1+\bar{\alpha}^{2})\beta^{2}+4(1+\bar{\alpha}^{2})\gamma\right)\mu^{4}&\\\
-\left(2(1+2\gamma)\beta+2(1+3\bar{\alpha}^{2})\beta^{2}+2(1+2\bar{\alpha}^{2})\gamma-(1+\bar{\alpha}^{2})\gamma^{2}\right)\mu^{5}&\\\
+2\bar{\alpha}^{2}\gamma^{2}\mu^{6}+\bar{\alpha}^{2}\gamma^{2}\mu^{7}&\\\
=3\left(\frac{r_{\rm
V}}{r}\right)^{3}(1+\bar{\alpha}^{2})(1+\mu)^{2}(1-\gamma\mu^{2})&\,.\end{split}$
(B.19)
In here, we defined the Vainshtein radius
$r_{\rm V}=\left(\frac{r_{\rm S}}{m_{\mathrm{FP}}^{2}}\right)^{1/3}$ (B.20)
that we already encountered before. Our result coincides with the one obtained
in Ref. [90].
Eq. (B.19) represents a seventh order algebraic polynomial for $\mu$ and as
such has up to seven solutions. As argued in Ref. [93] for $\gamma>0$, three
solutions are real-valued for all $r$, while two solutions are real only up to
a certain critical radius that depends on the other parameters. The last two
solutions are complex-valued. The three everywhere real-valued solutions are
shown in Fig. 12. In the opposite case $\gamma<0$, two of the real-valued
solutions become complex-valued and the remaining real-valued solutions are
neither asymptotically bidiagonal nor restore GR inside the Vainshtein regime.
The case $\gamma=0$ is special because the polynomial is only of degree five.
There is one solution that realises the Vainshtein mechanism for small radii,
which however is not asymptotically bidiagonal and it is complex-valued for
very small radii. Although there might be special situations in which the
Vainshtein mechanism works even for this case [160, 93], we exclude $\gamma=0$
from our analysis and restrict to $\gamma>0$.
We can infer further information about the general behavior of $\mu$. For
radii much larger than the Vainshtein radius, the term $(r_{\rm V}/r)^{3}$ on
the right hand side of eq. B.19 is small, which implies that the Stückelberg
field is small as well, $\mu\ll 1$. This gives rise to three asymptotic values
for $\mu$ as can be seen in Fig. 12. One of these is given by $\mu=0$ such
that we can neglect nonlinearities in $\mu$. This solution corresponds to the
one obtained when linearising the field equations as in section B.1.1. For
small radii, the term $(r_{\rm V}/r)^{3}$ on the right hand side is large.
This allows to identify two classes of solutions. Either, $\mu$ on the right
hand side compensates that large term. This is the case for $\mu=-1$ or
$\mu=\pm 1/\sqrt{\gamma}$. Note that the second solution exists only for
$\gamma>0$. These are the three asymptotic values that show up in Fig. 12. The
other possibility is that $\mu$ diverges as $r\rightarrow 0$.
Figure 12: Plot of the Stückelberg field $\mu$ as a function of $r/r_{\rm V}$
for the exemplary parameter values $\bar{\alpha}=1$, $\beta=1$, and
$\gamma=16$. The Vainshtein-Yukawa solution (solid black) smoothly evolves
from zero at large distances to a non-zero value inside the Vainshtein regime
such as to restore GR on these scales. The other two solutions (red dotted and
green dashed) also restore GR inside the Vainshtein regime, but they do not
tend to zero asymptotically. For the same set of parameters, there are two
more solutions that are real valued only for small radii $r$.
#### B.1.3 Matching the analytic solutions
It remains to determine the constants of integration $C_{1}$ and $C_{2}$. We
do this by matching the analytic solutions in the regime where both of them
are valid, i.e. $r_{\rm V}\ll r\ll m_{\mathrm{FP}}^{-1}$. Linearising the
nonlinear solutions eqs. B.14, B.17 and B.19 for $r\gg r_{\rm V}$, we obtain
$\displaystyle\mu=-\frac{r_{\rm
S}}{3m_{\mathrm{FP}}^{2}r^{3}}\,,\qquad\lambda_{\rm
g}=\frac{(3+2\bar{\alpha}^{2})r_{\rm
S}}{3(1+\bar{\alpha}^{2})r}\,,\qquad\lambda_{\rm f}=\frac{r_{\rm
S}}{3(1+\bar{\alpha}^{2})r}\,.$ (B.21)
Here, we linearised the Stückelberg field around $\mu=0$. Expanding on the
other hand the linear solutions (B.13) for $r\ll m_{\mathrm{FP}}^{-1}$ yields
$\displaystyle\mu=-\frac{C_{1}}{4m_{\mathrm{FP}}^{2}r^{3}}\,,\qquad\lambda_{\rm
g}=\frac{\bar{\alpha}^{2}C_{1}+2(1+\bar{\alpha}^{2})C_{2}}{2(1+\bar{\alpha}^{2})r}\,,\qquad\lambda_{\rm
f}=-\frac{C_{1}-2(1+\bar{\alpha}^{2})C_{2}}{2(1+\bar{\alpha}^{2})r}\,.$ (B.22)
These solutions coincide, if we choose the constants of integration to be
$\displaystyle C_{1}=\frac{4r_{\rm S}}{3}\,,\qquad C_{2}=\frac{r_{\rm
S}}{1+\bar{\alpha}^{2}}\,.$ (B.23)
### B.2 Existence of Vainshtein-Yukawa branch
So far, we have studied analytic solutions on two different scales. For large
radii, $r\gg r_{\rm V}$, the gravitational potential is a combination of the
$1/r$-law and a Yukawa-type potential:
$\nu_{\rm g}=-\frac{r_{\rm
S}}{(1+\bar{\alpha}^{2})}\left(\frac{1}{r}+\frac{4\bar{\alpha}^{2}}{3}\frac{e^{-m_{\mathrm{FP}}\,r}}{r}\right)$
(B.24)
as follows from eqs. B.13 and B.23. On smaller length scales, $r_{\rm S}\ll
r\ll r_{\rm V}$ the Stückelberg field $\mu$ is nonlinear. The nonlinearities
are such that the gravitational potential is given by
$\nu_{\rm g}=-\frac{r_{\rm S}}{r}$ (B.25)
as follows from eq. B.16 on a solution where $\mu$ is constant. This is
summarised in eq. 4.4. For the screening mechanism to work it is necessary
that the solution of the Stückelberg field realising these two regimes exists
and is real-valued for every $r$ without branch cuts.
Figure 13: Exclusion plots for a working Vainshtein mechanism for the
$\beta_{1}\beta_{2}\beta_{3}$-model. Left panel: In the blue-shaded region
$\beta<d_{1}/d_{2}$ is implied while in the yellow-shaded region $d_{2}<0$
holds. In the overlap, there is no everywhere real solution for the
Stückelberg field $\mu$ that incorporates the Vainshtein mechanism at small
radii while giving rise to the correct asymptotics. Right panel: In the green-
shaded region $\mu\ll 1$ is no solution for large radii while in the purple-
shaded region $\gamma<1$. The yellow-shaded region shows the overlap of the
bounds presented in the left panel. In the gray shaded region the Higuchi
bound is violated. In summary, only outside the shaded regions, the bimetric
model incorporates a working Vainshtein mechanism.
This point was studied in detail in [90]. Firstly, the nonlinear equation
(B.19) determining $\mu$ must give rise to a solution with $\mu\ll 1$ for
$r\gg r_{\rm V}$, which matches the linearised solution. This is the case only
if
$\beta<\sqrt{\gamma}$ (B.26)
is satisfied. This bound is stated in eq. 4.6 and ensures a consistent
asymptotic behavior of the nonlinear solutions. This is the condition ensuring
consistent asymptotics as stated in eq. 4.6. Secondly, the linearised solution
can be smoothly connected only to the nonlinear solution with
$\mu=-1/\sqrt{\gamma}$, which is required to be larger than the solution with
$\mu=-1$. Therefore, a consistent screening regime requires the parameters to
satisfy
$\gamma>1$ (B.27)
as stated in eq. 4.5. Thirdly, there must be a solution that is real-valued
for every $r$ and interpolates between the two aforementioned asymptotic
limits without branch cuts. This is the case if the parameters satisfy the
following bound:
$\beta>\frac{d_{1}}{d_{2}}\quad{\rm if}\quad d_{2}<0\,,$ (B.28)
where
$\displaystyle\begin{split}d_{1}&=-1+6(1+\bar{\alpha}^{2})\sqrt{\gamma}(1+\gamma)-(13+12\bar{\alpha}^{2})\gamma\,,\\\
d_{2}&=1+3\bar{\alpha}^{2}-6(1+\bar{\alpha}^{2})\sqrt{\gamma}+3(1+\bar{\alpha}^{2})\gamma\,.\end{split}$
(B.29)
The bound in eq. B.27 implies $\beta_{3}>0$. Combining eqs. B.26 and B.27 then
implies $\beta_{2}<0$. A theoretically consistent expansion history requires
$\beta_{1}>0$. Therefore, the $\beta_{1}\beta_{2}\beta_{3}$-model is the only
bimetric model with up to three free parameters that allows for Vainshtein
screening and has a consistent background cosmology. In fig. 13 the bounds in
eqs. B.26, B.27 and B.28 are presented in terms of the physical parameters
$\bar{\alpha}$ and $m_{\mathrm{FP}}^{2}/\Lambda$ for that model. To express
the interaction in terms of the physical parameters, we use the parameter
relations reported in appendix A. In the left panel the violation of the bound
(B.28) is shown. In the overlap of the blue- and yellow-shaded regions, the
Vainshtein-Yukawa solution does not exist. In the right panel, we combine this
bound with the other two. The shaded regions represent, where the bounds are
violated and hence where the Vainshtein screening mechanism does not work.
Only the region left white gives rise to screening. In fig. 7, the Vainshtein
bounds are combined with the bounds ensuring a consistent background
cosmology.
## References
* [1] LIGO Scientific, Virgo collaboration, _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116 (2016) 061102 [1602.03837].
* [2] A. Einstein, _Über Gravitationswellen_ , _Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. )_ 1918 (1918) 154.
* [3] LIGO Scientific, Virgo collaboration, _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ , _Phys. Rev. Lett._ 119 (2017) 161101 [1710.05832].
* [4] LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL collaboration, _Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A_ , _Astrophys. J. Lett._ 848 (2017) L13 [1710.05834].
* [5] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
* [6] ACT collaboration, _The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters_ , 2007.07288.
* [7] K.C. Wong et al., _H0LiCOW – XIII. A 2.4 per cent measurement of H0 from lensed quasars: 5.3 $\sigma$ tension between early- and late-Universe probes_, _Mon. Not. Roy. Astron. Soc._ 498 (2020) 1420 [1907.04869].
* [8] A.G. Riess, S. Casertano, W. Yuan, L.M. Macri and D. Scolnic, _Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond $\Lambda$CDM_, _Astrophys. J._ 876 (2019) 85 [1903.07603].
* [9] W.L. Freedman, B.F. Madore, T. Hoyt, I.S. Jang, R. Beaton, M.G. Lee et al., _Calibration of the Tip of the Red Giant Branch (TRGB)_ , 2002.01550.
* [10] D. Pesce et al., _The Megamaser Cosmology Project. XIII. Combined Hubble constant constraints_ , _Astrophys. J. Lett._ 891 (2020) L1 [2001.09213].
* [11] J.L. Bernal, L. Verde and A.G. Riess, _The trouble with $H_{0}$_, _JCAP_ 10 (2016) 019 [1607.05617].
* [12] S. Weinberg, _The Cosmological Constant Problem_ , _Rev. Mod. Phys._ 61 (1989) 1.
* [13] J. Martin, _Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)_ , _Comptes Rendus Physique_ 13 (2012) 566 [1205.3365].
* [14] G. Dvali and C. Gomez, _Quantum Compositeness of Gravity: Black Holes, AdS and Inflation_ , _JCAP_ 01 (2014) 023 [1312.4795].
* [15] G. Dvali and C. Gomez, _Quantum Exclusion of Positive Cosmological Constant?_ , _Annalen Phys._ 528 (2016) 68 [1412.8077].
* [16] G. Dvali, C. Gomez and S. Zell, _Quantum Break-Time of de Sitter_ , _JCAP_ 06 (2017) 028 [1701.08776].
* [17] G. Dvali and C. Gomez, _On Exclusion of Positive Cosmological Constant_ , _Fortsch. Phys._ 67 (2019) 1800092 [1806.10877].
* [18] Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, _The Landscape, the Swampland and the Era of Precision Cosmology_ , _Fortsch. Phys._ 67 (2019) 1800075 [1808.09440].
* [19] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, _De Sitter Space and the Swampland_ , 1806.08362.
* [20] P. Agrawal, G. Obied, P.J. Steinhardt and C. Vafa, _On the Cosmological Implications of the String Swampland_ , _Phys. Lett. B_ 784 (2018) 271 [1806.09718].
* [21] S.K. Garg and C. Krishnan, _Bounds on Slow Roll and the de Sitter Swampland_ , _JHEP_ 11 (2019) 075 [1807.05193].
* [22] H. Ooguri, E. Palti, G. Shiu and C. Vafa, _Distance and de Sitter Conjectures on the Swampland_ , _Phys. Lett. B_ 788 (2019) 180 [1810.05506].
* [23] A. Joyce, L. Lombriser and F. Schmidt, _Dark Energy Versus Modified Gravity_ , _Ann. Rev. Nucl. Part. Sci._ 66 (2016) 95 [1601.06133].
* [24] D. Lovelock, _The Einstein tensor and its generalizations_ , _J. Math. Phys._ 12 (1971) 498.
* [25] G.W. Horndeski, _Second-order scalar-tensor field equations in a four-dimensional space_ , _Int.J.Theor.Phys._ 10 (1974) 363.
* [26] C. Deffayet, G. Esposito-Farese and A. Vikman, _Covariant Galileon_ , _Phys. Rev. D_ 79 (2009) 084003 [0901.1314].
* [27] G. Tasinato, _Cosmic Acceleration from Abelian Symmetry Breaking_ , _JHEP_ 04 (2014) 067 [1402.6450].
* [28] L. Heisenberg, _Generalization of the Proca Action_ , _JCAP_ 05 (2014) 015 [1402.7026].
* [29] P. Creminelli and F. Vernizzi, _Dark Energy after GW170817 and GRB170817A_ , _Phys. Rev. Lett._ 119 (2017) 251302 [1710.05877].
* [30] J. Sakstein and B. Jain, _Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories_ , _Phys. Rev. Lett._ 119 (2017) 251303 [1710.05893].
* [31] J.M. Ezquiaga and M. Zumalacárregui, _Dark Energy After GW170817: Dead Ends and the Road Ahead_ , _Phys. Rev. Lett._ 119 (2017) 251304 [1710.05901].
* [32] T. Baker, E. Bellini, P. Ferreira, M. Lagos, J. Noller and I. Sawicki, _Strong constraints on cosmological gravity from GW170817 and GRB 170817A_ , _Phys. Rev. Lett._ 119 (2017) 251301 [1710.06394].
* [33] C. de Rham, G. Gabadadze and A.J. Tolley, _Resummation of Massive Gravity_ , _Phys.Rev.Lett._ 106 (2011) 231101 [1011.1232].
* [34] S. Hassan and R.A. Rosen, _Resolving the Ghost Problem in non-Linear Massive Gravity_ , _Phys.Rev.Lett._ 108 (2012) 041101 [1106.3344].
* [35] C. de Rham, J.T. Deskins, A.J. Tolley and S.-Y. Zhou, _Graviton Mass Bounds_ , _Rev. Mod. Phys._ 89 (2017) 025004 [1606.08462].
* [36] S. Hassan and R.A. Rosen, _Bimetric Gravity from Ghost-free Massive Gravity_ , _JHEP_ 1202 (2012) 126 [1109.3515].
* [37] K. Max, M. Platscher and J. Smirnov, _Gravitational Wave Oscillations in Bigravity_ , _Phys. Rev. Lett._ 119 (2017) 111101 [1703.07785].
* [38] K. Max, M. Platscher and J. Smirnov, _Decoherence of Gravitational Wave Oscillations in Bigravity_ , _Phys. Rev. D_ 97 (2018) 064009 [1712.06601].
* [39] M. Fierz and W. Pauli, _On relativistic wave equations for particles of arbitrary spin in an electromagnetic field_ , _Proc.Roy.Soc.Lond._ A173 (1939) 211.
* [40] V. Zakharov, _Linearized gravitation theory and the graviton mass_ , _JETP Lett._ 12 (1970) 312.
* [41] H. van Dam and M. Veltman, _Massive and massless Yang-Mills and gravitational fields_ , _Nucl.Phys._ B22 (1970) 397.
* [42] A. Vainshtein, _To the problem of nonvanishing gravitation mass_ , _Phys.Lett._ B39 (1972) 393.
* [43] M. Ostrogradsky, _Mémoires sur les équations différentielles, relatives au problème des isopérimètres_ , _Mem. Acad. St. Petersbourg_ 6 (1850) 385.
* [44] A. Pais and G. Uhlenbeck, _On Field theories with nonlocalized action_ , _Phys. Rev._ 79 (1950) 145.
* [45] D. Boulware and S. Deser, _Can gravitation have a finite range?_ , _Phys.Rev._ D6 (1972) 3368.
* [46] C. de Rham and G. Gabadadze, _Generalization of the Fierz-Pauli Action_ , _Phys.Rev._ D82 (2010) 044020 [1007.0443].
* [47] C. de Rham, G. Gabadadze and A.J. Tolley, _Ghost free Massive Gravity in the Stückelberg language_ , _Phys.Lett._ B711 (2012) 190 [1107.3820].
* [48] S. Hassan, R.A. Rosen and A. Schmidt-May, _Ghost-free Massive Gravity with a General Reference Metric_ , _JHEP_ 1202 (2012) 026 [1109.3230].
* [49] S. Hassan and R.A. Rosen, _Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity_ , _JHEP_ 1204 (2012) 123 [1111.2070].
* [50] S. Hassan, A. Schmidt-May and M. von Strauss, _Proof of Consistency of Nonlinear Massive Gravity in the Stückelberg Formulation_ , _Phys.Lett._ B715 (2012) 335 [1203.5283].
* [51] S. Hassan and A. Lundkvist, _Analysis of constraints and their algebra in bimetric theory_ , _JHEP_ 08 (2018) 182 [1802.07267].
* [52] V. Errasti Díez, M. Maier, J.A. Méndez-Zavaleta and M. Taslimi Tehrani, _Lagrangian constraint analysis of first-order classical field theories with an application to gravity_ , _Phys. Rev. D_ 102 (2020) 065015 [2007.11020].
* [53] G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava and A.J. Tolley, _Massive Cosmologies_ , _Phys. Rev._ D84 (2011) 124046 [1108.5231].
* [54] A.E. Gümrükçüoğlu, C. Lin and S. Mukohyama, _Open FRW universes and self-acceleration from nonlinear massive gravity_ , _JCAP_ 1111 (2011) 030 [1109.3845].
* [55] A.E. Gümrükçüoğlu, C. Lin and S. Mukohyama, _Cosmological perturbations of self-accelerating universe in nonlinear massive gravity_ , _JCAP_ 1203 (2012) 006 [1111.4107].
* [56] A. De Felice, A.E. Gümrükçüoğlu and S. Mukohyama, _Massive gravity: nonlinear instability of the homogeneous and isotropic universe_ , _Phys.Rev.Lett._ 109 (2012) 171101 [1206.2080].
* [57] K. Hinterbichler, _Theoretical Aspects of Massive Gravity_ , _Rev.Mod.Phys._ 84 (2012) 671 [1105.3735].
* [58] C. de Rham, _Massive Gravity_ , _Living Rev.Rel._ 17 (2014) 7 [1401.4173].
* [59] M.S. Volkov, _Cosmological solutions with massive gravitons in the bigravity theory_ , _JHEP_ 1201 (2012) 035 [1110.6153].
* [60] M. von Strauss, A. Schmidt-May, J. Enander, E. Mörtsell and S. Hassan, _Cosmological Solutions in Bimetric Gravity and their Observational Tests_ , _JCAP_ 1203 (2012) 042 [1111.1655].
* [61] D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, _FRW Cosmology in Ghost Free Massive Gravity_ , _JHEP_ 1203 (2012) 067 [1111.1983].
* [62] M.S. Volkov, _Exact self-accelerating cosmologies in the ghost-free bigravity and massive gravity_ , _Phys.Rev._ D86 (2012) 061502 [1205.5713].
* [63] M.S. Volkov, _Exact self-accelerating cosmologies in the ghost-free massive gravity – the detailed derivation_ , _Phys.Rev._ D86 (2012) 104022 [1207.3723].
* [64] Y. Akrami, T.S. Koivisto and M. Sandstad, _Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality_ , _JHEP_ 1303 (2013) 099 [1209.0457].
* [65] A. De Felice, A.E. Gümrükçüoğlu, S. Mukohyama, N. Tanahashi and T. Tanaka, _Viable cosmology in bimetric theory_ , _JCAP_ 1406 (2014) 037 [1404.0008].
* [66] M.S. Volkov, _Self-accelerating cosmologies and hairy black holes in ghost-free bigravity and massive gravity_ , _Class.Quant.Grav._ 30 (2013) 184009 [1304.0238].
* [67] F. Könnig, A. Patil and L. Amendola, _Viable cosmological solutions in massive bimetric gravity_ , _JCAP_ 1403 (2014) 029 [1312.3208].
* [68] E. Mörtsell and S. Dhawan, _Does the Hubble constant tension call for new physics?_ , _JCAP_ 1809 (2018) 025 [1801.07260].
* [69] M. Lüben, A. Schmidt-May and J. Smirnov, _Vainshtein Screening in Bimetric Cosmology_ , _Phys. Rev. D_ 102 (2020) 123529 [1912.09449].
* [70] D. Comelli, M. Crisostomi and L. Pilo, _Perturbations in Massive Gravity Cosmology_ , _JHEP_ 1206 (2012) 085 [1202.1986].
* [71] N. Khosravi, H.R. Sepangi and S. Shahidi, _Massive cosmological scalar perturbations_ , _Phys. Rev. D_ 86 (2012) 043517 [1202.2767].
* [72] M. Berg, I. Buchberger, J. Enander, E. Mörtsell and S. Sjörs, _Growth Histories in Bimetric Massive Gravity_ , _JCAP_ 1212 (2012) 021 [1206.3496].
* [73] Y. Sakakihara, J. Soda and T. Takahashi, _On Cosmic No-hair in Bimetric Gravity and the Higuchi Bound_ , _PTEP_ 2013 (2013) 033E02 [1211.5976].
* [74] F. Könnig and L. Amendola, _Instability in a minimal bimetric gravity model_ , _Phys.Rev._ D90 (2014) 044030 [1402.1988].
* [75] D. Comelli, M. Crisostomi and L. Pilo, _FRW Cosmological Perturbations in Massive Bigravity_ , _Phys.Rev._ D90 (2014) 084003 [1403.5679].
* [76] A.R. Solomon, Y. Akrami and T.S. Koivisto, _Linear growth of structure in massive bigravity_ , _JCAP_ 1410 (2014) 066 [1404.4061].
* [77] F. Könnig, Y. Akrami, L. Amendola, M. Motta and A.R. Solomon, _Stable and unstable cosmological models in bimetric massive gravity_ , _Phys.Rev._ D90 (2014) 124014 [1407.4331].
* [78] M. Lagos and P.G. Ferreira, _Cosmological perturbations in massive bigravity_ , _JCAP_ 1412 (2014) 026 [1410.0207].
* [79] F. Könnig, _Higuchi Ghosts and Gradient Instabilities in Bimetric Gravity_ , _Phys.Rev._ D91 (2015) 104019 [1503.07436].
* [80] Y. Akrami, S.F. Hassan, F. Könnig, A. Schmidt-May and A.R. Solomon, _Bimetric gravity is cosmologically viable_ , _Phys. Lett._ B748 (2015) 37 [1503.07521].
* [81] K. Aoki, K.-i. Maeda and R. Namba, _Stability of the Early Universe in Bigravity Theory_ , _Phys. Rev._ D92 (2015) 044054 [1506.04543].
* [82] E. Mörtsell and J. Enander, _Scalar instabilities in bimetric gravity: The Vainshtein mechanism and structure formation_ , _JCAP_ 1510 (2015) 044 [1506.04977].
* [83] K. Aoki and S. Mukohyama, _Massive gravitons as dark matter and gravitational waves_ , _Phys. Rev. D_ 94 (2016) 024001 [1604.06704].
* [84] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermaäe et al., _Bigravitational origin of dark matter_ , _Phys. Rev._ D94 (2016) 084055 [1604.08564].
* [85] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermäe et al., _Heavy spin-2 Dark Matter_ , _JCAP_ 1609 (2016) 016 [1607.03497].
* [86] X. Chu and C. Garcia-Cely, _Self-interacting Spin-2 Dark Matter_ , _Phys. Rev._ D96 (2017) 103519 [1708.06764].
* [87] D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, _Spherically Symmetric Solutions in Ghost-Free Massive Gravity_ , _Phys.Rev._ D85 (2012) 024044 [1110.4967].
* [88] M. Platscher, J. Smirnov, S. Meyer and M. Bartelmann, _Long Range Effects in Gravity Theories with Vainshtein Screening_ , _JCAP_ 1812 (2018) 009 [1809.05318].
* [89] J. Enander and E. Mörtsell, _Strong lensing constraints on bimetric massive gravity_ , _JHEP_ 1310 (2013) 031 [1306.1086].
* [90] J. Enander and E. Mörtsell, _On stars, galaxies and black holes in massive bigravity_ , _JCAP_ 1511 (2015) 023 [1507.00912].
* [91] M. Hohmann, _Post-Newtonian parameter $\gamma$ and the deflection of light in ghost-free massive bimetric gravity_, _Phys. Rev. D_ 95 (2017) 124049 [1701.07700].
* [92] M. Lüben, E. Mörtsell and A. Schmidt-May, _Bimetric cosmology is compatible with local tests of gravity_ , _Class. Quant. Grav._ 37 (2020) 047001 [1812.08686].
* [93] E. Babichev and M. Crisostomi, _Restoring general relativity in massive bigravity theory_ , _Phys. Rev._ D88 (2013) 084002 [1307.3640].
* [94] A. De Felice, T. Nakamura and T. Tanaka, _Possible existence of viable models of bi-gravity with detectable graviton oscillations by gravitational wave detectors_ , _PTEP_ 2014 (2014) 043E01 [1304.3920].
* [95] G. Cusin, R. Durrer, P. Guarato and M. Motta, _Gravitational waves in bigravity cosmology_ , _JCAP_ 05 (2015) 030 [1412.5979].
* [96] L. Amendola, F. Könnig, M. Martinelli, V. Pettorino and M. Zumalacárregui, _Surfing gravitational waves: can bigravity survive growing tensor modes?_ , _JCAP_ 05 (2015) 052 [1503.02490].
* [97] M. Johnson and A. Terrana, _Tensor Modes in Bigravity: Primordial to Present_ , _Phys. Rev. D_ 92 (2015) 044001 [1503.05560].
* [98] Y. Sakakihara and J. Soda, _Primordial Gravitational Waves in Bimetric Gravity_ , _JCAP_ 09 (2015) 015 [1504.04969].
* [99] M. Fasiello and R.H. Ribeiro, _Mild bounds on bigravity from primordial gravitational waves_ , _JCAP_ 07 (2015) 027 [1505.00404].
* [100] G. Cusin, R. Durrer, P. Guarato and M. Motta, _Inflationary perturbations in bimetric gravity_ , _JCAP_ 09 (2015) 043 [1505.01091].
* [101] Y. Sakakihara and T. Tanaka, _Primordial fluctuations from inflation in dRGT bimetric theory of gravity_ , _JCAP_ 09 (2016) 033 [1605.05790].
* [102] M. Biagetti, E. Dimastrogiovanni and M. Fasiello, _Possible signatures of the inflationary particle content: spin-2 fields_ , _JCAP_ 10 (2017) 038 [1708.01587].
* [103] E. Dimastrogiovanni, M. Fasiello and G. Tasinato, _Probing the inflationary particle content: extra spin-2 field_ , _JCAP_ 08 (2018) 016 [1806.00850].
* [104] J.B. Jiménez, J.M. Ezquiaga and L. Heisenberg, _Probing cosmological fields with gravitational wave oscillations_ , _JCAP_ 04 (2020) 027 [1912.06104].
* [105] M. Lüben, A. Schmidt-May and J. Weller, _Physical parameter space of bimetric theory and SN1a constraints_ , _JCAP_ 09 (2020) 024 [2003.03382].
* [106] S. Dhawan, A. Goobar, E. Mörtsell, R. Amanullah and U. Feindt, _Narrowing down the possible explanations of cosmic acceleration with geometric probes_ , _JCAP_ 07 (2017) 040 [1705.05768].
* [107] M. Lindner, K. Max, M. Platscher and J. Rezacek, _Probing alternative cosmologies through the inverse distance ladder_ , _JCAP_ 10 (2020) 040 [2002.01487].
* [108] E. Fischbach and C. Talmadge, _The Search for Non-Newtonian Gravity_ , Springer-Verlag New York (1999), 10.1007/978-1-4612-1438-0.
* [109] E. Adelberger, B.R. Heckel and A. Nelson, _Tests of the gravitational inverse square law_ , _Ann. Rev. Nucl. Part. Sci._ 53 (2003) 77 [hep-ph/0307284].
* [110] E. Adelberger, J. Gundlach, B. Heckel, S. Hoedl and S. Schlamminger, _Torsion balance experiments: A low-energy frontier of particle physics_ , _Prog. Part. Nucl. Phys._ 62 (2009) 102.
* [111] J. Murata and S. Tanaka, _A review of short-range gravity experiments in the LHC era_ , _Class. Quant. Grav._ 32 (2015) 033001 [1408.3588].
* [112] B. Bertotti, L. Iess and P. Tortora, _A test of general relativity using radio links with the Cassini spacecraft_ , _Nature_ 425 (2003) 374.
* [113] C.M. Will, _The Confrontation between General Relativity and Experiment_ , _Living Rev. Rel._ 17 (2014) 4 [1403.7377].
* [114] A. Schmidt-May and M. von Strauss, _Recent developments in bimetric theory_ , _J. Phys._ A49 (2016) 183001 [1512.00021].
* [115] S. Hassan and R.A. Rosen, _On Non-Linear Actions for Massive Gravity_ , _JHEP_ 1107 (2011) 009 [1103.6055].
* [116] S. Hassan and M. Kocic, _On the local structure of spacetime in ghost-free bimetric theory and massive gravity_ , _JHEP_ 05 (2018) 099 [1706.07806].
* [117] C. de Rham, L. Heisenberg and R.H. Ribeiro, _On couplings to matter in massive (bi-)gravity_ , _Class.Quant.Grav._ 32 (2015) 035022 [1408.1678].
* [118] C. de Rham, L. Heisenberg and R.H. Ribeiro, _Ghosts & Matter Couplings in Massive (bi-&multi-)Gravity_, _Phys.Rev._ D90 (2014) 124042 [1409.3834].
* [119] L. Heisenberg, _Quantum corrections in massive bigravity and new effective composite metrics_ , _Class.Quant.Grav._ 32 (2015) 105011 [1410.4239].
* [120] L. Heisenberg, _More on effective composite metrics_ , _Phys. Rev._ D92 (2015) 023525 [1505.02966].
* [121] K. Hinterbichler and R.A. Rosen, _Note on ghost-free matter couplings in massive gravity and multigravity_ , _Phys. Rev._ D92 (2015) 024030 [1503.06796].
* [122] M. Lüben and A. Schmidt-May, _Ghost-Free Completion of An Effective Matter Coupling in Bimetric Theory_ , _Fortsch. Phys._ 66 (2018) 1800031 [1804.04671].
* [123] S. Hassan, A. Schmidt-May and M. von Strauss, _On Consistent Theories of Massive Spin-2 Fields Coupled to Gravity_ , _JHEP_ 1305 (2013) 086 [1208.1515].
* [124] H. Nersisyan, Y. Akrami and L. Amendola, _Consistent metric combinations in cosmology of massive bigravity_ , _Phys. Rev._ D92 (2015) 104034 [1502.03988].
* [125] G. Cusin, R. Durrer, P. Guarato and M. Motta, _A general mass term for bigravity_ , _JCAP_ 1604 (2016) 051 [1512.02131].
* [126] M. Fasiello and A.J. Tolley, _Cosmological Stability Bound in Massive Gravity and Bigravity_ , _JCAP_ 1312 (2013) 002 [1308.1647].
* [127] A. Higuchi, _Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time_ , _Nucl.Phys._ B282 (1987) 397.
* [128] M. Högås, F. Torsello and E. Mörtsell, _On the Stability of Bimetric Structure Formation_ , _Submitted to: JCAP_ (2019) [1910.01651].
* [129] L. Chen, Q.-G. Huang and K. Wang, _Distance priors from planck final release_ , _Journal of Cosmology and Astroparticle Physics_ 2019 (2019) 028?028.
* [130] SDSS collaboration, _Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples_ , _Astron. Astrophys._ 568 (2014) A22 [1401.4064].
* [131] WMAP collaboration, _Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation_ , _Astrophys. J. Suppl._ 180 (2009) 330 [0803.0547].
* [132] Z. Zhai, C.-G. Park, Y. Wang and B. Ratra, _Cmb distance priors revisited: effects of dark energy dynamics, spatial curvature, primordial power spectrum, and neutrino parameters_ , _Journal of Cosmology and Astroparticle Physics_ 2020 (2020) 009?009.
* [133] W. Hu and N. Sugiyama, _Small scale cosmological perturbations: An Analytic approach_ , _Astrophys. J._ 471 (1996) 542 [astro-ph/9510117].
* [134] F. Beutler, C. Blake, M. Colless, D.H. Jones, L. Staveley-Smith, L. Campbell et al., _The 6df galaxy survey: baryon acoustic oscillations and the local hubble constant_ , _Monthly Notices of the Royal Astronomical Society_ 416 (2011) 3017?3032.
* [135] A.J. Ross, L. Samushia, C. Howlett, W.J. Percival, A. Burden and M. Manera, _The clustering of the sdss dr7 main galaxy sample ? i. a 4 per cent distance measure at redshift 0.15_ , _Monthly Notices of the Royal Astronomical Society_ 449 (2015) 835?847.
* [136] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J.A. Blazek et al., _The clustering of galaxies in the completed sdss-iii baryon oscillation spectroscopic survey: cosmological analysis of the dr12 galaxy sample_ , _Monthly Notices of the Royal Astronomical Society_ 470 (2017) 2617?2652.
* [137] J.E. Bautista, M. Vargas-Magaña, K.S. Dawson, W.J. Percival, J. Brinkmann, J. Brownstein et al., _The sdss-iv extended baryon oscillation spectroscopic survey: Baryon acoustic oscillations at redshift of 0.72 with the dr14 luminous red galaxy sample_ , _The Astrophysical Journal_ 863 (2018) 110.
* [138] G.-B. Zhao, Y. Wang, S. Saito, H. Gil-Marín, W.J. Percival, D. Wang et al., _The clustering of the sdss-iv extended baryon oscillation spectroscopic survey dr14 quasar sample: a tomographic measurement of cosmic structure growth and expansion rate based on optimal redshift weights_ , _Monthly Notices of the Royal Astronomical Society_ 482 (2018) 3497?3513.
* [139] G. Schwarz, _Estimating the Dimension of a Model_ , _Annals Statist._ 6 (1978) 461.
* [140] R.E. Kass and A.E. Raftery, _Bayes Factors_ , _J. Am. Statist. Assoc._ 90 (1995) 773.
* [141] E. Aubourg et al., _Cosmological implications of baryon acoustic oscillation measurements_ , _Phys. Rev. D_ 92 (2015) 123516 [1411.1074].
* [142] E. Di Valentino, _A (brave) combined analysis of the $H_{0}$ late time direct measurements and the impact on the Dark Energy sector_, 2011.00246.
* [143] J. Hoskins, R. Newman, R. Spero and J. Schultz, _Experimental tests of the gravitational inverse square law for mass separations from 2-cm to 105-cm_ , _Phys. Rev. D_ 32 (1985) 3084.
* [144] R. Spero, J. Hoskins, R. Newman, J. Pellam and J. Schultz, _Test of the Gravitational Inverse-Square Law at Laboratory Distances_ , _Phys. Rev. Lett._ 44 (1980) 1645.
* [145] C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt et al., _Sub-millimeter tests of the gravitational inverse-square law_ , _Phys. Rev._ D70 (2004) 042004 [hep-ph/0405262].
* [146] S. Smullin, A. Geraci, D. Weld, J. Chiaverini, S.P. Holmes and A. Kapitulnik, _New constraints on Yukawa-type deviations from Newtonian gravity at 20 microns_ , _Phys. Rev. D_ 72 (2005) 122001 [hep-ph/0508204].
* [147] A. Cornaz, B. Hubler and W. Kuendig, _Determination of the gravitational constant at an effective interaction distance of 112-m_ , _Phys. Rev. Lett._ 72 (1994) 1152.
* [148] B. Hubler, A. Cornaz and W. Kuendig, _Determination of the gravitational constant with a lake experiment: New constraints for non-Newtonian gravity_ , _Phys. Rev. D_ 51 (1995) 4005.
* [149] J. Thomas, P. Kasameyer, O. Fackler, D. Felske, R. Harris, J. Kammeraad et al., _Testing the inverse-square law of gravity on a 465-m tower_ , _Phys. Rev. Lett._ 63 (1989) 1902.
* [150] J. Kammeraad, P. Kasameyer, O. Fackler, D. Felske, R. Harris, M. Millett et al., _New results from Nevada: A Test of Newton’s law using the BREN tower and a high density ground gravity survey_ , in _10th Moriond Workshop: New and Exotic Phenomena_ , pp. 245–254, 1990.
* [151] A.J. Romaides, R.W. Sands, E. Fischbach and C.L. Talmadge, _Final results from the WABG tower gravity experiment_ , _Phys. Rev. D_ 55 (1997) 4532.
* [152] D.E. Smith, D.C. Christodoulidis, R. Kolenkiewicz, P.J. Dunn, S.M. Klosko, M.H. Torrence et al., _A global geodetic reference frame from LAGEOS ranging (SL5.1AP)_ , _Journal of Geophysical Research: Solid Earth_ 90 (1985) 9221.
* [153] R.H. Rapp, _An estimate of equatorial gravity from terrestrial and satellite data_ , _Geophysical Research Letters_ 14 (1987) 730.
* [154] J. Dickey et al., _Lunar Laser Ranging: A Continuing Legacy of the Apollo Program_ , _Science_ 265 (1994) 482.
* [155] C. Talmadge, J. Berthias, R. Hellings and E. Standish, _Model Independent Constraints on Possible Modifications of Newtonian Gravity_ , _Phys. Rev. Lett._ 61 (1988) 1159.
* [156] E. Babichev, C. Deffayet and R. Ziour, _Recovering General Relativity from massive gravity_ , _Phys. Rev. Lett._ 103 (2009) 201102 [0907.4103].
* [157] E. Babichev, C. Deffayet and R. Ziour, _The Vainshtein mechanism in the Decoupling Limit of massive gravity_ , _JHEP_ 05 (2009) 098 [0901.0393].
* [158] E. Babichev, C. Deffayet and R. Ziour, _The Recovery of General Relativity in massive gravity via the Vainshtein mechanism_ , _Phys. Rev. D_ 82 (2010) 104008 [1007.4506].
* [159] E. Babichev and C. Deffayet, _An introduction to the Vainshtein mechanism_ , _Class.Quant.Grav._ 30 (2013) 184001 [1304.7240].
* [160] M.S. Volkov, _Hairy black holes in the ghost-free bigravity theory_ , _Phys.Rev._ D85 (2012) 124043 [1202.6682].
* [161] T. Hiramatsu, W. Hu, K. Koyama and F. Schmidt, _Equivalence Principle Violation in Vainshtein Screened Two-Body Systems_ , _Phys. Rev. D_ 87 (2013) 063525 [1209.3364].
* [162] J.K. Bloomfield, C. Burrage and A.-C. Davis, _Shape dependence of Vainshtein screening_ , _Phys. Rev. D_ 91 (2015) 083510 [1408.4759].
* [163] H. Geng and A. Karch, _Massive islands_ , _JHEP_ 09 (2020) 121 [2006.02438].
* [164] G. ’t Hooft, _Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking_ , _NATO Sci. Ser. B_ 59 (1980) 135.
* [165] A. Lewis, _GetDist: a Python package for analysing Monte Carlo samples_ , 1910.13970.
* [166] M. Platscher and J. Smirnov, _Degravitation of the Cosmological Constant in Bigravity_ , _JCAP_ 03 (2017) 051 [1611.09385].
|
# Magnetic white dwarfs in post-common-envelope binaries
S. G. Parsons1, B. T. Gänsicke2, M. R. Schreiber3,4, T. R. Marsh2, R. P.
Ashley2, E. Breedt5, S. P. Littlefair1 and H. Meusinger6,7
1 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK
2 Department of Physics, University of Warwick, Coventry CV4 7AL, UK
3 Departamento de Física, Universidad Técnica Santa María, Avenida España
1680, Valparaíso, Chile
4 Millennium Nucleus for Planet Formation (NPF), Valparaíso, Chile
5 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
6 Thüringer Landessternwarte, Sternwarte 5, 07778, Tautenburg, Germany
7 Universität Leipzig, Fakultät für Physik und Geowissenschaften, Linnestraße
5, 04103, Leipzig, Germany<EMAIL_ADDRESS>
(Accepted 2021 January 27. Received 2021 January 27; in original form 2020
December 17)
###### Abstract
Magnitude-limited samples have shown that 20-25 per cent of cataclysmic
variables contain white dwarfs with magnetic fields of Mega Gauss strength, in
stark contrast to the approximately 5 per cent of single white dwarfs with
similar magnetic field strengths. Moreover, the lack of identifiable
progenitor systems for magnetic cataclysmic variables leads to considerable
challenges when trying to understand how these systems form and evolve. Here
we present a sample of six magnetic white dwarfs in detached binaries with
low-mass stellar companions where we have constrained the stellar and binary
parameters including, for the first time, reliable mass estimates for these
magnetic white dwarfs. We find that they are systematically more massive than
non-magnetic white dwarfs in detached binaries. These magnetic white dwarfs
generally have cooling ages of more than 1 Gyr and reside in systems that are
very close to Roche-lobe filling. Our findings are more consistent with these
systems being temporarily detached cataclysmic variables, rather than pre-
cataclysmic binaries, but we cannot rule out the latter possibility. We find
that these systems can display unusual asymmetric light curves that may offer
a way to identify them in larger numbers in future. Seven new candidate
magnetic white dwarf systems are also presented, three of which have
asymmetric light curves. Finally, we note that several newly identified
magnetic systems have archival spectra where there is no clear evidence of
magnetism, meaning that these binaries have been previously missed.
Nevertheless, there remains a clear lack of younger detached magnetic white
dwarf systems.
###### keywords:
binaries: close – stars: white dwarfs – stars: late-type – stars: magnetic
fields — stars: novae, cataclysmic variables
††pubyear: 2021††pagerange: Magnetic white dwarfs in post-common-envelope
binaries–9
## 1 Introduction
Magnitude-limited surveys have revealed that approximately five per cent of
isolated white dwarfs possess megagauss magnetic fields (Kepler et al., 2013),
generally identified spectroscopically via the Zeeman splitting of their
absorption lines. The magnetic incidence is slightly higher in volume limited
samples (Tremblay et al., 2020; McCleery et al., 2020) and polarimetric
observations suggest that almost one in five white dwarfs could possess at
least kilogauss magnetic fields (Landstreet & Bagnulo, 2019; Bagnulo &
Landstreet, 2020).
A higher magnetic incidence is seen among the white dwarfs in mass
transferring cataclysmic variable (CVs). 20-25 per cent of these systems
appear to host a magnetic white dwarf (Ferrario et al., 2015), but the true
incidence could be far higher, with magnetic systems making up 36 per cent of
CVs within 150 pc (Pala et al., 2020). However, despite being the direct
progenitors of CVs, the incidence of magnetism among detached white dwarf plus
main-sequence star binaries is extremely low (Liebert et al., 2005; Liebert et
al., 2015), with only a handful of detached magnetic systems serendipitously
identified (e.g. Reimers et al., 1999) among the hundreds of known detached
post-common-envelope binaries (PCEBs) (Schreiber et al., 2010; Rebassa-
Mansergas et al., 2016). Moreover, all of these detached magnetic white dwarf
binaries contain cool ($T_{\mathrm{eff}}\lesssim 10,000$ K), hence old, white
dwarfs accreting material from the wind of their main-sequence companions and
so the progenitors of magnetic CVs remain elusive.
This issue was addressed by Tout et al. (2008), who developed the theory that
the magnetic fields of white dwarfs are generated via a dynamo operating
during a common-envelope phase. In this scenario all magnetic white dwarfs
would have originally been part of binary systems which were close enough that
once the progenitor of the white dwarf evolved off the main-sequence it
engulfed a companion star. Tout et al. (2008) hypothesised that the closer the
two stars got within the common-envelope, the stronger the resulting magnetic
field of the white dwarf would be, with the highest field strengths resulting
from systems that merge during the common envelope, creating isolated, high-
field magnetic white dwarfs. This idea also means that those binaries that
survive the common-envelope phase with a magnetic white dwarf must be very
close and would rapidly evolve into CVs, spending very little time as detached
pre-CVs.
However, recently Belloni & Schreiber (2020) demonstrated that this model
cannot be correct. Using population synthesis they showed that this process
would lead to an extremely high magnetic incidence among white dwarfs in close
binaries, in fact the overwhelming majority of these binaries should be
strongly magnetic. The predicted magnetic field strengths were also too low
compared with those measured for white dwarfs in CVs, with this process only
able to create white dwarfs with field strengths of around 5 MG, compared to
observations of CVs with typical magnetic field strengths of 7-230 MG
(Ferrario et al., 2015). Moreover, this model still produces a substantial
number of hot (young) magnetic white dwarfs in close detached binaries, a
population that is completely absent among the observed samples. It therefore
remains unclear how magnetic CVs are created and why there is an absence of
young magnetic white dwarfs in detached pre-CVs.
The small number of old magnetic white dwarfs in detached binaries represent a
crucial population for understanding how magnetic CVs are formed and evolve.
If they are genuinely pre-CV systems (i.e. they have been detached since the
end of the common envelope phase) then this would lend support to the idea
that magnetic CVs descent from detached magnetic PCEBs and these systems would
be pre-polars (PREPs, Schwope et al. 2009). On the other hand, if they are
magnetic CVs that have temporarily detached (so-called low accretion rate
polars, LARPs, Schwope et al. 2002), then this would imply a complete lack of
magnetic white dwarfs in pre-CVs, meaning that the magnetic fields of the
white dwarfs in CVs would need to be generated during the CV phase. However,
two factors have prevented a detailed investigation of this population,
firstly, only ten systems are currently known (Ferrario et al., 2015) which
limits any conclusions that can be drawn since the sample is subject to heavy
selection effects. Secondly, the stellar and binary parameters of these
systems are poorly constrained, since these binaries are single-lined and the
spectra often heavily contaminated by strong cyclotron emission, making direct
measurements difficult. Precise measurements of the fundamental parameters of
these binaries, such as the white dwarf masses and Roche-lobe filling factors,
would be a powerful result that could distinguish between the pre-CV or
temporarily detached CV scenarios.
In this paper we identify four new magnetic white dwarfs in detached binaries.
We analyse these binaries in detail, as well as two previously identified
detached magnetic white dwarf binaries, to precisely determine the stellar and
binary parameters for all six systems. We then discuss what these measurements
imply for the evolutionary history of these systems.
## 2 Target selection
The small number of previously discovered magnetic white dwarfs in detached
binaries with low mass stellar companions were predominantly identified via
cyclotron emission lines in their optical spectra, which mark them out from
the otherwise similar spectra of non-magnetic white dwarfs in detached PCEBs.
While these cyclotron lines allow a confident classification of a system as
possessing a magnetic white dwarf, it is likely that not all magnetic systems
exhibit strong cyclotron lines at optical wavelengths, as the strength of the
cyclotron lines is related to the accretion rate of material onto the white
dwarf. In these detached binaries it is thought that the accretion occurs via
the white dwarf capturing material from the wind of the main-sequence star.
This makes the strength of the emission lines sensitive to the binary
separation. A wider orbit will decrease the accretion rate, making the
cyclotron lines weaker and possibly undetectable. Moreover, depending upon the
orientation of the binary and the location of the magnetic poles on the white
dwarf, the strength of the cyclotron lines can vary substantially over the
binary orbit, potentially even disappearing entirely if the magnetic poles
rotate out of view (e.g. Reimers et al., 1999; Schmidt et al., 2005a).
Finally, most known magnetic white dwarfs in detached binaries have field
strengths of several tens of megagauss, where the cyclotron lines appear in
the optical. However, lower magnetic field strengths will shift these lines
towards longer wavelengths, meaning that signs of magnetism may no longer be
evident in the optical spectrum. This is the case for the eclipsing magnetic
white dwarf in SDSS J030308.35+005444.1, which possesses a magnetic field with
a strength of only 8 MG (Parsons et al., 2013). In this case the cyclotron
lines appear at infrared wavelengths and the optical spectrum shows no clear
indication of magnetism (Pyrzas et al., 2009). Therefore, it is possible to
miss magnetic white dwarf systems if only a single low resolution optical
spectrum is available.
While the optical spectrum of SDSS J030308.35+005444.1 shows no cyclotron
lines, it does hint that there is something unusual about the white dwarf. The
temperature of this white dwarf is thought to be around 9 000 K (Pyrzas et
al., 2009; Parsons et al., 2013). A non-magnetic white dwarf of this
temperature should possess strong Balmer absorption lines that would be easily
detectable in the Sloan Digital Sky Survey (SDSS) spectrum of this source,
even after accounting for the dilution from the M dwarf companion. However,
there are no obvious Balmer lines in the SDSS spectrum and the white dwarf was
classified as a featureless DC white dwarf (Rebassa-Mansergas et al., 2016).
High resolution data revealed that the lack of Balmer absorption lines is due
to a combination of Zeeman splitting and additional Balmer emission from the
white dwarf (Parsons et al., 2013), hence magnetism is clearly the cause of
the lack of Balmer absorption lines in the SDSS spectrum of this system. Apart
from cyclotron emission lines, very few white dwarfs in these systems show
(Zeeman split) Balmer absorption lines. Therefore, it is worth investigating
other systems previously classified as DC+dM binaries for signs of magnetism.
On this premise, the sample presented in this paper covers the majority of
systems classified as close DC+dM binaries in the SDSS white dwarf main-
sequence binary catalogue of Rebassa-Mansergas et al. (2016). Our sample also
contains two previously known, but poorly studied, detached magnetic white
dwarf binaries (IL Leo and HS 0922+1333), which were included in order to
understand the general properties of this population. We also included two
systems with unusual Catalina Real Time Transient Survey (CRTS) light curves
(SDSS J0750+4943 and SDSS J2229+1853), where magnetism was suspected as the
cause of the highly asymmetric shape of the light curves (Parsons et al.,
2015). Since the majority of our observations were obtained in the southern
hemisphere, we were unable to observe northern DC+dM binaries in the Rebassa-
Mansergas et al. (2016) catalogue, therefore our sample is not a complete
study of all known DC+dM binaries. In total this paper presents new
spectroscopic data for 11 systems.
Finally, seven new candidate detached magnetic white dwarf binaries were
discovered during a search for unusual quasars in Kohonen maps of SDSS spectra
(Meusinger et al., 2012; Meusinger et al., 2016). These systems are discussed
in Section 6.
## 3 Observations and their reduction
A full list of our observations is given in Table 1.
Table 1: Journal of spectroscopic observations. For X-shooter observations we list the number of spectra obtained and exposure times in the UVB, VIS and NIR arms. Telescope/ | Date | No. of | Exopsure | Transparency | seeing
---|---|---|---|---|---
Instrument | | spectra | times (s) | | (arcsec)
SDSS J022503.02+005456.2: | | |
VLT/X-shooter | 31 Aug 2017 | 5/8/13 | 600/350/240 | Clear | 0.5
VLT/X-shooter | 16 Sep 2017 | 5/8/13 | 600/350/240 | Clear | 0.6
SDSS J075015.11+494333.2: | | |
INT/IDS | 11 Feb 2015 | 13 | 900 | Thin clouds | 1.0
SDSS J084841.17+232051.7: | | |
VLT/X-shooter | 15 Apr 2017 | 6/11/31 | 600/300/120 | Clear | 0.5
VLT/X-shooter | 18 Apr 2018 | 15/27/75 | 600/300/120 | Clear | 0.9
SDSS J085336.03+072033.5: | | |
VLT/X-shooter | 17 Apr 2017 | 10/18/30 | 600/300/200 | Clear | 0.8
HS 0922+1333: | | |
VLT/X-shooter | 15 Apr 2017 | 10/18/30 | 600/300/200 | Clear | 0.5
IL Leo: | | |
VLT/X-shooter | 16 Apr 2017 | 20/18/22 | 300/334/300 | Clear | 0.6
SDSS J114030.06+154231.5: | | |
VLT/X-shooter | 16 Apr 2017 | 10/18/30 | 600/300/200 | Clear | 0.6
VLT/X-shooter | 17 Apr 2017 | 5/9/15 | 600/300/200 | Clear | 0.8
VLT/X-shooter | 15 May 2018 | 5/9/15 | 600/300/200 | Clear | 0.5
SDSS J131632.04-003758.0: | | |
VLT/X-shooter | 16 Apr 2017 | 5/9/15 | 600/300/200 | Clear | 0.6
SDSS J145238.12+204511.9: | | |
VLT/X-shooter | 16 Apr 2017 | 10/18/30 | 600/300/200 | Clear | 0.6
SDSS J220848.32+003704.6: | | |
VLT/X-shooter | 30 Aug 2017 | 5/8/13 | 600/350/240 | Thin clouds | 0.7
VLT/X-shooter | 4 Sep 2017 | 5/8/13 | 600/350/240 | Clear | 0.6
SDSS J222918.95+185340.2: | | |
INT/IDS | 25 Jul 2015 | 3 | 900 | Clear | 1.2
VLT/X-shooter | 30 Aug 2017 | 10/18/44 | 600/300/135 | Thin clouds | 0.6
VLT/X-shooter | 31 Aug 2017 | 10/18/44 | 600/300/135 | Clear | 0.5
### 3.1 VLT/X-shooter spectroscopy
The bulk of our spectroscopic observations were performed with the medium
resolution echelle spectrograph X-shooter (Vernet et al., 2011), which is
mounted at the Cassegrain focus of the VLT-UT2 at Paranal, Chile. X-shooter
covers the spectral range from the atmospheric cutoff in the UV to the near-
infrared K band in three separate arms, known as the UVB (0.30$-$0.56
microns), VIS (0.56$-$1.01 microns) and NIR (1.01$-$2.40 microns). Separate
slit widths can be set for each arm and our observations were performed with
slit widths of 1.0, 0.9 and 0.9 arcsec in the UVB, VIS and NIR arms
respectively. We also binned the detector in the VIS arm by a factor of 2 in
the spatial direction, while the UVB arm was binned by a factor of 2 in both
the spatial and dispersion directions This results in a resolution of
R$\sim$5000 in the UVB and NIR arms and R$\simeq$8000 in the VIS arm. All of
the data were reduced using the standard X-shooter pipeline release (version
3.5.0) within esoreflex.
The VIS arm spectra were telluric corrected using molecfit (Smette et al.,
2015; Kausch et al., 2015). Since our observations were performed in stare
mode (in order to minimise overheads in the optical) the NIR spectra have poor
sky subtraction and therefore no telluric correction was attempted for these
data. The fit to the telluric features was also used to correct for small
(typically $\sim$1$\mathrm{km\,s^{-1}}$) systemic velocity offsets in the data
via the method described in Parsons et al. (2017a). All spectra were then
placed on a heliocentric wavelength scale.
### 3.2 INT/IDS spectroscopy
SDSS J0750+4943 and SDSS J2229+1853 were both observed with the Intermediate
Dispersion Spectrograph (IDS) mounted on the Cassegrain focus of the 2.5m
Isaac Newton Telescope (INT) at La Palma, Spain. Both objects were observed
using the R300V grating centred at 6460Å and a slit width of 1″, resulting in
a dispersion of 1.9Å/pixel over a wavelength range of 4000-8700Å.
The data were reduced using the pamela111pamela is available through the
starlink software package (Currie et al., 2014) and molly222molly is available
from http://cygnus.astro.warwick.ac.uk/phsaap/software/ packages. The spectra
were first bias subtracted and flat-fielded, then optimally extracted. The
wavelength calibration was performed using CuAr+CuNe lamps exposed regularly
throughout each night. The observations of spectral standard stars were used
to flux calibrate our spectra, but no telluric correction was applied to the
IDS spectra.
## 4 Spectroscopic analysis
The sample of systems analysed in this paper can be split into two groups,
those containing a magnetic white dwarf and those with a non-magnetic white
dwarf. Unless otherwise stated, the spectra of the M dwarfs in all the systems
presented in this paper (both magnetic and non-magnetic) were analysed
identically. The normalised averaged spectra for all of the binaries analysed
in this paper are shown in Figures 1 and 2 for the non-magnetic and magnetic
systems respectively.
Figure 1: Normalised averaged spectra of the non-magnetic white dwarf plus M
dwarf binaries analysed in this paper. The spectra are completely dominated by
the M dwarf components, the white dwarf is only evident from a small excess at
the shortest wavelengths. Emission lines from the M dwarf are seen in all
systems indicating that these are active stars (since irradiation effects are
negligible in these systems). Figure 2: Normalised averaged spectra of the
magnetic white dwarf plus M dwarf binaries analysed in this paper. The M dwarf
components are visible in all the systems apart from IL Leo. Cyclotron
emission lines are seen in all systems, although the strength of these lines
varies substantially among systems. Zeeman-split hydrogen Balmer absorption
lines from the white dwarf are evident in the spectrum of IL Leo.
### 4.1 Radial velocity measurements
For all but one system analysed in this paper there are no clean features from
the white dwarf to help constrain its radial velocity, the exception being
SDSS J0848+2320, which contains a DZ white dwarf with clear calcium H/K
absorption lines (this white dwarf was originally classified as a DC white
dwarf based on the SDSS spectrum but our higher resolution data revealed that
it is actually a DZ). Therefore, in order to constrain the binary and physical
parameters, radial velocity measurements were obtained for only the M dwarf
components in these systems, except for SDSS J0848+2320, where we used
features from both stars.
Radial velocities were measured in each spectrum by fitting the sodium
absorption doublet from the M dwarf at $\simeq$8200Å. The lines were fitted
with a combination of a straight line and two Gaussian components. In order to
improve accuracy, we fitted all spectra of the same system simultaneously,
keeping the width and depth of the Gaussian components the same for all
spectra and only allowed the velocity of the Gaussians to vary between spectra
(note that we specifically targeted the sodium absorption doublet because it
is free from any cyclotron emission in all of the magnetic systems). The best
fitting values and their uncertainties were found using the Levenberg-
Marquardt method (Press et al., 1986) and are listed in Table 5. The same
method was used to fit the Ca ii absorption line from the white dwarf in SDSS
J0848+2320, although in this case a single Gaussian component was used. These
values are listed in Table 9.
Orbital periods, radial velocity semi-amplitudes of the main-sequence stars
and systemic velocities were determined by fitting the velocity measurements
with a constant plus sine wave over a range of periods and computing the
$\chi^{2}$ of the resulting fit. We also included any archival velocity
measurements from Rebassa-Mansergas et al. (2016). The orbital periods for
most of the systems analysed in this paper have been previously measured, but
our additional high-precision velocity measurements greatly improve the
precision of the ephemerides as well as radial velocity semi-amplitude
measurements. These updated values are listed in Table 2. Radial velocity
curves are plotted in Figure 3.
Figure 3: Radial velocity measurements and fits (red lines). Black points are
new measurements for the M dwarfs in these systems, previous measurements are
shown in grey (taken from Rebassa-Mansergas et al. 2016). Systems containing
magnetic white dwarfs are highlighted with an asterisk (∗). For SDSS
J0848+2320 we also show the velocity measurements for the white dwarf in blue
and the fit to these (red, dashed line).
#### 4.1.1 Radial velocity measurements for IL Leo
Figure 4: Trailed spectra of the H$\alpha$ emission line in the magnetic
system IL Leo with time running upwards. The left-hand panel shows the
VLT/Xshooter spectra. The centre panel shows our best fit model to the
irradiation-driven emission line from the M dwarf component. The right-hand
panel shows the residuals of the fit.
There are no absorption features from the M dwarf in the magnetic system IL
Leo, likely due to its faintness and the strong cyclotron lines in this system
(see Figure 2). Therefore, a different method was needed to constrain the
physical and binary parameters of this system. The only feature visible from
the M dwarf in this system is a H$\alpha$ emission line, which varies in
strength over the orbit, peaking in strength at an orbital phase of 0.5,
implying that it is caused by irradiation from the white dwarf. A reliable fit
to this emission line in individual spectra is difficult due to the weakness
of this line. Therefore, we decided to simultaneously fit all the spectra of
IL Leo around the H$\alpha$ line with a model in which the position of the
emission line changes sinusoidally according to
$\gamma+K_{\mathrm{em}}\sin{(2\pi\phi)}$, where $\gamma$ is the systemic
velocity, $K_{\mathrm{em}}$ is the radial velocity semi-amplitude of the
emission line and $\phi$ is the orbital phase. The strength of the line also
varies according to ($1-\cos{\phi})/2$ to simulate the behaviour of the line
due to irradiation (see Parsons et al. 2017b for a detailed description of the
model). The best fit model is shown in Figure 4 and yields a radial velocity
semi-amplitude of $K_{\mathrm{em}}=384.2\pm 4.5$ $\mathrm{km\,s^{-1}}$.
However, since this line is irradiation-driven, the emission is concentrated
on the inner hemisphere of the M dwarf. As such, the velocity of this emission
line does not track the centre-of-mass of the star and so this is only a lower
limit on the true radial velocity semi-amplitude of the star.
IL Leo is the only system to show clear Balmer absorption lines from the white
dwarf in its spectrum (see Figure 2). However, these lines can not be used to
measure the radial velocity of the white dwarf due to a combination of the low
signal-to-noise ratio of the spectra at these wavelengths (the lines are only
apparent when all the spectra of this system are combined) as well as the fact
that the lines are affected by the magnetic field of the white dwarf. The
lines are clearly Zeeman split and, since the field strength varies over the
surface of the white dwarf, the amount each component is shifted will vary
throughout the orbit, as different regions of the white dwarf rotate into
view, making any velocity measurements unreliable.
### 4.2 Rotational broadening measurements
The rotational broadening of the M dwarf in these close binaries is sensitive
to the binary and stellar parameters and thus it can be used to place useful
constraints on the stellar masses for example (see Section 5). Therefore, we
tried to measure the rotational broadening in all our systems. Unfortunately
there are no absorption features from the M dwarf in IL Leo so it was not
possible to measure the rotational broadening in this system. Moreover, SDSS
J0750+4943 only has data from the INT/IDS. The resolution of these data is
insufficient to measure a reliable rotational broadening, so this system also
lacks a measurement. Rotational broadening measurements were performed for all
the other systems we observed.
The rotational broadening was measured by artificially broadening the lines of
template stars to fit the observed line profiles of our systems, taking into
account any additional smearing of the lines from the velocity shift of the
main-sequence star during an exposure. Template M dwarf spectra were taken
from Parsons et al. (2017a). All of these template stars were observed with
X-shooter using an identical instrumental setup to the binaries presented in
this paper. Note that we applied a high-pass filter to both the observed and
broadened template spectra before comparing them in order to prevent the
continuum dominating the rotational broadening determination. We fitted just
the wavelength range around the sodium 8200 Å absorption doublet, since this
wavelength range is free from any cyclotron emission in all the magnetic
systems. The best fit rotational broadening measurements for our systems are
listed in Table 2. We reached a precision of better than 2.5
$\mathrm{km\,s^{-1}}$ in all our systems, although in the case of the longer
period systems (where the rotational broadening is much smaller) this
precision still results in a large fractional uncertainty.
## 5 Stellar and binary parameters
In this section we summarise the methods used to constrain the stellar and
binary parameters in both the magnetic and non-magnetic systems. All stellar
and binary parameters for our systems are given in Table 2, along with
literature values for similar systems that have been studied previously.
### 5.1 M dwarf masses and radii
We estimated the masses of the M dwarfs in all our systems using the mass-
luminosity relation for late-type stars from Mann et al. (2019). This relates
the masses of M dwarfs to their absolute $K_{S}$-band magnitudes. All of our
systems have good Gaia early DR3 parallaxes (Gaia Collaboration et al., 2020)
and 2MASS $K_{S}$-band magnitudes (Skrutskie et al., 2006), with the exception
of IL Leo, which lacks a $K_{S}$-band measurement due to its faintness.
Moreover, IL Leo is the only system in which the M dwarf does not
overwhelmingly dominate the $K_{S}$-band flux. As such, even if the system had
a $K_{S}$-band measurement the mass-luminosity relation would be difficult to
use in this case since the white dwarf contribution would need to be
subtracted beforehand. Therefore, the parameters of IL Leo were constrained
using a different technique that is summarised in Section 7.2.4. For all other
systems the $K_{S}$-band flux is completely dominated by the M dwarf. The
field strengths of the white dwarfs in all the magnetic systems we observed
are at least 60 MG (except IL Leo) and thus the cyclotron emission will be
limited to wavelengths shorter than 1.8 microns ($H$ band).
The mass-luminosity relation from Mann et al. (2019) yields masses with
statistical uncertainties of 2-3 per cent. We used the distance estimates to
our systems from Bailer-Jones et al. (2018) and propagated the uncertainties
in both the distances and 2MASS $K_{S}$-band magnitudes when calculating the M
dwarf masses. Typical uncertainties in the M dwarf masses are $\sim$5 per
cent. However, we note that the Mann et al. (2019) relation is based on M
dwarfs in wide, astrometric binaries and may not be as suitable for the
rapidly rotating and Roche-distorted M dwarfs in the close systems studied in
this paper. Indeed Parsons et al. (2018) found that this relation tended to
slightly underestimate the masses of M dwarfs in close binaries with white
dwarfs possibly due to the enhanced number of starspots expected on the
tidally locked and hence rapidly rotating M dwarfs in close binaries. Parsons
et al. (2018) found that the masses estimated from the mass-luminosity
relation could be up to 5 per cent underestimated. We therefore set a lower
limit of 5 per cent on the uncertainties of the M dwarf masses in this paper
to account for this systematic uncertainty. The effects of metallicity are
expected to be negligible for values in the solar neighbourhood (Mann et al.,
2019), hence we assume solar metallicity for all our targets.
We estimated the radii of the M dwarfs using a semi-empirical mass-radius
relationship for M dwarfs, based on a combination of a large dataset of M
dwarf radius measurements (Morrell & Naylor, 2019) and the mass-luminosity
relation from Mann et al. (2019). This semi-empirical mass-radius relationship
will be presented in more detail in a future publication (Brown et al., in
prep) and leads to more accurate radius predictions compared to theoretical
mass-radius relationships, which tend to under-predict the radii of low-mass
stars by 5-10 per cent (López-Morales & Ribas, 2005; Parsons et al., 2018).
Any effects from metallicity are likely to be at a level smaller than the 5
per cent systematic error we placed on the M dwarf masses (which are
propagated to the radius estimates).
### 5.2 White dwarf masses
The mass of the white dwarfs in these binary systems can be constrained using
the rotational broadening measurements ($V_{\mathrm{rot}}\sin{i}$) via
$V_{\mathrm{rot}}\sin{i}=K_{\mathrm{MS}}(1+q)\frac{R_{\mathrm{MS}}}{a},$ (1)
where $K_{\mathrm{MS}}$ is the radial velocity semi-amplitude of the M dwarf,
$q=M_{\mathrm{WD}}/M_{\mathrm{MS}}$, the binary mass ratio, $R_{\mathrm{MS}}$
is the radius of the M dwarf and $a$ is the semi-major axis (Marsh et al.,
1994). This relation is only valid if the M dwarf is synchronously rotating,
which is expected to be the case in these extremely close binary systems.
Table 2: Stellar and binary parameters for PCEBs containing cool white dwarfs. Systems analysed in this paper are highlighted in italic. M dwarf masses and Roche-lobe filling factors for the systems analysed in this paper were determined assuming that the star is in thermal equilibrium, hence the uncertainties on these values are purely statistical. T0 corresponds to the inferior conjunction of the main-sequence star. $G$ is the Gaia magnitude, which (along with the parallax measurements) are taken from early DR3. RLFF is the Roche-lobe filling factor of the M dwarf component. References: (1) Parsons et al. (2012c), (2) this paper, (3) Zorotovic et al. (2016), (4) Parsons et al. (2013), (5) Schmidt et al. (2005a), (6) Reimers & Hagen (2000), (7) Tovmassian & Zharikov (2007), (8) Reimers et al. (1999), (9) Schwarz et al. (2001), (10) Schmidt et al. (2007), (11) Schwope et al. (2009), (12) Schmidt et al. (2005b), (13) Burleigh et al. (2006), (14) Farihi et al. (2008), (15) Breedt et al. (2012), (16) Szkody et al. (2003), (17) Southworth et al. (2015), (18) Szkody et al. (2008), (19) Kafka et al. (2010). Name | $G$ | $\pi$ | Porb | T0 | KMS | $\gamma_{\mathrm{MS}}$ | $v\sin{i}$ | Spectral | MMS | MWD | Teff,WD | $B$111Parentheses denote secondary pole | RLFF | Reference
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (mag) | (mas) | (days) | (HJD) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) | type | ($\mathrm{M_{\odot}}$) | ($\mathrm{M_{\odot}}$) | (K) | (MG) | |
DC/Z+dM binaries: | | | | | | | | | | |
SDSS J0138-0016 | 16.43 | $20.36\pm 0.07$ | 0.07276491(2) | 2455867.507405(6) | $346.7\pm 2.3$ | $84.5\pm 1.2$ | - | M5.0 | $0.132\pm 0.003$ | $0.53\pm 0.01$ | $3570\pm 100$ | - | $0.91\pm 0.03$ | (1)
SDSS J0225+0054 | 18.99 | $4.49\pm 0.27$ | 0.9210733(2) | 2456407.7947(51) | $81.2\pm 2.0$ | $-4.0\pm 1.3$ | $5.5\pm 2.3$ | M4.5 | $0.166\pm 0.015$ | $0.55\pm 0.40$ | $6600\pm 300$ | - | $0.20\pm 0.02$ | (2)
SDSS J0848+2320 | 17.05 | $3.61\pm 0.10$ | 0.3717195(1) | 2458043.67163(5) | $167.0\pm 0.5$ | $29.5\pm 0.5$ | $69.5\pm 2.5$ | M3.0 | $0.430\pm 0.022$ | $0.44\pm 0.02$ | $7600\pm 500$ | - | $0.58\pm 0.02$ | (2)
SDSS J1140+1542 | 16.90 | $3.03\pm 0.11$ | 3.11329(3) | 2455807.491(19) | $125.2\pm 0.9$ | $-48.5\pm 0.5$ | $7.3\pm 1.5$ | M2.5 | $0.475\pm 0.024$ | $>$1.22 | $8900\pm 900$ | - | $0.15\pm 0.01$ | (2)
SDSS J1316$-$0037 | 17.66 | $5.42\pm 0.14$ | 0.4027340(2) | 2456241.50583(95) | $186.5\pm 1.1$ | $-15.5\pm 2.0$ | $26.5\pm 2.0$ | M4.0 | $0.202\pm 0.010$ | $0.64\pm 0.11$ | $7300\pm 200$ | - | $0.38\pm 0.02$ | (2)
SDSS J2208+0037 | 18.93 | $4.45\pm 0.34$ | 0.10337209(1) | 2457588.67776(4) | $233.9\pm 0.6$ | $11.3\pm 0.5$ | $68.2\pm 2.2$ | M5.0 | $0.150\pm 0.032$ | $0.46\pm 0.24$ | $6100\pm 400$ | - | $0.81\pm 0.17$ | (2,3)
Magnetic WD+dM binaries: | | | | | | | | | | |
SDSS J0303+0054 | 17.36 | $8.30\pm 0.10$ | 0.1344376668(1) | 2453991.617307(2) | $339.9\pm 0.3$ | $14.9\pm 0.2$ | $81.7\pm 1.1$ | M4.5 | 0.181-0.205 | 0.825-0.853 | $\sim$9150 | 8 | $0.84\pm 0.03$ | (4)
SDSS J0750+4943 | 16.08 | $4.66\pm 0.05$ | 0.17300847(5) | 2457065.49219(70) | $154.6\pm 3.9$ | $28.5\pm 4.3$ | - | M2.5 | $0.460\pm 0.023$ | $0.94\pm 0.12$ | $15000_{-1000}^{+1400}$ | 98 or 196 | $0.98\pm 0.02$ | (2)
SDSS J0837+3830222System shows some evidence of (episodic) Roche-lobe overflow | 18.86 | $0.98\pm 0.23$ | 0.1325 | - | - | - | - | M5.0 | $0.447\pm 0.083$ | - | - | 65 | $>$0.97 | (5)
SDSS J0853+0720 | 18.02 | $3.74\pm 0.15$ | 0.15021618(2) | 2456138.68614(20) | $236.3\pm 0.6$ | $-33.2\pm 0.5$ | $62.4\pm 2.3$ | M4.0 | $0.221\pm 0.016$ | $0.83\pm 0.15$ | $9000\pm 300$ | 84 | $0.82\pm 0.05$ | (2)
HS 0922+1333 | 16.46 | $6.06\pm 0.09$ | 0.1683125(5) | 2457859.61706(10) | $129.9\pm 2.3$ | $22.8\pm 2.4$ | $58.6\pm 2.2$ | M3.5 | $0.366\pm 0.018$ | $0.71\pm 0.07$ | $<$8500 | 66(81) | $0.95\pm 0.03$ | (2,6,7)
WX LMi | 16.34 | $10.31\pm 0.05$ | 0.1159229(13) | 2451157.6439(9) | - | - | - | M4.5 | $0.225\pm 0.003$ | - | - | 60(68) | $>$0.91 | (8,9)
IL Leo333Potential period bounce cataclysmic variable | 19.30 | $1.93\pm 0.23$ | 0.05709(18) | 2457860.61634(15) | $>$ $384.2\pm 4.5$ | $-43.2\pm 5.0$ | - | $>$M6 | $<$0.090 | $>$0.48 | $<$11000 | 42 | $>$0.95 | (2,10)
SDSS J1059+2727 | 20.30 | $1.30\pm 0.94$ | $>$0.125 | - | - | - | - | M4.0 | - | - | $<$8500 | 57 | - | (10)
SDSS J1206+5100 | 18.64 | $3.30\pm 0.16$ | 0.1576726(6) | 2458254.7030(5) | - | - | - | M3.5 | $0.208\pm 0.023$ | - | $\sim$9000 | 108 or 216 | $>$0.727 | (11)
SDSS J1212+0136222System shows some evidence of (episodic) Roche-lobe overflow,333Potential period bounce cataclysmic variable | 17.98 | $6.47\pm 0.13$ | 0.0614081(7) | 2453686.5276(1) | $>$ $355\pm 6$ | - | - | L5 | $<$0.080 | $>$0.41 | $\sim$9500 | 7 | $>$0.90 | (12,13,14)
SDSS J1250+1549333Potential period bounce cataclysmic variable | 18.20 | $7.58\pm 0.14$ | 0.05995(1) | 2455646.7056(1) | $>$ $360.9\pm 2.5$ | $12.8\pm 2.1$ | - | M8 | $<$0.085 | $>$0.42 | $\sim$10000 | 20 | $>$0.90 | (15)
PZ Vir | 20.48 | $1.78\pm 1.64$ | 0.11022(7) | - | - | - | - | M6.0 | - | - | - | 64 | - | (15,16)
SDSS J1452+2045 | 17.89 | $7.97\pm 0.12$ | 0.106265437(9) | 2456207.8708(1) | $362.0\pm 0.4$ | $-39.5\pm 0.5$ | $82.4\pm 1.5$ | M5.0 | $0.150\pm 0.008$ | $0.83\pm 0.08$ | $<$6500 | 300 | $0.84\pm 0.03$ | (2,3)
SDSS J1514+0744333Potential period bounce cataclysmic variable | 18.67 | $5.53\pm 0.18$ | 0.061610(1) | 2455646.89093(3) | $>$ $362.8\pm 0.7$ | $-28.0\pm 0.8$ | - | L3 | $<$0.085 | $>$0.44 | $\sim$10000 | 36 | $>$0.90 | (15)
MQ Dra | 17.34 | $5.51\pm 0.06$ | 0.182985(5) | 2454156.9138(1) | - | - | - | M5.0 | $0.238\pm 0.008$ | - | $\sim$8000 | 60 | $>$0.70 | (16,18)
SDSS J2048+0050222System shows some evidence of (episodic) Roche-lobe overflow | 18.27 | $1.66\pm 0.16$ | 0.175 | - | $110.4\pm 3.9$ | $-36.8\pm 1.5$ | - | M3.0 | $0.446\pm 0.044$ | - | - | 62 | $>$0.96 | (5,19)
SDSS J2229+1853 | 16.51 | $3.44\pm 0.06$ | 0.1891844(1) | 2457996.12995(3) | $203.1\pm 0.2$ | $12.9\pm 0.5$ | $105.5\pm 1.2$ | M3.0 | $0.460\pm 0.023$ | $0.76\pm 0.05$ | $<$8500 | 84 | $0.96\pm 0.03$ | (2)
With a measurement of the rotational broadening, radial velocity semi-
amplitude, mass and radius of the M dwarf then Equation 1 can be used to solve
for the mass of the white dwarf, since $a$ is only dependent upon the orbital
period ($P$) and stellar masses,
$a^{3}=\frac{G(M_{\mathrm{WD}}+M_{\mathrm{MS}})P^{2}}{4\pi^{2}},$ (2)
thus combining these two equations yields the white dwarf mass. Uncertainties
in all the parameters were propagated through these equations to determine the
statistical error on the white dwarf mass. The final error on the white dwarf
mass is strongly dependent on how precisely the rotational broadening can be
measured. For systems where the M dwarf is more slowly rotating this results
in substantial uncertainty on the white dwarf mass (e.g. for SDSS J0225+0054,
which has a period of 21.7 hours).
We used the stellar masses, orbital period and M dwarf radius to estimate the
Roche-lobe filling factor (RLFF) for each of our systems. These filling
factors (listed in Table 2) are given as the ratio of the radius of the M
dwarf to its volume-averaged Roche-lobe radius, as opposed to the linear RLFF
which is given by the distance from the centre of mass of the M dwarf to its
surface in the direction of the L1 point divided by the distance to the L1
point. Uncertainties in the stellar and binary parameters were propagated to
determine the error on the RLFF.
### 5.3 White dwarf temperatures
Figure 5: Left: The mass-luminosity relation for M dwarf stars using SDSS
$u$-band magnitudes. The sample of M dwarfs was taken from Morrell & Naylor
(2019). Masses were determined using the $K_{S}$-band mass-luminosity relation
from Mann et al. (2019). A fifth order polynomial fit is shown in red. The
shaded region marks the 16th to 84th percentile region of the measurements.
Right: The change in the $u$-band absolute magnitude of an M dwarf when a
white dwarf is placed next to it. Solid blue lines indicate white dwarfs with
different temperatures (labelled on the right) and surface gravities of
$\log{g}=8.0$, while the dashed blue lines are for white dwarfs with
$\log{g}=8.5$. The black points mark the values for the non-magnetic systems.
The magnetic system SDSS J0853+0720 is indicated by the red diamond.
Figure 6: Average spectra (black lines) of the non-magnetic systems in our
sample. For each system the best fit M dwarf template spectrum is shown in
green and a white dwarf model with the temperature determined from the
relation shown in Figure 5 and scaled to the distance of the system is shown
in red. The M dwarf subtracted spectrum is plotted below each spectrum with
the scaled white dwarf model over-plotted. The white dwarf models were not
fitted to the spectroscopic data, but the good agreement between the models
and the data supports the reliability of the method used to determine the
white dwarf temperatures. The overestimation of the white dwarf models at the
shortest wavelengths is likely due to slit losses in the X-shooter data.
The temperature of the white dwarf is a useful measurement to make, since it
helps constrain the age of the white dwarf and thus the system. However, this
is a difficult measurement to make because we specifically selected systems
containing featureless DC white dwarfs, since we suspected that these could be
hiding magnetic fields. This means that none of the white dwarfs in our sample
have clear white dwarf features that can be fitted to constrain their
temperature. Moreover, the vast majority of our systems lack any ultraviolet
data, preventing us from constraining the white dwarf temperatures at shorter
wavelengths. The situation is made even harder for the magnetic systems, since
the cyclotron emission dilutes and distorts any underlying emission from the
photosphere of the white dwarf.
The white dwarfs do contribute a small but detectable amount of optical flux
in these systems. Most of these systems were originally identified through
excess blue flux in their SDSS spectra (Rebassa-Mansergas et al., 2016) and so
we can use this to constrain the white dwarf temperatures. However, the M
dwarf contribution must be accounted for if accurate white dwarf temperatures
are to be obtained.
We constructed a mass-luminosity relation for M dwarf stars using SDSS
$u$-band data. The advantage of using the $u$-band is that there are
relatively few features in the spectra of M dwarfs in this wavelength range.
This mitigates the substantial scatter seen in the $V$-band luminosities of M
dwarfs caused by the large metallicity spread of field M dwarfs (Delfosse et
al., 2000). The disadvantage of using the $u$-band being the intrinsic
faintness of low-mass stars at these wavelengths. Fortunately the sample of M
dwarfs presented by Morrell & Naylor (2019) contains many bright, nearby
stars. Cross-matching this catalogue with SDSS gives 4564 M dwarfs with
reliable $u$-band magnitudes and parallaxes. Since all of these M dwarfs also
have $K_{S}$-band magnitudes we computed their masses using the Mann et al.
(2019) mass-luminosity relation and then constructed our own mass-luminosity
relation from the $u$-band data, which is shown in the left-hand panel of
Figure 5. The amount of scatter in this relation is similar to the
$K_{S}$-band mass-luminosity relation (Mann et al., 2019), although there are
relatively few measurements for stars below 0.15 $\mathrm{M_{\odot}}$ in the
$u$-band. A fifth order polynomial fit to this relation gives
$\displaystyle M_{u}$ $\displaystyle=$ $\displaystyle(26.52\pm
0.09)-(106.61\pm 1.29)M_{*}$ (3) $\displaystyle+(450.63\pm
6.86)M_{*}^{2}-(1058.07\pm 17.50)M_{*}^{3}$ $\displaystyle+(1255.07\pm
21.50)M_{*}^{4}-(603.48\pm 10.23)M_{*}^{5}$
where $M_{*}$ is the mass of the M dwarf, valid between 0.1
$\mathrm{M_{\odot}}$ and 0.65 $\mathrm{M_{\odot}}$.
We used this relation to estimate the M dwarf contribution to the $u$-band
fluxes in all the non-magnetic systems. The excess $u$-band flux was then
determined to be from the white dwarf. Since both the distance and white dwarf
mass are already known, the only other parameter that affects the $u$-band
luminosity of a white dwarf is its temperature, assuming a standard mass-
radius relationship (Fontaine et al., 2001) and a hydrogen atmosphere. The
effect of adding a white dwarf with a varying temperature is shown in the
right-hand panel of Figure 5, demonstrating that this is a particularly useful
technique for measuring the temperatures of very cool white dwarfs in binaries
with low-mass stars. In general this technique allows us to constrain the
white dwarf temperature to better than 500 K, although the uncertainty rises
as the M dwarf mass increases and this approach is not ideal for systems with
M dwarfs more massive than around 0.5 $\mathrm{M_{\odot}}$.
We tested the reliability of this method by scaling a white dwarf model
(Koester, 2010) to the distance of each system with the parameters determined
from the $u$-band fit and comparing this model to the X-shooter spectrum of
each system. The result of this is shown in Figure 6. For each system we plot
part of the X-shooter spectrum along with the best fit M dwarf template
spectrum (in green) and the scaled white dwarf model (in red). We also show
the M dwarf subtracted spectra with white dwarf models over-plotted. In each
case the predicted white dwarf flux matches the residual flux after the M
dwarf is subtracted, demonstrating that the white dwarf parameters estimated
from the $u$-band flux are reliable. Note that due to slit losses the white
dwarf models tend to over-predict the flux at the shortest wavelengths. This
imperfect calibration is why we have not attempted to constrain the white
dwarf temperatures from the X-shooter data directly.
We were able to use the same approach to measure the temperature of the
magnetic white dwarf in SDSS J0853+0720 because the SDSS data for this system
were obtained when the magnetic pole was not visible, although we note that
the tracks in the right-hand panel of Figure 5 are for non-magnetic white
dwarfs, which may not be appropriate for the magnetic white dwarf in SDSS
J0853+0720. The use of this method was not possible on any of the other
magnetic systems. In these cases we used the spectra taken when the cyclotron
lines were at their weakest and subtracted off the M dwarf component (by
scaling the best fit template spectrum used in the rotational broadening
measurements). We then scaled white dwarf model spectra (Koester, 2010) to the
distance of the system, keeping the mass fixed at the measured value and using
a mass-radius relationship (Fontaine et al., 2001). We varied the temperature
of the white dwarf until the flux was clearly over predicted between the
cyclotron lines (see Figure 7 for example) and used this to set an upper limit
on the white dwarf temperature. However, since the white dwarf models we used
are for non-magnetic white dwarfs and the M dwarf components are never
perfectly removed, the temperature estimate for all the magnetic systems
should be taken with some caution.
Figure 7: Spectrum of the magnetic system SDSS J2229+1853 taken at minimum
light (when the cyclotron lines are at their weakest) with the M dwarf
component removed. Overplotted are non-magnetic white dwarf models with
varying temperatures (8000 K, 7000 K and 6000 K) and the same mass as measured
for the white dwarf, scaled to the distance of the system (assuming a standard
mass-radius relationship, Fontaine et al. 2001). We place an upper limit on
the white dwarf temperature, above which the continuum flux (between cyclotron
lines) would be overestimated. Figure 8: Phase-folded trailed spectra of SDSS
J0750+4943 with the M dwarf component removed. Figure 9: Phase-folded trailed
spectra of SDSS J0853+0720 with the M dwarf component removed. Figure 10:
Phase-folded trailed spectra of HS 0922+1333 with the M dwarf component
removed. Figure 11: Phase-folded trailed spectra of IL Leo. Since the M dwarf
component is so weak in the optical there is no need to remove it in this
case. Figure 12: Phase-folded trailed spectra of SDSS J1452+2045 with the M
dwarf component removed. Figure 13: Phase-folded trailed spectra of SDSS
J2229+1853 with the M dwarf component removed.
### 5.4 White dwarf magnetic field strengths
We estimate the magnetic field strengths of the white dwarfs in all the
magnetic systems using the wavelengths of the cyclotron lines. The wavelength,
$\lambda$, of the cyclotron harmonic number $n$ is related to the magnetic
field $B$ via
$\lambda_{n}=\left(\frac{10700}{n}\right)\left(\frac{10^{8}}{B}\right)\,$\AA$,$
(4)
where $B$ is measured in Gauss. This equation is valid for the plasma
temperatures expected for white dwarf atmospheres (Wickramasinghe & Ferrario,
2000). In general the strongest cyclotron lines are easy to see in the
spectra, but field estimates usually require the identification of at least
two cyclotron lines in order to uniquely determine the harmonic numbers. Since
the M dwarfs typically dominate the spectrum, they can overwhelm weaker lines
causing them to be overlooked. Therefore, we removed the M dwarf contributions
from all the spectra, using the best fit template spectrum for the rotational
broadening measurements. Figures 8-13 show the resulting M dwarf subtracted
spectra, plotted as trailed spectra. This process revealed weaker lines in
many systems and allowed us to uniquely determine the field strengths via
Equation 4 in most systems. Unfortunately there is only one clear cyclotron
line visible in the spectra of SDSS J0750+4943 and SDSS J1452+2045. For SDSS
J1452+2045 the lack of any other visible cyclotron lines in the optical allows
us to determine the harmonic number of the only visible line and hence the
field strength, but for SDSS J0750+4943 there is some ambiguity in the
harmonic number of the line, meaning that there are two possible fields
strengths for this system. Some magnetic white dwarfs in our sample show poles
with different field strengths, indicated in Table 2.
## 6 Candidate magnetic systems
During a search for quasars in the SDSS spectroscopic database seven new
candidate magnetic white dwarfs in detached binaries were serendipitously
discovered. The SDSS spectra of these systems are plotted in Figure 14. All of
these systems show broad emission features similar to cyclotron lines. The
wavelengths of the cyclotron harmonics, estimates of the magnetic field
strengths implied by these as well as Gaia early DR3 astrometric data are
listed in Table 3. For the two systems where only a single cyclotron harmonic
is detected we are unable to uniquely identify the harmonic number of the
cyclotron lines, although the lack of any other cyclotron features means that
these must be either the $n=1$ or $n=2$ harmonics. We also analysed the Zwicky
Transient Facility (ZTF) Public Data Release 3 light curves of these systems
(Masci et al., 2019). Three systems show clear, periodic variability, likely
related to the orbital period of the systems. However, we consider these
periods preliminary. Follow up data are required in order to confirm that
these are indeed magnetic white dwarfs in detached binaries and to establish
the binary and stellar parameters, although the faintness of these systems
makes this challenging.
Figure 14: SDSS spectra of seven candidate close, detached binaries containing magnetic white dwarfs. The spectra are phase-average and have been smoothed by a 5-pixel boxcar. All of these systems appear to show strong cyclotron features. See Table 3 for details of these systems. Table 3: Candidate close, detached binaries containing magnetic white dwarfs from SDSS spectroscopy. The Gaia $G$ band magnitudes and parallaxes are taken from early DR3. The magnetic field strengths are purely based on the measured wavelengths of the cyclotron harmonics and were derived using Equation 4, as such they should be considered estimates. The photometric variability column lists any clear periodicity seen in the ZTF light curves of these sources. Name | $G$ | $\pi$ | B | Cyclotron harmonics | Photometric
---|---|---|---|---|---
| (mag) | (mas) | (MG) | detected (Å) | variability
SDSS J004924.50+222617.9 | 18.52 | $1.53\pm 0.25$ | 78 | 6840, 4560 ($n=2,3$) | 355 mins
SDSS J010844.12+022638.9 | 20.60 | - | 57 | 6450, 4850 ($n=3,4$) | 81 mins
SDSS J011331.69-033233.5 | 19.33 | $2.67\pm 0.34$ | 84 | 6380, 4250 ($n=2,3$) | 263 mins
SDSS J083615.53+223204.3 | 20.89 | - | 184 or 92 | 5830 ($n=1$ or $n=2$) | No data
SDSS J100615.27+242612.1 | 20.02 | $0.42\pm 0.53$ | 82 | 6530, 4350 ($n=2,3$) | No data
SDSS J130147.97+434549.4 | 19.96 | $2.19\pm 0.43$ | 198 or 99 | 5400 ($n=1$ or $n=2$) | No data
SDSS J223606.51+291559.9 | 19.67 | $0.60\pm 0.39$ | 83 | 6450, 4300 ($n=2,3$) | Non-variable
## 7 Notes on individual systems
In this section we outline results for the systems studied in detail by us
using X-shooter and/or IDS spectroscopy.
### 7.1 Non-magnetic systems
#### 7.1.1 SDSS J0225+0054
With a period of 21.9 hours Nebot Gómez-Morán et al. (2011), this is one of
the longest period systems observed in our sample. The system is well detached
and shows no signs of magnetism in its spectrum, although our data only cover
a limited phase range. However, given the low temperature of the white dwarf
($6600\pm 300$ K) it is perhaps unsurprising that it appears as a DC white
dwarf in SDSS. The stellar parameters are relatively similar to the eclipsing
system SDSS J1210+3347 (Pyrzas et al., 2012), which hosts a DZ white dwarf
showing a large number of metal absorption lines. The much longer period of
SDSS J0225+0054 likely means that the accretion rate onto the white dwarf is
too low to generate detectable metal lines. The long period of this system
also means that the rotational broadening of the M star is small ($5.5\pm 2.3$
$\mathrm{km\,s^{-1}}$) leading to a substantial uncertainty on the white dwarf
mass.
#### 7.1.2 SDSS J0848+2320
The SDSS spectra of this system showed substantial radial velocity variations
indicating that it was a close binary. Our X-shooter data revealed an orbital
period of 8.9 hours. No Balmer lines are seen from the white dwarf, but clear
Ca ii absorption is seen at 3934Å moving in anti-phase with the M dwarf
features, indicating this line originates from the white dwarf. This system
therefore made a excellent target to check the reliability of using the
rotational broadening to constrain the mass ratio (hence white dwarf mass),
since we have an independent measurement of the mass ratio directly from the
radial velocity data. The mass ratio derived from the rotational broadening
($q=0.94\pm 0.06$) is in good agreement with the directly measured value from
the radial velocities ($q=0.97\pm 0.03$), which lends confidence to the
parameters derived via this technique in the other systems.
The spectra show no clear evidence of magnetism from the white dwarf. The
sharpness of the Ca ii absorption line also argues against a strongly magnetic
white dwarf in this system.
#### 7.1.3 SDSS J1140+1542
The SDSS spectra of this system showed some radial velocity variations
indicating that it was a close binary. Our X-shooter data confirmed that this
is indeed the case and, when combined with previous velocity measurements,
yielded a period of 3.1 days, by far the longest period system in our sample.
As such, the system is well detached and shows no evidence of magnetism from
the white dwarf, although our phase coverage is quite sparse. The very small
rotational broadening in this system means that it cannot place useful
constraints on the white dwarf mass. However, the minimum white dwarf mass
implied by the orbital period, radial velocity semi-amplitude and mass of the
M dwarf is 1.22 $\mathrm{M_{\odot}}$, making this one of the most massive
white dwarfs in a detached PCEB.
#### 7.1.4 SDSS J1316-0037
This 9.7 hour binary (Nebot Gómez-Morán et al., 2011) shows no obvious
features from the white dwarf in its X-shooter spectrum. It is well detached,
with no clear signs of magnetism from the white dwarf, although our phase
coverage is low. We note that there is a slight discrepancy between the M
dwarf mass that we report and the value in Nebot Gómez-Morán et al. (2011).
The mass reported in Nebot Gómez-Morán et al. (2011) is based on a mass-
spectral type relationship (Rebassa-Mansergas et al., 2007), which does not
yield precise masses and should only be considered an estimate. Our M dwarf
masses are based on the $K_{s}$ band luminosities of these stars, which yields
more precise and accurate masses when compared to independent measurements
(Mann et al., 2019; Parsons et al., 2018).
#### 7.1.5 SDSS J2208+0037
With a period of only 2.5 hours (Zorotovic et al., 2016) this system sits
right in the middle of the CV period gap. This system is much closer to Roche-
lobe filling than the other non-magnetic systems. Interestingly, this system
has many traits in common with the magnetic systems, such as a short period
and high Roche-lobe filling factor, as well as a cool white dwarf ($6100\pm
400$ K). The only obvious difference is the low white dwarf mass $0.46\pm
0.24$ $\mathrm{M_{\odot}}$, which is substantially smaller than the white
dwarf masses measured in the magnetic systems. Despite the similarities with
the magnetic white dwarf systems, there is no clear evidence of magnetism from
the white dwarf in the X-shooter data.
### 7.2 Magnetic systems
#### 7.2.1 SDSS J0750+4943
This system was initially identified as a white dwarf plus main-sequence
binary from its SDSS colours (Rebassa-Mansergas et al., 2013) and found to be
a close binary with a period of 4.2 hours from its CRTS light curve (Parsons
et al., 2015). The shape of the CRTS light curve was reminiscent of those with
strongly Roche-distorted M dwarfs, but was highly asymmetric, distinguishing
it from most other systems. Our spectroscopic observations revealed that this
system contains a magnetic white dwarf and the strong cyclotron line at
$\sim$5450 Å (which is only visible for half of the period) was the cause of
the asymmetric light curve. This demonstrates that these detached magnetic
systems can be potentially identified through their unusual light curves.
Inspection of the spectra taken when the cyclotron emission is absent show no
clear features from the white dwarf.
We were able to measure a radial velocity semi-amplitude of the M dwarf from
our INT/IDS spectra, but not a rotational broadening. However, SDSS J0750+4943
is the only system in our sample with both a precise Gaia parallax and high
quality ultraviolet fluxes from the Galaxy Evolution Explorer (GALEX) mission
(Martin et al., 2005). As such, we constrained both the surface gravity and
temperature of the white dwarf by fitting its ultraviolet fluxes. We used non-
magnetic white dwarf models (Koester, 2010) and assumed a standard mass-radius
relationship for carbon-oxygen core white dwarfs (Fontaine et al., 2001) in
order to scale the model spectrum. We fitted the data using the Markov Chain
Monte Carlo method (Press et al., 2007) implemented using the python package
emcee (Foreman-Mackey et al., 2013). We included a prior on the parallax based
on the Gaia measurement and a prior on the reddening of $E(B-V)=0.023\pm
0.019$ mags, based on Gaia-2MASS dust maps (Lallement et al., 2019). The
result of this fit is shown in Figure 15.
We note that it is unclear how strong any cyclotron emission is in the
ultraviolet. Because only a single cyclotron line is visible in the IDS
spectrum we can only constrain the white dwarf field strength to be either 196
MG (the 5450 Å line is the fundamental) or 98 MG (the 5450 Å line is the $n=2$
harmonic). We can exclude the possibility of the cyclotron line being a higher
harmonic as this would result in additional harmonics appearing in the IDS
spectrum, which are not seen. If the field strength is the higher value then
several cyclotron harmonics are likely to contribute to the ultraviolet flux.
On the other hand this is less likely to be an issue if the field strength is
the lower value. This source was observed by GALEX in the NUV band for roughly
30 minutes. Inspection of the light curve of the system (produced using the
python package gphoton, Million et al. 2016) over the 30 minute timespan shows
not obvious variation, which might be expected if there was a substantial
cyclotron component. However, this only covers a small fraction of the orbit
so we cannot rule out contamination by cyclotron emission. Therefore, given
this, and the fact that we modelled the white dwarf using non-magnetic white
dwarf models, the parameters of this system should be interpreted with great
caution.
Figure 15: Top: Fit to the GALEX FUV and NUV measurements for the magnetic
system SDSS J0750+4943 using non-magnetic DA white dwarf models. Bottom:
Posterior probability distributions for model parameters obtained from the
fit. Priors we placed on both the parallax and reddening.
#### 7.2.2 SDSS J0853+0720
This system was previously classified as a DC+dM binary with a period of 3.6
hours (Nebot Gómez-Morán et al., 2011). The SDSS spectrum showed no clear
evidence that the white dwarf was magnetic. However, our X-shooter data
revealed two cyclotron lines. A strong line is seen at 4250 Å and a weaker
line is evident at 6300 Å once the M dwarf component was subtracted (see
Figure 9), which are the the $n=3$ and $n=2$ harmonics of an 84 MG field. Both
of these cyclotron lines become faint after phase 0.5 and so it is likely that
the SDSS spectrum was obtained around this phase and thus missed the cyclotron
lines. No clear white dwarf features are visible in the X-shooter spectra at
the phase when the cyclotron emission is weakest. This result confirms our
suspicion that magnetic white dwarfs could be hiding as DC white dwarfs within
the population of white dwarf plus main-sequence star systems discovered in
SDSS. This is further reinforced by the temperature estimate of $9000\pm 300$
K for this white dwarf. At this temperature the white dwarf should still show
strong Balmer absorption lines, even against the M dwarf. Zeeman splitting has
resulted in these lines becoming too weak to be detected against the M dwarf.
#### 7.2.3 HS 0922+1333
Originally discovered in the Hamburg Quasar Survey (Reimers & Hagen, 2000), HS
0922+1333 was the second detached magnetic white dwarf system discovered. It
shows evidence of two poles with different field strengths of 66 MG (visible
in Figure 10 around phase 0.9 where the $n=2$ harmonic at 8100 Å, $n=3$
harmonic at 5400 Å and $n=4$ harmonic at 4050 Å are all visible) and 81 MG
(visible around phase 0.5 where the $n=3$ harmonic at 4400 Å is seen along
with a weak $n=2$ component at 6600 Å). The orbital period was refined by
Tovmassian & Zharikov (2007), who also found tentative evidence of Roche-lobe
overflow via a stream of material leaving the L1 point. We see no such feature
in our X-shooter data, although our data cover less than half the orbit, so it
is conceivable that we missed this feature.
Our new data improve the radial velocity semi-amplitude measurement for the M
dwarf and, for the first time, yield a measure of its rotational broadening.
Using these measurements we find a white dwarf mass of $0.71\pm 0.07$
$\mathrm{M_{\odot}}$, somewhat higher than the average white dwarf mass in
detached white dwarf plus main-sequence binaries, but fairly typical for white
dwarfs in CVs (Zorotovic et al., 2011).
#### 7.2.4 IL Leo
Figure 16: Average spectrum of IL Leo (black line). Multiple Zeeman split
hydrogen Balmer absorption lines from the white dwarf are seen. The
wavelengths of the Zeeman split components (Friedrich et al., 1996) are shown
in grey as a function of the magnetic field strength. The field strength of
the white dwarf measured from the cyclotron emission is 42 MG. The $n=5$ (5160
Å) and $n=6$ (4300 Å) cyclotron harmonics are visible in this plot, although
they are heavily distorted by the absorption lines.
Also known as SDSS J1031+2028, this system was originally discovered from its
unusual SDSS spectrum after being targeted as a quasar candidate (Schmidt et
al., 2007). IL Leo is unique among our sample as the only system where the
main-sequence component does not dominate the optical flux. In fact, the faint
H$\alpha$ emission line shown in Figure 4 is the only features visible in the
X-shooter data from the main-sequence star. Furthermore, IL Leo is the only
magnetic system in our sample that clearly shows Zeeman split Balmer
absorption lines (see Figure 16). The X-shooter data show strong cyclotron
emission from a 42 MG magnetic field. The $n=3$ harmonic at 8600 Å, the $n=4$
harmonic at 6450 Å, $n=5$ harmonic at 5160 Å and $n=6$ harmonic at 4300 Å are
all visible (best seen in Figure 11). The wavelengths of the Zeeman split
Balmer absorption lines appear consistent with this field strength, but the
high field strength implies that many Zeeman components will be smeared out by
the variation of field strength across the white dwarf surface (factor of
$\simeq$2 for a dipole configuration). It is therefore difficult to measure
the field strength directly from the absorption lines, thus we adopt the field
strength as measured from the cyclotron emission.
The lack of a direct radial velocity semi-amplitude for the main-sequence star
(the H$\alpha$ emission only places a lower limit on this) and no rotational
broadening measurement makes it difficult to place any meaningful constraints
on the stellar parameters. However, the short period of the system, just 82
minutes, does offer some limits on the system parameters.
Assuming that the main-sequence star follows the same mass-radius relationship
from Section 5.1 then it must be less massive than 0.09 $\mathrm{M_{\odot}}$
in order to fit within its Roche-lobe at this orbital period, placing the star
close to the brown dwarf regime. Moreover, the Roche-lobe filling factor must
be $>$0.95. Smaller filling factors than this result in star that is
substantially undersized for its mass. This is true for any mass above
$\sim$0.07 $\mathrm{M_{\odot}}$, below this mass the radius is sensitive to
the age of the system and in principle a very old ($>$5 Gyr) brown dwarf could
have a smaller filling factor than this. However, we consider the possibility
of such a low mass companion unlikely, since the strength of the cyclotron
lines imply that the white dwarf is capturing a substantial amount of wind
material. Brown dwarfs are not expected to produce strong winds and so it is
unlikely that a brown dwarf, well within its Roche-lobe, could generate enough
wind material to produce the strong cyclotron features seen from the white
dwarf.
The H$\alpha$ emission line from the companion seen in the X-shooter data of
IL Leo (Figure 4) may be driven by irradiation from some accretion source in
the system, since the temperature of the white dwarf is too low to generate
such a strong emission line (see for example the white dwarf plus brown dwarf
binary WD 1032+011, which has a slightly longer period but contains a similar
temperature white dwarf to IL Leo, but shows no evidence of any emission lines
from the brown dwarf Casewell et al. 2020). This may be evidence of Roche-lobe
overflow in IL Leo, although there is no clear emission from an accretion
stream in the X-shooter data.
Finally, for a given set of stellar masses and a Roche-lobe filling factor,
the radial velocity of the H$\alpha$ emission line can be corrected to the
centre-of-mass of the main-sequence star following the technique outlined in
Parsons et al. (2012a). This places a lower limit on the mass of the white
dwarf, below which the measured radial velocity would imply an unfeasible
inclination regardless of the companion mass (more specifically, the system
should be eclipsing, which IL Leo is not). Unfortunately the limit on the
white dwarf mass is only $>$0.48 $\mathrm{M_{\odot}}$, which is not very
informative.
#### 7.2.5 SDSS J1452+2045
The period of this system was measured by Zorotovic et al. (2016) to be 2.6
hours, placing it right in the middle of the period gap. The SDSS spectrum of
this system shows no evidence of magnetism and the system was classified as a
DC+dM binary. However, our X-shooter data revealed a strong double-peaked
cyclotron line at around 3600 Å, which is outside of the SDSS spectral range,
hence the reason that the magnetic nature of the white dwarf was previously
missed. Remarkably, no other cyclotron lines are visible implying that the
line at 3600 Å is the fundamental from a 300 MG field (if this line was a
higher harmonic then additional cyclotron lines should be visible in the
X-shooter data). This is comfortably the highest field strength among our
sample and is higher than any known white dwarf in a CV (Ferrario et al.,
2015), although there are a number of single magnetic white dwarfs with larger
field strengths than this. No clear white dwarf features are visible in the
spectra at any orbital phase.
This system, along with the 8 MG eclipsing system SDSS J0303+0054,
demonstrates a key limitation of identifying magnetic systems from cyclotron
emission at optical wavelengths. Only a limited range of field strengths will
create strong cyclotron emission at optical wavelengths. Very high fields
(e.g. SDSS J1452+2045) will show cyclotron emission in the ultraviolet, while
very low field strengths (e.g. SDSS J0303+0054) will only show cyclotron
emission at infrared wavelengths.
With a mass of $0.83\pm 0.08$ $\mathrm{M_{\odot}}$ the white dwarf in SDSS
J1452+2045 is more massive than those typically found in detached binaries
with main-sequence stars (Zorotovic et al., 2011). However, assuming that the
main-sequence star follows the same mass-radius relationship from Section 5.1
then the system is quite well detached (filling factor of 0.84).
#### 7.2.6 SDSS J2229+1853
Like SDSS J0750+4943 this system was found to be a close binary with an
unusual asymmetric light curve (Parsons et al., 2015). Our follow-up data
confirm the magnetic nature of the white dwarf via the identification of very
strong cyclotron lines. The $n=2$ harmonic at 6400 Å, $n=3$ harmonic at 4250 Å
and $n=4$ harmonic at 3200 Å from an 84 MG field are visible in Figure 13),
although the field is clearly complex as the $n=2$ harmonic weakens as $n=4$
harmonic strengthens. Higher phase resolution or spectropolarimetric
observations are required to better understand the structure of the magnetic
field of the white dwarf in this system. With a period of 4.5 hours SDSS
J2229+1853 has the longest period of all known detached magnetic white dwarf
binaries and appears to be extremely close to Roche-lobe filling. No clear
white dwarf features are visible in the X-shooter spectra at any orbital
phase.
## 8 Discussion
### 8.1 The search for low field detached magnetic white dwarf binaries
Our target selection was driven by the discovery of the low field magnetic
white dwarf in SDSS J0303+0054 (Parsons et al., 2013). The spectrum of the
white dwarf in this binary lacks any hydrogen Balmer absorption lines, despite
its temperature being high enough to produce lines that should be visible in
the SDSS spectrum for example. Detailed analysis showed that the lack of
absorption lines is the result of a combination of Zeeman splitting (which
weakens the lines) and an additional (Zeeman split) emission component
originating from the white dwarf, which fills the absorption lines. Given that
emission lines are often seen from cool white dwarfs accreting from the wind
of companion stars (e.g. Tappert et al., 2011; Parsons et al., 2012c, 2017a;
Longstaff et al., 2019) we hypothesised that there could be other magnetic
white dwarfs hiding among the sample of white dwarf plus main-sequence star
binaries classified as DC+dM systems. Importantly, these systems would have
weaker magnetic fields than those seen in previously discovered detached
magnetic white dwarf binaries since the SDSS spectra of DC+dM binaries show no
evidence of cyclotron emission. Like SDSS J0303+0054, this emission would
instead occur at infrared wavelengths.
Our high-quality X-shooter spectra of DC+dM binaries did reveal two previously
unknown magnetic systems (SDSS J0853+0720 and SDSS J1452+2045), but the
magnetic nature of the white dwarf in both of these binaries was revealed via
cyclotron emission (which is not seen in the SDSS spectra of these sources),
rather than Zeeman split emission from the white dwarf. In fact, SDSS
J0303+0054 remains the only detached magnetic white dwarf binary to show
Zeeman split emission. None of the other DC+dM binaries show any emission
lines from the white dwarf (either Zeeman split or not), leading us to
conclude that they are unlikely to possess even weak magnetic fields. The
Balmer absorption lines from these white dwarfs are weak due to their low
temperatures and are therefore undetectable next to their brighter main-
sequence companions.
It is likely that the higher magnetic field strengths of white dwarfs in
detected magnetic systems compared to SDSS J0303+0054 (10s of MG, compared to
8 MG for SDSS J0303+0054) means that any hydrogen emission from the white
dwarfs in these binaries is so strongly Zeeman split that it is undetectable.
Hence the reason that the magnetic white dwarf in SDSS J0303+0054 is the only
one to currently show Zeeman split emission lines. The low magnetic field
strength of the white dwarf in SDSS J0303+0054 remains an outlier.
### 8.2 General properties of detached magnetic white dwarf binaries
The detached magnetic white dwarf systems share a number of common properties.
In general they contain cool white dwarfs, with temperatures lower than the
majority of detached non-magnetic white dwarfs and cooler than the white
dwarfs in polars (Schwope et al., 2009). Additionally, these detached magnetic
white dwarf systems are all very close to Roche-lobe filling (see Figure 17).
However, given that the magnetic nature of the white dwarfs in virtually all
of these systems was uncovered via the identification of cyclotron emission,
these may be selection effects. Wind accretion will be largest in those
systems closest to Roche-lobe overflow, leading to stronger cyclotron lines.
Moreover, the cyclotron lines are easier to see when the white dwarfs are
cooler, since the cyclotron emission dominates over the photosphere. On the
other hand, Zeeman splitting of the hydrogen Balmer lines would be easier to
see in hotter white dwarfs and would also not depend upon having any wind
accretion from the main-sequence star. Therefore, the lack of any magnetic
white dwarfs hotter than $\sim$10000 K among the thousands of white dwarf plus
main-sequence star systems spectroscopically observed by SDSS (Rebassa-
Mansergas et al., 2016) strongly implies that these systems are extremely rare
or perhaps do not exist at all. Interestingly, of the 10 systems originally
listed in Rebassa-Mansergas et al. (2016) as DC+dM binaries that have been
observed in detail with X-shooter, 4 are magnetic, implying that magnetic
systems appear to be much more common among cooler, hence older, white dwarf
systems. This is similar to the suggestion of an increase in the incidence of
magnetism among cooler single white dwarfs (Liebert et al., 2003; Kawka &
Vennes, 2014; Hollands et al., 2015)
Figure 17: Roche-lobe filling factors for detached white dwarf plus main-
sequence binaries as a function of the white dwarf cooling age. Arrows denote
lower limits. DA+dM systems taken from Rebassa-Mansergas et al. (2016).
Cooling ages were determined using the cooling sequence from Fontaine et al.
(2001). On average magnetic systems have larger filling factors and longer
cooling ages than non-magnetic systems. Figure 18: The measured accretion
rates for a sample of nearby cataclysmic variables (non-magnetic in black,
magnetic in green), detached magnetic white dwarf binaries (red) and detached
non-magnetic white dwarf binaries (blue).
### 8.3 Accretion rates
We estimated the accretion rate of material onto the white dwarfs in the
magnetic systems following the approach of Schwope et al. (2009) whereby the
cyclotron luminosity is equated to the accretion luminosity. We use the Gaia
distance estimates and measure the peak integrated flux of the cyclotron lines
in our M dwarf subtracted spectra (using the spectra taken when the cyclotron
lines are at their strongest). We removed any contribution from the white
dwarf’s photosphere by subtracting off a model white dwarf spectrum (non-
magnetic, hydrogen atmosphere, Koester 2010) with a temperature set to our
upper limits and scaled to the distance of the system. In most cases
subtracting off this contribution has a negligible effect on the accretion
rate estimates. Since our data cover such a wide wavelength range (with the
exception of SDSS J0750+4943) it is likely that we have included most of the
cyclotron flux. However, it is possible that there is significant flux in
cyclotron lines outside of our wavelength range in some cases. Moreover,
cyclotron emission is generally anisotropic and beamed adding further
uncertainty to our accretion rate measurements. In addition, a fraction of the
accretion luminosity may be emitted at X-ray wavelengths (e.g. Vogel et al.,
2007). Hence the accretion rates should be interpreted as only approximate.
The estimated accretion rates are given in Table 4. Figure 18 shows these
accretion rates relative to those measured for CVs within 150 pc (Pala et al.,
2020) as well as detached binaries with non-magnetic white dwarfs (Debes,
2006; Tappert et al., 2011; Pyrzas et al., 2012; Parsons et al., 2012b; Drake
et al., 2014). The accretion rates of these detached magnetic white dwarf
binaries are clearly below those seen in CVs, but are substantially higher
than the accretion rates seen in pre-cataclysmic binaries with non-magnetic
white dwarfs.
The higher accretion rates seen in detached magnetic white dwarf binaries
compared to non-magnetic white dwarf binaries with similar orbital periods
implies that the magnetic white dwarfs are able to capture a significantly
larger fraction of the wind material from their companion. This supports the
idea that angular momentum loss via magnetic braking may be reduced in
binaries containing a strongly magnetic white dwarf because the wind material
from the companion is trapped within the magnetosphere of the white dwarf and
hence is unable to carry away as much angular momentum (Li et al., 1994;
Webbink & Wickramasinghe, 2002). Population synthesis calculations have
recently shown that reduced magnetic braking is required to reproduce the
observed orbital period distribution, space density and mass transfer rates of
magnetic CVs (Belloni et al., 2020).
Table 4: Estimated accretion rates in detached magnetic white dwarf binaries. It is likely that our data do not capture the full cyclotron luminosity and therefore these accretion rates are only approximate. Name | $\dot{M}$ | Reference
---|---|---
| ($10^{-13}$$\mathrm{M_{\odot}}$/yr) |
SDSS J0750+4943 | 6.0 | This paper
SDSS J0837+3830 | 2.0 | Schmidt et al. (2005a)
SDSS J0853+0720 | 0.8 | This paper
HS 0922+1333 | 3.7 | This paper
WX LMi | 1.5 | Vogel et al. (2007)
IL Leo | 3.8 | This paper
SDSS J1059+2727 | $0.6-5.0$ | Schmidt et al. (2007)
SDSS J1206+5100 | 0.1 | Schwope et al. (2009)
SDSS J1452+2045 | 1.3 | This paper
MQ Dra | 0.6 | Schmidt et al. (2005a)
SDSS J2048+0050 | 0.5 | Schmidt et al. (2005a)
SDSS J2229+1853 | 6.5 | This paper
### 8.4 Evolutionary status
The evolutionary status of detached binaries containing magnetic white dwarfs
has been addressed by many authors (e.g. Schwope et al., 2009; Breedt et al.,
2012), but a general consensus has yet to be reached. The largest question
remains whether these systems are CVs that are in extreme low states or have
detached all together (low accretion rate polars, LARPs), or if they are pre-
CVs, where the main-sequence star is yet to fill its Roche-lobe (pre-polars,
PREPs). Several systems appear to show signs of episodic Roche-lobe overflow.
For example, Tovmassian & Zharikov (2007) saw evidence of a high velocity
H$\alpha$ component in HS 0922+1333, which they interpret as an accretion
stream, although no such feature is evident in our X-shooter data. A similar
feature was found in SDSS J2048+0050 by Kafka et al. (2010), which was not
seen in previous data (Schmidt et al., 2005a). Inspection of the long term
CRTS light curves of these systems also reveals a clear brightening event in
SDSS J0837+3830 (see Figure 19), where it is likely that the accretion rate
increased dramatically for around a year, likely as a result of Roche-lobe
overflow, before returning to its original brightness. It therefore appears
more likely that these systems are CVs that have temporarily detached. This is
also supported by our white dwarf mass estimates. For the 6 detached magnetic
systems with reliable mass estimates we obtain an average white dwarf mass of
$\langle M_{\mathrm{WD}}\rangle=0.82\pm 0.07$ $\mathrm{M_{\odot}}$, which is
higher than the average mass of white dwarfs in pre-CVs ($0.67\pm 0.21$
$\mathrm{M_{\odot}}$, Zorotovic et al. 2011), but is consistent with both non-
magnetic CV white dwarf masses ($0.83\pm 0.23$ $\mathrm{M_{\odot}}$, Zorotovic
et al. 2011; Schreiber et al. 2016) and magnetic CV white dwarf masses
($0.77\pm 0.02$ $\mathrm{M_{\odot}}$, Shaw et al. 2020, $0.84\pm 0.17$
$\mathrm{M_{\odot}}$ de Martino et al. 2020). Although it is worth noting that
in general isolated magnetic white dwarfs appear to have higher masses than
their non-magnetic counterparts (Ferrario et al., 2015; McCleery et al., 2020)
Figure 19: CRTS light curve of the magnetic system SDSS J0837+3830 showing a
period of increased brightness likely as a result of accretion via Roche-lobe
overflow.
On the other hand, our measured Roche-lobe filling factors are as low as 0.84,
which would place the main-sequence star quite far from Roche-lobe filling.
Indeed, it will take almost 1 Gyr for SDSS J0303+0054 to come into contact
(Parsons et al., 2013), which would argue more towards a pre-CV classification
for this system. However, the donor stars in CVs are pushed out of thermal
equilibrium and thus have radii larger than isolated stars of the same mass.
Therefore, if the system becomes detached long enough for the donor to relax
back to thermal equilibrium then it can substantially underfill the Roche-lobe
(this is thought to be the cause of the orbital period gap for example).
Figure 20 shows what the expected Roche-lobe filling factors that CVs would
drop to if the donor star was allowed to relax down to an equilibrium radius
(black line), using the standard evolutionary track from Knigge et al. (2011)
and an equilibrium radius given by the mass-radius relation from Section 5.1.
Hence any system on or above this line could, in principle, have been Roche-
lobe filling in the past and the donor has now relaxed. All of the magnetic
systems with reliable Roche-lobe filling factors are on or above this line,
hence there are no systems were we can rule out a previous phase of mass
transfer via Roche-lobe overflow. Also note that the radii of the main-
sequence stars in all these systems were determined from the semi-empirical
mass-radius relationship and hence we have assumed that the main-sequence
stars in these systems are in thermal equilibrium. If this is not the case
then our Roche-lobe filling factors will be underestimated, hence the points
in Figure 20 are likely lower limits, rather than exact values (the exception
being the eclipsing system SDSS J0303+0054, which has a direct radius
measurement).
Although we cannot rule out the possibility that these systems are in fact
pre-CVs, all the evidence supports the interpretation that these are CVs that
are in extreme low states or completely detached. This would imply that we are
yet to identify any magnetic white dwarfs in pre-CVs, hence the origin of
magnetic CVs remains unclear.
It is worth noting that the population of detached magnetic white dwarfs is
still very biased. The vast majority of these systems were identified from
cyclotron lines in their optical spectra, which are stronger in systems closer
to Roche-lobe filling for example. Cyclotron emission is also easier to detect
when the photosphere of the white dwarf is cool and hence faint, biasing us
towards systems with cooler white dwarfs. On the other hand, Zeeman splitting
of the hydrogen Balmer lines is easier to identify in hotter white dwarfs and
is independent of the accretion rate of wind material onto the white dwarf
(hence independent of Roche-lobe filling factor). Since these should be
straightforward to identify, the complete lack of young, well detached
strongly magnetic white dwarf binaries appears to be real.
Figure 20: Roche-lobe filling factors for detached white dwarf plus main-
sequence binaries as a function of their orbital periods. The black lines
correspond to the filling factor that a previously Roche-lobe filling system
would have if the donor relaxed to its equilibrium radius. Therefore, any
systems above these lines could be detached cataclysmic variables.
Finally, we note that IL Leo stands out from the other magnetic white dwarf
binaries analysed in this paper because of its short orbital period and low
companion mass. However, IL Leo does share a number of similar properties to
three other systems: SDSS J1212+0136 (Burleigh et al., 2006), SDSS J1250+1549
and SDSS J1514+0744 (Breedt et al., 2012). All of these systems contain
magnetic white dwarfs and very low mass (or potentially sub-stellar)
companions in binaries with periods very close to (but slightly above) the CV
period minimum. The evolutionary status of these short period magnetic white
dwarf binaries remains unclear (Breedt et al., 2012). Like the longer period
systems they may be pre-CV systems yet to come into contact or temporarily
detached CVs, but there is also the possibility that these very short period
magnetic white dwarf binaries are genuine CVs in which the donor star has
become a brown dwarf and are now evolving towards longer periods, so-called
"period-bounce" CVs. One example of a definite polar with a very low mass
companion that entered such an extended low state is EF Eri
(P${}_{\mathrm{orb}}=81$ min), which was an X-ray bright CV (Charles & Mason,
1979) until it switched off around 1997 (Beuermann et al., 2000) and has not
been active since then. The accretion rates in period-bounce systems are low
enough that they could appear similar to detached systems. There is some
evidence from X-ray observations that SDSS J1212+0136 may be Roche-lobe
filling (Stelzer et al., 2017), which lends support to this interpretation.
Therefore, these systems may form a distinct population from the longer period
detached magnetic white dwarf binaries that have been the focus of this paper.
However, IL Leo also differs from these systems in a number of ways. Its
optical spectrum is dominated by strong cyclotron emission, unlike the other
very short period systems, in which the photosphere of the white dwarf
dominates. The white dwarf in IL Leo also has a stronger magnetic field and
the orbital period of IL Leo is around 5 minutes shorter than these other
systems. IL Leo may represent a slightly earlier phase in the evolution of
these systems where the accretion rate is higher, generating stronger
cyclotron emission. As the binary evolves towards longer periods the accretion
rate will drop and in the future IL Leo may well resemble these other very
short period magnetic white dwarf binaries.
### 8.5 Identifying detached magnetic white dwarf binaries
The small number of currently known magnetic white dwarfs in detached binaries
limits the conclusions that we can draw from the population. Moreover, only
one eclipsing system is currently known (Parsons et al., 2013). The discovery
of new eclipsing systems would be particularly useful for distinguishing
between the pre-CV and detached CV scenarios, since eclipsing systems allow a
direct measurement of the radius of the main-sequence star (hence Roche-lobe
filling factor) and can place tight constraints on the white dwarf mass, even
though these systems are single lined. However, virtually all magnetic white
dwarfs in detached binaries have so far been discovered serendipitously from
their unusual optical spectra. As such, the sample is mostly limited to
systems observed spectroscopically by SDSS.
Despite their rather unusual spectra, featuring strong cyclotron emission,
detached magnetic white dwarf systems are not easily identifiable from their
colours or position in the Hertzsprung-Russell Diagram. Figure 21 shows the
location of these binaries in the Hertzsprung-Russell Diagram relative to non-
magnetic white dwarf plus main-sequence binaries. Unfortunately the magnetic
systems are indistinguishable from the non-magnetic population, with both
types of systems found between the white dwarf cooling track and the main-
sequence track.
Figure 21: Gaia Hertzsprung-Russell Diagram showing the location of detached
magnetic (blue diamonds) and non-magnetic (red circles) white dwarf plus main-
sequence binaries. A random sample of sources within 500 pc of the Sun are
shown in the background. Figure 22: Phase-folded light curves of six detached
magnetic white dwarf binaries displaying asymmetric light curves. Data for
SDSS J0750+4943 and SDSS J2229+1853 are CRTS $V$ band data, all other light
curves are ZTF $r$ band data. The asymmetric light curve shapes are a result
of the magnetic pole(s) of the white dwarf rotating into view, on top of
variations from the Roche-distorted main-sequence star.
Two systems in this study have demonstrated a useful method for identifying
these systems via their light curves. SDSS J0750+4943 and SDSS J2229+1853 were
both originally flagged as potential magnetic systems from their unusual
asymmetric CRTS light curves (see Figure 22 and Parsons et al. 2015). The
double-peaked shape of the phase-folded light curve is reminiscent of
ellipsoidal modulation, caused by the Roche-distorted main-sequence star
presenting a different surface area throughout the orbit. However, in both
cases one peak is substantially stronger than the other and the peaks have
been shifted from the quadrature phases. This distortion of the light curve is
due to the magnetic pole(s) of the synchronously rotating white dwarf moving
into view, adding cyclotron emission to the light curve. This likely requires
the system to be viewed from a relatively high inclination, in order for the
magnetic pole(s) to disappear from view and generate the largest variations.
Large scale multi-epoch photometric surveys such as ZTF and the Legacy Survey
of Space and Time (LSST) are likely to identify many objects with similar
looking light curves, offering us a chance to identify larger number of these
types of systems without the need for prior spectroscopic observations. Four
ZTF light curves of detached magnetic binaries (both confirmed and candidate
systems) are also shown in Figure 22 along with two CRTS light curves
demonstrating that these asymmetric light curves appear to be a common feature
of these systems.
## 9 Conclusions
We have identified four new and seven candidate magnetic white dwarfs in
detached binaries with main-sequence star companions. Using phase-resolved
medium resolution spectroscopy we have analysed these four new systems in
detail, along with two previously known magnetic white dwarf binaries and five
non-magnetic DC white dwarf plus main-sequence star binaries. For each system
we have constrained the stellar and binary parameters including, for the first
time, the masses of the magnetic white dwarfs. We found that these magnetic
white dwarfs are systematically more massive than non-magnetic white dwarfs in
pre-cataclysmic binaries, but are consistent with the measured masses of white
dwarfs in cataclysmic variables. We also found that, while some systems could
be quite far from Roche-lobe filling, it is possible that all of these systems
were Roche-lobe filling in the past, but the main-sequence stars are now
closer to thermal equilibrium, although we cannot conclusively rule out the
possibility that these systems are pre-cataclysmic binaries. Even if these
systems are pre-cataclysmic binaries, they all have cooling ages of around a
Gyr or more. Young detached magnetic white dwarf binaries remain elusive.
Finally, we note that several of our newly identified magnetic white dwarf
binaries have archival spectroscopy that show no clear evidence of magnetism,
thus demonstrating that these systems can be missed when using spectroscopic
data alone. We demonstrate that these magnetic white dwarf systems may be
identifiable from their unusual light curves.
## Data Availability Statement
Raw and reduced X-shooter data are available through the ESO archive at
http://archive.eso.org/cms.html. Raw IDS spectra are available through the
Isaac Newton Group (ING) archive at
http://casu.ast.cam.ac.uk/casuadc/ingarch/query, reduced IDS data are
available on request.
## Acknowledgements
We thank the referee for their useful comments and suggestions. SGP
acknowledges the support of a Science and Technology Facilities Council (STFC)
Ernest Rutherford Fellowship. BTG was supported by the UK STFC grant
ST/T000406/1 and by a Leverhulme Research Fellowship. TRM was supported by
STFC grant ST/T000406/1. MRS thanks for support from FONDECYT (grant 1181404)
and the millennium Nucleus for Planet Formation (NPF). The results presented
in this paper are based on observations collected at the European Southern
Observatory under programme IDs 099.D-0252 and 0101.D-0144.
## References
* Bagnulo & Landstreet (2020) Bagnulo S., Landstreet J. D., 2020, arXiv e-prints, p. arXiv:2010.05795
* Bailer-Jones et al. (2018) Bailer-Jones C. A. L., Rybizki J., Fouesneau M., Mantelet G., Andrae R., 2018, AJ, 156, 58
* Belloni & Schreiber (2020) Belloni D., Schreiber M. R., 2020, MNRAS, 492, 1523
* Belloni et al. (2020) Belloni D., Schreiber M. R., Pala A. F., Gänsicke B. T., Zorotovic M., Rodrigues C. V., 2020, MNRAS, 491, 5717
* Beuermann et al. (2000) Beuermann K., Wheatley P., Ramsay G., Euchner F., Gänsicke B. T., 2000, A&A, 354, L49
* Breedt et al. (2012) Breedt E., Gänsicke B. T., Girven J., Drake A. J., Copperwheat C. M., Parsons S. G., Marsh T. R., 2012, MNRAS, 423, 1437
* Burleigh et al. (2006) Burleigh M. R., et al., 2006, MNRAS, 373, 1416
* Casewell et al. (2020) Casewell S. L., et al., 2020, MNRAS, 497, 3571
* Charles & Mason (1979) Charles P. A., Mason K. O., 1979, ApJ, 232, L25
* Currie et al. (2014) Currie M. J., Berry D. S., Jenness T., Gibb A. G., Bell G. S., Draper P. W., 2014, in Manset N., Forshay P., eds, Astronomical Society of the Pacific Conference Series Vol. 485, Astronomical Data Analysis Software and Systems XXIII. p. 391
* Debes (2006) Debes J. H., 2006, ApJ, 652, 636
* Delfosse et al. (2000) Delfosse X., Forveille T., Ségransan D., Beuzit J. L., Udry S., Perrier C., Mayor M., 2000, A&A, 364, 217
* Drake et al. (2014) Drake J. J., Garraffo C., Takei D., Gaensicke B., 2014, MNRAS, 437, 3842
* Farihi et al. (2008) Farihi J., Burleigh M. R., Hoard D. W., 2008, ApJ, 674, 421
* Ferrario et al. (2015) Ferrario L., de Martino D., Gänsicke B. T., 2015, Space Sci. Rev., 191, 111
* Fontaine et al. (2001) Fontaine G., Brassard P., Bergeron P., 2001, PASP, 113, 409
* Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
* Friedrich et al. (1996) Friedrich S., Oestreicher R., Schweizer W., 1996, A&A, 309, 227
* Gaia Collaboration et al. (2020) Gaia Collaboration Brown A. G. A., Vallenari A., Prusti T., de Bruijne J. H. J., Babusiaux C., Biermann M., 2020, arXiv e-prints, p. arXiv:2012.01533
* Hollands et al. (2015) Hollands M. A., Gänsicke B. T., Koester D., 2015, MNRAS, 450, 681
* Kafka et al. (2010) Kafka S., Tappert C., Honeycutt R. K., 2010, MNRAS, 403, 755
* Kausch et al. (2015) Kausch W., et al., 2015, A&A, 576, A78
* Kawka & Vennes (2014) Kawka A., Vennes S., 2014, MNRAS, 439, L90
* Kepler et al. (2013) Kepler S. O., et al., 2013, MNRAS, 429, 2934
* Knigge et al. (2011) Knigge C., Baraffe I., Patterson J., 2011, ApJS, 194, 28
* Koester (2010) Koester D., 2010, Mem. Soc. Astron. Italiana, 81, 921
* Lallement et al. (2019) Lallement R., Babusiaux C., Vergely J. L., Katz D., Arenou F., Valette B., Hottier C., Capitanio L., 2019, A&A, 625, A135
* Landstreet & Bagnulo (2019) Landstreet J. D., Bagnulo S., 2019, A&A, 628, A1
* Li et al. (1994) Li J. K., Wu K. W., Wickramasinghe D. T., 1994, MNRAS, 268, 61
* Liebert et al. (2003) Liebert J., Bergeron P., Holberg J. B., 2003, AJ, 125, 348
* Liebert et al. (2005) Liebert J., et al., 2005, AJ, 129, 2376
* Liebert et al. (2015) Liebert J., Ferrario L., Wickramasinghe D. T., Smith P. S., 2015, ApJ, 804, 93
* Longstaff et al. (2019) Longstaff E. S., Casewell S. L., Wynn G. A., Page K. L., Williams P. K. G., Braker I., Maxted P. F. L., 2019, MNRAS, 484, 2566
* López-Morales & Ribas (2005) López-Morales M., Ribas I., 2005, ApJ, 631, 1120
* Mann et al. (2019) Mann A. W., et al., 2019, ApJ, 871, 63
* Marsh et al. (1994) Marsh T. R., Robinson E. L., Wood J. H., 1994, MNRAS, 266, 137
* Martin et al. (2005) Martin D. C., et al., 2005, ApJ, 619, L1
* Masci et al. (2019) Masci F. J., et al., 2019, PASP, 131, 018003
* McCleery et al. (2020) McCleery J., et al., 2020, MNRAS, 499, 1890
* Meusinger et al. (2012) Meusinger H., Schalldach P., Scholz R. D., in der Au A., Newholm M., de Hoon A., Kaminsky B., 2012, A&A, 541, A77
* Meusinger et al. (2016) Meusinger H., Schalldach P., Mirhosseini A., Pertermann F., 2016, A&A, 587, A83
* Million et al. (2016) Million C., et al., 2016, ApJ, 833, 292
* Morrell & Naylor (2019) Morrell S., Naylor T., 2019, MNRAS, 489, 2615
* Nebot Gómez-Morán et al. (2011) Nebot Gómez-Morán A., et al., 2011, A&A, 536, A43
* Pala et al. (2020) Pala A. F., et al., 2020, MNRAS, 494, 3799
* Parsons et al. (2012a) Parsons S. G., et al., 2012a, MNRAS, 419, 304
* Parsons et al. (2012b) Parsons S. G., et al., 2012b, MNRAS, 420, 3281
* Parsons et al. (2012c) Parsons S. G., et al., 2012c, MNRAS, 426, 1950
* Parsons et al. (2013) Parsons S. G., Marsh T. R., Gänsicke B. T., Schreiber M. R., Bours M. C. P., Dhillon V. S., Littlefair S. P., 2013, MNRAS, 436, 241
* Parsons et al. (2015) Parsons S. G., et al., 2015, MNRAS, 449, 2194
* Parsons et al. (2017a) Parsons S. G., et al., 2017a, MNRAS, 470, 4473
* Parsons et al. (2017b) Parsons S. G., et al., 2017b, MNRAS, 471, 976
* Parsons et al. (2018) Parsons S. G., et al., 2018, MNRAS, 481, 1083
* Press et al. (1986) Press W. H., Flannery B. P., Teukolsky S. A., 1986, Numerical recipes. The art of scientific computing
* Press et al. (2007) Press W. H., Teukolsky A. A., Vetterling W. T., Flannery B. P., 2007, Numerical recipes. The art of scientific computing, 3rd edn.. Cambridge: University Press
* Pyrzas et al. (2009) Pyrzas S., et al., 2009, MNRAS, 394, 978
* Pyrzas et al. (2012) Pyrzas S., et al., 2012, MNRAS, 419, 817
* Rebassa-Mansergas et al. (2007) Rebassa-Mansergas A., Gänsicke B. T., Rodríguez-Gil P., Schreiber M. R., Koester D., 2007, MNRAS, 382, 1377
* Rebassa-Mansergas et al. (2013) Rebassa-Mansergas A., Agurto-Gangas C., Schreiber M. R., Gänsicke B. T., Koester D., 2013, MNRAS, 433, 3398
* Rebassa-Mansergas et al. (2016) Rebassa-Mansergas A., Ren J. J., Parsons S. G., Gänsicke B. T., Schreiber M. R., García-Berro E., Liu X. W., Koester D., 2016, MNRAS, 458, 3808
* Reimers & Hagen (2000) Reimers D., Hagen H. J., 2000, A&A, 358, L45
* Reimers et al. (1999) Reimers D., Hagen H. J., Hopp U., 1999, A&A, 343, 157
* Schmidt et al. (2005a) Schmidt G. D., et al., 2005a, ApJ, 630, 1037
* Schmidt et al. (2005b) Schmidt G. D., Szkody P., Silvestri N. M., Cushing M. C., Liebert J., Smith P. S., 2005b, ApJ, 630, L173
* Schmidt et al. (2007) Schmidt G. D., Szkody P., Henden A., Anderson S. F., Lamb D. Q., Margon B., Schneider D. P., 2007, ApJ, 654, 521
* Schreiber et al. (2010) Schreiber M. R., et al., 2010, A&A, 513, L7
* Schreiber et al. (2016) Schreiber M. R., Zorotovic M., Wijnen T. P. G., 2016, MNRAS, 455, L16
* Schwarz et al. (2001) Schwarz R., Schwope A. D., Staude A., 2001, A&A, 374, 189
* Schwope et al. (2002) Schwope A. D., Brunner H., Hambaryan V., Schwarz R., 2002, in Gänsicke B. T., Beuermann K., Reinsch K., eds, Astronomical Society of the Pacific Conference Series Vol. 261, The Physics of Cataclysmic Variables and Related Objects. p. 102
* Schwope et al. (2009) Schwope A. D., Nebot Gomez-Moran A., Schreiber M. R., Gänsicke B. T., 2009, A&A, 500, 867
* Shaw et al. (2020) Shaw A. W., et al., 2020, MNRAS, 498, 3457
* Skrutskie et al. (2006) Skrutskie M. F., et al., 2006, AJ, 131, 1163
* Smette et al. (2015) Smette A., et al., 2015, A&A, 576, A77
* Southworth et al. (2015) Southworth J., Tappert C., Gänsicke B. T., Copperwheat C. M., 2015, A&A, 573, A61
* Stelzer et al. (2017) Stelzer B., de Martino D., Casewell S. L., Wynn G. A., Roy M., 2017, A&A, 598, L6
* Szkody et al. (2003) Szkody P., et al., 2003, ApJ, 583, 902
* Szkody et al. (2008) Szkody P., Linnell A. P., Campbell R. K., Plotkin R. M., Harrison T. E., Holtzman J., Seibert M., Howell S. B., 2008, ApJ, 683, 967
* Tappert et al. (2011) Tappert C., Gänsicke B. T., Rebassa-Mansergas A., Schmidtobreick L., Schreiber M. R., 2011, A&A, 531, A113
* Tout et al. (2008) Tout C. A., Wickramasinghe D. T., Liebert J., Ferrario L., Pringle J. E., 2008, MNRAS, 387, 897
* Tovmassian & Zharikov (2007) Tovmassian G. H., Zharikov S. V., 2007, A&A, 468, 643
* Tremblay et al. (2020) Tremblay P. E., et al., 2020, MNRAS, 497, 130
* Vernet et al. (2011) Vernet J., et al., 2011, A&A, 536, A105
* Vogel et al. (2007) Vogel J., Schwope A. D., Gänsicke B. T., 2007, A&A, 464, 647
* Webbink & Wickramasinghe (2002) Webbink R. F., Wickramasinghe D. T., 2002, MNRAS, 335, 1
* Wickramasinghe & Ferrario (2000) Wickramasinghe D. T., Ferrario L., 2000, PASP, 112, 873
* Zorotovic et al. (2011) Zorotovic M., Schreiber M. R., Gänsicke B. T., 2011, A&A, 536, A42
* Zorotovic et al. (2016) Zorotovic M., et al., 2016, MNRAS, 457, 3867
* de Martino et al. (2020) de Martino D., Bernardini F., Mukai K., Falanga M., Masetti N., 2020, Advances in Space Research, 66, 1209
## Appendix A M dwarf velocity measurements
Table 5: M dwarf velocity measurements presented in this paper HJD(mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$)
---|---|---
SDSS J022503.02+005456.2: |
2457996.8026337 | 71.2 | 3.2
2457996.8073199 | 70.6 | 4.2
2457996.8120076 | 71.5 | 3.8
2457996.8166954 | 70.3 | 4.0
2457996.8213834 | 75.7 | 6.1
2457996.8260709 | 72.1 | 3.3
2457996.8307592 | 71.2 | 3.3
2457996.8354470 | 69.7 | 2.6
2458012.7879503 | -14.0 | 2.7
2458012.7926343 | -20.7 | 2.0
2458012.7973319 | -23.7 | 1.8
2458012.8020175 | -26.3 | 2.9
2458012.8067054 | -23.6 | 2.2
2458012.8113943 | -29.7 | 2.0
2458012.8160809 | -28.1 | 2.7
2458012.8207624 | -34.4 | 2.2
SDSS J075015.11+494333.2: |
2457065.3639952 | 195.2 | 6.6
2457065.3746437 | 187.9 | 10.6
2457065.3852192 | 119.4 | 7.0
2457065.3957878 | 84.5 | 5.1
2457065.4066982 | 18.2 | 4.9
2457065.4172678 | -38.4 | 4.9
2457065.4292545 | -80.9 | 4.4
2457065.4417256 | -125.1 | 4.0
2457065.4522908 | -124.3 | 4.0
2457065.4638623 | -95.5 | 3.8
2457065.4749185 | -69.6 | 3.5
2457065.4870584 | 4.3 | 4.0
2457065.4977579 | 54.3 | 4.0
SDSS J084841.17+232051.7: |
2457859.4829554 | 35.7 | 0.5
2457859.4870521 | 23.4 | 0.5
2457859.4911656 | 11.7 | 0.5
2457859.4952740 | -0.4 | 0.5
2457859.4994626 | -11.8 | 0.5
2457859.5035729 | -23.5 | 0.5
2457859.5076811 | -35.0 | 0.5
2457859.5119777 | -47.2 | 0.5
2457859.5160924 | -56.3 | 0.5
2457859.5202022 | -65.2 | 0.5
2457859.5243221 | -73.6 | 0.5
2458224.5455325 | -59.2 | 0.6
2458224.5496477 | -68.4 | 0.6
2458224.5537677 | -77.7 | 0.6
2458224.5578703 | -85.7 | 0.7
2458224.5619978 | -95.0 | 0.6
2458224.5661160 | -102.1 | 0.6
2458224.5702439 | -109.3 | 0.6
2458224.5743663 | -116.5 | 0.6
2458224.5784726 | -120.0 | 0.7
2458226.5350704 | -30.9 | 0.6
2458226.5392074 | -19.0 | 0.6
2458226.5433374 | -10.2 | 0.6
2458226.5474558 | 2.1 | 0.6
2458226.5515812 | 12.1 | 0.7
2458226.5556985 | 23.5 | 0.7
2458226.5598027 | 32.1 | 0.8
2458226.5639267 | 44.7 | 0.7
2458226.5680457 | 56.4 | 0.9
2458227.5120304 | -39.8 | 0.5
2458227.5161571 | -48.8 | 0.5
Table 6: continued HJD(mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$)
---|---|---
2458227.5202809 | -58.6 | 0.5
2458227.5244036 | -67.7 | 0.5
2458227.5285246 | -77.5 | 0.5
2458227.5326328 | -88.5 | 0.6
2458227.5367557 | -93.0 | 0.5
2458227.5408717 | -100.8 | 0.5
2458227.5450031 | -108.2 | 0.5
SDSS J085336.03+072033.5: |
2457860.4815001 | 125.7 | 0.9
2457860.4856083 | 153.1 | 0.9
2457860.4897230 | 173.5 | 1.0
2457860.4938327 | 190.4 | 1.0
2457860.4979331 | 199.5 | 1.0
2457860.5020421 | 204.5 | 0.9
2457860.5061580 | 198.0 | 0.9
2457860.5102652 | 184.4 | 1.0
2457860.5143863 | 168.7 | 0.9
2457860.5208586 | 131.6 | 0.9
2457860.5249769 | 99.5 | 0.9
2457860.5290831 | 65.7 | 0.9
2457860.5331914 | 24.6 | 0.9
2457860.5373009 | -14.2 | 0.9
2457860.5414082 | -56.5 | 1.0
2457860.5455213 | -97.3 | 1.0
2457860.5496251 | -135.5 | 1.0
2457860.5537334 | -169.8 | 0.9
HS 0922+1333: |
2457859.5338685 | 18.9 | 0.5
2457859.5379599 | -1.9 | 0.5
2457859.5420732 | -19.6 | 0.5
2457859.5461804 | -38.1 | 0.5
2457859.5503025 | -56.3 | 0.5
2457859.5544109 | -71.4 | 0.5
2457859.5585068 | -85.1 | 0.5
2457859.5626153 | -94.7 | 0.5
2457859.5667225 | -102.8 | 0.5
2457859.5733788 | -105.7 | 0.5
2457859.5774744 | -106.1 | 0.5
2457859.5815834 | -102.2 | 0.5
2457859.5856908 | -95.5 | 0.5
2457859.5898001 | -86.2 | 0.5
2457859.5939076 | -75.6 | 0.5
2457859.5980182 | -62.4 | 0.5
2457859.6021255 | -46.4 | 0.5
2457859.6062442 | -29.7 | 0.5
SDSS J114030.06+154231.5: |
2457859.6166398 | 53.9 | 0.5
2457859.6207427 | 53.7 | 0.5
2457859.6248572 | 54.5 | 0.5
2457859.6289670 | 55.0 | 0.5
2457859.6330879 | 55.1 | 0.5
2457859.6371966 | 55.1 | 0.5
2457859.6413074 | 55.5 | 0.5
2457859.6456340 | 55.9 | 0.5
2457859.6497401 | 56.0 | 0.5
2457859.6562043 | 58.7 | 0.5
2457859.6603112 | 60.0 | 0.5
2457859.6644279 | 59.4 | 0.5
2457859.6685353 | 58.9 | 0.5
2457859.6726448 | 60.0 | 0.5
2457859.6767523 | 60.5 | 0.5
2457859.6808483 | 61.5 | 0.5
2457859.6849592 | 61.6 | 0.5
Table 7: continued HJD(mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$)
---|---|---
2457859.6890666 | 63.7 | 0.5
2457860.6527829 | -34.0 | 0.5
2457860.6568901 | -34.4 | 0.5
2457860.6610006 | -36.1 | 0.5
2457860.6651079 | -38.0 | 0.5
2457860.6692269 | -38.6 | 0.5
2457860.6733380 | -40.4 | 0.5
2457860.6774386 | -42.4 | 0.5
2457860.6815422 | -42.7 | 0.5
2457860.6856520 | -44.3 | 0.5
2458254.5282508 | -49.6 | 0.5
2458254.5323637 | -49.1 | 0.5
2458254.5364850 | -49.0 | 0.5
2458254.5405903 | -48.0 | 0.5
2458254.5447134 | -47.2 | 0.5
2458254.5488322 | -45.9 | 0.5
2458254.5529627 | -45.2 | 0.5
2458254.5570851 | -45.0 | 0.5
2458254.5611942 | -43.1 | 0.5
SDSS J131632.04-003758.0: |
2457859.6997413 | 9.7 | 0.6
2457859.7038488 | 20.8 | 0.6
2457859.7079576 | 32.9 | 0.6
2457859.7120561 | 45.5 | 0.6
2457859.7161644 | 56.3 | 0.6
2457859.7202732 | 66.9 | 0.5
2457859.7243900 | 76.8 | 0.6
2457859.7285012 | 86.4 | 0.6
2457859.7326015 | 97.0 | 0.6
SDSS J145238.12+204511.9: |
2457859.7397896 | -398.8 | 1.2
2457859.7438907 | -396.9 | 1.2
2457859.7480076 | -365.3 | 1.3
2457859.7521164 | -319.5 | 1.2
2457859.7562253 | -254.9 | 1.3
2457859.7603342 | -179.0 | 1.2
2457859.7644433 | -93.6 | 1.3
2457859.7685519 | -8.9 | 1.3
2457859.7726606 | 77.4 | 1.2
2457859.7790416 | 197.4 | 1.1
2457859.7831448 | 257.7 | 1.2
2457859.7872438 | 298.6 | 1.2
2457859.7913531 | 319.6 | 1.2
2457859.7954723 | 322.0 | 1.2
2457859.7995928 | 298.8 | 1.1
2457859.8037007 | 259.7 | 1.2
2457859.8078105 | 200.8 | 1.2
2457859.8119183 | 131.8 | 1.3
SDSS J220848.32+003704.6: |
2457995.5544827 | 72.3 | 2.5
2457995.5591719 | 131.3 | 2.5
2457995.5638697 | 185.2 | 2.1
2457995.5685470 | 220.8 | 2.0
2457995.5732435 | 251.9 | 2.7
2457995.5779321 | 243.1 | 1.9
2457995.5826196 | 225.5 | 2.0
2457995.5873061 | 194.5 | 2.6
2458000.6254337 | 142.5 | 3.3
2458000.6301217 | 205.8 | 3.3
2458000.6348104 | 232.0 | 2.9
2458000.6394954 | 240.4 | 2.4
2458000.6488727 | 219.7 | 2.2
2458000.6441842 | 248.8 | 2.3
Table 8: continued HJD(mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$)
---|---|---
2458000.6535578 | 188.9 | 2.1
2458000.6582466 | 137.7 | 2.1
SDSS J222918.95+185340.2: |
2457995.6374760 | 136.8 | 0.8
2457995.6415812 | 116.6 | 0.9
2457995.6456889 | 88.4 | 1.1
2457995.6498000 | 58.9 | 0.9
2457995.6539078 | 31.1 | 0.9
2457995.6580156 | 1.8 | 1.0
2457995.6621389 | -28.0 | 0.9
2457995.6662474 | -55.3 | 0.9
2457995.6703563 | -80.2 | 0.9
2457995.6771421 | -116.8 | 0.8
2457995.6812423 | -136.9 | 0.8
2457995.6853550 | -154.9 | 0.8
2457995.6894628 | -166.9 | 0.8
2457995.6935752 | -178.3 | 0.8
2457995.6976816 | -182.1 | 0.9
2457995.7017892 | -189.8 | 0.8
2457995.7058996 | -188.7 | 0.9
2457995.7100199 | -183.5 | 0.8
2457996.6735976 | -133.3 | 0.8
2457996.6777038 | -116.0 | 0.9
2457996.6818124 | -90.8 | 0.9
2457996.6859214 | -64.8 | 0.9
2457996.6900201 | -40.1 | 0.9
2457996.6941275 | -13.4 | 0.8
2457996.6982373 | 15.8 | 0.8
2457996.7023461 | 40.9 | 0.7
2457996.7064656 | 68.6 | 0.8
2457996.7131085 | 110.7 | 0.8
2457996.7172065 | 132.5 | 0.8
2457996.7213169 | 156.5 | 0.8
2457996.7254258 | 172.6 | 0.8
2457996.7295491 | 189.5 | 0.9
2457996.7336534 | 198.8 | 0.9
2457996.7377623 | 205.8 | 0.9
2457996.7418711 | 213.9 | 0.9
2457996.7459798 | 213.8 | 0.9
## Appendix B White dwarf velocity measurements
Table 9: White dwarf velocity measurements for SDSS J084841.17+232051.7 from the Ca ii 3934Å absorption line HJD(mid-exposure) | RV ($\mathrm{km\,s^{-1}}$) | Err ($\mathrm{km\,s^{-1}}$)
---|---|---
2457859.4846316 | 46.5 | 4.5
2457859.4919063 | 53.8 | 4.9
2457859.4991749 | 68.6 | 4.9
2457859.5064655 | 88.9 | 4.9
2457859.5137851 | 109.6 | 4.9
2457859.5210477 | 131.8 | 5.1
2458224.5472608 | 119.8 | 4.9
2458224.5546431 | 140.2 | 4.5
2458224.5620364 | 150.8 | 4.6
2458224.5694316 | 160.8 | 4.7
2458224.5768256 | 174.5 | 4.9
2458226.5367985 | 84.6 | 4.7
2458226.5441880 | 74.3 | 4.6
2458226.5515741 | 51.4 | 4.9
2458226.5589551 | 34.5 | 4.8
2458226.5663385 | 13.4 | 5.1
2458227.5137585 | 107.5 | 4.9
2458227.5211569 | 125.4 | 5.0
2458227.5285455 | 140.4 | 5.1
2458227.5359294 | 149.5 | 5.0
2458227.5433194 | 168.6 | 4.7
|
# A decade of radial-velocity monitoring of Vega and new limits on the
presence of planets
Spencer A. Hurt Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA Samuel N. Quinn Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA David W. Latham Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Andrew Vanderburg Department of Astronomy, University of Wisconsin─-Madison, 475 North Charter Street, Madison, WI 53706, USA Gilbert A. Esquerdo Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Michael L. Calkins Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Perry Berlind Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Ruth Angus Department of Astrophysics, American Museum of Natural History, 200 Central Park West, Manhattan, NY, USA Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, Manhattan, NY, USA Christian A. Latham Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA George Zhou NASA Hubble Fellow Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
###### Abstract
We present an analysis of $1524$ spectra of Vega spanning $10$ years, in which
we search for periodic radial velocity variations. A signal with a periodicity
of $0.676$ days and a semi-amplitude of $\mathord{\sim}10$ m s-1 is consistent
with the rotation period measured over much shorter time spans by previous
spectroscopic and spectropolarimetric studies, confirming the presence of
surface features on this A0 star. The timescale of evolution of these features
can provide insight into the mechanism that sustains the weak magnetic fields
in normal A type stars. Modeling the radial velocities with a Gaussian process
using a quasi-periodic kernel suggests that the characteristic spot evolution
timescale is $\mathord{\sim}180$ days, though we cannot exclude the
possibility that it is much longer. Such long timescales may indicate the
presence of failed fossil magnetic fields on Vega. TESS data reveal Vega’s
photometric rotational modulation for the first time, with a total amplitude
of only $10$ ppm, and a comparison of the spectroscopic and photometric
amplitudes suggest the surface features may be dominated by bright plages
rather than dark spots. For the shortest orbital periods, transit and radial
velocity injection recovery tests exclude the presence of transiting planets
larger than $2$ $\,R_{\rm\earth}$ and most non-transiting giant planets. At
long periods, we combine our radial velocities with direct imaging from the
literature to produce detection limits for Vegan planets and brown dwarfs out
to distances of $15$ au. Finally, we detect a candidate radial velocity signal
with a period of $2.43$ days and a semi-amplitude of $6$ m s-1. If caused by
an orbiting companion, its minimum mass would be $\mathord{\sim}20$
$\,M_{\rm\earth}$; because of Vega’s pole-on orientation, this would
correspond to a Jovian planet if the orbit is aligned with the stellar spin.
We discuss the prospects for confirmation of this candidate planet.
††software: astropy (Astropy Collaboration et al. 2018), batman (Kreidberg
2015), emcee (Foreman-Mackey et al. 2013), matplotlib (Hunter 2007), numpy
(Harris et al. 2020), PyMC3 (Salvatier et al. 2016), radvel (Fulton et al.
2018), SOAP 2.0 (Dumusque et al. 2015), Transit Least Squares (Hippke & Heller
2019)††facilities: FLWO:1.5m (TRES), TESS
## 1 Introduction
The search for exoplanets has traditionally focused on low-mass (FGKM) stars,
as intermediate mass stars pose several observational challenges. For example,
their larger size and mass translate to smaller transit and radial velocity
signals for a given planet size, and rapid rotation and reduced radial
velocity information content prevent precise Doppler spectroscopy (e.g.,
Beatty & Gaudi 2015). However, careful target selection and expanded
observational techniques have helped to overcome these difficulties and have
led to new opportunities for planet characterization. When A-type stars leave
the main sequence, they cool and spin down, providing RV surveys a means to
explore planet populations around intermediate mass stars. Observations of
post-main sequence stars show that ‘retired A stars’ are more likely than Sun-
like stars to host massive planets (Johnson et al. 2010a, b; Ghezzi et al.
2018). However, these analyses suggest a low occurrence rate of planets on
close-in orbits. Data from NASA’s Transiting Exoplanet Survey Satellite (TESS;
Ricker et al. 2015), reveal that the same is true of A-type main sequence
stars: the occurrence rate of hot Jupiters orbiting A stars is low, and not
dissimilar to that of hot Jupiters orbiting solar-type stars (Zhou et al.
2019).
A-type stars are often over-represented in wide-field transit surveys due to
their intrinsic brightness, so despite the challenges of detecting and
characterizing hot Jupiters orbiting A stars, many have now been discovered.
While A-type stars are often rapidly rotating and this poses problems for mass
measurements, it can facilitate characterization sometimes not possible for
other stars. Stellar obliquity, for example, can be measured from
spectroscopic transit observations or transits across a gravity darkened
stellar surface. For hot Jupiter hosts above the Kraft break (Kraft 1967),
spin-orbit misalignment is often observed (such that stellar obliquities
appear to be consistent with an isotropic distribution; Winn et al. 2010;
Schlaufman 2010; Albrecht et al. 2012), and A stars appear to be no exception
(e.g., Collier Cameron et al. 2010; Zhou et al. 2016, 2019; Ahlers et al.
2020a, b). It is unclear if this is due to primordial misalignment or orbital
migration and whether this stellar obliquity extends to small planets or long
period planets.
Due to their intrinsic brightness, A stars are good targets for imaging
surveys, for which the brightness of the stars is among the primary concerns.
A stars are also intrinsically young (since their main-sequence lifetimes are
short), which enhances the likelihood of detecting debris disks before they
disperse or self-luminous planets and brown dwarfs before they cool. Vega is a
$0$th magnitude A0V star (see Table 1 for additional stellar parameters) and
the anchor of the Vega magnitude system. Given its brightness and its special
history as a spectrophotometric calibrator (see, e.g., Hayes & Latham 1975),
it is a particularly well observed star.
Ever since the IRAS discovery of a circumstellar disk (Aumann et al. 1984),
Vega has been a frequent target of imaging studies. Though early detections of
the dust around Vega appeared to show a clumpy, asymmetrical formation
(Holland et al. 1998; Koerner et al. 2001; Wilner et al. 2002; Marsh et al.
2006), more recent data reveal that the disk is smoother than originally
thought (Su et al. 2005; Sibthorpe et al. 2010; Piétu et al. 2011; Hughes et
al. 2012; Holland et al. 2017). ALMA observations have resolved the structure
of Vega’s outer dust belt, which extends to $150$–$200$ au and has a steep
inner edge at $60-80$ au (Matrà et al. 2020). Spitzer observations detect mid-
IR excess in the disk consistent with an asteroid belt located at $14$ au (Su
et al. 2013). And near-IR excess has been detected close to the star,
corresponding to hot dust (Absil et al. 2006; Defrère et al. 2011).
Table 1: Stellar Parameters of Vega
Parameter | Value | Units | Source
---|---|---|---
R.A. | $18:36:56.336$ | hh:mm:ss | (1)
Dec. | $38:47:01.280$ | dd:mm:ss | (1)
$\mu_{\alpha}$ | $200.94$ | mas yr-1 | (1)
$\mu_{\delta}$ | $286.23$ | mas yr-1 | (1)
Parallax | $130.23\pm 0.36$ | mas | (1)
Distance | $7.68\pm 0.02$ | pc | (1)
Inclination | $6.2\pm 0.4$ | degrees | (2)
Rotational period | $0.71\pm 0.02$ | days | (2)
$T\textsubscript{eff}$ (Apparent) | $9660\pm 90$ | K | (2)
$T\textsubscript{eff}$ (Pole) | $10070\pm 90$ | K | (2)
$T\textsubscript{eff}$ (Equator) | $8910\pm 130$ | K | (2)
$R\textsubscript{pol}$ | $2.418\pm 0.012$ | $\,R_{\sun}$ | (2)
$R\textsubscript{eq}$ | $2.726\pm 0.006$ | $\,R_{\sun}$ | (2)
Mass | $2.15\begin{subarray}{c}+0.10\\\ -0.15\end{subarray}$ | $\,M_{\sun}$ | (2)
Upper Age Estimate | $700\begin{subarray}{c}+150\\\ -75\end{subarray}$ | Myr | (2)
Lower Age Estimate | $471\pm 57$ | Myr | (3)
Notes: (1) van Leeuwen (2007); (2) Monnier et al. (2012); (3) Yoon et al.
(2010)
While Yelverton et al. (2020) show that there are no clear or strong planet-
debris disk correlations, systems such as HR $8799$ (Marois et al. 2008,
2010), $\beta$ Pic (Lagrange et al. 2009, 2010), and 51 Eridani (Macintosh et
al. 2015) provide examples of stars hosting both imaged planets and a disk.
Furthermore, features in a disk can be used to investigate the possible
presence of planets and their properties, and Vega’s disk is a complex system
that contains features that could arise from a planetary system. For a star of
Vega’s age, circumstellar disks are maintained by debris from colliding
planetesimals (Wyatt 2008). The warm and cold belts are potential sources for
this debris, but radiation pressure, stellar winds, and Poynting Robertson
drag forces mean a high dust production rate is necessary to maintain the
disk. Defrère et al. (2011) conclude that major dynamic perturbations are
necessary to produce the quantities of dust observed, suggesting a system of
giant planets migrating outwards. Raymond & Bonsor (2014) use dynamical
simulations to find that low-mass, closely-spaced planets could efficiently
scatter exocomets inwards, accounting for the hot dust. A possible
configuration includes a system of planets between $5$ and $60$ au, wherein
the outermost planets have masses less than about $20$ $\,M_{\rm\earth}$, and
they suggest that a Jupiter mass planet beyond $15$ au would disrupt the
inward-scattering chain.
Gaps in the disk can also be used to infer the presence of planets. Observed
depletion of dust between detected belts has been suggested to indicate the
presence of multiple long-period giant planets (Su et al. 2013), a single $3$
$\,M_{\rm J}$ planet at $75$ au (Zheng et al. 2017), or a chain of low-mass
planets or a Saturn-mass planet (Bonsor et al. 2018). Matrà et al. (2020)
argue that the steep inner edge of the cold belt cannot be explained by
collisional evolution, which would result in a shallow inner slope. It could
be explained by a chain of planets in which the outermost is located near $70$
au and has a mass greater than $6$ $\,M_{\rm\earth}$, or by a single planet
with a mass of $5$ $\,M_{\rm J}$ and an orbital radius of $50-60$ au. Clearly
not all of these architectures can be present simultaneously, but the hot
dust, gap structures, and characteristics of the cold belt all allow for the
existence of a system of planets.
Figure 1: Relative radial velocities of Vega, derived from TRES spectra.
We view Vega approximately pole-on, allowing a coplanar planet on a near-
circular orbit to always be observed near its maximum separation, which
further improves Vega’s suitability for direct imaging. In the most recent
published search, Meshkat et al. (2018) explored the inner $15$ au using the
P1640 instrument on Palomar’s 5.1-m Hale telescope and placed mass detection
limits of $\mathord{\sim}20-85$ $\,M_{\rm J}$. Heinze et al. (2008) provide
MMT observations to constrain objects between $20$ and $80$ au, finding an
upper limit of $\mathord{\sim}5-20$ $\,M_{\rm J}$. And with Spitzer
observations, Janson et al. (2015) rule out any planets with a mass greater
than $\mathord{\sim}1-4$ $\,M_{\rm J}$ between $100$ and $200$ au. These
represent the lowest mass limits in the literature for objects orbiting Vega.
Table 2: TRES Radial Velocities of Vega BJD | RV | $\sigma$
---|---|---
| ( m s-1) | ( m s-1)
2456025.026123 | -10.0 | 23.1
2456026.017315 | -20.8 | 22.8
2456027.016471 | -16.1 | 25.4
2456028.020389 | -48.7 | 24.9
2456029.020685 | 17.2 | 24.4
2456030.020025 | -69.8 | 21.3
2456031.019574 | -37.6 | 19.8
2456033.007404 | -59.8 | 19.7
2456034.010060 | 2.9 | 16.0
The inner working angles of direct imaging instruments prevent good limits on
the existence of planets for scales similar to or smaller than the inner solar
system. Radial velocities can be used to complement imaging at small
separations and for somewhat lower masses at large separations. The problem of
rapid rotation remains, of course. Spectropolarimetric observations measure a
rotation period of $0.678\begin{subarray}{c}+0.036\\\ -0.029\end{subarray}$
days (Alina et al. 2012). A spectroscopic analysis by Böhm et al. (2015)
indicates stellar activity modulated at the same period. These results suggest
a stellar rotational velocity of nearly $200$ km s-1, but the pole-on
orientation leads to projected rotation of only about $20$ km s-1, which does
not severely limit radial velocity precision. Indeed, velocities from Böhm et
al. (2015) display scatter on the order of only 10 m s-1. One concern is that
because of the stellar orientation, radial velocities will be most sensitive
to planets misaligned with the star. However, at least for giant planets on
short periods, misaligned orbits appear to be the rule rather than the
exception for A-star hosts.
In this work, we present our analysis of $1524$ spectra of Vega, which can be
used to study stellar activity and search for planets smaller than $1$
$\,M_{\rm J}$ on short periods and massive planets out to 15 au. In Section 2,
we present the observations and data reduction. In Section 3, we explore the
radial velocities for periodic signals and discuss their origin. In Section 4,
we calculate detection limits. Lastly, in Section 5, we discuss our results.
## 2 Observations
### 2.1 TRES Spectroscopy
We obtained high-resolution spectra of Vega with the Tillinghast Reflector
Echelle Spectrograph (TRES; Fűrész 2008), which is mounted on the 1.5-m
Tillinghast Reflector at Fred L. Whipple Observatory on Mount Hopkins, AZ. It
has a resolving power of $R\mathord{\sim}44,000$, and a wavelength coverage of
$3850$–$9100$ Å. We obtained a total of 1524 spectra spanning the 10-year
period between UT 2009 June 13 and 2019 October 24. Typical exposure times
ranged from a few seconds to a few tens of seconds, achieving signal-to-noise
ratios (SNR) between about 300 and 1000 per resolution element.
Figure 2: Top: The TESS light curve of Vega, after decorrelation against the
spacecraft quaternion time series, as described in § 2.2. Individual 30-minute
cadence measurements are shown as small, light purple circles, while data in
6.5-hour bins are shown as large, dark purple circles. The median 6.5-hour
standard deviation is 5.0 ppm. Lower Left: Lomb-Scargle periodogram of the
TESS light curve, showing a peak at 0.68 days, consistent with the rotation
period of the star. Lower Right: The binned, phase-folded TESS light curve,
which shows a total variation of $\mathord{\sim}10$ ppm.
We optimally extracted and reduced the spectra following the procedures
outlined in Buchhave et al. (2010), and while we begin by following the radial
velocity measurement outlined therein, our final velocity extraction includes
a few key differences. We first cross-correlate each spectrum of Vega against
the strongest exposure, treating each spectral order separately. The relative
RV for each exposure is taken to be the location of the peak of the summed CCF
(across all orders) from that spectrum. The internal RV uncertainty for each
observation is taken to be the standard deviation of the locations of the CCF
peak of each order for that spectrum. Next, we shift and median combine the
1524 spectra to generate a master template spectrum. In the case of Vega, the
template SNR is 26,000 per resolution element. Though exposures of Vega are
typically short enough that cosmic ray rejection is not very important, we
identify outlier pixels and replace them with the median spectrum at that
location. Finally, we re-run the order-by-order cross-correlation, this time
against the high-SNR template.
TRES was not designed for long-term stability at the level of meters per
second, and has at times experienced drifts and jumps in its instrumental zero
point as large as a few tens of m s-1. Changes of this magnitude can mimic or
mask the presence of long-period companions. To combat this problem, we track
the zero point with nightly observations of several RV standard stars,
allowing us to measure and correct for zero point changes over time. From the
standard deviation of the RV standards, we also estimate the instrumental RV
noise floor, to be added in quadrature with the internal error estimates
described above. The TRES instrumental precision at the beginning of our data
set was $\mathord{\sim}50$ m s-1, but by the end of 2010 had improved to
$\mathord{\sim}10$–$15$ m s-1, thanks in large part to hardware upgrades.
Though these introduced some zero point changes, they are corrected for in the
same way as other zero point shifts. Our final, zero-point-corrected, relative
RVs are shown in Figure 1 and presented in Table 2.
### 2.2 TESS Photometry
Vega was observed by NASA’s TESS mission (Ricker et al. 2015) in Sector 14,
between UT 2019 July 18 and 2019 August 15, and in Sector 26, between UT 2020
June 8 and 2020 July 04. The star is bright enough to fill the full well depth
of hundreds of pixels, but the TESS detector is designed to preserve the flux
by spilling into neighboring pixels. As long as the star is not too close to
the edge of the detector, the full frame images (FFIs), returned at 30-minute
cadence, can be used to extract photometry from an area encompassing all of
the light. The FFIs were processed by the Science Processing Operations Center
(SPOC) at NASA Ames (Jenkins et al. 2015, 2016), adapted from the pipeline
originally developed for the Kepler mission (Jenkins et al. 2010). We then
performed photometry using apertures shaped to trace the distribution of
charge on the TESS images, including 3,625 pixels in Sector 14 and 4,335
pixels in Sector 26. We corrected the flux for sources within the aperture
using TESS magnitudes listed in the TESS Input Catalog. To account for
systematics caused by the motion of the spacecraft, we followed the procedure
outlined in Vanderburg et al. (2019). Namely, we assumed that systematics
caused by changes in spacecraft pointing can be corrected by decorrelating
against the background scattered light signal and combinations of the
spacecraft quaternions, which encode the pointing of the spacecraft every 2
seconds. We produced quaternion time series by calculating the standard
deviation and mean quaternions during each 30-minute exposure. Ultimately, we
found the best performance (i.e., lowest resulting light curve scatter) when
decorrelating against the scattered light signal and the first, second, and
third order of these quaternion time series. We exclude a few hours of data
most strongly affected by scattered light at the beginning of each orbit in
Sector 26, and we use a spline to flatten remaining low frequency systematics
that occur on the timescale of the spacecraft orbit.
Figure 3: GLS periodogram (top) and window function (bottom) for TRES radial
velocities. The horizontal lines in the periodogram correspond to FAPs of
$0.05$, $0.01$, and $0.001$. The peak located at $0.677$ days, corresponding
the the rotational period, is marked in red.
The resulting light curve is shown in Figure 2, and is remarkably quiet. The
median standard deviation in 6.5-hour bins is only 5.0 ppm. We run Transit
Least Squares (TLS, Hippke & Heller 2019) in search of transiting planets but
we find no evidence for transit-like features, and in Section 4.2 produce
detection limits using transit injection recovery. The maximum power in a
Lomb-Scargle periodogram occurs for a period of 0.68 days (Figure 2, lower
left panel), which is consistent with previous measurements of the rotation
period of Vega. We do detect the signal in both sectors individually at lower
significance, but because of the year-long gap between sectors we cannot
confirm whether the signal is stable in phase for the full time span. It is
interesting to note that while the period is still significant ($>3\sigma$) if
we exclude photometric outliers, the significance is highest when they are
included. This suggests that the apparent outliers vary in phase with the rest
of the data and may be associated with stellar activity rather than
systematics related to the instrument or data processing. This could indicate
that some surface features are evolving in brightness on very short
timescales, while others appear more stable over the course of a month, and
perhaps over the full year spanned by the TESS data.
## 3 Analysis of Radial Velocity Signals
In this section, we present analyses of the TRES data to search for and
characterize radial velocity signals arising from rotation and orbiting
companions. We begin our search for periodic signals in the radial velocity
data using a generalized Lomb Scargle periodogram (GLS, Zechmeister & Kürster
2009) and the window function, shown in Figure 3. The periodogram contains
many significant peaks with false alarm probabilities (FAPs) below $0.001$,
most of which fall at short periods. The window function reflects our nightly
observing cadence, with strong power at a sampling rate of $1\ \rm{day}^{-1}$.
Given that the majority of signals in the periodogram fall below $1$ day, the
true frequency of variability is likely above the Nyquist frequency of .5
days-1, and the rest of the peaks are aliases. We discuss identification of
the true peak below and then model this signal and others with Gaussian
processes and Keplerians.
Table 3: Significant Periodogram Signals (FAP < 0.1%) Period | Frequency | Notes | Amplitude | FAP
---|---|---|---|---
(days) | (days-1) | | ( m s-1) |
0.105 | 9.48 | $\mathrm{F_{rot}}+8$ | 7.16 | 7.71e-6
0.111 | 9.00 | $\cdots$ | 7.12 | 2.49e-5
0.118 | 8.48 | $\mathrm{F_{rot}}+7$ | 8.21 | 3.80e-8
0.134 | 7.48 | $\mathrm{F_{rot}}+6$ | 7.60 | 3.37e-6
0.154 | 6.48 | $\mathrm{F_{rot}}+5$ | 7.59 | 7.63e-6
0.182 | 5.48 | $\mathrm{F_{rot}}+4$ | 7.58 | 1.06e-6
0.221 | 4.52 | $\mathrm{\left|F_{rot}-6\right|}$ | 8.31 | 2.35e-8
0.223 | 4.48 | $\mathrm{F_{rot}}+3$ | 9.15 | 8.23e-11
0.263 | 3.80 | $\cdots$ | 6.88 | 9.05e-5
0.284 | 3.52 | $\mathrm{\left|F_{rot}-5\right|}$ | 8.17 | 5.79e-9
0.288 | 3.48 | $\mathrm{F_{rot}}+2$ | 9.43 | 1.06e-10
0.397 | 2.51 | $\mathrm{\left|F_{rot}-4\right|}$ | 7.40 | 4.36e-7
0.403 | 2.48 | $\mathrm{F_{rot}}+1$ | 8.72 | 8.48e-10
0.677 | 1.48 | $\mathrm{F_{rot}}$ | 9.06 | 9.86e-10
2.10 | 0.478 | $\mathrm{F_{rot}}-1$ | 7.29 | 2.31e-5
### 3.1 A Signal Arising from the Rotation of Active Regions
#### 3.1.1 Detection of the Activity Signal
One of the strongest peaks in the RV periodogram is located at $0.677$ days,
which corresponds to the previously reported rotation period (Alina et al.
2012; Böhm et al. 2015). Table 3 lists all of the significant peaks in the
periodogram, and shows how they relate to the rotation period. Nearly every
signal is an alias of the $0.677$-day periodicity, with harmonics also present
but at lower significance. While we have good reason to suspect that this
value corresponds to the true signal, we conduct a more rigorous test of the
aliases using the methodology of Dawson & Fabrycky (2010). This procedure
simulates a sinusoidal signal at the candidate period using the time stamps of
the observed data. The amplitudes and phases of each peak in the periodogram
are then compared between the synthetic and actual data. The simulated signal
that best reproduces the results from the observations is chosen as the true
period. While the signal we observe is likely caused by activity, and
therefore is somewhat irregular, this test does confirm $0.677$ days as the
best match to the observed periodogram.
Figure 4: Stacked periodogram of the TRES RVs, showing the periodicity at
0.677 days. The strength of the peak falls off around 1000 observations, after
which multiple peaks emerge, suggesting that this signal originates from
active surface regions rather than an orbiting companion.
A useful diagnostic for distinguishing between activity and orbiting
companions is the stacked periodogram, which is described by Mortier & Collier
Cameron (2017). As the number of observations included in a periodogram
increases, the power of a peak corresponding to a planetary signal should
monotonically increase, within the limits of the noise, while the power of a
peak corresponding to rotation of features on the stellar surface may not, due
to changes in the phase of the signal driven by the evolution of active
regions. In Figure 4, we present a stacked periodogram for our TRES radial
velocities. For the first $\mathord{\sim}1000$ observations, the power at
$0.677$ days increases, but afterwards, the power falls off. This supports the
conclusion that the periodicity originates from activity, and its variation
over the course of hundreds of observations is consistent with a long
timescale of evolution.
#### 3.1.2 Modeling the Activity Signal
We first attempt to model the stellar activity with a Keplerian orbit using
the radvel Python package (Fulton et al. 2018). However, Markov chain Monte
Carlo (MCMC) samples failed to converge. Additionally, a periodogram of the
residuals to the best fit showed significant power remaining near $0.677$
days. This suggests that a Keplerian model does not adequately fit the signal,
which is expected given the structure seen in the stacked periodogram. It
would have been surprising for the stellar surface to be described well by a
single Keplerian (corresponding to a single active region) over the span of a
decade.
We next consider a model in which active regions are very long-lived, but
multiple regions may exist with slightly different periods due to differential
rotation. For this exercise, we follow an iterative whitening procedure in
which we fit a Keplerian with initial conditions determined from the
periodogram following Delisle et al. (2016). After refining that fit, we
repeat the process with the strongest peak in the periodogram of the
residuals, until there are no signals remaining with ${\rm FAP}<0.1\%$. This
procedure returns only two signals, $0.676085$ and $0.676684$ days, both
coincident with the rotation period. This makes sense given that the same
hemisphere of the star is almost always visible. Very low latitude
(equatorial) active regions, visible for only a portion of the rotational
phase, may appear at harmonics of the rotation period. Otherwise, variability
caused by static surface features on Vega should be well described by a sum of
Keplerians at the rotation period, with small differences in period caused by
differential rotation. In Table 4 we present the parameters of the Keplerians
used to model these two dominant signals. Some power remains in the
periodogram of residuals near the rotation period, its aliases, and its
harmonics, which may indicate the presence of additional surface features.
However, it may also indicate that the surface of Vega is not stable on decade
timescales, in which case we need to apply a model that can better address its
time variability.
Quasi-periodic Gaussian processes (Rasmussen & Williams 2006; Roberts et al.
2012) are known to represent the effects of active regions rotating in and out
of view (Haywood et al. 2014; Rajpaul et al. 2015; Angus et al. 2018). Using
radvel, we model the correlated stellar noise with a kernel matrix whose
elements are defined as
$C_{ij}=\eta_{1}^{2}\exp{\left(-\frac{\left|t_{i}-t_{j}\right|}{\eta_{2}^{2}}-\frac{\sin{\left(\frac{\pi\left|t_{i}-t_{j}\right|}{\eta_{3}}\right)}}{2\eta_{4}^{2}}\right)},$
(1)
where $t_{i}$ and $t_{j}$ are observations made at any two times; $\eta_{1}$
is the amplitude hyperparamter; $\eta_{2}$ is the exponential decay timescale,
which is physically related to the lifetime of the active regions; $\eta_{3}$
is the period of the variability, corresponding to the rotational period; and
$\eta_{4}$ is the length scale of the periodic component, describing the high-
frequency variation in the stellar rotation. It is important to note, however,
that these physical interpretations are often not straightforward,
particularly given the degeneracies between the hyperparameters.
Table 4: Stellar activity RV models
Parameter | Units | Value
---|---|---
Multiple Keplerians |
$P_{1}$ | days | $0.676682\pm 0.000006$
$T_{c1}$ | BJD | $2457187.123\pm 0.013$
$e_{1}$ | $\cdots$ | $0.438\pm 0.052$
$\omega_{1}$ | deg | $344\pm 12$
$K_{1}$ | m s-1 | $11.6\pm 1.2$
$P_{2}$ | days | $0.676086\pm 0.000006$
$T_{c2}$ | BJD | $2457187.195\pm 0.051$
$e_{2}$ | $\cdots$ | $0.46\pm 0.22$
$\omega_{2}$ | deg | $179\pm 15$
$K_{2}$ | m s-1 | $11.1\pm 3.8$
$\gamma$ | m s-1 | $-33.69\pm 0.69$
$\sigma_{\rm jit}$ | m s-1 | $13.1\pm 1.0$
Quasi-Periodic Gaussian Process |
$\eta_{1}$ | m s-1 | $14.4^{+2.2}_{-1.8}$
$\eta_{2}$ | days | $179^{+65}_{-58}$
$\eta_{3}$ | days | $0.6764^{+0.13}_{-0.00021}$
$\eta_{4}$ | $\cdots$ | $0.44^{+0.16}_{-0.15}$
$\gamma$ | m s-1 | $-33.9^{+2.9}_{-2.8}$
$\sigma_{\rm jit}$ | m s-1 | $12.9\pm 1.4$
Note. — The full set of velocities is available as a machine-readable table. A
portion is shown here for form and content.
No trend is apparent in our data or the residuals of early fits and we choose
to exclude a linear or quadratic term in our model. However, we do include
offset and jitter terms to account for instrumental offset and noise. Priors
are only placed on parameters to keep them within physically possible limits.
The complete log-likelihood of this model is
$\ln\mathcal{L}=-\frac{1}{2}r^{T}K^{-1}r-\frac{1}{2}\ln{\left(\det
K\right)}-\frac{n}{2}\ln{\left(2\pi\right)},$ (2)
where $r$ is the vector of residuals, $K$ is the covariance matrix, and $n$ is
the number of data points.
We perform an affine-invariant Markov chain Monte Carlo (MCMC) exploration of
the parameter space using the ensemble sampler emcee (Foreman-Mackey et al.
2013, 2019). Our MCMC analysis used 8 ensembles of 50 walkers and converged
after $4950000$ steps, achieving a maximum Gelman-Rubin statistic (Gelman &
Rubin 1992) of $1.006$. The resulting posterior distributions are shown in
Table 4. We do note that this model is over-fitting the data, with a reduced
$\chi^{2}$ statistic of $0.869$; either the model is fitting noise or our
cadence is not sufficient to constrain high-frequency variations in the
rotation signal. In either case, the Gaussian process is likely too complex of
model for our data. We check our results with a second GP fit of the radial
velocities using a quasi-periodic kernel with the PyMC3 Python package
(Salvatier et al. 2016), which returns results well within $1\sigma$ of those
in Table 4.
### 3.2 Additional Signals of Interest
The activity signal at the period of stellar rotation is by far the strongest
we observe, but other candidate signals have been reported in previous Vega
data sets, and we also search for signals of lower significance in the TRES
RVs.
#### 3.2.1 Previously Suggested Signals
Böhm et al. (2015) used their SOPHIE spectra to search for additional short-
period signals and reported a possible detection at $1.77\ {\rm d}^{-1}$
($0.56$ days) with an amplitude of $6$ m s-1. If it has a planetary origin, it
would correspond to roughly a Saturn-mass planet well aligned with the stellar
spin near the co-rotation radius of the star. However, no such periodicity
appears in our TRES observations. When we inject this signal into our data, we
find that it would have been clearly detected, which suggests that the signal
in the SOPHIE data set does not correspond to a planet. While we cannot
conclude with certainty what the source of that signal was, the SOPHIE data
only span $5$ nights and cover about $7$ hours each night, so one possibility
is that it may have resulted from a combination of short-timescale stellar
variability and the sampling.
#### 3.2.2 Signals at Long Periods
Because our Gaussian process model of the $0.677$-day signal over-fits the
data, it is difficult to use the residuals of that model to search for—or to
jointly fit—additional signals. However, we note that in the original
periodogram of our radial velocities, the strongest peak with a timescale
longer than a few days is located at $197.3\pm 4.8$ days. This signal is not
formally significant, though a Keplerian fit converges (to $194$ days) and
implies a minimum mass similar to that of Saturn. It is possible that the
presence of the high frequency activity signal is suppressing its significance
and that careful modeling of the activity may enhance it. We offer several
possible explanations for this peak and investigate the effect of stellar
activity modeling.
The first explanation is that it simply arises from white noise. Since it is
not formally significant, it would not be unusual to observe it by chance. A
second, similar interpretation is that the roughly half-year period may be an
artifact of the observing cadence, given that our observations are necessarily
seasonal, with breaks at the same time each year when Vega is behind the Sun
and also during the August monsoon season in Arizona. There is indeed a small
peak in the window function near the same period ($182.8\pm 3.5$ days; Figure
3). Our RV standard stars, which have been observed with similar cadence for
similar time spans, show similar window functions but do not show similar
periodogram peaks, which seems to suggest the window function is not to blame.
However, it is possible that Vega could be susceptible to systematics not seen
in the standard stars, which have exposure times an order of magnitude longer.
For example, in the uncorrected standard star RVs, we observe a slight
seasonal periodicity that is corrected in our zero point analysis. If the
tracking during very short exposures of Vega is imperfect, this could lead to
(systematic) differences in the illumination of the fiber as a function of
position on the sky, and ultimately leave uncorrected instrumental effects
that vary with season. As a result, we cannot rule out instrumental and window
function effects as the source of this signal.
A third possibility is that although the timescales are very different, the
signals at $0.677$ days and $194$ days may both originate from activity; the
timescale of evolution of the former ($179^{+65}_{-58}$ days) derived from the
GP fit is consistent with the latter. Because the GP activity model is
flexible enough to absorb the long-period signal even if it is real, we
examine the effect of the multiple-Keplerian model on this signal. If active
regions are long-lived and exist with very slightly different periods due to
differential rotation, then beating between the two frequencies will lead to
evolution of the observed variation on long timescales. Modeling and removing
the signal from the static active regions should reduce the significance of
signals related to the beat frequency, but should not generally absorb
unrelated signals like the GP does. After fitting the two dominant activity
signals near $0.677$ days, the FAP of the peak at $194$ days is reduced (to
$\mathord{\sim}5\%$). However, fitting one more Keplerian corresponding to the
next most significant harmonic ($P_{\rm rot}/2$) removes the 194-day signal
completely.
Given the very modest significance of the $194$-day signal, the existence of
plausible alternative explanations, and its disappearance when removing
Keplerians associated with the rotation period, we do not consider this to be
a planetary candidate. There are no other long-period signals of note in our
RVs.
Table 5: Candidate Planetary Companion to Vega Parameter | Units | Value
---|---|---
| | Eccentric | Circular
$P$ | days | $2.42977\pm 0.00016$ | $2.42971\pm 0.00018$
$T_{c}$ | BJD | $2457186.51\pm 0.18$ | $2457186.631\pm 0.067$
$e$ | $\cdots$ | $0.25\pm 0.15$ | $0$
$\omega$ | deg | $304\pm 40$ | $\cdots$
$K$ | m s-1 | $6.4\pm 1.1$ | $6.1\pm 1.1$
$\gamma$ | m s-1 | $-34.30\pm 0.75$ | $-34.34\pm 0.75$
$\sigma$ | m s-1 | $9.7\pm 1.0$ | $9.7\pm 1.0$
$m_{p}\sin{i}$ | $\,M_{\rm\earth}$ | $21.9\pm 5.1$ | $21.2\pm 3.8$
#### 3.2.3 A Candidate Planetary Companion
Interestingly, after removing two Keplerian signals associated with the
stellar rotation, the strongest remaining peak has a period of $2.43$ days and
a formal FAP of only $<0.01$. Unlike the $194$-day signal, it is robust to the
inclusion of additional Keplerians near the rotation period and its harmonics;
the FAP of the 2.43-day signal remains below $1\%$ whether we account for
stellar noise or not, and even if we include more complex static models of the
stellar surface with as many as 5 Keplerian components. We do not find any
relationship between this period and the rotation period and its aliases
listed in Table 3, and the uncertainties on the periods are much smaller than
the differences between them. A circular orbit with a period of $2.43$ days
has an RV semi-amplitude of about $6$ m s-1, corresponding to a minimum mass
of about $20$ $\,M_{\rm\earth}$. According to the difference in the Bayesian
Information Criterion ($\Delta{\rm BIC}$), a circular orbit is statistically
preferred, but we present both eccentric and circular solutions in Table 5.
Figure 5 shows the phase folded RVs of the candidate planetary signal. While
we cannot conclusively rule out false positive scenarios, we discuss in
Section 5 ways in which we might confirm the candidate with future
observations and analyses.
Figure 5: Left: TRES RVs (small light blue circles) phase-folded to the
$2.43$-day period of a candidate planet orbiting Vega after removal of
activity signal by fitting Keplerians at the rotation period. Data binned in
phase are shown as orange circles. Right: The same orbit on a different
velocity scale to better show the binned data. The purple open circles
represent phased, binned data from the intermediate steps of whitening the RVs
by removing Keplerians associated with the rotation period. A consistent orbit
is seen when removing 0, 1, 2, 3, 4, or 5 activity signals.
## 4 Detection limits
### 4.1 TRES Radial Velocities
#### 4.1.1 Isotropic Orbits
We begin the calculation of our detection limits by randomly generating
$\mathord{\sim}10^{6}$ models corresponding to planets with semi-major axes
between $0$ and $15$ au and masses ranging from $0$ to $100$ $\,M_{\rm J}$.
Each orbit is assigned an inclination (drawn from a uniform distribution in
$\cos{i}$), eccentricity (drawn from a beta distribution described by Kipping
2013, with parameters $a=0.867$ and $b=3.03$), an argument of periastron
(drawn from a uniform distribution), and a time of periastron passage (drawn
from a uniform distribution). With radvel, we calculate the expected radial
velocities for each orbit at our times of observation. We then add noise
scaled to the observed uncertainties at each time stamp. Using radvel, we fit
a flat line to the synthesized RVs and calculate the $\chi^{2}$ statistic and
its associated $p$ value for this model. Low $p$ values indicate that the
synthesized signal is unlikely to arise from white noise—i.e., it is
distinguishable from a flat line and we therefore consider it detected. To set
a threshold for detection, we follow Latham et al. (2002), who demonstrate
that $p<0.001$ is a conservative threshold below which signals are almost
entirely real. This makes sense, since data comprising only white noise will
exhibit $p<0.001$ only $0.1\%$ of the time. While we find that other methods
of injection recovery—such as a requirement that $\Delta{\rm BIC}$ between the
detected signal and a flat line model be greater than $10$—yield more
sensitive limits, we adopt the conservative $p(\chi^{2})<0.001$ because we do
not address correlated noise in our simulations; in the likely event that we
cannot perfectly model the stellar activity, it may hinder our ability to
detect some signals. The top panel of Figure 6 shows the distribution of
detection probabilities for isotropically oriented orbits, from which it can
be seen that we are sensitive to sub-Saturn masses orbiting at $0.1$ au and
masses as low as about $2\ \,M_{\rm J}$ at $6$ au, which corresponds to an
orbital period of $10$ years around Vega. Beyond $6$ au, our detection limits
degrade more quickly, as our data no longer cover an entire orbit.
Nonetheless, we are sensitive to the most massive giant planets all the way
out to $15$ au. Given the roughly isotropic distribution of stellar
obliquities for hot stars hosting transiting hot Jupiters, the decision to
draw inclinations from an isotropic distribution may be the most realistic
assumption for short periods. However, it is unclear whether the same is true
at long periods. We therefore also explore our detection limits for well-
aligned orbits in the following section.
#### 4.1.2 Well-Aligned Orbits
For our purposes, we will consider a well-aligned orbit to fall within
$5\degree$ of the stellar spin axis. Assuming Vega has an inclination of
$6.5\degree$, this means that a well aligned planet would have an inclination
between $1.5\degree$ and $11.5\degree$. To calculate detection limits for
well-aligned orbits, we follow the same steps as in the previous section, but
instead assign inclinations drawn from a uniform distribution between
$1.5\degree$ and $11.5\degree$. The resulting detection probabilities are
shown in the middle panel of Figure 6. While drawing from a distribution of
well aligned orbits (highly inclined to the line of sight) clearly reduces
detection probabilities, we are still sensitive to the most massive giant
planets as widely separated as $6$ au; beyond this distance, we are only
sensitive to brown dwarfs or stars.
Figure 6: Top: TRES detection probabilities for objects in isotropically
distributed orbits. Middle: Detection probabilities for objects in orbits
well-aligned with Vega’s spin-axis. Bottom: TRES detection probabilities
combined with Palomar P1640 detection limits from Meshkat et al. (2018) for
objects in well-aligned orbits.
#### 4.1.3 Including Direct Imaging Limits
The direct imaging limits from Meshkat et al. (2018) also explore the inner
$15$ au around Vega for planets. While our RV detection limits fall off
exterior to $\mathord{\sim}6$ au, Meshkat et al. (2018) have limited
sensitivities interior to this boundary. We also note that direct imaging and
radial velocities are most sensitive to planets at opposite inclinations.
Consequently, a brown dwarf that is missed by RVs because it is highly
inclined to the line of sight may be detected by direct imaging; one that is
too close in projection for direct imaging can be detected by RVs. By
combining our results, we are able to provide a more comprehensive limit on
the presence of widely separated companions.
We calculate the projected separation for each randomly drawn sample in
Sections 4.1.1 and 4.1.2 on UT 2016 Aug 19 and 2017 June 05 (two of the
observing times in Meshkat et al. 2018) using Equation (7) in Kane & Gelino
(2011). We consider the planet to be detectable in the data presented by
Meshkat et al. (2018) if its mass falls above the five-sigma H-band mass
limits for either of the observation times. We then use the same criterion as
before to determine if it is detectable in our radial velocities. For
isotropically distributed orbits, our RVs are more sensitive than the imaging
at all separations, but for long-period orbits aligned with the stellar spin,
imaging is more sensitive. The detection probabilities for well-aligned orbits
using RVs and direct imaging are shown in the bottom panel of Figure 6.
### 4.2 TESS Photometry
Using the Python package batman (Kreidberg 2015), we randomly generate
$\mathord{\sim}10^{5}$ transit models for planets with periods ranging from
0.5 to 30 days and radii between 1 and 8 $\,R_{\rm\earth}$. Each orbit is
assumed to be circular and semi-major axes are calculated using Kepler’s law,
assuming that the planet’s mass is negligible compared to Vega. Transit times
are randomly assigned from a uniform distribution and inclinations are drawn
from a uniform distribution in $\cos{i}$ such that
$0<\cos{i}<\frac{R_{\star}-R_{p}}{a},$ (3)
where $a$ is the semi-major axis and $R_{\star}$ and $R_{p}$ are respectively
the stellar and planetary radii, ensuring that the planet transits.
Additionally, each synthetic transit follows quadratic limb darkening laws for
an A0 star in the TESS bandpass with $u_{1}=0.1554$ and $u_{2}=0.2537$ (Claret
2018).
We then inject each model into our light curve and conduct a transit search
using TLS. We consider a transit to be detectable if the best period recovered
by TLS is within $1\%$ of the injected period and if at least one of the
transit times is within the same margin of an injected transit. Additionally,
because a transit could appear within the gaps of our light curve, any
recovered integer multiples of the injected period are considered detectable.
We also require a transit to be distinguishable from a false positive to be
considered a true recovery, ensuring that the signal would not be dismissed
due to low signal in a real search. Using synthetic light curves and transits,
Hippke & Heller (2019) find that a signal detection efficiency (SDE) threshold
of 7 corresponds to a false positive rate of $1\%$ for TLS. Therefore, for a
transit to be detectable, we also require that the best period have an SDE of
7 or greater.
Figure 7: Transit detection limits for TESS observations of Vega. For periods
shorter than a few days, we can rule out most transiting planets with radii
greater than 2 $\,R_{\rm\earth}$, and for periods out to 15 days, planets with
radii greater than 3 $\,R_{\rm\earth}$. Planets with longer periods do not
transit frequently enough to be consistently ruled out. However, we note that
for many of the larger injected planets not recovered according to our
criteria, single transit events would be easily visible by eye, so our
detection map should be viewed as a conservative limit.
The results of our injection recovery test are shown in Figure 7. At the
shortest periods, we are sensitive to transiting planets as small as $2$
$\,R_{\rm\earth}$, and as small as $3$ $\,R_{\rm\earth}$ for orbital periods
similar to the duration of a TESS orbit ($\lesssim 14$ days). For periods
longer than a couple weeks, our formal sensitivity drops, as fewer transits
are expected; some planets may exhibit only one or two transits total, and
some might not even be observed once by TESS due to data gaps between orbits.
Many of these will not pass our automated TLS criteria for detection. On the
other hand, the transits of even relatively small planets orbiting Vega would
be easily visible by eye in the quiet TESS light curve. A $4$
$\,R_{\rm\earth}$ transit, for example, would be about 180 ppm. It is clear
that there are no Neptune-sized planets transiting in the TESS data even once.
While the formal detection limits shown in Figure 7 are illustrative of the
types of transiting planets we could detect most easily, they should be taken
as a conservative estimate.
## 5 Discussion
### 5.1 Stellar Activity
Figure 8: SOAP 2.0 simulations of the radial velocity and photometric
variation induced by spots (blue) and plages (red) at different latitudes. The
top row simulates a spot or plage on Vega’s equator, the second row a latitude
of $15\degree$, the third a latitude of $45\degree$, and the last a latitude
of $80\degree$. Dark spots deviate from Vega’s effective temperature by 750 K
and the sizes of active regions are varied to obtain an RV amplitude similar
to the observed values of $\mathord{\sim}15$ ms-1. While spots and plages are
both able to reproduce the RV variation, the change in flux is much lower for
plages. Vega’s surface may be more complicated than this toy model, but this
suggests that plages are more likely the dominant driver of the radial
velocity signal and the quiet TESS light curve.
Rotational modulation has consistently been found in spectroscopic
observations of Vega (Böhm et al. 2015; Petit et al. 2017), providing evidence
for active regions on the surface of the star. While chemically peculiar A
stars are known to have star spots, normal A stars conventionally do not
exhibit the same traits. Vega is the first normal A-type star observed to have
a weak surface magnetic field (Lignières et al. 2009; Petit et al. 2010),
accounting for the rotational modulation. However, the origins of these
magnetic fields are uncertain. One possible mechanism is dynamo action, where
thin convective layers host a dynamo driven by convective motion, generating a
magnetic field. On the other hand, the fields could be ‘failed fossils,’ which
are generated in the early phases of a star’s life and dynamically evolve
towards fossil equilibrium (e.g., Braithwaite & Cantiello 2013). These models
can be distinguished by the time variation of the magnetic field. Dynamos are
intrinsically variable, leading to spot lifetimes similar to the rotation
period. In contrast, failed fossil fields would evolve much more slowly.
Vega appears to have a very complex magnetic field. Böhm et al. (2015) find no
evidence for rapid variability consistent with dynamo action in their data,
suggesting the presence of star spots that last for over five days. However,
applying Doppler imaging techniques to the same data set, Petit et al. (2017)
do find rapidly evolving surface features in combination with stable
structures. Cantiello & Braithwaite (2019) suggest that this rapid variation
is a sign of dynamo-generated magnetic fields.
While our spectroscopic observations are likely not high-cadence enough to
characterize the short term evolution of co-rotating structures, they stretch
over a span of time much longer than the five consecutive nights used by Böhm
et al. (2015) and Petit et al. (2017), which may provide insight regarding the
long-term evolution. In Section 3, we described that our RVs can be modeled by
either a Gaussian process with a quasi-periodic kernel or the combination of
Keplerian signals very close to the rotation period. In both cases, the
implied timescale for evolution is long: the GP fit suggests an exponential
decay timescale of half a year, while the Keplerian model that adequately
reproduces the signal seen in our ten-year data set suggests that some surface
features may be even more long-lived. It is hard to distinguish between the
two models, but both imply the presence of surface features with lifetimes
longer than expected for dynamo fields. Even so, this is not necessarily at
odds with the complex and time-varying reconstructed surface map presented by
Petit et al. (2017), who identified both variable and stable surface features
in their data. We speculate that their stable features may be the ones
responsible for our long-lived RV signal, while their rapidly varying features
contribute high frequency noise to our data that we cannot characterize due to
our limited observing cadence. We do, however, measure stellar jitter with a
magnitude comparable to that of the coherent modulation, which may arise from
rapidly varying surface features. If both types of features exist, it may
imply magnetic structures influenced both by a failed fossil field and a
subsurface dynamo. However, long term spectropolarimetric observations of Vega
are necessary to conclusively determine the behavior of its active regions.
Another interesting feature of Vega’s activity is its low photometric
variation, particularly given its maximum radial velocity amplitude of
$14.4^{+2.2}_{-1.8}$ m s-1. A dark spot inducing a $10$ m s-1 variation on a
star rotating with ${v\sin{i_{\star}}}=20$ km s-1 should also induce a
photometric signal of $A_{\rm RV}/{v\sin{i_{\star}}}\mathord{\sim}500$ ppm
(Vanderburg et al. 2016); the observed amplitude of only $10$ ppm provides
independent evidence that dark spots do not dominate the surface. To explore
this further, we use SOAP 2.0 (Dumusque et al. 2015) to compare the effects of
star spots and plages at different latitudes on a Vega-like star (Figure 8).
We vary the size of the active region to reproduce an RV signal with an
amplitude of $\mathord{\sim}15$ m s-1 and compare the expected photometric
signals. We find that a dark spot would induce photometric variations ranging
from a hundred to over 700 ppm. On the other hand, a high-latitude plage can
reproduce photometric variations less than 10 ppm, consistent with the
observed TESS light curve. This is because plages are only marginally hotter
than the rest of the stellar surface, resulting in little flux variation, but
are still surrounded by local magnetic fields that suppress convective
blueshift and cause regions to appear redshifted, creating an RV signal
(Dumusque et al. 2015). Given that Petit et al. (2017) detected many bright
and dark regions on Vega’s surface, the picture is clearly not as simple as
our simulations, but it suggests that the observed variations are primarily
driven by plages.
### 5.2 Candidate Planets Orbiting Vega
Any planetary signals in our radial velocities may be difficult to detect,
since the dominant signal arises from stellar activity and its removal is not
straightforward. Nevertheless, we are able to rule out the presence of a
candidate signal near 0.53 or 0.56 days previously reported by Böhm et al.
(2015), and we do identify two signals worthy of further investigation. The
first, with a period of $196.4^{+1.6}_{-1.9}$ days, would have a semi-major
axis of $0.853^{+0.020}_{-0.022}$ au and a minimum mass of $0.252\pm 0.066$
$\,M_{\rm J}$, falling well below the detection limits of previous direct
imaging surveys. The power of the signal monotonically increases with the
addition of new data, which is what one would expect for a real orbiting
companion. However, it is not formally significant, is located near a small
peak in the window function, and its significance is reduced further when the
activity signal is modeled. We conclude that there is not good evidence for a
planet at this period.
The second interesting signal emerged from the velocity residuals when
modeling the activity with multiple Keplerians near the rotation period. We
identify a short-period signal with a formal false alarm probability less than
$1\%$, a best-fit period of $2.43$ days, and a semi-amplitude of $6$ m s-1,
implying a minimum mass of $\mathord{\sim}20$ $\,M_{\rm\earth}$. A true mass
this low would require a polar orbit, but A stars hosting short-period planets
do display a wide range of stellar obliquities, and even a highly inclined
orbit that is well aligned with the stellar spin ($i=6.2$ deg) would
correspond to a planetary mass companion ($\mathord{\sim}0.6$ $\,M_{\rm J}$).
Continued radial velocity observations could provide further insight into the
presence of a planet orbiting Vega, but more work needs to be done to account
for the activity signal; further spectroscopic observations should be planned
carefully, for example to achieve a cadence that resolves the rotational and
orbital periods, allowing better simultaneous modeling of planet and stellar
activity.
If there is a planet with a period of $2.43$ days orbiting Vega, there are a
few ways one might confirm its existence. TESS data rule out transits of
objects larger than about $2$ $\,R_{\rm\earth}$ at this period, so unless it
is very dense, this candidate planet does not transit. It has taken 1500
spectra obtained over 10 years to detect the RV signal at low significance, so
it may require a large investment of spectroscopic resources to increase the
SNR of the detection even if it is real. A high cadence campaign spanning many
orbital periods (ideally from multiple longitudes to achieve uninterrupted
data) might be best suited for mitigating stellar activity and limiting its
evolution during observations. Alternatively, it may be possible to directly
detect the planet using high resolution spectroscopy and cross-correlation
against a template of elemental or molecular species in the planetary
atmospheres (e.g., Snellen et al. 2010). With such a short period around such
a hot star, the planet would have an equilibrium temperature of
$\mathord{\sim}3250$ K (assuming a Bond albedo of 0.25), and would be the
second hottest known exoplanet after KELT-9b (Gaudi et al. 2017). KELT-9b and
several other hot Jupiters have had atmospheric lines successfully detected
with high resolution spectroscopy (e.g., Brogi et al. 2012; Birkby et al.
2013; Yan et al. 2019). Importantly, these detections are not reliant on a
transiting geometry. To predict the expected signal, we need to estimate the
star-to-planet contrast ratio, and therefore the planetary radius. Nearly all
highly irradiated giant planets have inflated radii (some as large as
$\mathord{\sim}2$ $\,R_{\rm J}$), so $1$ $\,R_{\rm J}$ would be a conservative
estimate. As outlined in Birkby (2018), the signal to noise of the planetary
signal is described by
$\rm
SNR_{planet}=\left(\frac{S_{p}}{S_{\star}}\right)SNR_{star}\sqrt{N_{lines}},$
(4)
where $S_{p}/S_{\star}$ is the planet-to-star contrast ratio. Adopting
$R_{p}=1$ $\,R_{\rm J}$ and assuming blackbody radiation, the planet-to-star
contrast ratio for the candidate Vega planet would be $2\times 10^{-5}$. As a
reference, assuming $1$ $\,R_{\rm J}$ for tau Boötis b (the first non-
transiting planet with a detection using this technique; Brogi et al. 2012),
we would estimate a contrast of $2.6\times 10^{-5}$. The total SNR of our Vega
spectra is about 26,000, implying ${\rm
SNR_{planet}}\mathord{\sim}0.5\sqrt{\rm N_{lines}}$. If we can resolve
$\mathord{\sim}100$ lines in the planetary spectrum, it is possible that we
could detect such a planet with our TRES data. However, a high confidence
detection may have to wait for a data set better suited to this purpose. The
SOPHIE data obtained by Böhm et al. (2015) have higher resolving power and
total SNR than our TRES data but cover a narrower wavelength range. TRES
covers the red optical where contrasts are not as extreme and where lines from
molecules such as TiO, CO, and ${\rm H_{2}O}$ may be present in the planetary
atmosphere. On the other hand, Fe I has been detected on the daysides of
ultra-hot Jupiters like KELT-9b and WASP-33b, so direct detection of the
planet is possible even in the blue optical. Ultimately, it may be necessary
to move to the near infrared where contrasts are further reduced and the
presence of strong molecular bands of CO and ${\rm H_{2}O}$ can boost the SNR.
### 5.3 Current and Future Limits on Planetary Companions
Using 1524 TRES RVs, we are able to place new detection limits on planets
orbiting within 15 au of Vega, a region in which direct imaging surveys have
only ruled out brown dwarfs. Assuming no preference for the orientation of the
orbital plane—consistent with the observed distribution for short-period
planets orbiting hot stars—we can rule out nearly all hot Jupiters. We are
sensitive to Saturn-mass objects at 0.1 au, Jupiters at 1 au, 5 $\,M_{\rm J}$
at 10 au, and 13 $\,M_{\rm J}$ at 15 au. For a planet well aligned with the
stellar spin, these masses increase by a factor of about 8 and even massive
planets would therefore be difficult to detect beyond $\mathord{\sim}3$ au.
Nevertheless, these limits indicate that brown dwarf companions in any
orientation are exceedingly unlikely within 3 au. While the majority of
planets that have been hypothesized from the observed architecture of Vega’s
disk would reside beyond 15 au, Raymond & Bonsor (2014) find that systems of
planets with masses ranging from $\mathord{\sim}1$ $\,M_{\rm J}$ at $5$–$10$
au to Neptune masses at tens of au could replenish the hot dust in the inner
belt. Our data are not quite sensitive enough to constrain planets this small
at these distances, with our detection probabilities falling off steeply below
about $2$ $\,M_{\rm J}$ at 5 au, even for planets drawn from an isotropic
inclination distribution.
With TESS data, we are able to place constraints on the size of any transiting
planets. While most planets would not transit, requiring both a very high
stellar obliquity and a semi-major axis $\lesssim 0.2$ au, the extremely
precise photometry obtained by TESS results in very sensitive detection
limits, despite the large stellar radius. We can rule out any transiting hot
Jupiters, along with most other planets with radii greater than $3$
$\,R_{\rm\earth}$. Further observations could place even more extensive limits
on transiting planets while helping us to better understand the active regions
on the star. The TESS extended mission will return to the northern hemisphere
in its second year of operations, which may be the next opportunity for
additional uninterrupted measurements at a similar precision. The 10-minute
full frame images used in the extended mission will also improve the time
resolution by a factor of three, opening the possibility of further
characterization of high frequency stellar variations.
Current and future space telescopes offer additional opportunities to search
for planets around Vega. Meshkat et al. (2018) show that planned James Webb
Space Telescope (JWST) NIRCam GTO observations of Vega (Beichman et al. 2010)
will have greater sensitivity than previous surveys, perhaps extending to
Saturn-mass planets. Additionally, MIRI GTO observations are expected to
resolve the potential asteroid belt analog (Beichman et al. 2017), providing
further insight into the disk structure of Vega and its implications for a
planetary system. However, JWST will only be able to search the region beyond
$1.5$″ (11 au) from Vega for planets; any closer, and the star will saturate
the instrument. The Nancy Grace Roman Space Telescope coronagraph instrument
(CGI) is more promising for close companions, as it is intended to observe
small fields around bright stars. Because Vega is so bright, Roman would not
need to observe a bright PSF reference star, potentially allowing better CGI
stability and improving contrast for point sources. On the other hand, the CGI
is optimized for stars with angular diameters less than $\mathord{\sim}2$ mas;
with an angular diameter over $3$ mas, there may be some light leakage when
observing Vega. While it is uncertain how these factors would balance out, if
the conventional Roman CGI sensitivity limits apply to the star, the mission
could observe Jupiter-sized planets within $0.2$″ (close to $1$ au) (Nemati et
al. 2017; Krist et al. 2018). Combining Roman and JWST observations, future
space missions promise to place new direct imaging limits on planets
throughout the entire Vega system.
As a nearby star, Vega is also an interesting candidate for astrometric
detection of companions, though the best precision—i.e., with Gaia—may not be
possible. Sahlmann et al. (2016, 2018) describe protocols to observe very
bright stars with Gaia, but Vega is far beyond the bright limit for standard
Gaia processing and they do not quantify what the uncertainties for such
measurements may be. If astrometric precision for Vega were to rival the
typical performance (e.g., $35\ \mu$as per measurement), we might expect Gaia
to be sensitive to Jupiter masses beyond about 1.5 au (Ranalli et al. 2018).
However, we do not know the precision with which Gaia can observe a star like
Vega, and it is unlikely to be close to the instrumental floor.
Radial velocities remain the most sensitive technique for companions within 1
au for the foreseeable future.
## 6 Conclusions
Using 1524 TRES spectra and two sectors of TESS photometry, we search for
planets orbiting Vega. We do not discover any transiting planets, but do
detect a candidate in our radial velocities with a period of $2.43$ days and a
semi-amplitude of $6$ m s-1, implying a minimum mass of about $20$
$\,M_{\rm\earth}$. Further observations and analysis will be required to
confirm or refute this candidate. We use our data to derive limits on the
presence of transiting planets within 0.2 au and non-transiting planets within
15 au. For orbits well aligned to the stellar spin, we are only sensitive to
the most massive planets inside about 1 au, but for misaligned orbits, we are
sensitive to sub-Saturn masses at small separations and the most massive
planets out to about 10 au. Combining our radial velocity limits with those
from previous direct imaging, we place new detection limits on brown dwarfs
out to 15 au. The TESS light curve is remarkably quiet and shows no signs of a
transiting planet. With transit injection recovery tests, we can rule out most
planets with radii greater than $3$ $\,R_{\rm\earth}$ and periods between 0.5
and 15 days.
We also identify rotational modulation in our data, which dominates the radial
velocities but is weak in our photometry, consistent with variation driven
primarily by bright plages, rather than dark spots. We model this signal with
a quasi-periodic Gaussian process and with multiple Keplerians, both of which
suggest that the structures on Vega’s surface evolve on timescales much longer
than the rotation period, implying that at least some of the surface features
may be fueled by a failed fossil magnetic field.
Future high resolution spectroscopy offers a path forward for the direct or
indirect detection of short-period planets orbiting Vega while simultaneously
characterizing the stellar surface features and the underlying mechanisms
driving them. Future photometry––such as the extended TESS mission will
further constrain the presence of transiting planets. At wider separations,
JWST and the Nancy Grace Roman Space Telescope promise to provide new
constraints on planets via direct imaging.
We thank Tiffany Meshkat for providing the P1640 direct imaging detection
limits, Vanessa Bailey for insight into observing Vega with the Nancy Grace
Roman Space Telescope CGI, Alessandro Sozzetti for information regarding Gaia
astrometric performance for bright stars, and Doug Gies for discussion of A
star surface features. We gratefully acknowledge the contributions made by
visiting TRES observers over the years: Robert Stefanik, Gabor Fűrész, Bence
Béky, Sumin Tang, Zach Berta-Thompson, Daniel Yahalomi, and Warren Brown. This
paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. We thank the TESS Architects
(George Ricker, Roland Vanderspek, Dave Latham, Sara Seager, Josh Winn, Jon
Jenkins) and the many TESS team members for their efforts to make the mission
a continued success. This work utilized resources from the University of
Colorado Boulder Research Computing Group, which is supported by the National
Science Foundation (awards ACI-1532235 and ACI-1532236), the University of
Colorado Boulder, and Colorado State University.
## References
* Absil et al. (2006) Absil, O., di Folco, E., Mérand, A., et al. 2006, A&A, 452, 237, doi: 10.1051/0004-6361:20054522
* Ahlers et al. (2020a) Ahlers, J. P., Kruse, E., Colón, K. D., et al. 2020a, ApJ, 888, 63, doi: 10.3847/1538-4357/ab59d0
* Ahlers et al. (2020b) Ahlers, J. P., Johnson, M. C., Stassun, K. G., et al. 2020b, arXiv e-prints, arXiv:2004.14812. https://arxiv.org/abs/2004.14812
* Albrecht et al. (2012) Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012, ApJ, 757, 18, doi: 10.1088/0004-637X/757/1/18
* Alina et al. (2012) Alina, D., Petit, P., Lignières, F., et al. 2012, in American Institute of Physics Conference Series, Vol. 1429, American Institute of Physics Conference Series, ed. J. L. Hoffman, J. Bjorkman, & B. Whitney, 82–85, doi: 10.1063/1.3701905
* Angus et al. (2018) Angus, R., Morton, T., Aigrain, S., Foreman-Mackey, D., & Rajpaul, V. 2018, MNRAS, 474, 2094, doi: 10.1093/mnras/stx2109
* Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., SipHocz, B. M., et al. 2018, aj, 156, 123, doi: 10.3847/1538-3881/aabc4f
* Aumann et al. (1984) Aumann, H. H., Gillett, F. C., Beichman, C. A., et al. 1984, ApJ, 278, L23, doi: 10.1086/184214
* Beatty & Gaudi (2015) Beatty, T. G., & Gaudi, B. S. 2015, PASP, 127, 1240, doi: 10.1086/684264
* Beichman et al. (2017) Beichman, C. A., Rieke, G., Bouwman, J., et al. 2017, Coronagraphic Imaging of Young Planets and Debris Disk with NIRCam and MIRI, JWST Proposal. Cycle 1
* Beichman et al. (2010) Beichman, C. A., Krist, J., Trauger, J. T., et al. 2010, PASP, 122, 162, doi: 10.1086/651057
* Birkby (2018) Birkby, J. L. 2018, Spectroscopic Direct Detection of Exoplanets (Springer), 16, doi: 10.1007/978-3-319-55333-7_16
* Birkby et al. (2013) Birkby, J. L., de Kok, R. J., Brogi, M., et al. 2013, MNRAS, 436, L35, doi: 10.1093/mnrasl/slt107
* Böhm et al. (2015) Böhm, T., Holschneider, M., Lignières, F., et al. 2015, A&A, 577, A64, doi: 10.1051/0004-6361/201425425
* Bonsor et al. (2018) Bonsor, A., Wyatt, M. C., Kral, Q., et al. 2018, MNRAS, 480, 5560, doi: 10.1093/mnras/sty2200
* Braithwaite & Cantiello (2013) Braithwaite, J., & Cantiello, M. 2013, MNRAS, 428, 2789, doi: 10.1093/mnras/sts109
* Brogi et al. (2012) Brogi, M., Snellen, I. A. G., de Kok, R. J., et al. 2012, Nature, 486, 502, doi: 10.1038/nature11161
* Buchhave et al. (2010) Buchhave, L. A., Bakos, G. Á., Hartman, J. D., et al. 2010, ApJ, 720, 1118, doi: 10.1088/0004-637X/720/2/1118
* Cantiello & Braithwaite (2019) Cantiello, M., & Braithwaite, J. 2019, ApJ, 883, 106, doi: 10.3847/1538-4357/ab3924
* Claret (2018) Claret, A. 2018, A&A, 618, A20, doi: 10.1051/0004-6361/201833060
* Collier Cameron et al. (2010) Collier Cameron, A., Guenther, E., Smalley, B., et al. 2010, MNRAS, 407, 507, doi: 10.1111/j.1365-2966.2010.16922.x
* Dawson & Fabrycky (2010) Dawson, R. I., & Fabrycky, D. C. 2010, ApJ, 722, 937, doi: 10.1088/0004-637X/722/1/937
* Defrère et al. (2011) Defrère, D., Absil, O., Augereau, J. C., et al. 2011, A&A, 534, A5, doi: 10.1051/0004-6361/201117017
* Delisle et al. (2016) Delisle, J. B., Ségransan, D., Buchschacher, N., & Alesina, F. 2016, A&A, 590, A134, doi: 10.1051/0004-6361/201527944
* Dumusque et al. (2015) Dumusque, X., Boisse, I., & Santos, N. C. 2015, SOAP 2.0: Spot Oscillation And Planet 2.0. http://ascl.net/1504.021
* Fűrész (2008) Fűrész, G. 2008, PhD thesis, University of Szeged, Hungary
* Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
* Foreman-Mackey et al. (2019) Foreman-Mackey, D., Farr, W., Sinha, M., et al. 2019, The Journal of Open Source Software, 4, 1864, doi: 10.21105/joss.01864
* Fulton et al. (2018) Fulton, B. J., Petigura, E. A., Blunt, S., & Sinukoff, E. 2018, PASP, 130, 044504, doi: 10.1088/1538-3873/aaaaa8
* Gaudi et al. (2017) Gaudi, B. S., Stassun, K. G., Collins, K. A., et al. 2017, Nature, 546, 514, doi: 10.1038/nature22392
* Gelman & Rubin (1992) Gelman, A., & Rubin, D. B. 1992, Statistical Science, 7, 457, doi: 10.1214/ss/1177011136
* Ghezzi et al. (2018) Ghezzi, L., Montet, B. T., & Johnson, J. A. 2018, ApJ, 860, 109, doi: 10.3847/1538-4357/aac37c
* Harris et al. (2020) Harris, C. R., Jarrod Millman, K., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
* Hayes & Latham (1975) Hayes, D. S., & Latham, D. W. 1975, ApJ, 197, 593, doi: 10.1086/153548
* Haywood et al. (2014) Haywood, R. D., Collier Cameron, A., Queloz, D., et al. 2014, MNRAS, 443, 2517, doi: 10.1093/mnras/stu1320
* Heinze et al. (2008) Heinze, A. N., Hinz, P. M., Kenworthy, M., Miller, D., & Sivanandam, S. 2008, ApJ, 688, 583, doi: 10.1086/592100
* Hippke & Heller (2019) Hippke, M., & Heller, R. 2019, A&A, 623, A39, doi: 10.1051/0004-6361/201834672
* Holland et al. (1998) Holland, W. S., Greaves, J. S., Zuckerman, B., et al. 1998, Nature, 392, 788, doi: 10.1038/33874
* Holland et al. (2017) Holland, W. S., Matthews, B. C., Kennedy, G. M., et al. 2017, MNRAS, 470, 3606, doi: 10.1093/mnras/stx1378
* Hughes et al. (2012) Hughes, A. M., Wilner, D. J., Mason, B., et al. 2012, ApJ, 750, 82, doi: 10.1088/0004-637X/750/1/82
* Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
* Janson et al. (2015) Janson, M., Quanz, S. P., Carson, J. C., et al. 2015, A&A, 574, A120, doi: 10.1051/0004-6361/201424944
* Jenkins et al. (2010) Jenkins, J. M., Caldwell, D. A., Chandrasekaran, H., et al. 2010, ApJ, 713, L87, doi: 10.1088/2041-8205/713/2/L87
* Jenkins et al. (2015) Jenkins, J. M., Twicken, J. D., Batalha, N. M., et al. 2015, AJ, 150, 56, doi: 10.1088/0004-6256/150/2/56
* Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Proc. SPIE, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, 99133E, doi: 10.1117/12.2233418
* Johnson et al. (2010a) Johnson, J. A., Howard, A. W., Bowler, B. P., et al. 2010a, PASP, 122, 701, doi: 10.1086/653809
* Johnson et al. (2010b) Johnson, J. A., Bowler, B. P., Howard, A. W., et al. 2010b, ApJ, 721, L153, doi: 10.1088/2041-8205/721/2/L153
* Kane & Gelino (2011) Kane, S. R., & Gelino, D. M. 2011, ApJ, 729, 74, doi: 10.1088/0004-637X/729/1/74
* Kipping (2013) Kipping, D. M. 2013, MNRAS, 434, L51, doi: 10.1093/mnrasl/slt075
* Koerner et al. (2001) Koerner, D. W., Sargent, A. I., & Ostroff, N. A. 2001, ApJ, 560, L181, doi: 10.1086/324226
* Kraft (1967) Kraft, R. P. 1967, ApJ, 150, 551, doi: 10.1086/149359
* Kreidberg (2015) Kreidberg, L. 2015, PASP, 127, 1161, doi: 10.1086/683602
* Krist et al. (2018) Krist, J., Effinger, R., Kern, B., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10698, Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, 106982K, doi: 10.1117/12.2310043
* Lagrange et al. (2009) Lagrange, A. M., Gratadour, D., Chauvin, G., et al. 2009, A&A, 493, L21, doi: 10.1051/0004-6361:200811325
* Lagrange et al. (2010) Lagrange, A. M., Bonnefoy, M., Chauvin, G., et al. 2010, Science, 329, 57, doi: 10.1126/science.1187187
* Latham et al. (2002) Latham, D. W., Stefanik, R. P., Torres, G., et al. 2002, AJ, 124, 1144, doi: 10.1086/341384
* Lignières et al. (2009) Lignières, F., Petit, P., Böhm, T., & Aurière, M. 2009, A&A, 500, L41, doi: 10.1051/0004-6361/200911996
* Macintosh et al. (2015) Macintosh, B., Graham, J. R., Barman, T., et al. 2015, Science, 350, 64, doi: 10.1126/science.aac5891
* Marois et al. (2008) Marois, C., Macintosh, B., Barman, T., et al. 2008, Science, 322, 1348, doi: 10.1126/science.1166585
* Marois et al. (2010) Marois, C., Zuckerman, B., Konopacky, Q. M., Macintosh, B., & Barman, T. 2010, Nature, 468, 1080, doi: 10.1038/nature09684
* Marsh et al. (2006) Marsh, K. A., Dowell, C. D., Velusamy, T., Grogan, K., & Beichman, C. A. 2006, ApJ, 646, L77, doi: 10.1086/506520
* Matrà et al. (2020) Matrà, L., Dent, W. R. F., Wilner, D. J., et al. 2020, arXiv e-prints, arXiv:2006.16257. https://arxiv.org/abs/2006.16257
* Meshkat et al. (2018) Meshkat, T., Nilsson, R., Aguilar, J., et al. 2018, AJ, 156, 214, doi: 10.3847/1538-3881/aae14f
* Monnier et al. (2012) Monnier, J. D., Che, X., Zhao, M., et al. 2012, ApJ, 761, L3, doi: 10.1088/2041-8205/761/1/L3
* Mortier & Collier Cameron (2017) Mortier, A., & Collier Cameron, A. 2017, A&A, 601, A110, doi: 10.1051/0004-6361/201630201
* Nemati et al. (2017) Nemati, B., Krist, J. E., & Mennesson, B. 2017, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10400, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1040007, doi: 10.1117/12.2274396
* Petit et al. (2017) Petit, P., Hébrard, E. M., Böhm, T., Folsom, C. P., & Lignières, F. 2017, MNRAS, 472, L30, doi: 10.1093/mnrasl/slx132
* Petit et al. (2010) Petit, P., Lignières, F., Wade, G. A., et al. 2010, A&A, 523, A41, doi: 10.1051/0004-6361/201015307
* Piétu et al. (2011) Piétu, V., di Folco, E., Guilloteau, S., Gueth, F., & Cox, P. 2011, A&A, 531, L2, doi: 10.1051/0004-6361/201116796
* Rajpaul et al. (2015) Rajpaul, V., Aigrain, S., Osborne, M. A., Reece, S., & Roberts, S. 2015, MNRAS, 452, 2269, doi: 10.1093/mnras/stv1428
* Ranalli et al. (2018) Ranalli, P., Hobbs, D., & Lindegren, L. 2018, A&A, 614, A30, doi: 10.1051/0004-6361/201730921
* Rasmussen & Williams (2006) Rasmussen, C. E., & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning (MIT Press)
* Raymond & Bonsor (2014) Raymond, S. N., & Bonsor, A. 2014, MNRAS, 442, L18, doi: 10.1093/mnrasl/slu048
* Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
* Roberts et al. (2012) Roberts, S., Osborne, M., Ebden, M., et al. 2012, Philosophical Transactions of the Royal Society of London Series A, 371, 20110550, doi: 10.1098/rsta.2011.0550
* Sahlmann et al. (2016) Sahlmann, J., Martín-Fleitas, J., Mora, A., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, 99042E, doi: 10.1117/12.2231240
* Sahlmann et al. (2018) Sahlmann, J., Mora, A., Martín-Fleitas, J. M., et al. 2018, in IAU Symposium, Vol. 330, Astrometry and Astrophysics in the Gaia Sky, ed. A. Recio-Blanco, P. de Laverny, A. G. A. Brown, & T. Prusti, 343–344, doi: 10.1017/S1743921317005592
* Salvatier et al. (2016) Salvatier, J., Wieckiâ, T. V., & Fonnesbeck, C. 2016, PyMC3: Python probabilistic programming framework. http://ascl.net/1610.016
* Schlaufman (2010) Schlaufman, K. C. 2010, ApJ, 719, 602, doi: 10.1088/0004-637X/719/1/602
* Sibthorpe et al. (2010) Sibthorpe, B., Vandenbussche, B., Greaves, J. S., et al. 2010, A&A, 518, L130, doi: 10.1051/0004-6361/201014574
* Snellen et al. (2010) Snellen, I. A. G., de Kok, R. J., de Mooij, E. J. W., & Albrecht, S. 2010, Nature, 465, 1049, doi: 10.1038/nature09111
* Su et al. (2005) Su, K. Y. L., Rieke, G. H., Misselt, K. A., et al. 2005, ApJ, 628, 487, doi: 10.1086/430819
* Su et al. (2013) Su, K. Y. L., Rieke, G. H., Malhotra, R., et al. 2013, ApJ, 763, 118, doi: 10.1088/0004-637X/763/2/118
* van Leeuwen (2007) van Leeuwen, F. 2007, A&A, 474, 653, doi: 10.1051/0004-6361:20078357
* Vanderburg et al. (2016) Vanderburg, A., Plavchan, P., Johnson, J. A., et al. 2016, MNRAS, 459, 3565, doi: 10.1093/mnras/stw863
* Vanderburg et al. (2019) Vanderburg, A., Huang, C. X., Rodriguez, J. E., et al. 2019, ApJ, 881, L19, doi: 10.3847/2041-8213/ab322d
* Wilner et al. (2002) Wilner, D. J., Holman, M. J., Kuchner, M. J., & Ho, P. T. P. 2002, ApJ, 569, L115, doi: 10.1086/340691
* Winn et al. (2010) Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, ApJ, 718, L145, doi: 10.1088/2041-8205/718/2/L145
* Wyatt (2008) Wyatt, M. C. 2008, ARA&A, 46, 339, doi: 10.1146/annurev.astro.45.051806.110525
* Yan et al. (2019) Yan, F., Casasayas-Barris, N., Molaverdikhani, K., et al. 2019, A&A, 632, A69, doi: 10.1051/0004-6361/201936396
* Yelverton et al. (2020) Yelverton, B., Kennedy, G. M., & Su, K. Y. L. 2020, MNRAS, 495, 1943, doi: 10.1093/mnras/staa1316
* Yoon et al. (2010) Yoon, J., Peterson, D. M., Kurucz, R. L., & Zagarello, R. J. 2010, ApJ, 708, 71, doi: 10.1088/0004-637X/708/1/71
* Zechmeister & Kürster (2009) Zechmeister, M., & Kürster, M. 2009, A&A, 496, 577, doi: 10.1051/0004-6361:200811296
* Zheng et al. (2017) Zheng, X., Lin, D. N. C., Kouwenhoven, M. B. N., Mao, S., & Zhang, X. 2017, ApJ, 849, 98, doi: 10.3847/1538-4357/aa8ef3
* Zhou et al. (2016) Zhou, G., Rodriguez, J. E., Collins, K. A., et al. 2016, AJ, 152, 136, doi: 10.3847/0004-6256/152/5/136
* Zhou et al. (2019) Zhou, G., Huang, C. X., Bakos, G. Á., et al. 2019, AJ, 158, 141, doi: 10.3847/1538-3881/ab36b5
|
# Refining Network Alignment
to Improve Matched Neighborhood Consistency
Mark Heimann 111Now at Lawrence Livermore National Laboratory.
Computer Science & Engineering, University of Michigan. Email: {mheimann,
shinech, vfatemeh<EMAIL_ADDRESS>Xiyuan Chen11footnotemark: 1 222Now at
Stanford University.
Fatemeh Vahedian11footnotemark: 1
Danai Koutra11footnotemark: 1
###### Abstract
Network alignment, or the task of finding meaningful node correspondences
between nodes in different graphs, is an important graph mining task with many
scientific and industrial applications. An important principle for network
alignment is _matched neighborhood consistency_ (MNC): nodes that are close in
one graph should be matched to nodes that are close in the other graph. We
theoretically demonstrate a close relationship between MNC and alignment
accuracy. As many existing network alignment methods struggle to preserve
topological consistency in difficult scenarios, we show how to refine their
solutions by improving their MNC. Our refinement method, RefiNA, is
straightforward to implement, admits scalable sparse approximation, and can be
paired _post hoc_ with _any_ network alignment method. Extensive experiments
show that RefiNA increases the accuracy of diverse unsupervised network
alignment methods by up to 90%, making them robust enough to align graphs that
are $5\times$ more topologically different than were considered in prior work.
## 1 Introduction
Network alignment, or the task of finding correspondences between the nodes of
multiple networks, is a fundamental graph mining task with applications to
user identity linkage [20], discovery of novel biological functions [15],
computer vision [10], schema matching in databases [21], and other problems of
academic and industrial interest. Methods to solve this problem vary widely in
techniques, ranging from nonlinear relaxations of a computationally hard
optimization problem [1], to belief propagation [2], genetic algorithms [25],
spectral methods [26], node embedding [13], and more.
Matching the topological structure of graphs is difficult, closely related to
the canonical graph isomorphism problem [1]. As a result, many network
alignment approaches rely heavily on node or edge side information [32, 22]
(in some cases ignoring the graph structure altogether), or on known ground-
truth alignments—used as anchor links to connect the two graphs [20, 34] or to
supervise the training of deep neural networks [9]. However, in many settings
[13, 5, 8], side information or anchor links may not be available.
Figure 1: RefiNA refines an initial network alignment solution, which maps
node A and its neighbors in $G_{1}$ far apart in $G_{2}$. The refined network
alignment solution has higher _matched neighborhood consistency_ : neighbors
of A are aligned to neighbors of a, to which A itself is aligned.
In this paper, we focus on the challenging problem of unsupervised topological
network alignment. With neither anchor links to seed the alignment process nor
side information to guide it, the main objective for this task is to preserve
some kind of topological consistency in the alignment solution. We
theoretically analyze the principle of _matched neighborhood consistency_
(MNC), or how well a node’s neighborhood maps onto the neighborhood of its
counterpart in the other graph (illustrated in Fig. 1), and show its
connection to alignment accuracy. On the other hand, we find that when network
alignment methods are inaccurate, the MNC of their solutions breaks down
(e.g., Fig. 1 left).
To address this, we introduce RefiNA, a method for refining network alignment
solutions _post hoc_ by iteratively updating nodes’ correspondences to improve
their MNC. By strategically limiting the possible correspondences per node to
update in each iteration, we can sparsify the computations to make RefiNA
scalable to large graphs. Experimentally, we show that RefiNA significantly
improves a variety of network alignment methods on many highly challenging
datasets, even when starting a network alignment solution with limited
accuracy. Also, it can be succinctly expressed as matrix operations in a few
lines of code, making it easy for practitioners to adopt. In this compact
formulation, we incorporate several useful insights for network alignment, and
our techniques have interesting connections to other successful graph-based
methods.
Our contributions are as follows:
* •
New Algorithm: We propose RefiNA, a post-processing step that can be applied
to the output of any network alignment method. Its compact design incorporates
several important insights for network alignment, and it permits a sparse
approximation that is scalable to large graphs.
* •
Theoretical Connections: We show a rigorous connection between _matched
neighborhood consistency_ , the property that RefiNA improves, and alignment
accuracy. We also provide network alignment insights justifying each of
RefiNA’s design choices and technical connections to other graph-based
methods.
* •
Experiments: We conduct thorough experiments on real and simulated network
alignment tasks and show that RefiNA improves the accuracy of many
methodologically diverse network alignment methods by up to 90%, making them
robust enough to recover matchings in $5\times$ noisier datasets than those
considered in prior work. We extensively drill down RefiNA to justify the
insights that inspire its design.
We provide our code and additional supplementary material at
https://github.com/GemsLab/RefiNA.
## 2 Related Work
Network alignment has numerous applications from matching schemas in databases
[21] and objects in images [10], to revealing biological functions shared by
different organisms [15], to integration of multiple data sources to create a
holistic worldview network [6]. as been widely studied and several methods
have been proposed to solve this problem. For example, the intuition of
NetAlign [2] as a message-passing algorithm is to “complete squares” by
aligning two nodes that share an edge in one graph to two nodes that share an
edge in another graph. Similarly, FINAL [32] has an objective of preserving
topological consistency between the graphs that may be augmented with node and
edge attribute information, if available. MAGNA [25] is a genetic algorithm
that can evolve network populations to maximize topological consistency.
Recent works leverage kernel methods [35] or optimal transport [31] but suffer
from high (cubic) computational complexity.
Networks can also be aligned by comparing nodes directly. Early works hand-
engineered node features: GRAAL [15] computes a graphlet degree signature,
while GHOST [23] defines a multiscale spectral signature. UniAlign [14] and
HashAlign [11] extract features for each node from graph statistics such as
degree and various node centralities. Recent works instead leverage node
embeddings that are very expressive, but must be comparable across networks
[13]. REGAL [13] computes node embeddings that capture structural roles that
are not specific to a particular network. Other methods align the embedding
spaces of different networks using techniques from unsupervised machine
translation: DANA [5] uses adversarial training [17], and CONE-Align [4] uses
non-convex alternating optimization methods. All these approaches find match
nodes with embedding-based nearest-neighbor search, which does not enforce
alignment consistency and can benefit from our refinement. Our contributions
are orthogonal to a recent effort to accelerate embedding-based node matching
via graph compression [24].
Some graph neural matching network models, often in specific applications like
social networks [19], computer vision [9], or knowledge graphs [30], use
objectives that enforce matching consistency between neighboring nodes. This
is similar to our approach RefiNA; indeed, in the supplementary §B we show
interesting technical connections between RefiNA and graph neural networks.
However, RefiNA, like Simple Graph Convolution [29] is much faster and simpler
to implement and apply than these graph neural networks, and it does not need
known node matchings to supervise training.
## 3 Theoretical Analysis
We first introduce key alignment concepts and notation. Then, we theoretically
justify the topological consistency principle that is the basis of our
refinement approach, RefiNA (§4).
### 3.1 Preliminaries
Table 1: Major symbols and definitions. Symbols | Definitions
---|---
$G_{\ell}=(\mathcal{V}_{\ell},\mathcal{E}_{\ell})$ | $\ell^{th}$ graph with nodeset $\mathcal{V}_{\ell}$, edgeset $\mathcal{E}_{\ell}$
$\mathbf{A}_{\ell}$ | Adjacency matrix of $G_{\ell}$
$n_{\ell},\bar{d}^{\ell}$ | Number of nodes and average degree in $G_{\ell}$
$\pi(\cdot)$ | An alignment between graphs $G_{1}$ and $G_{2}$; a function mapping a node in $\mathcal{V}_{1}$ to a node in $\mathcal{V}_{2}$
$\mathbf{M}$ | $n_{1}\times n_{2}$ matrix specifying correspondences of nodes in $\mathcal{V}_{1}$ to those in $\mathcal{V}_{2}$
$\mathcal{N}_{G_{\ell}}(u)$ | Neighbors of node $i$ in graph $G_{\ell}$
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(u)$ | “Mapped neighborhood” of node $i$; counterparts in $G_{2}$ (mapped by $\pi$) of nodes in $\mathcal{N}_{G_{1}}(i)$
#### 3.1.1 Graphs.
Following the network alignment literature [13, 25], we consider two
unweighted and undirected graphs $G_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ and
$G_{2}=(\mathcal{V}_{2},\mathcal{E}_{2})$ with their corresponding nodesets
$\mathcal{V}_{1},\mathcal{V}_{2}$ and edgesets
$\mathcal{E}_{1},\mathcal{E}_{2}$. We denote their adjacency matrices as
$\mathbf{A}_{1}$ and $\mathbf{A}_{2}$. Since they are symmetric,
$\mathbf{A}_{1}^{\top}=\mathbf{A}_{1}$ and
$\mathbf{A}_{2}^{\top}=\mathbf{A}_{2}$, and we simplify our notation below.
#### 3.1.2 Alignment.
A node alignment is a function $\pi:\mathcal{V}_{1}\rightarrow\mathcal{V}_{2}$
that maps the nodes of $G_{1}$ to those of $G_{2}$. It is also commonly
represented as a $|\mathcal{V}_{1}|\times|\mathcal{V}_{2}|$ alignment matrix
$\mathbf{M}$, where $\mathbf{M}_{ij}$ is the (real-valued or binary)
similarity between node $i$ in $G_{1}$ and node $j$ in $G_{2}$. $\mathbf{M}$
can be used to encode a mapping $\pi$, e.g., greedy alignment
$\pi(i)=\arg\max_{j}\mathbf{M}_{ij}$. We note that alignment between two
graphs should be sought if the nodes of the two graphs meaningfully
correspond.
#### 3.1.3 Neighborhood and Consistency.
Let $\mathcal{N}_{G_{1}}(i)=\\{j\in\mathcal{V}_{1}:(i,j)\in\mathcal{E}_{1}\\}$
be the neighbors of node $i$ in $G_{1}$, i.e., the set of all nodes with which
$i$ shares an edge. We define node $i$’s “mapped neighborhood” in $G_{2}$ as
the set of nodes onto which $\pi$ maps $i$’s neighbors:
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)=\\{j\in\mathcal{V}_{2}:\exists
k\in\mathcal{N}_{G_{1}}(i)\text{ s.t. }\pi(k)=j\\}$. For example, in Fig. 1
(first panel), node $A$’s neighbors in $G_{1}$ are $B,G,\text{ and }D$, which
are respectively mapped to nodes $f,g,\text{ and }e$, so
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(A)=\\{f,g,e\\}.$
We call the neighbors of $i$’s counterpart
$\mathcal{N}_{G_{2}}\big{(}\pi(i)\big{)}$. In Fig. 1, node $A$’s counterpart
is node $a$, whose neighbors are $b,g,\text{ and }d$. Thus,
$\mathcal{N}_{G_{2}}\big{(}\pi(A)\big{)}=\\{b,g,d\\}$.
The matched neighborhood consistency (MNC) [4] of node $i$ in $G_{1}$ and node
$j$ in $G_{2}$ is the Jaccard similarity of the sets
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)$ and $\mathcal{N}_{G_{2}}(j)$:
(3.1)
$\small\text{MNC}(i,j)=\frac{|\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)\cap\mathcal{N}_{G_{2}}(j)|}{|\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)\cup\mathcal{N}_{G_{2}}(j)|}.\\\
$
### 3.2 Theoretical Justification of MNC.
Several unsupervised network alignment algorithms attempt to enforce some
notion of topological consistency in their objective functions (§2). We
justify this intuition by showing that a specific form of topological
consistency, matched neighborhood consistency or MNC, has a close relationship
with alignment accuracy.
Unsupervised network alignment is commonly evaluated on graphs that are
isomorphic up to noisy or missing edges [13, 5, 25, 14, 32, 33]. When edges
are removed from one graph independently with probability $p$, we show that
accurate alignment entails high MNC.
###### Theorem 3.1
For isomorphic graphs $G_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ and
$G_{2}=(\mathcal{V}_{2},\mathcal{E}_{2})$, let $\pi(\cdot)$ be the
isomorphism. Let $\overline{G}_{2}=(\mathcal{V}_{2},\tilde{\mathcal{E}_{2}})$
be a noisy version of $G_{2}$ created by removing each edge from
$\mathcal{E}_{2}$ independently with probability $p$. Then for any node i in
$G_{1}$ and its counterpart $\pi(i)$ in $\overline{G}_{2}$,
$\mathbb{E}\big{(}\text{MNC}(i,\pi(i))\big{)}=1-p$.
However, this does not prove that a solution with perfect MNC will have
perfect accuracy. In fact, we can construct counterexamples, such as two
“star” graphs, each consisting of one central node connected to $n-1$
peripheral nodes (of degree one). Whatever the true correspondence of the
peripheral nodes, aligning them to each other in any order would lead to
perfect MNC. Prior network alignment work [14] has observed a few such special
cases and in fact gives up on trying to distinguish them from the graph
topology. We formalize this concept of _structural indistinguishability_ :
###### Definition 1
Let $\mathcal{N}_{k}(u)$ be the subgraph induced by all nodes that are $k$ or
fewer hops/steps away from node $u$. Two nodes $u$ and $v$ are structurally
indistinguishable if for all $k$, $\mathcal{N}_{k}(u)$ and
$\mathcal{N}_{k}(v)$ are isomorphic.
Our next result proves that for isomorphic graphs, structurally
indistinguishable nodes are the only possible failure case for a solution with
perfect MNC.
###### Theorem 3.2
For isomorphic graphs $G_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ and
$G_{2}=(\mathcal{V}_{2},\mathcal{E}_{2})$, suppose there exists $\pi(\cdot)$
that yields MNC = 1 for all nodes. Then, if $\pi$ misaligns a node $v$ to some
node $v^{*}$ instead of the true counterpart $v^{\prime}$, it is because
$v^{*}$ is structurally indistinguishable from $v^{\prime}$.
Proof idea. We give all the proofs in the supplementary §A. At a high level,
MNC measures the extent to which the alignment preserves edges in a node’s
neighborhood, and isomorphic graphs have a (not necessarily unique) perfectly
edge-preserving node matching.
Formalizing the intuitions of prior work: MNC was proposed as intuition for
modeling intra-graph node proximities when performing cross-graph matching
[4]. It was also used (not by that name) as a heuristic in embedding-based
network alignment [8]. Our analysis provides theoretical justification for
both works.
## 4 Method
We consider an unsupervised network alignment setting, where an initial
solution, $\mathbf{M}_{0}$, is provided by any network alignment method (§2).
We do _not_ take any of these initial node alignments as ground truth; on the
contrary, we seek to improve their correctness by leveraging insights from our
theory in §3.2. Thus, our problem differs from semi-supervised network
alignment that is seeded with known ground-truth node correspondences [20,
34]. Formally, we state it as:
###### Problem 1
Given a sparse initial alignment matrix $\mathbf{M}_{0}$ between the nodes of
two graphs $G_{1}$ and $G_{2}$, we seek to _refine_ this initial solution into
a new, real-valued matrix $\mathbf{M}$ of refined similarity scores that
encodes a more accurate alignment.
### 4.1 RefiNA and Connections to MNC.
Our theoretical results pave a path to solving Problem 1 by increasing matched
neighborhood consistency. While our results characterize “ideal” cases
(perfect accuracy or MNC), heuristic solutions in prior works have found that
increasing MNC tends to increase accuracy [4, 8]. Given an alignment matrix
returned by a network alignment method, how can we improve its MNC in a
principled way? We first derive a matrix-based form for MNC, which we prove in
the supplementary §A:
###### Theorem 4.1
The MNC of a binary alignment matrix $\mathbf{M}$ can be written as a matrix
$\mathbf{S}^{\text{MNC}}$ such that
$\text{MNC}(i,j)=\mathbf{S}^{\text{MNC}}_{ij}$ as:
(4.2)
$\mathbf{S}^{\text{MNC}}=\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}\;\oslash\;\\\
(\mathbf{A}_{1}\mathbf{M}\mathbf{1}^{n_{2}}\otimes\mathbf{1}^{n_{2}}+\mathbf{1}^{n_{1}}\otimes\mathbf{A}_{2}\mathbf{1}^{n_{2}}-\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})$
where $\oslash$ is elementwise division and $\otimes$ is outer product.
We can then compute refined alignments $\mathbf{M}^{\prime}$ by multiplicative
updating each node’s alignment score (in $\mathbf{M}$) with its matched
neighborhood consistency:
(4.3) $\mathbf{M}^{\prime}=\mathbf{M}\circ\mathbf{S}^{\text{MNC}}$
where $\circ$ denotes Hadamard product. Repeating over several iterations can
take advantage of an improving alignment solution. The high-level idea of our
proposed refinement scheme is to iteratively increase alignment scores for
nodes that have high MNC.
While we could just repeatedly iterate Eq. (4.3), we can introduce some
mechanisms leverage important insights without increasing complexity:
$\bullet$ I1: Prioritize high-degree nodes. Higher degree nodes are easier to
align [14], and so it is desirable to give them higher scores particularly in
early iterations. To do so, we use only the (elementwise) numerator of Eq.
(4.1), which counts nodes’ number of matched neighbors (excluding the
normalizing denominator increases the score for high degree nodes). Thus,
instead of using Eq. (4.3), we simplify our update rule to:
(4.4)
$\mathbf{M}^{\prime}=\mathbf{M}\circ\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}$
Additionally, this spares us the extra matrix operations required to compute
the denominator of Eq. (4.1).
$\bullet$ I2: Do not overly rely on the initial solution. At every iteration,
we add a small $\epsilon$ to every element of $\mathbf{M}$. This gives each
pair of nodes a token match score whether or not the initial alignment
algorithm identified them as matches, which gives us a chance to correct the
initial solution’s false negatives.
$\bullet$ I3: Allow convergence. Finally, to keep the scale of the values of
$\mathbf{M}$ from exploding, we we row-normalize $\mathbf{M}$ followed by
column-normalizing it at every iteration. Previous iterative graph matching
methods such as graduated assignment [10] require full normalization per
iteration using the Sinkhorn algorithm [27] to produce a doubly stochastic
matrix. In contrast, with RefiNA a single round of normalization suffices,
avoiding large computational expense (§5).
Putting it all together, we give the pseudocode of our method RefiNA in
Algorithm 1. RefiNA is powerful yet conceptually simple and straightforward to
implement. It requires only a few lines of code, with each line implementing
an important insight.
### 4.2 Connections to Other Graph Problems.
In the supplementary §B, we further justify RefiNA conceptually by comparing
and contrasting it to several other graph techniques. We find that RefiNA’s
update can be viewed as a more flexible version of the seed-and-extend node
matching heuristic [15, 23], and that it performs a graph filtering operation
akin to a graph neural network [29]. In particular, a similar neighborhood
aggregation operation is used by graph neural tangent kernels [7] for graph-
level comparison. This further highlights the connection between the
individual node similarities found in graph matching [32, 13] and aggregated
node similarities in graph kernels [32, 12].
Algorithm 1 RefiNA ($\mathbf{A}_{1},\mathbf{A}_{2},\mathbf{M}_{0},K,\epsilon$)
1:Input: adjacency matrices $\mathbf{A}_{1},\mathbf{A}_{2}$, initial alignment
matrix $\mathbf{M}_{0}$, number of iterations $K$, token match score
$\epsilon$
2:for $k=1\to K$ do $\triangleright$ Refinement iterations
3:
$\mathbf{M}_{k}=\mathbf{M}_{k-1}\circ\mathbf{A}_{1}\mathbf{M}_{k-1}\mathbf{A}_{2}$
$\triangleright$ MNC update
4: $\mathbf{M}_{k}=\mathbf{M}_{k}+\epsilon$ $\triangleright$ Add token match
scores
5: $\mathbf{M}_{k}=\text{Normalize}(\mathbf{M}_{k})$ $\triangleright$ By row
then column
6:end for
7:return $\mathbf{M}_{K}$
### 4.3 Optimizations: Sparse RefiNA.
The dense matrix updates lead to quadratic computational complexity in the
number of nodes. To scale RefiNA to large graphs, we sparsify it by updating
only a small number of alignment scores for each node. Intuitively, we forgo
updating scores of likely non-aligned node pairs (these updates would be small
anyway). Concretely, sparse RefiNA replaces Line 3 of Algorithm 1 with
$\mathbf{M}_{k|\mathbf{U}_{k}}=\mathbf{M}_{k-1|\mathbf{U}_{k}}\circ\mathbf{U}_{k}$
where the update matrix
$\mathbf{U}_{k}=\text{top-}\alpha(\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})$ is
a sparse version of $\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}$ containing only
the largest $\alpha$ entries per row. $\mathbf{M}_{k|\mathbf{U}_{k}}$ selects
the elements of $\mathbf{M}_{k}$ (pairs of nodes) corresponding to nonzero
elements in $\mathbf{U}_{k}$. These are the only elements on which we perform
an MNC-based update, and the only ones to receive a token match score (Line
4): $\mathbf{M}_{k|\mathbf{U}_{k}}=\mathbf{M}_{k|\mathbf{U}_{k}}+\epsilon$.
As the elements to update are selected by the size of their update scores,
which are computed using the previous solution $\mathbf{M}$, sparse RefiNA
relies somewhat more strongly on the initial solution. However, we still
comply with I2 by updating $\alpha>1$ possible alignments for each node. This
has sub-quadratic time and space requirements (cf. supplemental §C) and
accuracy comparable to dense updates (§ 5.4).
### 4.4 Assumption and Limitations.
Network alignment typically relies on some kind of topological consistency
assumption (§2). Likewise, RefiNA assumes that improving alignment MNC will
improve its quality. Our theory (§3.2) shows this assumption is well-founded
for the quasi-isomorphic graphs studied by many prior works [13, 5, 25, 14,
32, 33], and our experiments (§5) support RefiNA’s efficacy in even more use
cases. However, all assumptions have limitations, so we discuss two
limitations of ours. First, as Theorem 3.2 notes, misaligned nodes can have
high MNC. In practice, we find that this is comparatively rare, and we shed
insight into when it may occur (§5.3.1). Second, the correct alignment may
have low MNC, as is the case in some datasets consisting of multiple social
networks sharing a limited number of common users [32, 20]. Recent work
characterizes these datasets as having low _structural credibility_ :
topological consistency widely breaks down, and supervision and/or rich
attribute information is generally needed for successful network alignment
[28]. Future work could extend RefiNA to use such information.
## 5 Experiments
In this section, we first demonstrate RefiNA’s ability to improve diverse
unsupervised network alignment methods in a variety of challenging scenarios
at reasonable computational cost. We next perform a deep study of RefiNA that
verifies its various design insights.
### 5.1 Experimental Setup
#### 5.1.1 Data.
We choose network datasets from a variety of domains (e.g. biological, social)
(Tab. 2). We consider two scenarios for network alignment with ground truth
node correspondence: (1) Simulated noise. For each network with adjacency
matrix $\mathbf{A}$, we create a permuted copy
$\tilde{\mathbf{A}}=\mathbf{P}\mathbf{A}\mathbf{P}^{\top}$, where $\mathbf{P}$
is a random permutation matrix, and add noise by removing each edge with
probability $p\in[0.05,0.10,0.15,0.20,0.25]$. The task is to align each
network to its _noisy_ permuted copy ${\tilde{\mathbf{A}}}^{(p)}$, with the
ground truth alignments being given by $\mathbf{P}$. (2) Real noise. Our PPI-Y
dataset is commonly studied in biological network alignment [25]. The task is
to align the largest connected component of the yeast (_S.cerevisiae_) PPI
network to its copies augmented with 5, 10, 15, 20, and 25 percent additional
low-confidence PPIs (added in order of their confidence) [25].
We also show (§5.2.2) that RefiNA helps find more meaningful alignments even
for networks defined on differing underlying nodes, by aligning graphs with no
ground truth node correspondence: the PPI networks of (3) separate species (SC
& DM). We align a yeast network (_S.cerevisiae_ , a larger network than PPI-Y)
to a fruit fly network (_D.melanogaster_).
Table 2: Description of the datasets used. Name | Nodes | Edges | Description
---|---|---|---
Arenas Email [16] | 1 133 | 5 451 | communication
Hamsterster [16] | 2 426 | 16 613 | social
PPI-H [3] | 3 890 | 76 584 | PPI (human)
Facebook [18] | 4 039 | 88 234 | social
PPI-Y [25] | 1 004 | 8 323 | PPI (yeast)
LiveMocha [16] | 104 103 | 2 193 083 | social
SC & DM [26] | 5 926 | 88 779 | PPI (yeast)
1 124 | 9 763 | PPI (fruit fly)
#### 5.1.2 Metrics.
For graphs with ground-truth node correspondence, our main measure of
alignment success is accuracy: the proportion of correctly aligned nodes.
While this is the primary goal of network alignment, we also give matched
neighborhood consistency (MNC, Eq. (3.1)) as a secondary metric (e.g. in Fig.
5a-b), in light of its importance in our analysis (§3.2).
When the networks have no ground-truth node correspondence, accuracy is not
defined, so we assess two properties of the conserved network
$\overline{\mathbf{A}}=\mathbf{M}^{\top}\mathbf{A}_{1}\mathbf{M}\circ\mathbf{A}_{2}$
(the overlap of the aligned adjacency matrices):
* •
Normalized Overlap (N-OV): the percentage of conserved edges:
$100*\frac{\text{nnz}(\overline{\mathbf{A}})}{\max(\text{nnz}(\mathbf{A}_{1}),\text{nnz}(\mathbf{A}_{2}))}$
* •
Largest Conserved Connected Component (LCCC): the number of edges in the
largest connected component of $\overline{\mathbf{A}}$.
These respectively measure the alignment solution’s ability to conserve edges
and large substructures, which in biological networks may correspond to
functions that are shared across species [26]. Thus, larger values may
indicate a more biologically interesting alignment.
#### 5.1.3 Base Network Alignment Methods.
To illustrate RefiNA’s flexibility, we apply it to a set of base network
alignment methods using a diverse range of techniques (belief propagation,
spectral methods, genetic algorithms, and node embeddings): (1) NetAlign [2],
(2) FINAL [32], (3) REGAL [13], (4) CONE-Align [4], and (5) MAGNA [25].
(a)
(a) Arenas
(b) Hamsterster
(c) PPI-H
(d) Facebook
(e) PPI-Y
Figure 2: Alignment accuracy vs graphs’ topological difference. Both sparse
and dense refinement with RefiNA improve all base methods’ alignment accuracy
and robustness, often quite dramatically. (We run MAGNA only on PPI-Y, the
dataset with real noise, due to its high runtime—cf. Fig. 3LABEL:sub@fig:run-
magna.)
(a)
(a) Arenas
(b) Hamsterster
(c) PPI-H
(d) Facebook
(e) PPI-Y
Figure 3: Runtime for different refinement variations. Sparse refinement is
appreciably faster than dense refinement. Both refinements offer a modest
computational overhead, often on par with or faster than the time taken to
perform the original network alignment.
$\bullet$ Base Method Settings. We configure all base methods following the
literature (cf. the supplementary §D).
$\bullet$ RefiNA Settings. By default, we use $K=100$ refinement iterations
(cf. supplementary §E) and token match score
$\epsilon=\frac{1}{10^{\overline{p}}}$ where
$\overline{p}=\min\\{p\in\mathbb{N}:10^{p}>n\\}$, so that a node’s token match
scores will sum to less than 1. For sparse refinement, we update $\alpha=10$
entries per node. We justify these choices by sensitivity analysis.
### 5.2 Analysis of Alignment Performance
#### 5.2.1 Finding Known Correspondences.
In Fig. 2 we plot the accuracy of all methods with (solid/dashed lines) and
without (dotted lines) refinement. For simulated experiments (2a-2d) we report
average and standard deviation of accuracy over five trials.
Results. We see dramatic improvements in alignment accuracy for _all_ methods
with refinement. A striking example is NetAlign on the PPI-H dataset with 5%
noise. Initially finding just 4% of alignments, with RefiNA it achieves 94%
accuracy–going from a nearly completely incorrect to nearly completely
correct. Similarly, CONE-Align achieves well over 90% accuracy on Arenas and
PPI-H with refinement _and_ sees only a slight drop even at the highest noise
levels.
###### Observation 1
RefiNA greatly improves the accuracy of diverse network alignment methods
across datasets.
We are also able to align networks that are much noisier than previous works
had considered: our _lowest_ noise level is 5%, the _highest_ noise level in
[13]. RefiNA’s benefit is appreciable at up to 10% noise on most datasets for
FINAL, 15% noise for NetAlign, 20% for REGAL, and the full 25% noise for CONE-
Align.
###### Observation 2
Refinement can make network alignment methods considerably more robust to
noise.
Fig. 2 and the supplementary Tab. 3 (§F) show that network alignment methods
vary in how much their accuracy increases from RefiNA, and also the maximum
noise level at which they appreciably improve. This variation shows that
RefiNA is not ignoring the initial solution. Precisely characterizing the
“refinability” of initial solutions is an interesting question for future
work. Here, our goal is not to rank “better” or “worse” existing methods;
instead, comparing all methods to _their own_ unrefined solutions shows the
value of RefiNA.
###### Observation 3
Different methods benefit differently from refinement, but all benefit
considerably.
Compared to dense refinement, sparse refinement is slightly less accurate
overall. However, the gap between the two variants decreases as the noise
level increases, and in some cases (e.g., REGAL on Arenas and Hamsterster at
high noise levels) sparse refinement even performs better, possibly indicating
a regularizing effect of forgoing updates of low-confidence node pairs. Sparse
refinement is faster than dense refinement, as Fig. 3 shows, but both offer a
reasonable overhead compared to the initial alignment time. In general, they
are modestly slower than NetAlign and REGAL, two famously scalable methods, on
par with CONE-Align, faster than FINAL, and much faster than MAGNA.333On
account of MAGNA’s high runtime, we only run it on PPI-Y (where it takes 9612
seconds on average.)
###### Observation 4
RefiNA has a manageable computational overhead, particularly with sparse
updates.
#### 5.2.2 Conserving Meaningful Structures.
We now use RefiNA to help REGAL, NetAlign, and FINAL 444CONE-Align’s subspace
alignment operates on embeddings for the same number of nodes, which these
graphs do not have. align the SC-DM dataset consisting of the PPI networks of
different species: graphs whose nodes do not necessarily have an underlying
correspondence.
Results. Fig. 4 shows that using RefiNA, both measures of structural
conservation increase significantly for all methods, indicating more
successful alignments. In fact, these results are on par with ones obtained
using amino acid sequence information in addition to graph topology [26].
Interestingly, sparse refinement brings even greater improvements than dense.
###### Observation 5
RefiNA is useful even for graphs with no underlying node correspondence.
(a)
(b)
(a) Normalized Overlap
(b) Size of LCCC
Figure 4: Alignment of PPI networks of different species with no ground truth
node alignment. RefiNA helps base methods find potentially more biologically
interesting solutions by multiple metrics.
### 5.3 Drilldown: Network Alignment Insights.
In this section, we perform a drilldown of (dense) RefiNA that gives a deeper
understanding into its specific design choices in light of the three insights
we laid out in §4.
(a) Distribution of MNC and accuracy by node degree before refinement:
NetAlign, Arenas 5% noise.
(b) Distribution of MNC and accuracy by node degree after refinement:
NetAlign, Arenas 5% noise..
(c) Accuracy with varying token match score $\epsilon$: Arenas, 5% noise. The
methods almost overlap for $\epsilon>0$.
(d) Accuracy/runtime per iteration with limited vs full Sinkhorn
normalization: NetAlign, Hamsterster 5% noise.
Figure 5: Drilldown of RefiNA in terms of our three insights that inspired its
design. LABEL:sub@fig:deg-before-LABEL:sub@fig:deg-after For I1, before and
after alignment, high degree nodes are more likely to have high MNC and be
correctly aligned. LABEL:sub@fig:sensitivity:eps For I2, our token match score
admits a wide range of values that yield good refinement performance.
LABEL:sub@fig:sensitivity:normalization For I3, our proposed normalization
effectively avoids the computational expense of full Sinkhorn normalization.
#### 5.3.1 Aligning High Degree Nodes _( I1)_.
In Figs. 5a-b, we analyze the claim that high-degree nodes are easier to
align, which inspires RefiNA’s design, for NetAlign on the Arenas dataset for
brevity. We split the nodes into three groups by degree:
$[0,\frac{d_{\text{max}}}{3}),[\frac{d_{\text{max}}}{3},\frac{2d_{\text{max}}}{3}),[\frac{2d_{\text{max}}}{3},d_{\text{max}}]$
($d_{\text{max}}$ is the maximum node degree) and plot the distribution of MNC
in each group among correct and incorrectly aligned nodes.
Results. High-degree nodes are rarely misaligned even before refinement, and
the few nodes that are still misaligned afterward have low degrees. These
often still have a high MNC, probably because it is easier for smaller node
neighborhoods to be (nearly) isomorphic, resulting in (near) structural
indistinguishability (the challenge indicated by Theorem 3.2.)
###### Observation 6
It is easier to align high-degree nodes, verifying I1 that motivates RefiNA’s
update formulation to encourage early alignment of high-degree nodes.
#### 5.3.2 Token Match Score _( I2)_.
We now study the effects of the token match score $\epsilon$, used to overcome
some limitations of an erroneous initial solution (our second network
alignment insight in §4). We use the Arenas dataset with 5% noise averaged
over five trials (we observe similar trends on other datasets), varying
$\epsilon$ from $10^{-2}$ to $10^{-6}$ and $\epsilon=0$ (no token match
score).
Results. In Fig. 5c, we see that the performance can dramatically drop with
too large $\epsilon$, where the token match scores overwhelm the alignment
information. We need a positive token match score for the multiplicative
update rule to work: with $\epsilon=0$, RefiNA fails to discover new
alignments and only achieves the initial solutions’ accuracy (cf. Fig. 2a).
However, RefiNA works with a wide range of small positive choices of
$\epsilon$, including $\epsilon=10^{-4}$ as recommended by our criterion in §
5.1.
###### Observation 7
RefiNA is robust to the token match score, as long as it is not so large that
it would drown out the actual node similarity scores.
#### 5.3.3 Alignment Matrix Normalization _( I3)_.
RefiNA normalizes the rows and columns of the alignment matrix, per our third
insight in §4. Sinkhorn’s algorithm iteratively repeats this process,
converging to a doubly stochastic matrix [27]. Methods such as graduated
assignment [10] nest iterative Sinkhorn normalization at every iteration of
optimization. We refine NetAlign’s solution on the Hamsterster dataset with 5%
noise, both as proposed and using Sinkhorn’s algorithm (up to 1000 iterations
or a tolerance of $10^{-2}$).
Results. We see in Fig. 5d that Sinkhorn’s algorithm takes longer to converge
(particularly as refinement continues) and dominates the running time.
Meanwhile, our proposed normalization yields virtually the same accuracy,
while keeping computation time low ($\ll 1$ second per iteration) and
relatively fixed.
###### Observation 8
RefiNA is advantageous by being able to eschew the full Sinkhorn procedure.
(a) Accuracy/runtime vs sparsity level $\alpha$: CONE-Align, Facebook 5%
noise.
(b) Top-$k$ accuracy on a time budget: REGAL, LiveMocha 5% noise.
Figure 6: Sparse RefiNA. LABEL:sub@fig:sensitivity:sparse-accrun Varying the
update sparsity allows us to trade off accuracy and runtime.
LABEL:sub@fig:sensitivity:topalpha On extremely large graphs, RefiNA uncovers
additional alignment (and near-alignments).
### 5.4 Sparse Updates and Scalability
#### 5.4.1 Sparsity Level of Refinement.
By varying $\alpha$, the number of updates per node, we can interpolate
between the sparse and dense versions of RefiNA. We study this on Facebook
with 5% noise.
Results. Updating just one alignment per node leads to poor performance, as
per I2: we should not blindly trust the initial solution by using only its top
choice. However, the initial solution provides enough information that we can
use only the top few choices, with marginal accuracy returns compared to the
extra runtime required by using more top choices. Thus, sparse RefiNA offers a
favorable balance of accuracy and speed.
#### 5.4.2 “Soft” Alignments, Large Graphs.
To utilize the scalability of sparse RefiNA, we use it on a large dataset:
LiveMocha, which has over 100K nodes and a _million_ edges. We simulate an
alignment scenario with 5% noise. We only use REGAL, the most scalable base
alignment method we consider, together with sparse refinement (dense
refinement runs out of memory). We consider a _budgeted computation_ scenario,
running RefiNA for as long as REGAL takes (2600s). Meanwhile, we explore the
top-$k$ accuracy: the proportion of correct alignments in the top $k$ choices
per node [13] ranked by the real-valued entries of $\mathbf{M}$.
Results. In Fig. 6b, we see that RefiNA scales to large graphs and more than
doubles the accuracy of REGAL in the same amount of time it took REGAL to
obtain its initial solution. The top-$k$ scores also increase as we consider
more top matches per node (up to the sparsity parameter 10), which may be
useful for some applications [13].
###### Observation 9
Sparse refinement offers a favorable efficiency tradeoff and scales to large
graphs.
## 6 Conclusion
We have proposed RefiNA, a powerful technique for refining existing network
alignment methods that greatly improves accuracy and robustness. RefiNA is
simple to implement and apply _post hoc_ to a variety of methods. Its compact
formulation encodes several insights for network alignment, supported by our
extensive theoretical and empirical analysis, and is highly scalable with
sparse computation. We hope that all these positive qualities will make it
attractive to practitioners.
## Acknowledgements
This work is supported by NSF Grant No. IIS 1845491, Army Young Investigator
Award No. W9-11NF1810397, and Adobe, Amazon, Facebook, and Google faculty
awards.
## References
* [1] Yonathan Aflalo, Alexander Bronstein, and Ron Kimmel. On convex relaxation of graph isomorphism. PNAS, 2015.
* [2] Mohsen Bayati, Margot Gerritsen, David F Gleich, Amin Saberi, and Ying Wang. Algorithms for large, sparse network alignment problems. In ICDM, 2009.
* [3] Bobby-Joe Breitkreutz, Chris Stark, Teresa Reguly, Lorrie Boucher, Ashton Breitkreutz, Michael Livstone, Rose Oughtred, Daniel H Lackner, Jürg Bähler, Valerie Wood, et al. The biogrid interaction database: 2008 update. Nucleic acids research, 2008.
* [4] Xiyuan Chen, Mark Heimann, Fatemeh Vahedian, and Danai Koutra. Cone-align: Consistent network alignment with proximity-preserving node embedding. In CIKM, 2020.
* [5] Tyler Derr, Hamid Karimi, Xiaorui Liu, Jiejun Xu, and Jiliang Tang. Deep adversarial network alignment. arXiv preprint arXiv:1902.10307, 2019.
* [6] Joel Douglas, Ben Zimmerman, Alexei Kopylov, Jiejun Xu, Daniel Sussman, and Vince Lyzinski. Metrics for evaluating network alignment. GTA3 at WSDM, 2018.
* [7] Simon S Du, Kangcheng Hou, Russ R Salakhutdinov, Barnabas Poczos, Ruosong Wang, and Keyulu Xu. Graph neural tangent kernel: Fusing graph neural networks with graph kernels. In NeurIPS, 2019.
* [8] Xingbo Du, Junchi Yan, and Hongyuan Zha. Joint link prediction and network alignment via cross-graph embedding. In IJCAI, 2019.
* [9] Matthias Fey, Jan E Lenssen, Christopher Morris, Jonathan Masci, and Nils M Kriege. Deep graph matching consensus. In ICLR, 2020.
* [10] Steven Gold and Anand Rangarajan. A graduated assignment algorithm for graph matching. TPAMI, 1996.
* [11] Mark Heimann, Wei Lee, Shengjie Pan, Kuan-Yu Chen, and Danai Koutra. Hashalign: Hash-based alignment of multiple graphs. In PAKDD, 2018.
* [12] Mark Heimann, Tara Safavi, and Danai Koutra. Distribution of node embeddings as multiresolution features for graphs. In ICDM, 2019.
* [13] Mark Heimann, Haoming Shen, Tara Safavi, and Danai Koutra. Regal: Representation learning-based graph alignment. In CIKM, 2018.
* [14] Danai Koutra, Hanghang Tong, and David Lubensky. Big-align: Fast bipartite graph alignment. In ICDM, 2013.
* [15] Oleksii Kuchaiev, Tijana Milenković, Vesna Memišević, Wayne Hayes, and Nataša Pržulj. Topological network alignment uncovers biological function and phylogeny. Journal of the Royal Society Interface, 2010.
* [16] Jérôme Kunegis. Konect: the koblenz network collection. In WWW. ACM, 2013.
* [17] Guillaume Lample, Alexis Conneau, Marc’Aurelio Ranzato, Ludovic Denoyer, and Hervé Jégou. Word translation without parallel data. In ICLR, 2018.
* [18] Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014.
* [19] Chaozhuo Li, Senzhang Wang, Yukun Wang, Philip Yu, Yanbo Liang, Yun Liu, and Zhoujun Li. Adversarial learning for weakly-supervised social network alignment. In AAAI, 2019.
* [20] Li Liu, William K Cheung, Xin Li, and Lejian Liao. Aligning users across social networks using network embedding. In IJCAI, 2016.
* [21] Sergey Melnik, Hector Garcia-Molina, and Erhard Rahm. Similarity flooding: A versatile graph matching algorithm and its application to schema matching. In ICDE, 2002.
* [22] Lei Meng, Joseph Crawford, Aaron Striegel, and Tijana Milenkovic. Igloo: Integrating global and local biological network alignment. In MLG at KDD, 2016.
* [23] Rob Patro and Carl Kingsford. Global network alignment using multiscale spectral signatures. Bioinformatics, 2012.
* [24] Kyle K Qin, Flora D Salim, Yongli Ren, Wei Shao, Mark Heimann, and Danai Koutra. G-crewe: Graph compression with embedding for network alignment. In CIKM, 2020.
* [25] Vikram Saraph and Tijana Milenković. Magna: maximizing accuracy in global network alignment. Bioinformatics, 2014.
* [26] Rohit Singh, Jinbo Xu, and Bonnie Berger. Global alignment of multiple protein interaction networks with application to functional orthology detection. PNAS, 2008.
* [27] Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat., 1964.
* [28] Chenxu Wang, Yang Wang, Zhiyuan Zhao, Dong Qin, Xiapu Luo, and Tao Qin. Credible seed identification for large-scale structural network alignment. DAMI, 2020.
* [29] Felix Wu, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. Simplifying graph convolutional networks. In ICML, 2019.
* [30] Yuting Wu, Xiao Liu, Yansong Feng, Zheng Wang, and Dongyan Zhao. Neighborhood matching network for entity alignment. In ACL, 2020.
* [31] Hongteng Xu, Dixin Luo, Hongyuan Zha, and Lawrence Carin Duke. Gromov-wasserstein learning for graph matching and node embedding. In ICML, 2019.
* [32] Si Zhang and Hanghang Tong. Final: Fast attributed network alignment. In KDD, 2016.
* [33] Si Zhang, Hanghang Tong, Jie Tang, Jiejun Xu, and Wei Fan. ineat: Incomplete network alignment. In ICDM, 2017.
* [34] Si Zhang, Hanghang Tong, Jiejun Xu, Yifan Hu, and Ross Maciejewski. Origin: Non-rigid network alignment. In Big Data, 2019.
* [35] Zhen Zhang, Yijian Xiang, Lingfei Wu, Bing Xue, and Arye Nehorai. KerGM: Kernelized Graph Matching. In NeurIPS, 2019.
## A Proofs of Theorems in § 3.2-4
* Proof.
(Theorem 3.1: Perfect alignment accuracy implies high MNC). By Eq. (3.1),
$\text{MNC}(i,\pi(i))=\frac{|\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)\cap\mathcal{N}_{\overline{G}_{2}}(\pi(i))|}{|\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)\cup\mathcal{N}_{\overline{G}_{2}}(\pi(i))|}$.
By definition,
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)=\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)$;
it does not change as neither $\pi$ nor $G_{1}$’s adjacency matrix
$\mathbf{A}_{1}$ is affected by the noise. However,
$\mathcal{N}_{\overline{G}_{2}}(\pi(i))\subseteq\mathcal{N}_{G_{2}}(\pi(i))$,
since under edge removal $\pi(i)$ can only lose neighbors in
$\overline{G}_{2}$ compared to $G_{2}$.
Now $\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)=\mathcal{N}_{G_{2}}(\pi(i))$ since
by definition an isomorphism is edge preserving, and so
$\mathcal{N}_{G_{2}}(\pi(i))=\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)$, which is
the same as $\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)$. Thus,
$\mathcal{N}_{\overline{G}_{2}}(\pi(i))\subseteq\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)$.
We can simplify
$\text{MNC}(i,\pi(i))=\frac{|\mathcal{N}_{\overline{G}_{2}}(\pi(i))|}{|\tilde{\mathcal{N}}_{\overline{G}_{2}}^{\pi}(i)|}=\frac{|\mathcal{N}_{\overline{G}_{2}}(\pi(i))|}{|\mathcal{N}_{G_{2}}(\pi(i))|}.$
However, every node $j^{\prime}\in\mathcal{N}_{G_{2}}(\pi(i))$ is also in
$\mathcal{N}_{\overline{G}_{2}}(\pi(i))$ as long as the edge
$\big{(}\pi(i),j^{\prime})\big{)}\in\mathcal{E}_{2}$ has not been removed from
$\tilde{\mathcal{E}}_{2}$, which happens with probability $p$. So,
$\mathbb{E}\Big{(}\frac{|\mathcal{N}_{\overline{G}_{2}}(\pi(i))|}{|\mathcal{N}_{G_{2}}(\pi(i))|}\Big{)}=\mathbb{E}\big{(}\text{MNC}(i,\pi(i))\big{)}=1-p$.
* Proof.
(Theorem 3.2: Perfect MNC implies perfect accuracy except for structurally
indistinguishable nodes). Since for isomorphic graphs, a node $v$ is
structurally indistinguishable from its true counterpart $v^{\prime}$, and
since graph isomorphism is transitive, it suffices to show that $v^{*}$ is
also structurally indistinguishable from $v$. Suppose for some $k$,
$\mathcal{N}_{k}(v)$ is not isomorphic to $\mathcal{N}_{k}(v^{*})$. Then by
definition there exists neighboring nodes $a,b\in\mathcal{N}_{k}(v)$ where
either $\pi(a)$ or $\pi(b)$ is not in $\mathcal{N}_{k}(v^{*})$, or $\pi(a)$
and $\pi(b)$ do not share an edge.
In case 1, without loss of generality $\pi(b)\notin\mathcal{N}_{k}(v^{*})$.
Then no bijective mapping exists between a shortest path between $v^{*}$ and
$\pi(b)$ and a shortest path from $v^{*}$ to $\pi(b)$. There will thus be
neighbors on one path whose counterparts are not neighbors on the other path,
making MNC less than 1: a contradiction.
In case 2, since $\pi(b)$ is the counterpart of a neighbor of $a$, it must
also be a neighbor of the counterpart of $a$, which is a contradiction of the
assumption that $\pi(a)$ and $\pi(b)$ do not share an edge, or else
MNC($a,\pi(a)$) $<$ 1, another contradiction. Thus, we conclude that the
$k$-hop neighborhoods are isomorphic, that is, $v$ and $v^{*}$ are
structurally indistinguishable.
* Proof.
(Theorem 4.1: Matrix form of MNC).
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)=\\{\ell:\exists
k\in\mathcal{V}_{1}\text{ s.t. }\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}\neq
0\\}$, and of course
$\mathcal{N}_{G_{2}}(j)=\\{\ell:\mathbf{A}_{2_{j\ell}}\neq 0\\}$. Since the
product of two numbers is nonzero if and only if both numbers are nonzero,
$\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)\cap\mathcal{N}_{G_{2}}(j)=\\{\ell:\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}\mathbf{A}_{2_{j\ell}}\neq
0\\}$. For binary $\mathbf{A}_{1},\mathbf{A}_{2},$ and $\mathbf{M}$, the
cardinality of this set, which is the numerator of Eq. (3.1), is
$\sum_{k\in\mathcal{V}_{1},\ell\in\mathcal{V}_{2}}\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}\mathbf{A}_{2_{j\ell}}=(\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})_{ij}$.
Meanwhile, the denominator of Eq. (3.1) is
$|\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)\cup\mathcal{N}_{G_{2}}(j)|=|\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)|+|\mathcal{N}_{G_{2}}(j)|-|\tilde{\mathcal{N}}_{G_{2}}^{\pi}(i)\cap\mathcal{N}_{G_{2}}(j)|$.
Plugging in for each individual term, we obtain
$\sum_{k\in\mathcal{V}_{1}}\sum_{\ell\in\mathcal{V}_{2}}\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}+\sum_{\ell\in\mathcal{V}_{2}}\mathbf{A}_{2_{j\ell}}-\sum_{k\in\mathcal{V}_{1}}\sum_{\ell\in\mathcal{V}_{2}}\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}\mathbf{A}_{2_{j\ell}}.$
Substituting matrix products, this becomes
$\sum_{\ell\in\mathcal{V}_{2}}(\mathbf{A}_{1}\mathbf{M})_{i\ell}+\sum_{\ell\in\mathcal{V}_{2}}\mathbf{A}_{2_{j\ell}}-(\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})_{ij}$.
Using all-$1$ vectors to sum over columns, this is
$(\mathbf{A}_{1}\mathbf{M}\mathbf{1}^{n_{2}})_{i}+(\mathbf{A}_{2}\mathbf{1}^{n_{2}})_{j}-(\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})_{ij}.$
Then, expanding the two left vectors into matrices with outer product:
$(\mathbf{A}_{1}\mathbf{M}\mathbf{1}^{n_{2}}\otimes\mathbf{1}^{n_{2}})_{ij}+(\textbf{1}^{n_{1}}\otimes\mathbf{A}_{2}\mathbf{1}^{n_{2}})_{ij}-(\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2})_{ij}.$
Adding everything together, the denominator is the $ij$-th entry of the matrix
$\mathbf{A}_{1}\mathbf{M}\mathbf{1}^{n_{2}}\otimes\mathbf{1}^{n_{2}}+\textbf{1}^{n_{1}}\otimes\mathbf{A}_{2}\mathbf{1}^{n_{2}}-\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}$.
## B Connections to Other Graph Methods
We show additional connections between RefiNA and other diverse graph methods:
first, seed-and-extend as an alignment strategy, and second a graph filtering
perspective on RefiNA’s update rule similar to the analysis of graph neural
networks.
Seed-and-extend alignment heuristic. Many global network alignment methods
[15, 23] use this heuristic to find node correspondences between two or
multiple networks. Given initial pairwise similarities (or alignment costs)
between nodes of the compared graphs as $\mathbf{M}$, a pair of nodes $i$ and
$j$ with high probability to be aligned (e.g., whose similarity according to
$\mathbf{M}$ is above some confidence threshold) are set as the seed regions
of the alignment. After the seed $(i,j)$ is selected, the $r\textendash$hop
neighborhoods of $i$ and $j$ (i.e., $\mathcal{N}_{r,G_{1}}(i)$ and
$\mathcal{N}_{r,G_{2}}(j)$) in their respective graphs are built. Next, the
selected seed $(i,j)$ is extended for the final alignment
$\mathbf{M}^{\prime}$ by greedily matching nodes in $\mathcal{N}_{r,G_{1}}(i)$
and $\mathcal{N}_{r,G_{2}}(j)$, searching for the pairs
$(i^{\prime},j^{\prime}):i^{\prime}\in\mathcal{N}_{r,G_{1}}(i)$ and
$j^{\prime}\in\mathcal{N}_{r,G_{2}}(j)$ that are not already aligned and can
be aligned with the maximum value of similarity according to $\mathbf{M}$. The
process can be written as $\forall
k\in\mathcal{N}_{r,G_{1}}(i),\mathbf{M}^{\prime}_{k\ell}\neq 0\text{ if
}\ell=\arg\max_{\ell\in\mathcal{N}_{r,G_{2}}(j)}\mathbf{M}_{k\ell}$.
By setting $r=1$ to consider each seed’s direct neighbors, and
$\mathbf{M}^{\prime}_{k^{*}\ell^{*}}\neq 0$ if
$k^{*},\ell^{*}=\arg\max_{k\in\mathcal{V}_{1},\ell\in\mathcal{V}_{2}}\mathbf{A}_{1_{ik}}\mathbf{M}_{k\ell}\mathbf{A}_{2_{j\ell}}$,
we see that the seed-and-extend heuristic analyzes the same set of elements
used to compute the update in RefiNA (Eq. (4.4)). However, instead of summing
them to update the similarity of seed nodes $i$ and $j$, it takes the argmax
over them to adaptively select the next pair of alignments. Thus, seed-and-
extend aligns less well with I1 by relying heavily on a correct initial
solution, as the early alignments are irrevocable and used to restrict the
scope of subsequent alignments.
Graph Filtering. The matching matrix $\mathbf{M}$ can also be interpreted as a
high-dimensional feature matrix. For example, for each node in $G_{1}$, a row
of $\mathbf{M}$ may be regarded as an $n_{2}$-dimensional feature vector
consisting of the node correspondences to each of the nodes in $G_{2}$, and
similarly the $n_{2}\times n_{1}$ matrix $\mathbf{M}^{\top}$ contains
$n_{1}$-dimensional cross-network correspondence features for each node in
$G_{2}$. For challenging alignment scenarios, these are likely highly noisy
features. However, recent works have shown that multiplying a node feature
matrix by the graph’s adjacency matrix corresponds to a low-pass filtering
operation, which is of interest in explaining the mechanisms of graph neural
networks [29].
We can write our update rule in Eq. (4.4) as a feature matrix left multiplied
by adjacency matrices:
$\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}=\mathbf{A}_{1}(\mathbf{A}_{2}\mathbf{M}^{\top})^{\top}$
(for undirected graphs, $\mathbf{A}_{2}^{\top}=\mathbf{A}_{2}$), where
$\mathbf{A}_{2}\mathbf{M}^{\top}$ produces a filtered set of
$n_{1}$-dimensional features. By taking the transpose, these may be
interpreted as $n_{2}$-dimensional features for each node of $G_{1}$, which
are then filtered again by left multiplication with $\mathbf{A}_{1}$.555Graph
convolutional networks use the augmented normalized adjacency matrix
$\tilde{\mathbf{D}}^{-\frac{1}{2}}\tilde{\mathbf{A}}\tilde{\mathbf{D}}^{-\frac{1}{2}}\text{
where }\tilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}$, which we have not found
helpful.
Interpreting RefiNA’s updates as graph filtering explains its strong
performance, as well as the success of recent supervised graph matching work
[34, 9] using graph neural networks. Of course RefiNA does not have the
learnable parameters and nonlinearities of a graph neural network. However,
just as SGC [29] recently compares favorably to graph neural networks by
replacing their deep nonlinear feature extraction with repeated multiplication
by the adjacency matrix, we find that that unsupervised “alignment filtering”
is highly effective.
Graph Kernels. Just as nodes can be matched across networks using suitable
cross-network node similarities, entire graphs can be compared (for the
purposes of performing machine learning tasks on graph-structured data points)
by aggregating their nodes’ similarities [12, 7]. Some base network alignment
methods we studied are known to have close connections to graph kernels. For
example, FINAL’s pooled cross-network node similarities are closely related to
a graph kernel based on random walks [32]; likewise, the xNetMF node
embeddings that REGAL uses to match nodes one by one [13] can also be used to
construct a feature map for an entire graph, where the dot product between two
graphs’ feature maps approximates the mean-pooled (kernelized) node embedding
similarities [12]. We now show that the update rule of RefiNA is also related
to the graph neural tangent kernel (GNTK) [7], a graph kernel designed to
achieve the effect of an infinitely wide graph neural network trained by
gradient descent.
GNTK starts by computing a cross-network node similarity matrix, the same as
our matrix $\mathbf{M}$. (As proposed in [7], this matrix is computed using
input node features that are common in graph classification benchmark
datasets, but the formulation can be more general.) To achieve the effect of a
graph neural network’s feature aggregation, the node similarities are
iteratively propagated across graphs:
$\mathbf{M}^{\prime}_{ij}=c_{i}c_{j}\sum\limits_{u\in\\{\mathcal{N}_{G_{1}}(i)\cup
i\\}}\sum\limits_{v\in\\{\mathcal{N}_{G_{2}}(j)\cup j\\}}\mathbf{M}_{uv}$
Up to scaling factors $c_{i}$ and $c_{j}$ designed to model the effects of
particular graph neural network architectures, this term is elementwise
equivalent to our update
$\mathbf{M}^{\prime}=\mathbf{A}_{1}\mathbf{M}\mathbf{A}_{2}$, where the
graphs’ adjacency matrices have been augmented with self-loops as is
commonplace for graph neural networks. At each round of propagation, GNTK
post-processes $\mathbf{M}$ by performing $R$ transformations corresponding to
$R$ fully-connected neural network layers with ReLU activation, whereas we
simply add token match scores and normalize.
The analysis for GNTK connects the training of kernel machines (the final
kernel value is obtained by applying a READOUT operation to the refined node
similarities, a simple example being to sum them all) with that of graph
neural networks, in either case using graph-level supervision [7]. While this
is of course a quite different setup from the unsupervised node-level matching
problem RefiNA tackles, a very interesting direction for future work is to see
if RefiNA and/or GNTK can benefit from further methodological exchange (at
minimum, our sparse refinement techniques could approximate the computation of
GNTK between large graphs). Such analysis may also reveal more about the
general connection between network alignment based on individual node-level
similarities and network classification based on aggregated node-level
similarities.
## C Complexity Analysis of RefiNA
Here, we analyze the precise computational complexity of RefiNA with both
sparse and dense refinement.
### C.1 Complexity of Dense Refinement.
To simplify notation, we assume that both graphs have $n$ nodes [13]. For $K$
iterations, our algorithm computes the left and right multiplication of a
dense $n\times n$ matching matrix with two adjacency matrices of graphs with
average degree (number of nonzero entries per row of the adjacency matrix)
$\bar{d}_{1}$ and $\bar{d}_{2}$, respectively. Thus, the time complexity of
this update step is $O(n^{2}(\bar{d}_{1}+\bar{d}_{2}))$. Normalizing the
matrix at each iteration and adding in token match scores requires $O(n^{2})$
time. Therefore, the overall time complexity is
$O\Big{(}Kn^{2}(\bar{d}_{1}+\bar{d}_{2})\Big{)}$. While the number of
iterations and average node degree are in practice constant or asymptotically
smaller than the graph size, the quadratic time and space complexity of
RefiNA’s dense matrix operations makes it harder to apply to large graphs.
### C.2 Complexity of Sparse Refinement.
For $K$ iterations, we compute a sparse update matrix with $O(n\alpha)$
nonzero entries by multiplying matrices with $O(n\bar{d}_{1})$, $O(Kn\alpha)$,
and $O(n\bar{d}_{2})$ nonzero entries, respectively. It takes
$O(nK\alpha\bar{d}_{1})$ time to compute
$\tilde{\mathbf{A}_{1}}=\mathbf{A}_{1}\mathbf{M}_{k-1}$ and
$O(nK\alpha\bar{d}_{1}\bar{d}_{2})$ time to compute
$\tilde{\mathbf{A}_{1}}\mathbf{A}_{2}$. We then compute $\mathbf{M}_{k}$ by
updating $O(n\alpha)$ entries in $\mathbf{M}_{k-1}$ per iteration. Thus,
$\mathbf{M}_{k}$ may have $O(Kn\alpha)$ nonzero entries and requires
$O(Kn\alpha)$ time to update and normalize. Altogether, the runtime is now
$O(nK^{2}\alpha\bar{d}_{1}\bar{d}_{2})$, i.e. linear in the number of nodes.
(This is a worst-case analysis; in practice, the runtime scales close to
linearly with $K$.) We can also avoid storing a dense matching matrix, leading
to subquadratic space complexity.
## D Baseline Hyperparameter Settings
For REGAL’s own xNetMF embeddings, we used default embedding dimension
$\lfloor 10\log_{2}(n_{1}+n_{2})\rfloor$ [13], maximum neighborhood distance
$2$, neighborhood distance discount factor $\delta=0.1$, and resolution
parameter $\gamma_{\text{struc}}=1$, all recommended parameters. For the
embeddings used in CONE-Align, we set embedding dimension $d=128$, context
window size $w=10$, and negative sampling parameter $\alpha=1$. We used
${n_{0}}=10$ iterations and regularization parameter $\lambda_{0}=1.0$ for the
convex initialization of the subspace alignment, which we performed with
$T=50$ iterations of Wasserstein Procrustes optimization with batch size
$b=10$, learning rate $\eta=1.0$, and regularization parameter $\lambda=0.05$
as were suggested by the authors [4]. For REGAL and CONE-Align, we computed
embedding similarity with dot product followed by softmax normalization [9],
using $k$-$d$ trees to perform fast 10-nearest neighbor search for REGAL on
LiveMocha [13].
NetAlign and FINAL require a matrix of prior alignment information, which we
computed from pairwise node degree similarity. Then following [13, 5], we
constructed this matrix by taking the top
$k=\lfloor\log_{2}\big{(}n_{1}+n_{2})/2\big{)}\rfloor$) entries for each node
in $G_{1}$; that is, the top $k$ most similar nodes in $G_{2}$ by degree.
For MAGNA, starting from a random initial population with size 15000, we
simulated the evolution for 2000 steps following [25] using edge correctness
as the optimizing measure as it is similar to the objectives of our other
methods. We used 3 threads to execute the alignment procedure.
We consider the output of all methods to be a binary matrix $\mathbf{M}$
consisting of the “hard” (one-to-one) alignments they find, to treat methods
consistently and to show that RefiNA can refine the most general network
alignment solution. It is worth noting that some methods (e.g. REGAL, CONE-
Align) can also produce “soft” alignments (real-valued node similarity scores)
and our formulation is capable of using those.
Computing Environment. We performed all experiments on an Intel(R) Xeon(R) CPU
E5-1650 at 3.50GHz, 256GB RAM.
## E Convergence Analysis: Accuracy & MNC
(a)
(a) Arenas 5% noise
(b) Hamsterster 5% noise
(c) PPI-H 5% noise
(d) Facebook 5% noise
(e) PPI-Y 5% added edges
(f) Arenas 25% noise
(g) Hamsterster 25% noise
(h) PPI-H 25% noise
(i) Facebook 25% noise
(j) PPI-Y 25% added edges
Figure 7: Analysis of RefiNA as a function of number of iterations (0 =
performance of base method before refinement). Arenas, PPI-H, and Hamsterster
datasets show accuracy and MNC, and Facebook and PPI-Y datasets show accuracy
of sparse and dense RefiNA versions. Convergence rates are different for
different methods, but often happen well before 100 iterations for both sparse
and dense RefiNA. Accuracy and MNC consistently increase and follow similar
trends, as per their theoretical connection.
One of the parameters of RefiNA is $K$, the number of iterations for which the
initial matching is refined. In Fig. 7, we plot the performance at each
iteration, up to our maximum value of 100, for all methods and datasets (at
lowest and highest noise levels, or number of added low-confidence PPIs in the
case of PPI-Y). For brevity, we show accuracy and MNC for dense refinement
only on the first three datasets (Arenas, Hamsterster, and PPI-H), and the
per-iteration accuracy only for sparse and dense refinement on the remaining
datasets.
Results. We see that accuracy and MNC tend to trend similarly, as do sparse
and dense refinement. In both cases, we see similar trends for the same
colored curves. As for RefiNA variations, dense refinement of CONE-Align grows
in accuracy slightly more steeply than sparse refinement. For MNC, we see that
accuracy and MNC tend to have very similar values at 5% noise (thus the lines
of the same color are close to overlapping). At 25% noise, MNC is lower than
accuracy for highly accurate methods like CONE-Align—this is to be expected
because Theorem 3.1 proved that the expected average MNC under even for a
perfect alignment solution is bounded by the noise ratio. For the remaining
methods which struggle to achieve high accuracy, MNC is higher than accuracy.
As Theorem 3.2 showed, it is possible to find a high-MNC alignment that is
still not perfectly accurate (due to structural indistinguishability). Indeed,
here we see this happening in some especially challenging settings starting
from less favorable initial solutions. Nevertheless, in most cases, improving
MNC is a successful strategy for accurate network alignment.
Convergence rates differ for different methods and datasets. For example,
FINAL is generally the slowest method to converge, taking close to 70
iterations to fully converge on the Arenas dataset, with NetAlign taking
around 20, REGAL around 10, and CONE-Align around 5. On the PPI-H dataset with
5% noise, FINAL sees slow progress for nearly the full 100 iterations before
making rapid progress at the end.
In general, we find that a modest number of iterations is generally
sufficient. While we allow all methods 100 iterations of refinement in our
experiments, the elbow-shaped curves indicate that in practice, convergence
often happens in far fewer than 100 iterations (with some exceptions for a few
methods on the PPI-Y and Facebook datasets). In practice, early convergence
can be ascertained by small changes in the discovered alignments.
###### Observation 10
Accuracy and MNC, sparse and dense refinement tend to trend similarly on each
method on each dataset. Although convergence rates differ for different
methods and datasets, convergence is quite fast in practice.
## F Improvement thanks to RefiNA
To further illustrate that different methods benefit differently from
refinement, we summarize our results from Fig. 2 in tabular format in Tab. 3,
where we show the maximum improvement in mean accuracy thanks to RefiNA for
each base method on each dataset (across refinement types and noise levels,
averaged over five trials). We also show the maximum noise level at which
RefiNA is able to yield a noticeable improvement (3% or more). As discussed in
§5, while all network alignment methods benefit from RefiNA, this does not
eliminate the effect of the initial solution. We have also observed that
RefiNA poorly refines a randomly initialized alignment matrix on our datasets.
Again, this indicates that our contributions complement rather than replace
existing network alignment solutions.
Table 3: Maximum improvement thanks to RefiNA for each base method on each dataset (with abbreviated name), and maximum noise level at which RefiNA brings noticeable improvement. For all methods and all datasets, RefiNA brings dramatic increases in accuracy and can often bring significant improvement even at very high noise levels. | REGAL | NetAlign | FINAL | CONE-Align | MAGNA
---|---|---|---|---|---
AR | | 82.18%
---
20% noise
| 84.63%
---
10% noise
| 86.93%
---
5% noise
| 58.02%
---
25% noise
| N/A
---
P-H | | 80.53%
---
20% noise
| 90.02%
---
10% noise
| 79.57%
---
5% noise
| 83.28%
---
25% noise
| N/A
---
HA | | 47.86%
---
20% noise
| 55.85%
---
15% noise
| 52.68%
---
10% noise
| 39.50%
---
25% noise
| N/A
---
FB | | 61.22%
---
15% noise
| 23.76%
---
15% noise
| 17.03%
---
5% noise
| 48.51%
---
25% noise
| N/A
---
P-Y | | 32.17%
---
20% noise
| 34.67%
---
25% noise
| 18.83%
---
25% noise
| 45.72%
---
25% noise
| 6.77%
---
25% noise
|
Jackie Shen<EMAIL_ADDRESS>
# Nine Challenges in Modern Algorithmic Trading and Controls
Jackie Shen Deep QuanTech, LLC, New York, NY 10025, USA
###### Abstract
This editorial article partially informs the algorithmic trading community
about launching of the new journal Algorithmic Trading and Controls (ATC). ATC
is an online open-access journal that publishes novel works on algorithmic
trading and its control methodologies. In this inaugural article, we discuss
nine major challenges that contemporary Algo trading faces. There is nothing
superstitiously magical about the number “nine,” but so is any other one.
Several of these challenges are at the strategy level, including for example,
trading of illiquid securities or optimal portfolio execution. Others are more
at the level of risk management and controls, such as on how to develop
automated controls, testing and simulations. The editorial views could be
inevitably personal and biased, but have been explored with the most innocent
intention of contributing to this important field in modern financial services
and technologies.
###### keywords:
Algos, liquidity, portfolio, correlation, special days, derivative pricing,
universe, clustering, machine learning, auctions, shortfall, transaction cost,
unit test, regression test, simulation, automated controls
## 1 Introduction
By this inaugural article, the new journal Algorithmic Trading and Controls
(ATC) is formally rolled out.
ATC is a discretionary, moderately for-profit, and online open-access journal
that publishes novel works on algorithmic trading (a.k.a. “Algo” or “Algo
trading”) and related control frameworks. Automated Algo trading generates
both revenues and risks, and hence the importance of automated controls should
never be underestimated. We refer interested readers to the website of ATC for
more details: https://atc.deepquantech.com.
In this editorial article, we discuss nine major challenges that contemporary
Algo trading faces. Before diving into the details, we first clarify that as
for the journal ATC, we restrict the notion of “Algo trading” to two most
common and influential activites: optimal trade execution and market making.
The latter covers both exchange market making and automated over-the-counter
(OTC) liquidity provision such as Request-For-Quote (RFQ). Both the journal
ATC and this editorial article do not intend to cover purely opportunistic
trading activities that seek alphas or arbitraging opportunities for a
principal account. The signals and strategies involved in such trading
activities are confidential and proprietary, and by default prohibited from
publishing.
In terms of asset classes, we focus more on well-regulated markets such as of
equities, futures, rates products, or liquid foreign exchanges (FX). Whenever
applicable, we also comment on others such as OTC derivatives or credits. This
article also focuses primarily on the markets in North America, esp. in the
United States.
The nine challenges being explored are as follows.
1. [1]
Trading Illiquid Securities
2. [2]
Optimal Portfolio Execution
3. [3]
Clustering of the Tading Universe
4. [4]
Handling of Special Days
5. [5]
Real-Time Pricing of Derivatives
6. [6]
Trading in (Close) Auctions
7. [7]
Transaction Cost Models
8. [8]
Automated Controls for Automated Trading
9. [9]
Full-Scale Testing and Simulation
Some of these challenges call for either strategy revamping or more
intelligent data analytics, while others are more concerned with risk controls
or robust testing and simulation frameworks. Both are in scope for this new
exciting journal and require broker-dealers or trading houses to make further
investments in talents, analytics, infrastructures, and the Three Lines of
Defense.
Finally, an editorial article of such nature could be inevitably personal and
biased. But its purpose is to initiate further healthy dialogues within the
Algo trading community, which includes fund managers, broker-dealers, trading
agents or houses, and regional supervisory functions such as the Federal
Reserve Board or the Prudential Regulation Authority.
Algo trading has been playing an increasingly vital role in the modern
landscape of financial technologies and services, and profoundly impacting
both the main street and Wall Street. As a result, it can no longer be
operated in the conventional style of black boxes, and must start to promote a
healthy culture of open discussions, information sharing, and collaborations.
## 2 The Nine Challenges for ATC
We now elaborate on the nine challenges. The order is somewhat at will due to
the relative independence.
### 2.1 Trading Illiquid Securities
On the market side, the limit order books of an illiquid name are typically
“thin” and active market participants are also very limited, e.g., mostly
registered market makers who are obliged to post two-sided quotes. Such poor
liquidity qualities make these names highly sensitive to supply-demand
imbalances.
As a result, most illiquid names invariably share these macro characteristics:
lower average daily volumes (ADV), wider average bid-offer spreads, and higher
volatilities. All these negative factors aggravate implementation shortfalls
(IS) whenever trading these names.
On the trading side, the constraint of completion (within a given execution
horizon) raises a major hurdle for Algos. Any passive waiting and inaction
“now” may pile up positions for “later” stages and hence increase the
squeezing risk and impact costs. For this reason, traders may better turn to
Algos with flat execution profiles such as VWAP/TWAP or Participation of
Volumes (PoV). Further diligence must be exercised because the intraday volume
profiles of illiquid names demand either longer time bins to accumulate
sufficient volumes, or otherwise to be interpreted in probabilistic manners.
For equities, dark pools (wherever available in the regional markets) offer
alternative venues to minimize information leakage or seek extra liquidities.
They are indeed indispensable for many Algos crafted specifically for illiquid
or small-cap names. When placing pegging orders in dark pools (e.g., pegged to
the mid of the National Best Bid and Offer (NBBO)), Algo designers must pay
extra attention to monitor the dynamics and toxicity of the pegged prices. In
the scenario of a single market maker, for example, the prices may merely
reflect the inventory pressure being experienced by the market maker at the
moment, and may not necessarily reflect the “true” values (TV).
For some less liquid fixed-income products, execution usually hybrids agency
and principal trading. An execution Algo constantly monitors external venues
as well as the estimated TVs and spreads of the agent herself. Whenever the
agent can offer prices improved from external venues, the Algo may execute
some portions in the principal capacity to benefit a client. Such practice
must be governed by the Principle of Best Executions universally required by
regulators (e.g., MiFID of the European Securities and Markets Authority
(ESMA)). This demands robust market data connections with low latency, as well
as accurate real-time computation of the TVs and spreads, etc.
To automate and integrate all the aforementioned logic and signals remains a
major challenge for Algo developers, in order to most effectively trade small-
cap names or many illiquid fixed-income products. For instance, to automate
the trading of municipal bonds, many of which are illiquid, the main hurdle
turns out to be very elementary - how to properly estimate their TVs in real
time when actual trades are very sparse and hence credible footholds for
pricing are simply not there.
### 2.2 Optimal Portfolio Execution
Unstructured portfolios can be executed using asynchronous single-name Algos.
By “unstructured,” we informally refer to baskets that have been formed
without any systematic objectives in mind, e.g., delta neutrality or long-
short balance.
For a sizable portfolio whose execution spans over a sufficiently long time
window, execution risk can be reduced if the individual names happen to hedge
among themselves to some degree. This is usually the case for a structured
portfolio, for instance, one resulted from index rebalancing when some new
names are to be acquired and an equal amount of selected old names to be
liquidated.
In the modern portfolio theory, hedging is quantified by the correlations and
volatilities of the names. They can be calibrated directly from the
historically observed returns, or indirectly assembled from multi-factor
models such as the Barra™ or Axioma™.
Conventionally these are often end-of-day (EOD) models. One major challenge is
how to revamp the EOD risk models to land on effective real-time risk models.
A popular practice is to use the EOD correlation models as the backbone,
assuming that correlations vary slowly over longer horizons. The con of this
assumption is that EOD correlations are typically calibrated over longer
horizons and may miss any emerging correlating patterns for the purpose of
intraday trading. Such scenarios can emerge when a given name starts to break
away from a cohort due to some major breakthroughs of products or services,
e.g., a pharmaceutical company with an important new drug approved, or a
public company announced to be included in a popular market index. The
information has been released, but classical EOD models may act too slow to
reflect it in a timely manner. Using weighting schemes like the exponential
moving average (EWA)) helps catch up in reaction but still acts too passively
for the purpose of intraday trading.
Calibrating intraday volatilities imposes another challenge. For any given
single name, static profiling is to establish a static volatility curve
$\sigma(t)$ so that, for example, $\sigma(10:18\mathrm{am})$ denotes the
average volatility at 10:18am. Such profiles can be prepared overnight and
stand by ready before a day-trading session commences. It is a stable and
predictable tool, but may lose touch with the intraday reality of a particular
given day. A more ideal solution would turn to dynamic profiling when the
entire intraday curve is not pre-calculated but gradually rolled out. At each
“current” time $t$, the future profile $\sigma(t:EOD)$ can be modeled as a
stochastic process or updated belief based on what has been observed in the
market “so far.” This can be computationally more expensive but surf well with
real-time market waves.
Away from risk correlations, correlations among impact costs have been largely
muted in both academic and industrial works. The assumption of independent
impact costs might approximate well for most individual names. But there are
scenarios well worth further data-driven studies. For example, suppose that a
portfolio contains both a single common stock named SABC and an exchange-
traded fund (ETF) named FXYZ that has SABC as one of its sizable constituents
(e.g., SABC=Exxon Mobil Corp. and FXYZ=XLE - Energe Select SPDR Fund). Thanks
to index arbitragers, any sudden push-ups of SABC can be transferred to FXYZ
almost instantaneously, and vice versa. As a result, the impact costs of
trading sizable amounts of SABC and FXYZ could be intricately entangled,
potentially leading to non-negligible and verifiable observables.
To organically integrate all these risk and cost analytics into coherent
portfolio execution models imposes another major challenge. Such Algo models
must be mathematically tractable and computational feasible and efficient.
Among them, self-contained dynamic programming models are notoriously harder.
Finally, as broker-dealers and trading agents become more enthusiastic in
integrating and unifying their platforms and offering cross-asset trading
Algos, it is another challenge to optimally trade portfolios containing
multiple but correlated asset classes, e.g., common stocks, ETFs, futures,
options, or general FICC products.
### 2.3 Clustering/Classification of the Universe
The security universe covered by a typical investment bank, execution agent,
or registered market maker is usually vast, potentially including tens of
thousands of different names depending on asset classes. This is especially
true for global equities trading.
Clustering helps organize trading universes and drastically reduces operating
complexities. In a stationary trading environment, clustering can
substantially improve operational efficiency by sharing a set of common
strategies, parameter factories, implementations, or risk controls within
individual clusters. In an emergency scenario, clustering can also offer a
default framework or procedure for automated handling, e.g., instantaneous
imputation of certain risk characteristics when data servers or connections
are experiencing unauthorized naps or unexpected glitches.
Conventional frameworks or risk models have already been able to offer some
rudimentary schemes of clustering or classification, e.g., via the directions
of sectors, industrial groups, or fundamental risk drivers such as market
capital sizes, value vs. growth, etc.
Modern machine learning (ML) techniques probably can offer more. For the
purpose of intraday optimal execution or continuous market making, an
overnight process of clustering and classification is more ideal than the
traditionally static segmentation schemes, such as those merely driven by
sectors or industrial groups. Here the main challenges are to sort out all
security characteristics, be them fundamental or technical, that are most
relevant to intraday trading for either optimal execution or market making.
ML-driven clustering may also have to accommodate conventional risk models and
to set proper clustering objectives. Unlike prevailing risk models, non-
numerical characteristics can also be accommodated using modern ML techniques,
such as categorical feature variables or those derived from alternative data
like investment sentiments on social media.
### 2.4 Handling of Special Days
There does not seem to exist a formal theory about special days, but
industrial practitioners know that they ought to be treated differently. In
the modern era when technologies can facilitate quicker and more adaptive
responses to market dynamics, indeed they deserve more customized and
responsive strategies.
Well-known examples include the Fed Announcement Days, Half Trading Days,
Month-End or Quarter-End Days, Index Rebalancing Days, Triple Witching Days,
etc. Trading patterns are more pronounced on certain special days than on some
others. Also days like Index Rebalancing may observe more salient impacts on
certain specific individual names, i.e., the new joiners and dropouts for an
index.
On special days, trading patterns may differ in all sessions, including for
instance, the continuous core session, and open and close auctions. As a
result, the corresponding trading parameters should be prepared using proper
statistical methods or machine learning techniques. During real-time
implementation of a special day, trade beliefs and forecasting must be further
updated based on the specially calibrated models, parameter factories, and the
actually observed market dynamics. Each special day may assume its very
special identities.
An equally stimulating notion, though less popularly implemented in the Algo
trading industry, is “Special Periods.” Special periods may be anchored around
special days, for example, the first trading week or month of an Initial
Public Offering (IPO) for equities, or the week near a roll date of a major
index futures for futures, etc. They may signify the periods of known
transitions and uncertainties, and hence are more tractable than latent
periods in general regime-switching models.
Special days or periods represent the non-stationary moments of the life
cycles of securities or trading environments, and may present good
opportunities for those who master them. Trading with specialized and
effective strategies is certainly not a trivial task, and requires special
investments in talents and analytics.
### 2.5 Real-Time Pricing of Derivatives
Traditionally in most investment banks, pricing models have been developed for
booking trades manually. Traders have to add further overhead premiums such as
spreads or commission fees on top of the “true” prices projected by pricing
models. Pricing models and the associated risk analytics are also utilized by
mid- and back-offices for monitoring the end-of-day (EOD) positions and
aggregated risks.
With increasing demands on derivatives and more efficient exchange or OTC
trading, automated trading and clearing of derivatives have gradually become a
priority for many investment banks. Among all building blocks, pricing models
stand out as the core pillars.
Here the main challenges include: (A) speeding up computation for pricing and
associated risks, and (B) revamping pricing infrastructures, including data
connections and servers, to facilitate fast and robust real-time price
queries. The two are clearly intertwined.
Conventional EOD reading and construction of pricing curves, e.g., interest
rate curves or credit curves, are too sluggish for intraday and real-time
trading. Ideally a pricing engine must be able to query market data (e.g.,
money markets, bills, notes, bonds, or swaps and swaptions on rates or
credits) in real time, and then to construct on the fly the implied rates or
credit curves. This demands rewiring or upgrade of market data subscriptions
and connections, as well as computing engines for curve calibration.
Furthermore, the heavy machinery of partial differential equations (PDE) for
option pricing or Monte-Carlo (MC) simulations for exotics has to be re-
designed, in order to substantially catch up in speed for real-time dynamic
environments. Taylor or asymptotic expansion, spline interpolation or
extrapolation, and more general approximation techniques probably have to be
adopted to reduce frequent calls of the heavy artillery. The second- or
minute-time windows of computation must be compressed towards the scale of
milli- or micro-seconds for Algo trading, at the sacrifice of some accuracy.
On the control side, both the Tech Risk Management (TRM) and Model Risk
Management (MRM) must step up in scrutinizing the soundness and robustness of
these novel infrastructures and real-time pricing logic. Previous approvals on
EOD pricing models do not automatically transfer to real-time models!
Needless to say, automated derivative trading will become the most exciting
area for most investment banks or trading firms. It requires serious
investments in the best infrastructures, analytics, and above all, IT talents.
### 2.6 Trading in (Close) Auctions
There are open and close auctions in the US national market system (NMS).
Intraday auctions also exist in some regions such as Europe. Auctions provide
important alternatives for liquidity sourcing and price formation, and play a
critical role in modern-day trading.
Among all, close auctions are becoming the most prominent trading sessions
across global markets. Perhaps it can be best justified by this simple keyword
- “completion,” that universally governs intraday trading activities, be them
high-touch or low-touch.
When traders or portfolio managers are mandated to liquidate or acquire
certain positions before any given EOD, the close auction offers the final
substantive pool of liquidities. This applies, for instance, to index fund
managers who attempt to minimize fund tracking errors on an index rebalancing
day, or to traders on a central risk book who attempt to stay compliant with a
firm’s internal EOD risk limits and allocations.
Close auction volumes have been on steady rise and no serious liquid-seeking
traders can afford to miss them. For developed markets such as in US or
Europe, average close auction volumes have already stepped into the double-
digit zone (as a percentage of the average daily volume (ADV)).
Algos offer a variety of options to traders or clients for effectively tapping
auction liquidities. Unless explicitly instructed to complete execution before
the close, in theory any Algo can offer participation in close auctions.
Taking VWAP for instance, a natural way is to treat the close auction volume
as a Dirac “point” mass and then to allocate the auction participation
proportionally.
In addition, there also exist dedicated auction Algos that are marketed under
the name of “Target Close (TC).” Different broker-dealers or execution agents
may design it with their own objectives and customizable options. Each TC Algo
attempts to benchmark against the close price while maximally curbing
potential price impact or information leakage. This means that some portions
may have to be traded in the continuous core session just before a close
auction.
To best serve the interests of trading clients, all these Algos that tap close
liquidities must develop forecasting models on auction volumes and prices, as
well as their pre-close dynamics. They must properly digest information such
as imbalance and indicative prices that is being continuously disseminated to
the public after a certain time before a close auction (e.g., 3:45pm in US).
Modern data analytics and machine learning methods can probably improve these
predictive models. Traditionally only straightforward statistics have been
explored. The main challenges here are that each primary exchange has its own
auction roll-out procedures and rules, and that some specific operations could
perplex modeling efforts, e.g., special orders like NYSE’s Closing D Orders.
Finally, it is also nontrivial to seamlessly integrate these predictive
analytics into a self-contained and objective-driven optimization problem.
### 2.7 Transaction Cost Models (TCM)
In the current article, TCM is restricted to pre-trade forecasting models for
estimating the transaction costs of trading any proposed positions. We shall
reserve TCA, Transaction Cost Analysis, for any post-trade analysis on the
costs of actually executed trades. The costs due to fees and commissions are
out of scope, since they are either published or contracted. In addition,
transaction costs here mainly refer to the impact costs, not the market risk
costs associated with innate market fluctuations.
In an editorial article like this, savvy readers might have been searching for
the keyword “TCM” from the very start. Indeed, TCM is probably the most
celebrated metric in Algo Trading, though this does not mean that it has been
thoroughly understood.
In fact, TCM to the Algo community behaves a bit like the concept of “gravity”
to the society. For thousands of years, human beings have been aware of the
existence of gravity and successfully applied it to important social-economic
activities such as measuring the weight of grains for taxing purposes or the
weight of gold and silver as currencies. The true enlightenment of gravity,
however, did not emerge until Newton and Einstein uncovered the laws behind.
In the earlier years, broker-dealers or trading agents did openly reveal their
TCM models either formally or informally. But the trend is that these models
sink deeper and deeper underwater, and become proprietary and confidential.
This is especially true for many emerging TCM models for FICC, such as those
for bonds or Foreign Exchanges (FX). Freely accessible TCM models are very
rare. For instance, only the Kissell Research Group (KRG) still maintains an
open and free TCM model under the brand name of “I-Star,” at the time when
this article is published.
In theory, for any given security there should be a single ground-truth TCM
model, which should be kept open, transparent, and accessible to any traders
or fund managers. The fact that different firms develop their own confidential
and proprietary TCM models perhaps already suggests something disturbing. Or
rather, it may have also signalled the very complexity and ambiguity of the
notion of TCM. Even for post-trade TCA when trade data have been completely
observed, it is not so straightforward to carve out the net impact costs.
Let us dip into some light details. Most TCM models seek a function form of:
$TCM=\Phi(Q,[T_{0},T_{1}]\mid s),$
where $s$ denotes a given security, $Q$ a targeted buy/sell position in $s$,
and $[T_{0},T_{1}]$ a designated execution window, e.g., $Q=60,000$ shares,
$T_{0}=10:00$ am, and $T_{1}=2:00$ pm. In terms of analytics, the security $s$
supplies all the cost and risk parameters, e.g., average spread $\theta_{s}$,
average daily volume $ADV_{s}$, and average volatility $\sigma_{s}$. Hence the
expanded function form is given by:
$TCM=\Phi(Q,[T_{0},T_{1}],\theta_{s},ADV_{s},\sigma_{s}).$
It is a convenient format for pre-trade cost forecasting, as well as for
portfolio optimization when trading costs are factored in. But it does not
differentiate among the actual Algos. Such a model often implicitly assumes
the VWAP or PoV (Participation of Volume) Algo. A more ideal model should
indicate such dependency, i.e.,
$TCM=\Phi(Q,[T_{0},T_{1}],\theta_{s},ADV_{s},\sigma_{s}\mid Algo).$
For example, the net impact cost is very different for a typical front-loading
Implementation Shortfall (IS) Algo that is benchmarked against the arrival
price, as versus a more flat-loading VWAP Algo. Using an Algo-independent TCM
model to project IS impact costs is doomed to be inaccurate.
The reality is that few broker-dealers or trading houses provide Algo-specific
TCM models, to our best knowledge.
Furthermore, TCM modelling also faces some theoretical challenges. Consider a
schedualing-based Algo, be it a non-dynamic VWAP or IS Algo, for which the
execution path $\displaystyle q_{t\in[T_{0},T_{1}]}$ is pre-scheduled by a
suitable optimization model and satisfies
$\int_{T_{0}}^{T_{1}}q_{t}dt=Q,\quad\mbox{with $q_{t}$ denoting trading
speed.}$
It is generally held true that the final netted IS, expressed as the basis-
point spread over the arrival price, bears the form of:
$IS=C+Z.$
Here $\displaystyle C=TCM=\mathcal{F}(q_{t\in[T_{0},T_{1}]}\mid s)$ is a
deterministic functional of the execution path $\displaystyle
q_{t\in[T_{0},T_{1}]}$, for the given security $s$, and $Z$ is a zero-mean
random component resulted from the innate stochastic price fluctuations of the
market (e.g., Brownians as in most published works including Bertsimas and Lo,
Almgren and Chriss, or Shen, just to name a few). In particular, one has the
convenient interpretation of the TCM: $TCM=C=E[IS]$.
In reality, even for a given deterministic schedule $\displaystyle
q_{t\in[T_{0},T_{1}]}$, $C$ is still stochastic. The cost component $C$
involves the complex interactions of numerous real-time factors, including the
dynamics of the limit order books, the strategy of allocating marketable vs.
limit orders, and the usage of dark or grey venues and different order types.
Some of these variables are latent and not directly observable, e.g., the
liquidity in a dark or grey venue, or the waiting queues of limit orders.
In addition, it is also less obvious why the two random components $C$ and $Z$
should be independent. Here perhaps one needs a bit bold revolution that is
parallel to the “Theory of Relativity,” in spirit. For a sizable trading path
ripping through a given market (e.g., with an average market participation
rate of 30%, say), why should one still believe in the existence of an
“absolute” market where the rest participants still trade according to a pre-
designed and undisturbed Brownian motion?
There is still some long way to go before reaching a more coherent and matured
theory of TCM and more rigorous computational implementations. Notice that the
R-Squared scores are universally low for TCM models of the current generation,
as low as in the teens or with single digits. TCM 2.0, which is yet to come,
can perhaps substantially benefit from modern data analytics as well as
machine learning techniques. A coherent theory clearly holds the key. On the
other hand, for many less liquid FICC securities or ones whose markets are
still at their infancy stages, it will take some extra miles to materialize
even TCM 1.0.
### 2.8 Automated Controls of Automated Trading
Algos differ from many conventional financial products or services. First and
foremost, Algos directly access the National Market Systems (US), Regulated
Markets (MiFID, Europe), or general national exchanges. Any serious system
glitches, operational incidents, or design flaws could generate broad impacts
on the regional security prices, major index levels, associated derivative
markets, or even the net asset values (NAV) of pension or retirement funds.
Algos expose their owners, agents, or clients to all major types of risks, be
them investment banks, broker-dealers, trading firms or various funds. The
main risk types include, for example,
1. (a)
regulatory risk for infringing rules, laws, or regulations on securities,
markets, and trading,
2. (b)
financial risks for suffering substantial principal losses as a result of
erroneous trading activities,
3. (c)
reputational risks for violating the core principles of financial integrity,
or for offering poorly managed products and services to clients, and
4. (d)
operational and technological risks for inadequately testing and monitoring
trading systems, networks, or servers and data centers, etc.
Because of the autonomous nature, most behaviors of Algos have to be
controlled in an automated or low-touch way, instead of via manual or high-
touch interventions. The latter applies only to ultimate controls such as the
Emergency Shutdown Procedure (a.k.a. the “Kill Switch”) when the entire Algo
system or exchange connections have to be shut down via manual commands (as in
Linux) or clicking on-screen “panic” buttons.
Controls throughout the life cycles of orders, e.g., incoming parent orders
and child orders at different stages, have to be automated and embedded within
the order or execution management systems (OMS/EMS). There should be blocks of
control codes or scripts residing within the OMS/EMS that can automatically
police order activities, e.g., parent order acceptance, child order
generation, order splitting and routing, and messaging with external exchanges
or venues.
No senior management teams or clients can feel truly at ease with black-box
Algo systems unless it is confirmed that these systems are largely self-
regulatory and that comprehensive controls are automated and algorithmic as
well.
Controls could be kept simple for a small proprietary trading firm who focuses
on only a very limited set of securities using a limited set of Algos. For a
large-scale investment bank, broker-dealer, or trading firm, however, it is a
daunting task to develop a rigorous and effective control framework that is
transparent and auditable, e.g., by either the internal audit teams or
external regulators. These firms trade hundreds or thousands of names across
multiple asset classes on each business day, relying on tens or hundreds of
Algos.
At the minimum, such a control framework means
1. (i)
to establish control governance structures or committees within a given firm,
2. (ii)
to develop formal control policies and procedures,
3. (iii)
to construct and maintain a detailed control inventories, including some key
pieces such as identified risks, proposed controls, actual implementations
within the OMS/EMS or beyond, and unit or regression tests that prove the
effectiveness of the implemented controls,
4. (iv)
to clearly delegate and orchestrate the responsibilities within the Three
Lines of Defense, including Algo desks, risk management, and independent
internal or external audit teams, and
5. (v)
to monitor and document the entire life cycles of the controls, including (a)
any onboarding requirements for new Algos and associated controls, (b) change
management of existing controls, (c) effectiveness and breaching incidents of
the established controls, and (d) periodic reviews of the controls.
To establish a matured control framework often requires multiple years of
commitment and investment from investment banks or trading firms!
To better elucidate the above discussion, which is somewhat abstract, let us
walk through a relatively “simple” example.
Suppose for a given Algo named OGLA, among its 280 proposed controls, there is
one specific control with identification number CTL-ID9988 which is to limit
an incoming parent order to a compliance limit of $\Theta=128$ MM USD. Any
incoming order exceeding this limit will be rejected and returned to the
trader or client who has submitted it.
This control reads very self-explanatory and almost trivial. But one should
not be fooled by its illusory simplicity! From the control-framework point of
view, one can and should challenge it from multiple facets.
* •
(Ownership) Who defines this limit of 128 MM? And who are the validators and
approvers?
* •
(Documentation) What is the rationale in the historical context of this Algo
named OGLA? And where is this rationale documented?
* •
(Data Security) Within the trading system of Algo OGLA, where is this limit
number of “128 MM USD” stored? And who has the right to access and overwrite
it?
* •
(Ongoing Monitoring) In the past quarters, what is the rejection rate under
this limit? If the rejection rate has been consistently high, which may have
signalled a systematic increment of trading scales from clients (instead of
due to fat fingers), should the Algo desk consider to increase this limit for
legitimate business? In the same fashion, if the maximum position has been
only 30% of this limit consistently in the past quarters, should the Algo desk
consider to lower it in order to more effectively curb fat-finger errors?
* •
(Exception Handling) When there is a legitimate reason for a trade to go above
this limit, e.g., when both the external client and the internal sales
trader(s) have manually communicated about and confirmed such a trade size,
what is the emergency procedure for such a trade to legitimately pass the
limit check, instead of being rejected outright?
The example has been fabricated but the above points are profoundly real. The
actual impact could go way beyond the couple of lines that implement such a
deceptively trivial control:
⬇
if(order.notional > CTL_ID9988.limit)
order.status = REJECTED; …
Another major challenge for developing a coherent control framework is that
Algos by nature are dynamic. In response to emerging trading environments, new
client requirements, or novel IT developments, Algo systems are in a constant
state of morphing and revamping. It is highly burdensome to scrutinize and
approve frequent but legitimate (and occasionally very urgent) changes while
maintaining a consistent policy.
One partial solution is perhaps to replace human approvers and validators by
automated algorithms, e.g., via machine learning techniques or artificial
intelligence. But this could take away the already shrinking pool of jobs for
working daddies and mommies - a ubiquitous wrestle between humans and AIs in
the modern era.
### 2.9 Full-Scale Testing and Simulation
If analytics and strategies define the mind and soul of an Algo, lines after
lines of codes then build up the very flesh and body. Ensuring the healthiness
of the body is the highest priority for Algo development and maintenance.
The codes embody all the critical functions of Algos, such as messaging with
external venues or clients, listening to real-time market trade and quote
(TAQ) data, and querying reference data, profiling data, or parameter
factories. Most importantly, they implement all the core EMS/OMS logic. The
codes as a whole constitute into a complex ecosystem of interacting units.
In general, objective oriented programming (OOP) offers an effective framework
for large-scale code design, components structuring, sharing of common
functionalities, and multi-developer collaboration. For example, C++ and Java
have been broadly employed as the mainstream languages for Algo development.
But sound OOP structuring does not always guarantee bulletproof shelters from
coding errors.
As time passes, any given Algo system has to evolve in order to fix bugs,
incorporate novel strategies, or offer new functions and features. Then logic
and strategies become more and more involved and coding structures more
complex. As a result, an Algo system may become increasingly vulnerable to
programming bugs and flaws.
The human factor is a significant source of such potential errors. Developers
or strategists come and go. Consequently coding styles may change and many
hidden intentions of initial designs (e.g., on classes, variables or
functions) may gradually get lost or misused. Formal documentation of all
coding details is virtually impossible, while informal in-line commenting is
also insufficient to maintain code sanity.
The other major source of errors arises from all the revamping efforts for
expanding new features, functionalities, or products. It is highly nontrivial
to ensure an organic integration of the new and old codes, especially for
large-scale or in-depth projects such as platform migrations or adopting
complex quantitative models. Here are two example scenarios when due diligence
must be exercised. In reality, one must face all types of challenging
scenarios.
1. (a)
For instance, inserting a new member function into an existing class which
modifies an existing global variable could turn very risky without careful
examination on how the variable has been utilized in the existing framework.
This is especially true when this global variable has been used somewhere else
as a control signal for making trade decision such as order splitting or
cancellation.
2. (b)
New Algo products are often built upon existing modules. These units must be
organically integrated, instead of being linked perfunctorily. Previously they
may have been operating independently. Once being encapsulated under the hood
of some new parent logic, these units may have to run in parallel or series.
As a result, a responsible developer would have to carefully examine the
signals that these modules all listen to, the controls all governed by, as
well as the complete flow of cause-effect events. Otherwise, serious glitches
could surface under certain trading environments that happen to awaken some
previously dormant bugs.
Unit Tests are designed to verify that individual member functions or task
blocks have been coded up as intended. Regression Tests make sure that these
units or other general functionalities remain stable and predictable during
rounds of code changes. Regression tests are especially critical for hard
compliance controls such as on notional or credit limits.
These popular and automated tests, if sufficiently comprehensive and accurate,
can indeed deliver a high-level of assurance on the soundness of an Algo
system.
But an Algo system is not merely an inorganic stack of individual units or
functionalities. In general, neither unit tests nor regression tests can go
all the way bottom-up to cover the entire dynamic decision trees embedded
within an Algo system. Most often bugs sneak around in these decision trees
where no tests have ever probed.
As a result, unit and regression tests must be augmented by full-scale
simulations of an entire Algo system. This is where the ultimate challenge
lies.
First of all, it is highly nontrivial to simulate the dynamics and all
possible scenarios of the markets. For example, for testing purposes, one must
be able to simulate a sudden trade halt (e.g., as triggered by a circuit
breaker rule) and to test/simulate how an Algo system handles the entire life
cycle of such a halt. Similarly, for a global market system whose trading
hours revolve just as the Earth does, e.g., the FX market, the simulation
system must be able to highlight and react to the particular market open and
close periods of other regional markets and the companion liquidity spikes.
These are merely two examples.
Even with well-designed mocked market dynamics and event sequencing, the other
side of the challenge is to require a simulation system to go through all
scenario paths that an Algo system can possibly wander through. The dynamic
actions of an Algo may involve numerous control or switch statements, e.g.,
typically coded up by lines like “if-elseif’s-else” or “switch.” In addition,
they are often cascaded from parent-level requests to children-level
responses. The net effect is that a typical Algo amounts to a growing decision
tree with numerous branches along the time axis. Failure to simulate through
any particular path may expose the Algo to a potentially unregistered bug. But
it is a daunting task to ensure that all probably paths be fully visited and
simulated.
Finally, Algo developers should pay extra attention to the testing and
simulation of system capacity. An Algo system or action that runs smoothly for
5 test names in simulation does not necessarily prove that it will behave so
in a real trading environment for 500 synchronous names. When this happens,
the financial risk could turn very high when an Algo system fails for critical
actions like the “Kill Switch.”
## 3 Conclusion
Algorithmic trading and its effective controls have been playing a fundamental
role in contemporary financial technologies and services. Optimized trade
execution for pension funds, retirements funds and non-profit endowment funds,
for example, directly impacts the life of hundreds of millions of main-street
citizens. Similarly, automated market making is also vital for maintaining
orderly and robust markets by stable liquidity pooling.
Therefore, it is beneficial for the entire Algo trading community to nurture
an open and collaborative culture, as well as to ensure the healthiness and
further advancement of ATC.
This editorial article has been written in this very spirit. We have
summarized the current status and challenges facing the nine important facades
of ATC. When turned over, the coin of challenges also reveals the other side
of exciting opportunities and competing edges in the Algo trading industry.
Trading institutions who are committed in making further investments in
talents, analytics, technologies, and control frameworks will finally excel.
Finally, we emphasize that by no means these challenges are claimed to
constitute into an exclusive list. There exist also some other important ones,
for example, with regard to the robustness of data servers and connections,
effectiveness of integrating modern data analytics, handling less liquid asset
classes or OTC derivatives, and developing ultra low-latency trading systems.
We also remind readers that the editorial views herein could be inevitably
personal and biased.
## Acknowledgments
The author is very grateful to all the former colleagues at the equities Algo
trading teams at both J.P. Morgan and Barclays, all the risk and control teams
at Goldman Sachs for electronic trading on all asset classes, as well as all
the consultant colleagues from E&Y and KPMG who have played critical roles in
disseminating and fusing knowledge and practices on electronic trading. The
views in this article are however personal, and by no means imply any
endorsement by or representation of these institutions or colleagues.
Version History: v.202101 - Initial Publication
|
# Space Weathering Within C-Complex Main Belt Asteroid Families
Cristina A. Thomas Northern Arizona University, Department of Astronomy and
Planetary Science
PO Box 6010, Flagstaff, AZ 86011 USA David E. Trilling Northern Arizona
University, Department of Astronomy and Planetary Science
PO Box 6010, Flagstaff, AZ 86011 USA Andrew S. Rivkin Johns Hopkins
University Applied Physics Laboratory Tyler Linder University of North
Dakota
###### Abstract
Using data from the Sloan Digital Sky Survey (SDSS) Moving Object Catalog, we
study color as a function of size for C-complex families in the Main Asteroid
Belt to improve our understanding of space weathering of carbonaceous
materials. We find two distinct spectral slope trends: Hygiea-type and Themis-
type. The Hygiea-type families exhibit a reduction in spectral slope with
increasing object size until a minimum slope value is reached and the trend
reverses with increasing slope with increasing object size. The Themis family
shows an increase in spectral slope with increasing object size until a
maximum slope is reached and the spectral slope begins to decrease slightly or
plateaus for the largest objects. Most families studied show the Hygiea-type
trend. The processes responsible for these distinct changes in spectral slope
affect several different taxonomic classes within the C-complex and appear to
act quickly to alter the spectral slopes of the family members.
## 1 Introduction
Large spectral and spectrophotometric datasets enable powerful studies of the
compositions and surface properties of small body populations in our Solar
System. In particular, the Sloan Digital Sky Survey (SDSS) Moving Object
Catalog is an excellent resource that contains photometric observations of
over 100,000 unique known moving objects. Previous studies have used the SDSS
Moving Object Catalog photometry to investigate the distribution of taxonomic
types across the Main Belt (Carvano et al., 2010; DeMeo & Carry, 2013), the
colors of the Main Belt families (Parker et al., 2008), and space weathering
trends both between families (Nesvornỳ et al., 2005b) and within certain
families (Thomas et al., 2012). All of these works have demonstrated that the
SDSS photometry can be reliably used to distinguish between taxonomic classes
and determine spectral slopes. The ability of this dataset to determine
spectral slopes for large numbers of objects is particularly important to the
study of spectral trends associated with space weathering.
Spectral changes due to space weathering have been extensively studied for
S-complex asteroids (e.g. Binzel et al., 2004; Thomas et al., 2011, 2012;
Gaffey, 2010), ordinary chondrite meteorites (e.g., Strazzulla et al., 2005;
Moroz et al., 1996), the Hayabusa samples (e.g., Noguchi et al., 2011), and
other silicate-rich materials (e.g., Sasaki et al., 2001; Pieters et al.,
2000; Brunetto & Strazzulla, 2005; Loeffler et al., 2009). Many investigations
have concluded that silicate-rich materials display similar signs of space
weathering including increased spectral slope, decreased band depth, and
decreased albedo. In the past decades our understanding of how space
weathering affects the physical properties of S-complex material has
continuously improved, while our knowledge regarding the C-complex materials
has lagged behind.
Nesvornỳ et al. (2005b) examined space weathering for the S and C-complex
families across the Main Belt by investigating the change in principal color
components compared to the age of the family. Their analysis supported the
conclusion that S-complex materials show an increased slope with increasing
age and was the first to suggest that the spectral slopes of C-complex
materials show a blueing trend with increased age. Complementary studies of
carbonaceous chondrites have shown that laboratory space weathering simulation
experiments on different meteorite types have resulted in conflicting observed
spectral slope changes. The laboratory observations were summarized in Lantz
et al. (2017): irradiation of CV and CO chondrites results in redder spectral
slopes, while CI and CM chondrite spectra become bluer. All of these published
C-complex trends match a key expectation of space weathering induced spectral
changes that the trend should continue with time up to the point when
saturation is reached and no further change occurs (e.g., Pieters et al.,
2000).
We examine space weathering trends within Main Belt asteroid families using
the Sloan Digital Sky Survey (SDSS) Moving Object Catalog. By considering each
family independently, we remove variation in composition from our analysis of
changing spectral properties. The assumption that composition is consistent
within each family is common, but imperfect (e.g., Masiero et al., 2015). For
most families included in this study, we do not have evidence to suggest
compositional variation within the family. The case of the partially
differentiated Themis parent body is discussed further in §4. We investigate
trends of spectral slope with respect to asteroid size for all known C-complex
asteroid families with sufficient data. We assume that smaller objects have
younger surfaces on average since the expected collisional lifetime (O’Brien &
Greenberg, 2005; de Elia & Brunini, 2007), the timescale for regolith refresh
via seismic shaking (Richardson et al., 2005), and the potential to experience
YORP spin-up and failure (Rubincam, 2000; Graves et al., 2018) are dependent
on the size of the asteroid.
## 2 Data & Analysis
The Sloan Digital Sky Survey (SDSS) Moving Object Catalog (MOC, Ivezic et al.
(2002)) consists of near simultaneous observations in five photometric filters
($u$, $g$, $r$, $i$, and $z$) with effective center wavelengths of 3551, 4686,
6166, 7480 and 8932 $\AA$, respectively (e.g., Ivezić et al., 2001; Juric et
al., 2002). The use of the $g$, $r$, $i$, and $z$ filters provides sufficient
wavelength coverage to determine likely taxonomic classifications (e.g.,
Carvano et al., 2010; DeMeo & Carry, 2013) and study spectral slope trends
(e.g., Thomas et al., 2012; Graves et al., 2018). We use the fourth and final
data release of the SDSS MOC which contains photometry of over 470,000 moving
objects including over 100,000 unique known objects. Our previous work (Thomas
et al., 2012) used the older third data release due to non-photometric data in
the fourth release. We have modified our analysis procedure to account for the
non-photometric data.
We investigated each C-complex asteroid family included in the Nesvorny (2015)
version 3.0 catalog. The Nesvorny family lists include the calculated $C_{j}$
parameter for each asteroid to identify potential dynamical interlopers in the
family according to their V-shape criterion (Nesvornỳ et al., 2015). Objects
with $\lvert C_{j}\rvert>1$ are suspected to be dynamical interlopers and are
removed from the sample. This criterion was defined in Nesvornỳ et al. (2015)
and its usage is supported by analysis of the Eos family in Vokrouhlickỳ et
al. (2006). Once these potential interlopers have been removed, we identify
the family asteroids within the SDSS MOC. The Nesvorny family lists include
asteroid numbers, while the SDSS MOC include columns for asteroid number and
preliminary designation or name. Additionally, the last update to the SDSS MOC
predates the Nesvorny catalogs by several years and many objects in the MOC
were numbered in the intervening years. To properly identify any given family
member within the SDSS dataset, we modified the check observability python
script which uses the callhorizons111https://github.com/mommermi/callhorizons
module to query JPL Horizons. For each asteroid number, our modified routine
returns the name (when available) and preliminary designation. To guarantee
that each object observation is only returned once, we search the MOC for all
asteroid names and preliminary designations within each family.
Table 1: C-Complex Families in the SDSS Moving Object Catalog
Asteroid | Asteroid | Nesvorny | Number of | Bin | Spectral | Albedo | Family
---|---|---|---|---|---|---|---
Number | Name | ID | Objects | Size | Type | | Age
Complete Families
10 | Hygiea | 601 | 326 | 65 | C | 0.068aafootnotemark: | 2 $\pm$1 Gyccfootnotemark:
24 | Themis | 602 | 448 | 90 | C | 0.066aafootnotemark: | 2.3 Gyddfootnotemark:
128 | Nemesis | 504 | 52 | 10 | C | 0.071aafootnotemark: | 200 $\pm$100 Myccfootnotemark:
145 | Adeona | 505 | 244 | 50 | Ch | 0.059aafootnotemark: | 700 $\pm$500 Myccfootnotemark:
668 | Dora | 512 | 131 | 26 | Ch | 0.06bbfootnotemark: | 500 $\pm$200 Myeefootnotemark:
Incomplete Families
31 | Euphrosyne | 901 | 131 | 26 | Cb | 0.056aafootnotemark: | $<$1.5 Gyeefootnotemark:
163 | Erigone | 406 | 130 | 26 | Ch | 0.051aafootnotemark: | 300 $\pm$100 Myfffootnotemark:
490 | Veritas | 609 | 99 | 20 | Ch | 0.066aafootnotemark: | 8.3 $\pm$0.5 Myfffootnotemark:
1726 | Hoffmeister | 519 | 74 | 15 | C | 0.06bbfootnotemark: | 300 $\pm$ 200 Myccfootnotemark:
aMasiero et al. (2013), bDeMeo & Carry (2013), cNesvornỳ et al. (2005a),
dMarzari et al. (1995),
eBrož et al. (2013),fNesvornỳ et al. (2015)
For each asteroid family, we remove unreliable SDSS MOC data according to
various criteria. We remove all observations with apparent magnitudes greater
than 22.0, 22.2, 22.2, 21.3, and 20.5 in any of the $u$, $g$, $r$, $i$, and
$z$ filters, respectively. These are the limiting magnitudes for 95%
completeness (Ivezić et al., 2001; DeMeo & Carry, 2013). To address the non-
photometric nights included within the fourth release of the SDSS MOC, we
remove observations with flags relevant to moving objects and good photometry:
$edge$, $badsky$, $peakstooclose$, $notchecked$, $binned4$, $nodeblend$,
$deblenddegenerate$, $badmovingfit$, $toofewgooddetections$, and $stationary$
(as done in DeMeo & Carry, 2013). These flags address a variety of issues
including if the object was too close to the edge of the frame, if the local
sky was poorly determined, if the peak of the object was too close to another
object to be deblended, or if the object was not detected to move. Additional
information for each of these flags can be found in the documentation
associated with the fourth SDSS MOC
release222http://faculty.washington.edu/ivezic/sdssmoc/sdssmoc.html.
We exclude the $u$ filter observations due to the significantly higher errors
on the observations and place a limit on photometric uncertainty on the
remaining 4 filters of 0.06 magnitudes. This is a more restrictive uncertainty
limit than Thomas et al. (2012) and DeMeo & Carry (2013). The selection of
0.06 magnitudes as the uncertainty limit was driven by the spectral slope
analysis of the C-complex families. Spectrophotometry of a C-complex object
will have a small spectral slope with calculated slope errors that will be
notably impacted by large photometric errors. We selected the photometric
uncertainty limit to balance the need for accurate photometry while
maintaining an adequate sample size in the family populations. We restrict our
analysis to the 9 families that have SDSS photometry with a minimum of 50
unique objects.
Table 2 includes information on all of the C-complex families in our analysis
including the parent asteroid number and name, the Nesvorny catalog number,
the number of family objects in our analysis, the bin size for each family,
the spectral type, albedo, and the age of the family. To ensure that any
observed spectral trends are not the result of observational biases due to the
incompleteness of a family, we examine the size frequency distribution of each
of these 9 C-Complex families. A family is classified as complete in this
analysis if the completeness limit for the family is at an H magnitude larger
(diameter smaller) than any observed changes in our spectral slope trends
(Section 3). Five families are classified as complete. For those five
families, we repeat our analysis to ensure that the inclusion of data from
objects beyond the completeness limit does not impact our results. Details
regarding how we determine completeness and our additional analysis of the
five complete families are included in Appendix A. We include all 9 families
in our analysis because the incomplete families show spectral trends
consistent with the complete ones, which suggests that the observed trends are
properties of the families and are not a product of observational bias.
To calculate the slope for each observation, we convert the SDSS magnitudes to
reflectance values. We remove the known Sloan filter solar
colors333http://classic.sdss.org/dr6/algorithms/sdssUBVRITransform.html from
each of the color indices: $g-r=0.44$, $r-i=0.11$, and $i-z=0.03$. The Sun-
removed color indices are then converted into reflectance values, which are
normalized to the $r$ band (6166 $\AA$). Each slope is calculated as a linear
fit to the $g$, $r$, $i$, and $z$ reflectance values using the central
wavelength of the filters. Slope values are given in %/micron. The errors
associated with these reflectance values are calculated using standard error
propagation techniques. The error of the slope is determined by a Monte Carlo
calculation. Each calculated reflectance value is modified with an offset
equal to the propagated reflectance error multiplied by a random number found
from a Gaussian distribution between -1 and 1. For each observation, this
calculation is done 20,000 times and a slope is determined for each modified
spectrum. The slope error is calculated as the standard deviation of the
modified slopes generated by this process. To avoid weighting our analysis on
objects that were observed many times by the SDSS, we average the slopes and
propagate the slope errors for all objects with multiple observations. This is
the same process as used in Thomas et al. (2012). In the Themis family, we
rejected two likely interlopers within the family for having slopes many
standard deviations higher than the surrounding sample.
We investigate overall trends in spectral slope with respect to object size by
using a running box mean as was done in Binzel et al. (2004), Thomas et al.
(2011), and Thomas et al. (2012). The bin size for each family is selected to
be approximately 20% of the total number of objects. We use a bin size of
approximately half the fractional size of the bins chosen by Binzel et al. (50
for 145 objects) and Thomas et al. (35 for 90 objects; 150 for 402 objects).
The narrower bin size enables the study of more structure within our slope
trends. The error for the average slope value of each bin is the propagation
of each individual slope error. We use Solar System absolute magntude, $H$, as
our primary indicator of object size. We approximate object diameters using
the H magnitudes provided by the SDSS MOC and the known average albedos of the
families from Masiero et al. (2013) if available. If no family average albedo
is available, we use the average albedo by taxonomic complex for Main Belt
objects given by DeMeo & Carry (2013) (C-complex $p_{V}=0.06$). We use the
taxonomic classifications given by Masiero et al. (2015) for each family.
Figure 1: Spectrophotometric slope versus H magnitude and approximate object
diameter for the nine C-complex asteroid families that have at least 50 unique
objects with a photometry magnitude error limit of 0.06 magnitudes. Each
figure includes the family name, taxonomic type, number of objects in the
final sample, the bin size for the running box mean calculation, and whether
the family is complete or incomplete. The range of H magnitude is held
constant for each family and is shown along the bottom axis for each figure.
The top axis shows the approximate diameters for the corresponding H
magnitudes using the average albedos for each family given in Table 2. The
Hygiea-type families (blue) show a reduction in spectral slope with increasing
object size until a minimum slope value is reached and the trend reverses with
increasing slope with increasing object size. Section 3 and Figure 4 contain
further discussion regarding the classification of Hoffmeister as a Hygiea-
type family. The Themis family (red) displays an increase in spectral slope
with increasing object size until a maximum slope is reached and the spectral
slope begins to decrease slightly for the largest objects.
## 3 Results
We see two distinct trends for the C-complex families: Hygiea-type and Themis-
type. Figure 1 shows a panel of the 9 complete and incomplete C-complex
families analyzed in this study. Each figure in the panel shows the binned
spectrophotometric slopes versus H magnitude. Each figure also includes
estimated diameter (calculated using the albedos in Table 2) along the top
axis. For clarity, each family is labeled with their taxonomic type, the
number of unique objects in the analysis, the bin size used, and whether the
family is complete or incomplete. The Hygiea-type trend shows a clear
reduction in spectral slope (blueing) with increasing object size until a
minimum slope value is reached. The trend then reverses and the spectral slope
increases (reddening) with increasing object size until the largest objects in
the family are sampled (e.g., Hygiea family) or the spectral slope plateaus
(e.g., Nemesis family). The spectral slopes tend to be approximately equal for
the smallest and largest objects in the sample. We include the slope trends
for the incomplete families in our analysis since all the families are
consistent with the Hygiea-type trend. The Themis-type trend has a clear
increase in spectral slope (reddening) with increasing object size until a
maximum slope value is reached. At our nominal error limit (Figure 1), the
trend then reverses and the spectral slope decreases slightly. For a higher
error limit (0.08 mag, Figure 2), the spectral slope plateaus once the maximum
slope value is reached.
Figure 2: Wtih a less selective error limit of 0.08 magnitudes, the Themis
family shows a clear increase in spectral slope with increasing object size
until the spectral slope plateaus at objects with diameters of approximately 5
km.
Since smaller objects likely have younger surfaces on average due to their
shorter collisional disruption (e.g., de Elia & Brunini, 2007), surface
refresh via seismic shaking (Richardson et al., 2005), and YORP spin-up and
failure (Graves et al., 2018) timescales, we interpret these spectral slope
changes to be due to space weathering of the C-complex asteroid surfaces. We
anticipate that non-catastrophic surface refresh processes are the primary
methods of exposing fresh regolith and resetting the surface age.
To further investigate the space weathering hypothesis for the Hygiea-type
families, we examine the relationship between the location of the family in
the Main Belt and the size at which the minimum spectral slope is reached.
Figure 3 shows the H magnitude of the bin with the minimum spectral slope
versus the semi-major axis of the asteroid family parent body. The figure also
displays approximate diameter calculated from H magnitude using the average
C-complex albedo from DeMeo & Carry (2013). The transition from slope blueing
to slope reddening occurs at smaller sizes for families that are closer to the
Sun. This correlation suggests that the process responsible for this
transition occurs at a younger surface age for objects with smaller semi-major
axes. Given this relationship, it is likely that the increased solar wind
flux, micrometeorite flux, and micrometeorite velocities at smaller
heliocentric distances are a contributing factor to the timescales associated
with the observed spectral slope trend.
Figure 3: For Hygiea-type families, the binned H magnitude of the slope
minimum versus the semi-major axis of the family parent body. The right axis
has the approximate diameter calculated using the average C-complex albedo,
$p_{V}=0.06$ (DeMeo & Carry, 2013). We correlate object size to average
surface age as indicated. The transition from slope blueing to slope reddening
occurs at smaller sizes (and, therefore, younger average surface ages) for
families that are closer to the Sun. We link smaller semi-major axis with a
faster weathering process due to the increased solar wind, micrometeorite
flux, and micrometeorite velocities.
Seven of the nine families investigated show the Hygiea-type spectral slope
trend with the nominal SDSS magnitude error limit. Since the imposed error
limit preferentially removes smaller objects, which tend to have higher
observational errors, we consider if this restrictive limit has prevented the
observation of the V-shape in the remaining two families. When we use a less
selective error limit of 0.08 magnitudes, the Hoffmeister family clearly shows
the same Hygiea-type shape (Figure 4). Therefore, we classify this family as a
member of the Hygiea-type group. Despite this classification, we do not add
Hoffmeister to the sample in Figure 3 because the different error limits
result in significantly different sample and bin sizes which affects the
position of the average H magnitude for the blueing to reddening transition.
As previously noted, using a less restrictive error limit for Themis (Figure
2) slightly changes the spectral slope trend at the largest sizes, but the
trend remains unchanged at the smallest sizes. Additionally, if we use the
trend in Figure 3 as a guide for where we would expect to observe the Hygiea-
type slope transition in Themis, the slope minimum would occur at H$\sim$14.3
which is present in the H magnitude range included in the 0.06 magnitude error
limit analysis (Figure 1). We conclude that the Themis family shows a trend
that is distinctly different than the Hygiea-type slope trend present in the
other families.
We note that the Hygiea and Themis families show different spectral slope
trends with respect to object size despite having several similarities that
are critical to the space weathering process such as heliocentric distance,
average albedo (Masiero et al., 2015), and family age (Nesvornỳ et al., 2005a;
Marzari et al., 1995). The families are the two largest families in our sample
with sample sizes of hundreds of objects and the key features of their slope
trends are robust to changes in the bin size and the applied error limit. The
observed difference in slope trends suggests that the spectral slopes changes
in each family are likely due to crucial differences in surface composition.
Previous work (e.g., Nesvornỳ et al., 2005b) examined space weathering across
the Main Belt by comparing objects from different families directly to each
other. Our analysis finds that the range of spectral slopes varies greatly
from family to family. Therefore, any comparison of slopes between families
would see spectral slope variation that is likely related to the composition
and specific spectral characteristics of the families themselves instead of
being representative of the age of the asteroid surfaces. We find no clear
correlations between the calculated spectral slopes or the depth of the
V-shaped Hygiea-type trend for each family with their ages or heliocentric
distances.
To complement our spectral slope analysis, we also attempted to investigate
albedo changes with respect to size using data from the Wide-field Infrared
Survey Explorer (WISE) spacecraft (Wright et al., 2010). We used the Nesvorny
(2015) lists described in Section 2 to identify family members within the
Mainzer et al. (2016) Planetary Data System catalog. We restricted the WISE
results to those data marked with the “DVBI” fit code indicating that the
diameter, visible geometric albedo, NEATM beaming parameter, and infrared
geometric albedo were allowed to vary in the thermal fit. Unfortunately, the
Nesvorny (2015) objects are all optically selected and the results from the
combination of this data with the NEOWISE catalog showed a clear bias against
small dark objects. The lack of data was especially apparent for objects that
had diameters smaller than the size at which the Hygiea-type families reach
the local slope minimum and the spectral slope trend changes direction.
Masiero et al. (2013) used the hierarchical clustering method (HCM) on the
known Main Belt asteroids detected by WISE to identify family members. When we
use the Masiero et al. (2013) family lists to examine potential albedo trends
using strict error limits ($\sigma_{p_{V}}=0.05$), the sample size remaining
is too small in the relevant size range even for the largest families to
examine the families for trends. While we cannot examine any albedo trends in
detail, we note that the standard deviations on the average C-complex family
albedos (Masiero et al., 2013, 2015) are smaller than for comparably sized
S-complex families (e.g., Hygiea $p_{V}=0.070\pm 0.018$ vs. Flora
$p_{v}=0.305\pm 0.064$) so it is unlikely that there is significant variation
of the albedos within each family.
Figure 4: The Hoffmeister family does not show the Hygiea-type trend with the
nominal magnitude error cutoff. If we use a less selective error limit of 0.08
magnitudes, the Hoffmeister family shows the V-shaped Hygiea-type trend.
## 4 Discussion
Analyses of S-complex objects have shown clear space weathering trends from
examining the relationship between object size and spectral slope (Binzel et
al., 2004; Thomas et al., 2012). By examining this trend within known,
characterized families we reduce potential spectral variation due to
compositional differences. The two types of slope trends seen in the C-complex
families show trademarks of being products of the competing processes of space
weathering and surface refreshing. Most importantly, the trends show a clear
dependence of spectral slope on the size of the object, which we use as a
proxy for average surface age, within each family. We also see evidence that
the Hygiea-type trend observed in 8 of our 9 families is correlated with the
distance from the Sun of the family’s parent body. We note that this
transition happens within a narrow range of estimated object diameters
($\sim$4-10km). The transition from reddening to blueing (or plateau) in the
Themis family also occurs within this same size range. Therefore, the average
surface age of objects of this size is important to understanding the
processes responsible for these trends and both the weathering and surface
refresh timescales. Additionally, the noted diameter range of the slope
transitions for the 9 C-complex families includes the diameter ($\sim$5 km)
associated with space weathering saturation for S-types in the near-Earth
object population (Binzel et al., 2004) and the Koronis family (Thomas et al.,
2011, 2012).
The processes responsible for this Hygiea-type space weathering trend affect
several different taxonomic classes within the C-complex. The C, Cb, and Ch
classes are represented in our sample. These three classes are distinguished
in the visible wavelength region by the curvature of the overall spectrum (Bus
& Binzel, 2002). The C-class spectra are more concave than the other two
classes and contain a weak ultraviolet absorption at the short wavelength end
of the spectrum, Ch-class spectra contain the 0.7 $\mu$m absorption feature
that has been connected to hydrated silicates, and the Cb-class spectra are
notably flatter than the other two groups. These spectral differences are
indicative of differing compositions. We observe this Hygiea-type space
weathering process in a large fraction of our analyzed families of different
spectral types, which suggests that the process or processes responsible for
this weathering pattern is common among Main Belt C-complex objects.
Table 2 includes the calculated dynamical ages of the families in our
analysis. The Veritas family is the youngest on our list by a significant
margin with an age of 8.3 $\pm$ 0.5 million years (Nesvornỳ et al., 2015). The
spectral slope trend seen in the Veritas family is just as distinct as the
trends seen in significantly older families. The presence of the Hygiea-type
trend in the Veritas family suggests that the timescale necessary to reach
spectral maturity is short especially considering that Veritas is located in
the outer Main Belt ($a=3.17$ au).
Seeing these spectral slope trends in various families with an extremely wide
range of ages indicates that there must be processes that are actively
refreshing the surfaces of the smaller family members. The transition in slope
trend for Hygiea-type families occurs for objects with diameters of
approximately 4-10 kilometers. One possibility for refreshing the surface is
regolith movement from small impactors and the resulting seismic shaking of
the object. The Richardson et al. (2005) model shows that a 20 cm impactor
could cause “vertical launching” (greater than 1 $g_{ast}$ accelerations) on
the surface of a 5 km object under restrictive seismic propagation conditions
with cohesive regolith. Rivkin et al. (2011) used the O’Brien & Greenberg
(2005) Main Belt impact recurrence times and scaled the Richardson et al.
(2005) results to estimate the global regolith refresh timescale to be $\sim
10^{6}$ years for the surface of a 5 km asteroid. This refresh timescale is
orders of magnitude smaller than the ages of all families presented here
(except young Veritas) and is comparable to the fast weathering rates
suggested by Vernazza et al. (2009). Another process that could be responsible
for resetting the surface age is YORP spin up and failure (e.g., Graves et
al., 2018). Rubincam (2000) estimated the time required to double the rotation
rate of several theoretical asteroids via YORP thermal torque. One
hypothetical object included in the study was “pseudo-Deimos” which was given
the same shape, moment of inertia, and albedo as actual Deimos, but placed in
the Main Belt with a semi-major axis of 3 AU. “Pseudo-Deimos” can be used to
estimate the time required to double the rotation rate for asteroids in the
Hygiea and Themis families given the selected semi-major axis and similar
albedo (Deimos $p_{V}$=0.068$\pm$0.007, Thomas et al., 1996). The results
presented in Figure 6 of Rubincam (2000) indicate that an 8 km diameter object
(the approximate object size at the transition in the slope trend for both the
Hygiea and Themis families) would double its rotation rate in less than 100
Myrs. This timescale is an upper limit for the objects in our C-complex
families since the spin up process would happen more rapidly at smaller semi-
major axes. This spin up timescale is longer than that estimated from the
Richardson et al. (2005) seismic shaking model, but is still significantly
shorter than the ages of most of the families discussed. We expect that the
asteroid surfaces are being refreshed via a combination of these two
mechanisms.
One additional possibility to explain the change in spectral slope for the
largest asteroids within the families is variation in grain size. Vernazza et
al. (2016) concluded that observed variation in spectral slopes for large Ch
and Cgh-type asteroids is due to the anti-correlation between regolith grain
size and the object diameter and, therefore, surface gravity. The largest
asteroids (D$>$100 km) in their sample have redder spectral slopes and are
more consistent with a fine-grained CM chondrite-like regolith compared to the
more neutral spectral slopes of the smaller objects (D$<$60 km) which were
linked to a coarse-grained CM chondrite-like regolith.
Studies of carbonaceous chondrite meteorites have shown that laboratory space
weathering simulation experiments on different meteorite types have resulted
in divergent observed spectral and albedo changes. Lantz et al. (2017) and
Lantz et al. (2018) clearly summarize the observed characteristics.
Irradiation of CV and CO chondrites has resulted in spectra with redder slopes
that are more concave and have lower albedos, while CI and CM meteorite
spectra become bluer and more convex while their albedos get higher (e.g.,
Brunetto et al., 2014, 2015; Lantz et al., 2015; Matsuoka et al., 2015).
Similarly, experiments with Tagish Lake show a flattening and brightening of
the spectrum with increased irradiation (Vernazza et al., 2013). In all of
these laboratory investigations of space weathering processes with meteorites,
increased irradiation causes a consistent increase or a decrease in spectral
slope. The vast majority of past work involving both laboratory experiments
and telescopic observations have concluded that the trend in spectral slope
only moves in one direction. The results from the families analyzed in this
study directly contradict those steady space weathering trends. Our analysis
shows spectral slope trends that change direction after some time period.
Irradiation experiments by Kaluna et al. (2017) of aqueously altered minerals
have demonstrated that a change in direction of the spectral slope evolution
is possible. The spectral slopes of their Fe-rich assemblage (consisting of
cronstedtite, pyrite, and siderite) initially reddened and then became bluer
with increased irradiation. To understand the physical processes responsible
for their spectral trends, Kaluna et al. (2017) used scanning and transmission
electron microscopy to examine their samples and radiative transfer modeling
to model the effects of the irradiation products. For the Fe-rich assemblage,
they observed nanophase iron and micron sized carbon-rich particles and
estimated the optical effects of these components from the model. Kaluna et
al. (2017) conclude that the nanophase iron particles were responsible for the
spectral reddening and the micron sized carbon particles caused the spectral
blueing. They explain the slope transition as the carbon particles likely
dominating the optical properties once a critical amount is present on the
surface. This spectral analysis was performed on data with visible and near-
infrared wavelengths, but the slope variations shown in their Figure 3
indicate that the observed slope trend would also be present in visible
wavelength data, such as that provided by SDSS.
Since we also observed a change in direction of the spectral slope, we are
likely observing the results of two competing surface process similar to what
was observed by Kaluna et al. (2017). However, the Hygeia-type spectral slope
trends are the opposite of those observed in the Kaluna et al. (2017)
experiments: we observe blueing of the spectral slope followed by slope
reddening. It is possible that our trends are also the result of the competing
optical effects of nanophase iron and micron sized carbon-rich particles under
different conditions. Many of the compositional components expected in the
C-complex would have sufficient iron and carbon to create these irradiation
products. Additional laboratory experiments may help explain the physical
processes behind this Hygiea-type space weathering spectral slope trend.
The Themis family is the only family in our C-complex sample that does not
show a Hygiea-type trend. It is not surprising that Themis is different given
its unique composition and evolution. Water ice has been detected on the
surface of Themis (Rivkin & Emery, 2010; Campins et al., 2010) and past work
suggests that the parent body experienced some thermal evolution (Castillo-
Rogez & Schmidt, 2010; Marsset et al., 2016). A partially differentiated
Themis parent body would result in a family that contained various
compositions and, therefore, spectral properties. The unique Themis spectral
slope trend could be due to space weathering of the thermally evolved
asteroids. It is also possible that the observed Themis spectral slope trend
is indicative of variation in composition, but there is no known reason why
composition would vary with object size within the family. The change in slope
direction for Themis is not as distinct as in the Hygiea-type trends and is
absent in our analysis with higher photometric error limits. Our data is also
mostly consistent with Kaluna et al. (2016), who conclude that Themis family
visible wavelength spectra become redder due to space weathering.
There are numerous C-complex families that are not included in this analysis
because there were not an adequate number of family members in the SDSS MOC
that met our strict magnitude error limits. We anticipate that future large
photometric catalogs, such as the one that will be generated by the Rubin
Observatory (formerly LSST), will enable the study of space weathering trends
in C-complex families with greater detail. The expected single exposure
limiting magnitudes (5$\sigma$) for the Rubin Observatory are 23.4, 24.8,
24.3, 23.9, and 23.3 in any of the $u$, $g$, $r$, $i$, and $z$ filters,
respectively444https://smtn-002.lsst.io/. These exposure depths are all at
least 2 magnitudes fainter than the 95% completeness limit for SDSS, which
suggests the upcoming survey will be able to observe objects with diameters
$\sim$2.5 smaller than SDSS. In addition to adding new families to this type
of study, we anticipate spectrophotometry for objects down to $\sim$1 km in
diameter for several of the C-complex families included in our analysis.
## 5 Conclusions
Our analysis of space weathering trends in C-complex families shows two
distinct spectral slope trends: Hygiea-type and Themis-type. Eight of the nine
families studied show the Hygiea-type space weathering trend. The processes
responsible for the distinct changes in spectral slope affect several
different taxonomic classes within the C-complex and appear to act quickly to
alter the spectral slopes of the family members. The Hygiea-type trend appears
to be correlated with the distance from the Sun of the family indicating that
the observed trends are related to space weathering of the objects. The Themis
family is the only family to not show the V-shaped Hygiea-type slope trend. We
encourage further laboratory experiments to explain the processes driving
these two space weathering trends.
Funding for the creation and distribution of the SDSS Archive has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Aeronautics and Space Administration, the National Science
Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and
the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is
managed by the Astrophysical Research Consortium (ARC) for the Participating
Institutions. The Participating Institutions are The University of Chicago,
Fermilab, the Institute for Advanced Study, the Japan Participation Group, The
Johns Hopkins University, the Korean Scientist Group, Los Alamos National
Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-
Institute for Astrophysics (MPA), New Mexico State University, University of
Pittsburgh, University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington. Some of this work was
carried out while TL was supported by the NSF Research Experiences for
Undergraduates program at NAU under NSF award AST-1004107.
## Appendix A Completeness Limits for Families
We examine the size frequency distribution (SFD) of each of the 9 C-Complex
families to determine if our observed spectral trends are the result of
observational biases. We present the size frequency distribution of each
family in two different methods (Figure 5): (1) using the Nesvorny (2015)
family list with H magnitudes from the MPC and (2) using the list of family
objects identified in the SDSS MOC with H magnitudes from ASTORB. For the
asteroids identified in the SDSS MOC, we removed those objects with observed
magnitudes beyond the limiting magnitudes for 95% completeness (greater than
22.0, 22.2, 22.2, 21.3, and 20.5 in any of the $u$, $g$, $r$, $i$, and $z$
filters, Ivezić et al., 2001). We used the provided catalog H magnitudes for
each of these datasets. The H magnitudes are similar, but are not identical
for many objects. Each size frequency distribution uses H magnitude bins of
0.2 magnitudes, which mitigates many of the H magnitude differences in the two
datasets. Our family analysis removes additional photometry from the SDSS MOC,
but we choose not to remove additional data from this SFD investigation since
this subset of the data best represents the completeness of the sample.
Figure 5: Size frequency distributions (SFDs) for each family are used to
determine completeness. The figure panel contains two SFDs for each family:
(1) from the Nesvorny (2015) list and (2) using the list of family objects
identified in the SDSS MOC. We use the SFD from the SDSS MOC to determine if
the family is complete. The completeness limit is estimated from the maximum
value of the SFD and a family is determined to be complete if the completeness
limit (dashed line) is at a larger H magnitude (smaller diameter) than H
magnitude of the change in the spectral slope trend (blue for Hygiea-type, red
for Themis).
We use the size frequency distribution from the families within the SDSS MOC
to determine if a family is complete. We classify a family as complete if the
completeness limit for the family is at an H magnitude larger (diameter
smaller) than any changes in our spectral slope trends. We estimate the
completeness limit for each family as the maximum value of the SDSS MOC family
SFD. This process slightly overestimates (with respect to H magnitude) the
true completeness limit for the families. Figure 5 shows the determined
completeness limit as a dashed line and the H magnitude at which the spectral
slope changes as a blue (Hygiea-type) or red (Themis-type) line. We also
classify a family as incomplete if the spectral slope change is within
$\sim$0.2 (the SFD bin size) magnitudes of the completeness limit (Erigone,
Veritas, Hoffmeister). We use the Nesvorny (2015) SFDs for each family to
compare the SDSS MOC results to that of the entire family.
The slope trends presented in Figure 1 for the complete families includes
slopes for objects beyond the defined completeness limit. Due to the running
box mean calculation, slopes for objects beyond the family’s completeness
limit will be included in calculated average slopes near and under that limit.
We verify the robustness of our slope trends to potential observational biases
introduced through extending beyond the completeness limit by repeating our
analysis for the 5 complete families using only family members with H
magnitudes smaller than the completeness limit. We find that the recalculated
slope trends of the complete families are consistent with the trends presented
this work and that the Hygiea-type and Themis-type trends remain distinct from
each other (Figure 6).
Figure 6: Spectrophotometric slope versus H magnitude and approximate object
diameter for the five complete C-complex asteroid families using only objects
with H magnitudes below the defined completeness limit for the family. Each
figure includes the family name, taxonomic type, number of objects in the
sample, and the bin size for the running box mean calculation. The range of H
magnitude is held constant for each family and is shown along the bottom axis
for each figure. The top axis shows the approximate diameters for the
corresponding H magnitudes. For comparison, this figure uses the same range of
slope values for each family as seen in Figure 1. The Hygiea-type families
(blue) and Themis family (red) show a similar spectral trends to those in
Figure 1.
## References
* Binzel et al. (2004) Binzel, R. P., Rivkin, A. S., Stuart, J. S., et al. 2004, Icarus, 170, 259
* Brož et al. (2013) Brož, M., Morbidelli, A., Bottke, W., et al. 2013, Astronomy & Astrophysics, 551, A117
* Brunetto et al. (2015) Brunetto, R., Loeffler, M. J., Nesvornỳ, D., Sasaki, S., & Strazzulla, G. 2015, Asteroids iv, 597
* Brunetto & Strazzulla (2005) Brunetto, R., & Strazzulla, G. 2005, Icarus, 179, 265
* Brunetto et al. (2014) Brunetto, R., Lantz, C., Ledu, D., et al. 2014, Icarus, 237, 278
* Bus & Binzel (2002) Bus, S. J., & Binzel, R. P. 2002, Icarus, 158, 146
* Campins et al. (2010) Campins, H., Hargrove, K., Pinilla-Alonso, N., et al. 2010, Nature, 464, 1320
* Carvano et al. (2010) Carvano, J., Hasselmann, P., Lazzaro, D., & Mothé-Diniz, T. 2010, Astronomy & Astrophysics, 510, A43
* Castillo-Rogez & Schmidt (2010) Castillo-Rogez, J. C., & Schmidt, B. 2010, Geophysical Research Letters, 37
* de Elia & Brunini (2007) de Elia, G. C., & Brunini, A. 2007, Astronomy & Astrophysics, 466, 1159
* DeMeo & Carry (2013) DeMeo, F., & Carry, B. 2013, Icarus, 226, 723
* Gaffey (2010) Gaffey, M. J. 2010, Icarus, 209, 564
* Graves et al. (2018) Graves, K. J., Minton, D. A., Hirabayashi, M., DeMeo, F. E., & Carry, B. 2018, Icarus, 304, 162
* Ivezic et al. (2002) Ivezic, Z., Juric, M., Lupton, R. H., Tabachnik, S., & Quinn, T. 2002, in Survey and Other Telescope Technologies and Discoveries, Vol. 4836, International Society for Optics and Photonics, 98–104
* Ivezić et al. (2001) Ivezić, Ž., Tabachnik, S., Rafikov, R., et al. 2001, The Astronomical Journal, 122, 2749
* Juric et al. (2002) Juric, M., Ivezic, Z., Lupton, R. H., et al. 2002, ASTRONOMICAL JOURNAL-AMERICAN ASTRONOMICAL SOCIETY, 124, 1776
* Kaluna et al. (2017) Kaluna, H., Ishii, H., Bradley, J., Gillis-Davis, J., & Lucey, P. 2017, Icarus, 292, 245
* Kaluna et al. (2016) Kaluna, H. M., Masiero, J. R., & Meech, K. J. 2016, Icarus, 264, 62
* Lantz et al. (2018) Lantz, C., Binzel, R., & DeMeo, F. 2018, Icarus, 302, 10
* Lantz et al. (2017) Lantz, C., Brunetto, R., Barucci, M., et al. 2017, Icarus, 285, 43
* Lantz et al. (2015) —. 2015, Astronomy & Astrophysics, 577, A41
* Loeffler et al. (2009) Loeffler, M., Dukes, C., & Baragiola, R. 2009, Journal of Geophysical Research: Planets, 114
* Mainzer et al. (2016) Mainzer, A., Bauer, J., Cutri, R., et al. 2016, NASA Planetary Data System, 247
* Marsset et al. (2016) Marsset, M., Vernazza, P., Birlan, M., et al. 2016, Astronomy & Astrophysics, 586, A15
* Marzari et al. (1995) Marzari, F., Davis, D., & Vanzani, V. 1995, Icarus, 113, 168
* Masiero et al. (2015) Masiero, J. R., DeMeo, F. E., Kasuga, T., & Parker, A. H. 2015, Asteroids IV, 323
* Masiero et al. (2013) Masiero, J. R., Mainzer, A., Bauer, J., et al. 2013, The Astrophysical Journal, 770, 7
* Matsuoka et al. (2015) Matsuoka, M., Nakamura, T., Kimura, Y., et al. 2015, Icarus, 254, 135
* Moroz et al. (1996) Moroz, L., Fisenko, A., Semjonova, L., Pieters, C., & Korotaeva, N. 1996, Icarus, 122, 366
* Nesvorny (2015) Nesvorny, D. 2015, NASA Planetary Data System, 234
* Nesvornỳ et al. (2005a) Nesvornỳ, D., Bottke, W. F., Vokrouhlickỳ, D., Morbidelli, A., & Jedicke, R. 2005a, Proceedings of the International Astronomical Union, 1, 289
* Nesvornỳ et al. (2015) Nesvornỳ, D., Brož, M., Carruba, V., et al. 2015, Asteroids IV, 297
* Nesvornỳ et al. (2005b) Nesvornỳ, D., Jedicke, R., Whiteley, R. J., & Ivezić, Ž. 2005b, Icarus, 173, 132
* Noguchi et al. (2011) Noguchi, T., Nakamura, T., Kimura, M., et al. 2011, Science, 333, 1121
* O’Brien & Greenberg (2005) O’Brien, D. P., & Greenberg, R. 2005, Icarus, 178, 179
* Parker et al. (2008) Parker, A., Ivezić, Ž., Jurić, M., et al. 2008, Icarus, 198, 138
* Pieters et al. (2000) Pieters, C. M., Taylor, L. A., Noble, S. K., et al. 2000, Meteoritics & Planetary Science, 35, 1101
* Richardson et al. (2005) Richardson, J. E., Melosh, H. J., Greenberg, R. J., & O’Brien, D. P. 2005, Icarus, 179, 325
* Rivkin & Emery (2010) Rivkin, A. S., & Emery, J. P. 2010, Nature, 464, 1322
* Rivkin et al. (2011) Rivkin, A. S., Thomas, C. A., Trilling, D. E., Enga, M.-t., & Grier, J. A. 2011, Icarus, 211, 1294
* Rubincam (2000) Rubincam, D. P. 2000, Icarus, 148, 2
* Sasaki et al. (2001) Sasaki, S., Nakamura, K., Hamabe, Y., Kurahashi, E., & Hiroi, T. 2001, Nature, 410, 555
* Strazzulla et al. (2005) Strazzulla, G., Dotto, E., Binzel, R., et al. 2005, Icarus, 174, 31
* Thomas et al. (2011) Thomas, C. A., Rivkin, A. S., Trilling, D. E., Enga, M.-t., & Grier, J. A. 2011, Icarus, 212, 158
* Thomas et al. (2012) Thomas, C. A., Trilling, D. E., & Rivkin, A. S. 2012, Icarus, 219, 505
* Thomas et al. (1996) Thomas, P., Adinolfi, D., Helfenstein, P., Simonelli, D., & Veverka, J. 1996, Icarus, 123, 536
* Vernazza et al. (2009) Vernazza, P., Binzel, R., Rossi, A., Fulchignoni, M., & Birlan, M. 2009, Nature, 458, 993
* Vernazza et al. (2013) Vernazza, P., Fulvio, D., Brunetto, R., et al. 2013, Icarus, 225, 517
* Vernazza et al. (2016) Vernazza, P., Marsset, M., Beck, P., et al. 2016, The Astronomical Journal, 152, 54
* Vokrouhlickỳ et al. (2006) Vokrouhlickỳ, D., Brož, M., Morbidelli, A., et al. 2006, Icarus, 182, 92
* Wright et al. (2010) Wright, E. L., Eisenhardt, P. R., Mainzer, A. K., et al. 2010, The Astronomical Journal, 140, 1868
|
# E-commerce warehousing: learning a storage policy
Adrien Rimélé<EMAIL_ADDRESS>Philippe Grangier Michel Gamache
Michel Gendreau Louis-Martin Rousseau Department of Mathematics and
Industrial Engineering, Polytechnique Montreal CIRRELT, Research Centre on
Enterprise Networks, Logistics and Transportation GERAD, Group for Research
in Decision Analysis IVADO Labs
###### Abstract
E-commerce with major online retailers is changing the way people consume. The
goal of increasing delivery speed while remaining cost-effective poses
significant new challenges for supply chains as they race to satisfy the
growing and fast-changing demand. In this paper, we consider a warehouse with
a Robotic Mobile Fulfillment System (RMFS), in which a fleet of robots stores
and retrieves shelves of items and brings them to human pickers. To adapt to
changing demand, uncertainty, and differentiated service (e.g., prime vs.
regular), one can dynamically modify the storage allocation of a shelf. The
objective is to define a dynamic storage policy to minimise the average cycle
time used by the robots to fulfil requests. We propose formulating this system
as a Partially Observable Markov Decision Process, and using a Deep Q-learning
agent from Reinforcement Learning, to learn an efficient real-time storage
policy that leverages repeated experiences and insightful forecasts using
simulations. Additionally, we develop a rollout strategy to enhance our method
by leveraging more information available at a given time step. Using
simulations to compare our method to traditional storage rules used in the
industry showed preliminary results up to 14% better in terms of travelling
times.
###### keywords:
Decision processes , Supply chain management , E-commerce , Storage Policy ,
Reinforcement Learning
††journal: European Journal of Operations Research
## 1 Introduction
### 1.1 Warehousing in e-commerce
Warehousing occupies a central role in a supply chain. According to Gu et al.
[2007], the primary purposes of a warehouse are to act as a buffer to adapt to
the variability of the production flow; to consolidate products; and to add
marginal value such as pricing, labelling or customisation.
The recent and massive development of e-commerce introduces great challenges
in the _Business to Customer_ segment. E-commerce involves enormous volumes of
orders, and online sales keep growing in number: in 2016, e-commerce sales
grew by 23.7%, accounting for 8.7% of the total market [Boysen et al., 2019b].
These orders consist, on average, of very few items: the same authors mention
that the average number of items per order at Amazon is only 1.6. E-commerce
also faces great demand uncertainty, with fast-changing demand trends, and the
need to satisfy orders quickly [Yaman et al., 2012, Boysen et al., 2019b, a].
Amazon, for instance, offers customers _Prime_ , a differentiated service that
used to offer delivery within two days and is now, under certain conditions,
promising same-day delivery. Moreover, with easily accessible alternatives
available to customers, online retailers face high competitive pressure, and
the link between logistics performance and customer loyalty is much stronger
compared to other industries [Ramanathan, 2010].
Parts-to-picker Automated Storage and Retrieval Systems (AS/RS), with the
typical single, aisle-captive crane retrieving bins from a static rack, have
been used for several decades and have proven their superior operational
efficiency compared to manual systems, at the cost of an essential initial
investment [Roodbergen & Vis, 2009]. However, some limitations of AS/RS become
apparent when facing new challenges posed by the rise of e-commerce: cranes
are sensitive points of failure; expandability is complicated and expensive;
batching and zoning require consolidation and a delay-sensitive dependency
between operators; and a single crane per aisle can only perform so many
cycles. According to Davarzani & Norrman [2015], “adaptation is urgent” to
keep up with the growing demand.
### 1.2 Robotic Mobile Fulfillment System
Kiva Systems created the first RMFS in 2006, this company was later bought by
Amazon and re-branded Amazon Robotics in 2012 [Wurman et al., 2008, Azadeh et
al., 2017, Boysen et al., 2019a]. However, RMFSs, also referred to as rack-
moving mobile robot based warehouses, or Kiva warehouses, are not limited to
Amazon warehouses. Other companies have developed their own systems: for
instance, Alibaba’s Zhu Que robots, the Quicktron robots (Huawei), the Open
Shuttles by Knapp, CarryPick by Swisslog, Butler by GreyOrange, the Locus
Robotics system and others [Kirks et al., 2012, Boysen et al., 2019a].
This new type of automated warehouse involves a large fleet of small robots
that move freely around the storage area to retrieve shelves full of items and
bring them to operators for picking (possibly after waiting in a queue) before
returning to a storage location. Interestingly, when a robot is loaded, it
must travel along the aisles, but when unloaded, it can pass under stored
shelves to take shortcuts. This type of operations is equivalent to a mini-
load with dual-command cycles system with a fleet of non-aisle-captive robots
[Roodbergen & Vis, 2009]. For a more detailed description of an RMFS and its
advantages compared to traditional systems, the interested reader is referred
to the following references: Wurman et al. [2008]; D’Andrea & Wurman [2008];
Enright & Wurman [2011]; and Azadeh et al. [2017].
Rimélé et al. [2020] present a mathematical modelling framework that
formalises optimisation opportunities in an RMFS. Because e-commerce promises
fast delivery of small orders, the fulfilment of requests follows what can be
called an _order streaming_ logic [Banker, 2018], which “drops orders […] to
the floor as soon as they are received”. Because of this specificity, the
option of batching requests is minimal, and the system needs to account for
uncertainty and adapt online to newly revealed orders. Among the operational
decisions, the real-time storage allocation problem can leverage the great
storage flexibility of the system. When a robot leaves a picking station, it
does not need to store the shelf in its initial location but can choose _any_
free location. This allows for the real-time adaptation of the storage layout
in response to the demand, thereby minimising the travelling time of the
robots, which translates into increasing an upper bound of the throughput.
### 1.3 Contributions and paper structure
This work focuses on a specific operational control aspect of a Robotic Mobile
Fulfillment System for e-commerce, namely the problem of learning an effective
dynamic storage policy. When a robot is available at an operator’s station, a
decision must be made in real-time about which storage location to use to
minimise the average cycle time of the robots. To make such a decision, one
can use information about the next request to process, statistical insights
about the demand, and other input regarding the current state of the
warehouse’s layout. This information at a given time step defines a state’s
representation in an environment that we approximate as a partially observable
Markov decision process (POMDP). We propose the application of a variant of a
Q-learning agent [Watkins, 1989, Hessel et al., 2018, Mnih et al., 2015], a
Reinforcement Learning (RL) algorithm, to learn an efficient dynamic storage
policy that aims at minimising the average cycle time. More precisely, we
implement a Differential Double Q-learning agent with experience relays to
minimise the average travelling time of the robots. Inspired by _Class-based_
storage policies, the decision on where to return a shelf is made based on a
discretisation of the storage area into zones. A discrete event simulator is
used to evaluate the performance of our method compared to baselines on
different levels of skewed demand distributions. To further improve the
performance of our method, we propose an extension using a rollout strategy
over a fixed horizon of already-revealed requests. This strategy is also
applicable to the baseline methods from the literature.
The presentation of our proposed approach is organised as follows. Section 2
first formalises the optimisation problem of dynamically allocating storage
locations to shelves, then presents a literature review related to storage
policies in automated warehouses and operational control applications specific
to RMFS. Section 3 presents our methodology, with a discretisation of the
storage area into zones, the system representation as a POMDP, and the
reinforcement learning variant of a Q-learning agent used to learn an
efficient storage policy, as well as the rollout strategy to further improve
performance. Section 4 explains how we generated data in our simulations and
shows results obtained by our method compared with baselines from the
literature. Finally, Section 5 summarises the obtained results and discusses
future research avenues.
## 2 Problem definition and literature
### 2.1 Decision-making framework
In this section, we present the storage allocation decision-making framework,
as well as the relevant literature on the topic.
For the case of an e-commerce Amazon-type warehouse, a modelling of
operational decisions is proposed by Rimélé et al. [2020], with a stochastic
dynamic optimisation formulation. In this formulation, an incoming request
corresponds to a single item type and consolidation of orders, if any, is
assumed to take place in the downstream sections of the warehouse. Orders are
revealed online and are considered available for fulfilment as soon as they
arrive, with no batching. An _event_ corresponds to a robot being available
for a task. The tasks of a robot correspond to a sequence of _dual-command
(DC) cycles_ for which the total time can be decomposed into: (i) storage
access time; (ii) interleaving (or transition) time to retrieval location;
(iii) retrieval time; and (iv) waiting time at the picking station’s queue
(see arcs on the left side of Figure 1). To create those cycles, storage and
retrieval decisions must be made. After the picking operation, the robot is
available at the picking station, and it has to perform a _storage task_ ,
which consists in selecting an available location for storage. When the shelf
has been stored, the robot needs to perform a _retrieval task_ , which
consists in choosing which order to fulfil next, from which shelf, and which
picking station. Besides those two typical tasks, we also consider an
_opportunistic task_ , a task that can only occur in a particular situation
where a requested item is stored on a shelf currently being carried by the
robot. In this case, the robot can go directly to the back of the waiting line
at the operator, without passing by the storage area (right side of Figure 1).
Figure 1: Dual-command cycle (left) and opportunistic task (right)
In this work, we focus on optimising storage decisions only. It is assumed
that an external system provides the sequence of orders and the decisions on
which shelf and picking station to use. To be more precise, this system simply
considers orders already revealed and yet to be fulfilled and dynamically
sequences them by increasing deadlines, as an emergency rule. If an order
requires a shelf carried by another robot, this order is temporarily skipped
to be processed by that other robot as an opportunistic task. Also,
opportunistic tasks are performed as often as possible. The objective is to
minimise the average travelling time, which is directly related to the maximal
throughput capacity of the warehouse [Park, 2012]. We make other
simplification assumptions similar to the ones enumerated in the simulation
study in Rimélé et al. [2020]. We consider one picking station only, an item
is associated with exactly one shelf in the considered storage area,
replenishment is explicitly ignored even if some orders could correspond to
replenishment tasks, travelling and processing times are deterministic and,
finally, congestion of robots in the aisles is ignored.
### 2.2 Storage policies
Commonly used and studied storage policies for AS/RS are _Random_ storage
policy or policies based on the container’s turnover rate. The former gives a
container an equal probability to occupy any available location. While this
method is space-efficient and easy to implement in practice, it can obviously
result in poor accessibility and low productivity during operations. The
latter method is typically expressed as a full-turnover-based storage
assignment or a class-based storage assignment. A full-turnover-based
assignment locates the most-requested containers closest to the Input/Output
(I/O) point; however, a practical concern appears when new containers enter
and leave the system, or when the turnover rates change. In _Class-based_
storage, storage locations are clustered into a given number of classes based
on their proximity to the I/O point (see left side of Figure 2). Then, a
container is assigned to a zone based on its turnover rate; its storage within
this zone is random. While this method benefits from both _Random_ storage and
full-turnover-based assignments, it requires the definition of the number of
classes and their dimensions. Numerous analytical and simulation studies such
as Hausman et al. [1976], Graves et al. [1977], Schwarz et al. [1978], Bozer &
White [1984], and Gagliardi et al. [2012] show that _Class-based_ storage
assignment demonstrates the best performance and practicality. However, while
those applications generally apply to both single and dual-command cycles
system, it seems they have been more driven by the former. In another
simulation study, van den Berg & Gademann [2000] find out that for dual-
command cycles, selecting the storage location that minimises the cumulative
distance from the I/O point to the storage location, plus the distance from
the storage location to the following retrieval location, gives the best
average completion time. This decision rule is often referred to as the
_Shortest Leg_ storage policy.
### 2.3 RMFS-related work
Some recent work focuses more specifically on RMFS, either on storage
decisions or closely related topics like performance measures and others.
Lamballais et al. [2017] use queueing network models to analytically estimate
maximum order throughput, average cycle times, and robot utilisation. Using
these models allows one to quickly evaluate different layouts or robot zoning
strategies. Among their results, they show the good accuracy of their models
compared to simulations. They also show that the throughput is quite
insensitive to the length-to-width ratio of the storage area, but is more
strongly affected by the location of the picking stations. Also using queueing
network models, Roy et al. [2019] study the effect of dedicating robots
exclusively to either retrieval tasks or replenishment tasks, or a combination
of both. They find that the latter reduces the picking time by up to one
third, but increases the replenishment time up to three times. They also study
the allocation to multiple classes and conclude that allocating robots to
least congested classes results in a similar performance as a dedicated zone
policy.
Boysen et al. [2017] study the problem of synchronising the processing of
orders at a picking station with the visits of the shelves carried by robots.
By considering a given set of orders, and a capacity to process orders
simultaneously, their objective is to minimise the number of visits of shelves
to the picking station. They propose a MIP model, a decomposition approach,
and different heuristics and show that they can halve the robot fleet size.
Regarding storage allocation, Weidinger et al. [2018] define a _Rack
Assignment Problem_ , which assigns each stopover (return of a rack) of a
given set (batching) to an open storage location. They consider the time at
which each rack visits the picking station to be known. They propose a MIP
model as a special case of an interval scheduling problem, with the surrogate
objective of minimising the total loaded distance. The model formalises the
fact that two stopovers can occupy the same storage location only if their
storage intervals cannot overlap. They do not consider robots individually,
but instead they expect that a robot will always be available for the task.
They propose solving their model using an Adaptive Large Matheuristic Search
(ALMS), and compare their results with other storage policies, such as those
presented in Section 2.2. On their surrogate objective, ALMS demonstrates good
performance, outperforming the other policies on the total travel time
(unloaded robots are considered to travel twice as fast as loaded ones).
Similar to the finding of van den Berg & Gademann [2000], the authors note the
very good performance of the _Shortest Leg_ policy; compared to their method,
it only increases the travel distance by 3.49% and the size of the robot fleet
by 2.17%, without batching.
Yuan et al. [2018] propose a fluid model to assess the performance of
velocity-based storage policies in terms of travelling time to store and
retrieve shelves. They find that when the items are stowed randomly within a
shelf, a 2-class or 3-class storage policy reduces the travelling time between
6 to 12%. This theoretical result is validated by simulation. Additionally,
they find that if items are stowed together based on their velocity, the
travelling distances considered can be reduced by up to 40%.
In situations when batching or zoning requires consolidation, different
options can be considered. Boysen et al. [2019b] consider a manual
consolidation with put walls (sorting shelves). They propose a MIP to optimise
the release sequence of bins from intermediate storage so that orders are
quickly sorted on the put wall and idle times of operators are minimised. In a
case where the intermediate storage uses an AS/RS that serves the
consolidation area with a conveyor belt, Boysen et al. [2018] propose a MIP
model for the minimum order spread sequencing problem, which consists in
minimising the number of conveyor segments between the first and last
occurrence of an order.
Merschformann et al. [2019] use a discrete event simulator presented in
Merschformann et al. [2018], which considers most, if not all, of the
operational decisions occurring in an RMFS. They test a multitude of
combinations of decision rules concerning, for instance, the assignment to a
picking or replenishment station, the robot selection, storage allocation,
etc. They observe significant performance differences between distinct
combinations, as well as strong cross-dependencies between some decision
rules, suggesting there may be some benefit to exploring integrated
approaches.
## 3 Methodology
This section presents our method for solving the dynamic storage allocation
problem in an RMFS. We first define the notion of zones that we use in our
storage policy. Then, we explain the representation of the system as a POMDP
and the features extraction. In Section 3.3, we describe the variant of
Q-learning, a Reinforcement Learning agent, that we use. Section 3.4 presents
implementation details essential to the success of the method. Finally,
Section 3.5 describes a look-ahead rollout strategy to enhance our storage
policy, by making better-informed decisions using already-revealed orders.
### 3.1 Storage zones definition
In this work, inspired by _Class-based_ storage policies from the literature,
we propose optimising a policy that will allocate shelves to classes instead
of exact locations. Contrary to the standard _Class-based_ storage assignment
presented in Section 2.2, the online assignment to a class is not only made
based on the turnover rate of the shelf to store, but also on a set of diverse
features. Since we are interested in minimising dual-command cycle completion
times, as opposed to the access to storage or retrieval time alone (like in
single command cycles), the interleaving time between a storage location and
the next retrieval location is essential. With the usual definition of classes
distributed with an increasing distance from the picking station, such as in
Figure 2 (left), the access and retrieval times depend directly on the
assigned class, but the interleaving time is quite random. For instance, in
the ideal case where we store and retrieve shelves from the same storage class
2, we see that the cycle can either be very good (if the two locations are
adjacent), or very bad if the locations are at the two extremities. To better
account for the importance of interleaving time, we propose another definition
of classes, which we now call _zones_ as represented in Figure 2 (right).
These zones are now clustered into regular shapes such that locations within
the same zone are close to each other and such that the zones are distributed
in an increasing distance from the picking station. With this new zone
definition, a storage and a retrieval from the same zone would entail a short
interleaving time and, thus, a good cycle. Note that the selection of the
exact storage location within a zone is arbitrarily made by choosing the open
location closest to the picking station.
Figure 2: Concentric class-based storage layout (left) vs. proposed zones
layout (right)
### 3.2 System representation
Rimélé et al. [2020] model the storage and retrieval operations in an AS/RS as
a Markov Decision Process (MDP), which gives a full representation of the
warehouse, including all of the revealed orders, as a state of the system when
an event occurs (corresponding to the need for a storage decision here).
Instead of giving such a complete representation of the current state of the
warehouse, we propose a _Partially Observable_ MDP (POMDP), where only some
characteristics of the real state are used as state representation. In the
traditional _Class-based_ storage policy, for instance, the representation of
the state is simply the turnover rate of the shelf to store, and based on this
information the robot knows which class to assign it to.
We decided to represent a state with some more characteristics (features),
given below:
* 1.
Average turnover rate of the shelf to store
* 2.
Relative rank of the shelf’s turnover rate
* 3.
Zone of the next retrieval task (one-hot encoding, one feature per zone)
* 4.
Occupation levels of each zone (one feature per zone)
* 5.
For each zone, the number of robots that are currently moving toward the zone
to retrieve a shelf
An extra feature is used to encode whether the next task is an opportunistic
one (the shelf is not stored but is already being carried by the robot). In
total, if $n$ denotes the number of zones, there are $3n+3$ scalars in the
features vector.
The intuitive idea is that there must be a balance between greedily minimising
the immediate cycle time, similar to the _Shortest Leg_ storage policy, and
minimising the future access time, similar to a turnover-based storage policy
as presented in Section 2.2. The objective is to learn a policy that, based on
the above set of features, will aim at minimising the average cycle time.
### 3.3 Reinforcement learning method
We propose using a Reinforcement Learning (RL) method to learn an efficient
storage policy. The field of RL offers a diversity of methods to solve
decision problems whose objectives are defined recursively by a Bellman
equation, introduced in dynamic programming (DP). Compared to traditional DP
methods that require complete information about the model and probability
distributions, RL methods can learn from repeated experiences, even in large
state and action space, when using function approximations. As presented by
Sutton & Barto [2018], in RL the decision-maker, or agent, interacts with its
environment. At every time step $t$, and based on the current representation
of the state $S_{t}$, the agent selects an action $A_{t}$, which impacts the
environment and receives from it a reward $R_{t+1}$ and a new state
representation $S_{t+1}$. Based on repeated experiences, the goal of the agent
is to learn a policy that maximises the onward cumulative sum of rewards. Note
that the agent is not necessarily given the complete state characteristics; it
can only be given a partial representation through a POMDP. The state
representation defined in Section 3.2 is the partially observable state given
to the agent in our approach. In value-based RL, given a policy $\pi$, the
agent learns the State-Action value function
$Q_{\pi}(s,a)=\mathbb{E}\left[R_{1}+R_{2}+...\>|\>S_{0}=s,A_{0}=a,\pi\right]$
that represents the expected (possibly discounted) cumulative reward at state
$s$ if action $a$ is taken, if policy $\pi$ is followed afterwards. The
optimal value function $Q^{*}$ is such that
$Q^{*}(s,a)=\textup{max}_{\pi}Q_{\pi}(s,a)$ for all state and action pairs,
and obtaining a decision policy from a value function simply consists in
acting greedily with regard to action values :
$A_{\pi}(s)=\textup{argmax}_{a}Q_{\pi}(s,a)$.
In Q-learning [Watkins, 1989], the value function is iteratively updated by
sampling based on the Bellman equation: $Q(S_{t},A_{t})\leftarrow
Q(S_{t},A_{t})+\alpha[R+\gamma\>\textup{max}_{a}Q(S_{t+1},a)-Q(S_{t},A_{t})]$,
so the update uses the received reward and the current estimate of the next
state value (bootstrap) as its target. Importantly, for convergence to
optimality, the Q-learning agent needs to correctly learn the values of its
actions (exploitation) while trying new actions, possibly more promising
(exploration). This trade-off is central to any RL method, and here we will
consider an $\epsilon$-greedy selection rule for this purpose ($\epsilon$ % of
the time, the greedy action is selected, otherwise a random action is taken).
Even if our state space has a small number of dimensions and the number of
actions is also small, calculating the value function individually for every
state-action pair with a tabular method would be out of reach considering the
exponential number of combinations. Instead, Deep Q-learning uses a neural
network (NN) as a function approximator that takes the state features as
inputs and outputs the estimated values for each possible action [Mnih et al.,
2015]. Using function approximators allows to escape the curse of
dimensionality by leveraging information about similar states and by
generalising to unseen state-action pairs using only a limited number of
parameters. Let us denote by $\mathbf{w_{t}}$ the current parameters, weights,
of the neural network function approximator and by $Q(s,a,\mathbf{w_{t}})$ the
value of state $s$ \- action $a$ estimated by this neural network. Because a
neural network is differentiable, we can minimise the square loss error
between the current estimate value and the target by taking a step in the
opposite direction of the value function’s gradient:
$\mathbf{w_{t+1}}\leftarrow\mathbf{w_{t}}+\alpha[R_{t+1}+\gamma\>\textup{max}_{a}Q(S_{t+1},a,\mathbf{w_{t}})-Q(S_{t},A_{t},\mathbf{w_{t}})]\nabla
Q(S_{t},A_{t},\mathbf{w_{t}})$.
In the case of an infinite time horizon system, such as ours where retrieval
tasks never end, rewards cannot simply be accumulated because the state values
would never converge but rather would diverge to infinity. The most common
solution is to apply a discount factor $\gamma$ in the Bellman equations, such
that for a given policy $\pi$ we have:
$Q_{\pi}(s)=\mathbb{E}_{\pi}[R_{t+1}+\gamma\>Q_{\pi}(S_{t+1})\>|\>S_{t}=s]$,
with $0<\gamma<1$. The justification for using a discount factor is first for
the mathematical convenience of convergence of the cumulative finite rewards.
Also, in some systems, a discount factor can represent an economical discount
rate, or at least the idea that rewards received now are worth more than they
would be later. Yet in our case, no incentive should be given for early
performance since operations throughout the day have equal importance and no
termination occurs. An alternative to discounted rewards is what is called
_differential returns_ or _average rewards_ [Sutton & Barto, 2018]. Introduced
in Dynamic Programming [Bertsekas, 1976] and later in Reinforcement Learning
(R-Learning in Watkins [1989]), the idea is to optimise the average sum of
reward $\lim_{n\to\infty}\frac{\sum_{t=1}^{n}R_{t}}{n}$ by considering return
values equal to $R_{t}-\bar{R}$, with $\bar{R}$ the average reward of policy
$\pi$. This setting has the interesting property of having the sum of returns
converging (to 0) while maximising the average reward in the process. When
action $A_{t}$ of policy $\pi$ is taken, the state-action value update rule
becomes $Q(S_{t},A_{t})\leftarrow
Q(S_{t},A_{t})+\alpha[R-\bar{R}+\textup{max}_{a}Q(S_{t+1},a)-Q(S_{t},A_{t})]$.
Q-learning is known to suffer from an overestimation of actions’ values. Two
main reasons explain this phenomenon. First, in its update mechanism, the
maximum value of the next state-action pair is used as an approximation for
the maximum expected value. Second, both the action selection and value
estimation come from the same maximum operator that suffers from estimation
errors, and thus favours overestimated values. To answer this problem, van
Hasselt [2010] proposes a Double Q-learning method in the tabular case, which
was later extended to Deep Q-learning in van Hasselt et al. [2016]. For Deep
Q-learning, the general idea is to separate the action selection from the
value estimation using two neural networks, the online NN and the target NN.
The authors demonstrate that this approach eliminates the positive bias on the
action values and results, in general, in better performance. The online NN is
updated as before, but the target NN is not. Instead, it receives a copy of
the weights of the online NN every $K$ iterations.
Algorithm 1 combines the different elements presented above into one agent,
denoted as _Double Differential Deep Q-Learning_. Step 1 initialises a
starting state, the average reward (arbitrarily set, for instance, to 0) and
the weights of both the online and target neural networks, equal to each
other. For every iteration, Step 3 selects an action, greedily or not, takes
it, and observes the reward and the next state. Step 4 calculates the target
value for state $S_{t}$ \- action $A_{t}$ pair. Step 5 does the gradient
descent to update the online network. If the action selected $A_{t}$ was
greedy with respect to the online network, Steps 6 to 8 update the average
reward $\bar{R}$. Finally, every $K$ iterations, the target network receives a
copy of the online network weights (Steps 9 to 11).
1: initialise state $S_{0},\bar{R},\mathbf{w_{t}}=\mathbf{w^{-}_{t}}$
2: for $t=0$ to $\infty$ do
3: take action $A_{t}$ ($\epsilon\text{-greedy}$ wrt
$Q(S_{t},.,\mathbf{w_{t}})$), observe $R_{t+1}$, $S_{t+1}$
4: $Y_{t}\leftarrow
R_{t+1}-\bar{R}+Q(S_{t+1},\operatorname*{argmax}_{a}Q(S_{t+1},a,\mathbf{w_{t}}),\mathbf{w^{-}_{t}})$
5:
$\mathbf{w_{t+1}}\leftarrow\mathbf{w_{t}}+\alpha[Y_{t}-Q(S_{t},A_{t},\mathbf{w_{t}})]\nabla
Q(S_{t},A_{t},\mathbf{w_{t}})$
6: if $A_{t}==\operatorname*{argmax}_{a}Q(S_{t},a,\mathbf{w_{t}})$ then
7: $\bar{R}\leftarrow\bar{R}+\beta[Y_{t}-Q(S_{t},A_{t},\mathbf{w_{t}})]$
8: end if
9: if $t\>\text{mod}\>K==0$ then
10: $\mathbf{w^{-}_{t}}\leftarrow\mathbf{w_{t}}$
11: end if
12: end for
Algorithm 1 Double Differential Deep Q-Learning
### 3.4 Implementation details
In our experiments, we use a fully connected feed-forward neural network as a
function approximator. While the neural network has one output per storage
zone, which represents the expected cumulative cycle times onwards (relative
to the average cycle time), if the storage zone is selected, some zones may
not be selected if they are currently full. For this reason, when we select an
action based on its value out of the neural network, we first need to filter
out full zones.
Another point worth mentioning deals with backpropagation when an
opportunistic task is performed. Such a task is enforced when possible, which
occurs when the next retrieval zone is not a zone but the picking station
(encoded in the features extraction). Even though the agent does not require
any action to be taken, it is essential to backpropagate the observed reward
in the neural network for all of the actions. The reason for that is
bootstrapping. If a non-opportunistic task transitions to a state that will
enforce an opportunistic task, the state-action value update in Q-learning
uses the observed cycle time as well as the estimated value of the future
state, which, in this case, requires an opportunistic task. The neural network
must then be accurate for such a state and learn that the values of all the
storage zones are to be equal to the value of an opportunistic task.
As in Mnih et al. [2015], we use experience replay. It consists in storing
simulation experiences (vector $S_{t},A_{t},R_{t+1},S_{t+1}$) into a fixed
size replay memory. Instead of training after each observed experience, we
wait for some iterations before sampling a batch of experiences from the
replay memory and then train the neural networks with mini-batches. This batch
training affords more training stability, allows the reuse of past
experiences, and speeds up the training step.
Finally, to make better use of available information, particularly requests
that are already revealed at a given time step, we propose an extension to
using only Q-values. A look-ahead rollout strategy is proposed. We present
this method in Section 3.5 and the results of our Q-learning application with
and without rollouts is presented in Section 4.
### 3.5 Look-ahead rollout strategy
Monte Carlo Tree Search (MCTS) methods are exploration techniques to improve
policies, based on previous work on Monte Carlo methods. Coulom [2007] first
describes a tree search method applied to games. Several variants have been
adapted for numerous applications, such as the famous AlphaGo Zero from Silver
et al. [2017]. The general idea behind MCTS is to explore the action space by
simulations following a tree structure, further exploring the promising areas
of the tree.
For the RMFS real-time storage assignment, the objective of a look-ahead
search is twofold. First, as in other applications, it helps the agent take
better actions by exploring numerous possible scenarios. Second, and more
specific to our application, the look-ahead can leverage information that is
already revealed but not necessarily given in the state representation. This
is the case of future orders. So far, in Section 3.2, only information about
the next order the robot needs to retrieve is included in the state features.
However, at a given time step, while not all future orders have been revealed,
some have. The look-ahead can then use _only_ those already-revealed orders
(even if the real future sequence will differ) to take a better informed first
action.
Figure 3 illustrates this point. The orders $o_{i}$ are sorted according to
the sequence in which they will actually be processed, but at time $t_{0}$
only the _green_ orders $\\{o_{1},\>o_{2},\>o_{4},\>o_{5}\\}$ are known. The
_orange_ orders $\\{o_{3},\>o_{6},\>o_{8}\\}$ will be revealed later at time
$t_{1}>t_{0}$, for instance by the time order $o_{2}$ will have been
processed. In this case, order $o_{3}$ is be inserted before $o_{4}$, for
priority reasons. In this case, even if new orders will later be inserted into
the list, at time $t_{0}$ only requests $\\{o_{1},\>o_{2},\>o_{4},\>o_{5}\\}$
are used in the rollout, the only exceptions being potential other _green_
requests not depicted on Figure 3.
Figure 3: Revealed orders at decision time
Bertsekas [2019] presents different approaches for MCTS implementation. One of
these approaches is a single-step look-ahead with truncated rollout and cost
approximation. Figure 4 presents the implementation of this type of rollout
adapted to our problem. At first, all feasible actions at a given state
$S_{0}$ (time $t_{0}$) are considered and their values $V(S_{0},A_{i})$ and
visit counts $n_{A_{i}}$ are set to $0$. At each iteration, an action is
randomly selected (action $A_{1}$ in Figure 4) and a simulation of $h$
(horizon) consecutive storage tasks is run. This finite simulation uses orders
revealed by time $t_{0}$. The action selection in the simulation uses the
agent’s current policy. The sum of reward along the simulation is computed,
and the value of the last state $S_{h}$ is, by bootstrapping, set as the
Q-value of the agent. Then, the value of the selected action is updated to its
experienced average, and the visit counter is incremented. The process is
repeated for a given number of iterations, and finally, the action presenting
the best (min) value is selected. Note that in our specific usage of this
look-ahead rollout strategy, each feasible action needs to be selected only
once. Indeed, both the policy and the transitions within the rollout are
deterministic because of the Q-learning agent’s policy and because we do not
simulate unknown future orders.
Figure 4: Look-ahead rollout strategy
## 4 Simulation study
In this section, we present a simulation study comparing our proposed methods
to typical storage policies found in the literature. We start by defining the
framework of the study, before presenting our results.
### 4.1 Parameters
Kiva and Amazon have claimed travel speeds up to 3 to 4 miles per hour;
however, to account for turns and congestion, we assume that the robots travel
at a constant speed of 1.4 miles per hour (about 0.6 m/s). In our experiments,
we assume an average picking time by the human operators of 8 sec and a
loading and unloading time by the robots of 3 sec. Our study considers a
relatively small storage area, with 36 storage locations (6 zones of size 2 by
3 shelves) for 34 shelves, 5 automated robots and one picking station.
Incoming orders in warehousing systems are typically skewed, with a small
percentage of the items representing the vast majority of the orders and the
others qualified as slow movers. Hausman et al. [1976] propose modelling such
a demand distribution with the definition of an _ABC_ curve:
$G(x)=x^{s}\quad\text{for }0<s\leq 1$, which represents the proportion of
cumulative demand (%) of the first $x\%$ of the shelves. The smaller the
skewness parameter $s$, the more skewed the demand distribution. We follow
this suggestion by associating with every item type $i$ (equivalent to the
corresponding shelf), at every discretised period, a Poisson distribution of
average
$\lambda_{i}=\left(\left(\frac{i}{m}\right)^{s}-\left(\frac{i-1}{m}\right)^{s}\right)\times\frac{n}{N}$,
where $m$ is the number of item types, $n$ the total number of orders in the
time horizon, and $N$ the number of discretised periods. Then, for every
generated order $k$, a completion time $\delta_{k}$ is randomly drawn from a
uniform distribution of interval $[1;\alpha\times T]$ where $\alpha$ defines
the tightness of the completion times. Note that in the case where a drawn
demand for a given item type is greater than one, the resulting orders are
automatically grouped, corresponding to an opportunistic task of type 2 in
Rimélé et al. [2020]. The values used here are: $m=34$, $n=30000$, $N=1440$
(corresponding to discretised periods every 1 min) for a time horizon of 24h,
and $\alpha=0.4$. The values of the skewness parameters will be varied to
study their impact on the different storage policies.
Figure 5: Simulation study - Plan view of the storage area
After some trial and error, we set the parameter values for the RL agent as
follows. Each training phase is conducted over 10000 episodes, corresponding
to 24h of simulated incoming orders. Each of these training episodes uses its
own generated instance of demand, while the comparisons between the baselines
and the test of the RL policy are made on another shared instance. The
exploration rate $\epsilon$ is kept constant at $0.1$. The neural networks are
created using the Keras library. We use the mean square error loss function
and the Adam optimiser for gradient descent. Both neural networks are feed-
forward, fully connected with 3 inner layers of 32 neurons each and Relu
activation functions. The output layer presents one neuron per zone of storage
(6), and these neurons are linearly activated. The learning rate is set to
$0.00025$, the replay memory has a capacity of $1000$, and batch training is
performed every $100$ iterations. This batch consists of $256$ experiences
sampled from the replay memory, and they are fitted to the model in mini-
batches of $64$. The target network’s weights are updated every $500$
iterations.
### 4.2 Results
This section presents results obtained by the proposed approach and the
standard baselines. First, experiments were run using the method presented in
Section 3.3 on different instances corresponding to different values of demand
skewness parameter $s$. Then, the rollout strategy presented in Section 3.5
was applied on both the Q-learning agent and the best-performing baseline, and
gains that can be obtained by using additional available information were
observed. The three baselines tested were the following: _Random_ storage
policy, _Class-based_ and _Shortest Leg_.
The _Class-based_ storage policy uses the concentric classes definition,
depicted in Figure 2 (left) and conventionally described in the literature.
Since our method allocates shelves to zones instead of exact locations, the
_Shortest Leg_ storage policy is implemented similarly. When a storage task
needs to be performed and the location of the next retrieval is known, the
exact potential location within each zone can be identified because of the
arbitrary choice of selecting the location closest to the picking station.
Knowing the candidate location from each zone (if any), the _leg_ distance can
be computed as being the distance from the picking station to the storage
location plus the distance from the storage location to the next retrieval.
The selected zone corresponds to the smallest leg value. This policy results
in greedily minimising the immediate cycle times.
Table 1 presents the average travelling times, $t(s)$, obtained from the three
baselines and the RL agent, without rollouts, for several skewness parameter
values ranging from $s=0.4$ (very skewed) to $s=1$ (flat distribution). For
each storage policy, the corresponding performance gain, g(%), compared to the
_Random_ storage policy is also computed; this represents the percentage
decrease in travelling time. As expected, the performance of the _Class-based_
policy increases when the skewness parameter value $s$ decreases. In
concordance with the literature, we observe the good performance of the
_Shortest Leg_ policy, which remains rather constant, between 10 and 11% of
improvement, depending on the skewness parameter values. In fact, it is only
surpassed once by the _Class-based_ policy for the most skewed distribution
$s=0.4$, but it offers the net advantage of being independent of the
distribution pattern. The Q-learning agent consistently performs better than
the baselines, between 3.21% and 4.83% of additional gain compared to the
baselines relative to _Random_ storage. Similar to the SL policy, its
performance remains rather independent of the skewness of the distribution.
Figure 6 presents a typical training curve, here for parameter $s=0.6$. The
_x_ axis shows the number of training episodes ranging from 0 to 10000 and _y_
corresponds to the relative gain in travelling time compared to the _Random_
storage policy. The horizontal dashed lines in the graph represent the
performance of the three baseline policies. The test policy curve represents
the performance of the RL policy in training, which is tested every 100
training episodes. With our training settings, we notice the non-linear
evolution of the test policy curve until it reaches some plateau value around
5000 episodes.
Storage policy | Random | Class-based | SL | Agent
---|---|---|---|---
t(s) | g(%) | t(s) | g(%) | t(s) | g(%) | t(s) | g(%)
s=0.4 | 37.97 | - | 33.41 | 12.01 | 33.61 | 11.48 | 32.19 | 15.22
s=0.5 | 38.62 | - | 34.65 | 10.27 | 34.18 | 11.48 | 32.90 | 14.79
s=0.6 | 38.69 | - | 35.34 | 8.65 | 34.79 | 10.09 | 32.92 | 14.92
s=0.7 | 38.73 | - | 36.01 | 7.02 | 34.70 | 10.39 | 32.91 | 15.02
s=0.8 | 38.94 | - | 36.48 | 6.30 | 34.76 | 10.73 | 33.04 | 15.15
s=0.9 | 38.98 | - | 37.41 | 4.03 | 34.53 | 11.42 | 33.03 | 15.27
s=1.0 | 39.06 | - | 38.26 | 2.05 | 34.76 | 11.01 | 33.4 | 14.48
Table 1: Average travelling times t(s) and performance gains g(%) depending on
distribution skewness parameter value s - without rollouts Figure 6: Typical
instance of a training curve (here for s=0.6)
Over a second phase, we retained the overall best performing-baseline (SL) and
our Q-learning agent to enhance them using the look-ahead rollout strategy
presented in Section 3.5. Again, we tested the resulting policies on different
skewness parameter $s$ values, but we also considered different horizon values
$h$, which define how far ahead the rollout is run. The only difference in the
application of the rollout strategy to the SL baseline compared to the
Q-learning agent is the absence of a Q-value at the last step of the rollout.
In this case, the SL policy is run over the next $h$ already-revealed orders,
and the average of the resulting cycle times is computed, for each first
feasible action. Table 2 presents the obtained results in terms of performance
gains (%) compared to _Random_ storage. First, we notice that for all
parameter values, the performance is significantly improved. We can see that
for both policies, the performance varies depending on the horizon of the
rollout that is considered. As a general statement, it appears that horizon
values 20 and 30 give the best results, with some exceptions. The fact that
the performance increases with the horizon value before decreasing again is
not too surprising. While using revealed orders in the rollouts is beneficial
to making better-informed decisions, the longer the rollout, the more likely
it is that new, unknown orders will be later inserted between the orders
considered in the rollout. When this phenomenon starts to occur too often, it
deteriorates the insight of the rollout and results in worse policies. The
values in bold in the table designate the best results for each parameter $s$
for both policies. For the SL storage policy, the application of the rollout
strategy increases its performance gains between 5.26% ($s=1.0$) and 7.11%
($s=0.7$). For the Q-learning agent, the performance gains range between 2.78%
($s=0.8$) and 4.54% ($s=0.4$). It appears that using rollouts benefits
strongly skewed distributions slightly more than less skewed ones, which is
most likely explained by more opportunities associated with fast and slow-
moving shelves. Also, the rollout strategy benefits the SL policy more so than
the agent. The differences in performance gains between the two policies with
rollouts are now 2.29, 0.83, 1.47, 1.07, 1.55, 2.2 and 1.29% for increasing
$s$ values. Using rollouts gives a policy insights about future behaviours by
simulating possible trajectories. While the Q-learning agent still benefits
from this insight, with the Bellman equation it is, by design, already taking
into account future impacts of a decision. On the other hand, the SL policy
acts perfectly greedily; it has more to gain by looking into possible future
outcomes. While the combination of the Q-learning agent with rollouts still
performs best with an average of 1.53% of performance gain compared to SL with
rollouts, this assessment has interesting practical value because of the
relative simplicity of implementation of SL with rollouts and its highly
competitive performance. Finally, when comparing the results of the complete
method with the best baseline commonly used from the literature (SL without
rollouts), an average performance gain of 7.58% is found.
| SL + rollouts | Agent + rollouts
---|---|---
h value | 5 | 10 | 20 | 30 | 40 | 5 | 10 | 20 | 30 | 40
s=0.4 | 14.01 | 16.83 | 17.47 | 17.07 | 17.22 | 16.26 | 17.01 | 19.76 | 18.84 | 18.80
s=0.5 | 14.91 | 16.03 | 17.57 | 17.65 | 16.00 | 15.17 | 17.14 | 18.48 | 18.41 | 17.89
s=0.6 | 14.97 | 15.47 | 16.06 | 16.92 | 16.05 | 15.71 | 17.45 | 18.39 | 17.99 | 18.16
s=0.7 | 14.41 | 15.45 | 17.50 | 16.22 | 15.85 | 14.96 | 16.82 | 18.13 | 18.57 | 18.11
s=0.8 | 14.23 | 15.51 | 16.38 | 15.42 | 15.46 | 15.13 | 16.32 | 17.23 | 17.93 | 16.96
s=0.9 | 14.22 | 15.49 | 16.34 | 16.79 | 16.52 | 15.11 | 17.04 | 17.83 | 18.95 | 18.99
s=1.0 | 15.15 | 15.60 | 16.27 | 15.64 | 15.61 | 15.27 | 16.81 | 17.34 | 17.56 | 17.17
Table 2: Performance gains g(%) depending on distribution skewness parameter
value s and horizon h - with look-ahead rollouts
## 5 Conclusions
This work tackles the problem of dynamically allocating shelves to storage
locations within a Robotic Mobile Fulfillment System. Because of the very
nature of the e-commerce market, new orders need to be considered, if not
fulfilled, as soon as they are revealed, which limits opportunities for
batching. Typical methods from the literature correspond to decision rules
which rely either on the distinct turnover rates of the containers or on
minimising immediate cycles greedily. After making several assumptions about
other operational decision rules in such warehouses, as well as the physical
characteristics of the warehouse, we propose defining a storage policy using a
POMDP and a Q-learning agent from Reinforcement Learning, to minimise the
average travelling cycle time. This agent learns from repeated experiences
which storage decision should be made based on a set of features representing
the current state of the warehouse. Using zone-based storage allocation, we
compared our method to decision rule baselines, including the _Class-based_
storage policy and _Shortest Leg_ storage policy. A performance gain between
3.5% and 5% higher than the best baseline relative to _Random_ storage was
observed. In a second phase, the objective was to leverage additional
information regarding orders that are already revealed at a given time step.
While new orders will later appear and be inserted between those revealed
orders, the latter can still be used in a look-ahead rollout strategy applied
at the decision-making step. This rollout simulates finite horizon
trajectories using the storage policy at hand for action selection within the
trajectory, as well as a Q-value estimation at the last step, acting as an
estimation of future expected objective value. Such a rollout strategy has the
benefit of being applicable to any storage policy, which allowed us to make
fair comparisons between the Q-learning agent and the _Shortest Leg_ policy,
both using rollouts. After selecting the proper horizon length, using look-
ahead rollouts increased the performance gains of the SL and Q-learning
policies by about 6 and 4%, respectively. Even though the Q-learning with
rollouts policy gave the best results, the higher positive impact of rollouts
on the SL policy makes it an interesting contender, due to its relative
simplicity of application. Comparing the policy obtained from Q-learning with
rollouts to the best baseline, we observed an average performance gain of
7.5%, which, we think, demonstrates the potential for more research in this
direction.
Future research may build on this work to extend it further. First, storage
allocation could select exact open locations instead of using the notion of
zones as described here. Instead of using single-step look-ahead rollouts, one
could implement a complete Monte Carlo Tree Search algorithm to better explore
action selection. The expensive computational cost of such an approach may
limit online deployment but, coupled with a policy gradient agent instead of a
Q-learning one, the tree search could be used extensively in the training
phase only for fast future deployment. Another aspect worth looking into would
be the waiting time of robots at the picking stations. In this work, we only
consider the travelling time, which is a lower bound to the full cycle time.
Aiming at minimising the travelling time along with implementing a distributed
arrival of robots at the picking stations could result in higher throughput,
but how to do that exactly appears to be quite challenging.
## Acknowledgements
We are grateful to the Mitacs Accelerate Program for providing funding for
this project. We also wish to gratefully acknowledge the support and valuable
insights of our industrial partners, JDA Software and Element AI.
## References
* Azadeh et al. [2017] Azadeh, K., de Koster, R., & Roy, D. (2017). Robotized Warehouse Systems: Developments and Research Opportunities. SSRN Electronic Journal, (pp. 1–55). doi:10.2139/ssrn.2977779.
* Banker [2018] Banker, S. (2018). New Solution Changes the Rules of Warehouse Automation. Forbes Business, . URL: https://www.forbes.com/sites/stevebanker/2018/06/12/new-solution-changes-the-rules-of-warehouse-automation.
* van den Berg & Gademann [2000] van den Berg, J. P., & Gademann, A. (2000). Simulation study of an automated storage/retrieval system. International Journal of Production Research, 38, 1339–1356. doi:10.1080/002075400188889.
* Bertsekas [1976] Bertsekas, D. (1976). Dynamic Programming and Stochastic Control. Mathematics in Science and Engineering, 125, 222–293.
* Bertsekas [2019] Bertsekas, D. (2019). Reinforcement learning and optimal control. Athena scientific.
* Boysen et al. [2017] Boysen, N., Briskorn, D., & Emde, S. (2017). Parts-to-picker based order processing in a rack-moving mobile robots environment. European Journal of Operational Research, 262, 550–562. doi:10.1016/j.ejor.2017.03.053.
* Boysen et al. [2018] Boysen, N., Fedtke, S., & Weidinger, F. (2018). Optimizing automated sorting in warehouses: The minimum order spread sequencing problem. European Journal of Operational Research, 270, 386–400. doi:10.1016/j.ejor.2018.03.026.
* Boysen et al. [2019a] Boysen, N., de Koster, R., & Weidinger, F. (2019a). Warehousing in the e-commerce era: A survey. European Journal of Operational Research, 277, 396–411. doi:10.1016/j.ejor.2018.08.023.
* Boysen et al. [2019b] Boysen, N., Stephan, K., & Weidinger, F. (2019b). Manual order consolidation with put walls: the batched order bin sequencing problem. EURO Journal on Transportation and Logistics, 8, 169–193. doi:10.1007/s13676-018-0116-0.
* Bozer & White [1984] Bozer, Y. A., & White, J. A. (1984). Travel-Time Models for Automated Storage/Retrieval Systems. IIE Transactions, 16, 329–338. doi:10.1080/07408178408975252.
* Coulom [2007] Coulom, R. (2007). Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search. In H. J. van den Herik, P. Ciancarini, & H. H. L. M. J. Donkers (Eds.), Computers and Games (pp. 72–83). Berlin, Heidelberg: Springer Berlin Heidelberg.
* D’Andrea & Wurman [2008] D’Andrea, R., & Wurman, P. (2008). Future challenges of coordinating hundreds of autonomous vehicles in distribution facilities. 2008 IEEE International Conference on Technologies for Practical Robot Applications, TePRA, (pp. 80–83). doi:10.1109/TEPRA.2008.4686677.
* Davarzani & Norrman [2015] Davarzani, H., & Norrman, A. (2015). Toward a relevant agenda for warehousing research: literature review and practitioners’ input. Logistics Research, 8. doi:10.1007/s12159-014-0120-1.
* Enright & Wurman [2011] Enright, J. J., & Wurman, P. R. (2011). Optimization and coordinated autonomy in mobile fulfillment systems. AAAI Workshop - Technical Report, WS-11-09, 33–38.
* Gagliardi et al. [2012] Gagliardi, J.-P., Renaud, J., & Ruiz, A. (2012). On storage assignment policies for unit-load automated storage and retrieval systems. International Journal of Production Research, 50, 879–892. doi:10.1080/00207543.2010.543939.
* Graves et al. [1977] Graves, S. C., Hausman, W. H., & Schwarz, L. B. (1977). Storage-Retrieval Interleaving in Automatic Warehousing Systems. Management Science, 23, 935–945. doi:10.1287/mnsc.23.9.935.
* Gu et al. [2007] Gu, J., Goetschalckx, M., & McGinnis, L. F. (2007). Research on warehouse operation: A comprehensive review. European Journal of Operational Research, 177, 1–21. doi:10.1016/j.ejor.2006.02.025.
* van Hasselt [2010] van Hasselt, H. (2010). Double Q-learning. Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010, (pp. 1–9).
* van Hasselt et al. [2016] van Hasselt, H., Guez, A., & Silver, D. (2016). Deep Reinforcement Learning with Double Q-Learning. Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI-16), (pp. 2094–2100).
* Hausman et al. [1976] Hausman, W. H., Schwarz, L. B., & Graves, S. C. (1976). Optimal Storage Assignment in Automatic Warehousing Systems. Management Science, 22, 629–638. doi:10.1287/mnsc.22.6.629.
* Hessel et al. [2018] Hessel, M., Modayil, J., Van Hasselt, H., Schaul, T., Ostrovski, G., Dabney, W., Horgan, D., Piot, B., Azar, M., & Silver, D. (2018). Rainbow: Combining improvements in deep reinforcement learning. In 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 (pp. 3215–3222).
* Kirks et al. [2012] Kirks, T., Stenzel, J., Kamagaew, A., & en Hompel, M. (2012). Cellular Transport Vehicles for Flexible and Changeable Facility Logistics Systems. Logistics Journal, 2192(9084).
* Lamballais et al. [2017] Lamballais, T., Roy, D., & de Koster, R. (2017). Estimating performance in a Robotic Mobile Fulfillment System. European Journal of Operational Research, 256, 976–990. doi:10.1016/j.ejor.2016.06.063.
* Merschformann et al. [2019] Merschformann, M., Lamballais, T., de Koster, R., & Suhl, L. (2019). Decision rules for robotic mobile fulfillment systems. Operations Research Perspectives, 6, 100128. doi:10.1016/j.orp.2019.100128.
* Merschformann et al. [2018] Merschformann, M., Xie, L., & Li, H. (2018). RAWSim-O: A simulation framework for robotic mobile fulfillment systems. Logistics Research, 11, 1–11. doi:10.23773/2018{\\_}8.
* Mnih et al. [2015] Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, A., Riedmiller, M., Fidjeland, A. K., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., & Hassabis, D. (2015). Human-level control through deep reinforcement learning. Nature, 518, 529–533. doi:10.1038/nature14236.
* Park [2012] Park, B. C. (2012). Order Picking: Issues, Systems and Models. In Warehousing in the Global Supply Chain (pp. 1–30). London: Springer London. doi:10.1007/978-1-4471-2274-6{\\_}1.
* Ramanathan [2010] Ramanathan, R. (2010). The moderating roles of risk and efficiency on the relationship between logistics performance and customer loyalty in e-commerce. Transportation Research Part E: Logistics and Transportation Review, 46, 950–962. doi:10.1016/j.tre.2010.02.002.
* Rimélé et al. [2020] Rimélé, A., Gamache, M., Gendreau, M., Grangier, P., & Rousseau, L.-M. (2020). Robotic Mobile Fulfillment Systems: a Mathematical Modelling Framework for E-commerce Applications. Cahiers du Cirrelt, CIRRELT-2020-42.
* Roodbergen & Vis [2009] Roodbergen, K. J., & Vis, I. F. A. (2009). A survey of literature on automated storage and retrieval systems. European Journal of Operational Research, 194, 343–362. doi:10.1016/j.ejor.2008.01.038.
* Roy et al. [2019] Roy, D., Nigam, S., de Koster, R., Adan, I., & Resing, J. (2019). Robot-storage zone assignment strategies in mobile fulfillment systems. Transportation Research Part E: Logistics and Transportation Review, 122, 119–142. doi:10.1016/j.tre.2018.11.005.
* Schwarz et al. [1978] Schwarz, L. B., Graves, S. C., & Hausman, W. H. (1978). Scheduling Policies for Automatic Warehousing Systems: Simulation Results. A I I E Transactions, 10, 260–270. doi:10.1080/05695557808975213.
* Silver et al. [2017] Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L., Lai, M., Bolton, A., Chen, Y., Lillicrap, T., Hui, F., Sifre, L., Van Den Driessche, G., Graepel, T., & Hassabis, D. (2017). Mastering the game of Go without human knowledge. Nature, 550, 354–359. doi:10.1038/nature24270.
* Sutton & Barto [2018] Sutton, R. S., & Barto, A. G. (2018). Reinforcement Learning: An Introduction. MIT press.
* Watkins [1989] Watkins, C. (1989). Learning from delayed rewards. Ph.D. thesis King’s College.
* Weidinger et al. [2018] Weidinger, F., Boysen, N., & Briskorn, D. (2018). Storage assignment with rack-moving mobile robots in KIVA warehouses. Transportation Science, 52, 1479–1495. doi:10.1287/trsc.2018.0826.
* Wurman et al. [2008] Wurman, P. R., D’Andrea, R., & Mountz, M. (2008). Coordinating Hundreds of Cooperative, Autonomous Vehicles in Warehouses. AI Magazine, 29, 9–9. doi:10.1609/AIMAG.V29I1.2082.
* Yaman et al. [2012] Yaman, H., Karasan, O. E., & Kara, B. Y. (2012). Release time scheduling and hub location for next-day delivery. Operations Research, 60, 906–917. doi:10.1287/opre.1120.1065.
* Yuan et al. [2018] Yuan, R., Graves, S. C., & Cezik, T. (2018). Velocity-Based Storage Assignment in Semi-Automated Storage Systems. Production and Operations Management, 28, 354–373. doi:10.1111/poms.12925.
|
# On the Frank-Wolfe algorithm for non-compact constrained optimization
problems ††thanks: The first author was supported in part by CNPq grants
305158/2014-7 and 302473/2017-3, FAPEG/PRONEM- 201710267000532 and CAPES. The
second author was supported in part by Fundação de Apoio à Pesquisa do
Distrito Federal (FAP-DF) by the grant 0193.001695/2017 and PDE 05/2018
O. P. Ferreira Instituto de Matemática e Estatística, Universidade Federal de
Goiás, CEP 74001-970 - Goiânia-GO, Brazil<EMAIL_ADDRESS>W. S. Sosa
Programa de Pós-Graduação em Economia, Universidade Católica de Brasília, CEP
70790-160 - Brasília-DF, Brazil, E-mail<EMAIL_ADDRESS>
###### Abstract
This paper is concerned with the Frank–Wolfe algorithm for a special class of
non-compact constrained optimization problems. The notion of asymptotic cone
is used to introduce this class of problems as well as to establish that the
algorithm is well defined. These problems, with closed and convex constraint
set, are characterized by two conditions on the gradient of the objective
function. The first establishes that the gradient of the objective function is
Lipschitz continuous, which is quite usual in the analysis of this algorithm.
The second, which is new in this subject, establishes that the gradient
belongs to the interior of the dual asymptotic cone of the constraint set.
Classical results on the asymptotic behavior and iteration-complexity bounds
for the sequence generated by the Frank–Wolfe algorithm are extended to this
new class of problems. Examples of problems with non-compact constraints and
objective functions satisfying the aforementioned conditions are also
provided.
Keywords: Frank-Wolfe method; constrained optimization problem; non-compact
constraint.
AMS subject classification: 90C25, 90C60, 90C30, 65K05.
## 1 Introduction
The Frank–Wolfe algorithm or conditional gradient method is one of the oldest
methods for finding minimizers of differentiable functions on compact convex
sets. It was initially proposed in 1956 [1] for solving quadratic programming
problems with linear constraints (see also [2, 3]), and it has since attracted
considerable attention owing to its simplicity and ease of implementation, as
it only requires access to a linear minimization oracle over the constraint
set. In particular, it allows low storage cost and readily exploits
separability and sparsity; therefore, it can be effectively applied to large-
scale problems. With the emergence of machine-learning applications, this
method has recently gained increasing popularity [4, 5, 6]. Accordingly, it
has been extensively studied, and several variants thereof have been developed
[7, 8, 9, 10, 11, 12, 13, 14] and references therein.
The aim of this study is to extend the Frank–Wolfe algorithm to a special
class of constrained optimization problems ${\rm Minimize}_{x\in{C}}f(x)$,
where $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a continuously differentiable
function, and $C\subset{\mathbb{R}}^{n}$ is a closed and convex but not
necessarily compact set. In addition to the classical assumptions (i.e., the
gradient of $f$ is Lipschitz continuous), we assume that $\nabla
f(x)\in\mbox{int}(C_{\infty})^{*}$ for all $x\in C$, where $\nabla f$ and
$\mbox{int}(C_{\infty})^{*}$ denote the gradient of $f$ and the interior of
the positive dual asymptotic cone of $C$, respectively. For this class of
functions, with classical assumptions, we also extend previous results on the
asymptotic behavior and iteration-complexity bounds for the sequence generated
by the Frank–Wolfe algorithm.
This paper is organized as follows. In Section 2, notations and auxiliary
results are presented. In Section 3, we formulate the Frank–Wolfe algorithm.
In Section 3.1, we establish that the sequence generated by this algorithm is
well defined. Section 3.2 is devoted to the study of the asymptotic-
convergence properties of Algorithm 1, and Section 3.3 to the study of the
iteration-complexity bounds. In Section 4, we present some examples. We
conclude the paper in Section 5.
## 2 Preliminaries
Herein, we present notations, definitions, and auxiliary results. Let
${\mathbb{R}}^{n}$ be the $n$-dimensional Euclidean space with the usual inner
product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$. We denote by
${\mathbb{R}}^{m\times n}$ the set of all $m\times n$ matrices with real
entries (${\mathbb{R}}^{n}\equiv{\mathbb{R}}^{n\times 1}$), by $e^{i}$ the
$i$-th canonical unit vector in ${\mathbb{R}}^{n}$, and by ${\rm I_{n}}$ the
$n\times n$ identity matrix. A set ${\cal K}\subseteq{\mathbb{R}}^{n}$ is
called a cone if for any $\alpha>0$ and $x\in\cal{K}$, we have $\alpha
x\in\cal{K}$. A cone ${\cal K}\subseteq{\mathbb{R}}^{n}$ is called convex if
for any $x,y\in\cal{K}$, we have $x+y\in\cal{K}$. The positive dual cone of a
cone ${\cal{K}}\subseteq{\mathbb{R}}^{n}$ is the cone
${\cal{K}}^{*}\\!\\!:=\\!\\{x\in{\mathbb{R}}^{n}:~{}x^{{\scriptscriptstyle\mathrm{T}}}y\geq\\!0,~{}\forall\,y\\!\in\\!{\cal{K}}\\}$,
and its interior is denoted by
$\mbox{int}{\cal{K}}^{*}\\!\\!:=\\!\\{x\in{\mathbb{R}}^{n}:~{}x^{{\scriptscriptstyle\mathrm{T}}}y>0,~{}\forall\,y\\!\in\\!{\cal{K}}{\setminus\\{0\\}}\\}$.
Let $C\subset{\mathbb{R}}^{n}$ be a closed convex set; then, we define the
asymptotic cone of $C$ by
$C_{\infty}:=\Big{\\{}d\in{\mathbb{R}}^{n}:~{}\exists~{}(t_{k})_{k\in\mathbb{N}}\subset(0,\infty),~{}\exists~{}(x^{k})_{k\in\mathbb{N}}\subset
C;\lim_{k\to\infty}t_{k}=0,\lim_{k\to\infty}t_{k}x_{k}=d\Big{\\}},$
or equivalently, $C_{\infty}:=\\{d\in{\mathbb{R}}^{n}:~{}x+td\in
C,\forall~{}x\in C,~{}\forall~{}t\geq 0\\}$ [15, pp. 39]. Let
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be a continuously differentiable function.
We consider the problem of determining an optimum point of $f$ in a closed
convex set ${C}\subset{\mathbb{R}}^{n}$, that is, a point $x^{*}\in{C}$ such
that $f(x^{*})\leq f(x)$ for all $x\in{C}$. We denote this constrained problem
as
$\displaystyle{\rm Minimize}_{x\in{C}}f(x).$ (1)
The optimal value of $f$ on ${C}$ is denoted by $f^{*}$, that is,
$f^{*}:=\inf_{x\in{C}}f(x)$. The first-order optimality condition for problem
(1) is stated as
$\nabla f(x^{*})^{T}(x-x^{*})\geq 0,\qquad\forall~{}x\in{C}.$ (2)
In general, the condition (2) is necessary but not sufficient for optimality.
Thus, a point $x^{*}\in C$ satisfying condition (2) is called a stationary
point to problem (1).
###### Definition 2.1.
Let $C\subset{\mathbb{R}}^{n}$ be a convex set. A function
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is called $M$-strongly convex with
parameter $M\geq 0$ on $C$ if the inequality $f(tx+(1-t)y)\leq
tf(x)+(1-t)f(y)-\frac{1}{2}Mt(1-t)\|y-x\|^{2}$ holds for all $t\in[0,1]$ and
$x,y\in{C}$. In particular, for $M=0$, $f$ is called convex rather than
$0$-strongly convex.
The following results provide a useful characterization of convex/strongly
convex differentiable functions, see the proof in [16, Theorem 4.1.1, pp.
183].
###### Proposition 2.2.
Let $C\subset{\mathbb{R}}^{n}$ be a convex set,
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be a continuously differentiable function,
and $M\geq 0$. Then, $f$ is $M$-strongly convex in $C$ if and only if
$f(y)\geq f(x)+\nabla f(x)^{T}(y-x)+M\|x-y\|^{2}/2$ for all $x,y\in C$.
###### Remark 1.
It is well known that if $f$ is $M$-strong convex, then (2) is sufficient for
optimality, that is, any point $x^{*}\in C$ satisfying (2) is a minimizer to
problem (1).
The proof of the next lemma can be found in [17, Lemma 6].
###### Lemma 2.3.
Let $\\{a_{k}\\}$ be a nonnegative sequence of real numbers. If $\Gamma
a_{k}^{2}\leq a_{k}-a_{k+1}$ for some $\Gamma>0$ and for any $k=1,...,\ell$,
then $a_{\ell}\leq a_{0}/(1+\Gamma a_{0}\ell)<1/(\Gamma\ell)$.
## 3 Frank–Wolfe algorithm
Herein, we formulate the Frank–Wolfe algorithm to solve problem (1). To this
end, we henceforth assume that the constraint set $C\subset{\mathbb{R}}^{n}$
is closed and convex (not necessarily compact), the objective function
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ of problem (1) is continuously
differentiable, and its gradient satisfies the following condition:
* (A)
$\|\nabla f(x)-\nabla f(y)\|\leq L\|x-y\|$ for all $x,y\in{C}$ and $L>0$.
###### Remark 2.
In Section 4, we present examples of problem (1) with objective function
satisfying (A).
To formulate the Frank–Wolfe algorithm, we should assume that there exists a
linear-optimization oracle (LO oracle) capable of minimizing linear functions
over $C$.
###### Algorithm 1.
CondGC,f method
0.
Select $x^{0}\in{C}$. Set $k=0$.
1.
Use an “LO oracle” to compute an optimal solution $p^{k}$ and the optimal
value $v_{k}^{*}$ as
$p^{k}\in{\rm argmin}_{p\in{C}}\nabla f(x^{k})^{T}(p-x^{k}),\qquad
v_{k}^{*}:=\nabla f(x^{k})^{T}(p^{k}-x^{k}).$ (3)
2.
If $v^{*}_{k}=0$, then stop; otherwise, compute the step size
$\lambda_{k}\in(0,1]$ as
$\lambda_{k}:=\mbox{min}\left\\{1,\frac{|v_{k}^{*}|}{L\|p^{k}-x^{k}\|^{2}}\right\\}={\rm
argmin}_{\lambda\in(0,1]}\left\\{v_{k}^{*}\lambda+\frac{L}{2}\|p(x^{k})-x^{k}\|^{2}\lambda^{2}\right\\},$
(4)
and set the next iterate $x^{k+1}$ as
$x^{k+1}:=x^{k}+\lambda_{k}(p^{k}-x^{k}).$ (5)
3.
Set $k\leftarrow k+1$, and go to step 1.
We conclude this section by stating a basic inequality for functions
satisfying assumption (A) [18, Lemma 2.4.2].
###### Lemma 3.1.
Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a continuously differentiable function
satisfying condition (A), $x\in{C}$, and $\lambda\in[0,1]$. Then,
$f(x+\lambda(p-x))\leq f(x)+\nabla
f(x)^{{\scriptscriptstyle\mathrm{T}}}(p-x)\lambda+\frac{L}{2}\|p-x\|^{2}\lambda^{2},\qquad\forall~{}p\in{C}.$
(6)
### 3.1 Well-definedness
Herein, we establish that the sequence $(x^{k})_{k\in\mathbb{N}}$ generated by
Algorithm 1 is well defined. To this end, we assume that the gradient of the
objective function $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ of problem (1)
satisfies the following condition:
* (B)
$\nabla f(x)\in\mbox{int}(C_{\infty})^{*}$, for all $x\in C$.
###### Remark 3.
It follows from [15, Proposition 2.2.3] that a closed convex set
$C\subset{\mathbb{R}}^{n}$ is compact if and only if $C_{\infty}=\\{0\\}$.
Then, $\mbox{int}(C_{\infty})^{*}={\mathbb{R}}^{n}$, and therefore (B) holds
trivially if $C$ is compact. In Section 4, we present examples of problem (1)
with $\nabla f$ satisfying (B) with unbounded constraint set $C$.
We now use ${\bf(B})$ to prove a general result that implies the existence of
a solution to the problem (3). It is worth mentioning that ${\bf(A})$ is not
required in the proof.
###### Proposition 3.2.
The following three assertions hold:
(i)
For each $x\in C$, the set $D_{x}:=\left\\{p\in C:~{}\nabla f(x)^{T}(p-x)\leq
0\right\\}$ is compact.
(ii)
For each $x\in C$, the linear problem
$\displaystyle{\rm Minimize}_{p\in{C}}\nabla f(x)^{T}(p-x)$ (7)
has a solution.
(iii)
If $D\subset C$ is a bounded set, then the set
$\bigcup_{x\in D}\Big{\\{}q_{x}\in C:~{}q_{x}\in{\rm argmin}_{p\in{C}}\nabla
f(x)^{T}(p-x)\Big{\\}},$ (8)
is also bounded.
###### Proof.
To prove (i), let $x\in C$. We assume toward a contradiction that $D_{x}$ is
unbounded. Thus, let $(q^{k})_{k\in\mathbb{N}}\subset D_{x}$ such that
$\lim_{k\to\infty}\|q^{k}\|=\infty$, and let
$(t_{k})_{k\in\mathbb{N}}\subset(0,\infty)$ be the sequence defined by
$t_{k}:=1/\|q^{k}\|$ for all $k=0,1\ldots$. Then, $\lim_{k\to\infty}t_{k}=0$.
As $t_{k}q^{k}=q^{k}/\|q^{k}\|$, we conclude that $\|t_{k}p_{k}\|=1$ for all
$k=0,1,\ldots$. Hence, there exist subsequences
$(q^{k_{j}})_{j\in\mathbb{N}}\subset D_{x}$ and
$(t_{k_{j}})_{j\in\mathbb{N}}\subset(0,\infty)$ such that
$\lim_{k_{j}\to\infty}t_{k_{j}}q^{k_{j}}=d\in C_{\infty}$. Thus, the
definition of $D_{x}$ implies
$\nabla f(x)^{T}\left(t_{k_{j}}q^{k_{j}}-t_{k_{j}}x\right)=\nabla
f(x)^{T}\left(\frac{q^{k_{j}}}{\|q^{k_{j}}\|}-\frac{x}{\|q^{k_{j}}\|}\right)\leq
0.$
Taking the limit in the last inequality as $j$ tends to $\infty$, we conclude
that $\nabla f(x)^{T}d\leq 0$, which is absurd. Indeed, assumption (B) implies
that $\nabla f(x)\in\mbox{int}(C_{\infty})^{*}$, and as $d\in C_{\infty}$, we
have $\nabla f(x)^{T}d>0$. Therefore, (i) is proved. To prove (ii), it is
sufficient to note that the problem (7) has $D_{x}$ as a sublevel set, which
by (i) is compact. We now prove (iii). We assume toward a contradiction that
the set in (8) is unbounded. Then, there exist sequences
$(x^{k})_{k\in\mathbb{N}}\subset D$ and $(q_{x^{k}})_{k\in\mathbb{N}}\subset
C$ such that $\lim_{k\to\infty}\|q_{x^{k}}\|=\infty$. Thus, as $D$ is bounded,
we have
$\lim_{k\to\infty}\tau_{k}=0,\qquad\mbox{where}\quad\tau_{k}:=\frac{1}{\|q_{x^{k}}-x^{k}\|},\qquad
k=0,1,\ldots.$ (9)
However, as $C$ is convex, and $(x^{k})_{k\in\mathbb{N}}$ and
$\\{q_{x^{k}}\\}$ belong to $C$, we have
$x^{k}+t\left(q_{x^{k}}-x^{k}\right)\in C,\qquad k=0,1,\ldots$ (10)
for any $t\in(0,1)$. As
$\tau_{k}(q_{x^{k}}-x^{k})=(q_{x^{k}}-x^{k})/\|q_{x^{k}}-x^{k}\|$, we have
$\|\tau_{k}(q_{x^{k}}-x^{k})\|=1$ for all $k=0,1,\ldots$. Thus, there exist
subsequences $(x^{k_{j}})_{j\in\mathbb{N}}\subset D$,
$(q_{x^{k_{j}}})_{j\in\mathbb{N}}\subset C$, and
$(t_{k_{j}})_{j\in\mathbb{N}}\subset(0,\infty)$ such that
$\lim_{k_{j}\to\infty}\tau_{k_{j}}(q_{x^{k_{j}}}-x^{k_{j}})=v.$ (11)
As $(x^{k_{j}})_{j\in\mathbb{N}}\subset D$ and $D$ is bounded, (9), (10), and
(11) yield
$\lim_{k_{j}\to\infty}\tau_{k_{j}}\left[x^{k_{j}}+t\left(q_{x^{k_{j}}}-x^{k_{j}}\right)\right]=\lim_{k_{j}\to\infty}\left[\tau_{k_{j}}x^{k_{j}}+t\tau_{k_{j}}\left(q_{x^{k_{j}}}-x^{k_{j}}\right)\right]=tv\in
C_{\infty}.$ (12)
As $q_{x^{k_{j}}}\in{\rm argmin}_{p\in{C}}\nabla
f(x^{k_{j}})^{T}(p-x^{k_{j}})$ and $x^{k_{j}}\in C$, we have $\nabla
f(x^{k_{j}})^{T}(q_{x^{k_{j}}}-x^{k_{j}})\leq 0$ for all $j=0,1,\ldots$. Then,
as $(t_{k_{j}})_{j\in\mathbb{N}}\subset(0,\infty)$, we have
$\nabla f(x^{k_{j}})^{T}\left(\tau_{k_{j}}(q_{x^{k_{j}}}-x^{k_{j}})\right)\leq
0,\qquad j=0,1,\ldots.$
Considering that $D$ is bounded and $(x^{k_{j}})_{j\in\mathbb{N}}\subset D$,
we can assume without loss of generality that
$\lim_{k_{j}\to\infty}x^{k_{j}}={\bar{x}}.$ Thus, taking the limit in the last
inequality as $j$ tends to $\infty$ and using (11), we obtain $\nabla
f({\bar{x}})^{T}v\leq 0$, which is absurd because, by (12), we have $v\in
C_{\infty}$, and by assumption (B), we have $\nabla
f({\bar{x}})\in\mbox{int}(C_{\infty})^{*}$. Therefore, the proof of (iii) is
complete. ∎
In the next lemma, we establish that the sequence $(x^{k})_{k\in\mathbb{N}}$
generated by Algorithm 1 is well defined. We also obtain results related to
the optimal value $v_{k}^{*}$ defined in (3).
###### Lemma 3.3.
The sequences $\\{p^{k}\\}_{k\in\mathbb{N}}\subset C$ and
$\\{x^{k}\\}_{k\in\mathbb{N}}\subset C$ are well defined. Moreover, the
following assertions hold:
(i)
$v_{k}^{*}\leq 0$ for all $k=0,1,\ldots$;
(ii)
$v_{k}^{*}=0$ if and only if $x^{k}$ is a stationary point of problem (1);
(iii)
$v_{k}^{*}<0$ if and only if $\lambda_{k}>0$ and $p^{k}\neq x^{k}$.
###### Proof.
For each $x^{k}\in C$, it follows from Proposition 3.2 that $p^{k}$ and
$v_{k}^{*}$ in (3) can be computed and $p^{k}\in C$. As $x^{0}\in C$ and
$0\leq\lambda\leq 1$, by using (5) and an inductive argument, we conclude that
$(p^{k})_{k\in\mathbb{N}}$, $(v_{k}^{*})_{k\in\mathbb{N}}$, and
$(x^{k})_{k\in\mathbb{N}}$ are well defined, and that
$(p^{k})_{k\in\mathbb{N}}$ and $(x^{k})_{k\in\mathbb{N}}$ belong to $C$. To
prove (i), it suffices to note that the optimality in (3) implies
$v_{k}^{*}\leq\nabla f(x^{k})^{T}(x^{k}-x^{k})=0$ for all $k=0,1,\ldots$. To
prove(ii), we note that (3) implies that $v_{k}^{*}\leq\nabla
f(x^{k})^{T}(p-x^{k})$ for all $p\in C$. Thus, if $v_{k}^{*}=0$, we conclude
that $0\leq\nabla f(x^{k})^{T}(p-x^{k})$ for all $p\in C$. Hence, $x^{k}$
satisfies (2), that is, $x^{k}$ is a stationary point of problem (1).
Conversely, if $x^{k}$ is a stationary point of problem (1), then (2) implies
$0\leq\nabla f(x^{k})^{T}(p-x^{k})$ for all $p\in C$. As $p^{k}\in C$, we
conclude that $0\leq\nabla f(x^{k})^{T}(p^{k}-x^{k})=v_{k}^{*}$. Therefore, by
(i), we have $v_{k}^{*}=0$. We now prove (iii). It is immediate from (i) and
(4) that $v_{k}^{*}<0$ if and only if $\lambda_{k}>0$ and $p^{k}\neq x^{k}$;
this concludes the proof. ∎
It follows from Lemma 3.3 that Algorithm 1 generates either an infinite
sequence or a finite sequence $\\{x^{k}\\}_{k\in\mathbb{N}}\subset C$, the
last iterate of which is a stationary point of problem (1). Henceforth, let
$\\{p^{k}\\}_{k\in\mathbb{N}}\subset C$ and
$\\{x^{k}\\}_{k\in\mathbb{N}}\subset C$ be sequences generated by Algorithm 1;
we assume that these sequences are infinite.
### 3.2 Asymptotic convergence analysis
Herein, we study the asymptotic convergence of Algorithm 1. We first prove an
important inequality.
###### Lemma 3.4.
The following inequality holds:
$f(x^{k+1})\leq f(x^{k})-\frac{1}{2}|v_{k}^{*}|\lambda_{k},\qquad
k=0,1,\ldots.$ (13)
Consequently, $f(x^{k})>f(x^{k+1})$ for all $k=0,1,\ldots$.
###### Proof.
Let $x^{k}\in{C}$ be defined as in Algorithm 1, and $v_{k}^{*}$ as in (3). We
first recall that we have assumed that $(x^{k})_{k\in\mathbb{N}}$ is infinite.
Thus, Lemma 3.3 implies that $v_{k}^{*}<0$ and $p^{k}\neq x^{k}$. Applying
Lemma 3.1 with $x=x^{k}$, $p=p^{k}$, and $\lambda=\lambda_{k}$, we have
$f(x^{k+1})\leq
f(x^{k})+v_{k}^{*}\lambda_{k}+\frac{L}{2}\|p^{k}-x^{k}\|^{2}\lambda_{k}^{2}.$
(14)
We separately consider two cases:
$\lambda_{k}=|v_{k}^{*}|/{(L\|p^{k}-x^{k}\|^{2})}$ and $\lambda_{k}=1$. In the
former, it follows from (14) that
$f(x^{k+1})\leq f(x^{k})-\frac{1}{2}|v_{k}^{*}|\lambda_{k}.$ (15)
If now $\lambda_{k}=1$, then (14) becomes $f(x^{k+1})\leq
f(x^{k})-|v_{k}^{*}|+L\|p^{k}-x^{k}\|^{2}/2$, and (4) yields
$\lambda_{k}=1\leq|v_{k}^{*}|/(L\|p^{k}-x^{k}\|^{2})$. Thus, we obtain
$f(x^{k+1})\leq f(x^{k})-(|v_{k}^{*}|/2)\lambda_{k}.$ Therefore, combining
this inequality with (15) yields (13). As we assumed that $v_{k}^{*}<0$ for
all $k=0,1,\ldots$, the second part follows, and the proof is complete. ∎
The next result shows a partial asymptotic-convergence property of the
Frank–Wolfe algorithm; it requires neither convexity nor strong convexity on
$f$.
###### Theorem 3.5.
Each limit point ${\bar{x}}\in C$ of $(x^{k})_{k\in\mathbb{N}}$ is stationary
for the problem (1).
###### Proof.
Let ${\bar{x}}\in C$ be a limit point of the sequence
$(x^{k})_{k\in\mathbb{N}}$, and let $(x^{k_{j}})_{j\in\mathbb{N}}$ be a
subsequence of $(x^{k})_{k\in\mathbb{N}}$ such that
$\lim_{j\to\infty}x^{k_{j}}={\bar{x}}$. Hence,
$\lim_{j\to\infty}f(x^{k_{j}})=f({\bar{x}})$. As Lemma 3.4 implies that
$(f(x^{k}))_{k\in\mathbb{N}}$ is a decreasing sequence, we conclude that
$(f(x^{k}))_{k\in\mathbb{N}}$ converges to $f({\bar{x}})$. Thus, in
particular,
$\lim_{k\to\infty}[f(x^{k})-f(x^{k+1})]=0.$ (16)
Moreover, as $(x^{k_{j}})_{j\in\mathbb{N}}$ is bounded, by combining the
inclusion in (3) with (iii) of Proposition 3.2, it follows that
$(p^{k_{j}})_{j\in\mathbb{N}}$ is also bounded. Let
$(p^{k_{\ell}})_{\ell\in\mathbb{N}}$ be a subsequence of
$(p^{k_{j}})_{j\in\mathbb{N}}$ such that
$\lim_{\ell\to\infty}p^{k_{\ell}}={\bar{p}}$. If ${\bar{x}}={\bar{p}}$, then
by (3) and the continuity of $\nabla f$, we have
$\lim_{\ell\to\infty}v_{k_{\ell}}^{*}=\nabla
f(x^{k_{\ell}})^{T}(p^{k_{\ell}}-x^{k_{\ell}})=0$. We now assume that
${\bar{x}}\neq{\bar{p}}$. Hence, combining Lemma 3.4 with (16), we conclude
that $\lim_{\ell\to\infty}|v_{k_{\ell}}^{*}|\lambda_{k_{\ell}}=0$, whereas (4)
yields
$|v_{k_{\ell}}^{*}|\lambda_{k_{\ell}}=\mbox{min}\left\\{|v_{k_{\ell}}^{*}|,\frac{|v_{k_{\ell}}^{*}|^{2}}{L\|p^{k_{\ell}}-x^{k_{\ell}}\|^{2}}\right\\}.$
(17)
As $\lim_{\ell\to\infty}x^{k_{\ell}}={\bar{x}}$,
$\lim_{\ell\to\infty}p^{k_{\ell}}={\bar{p}}$, and ${\bar{x}}\neq{\bar{p}}$, we
obtain that
$\lim_{\ell\to\infty}\|p^{k_{\ell}}-x^{k_{\ell}}\|=\|{\bar{x}}-{\bar{p}}\|\neq
0$. Thus, as $\lim_{\ell\to\infty}|v_{k_{\ell}}^{*}|\lambda_{k_{\ell}}=0$, it
follows from (17) that $\lim_{\ell\to\infty}|v_{k_{\ell}}^{*}|=0$. Moreover,
the optimality of $v_{k_{\ell}}^{*}$ in (3) implies
$v_{k_{\ell}}^{*}\leq\nabla
f(x^{k_{\ell}})^{{\scriptscriptstyle\mathrm{T}}}(p-x^{k_{\ell}}),\qquad\forall~{}p\in
C.$ (18)
As $\lim_{\ell\to\infty}v_{k_{\ell}}^{*}=0$, taking the limit in (18) and
using the continuity of $\nabla f$, we have $\nabla
f(\bar{x})^{{\scriptscriptstyle\mathrm{T}}}(p-\bar{x})\geq 0$ for all $p\in
C$. Therefore, $\bar{x}$ is stationary for the problem (1). ∎
We now show that if $f$ is assumed convex, we can improve the previous result.
###### Theorem 3.6.
The following assertions hold:
(i)
If $f$ is a convex function and $x^{*}$ is a cluster point of the sequence
$(x^{k})_{k\in\mathbb{N}}$, then $x^{*}$ is a solution for the problem (1).
(ii)
If $f$ is an $M$-strongly convex function, then $(x^{k})_{k\in\mathbb{N}}$
converges to a point $x^{*}\in C$ that is a solution of problem (1). Moreover,
$\|x^{k}-x^{*}\|\leq\sqrt{2(f(x^{k})-f(x^{*}))/M}$ for all $k=0,1,\ldots$.
###### Proof.
To prove(i), we assume that $f$ is convex and $x^{*}$ is a cluster point of
$(x^{k})_{k\in\mathbb{N}}$. As $f$ is convex, applying Proposition 2.2 with
$M=0$, we obtain that $f(p)\geq f(x^{*})+\nabla f(x^{*})^{T}(p-x^{*})$ for all
$p\in C$. Therefore, considering that Theorem 3.5 implies that $\nabla
f(x^{*})^{{\scriptscriptstyle\mathrm{T}}}(p-x^{*})\geq 0$ for all $p\in C$, we
conclude that $f(p)\geq f(x^{*})$ for all $p\in C$. Then, $x^{*}$ is a
solution for the problem (1). To prove (ii), we first note that as $f$ is
$M$-strongly convex, the level set ${\cal L}_{f(x^{0})}:=\\{x\in C:~{}f(x)\leq
f(x^{0})\\}$ is bounded. Lemma 3.4 now implies that
$(x^{k})_{k\in\mathbb{N}}\subset{\cal L}_{f(x^{0})}$. Hence,
$(x^{k})_{k\in\mathbb{N}}$ is also bounded. Let $x^{*}\in C$ be a cluster
point of $(x^{k})_{k\in\mathbb{N}}$. It follows from (i) that $x^{*}$ is a
solution of problem (1). Furthermore, combining (2) with Proposition 2.2, we
obtain
$f(x^{k})-f(x^{*})\geq\frac{M}{2}\|x^{*}-x^{k}\|^{2},\qquad k=0,1,\ldots.$
(19)
As Lemma 3.4 implies that $(f(x^{k}))_{k\in\mathbb{N}}$ is a decreasing
sequence, we have $\lim_{k\to+\infty}f(x^{k})=f(x^{*})$. Therefore, (19)
implies that $(x^{k})_{k\in\mathbb{N}}$ converges to $x^{*}$. Finally, we note
that (19) is equivalent to the inequality in (ii), and the proof is complete.
∎
### 3.3 Iteration-complexity analysis
Herein, we derive two iteration-complexity bounds for the sequence
$(x^{k})_{k\in\mathbb{N}}$ generated by Algorithm 1. To this end, we assume
that $\lim_{k\to\infty}x_{k}=x^{*}$. Then, (iii) of Proposition 3.2 implies
that $(p^{k})_{k\in\mathbb{N}}$ is bounded. Therefore, we define
$0<\sigma:=\sup_{k}\\{\|p^{k}-x^{k}\|:~{}k=0,1,\ldots\\}<\infty.$ (20)
We also define the following constants:
$\Gamma:=\min\left\\{\frac{1}{2\gamma\sigma},\frac{1}{2L\sigma^{2}}\right\\}>0,\qquad\gamma:=\max\left\\{\|\nabla
f(x^{k})\|:~{}k=0,1,\ldots\right\\}>0.$ (21)
###### Theorem 3.7.
The following assertions hold:
(i)
If $f$ is a convex function, then $f(x^{k})-f^{*}\leq\Gamma^{-1}/k$ for all
$k=1,2,\ldots$.
(ii)
If $f$ is $M$-strongly convex, then $\|x^{k}-x^{*}\|\leq\sqrt{2/(\Gamma
M)}/\sqrt{k}$ for all $k=1,2,\ldots$.
###### Proof.
To prove (i), we first prove the following inequality:
$\Gamma{v_{k}^{*}}^{2}\leq f(x^{k})-f(x^{k+1}),\qquad\forall~{}k=0,1,\dots,$
(22)
where $\Gamma$ is defined in (21). By using Lemma 3.4, with $\sigma$ defined
in(20), and considering that ${v_{k}^{*}}<0$, we conclude after some algebraic
manipulations that
$\min\left\\{\frac{1}{2|v_{k}^{*}|},\frac{1}{2L\sigma^{2}}\right\\}{v_{k}^{*}}^{2}\leq
f(x^{k})-f(x^{k+1}),\qquad\forall~{}k=0,1,...$ (23)
Further, by combining (3) with (20) and the second equality in (21), we obtain
$0<|v_{k}^{*}|\leq\|\nabla f(x^{k})\|\|x^{k}-p^{k}\|\leq\gamma\sigma$ for all
$k=0,1,\ldots$, which implies
$\frac{1}{\gamma\sigma}\leq\frac{1}{|v_{k}^{*}|},\qquad k=0,1,\ldots.$
Thus, (22) follows from (23), the previous inequality, and (21). It now
follows from (ii) of Theorem 3.6 that $x^{*}$ is a solution of problem (1). As
$f$ is convex, we have $f^{*}=f(x^{*})\geq f(x^{k})+\nabla
f(x^{k})^{T}(x^{*}-x^{k})$ for all $k$. Thus, (3) implies that
$f^{*}-f(x^{k})\geq\nabla f(x^{k})^{T}(x^{*}-x^{k})\geq
f(x^{k})^{T}(p^{k}-x^{k})=v_{k}^{*}$ for all $k$. As $f^{*}\leq f(x^{k})$ for
all $k$, we conclude that $v_{k}^{*}\leq f^{*}-f(x^{k})\leq 0$ for all $k$.
Therefore, we obtain $(f(x^{k})-f^{*})^{2}\leq v_{k}^{*2}$ for all
$k=0,1,\ldots$; this, combined with (22), yields
$(f(x^{k})-f^{*})-(f(x^{k+1})-f^{*})\geq\Gamma(f(x^{k})-f^{*})^{2},\qquad
k=0,1,\ldots.$ (24)
As $\Gamma>0$, if we define $a_{k}:=f(x^{k})-f^{*}$, we conclude from (24)
that $a_{k}-a_{k+1}\geq\Gamma a_{k}^{2}$. Thus, (i) follows by applying Lemma
2.3. To prove (ii), we first note that the $M$-strong convexity of $f$ and (2)
imply that $\|x^{k}-x^{*}\|\leq\sqrt{2(f(x^{k})-f(x^{*}))/M}$ for all
$k=0,1,\ldots$. Therefore, (ii) follows by using the inequality in (i), and
the proof is complete. ∎
## 4 Examples
Herein, we present examples of problem (1), with $f$ satisfying (A) and (B),
and $C$ unbounded. We first present a general class of functions $f$
satisfying (A) and (B). To this end, let ${\cal{K}}\subset\mathbb{R}^{n}$ be a
closed convex cone such that
$\mbox{int}{\cal{K}}\cap\mbox{int}{\cal{K}}^{*}\neq\varnothing$,
$G:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a differentiable function, and
$G^{\prime}$ be its Jacobian. We assume that for constants $L_{1}>0$ and
$L_{2}>0$, the function $G$ satisfies the following conditions:
(C1)
$\|G(x)-G(y)\|\leq L_{1}\|x-y\|$ for all $x,y\in{\cal{K}}$;
(C2)
$\|G^{\prime}(x)x-G^{\prime}(y)y\|\leq L_{2}\|x-y\|$ for all
$x,y\in{\cal{K}}$;
(C3)
$G(x)+G^{\prime}(x)x\in{\cal{K}}^{*}$ for all $x\in{\cal{K}}$.
Let $a\in\mbox{int}{\cal{K}}^{*}$, and a function
$f:\mathbb{R}^{n}\to\mathbb{R}$ be defined by
$f(x):=a^{T}x+G(x)^{T}x.$ (25)
###### Lemma 4.1.
Let $C\subset{\cal{K}}$ be closed and convex. Then, the gradient $\nabla f$ of
$f$ defined in (25) satisfies conditions (A) and (B) on $C$.
###### Proof.
We first note that the gradient $\nabla f$ of the function $f$ defined in (25)
is given by
$\nabla f(x)=a+G(x)+G^{\prime}(x)x.$ (26)
Thus, using (C1) and (C2), we conclude that $\|\nabla f(x)-\nabla
f(y)\|\leq(L_{1}+L_{2})\|x-y\|$ for all $x,y\in C$. Hence, $\nabla f$
satisfies (A). Finally, as $a\in\mbox{int}{\cal{K}}^{*}$, (C3) implies that
$\nabla f(x)\in\mbox{int}{\cal{K}}^{*}$ for all $x\in C$. Furthermore, as
$C\subset{\cal{K}}$, we obtain that $C_{\infty}\subset{\cal{K}}$. Hence,
${\cal{K}}^{*}\subset C_{\infty}^{*}$, which implies that
$\mbox{int}{\cal{K}}^{*}\subset\mbox{int}(C_{\infty}^{*})$. Thus, we conclude
that $\nabla f(x)\in\mbox{int}(C_{\infty})^{*}$ for all $x\in C$. Therefore,
$\nabla f$ satisfies (B) on $C$. ∎
We now recall a well-known result about Lipschitz functions, which is an
immediate consequence of the mean-value inequality.
###### Theorem 4.2.
Let $C\subset{\mathbb{R}}^{n}$ be a convex set,
$F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ be a continuously differentiable
function, and $F^{\prime}$ be its Jacobian. We assume that there exists a
constant $L\geq 0$ such that $\|F^{\prime}(x)\|\leq L$ for all $x\in C$. Then,
$F$ is Lipschitz continuous with constant $L$ on $C$, that is,
$\|F(x)-F(y)\|\leq L\|x-y\|$ for all $x,y\in C$.
Moreover, the following characterization of convex functions is required; its
proof can be found in [16, Theorem 4.1.1, pp. 190].
###### Theorem 4.3.
Let $C\subset{\mathbb{R}}^{n}$ be a convex set,
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be a twice continuously differentiable
function, $\nabla^{2}f$ be its Hessian, and $M\geq 0$ be a constant. Then, $f$
is $M$-strongly convex on $C$ if and only if $v^{T}\nabla^{2}f(x)v\geq
M\|v\|^{2}$ for all $x\in C$ and $v\in{\mathbb{R}}^{n}$.
In the following, we present specific examples of functions $G$ satisfying
(C1)–(C3) on the cone ${\cal{K}}=\mathbb{R}^{n}_{+}$ such $f$ in (25) is
convex and $\nabla f$ satisfies (A) and (B) for any closed convex set
$C\subset\mathbb{R}^{n}_{+}$.
###### Example 4.4.
Let $Q=(q_{ij})\in{\mathbb{R}}^{n\times n}$ with entries $q_{ij}\geq 0$ for
all $i,j$ and $a\in\mathbb{R}^{n}_{++}$. Let
$G:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear function defined by $G(x)=Qx$.
Direct calculations show that
$\|G(x)-G(y)\|=\|G^{\prime}(x)x-G^{\prime}(y)y\|\leq\|Q\|\|x-y\|$ for all
$x,y\in\mathbb{R}^{n}$. Thus, $G$ satisfies (C1) and (C2), with
$L_{1}=L_{2}=\|Q\|$, for any cone ${\cal{K}}$. As $q_{ij}\geq 0$ and
$a\in\mathbb{R}^{n}_{++}$, $G$ also satisfies (C3) in
${\cal{K}}=\mathbb{R}^{n}_{+}$. Moreover, if $Q$ satisfies the condition
$v^{T}Qv\geq M\|v\|^{2}$ for all $v\in{\mathbb{R}}^{n}$ and some $M\geq 0$,
then Theorem 4.3 implies that the problem (1) with the associated quadratic
function $f(x):=a^{T}x+x^{T}Qx$ is $M$-strong convex. Moreover, Lemma 4.1
implies that $\nabla f(x)=a+Qx$ satisfies (A) and (B) for any closed and
convex set $C\subset\mathbb{R}^{n}_{+}$. For instance,
$C:=\left\\{x\in\mathbb{R}^{p}:~{}x_{1}\geq x_{2}\geq\cdots\geq x_{p}\geq
0\right\\}$, the monotone nonnegative cone.
###### Example 4.5.
Let $e:=(1,\ldots,n)\in\mathbb{R}^{n}$ be a vector, and $\alpha>0$ and
$\beta>0$ be constants satisfying $2\alpha>3\beta^{3/2}\sqrt{n}$. Then, the
function $G:\mathbb{R}^{n}\to\mathbb{R}^{n}$ defined by
$G(x):=\alpha x+\frac{\beta}{\sqrt{1+\beta x^{T}x}}e,$ (27)
satisfies (C1)–(C3) in ${\cal{K}}=\mathbb{R}^{n}_{+}$. Moreover, the problem
(1) with the associated function
$f(x):=a^{T}x+\alpha x^{T}x+\frac{\beta}{\sqrt{1+\beta x^{T}x}}e^{T}x,$ (28)
with $a\in\mathbb{R}^{n}_{++}$ is $M$-strong convex. Moreover, $\nabla f$
satisfies (A) and (B). Indeed, we first note that $f(x):=a^{T}x+G(x)^{T}x$.
Some calculations show that
$G^{\prime}(x)=\alpha{\rm I_{n}}+\frac{-\beta^{2}}{(1+\beta
x^{T}x)^{3/2}}ex^{T},\qquad G(x)+G^{\prime}(x)x=2\alpha
x+\frac{\beta}{(1+\beta x^{T}x)^{3/2}}e,$ (29)
where ${\rm I_{n}}\in{\mathbb{R}}^{n\times n}$ is the identity matrix. Thus,
$\nabla f(x)=a+G(x)+G^{\prime}(x)x$. Hence, after some calculations, we have
$\nabla^{2}f(x)=2\alpha{\rm I_{n}}+\frac{-3\beta^{2}}{(1+\beta
x^{T}x)^{5/2}}ex^{T}.$ (30)
The first equality in (29) yields
$\|G^{\prime}(x)\|\leq\beta^{3/2}\sqrt{n}+\alpha$ for all $x\in C$, and (30)
implies
$0<\left(2\alpha-3\beta^{3/2}\sqrt{n}\right)|v\|^{2}\leq
v^{T}\nabla^{2}f(x)v\leq\left(2\alpha+3\beta^{3/2}\sqrt{n}\right)\|v\|^{2},$
(31)
for all $x\in C$ and all $v\in{\mathbb{R}}^{n}$. As
$\|G^{\prime}(x)\|\leq\beta^{3/2}\sqrt{n}+\alpha$, it follows from Theorem 4.2
that $G$ also satisfies (C1) with $L_{1}=\beta^{3/2}\sqrt{n}+\alpha$. In
particular, (31) implies that $\|\nabla^{2}f(x)\|\leq
2\alpha+3\beta^{3/2}\sqrt{n}$, for all $x\in C$. Hence, as
$\|G^{\prime}(x)\|\leq\beta^{3/2}\sqrt{n}+\alpha$ for all $x\in C$, Theorem
4.2 also implies that
$\|G^{\prime}(x)x-G^{\prime}(y)y\|\leq\|G(x)-G(y)\|+\|\nabla f(x)-\nabla
f(y)\|\leq\left(4\beta^{3/2}\sqrt{n}+3\alpha\right)\|x-y\|,$
for all $x,y\in C$. Thus, $G$ also satisfies (C2) with
$L_{2}=4\beta^{3/2}\sqrt{n}+3\alpha$. The second inequality in (29) implies
that $G(x)+G^{\prime}(x)x\in{\mathbb{R}^{n}_{++}}$ for all
$x\in\mathbb{R}^{n}_{+}$. As $(\mathbb{R}^{n}_{+})^{*}=\mathbb{R}^{n}_{+}$,
$G$ satisfies (C3). Therefore, Lemma 4.1 implies that $\nabla f$ of $f$ in
(28) satisfies conditions (A) and (B) for any closed and convex set
$C\subset\mathbb{R}^{n}_{+}$. For instance,
$C:=\left\\{x\in\mathbb{R}^{p}:~{}x_{1}\geq x_{2}\geq\cdots\geq x_{p}\geq
0\right\\}$, the monotone nonnegative cone. Finally, using (31), it follows
from Theorem 4.3 that $f$ in (28) is $M$-strong convex with
$M=2\alpha-3\beta^{3/2}\sqrt{n}$.
In the next example, we present directly a convex function $f$ satisfying (A)
and (B) in a closed convex set $C\subset\mathbb{R}^{n}_{+}$
###### Example 4.6.
Let $\beta>0$, $a\in\mathbb{R}^{n}_{++}$, and $f:\mathbb{R}^{n}\to\mathbb{R}$
be defined by
$f(x):=a^{T}x+\sqrt{1+\beta x^{T}x}.$ (32)
Let $C:=\left\\{x\in\mathbb{R}^{p}:~{}x_{1}\geq x_{2}\geq\cdots\geq x_{p}\geq
0\right\\}$ be the monotone nonnegative cone. We note that, in this case, the
gradient and the Hessian of $f$ are given by
$\nabla f(x)=a+\frac{\beta}{\sqrt{1+\beta
x^{T}x}}x,\qquad\nabla^{2}f(x)=\frac{\beta}{\sqrt{1+\beta x^{T}x}}{\rm
I_{n}}-\frac{\beta^{2}}{(1+\beta x^{T}x)^{3/2}}xx^{T},$
respectively, where ${\rm I_{n}}\in{\mathbb{R}}^{n\times n}$ is the identity
matrix. Some calculations show that
$\frac{\beta}{(1+\beta x^{T}x)^{3/2}}v^{T}v\leq v^{T}\nabla^{2}f(x)v\leq\beta
v^{T}v,\qquad\forall~{}v\in\mathbb{R}^{n},$
which implies that $\nabla^{2}f(x)$ is positive definite and
$\|\nabla^{2}f(x)\|\leq\beta$. Thus, using Theorems 4.3 and 4.2, we conclude
that $f$ is convex, and $\nabla f$ is Lipschitz continuous with constant
$\beta$. Moreover, for any closed and convex set $C\subset\mathbb{R}^{n}_{+}$,
we have $C_{\infty}\subset\mathbb{R}^{n}_{+}$. Hence,
$\mathbb{R}^{n}_{+}=(\mathbb{R}^{n}_{+})^{*}\subset C_{\infty}^{*}$, which
implies that $\mathbb{R}^{n}_{++}\subset int(C_{\infty}^{*})$. As $\nabla
f(x)\in\mathbb{R}^{n}_{++}$ for all $x\in C$, we conclude that $\nabla f(x)\in
int(C_{\infty}^{*})$ for all $x\in C$. Finally, the convexity of $g$ implies
that $C$ is convex. Therefore, the problem (1) with the objective function
(32) is convex, and $f$ satisfies conditions (B) and (A).
Let us present two more examples of convex functions and the respective convex
sets satisfying conditions (B) and (A).
(i)
Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be defined by
$f(x)=\ln(e^{x_{1}}+e^{x_{2}}+\ldots+e^{x_{n})}$. This function satisfies (A)
[19, Example 5.15, pp. 115]. Some calculations show that it also satisfies (B)
for any convex set $C\subset\mathbb{R}^{n}_{+}$.
(ii)
Let $C\subset\mathbb{R}^{n}_{++}$ be closed and convex, and
$d_{C}(x):=\min_{y\in C}\|x-y\|$ for $x\in\mathbb{R}^{n}$. We define the
convex function $\psi_{C}:\mathbb{R}^{n}\to\mathbb{R}$ by
$\psi_{C}(x):=\frac{1}{2}\|x\|^{2}-\frac{1}{2}d^{2}_{C}(x)$ [19, Example
2.17.4, pp. 22]. We can prove that $\nabla\psi_{C}(x)=P_{C}(x)$, where $P_{C}$
denotes the orthogonal projection onto $C$ [19, Example 3.49, pp. 61]. As
$\mathbb{R}^{n}_{++}\subset int(C_{\infty}^{*})$ and
$C\subset\mathbb{R}^{n}_{++}$, we conclude that $\nabla\psi_{C}(x)\in
C_{\infty}^{*}$, which implies that $\psi_{C}$ satisfies (B). Moreover, the
nonexpansivity of the projection implies that $\psi_{C}$ also satisfies (A).
We conclude this section by presenting examples of unbounded convex sets that
appear as constraints in optimization problems.
(a)
$C:=\\{x\in\mathbb{R}^{n}_{+}:~{}1\leq x_{1}\ldots x_{n}\\}$;
(b)
$C:=\\{x\in\mathbb{R}^{n}_{+}:~{}1\leq x_{1}+\ldots+x_{n}\\}$;
(c)
$C:=\\{x\in\mathbb{R}^{n}_{+}:~{}b\leq Ax\\}$, where
$A=(a_{ij})\in{\mathbb{R}}^{n\times n}$ with $a_{ij}>0$ and
$b\in\mathbb{R}^{n}_{++}$.
Let $\Omega\subset\mathbb{R}^{n}$ be a closed, convex set, and
$g:\Omega\to\mathbb{R}$ be a convex function. The epigraph of $g$ is defined
by $\mbox{epi}(g):=\\{(x,t)\in\Omega\times\mathbb{R}:~{}g(x)\leq t\\}.$ The
set $C=\mbox{epi}(g)\subset{\mathbb{R}}^{n}\times{\mathbb{R}}$ is convex and
unbounded. We now provide specific examples.
(1)
(Lorentz cone)
$\left\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}:~{}\|x\|_{2}\leq t\right\\}$,
where $\|\cdot\|_{2}$ denotes the $2$-norm;
(2)
$\left\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}:~{}\|x\|_{1}\leq t\right\\}$,
where $\|\cdot\|_{1}$ denotes the $1$-norm;
(3)
$\left\\{(x_{1},\ldots,x_{n},t)\in\
\mathbb{R}^{n}_{++}\times\mathbb{R}:~{}1/(x_{1}\ldots x_{n})\leq t\right\\}$.
We point out that projecting on the sets $(1)$, $(2)$, and $(3)$ is not a
particularly expensive task [19, Chapter 6]. Finally, we present an example on
the cone of positive semidefinite matrices.
###### Example 4.7.
The cone of positive semidefinite (resp., definite) $n\times n$ symmetric
matrices is denoted by ${\mathbb{S}}^{n}_{+}$ (resp., ${\mathbb{S}}^{n}_{++}$)
and is selfdual, that is, ${{\mathbb{S}}^{n}_{+}}^{*}={\mathbb{S}}^{n}_{+}$.
The trace of $X=(X_{ij})\in{\mathbb{S}}^{n}$ is denoted by
$trX:=\sum_{i=1}^{p}X_{ii}$. Given $X$ and $Y$ in ${\mathbb{S}}^{n}$, their
inner product is defined as $\langle
X,Y\rangle:=trXY=\sum_{i=1,j=1}^{n,m}X_{ij}Y_{ij}$, whereas the norm of $X$ is
defined by $\|X\|:=\langle X,X\rangle^{1/2}$. Let $\beta>0$ and $\gamma>0$ be
such that $2\beta>\gamma$, and let $f:{\mathbb{S}}^{n}_{+}\to\mathbb{R}$ be
defined by $f(X):=\beta\|X||^{2}+\gamma\ln(1+trX).$ We consider the set
$C:=\\{X\in{\mathbb{S}}^{n}_{+}:~{}g(X)\leq 0\\}$, where
$g:{\mathbb{S}}^{n}_{+}\to\mathbb{R}$ is a convex function. The gradient of
$f$ is given by
$\nabla f(X)=2\beta X+\frac{1}{1+trX}{\rm
I_{n}}\in{\mathbb{S}}^{n}_{++},\qquad\forall~{}X\in{\mathbb{S}}^{n}_{+}.$
Calculating the Hessian of $F$, we obtain
$\langle\nabla^{2}f(X)V,V\rangle=2\beta\|V\|^{2}-\frac{\gamma}{\left(1+trX\right)^{2}}(trV)^{2},\quad\forall~{}X\in{\mathbb{S}}^{n}_{+},\quad\forall~{}V\in{\mathbb{S}}^{n}.$
Moreover, $0<(2\beta-\gamma)\|V\|^{2}\langle\nabla f(X)V,V\rangle\leq
2\beta\|V\|^{2}$ for all $X\in{\mathbb{S}}^{n}_{+}$ and $0\neq
V\in{\mathbb{S}}^{n}$. Thus, Theorems 4.2 and 4.3 imply that $\nabla f$ is
Lipschitz continuous with constant $2\beta$, and $f$ is strongly convex with
constant $2\beta-\gamma$, respectively. Furthermore, as
$C\subset{\mathbb{S}}^{n}_{+}$, we conclude that
$C_{\infty}\subset{\mathbb{S}}^{n}_{+}$. Hence,
${\mathbb{S}}^{n}_{+}={{\mathbb{S}}^{n}_{+}}^{*}\subset C_{\infty}^{*}$, which
implies that ${\mathbb{S}}^{n}_{++}\subset int(C_{\infty}^{*})$. As $\nabla
f(x)\in{\mathbb{S}}^{n}_{++}$ for all $x\in C$, we conclude that $\nabla
f(x)\in int(C_{\infty}^{*})$ for all $x\in C$. Moreover the convexity of $g$
implies that $C$ is convex. Therefore, the problem (1) is convex, and $f$
satisfies conditions (A) and (B). In particular, if
$g:{\mathbb{S}}^{n}_{++}\to\mathbb{R}$ given by $g(x)=1/det(X)$ or
$g(x)=1/tr(X)$, then the set $C\subset{\mathbb{S}}^{n}_{++}$ is convex and
unbounded.
## 5 Conclusions
In this study, we considered the classical Frank–Wolf algorithm for nonempty,
closed, convex, not necessarily compact constraints. To study its convergence
properties, we used recession techniques. The examples in Section 4
demonstrated that the Frank–Wolf algorithm can indeed be applied to several
optimization problems (not necessarily convex) with non-compact constraint
sets.
## Funding
The first author was supported in part by CNPq grants 305158/2014-7 and
302473/2017-3, FAPEG/PRONEM- 201710267000532, and CAPES. The second author was
supported in part by Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF)
through grants 0193.001695/2017 and PDE 05/2018. This research was partly
carried out during a visit of the second author to the Center for Mathematical
Research (CRM) (while Western Catalonia was in a state of alert), in the
framework of the Research-in-pairs call in 2020. The CRM is a paradise for
research, and the second author appreciates its hospitality and support.
## References
* [1] Frank M, Wolfe P. An algorithm for quadratic programming. Nav Res Log. 1956;:95–110.
* [2] Demyanov VF, Rubinov AM. Approximate methods in optimization problems. American Elsevier Publishing Co., Inc., New York; 1970.
* [3] Levitin E, Polyak B. Constrained minimization methods. USSR Computational Mathematics and Mathematical Physics. 1966;6(5):1–50.
* [4] Jaggi M. Revisiting frank-wolfe: Projection-free sparse convex optimization. Proceedings of the 30th International Conference on International Conference on Machine Learning - Volume 28. 2013;ICML’13:I–427–I–435.
* [5] Lacoste-Julien S, Jaggi M. On the global linear convergence of frank-wolfe optimization variants. arXiv e-prints. 2015;arXiv:1511.05932.
* [6] Lan G. The complexity of large-scale convex programming under a linear optimization oracle. arXiv e-prints.
* [7] Beck A, Teboulle M. A conditional gradient method with linear rate of convergence for solving convex linear systems. Math Methods Oper Res. 2004;59(2):235–247.
* [8] Harchaoui Z, Juditsky A, Nemirovski A. Conditional gradient algorithms for norm-regularized smooth. convex optimization. Math Program. 2015;152(1-2, Ser. A):75–112.
* [9] Boyd N, Schiebinger G, Recht B. The alternating descent conditional gradient method for sparse inverse problems. SIAM J Optim. 2017;27(2):616–639.
* [10] Luss R, Teboulle M. Conditional gradient algorithms for rank-one matrix approximations with a sparsity constraint. SIAM Rev. 2013;55(1):65–98.
* [11] Freund RM, Grigas P, Mazumder R. An extended Frank-Wolfe method with “in-face” directions, and its application to low-rank matrix completion. SIAM J Optim. 2017;27(1):319–346.
* [12] Konnov IV. Simplified versions of the conditional gradient method. Optimization. 2018;67(12):2275–2290.
* [13] Ghadimi S. Conditional gradient type methods for composite nonlinear and stochastic optimization. Math Program. 2019;173(1-2, Ser. A):431–464.
* [14] Lan G, Zhou Y. Conditional gradient sliding for convex optimization. SIAM J Optim. 2016;26(2):1379–1409.
* [15] Hiriart-Urruty JB, Lemaréchal C. Fundamentals of convex analysis. Springer-Verlag, Berlin; 2001. Grundlehren Text Editions; abridged version of ıt Convex analysis and minimization algorithms. I [Springer, Berlin, 1993\.
* [16] Hiriart-Urruty JB, Lemaréchal C. Convex analysis and minimization algorithms. I. (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]; Vol. 305). Springer-Verlag, Berlin; 1993\.
* [17] Polyak BT. Introduction to optimization. Optimization Software, New York; 1987. Translations Series in Mathematics and Engineering.
* [18] Dennis JE Jr, Schnabel RB. Numerical methods for unconstrained optimization and nonlinear equations. (Classics in Applied Mathematics; Vol. 16). Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; 1996.
* [19] Beck A. First-order methods in optimization. (MOS-SIAM Series on Optimization; Vol. 25). Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA; 2017.
|
11institutetext: Valentina Lenarduzzi 22institutetext: LUT University, Finland
22email<EMAIL_ADDRESS>33institutetext: Savanna Lujan and Nyyti
Saarimäki 44institutetext: Tampere University, Finland
44email<EMAIL_ADDRESS>
44email<EMAIL_ADDRESS>55institutetext: Fabio Palomba
66institutetext: SeSa Lab - University of Salerno, Italy
66email<EMAIL_ADDRESS>
# A Critical Comparison on Six Static Analysis Tools: Detection, Agreement,
and Precision
Valentina Lenarduzzi Savanna Lujan Nyyti Saarimäki Fabio Palomba
(Received: date / Accepted: date)
###### Abstract
Background. Developers use Automated Static Analysis Tools (ASATs) to control
for potential quality issues in source code, including defects and technical
debt. Tool vendors have devised quite a number of tools, which makes it harder
for practitioners to select the most suitable one for their needs. To better
support developers, researchers have been conducting several studies on ASATs
to favor the understanding of their actual capabilities.
Aims. Despite the work done so far, there is still a lack of knowledge
regarding (1) which source quality problems can actually be detected by static
analysis tool warnings, (2) what is their agreement, and (3) what is the
precision of their recommendations. We aim at bridging this gap by proposing a
large-scale comparison of six popular static analysis tools for Java projects:
Better Code Hub, CheckStyle, Coverity Scan, Findbugs, PMD, and SonarQube.
Method. We analyze 47 Java projects and derive a taxonomy of warnings raised
by 6 state-of-the-practice ASATs. To assess their agreement, we compared them
by manually analyzing - at line-level - whether they identify the same issues.
Finally, we manually evaluate the precision of the tools.
Results. The key results report a comprehensive taxonomy of ASATs warnings,
show little to no agreement among the tools and a low degree of precision.
Conclusions. We provide a taxonomy that can be useful to researchers,
practitioners, and tool vendors to map the current capabilities of the tools.
Furthermore, our study provides the first overview on the agreement among
different tools as well as an extensive analysis of their precision.
###### Keywords:
Static analysis tools Software Quality Empirical Study.
## 1 Introduction
Automated Static Analysis Tools (ASATs) are instruments that analyze
characteristics of the source code without executing it, so that they can
discover potential source code quality issues Ernst et al. (2015). These tools
are getting more popular as they are becoming easier to use—especially in
continuous integration pipelines Zampetti et al. (2017)—and there is a wide
range to choose from Vassallo et al. (2019). However, as the number of
available tools grows, it becomes harder for practitioners to choose the tool
(or combination thereof) that is most suitable for their needs Thomas et al.
(2016).
To help practitioners with this selection process, researchers have been
conducting empirical studies to compare the capabilities of existing ASATs
Mantere et al. (2009); Wilander and Kamkar (2002). Most of these
investigations have focused on (1) the features provided by the tools, e.g.,
which maintainability dimensions can be tracked by current ASATs, (2)
comparing specific aspects considered by the tools, such as security Antunes
and Vieira (2009); McLean (2012) or concurrency defects Al Mamun et al.
(2010), and (3) assessing the number of false positives given by the available
static analysis tools Johnson et al. (2013).
Recognizing the effort spent by the research community, which led to notable
advances in the way tool vendors develop ASATs, we herein notice that our
knowledge on the capabilities of the existing SATs is still limited. More
specifically, in the context of our research we point our that three specific
aspects are under-investigated: (1) which source quality problems can actually
be detected by static analysis tools, (2) what is the agreement among
different tools with respect to source code marked as potentially problematic,
and (3) what is the precision with which a large variety of the available
tools provide recommendations. An improved knowledge of these aspects would
not only allow practitioners to take informed decisions when selecting the
tool(s) to use, but also researchers/tool vendors to enhance them and improve
the level of support provided to developers.
In this paper, we aim to address this gap of knowledge by designing and
conducting a large-scale empirical investigation into the detection
capabilities of six popular state-of-the-practice ASATs, namely SonarQube,
Better Code Hub, Coverity Scan, Findbugs, PMD, and CheckStyle.111ASATs verify
code compliance with a specific set of warnings that, if violated, can
introduce an issue in the code. This issue can be accounted for as “source
code quality issue”: as such, in the remaining the paper we use this term when
referring to the output of the considered tools. Specifically, we run the
considered tools against a corpus of 47 projects from the Qualitas Corpus
dataset and (1) depict which source quality problems can actually be detected
by the tools, (2) compute the agreement among the recommendations given by
them at line-level, and (3) manually compute the precision of the tools.
The key results of the study report a taxonomy of static analysis tool
warnings, but also show that, among the considered tools, SonarQube is the one
able to detect most of the quality issues that can be detected by the other
ASATs. However, when considering the specific quality issues detected, there
is little to no agreement among the tools, indicating that different tools are
able to identify different forms of quality problems. Finally, the precision
of the considered tools ranges between 18% and 86%, meaning that the practical
usefulness of some tools is seriously threatened by the presence of false
positives—this result corroborates and enlarges previous findings Johnson et
al. (2013) on a larger scale and considering a broader set of tools.
Based on our findings, our paper finally discusses and distills a number of
lessons learned, limitations, and open challenges that should be considered
and/or addressed by both the research community and tool vendors.
Structure of the paper. The study setting is described in Section 2. The
results are presented in Section 3 and discussed in Section 4. Section 5
identifies the threats to the validity of our study, while Section 6 presents
related works on static analysis tools. Finally, in Section 7 we draw
conclusions and provide an outlook on our future research agenda.
## 2 Empirical Study Design
We designed our empirical study as a case study based on the guidelines
defined by Runeson and Höst Runeson and Host (2009). The following sections
describe the goals and specific research questions driving our empirical study
as well as the data collection and analysis procedures.
### 2.1 Goal and Research Questions
The _goal_ of our empirical study is to compare state-of-the-practice ASATs
_with the aim of_ assessing their capabilities when detecting source code
quality issues with respect to (1) the types of problems they can actually
identify; (2) the agreement among them, and (3) their precision. Our ultimate
_purpose_ is to enlarge the knowledge available on the identification of
source code quality issues with ASATs from the _perspective_ of both
researchers and tool vendors. The former are interested in identifying areas
where the state-of-the-art tools can be improved, thus setting up future
research challenges, while the latter are instead concerned with assessing
their current capabilities and possibly the limitations that should be
addressed in the future to better support developers.
More specifically, our goal can be structured around three main research
questions (RQs). As a first step, we aimed at understanding what kind of
issues different tool warnings detect when run on source code. An improved
analysis of this aspect may extend our knowledge on whether and how various
types of quality issues are identified by the existing tools. Hence, we asked:
RQ1. _What source code quality issues can be detected by Automated Static
Analysis Tools?_
Once we had characterized the tools with respect to what they are able to
identify, we proceeded with a finer-grained investigation aimed at measuring
the extent to which ASATs agree with each other. Regarding this aspect,
further investigation would not only benefit tool vendors who want to better
understand the capabilities of their tools compared to others, but would also
benefit practitioners who would like to know whether it is worth using
multiple tools within their code base. Moreover, we were interested in how the
issues from different tools overlap with each other. We wanted to determine
the type and number of overlapping issues, but also whether the overlapping is
between all tools or just a subset.
RQ2. _What is the agreement among different Automated Static Analysis Tools
when detecting source code quality issues?_
Finally, we focused on investigating the potential usefulness of the tools in
practice. While they could output numerous warnings that alert developers of
the presence of potential quality problems, it is still possible that some of
these warnings might represent false positive instances, i.e., that they
wrongly recommend source code entities to be refactored/investigated. Previous
studies have highlighted the presence of false positives as one of the main
problems of the tools currently available Johnson et al. (2013); our study
aims at corroborating and extending the available findings, as further
remarked in Section 2.4.
RQ3. _What is the precision of Static Analysis Tools?_
All in all, our goal was to provide an updated view on this matter and
understand whether, and to what extent, this problem has been mitigated in
recent years or whether there is still room for improvement.
### 2.2 Context of the Study
The _context_ of our study consisted of software systems and tools. In the
following, we describe our selection.
#### Project Selection
We selected projects from the Qualitas Corpus collection of software systems
(Release 20130901), using the compiled version of the Qualitas Corpus Terra et
al. (2013).
The dataset contains 112 Java systems with 754 versions, more than 18 million
LOCs, 16,000 packages, and 200,000 classes analyzed. Moreover, the dataset
includes projects from different contexts such as IDEs, databases, and
programming language compilers. More information is available in Terra et al.
(2013). In our study, we considered the “r” release of each of the 112
available systems. Since two of the automated static analysis tools
considered, i.e., Coverity Scan and Better Code Hub, require permissions in
the GitHub project or the upload of a configuration file, we privately
uploaded all 112 projects to our GitHub account in order to enable the
analysis222The GitHub projects, with the related configuration adopted for
executing the tools, will be made public in the case of acceptance of this
paper..
#### ASATs Selection
We selected the six tools described below. The choice of focusing on those
specific tools was driven by the familiarity of the authors with them: this
allowed us to (1) use/run them better (e.g., by running them without errors)
and (2) analyze their results better, for instance by providing qualitative
insights able to explain the reasons behind the achieved results. The analysis
of other tools is already part of our future research agenda.
SonarQube333http://www.sonarsource.org/ is one of the most popular open-source
static code analysis tools for measuring code quality issues. It is provided
as a service by the sonarcloud.io platform or it can be downloaded and
executed on a private server. SonarQube computes several metrics such as
number of lines of code and code complexity, and verifies code compliance with
a specific set of “coding warnings” defined for most common development
languages. If the analyzed source code violates a coding warning, the tool
reports an “issue”. The time needed to remove these issues is called
remediation effort.
SonarQube includes reliability, maintainability, and security warnings.
Reliability warnings, also named Bugs, create quality issues that “represent
something wrong in the code” and that will soon be reflected in a bug. Code
smells are considered “maintainability-related issues” in the code that
decrease code readability and code modifiability. It is important to note that
the term “code smells” adopted in SonarQube does not refer to the commonly
known term code smells defined by Fowler et al. Fowler and Beck (1999), but to
a different set of warnings.
Coverity Scan444https://scan.coverity.com/ is another common open-source
static analysis tool. The code build is analyzed by submitting the build to
the server through the public API. The tool detects defects and
vulnerabilities that are grouped by categories such as: resource leaks,
dereferences of NULL pointers, incorrect usage of APIs, use of uninitialized
data, memory corruptions, buffer overruns, control flow issues, error handling
issues, incorrect expressions, concurrency issues, insecure data handling,
unsafe use of signed values, and use of resources that have been
freed555https://scan.coverity.com/faq\\#what-is-coverity-scan. For each of
these categories, there are various issue types that explain more details
about the defect. In addition to issue types, issues are grouped based on
impact: low, medium, and high. The static analysis applied by Coverity Scan is
based on the examination of the source code by determining all possible paths
the program may take. This gives a better understanding of the control and
data flow of the code666https://devguide.python.org/coverity.
Better Code Hub777https://bettercodehub.com/ is also a commonly used static
analysis tool that assesses code quality. The analysis is done through the
website’s API, which analyzes the repository from GitHub. The default
configuration file can be modified for customization purposes. Code quality is
generally measured based on structure, organization, modifiability, and
comprehensibility.
This is done by assessing the code against ten guidelines: write short units
of code, write simple units of code, write code once, keep unit interfaces
small, separate concern in modules, couple architecture components loosely,
keep architecture components balanced, keep your code base small, automate
tests, and write clean code. Out of the ten guidelines, eight guidelines are
grouped based on type of severity: medium, high, and very high. Compliance is
rated on a scale from 1-10 based on the
results888https://pybit.es/bettercodehub.html.
Better Code Hub static analysis is based on the analysis of the source code
against heuristics and commonly adopted coding conventions. This gives a
holistic view of the health of the code from a macroscopic perspective.
Checkstyle999https://checkstyle.org is an open-source developer tool that
evaluates Java code quality. The analysis is done either by using it as a side
feature in Ant or as a command line tool. Checkstyle assesses code according
to a certain coding standard, which is configured according to a set of
checks. Checkstyle has two sets of style configurations for standard checks:
Google Java Style101010https://checkstyle.sourceforge.io/google_style.html and
Sun Java Style111111https://checkstyle.sourceforge.io/sun_style.html. In
addition to standard checks provided by Checkstyle, customized configuration
files are also possible according to user preference.
121212https://checkstyle.sourceforge.io/index.html
These checks are classified under 14 different categories: annotations, block
checks, class design, coding, headers, imports, javadoc comments, metrics,
miscellaneous, modifiers, naming conventions, regexp, size violations, and
whitespace. Moreover, the violation of the checks are grouped under two
severity levels: error and
warning131313https://checkstyle.sourceforge.io/checks.html, with the first
reporting actual problems and the second possible issues to be verified.
FindBugs141414http://findbugs.sourceforge.net is a static analysis tool for
evaluating Java code, more precisely Java bytecode. The analysis is done using
the GUI, which is engaged through the command line. The analysis applied by
the tool is based on detecting bug patterns. According to FindBugs, the bug
patterns arise for the following main reasons: difficult language features,
misunderstood API features, misunderstood invariants when code is modified
during maintenance, and garden variety
mistakes.151515http://findbugs.sourceforge.net/findbugs2.html
161616http://findbugs.sourceforge.net/factSheet.html Such bug patterns are
classified under 9 different categories: bad practice, correctness,
experimental, internationalization, malicious code vulnerability,
multithreaded correctness, performance, security, and dodgy code. Moreover,
the bug patterns are ranked from 1-20. Rank 1-4 is the scariest group, rank
5-9 is the scary group, rank 10-14 is the troubling group, and rank 15-20 is
the concern group171717http://findbugs.sourceforge.net/bugDescriptions.html.
PMD181818https://pmd.github.io/latest/ is a static analysis tool mainly used
to evaluate Java and Apex, even though it can also be applied to six other
programming languages. The analysis is done through the command line using the
binary distributions. PMD uses a set of warnings to assess code quality
according to the main focus areas: unused variables, empty catch blocks,
unnecessary object creation, and more. There are a total of 33 different
warning set configurations191919https://github.com/pmd/pmd/tree/master/pmd-
java/src/main/resources/rulesets/java for Java projects. The warning sets can
also be customized according to the user
preference202020https://pmd.github.io/latest/index.html. These warnings are
classified under 8 different categories: best practices, code style, design,
documentation, error prone, multi threading, performance, and security.
Moreover, the violations of warnings are measured on a priority scale from
1-5, with 1 being the most severe and 5 being the
least212121https://pmd.github.io/latest/pmd$_$rules$_$java.html.
### 2.3 Study Setup and Data Collection
This section describes the study setup used to collect the data from each tool
and the data collection process. We analyzed a single snapshot of each
project, considering the release available in the dataset for each of 112
systems.
SonarQube. We first installed SonarQube LTS 6.7.7 on a private server having
128 GB RAM and 4 processors. However, because of the limitations of the open-
source version of SonarQube, we are allowed to use only one core, therefore
more cores would have not been beneficial for our scope. We decided to adopt
the LTS version (Long-Time Support) since this is the most stable and best-
supported version of the tool.
We executed SonarQube on each project using SDK 8 (Oracle) and the sonar-
scanner package version 4.2. Each project was analyzed using the original
sources and the binaries provided in the dataset. Moreover, we configured the
analysis (in the file sonar-project.properties) reporting information
regarding the project key, project name, project version, source directories,
test directories, binary directories, and library directories. It is important
to note that the analysis performed using the original binaries reduced the
issues of compilation errors and missing libraries. Moreover, it also helped
to reduce issues related to false
positives222222https://docs.sonarqube.org/latest/analysis/languages/java/.
Once the configuration file was set up, the analysis began by running sonar
scanner in the root directory of the project:
$ sonar-scanner.
After all the projects had been analyzed, we extracted the data related to the
projects using the ”SonarQube Exporter”
tool232323https://github.com/pervcomp/SonarQube-Exporter/releases. This makes
it possible to extract SonarQube project measures and issues in CSV format.
Under the target directory, the extraction directory was made. The CSV files
were then extracted from the server by running the following command:
$java-jar csv-from-sonar-0.4.1-jar-with-dependencies.jar
All projects starting with “QC:” were extracted from the server by clicking
“Save issues” and “Save measures”. This exported CSV files into the extraction
directory for each project.
Coverity Scan. The projects were registered in Coverity Scan (version 2017.07)
by linking the github account and adding all the projects to the profile.
Coverity Scan was set up by downloading the tarball file from
https://scan.coverity.com/download and adding the bin directory of the
installation folder to the path in the .bash_profile. Afterwards the building
process began, which was dependent on the type of project in question.
Coverity Scan requires to compile the sources with a special command.
Therefore, we had to compile them, instead of using the original binaries. For
our projects, the following commands were used in the directory of the project
where the build.xml and pom.xml files reside:
* •
ant (build.xml) for building and deploying Java projects. The projects had
various Java versions, so the appropriate Java version was installed according
to the documentation (if available) for building.
* –
$ cov-build –dir cov-int ant jar
* –
$ cov-build –dir cov-int ant
* •
maven (pom.xml) for building projects. The appropriate maven version was used
according to the documentation.
* –
$ cov-build –dir cov-int mvn install
* –
$ cov-int –dir cov-int mvn clean install
After building the project, if over 85% of tests were successfully executed, a
tar archive of the cov-int build folder was created by running:
$ tar czvf myproject.tgz cov-int
Once the tar archive was created, the file was submitted through the project
dashboard at https://scan.coverity.com/projects/myproject to “Submit Build”.
The analysis was then performed, and the results were displayed on the
dashboard. The data regarding the analysis was extracted from the dashboard
from “View Defects”, where the detailed report regarding the analysis results
was located. Under Outstanding Issues -$>$ Export CSV, the CSV file containing
all issues of the project was downloaded for each project.
Better Code Hub. The .bettercodehub.yml files were configured by defining the
component_depth, languages, and exclusions. The exclusions were defined so
that they would exclude all directories that were not source code, since
Better Code Hub only analyzes source code. The analysis was conducted between
January 2019 and February 2019, so the Better Code Hub version hosted during
that time was used.
Once the configuration file had been created, it was saved in the root
directory of the project. The local changes to the project were added,
committed, and pushed to GitHub. Afterwards, the project analysis started from
Better Code Hub’s dashboard, which is connected to the GitHub account
https://bettercodehub.com/repositories. The analysis started by clicking the
“Analyze” button on the webpage. The results were then shown on the webpage.
Once all the projects had been analyzed, the data was sent in one CSV file.
All projects containing more than 100,000 lines of code were not analyzed, due
to Better Code Hub’s limit.
Checkstyle. The JAR file for the Checkstyle analysis was downloaded directly
from Checkstyle’s website242424https://checkstyle.org/#Download in order to
engage the analysis from the command line. The executable JAR file used in
this case was checkstyle$>$8.30-all.jar. In addition to downloading the JAR
executable, Checkstlye offers two different types of warning sets for the
analysisCheckstyleDownload.
For each of the warning sets, the configuration file was downloaded directly
from Checkstyle’s
website252525https://github.com/checkstyle/checkstyle/tree/master/src/main/resources.
In order to start the analysis, the files checkstyle-8.30-all.jar and the
configuration file in question were saved in the directory where all the
cloned repositories from Java Qualitas Corpus resided. To make the analysis
more swift, a bash script was made to execute the analysis for each project in
one go. This can be seen in Listing 1.
Listing 1: Checkstyle bash script tailored towards each warning set.
⬇
#!/bin/bash
while read in; do java -jar checkstyle-8.30-all.jar
-c /RULESET -f xml ”$in”/ > ”$in”_CS_RULESET.xml ;
done < projectList.txt””
where RULESET represents the warning set used for the analysis, "$\$$in"
represents the project name which is imported from projectList.txt, and
"$\$$in"_CS_RULESET.xml represents the export file name of the analysis
results in XML format. The text file projectList.txt consists of all the
project names, in order to execute the analysis for all projects in one go. An
example of how the projects were analyzed with Checkstyle according to the
warning set Google Java
Style262626https://github.com/checkstyle/checkstyle/blob/master/src/main/resources/google_checks.xml
is show in Listing 2.
Listing 2: Example of Checkstyle bash script for Google Java Style
configuration.
⬇
#!/bin/bash
while read in; do java -jar checkstyle-8.30-all.jar
-c /google_checks.xml -f xml ”$in”/ >
”$in”_CS_Google_Checks.xml ;
done < projectList.txt””
Findbugs. FindBugs 3.0.1 was installed by running brew install findbugs in the
command line. Once installed, the GUI was then engaged by writing spotbugs.
From the GUI, the analysis was executed through
$\texttt{File}\rightarrow\texttt{New\hskip 3.55658ptProject}$. The classpath
for the analysis was identified to be the location of the project directory.
Moreover, the source directories were identified to be the project JAR
executables. Once the classpath and source directories were identified, the
analysis was engaged by clicking Analyze in the GUI. Once the analysis
finished, the results were saved through File $\rightarrow$ Save as using the
XML file format.
PMD. PMD 6.23.0 was downloaded from
GitHub272727https://github.com/pmd/pmd/releases/download/pmd$_$releases\%2F6.23.0/pmd-
bin-6.23.0.zip as a zip file. After unzipping, the analysis was engaged by
identifying several parameters: project directory, export file format, warning
set, and export file name. In addition to downloading the zip file, PMD offers
32 different types of warning sets for Java written
projects282828https://github.com/pmd/pmd/tree/master/pmd-
java/src/main/resources/rulesets/java. We developed a bash script, shown in
Listing 3, to engage the analysis for each project in one go.
Listing 3: PMD bash script tailored towards each warning set.
⬇
#!/bin/bash
while read in;
do $HOME/pmd-bin-6.23.0/bin/run.sh pmd -dir ”$in”/
-f xml -R rulesets/java/RULESET
-reportfile ”$in”_PMD_RULESET.xml;
done < projectList.txt””
The parameter HOME represents the full path where the binary resides, "$in"
represents the project name which is imported from projectList.txt, RULESET
represents the warning set used for the analysis, and
"$in"$\\_$PMD$\\_$RULESET.xml represents the export file name of the analysis
results in XML format. Just like in Listing 1, projectList.txt consists of all
the project names. An example of how the projects were analyzed for the
warning set Clone
Implementation292929https://github.com/pmd/pmd/blob/master/pmd-
java/src/main/resources/rulesets/java/clone.xml is show in Listing 4.
Listing 4: Example of PMD bash script for Clone Implementation configuration.
⬇
#!/bin/bash
while read in;
do $HOME/pmd-bin-6.23.0/bin/run.sh pmd -dir ”$in”/
-f xml -R rulesets/java/clone.xml
-reportfile ”$in”_PMD_Clone.xml;
done < projectList.txt””
### 2.4 Data Analysis
In this section, we describe the analysis methods employed to address our
research questions (RQs).
Source code quality issues identified by the tools (RQ1). In order to
determine the tool detection capabilities and warnings overlaps, we first
identified warnings that can be detected by each tool, also considering type
(if available), category, and severity. Then, we calculated how many warnings
are violated in our projects.
Agreement among the tools (RQ2). We expected that similar warnings should be
violated in the same class, and in particular in the same position of the
code. For the tools that provided information on the exact position (the lines
of code where the warning is detected), we analyzed the agreement using the
bottom-up approach. Therefore, we examined the overlapping positioning (start
and end lines) of the warnings and identifying whether the warnings highlight
the same kind of issue. Essentially pairing the warnings based on position and
checking whether they have similar definitions.
In order to check whether the warning pairs identified in RQ1 appear in the
same position in the code, we scanned and checked in which classes each
warning was violated. Then we counted how many times the warning pairs were
violated in the same class. Only warning pairs identified across the tools
were considered. warnings detected only by one tool were not considered.
For example, there was a warning pair identified between SonarQube and Better
Code Hub: ”SQ_1 and BCH_1”. Considering the warning pair detected by SonarQube
(SQ) and Better Code Hub (BCH), we expected that these two warnings would be
violated at least in the same class, since they identify the same kind of
issue. So if for instance SQ_1 is violated 100 times and BCH_1 130 times, we
would expect to find the warning pair violated at least 100 times in the same
class. Hence, we can calculate the agreement between the warnings in the
warning pair as follows:
$Agreement_{r}=\frac{\\#SR}{\\#issues_{r}}$ (1)
where $\\#SR$ is the number of instances in which a warning pair is detected
together in the same class, and $\\#issues_{r}$ is the overall number of
issues of warning $r$ detected in the warning pair.
Agreement is calculated separately for both warnings in a warning pair.
Perfect overlap is obtained if the agreement for both warnings is equal to
one. This means that all the issues generated by those warnings are always
detected in the same classes, and no issues are detected separately in a
different class.
After analyzing using the top-down approach from the definition level to the
class level, we continued the analysis using the bottom-up approach from the
line level to the definition level. For each class, we considered the start
and end lines of the issue and compared the degree to which the warnings
overlap according to the lines affected.
As the granularity of the warnings varies between the tools, we checked what
fraction of the affected lines are overlapping between the warnings, instead
of requiring sufficient overlap between both warnings. This was done by
selecting one issue at a time (reference) and comparing all other issues
(comparison) to that. If the lines affected by the comparison warning were
residing within the lines affected by the reference warning, we defined the
warnings as overlapped. To quantify the degree of overlapping, we used the
percentage of the lines affected by the comparison issue that overlapped with
the reference issue. The results were grouped based on four percentage
thresholds: 100%, 90%, 80% and 70%.
Figure 1: Determining overlapping warnings at “line-level”.
The concept is visualized in Figure 1. The lines represent the issues in a
code file, indicating the start and end of the affected code lines. The
percentages represent the ratio of the lines affected by the warning that lie
within the lines affected by the reference warning. Depending on the threshold
used, different warnings are selected based on the overlapping percentage.
Unfortunately, only SonarQube, Better Code Hub, Checkstyle, PMD, and FindBugs
provide information about the actual ”position” of the detected warnings. They
report information about the ”start line” and ”end line” for each class.
Regarding Coverity Scan, this information is only available in the web
interface, but not in the APIs. Moreover, Coverity Scan licence does not allow
to crawl the web interface. Since we have detected 8,828 Coverity Scan
warnings violated in our projects, it would not have been visible to report
this information manually.
Precision of the tools (RQ3). In our last research question, we aimed at
assessing the precision of the considered tools. From a theoretical point of
view, precision is defined as the ratio between the true positive source code
quality issues identified by a tool and the total number of issues it detects,
i.e., true positives plus false positive items (TPs + FPs). Formally, for each
tool we computed precision as follows:
$precision=\frac{TPs}{TPs+FPs}$ (2)
It is worth remarking that our focus on precision is driven by recent findings
in the field that showed that the presence of false positives is among the
most critical barriers to the adoption of static analysis tools in practice
Johnson et al. (2013); Vassallo et al. (2019). Hence, our analysis provides
research community, practitioners, and tool vendors with indications on the
actual precision of the currently available tools—and aims at possibly
highlighting limitations that can be addressed by further studies. It is also
important to remark that we do not assess recall, i.e., the number of true
positive items identified over the total number of quality issues present in a
software project, because of the lack of a comprehensive ground truth. We plan
to create a similar dataset and perform such an additional evaluation as part
of our future research agenda.
When assessing precision, a crucial detail is related to the computation of
the set of true positive quality issues identified by each tool. In the
context of our work, we conducted a manual analysis of the warnings
highlighted by the six considered tools, thus marking each of them as true or
false positive based on our analysis of (1) the quality issue identified and
(2) the source code of the system where the issue was detected. Given the
expensive amount of work required for a manual inspection, we could not
consider all the warnings output by each tool, but rather focused on
statistically significant samples. Specifically, we took into account a 95%
statistically significant stratified sample with a 5% confidence interval of
the 65,133, 8,828, 62,293, 402,409, 33,704, and 467,583 items given by Better
Code Hub, Coverity Scan, SonarQube, Checkstyle, FindBugs, and PMD
respectively: This step led to the selection of a set of 375 items from Better
Code Hub, 367 from Coverity Scan, 384 from SonarQube, 384 from Checkstyle, 379
from FindBugs, and 380 from PMD.
To increase the reliability of this manual analysis, two of the authors of
this paper (henceforth called the inspectors) first independently analyzed the
warning samples. They were provided with a spreadsheet containing six columns:
(1) the name of the static analysis tool the row refers to, i.e., Better Code
Hub, Coverity Scan, Sonarqube, Checkstyle, FindBugs, and PMD; (2) the full
path of the warning identified by the tool that the inspectors had to verify
manually; and (3) the warning type and specification, e.g., the code smell.
The inspectors’ task was to go over each of the warnings and add a seventh
column in the spreadsheet that indicated whether the warning was a true or a
false positive. After this analysis, the two inspectors had a four-hour
meeting where they discussed their work and resolved any disagreements: All
the items marked as true or false positive by both inspectors were considered
as actual true or false positives; in the case of a disagreement, the
inspectors re-analyzed the warning in order to provide a common assessment.
Overall, after the first phase of inspection, the inspectors reached an
agreement of 0.84—which we computed using Krippendorff’s alpha Krα
Krippendorff (2018) and which is higher than 0.80, which has been reported to
be the standard reference score for Krα Antoine et al. (2014).
In Section 6, we report the precision values obtained for each of the
considered tools and discuss some qualitative examples that emerged from the
manual analysis of the sample dataset.
### 2.5 Replicability
In order to allow the replication of our study, we have published the raw data
in a replication package303030 https://figshare.com/s/3c40dce067507b3e6c63.
## 3 Analysis of the Results
In this section, we report and discuss the results obtained when addressing
our research questions (RQs).
RQ1. What quality issues can be detected by Static Analysis Tools? Here we
analyzed how many warnings are actually output by the considered static
analysis tools as well as the types of issues they are able to discover.
Static Analysis Tools Detection Capability. We report the detection capability
of each tool in terms of how many warnings can be detected, and the
classification of internal warnings (e.g., type and severity). Moreover, we
report the diffusion of the warning in the selected projects.
Better Code Hub detects a total of 10 warnings, of which 8 are grouped based
on type and severity. Better Code Hub categorizes the 8 warnings under 3
types: RefactoringFileCandidateWithLocationList, RefactoringFileCandidate, and
RefactoringFileCandidateWithCategory. Of these 8 warnings, one is of
RefactoringFileCandidateWithLocationList type, six are of
RefactoringFileCandidate type, and one is of
RefactoringFileCandidateWithCategory type. In addition to the types, Better
Code Hub assigns three possible severities to the warnings: Medium, High, and
Very High. Of these eight warnings, four were classified as Medium severity,
four as High severity, and eight as Very High severity. Some of the warnings
have more than one severity possibly assigned to them.
Checkstyle detects a total of 173 warnings which are grouped based on type and
severity. Checkstyle categorizes the 173 warnings under 14 types: Annotations,
Block Checks, Class Design, Coding, Headers, Imports, Javac Comments, Metrics,
Miscellaneous, Modifiers, Naming Conventions, Regexp, Size Violations, and
Whitespace. Of these 173 warnings, 8 are of Annotations type, 6 are of Block
Checks type, 9 are of Class Design type, 52 are of Coding type, 1 is of
Headers type, 8 are of Imports type, 19 are of Javac Comments type, 6 are of
Metrics type, 16 are of Miscellaneous type, 4 are of Modifiers type, 16 are of
Naming Conventions type, 4 are of Regexp type, 8 are of Size Violations type,
and 16 are of Whitespace type. In addition to these types, Checkstyle groups
these checks under four different severity levels: Error, Ignore, Info, and
Warning. The distribution of the checks with respect to the severity levels is
not provided in the documentation.
Coverity Scan’s total scope of detectable warnings as well as the
classification is not known, since its documentation requires being a client.
However, within the scope of our results, Coverity Scan detected a total of
130 warnings. These warnings were classified under three severity levels: Low,
Medium, and High. Of these 130 warnings, 48 were classified as Low severity,
87 as Medium severity, and 12 as High severity. Like Better Code Hub, some of
Coverity Scan’s warnings have more than one severity type assigned to them.
Findbugs detects a total of 424 warnings grouped based on type and severity.
It categorizes the 424 warnings under 9 types: Bad practice, Correctness,
Experimental, Internationalization, Malicious code vulnerability,
Multithreaded correctness, Performance, Security, and Dodgy code. Of these 424
warnings, 88 are of Bad practice type, 149 are of Correctness, 3 are of
Experimental, 2 are of Internationalization, 17 are of Malicious code
vulnerability, 46 are of Multithreaded correctness, 29 are of Performance, 11
are of Security, and 79 are of Dodgy code. In addition to these types,
Findbugs ranks these ’bug patterns’ from 1-20. Rank 1-4 is the scariest group,
rank 5-9 is the scary group, rank 10-14 is the troubling group, and rank 15-20
is the concern group.
PMD detects a total of 305 warnings which are grouped based on type and
severity. PMD categorizes the 305 warnings under 8 types: Best Practices, Code
Style, Design, Documentation, Error Prone, Multithreading, Performance, and
Security. Of these 305 warnings, 51 are of Best Practices, 62 are of Code
Style, 46 are of Design, 5 are of Documentation, 98 are of Error Prone, 11 are
of Multithreading, 30 are of Performance, and 2 are of Security type. In
addition to the types, PMD categorizes the warnings according to five priority
levels (from P1 “Change absolutely required” to P5 “Change highly optional”).
Rule priority guidelines for default and custom-made warnings can be found in
the PMD project documentation.313131https://pmd.github.io/latest/
SonarQube LTS 6.7.7 detects a total of 413 warnings which are grouped based on
type and severity. SonarQube categorizes the 413 warnings under 3 types: Bugs,
Code Smells, and Vulnerabilities. Of these 413 warnings, 107 warnings are
classified as Bugs, 272 as Code Smells, and 34 as Vulnerabilities. In addition
to the types, SonarQube groups the warnings under 5 severity typers: Blocker,
Critical, Major, Minor, and Info. Considering the assigned severity levels,
SonarQube detects 36 Blocker, 61 Critical, 170 Major, 141 Minor, and 5 Info
warnings. Unlike Better Code Hub and Coverity Scan, SonarQube has only one
severity and classification type assigned to each rule.
Table 1: RQ1. Rule diffusion across the 47 projects.
Project Name | #Classes | #Methods | SQ | BCH | Coverity | Checkstyle | PMD | FindBugs | Total
---|---|---|---|---|---|---|---|---|---
AOI | 865 | 2568 | 10865 | 924 | 123 | 250201 | 108458 | 1979 | 372550
Collections | 646 | 2019 | 4545 | 584 | 25 | 85501 | 39893 | 185 | 130733
Colt | 627 | 1482 | 7452 | 560 | 62 | 172034 | 47843 | 4 | 227955
Columba | 1288 | 2941 | 7030 | 662 | 70 | 166062 | 49068 | 1345 | 224237
DisplayTag | 337 | 683 | 853 | 452 | 22 | 32033 | 10137 | 32 | 43529
Drawswf | 1031 | 1079 | 3493 | 559 | 65 | 368052 | 22264 | 69 | 394502
Emma | 509 | 962 | 4451 | 648 | 55 | 68838 | 16524 | 172 | 90688
Findbugs | 1396 | 4691 | 12496 | 600 | 134 | 320087 | 90309 | 1068 | 424694
Freecol | 1569 | 4857 | 5963 | 607 | 337 | 127363 | 79588 | 704 | 214562
Freemind | 1773 | 3460 | 5698 | 662 | 112 | 128590 | 50873 | 1536 | 187471
Ganttproject | 1093 | 2404 | 12349 | 642 | 64 | 71872 | 36689 | 898 | 122514
Hadoop | 3880 | 10701 | 24125 | 682 | 665 | 284315 | 228966 | 1547 | 540300
HSQLDB | 1284 | 5459 | 14139 | 620 | 178 | 192010 | 109625 | 182 | 316754
Htmlunit | 3767 | 9061 | 5176 | 924 | 141 | 92998 | 59807 | 467 | 159513
Informa | 260 | 644 | 992 | 594 | 56 | 11276 | 9364 | 217 | 22499
Jag | 1234 | 1926 | 6091 | 301 | 56 | 24643 | 19818 | 408 | 51317
James | 4138 | 2197 | 6091 | 656 | 82 | 336107 | 29253 | 25 | 372214
Jasperreports | 2380 | 4699 | 17575 | 702 | 226 | 643076 | 96000 | 1420 | 758999
Javacc | 269 | 689 | 3693 | 504 | 29 | 24936 | 17784 | 39 | 46985
JBoss | 7650 | 18239 | 42190 | 415 | 51 | 1084739 | 377357 | 1158 | 1505910
JEdit | 2410 | 4918 | 15464 | 630 | 134 | 434183 | 93605 | 74 | 544090
JExt | 2798 | 2804 | 7185 | 585 | 339 | 276503 | 42693 | 125 | 327430
JFreechart | 1152 | 3534 | 6708 | 660 | 88 | 154064 | 89284 | 849 | 251653
JGraph | 314 | 1350 | 2577 | 666 | 128 | 98119 | 22516 | 41 | 124047
JGgraphPad | 433 | 916 | 2550 | 599 | 10 | 62230 | 18777 | 75 | 84241
JGgraphT | 330 | 696 | 922 | 562 | 35 | 23808 | 11147 | 15 | 36489
JGroups | 1370 | 4029 | 14497 | 602 | 391 | 265886 | 89601 | 1560 | 372537
JMoney | 183 | 455 | 575 | 426 | 52 | 15377 | 5639 | 118 | 22187
Jpf | 143 | 443 | 522 | 558 | 21 | 14736 | 7054 | 20 | 22911
JRefactory | 4210 | 5132 | 18165 | 580 | 129 | 207911 | 116452 | 2633 | 345870
Log4J | 674 | 1028 | 2042 | 625 | 125 | 40206 | 15463 | 162 | 58623
Lucene | 4454 | 10332 | 11332 | 707 | 85 | 627683 | 233379 | 585 | 873771
Marauroa | 266 | 777 | 1228 | 547 | 33 | 53616 | 10681 | 148 | 66253
Maven | 1730 | 4455 | 3110 | 642 | 121 | 225017 | 46620 | 1242 | 276752
Megamek | 3225 | 8754 | 14974 | 600 | 321 | 346070 | 174680 | 3430 | 540075
Myfaces_core | 1922 | 5097 | 22247 | 312 | 121 | 619072 | 174790 | 790 | 817332
Nekohtml | 82 | 269 | 623 | 460 | 13 | 12987 | 3979 | 56 | 18118
PMD | 1263 | 3116 | 8818 | 616 | 50 | 109519 | 47664 | 543 | 167210
POI | 2276 | 8648 | 19463 | 903 | 771 | 476488 | 162045 | 792 | 660462
Proguard | 1043 | 1815 | 3203 | 646 | 23 | 115466 | 37221 | 6 | 156565
Quilt | 394 | 638 | 1075 | 386 | 13 | 16840 | 7488 | 170 | 25972
Sablecc | 251 | 886 | 4385 | 520 | 25 | 30840 | 19756 | 101 | 55627
Struts | 2598 | 6719 | 8878 | 616 | 57 | 231912 | 106513 | 253 | 348229
Sunflow | 227 | 670 | 1549 | 478 | 46 | 32251 | 20937 | 63 | 55324
Trove | 421 | 477 | 454 | 216 | 78 | 15507 | 5430 | 416 | 22101
Weka | 2147 | 6286 | 32258 | 604 | 1437 | 365535 | 195774 | 4118 | 599726
Xalan | 2174 | 4758 | 18362 | 844 | 232 | 330254 | 121685 | 1864 | 473241
Total | 74,486 | 169,763 | 418,433 | 27,888 | 7,431 | 9,686,813 | 3,380,493 | 33,704 | 13,554,762
Static Analysis Tools Warnings Detected in our projects. We obtained results
only for 47 projects out of the 112 contained in the Qualitas Corpus dataset,
applying the warnings defined by Better Code Hub, Checkstyle, Coverity Scan,
Findbugs, PMD, and SonarQube. Unfortunately, the used versions of Better Code
Hub and Coverity Scan were not able to analyze all the dataset. So, we
considered only the projects analyzed by all the six tools.
Table 2: RQ1. Detection Capability and Detected Warnings in the 47 projects.
Tool | Detection Capability | Detected warning
---|---|---
# rule | type | severity | # rule | # occurrences | type | severity
Better Code Hub | 10 | 3 | 3 | 8 | 27,888 | 3 | 3
Checkstyle | 173 | 14 | 4 | 88 | 9,686,813 | 12 | 2
Coverity | | | | 130 | 7,431 | 26 | 3
Findbugs | 424 | 9 | 4 | 255 | 33,704 | 9 | 3
PMD | 305 | 8 | 5 | 275 | 3,380,493 | 7 | 5
SonarQube | 413 | 3 | 5 | 180 | 418,433 | 3 | 5
Table 3: RQ1. The top-10 warnings detected by Better Code Hub, Checkstyle, Coverity Scan, FindBugs, PMD, and SonarQube. Id | Better Code Hub Detected Rule | #
---|---|---
| WRITE_CLEAN_CODE | 16,055
| WRITE_CODE_ONCE | 14,692
| WRITE_SHORT_UNITS | 6,510
| AUTOMATE_TESTS | 6,475
| WRITE_SIMPLE_UNITS | 6,362
| SMALL_UNIT_INTERFACES | 6,352
| SEPARATE_CONCERNS_IN_MODULES | 5,880
| COUPLE_ARCHITECTURE_COMPONENTS_LOOSELY | 2,807
Id | Checkstyle Detected Rule | #
| IndentationCheck | 3,997,581
| FileTabCharacterCheck | 2,406,876
| WhitespaceAroundCheck | 865,339
| LeftCurlyCheck | 757,512
| LineLengthCheck | 703,429
| RegexpSinglelineCheck | 590,020
| FinalParametersCheck | 406,331
| ParenPadCheck | 333,007
| NeedBracesCheck | 245,110
| MagicNumberCheck | 223,398
Id | Coverty Scan Detected Rule | #
| Dereference null return value | 1,360
| Dm: Dubious method used | 689
| Unguarded read | 556
| Explicit null dereferenced | 514
| Resource leak on an exceptional path | 494
| Dereference after null check | 334
| Resource leak | 301
| DLS: Dead local store | 293
| Missing call to superclass | 242
| Se: Incorrect definition of serializable class | 224
Id | FindBugs Detected Rule | #
| BC_UNCONFIRMED_CAST | 2,840
| DM_NUMBER_CTOR | 2,557
| BC_UNCONFIRMED_CAST_OF_RETURN_VALUE | 2,424
| DM_DEFAULT_ENCODING | 1,946
| RCN_REDUNDANT_NULLCHECK_OF_NONNULL_VALUE | 1,544
| DLS_DEAD_LOCAL_STORE | 1,281
| DM_FP_NUMBER_CTOR | 959
| SE_NO_SERIALVERSIONID | 944
| REC_CATCH_EXCEPTION | 887
| SE_BAD_FIELD | 878
Id | PMD Detected Rule | #
| LawOfDemeter | 505,947
| MethodArgumentCouldBeFinal | 374,159
| CommentRequired | 368,177
| LocalVariableCouldBeFinal | 341,240
| CommentSize | 153,464
| DataflowAnomalyAnalysis | 152,681
| ShortVariable | 136,162
| UselessParentheses | 128,682
| BeanMembersShouldSerialize | 111,400
| ControlStatementBraces | 110,241
Id | SonarQube Detected Rule | #
S1213 | The members of an interface declaration or class should appear in a pre-defined order | 30,888
S125 | Sections of code should not be ”commented out” | 30,336
S00122 | Statements should be on separate lines | 26,072
S00116 | Field names should comply with a naming convention | 25,449
S00117 | Local variable and method parameter names should comply with a naming convention | 23,497
S1166 | Exception handlers should preserve the original exceptions | 21,150
S106 | Standard outputs should not be used directly to log anything | 19,713
S1192 | String literals should not be duplicated | 19,508
S134 | Control flow statements ”if”,”for”,”while”,”switch” and ”try” should not be nested too deeply | 17,654
S1132 | Strings literals should be placed on the left side when checking for equality | 13,576
In total, the projects were infected by 936 warnings violated 13,554,762
times. 8 (out of 10) warnings were detected by Better Code Hub 27,888 times,
88 (out of 173) warnings were detected by Checkstyle 9,686,813 times, 130
warnings were detected by Coverity Scan 7,431 times, 255 (out of 424) warnings
were detected by Findbugs 33,704 time, 275 (out of 305) warnings were detected
by PMD 3,380,493 times, and 180 (out of 413) warnings were detected by
SonarQube 418.433 times (Table 1 and Table 2). It is important to note that in
Table 2, the detection capability is empty for Coverity. As mentioned earlier,
the full detection capability is only provided to clients and not on the
public API. We also computed how often warnings were violated by grouping them
based on type and severity. The full results of this additional analysis are
reported in our replication packagepackage.
Given the scope of warnings that were detected, our projects were affected by
all warnings that are detectable by Better Code Hub and by some warnings that
are detectable by Coverity Scan and SonarQube (Table 3). For the sake of
readability, we report only the Top-10 warnings detected in our projects by
the six tools. The complete list is available in the replication
packagepackage.
Finding 1. Our analysis provided a mapping of the warnings that currently
available tools can identify in software projects. Overall, the amount of
warnings detected by the six automated static analysis tools is significant
(936 warnings detected 13,554,762 times); hence, we can proceed with the
analysis of the remaining RQs.
Table 4: RQ2. Rule pairs that overlap at the “class-level”. Rule pairs | # occurrences | # max occurrences | %
---|---|---|---
Checkstyle - PMD | 4,872 | 3,380,493 | 0.144
SonarQube - PMD | 4,126 | 418,433 | 0.98
Findbugs - PMD | 3,161 | 33,704 | 9.378
SonarQube - Checkstyle | 1,495 | 418,433 | 0.357
Findbugs - Checkstyle | 1,265 | 33,704 | 3.753
BCH-PMD | 1,017 | 27,888 | 3.646
SonarQube - Findbugs | 849 | 33,704 | 2.518
BCH-SonarQube | 517 | 27,888 | 1.853
BCH-Checkstyle | 440 | 27,888 | 1.577
BCH-Findbugs | 235 | 27,888 | 0.842
Coverity - BCH | 117 | 7,431 | 1.574
Coverity - Checkstyle | 128 | 7,431 | 1.723
Coverity - Findbugs | 120 | 7,431 | 1.615
Coverity - PMD | 128 | 7,431 | 1.723
Coverity - SonarQube | 128 | 7,431 | 1.723
Total | 18,598 | 4,457,178 | 0.417
RQ2. What is the agreement among different Static Analysis Tools? Our second
research question focused on the analysis of the agreement between the static
analysis tools.
Agreement based on the overlapping at “class-level”. In order to include
Coverity Scan in this analysis, we first evaluated the detection agreement at
“class-level”, considering each class where the warnings detected by the other
five tools overlapped at 100% and where at least one warning of Coverity Scan
was violated in the same class.
To calculate the percentage of warnings pairs (columns “%”, Table 4) that
appear together, we checked the occurrences of both tools in our projects,
then we considered only the minimum value. For example, in Table 4,
calculating the percentage between Checkstyle - PMD warning pairs, we have
9,686,813 warnings Checkstyle detected and 3,380,493 PMD ones detected. The
combination of these warnings should be maximum 3,380,493 (the minimum value
between the two). We calculated the percentage considering the column “#
occorrences” and the column “# possible occorrences”.
Table 5: RQ2. Rule pairs that overlap at the 100% threshold considering the “line-level”. Rule pairs | # occurrences | # possible occurrences | %
---|---|---|---
Checkstyle - PMD | 4,872 | 3,380,493 | 0.144
SonarQube - PMD | 4,126 | 418,433 | 0.98
Findbugs - PMD | 3,161 | 33,704 | 9.378
SonarQube - Checkstyle | 1,495 | 418,433 | 0.357
Findbugs - Checkstyle | 1,265 | 33,704 | 3.753
BCH-PMD | 1,017 | 27,888 | 3.646
SonarQube - Findbugs | 849 | 33,704 | 2.518
BCH-SonarQube | 517 | 27,888 | 1.853
BCH-CheckSyle | 440 | 27,888 | 1.577
BCH-Findbugs | 235 | 27,888 | 0.842
Total | 17,977 | 4,430,023 | 0.4%
The warnings overlap at the “class-level” is always low, as reported in Table
4. This means that a piece of code violated by a warning detected by one tool
is almost never violated by another warning detected by another tool. In the
best case (Table 4), only 9.378% of the possible warning (Findbugs-PMD).
Moreover, we did find no warnings pair at “class-level” considering more that
two tools (e.g. Checkstyle-Findbugs-PMD).
For each warnings pair we computed the detection agreement at class level. For
the sake of readability, we report these results in Appendix A. Specifically,
the three tables (Table 9, Table 10, and Table 8) overview the detection
agreement of each warning pair, according to the procedure described in
Section 2.4. As further explained in the appendix, for reasoning of space we
only showed the 10 most recurrent pairs, putting the full set of results in
our replication package package. In these tables, the third and fourth columns
(eg. “#BCH pairs” and “# CHS pairs”, Table 9) report how many times a warning
instance from a tool exists with another one. The tools have separate values
for the number of co-occurrences as the number of instances differs as well,
for example, could be that a large rule contains several instances of the
comparison rule. Then for other rule this counts as one co-occurrence while
for the other rule each included rule grows the number. This makes sure the
agreement is between 0 and 1 for both rules. The remaining two columns report
the agreement of each tools considered in the warning pairs (eg. “#BCH Agr.”
and “# CHS Agr.”, Table 9). Results showed for all the warning pairs that the
agreement at “class-level” is very low, as none of the most recurrent warning
pairs agree well. The results also highlighted the difference in the
granularity of the warnings.
Agreement based on the overlapping at the “line-level”. Since we cannot
compare at “line-level” the warnings detected by Coverity Scan, we could only
consider the remaining five static analysis tools. Using the bottom-up
approach (Figure 1), several rule pairs were found according to the 100%, 90%,
80%, and 70% thresholds. Using the threshold of 100% which indicates that a
rule completely resides within the reference rule, we found 17,977 rule pairs,
as reported in Table 5. Using the thresholds of 90%, 80%, and 70% the
following rule pairs were found respectively: 17,985, 18,004, and 18,025
(Table 6). These warnings resided partially within the reference rule.
Table 6: RQ2. Rule pairs that overlap at the different thresholds, i.e., 90/80/70%, considering the “line-level”. Rule pairs | # occurrences
---|---
90% | 80% | 70% | possible
#(%) | #(%) | #(%)
Checkstyle - PMD | 4,872 (0.144) | 4,874 (0.144) | 4,876 (0.144) | 3,380,493
SonarQube - PMD | 4,126 (0.986) | 4,130 (0.987) | 4,139 (0.989) | 418,433
Findbugs - PMD | 3,167 (9.39) | 3,173 (9.41) | 3,173 (9.41) | 33,704
SonarQube - Checkstyle | 1,495 (0.357) | 1,496 (0.357) | 1,496 (0.357) | 418,433
Findbugs - Checkstyle | 1,265 (3.753) | 1,265 (3.753) | 1,265 (3.753) | 33,704
BCH-PMD | 1,017 (3.646) | 1,019 (3.646) | 1,024 (3.647) | 27,888
SonarQube - Findbugs | 849 (2.519) | 849 (2.519) | 849 (2.519) | 33,704
BCH-SonarQube | 517 (1.853) | 521 (1.868) | 522 (1.868) | 27,888
BCH-CheckSyle | 440 (1.577) | 440 (1.577) | 441 (1.578) | 27,888
BCH-Findbugs | 237 (0.849) | 237 (0.849) | 240 (0.860) | 27,888
Total | 18,004 (0.4) | 18,004 (0.4) | 18,025 (0.4) | 4,430,023
Similarly to what happened with the agreement at “class-level”, it is
important to note that the overlap at the “line-level” is always low. Results
show that, also in this case, only 9.378% of the possible rule occurrences are
detected in the same line by the same two tools (Findbugs and PMD). In
addition, also in this case we did find no pair warnings at “line-level”
considering more that two tools (e.g. Checkstyle-Findbugs-PMD).
When considering the agreement for each warning pair at “line-level‘, we could
not obtain any result because of computational reasons. Indeed, the analysis
at line-level of 936 warning types that have been violated 13,554,762 times
would have required a prohibitively expensive amount of time/space—according
to our estimations, it would have been taken up to 1.5 years—and, therefore,
we preferred excluding it.
Finding 2. The warnings overlapping among the different tools is very low
(less than 0.4%). The warning pairs Checkstyle - PMD as the lowest overlap
(0.144%) and Findbugs - PMD the highest one (9.378%). Consequently also the
detection agreement is very low.
RQ3. What is the precision of the static analysis tools?
In the context of our last research question, we focused on the precision of
the static analysis tools when employed for TD detection. Table 7 reports the
results of our manual analyses. As shown, the precision of most tools is quite
low, e.g., SonarQube has a precision of 18%, with the only exception of
CheckStyle whose precision is equal to 86%.
Table 7: RQ3. Precision of the considered SATs over the manually validated sample set of warnings. SAT | # warnings | # True Positives | Precision
---|---|---|---
Better Code Hub | 375 | 109 | 29%
Checkstyle | 384 | 330 | 86%
Coverity Scan | 367 | 136 | 37%
Findbugs | 379 | 217 | 57%
PMD | 384 | 199 | 52%
SonarQube | 384 | 69 | 18%
In general, based on our findings, we can first corroborate previous findings
in the field Antunes and Vieira (2009); Johnson et al. (2013); McLean (2012)
and the observations reported by Johnson et al. Johnson et al. (2013), who
found through semi-structured interviews with developers that the presence of
false positives represents one of the main issues that developers face when
using static analysis tools in practice. With respect to the qualitative
insights obtained by interviewing developers Johnson et al. (2013), our work
concretely quantifies the capabilities of the considered static analysis
tools.
Looking deeper into the results, we could delineate some interesting
discussion points. First, we found that for Better Code Hub and Coverity Scan
almost two thirds of the recommendations represented false alarms, while the
lowest performance was achieved by SonarQube. The poor precision of the tools
is likely due to the high sensitivity of the warnings adopted to search for
potential issues in the source code, e.g., threshold values that are too low
lead to the identification of false positive TD items. This is especially true
in the case of SonarQube: In our dataset, it outputs an average of 47.4
violations per source code class, often detecting potential TD in the code too
hastily.
A slightly different discussion is the one related to the other three static
analysis tools, namely PMD, Findbugs, and Checkstyle.
As for the former, we noticed that it typically fails when raising warnings
related to naming conventions. For instance, this is the case of the
’AbstractName’ warning: it suggests the developer that an abstract class
should contain the term Abstract in the name. In our validation, we discovered
that in several cases the recommendation was wrong because the contribution
guidelines established by developers explicitly indicated alternative naming
conventions. A similar problem was found when considering FindBugs. The
precision of the tool is 57% and, hence, almost half of the warnings were
labeled as false positives. In this case, one of the most problematic cases
was related to the ’BC_UNCONFIRMED_CAST’ warnings: these are raised when a
cast is unchecked and not all instances of the type casted from can be cast to
the type it is being cast to. In most cases, these warnings have been labeled
as false positives because, despite casts were formally unchecked, they were
still correct by design, i.e., the casts could not fail anyway because
developers have implicitly ensured that all of them were correct.
Finally, Checkstyle was the static analysis tool having the highest precision,
i.e., 86%. When validating the instances output by the tool, we realized that
the warnings raised are related to pretty simple checks in source code that
cannot be considered false positives, yet do not influence too much the
functioning of the source code. To make the reasoning clearer, let consider
the case of the ’IndentationCheck’ warning: as the name suggests, it is raised
when the indentation of the code does not respect the standards of the
project. In our sample, these warnings were all true positives, hence
contributing to the increase of the precision value. However, the
implementation of these recommendations would improve the documentation of the
source code but not dealing with possible defects or vulnerabilities. As such,
we claim that the adoption of Checkstyle would be ideal when used in
combination with additional static analysis tools.
To broaden the scope of the discussion, the poor performance achieved by the
considered tools reinforces the preliminary research efforts to devise
approaches for the automatic/adaptive configuration of static analysis tools
Nadi et al. (2014); Di Nucci et al. (2017) as well as for the automatic
derivation of proper thresholds to use when locating the presence of design
issues in source code Aniche et al. (2016); Fontana Arcelli et al. (2015b). It
might indeed be possible that the integration of those approaches into the
inner workings of the currently available static analysis tools could lead to
a reduction of the number of false positive items. In addition, our findings
also suggest that the current static analysis tools should not limit
themselves to the analysis of source code but, for instance, complementing it
with additional resources like naming conventions actually in place in the
target software system.
Finding 3. Most of the considered SATs suffer from a high number of false
positive warnings, and their precision ranges between 18% and 57%. The only
expection is Checkstyle (precision=86%), even though most of the warnings it
raises are related to documentation issues rather than functional problems
and, as such, its adoption should be complemented with other static analysis
tools.
## 4 Discussion and Implications
The results of our study provide a number of insights that can be used by
researchers and tool vendors to improve SATs. Specifically, these are:
There is no silver bullet. According to the results obtained in our study, and
specifically for RQ1, different SAT warnings are able to cover different
issues, and can therefore find different forms of source code quality
problems: Hence, we can claim that _there is no silver bullet that is able to
guarantee source code quality assessment on its own_. On the one hand, this
finding highlights that practitioners interested in detecting quality issues
in their source code might want to combine multiple SATs to find a larger
variety of problems. On the other hand, and perhaps more importantly, our
results suggest that the research community should have an interest in and be
willing to devise more advanced algorithms and techniques, e.g., ensemble
methods or meta-models Catolino and Ferrucci (2019); Catolino et al. (2019,
2018); Palomba et al. (2017), that can (1) combine the results from different
static analysis tools and (2) account for possible overlaps among the rules of
different SATs. This would allow the presentation of more complete reports
about the code quality status of the software systems to their developers.
Learning to deal with false positives. One of the main findings of our study
concerns with the low performance achieved by all static analysis tools in
terms of precision of the recommendations provided to developers (RQ3). Our
findings represent the first attempt to concretely quantify the capabilities
of the considered SATs in the field. Moreover, our study provides two
practical implications: (1) It corroborates and triangulates the qualitative
observations provided by Johnson et al. Johnson et al. (2013), hence
confirming that the real usefulness of static analysis tools is threatened by
the presence of false positives; (2) it supports the need for more research on
how to deal with false positives, and particularly on how to filter likely
false alarms Fontana Arcelli et al. (2015a) and how to select/prioritize the
warnings to be presented to developers Kim and Ernst (2007); Liang et al.
(2010); Lenarduzzi et al. (2019a). While some preliminary research efforts on
the matter have been made, we believe that more research should be devoted to
these aspects. Finally, our findings may potentially suggest the need for
further investigation into the effects of false positives in practice: For
example, it may be worthwhile for researchers to study what the maximum number
of false positive instances is that developers can deal with, e.g., they
should devise a critical mass theory for false positive ASAT warnings Oliver
and Marwell (2001) in order to augment the design of existing tools and the
way they present warnings to developers.
Complementing static analysis tools. The findings from our RQ1 and RQ2
highlight that most of the issues reported by the state-of-the-art static
analysis tools are related to rather simple problems, like the writing of
shorter units or the automation of software tests. These specific problems
could possibly be avoided if current static analysis tools would be
complemented with effective tools targeting (1) automated refactoring and (2)
automatic test case generation. In other words, our findings support and
strongly reinforce the need for a joint research effort among the communities
of source code quality improvement and testing, which are called to study
possible synergies between them as well as to devise novel approaches and
tools that could help practitioners complement the outcome provided by static
analysis tools with that of other refactoring and testing tools. For instance,
with effective refactoring tools, the number of violations output by SATs
would be notably reduced, possibly enabling practitioners to focus on the most
serious issues.
## 5 Threats to Validity
A number of factors might have influenced the results reported in our study.
This section discusses the main threats to validity and how we mitigated them.
Construct Validity. Threats in this category concern the relationship between
theory and observation. A first aspect is related to the dataset used. In our
work, we selected 112 projects from the Qualitas Corpus Terra et al. (2013),
which is one of the most reliable data sources in software engineering
research Tempero et al. (2010). Another possible threat relates to the
configuration of the SATs employed. None of the considered projects had all
the static analysis tools configured and so we had to manually introduce them;
in doing so, we relied on the default configuration of the tools since we
could not rely on different configurations given directly by the developers of
the projects. Nevertheless, it is important to point out that this choice did
not influence our analyses: indeed, we were interested in comparing the
capabilities of existing tools independently from their practical usage in the
considered systems. The problem of configuring the tools therefore does not
change the answers to our research questions.
Internal Validity. As for potential confounding factors that may have
influenced our findings, it is worth mentioning that some issues detected by
SonarQube were duplicated: in particular, in some cases the tool reported the
same issue violated in the same class multiple times. To mitigate this issue,
we manually excluded those cases to avoid interpretation bias; we also went
over the rules output by the other static analysis tools employed to check for
the presence of duplicates, but we did not find any.
External Validity. Threats in this category are concerned with the
generalization of the results. While we cannot claim that our results fully
represent every Java project, we considered a large set of projects with
different characteristics, domains, sizes, and architectures. This makes us
confident of the validity of our results in the field, yet replications
conducted in other contexts would be desirable to corroborate the reported
findings.
Another discussion point is related to our decision to focus only on open-
source projects. In our case, this was a requirement: we needed to access the
code base of the projects in order to configure the static analysis tools.
Nevertheless, open-source projects are comparable—in terms of source code
quality—to closed-source or industrial applications Lenarduzzi et al. (2019d);
hence, we are confident that we might have obtained similar results by
analyzing different projects. Nevertheless, additional replications would
provide further complementary insights and are, therefore, still desirable.
Finally, we limited ourselves to the analysis of Java projects, hence we
cannot generalize our results to projects in different programming languages.
Therefore, further replications would be useful to corroborate our results.
Conclusion Validity. With respect to the correctness of the conclusions
reached in this study, this has mainly to do with the data analysis processes
used. In the context of RQ1 and RQ3, we conducted iterative manual analyses in
order to build the taxonomy and study the precision of the tools,
respectively. While we cannot exclude possible imprecision, we mitigated this
threat by involving more than one inspector in each phase, who first conducted
independent evaluations that were later merged and discussed. Perhaps more
importantly, we made all data used in the study publicly available with the
aim of encouraging replicability, other than a further assessment of our
results.
In RQ2 we proceeded with an automatic mechanism to study the agreement among
the tools. As explained in Section 2.3, different static analysis tools might
possibly output the same warnings in slightly different positions of the
source code, e.g., highlighting the violation of a rule at two subsequent
lines of code. To account for this aspect, we defined thresholds with which we
could manage those cases where the same warnings were presented in different
locations. In this case, too, we cannot exclude possible imprecision; however,
we extensively tested our automated data analysis script. More specifically,
we manually validated a subset of rules for which the script indicated an
overlap between two tools with the aim of assessing whether it was correct or
not. This manual validation was conducted by one of the authors of this paper,
who took into account a random sample of 300 candidate overlapping rules. In
this sample, the author could not find any false positives, meaning that our
script correctly identified the agreement among tools. This further analysis
makes us confident of the validity of the findings reported for RQ2.
## 6 Related Work
Automated Static Analysis Tools (ASATs) are getting more popular Vassallo et
al. (2019); Lenarduzzi et al. (2019c) as they are becoming easier to use
Zampetti et al. (2017). The use of static analysis tools has been studied by
several researchers in the last years Wagner et al. (2005); Nagappan and Ball
(2005); Zheng et al. (2006); Nanda et al. (2010). In this section, we report
the relevant work on static analysis tools focusing on their usage Saarimäki
et al. (2019); Lenarduzzi et al. (2019b, 2020), warnings and the detected
problems Flanagan et al. (2002); Heckman and Williams (2011); Beller et al.
(2016).
Developers can use ASATs, such as SonarQubeSonar and CheckStyle323232
https://checkstyle.sourceforge.io/, to evaluate software source code, finding
anomalies of various kinds in the code Rutar et al. (2004); Tomas et al.
(2013). Moreover, ASATs are widely adopted in many research studies in order
to evaluate the code quality Johnson et al. (2013); Schnappinger et al.
(2019); Marcilio et al. (2019) and identify issues in the code Saarimäki et
al. (2019); Lenarduzzi et al. (2019b, 2020). Some studies demonstrated that
some rules detected by ASATs can be effective for identifying issues in the
code Zheng et al. (2006); Lenarduzzi et al. (2019a, 2020). However, evaluating
the performance in defect prediction, results are discordant comparing
different tools (e.g. FindBugsFindbugs and PMDPMD) Rahman et al. (2014).
Rutar et al. Rutar et al. (2004) compared five bug-finding tools for Java
(Bandera333333http://bandera.projects.cs.ksu.edu/,
ESC/Java2343434https://kindsoftware.com/products/opensource/ESCJava2/,
FindBugs353535 http://findbugs.sourceforge.net/, JLint363636
http://jlint.sourceforge.net/, and PMD373737 https://pmd.github.io/), that use
syntactic bug pattern detection, on five projects, including JBoss
3.2.3383838http://www.jboss.org/ and Apache Tomcat
5.019393939http://jakarta.apache.org/tomcat. They focused on the different
warnings (also called rules) provided by each tool, and their results
demonstrate some overlaps among the types of errors detected, which may be due
to the fact that each tool applies different trade-offs to generate false
positives and false negatives. Overall, they stated that warnings provided by
the different tools are not correlated with each other. Complementing the work
by Rutar et al. Rutar et al. (2004), we calculated the agreement of ASATs on
TD identification. In addition, we investigated the precision with which these
tools output warnings. Finally, we also investigated the types of TD items
that can actually be detected by existing ASATs.
Tomas et al. Tomas et al. (2013) performed a comparative analysis by means of
a systematic literature review. In total, they compared 16 Java code static
analysis tools, including JDepend404040https://github.com/clarkware/jdepend,
FindbugsFindbugs, PMDPMD, and SonarQubeSonar. They focused on internal quality
metrics of a software product and software tools of static code analysis that
automate measurement of these metrics. As results, they reported the tools’
detection strategies and what they detect. For instance, most of them automate
the calculation of internal quality metrics, the most common ones being code
smells, complexity, and code size Tomas et al. (2013). However, they did not
investigate agreement between the tools’ detection rules.
Avgeriou et al. Avgeriou et al. (2021) identified the available static
analysis tools for the Technical Debt detection. They compared features and
popularity of nine tools investigating also the empirical evidence on their
validity. Results can help practitioners and developers to select the suitable
tool against the other ones according to the measured information that
satisfied better their needs. However, they did not evaluate their agreement
and precision in the detection.
Focusing on developers’ perception on the usage of static analysis tools,
ASATs can help to find bugs Johnson et al. (2013). However, developers are not
sure about the usefulness of the rules Taibi et al. (2017); Vassallo et al.
(2018); Sadowski et al. (2018), they do pay attention to different rules
categories and priorities and remove violations related to rules with high
severity Vassallo et al. (2018) in order to avoid the possible risk of faults
Taibi et al. (2017). Moreover, false positives and the way in which the
warnings are presented, among other things, are barriers to their wider
adoption Johnson et al. (2013). Some studies highlighted the need to reduce
the number of detectable rules Muske et al. (2018); Bodden (2018) or summarize
them based on similarities Vassallo et al. (2018). ASATs are able to detect
many defects in the code. However, some tools do not capture all the possible
defect even if they could be detected by the tools Thung et al. (2015). Even
if some studies since the beginning of 2010 highlighted the need to better
clarify the precision of the tools, differentiating false positives from
actionable rules Liang et al. (2010); Ruthruff et al. (2008), many studies
deal with the many false positives produced by different tools, such as
FindBugsFindbugs Thung et al. (2015); Ayewah and Pugh (2010); Ayewah et al.
(2008), JLintJLint, PMDPMD, CheckStyleCheckstyle, and
JCSC414141http://jcsc.sourceforge.net Thung et al. (2015).
At the best of our knowledge, our work is the first that investigate in
details which source quality problems can actually be detected by the
available tools, trying to make a comparison based on the description, what is
their agreement, and what is the precision of their recommendations.
## 7 Conclusion
In this paper, we performed a large-scale comparison of six popular Static
Analysis Tools (Better Code Hub, CheckStyle, Coverity Scan, Findgugs, PMD, and
SonarQube) with respect to the detection of static analysis warnings. We
analyzed 47 Java projects from the Qualitas Corpus dataset, and derived
similar warnings that can be detected by the tools. We also compared their
detection agreement at “line-level” and “class-level”, and manually analyzed
their precision. To sum up, the contributions of this paper are:
1. 1.
A comparison of the warnings that can be detected by the tools (taxonomy),
which may be useful for researchers and tool vendors to understand which
warnings should be considered during refactoring;
2. 2.
An analysis of the agreement among the tools, which can inform tool vendors
about the limitations of the current solutions available the market;
3. 3.
The first quantification of the precision of six static analysis tools (Better
Code Hub, CheckStyle, Coverity Scan, Findgugs, PMD, and SonarQube).
Our future work includes an extension of this study with the evaluation of the
recall, and the in-vivo assessment of the tools.
## References
* Al Mamun et al. (2010) Al Mamun MA, Khanam A, Grahn H, Feldt R (2010) Comparing four static analysis tools for java concurrency bugs. In: Third Swedish Workshop on Multi-Core Computing (MCC-10)
* Aniche et al. (2016) Aniche M, Treude C, Zaidman A, Van Deursen A, Gerosa MA (2016) Satt: Tailoring code metric thresholds for different software architectures. In: International Working Conference on Source Code Analysis and Manipulation (SCAM), pp 41–50
* Antoine et al. (2014) Antoine JY, Villaneau J, Lefeuvre A (2014) Weighted krippendorff’s alpha is a more reliable metrics for multi-coders ordinal annotations: experimental studies on emotion, opinion and coreference annotation. In: 14th Conference of the European Chapter of the Association for Computational Linguistics”, pp 550–559
* Antunes and Vieira (2009) Antunes N, Vieira M (2009) Comparing the effectiveness of penetration testing and static code analysis on the detection of sql injection vulnerabilities in web services. In: International Symposium on Dependable Computing
* Avgeriou et al. (2021) Avgeriou P, Taibi D, Ampatzoglou A, Arcelli Fontana F, Besker T, Chatzigeorgiou A, Lenarduzzi V, Martini A, Moschou N, Pigazzini I, Saarimäki N, Sas D, Soares de Toledo S, Tsintzira A (2021) An overview and comparison of technical debt measurement tools. IEEE Software
* Ayewah and Pugh (2010) Ayewah N, Pugh W (2010) The google findbugs fixit. In: 19th International Symposium on Software Testing and Analysis, p 241–252
* Ayewah et al. (2008) Ayewah N, Hovemeyer D, Morgenthaler JD, Penix J, Pugh W (2008) Using static analysis to find bugs. IEEE Softw 25:22–29
* Beller et al. (2016) Beller M, Bholanath R, McIntosh S, Zaidman A (2016) Analyzing the state of static analysis: A large-scale evaluation in open source software. In: 23rd International Conference on Software Analysis, Evolution, and Reengineering (SANER), vol 1, pp 470–481
* Bodden (2018) Bodden E (2018) Self-adaptive static analysis. In: 40th International Conference on Software Engineering: New Ideas and Emerging Results, p 45–48
* Catolino and Ferrucci (2019) Catolino G, Ferrucci F (2019) An extensive evaluation of ensemble techniques for software change prediction. Journal of Software: Evolution and Process p e2156
* Catolino et al. (2018) Catolino G, Palomba F, De Lucia A, Ferrucci F, Zaidman A (2018) Enhancing change prediction models using developer-related factors. Journal of Systems and Software 143:14–28
* Catolino et al. (2019) Catolino G, Palomba F, Fontana Arcelli F, De Lucia A, Zaidman A, Ferrucci F (2019) Improving change prediction models with code smell-related information. arXiv preprint arXiv:190510889
* Di Nucci et al. (2017) Di Nucci D, Palomba F, Oliveto R, De Lucia A (2017) Dynamic selection of classifiers in bug prediction: An adaptive method. Transactions on Emerging Topics in Computational Intelligence 1(3):202–212
* Ernst et al. (2015) Ernst NA, Bellomo S, Ozkaya I, Nord RL, Gorton I (2015) Measure it? Manage it? Ignore it? Software practitioners and technical debt. Symposium on the Foundations of Software Engineering pp 50–60
* Flanagan et al. (2002) Flanagan C, Leino KRM, Lillibridge M, Nelson G, Saxe JB, Stata R (2002) Extended static checking for java. In: Conference on Programming Language Design and Implementation, p 234–245
* Fontana Arcelli et al. (2015a) Fontana Arcelli F, Ferme V, Zanoni M (2015a) Filtering code smells detection results. In: International Conference on Software Engineering-Volume 2, pp 803–804
* Fontana Arcelli et al. (2015b) Fontana Arcelli F, Ferme V, Zanoni M, Yamashita A (2015b) Automatic metric thresholds derivation for code smell detection. In: International workshop on emerging trends in software metrics, pp 44–53
* Fowler and Beck (1999) Fowler M, Beck K (1999) Refactoring: Improving the design of existing code. Addison-Wesley Longman Publishing Co, Inc
* Heckman and Williams (2011) Heckman S, Williams L (2011) A systematic literature review of actionable alert identification techniques for automated static code analysis. Information and Software Technology 53(4):363 – 387, special section: Software Engineering track of the 24th Annual Symposium on Applied Computing
* Johnson et al. (2013) Johnson B, Song Y, Murphy-Hill E, Bowdidge R (2013) Why don’t software developers use static analysis tools to find bugs? In: 2013 35th International Conference on Software Engineering (ICSE), IEEE, pp 672–681
* Kim and Ernst (2007) Kim S, Ernst MD (2007) Which warnings should i fix first? In: Joint meeting of the European software engineering conference and the ACM SIGSOFT symposium on The foundations of software engineering, ACM, pp 45–54
* Krippendorff (2018) Krippendorff K (2018) Content analysis: An introduction to its methodology. Sage publications
* Lenarduzzi et al. (2019a) Lenarduzzi V, Lomio F, Huttunen H, Taibi D (2019a) Are sonarqube rules inducing bugs? International Conference on Software Analysis, Evolution and Reengineering (SANER) Preprint: arXiv:190700376
* Lenarduzzi et al. (2019b) Lenarduzzi V, Martini A, Taibi D, Tamburri DA (2019b) Towards surgically-precise technical debt estimation: Early results and research roadmap. In: International Workshop on Machine Learning Techniques for Software Quality Evaluation, MaLTeSQuE 2019, pp 37–42
* Lenarduzzi et al. (2019c) Lenarduzzi V, Sillitti A, Taibi D (2019c) A survey on code analysis tools for software maintenance prediction. In: Software Engineering for Defence Applications - SEDA 2018, Springer-Verlag, Advances in Intelligent Systems and Computing (AISC), vol 925
* Lenarduzzi et al. (2019d) Lenarduzzi V, Tosi D, Lavazza L, Morasca S (2019d) Why do developers adopt open source software? past, present and future. In: Open Source Systems, Springer International Publishing, pp 104–115
* Lenarduzzi et al. (2020) Lenarduzzi V, Saarimäki N, Taibi D (2020) Some sonarqube issues have a significant but smalleffect on faults and changes. a large-scale empirical study. Journal of Systems and Software 170
* Liang et al. (2010) Liang G, Wu L, Wu Q, Wang Q, Xie T, Mei H (2010) Automatic construction of an effective training set for prioritizing static analysis warnings. In: Int. Conf. on Automated software engineering, pp 93–102
* Mantere et al. (2009) Mantere M, Uusitalo I, Roning J (2009) Comparison of static code analysis tools. In: Int. Conf. on Emerging Security Information, Systems and Technologies, pp 15–22
* Marcilio et al. (2019) Marcilio D, Bonifácio R, Monteiro E, Canedo E, Luz W, Pinto G (2019) Are static analysis violations really fixed? a closer look at realistic usage of sonarqube. In: 27th International Conference on Program Comprehension, p 209–219
* McLean (2012) McLean RK (2012) Comparing static security analysis tools using open source software. In: Int. Conf. on Software Security and Reliability, pp 68–74
* Muske et al. (2018) Muske T, Talluri R, Serebrenik A (2018) Repositioning of static analysis alarms. In: 27th International Symposium on Software Testing and Analysis, p 187–197
* Nadi et al. (2014) Nadi S, Berger T, Kästner C, Czarnecki K (2014) Mining configuration constraints: Static analyses and empirical results. In: International Conference on Software Engineering, ACM, pp 140–151
* Nagappan and Ball (2005) Nagappan N, Ball T (2005) Static analysis tools as early indicators of pre-release defect density. In: 27th International Conference on Software Engineering (ICSE), pp 580–586
* Nanda et al. (2010) Nanda MG, Gupta M, Sinha S, Chandra S, Schmidt D, Balachandran P (2010) Making defect-finding tools work for you. In: 32nd ACM/IEEE International Conference on Software Engineering - Volume 2, p 99–108
* Oliver and Marwell (2001) Oliver PE, Marwell G (2001) Whatever happened to critical mass theory? a retrospective and assessment. Sociological Theory 19(3):292–311
* Palomba et al. (2017) Palomba F, Zanoni M, Fontana Arcelli F, De Lucia A, Oliveto R (2017) Toward a smell-aware bug prediction model. Transactions on Software Engineering 45(2):194–218
* Rahman et al. (2014) Rahman F, Khatri S, Barr ET, Devanbu P (2014) Comparing static bug finders and statistical prediction. In: 36th International Conference on Software Engineering, p 424–434
* Runeson and Host (2009) Runeson P, Host M (2009) Guidelines for conducting and reporting case study research in software engineering. Empirical Softw Engg 14(2):131–164
* Rutar et al. (2004) Rutar N, Almazan CB, Foster JS (2004) A comparison of bug finding tools for java. In: Symposium on Software Reliability Engineering, pp 245–256
* Ruthruff et al. (2008) Ruthruff JR, Penix J, Morgenthaler JD, Elbaum S, Rothermel G (2008) Predicting accurate and actionable static analysis warnings: An experimental approach. In: 30th International Conference on Software Engineering, p 341–350
* Saarimäki et al. (2019) Saarimäki N, Lenarduzzi V, Taibi D (2019) On the diffuseness of code technical debt in open source projects. In: International Conference on Technical Debt (TechDebt 2019)
* Sadowski et al. (2018) Sadowski C, Aftandilian E, Eagle A, Miller-Cushon L, Jaspan C (2018) Lessons from building static analysis tools at google. Commun ACM 61(4):58–66
* Schnappinger et al. (2019) Schnappinger M, Osman MH, Pretschner A, Fietzke A (2019) Learning a classifier for prediction of maintainability based on static analysis tools. In: 27th International Conference on Program Comprehension, p 243–248
* Taibi et al. (2017) Taibi D, Janes A, Lenarduzzi V (2017) How developers perceive smells in source code: A replicated study. Information and Software Technology 92:223 – 235
* Tempero et al. (2010) Tempero E, Anslow C, Dietrich J, Han T, Li J, Lumpe M, Melton H, Noble J (2010) The qualitas corpus: A curated collection of java code for empirical studies. Asia Pacific Software Engineering Conference pp 336–345
* Terra et al. (2013) Terra RM, Miranda LF, Valente MT, da Silva Bigonha R (2013) Qualitas.class corpus: a compiled version of the qualitas corpus. ACM SIGSOFT Software Engineering Notes 38:1–4
* Thomas et al. (2016) Thomas TW, Lipford H, Chu B, Smith J, Murphy-Hill E (2016) What questions remain? an examination of how developers understand an interactive static analysis tool. In: Symposium on Usable Privacy and Security
* Thung et al. (2015) Thung F, Lucia, Lo D, Jiang L, Rahman F, Devanbu PT (2015) To what extent could we detect field defects? an extended empirical study of false negatives in static bug-finding tools. Automated Software Engg 22(4):561–602
* Tomas et al. (2013) Tomas P, Escalona MJ, Mejias M (2013) Open source tools for measuring the Internal Quality of Java software products. A survey. Computer Standards and Interfaces 36(1):244–255
* Vassallo et al. (2018) Vassallo C, Panichella S, Palomba F, Proksch S, Zaidman A, Gall HC (2018) Context is king: The developer perspective on the usage of static analysis tools. In: 25th International Conference on Software Analysis, Evolution and Reengineering (SANER), pp 38–49
* Vassallo et al. (2019) Vassallo C, Panichella S, Palomba F, Proksch S, Gall HC, Zaidman A (2019) How developers engage with static analysis tools in different contexts. Empirical Software Engineering pp 1–39
* Wagner et al. (2005) Wagner S, Jürjens J, Koller C, Trischberger P (2005) Comparing bug finding tools with reviews and tests. In: International Conference on Testing of Communicating Systems, p 40–55
* Wilander and Kamkar (2002) Wilander J, Kamkar M (2002) A comparison of publicly available tools for static intrusion prevention. In: Workshop on Secure IT Systems
* Zampetti et al. (2017) Zampetti F, Scalabrino S, Oliveto R, Canfora G, Di Penta M (2017) How open source projects use static code analysis tools in continuous integration pipelines. In: Int. Conf. on Mining Software Repositories, pp 334–344
* Zheng et al. (2006) Zheng J, Williams L, Nagappan N, Snipes W, Hudepohl JP, Vouk MA (2006) On the value of static analysis for fault detection in software. IEEE Transactions on Software Engineering 32(4):240–253
## Appendix A
In the following, we report more results achieved in the context of RQ2. For
each warnings pair we computed the detection agreement at class level as
reported in Table 9, Table 10, and Table 8, according to the process described
in Section 2.4. It is worth remarking that, for the sake of readability, we
only show the 10 most recurrent pairs. The results for the remaining
thresholds are reported in the replication packagepackage.
Table 8: RQ2. The 10 most recurrent rule pairs detected in the same class by
the considered SATs and their corresponding agreement values.
SonarQube | FindBugs | # SQ pairs | # FB pairs | SQ Agr. | FB Agr.
---|---|---|---|---|---
CommentedOutCodeLine | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 2 | 1 | 0.000 | 1.000
S1155 | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 1 | 1 | 0.000 | 1.000
S1186 | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 1 | 1 | 0.000 | 1.000
S135 | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 3 | 1 | 0.001 | 1.000
S1312 | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 1 | 1 | 0.000 | 1.000
complex_class | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 1 | 1 | 0.000 | 1.000
S1195 | DM_DEFAULT_ENCODING | 1 | 1 | 1.000 | 0.001
S1195 | EI_EXPOSE_REP | 1 | 1 | 1.000 | 0.001
S1301 | LG_LOST_LOGGER_DUE_TO_WEAK_REFERENCE | 1 | 1 | 0.001 | 1.000
S1148 | FI_NULLIFY_SUPER | 1 | 1 | 0.000 | 1.000
SonarQube | PMD | # SQ pairs | # PMD pairs | SQ Agr. | PMD Agr.
S1192 | FinalizeOnlyCallsSuperFinalize | 1 | 1 | 0.000 | 1.000
S2110 | JUnit4SuitesShouldUseSuiteAnnotation | 1 | 1 | 1.000 | 0.001
S2110 | AvoidCatchingGenericException | 1 | 5 | 1.000 | 0.000
S2110 | AvoidCatchingNPE | 1 | 4 | 1.000 | 0.011
S2110 | AvoidPrintStackTrace | 1 | 1 | 1.000 | 0.000
S2110 | CloseResource | 1 | 2 | 1.000 | 0.000
S2110 | CommentSize | 1 | 2 | 1.000 | 0.000
S2110 | DataflowAnomalyAnalysis | 1 | 5 | 1.000 | 0.000
S2110 | JUnit4TestShouldUseBeforeAnnotation | 1 | 1 | 1.000 | 0.001
S2110 | ShortVariable | 1 | 25 | 1.000 | 0.000
* •
BCH means Better Code Hub, and SQ means SonarQube, while Ag. means Agreement
Table 9: RQ2. The 10 most recurrent rule pairs detected in the same class by the considered SATs and their corresponding agreement values. BetterCodeHub | CheckStyle | # BCH pairs | # CHS pairs | BCH Agr. | CHS Agr.
---|---|---|---|---|---
WRITE_SIMPLE_UNITS | AvoidEscapedUnicodeCharactersCheck | 1 | 29859 | 0.000 | 0.529
WRITE_SHORT_UNITS | AvoidEscapedUnicodeCharactersCheck | 1 | 29859 | 0.000 | 0.529
WRITE_CODE_ONCE | AvoidEscapedUnicodeCharactersCheck | 1 | 29859 | 0.000 | 0.529
WRITE_SIMPLE_UNITS | OperatorWrapCheck | 1 | 60694 | 0.000 | 0.312
WRITE_SHORT_UNITS | OperatorWrapCheck | 1 | 60694 | 0.000 | 0.312
WRITE_CODE_ONCE | OperatorWrapCheck | 1 | 60694 | 0.000 | 0.312
WRITE_SHORT_UNITS | IllegalTokenTextCheck | 4 | 418 | 0.001 | 0.306
WRITE_SIMPLE_UNITS | IllegalTokenTextCheck | 4 | 418 | 0.001 | 0.306
AUTOMATE_TESTS | IllegalTokenTextCheck | 2 | 418 | 0.000 | 0.306
WRITE_CODE_ONCE | AtclauseOrderCheck | 30 | 210 | 0.002 | 0.306
BetterCodeHub | CoverityScan | # BCH pairs | # CS pairs | BCH Agr. | CS Agr.
AUTOMATE_TESTS | Exception leaked to user interface | 2 | 1 | 0.000 | 1.000
SEPARATE_CONCERNS_IN_MODULES | Unsafe reflection | 2 | 2 | 0.000 | 1.000
COUPLE_ARCHITECTURE_COMPONENTS_LOOSELY | AT: Possible atomicity violation | 2 | 1 | 0.001 | 1.000
WRITE_CLEAN_CODE | Use of hard-coded cryptographic key | 20 | 1 | 0.001 | 1.000
WRITE_SIMPLE_UNITS | Exception leaked to user interface | 2 | 1 | 0.000 | 1.000
WRITE_SHORT_UNITS | Exception leaked to user interface | 2 | 1 | 0.000 | 1.000
WRITE_CODE_ONCE | Exception leaked to user interface | 2 | 1 | 0.000 | 1.000
WRITE_SHORT_UNITS | Unsafe reflection | 2 | 2 | 0.000 | 1.000
WRITE_SIMPLE_UNITS | Unsafe reflection | 2 | 2 | 0.000 | 1.000
AUTOMATE_TESTS | Dead default in switch | 2 | 1 | 0.000 | 1.000
BetterCodeHub | FindBugs | # BCH pairs | # FB pairs | BCH Agr. | FB Agr.
WRITE_CODE_ONCE | ICAST_BAD_SHIFT_AMOUNT | 12 | 8 | 0.001 | 1.000
WRITE_SIMPLE_UNITS | SA_LOCAL_SELF_ASSIGNMENT_INSTEAD_OF_FIELD | 2 | 1 | 0.000 | 1.000
AUTOMATE_TESTS | FI_MISSING_SUPER_CALL | 3 | 1 | 0.000 | 1.000
SMALL_UNIT_INTERFACES | ICAST_BAD_SHIFT_AMOUNT | 12 | 8 | 0.002 | 1.000
WRITE_SHORT_UNITS | INT_VACUOUS_BIT_OPERATION | 6 | 1 | 0.001 | 1.000
WRITE_SHORT_UNITS | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 4 | 1 | 0.001 | 1.000
SEPARATE_CONCERNS_IN_MODULES | EQ_COMPARING_CLASS_NAMES | 2 | 1 | 0.000 | 1.000
AUTOMATE_TESTS | NP_ALWAYS_NULL_EXCEPTION | 2 | 1 | 0.000 | 1.000
WRITE_CLEAN_CODE | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 4 | 1 | 0.000 | 1.000
AUTOMATE_TESTS | ICAST_BAD_SHIFT_AMOUNT | 3 | 8 | 0.000 | 1.000
BetterCodeHub | PMD | # BCH pairs | # PMD pairs | BCH Agr. | PMD Agr.
SEPARATE_CONCERNS_IN_MODULES | FinalizeOnlyCallsSuperFinalize | 2 | 1 | 0.000 | 1.000
WRITE_CODE_ONCE | FinalizeOnlyCallsSuperFinalize | 4 | 1 | 0.000 | 1.000
SEPARATE_CONCERNS_IN_MODULES | AvoidMultipleUnaryOperators | 3 | 2 | 0.001 | 1.000
COUPLE_ARCHITECTURE_COMPONENTS_LOOSELY | AvoidMultipleUnaryOperators | 3 | 2 | 0.001 | 1.000
SMALL_UNIT_INTERFACES | FinalizeOnlyCallsSuperFinalize | 4 | 1 | 0.001 | 1.000
WRITE_CLEAN_CODE | InvalidLogMessageFormat | 2 | 1 | 0.000 | 1.000
AUTOMATE_TESTS | FinalizeOnlyCallsSuperFinalize | 2 | 1 | 0.000 | 1.000
AUTOMATE_TESTS | EmptyStatementBlock | 2 | 160 | 0.000 | 0.748
WRITE_SHORT_UNITS | EmptyStatementBlock | 2 | 160 | 0.000 | 0.748
WRITE_SIMPLE_UNITS | EmptyStatementBlock | 4 | 160 | 0.001 | 0.748
CheckStyle | CoverityScan | # CHS pairs | # CS pairs | CHS Agr. | CS Agr.
FinalParametersCheck | Unsafe reflection | 313 | 2 | 0.001 | 1.000
InvalidJavadocPositionCheck | IP: Ignored parameter | 4 | 2 | 0.000 | 1.000
NoWhitespaceAfterCheck | IP: Ignored parameter | 1 | 2 | 0.000 | 1.000
MissingJavadocMethodCheck | IP: Ignored parameter | 112 | 2 | 0.001 | 1.000
MagicNumberCheck | IP: Ignored parameter | 50 | 2 | 0.000 | 1.000
LineLengthCheck | IP: Ignored parameter | 6 | 2 | 0.000 | 1.000
JavadocVariableCheck | IP: Ignored parameter | 36 | 2 | 0.000 | 1.000
JavadocStyleCheck | IP: Ignored parameter | 84 | 2 | 0.001 | 1.000
JavadocMethodCheck | IP: Ignored parameter | 66 | 2 | 0.001 | 1.000
IndentationCheck | IP: Ignored parameter | 1152 | 2 | 0.000 | 1.000
CheckStyle | FindBugs | # CHS pairs | # FB pairs | CHS Agr. | FB Agr.
AvoidStarImportCheck | SA_FIELD_SELF_COMPUTATION | 8 | 2 | 0.000 | 1.000
FinalParametersCheck | NP_NONNULL_FIELD_NOT_INITIALIZED_IN_CONSTRUCTOR | 3 | 1 | 0.000 | 1.000
RedundantModifierCheck | IC_SUPERCLASS_USES_SUBCLASS_DURING_INITIALIZATION | 1 | 1 | 0.000 | 1.000
DesignForExtensionCheck | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 8 | 1 | 0.000 | 1.000
NeedBracesCheck | TQ_EXPLICIT_UNKNOWN_SOURCE_VALUE_REACHES_NEVER… | 28 | 1 | 0.000 | 1.000
JavadocVariableCheck | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 8 | 1 | 0.000 | 1.000
JavadocVariableCheck | IC_SUPERCLASS_USES_SUBCLASS_DURING_INITIALIZATION | 1 | 1 | 0.000 | 1.000
WhitespaceAroundCheck | BIT_IOR | 10 | 1 | 0.000 | 1.000
JavadocPackageCheck | IC_SUPERCLASS_USES_SUBCLASS_DURING_INITIALIZATION | 1 | 1 | 0.000 | 1.000
MissingJavadocMethodCheck | FI_MISSING_SUPER_CALL | 149 | 1 | 0.001 | 1.000
Table 10: RQ2. The 10 most recurrent rule pairs detected in the same class by the considered SATs and their corresponding agreement values. CheckStyle | PMD | # CHS pairs | # PMD pairs | CHS Agr. | PMD Agr.
---|---|---|---|---|---
FinalParametersCheck | AvoidMultipleUnaryOperators | 51 | 2 | 0.000 | 1.000
RegexpSinglelineCheck | AvoidMultipleUnaryOperators | 75 | 2 | 0.000 | 1.000
NoFinalizerCheck | FinalizeOnlyCallsSuperFinalize | 1 | 1 | 0.015 | 1.000
VisibilityModifierCheck | InvalidLogMessageFormat | 15 | 1 | 0.000 | 1.000
AbbreviationAsWordInNameCheck | FinalizeOnlyCallsSuperFinalize | 1 | 1 | 0.000 | 1.000
NoWhitespaceAfterCheck | AvoidMultipleUnaryOperators | 7 | 2 | 0.000 | 1.000
VariableDeclarationUsageDistanceCheck | AvoidMultipleUnaryOperators | 7 | 2 | 0.001 | 1.000
NeedBracesCheck | AvoidMultipleUnaryOperators | 90 | 2 | 0.000 | 1.000
JavadocParagraphCheck | FinalizeOnlyCallsSuperFinalize | 1 | 1 | 0.000 | 1.000
NonEmptyAtclauseDescriptionCheck | FinalizeOnlyCallsSuperFinalize | 10 | 1 | 0.000 | 1.000
CoverityScan | FindBugs | #CS pairs | #FB pairs | CS Agr. | FB Agr.
UCF: Useless control flow | BC_UNCONFIRMED_CAST_OF_RETURN_VALUE | 1 | 2 | 1.000 | 0.001
Dead default in switch | DLS_DEAD_LOCAL_STORE | 1 | 5 | 1.000 | 0.004
DLS: Dead local store | NP_ALWAYS_NULL_EXCEPTION | 3 | 1 | 0.010 | 1.000
UCF: Useless control flow | BC_UNCONFIRMED_CAST | 1 | 1 | 1.000 | 0.000
Unsafe reflection | NP_LOAD_OF_KNOWN_NULL_VALUE | 2 | 2 | 1.000 | 0.012
UCF: Useless control flow | UCF_USELESS_CONTROL_FLOW | 1 | 1 | 1.000 | 0.011
OGNL injection | RCN_REDUNDANT_NULLCHECK_OF_NONNULL_VALUE | 1 | 1 | 1.000 | 0.001
Failure to call super.finalize() | FI_NULLIFY_SUPER | 1 | 1 | 0.500 | 1.000
REC: RuntimeException capture | RV_RETURN_VALUE_OF_PUTIFABSENT_IGNORED | 1 | 1 | 0.013 | 1.000
USELESS_STRING: Useless/non-informative string… | FI_MISSING_SUPER_CALL | 5 | 1 | 0.250 | 1.000
CoverityScan | PMD | # CS pairs | # PMD pairs | CS Agr. | PMD Agr.
Dead default in switch | AvoidInstantiatingObjectsInLoops | 1 | 1 | 1.000 | 0.000
TLW: Wait with two locks held | ShortVariable | 1 | 10 | 1.000 | 0.000
TLW: Wait with two locks held | UseCorrectExceptionLogging | 1 | 3 | 1.000 | 0.010
TLW: Wait with two locks held | UseConcurrentHashMap | 1 | 5 | 1.000 | 0.002
TLW: Wait with two locks held | UnusedImports | 1 | 21 | 1.000 | 0.000
TLW: Wait with two locks held | UnnecessaryFullyQualifiedName | 1 | 2 | 1.000 | 0.000
TLW: Wait with two locks held | TooManyMethods | 1 | 1 | 1.000 | 0.000
TLW: Wait with two locks held | TooManyFields | 1 | 1 | 1.000 | 0.001
TLW: Wait with two locks held | StdCyclomaticComplexity | 1 | 2 | 1.000 | 0.000
TLW: Wait with two locks held | ProperLogger | 1 | 2 | 1.000 | 0.001
FindBugs | PMD | # FB pairs | # PMD pairs | FB Agr. | PMD Agr.
DMI_EMPTY_DB_PASSWORD | LawOfDemeter | 1 | 9 | 1.000 | 0.000
TESTING | OnlyOneReturn | 1 | 169 | 1.000 | 0.002
ICAST_BAD_SHIFT_AMOUNT | DataflowAnomalyAnalysis | 8 | 100 | 1.000 | 0.001
BIT_IOR | SignatureDeclareThrowsException | 1 | 14 | 1.000 | 0.000
BC_IMPOSSIBLE_DOWNCAST_OF_TOARRAY | ForLoopCanBeForeach | 1 | 6 | 1.000 | 0.000
LG_LOST_LOGGER_DUE_TO_WEAK_REFERENCE | MethodArgumentCouldBeFinal | 1 | 62 | 1.000 | 0.000
QF_QUESTIONABLE_FOR_LOOP | MethodArgumentCouldBeFinal | 1 | 6 | 1.000 | 0.000
HRS_REQUEST_PARAMETER_TO_HTTP_HEADER | MethodArgumentCouldBeFinal | 1 | 6 | 1.000 | 0.000
ICAST_BAD_SHIFT_AMOUNT | ConfusingTernary | 8 | 4 | 1.000 | 0.000
BIT_IOR | ConfusingTernary | 1 | 3 | 1.000 | 0.000
SonarQube | BetterCodeHub | # SQ pairs | # BCH pairs | SQ Agr. | BCH Agr.
S2275 | COUPLE_ARCHITECTURE_COMPONENTS_LOOSELY | 1 | 2 | 1.000 | 0.001
S2275 | AUTOMATE_TESTS | 1 | 2 | 1.000 | 0.000
S888 | COUPLE_ARCHITECTURE_COMPONENTS_LOOSELY | 2 | 4 | 0.667 | 0.001
S888 | WRITE_CLEAN_CODE | 2 | 10 | 0.667 | 0.001
S888 | WRITE_CODE_ONCE | 2 | 26 | 0.667 | 0.002
S888 | AUTOMATE_TESTS | 2 | 2 | 0.667 | 0.000
S888 | SEPARATE_CONCERNS_IN_MODULES | 2 | 2 | 0.667 | 0.000
ObjectFinalizeOverridenCalls SuperFinalizeCheck | AUTOMATE_TESTS | 1 | 3 | 0.500 | 0.000
S2232 | SEPARATE_CONCERNS_IN_MODULES | 1 | 1 | 0.500 | 0.000
S2232 | AUTOMATE_TESTS | 1 | 1 | 0.500 | 0.000
SonarQube | CheckStyle | # SQ pairs | # CHS pairs | SQ Agr. | CHS Agr.
S2252 | CommentsIndentationCheck | 1 | 1 | 1.000 | 0.000
S2200 | NeedBracesCheck | 2 | 12 | 1.000 | 0.000
S2123 | RedundantModifierCheck | 2 | 56 | 1.000 | 0.002
S2123 | InvalidJavadocPositionCheck | 2 | 4 | 1.000 | 0.000
S2123 | RegexpSinglelineCheck | 2 | 42 | 1.000 | 0.000
S2123 | WhitespaceAroundCheck | 2 | 22 | 1.000 | 0.000
S2200 | EqualsHashCodeCheck | 2 | 1 | 1.000 | 0.002
S2200 | FileTabCharacterCheck | 2 | 8 | 1.000 | 0.000
S2200 | FinalParametersCheck | 2 | 9 | 1.000 | 0.000
S2200 | IndentationCheck | 2 | 1 | 1.000 | 0.000
SonarQube | CoverityScan | # SQ pairs | # CS pairs | SQ Agr. | CS Agr.
S1244 | Unexpected control flow | 2 | 2 | 0.001 | 1.000
S00105 | Use of hard-coded cryptographic key | 1 | 1 | 0.000 | 1.000
S1125 | Dead default in switch | 4 | 1 | 0.001 | 1.000
S1126 | Dead default in switch | 2 | 1 | 0.001 | 1.000
S1132 | Dead default in switch | 1 | 1 | 0.000 | 1.000
S1149 | Dead default in switch | 1 | 1 | 0.000 | 1.000
S1151 | Dead default in switch | 8 | 1 | 0.001 | 1.000
S1172 | Dead default in switch | 4 | 1 | 0.002 | 1.000
S1213 | Dead default in switch | 26 | 1 | 0.001 | 1.000
S1226 | Dead default in switch | 7 | 1 | 0.001 | 1.000
|
# Extended phase space thermodynamics for Lovelock black holes with non-
maximally symmetric horizons
N. Farhangkhah1111email address<EMAIL_ADDRESS>and Z. Dayyani2 1
Department of Physics, Shiraz Branch, Islamic Azad University, Shiraz 71993,
Iran
2 Physics Department and Biruni Observatory, College of Sciences, Shiraz
University, Shiraz 71454, Iran
###### Abstract
We study thermodynamics and critical behaviors of higher-dimensional Lovelock
black holes with non-maximally symmetric horizons in the canonical ensemble of
extended phase space. The effects from non-constancy of the horizon of the
black hole via appearing two chargelike parameters in thermodynamic quantities
of third-order Lovelock black hole are investigated. We find that Ricci flat
black holes with nonconstant curvature horizon show critical behavior. This is
an interesting feature that is not seen for any kind of black hole in Einstein
or Lovelock gravity in the literature. We examine how various interesting
thermodynamic phenomena such as standard first-order small-large black hole
phase transition, a reentrant phase transition, or zeroth order phase
transition happen for Ricci flat, spherical, or hyperbolic black holes with
nonconstant curvature horizon depending on the values of Lovelock coefficient
and chargelike parameters. While for a spherical black hole of third order
Lovelock gravity with constant curvature horizon phase transition is observed
only for $7\leq d\leq 11$, for our solution criticality and phase transition
exist in every dimension. With a proper choice of the free parameters, a
large-small-large black hole phase transition occurs. This process is
accompanied by a finite jump of the Gibbs free energy referred to as a zeroth-
order phase transition. For the case $\kappa=-1$ a novel behavior is found for
which three critical points could exist.
###### pacs:
04.50.-h,04.20.Jb,04.70.Bw,04.70.Dy
## I Introduction
Einstein’s theory of general relativity, is the most successful theory of
gravity. At very high energies close to the Planck scale, higher order
curvature terms can no longer be neglected. Over the past years, motivated by
Superstring/M-theory string , higher dimensional gravity has been a prevailing
subject of study. The most famous theory that generalizes general relativity
in higher dimensions is Lovelock theory Lovelock . This theory keeps the order
of the field equations down to second order in derivatives. The Lovelock
Lagrangian consists of a sum of dimensionally extended Euler densities, and
the second-order field equations in this model give rise to the ghost-free
nature of the theory. The best testing ground for any modified theory of
gravity will be to search for black hole solutions. Investigating different
aspects of black hole physics has raised many interests.
Since it was found that black holes in classical general theory of relativity
obey laws that are analogous to the laws of thermodynamics, a lot of
attentions have been attracted to study the thermodynamic properties of the
black holes. First, Hawking proposed that the area of the black hole event
horizon never decreases. This is analogous to the second law of thermodynamics
with area of event horizon playing the role of entropy in thermodynamics.
After that the black hole entropy was introduced as proportional to the black
hole surface area in Planck units by Bekenstein Beken . Hawking also suggested
the temperature for the black hole Hawking , and proposed that like any other
hot body, the black hole radiates in analogy to the zeroth law of
thermodynamics. Thus the black hole can be considered to be a thermodynamic
object and study of black hole thermodynamics has provided interesting
information about the underlying structure of the spacetime. The discovery of
the famous Hawking-Page phase transition in Schwarzschild Anti de Sitter (AdS)
black holes Haw-Page was a begin to a wide area of researches in the context
of black hole thermodynamics.
In thermal systems, Van der Waals equation modifies the equation of state for
an ideal gas to one that approximates the behavior of real fluids. Through
studying the thermodynamics of charged black holes, it was found that the
first order small-large phase transition for charged black hole in AdS space
is quite similar to the liquid-gas change of phase occurring in Van der Waals
fluids Chamb1 ; Chamb2 . It was showed that $Q-\Phi$ diagram of the charged
black holes is similar to the $P-V$ diagram of the van der Waals system.
Researches in this regards, led to the assumption of an extended thermodynamic
phase space. In this framework, black hole thermodynamics is studied in the
asymptotic AdS space with a negative cosmological constant $\Lambda$. The
cosmological constant is represented as a pressure $(\Lambda=-P/8\pi),$ and
the thermodynamically conjugate variable is the thermodynamic volume Dolan1 ;
Kastor1 ; Cvetic . There exist good reasons why the variation of $\Lambda$
should be included in thermodynamic considerations Gibb ; Crei : Firstly, in
theories where physical constants such as Yukawa couplings, gauge coupling
constants, or the cosmological constant are not fixed a priori, but arise as
vacuum expectation values and hence can vary, it is natural to include
variations of these ‘constants’ in the thermodynamic formulae such as the
first law. In fact such ‘constants’ are typically to be thought of as the
values at infinity of scalar fields. Secondly, in the presence of a
cosmological constant the first law of black hole thermodynamics becomes
inconsistent with the Smarr relation Smarr unless the variation of $\Lambda$
is included in the first law and then the black hole mass $M$ is identified
with enthalpy rather than internal energy. One can also define other
thermodynamic quantities of black holes such as adiabatic compressibility,
specific heat at constant pressure, or even the speed of sound Dolan2 ; Dolan3
. In Karch it is shown that from simple field theoretic considerations a
universal Smarr formula emerges in holographic descriptions of black holes
with large $N$ duals and considering $\Lambda$ as a dynamical variable can be
understood from the point of view of the dual holographic field theory.
From another point of view, one can introduce a new gauge field in the
Lagrangian in which $\Lambda$ appears as the conserved charge associated with
the global part of the gauge symmetry of this gauge field. This formulation
brings a new perspective to $\Lambda$ as a parameter of the solution, and can
naturally contribute to the first law of black hole thermodynamics just like
other solution parameters like mass, entropy, angular momentum, electric
charge etc. Hajian
The analysis of the $P-V$ critical behaviors in the extended phase space has
been under study extensively and generalized to higher dimensional charged
black holes Kubiz ; Belhaj1 ; Spall , rotating black holes Chabab ; Alta1 ,
and black holes with Born-Infeld filed Guna . If a monotonic variation of any
thermodynamic quantity results in two (or more) phase transitions such that
the final state is macroscopically similar to the initial state, the system
undergoes the reentrant phase transition Nara ; Alta1 . The reentrant phase
transition was found for the four-dimensional Born-Infeld-AdS black hole
spacetimes, Guna , and for the black holes of third-order Lovelock gravity
Frassino . The situation is accompanied by a discontinuity in the global
minimum of the Gibbs free energy, referred to as a zeroth-order phase
transition and seen in superfluidity and superconductivity Maslov .
Corrections to black hole thermodynamics from higher-curvature terms in
Lovelock theory have revealed interesting features. In Lovelock theory the
entropy is given by a complicated relationship depending on higher-curvature
terms, and is no longer proportional to the area of the horizon Iyer . Both
the first law and the associated Smarr formula in an extended phase space were
obtained exploiting the Killing potential formalism Kastor2 . It is also
proposed to introduce a quantity conjugated to the Lovelock coefficient in the
first law of black hole thermodynamics and in the Smarr relation. $P-V$
criticality has been searched for Gauss-Bonnet Cai ; Zou ; Sheykhi ; Hendi
and third-order Lovelock Xu ; Mo ; Belhaj2 black holes. It was shown in Cai
that $P-V$ criticality can be observed for spherical Gauss-Bonnet black holes
even when charge is absent. For third order Lovelock gravity, it is found that
for $\kappa=1$ only for dimensions $7\leq d\leq 11$ critical points exists. In
all the cases mentioned above, for $\kappa=0$ there is no critical point in
the extended thermodynamic phase space. All the works mentioned above have
considered black hole with maximally symmetric horizons. In Ref. Dotti a
novel class of black hole solution is derived with nonconstant curvature
horizon in Lovelock gravity. The properties of this kind of black hole are
investigated in second and third order Lovelock gravity Maeda ; Farhang ;
Farhang2 ; Ohashi ; Ray2 . A noteworthy change when considering nonconstant
curvature horizon is that new chargelike parameters appear in the metric
function with the advantage of higher curvature terms, and modify the
properties of the black holes. Specially, Ricci flat solutions of this kind of
black holes show interesting features and this motivates us to investigate the
effects of non constancy of the horizon on the $P-V$ criticality of such black
holes.
Our paper is organized as follows. We begin in Sec. II by reviewing the
solutions of Lovelock gravity with non-constant curvature horizons and the
extended phase space thermodynamics in Lovelock theory is discussed. In Secs.
III, IV and V, specifying to black holes of 3-order Lovelock gravity, critical
behavior of the black hole with nonconstant horizon but constant sectional
curvatures $\kappa=0,\pm 1$ is studied, and we will see how criticality and
phase transition can occur in various cases. Also we will calculate the
critical exponents for these black holes and show that they are in the same
university class as the van der Waals gas. Our results are summarized in the
concluding section VI.
## II
Extended thermodynamics of nonmaximally symmetric Lovelock AdS black holes
To start, we consider the physical action describing Lovelock gravity which is
in the following form:
$I=\int_{\mathcal{M}}d^{d}x\sqrt{-g}\left(-2\Lambda+\sum_{p=1}^{\overline{p}}\alpha_{p}\mathcal{L}^{(p)}\right).$
(1)
where $\Lambda$ is the cosmological constant and $\alpha_{p}$’s are the
Lovelock coupling constants with the choose of $\alpha_{1}=1$. The Einstein
term $\mathcal{L}^{(1)}$ equals to $R$ and the second order Lovelock term is
$\mathcal{L}^{(2)}=R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$
Also $\mathcal{L}^{(3)}$ is the third order Lovelock Lagrangian which is
described as
$\displaystyle\mathcal{L}^{(3)}$ $\displaystyle=$ $\displaystyle
2R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\rho\tau}R_{\phantom{\rho\tau}{\mu\nu}}^{\rho\tau}+8R_{\phantom{\mu\nu}{\sigma\rho}}^{\mu\nu}R_{\phantom{\sigma\kappa}{\nu\tau}}^{\sigma\kappa}R_{\phantom{\rho\tau}{\mu\kappa}}^{\rho\tau}+24R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\nu\rho}R_{\phantom{\rho}{\mu}}^{\rho}$
(2)
$\displaystyle+3RR^{\mu\nu\sigma\kappa}R_{\sigma\kappa\mu\nu}+24R^{\mu\nu\sigma\kappa}R_{\sigma\mu}R_{\kappa\nu}+16R^{\mu\nu}R_{\nu\sigma}R_{\phantom{\sigma}{\mu}}^{\sigma}-12RR^{\mu\nu}R_{\mu\nu}+R^{3}.$
We start with the following metric
$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}\gamma_{ij}(z)dz^{i}dz^{j},$ (3)
which is a warped product of a $2$-dimensional Riemannian submanifold $M^{2}$
and an $(d-2)$-dimensional submanifold $K^{(d-2)}$. In this relation $i,j$ go
from $2,...,d-1$. The submanifold $K^{(d-2)}$ with the unit metric
$\gamma_{ij}$ is assumed to be an Einstein manifold with nonconstant curvature
but having a constant Ricci scalar being
$\widetilde{R}=\kappa(d-2)(d-3),\text{ \ }$ (4)
with $\kappa$ being the sectional curvature. For the tensor components of the
submanifold $K^{(d-2)}$ a tilde is used. The Ricci and Riemann tensors of the
Einstein manifold are
$\displaystyle\text{\ \ \ \ \ \ \ }\widetilde{R}_{ij}$ $\displaystyle=$
$\displaystyle\kappa(d-3)\gamma_{ij},$ (5)
$\displaystyle\widetilde{{R}}{{}_{ij}}^{kl}$ $\displaystyle=$
$\displaystyle\widetilde{{C}}{{}_{ij}}^{kl}+\kappa({\delta_{i}}^{k}{\delta_{j}}^{l}-{\delta_{i}}^{l}{\delta_{j}}^{k})\text{\
},$ (6)
where $\widetilde{{C}}_{ij}^{kl}$ is the Weyl tensor of $K^{(d-2)}$.
Choosing $\overline{p}=3$ in the field equation, for the metric (3) to be a
solution of field equations in third order Lovelock theory in vacuum, it would
suffice that the Weyl tensor of the horizon satisfies the following
constraints
$\sum_{kln}\widetilde{{C}}{{}_{ki}}^{nl}\widetilde{{C}}{{}_{nl}}^{kj}=\frac{1}{d}{\delta_{i}}^{j}\sum_{kmpq}\widetilde{{C}}{{}_{km}}^{pq}\widetilde{{C}}{{}_{pq}}^{km}\equiv\eta_{2}{\delta_{i}}^{j},$
(7)
$\displaystyle\sum_{klnmp}2(4\widetilde{{C}}{{}^{nm}}_{pk}\widetilde{{C}}{{}^{kl}}_{ni}\widetilde{{C}}{{}^{pj}}_{ml}-\widetilde{{C}}{{}^{pm}}_{ni}\widetilde{C}^{jnkl}\widetilde{C}_{klpm})$
(8) $\displaystyle=$
$\displaystyle\frac{2}{d}{\delta_{i}}^{j}\sum_{klmpqr}\left(4\widetilde{{C}}{{}^{qm}}_{pk}\widetilde{{C}}{{}^{kl}}_{qr}\widetilde{{C}}{{}^{pr}}_{ml}-\widetilde{{C}}{{}^{pm}}_{qr}\widetilde{C}^{rqkl}\widetilde{C}_{klpm}\right)$
$\displaystyle\equiv$ $\displaystyle\eta_{3}{\delta_{i}}^{j}.$
The first constraint was originally introduced by Dotti and Gleiser in Dotti
and is due to the Gauss-Bonnet term, and the second one which is dictated by
the third order Lovelock term, is obtained in Farhang . These two new
chargelike parameters appear in the metric function with the advantage of
higher curvature terms, and modify the properties of the black holes.
Considering the case
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{(d-3)(d-4)}$ (9) $\displaystyle\alpha_{3}$
$\displaystyle=$ $\displaystyle\frac{\alpha^{2}}{72\binom{n-2}{4}},$ (10)
the metric function $f(r)$ is given by Farhang
$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\kappa+\frac{r^{2}}{\alpha}\left\\{1+\left(j(r)\pm\sqrt{\gamma(r)+j^{2}(r)}\right)^{1/3}-\gamma(r)^{1/3}\left(j(r)\pm\sqrt{\gamma(r)+j^{2}(r)}\right)^{-1/3}\right\\},$
$\displaystyle j(r)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{3\alpha}{2}\left(-\frac{2\Lambda}{(d-1)(d-2)}-\frac{m}{r^{d-1}}+\frac{\alpha^{2}\hat{\eta}_{3}}{3r^{6}}\right),\text{
\ }$ $\displaystyle\text{\ \ \ }\gamma(r)$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{2}\hat{\eta}_{2}}{r^{4}}\right)^{3},$ (11)
where we define $\hat{\eta}_{2}=\frac{(d-6)!\eta_{2}}{(d-2)!}$ and
$\hat{\eta}_{3}=\frac{(d-8)!\eta_{3}}{(d-2)!}$ for simplicity. Note that
$\alpha$ and $\hat{\eta}_{2}$ are positive parameters, while $\hat{\eta}_{3}$
can be positive or negative relating to the metric of the spacetime. We should
also mention that in order to have the effects of non-constancy of the
curvature of the horizon in third order Lovelock gravity, $d$ should be larger
than seven, since the constants $\hat{\eta}_{2}$ and $\hat{\eta}_{3}$ are
evaluating on the $(d-2)$-dimensional boundary
In what follows we treat the (negative) cosmological constant $\Lambda$ as
thermodynamic pressure and its conjugate quantity as thermodynamic volume
Kubiz
$P=-\frac{\Lambda}{8\pi},$ (12) $V=(\frac{\partial M}{\partial
P})_{S,\alpha}=\frac{\Sigma_{d-2}r_{h}^{d-1}}{d-1},$ (13)
We obtain the parameter $M$ in terms of the horizon radius $r_{h}$ by solving
$f(r)=0$ as below
$M=\frac{(d-2)\Sigma_{d-2}}{16\pi}[\frac{16\pi
P}{(d-1)(d-2)}r_{h}^{d-1}+\kappa
r_{h}^{d-3}+\alpha(\kappa^{2}+\hat{\eta}_{2}]r_{h}^{d-5}+\frac{\alpha^{2}}{3}(\kappa^{3}+3\hat{\eta}_{2}\kappa+\hat{\eta}_{3})r_{h}^{d-7}]$
(14)
which is interpreted as enthalpy rather than the internal energy of the
gravitational system. $\Sigma_{d-2}$ denotes the volume of the
$(d-2)$-dimensional hypersurface $K^{(d-2)}.$The Hawking temperature of such
black holes, related with the surface gravity on the horizon $r=r_{h}$ is
given by Farhang
$T=\frac{\frac{16\pi P}{(d-2)}r_{h}^{6}+(d-3)\kappa
r_{h}^{4}+(d-5)(\hat{\eta}_{2}+\kappa^{2})\alpha
r_{h}^{2}+(d-7)\frac{\alpha^{2}}{3}(\hat{\eta}_{3}+3\kappa\hat{\eta}_{2}+\kappa^{3})}{4\pi
r_{h}[r_{h}^{4}+2\kappa\alpha
r_{h}^{2}+\alpha^{2}(\hat{\eta}_{2}+\kappa^{2})]},$ (15)
and entropy can be derived by making use of the Wald prescription as
$S=\frac{(d-2)\Sigma_{d-2}r_{h}^{d-2}}{4}[\frac{1}{(d-2)}+\frac{2\kappa\alpha}{r_{h}^{2}(d-4)}+\frac{\alpha^{2}(\hat{\eta}_{2}+\kappa^{2})}{r_{h}^{4}(d-6)}].$
(16)
The first law, in the extended phase space, yields
$dM=TdS+VdP+\mathcal{A}d\alpha$ (17)
where $\mathcal{A}$ denote the quantities conjugated to the Lovelock
coefficient and is calculated as below
$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle(\frac{\partial
M}{\partial\alpha})_{S,P}=\frac{(d-2)\Sigma_{d-2}r_{h}^{d-7}}{48\pi}\\{3r^{2}(\kappa^{2}+\widehat{\eta}_{2})+2\alpha(\kappa^{3}+3\kappa\widehat{\eta}_{2}+\widehat{\eta}_{3})$
(18)
$\displaystyle-\frac{(\frac{2\kappa}{(d-4)}r^{2}+\frac{2\alpha(\kappa^{2}+\widehat{\eta}_{2})}{(d-6)})}{r^{4}+2\kappa\alpha
r^{2}+\alpha^{2}(\kappa^{2}+\widehat{\eta}_{2})}[\frac{48\pi
r^{6}P}{d-2}+3\kappa(d-3)r^{4}+3(d-5)\alpha(\kappa^{2}+\widehat{\eta}_{2})r^{2}+(d-7)\alpha^{2}(\kappa^{3}+3\kappa\widehat{\eta}_{2}+\widehat{\eta}_{3})]\\}.$
These thermodynamical quantities satisfy the generalized Smarr relation in the
extended phase space
$M=\frac{d-2}{d-3}TS-\frac{2}{d-3}VP+\frac{2}{d-3}\mathcal{A}\alpha$ (19)
One can rearrange Eq. (15) to get thermodynamic equation of the state for the
black hole in the following form,
$P=\frac{T}{v}-\frac{\kappa(d-3)}{\pi(d-2)v^{2}}+\frac{32\kappa\alpha
T}{(d-2)^{2}v^{3}}-\frac{16\alpha(d-5)(\hat{\eta}_{2}+\kappa^{2})}{\pi(d-2)^{3}v^{4}}+\frac{256\alpha^{2}T(\hat{\eta}_{2}+\kappa^{2})}{(d-2)^{4}v^{5}}-\frac{256\alpha^{2}(d-7)(\kappa^{3}+3\kappa\hat{\eta}_{2}+\hat{\eta}_{3})}{3\pi(d-2)^{5}v^{6}},$
(20)
in which we have introduced the parameter
$v=\frac{4r_{h}}{(d-2)}$ (21)
as an effective specific volume.
If we consider the Van der Waals equation given as
$P=\frac{T}{v-b}-\frac{a}{v^{2}},$ (22)
and make use of the series expansion
$(1-\frac{b}{v})^{-1}=\sum_{n=0}(\frac{b}{v})^{n},$ (23)
it is well seen that if we keep the higher order terms in the Taylor series
expansion, the Van der Waals equation is in correspondence with the equation
of state (20) including the terms which appear from Lovelock gravity.
The critical point occurs when $P=P(v)$ has an inflection point, i.e.,
$\frac{\partial P}{\partial v}=0,\text{ \ \ \ \ \ \ \
}\frac{\partial^{2}P}{\partial v^{2}}=0$ (24)
and $\frac{\partial^{2}P}{\partial v^{2}}$ changes signs around each of the
solution. One of the best ways to investigate the critical behavior and phase
transition of the system is to plot the isotherm diagrams and compare with Van
der Waals liquid-gas system. In what follows we shall investigate the $P-v$
criticality of the black hole with nonconstant horizon but constant sectional
curvatures $\kappa=0,\pm 1.$
## III Critical behavior of Lovelock Ricci flat black holes with $\kappa=0$
For $\kappa=0$, the equation of state can be written as
$P=\frac{T}{v}-\frac{16\alpha(d-5)\hat{\eta}_{2}}{\pi(d-2)^{3}v^{4}}+\frac{256\alpha^{2}\hat{\eta}_{2}T}{(d-2)^{4}v^{5}}-\frac{256\alpha^{2}(d-7)\hat{\eta}_{3}}{3\pi(d-2)^{5}v^{6}},$
(25)
To obtain the critical points, if exist, we should solve Eqs. (24) which could
be simplified as
$x^{3}+qx+s=0\text{ \ \ \ \ \ \ \ \ \ \ \ ,\ }x=v^{2}\text{\ \ \ \ \ \ \ \ \ \
\ \ \ }$ (26)
with the parameters $q$ and $s$ given by
$\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{5}{3}\hat{\eta}_{2}\mathcal{B}^{2}-3\mathcal{C}^{2},\text{
\ \ \ \ \
}s=\frac{35(d-7)\hat{\eta}_{3}}{27(d-5)}\mathcal{B}^{3}+2\mathcal{C}^{3}$
$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\frac{16\alpha}{(d-2)^{2}},\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
}\mathcal{C}=\frac{40\hat{\eta}_{3}(d-7)\alpha}{9\hat{\eta}_{2}(d-2)^{2}(d-5)}.$
(27)
As we mentioned before, $\hat{\eta}_{2}$ is a positive parameter but
$\hat{\eta}_{3}$ can take an arbitrary positive or negative value. It is well
known that the multiplication of three roots of Eq. (26) is proportional to
$-s$ which can be shown through a straightforward calculation to be
proportional to $-\hat{\eta}_{3}$. Thus for negative values of
$\hat{\eta}_{3},$ the multiplication of three roots of Eq. (26) is positive
and thus there exists at least one positive real root for this equation. This
fact leads to the existence of at least one real root for Eqs. (24). One
should note that in the expression for $P$ which is given by the relation
(25), the last term is dominant, as $v\rightarrow 0,$ which is positive for
negative values of $\hat{\eta}_{3}.$ The isotherm diagrams $P-v$ for a Ricci
flat black hole are displayed in Fig. 1 for two values of $d$. The plots
obviously show a first order phase transition in the system for $T<T_{c}$
which is really similar to the Van der Waals liquid-gas system. As it is seen,
for a fixed temperature lower than the critical one, in the small radius
region and large one the compression coefficient is positive, which shows
stable phases. Between them there is an unstable phase. therefore a
small/large black hole phase transition occurs.
It is worthwhile to emphasis that such a phase transition is never seen for
$\hat{\eta}_{2}=\hat{\eta}_{3}=0$. As is well known any planar black holes
with constant curvature horizon of Einstein or higher-order Lovelock gravity
in an arbitrary number of spacetime dimensions in vacuum or even in the
existence of Maxwell, Born-Infeld, or dilaton fields do not admit critical
behavior. This interesting behavior is due to the existence of
$\hat{\eta}_{2}$ and $\hat{\eta}_{3}$ which appear as a result of the
nonconstancy of the horizon and makes drastic changes to the equation of state
in the case $\kappa=0.$ Also there is no criticality for $\hat{\eta}_{2}\neq
0$ and $\hat{\eta}_{3}=0$. This reveals the effect of higher-curvature terms
in third-order Lovelock gravity, which cause novel changes in the properties
of the spacetime.
(a) $\alpha=1$, $\hat{\eta}_{2}=1$, $\hat{\eta}_{3}=-1$ and $d=11$
(b) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=-5$ and $d=8$
Figure 1: $P-v$ diagram of Lovelock black holes with $\kappa=0$ .
The solutions to Eq. (26) could be written as
$\displaystyle v_{c}$ $\displaystyle=$
$\displaystyle\sqrt{x}=[\sqrt[3]{-\frac{s}{2}+\sqrt{(\frac{q}{3})^{3}+(\frac{s}{2})^{2}}}+\sqrt[3]{-\frac{s}{2}-\sqrt{(\frac{q}{3})^{3}+(\frac{s}{2})^{2}}}]^{\frac{1}{2}},$
(28) $\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{(d-7)\hat{\eta}_{3}}{2(d-2)\pi\hat{\eta}_{2}v_{c}}+\frac{3}{80}\frac{(d-5)(d-2)}{\alpha\pi}v_{c}.$
(29)
For positive values of $\hat{\eta}_{3},$ the relation (28) makes a limitation
on $\hat{\eta}_{3}$ that depends on $\alpha$, $\hat{\eta}_{2}$ and number of
dimensions $d$ as
$\quad\hat{\eta}_{3}<\frac{2\sqrt{3}(d-5)\hat{\eta}_{2}^{3/2}}{5(d-7)}.$ (30)
For positive values of $\hat{\eta}_{3}$, satisfying the above constraint, Eq.
(26), has at least one real root introduced as $v_{c}$. To witness the $P-v$
criticality behavior we plot the $P-v$ diagram in Fig. 2. $P-v$ diagrams in
diverse dimensions are the same and so without loss of generality we present
them for $d=8$. Fig. 2(b) show that for some values of $\hat{\eta}_{3}$, in
every dimension, there are two critical points, one with negative (unphysical)
and the other with positive pressure. The isothermal plots in this case are
quite similar with the $P-v$ diagram of Born-Infeld-AdS black holes Guna . As
$\hat{\eta}_{3}$ is increased up to the limiting value obtained in the
relation (30), both values of critical pressure become positive as is seen in
2(a).
(a) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=5$ and $d=8$
(b) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=2$ and $d=8$
Figure 2: $P-v$ diagram of Lovelock black holes with positive $\hat{\eta}_{3}$
for $\kappa=0$
### III.1 Gibbs free energy
One of the most important items that helps us to determine phase transition of
a system refers to study its thermodynamic potential. Gibbs free energy
generally is computed from the Euclidean action with appropriate boundary term
Dayyanipld while the lowest Gibbs free energy is associated with global
stable state. In the canonical ensemble and extended phase space,
thermodynamic potential closely associates with the Gibbs free energy
$G=M-TS$. As is well known, to have a physical behavior, the second order
derivative of Gibbs energy with respect to the temperature should be negative
to have a positive heat capacity. Zeroth order phase transition occurs in the
system when Gibbs energy is discontinuous. This behavior was formerly observed
in superfluidity and superconductivity maslov . Any discontinuity in fist
(second) order derivatives of Gibbs energy leads to a first (second) order
phase transition in the system. We calculate the Gibbs free energy of the
black hole to elaborate the phase transition of the system as below,
$\displaystyle G$ $\displaystyle=$ $\displaystyle
G(P,T)=-\frac{Pr_{h}^{d-1}[5\alpha^{2}(d-2)\eta_{2}+(d-6)r_{h}^{4}]}{(d-6)(d-2)(d-1)\left(\alpha^{2}\eta_{2}+r_{h}^{4}\right)}$
(31)
$\displaystyle+\frac{\eta_{3}r_{h}^{d-7}[\alpha^{4}(d-2)\left(d^{2}-3d+2\right)\eta_{2}+5\alpha^{2}(d-6)\left(d^{2}-3d+2\right)r_{h}^{4})}{48\pi(d-6)(d-2)(d-1)\left(\alpha^{2}\eta_{2}+r_{h}^{4}\right)}$
$\displaystyle+\frac{r_{h}^{d-7}\left(9\alpha(d-6)\left(d^{2}-3d+2\right)\eta_{2}r_{h}^{6}-3\alpha^{3}(d-2)\left(d^{2}-3d+2\right)\eta_{2}^{2}r_{h}^{2}\right)}{48\pi(d-6)(d-2)(d-1)\left(\alpha^{2}\eta_{2}+r_{h}^{4}\right)}$
where $r_{h}$ should be understood as a function of pressure and temperature
via the equation of state. The Gibbs free energy corresponding to Fig. 1 is
depicted in Fig. 3. One can note that for negative $\eta_{3}$, the Gibbs
energy has a smooth behavior as a function of $T$, for $P>P_{c}$ whereas for
$P<P_{c}$ it exhibits the small/large black hole first order phase transition
and a usual swallowtail shape as we expect which is the characteristic of van
der Waals fluid.
(a) $\alpha=1$, $\hat{\eta}_{2}=1$, $\hat{\eta}_{3}=-1$ and $d=11$
(b) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=-5$ and $d=8$
Figure 3: Gibbs diagrams of Lovelock black holes for $\hat{\eta}_{3}<0$ and
$\kappa=0$
The corresponding Gibbs diagram to Fig. 2 with two critical points is
displayed in Fig. 4. In Fig. 4(a) when $P\leq P_{c1}$ the lower (upper) branch
is thermodynamically stable(unstable). There is only one physical branch and
Gibbs energy shows no phase transition in the system. However, a first order
phase transition may happen in the range of $P_{c1}<P<P_{c2}$ as shown with
solid red line in Fig. 4(a). In Fig. 4(b) we can see a first order phase
transition similar to Van der Waals fluid in the range $0<P<P_{c2}$ and thus
the second critical point (with $P=P_{c2}$) is physical. The critical point
with negative pressure does not globally minimize the Gibbs energy and hence
is not physical. Although the equation of state leads to one or two critical
point in this case, investigating the Gibbs diagrams represents only one
physical critical point.
(a) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=5$ and $d=8$. One has two
critical points with positive pressure while only the one with $P=P_{c2}$
corresponds to the first order phase transition between small and large black
holes. The other does not globally minimize the Gibbs free energy and hence is
unphysical.
(b) $\alpha=1$, $\hat{\eta}_{2}=2$, $\hat{\eta}_{3}=2$ and $d=8$. There are
two critical points with positive and negative pressure. As one expects only
$T=T_{c2}$ with positive pressure corresponds to the first order phase
transition. The other critical point ($T=T_{c1}$) is unphysical
Figure 4: Gibbs diagrams of Lovelock black holes for $\hat{\eta}_{3}>0$ and
$\kappa=0$. The curves are shifted and rescaled for more clarity.
### III.2 Critical exponents
Critical exponent characterizes the behavior of physical quantities in the
close vicinity of the critical point. We proceed to calculate the critical
exponents $\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$ and
$\delta^{\prime}$ for the phase transition of a $d$-dimensional Lovelock black
hole. In order to calculate the first critical exponent $\alpha^{\prime}$, we
consider the entropy $S$ given by Eq. (16) as a function of $T$ and $v$.
Making use of Eq.(21) we have
$S=S\left(T,v\right)=4^{1-d}(d-2)^{d-2}v^{d-2}+\frac{\alpha^{2}4^{5-d}(d-2)^{d-5}v^{d-6}\hat{\eta}_{2}}{d-6}$
(32)
It is clear that entropy does not depend on the temperature in this relation
and hence $C_{V}=0$. This indicates that relative critical exponent will be
zero
$C_{V}\propto\left(\frac{T}{T_{c}}-1\right)^{\alpha^{\prime}}\Rightarrow\alpha^{\prime}=0.$
(33)
To obtain the other exponents, we define the reduced thermodynamic variables
as
$p\equiv\frac{P}{P_{c}},\quad\nu\equiv\frac{v}{v_{c}},\quad\tau\equiv\frac{T}{T_{c}},$
and expansion parameters as
$t=\tau-1,\quad\omega=\nu-1=\dfrac{v}{v_{c}}-1.$ (34)
Then we can make Taylor expansion for the equation of state Eq.(25) as
$p=1+At-Bt\omega-C\omega^{3}+O\left(t\omega^{2},\omega^{4}\right),$ (35)
where $A$, $B$ and $C$ are constants depending on $d$, $\alpha$,
$\hat{\eta}_{2}$ and $\hat{\eta}_{3}$.
Denoting the volume of small and large black holes by $\omega_{s}$ and
$\omega_{l}$, respectively, differentiating Eq. (35) with respect to $\omega$
at a fixed $t<0$, and applying the Maxwell’s equal area law Spall one obtains
$\displaystyle p$ $\displaystyle=$ $\displaystyle 1+At-
Bt\omega_{l}-C\omega_{l}^{3}=1+At-Bt\omega_{s}-C\omega_{s}^{3}$ $\displaystyle
0$ $\displaystyle=$ $\displaystyle-
P_{c}\int_{\omega_{l}}^{\omega_{s}}\omega\left(Bt+3C\omega^{2}\right)d\omega,$
(36)
which leads to the unique non-trivial solution
$\omega_{l}=-\omega_{s}=\sqrt{-\frac{Bt}{C}}.$ (37)
Thus, the exponent $\beta^{\prime}$, which describes the behaviour of the
order parameter $\eta=v_{c}\left(\omega_{l}-\omega_{s}\right)$ on a given
isotherm, may be calculated through the use of Eq. (37) as:
$\eta=2v_{c}\omega_{l}=2\sqrt{-\frac{Bt}{C}}\quad\Longrightarrow\quad\beta^{\prime}=\frac{1}{2}.$
(38)
To calculate the exponent $\gamma^{\prime}$, we may determine the behavior of
the isothermal compressibility near the critical point
$\kappa_{T}=-\frac{1}{V}\frac{\partial V}{\partial
P}\Big{|}_{T}\varpropto\left|t\right|^{-\gamma^{\prime}}.$
Since $dv/d\omega=v_{c}$, the isothermal compressibility near the critical
point reduces to
$\kappa_{T}=-\frac{1}{V}\frac{\partial V}{\partial
P}\Big{|}_{T}\varpropto\frac{V_{c}}{BP_{c}t},$ (39)
which shows that $\gamma^{\prime}=1$. Finally the ‘shape‘ of the critical
isotherm $t=0$ is given by (35)
$p-1=-C\omega^{3},$ (40)
which indicates that $\delta^{\prime}=3$.
The critical exponents associated with this type of Lovelock black holes are
independent of metric parameters and the dimension of the spacetime. This is
consistent with the results of mean field theory that believe the critical
exponents are universal and do not depend on the details of the physical
system.
## IV Critical behavior of Lovelock spherical black holes with $\kappa=1$
When the topology of the black hole horizon is spherical the equation of state
is in the form
$P=\frac{T}{v}-\frac{d-3}{\pi(d-2)v^{2}}+\frac{32\alpha
T}{(d-2)^{2}v^{3}}-\frac{16\alpha(d-5)\left[\hat{\eta}_{2}+1\right]}{\pi(d-2)^{3}v^{4}}+\frac{256\alpha^{2}\left[\hat{\eta}_{2}+1\right]T}{(d-2)^{4}v^{5}}-\frac{256\alpha^{2}(d-7)\left[3\hat{\eta}_{2}+\hat{\eta}_{3}+1\right]}{3\pi(d-2)^{5}v^{6}}$
(41)
In different types of black holes, first order phase transition occurs for
black holes which have spherically symmetric horizon. Therefor it is important
to investigate this case $(\kappa=1)$ and compare our results with the other
types of black holes.
It is shown in Xu , that for third order Lovelock black holes with $\kappa=1$
and constant curvature horizon there exist two critical points for $8\leq
d\leq 11$ and no critical point exists for $d>11$.
For our solution with nonconstant curvature horizon, Eqs. (24) could be
simplified as a polynomial of degree 4 as:
$v^{4}+bv^{3}+cv^{2}+dv+e=0$ (42)
with
$\displaystyle b$ $\displaystyle=$ $\displaystyle
96\frac{\hat{\eta}_{2}(d-5)-2}{d-3}$ $\displaystyle c$ $\displaystyle=$
$\displaystyle
256\frac{5(d-7)\hat{\eta}_{3}+12(d-10)\hat{\eta}_{2}+2(d-25)}{d-3}$
$\displaystyle d$ $\displaystyle=$ $\displaystyle
8192\frac{9(d-7)\hat{\eta}_{3}-5(d-5)\hat{\eta}_{2}^{2}+(17d-139)\hat{\eta}_{2}+2(2d-19)}{d-3}$
$\displaystyle e$ $\displaystyle=$ $\displaystyle
327680(d-7)\frac{[(\hat{\eta}_{2}+1)\hat{\eta}_{3}+1]+3\hat{\eta}_{2}^{2}+4\hat{\eta}_{2}}{d-3}.$
(43)
A well-known calculation yields
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
b^{2}c^{2}d^{2}-4b^{2}c^{3}e-4b^{3}d^{3}+18b^{3}cde-27b^{4}e^{2}-4c^{3}d^{2}$
(44) $\displaystyle+16c^{4}e+18bcd^{3}-80bc^{2}de-6b^{2}d^{2}e+144b^{2}ce^{2}$
$\displaystyle-27d^{4}+144cd^{2}e-128c^{2}e^{2}-192bde^{2}+256e^{3}.$
For the equation (42) to have solution, $\Delta$ should be positive, which
again makes a constraint on the parameter $\hat{\eta}_{3}$ relating to
$\hat{\eta}_{2},$ $\alpha$ and $d.$ On the other hand, a look at the relation
(41) reveals that the dominant term is the last one, which is positive for
$\hat{\eta}_{3}\leq-(3\hat{\eta}_{2}+1)$. If $\hat{\eta}_{3}$ is chosen in
such a way that this inequality and also $\Delta>0$ hold, the pressure tends
to $+\infty$ as $v\rightarrow 0$ and Eq. (42) has one real root. Thus the
system demonstrates Van der Waals behavior. For the values of $\hat{\eta}_{3}$
satisfying $\hat{\eta}_{3}>-(3\hat{\eta}_{2}+1)$, and the constraint
$\Delta>0$, the pressure tends to $-\infty$ as $v\rightarrow 0$ and we may
have up to two critical points. Various physical situations are summarized in
table 1 for three values of $d$.
Table 1: Critical values in different dimensions for $\kappa=1$
$\ \ \ d\ \ \ $ | $\ \ \alpha\ \ $ | $\ \ \eta_{2}\ \ $ | $\ \ \ \eta_{3}\ \ \ \ $ | $\ \ \ \ \ v_{c1}\ \ \ \ \ $ | $\ \ \ \ \ \ T_{c1}\ \ \ \ \ $ | $\ \ \ \ \ P_{c1}\ \ \ \ \ $ | $\ \ \ \ \ v_{c2}\ \ \ \ $ | $\ \ \ \ \ T_{c2}\ \ \ $ | $\ \ \ \ \ P_{c2}\ \ $
---|---|---|---|---|---|---|---|---|---
$\ \ \ 8$ | $\ \ 1$ | $\ \ 1$ | $\ \ 0.8$ | $\ \ 0.5632$ | $\ \ 0.2027$ | $\ \ \ 0.0133$ | $\ \ 1.1376$ | $\ \ 0.2131$ | $\ \ 0.0630$
$\ \ \ 8$ | $\ \ 1$ | $\ \ 1$ | $-0.5$ | $\ \ 0.4369$ | $\ \ 0.1823$ | $\ -0.1481$ | $\ \ 1.2124$ | $\ \ 0.2101$ | $\ \ 0.0599$
$\ \ \ 8$ | $\ \ 1$ | $\ \ 1$ | $\ -5$ | $\ \ 1.3750$ | $\ \ 0.2023$ | $\ \ \ 0.0531$ | — | — | —
$\ \ \ 9$ | $\ \ 1$ | $\ \ 1$ | $-0.5$ | $\ \ 0.5914$ | $\ \ 0.2716$ | $\ \ \ 0.1007$ | $\ \ 0.8118$ | $\ \ 0.2728$ | $\ \ 0.1067$
$\ \ \ 9$ | $\ \ 1$ | $\ \ 1$ | $\ -3$ | $\ \ 0.2201$ | $\ \ 0.1707$ | $\ \ -3.0175$ | $\ \ 1.050$ | $\ \ 0.8346$ | $\ \ 0.0879$
$\ \ \ 9$ | $\ \ 1$ | $\ \ 1$ | $\ -5$ | $\ \ 1.1433$ | $\ \ 0.2516$ | $\ \ \ 0.0805$ | — | — | —
$\ \ 12$ | $\ \ 1$ | $\ \ 1$ | $\ -2$ | $\ \ 0.3700$ | $\ \ 0.4399$ | $\ \ \ 0.2573$ | $\ \ 0.5193$ | $\ \ 0.4422$ | $\ \ 0.2793$
$\ \ 12$ | $\ \ 1$ | $\ \ 1$ | $\ -3$ | $\ \ 0.1907$ | $\ \ 0.3460$ | $\ \ -2.0575$ | $\ \ 0.6479$ | $\ \ 0.4230$ | $\ \ 0.0234$
$\ \ 12$ | $\ \ 1$ | $\ \ 1$ | $\ -5$ | $\ \ 0.7605$ | $\ \ 0.4001$ | $\ \ \ 0.1960$ | — | — | —
(a) $d=8,\alpha=1$, $\hat{\eta}_{2}=1$ and $\hat{\eta}_{3}=-5$.
(b) $\alpha=1$, $\hat{\eta}_{2}=1$ and $\hat{\eta}_{3}=0.8$.
(c) $d=8,\alpha=1$, $\hat{\eta}_{2}=1$ and $\hat{\eta}_{3}=-0.5$.
Figure 5: $P-v$ Diagram of Lovelock black holes with spherical horizon,
$\kappa=1$, in $d=8$ which shows a Van der Waals behavior (a), and the
possibility of reentrant phase transition (b) and (c).
The corresponding $P-v$ diagrams for diverse choices of $\eta_{3}$ are
depicted in Fig 5. The diagrams are the same in any dimension $d$. So without
loss of generality we plot them for $d=8$. The interesting point that one
should note is that despite of the case for the black holes with constant
curvature horizon, there exists criticality for $d\geq 11$. As one can see,
for some choices of $\hat{\eta}_{3}$ two critical points are present, one with
negative (unphysical) and the other with positive pressure (5(c)). But one can
find values for free parameters for which two critical points with positive
pressure exist as is seen in Fig. 5(b). This behavior is reminiscent of the
interesting reentrant phase transition. So we go through the Gibbs plot for
these cases. The Gibbs free energy obeys the following thermodynamic relation
for any value of $\kappa$
$\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{Pr^{d-1}}{d-1}+\frac{(d-2)kr^{d-3}}{16\pi}+\frac{\alpha(d-2)r^{d-5}\left(\hat{\eta}_{2}+k^{2}\right)}{16\pi}+\frac{\alpha^{2}(d-2)r^{d-7}\left(3\hat{\eta}_{2}k+\hat{\eta}_{3}+k^{3}\right)}{48\pi}$
(45)
$\displaystyle-\frac{r^{d-7}\left(\alpha^{2}(d-4)(d-2)\left(\hat{\eta}_{2}+k^{2}\right)+2\alpha(d-6)(d-2)kr^{2}+(d-6)(d-4)r^{4}\right)}{48\pi(d-6)(d-4)(d-2)\left(\alpha^{2}\hat{\eta}_{2}+\left(\alpha
k+r^{2}\right)^{2}\right)}$
$\displaystyle\times\left\\{\alpha^{2}(d-7)(d-2)\left(3\hat{\eta}_{2}k+\hat{\eta}_{3}+k^{3}\right)+3\alpha(d-5)(d-2)r^{2}\left(\hat{\eta}_{2}+k^{2}\right)+3(d-3)(d-2)kr^{4}+48\pi
Pr^{6}\right\\}$
The behavior of Gibbs free energy for a positive value of $\hat{\eta}_{3}$ is
depicted in Fig. 6. Two positive critical pressures are $T_{c1}$ and $T_{c2}$.
Looking at Fig. 6(a) we observe that between $P_{c1}=0.013$ and
$P=0.052<P_{c2}$, the lower branch is globally stable and therefore no phase
transition could happen.
By increasing the pressure, zeroth and first order phase transition take place
between $P=0.053$ and $P=0.054$ (See Fig. 6(b)). The first order phase
transition occurs between small black hole (SBH) and large black hole (LBH).
As well as the first order phase transition, a finite jump in Gibbs free
energy is seen in the mentioned region which leads to a zeroth order phase
transition between SBH and LBH (or intermediate black hole). Interestingly
enough, in the above certain range of pressure, a reentrant phase transition
occurs. Heretofore, zeroth order and reentrant phase transition has been
detected in charged dilaton Dehy1 and Born-Infeld Kubiz ; Dehy2 black holes.
A reentrant phase transition is a Combination of two (or more) phase
transition, in such a way that the initial and final phase of system are
macroscopically the same.
By increasing the pressure, in the region $0.054<P\leq P_{c2}=0.063$ the
zeroth order phase transition disappears and only first order phase transition
could bee seen. Finally, the first order phase transition finishes at second
critical point $P=P_{c2}=0.063$ as is seen in Fig. 6(c). For $P>P_{c2}$, the
lower branch is the unique globally stable branch and we do not expect any
phase transition.
(a) The lower (upper) branch is thermodinamically stable(unstable). There is
only one physical branch and no phase transition occurs in the system.
(b) A first order phase transition occurs in $T_{1}$ and a zeroth order in
$T_{0}$. In addition a finite jump in Gibbs diagram leads to a zeroth order
phase transition between small and large (intermediate) black hole.
(c) The first order phase transition finishes in the critical point
$P=P_{c2}$. For $P>P_{c2}$, we see only one stable (lower) branc with no phase
transition.
Figure 6: Gibbs diagram with $d=8$, $\alpha=1$, $\hat{\eta}_{2}=1$ and
$\hat{\eta}_{3}=0.8$ .
(a) A zeroth order phase transition occurs in $T_{0}$. The only stable branch
is lower one which is showed by solid blue line. The diagram showes a finite
jump between stable and unstable branch and physically can not exist.
(b) The first order phase transition occurs in $T_{1}$ and a zeroth order in
$T_{0}$. A reentrant phase transition occurs between LBH /SBH /LBH.
(c) The first order phase transition finished in the critical point
$P=P_{c2}$. There is only one stable (lower) branch For $P>P_{c2}$ and so no
phase transition occurs.
Figure 7: Gibbs diagram with $d=8$, $\alpha=1$, $\hat{\eta}_{2}=1$ and
$\hat{\eta}_{3}=-0.5$.
As one can see in Fig. 5(c), there are two critical points with positive and
negative pressure. Here we would like to study the associated Gibbs free
energy in Fig. 7. In the region $0<P<0.032$, there exist a discontinuity in
the Gibbs diagram (See Fig. 7(a)). In this figure, the blue solid line shows
reasonable branch of Gibbs energy and dashed lines show unstable parts of
Gibbs energy with negative heat capacity. One may observe two physical phase
transitions for $P=0.04$ in Fig.7(b). A first order phase transition occurs at
$T_{1}$, where the Gibbs diagram is continuous but its derivative is not.
Also, there is a finite gap in Gibbs diagram in $T_{0}$ and so a zeroth order
phase transition occurs in this point. Fig.7(b) represents a reentrant phast
transition between LBH/ SBH/ LBH . More increasing of the pressure leads to
the disappearance of zeroth order phase transition. For example in $P=0.055$,
there is no zeroth order phase transition. Fig. 7(c) shows three different
states of Gibbs energy around the second critical pressure $(P_{c2}=0.06)$.
For $P=0.055<P_{c2}$, one sees a first order phase transition and for
$P=0.07>P_{c2}$, no phase transition happens in the system. While $T=T_{c2}$
is the critical point for which second order phase transition occurs, because
of the existence of only one stable branch for $P>P_{c2}$, no phase transition
may occur.
Figure 8: The schematic behavior of isobar $T$-$r_{h}$ diagram and the
corresponding $G-T$ curve (inset) of Lovelock black hole. As temperature
decreases the black hole follows direction of arrows. The first and zeroth
order phase transition are identified by black dotted and green dot-dashed
curve, respectively. A large/small/large(intermediate) corresponds to a
reentrant phase transition. The positive (negative) sign of heat capacity is
displayed by the blue solid (dash red) line. We have changed the size of
diagram and shifted for more clarity but this diagram can represent the
behavior of Figs. 6(b) and 7(b).
Now, we go through the study of the behavior of isobar $T-r_{h}$ diagram and
the corresponding $G-T$ curve (inset) for Lovelock black hole with nonconstant
curvature horizon. As it is illustrated in Fig. 8 by decreasing the radius of
horizon in the $G-T$ plane (the inset in Fig. 8), black hole follows the lower
solid blue branch until it reaches $T_{1}$ and changes the direction to switch
to left solid blue curve in $G-T$ plane with a first order LBH/SBH phase
transition. This is identified by black dotted line in $T-r_{h}$ plane. In the
case of more decreasing of $r_{h}$, the system experiences zeroth order phase
transition between small and large black hole at $T_{0}$ by a finite jump (
inset in Fig. 8) which is shown by dot-dashed green line in $T-r_{h}$ diagram
in Fig. 8. Eventually, the black hole tracks the blue solid line to the end.
We should mention that we have shifted the diagram for more clarity but this
diagram can represent the behavior of Figs. 6(b) and 7(b).
The same as what we observed in the case of Ricci flat black holes with
$\kappa=0$ in Sec. III, the entropy does not depend on the temperature and so
the first exponent $\alpha^{\prime}$ equals zero for $\kappa=1$. Following the
approach discussed in Sec.III.2, we can write the reduced equation of state as
$p=1+At-Bt\omega-C\omega^{3}+O\left(t\omega^{2},\omega^{4}\right).$ (46)
Therefore, it is easy to show that the critical exponents read
$\beta^{\prime}=\frac{1}{2},\quad\gamma^{\prime}=1,\quad\delta^{\prime}=3$
(47)
## V Critical behavior of Lovelock Hyperbolic black holes with $\kappa=-1$
When the topology of the black hole horizon is hyperbolic, the equation of
state reads
$P=\frac{T}{v}+\frac{d-3}{\pi(d-2)v^{2}}-\frac{32\alpha
T}{(d-2)^{2}v^{3}}-\frac{16\alpha(d-5)(\hat{\eta}_{2}+1)}{\pi(d-2)^{3}v^{4}}+\frac{256\alpha^{2}(\hat{\eta}_{2}+1)T}{(d-2)^{4}v^{5}}-\frac{256\alpha^{2}(d-7)(-3\hat{\eta}_{2}+\hat{\eta}_{3}-1)}{3\pi(d-2)^{5}v^{6}}.$
(48)
In addition, the Gibbs free energy of the black hole can be calculated from
Eq.(45) by substituting $\kappa=-1$.
For the black hole with hyperbolic horizon in different theories of
electrodynamics and general relativity, first order phase transition is rarely
seen. One critical point exists for second and third order Lovelock hyperbolic
black holes with constant curvature horizons Xu . So, it is interesting to
search for such a phase transition for Lovelock black holes with nonconstant
curvature horizon when $\kappa=-1$.
Similar to the case with $\kappa=1$ which we discussed in Sec. IV, the
equation of state, (Eqs. 24) leads to a polynomial of degree 4 and numeric
calculations show that depending on the values of the parameters $\alpha$,
$\hat{\eta}_{2}$, and $\hat{\eta}_{3}$ there may exist one or two physical
critical points. In addition, we find some values for the free parameters that
results in three critical points, with two of them having positive pressure
and one with negative pressure. We could not find any case with three positive
critical pressure.
Table 2: Critical values in different dimensions for $\kappa=-1$
$\ d$ $\ \alpha\ \ \ \ \ $ $\ \hat{\eta}_{2}$ $\ \ \ \ \ \hat{\eta}_{3}\ \ \ \
\ \ $ $T_{c1}$ $\ P_{c1}$ $T_{c2}$ $\ P_{c2}$ $\ T_{c3}$ $\ P_{c3}$ 8 1 3 1.1
0.18162 0.1770 — — — — 10 1 10 -3.5 0.3887 0.2782 — — — — 8 1 0.5 -0.5 0.0636
0.2268 0.9091 0.5400 — — 9 1 0.5 0.1 0.0823 0.3368 0.9251 0.38650 — — 9 1 0.7
3.2 0.0622 -306.37 0.8342 0.6184 1.2620 0.8414
In Fig. 9, we have only one critical point for chosen parameters $d$,
$\hat{\eta}_{2}$, $\hat{\eta}_{3}$ and $\alpha$. This case is exactly similar
to the Van der Waals phase transition with the same isotherm curves. The
swallowtail shapes of Gibbs diagras verify the first order phase transition
too. In Fig. 10, we choose the parameters so that we can observe two physical
critical points. The relevant Gibbs energy in Fig. 10(b) shows two critical
curves associated with $P_{c1}$ and $P_{c2}$. There is no phase transition for
$P_{c1}<P<P_{c2}$. Also the Gibbs energy and its derivatives with respect to
the temperature are continuous. We can see first order phase transition for
$P<P_{c1}$ or $P>P_{c2}$.
The interesting case is the case with three critical points for Lovelock black
holes with $\kappa=-1$ which occurs for some specific values of parameters.
The corresponding $P-v$ plot is depicted in Fig. 11. Two critical points with
positive pressure are shown in Fig. 11(a) and the corresponding Gibbs free
energy is depicted in Fig. 11(b). The third critical point is far from the
others and so we bring it in a separate diagram. As it is seen in Fig. 11(c),
the third critical point has negative pressure. Fig. 11(d) represents the
Gibbs free energy corresponding to this point. It is worth to note that for
$P\leq P_{c3}$ the Gibbs energy is completely unphysical with negative
compressibility while for $P>P_{c3}$ there exists some physical part in Gibbs
diagram. We should emphasis that the curves in Gibbs diagrams are rescaled and
shifted for more clarity.
(a) $P-v$ diagram
(b) Gibbs diagram.
Figure 9: Critical points: $\kappa=-1$, $\alpha=1$, $\hat{\eta}_{2}=10$,
$\hat{\eta}_{3}=-3.5$ and $d=10$. There exists a first order phase transition
with one physical critical point similar to the Van der Waals system.
(a) $P-v$ diagram
(b) Gibbs diagram
Figure 10: Critical points: $\kappa=-1$, $\alpha=1$, $\hat{\eta}_{2}=0.5$,
$\eta_{3}=-0.5$ and $d=8$. There are two physical critical point in the above
diagrams. We have changed the scale of Gibbs diagrams in the way that two
critical curves are visible in one diagram.
(a) $P-v$ diagram for two of the three critical points
(b) Gibbs diagram for two of the three critical points.
(c) $P-v$ diagram for the third critical point.
(d) Gibbs diagram for the third critical point.
Figure 11: Critical points: $\kappa=-1$, $\alpha=1$, $\hat{\eta}_{2}=0.7$,
$\hat{\eta}_{3}=3.2$ and $d=9$
Following the approach in Sec. III.2 and using reduced thermodynamic variables
and Taylor expansion for the equation of state, Eq.(48), we obtain critical
exponents as
$\alpha^{\prime}=0,\quad\beta^{\prime}=\frac{1}{2},\quad\gamma^{\prime}=1,\quad\delta^{\prime}=3$
(49)
which is consistent with Van der Waals exponents and the results of mean field
theory. It is worthwhile to emphasise that in a special case with
$\hat{\eta}_{2}=\hat{\eta}_{3}=0$, a peculiar isolated critical point emerges
for hyperbolic black holes and is characterized by non-standard critical
exponents, which is discussed in details in Ref. Frassino .
## VI Concluding Remarks
In this study we presented some thermodynamic behaviors of black holes of a
more general class of solutions possessing non-constant curvature horizons.
The horizon space of these kinds of black holes is nonmaximally symmetric
Einstein space. Nontrivial Weyl tensor of such exotic horizons is exposed to
the bulk dynamics through the higher-order Lovelock term. Investigating the
$P-v$ criticality behavior of such black holes of Lovelock gravity in the
extended phase space led to interesting and qualitatively new behaviors. By
introducing the conjugate quantity to Lovelock parameter $\alpha$, We showed
that the first law of thermodynamics and the Smarr formula hold. By
considering the thermodynamics of these kinds of black holes with nonmaximally
symmetric horizons in cubic Lovelock gravity, we have found some particularly
novel and interesting results. As it is well-known no criticality has been
found for all known types of Ricci flat black holes in Einstein or Lovelock
theories of gravity. Thus we went through Ricci flat black holes with
nonconstant curvature horizon and we found that there exists criticality in
every dimension $d>8$ for such black holes with $\kappa=0$. We obtained the
exact solutions by solving the cubic equation and showed that relating to the
values of chargelike parameters appearing in the metric function, Van der
Waals-like behavior and first order phase transition may happen. For some
values of $\hat{\eta}_{3}$, which is a chargelike parameter that is inserted
in the metric function due to the appearance of third-order curvature terms,
two critical point emerge. We have also computed the critical exponents of the
phase transition and found that in the canonical ensemble the thermodynamic
exponents coincide with those of the Van der Waals fluid.
For the black holes with spherical and constant curvature horizons critical
points do not exist for $d>11$. For our solutions with non constant curvature
horizon we carried out the study numerically and found that one or two
critical points exist in every dimension even dimensions higher than 11 with
the proper choices of the parameters. We saw how the value of the parameters
that are being emerged in the solutions as a result of the nonconstancy of the
curvature of the horizon, affect the types of phase transition. To disclose
the phase structure of the solutions and classify their types, we studied the
Gibbs free energy. For $\kappa=1$ two different behaviors have been found. For
some values of the free parameters, a first order phase transition occurs
between small and large black holes which is accompanied by a discontinuity in
the slop of Gibbs free energy at transition point. We showed that if the
parameters adopt some proper values, a large-small-large black hole transition
would happen. This process was shown to be accompanied by a finite jump of the
Gibbs free energy referred to as the zeroth-order phase transition. While for
the black holes with hyperbolic horizon in different theories of general
relativity, phase transition is rarely seen, we showed that for our solution
in the case $\kappa=-1$, in every dimension $d\geq 8$, various kinds of
interesting phase transitions happen. It is interesting enough to see that the
usual Van der Waals-like small/large black hole phase transition and reentrant
phase transition can occur for our solution. Also a novel behavior of
hyperbolic Lovelock black holes with nonconstant curvature horizon has been
found for which three critical points could exist with a proper choices of the
parameters in the solution. Finally it is important to mention that as we
carried out the study numerically, it is possible to find some other different
behaviors by other choices of the free parameters.
## References
* (1) M. B. Greens, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge University Press, Cambridge, England, 1987); D. Lust and S. Theusen, Lectures on String Theory (Springer, Berlin, 1989); J. Polchinski, String Theory (Cambridge University Press, Cambridge, England, 1998).
* (2) D. Lovelock, J. Math. Phys. 12, 498 (1971).
* (3) J. D. Bekenstein, Phys. Rev. D 7, 949 (1973).
* (4) S. W. Hawking, Comm. Math. Phys. 43, 199 (1975).
* (5) S. W. Hawking, D. N. Page, Commun. Math. Phys. 87, 577 (1983).
* (6) A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Phys. Rev. D 60, 064018 (1999).
* (7) A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Phys. Rev. D 60 104026 (1999).
* (8) B. P. Dolan, Class. Quant. Grav. 28, 235017 (2011).
* (9) D. Kastor, S. Ray, and J. Traschen, Class. Quant. Grav. 26, 195011 (2009).
* (10) M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, Phys. Rev. D 84, 024037 (2011).
* (11) G. W. Gibbons, R. Kallosh, and B. Kol, Moduli, Phys.Rev.Lett. 77 4992–4995 (1996).
* (12) J. Creighton and R. B. Mann, Phys.Rev. D 52 4569 (1995),
* (13) L. Smarr, Phys. Rev. Lett. 30 71 (1973) Erratum: [Phys. Rev. Lett. 30 521 (1973)].
* (14) B. P. Dolan, Class. Quant. Grav. 28, 125020 (2011).
* (15) B. P. Dolan, Phys. Rev. D 84, 127503 (2011).
* (16) A. Karch, B. Robinson, JHEP 12 073 (2015).
* (17) D. Chernyavsky, K. Hajian, Class. Quantum Grav. 35 125012 (2018); K. Hajian, H. Özşahin, and B. Tekin, [arXiv:2103.10983] (2021).
* (18) D. Kubiznak and R. B. Mann, JHEP 07 033 (2012).
* (19) A. Belhaj, M. Chabab, H. El Moumni and M. B. Sedra, Chin. Phys. Lett. 29, 100401 (2012).
* (20) E. Spallucci and A. Smailagic, Phys. Lett. B 723, 436 (2013).
* (21) A. Belhaj, M. Chabab, H. E. Moumni, L. Medari and M. B. Sedra, Chin. Phys. Lett. 30, 090402 (2013).
* (22) N. Altamirano, D. Kubiznak and R. B. Mann, Phys. Rev. D 88, 101502 (2013).
* (23) S. Gunasekaran, R. B. Mann and D. Kubiznak, JHEP 1211, 110 (2012).
* (24) T. Narayanan and A. Kumar, Physics Reports 249 135–218 (1994).
* (25) A. M. Frassino, D. Kubiznak, R. B. Mann and F. Simovi, JHEP 09, 080 (2014).
* (26) V. P. Maslov, Mathematical notes of the Academy of Sciences of the USSR 76, 697 (2004).
* (27) V. Iyer and R.M. Wald, Phys. Rev. D 50 846 (1994).
* (28) D. Kastor, S. Ray and J. Traschen, Gravity, Class. Quant. Grav. 27 235014 (2010).
* (29) R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, JHEP 1309, 005 (2013).
* (30) D. C. Zou, Y. Liu and B. Wang, Phys. Rev. D 90, 044063 (2014).
* (31) H. Yazdikarimi, A. Sheykhi, Z. Dayyani, Phys. Rev. D 99, 124017 (2019).
* (32) S. H. Hendi, S. Panahiyan, B. Eslam Panah, Prog. Theor. Exp. Phys. 2015 no.10, 103E01 (2015).
* (33) H. Xu, W. Xu and L. Zhao, Eur. Phys. J. C 74 3074 (2014).
* (34) J. X. Mo and W. B. Liu, Eur. Phys. J. C 74 2836 (2014).
* (35) A. Belhaj, M. Chabab, H. El Moumni, K. Masmar and M. B. Sedra, Int. J. Geom. Meth. Mod. Phys. 12 1550017 (2014).
* (36) G. Dotti and R. J. Gleiser, Phys. Lett. B 627, 174 (2005).
* (37) H. Maeda, Phys. Rev. D 81, 124007 (2010); H. Maeda, M. Hassaine, and C. Martinez, J. High Energy Phys. 08 123 (2010).
* (38) N. Farhangkhah and M. H. Dehghani, Phys. Rev. D 90, 044014 (2014).
* (39) N. Farhangkhah, Int. J. Mod. Phys. D 25, 1650030 (2016); N. Farhangkhah, Phys. Rev. D 97, 084031 (2018).
* (40) S. Ohashi and M. Nozawa, Phys. Rev. D 92, 064020 (2015).
* (41) S. Ray, Classical Quantum Gravity 32, 195022 (2015).
* (42) Z. Dayyani, A. Sheykhi and M. H. Dehghani, Phys Rev D 95, 084004 (2017).
* (43) V. P. Maslov, Mathematical Notes 76, 697 (2004).
* (44) A. Dehyadegari, A. Sheykhi and A. Montakhab, Phys. Rev. D 96, 084012 (2017).
* (45) A. Dehyadegari and A. Sheykhi, Phys. Rev. D 98, 024011 (2018).
|
# Automating LC–MS/MS mass chromatogram quantification: Wavelet transform
based peak detection and automated estimation of peak boundaries and signal-
to-noise ratio using signal processing methods.
Florian Rupprecht Sören Enge Kornelius Schmidt Wei Gao Clemens Kirschbaum
Robert Miller Technische Universität Dresden (TUD), Department of Psychology,
Dresden, Germany Medical School Berlin (MSB), Department of Psychology,
Faculty of Science, Berlin, Germany Health Technology Assessment & Outcomes
Research, Pfizer Germany GmbH, Linkstr. 10, 10785 Berlin, Germany
###### Abstract
Background and Objective: While there are many different methods for peak
detection, no automatic methods for marking peak boundaries to calculate area
under the curve (AUC) and signal-to-noise ratio (SNR) estimation exist. An
algorithm for the automation of liquid chromatography tandem mass spectrometry
(LC–MS/MS) mass chromatogram quantification was developed and validated.
Methods: Continuous wavelet transformation and other digital signal processing
methods were used in a multi-step procedure to calculate concentrations of 6
different analytes. To evaluate the performance of the algorithm, the results
of the manual quantification of 446 hair samples with 6 different steroid
hormones by two experts were compared to the algorithm results. Results: The
proposed approach of automating LC–MS/MS mass chromatogram quantification is
reliable and valid. The algorithm returns less non-detectables than human
raters. Based on signal to noise ratio, human non-detectables could be
correctly classified with a diagnostic performance of AUC = $0.95$.
Conclusions: The algorithm presented here allows fast, automated, reliable,
and valid computational peak detection and quantification in LC–MS/MS.
###### keywords:
Chromatography , Peak quantification , Peak boundaries , Signal processing ,
Automation , LC–MS/MS
††journal: Biomedical Signal Processing and Control
## 1 Introduction
A very time consuming part of LC–MS/MS quantification and related methods is
manual peak marking in the raw mass chromatograms. While many different
methods for peak detection have been developed [1], to our knowledge, there is
no open source method which also is able to mark peak boundaries for area
under the curve (AUC) measurement and perform signal-to-noise ratio (SNR)
estimation. This article describes an automated procedure capable of
performing these tasks, handles fluctuations in retention time (RT) and
perform calibration to a known reference standard.
There are three direct advantages of analyzing LC–MS/MS mass chromatograms by
implementing an automated process: Increased productivity and speed by
decreasing manual labour, increased objectivity and reliability resulting out
of the absence of human raters and, resulting from the non-proprietary nature
of the described method, increased replicability [2].
This article first describes underlying concepts and models applied by the
algorithm. Then a test with real world data is performed and a comparison of
the algorithm results to results manually quantified by human experts is made.
Finally the following two hypotheses were evaluated:
1. 1.
The algorithm returns analyte concentrations which correlate strongly with
those manually quantified by experts.
2. 2.
The algorithm returns analyte concentrations which are valid.
The manuscript is structured as follows. In section 2 the algorithm and its
theoretical background are described, in section 3 algorithm results are
compared to results manually compiled by human raters.
## 2 Theory and calculation
The following sections describe an algorithm for automatic LC–MS/MS mass
chromatogram quantification. The first section introduces a generic model of
chromatogram composition which forms a theoretical basis for the subsequent
sections about decomposition, transformation and processing of measured
chromatograms. The final sections then outline a multi-step procedure that is
necessary to calculate the absolute concentrations of an analyte.
### 2.1 A generic model of mass chromatograms
A chromatogram can be described as a time series of LC–MS/MS detector
intensities $y(t)\in\mathbb{R}$, where $t\in\mathbb{R^{+}}$ indicates time.
The relevant sequence containing the peak can be constructed from several
component series using the following additive model.
$y(t)=B(t)+P(t)+N(t)\,,$ (1)
where $B(t)=b+(t-t_{R})\cdot d$ is a linear trend component containing
background $b\in\mathbb{R^{+}}$ and linear drift $d\in\mathbb{R}$ of the
chromatogram, $P(t)$ is the peak distribution component, and $N(t)$ is an
irregular component containing noise (besides low frequency background noise,
this also can contain nearby peaks and low frequency noise or drift). These
components are illustrated in Figure 1.
Retention time (RT, $t_{R}$) is the time at which the peak component has its
maximum intensity. Peak height ($h$) is background ($b=B(t_{\mathrm{R}})$)
subtracted by signal intensity at RT $y(t_{\mathrm{R}})-b$. Slope ($s$) is the
slope of the background component. All these measures are annotated in Figure
1.
To measure analyte concentration, the area of the peak needs to be calculated.
Peak components are determined by performing several data transformations on
the chromatogram time series.
Figure 1: Generic model for chromatographic peaks. The actual LC–MS/MS sensor
data ($y(t)$) is combined background and drift ($B(t)$), peak distribution
($P(t)$) and noise ($N(t)$). Note that this Figure does not include peak skew
or nearby peaks. Peak range is the time interval between the lower bound
($t_{\mathrm{L}}$) and upper bound ($t_{\mathrm{U}}$). Retention time (RT,
$t_{\mathrm{R}}$) indicates maximum sensor intensity in the peak range. Peak
height ($h$) is background ($b=B(t_{\mathrm{R}})$) subtracted by sensor
intensity of the combined signal at RT $y(t_{\mathrm{R}})-b$. Slope ($s$) is
the slope of the background component. Note that in the implementation
background and slope are defined using the line between lower and upper bounds
of the peak range to simplify the process. This is not necessarily equivalent
to the definition in this model.
### 2.2 Data transformations
The following number of transformations are calculated for a given
chromatogram. Figure 2 illustrates the transformations and their order.
Figure 2: Chromatogram time series transformations. The LC–MS/MS
chromatographic time series $Y(t)$ is smoothed using a gaussian kernel. High-
pass filtered data is obtained by differentiating smoothed and original time
series. The second derivative of the smoothed data is approximated and the
continuous wavelet transformation (CWT) is calculated.
##### Smoothing / low-pass filtering
The data are smoothed using a 1-D gaussian kernel smoother [see 3, p. 40-42].
The following formula returns aggregation weights by which chromatogram
intensity values are smoothed. Smoothness can be adjusted using the manually
set hyperparameter $\sigma$.
$G(t)=\frac{1}{\sqrt[]{2\pi}\sigma}\exp{\left(-\frac{t^{2}}{2\sigma^{2}}\right)}$
(2)
##### High-pass filtering
By differentiation of the original $Y(t)$ and smoothed $S(t)$ time series, the
high-pass filtered data $H(t)=Y(t)-S(t)$ is received. This makes it dependent
on the $\sigma$ parameter of the gaussian kernel.
##### Derivation
The second derivative $D(t)=S^{\prime\prime}(t)$ of the smoothed data $S(t)$
is approximated by applying
$S^{\prime}\left(\frac{t_{i}+t_{i-1}}{2}\right)=\frac{S(t_{i})-S(t_{i-1})}{t_{i}-t_{i-1}}$
(3)
twice.
##### Wavelet decomposition
Continuous wavelet transformations are used to create time-frequency
breakdowns of the chromatograms [see 4].
Exponentially modified Gaussian distributions are commonly used to model
chromatographic peaks [5]. The second derivative of the gaussian distribution
function is used to build a mother wavelet. Chromatographic peaks have a lower
frequency than the interfering noise. This enables the separation and further
analysis of the noise and peak distributions by investigating CWT frequency
breakdowns.
Continuous wavelet transforms are used to divide the data111The original mass
chromatogram is unevenly spaced. While the transformations up to this point
can handle this type of data, the continuous wavelet transform implementation
is constrained to the use of evenly sampled data. The data are then resampled
in even time steps using linear interpolation. If data loss is a concern the
sampling resolution can be increased. into wavelets. The continuous wavelet
transform of a function $x(t)$ with scale $a\in\mathbb{R^{+}}$ and a
translation variable $b\in\mathbb{R}$ can be expressed by the following
integral:
$X_{w}(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}x(t)\psi\left(\frac{t-b}{a}\right)dt$
(4)
Chromatographic peaks are commonly modelled by exponentially modified gaussian
distributions [5]. The negative normalized second derivative of a Gaussian
function is used. It is usually called the _mexican hat wavelet_ [6]:
$\psi(t)=\frac{2}{\sqrt{3\sigma}\pi^{1/4}}\left(1-\left(\frac{t}{\sigma}\right)^{2}\right)\exp{\left(-\frac{t^{2}}{2\sigma^{2}}\right)}$
(5)
Note that the processed sensor data also has to be differentiated so that the
Gaussian derivative wavelet fits to the peak distribution. This furthermore
has the advantage, that both background and linear drift are eliminated from
the raw mass chromatogram.
Then the continuous wavelet transform is used to construct a time-frequency
representation of a signal. Figure 3 shows a chromatographic time series, its
second derivative, and the resulting CWT coefficient matrix.
Figure 3: Chromatographic time series, second derivative (scaled for
visibility), and the resulting coefficient matrix after wavelet
transformation. The time-intensity graph also contains smoothed and high-pass
filtered (HPF) time series, as well as relevant points of the retention time
fitness function (RT fit) which is referred to in section 2.3.4.
### 2.3 Chromatogram processing
Each single chromatogram is analyzed in multiple steps. Possible peaks, their
RTs, and area boundaries are identified first. Then signal-to-noise ratio
(SNR), as well as peak area, height, background and slope are calculated.
Lastly a fitness value is calculated by which the peak most likely associated
with the analyte is selected.
#### 2.3.1 Peak identification & boundary estimation
Peak RT ($t_{\mathrm{R}}$) and boundaries ($t_{\mathrm{L}}$, $t_{\mathrm{U}}$)
(see Figure 1) are estimated in multiple steps. Figure 4 serves as a visual
aid for picturing changes in the peak polygon in each step.
(a) Peak area after initial boundary estimation.
(b) Peak area after friction boundary correction.
(c) Peak area after partial convex hull boundary correction.
Figure 4: This Figure illustrates changes in the area polygon defined by
estimations for lower ($\widehat{t_{\mathrm{L}}}$) and upper
($\widehat{t_{\mathrm{U}}}$) peak boundaries after initial approximation
(4(a)) and the two subsequent correction steps. First a Newtonian friction
inspired approach is used in conjunction with the smoothed time series to
widen the bounds (4(b)). The resulting potentially self-overlapping area
polygon is then modified by first calculating the lower part of the convex
hull and removing separate polygons after (4(c)).
##### Peak retention time (RT)
Non-maximum suppression (NMS) [7] is used to identify local maxima in the CWT
coefficient matrix. NMS chooses points that have exclusively greater value
neighbouring points. Local maxima in the CWT matrix correspond to low
frequency components, peak components in this case, in the chromatogram data.
For this reason NMS returns a list of peak RT with associated time, scale and
coefficient values, each representing a potential peak in the data.
##### Peak boundary estimation
To determine the peak area, lower and upper bounds of the peak component are
needed. These are approximated in two steps.
While peak RT ($t_{R}$) is projected to a local maxima in the CWT coefficient
matrix, approximate inflection points of the peak distribution should
correspond to minima. The vector of CWT coefficients with the same scale as
the previously determined maximum is inspected for the closest minima next to
the maximum. This gives us estimates for lower and upper peak bounds.
##### Friction boundary correction
As the wavelet transformation models uniformly distributed gaussian
distributions and chromatographic peaks are often subject to skew, these
approximate boundaries need to be corrected.
A Newtonian friction inspired approach is performed. Similar to a physical
body gradually sliding down a hillside until a slope is reached at which
friction stops it from moving, the initially estimated boundaries of the peak
component are increased until a specified slope threshold is reached. This is
detailed in Algorithm 1. It is advised to normalize the threshold to the
intensity range of each analyzed chromatogram.
Algorithm 1 Peak boundary correction. The size function returns the size of a
list.
1:$Y$: Smoothed chromatogram data
2:$p$: List of peak boundaries
3:$t$: Threshold
4:$t\leftarrow(\textsc{max}(Y)-\textsc{min}(Y))\cdot t$
5:for each $(u,l)\in p$ do
6: while $u<\textsc{size}(Y)-1$ do
7: if $Y_{u}-Y_{u+1}>t$ then
8: $u\leftarrow u+1$
9: while $l>0$ do
10: if $Y_{l}-Y_{l-1}>t$ then
11: $l\leftarrow l-1$
##### Partial convex hull boundary correction
At this point resulting peak polygons may overlap themselves at their lower
and upper boundaries when there is significant noise in the data. To
circumvent this and to mimic the behavior of a human rater the _bottom_ of the
convex hull222When the points of the time series between temporal lower and
upper boundaries are considered points on a time-intensity plane, the convex
hull is the smallest convex polygon that contains them all. of the peak data
is calculated. A variation of Grahams scan [see 8] is implemented. It is
described in Algorithm 2. Next, all areas of the polygon separated from the
main peak area are removed. Peak boundaries are then set to the minimum and
maximum $x$ (time) value of the polygon vertices and peak RT to the maximum
$y$ (intensity) value.
Algorithm 2 Partial convex hull boundary correction. Modification of Grahams
scan [see 8]. The size function return the current size of the stack, push and
pop are standard stack operations of adding and removing an element and orient
returns the orientation $o$ (clockwise ($o>0$), counterclockwise ($o<0$) or
colinear ($o=0$)) of an ordered triplet.
1:$W$: Empty stack
2:$p$: Vector of points ordered by increasing $x$ coordinate
3:push($W,p_{1}$)
4:push($W,p_{2}$)
5:for $i\leftarrow 3$ to $N$ do
6: while $\textsc{size}(W)\geq
2\And\textsc{orient}(p_{i},W_{top},W_{second})\leq 0$ do
7: pop($W$)
8: push($p$)
#### 2.3.2 Peak area, height, background and slope
Peak area ($a\in\mathbb{R^{+}}$), height ($h\in\mathbb{R^{+}}$), background
($b\in\mathbb{R^{+}}$) and slope ($s\in\mathbb{R}$) can be calculated after
the peak boundaries are known (see Figure 1). Note that of these metrics only
peak area and height are used in the evaluative section of this paper.
Formulas for background and slope are described for completeness.
##### Area under the curve (AUC)
The AUC is defined as the area integral chromatographic data constraint to the
peak component subtracted by the area integral of the slope between lower and
upper peak bounds. This is equivalent to the peak polygon area and can easily
be calculated using gauss’s area formula:
$a=\frac{1}{2}{\Big{|}}\sum_{{i=1}}^{{n-1}}x_{i}y_{{i+1}}+x_{n}y_{1}-\sum_{{i=1}}^{{n-1}}x_{{i+1}}y_{i}-x_{1}y_{n}{\Big{|}}\,,$
(6)
where $x\in\mathbb{R^{+}}$ are the temporal and $y\in\mathbb{R^{+}}$ the
intensity dimension coordinates of all measurements between peak bounds.
##### Peak slope
The slope $s\in\mathbb{R}$ is defined through the time positions of lower and
upper peak boundaries:
$s=\frac{y(t_{\mathrm{U}})-y(t_{\mathrm{L}})}{t_{\mathrm{U}}-t_{\mathrm{L}}}\,,$
(7)
where $y(t)\in\mathbb{R^{+}}$ is the chromatogram time series and
$t_{\mathrm{U}}\in\mathbb{R^{+}}$ is the time of the lower peak boundary.
##### Peak background
The background $b\in\mathbb{R^{+}}$ is determined by the temporal position of
the peak maximum on the peak slope with the lower peak bound as the line
intercept:
$b=y(t_{\mathrm{L}})+(t_{\mathrm{R}}-t_{\mathrm{L}})\cdot s\,,$ (8)
where $t_{\mathrm{L}}\in\mathbb{R^{+}}$ is the time of the lower peak
boundary.
##### Peak height
The height $h\in\mathbb{R^{+}}$ is defined as the difference of the peak
maximum to its background:
$h=y(t_{\mathrm{R}})-b\,,$ (9)
where $t_{\mathrm{R}}\in\mathbb{R^{+}}$ the time of the peak maximum.
#### 2.3.3 Signal-to-noise ratio (SNR)
While there is a lack of definitive guidelines for SNR measurements, signal is
often defined as maximum peak height above baseline and noise as standard
deviation or root mean squared of a manually selected part of the chromatogram
where no peaks are present [9].
We propose to approximate SNR ($r\in\mathbb{R^{+}}$), by dividing standard
deviation of the high-pass data ($H(x)\in\mathbb{R^{+}}$, mean $\overline{H}$,
number of data points $N$) by doubled peak height ($h$):
$r=\frac{\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(H(x)-\overline{H})^{2}}}{2h}$ (10)
This method can be performed with very little or no data points outside of the
peak bounds. Note that this method of estimating SNR is dependent on the
smoothing hyperparameter $\sigma$. This means that SNR can only be compared
across analytes when the same $\sigma$ is used.
#### 2.3.4 Peak selection
The peak identification procedure performed previously returns multiple
possible peaks.
To select the appropriate one a peak fitness function
$f(t_{\mathrm{R}})=c\cdot{}g(t_{\mathrm{R}})$ (11)
is calculated for each peak depending on its RT
($t_{\mathrm{R}}\in\mathbb{R^{+}}$), its CWT coefficient
($c\in\mathbb{R^{+}}$) and a RT fitness function ($g(t_{\mathrm{R}})$). The
peak with the highest fitness value is selected and peaks with negative signed
fitness are rejected completely. Note that this can lead to cases where no
peak is detected in a chromatogram. These are considered non-detectables (ND).
The RT fitness function $g(t_{\mathrm{R}})$ is used for the selection of peaks
depending on available information about the expected peak RT. When no
information about the expected RT is known, the RT fitness function is
equivalent to $g(t_{\mathrm{R}})=1$. This leads to selection of the peak with
the highest CWT coefficient and rejects none. In other cases where an estimate
of the RT ($\widehat{t_{\mathrm{R}}}\in\mathbb{R^{+}}$) and a range
hyperparameter ($a\in\mathbb{R^{+}}$) is available, a quadratic fitness
function
$g(t_{\mathrm{R}})=1-\left(\frac{t_{\mathrm{R}}-\widehat{t_{\mathrm{R}}}}{a}\right)^{2}\,,$
(12)
is created to weigh peaks by proximity to expected RT and leads to rejected
peaks outside of the range.
### 2.4 Whole dataset quantification
A whole dataset is comprised out of multiple samples. Each sample contains
data associated with multiple measured analytes. Besides the chromatogram by
which an (unknown) analyte concentration is measured directly, it also
contains a chromatogram of an internal standard analyte (IS) with a known
analyte concentration. The use of stable isotopically labelled internal
standards is a commonly used technique that helps to control variability in
quantitative assays [10, 11]. There are also several calibration samples which
are used for concentration calibration. These samples contain a known
staggered amount of the analyte in addition to the known amount of IS. The
single chromatogram processing described in the previous section, is modified
by hyperparameters specified for each analyte and IS.
The overall process of whole dataset quantification consists out of three main
parts, RT calibration, concentration calibration, and final quantification.
These are described in the following sections. Figure 5 contains a system
diagram which gives an overview over all parts of the routine.
Figure 5: System diagram for a single analyte and its internal standard of a
complete dataset. Samples contain an IS chromatogram with a known
concentration and an analyte chromatogram which may have a known
concentration. There are processing parameters for both analyte and IS which
modify single chromatogram processing. RT calibration, concentration
calibration and the final quantification are three separate steps which each
process a different set of samples. Green boxes indicate outputs of processes.
#### 2.4.1 Retention time (RT) calibration
While the RT of an analyte should be known approximately, it has some variance
[12, 13]. The aim of this step is to analyze analyte chromatograms in which
the right peaks are easily identifiable, to provide better estimates for the
expected RT of a specific analyte and subsequently make peak selection easier
in all other samples of the analyte.
Out of the calibration samples all are selected which contain a sufficient
quantity of the calibrator analyte to be able to easily and distinctly detect
the mass chromatogram peaks. Selected samples have a known concentration ratio
equal or greater than a threshold hyperparameter $x$
($C_{\mathrm{A}}/C_{\mathrm{IS}}\geq x$, where
$C_{\mathrm{A}}\in\mathbb{R^{+}}$ is the known concentration of the analyte
and $C_{\mathrm{IS}}\in\mathbb{R^{+}}$ is the known concentration of the IS).
Both IS and analyte chromatogram are processed as described in previous
sections. When wrong peaks are present in the data a guessed rough estimate of
the expected RT and an acceptable range in the analyte parameters can be
specified to create a preliminary RT fitness function (see 2.3.4).
After obtaining all peak time positions of the selected samples, their
arithmetic mean is used as an expected RT difference.
Now a calibrated RT for both analyte and internal standard has been
calculated. The data showed that the standard deviation of both of them
individually is consistently greater than the standard deviation of their
difference (see results Table 1). In subsequent analyses the IS peak RT
($t_{\mathrm{R}}^{\mathrm{IS}}\in\mathbb{R^{+}}$) is retrieved first. IS peaks
are generally easier to identify because of their constantly high
concentration. Then the previously calculated mean RT difference
($t_{\Delta}\in\mathbb{R}$) is used to calculate the expected RT of the
analyte $t_{\mathrm{R}}^{\mathrm{A}}=t_{\mathrm{R}}^{\mathrm{IS}}+t_{\Delta}$.
In combination with a smaller acceptable RT range, this allows to build more
accurate, dynamic RT fitness functions for the next chromatogram processing
steps.
Analyte | ${t^{\mathrm{A}}_{\mathrm{R}}}$ | ${t^{\mathrm{IS}}_{\mathrm{R}}}$ | ${t_{\Delta}}$ | $\beta$
---|---|---|---|---
Cortisol | 4.687 (0.018) | 4.682 (0.019) | 0.005 (0.007) | 2.41
Cortisol Q | 4.688 (0.018) | 4.682 (0.019) | 0.006 (0.008) | 4.35
Cortisone | 4.506 (0.014) | 4.493 (0.013) | 0.013 (0.007) | 1.61
Testosterone | 5.841 (0.044) | 5.810 (0.040) | 0.036 (0.007) | 0.31
Progesterone | 7.108 (0.055) | 7.042 (0.054) | 0.066 (0.006) | 1.07
DHEA | 6.135 (0.046) | 6.098 (0.045) | 0.036 (0.007) | 0.76
Table 1: Calibration data for RT and concentration calibration. Measured RT
mean and standard deviation (in parentheses) of the RT calibration samples
(see section 2.4.1). Note that the standard deviation of analyte RT
(${t^{\mathrm{A}}_{\mathrm{R}}}$) and IS RT (${t^{\mathrm{IS}}_{\mathrm{R}}}$)
is consistently greater then the standard deviation of their difference
${t_{\Delta}}$. $\beta$ is the slope of the concentration calibration
regression model. Cortisol Q refers to the second most sensitive transition of
cortisol, DHEA to dehydroepiandrosterone.
#### 2.4.2 Concentration calibration
The calibration samples, as previously described, contain known concentrations
of the analyte as well as IS. While only calibration samples with a large
known concentration ratio were used when calibrating the RT, now all of them
are processed.
First, the IS chromatogram is processed like before. Then the analyte
chromatogram is processed using the dynamic RT fitness function which depends
on the IS RT and the mean RT difference calculated in the previous step. Then
the analyte peak area ($a_{\mathrm{A}}\in\mathbb{R+}$) is divided by IS peak
area ($a_{\mathrm{IS}}\in\mathbb{R+}$) to obtain a relative measured
concentration ($M=a_{\mathrm{A}}/a_{\mathrm{IS}}$). In combination with the
relative known concentration ($C=C_{\mathrm{A}}/C_{\mathrm{IS}}$) available
for each sample in this step, their linear relationship is modeled using
linear regression without an intercept.
$C=\beta M$ (13)
The resulting regression model can be used to estimate absolute concentrations
from measured relative concentrations whenever the linearity assumption holds.
See Figure 6 for a visual example of such a model.
Figure 6: Plot of the linear regression model for concentration calibration of
cortisol using our data.
#### 2.4.3 Absolute quantification
Absolute sample quantification can now be performed for all samples. The
chromatogram processing is performed just as in the previous, concentration
calibration step. To obtain absolute measured concentrations the concentration
calibration regression function of the previous step is applied to each
relative measured concentration.
### 2.5 Software availability
We supply333Code will be released upon acceptance of this paper. a open source
reference application which implements all parts of the algorithm. The
software reads .mzML data, an open source standard for mass spectrometry data
[14].
## 3 Method
##### Specimens Collection and Preprocessing
The raw mass chromatogram data were made available from the Dresden
longitudinal study of chronic stress and cognitive control (StressCog) [see
15]. Study approval was granted by the Dresden ethics committee (dossier
EK23012016, IRB00001473, and IORG0001076). In the recruitment process 8,400
eligible participants from the population registry of the City of Dresden
between 25 to 55 years old and within the metropolitan area, were selected and
contacted by invitation letter. The study Protocol encompassed the collection
of hair samples with a min lenght of 3 cm at the posterior vertex region of
the head which was required for analyte extraction [see 16]. See Schmidt et
al. [17] for more details of the sampling and data collection process.
The analyses in this paper were performed on a subset of 230 hair samples from
229 different human subjects (137 females; age range 25 - 55; mean age =
38.05; age SD = 8.51) of the StressCog study. Samples were processed following
the protocol described in Gao et al. [16] with slight adjustments. The non-
pulverized samples were washed with isopropanol and steroid hormones were
extracted from 7.5 mg hair by methanol incubation. Column switching on-line
solid phase extraction (SPE) was performed, followed by analyte detection on
an AB Sciex API 5000 QTrap mass spectrometer. When enough material for two LC-
MS/MS measurements was available the samples were split. This resulted in 446
samples which were processed by the LC-MS/MS (calibration samples excluded) to
obtain selected-reaction monitoring (SRM) chromatograms.
##### Analyte Quantitation
The performance of the algorithm was evaluated by comparing its results to the
manual quantification by two trained experts, referred to as expert 1 and
expert 2. They are both trained in the manual marking of chromatographic
peaks. Expert 1 was more experienced, and has manually marked approximately
60,000 samples with each 6 analytes before analyzing the data used in this
paper, while expert 2 has marked about a tenth of this sample count, 5,000
samples, with 5 different analytes.
225 SRM chromatograms were analyzed by one expert, 221 by two experts. 216 of
these LC-MS/MS samples were split from the same hair samples, which were
therefore measured and analyzed twice. Multiplied by 6 different investigated
analytes this results in 2676 individual measurements analyzed by one or both
experts. All samples were also analyzed using the algorithm. These numbers are
further visualized in supplementary Figure 9.
The measured analytes were cortisol, cortisone, testosterone, progesterone and
dehydroepiandrosterone (DHEA). The selection of analytes and the procedure
follows [16]. The second most sensitive transition of cortisol (indicated by
the Q suffix) was also included in the selection of analytes.
To be able to programmatically access the raw mass chromatograms, the data was
converted from a proprietary file format to .mzML, using the ProteoWizard
MSConvert software [18, 19]. Then the supplied software implementation (see
2.5) was used to process the data.
For the manual quantification data, the AUC of both analyte and IS were marked
and extracted by the experts using the AB SCIEX Analyst software version
1.6.2.
##### Algorithm settings
Algorithm hyperparameters were mostly the same across analytes with the
exception of fixed RT windows (expected IS RT and acceptable range) which had
to be specified for the IS of progesterone, testosterone and DHEA, because of
wrong IS peaks close to the correct one.
The absolute measured concentration (analyte AUC divided by IS AUC) was used
for all evaluative statistics. Manual compiled relative concentrations were
also scaled by the $\beta$ of the algorithms concentration calibration
regression. This increases comparability between analytes for calculations
which include all analytes in comparison to using the relative concentrations.
And by not using the manually compiled values to create a separate calibration
regression systematic errors are not overestimated.
##### Statistical analyses
Statistical analyses were performed with R version 3.6 [20], latent variable
models were estimated with lavaan [21].
## 4 Results
Table 2 contains descriptive statistics for measured relative concentrations
as well as SNR of each analyte.
Analyte | N | Concentration (pg/mg) | SNR (Algorithm)
---|---|---|---
| | Algorithm | Expert 1 | Analyte | IS
Cortisol | 389 | 8.07 | (24.56) | 8.10 | (25.53) | 15.12 | (5.57) | 25.23 | (1.74)
Cortisol Q | 390 | 8.01 | (23.73) | 8.34 | (24.62) | 11.61 | (5.06) | 25.32 | (1.64)
Cortisone | 446 | 16.11 | (16.49) | 15.91 | (16.33) | 20.77 | (4.80) | 24.19 | (2.55)
Testosterone | 178 | 4.09 | (26.43) | 4.00 | (25.46) | 5.35 | (6.51) | 32.46 | (8.29)
Progesterone | 415 | 8.09 | (64.21) | 7.73 | (61.16) | 13.39 | (8.65) | 30.81 | (4.34)
DHEA | 414 | 1.89 | (2.60) | 1.85 | (2.59) | 13.74 | (6.52) | 34.52 | (3.34)
Table 2: Descriptive statistics by analyte. Means followed by standard
deviations in parentheses. All cases in which both the expert and the
algorithm could detect a peak are included.
##### Non-detectables (ND)
Chromatograms where an expert cannot find a peak are marked as non-detectable
(ND). In these cases, the analyte concentration is presumed to be less than
the measuring instruments lower limit of detection. These cases were compared
to cases when the algorithm cannot mark a peak with $\mathrm{AUC}>0$ in the
data.
Contingency data for successful peak marking is displayed in Table 3.
Sensitivity is close to $100\%$ across experts and analytes with the exception
of testosterone where it is just over $95\%$. Specificity on the other hand is
generally very low. Testosterone had the highest rate of NDs with 261 of the
446 samples marked as ND by expert 1. The algorithms specificity for
testosterone NDs was $54\%$. Comparing all samples of all analytes measured by
expert 1, there were 235 false positives.
Analyte | Expert 1 ($n=446$) | Accuracy (FP class.)
---|---|---
| TP | FN | FP | TN | Sensitivity | Specificity | A+S | AUC | SNR
Cortisol | 389 | 2 | 16 | 39 | 99.49 % | 70.91 % | 1.00 | 0.98 | 0.87
Cortisol Q | 390 | 0 | 45 | 11 | 100.00 % | 19.64 % | 0.91 | 0.92 | 0.75
Cortisone | 446 | 0 | 0 | 0 | 100.00 % | - | - | - | -
Testosterone | 178 | 7 | 121 | 140 | 96.22 % | 53.64 % | 0.95 | 0.88 | 0.95
Progesterone | 415 | 0 | 30 | 1 | 100.00 % | 3.23 % | 0.88 | 0.88 | 0.81
DHEA | 414 | 0 | 23 | 9 | 100.00 % | 28.12 % | 0.95 | 0.94 | 0.89
All | 2232 | 9 | 235 | 200 | 99.60 % | 45.98 % | 0.95 | 0.95 | 0.91
Analyte | Expert 2 ($n=221$) | Accuracy (FP class.)
| TP | FN | FP | TN | Sensitivity | Specificity | A+S | AUC | SNR
Cortisol | 199 | 4 | 4 | 14 | 98.03 % | 77.78 % | - | - | -
Cortisol Q | 205 | 2 | 10 | 4 | 99.03 % | 28.57 % | 1.00 | 1.00 | 0.93
Cortisone | 220 | 0 | 1 | 0 | 100.00 % | 0.00 % | - | - | -
Testosterone | 106 | 5 | 45 | 65 | 95.50 % | 59.09 % | 0.98 | 0.86 | 0.97
Progesterone | 217 | 0 | 4 | 0 | 100.00 % | 0.00 % | - | - | -
DHEA | 212 | 2 | 4 | 3 | 99.07 % | 42.86 % | - | - | -
All | 1159 | 13 | 68 | 86 | 98.89 % | 55.84 % | 0.96 | 0.96 | 0.94
Table 3: Contingency data for peak detection. The algorithms sensitivity of
peak detection was very high. Specificity was very low, mainly due to false
positives (FP). Binomial regression was able to classify FPs (abbreviated FP
class.) with high accuracy (area under the receiver operating characteristic).
Equal group sizes were randomly sampled for hormones with at least 10 false
positives. A+S indicates the regression model with both AUC and SNR as
predictors.
It was investigated whether analyte peak AUC or analyte SNR could be used to
predict false positives. Three binomial regression models created. A combined
model, $Y_{\mathrm{A+S}}=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}$, where $Y$
is the probability of being an FP and $X_{1}$ and $X_{2}$ were AUC and SNR
respectively, as well as models which just had AUC and SNR as predictors. As
the number of FPs was greatly lower than the number of TPs, the same number of
TPs was randomly sampled to obtain equal group sizes. All three models were
calculated for all analytes with at least 10 FPs as well as for the combined
data. Overall accuracy (AUROC) for peak AUC was $0.91$, for SNR it was $0.95$
and the combined model had an accuracy of $0.95$. Accuracies for all
calculated models are displayed in Table 3. Figure 7(a) is a scatterplot of
the resampled cases with peak AUC and SNR on the axes. Figure 7(b) shows the
receiver operating characteristic (ROC) for both dimensions.
(a) Scatterplot of expert 1 true and false positives.
(b) ROC for classifying expert 1 FPs. Accuracy (AUROC) of AUC: 0.95, SNR:
0.89, combined model: 0.95. The AUC model graph and the the combined model
graph are oversecting.
Figure 7: Binominal regression classifiers for false positive peak detection.
AUC and SNR as measured by the algorithm were investigated for their ability
to predict false positives. These Figures show data of all analytes for expert
1. True positives were randomly sampled to obtain equal group sizes.
##### Correlation & differences
Pearson correlation coefficients of the log transformed concentrations of all
samples and analytes were all $r=0.98$ between algorithm and expert 1 and 2 as
well as between the experts. Table 4 shows single correlation coefficients for
all analytes and raters. The supplementary Figure 10 shows scatter plots for
all analytes and the supplementary Figure 11 shows Bland-Altman plots of the
differences between raters.
Analyte | Algorithm | Algorithm | Expert 1
---|---|---|---
| Expert 1 | Expert 2 | Expert 2
Cortisol | 0.981 | 0.981 | 0.985
Cortisol Q | 0.950 | 0.972 | 0.952
Cortisone | 0.998 | 0.983 | 0.988
Testosterone | 0.936 | 0.951 | 0.945
Progesterone | 0.970 | 0.964 | 0.970
DHEA | 0.953 | 0.898 | 0.952
All | 0.983 | 0.979 | 0.983
Table 4: Pearson correlation coefficients for log transformed hormone
concentrations.
##### Latent variable model
The data contained several LC–MS/MS measurements of the same original samples,
rated by the two experts and the algorithm (see supplementary Figure 9). An
analogous structural equation model was created. Expert and Algorithm ratings
were considered manifest variables, LC–MS/MS data was considered first order
latent variables and _true_ hormone concentration as second order latent
variable.
Figure 8: Path diagram of the measurement model. Rectangles represent manifest
variables (Analyte concentrations estimated by experts $E_{1}$ and $E_{2}$ as
well as by the algorithm $A$) for each LC–MS/MS sample. Rounded rectangles
represent latent factors. First order latent factors are the measured samples,
the second order latent factor is the hormone concentration. Residual
variances are not shown in the diagram.
Based on the latent state-trait theory [22], the total variance of analyte
concentration per specimen replicate was decomposed into estimates of (1)
reliability, (2) replicate specificity and (3) replicate consistency for each
quantitation method (i.e., algorithm, rater 1, and rater 2). The model’s
structure and its parameters are shown in Figure 8. Additional method-specific
variance components [see 23] were neither numerically identifiable nor
necessary to account for the covariance among cortisol cortisone, and DHEA
quantitations. The fit between the observed and the model-implied covariance
structure was close for these analytes ($\chi^{2}\geq 10.417$,
$\mathrm{df}=6$, $p\geq 0.108$, $\mathrm{RMSEA}\leq 0.059$, $\mathrm{SRMR}\leq
0.019$). Accordingly, the algorithm measured the same construct as the human
raters did. By contrast, the models for progesterone and testosterone
quantitation showed a considerable lack of fit ($\chi^{2}\geq 21.632$,
$\mathrm{df}=6$, $p\geq 0.001$, $\mathrm{RMSEA}\leq 0.137$, $\mathrm{SRMR}\leq
0.307$) due to method-related residual covariance among the human raters, and
replicate-related residual covariance among all methods, respectively. That
is, the quantitation of progesterone by the algorithm differed slightly from
the human raters, whereas there were differences in the within-replicate
variance of testosterone irrespective of the quantatitation method.
Reliability, specificity, and consistency of each quantitation method and
analyte are reported in Table 5.
| Consistency | Specificity | Reliability
---|---|---|---
| $A$ | $E_{1}$ | $E_{2}$ | $A$ | $E_{1}$ | $E_{2}$ | $A$ | $E_{1}$ | $E_{2}$
Cortisol | 0.801 | 0.787 | 0.732 | 0.185 | 0.182 | 0.169 | 0.986 | 0.969 | 0.901
Cortisol Q | 0.760 | 0.762 | 0.708 | 0.206 | 0.207 | 0.192 | 0.966 | 0.968 | 0.900
Cortisone | 0.774 | 0.757 | 0.633 | 0.220 | 0.215 | 0.180 | 0.994 | 0.971 | 0.813
Testosterone | 0.869 | 0.896 | 0.699 | 0.102 | 0.105 | 0.082 | 0.971 | 1.002 | 0.782
Progesterone | 0.488 | 0.495 | 0.439 | 0.498 | 0.506 | 0.448 | 0.986 | 1.001 | 0.888
DHEA | 0.819 | 0.839 | 0.663 | 0.140 | 0.144 | 0.114 | 0.959 | 0.982 | 0.776
Table 5: Consistency, specificity, and reliability of analyte quantitation
(2nd replicate) by the algorithm ($A$), and both expert raters ($H_{1}$,
$H_{2}$).
## 5 Discussion & Limitations
Measurements automatically calculated by the algorithm correlate very strongly
with those manually compiled by experts. The algorithm returns far less non-
detectables than expert raters. It was demonstrated that these peak detection
false positives can be classified by the (algorithm measured) peak AUC and
SNR. This leads us to recommend setting a minimum threshold for one or both of
these values for each analyte with high ND rates (e.g. testosterone).
Latent variable models showed that the algorithm is able to measure the
chromatograms trait very similar to the experts, while being closer to the
more experienced experts results. Consistency might be slightly overestimated
due to list-wise exclusion.
Subsequent studies should experiment with modifications of algorithm
parameters and the effect of disabling for example, the partial convex hull
peak boundary correction which was designed to imitate expert rating behavior
or use the smoothed peak area for calculating AUC. These modifications could
potentially decrease model error.
Furthermore, it should be investigated if the algorithm is reliable with other
types of data. These could be other LC–MS/MS mass chromatogram types as well
as other types of chromatograms and spectrograms. If nonlinear drift would
emerge as a problem when analyzing different types of data, we suggest
investigating the addition of a high-pass filter.
### 5.1 Conclusion
In conclusion, the algorithm presented here allows fast, automated, reliable
and valid computational peak detection and quantification in LC–MS/MS and is
also able to quantify SNR automatically.
## 6 Acknowledgements
Special thanks to the employees of Dresden LabService GmbH and Christian
Rupprecht (Department of Engineering Science, University of Oxford) for their
continuous support and suggestions. This work was supported by the German
Research Foundation (DFG, Grant No. SFB 940/2).
## 7 Conflict of interest
The authors have no conflict of interest to declare.
## References
## References
* Yang et al. [2009] C. Yang, Z. He, W. Yu, Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis, BMC Bioinformatics 10 (1) (2009) 4, doi:10.1186/1471-2105-10-4.
* Plesser [2018] H. E. Plesser, Reproducibility vs. Replicability: A Brief History of a Confused Terminology, Frontiers in Neuroinformatics 11 (2018) 76, doi:10.3389/fninf.2017.00076.
* Davies [2012] E. R. Davies, Computer and machine vision: theory, algorithms, practicalities, Academic Press, 2012\.
* Daubechies [1992] I. Daubechies, Ten Lectures on Wavelets, vol. 61, Society for Industrial and Applied Mathematics, doi:10.1137/1.9781611970104, 1992\.
* Grushka [1972] E. Grushka, Characterization of exponentially modified Gaussian peaks in chromatography, Analytical Chemistry 44 (11) (1972) 1733–1738, doi:10.1021/ac60319a011.
* Torrence and Compo [1998] C. Torrence, G. P. Compo, A practical guide to wavelet analysis, Bulletin of the American Meteorological society 79 (1) (1998) 61–78.
* Neubeck and Gool [2006] A. Neubeck, L. V. Gool, Efficient Non-Maximum Suppression, in: 18th International Conference on Pattern Recognition (ICPR'06), vol. 3, IEEE, IEEE, 850–855, doi:10.1109/icpr.2006.479, 2006\.
* Graham [1972] R. Graham, An efficient algorith for determining the convex hull of a finite planar set, Information Processing Letters 1 (4) (1972) 132–133, doi:10.1016/0020-0190(72)90045-2.
* Wells et al. [2011] G. Wells, H. Prest, C. W. Russ IV, Why use signal-to-noise as a measure of MS performance when it is often meaningless?, Advanstar Communications Inc. .
* Dolan [2012] J. W. Dolan, When Should an Internal Standard be Used?, LCGC North America 30 (6) (2012) 474–480.
* Sargent [2013] M. Sargent, Guide to achieving reliable quantitative LC-MS measurements, RSC Analytical Methods Committee, URL https://www.rsc.org/images/AMC%20LCMS%20Guide_tcm18-240030.pdf, 2013\.
* Klammer et al. [2007] A. A. Klammer, X. Yi, M. J. MacCoss, W. S. Noble, Improving Tandem Mass Spectrum Identification Using Peptide Retention Time Prediction across Diverse Chromatography Conditions, Analytical Chemistry 79 (16) (2007) 6111–6118, doi:10.1021/ac070262k.
* Guo et al. [1987] D. Guo, C. T. Mant, R. S. Hodges, Effects of ion-pairing reagents on the prediction of peptide retention in reversed-phase high-resolution liquid chromatography, Journal of Chromatography A 386 (1987) 205–222, doi:10.1016/s0021-9673(01)94598-4.
* Martens et al. [2010] L. Martens, M. Chambers, M. Sturm, D. Kessner, F. Levander, J. Shofstahl, W. H. Tang, A. Römpp, S. Neumann, A. D. Pizarro, L. Montecchi-Palazzi, N. Tasman, M. Coleman, F. Reisinger, P. Souda, H. Hermjakob, P.-A. Binz, E. W. Deutsch, mzML—a Community Standard for Mass Spectrometry Data, Molecular & Cellular Proteomics 10 (1), doi:10.1074/mcp.r110.000133.
* Miller et al. [2019] R. Miller, K. Schmidt, C. Kirschbaum, S. Enge, Chronic Stress and Executive Functioning: A longitudinal perspective, doi:10.17605/OSF.IO/KDMX5, URL osf.io/kdmx5, 2019.
* Gao et al. [2013] W. Gao, T. Stalder, P. Foley, M. Rauh, H. Deng, C. Kirschbaum, Quantitative analysis of steroid hormones in human hair using a column-switching LC–APCI–MS/MS assay, Journal of Chromatography B 928 (2013) 1–8, doi:10.1016/j.jchromb.2013.03.008.
* Schmidt et al. [2020] K. Schmidt, S. Enge, R. Miller, Reconsidering the construct validity of self-reported chronic stress: A multidimensional item response theory approach., Psychological Assessment 32 (11) (2020) 997–1014, doi:10.1037/pas0000829.
* Kessner et al. [2008] D. Kessner, M. Chambers, R. Burke, D. Agus, P. Mallick, ProteoWizard: open source software for rapid proteomics tools development, Bioinformatics 24 (21) (2008) 2534–2536, doi:10.1093/bioinformatics/btn323.
* Chambers et al. [2012] M. C. Chambers, B. Maclean, R. Burke, D. Amodei, D. L. Ruderman, S. Neumann, L. Gatto, B. Fischer, B. Pratt, J. Egertson, K. Hoff, D. Kessner, N. Tasman, N. Shulman, B. Frewen, T. A. Baker, M.-Y. Brusniak, C. Paulse, D. Creasy, L. Flashner, K. Kani, C. Moulding, S. L. Seymour, L. M. Nuwaysir, B. Lefebvre, F. Kuhlmann, J. Roark, P. Rainer, S. Detlev, T. Hemenway, A. Huhmer, J. Langridge, B. Connolly, T. Chadick, K. Holly, J. Eckels, E. W. Deutsch, R. L. Moritz, J. E. Katz, D. B. Agus, M. MacCoss, D. L. Tabb, P. Mallick, A cross-platform toolkit for mass spectrometry and proteomics, Nature Biotechnology 30 (10) (2012) 918–920, doi:10.1038/nbt.2377.
* R Core Team [2020] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/, 2020.
* Rosseel [2012] Y. Rosseel, lavaan: An R Package for Structural Equation Modeling, Journal of Statistical Software 48 (2) (2012) 1–36, URL http://www.jstatsoft.org/v48/i02/.
* Steyer et al. [1999] R. Steyer, M. Schmitt, M. Eid, Latent state–trait theory and research in personality and individual differences, European Journal of Personality 13 (5) (1999) 389–408, doi:10.1002/(SICI)1099-0984(199909/10)13:5¡389::AID-PER361¿3.0.CO;2-A.
* Geiser and Lockhart [2012] C. Geiser, G. Lockhart, A comparison of four approaches to account for method effects in latent state–trait analyses., Psychological methods 17 (2) (2012) 255, doi:10.1037/a0026977.
## Appendix A Supplement
Abbreviation | Explanation
---|---
AUC | Area under the curve
AUROC | Area under the ROC = Accuracy
IS | Internal standard
LC–MS/MS | Liquid chromatography tandem mass spectrometry
ROC | Receiver operating characteristic
SNR | Signal to noise ratio
Table 6: Abbreviations Figure 9: Visualization of sample number and
dependence. 230 different hair samples were split and prepared to 446 LC-MS/MS
samples. This Figure displays the resulting number of chromatograms and how
many of them were repeated measurements. $\Sigma$ indicates the total sum
including split samples and $\cap$ the number of intersecting original
samples.
(a) Cortisol.
(b) Cortisol Q.
(c) Cortisone.
(d) Testosterone.
(e) Progesterone.
(f) DHEA.
Figure 10: Scatter plots of expert 1 and algorithm rating for each analyte
concentration.
(a) Differences of algorithm and expert 1.
(b) Differences of algorithm and expert 2.
(c) Differences between the experts.
Figure 11: Bland-Altman plots of log transformed measures. 95% confidence
intervals for the prediction of mean difference are indicated by the red
lines. The blue line indicates mean difference. Note that the vertical axis is
scaled using the hyperbolic arc-sine function to better depict differences of
the skewed data.
|
# Resolving Structure in the Debris Disk around HD 206893 with ALMA
Ava Nederlander Astronomy Department and Van Vleck Observatory, Wesleyan
University, Middletown, CT 06459, USA A. Meredith Hughes Astronomy
Department and Van Vleck Observatory, Wesleyan University, Middletown, CT
06459, USA Anna J. Fehr Astronomy Department and Van Vleck Observatory,
Wesleyan University, Middletown, CT 06459, USA Kevin M. Flaherty Department
of Astronomy and Department of Physics, Williams College, Williamstown, MA
01267, USA Kate Y. L. Su Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721, USA Attila Moór Konkoly Observatory,
Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út
15-17, H-1121 Budapest, Hungary ELTE Eötvös Loránd University, Institute of
Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary Eugene Chiang
Department of Astronomy, University of California at Berkeley, CA 94720-3411,
USA Department of Earth and Planetary Science, University of California at
Berkeley, CA 94720-4767, USA Sean M. Andrews Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA David J.
Wilner Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden St.,
Cambridge, MA 02138, USA Sebastian Marino Institute of Astronomy, University
of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
(Received Jun 17, 2020; Accepted Jan 11, 2021)
###### Abstract
Debris disks are tenuous, dusty belts surrounding main sequence stars
generated by collisions between planetesimals. HD 206893 is one of only two
stars known to host a directly imaged brown dwarf orbiting interior to its
debris ring, in this case at a projected separation of 10.4 au. Here we
resolve structure in the debris disk around HD 206893 at an angular resolution
of 0$\farcs$6 (24 au) and wavelength of 1.3 mm with the Atacama Large
Millimeter/submillimeter Array (ALMA). We observe a broad disk extending from
a radius of $<51$ au to $194^{+13}_{-2}$ au. We model the disk with a
continuous, gapped, and double power-law model of the surface density profile,
and find strong evidence for a local minimum in the surface density
distribution near a radius of 70 au, consistent with a gap in the disk with an
inner radius of $63^{+8}_{-16}$ au and width $31^{+11}_{-7}$ au. Gapped
structure has been observed in four other debris disks – essentially every
other radially resolved debris disk observed with sufficient angular
resolution and sensitivity with ALMA – and could be suggestive of the presence
of an additional planetary-mass companion.
## 1 Introduction
Debris disks are a common outcome of the star and planet formation process,
and serve as a signpost of mature planetary systems. Bright debris disks are
observed around some 25-30% of main sequence stars (Trilling et al., 2008;
Thureau et al., 2014; Montesinos et al., 2016; Sibthorpe et al., 2018). The
true incidence is likely to be higher, since the sensitivity of current
observations limits our ability to detect debris disks to those systems that
are at least an order of magnitude more luminous than the disk generated by
the Solar System’s Kuiper Belt (Matthews et al., 2014; Hughes et al., 2018,
and references therein). While the dust in debris disks is certainly worthy of
study in its own right, it also presents an opportunity to trace the
properties of planetary systems.
As direct imaging searches for exoplanets have matured, a small number of
systems have been discovered in which both directly imaged companions and
debris are present (e.g., Marois et al., 2008; Lagrange et al., 2009; Rameau
et al., 2013; Mawet et al., 2015; Konopacky et al., 2016; Meshkat et al.,
2017). These systems are extremely valuable dynamical laboratories. While
debris dust is subject to a number of different forces – including radiation
pressure, stellar winds, gas drag, and gravity – the largest grains, imaged at
millimeter wavelengths by facilities like the Atacama Large
Millimeter/submillimeter Array (ALMA), are effectively impervious to all of
the major forces except for gravity and collisions (e.g., Su et al., 2005;
Strubbe & Chiang, 2006; Wyatt, 2008; Wilner et al., 2011; Löhne et al., 2012).
One useful dynamical concept is that of the “chaotic zone,” which is a term
from 3-body dynamics that refers to the radial extent around a planet within
which stable orbits for test particles do not exist (Wisdom, 1980; Lecar et
al., 2001). The extent of the chaotic zone depends on the mass of the planet
and its semimajor axis and eccentricity. For a planet sculpting a debris belt,
the separation between the planet’s location and the edge of the debris belt
will be equal to the extent of the planet’s chaotic zone (e.g., Quillen, 2006;
Chiang et al., 2009; Mustill & Wyatt, 2012; Morrison & Malhotra, 2015).
Therefore, by locating the edge of the debris belt and measuring its
relationship to the observed location of a directly imaged planet, we can
place a dynamical constraint on the mass of the planet. Such dynamical
constraints on directly imaged companions are valuable because otherwise the
masses of directly imaged companions are typically estimated using models of
the luminosity evolution of planets and brown dwarfs, which are typically
poorly calibrated and sensitive to the often highly uncertain age of the
system (e.g., Chabrier et al., 2000; Baraffe et al., 2003). Meaningful
dynamical constraints on the mass of a directly imaged companion have already
been shown to be possible with data from ALMA and the Submillimeter Array
(SMA) in the case of the HR 8799 system (Wilner et al., 2018).
HD 206893, an F5V star located 40.8 pc from Earth (Gaia Collaboration et al.,
2016, 2018), is one of two known systems in which a brown dwarf orbits
interior to a debris ring (Milli et al., 2017). The other is HR 2562
(Konopacky et al., 2016). The debris disk around HD 206893 has been previously
detected by Williams & Andrews (2006), who did not have sufficient angular
resolution to measure the disk structure. The disk outer radius has been
marginally resolved by Herschel at a wavelength of 70 $\mu$m, and analysis of
the Spectral Energy Distribution (SED) suggests an inner radius of $\sim$50 au
(Moór et al., 2011). However, previous studies of debris-bearing systems have
found that there is significant scatter in the relationship between spatially
resolved disk size and the size estimated from the SED (e.g., Booth et al.,
2013; Pawellek et al., 2014; Morales et al., 2016), rendering spatially
resolved observations critical for studying the interplay between the
companion and the disk inner edge.
The companion orbits the star at a projected separation of $10.4\pm 0.1$ au,
which is clearly interior to the projected debris ring radius (Milli et al.,
2017). While it has so far been detected in two epochs of observation 10
months apart, confirming common proper motion with the star, the constraints
on its orbital properties are not yet strong, and its eccentricity and
alignment with the disk plane are still highly uncertain. The initial mass
estimates of the companion based on its H-band flux range from 24 to 73 MJup,
depending in part on the poorly constrained age of the system (They assumed
0.2-2 Gyr, later refined to $250_{-200}^{+450}$ Myr by Delorme et al., 2017).
Its spectral type lies among L5-L9 field dwarfs, although it is currently the
reddest known object among young and dusty L dwarfs in the field; it is not
yet clear whether its large color excess arises from a dusty atmosphere, or
whether it is the result of disk reddening (Milli et al., 2017). Analysis of
additional SPHERE data find a best-fit mass in the range of 15-30 MJup, while
noting that the data are compatible with everything from a 12 MJup companion
at an age of 50 Myr to a 50 MJup Hyades-age brown dwarf (Delorme et al.,
2017). Follow-up observations combining direct imaging constraints with radial
velocity data suggest the presence of an additional $\sim$15 MJup companion at
a separation of 1.4–2.6 au from the star (Grandjean et al., 2019).
Current knowledge is consistent with a scenario in which the companion
truncates the disk, but a measurement of the disk inner radius is necessary to
confirm this scenario. Here we observe the disk with ALMA (Section 2), examine
its radial intensity profile (Section 3) and measure the location of the inner
radius of the debris belt (Section 4). We also report a detection of a local
minimum of the surface density near a radius of 70 au in the outer debris
belt. If the minimum is due to a gap, this would be the fourth debris disk to
exhibit gapped structure after HD 107146 (Ricci et al., 2015), HD 92945
(Marino et al., 2019), and HD 15115 (MacGregor et al., 2019) – a feature that
appears to be common in debris disks observed with sufficient sensitivity and
angular resolution to resolve substructure. The presence of gapped structure
has several possible explanations, with the most compelling being an
additional planetary-mass companion (Section 5). We summarize the conclusions
of our investigation in Section 6.
## 2 Observations
We observed the HD 206893 debris disk in six scheduling blocks between June
and September 2018 (ALMA project 2018.1.00193.S, PI Hughes). Three different
antenna configurations with a baseline lengths ranging from 15 to 1246 m are
included in the data. There are four spectral windows, each 1.875 GHz wide to
provide maximum sensitivity to dust continuum emission. The three continuum
windows were centered on frequencies of 228.5, 216.5, and 214.5 GHz, with
channel spacings of 15.6 MHz (20.3 km s-1). One spectral window was centered
on the rest frequency of the CO(2-1) molecular line (230.53800 GHz) with a
channel spacing of 976 kHz (1.27 km s-1). Table 1 lists the dates, times,
number of antennas, baseline lengths, time on-source, the average precipitable
water vapor, synthesized beam geometry, and rms noise values for each
scheduling block. The quasar J2148+0657 was used as both the flux and passband
calibrator for all scheduling blocks. J2131-1207 was used as the gain
calibrator for the two June tracks, and J2146-1525 was used as the gain
calibrator for all other tracks.
Calibration, reduction, and imaging were carried out using standard tasks from
the Common Astronomy Software Applications (CASA) package (McMullin et al.,
2007), version 5.1.1-5. The statistical weights for each visibility were
recalculated using the variance of visibilities on nearby points in the uv
plane, as described in Flaherty et al. (2017). The ALMA technical handbook
states that the expected absolute flux calibration uncertainty should be 5% to
10% for these Band 6 observations.
Table 1: ALMA Observations of HD 206893
Date/Time (UT) | # Antennas | Baseline lengths | On-source time | Average pwv | Beam Major Axis | Beam Minor Axis | Beam PA | rms noise
---|---|---|---|---|---|---|---|---
| | (m) | (min) | (mm) | (”) | (”) | (∘) | ($\mu$Jy beam-1)
Jun27/06:52 | 46 | 15-312 | 57.2 | 1.3 | 1.62 | 1.26 | -79.0 | 12.8
Jun27/07:49 | 46 | 15-312 | 57.0 | 1.1 | 1.62 | 1.25 | -68.8 | 11.7
Aug30/02:23 | 45 | 15-782 | 68.9 | 1.7 | 0.73 | 0.57 | 76.8 | 12.5
Aug30/03:34 | 45 | 15.1-782 | 68.9 | 1.6 | 0.75 | 0.58 | 80.7 | 12.3
Sep10/01:22 | 46 | 15-1213 | 71.8 | 2.1 | 0.56 | 0.43 | 53.5 | 13.0
Sep17/01:22 | 45 | 15-1246 | 69.0 | 0.6 | 0.49 | 0.34 | 54.7 | 10.0
Combined | $\cdots$ | 15-1246 | 392.8 | $\cdots$ | 0.71 | 0.58 | 66.6 | 5.5
## 3 Results
The four spectral windows were combined to generate images of the dust
continuum emission. Figure 1 shows a naturally weighted image of the combined
data set generated using the CASA task tclean. We applied a 200 k$\lambda$
taper to the interferometric data to bring out the large-scale structure of
the debris ring.
Figure 1: (Left) Naturally weighted ALMA image of the 1.3 mm continuum
emission from the HD 206893 system. (Right) Same image with a visibility-
domain taper of 200 k$\lambda$ applied to bring out the large-scale structure
of the source. In both panels, contour levels are [-2,2,4,6]$\times\sigma$,
where $\sigma$ is the rms noise in the image: 5.5 $\mu$Jy beam-1 for the
naturally weighted image and 6.0 $\mu$Jy beam-1 for the image with the taper.
The hatched ellipse in the lower left corner represents the size and
orientation of the synthesized beam: 0$\farcs$71$\times$0$\farcs$58 for the
naturally weighted image and 0$\farcs$9$\times$1$\farcs$0 for the tapered
image. The star symbol represents the pointing center of the observations,
i.e., the expected position of the star including a proper motion correction,
and the dot represents the position of the brown dwarf companion directly
imaged by Milli et al. (2017).
Using the MIRIAD task cgcurs, we measure an integrated flux density of $670\pm
30$ $\mu$Jy at a wavelength of 1.3 mm enclosed within the 3$\sigma$ contours
of the tapered image (not including the systematic flux calibration
uncertainty). Using the CASA viewer task, we measure the extent of the region
enclosed by the 3$\sigma$ contours, which extends to a diameter of 7$\farcs$3
(corresponding to a radial extent of 150 au) along the broadest dimension, and
5$\farcs$0 along the narrowest dimension. The morphology of the disk traced by
the best-detected (SNR$>$4) region can be described as an ellipse (i.e., an
inclined ring) with some extra central flux in the interior. There is a clear
deficit of flux between the outermost bright ring and the central flux
component, which exhibits a diameter along the major axis of approximately
2$\farcs$4 (corresponding to a radius of 49 au).
Figure 2 plots the azimuthally averaged intensity profile as a function of
linear separation from the central star, assuming that the disk has circular
geometry and deprojecting the intensity profile along elliptical contours in
the sky plane using the best-fit inclination of 45∘ and position angle of 60∘,
which are the values obtained from the best-fit Markov Chain Monte Carlo
(MCMC) model of a double power-law surface density profile with a radial gap,
as described in Section 4 below. The figure also shows the projected
separation of the brown dwarf (BD) companion, HD 206893 B, as a dashed
vertical line. The intensity profile reveals a central peak that is broader
than the synthesized beam (Full Width Half Maximum (FWHM) $\simeq 26$ au),
indicating some disk flux located at radii close to the star. There is a hint
of an inner edge to the disk outside the orbit of the BD with a local peak at
a radius of 38 au, but the slight bump is small compared to the uncertainty on
the flux density and is likely insignificant. There is a local minimum in flux
at a separation of 77 au, and the brightest peak occurs at a separation of 115
au – the estimated relative uncertainty on all of these radii is $\pm$5 au,
the average of the major axis and minor axis lengths of the synthesized beam
divided by the average signal-to-noise ratio (SNR) of the disk of $\sim$5\. We
further analyze the disk structure, including potential deviations from
axisymmetry, in Section 4 below.
Figure 2: Azimuthally averaged radial intensity profile of the naturally
weighted image, deprojected along elliptical contours assuming a circular
geometry and an inclination to the line of sight of 45∘ and position angle of
60∘. The blue shaded region represents the standard error of the mean for each
radial bin, where the standard deviation is calculated for all pixels within
the bin and then divided by the square root of the number of beams sampled.
The size of the synthesized beam is indicated by the gray Gaussian marked
“Beam.” The 10.4 au projected separation of HD 206893 B (Milli et al., 2017)
is marked with a vertical dashed line.
The surface brightness of the central peak, $19.7\pm 5.5$ $\mu$Jy beam-1, is
approximately consistent with the expected flux density of 12.6 $\mu$Jy for a
star with a temperature of 6500 K and radius 1.26 R⊙ (Delorme et al., 2017),
when approximating the star as a blackbody (in this case, the systematic flux
uncertainty provides the largest source of error in the comparison). While the
properties of stars at millimeter wavelengths are not well understood, and can
deviate significantly from the expected blackbody flux for a variety of
reasons including chromospheric emission and flaring (Cranmer et al., 2013;
White et al., 2018, 2019, 2020), this estimate provides a ballpark figure to
guide our interpretation of the emission morphology. As we demonstrate in
Section 4 below, there is a statistically significant contribution from a disk
component located around the central point source that is only marginally
resolved. When the disk is modeled with a gap, which allows for an inner disk
component that can contribute to the flux around the point source, the stellar
flux in the model is $17^{+5}_{-6}$ $\mu$Jy, consistent with expectations –
but when the disk is modeled without a gap, the inner radius of the disk gets
pushed to larger radii (presumably to avoid filling the gap with too much
flux) and as a result the stellar flux in the model becomes elevated to
$35^{+2}_{-3}$ $\mu$Jy in an attempt to reproduce the flux from the inner
regions of the debris belt. The measured position of the central flux density
peak is offset from the proper motion corrected Gaia position of the star by
0$\farcs$48, which is marginally significant at the 3$\sigma$ level (taking
into account only the relative position uncertainty of the angular resolution
$\theta$ divided by the SNR); however, subsequent analysis including detailed
modeling of both the disk and the star with MCMC techniques (Section 4 and
Table 3 below) yields no statistically significant difference between the
expected and observed position of the star. In this case the relative
uncertainty is the larger source of uncertainty compared to the theoretical
astrometric uncertainty of $\sim$40 mas, calculated for the frequency,
baseline length, and SNR of our observation based on the equation in Chapter
10.5.2 of the ALMA technical handbook.
We did not detect any gas emission within a range of $\pm 10$ km s-1 around
the systemic velocity of the source, with a 3-sigma upper limit of
approximately 40 mJy km s-1. The upper limit was measured by generating a
moment 0 map for the $\pm 10$ km s-1 velocity range and then integrating the
emission enclosed by the 2-sigma contours for the continuum emission imaged
with the 200 k$\lambda$ taper. We assumed a systemic velocity of -12.45 km s-1
in the heliocentric rest frame (Grandjean et al., 2019; Gaia Collaboration et
al., 2018), which translates to -5.07 km s-1 in the LSRK frame. Our upper
limit translates to a flux of $3.1\times 10^{-22}$ W m-2, which is orders of
magnitude above the predicted CO flux of $5.6\times 10^{-25}$ W m-2 from Kral
et al. (2017), indicating that the upper limit from this observation cannot
place useful constraints on the composition of the molecular gas.
## 4 Analysis
In this section, we analyze the visibility data from the HD 206893 millimeter
emission to characterize in detail the spatial distribution of dust in the
system. We adopt a modeling approach that combines a parametric ray-tracing
code to generate synthetic model images of an axisymmetric disk with an MCMC
fitting algorithm, allowing us to characterize the radial distribution of dust
in the system.
### 4.1 Modeling Formalism
To measure the disk structure, we generated synthetic model images of debris
disks with varying geometries that were compared with the interferometric
data. For each model image, we compared the data with the model in the
visibility domain and calculated a $\chi^{2}$ metric to evaluate the goodness
of fit between the model and the data.
To generate sky-projected model images of debris disks, we use a ray-tracing
code described in Flaherty et al. (2015)111https://github.com/kevin-
flaherty/disk_model3, which is a Python adaptation of an earlier IDL code by
Rosenfeld et al. (2013). The code translates parametric models of the density
and temperature of dust and gas to a grid of density and temperature that is
then rotated and integrated along the line of sight to produce a sky-projected
map of the millimeter intensity. While there is a fully 3-D version of the
code available for eccentric disks, we use the 2-D version that assumes a
circular, axisymmetric flux distribution with a linearly increasing scale
height, since there is no a priori evidence for deviations from axisymmetry
like a clear offset of the disk center from the star position (and indeed, the
assumption is verified a posteriori by the lack of residuals after the best-
fit model is subtracted from the data). We assume a dust opacity of 2.3 cm2
g-1 (Beckwith et al., 1990), yielding optically thin emission for the dust in
all of the models within the parameter space explored by the MCMC chains.
Since density and temperature are degenerate for optically thin emission, we
assume that these large grains are in blackbody equilibrium with the central
star and calculate a dust grain temperature of
$T_{\mathrm{dust}}=\left(\frac{L_{*}}{16\pi\sigma R^{2}}\right)^{1/4}$ (1)
where $L_{*}$ is the bolometric luminosity of the star, $\sigma$ is the
Stefan-Boltzmann constant, and $R$ is the distance of the dust grain from the
central star. We adopt a stellar luminosity of $L_{*}$ = 2.83 L☉ for HD 206893
(Delorme et al., 2017). We assumed several different possible functional forms
for the surface density of the disk; the functional forms for the surface
density are summarized in Table 4.
For example, the surface density in the disk follows a power-law distribution
where
$\Sigma(R)=\Sigma_{c}R^{p}$ (2)
between radii of $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$, where $\Sigma_{c}$
is a normalization for the total dust mass in the disk, $M_{\mathrm{disk}}$
(see Table 4 for definition), and $p$ is a power law index that we initially
set to a value of 1.0. While the value of $p$ is interesting from the
perspective of debris disk evolution, there is a well-known degeneracy between
$p$ and the location of the outer radius $R_{\mathrm{out}}$ (see, e.g.,
section 4.2.2. of Ricarte et al., 2013) and our data are of sufficiently
limited sensitivity and angular resolution that we chose to focus on
$R_{\mathrm{out}}$ rather than $p$. We chose a value of 1 as a middle-of-the-
road estimate bounded by theoretical predictions of 0 for collisional
evolution models that assume a collisional lifetime of the largest
planetesimal longer than the age of the system (Schüppler et al., 2016; Marino
et al., 2017), and 7/3 for those that assume the opposite (Kennedy & Wyatt,
2010).
As it became clear that the local minimum and maximum evident in the radial
surface density profile (Figure 2) would require a more complex radial surface
density distribution than the initial flat disk model, we explored two
families of more complicated models: models with broken power laws that
switched power law indices at one or two transition radii within the inner and
outer radius of the disk, and models that included a sharp radial gap in the
surface density profile at a particular radius. When analyzing surface
brightness features with modest signal-to-noise, it is often not possible to
determine the exact shape of the feature, but broken power laws and sharp gaps
are both common families of solutions assumed in the literature (see, e.g.,
Ricci et al., 2015) and adequately represent the extremes of an abrupt and
deep gap vs. a broad and shallow gap.
After generating the sky-projected images, we apply a primary beam correction
(multiplying by the primary beam) and then convert the model image into
synthetic model visibilities using the MIRIAD task uvmodel (Sault et al.,
1995), which we then compare with the data in the uv plane to calculate a
$\chi^{2}$ value as a goodness-of-fit test. Comparing data in the uv plane is
desirable both because uncertainties are well characterized in the uv plane
(unlike in the image domain, where the uncertainties are unknown and
correlated between pixels, and nonlinearities can be introduced by the CLEAN
process), and because it is agnostic to the choice of imaging parameters and
allows us to take full advantage of the range of baseline lengths sampled.
We fit the models to the data using an affine-invariant MCMC sampler (Goodman
& Weare, 2010) implemented in Python in the software package emcee (Foreman-
Mackey et al., 2013). The goodness of a fit between the synthetic and observed
visibilities are evaluated by a log-likelihood metric $\ln$ $\mathcal{L}$=
$-\chi^{2}/2$. The MCMC code directs an ensemble of walkers in an exploration
of parameter space, according to the calculated probability that a given
walker position (representing a single set of model parameters) provides a
better fit to the data than the previous walker position. After a “burn-in”
phase during which the walkers search downhill for the $\chi^{2}$ minimum, the
process results in a set of model parameters describing the different “steps”
that each walker makes, which can then be agglomerated into a marginalized
posterior probability distribution for each parameter (see Figs. 7, 9, and 8).
We performed several MCMC runs in order to investigate a variety of model
formalisms. Initially we varied eight parameters: the inner radius
($R_{\mathrm{in}}$), the distance between the inner part of the disk and the
outer part of the disk ($\Delta$R, which is related to the disk outer radius
as $R_{\mathrm{out}}=R_{\mathrm{in}}+\Delta R$), the mass of the disk
($M_{\mathrm{disk}}$), the flux density of the central star $F_{\mathrm{*}}$,
the position angle of the disk major axis (PA), the inclination of the disk
relative to the observer’s line of sight ($i$), and the position offset in
right ascension ($\Delta x$) and declination ($\Delta y$) of the star-disk
system relative to the pointing center of the interferometer. Subsequently, we
introduced a gap inner radius ($R_{\mathrm{in,Gap}}$) and width ($\Delta
R_{\mathrm{Gap}}$), and for the power law models, one or two transition radii
($Rt1$ and $Rt2$) with power law indices for the disk segments between the
transition radii ($pp1$, $pp2$, and $pp3$). All parameters were sampled
linearly except for disk mass which was sampled logarithmically, effectively
equivalent to assuming a log-uniform prior, and position angle and
inclination, which were sampled as $\cos{i}$ and $\cos{PA}$ to avoid
undersampling of the extrema.
The initial run revealed two issues that made us suspect that a more complex
distribution of material was necessary: first, the stellar flux was
approximately double the anticipated value based on the blackbody
approximation, and the models showed a much sharper central peak than we saw
in the data, suggesting that there must be some diffuse emission around the
central star. In addition, the models seemed to explore two regions of
parameter space: initially they explored a region of parameter space with a
small inner radius, which had a more reasonable stellar flux but left a ring
of negative residuals farther from the star, and then eventually they landed
in a region of parameter space where the disk inner radius was larger than
expected – too large for the inner edge to be carved by the brown dwarf,
though it is certainly possible that other unseen companions could be carving
the disk edge – and the stellar flux was too high. This behavior suggests that
there might actually be two edges to the disk: one close to the star,
accounting for the diffuse flux around the star, and another farther out.
Therefore, we explored a model with a radial gap in the disk. We assumed
uniform priors that required the inner radius of the gap to fall within the
radial extent of the disk ([$R_{in}$,$R_{out}$]) and that required the width
of the gap to be smaller than the total width of the disk ([0,$\Delta R$]).
The functional form of the gap is a top-hat: we assumed that the gap was
completely empty, and the edges of the gap are step functions.
In an attempt to remain agnostic about the functional form of the minimum in
the surface brightness, we explored both a power-law with an empty gap, and a
double power law, motivated by the set of functional forms assumed by Ricci et
al. (2015). However, we found that the double power law transition radius fell
on the peak in the surface brightness around 115 au, which meant that we were
not able to evaluate whether a break in the power-law surface density could
reproduce the observed surface brightness profile as well as a disk with a
gap. We therefore explored two additional profiles: a double power-law with a
radial gap, and a triple power-law with two transitional radii. All of the
models were consistent in preferring a dip in the radial surface density
profile near 75 au and a peak near 115 au, and the comparison between the
latter two functional forms demonstrates that an empty gap with sharp edges
yields comparable results to a more shallow power-law inflection point and is
preferred with modest statistical significance. Table 4 presents a summary of
the functional forms, surface density normalization, free parameters, and
best-fit lnprob values for each of the seven classes of models that we fit to
the data. For the remainder of the paper, we focus on the comparison between
the flat disk (which ignores the local maximum and minimum of the surface
density), the double power law with a gap, and the triple power law, since the
latter two were the models that best (statistically and by eye) reproduced the
features of the observations.
The limits of the priors for all parameters are listed in Table 2. For the
flat disk we used 16 walkers and ran the chain for 2000 steps beyond the burn-
in period. For the double power law with gap we used 30 walkers and ran the
chain for 2000 steps beyond the burn-in period. For the triple power law, we
used 30 walkers and ran the chain for 3000 steps beyond the burn-in period. In
all cases, the burn-in period was estimated by eye based on where the lnprob
values leveled off to a relatively constant maximum. We also followed up with
an autocorrelation analysis showing that while the autocorrelation time was
still rising by the end of each chain, all parameters had leveled off so that
the fractional error in the mean was no more than a few percent, and the
standard error of the mean was stable. We show some sample plots from the
double power-law model with a gap and the triple power-law model in the
Appendix. The best-fit models, as well as the median and uncertainties given
by the 16th and 84th percentiles of the posterior distribution, are presented
in Table 3. Figure 3 shows the radially averaged visibility profile for the
data, compared with the best-fit models for the flat disk, double power-law
with gap, and triple power-law models. The visibilities have been deprojected
assuming an inclination of 44∘ and position angle of 60∘ (the best-fit values
for the double power-law model).
Figure 3: Elliptically averaged visibility profile comparing the data (blue points) with the best-fit flat disk (red line), double power-law with a gap (green line) and triple power-law (orange line) models. The top panel shows the real part of the visibilities while the bottom panel shows the imaginary part of the visibilities. The visibilities have been deprojected assuming an inclination of 44∘ and a position angle of 60∘, the best-fit values for the double power-law model. Table 2: MCMC Priors Parameters | Flat Disk | Double Power Law with Gap | Triple Power Law |
---|---|---|---|---
$R_{in}$ (au) | [ 0, $1\times 10^{4}$] | [0, $1\times 10^{4}$] | [0, $1\times 10^{4}$] |
$\Delta$R(au) | [ 0, $1\times 10^{4}$] | [0, $1\times 10^{4}$] | [0, $1\times 10^{4}$] |
Log($M_{\mathrm{disk}}$) ($M_{\earth}$) | [ -10, -2] | [-10, -2] | [-10, -2] |
$F_{\mathrm{*}}$ ($\mu$Jy) | [0, $1\times 10^{8}$] | [0, $1\times 10^{8}$] | [0, $1\times 10^{8}$] |
cos(PA) (∘) | [ -1, 1] | [ -1, 1] | [-1, 1] |
cos($i$) (∘) | [-1, 1] | [-1, 1] | [-1, 1] |
$\Delta x$ ($\arcsec$) | [-5, 5] | [-5, 5] | [-5, 5] |
$\Delta y$ ($\arcsec$) | [-5, 5] | [-5, 5] | [-5, 5] |
$R_{\mathrm{in,Gap}}$ (au) | | [$R_{\mathrm{in}}$, $R_{\mathrm{out}}$] | |
$\Delta R_{\mathrm{Gap}}$ (au) | | [0, $\Delta R$] | |
pp1 | | -5, 5 | -5, 5 |
pp2 | | -5, 5 | -5, 5 |
pp3 | | | -5, 5 |
Rt1 | | $R_{in}$, $R_{out}$ | $R_{in}$, Rt2 |
Rt2 | | | Rt1, $R_{out}$ |
Figures 4, 5, and 6 show the tapered data image (left) compared with the best-
fit model (center), sampled at the same baseline separations and orientations
and imaged with the same parameters as the data, and the residuals (right).
The lack of significant ($>3\sigma$) residuals for these models shows that all
models adequately fit the data. The main difference visible in the best-fit
models is the distribution of flux around the star, towards the center of the
disk. For the model without a gap or power-law minimum, the stellar flux is
higher (by a factor of two) and the distribution of flux in the center of the
system is strongly and centrally peaked. For the models with a gap or power-
law break, the stellar flux is lower (more in line with expectations from the
blackbody estimate) and the emission is more diffuse around the star, since
the model with a gap incorporates disk emission extending throughout the inner
regions of the disk.
Since the models with a gap or power-law break used five more parameters than
the model without a gap, we expect them to be able to provide a better fit
simply due to the larger number of degrees of freedom. In order to
appropriately penalize the additional parameters when evaluating changes in
the goodness-of-fit parameter, we used the AIC, a form of the Aikake
Information Criterion, and the BIC, the Bayesian Information Criterion. The
BIC penalizes the use of additional parameters more than the AIC, and also
takes into account the sample size (which the AIC does not). A BIC score of
$\Delta$BIC $>$ 10 implies “very strong” evidence of a statistically improved
fit (Kass & Raftery, 1995).
When calculated, the $\Delta$BIC comparing the double power law with a gap to
the flat disk is 37.6, implying “very strong” evidence that the double power
law with a gap is a better fit to the data than the flat disk. The AIC also
returns a probability of $1.1\times 10^{-8}$ that the flat disk is a better
fit than the model with a gap. The corresponding BIC and AIC values for the
triple power law gap compared with the flat disk model are 44.2 and $3.1\times
10^{-7}$, also highly significant. With so many models, there is a large
number of potential comparisons we could make, but it is perhaps worth noting
that for any given model type (single power law, or double power law) adding a
radial gap provides a significantly better fit than the same model without the
gap, even taking into account the increase in the number of parameters (the
$\Delta$ BIC value for the flat power law with a gap compared to the flat
power law without a gap is 16, the value for the single power law with a gap
compared to the single power law without a gap is 23, and the value for the
double power law with a gap compared to the double power law without a gap is
5.9). These values corroborate the conclusion that we have statistically
significant evidence for a radial gap in the disk.
As for the shape of the gap, the double power law with a gap and the triple
power law both provide an excellent fit to the data using the same number of
parameters, though the double power-law with a gap does have the larger lnprob
value. On the basis of the difference in lnprob, the probability that the
double power law with a gap provides a better fit to the data than the triple
power law is 0.03, which we take as suggestive but not conclusive evidence
that a sharp, empty gap might provide a better fit to the data than a more
shallow transition between two power laws.
Figure 4: (Left) Naturally weighted ALMA image of the 1.3 mm continuum
emission from the HD 206893 system, with a taper of 200 k$\lambda$ applied to
bring out the large-scale structure of the source. (Center Left) Full
resolution model image for a flat disk showing the structure of the disk with
a stellar flux equal to zero. (Center Right) Model image sampled at the same
baseline lengths and orientations as the ALMA data, showing the best-fit model
without a gap in the middle of the dust disk. (Right) Residual image after
subtracting the model from the data in the visibility domain. Contour levels
and symbols are as in Figure 1. Figure 5: (Left) Naturally weighted ALMA
image of the 1.3 mm continuum emission from the HD 206893 system, with a taper
of 200 k$\lambda$ applied to bring out the large-scale structure of the
source. (Center Left) Full resolution image of a triple power-law model
showing the structure of the disk with a stellar flux equal to zero. (Center
Right) Model image sampled at the same baseline lengths and orientations as
the ALMA data, showing the best-fit model with a gap in the middle of the dust
disk. (Right) Residual image after subtracting the model from the data in the
visibility domain. Contour levels and symbols are as in Figure 1. Figure 6:
Same as Fig 5, but for a model of the disk that is parameterized as a double
power law and includes a gap in the radial dust distribution. Table 3: MCMC
Fitting Results
Parameter | Flat Disk | Double Power Law with Gap | Triple Power Law
---|---|---|---
| Best Fit | Median | Best Fit | Median | Best Fit | Median
$R_{in}$ (au) | 9 | $<44^{a}$ | 21a | $<51^{a}$ | 35 | $29^{+7}_{-20}$
$\Delta$R(au) | 155 | $151^{+14}_{-11}$ | 176 | $166^{+20}_{-16}$ | 159 | $164^{+18}_{-14}$
Log($M_{\mathrm{disk}}$) ($M_{\earth}$) | -1.66 | $-1.63^{+0.03}_{-0.03}$ | -1.61 | $-1.63^{+0.03}_{-0.02}$ | -1.74 | $-1.74^{+0.07}_{-0.07}$
$F_{\mathrm{*}}$ ($\mu$Jy) | 16 | $19^{+6}_{-6}$ | 17 | $18^{+5}_{-5}$ | 14 | $15^{+5}_{-6}$
PA (∘) | 66 | $63^{+3}_{-3}$ | 60 | $59^{+3}_{-3}$ | 60 | $63^{+3}_{-3}$
$i$ (∘) | 47 | $47^{+2}_{-2}$ | 45 | $44^{+3}_{-3}$ | 43 | $47^{+3}_{-3}$
$\Delta x$ ($\arcsec$) | 0.11 | $0.14^{+0.07}_{-0.07}$ | 0.16 | $0.11^{+0.07}_{-0.09}$ | 0.11 | $0.09^{+0.08}_{-0.09}$
$\Delta y$ ($\arcsec$) | 0.05 | $0.04^{+0.06}_{-0.06}$ | 0.04 | $0.03^{+0.05}_{-0.05}$ | 0.05 | $0.06^{+0.11}_{-0.07}$
$R_{\mathrm{in,Gap}}$ (au) | | | 67 | $63^{+8}_{-16}$ | |
$\Delta R_{\mathrm{Gap}}$ (au) | | | 32 | $31^{+11}_{-7}$ | |
pp1 | | | -2.0 | $-1.1^{+1.1}_{-0.8}$ | -2.7 | $-1.2^{+0.8}_{-1.1}$
pp2 | | | 2.8 | $3.0^{+1.0}_{-0.9}$ | 4.7 | $>0.23$
pp3 | | | | | -3.7 | $-3.0^{+1.3}_{-1.0}$
Rt1 | | | 97 | $102^{+16}_{-17}$ | 73 | $71^{+9}_{-33}$
Rt2 | | | | | 113 | $115^{+8}_{-7}$
Ln prob | -10733015.1 | | -10732991.8 | | -10732995.1 |
Note. — a The inner radius is unresolved in the models without a gap, so the
best-fit value of 9 au or 21 au is not meaningful. The upper limit of 44 au or
51 au represents the 99.7th percentile of the posterior distribution.
Table 4: Summary of different functional forms assumed for the surface density
profile
Model Type | Surface Density Profile | Normalization | Variable Parameters | best-fit lnprob
---|---|---|---|---
flat disk | $\Sigma(r)=\Sigma_{d}r$ | $\Sigma_{d}=\frac{3M_{\mathrm{dust}}}{2\pi(R_{\mathrm{out}}^{3}-R_{\mathrm{in}}^{3})}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$ | -10733015.1
flat disk with gap | $\Sigma(r)=\Sigma_{d}r$ | $\Sigma_{d}=\frac{3M_{\mathrm{dust}}}{2\pi(R_{\mathrm{out}}^{3}-R_{\mathrm{in}}^{3})}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, $R_{\mathrm{in,Gap}}$, $\Delta R_{\mathrm{Gap}}$ | -10733006.4
power law | $\Sigma(r)=\Sigma_{d}r^{pp}$ | $\Sigma_{d}=\frac{M_{\mathrm{dust}}(pp+2)}{2\pi(R_{\mathrm{out}}^{2+pp}-R_{\mathrm{in}}^{2+pp})}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, pp, rt | -10733011.5
power law with gap | $\Sigma(r)=\Sigma_{d}r^{pp}$ | $\Sigma_{d}=\frac{M_{\mathrm{dust}}(pp+2)}{2\pi(R_{\mathrm{out}}^{2+pp}-R_{\mathrm{in}}^{2+pp})}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, $R_{\mathrm{in,Gap}}$, $\Delta R_{\mathrm{Gap}}$, pp, rt | -10733006.3
double power law | $\Sigma(r)=\begin{cases}\Sigma_{t}\times r^{-pp1}~{}~{}\mbox{if}~{}~{}R_{\mathrm{in}}<r<R_{\mathrm{t}}\\\ \Sigma_{t}\times r^{-pp2}~{}~{}\mbox{if}~{}~{}R_{\mathrm{t}}<r<R_{\mathrm{out}}\end{cases}$ | $\Sigma_{d}=\frac{(M_{\mathrm{dust}}j_{1}j_{2}))}{2\pi(m_{1}j_{2}R_{\mathrm{t}}^{pp1}+m_{2}j_{1}R_{\mathrm{t}}^{pp2)}}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, pp1, pp2, rt | -10733005.7
double power law with gap | $\Sigma(r)=\begin{cases}\Sigma_{t}\times r^{-pp1}~{}~{}\mbox{if}~{}~{}R_{\mathrm{in}}<r<R_{\mathrm{t}}\\\ \Sigma_{t}\times r^{-pp2}~{}~{}\mbox{if}~{}~{}R_{\mathrm{t}}<r<R_{\mathrm{out}}\end{cases}$ | $\Sigma_{d}=\frac{(M_{\mathrm{dust}}j_{1}j_{2}))}{2\pi(m_{1}j_{2}R_{\mathrm{t}}^{pp1}+m_{2}j_{1}R_{\mathrm{t}}^{pp2}}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, $R_{\mathrm{in,Gap}}$, $\Delta R_{\mathrm{Gap}}$, pp1, pp2, rt | -10732991.8
triple power law | $\Sigma(r)=\begin{cases}\Sigma_{t1}\times r^{pp1}~{}~{}\mbox{if}~{}~{}R_{\mathrm{in}}<r<R_{\mathrm{t1}}\\\ \Sigma_{t1}\times r^{pp2}~{}~{}\mbox{if}~{}~{}R_{\mathrm{t1}}<r<R_{\mathrm{t2}}\\\ \Sigma_{t2}\times r^{pp2}~{}~{}\mbox{if}~{}~{}R_{\mathrm{t2}}<r<R_{\mathrm{t3}}\end{cases}$ | $\Sigma_{t1}=\frac{M_{\mathrm{dust}}j_{1}j_{2}j_{3}}{2\pi(R_{\mathrm{t1}}^{-pp1}j_{2}j_{3}m_{1}+R_{\mathrm{t2}}^{-pp2}j_{1}j_{3}m_{2}+\frac{R_{\mathrm{t2}}^{pp2-pp3}}{R_{\mathrm{t2}}^{p}p2}j_{1}j_{2}m_{3}}$ $\Sigma_{t2}=\Sigma_{t1}\left(\frac{R_{t2}}{R_{t1}}\right)^{pp2}$ | $R_{in}$, $\Delta$R, Log($M_{\mathrm{disk}}$), $F_{\mathrm{*}}$, PA, $i$, $\Delta x$, $\Delta y$, pp1, pp2, pp3, rt1, rt2 | -10732995.1
Note. — $j_{1}=2+pp1$, $j_{2}=2+pp2$, $j_{3}=2+pp3$,
$m_{1}=R_{\mathrm{t1}}^{j1}-R_{\mathrm{in}}^{j1}$,
$m_{2}=R_{\mathrm{t2}}^{j2}-R_{\mathrm{t1}}^{j2}$,
$m_{3}=R_{\mathrm{out}}^{j3}-R_{\mathrm{t2}}^{j3}$
Figure 7: Histograms of the marginalized posterior probability distributions
for the flat disk MCMC run. The central dashed line designates the median of
each distribution while the outer lines mark the 16th and 84th percentiles.
Figure 8: Histograms of the marginalized posterior probability distributions
for the double power law MCMC run with a gap. The central dashed line
designates the median of each distribution while the outer lines mark the 16th
and 84th percentiles. Figure 9: Histograms of the marginalized posterior
probability distributions for the triple power law MCMC run without a gap. The
central dashed line designates the median of each distribution while the outer
lines mark the 16th and 84th percentiles.
## 5 Discussion
### 5.1 Disk Structure Constraints
The results of our MCMC fits demonstrate that the HD 206893 disk is well
described by a broad underlying flux distribution extending from radii of
$<51$ au to $194^{+13}_{-2}$ au (this constraint on $R_{\mathrm{out}}$ is
measured from the posterior distribution of
$R_{\mathrm{out}}=R_{\mathrm{in}}+\Delta R$, which is not shown in Figure 8),
with a gap beginning at a radius of $63^{+8}_{-16}$ au and exhibiting a width
of $31^{+11}_{-7}$ au. There are no statistically significant ($>3\sigma$)
residuals, which indicates that the observations are well described by an
azimuthally symmetric and circular distribution of flux. Given the limited SNR
ratio of the data (the brightest parts of the dust disk are approximately at
the $5\sigma$ level on average), we can only rule out local (or point-to-
point) spatial variations in flux of $\gtrsim 50$% on spatial scales
comparable to the beam FWHM of $\sim$30 au or larger.
When we compare our derived disk size to the sample of spatially resolved
disks investigated by Matrà et al. (2018), we note that the value of R for HD
206893 according to the definition laid out in that paper would be
approximately 115 au. Of their sample of ten F stars, only one (HR 8799) has a
larger radius – and that star has a particularly unusual $\lambda$ Boo
spectrum that is in some ways more consistent with an A star spectral type.
The disk around HD 206893 is therefore among the largest known debris disks
around F stars, although comparable to that around HD 170773 (Sepulveda et
al., 2019).
Our model makes several simplifying assumptions that may or may not be
realistic: the completely empty gap (rather than allowing the depth of the gap
to vary), and the sharp edges of both the disk inner and outer radii and the
inner and outer edges of the gap. Better characterizing these features – the
depth of the gap, the sharpness of the edges – would be interesting from the
perspective of understanding debris disk evolution and interactions between
the disk and the planet. Unfortunately, this initial reconnaissance of the
disk structure is necessarily limited in angular resolution and sensitivity,
and the quality of the data do not justify a more complex model (as evidenced
by the lack of residuals). However, armed with this new knowledge of the
critical spatial and flux scales for the system, future observations at higher
angular resolution and sensitivity will be able to elucidate these details.
One of the key parameters of interest is the inner radius of the disk.
Previous models of the spatially unresolved SED had estimated that the
location of the inner radius should fall around 50 au (Moór et al., 2011),
estimated from the derived dust temperature of $49\pm 2$ K, although
comparisons between spatially resolved observations and SED estimates tend to
reveal a large amount of scatter, with spatially resolved measurements biased
towards larger radii than SED-based estimates (Booth et al., 2013; Pawellek et
al., 2014; Morales et al., 2016). In this case, the measured radius is smaller
than the SED-estimated radius, and appears to be spatially unresolved with a
radius of $<51$ au. This is good news for obtaining dynamical measurements of
the mass of the brown dwarf: an inner radius of 50 au (as suggested by the SED
models in Moór et al., 2011) is too large to be plausibly truncated by a brown
dwarf with a projected separation of 10 au, at least in the absence of extreme
eccentricity, which is ruled out by the combination of astrometric, radial
velocity, and direct imaging constraints explored by Grandjean et al. (2019).
We should note that there is some tension between a derived inner radius of
$<51$ au and the 50 K temperature constraint from the SED (Moór et al., 2011),
since even blackbody-like grains should be hotter than 50 K at distances
interior to $\sim$40 au from the central star. The temperature probed by the
SED is likely dominated by the bulk of the opacity, i.e., $\sim\mu$m-sized
grains in a collision-dominated debris disk. Small $\mu$m-sized grains are
expected to locate further from the star than those of blackbody-like grains
at the same temperature. The measured disk extent and the disk SED are largely
consistent. However, the morphology of the ALMA data seems to suggest that at
least some millimeter-size grains must be present within the central 1-2
beams, corresponding to $\lesssim 37$ au in radius (assuming a beam FWHM of
0$\farcs$6 and a diameter of 1.5 beams for the central flux component). Future
investigation is needed, particularly the detailed shape of the infrared SED
and the exact location of the inner disk edge.
The bad news is that the current data do not have sufficient angular
resolution to allow us to provide constraints on the mass of the BD companion
that are more restrictive than the constraints from Grandjean et al. (2019).
Our best-fit value for the inner radius of the triple power law model (35 au),
when interpreted in the context of chaotic zone theory and assuming a
semimajor axis of 13 au (from the MCMC fits to the combined astrometric, RV,
and direct imaging constraints in Grandjean et al., 2019), provides an upper
limit on the mass of the brown dwarf of $<1170$ MJup, assuming zero
eccentricity. The extent of the chaotic zone depends on the semimajor axis of
the companion’s orbit, as well as the mass ratio $\mu$ between the companion
and the central star. To derive the upper limit on the mass, we use the
functional form derived by Morrison & Malhotra (2015) for the outer extent of
the chaotic zone for a high-$\mu$ companion: $\Delta a\simeq
1.7\mu^{0.31}a_{p}$. It is possible to place somewhat better constraints
through a more detailed dynamical analysis, although even that is difficult
due to the likely presence of an additional companion causing the RV drift
detected by Grandjean et al. (2019). Marino et al. (2020) conduct just such an
analysis and show that the location of the inner edge is consistent with being
carved by the brown dwarf, though it could be farther out than expected based
on chaotic zone theory due to secular resonances with the companion
responsible for the RV drift. Since the location of the inner edge of the disk
is so steeply dependent on the companion mass, even a modest improvement in
the angular resolution will yield substantially improved constraints on the
dynamical mass measurement of the brown dwarf companion.
Another important constraint provided by the circumstellar disk measurement is
the inclination of the planetesimal belt. While the nearly face-on
configuration precludes meaningful constraints on the vertical structure, and
therefore the dispersion of inclinations, the average inclination is
constrained by our analysis to be $45^{\circ}\pm 4^{\circ}$. This range of
inclinations is somewhat inconsistent with the plausible range of 20∘-41∘ for
the brown dwarf companion reported by Grandjean et al. (2019), as well as the
$30^{\circ}\pm 5^{\circ}$ inclination of the stellar pole derived by Delorme
et al. (2017). Interestingly, the version of the Grandjean et al. (2019)
dynamical MCMC analysis that combines the constraints from radial velocity,
astrometry, stellar proper motion, and direct imaging results in a posterior
distribution for the inclination of the companion of $45^{\circ}\pm 3^{\circ}$
and a mass of $10^{+5}_{-4}$ MJup, though they note that the median $\chi^{2}$
on the RV data is six times higher than when they do not include the stellar
proper motion constraints, indicating that there may be some inconsistency in
the ability of their model to reproduce both the RV and stellar proper motion
data – perhaps providing additional support for the presence of another
companion in the system. At this point, while we cannot rule out mutual
inclination for the star and brown dwarf companion, coplanarity seems
plausible.
### 5.2 Gap Detection and Implications
The AIC and BIC comparison between the best-fit models provide strong evidence
for a gap or local minimum in the surface density near a radius of 80 au. This
marks the fourth detection of a radial gap in a broad debris belt, after HD
107146 (Ricci et al., 2015; Marino et al., 2018), HD 15115 (MacGregor et al.,
2019), and HD 92945 (Marino et al., 2019). While the sample size is not large,
there are only a couple of other debris disks with broad ($\Delta R/R\gtrsim
1$) debris belts that have been imaged with sufficient resolution and
sensitivity at millimeter wavelengths to detect substructure, including
$\beta$ Pic (Dent et al., 2014; Matrà et al., 2019) and AU Mic (MacGregor et
al., 2013), the latter of which in fact does exhibit tentative (AIC
$3.7\sigma$, $\Delta$BIC = 4.3) evidence for a gap in its edge-on planetesimal
belt (Daley et al., 2019). The newly detected gap in the HD 206893 debris disk
is therefore part of an emerging trend of gapped structure in broad debris
disks, which will be exciting to confirm and explore with future high-
resolution observations.
The cause of the gapped structure in debris disks is not well understood. The
main categories into which proposed mechanisms fall include: (1) gas-dust
dynamics, (2) inheritance from the protoplanetary disk phase, and (3)
dynamical interactions with a planet.
Lyra & Kuchner (2013) note that a robust clumping instability can organize
dust into multiple eccentric rings even in the absence of a planet. The main
prerequisite for the effect to operate is a gas-to-dust mass ratio $\gtrsim
1$. Given our upper limit on the CO flux density of 40 mJy km s-1, the
approximate upper limit on the CO mass, assuming LTE and an excitation
temperature of 30 K, is roughly $8.4\times 10^{-5}$ $M_{\earth}$, using
molecular data for CO v=0 drawn from the Cologne Database for Molecular
Spectroscopy (Endres et al., 2016). With the best-fit dust mass from Table 3
of 0.021 M⊕, the upper limit on the gas-to-dust mass ratio is $1.0\times
10^{-4}$ if we assume that the gas is dominated by CO, or 0.060 if we assume a
protoplanetary-like composition with a CO/H2 abundance ratio of $10^{-4}$,
corresponding to a mass ratio of $\sim$0.0014. Given these stringent limits,
it is unlikely that gas dynamics are responsible for the double-ringed
structure in this system.
One way in which gas dynamics could conceivably play a role despite the lack
of gas in the present-day debris disk is if the gapped structure is inherited
from the protoplanetary disk phase. The Disk Structures at High Angular
Resolution Project (DSHARP) has characterized the locations of rings and gaps
in a sample of 20 (18 of which are single) bright, nearby protoplanetary disks
(Huang et al., 2018). The outer dust radius of HD 206893 would place it among
the larger disks in the sample (3 of 20 DSHARP disks are as large or larger),
and the central gap radius of 78 au would place it roughly in the 79th
percentile of the 52 protoplanetary disk gaps identified by DSHARP. The
structure of the HD 206893 system is therefore generally consistent with the
distribution of disk radii and gap locations identified by DSHARP, though
perhaps on the larger side especially given its spectral type. It is also
worth noting that the DSHARP sample is biased towards systems with high dust
luminosities, and is almost certainly not representative. Of course, the
origin of the gaps and rings in protoplanetary disks is also not well
understood, so if the structure is inherited then at most we have gained some
evidence that the dust rings in DSHARP do correspond to planetesimal rings.
Proposed mechanisms for generating the gaps in protoplanetary disks include
chemical effects like snow lines (e.g., Banzatti et al., 2015; Okuzumi et al.,
2016), magnetohydrodynamic instabilities of various flavors along with
resulting gas pressure gradients (e.g., Johansen et al., 2009; Bai & Stone,
2014; Simon & Armitage, 2014; Flock et al., 2015; Lyra et al., 2015), and, of
course, one or more (proto)planets (e.g. Papaloizou & Lin, 1984; Bryden et
al., 1999; Nelson et al., 2000). At this point, there is enough evidence
pointing to the presence of planets in the disk at young ages to make it
likely that at least some of the gaps and rings in protoplanetary disks are
indeed caused by planets (e.g., Isella et al., 2018, 2019; Pinte et al., 2018;
Teague et al., 2018).
The extent to which disk structure is likely to be retained between the
protoplanetary and debris disk phase is still largely an open question. Most
theoretical studies of debris disk structure assume that the protoplanetary
disk dust profile is not retained, and make predictions for debris disk
structure that depend only on the locations of colliding planetesimals that
produce the dust, along with the masses and orbital properties of the larger
bodies that dynamically sculpt it (plus, for smaller grains, radiation
pressure, stellar winds, and sometimes Poynting-Robertson drag, depending on
the collision rate).
In the absence of a reservoir of gas whose mass is comparable to that of the
dust, it is difficult to break the radial symmetry of the disk without
planets. Even within the category of planet-sculpted gaps, there are multiple
theoretical considerations that point to different masses and orbital
properties of the underlying planetary system. However, the most
straightforward explanation is that a single additional unseen planet-mass
companion, located within the gap, is responsible for clearing the dust at the
location of the gap. In that case, the semimajor axis and width of the gap
encode the location and mass of the planet. Ignoring the presence of the brown
dwarf at 10 au and using the best-fit values for $R_{\mathrm{gap}}$ and
$\Delta R_{\mathrm{gap}}$ of 67 au and 32 au, respectively, the three-body
chaotic zone theory would predict that the gap is carved by a planet with
semimajor axis $a_{p}=80$ au and mass ratio $\mu=1.0\times 10^{-3}$,
corresponding to a mass of 1.4 MJup, assuming an orbital eccentricity of zero.
To derive these estimates, we use the general functional form from Morrison &
Malhotra (2015) where $\Delta a=C\mu^{\beta}a_{p}$, where $C$ and $\beta$ are
constants tabulated separately for the inner and outer chaotic zone width for
a range of ratios of the radius of the planet $R_{p}$ to the extent of the
Hill sphere $R_{H}$; we use the values tabulated for $R_{p}/R_{H}=0.001$. If
the planet has significant eccentricity, then the width of the gap would be
expected to increase as $1.8e^{1/5}\mu^{1/5}$, at least above a critical
eccentricity of $0.21\mu^{3/7}=0.004$, implying that a lower-mass planet on an
eccentric orbit could be responsible for carving the observed 31 au gap width
(Mustill & Wyatt, 2012). Propagating uncertainties through to the mass
estimate is non-trivial, not only due to the relative uncertainties on our
measured values of the gap radius and depth, but also due to uncertainties on
the parameters $C$ and $\beta$, the asymmetric nature of the chaotic zone, and
the degeneracy between model parmeters like $p$ and the gap properties, so the
value of 1.4 MJup should be considered a ballpark estimate subject to many
systematic and relative uncertainties.
More exotic possibilities have been invoked to explain the gapped structure in
the HD 107146 disk, notably the possibility that both the inner disk edge and
the gap could be carved by a single planet with mass comparable to the debris
disk and eccentricity between 0.4-0.5, located interior to the inner edge of
the broad debris belt (Pearce & Wyatt, 2015). That configuration was
ultimately ruled out for the HD 107146 system by Marino et al. (2018). It is
similarly unlikely to apply to the HD 206893 system, because the location of
the inner disk edge appears to be adjacent to the chaotic zone of the directly
imaged brown dwarf, and the brown dwarf’s mass is so much larger than that of
the disk that the apsidal anti-alignment described for low-mass companions in
Pearce & Wyatt (2015) would not occur. Another possibility, of course, is that
multiple planets could be carving the gap, in which case the mass of each
planet would be substantially smaller than the 1.4 MJup that we derive for the
single-planet case.
### 5.3 N-body Simulations of the Star, Brown Dwarf, and Putative Planet
The chaotic zone theory on which we base our estimate of planet mass is
fundamentally a 3-body result, but adding the dynamical influence of the brown
dwarf (effectively treating the central star as a low-$\mu$ binary system)
makes the system fundamentally a 4-body problem. We therefore conducted N-body
simulations with and without the brown dwarf (the latter to verify that the
gap depth and width were not significantly affected by the presence of the
brown dwarf) to investigate whether the orbital properties of the brown dwarf
have a detectable impact on the width of the gap.
In order to simulate gravitational interaction between the star, brown dwarf,
putative planet, and dust particles in the disk, we used the N-body software
REBOUND (Rein & Liu, 2012) with hybrid integrator MERCURIUS (Rein et al.,
2019), which switches from a fixed to variable timestep when any particle
comes within a certain distance of a massive particle, here chosen to be three
Hill radii. We chose the fixed timestep to be 4% of the brown dwarf’s initial
period, which is less than 8% of every dust particle’s period when the
simulation begins. Dust particles are treated in the test particle limit.
Particles are initially randomly distributed with a uniform distribution of
semimajor axes between 8 and 158 au, a uniform distribution of eccentricity
from 0 to 0.02, and a uniform distribution of inclination from 0$\circ$ to
10$\circ$. We placed 104 disk particles, adequate to sample the 150 au span of
the disk and recover smooth images of the final disk density distribution. We
assumed a stellar mass of 1.32 M$\sun$ and a stellar radius of 1.26 R$\sun$
(Delorme et al., 2017). The brown dwarf is placed on a orbit with semimajor
axis 10.4 au and assumed to have a radius of 1.3 RJup. We assumed a planet
mass of 1.4 MJup and a bulk density of 1.3 g cm-3. We placed the planet on a
circular orbit at 80 au. We integrate the system up to 10 Myr, which is
sufficiently long to capture many orbital timescales for all the bodies. We
then assign a weight to each particle based on its initial semimajor axis to
mock up a surface density that follows the best-fit double power-law model
parameters. We ran a set of simulations, varying the brown dwarf mass between
12 and 50 MJup (the range of plausible values from Grandjean et al., 2019)
with a 1.3 MJup planet. We also investigated the effects of placing the brown
dwarf on orbits with eccentricities of 0.03, 0.1, and 0.3, as well as
introducing an orbital inclination to the plane of the planetesimal disk of
5$\circ$ or 10$\circ$.
A summary of the properties of the brown dwarf and the measured gap width and
depth based on the final particle distribution is presented in Table 5. The
gap width and depth were estimated by fitting a power law to the initial
surface brightness distribution, subtracting the final profile from the fit to
the initial profile, excluding the “Trojan” particles at the center of the
gap, and fitting a top hat function to the result. Trojans were defined as any
particle with a semimajor axis that falls within 1 Hill Radius of the
semimajor axis of the planet. Two sample radial surface brightness profiles
are presented in Figure 10. The left panel shows the result for a low-mass (12
MJup) brown dwarf with zero eccentricity, and the right panel shows the result
for a high-mass (50 MJup) brown dwarf with an eccentricity of 0.3. In both
panels, the solid vertical lines represent the location of the brown dwarf
(near 10.4 au) and planet (near 79 au), and the adjacent dashed lines
represent the extent of each body’s chaotic zone, as calculated from the
formulae presented in Morrison & Malhotra (2015) for the zero-eccentricity
case and Mustill & Wyatt (2012) for the high-eccentricity case. While the top-
hat function fit to the surface brightness profile yields a slightly broader
width of 34 au than expected from the chaotic zone extent of 32 au, the
figures demonstrate that the gap edge is not well defined and the chaotic zone
approximation is a good estimate for where the surface brightness starts to
increase away from the influence of the companion.
We therefore find that the N-body simulations confirm that the chaotic zone
approximation is appropriate for most of the parameter space covered by
current estimates of the properties of the brown dwarf companion. Even a high
mass brown dwarf (50 MJup) with substantial eccentricity does not
significantly perturb the planet’s orbit (and therefore the width and location
of the gap estimated by chaotic zone theory). This result is consistent with a
back-of-the-envelope estimate which places the secular timescale at $<10$ Myr
and the maximum eccentricity of the outer planet of 0.06 for a brown dwarf-to-
planet mass ratio $>>1$. The depth of the gap for the case of 15 MJup is
roughly 94%, which is slightly shallower than the assumed 100% depth of the
gap in our ray-tracing code. Assuming a shallower gap would likely lead to a
broader estimated width of the gap in the MCMC code, indicating that our
estimate of the planet mass may be slightly too low. However, since the N-body
simulation includes Trojans that may or may not be realistic (for example, the
model does not include collisional evolution or the formation of the planet’s
core, either of which might deplete particles within the gap).
Finally, we then fixed the mass of the brown dwarf at 15 MJup, on a circular
orbit at 10.4 au, and ran a set of simulations varying the planet mass between
0.5 and 50 M⊕. We also investigated the effects of placing the planet with
eccentricities of 0.03, 0.1, and 0.3, as well as introducing an orbital
inclination to the plane of the disk of 5∘, 10∘, and 15∘. A summary of the
properties of the planet and the measured gap width and depth based on the
final particle distribution is presented in Table 6. The gap width calculated
by the top-hat function fig continues to yield results higher than the chaotic
zone estimate for all planet masses and eccentricities. However, the changes
in gap width generally conform to chaotic zone theory, as gap width increases
with planet mass to the 0.27 power, close to the formula from (Morrison &
Malhotra, 2015). The gap width increases proportionally to eccentricity to the
0.21 power, close to the formula from (Mustill & Wyatt, 2012). As expected, as
mass increased the depth of the gap increased as well as the width. As
eccentricity increased, the depth of the gap decreased. Inclining the planet’s
orbit had little effect on the width of the gap, although the depth of the gap
slightly decreased.
Figure 10: Surface brightness of particles in an N-body simulation of the HD
206893 system as a function of radius relative to the barycenter of the
system. These plots show the evolution of disk particles over 10 Myr, with a
planet on a circular orbit at 80 au. The surface brightness assumes an initial
surface density proportional to that derived for the double power-law model in
Table 3 and a dust temperature profile proportional to $R^{-1/2}$. The dashed
vertical lines represent the chaotic zone of the brown dwarf and the 1.4 MJup
planet, calculated according to formulae from Mustill & Wyatt (2012); Morrison
& Malhotra (2015). The left panel shows a system with a 12 MJup brown dwarf
with 0 eccentricity. The right panel shows a system with a 50MJup brown dwarf
with 0.3 eccentricity. Table 5: Gap Width and Depth with Varying Brown Dwarf
Parameters
BD Mass (MJup) | BD Eccentricity | BD Inclination($\circ$) | Gap Width (au) | Gap Depth (%)
---|---|---|---|---
12 | 0 | 0 | 34 | 95
15 | 0 | 0 | 35 | 94
30 | 0 | 0 | 35 | 97
50 | 0 | 0 | 34 | 96
50 | 0.03 | 0 | 34 | 98
50 | 0.1 | 0 | 34 | 96
50 | 0.3 | 0 | 35 | 99
50 | 0 | 5 | 34 | 95
50 | 0 | 10 | 34 | 97
No BD | | | 34 | 98
Table 6: Gap Width and Depth with Varying Planet Parameters
Planet Mass (M$\earth$) | Planet Eccentricity | Planet Inclination ($\circ$) | Gap Width (AU) | Gap Depth (%)
---|---|---|---|---
0.5 | 0 | 0 | Gap Undetectable
1 | 0 | 0 | 7 | 40
5 | 0 | 0 | 11 | 51
10 | 0 | 0 | 13 | 58
15 | 0 | 0 | 15 | 62
20 | 0 | 0 | 16 | 60
25 | 0 | 0 | 16 | 59
30 | 0 | 0 | 18 | 64
35 | 0 | 0 | 18 | 66
40 | 0 | 0 | 19 | 71
45 | 0 | 0 | 20 | 71
50 | 0 | 0 | 20 | 72
100 | 0 | 0 | 24 | 86
200 | 0 | 0 | 29 | 93
400 | 0 | 0 | 33 | 97
445 | 0 | 0 | 34 | 96
445 | 0.01 | 0 | 34 | 95
445 | 0.03 | 0 | 35 | 95
445 | 0.1 | 0 | 44 | 89
445 | 0.3 | 0 | Gap Undetectable |
445 | 0 | 5 | 34 | 96
445 | 0 | 10 | 35 | 94
445 | 0 | 15 | 34 | 92
## 6 Summary and Conclusions
As one of only two known systems to host a brown dwarf orbiting within a
debris ring, HD 206893 presents a rare and valuable opportunity to study
companion-disk interactions and place dynamical constraints on the mass of a
directly imaged companion. The ALMA observations at a wavelength of 1.3 mm
presented here spatially resolve the radial structure of the disk, revealing a
broad distribution of planetesimals extending from radii of $<51$ au to
$194^{+13}_{-2}$ au, with statistically significant (according to the AIC/BIC)
evidence for a gap in the dust disk with inner radius $63^{+8}_{-16}$ au and
width $31^{+11}_{-7}$ au.
The inner radius of the disk is not resolved by the current ALMA observation
of the system, allowing us to place only a modest upper limit on the mass of
the companion of $<1170$ MJup. The serendipitous discovery of a gap in the
disk marks the fourth time that a radial gap has been discovered within a
broad debris belt at millimeter wavelengths, among the $\sim$6 systems that
have so far been surveyed at sufficient resolution and sensitivity to detect
such a gap. While the origin of gapped structure is still unclear, in this
case the low limit on the gas mass in the system renders pressure gradients
due to residual disk gas an unlikely explanation, so the gap must either have
been inherited from the protoplanetary disk phase or carved by one or more
additional, unseen companions at larger separation in the system. If the gap
is carved by a single planet on a circular orbit, chaotic zone theory predicts
that it should have a mass of 1.4 MJup at a semimajor axis of 79 au.
Future ALMA observations at higher angular resolution have the potential to
not only place valuable dynamical constraints on the mass of the brown dwarf
companion by measuring the location of the inner disk edge, but can also
better constrain the properties of the putative planet by measuring the gap
width and depth. Surveying the radial structure of broad debris disks at
millimeter wavelengths has the potential to distinguish between scenarios in
which the gapped structure is inherited from the protoplanetary disk and
scenarios in which it is actively carved by unseen planets, while also
providing guidance and insight for future direct imaging surveys for planets
in debris-rich systems.
We express appreciation to Rebekah Dawson for advice about the N-body
simulations. A.N. is sponsored by Wesleyan University’s Research in the
Sciences Fellowship and Wesleyan University’s Quantitative Analysis Center
Apprenticeship. A.M.H. is supported by a Cottrell Scholar Award from the
Research Corporation for Science Advancement. A.M. acknowledges the support of
the Hungarian National Research, Development and Innovation Office NKFIH Grant
KH-130526. K.Y.L.S. acknowledges the partial support from NASA ADAP grant
NNX15AI86G. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2018.1.00193.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. This work has made use of data from the European Space Agency (ESA)
mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Software: Astropy (Astropy
Collaboration et al., 2018), CASA (McMullin et al., 2007), emcee (Foreman-
Mackey et al., 2013), Matplotlib (Hunter, 2007), MIRIAD (Sault et al., 1995),
NumPy (van der Walt et al., 2011), Pandas (Wes McKinney, 2010), Uncertainties,
http://pythonhosted.org/uncertainties In this Appendix, we show the corner
plots for the MCMC chains for the double power law model with a gap (Figure
11) and for the triple power law model (Figure 12). The corner plots show a
histogram for each parameter at the top of each column, and slices through
parameter space for the rest of the grid. The best-fit value of each parameter
is marked with a blue line.
Figure 11: Corner plot for the MCMC chain for the double power-law model with
a gap, after removing burn-in. Histograms for each parameter are plotted at
the top of the corresponding column, while the plots for the rest of the grid
show the distribution of walkers across slices through parameter space for the
corresponding pair of parameters. The best-fit value for each parameter is
plotted with a blue line. Figure 12: Corner plot for the MCMC chain for the
triple power-law model after removing burn-in. Symbols as in Figure 11.
## References
* Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
* Bai & Stone (2014) Bai, X.-N., & Stone, J. M. 2014, ApJ, 796, 31
* Banzatti et al. (2015) Banzatti, A., Pinilla, P., Ricci, L., et al. 2015, ApJ, 815, L15
* Baraffe et al. (2003) Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, A&A, 402, 701
* Beckwith et al. (1990) Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924
* Booth et al. (2013) Booth, M., Kennedy, G., Sibthorpe, B., et al. 2013, MNRAS, 428, 1263
* Bryden et al. (1999) Bryden, G., Chen, X., Lin, D. N. C., Nelson, R. P., & Papaloizou, J. C. B. 1999, ApJ, 514, 344
* Chabrier et al. (2000) Chabrier, G., Baraffe, I., Allard, F., & Hauschildt, P. 2000, ApJ, 542, 464
* Chiang et al. (2009) Chiang, E., Kite, E., Kalas, P., Graham, J. R., & Clampin, M. 2009, ApJ, 693, 734
* Cranmer et al. (2013) Cranmer, S. R., Wilner, D. J., & MacGregor, M. A. 2013, ApJ, 772, 149
* Daley et al. (2019) Daley, C., Hughes, A. M., Carter, E. S., et al. 2019, ApJ, 875, 87
* Delorme et al. (2017) Delorme, P., Schmidt, T., Bonnefoy, M., et al. 2017, A&A, 608, A79
* Dent et al. (2014) Dent, W. R. F., Wyatt, M. C., Roberge, A., et al. 2014, Science, 343, 1490
* Endres et al. (2016) Endres, C. P., Schlemmer, S., Schilke, P., Stutzki, J., & Müller, H. S. P. 2016, Journal of Molecular Spectroscopy, 327, 95
* Flaherty et al. (2015) Flaherty, K. M., Hughes, A. M., Rosenfeld, K. A., et al. 2015, ApJ, 813, 99
* Flaherty et al. (2017) Flaherty, K. M., Hughes, A. M., Rose, S. C., et al. 2017, ApJ, 843, 150
* Flock et al. (2015) Flock, M., Ruge, J. P., Dzyurkevich, N., et al. 2015, A&A, 574, A68
* Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
* Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
* Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
* Goodman & Weare (2010) Goodman, J., & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, 5, 65
* Grandjean et al. (2019) Grandjean, A., Lagrange, A. M., Beust, H., et al. 2019, A&A, 627, L9
* Huang et al. (2018) Huang, J., Andrews, S. M., Dullemond, C. P., et al. 2018, ApJ, 869, L42
* Hughes et al. (2018) Hughes, A. M., Duchêne, G., & Matthews, B. C. 2018, ARA&A, 56, 541
* Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90
* Isella et al. (2019) Isella, A., Benisty, M., Teague, R., et al. 2019, ApJ, 879, L25
* Isella et al. (2018) Isella, A., Huang, J., Andrews, S. M., et al. 2018, ApJ, 869, L49
* Johansen et al. (2009) Johansen, A., Youdin, A., & Klahr, H. 2009, ApJ, 697, 1269
* Kass & Raftery (1995) Kass, R. E., & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773
* Kennedy & Wyatt (2010) Kennedy, G. M., & Wyatt, M. C. 2010, MNRAS, 405, 1253
* Konopacky et al. (2016) Konopacky, Q. M., Rameau, J., Duchêne, G., et al. 2016, ApJ, 829, L4
* Kral et al. (2017) Kral, Q., Matrà, L., Wyatt, M. C., & Kennedy, G. M. 2017, MNRAS, 469, 521
* Lagrange et al. (2009) Lagrange, A. M., Gratadour, D., Chauvin, G., et al. 2009, A&A, 493, L21
* Lecar et al. (2001) Lecar, M., Franklin, F. A., Holman, M. J., & Murray, N. J. 2001, ARA&A, 39, 581
* Löhne et al. (2012) Löhne, T., Augereau, J. C., Ertel, S., et al. 2012, A&A, 537, A110
* Lyra & Kuchner (2013) Lyra, W., & Kuchner, M. 2013, Nature, 499, 184
* Lyra et al. (2015) Lyra, W., Turner, N. J., & McNally, C. P. 2015, A&A, 574, A10
* MacGregor et al. (2013) MacGregor, M. A., Wilner, D. J., Rosenfeld, K. A., et al. 2013, ApJ, 762, L21
* MacGregor et al. (2019) MacGregor, M. A., Weinberger, A. J., Nesvold, E. R., et al. 2019, ApJ, 877, L32
* Marino et al. (2017) Marino, S., Wyatt, M. C., Kennedy, G. M., et al. 2017, MNRAS, 469, 3518
* Marino et al. (2019) Marino, S., Yelverton, B., Booth, M., et al. 2019, MNRAS, 484, 1257
* Marino et al. (2018) Marino, S., Carpenter, J., Wyatt, M. C., et al. 2018, MNRAS, 479, 5423
* Marino et al. (2020) Marino, S., Zurlo, A., Faramaz, V., et al. 2020, MNRAS, 498, 1319
* Marois et al. (2008) Marois, C., Macintosh, B., Barman, T., et al. 2008, Science, 322, 1348
* Matrà et al. (2018) Matrà, L., Marino, S., Kennedy, G. M., et al. 2018, ApJ, 859, 72
* Matrà et al. (2019) Matrà, L., Wyatt, M. C., Wilner, D. J., et al. 2019, AJ, 157, 135
* Matthews et al. (2014) Matthews, B. C., Krivov, A. V., Wyatt, M. C., Bryden, G., & Eiroa, C. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 521
* Mawet et al. (2015) Mawet, D., David, T., Bottom, M., et al. 2015, ApJ, 811, 103
* McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
* Meshkat et al. (2017) Meshkat, T., Mawet, D., Bryan, M. L., et al. 2017, AJ, 154, 245
* Milli et al. (2017) Milli, J., Hibon, P., Christiaens, V., et al. 2017, A&A, 597, L2
* Montesinos et al. (2016) Montesinos, B., Eiroa, C., Krivov, A. V., et al. 2016, A&A, 593, A51
* Moór et al. (2011) Moór, A., Pascucci, I., Kóspál, Á., et al. 2011, ApJS, 193, 4
* Morales et al. (2016) Morales, F. Y., Bryden, G., Werner, M. W., & Stapelfeldt, K. R. 2016, ApJ, 831, 97
* Morrison & Malhotra (2015) Morrison, S., & Malhotra, R. 2015, ApJ, 799, 41
* Mustill & Wyatt (2012) Mustill, A. J., & Wyatt, M. C. 2012, MNRAS, 419, 3074
* Nelson et al. (2000) Nelson, R. P., Papaloizou, J. C. B., Masset, F., & Kley, W. 2000, MNRAS, 318, 18
* Okuzumi et al. (2016) Okuzumi, S., Momose, M., Sirono, S.-i., Kobayashi, H., & Tanaka, H. 2016, ApJ, 821, 82
* Papaloizou & Lin (1984) Papaloizou, J., & Lin, D. N. C. 1984, ApJ, 285, 818
* Pawellek et al. (2014) Pawellek, N., Krivov, A. V., Marshall, J. P., et al. 2014, ApJ, 792, 65
* Pearce & Wyatt (2015) Pearce, T. D., & Wyatt, M. C. 2015, MNRAS, 453, 3329
* Pinte et al. (2018) Pinte, C., Price, D. J., Ménard, F., et al. 2018, ApJ, 860, L13
* Quillen (2006) Quillen, A. C. 2006, MNRAS, 372, L14
* Rameau et al. (2013) Rameau, J., Chauvin, G., Lagrange, A. M., et al. 2013, ApJ, 772, L15
* Rein & Liu (2012) Rein, H., & Liu, S. F. 2012, A&A, 537, A128
* Rein et al. (2019) Rein, H., Hernandez, D. M., Tamayo, D., et al. 2019, MNRAS, 485, 5490
* Ricarte et al. (2013) Ricarte, A., Moldvai, N., Hughes, A. M., et al. 2013, ApJ, 774, 80
* Ricci et al. (2015) Ricci, L., Carpenter, J. M., Fu, B., et al. 2015, ApJ, 798, 124
* Rosenfeld et al. (2013) Rosenfeld, K. A., Andrews, S. M., Hughes, A. M., Wilner, D. J., & Qi, C. 2013, ApJ, 774, 16
* Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 433
* Schüppler et al. (2016) Schüppler, C., Krivov, A. V., Löhne, T., et al. 2016, MNRAS, 461, 2146
* Sepulveda et al. (2019) Sepulveda, A. G., Matrà, L., Kennedy, G. M., et al. 2019, ApJ, 881, 84
* Sibthorpe et al. (2018) Sibthorpe, B., Kennedy, G. M., Wyatt, M. C., et al. 2018, MNRAS, 475, 3046
* Simon & Armitage (2014) Simon, J. B., & Armitage, P. J. 2014, ApJ, 784, 15
* Strubbe & Chiang (2006) Strubbe, L. E., & Chiang, E. I. 2006, ApJ, 648, 652
* Su et al. (2005) Su, K. Y. L., Rieke, G. H., Misselt, K. A., et al. 2005, ApJ, 628, 487
* Teague et al. (2018) Teague, R., Bae, J., Bergin, E. A., Birnstiel, T., & Foreman-Mackey, D. 2018, ApJ, 860, L12
* Thureau et al. (2014) Thureau, N. D., Greaves, J. S., Matthews, B. C., et al. 2014, MNRAS, 445, 2558
* Trilling et al. (2008) Trilling, D. E., Bryden, G., Beichman, C. A., et al. 2008, ApJ, 674, 1086
* van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22
* Wes McKinney (2010) Wes McKinney. 2010, in Proceedings of the 9th Python in Science Conference, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61
* White et al. (2018) White, J. A., Aufdenberg, J., Boley, A. C., et al. 2018, ApJ, 859, 102
* White et al. (2019) —. 2019, ApJ, 875, 55
* White et al. (2020) White, J. A., Tapia-Vázquez, F., Hughes, A. G., et al. 2020, ApJ, 894, 76
* Williams & Andrews (2006) Williams, J. P., & Andrews, S. M. 2006, ApJ, 653, 1480
* Wilner et al. (2011) Wilner, D. J., Andrews, S. M., & Hughes, A. M. 2011, ApJ, 727, L42
* Wilner et al. (2018) Wilner, D. J., MacGregor, M. A., Andrews, S. M., et al. 2018, ApJ, 855, 56
* Wisdom (1980) Wisdom, J. 1980, AJ, 85, 1122
* Wyatt (2008) Wyatt, M. C. 2008, ARA&A, 46, 339
|
Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2017.DOI
This work was supported by the Natural Sciences and Engineering Research
Council of Canada, Prompt Québec, and Aerosystems International Inc.
Corresponding author: Alban Main de Boissiere (e-mail: alban.main-de-
boissiere.1@ens.etsmtl.ca).
# Bridging the gap between Human Action Recognition and Online Action
Detection
ALBAN MAIN DE BOISSIERE1 RITA NOUMEIR.1 Laboratoire de Traitement de
l’Information en Santé, École de Technologie Supérieure, Montreal, QC H3C 1K3
(email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Action recognition, early prediction, and online action detection are
complementary disciplines that are often studied independently. Most online
action detection networks use a pre-trained feature extractor, which might not
be optimal for its new task. We address the task-specific feature extraction
with a teacher-student framework between the aforementioned disciplines, and a
novel training strategy. Our network, Online Knowledge Distillation Action
Detection network (OKDAD), embeds online early prediction and online temporal
segment proposal subnetworks in parallel. Low interclass and high intraclass
similarity are encouraged during teacher training. Knowledge distillation to
the OKDAD network is ensured via layer reuse and cosine similarity between
teacher-student feature vectors. Layer reuse and similarity learning
significantly improve our baseline which uses a generic feature extractor. We
evaluate our framework on infrared videos from two popular datasets, NTU RGB+D
(action recognition, early prediction) and PKU MMD (action detection). Unlike
previous attempts on those datasets, our student networks perform without any
knowledge of the future. Even with this added difficulty, we achieve state-of-
the-art results on both datasets. Moreover, our networks use infrared from
RGB-D cameras, which we are the first to use for online action detection, to
our knowledge.
###### Index Terms:
Action detection, Action recognition, Early prediction, Infrared, Online
action detection, RGB+D
=-15pt
## I Introduction
Computer vision branches out to many subfields that are often studied
independently. In the video understanding domain, action recognition aims at
classifying entire segmented sequences. On the other end, action detection
embodies the ability to detect and classify multiple activities inside a
longer, unsegmented sequence.
The emergence of consumer-grade depth cameras (RGB+D) [26], [65] has sparked a
new research dynamic in action understanding. Real-time pose estimation
algorithms [46] are employed to extract 3D skeleton data from the depth
stream. Additionally, RGB and infrared streams are also available. Except for
the infrared, the various streams have been widely studied [54]. In essence,
the infrared and RGB representations are similar, with an advantage for the
former. Infrared videos are represented on a gray scale and are noisy, which
logically encourages the use of RGB data. But infrared is less affected by
illumination conditions and is usable in the dark, an important property for
security and night-vision applications.
Figure 1: Action recognition to online action detection framework. A teacher
is first trained. Its 3D CNN and classifier are reused by the students.
Teacher feature vector reconstruction is encouraged via knowledge
distillation. The online action detection student (OKDAD) embeds an additional
temporal proposal module.
Action recognition is usually conducted by analyzing a sequence in its
entirety before emitting a prediction. Early efforts leveraged recurrent
neural networks (RNN) to study skeleton data as time series [35], [44], [63].
This then shifted toward 2D convolutional neural networks (CNN) [24], [27],
[64] and graph convolutional networks (GCN) [59]. But temporal normalization
is always performed through sampling [35], [44], [63], image mapping and
rescaling [24], [27], [64] or global pooling [59]. Temporal normalization is
referred to as a step that normalizes the duration of a sequence. In other
words, the duration of the sequence is known before it is studied. As such,
most action recognition methods are considered offline, i.e. not applicable as
the action happens or in real-time.
Early action prediction aims at recognizing a human activity before it is
fully executed. The objective is therefore similar to the classification task
of online action recognition. While some approaches are in line with the
online paradigm [37], [43], recent attempts tackle the problem by dividing a
sequence into $N$ shorter segments before evaluating a subset of these [19],
[39], [55]. In other words, the duration of the sequence is still known and
used, which is considered offline.
Online action detection evaluates raw, unsegmented sequences containing
multiple actions [9]. The objective is to recognize an occurring activity and
classify it frame by frame, or by groups of frames, as it happens. It differs
from offline action detection where the sequence is studied in its entirety
before temporal segments are proposed, then classified. Following the works of
[8], we focus on infrared data from RGB-D cameras to further perpetuate their
legitimacy as a stream candidate for action recognition and detection.
Infrared from RGB-D cameras shows a strong potential for security and night-
vision applications. In [8], infrared yields great results for action
recognition. In this work, we go one step further and use infrared as a ready-
to-embed network for real-time online action detection.
This work is motivated by the apparent natural progression from action
recognition to online action detection that has not been previously exploited.
A strong online early prediction network should translate well to online
action detection. A network that is able to correctly classify an action as it
happens should also be able to differentiate an action vs. no action in time.
In other words, the online action detection problem can be broken down into
two parallel online early prediction tasks. The first task is to classify
video frames as containing an action or not. The second is to classify the
possible ongoing action. In essence, online action detection can be expressed
as binary online early prediction to detect actions and conventional online
early prediction to classify them.
As such, we focus our efforts on online early prediction which are most
frequently evaluated on the NTU-RGB+D dataset [44] instead of common online
action detection datasets [9], [20]. Additionally, our work heavily borrows
elements from [8] which uses infrared from RGB-D cameras.
We propose a framework (Fig. 1) to link the above-mentioned research problems,
using a 3D CNN and long short-term memory (LSTM) RNNs as building blocks. We
deploy an offline network to tackle both action recognition and early
prediction (teacher). We use it to distill knowledge to an early prediction
student network, which is online, likewise for an online action detection
student. The classification LSTM is used to recognize an ongoing action for
both students. The ”actionness” LSTM, used only for online action detection,
classifies a block of frames as containing an action or not. From here on out,
we refer to actionness as the likelihood of a frame-block being an action. We
train our network on infrared videos from RGB+D cameras.
Our main contributions are as follows: 1) A novel framework for both RGB-D
human action recognition and offline early prediction with infrared videos; 2)
Cosine similarity loss terms, which improve teacher accuracy; 3) A teacher-
student knowledge distillation framework for offline to online early
prediction based on cosine similarity; 4) An online action detection
architecture which builds upon the online student network; 5) State-of-the-art
results and extensive experiments on infrared videos from two popular
benchmark datasets [34], [44].
The project code will be publicly available upon acceptance of the paper.
Video demonstrations and further illustrations will also be available on the
project page.
## II Related work
### II-A Human action recognition
Pioneering approaches to video action recognition used handcrafted
spatiotemporal features, such as scale-invariant feature transform, histogram
of oriented gradients, and improved Dense Trajectories, which are still
competitive [52].
Recent efforts shifted toward deep learning. In [23], different temporal
fusing schemes are explored, with 2D CNNs as spatial feature extractors. In
[48], a two-stream network models spatiotemporal features via RGB images and
optical flow. Temporal dependencies may also be modeled via recurrent networks
[11], [62]. A 2D CNN outputs a feature vector for each frame, or group of
frames, which is then fed to an LSTM network. In [11], the CNN is pre-trained
and frozen during training, which might not be ideal as the CNN cannot learn
in the context of the video. Another family of networks is 3D CNNs [5], [50],
[51], [56]. Their major drawback is the number of trainable parameters. In
[51], an architecture called ResNet (2+1)D (R(2+1)D) uses skip connections as
its 2D counterpart [17]. This leads to fewer parameters to optimize while
retaining state-of-the-art performances. Additionally, spatial and temporal
convolutions are separated by nonlinear activation functions to allow for a
more complex representation.
Skeleton data are powerful [22], but superiority against video data is
unclear; rather, they seem to be complementary [8], [54]. First modern deep
learning attempts gravitated toward various forms of RNNs [35], [44], [63].
Then skeleton to 2D image mapping with 2D CNNs yielded better results [24],
[27], [64]. Nowadays, graph convolutional networks are the state of the art
[59], [45].
### II-B Early action prediction
Early action prediction follows the paradigm of action recognition, but on
partially observed sequences. The first attempts used handcrafted features in
the form of representation of visual words [30], hierarchical movemes [32] and
histogram of spatiotemporal features [42].
Deep learning approaches can be separated into two categories: online and
offline. Online approaches do not use the duration of a sequence as part of a
preprocessing step. In [37], a CNN+LSTM network is used. A novel loss is
introduced which encourages a monotonic ascending prediction score over time.
In [43], a CNN+LSTM network is used for very early prediction, but can at most
study 50 frames. A similar architecture is used in [29], but is offline
because of the bidirectional nature of the RNN.
The offline methods either preprocess data in the temporal domain [25], [55],
or the use information from the future [25], [29], [39]. In [25], partial and
full sequences are confronted in an adversarial learning context on
handcrafted skeleton images. In [55], a bidirectional CNN+LSTM teacher
distills its information to a unidirectional student network. The architecture
could be online, but a temporal normalization step is first performed.
### II-C Action detection
Action detection analyzes raw sequences and outputs temporal segments
containing activities with a class prediction. The online framework emits a
prediction frame by frame, or by short blocks of frames, without future
context. In an offline scenario, a sequence is studied in its entirety before
outputting predictions.
#### II-C1 Offline action detection
Early efforts focused on handcrafted features and sliding windows of different
sizes [4], [38], [53], [61]. Deep learning attempts follow the framework of
the R-CNN family for object detection in images [16], [15], [40]. The network
outputs temporal proposals, ranks them, then classifies them. The
architectures presented in [6], [7], [12], [13], [57] combine those tasks in
an end-to-end fashion. In [49], proposal precedes classification via
spatiotemporal attention LSTMs on skeleton data.
Closer to the online efforts, some architectures study sequences in a single
streamflow. In [1] and [2], a 3D CNN+RNN architecture studies videos by groups
of $\delta$ frames. The network in [1] could be used in real-time, but is
limited by a fixed maximum proposal size and demanding post-processing.
#### II-C2 Online action detection
De Geestet et al. outlined the challenges of online action detection [9] and
later proposed a two-stream LSTM architecture [10]. The first stream
interprets the input, the other the temporal dependencies between actions. In
[14], a Reinforced Encoder-Decoder (RED) network uses a CNN feature extractor
with an LSTM. The network is designed for anticipation, but can be used for
online detection. In [47], the precise start time of an action is emphasized
with adversarial networks. In [58], a temporal recurrent network (TRN) is
introduced with a prediction module. Using a pre-trained feature extractor
yields good results in [1], [14], [58], but we believe fine-tuning a network
in the context of its new task leads to improved performances, as demonstrated
in [36]
Figure 2: Online preprocessing on infrared videos. A crop around the subjects
is performed using 2D skeleton data. Figure 3: Teacher framework. a) A random
observation ratio is used during training. b) $T^{off}$ frames are sampled
from $T^{off}$ subwindows of even sizes. c) The normalized sequence is fed to
a ResNet (2+1)D. d) The feature vector is normalized by a 1D batch
normalization layer. e) The normalized teacher feature vector $x^{p}$ is used
for prediction.
### II-D Knowledge Distillation
Knowledge distillation regroups techniques that aim to transfer the knowledge
of a large pre-trained network to a smaller one. The concept was introduced in
[3]. Popularized in [18], ”softmax temperature” is proposed. The student
network learns from both the ground truth and smoothed soft labels from the
teacher. Minimizing the mean square error (MSE) between student and teacher
outputs is a possibility [41]. In [60], knowledge distillation shows a faster
learning time for the student network, which eventually outperforms the
teacher. In [55], the loss function minimizes the maximum mean discrepancy
(MMD) between teacher and student for early action prediction. Our approach
borrows elements from both transfer learning and knowledge distillation. Also,
we shift the focus from an offline teacher to online students. Because we do
not only reuse a network but attempt to distill the teacher’s representation,
our framework also falls under the knowledge distillation paradigm.
## III Action recognition to online action detection framework
We tackle multiple video understanding tasks, from human action recognition to
online action detection, and propose a framework to link those together. An
offline teacher is deployed for action recognition and early prediction.
During training, intraclass cosine similarity and interclass cosine
dissimilarity are encouraged. Knowledge distillation is employed to transfer
the representation of the teacher to an online early prediction student
network. Distillation is done via the reuse of layers and cosine similarity
between teacher and student feature vectors at different time points. The
student network is then transposed to an online action detection task. We
introduce a sigmoid-weighted temporal loss to train the online networks.
Figure 4: Student networks. A sequence is broken down into chunks of $\delta$
frames fed to a 3D CNN. The actionness LSTM proposes temporal segments. The
classification LSTM outputs a prediction when an action is being detected. The
network can be used for online early action prediction with the classification
LSTM only.
### III-A Preprocessing
#### III-A1 Cropping strategy
With online action detection as our final objective, we implement an online
cropping strategy on the infrared frames, inspired by [8]. This acts as a hard
attention mechanism by focusing the action on the subjects performing it. It
is used across the different networks. Also, the teacher and student networks
will study sequences differently. The teacher requires a temporal
normalization step, while the students do not, which leads to different
sampling strategies.
Because 3D CNNs embed many trainable parameters, the video frame resolution is
heavily downscaled to offset the memory needs. This results in a non-
negligible loss of information. For example, a small object might become
indistinguishable.
In a human action recognition context, the background provides little context
regarding the activity performed. Thus, we use the 3D skeleton coordinates on
the infrared frames to create a region of interest around the subject(s). As
such, we extract the maximal and minimal pixel values across all joints at
each time frame to capture the subject in a bounding box (Fig. 2).
Because a frame is cropped before the resizing operation, the downscaling
factor is less important, resulting in less information loss. However, because
the bounding box is recalculated for every frame, it moves across time. This
results in a fake camera movement and different scaling factors over time, as
shown Fig. 2. Performances are affected because pixels lose their temporal
coherence.
#### III-A2 Sampling strategies
We define offline and online sampling strategies for the teacher and online
networks respectively.
For action recognition, an action sequence is divided into $T^{off}$
subwindows of equal sizes, as done in [8]. A random frame is sampled from
each, creating a normalized sequence of length $T^{off}$ which is fed to the
teacher.
For the offline early prediction task, the total number of frames $N$ of a
sequence is adjusted depending on the observation ratio $r$, i.e., the
percentage of the sequence considered: $N_{partial}=\text{floor}(rN)$. For
instance, for a sequence of $N=100$ total frames and an observation ratio of
$r=80\%$, we will consider the $N_{partial}=80$ first frames. The same
strategy employed for action recognition is then used with the adjusted number
of frames. If $T^{off}$ is greater than $N_{partial}$, the first $N_{partial}$
frames are considered. The end of the resulting sequence is padded with black
frames to reach a size $T^{off}$. For example, if $T^{off}$ is set to 15, and
$N_{partial}=10$, then 5 black frames are added to get a normalized sequence
of size $T^{off}$.
For the online networks (online early prediction and online action detection),
the sequences are studied in their entirety. The frame rate is reduced by a
factor $s$, meaning we keep one of every $s$ frames.
### III-B Network architectures
The building blocks for our architectures are a 3D CNN, LSTMs, and fully
connected layers. They are reused across the different networks which improves
both results and learning times.
#### III-B1 Offline teacher
We use an 18-layer deep ResNet (2+1)D (R(2+1)D) as our backbone [56]. The
network is pre-trained on Kinetics-400 [5]. Our offline early prediction
framework is summarized Fig. 3. The R(2+1)D network outputs a feature vector
of length 512 which we then normalize with a 1D batch normalization layer
[21]. Batch normalization allows for all feature vectors to have roughly the
same Euclidean norm. As such, minimizing intraclass cosine similarity should
also reduce the MSE between feature vectors. We call $x^{p}$ the normalized
teacher feature vector. A final fully connected layer, the classifier, is used
to output a softmax-normalized class probability distribution.
#### III-B2 Online early prediction student
The online early prediction student network reuses the same R(2+1)D as the
teacher but adds a classification LSTM network on top (Fig. 4). For early
prediction alone, it does not embed the actionness LSTM. A sequence is divided
into $T^{on}$ subsequences of $\delta$ frames, with
$T^{on}=\text{ceil}(\frac{N}{\delta})$. For instance, with $\delta=5$, a
sequence of $N=98$ frames, then $T^{on}=20$. Each subsequence is fed to the
R(2+1)D network. At each time step $t\in\\{1,..,T^{on}\\}$, the computed
feature vector is used as input for the LSTM. The classification LSTM hidden
vector $h^{c}_{t}$ will then be fed to a ”reconstruction” fully connected
layer. From here on out, the superscript $c$ refers to the classification
LSTM. We call the outputted vector $x^{c}_{t}$ at each time step $t$. During
training, the goal will be to approximate $x^{p}_{t}$ with $x^{c}_{t}$. Here,
$x^{p}_{t}$ is the feature vector outputted by the teacher with
$N_{partial}=ts\delta$.
#### III-B3 OKDAD student
We introduce the Online Knowledge Distillation Action Detection (OKDAD)
network (Fig. 4). It builds upon the previous student network with an
additional actionness LSTM layer. The actionness module has two roles.
Firstly, it proposes temporal segments as the sequence happens. Secondly, it
resets the cell and hidden state vectors of the classification LSTM.
Intuitively, it forces the classification LSTM to forget about the past once
an action is over. As such, when a new action is discovered, the
classification LSTM is reduced to an online early prediction task.
The hidden state vector of the actionness LSTM $h^{a}_{t}$ (the superscript
$a$ refers to the actionness LSTM) is fed to a fully connected layer. Its
output is followed by the sigmoid function, to predict the probability that
the current frame block contains an action. With $y^{a}_{t}$ the actionness
probability, the classification LSTM is updated as follows:
$\displaystyle h^{c}_{t-1}$ $\displaystyle:=y^{a}_{t}h^{c}_{t-1}$ (1)
$\displaystyle c^{c}_{t-1}$ $\displaystyle:=y^{a}_{t}c^{c}_{t-1}.$ (2)
### III-C Knowledge distillation
#### III-C1 Teacher loss
Recall that we aim to distill teacher knowledge to an online action detection
network. We hypothesize that the reconstruction task of the student will be
easier if the teacher’s feature vectors have low intraclass and high
interclass distance. We introduce loss terms based on cosine similarity, as we
believe it is more appropriate than MSE for that task. In essence, it is more
practical to ”push apart” vectors of different classes with cosine similarity.
Additionally, because the batch normalization layer implies that feature
vectors have roughly the same Euclidean norm, encouraging cosine similarity
for vectors of same classes should also reduce MSE, without explicitly
penalizing it. Also, distance between vectors becomes harder to interpret in
high feature dimensional spaces, 512 in our case. Therefore, we propose a
novel loss function:
$\displaystyle\text{loss}=$ $\displaystyle\text{r}^{\gamma}[\text{cross-
entropy}+\text{similarity loss}+\text{dissimilarity loss}]$ (3)
$\displaystyle\text{loss}=$ $\displaystyle
r^{\gamma}[\frac{1}{|B|}\sum_{i=1}^{|B|}H(\hat{y}_{i},y^{p}_{i})$ (4)
$\displaystyle+\frac{\alpha}{N_{=}}\sum_{i=1}^{|B|}(\sum_{j=i+1}^{|B|}-\mathbbm{1}_{\hat{y}_{i}=\hat{y}_{j}}\ln(\frac{\cos({\angle({x^{p}_{i},x^{p}_{j}}}))+1}{2}))$
$\displaystyle+\frac{\beta}{N_{\neq}}\sum_{i=1}^{|B|}(\sum_{j=i+1}^{|B|}-\mathbbm{1}_{\hat{y}_{i}\neq\hat{y}_{j}}\ln(1-\frac{\cos({\angle({x^{p}_{i},x^{p}_{j}}}))+1}{2}))].$
$N_{=}$ and $N_{\neq}$ are respectively the number of pairs of a same class,
and different classes for a given batch:
$\displaystyle N_{=}$
$\displaystyle=\sum_{i=1}^{|B|}(\sum_{j=i+1}^{|B|}\mathbbm{1}_{\hat{y}_{i}=\hat{y}_{j}})$
(5) $\displaystyle N_{\neq}$
$\displaystyle=\sum_{i=1}^{|B|}(\sum_{j=i+1}^{|B|}\mathbbm{1}_{\hat{y}_{i}\neq\hat{y}_{j}}).$
(6)
And:
$\mathbbm{1}_{condition}=\left\\{\begin{array}[]{ll}1&\mbox{if condition is
true}\\\ 0&\mbox{otherwise.}\end{array}\right.$ (7)
Also, $B$ is a batch of sequences, $\angle$ is the angle between two vectors
$\hat{y}$ the ground truth, $y^{p}$ the teacher prediction, $x^{p}$ the
normalized teacher feature vector, $H$ the cross-entropy loss, $r$ the
observation ratio, $|B|$ the number of sequences in a batch, $\alpha$,
$\beta$, and $\gamma$ three hyperparameters.
The loss consists of three terms. The cross-entropy loss, $H$, encourages
correct predictions with a bias in favor of high confidence. The similarity
loss term reduces cosine similarity between same-class vectors in a batch. The
dissimilarity loss term decreases cosine dissimilarity between different-class
vectors in a batch. Hyperparameters $\alpha$ and $\beta$ weigh the importance
of similarity and distance loss terms respectively. Hyperparameter $\gamma$,
taken from $]0,+\infty[$, modulates the importance of a smaller observation
ratio. The smaller the value, the more penalty is applied.
#### III-C2 Online early prediction student loss
We want our student network to mimic the teacher feature vector $x^{p}_{t}$
generated at different time steps $t\in\\{1,..,T^{on}\\}$. We propose a loss
function that allows the network to learn autonomously while being guided by
the teacher’s correct predictions:
$\displaystyle\text{loss}=\frac{1}{|B|}\sum_{i=1}^{|B|}$ $\displaystyle
L_{s}(\hat{y}_{i},y^{c}_{i},x^{p}_{i},x^{c}_{i});$ (8)
$\displaystyle\text{loss}=\frac{1}{|B|}\sum_{i=1}^{|B|}$
$\displaystyle\left[\sum_{t=1}^{T^{on}_{i}}\sigma_{i,t}[H(\hat{y}_{i,t},y^{c}_{i,t})\right.$
$\displaystyle\left.-\eta(1-\epsilon_{i,t})\ln(\frac{\cos({\angle({x_{i,t}^{p},x_{i,t}^{c}}}))+1}{2})]\right].$
Again, $B$ is a set of training examples, i.e. a batch, and $i$ the index of
given element. Here, $\epsilon_{t}$ is the teacher error, such that
$\epsilon_{t}=1-P(y_{t}^{p}=\hat{y})$, where $P(y_{t}^{p}=\hat{y})$ is the
softmax-generated probability from the teacher for the correct label. Weighted
arithmetic average coefficients are sampled evenly from the sigmoid function
in the $[-2,2]$ range:
$\sigma_{t}=\frac{2}{T^{on}}\sigma(4\frac{t-1}{T^{on}-1}-2)$ and
$\sum_{t=1}^{T^{on}}\sigma_{t}=1$. The network is thus less penalized on early
outputs. Notice that $T^{on}$ is variable. If $T_{i}^{on}=1$, then we take
$\sigma_{i,t}=1$. The hyperparameter $\eta$ weighs the contribution of the
teacher.
The loss consists of two terms. The first one dynamically penalizes
classification based on total sequence length. The second one guides the
student’s reconstruction vector toward the teacher’s normalized feature
vector. To further encourage similar representations, the teacher’s
classification layer is reused by the student, and frozen during training.
#### III-C3 OKDAD student loss
During the training of the OKDAD network, two tasks are optimized. The first
is the temporal segment propositions. This is ensured by minimizing the binary
cross-entropy $H$ between predicted and true actionness at each frame-block.
The second task is early prediction for each action in a studied sequence,
with $A$ a set of actions and $|A|$ its cardinality. To do so, our sigmoid
weighted temporal loss $L_{s}$ is reused (8). We propose the following loss
function:
$\displaystyle\text{loss}=\frac{1}{|B|}\sum_{i=1}^{|B|}$
$\displaystyle\left[\frac{1}{T}\sum_{t=1}^{T}H(\hat{y}^{a}_{i,t},y^{a}_{i,t})\right.$
(9)
$\displaystyle\left.+\sum^{|A_{i}|}_{a=1}L_{s}(\hat{y}_{i,a}^{c},y_{i,a}^{c},x^{p}_{i,a},x^{c}_{i,a})\right]$
Intuitively, the OKDAD network learns to detect ongoing actions. When an
action is detected, classifying it as it happens is equivalent to the online
early prediction task. Recall the classification LSTM is only instantiated
when an action is ongoing. In this regard, the second loss term only penalizes
frame-blocks containing an action.
To summarize, the progression from action recognition to online action
detection is as follows. First, an action detection dataset is reduced to an
offline early prediction task which a teacher network learns to execute. The
knowledge is distilled to an online action detection network, which we showed
can be considered as an online early prediction task doubled with temporal
action proposals.
## IV Experiments
We test our method on infrared videos from RGB+D human action datasets: NTU
RGB+D [44] and PKU-MMD [34]. To the best of our knowledge, they are the only
RGB+D datasets including the infrared stream. NTU RGB+D is a human action
recognition dataset on which we evaluate our offline teacher and the
classification performances of our OKDAD architecture. PKU-MMD is a human
action detection dataset, on which we evaluate both the temporal action
proposal and classification tasks of OKDAD.
We focus our efforts on the online classification task. We believe a strong
online early prediction network should naturally perform well for temporal
action propositions, as it can be seen as a binary classification task.
### IV-A Implementation details
TABLE I: Early prediction results on NTU-RGB+D (accuracy in %). Bold values are the best. Observation ratio | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% | Avg.
---|---|---|---|---|---|---|---|---|---|---|---
KNN [19] | 7.4 | 9.5 | 12.2 | 16.0 | 20.8 | 25.9 | 30.8 | 34.4 | 36.1 | 37.0 | 23.01
RankLSTM [37] | 11.5 | 16.4 | 25.6 | 37.7 | 47.9 | 55.9 | 60.9 | 64.4 | 66.0 | 65.9 | 45.22
DeepSCN [31] | 16.8 | 21.4 | 30.5 | 39.9 | 48.7 | 54.6 | 58.1 | 60.1 | 60.0 | 58.6 | 44.87
MSRNN [19] | 15.1 | 20.3 | 29.5 | 41.3 | 51.6 | 59.1 | 63.9 | 67.3 | 68.8 | 69.2 | 48.61
PTSL [55] | 27.8 | 35.8 | 46.2 | 58.4 | 67.4 | 73.8 | 77.6 | 80.0 | 81.4 | 82.0 | 63.04
DBNet [39] | 27.9 | 33.3 | 47.2 | 56.9 | 68.5 | 74.5 | 78.5 | 80.5 | 81.6 | 81.5 | 63.04
Teacher | 10.5 | 29.5 | 49.9 | 66.7 | 76.0 | 81.1 | 83.9 | 85.4 | 86.3 | 86.4 | 65.57
Student | 19.3 | 28.1 | 38.6 | 55.5 | 67.9 | 75.5 | 80.6 | 83.7 | 85.5 | 86.6 | 62.13
An R(2+1)D network [56] is used across all networks. It outputs a 1D feature
vector of length 512. For the student, the classification and actionness RNNs
are single LSTMs with 2048 and 1024 features in the hidden state respectively.
The reconstruction layer is a single linear layer with an input size of 2048
and an output size of 512.
During teacher training, we sample the sequence observation ratio $r$
uniformly between $[r_{min},2]$. If the ratio is greater than one, it is set
to one. This favors training on entire sequences. We use $r_{min}=0.025$ and
use at least one frame. For best results, we use $\alpha=1$, $\beta=0.5$ and
$\gamma=\frac{2}{3}$. We set $T^{off}=15$ and train with a batch size of $16$
and a learning rate of $1e^{-4}$.
For online early prediction student training, we sample every $s$ frames from
the entire sequences, with $s=3$. Each subsequence contains $\delta=5$ frames;
as such, $s\delta=T^{off}=15$. We fix
$T^{on}=\text{ceil}(\frac{N_{max}}{s\delta})$ for all sequences, with
$N_{max}$ the maximum number of frames in a sequence across the entire
dataset. Thus, every sequence can be studied in its entirety while training by
batches. Shorter sequences are padded with black frames. Because the student
loss (equation 8) is dynamic, the predictions for the black frames are not
penalized.
The R(2+1)D network is fine-tuned during training. As such, the 3D CNN
backbone can shift its purpose from global to local feature extractor. The
classifier, reused from the teacher, is frozen. We set hyperparameter
$\eta=1$, batch size to $32$ and learning rate to $1e^{-3}$.
For OKDAD training, we keep $s$, $\delta$, $\eta$, batch size, learning rate,
and frozen parameters identical as when training for online early prediction.
We randomly sample subsequences of $T=40$ frame-blocks, which equals 20
seconds. The randomly sampled subsequences contain a mix of actions and non-
actions.
When evaluating the performances of OKDAD, predicted actionness with
probabilities over 0.75 are considered positive. Temporal segments are created
from adjacent positive predictions. Also, sequences are evaluated in their
entirety to mimic a real-time scenario. An entire sequence from PKU-MMD
averages around 3 minutes.
Previous works typically implement offline architectures for early prediction
and action detection on NTU-RGB+D and PKU-MMD. To compare our online networks
to these attempts, we calculate the offline classification accuracy by
weighing the predictions at different time steps with the sigmoid weights from
equation 8, with $T^{on}$ adapted to the observation ratio.
The Adam optimizer [28] is used systematically across all training.
### IV-B NTU RBG+D human action recognition dataset
NTU RGB+D is a state-of-the-art human action recognition dataset [44]. It
contains 60 action classes, from daily activities to medical conditions,
spread across 56,880 clips, 40 subjects, and 80 views. Each action contains up
to two subjects. Captured from different views and setups, this great
diversity makes NTU RGB+D a challenging dataset. Accuracy is used as the
evaluation metric.
There are two benchmark evaluations: cross-subject (CS) and cross-view (CV).
Following previous early prediction works on this dataset [39], [55], we only
consider the CS benchmark, which splits training and test sets across
different subjects. We sample 5% of our training set as our validation set, as
in [44].
TABLE II: Action detection results on PKU-MMD (in mAPa [34]). Bold values are the best. | Cross-Subject
---|---
$\theta$ | 0.1 | 0.3 | 0.5 | 0.7
RS+DR+DOF [34] | 0.647 | 0.476 | 0.199 | 0.026
CS+DR+DOF [34] | 0.649 | 0.471 | 0.199 | 0.025
TAP-B [49] | 0.544 | 0.514 | 0.461 | 0.327
TAP-B-M [49] | 0.557 | 0.53 | 0.431 | 0.242
RGB+D+F+S [36] | 0.903 | 0.895 | 0.833 | -
OKDAD $\alpha=1,\beta=0.5,\eta=0$ | 0.899 | 0.886 | 0.822 | 0.644
OKDAD $\alpha=1,\beta=0.5,\eta=1$ | 0.908 | 0.893 | 0.826 | 0.651
OKDAD $\alpha=0,\beta=0,\eta=0$ | 0.912 | 0.897 | 0.842 | 0.672
OKDAD $\alpha=0,\beta=0,\eta=1$ | 0.915 | 0.905 | 0.850 | 0.679
The results of our networks for different observation ratios are presented
Table I. The strong performances of the teacher are outlined. For ratios
greater than 20%, the network consistently outperforms the previous state of
the art by a large margin. At 50%, accuracy is improved by 7.5%, for a total
of 76.0%. At 100%, the accuracy is improved by 4.9%. On average, the teacher
performs 2.5% better than the best previous attempts. For very early
predictions (less than or equal to 20%), the teacher underperforms. At 10%, we
note a 17.4% difference, and 6.3% at 20% compared to [39].
Additionally, we report the offline performances of our online early
prediction student. Best results are achieved with $\eta=1$ for the student
with $\alpha=\beta=0$ for the teacher. The student is not able to match the
performance of the teacher, except for an observation ratio of 100% with 86.6%
for the student. Nonetheless, the student performs on par with the current
state of the art, systematically outperforming it for ratios greater than 50%.
At 60%, the student performs 1% better than its closest competitor (with
75.5%) and 5.1% better at 100% (with 86.6%). These results are particularly
encouraging considering the arguably harder online setting of our network.
TABLE III: Impact of cosine penalty on teacher network. Accuracy in % (Acc.), intraclass (Intra) and interclass (Inter) cosine similarity are reported. Bold values are the best. Observation ratio | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% | Avg.
---|---|---|---|---|---|---|---|---|---|---|---
No penalty teacher | Acc. | 8.9 | 31.3 | 51.6 | 66.7 | 74.7 | 79.5 | 82.6 | 84.0 | 84.6 | 84.8 | 64.87
Intra | 0.80 | 0.49 | 0.44 | 0.46 | 0.50 | 0.53 | 0.55 | 0.56 | 0.57 | 0.57 | 0.55
Inter | 0.83 | 0.30 | 0.11 | 0.08 | 0.09 | 0.10 | 0.11 | 0.12 | 0.13 | 0.13 | 0.20
Cos penalty teacher | Acc. | 10.5 | 29.5 | 49.9 | 66.7 | 76.0 | 81.1 | 83.9 | 85.4 | 86.3 | 86.4 | 65.57
Intra | 0.78 | 0.56 | 0.59 | 0.68 | 0.74 | 0.78 | 0.80 | 0.82 | 0.82 | 0.82 | 0.74
Inter | 0.73 | 0.25 | 0.10 | 0.07 | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 | 0.15
TABLE IV: Contribution of layer reuse and knowledge distillation with different teachers. Accuracy in % (Acc.), average cosine similarity (Sim.) and MSE between teacher and student feature vectors are reported. Bold values are the best. Observation ratio | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% | Avg.
---|---|---|---|---|---|---|---|---|---|---|---
Baseline | Acc. | 8.9 | 10.8 | 14.3 | 21.5 | 30.0 | 37.2 | 42.4 | 45.8 | 48.3 | 49.2 | 30.84
Student with cos penalty teacher | $\eta=0$ | Acc. | 18.5 | 26.3 | 37.4 | 53.9 | 67.0 | 73.9 | 78.5 | 81.7 | 83.4 | 84.8 | 60.54
Sim. | 0.94 | 0.91 | 0.83 | 0.79 | 0.77 | 0.77 | 0.76 | 0.75 | 0.75 | 0.75 | 0.80
MSE | 0.14 | 0.20 | 0.32 | 0.41 | 0.50 | 0.56 | 0.62 | 0.67 | 0.72 | 0.76 | 0.49
$\eta=1$ | Acc. | 18.0 | 27.4 | 36.9 | 52.3 | 65.8 | 73.8 | 79.1 | 82.5 | 84.4 | 85.5 | 60.57
Sim. | 0.97 | 0.95 | 0.92 | 0.91 | 0.92 | 0.93 | 0.93 | 0.93 | 0.94 | 0.94 | 0.93
MSE | 0.09 | 0.11 | 0.17 | 0.20 | 0.21 | 0.24 | 0.27 | 0.30 | 0.33 | 0.36 | 0.23
Student with no penalty teacher | $\eta=0$ | Acc. | 18.0 | 27.3 | 37.3 | 53.7 | 66.3 | 73.9 | 79.0 | 82.5 | 84.2 | 85.3 | 60.75
Sim. | 0.94 | 0.90 | 0.80 | 0.74 | 0.70 | 0.69 | 0.68 | 0.68 | 0.67 | 0.67 | 0.75
MSE | 0.63 | 0.71 | 0.87 | 0.81 | 0.83 | 0.85 | 0.86 | 0.89 | 0.91 | 0.93 | 0.83
$\eta=1$ | Acc. | 19.3 | 28.1 | 38.6 | 55.5 | 67.9 | 75.5 | 80.6 | 83.7 | 85.5 | 86.6 | 62.13
Sim. | 0.98 | 0.97 | 0.92 | 0.89 | 0.88 | 0.87 | 0.87 | 0.87 | 0.87 | 0.86 | 0.90
MSE | 0.48 | 0.51 | 0.54 | 0.43 | 0.40 | 0.40 | 0.41 | 0.42 | 0.44 | 0.46 | 0.45
### IV-C PKU-MMD action detection dataset
PKU-MMD is an RGB+D offline action detection dataset [34]. It contains 1,076
long sequences with approximately 20 actions from 51 classes per instance. It
totals 21,545 actions performed by 66 subjects. Mean Average Precision of
actions (mAPa) is used as the evaluation metric. To our knowledge, no online
action detection dataset with infrared data exists. To compare our results
with previous attempts, we evaluate the offline performances of OKDAD on the
cross-subject benchmark, while evaluating the sequences in an online fashion.
Table II shows the performances of our OKDAD network for different temporal
intersection over union thresholds $\theta$. Our method outperforms previous
attempts for all thresholds. Especially for $\theta=0.7$, OKDAD does not
experience a similar drop in performance as [34] and [49]. This suggests a
considerable improvement in accurate temporal boundary detection, while still
performing online.
We evaluate OKDAD with two different teachers, with and without knowledge
distillation for each. We find using a teacher without cosine similarity
penalties yields better results. However, in both cases, encouraging cosine
similarity between student and teacher vectors improves results noticeably.
This is in line with our findings section IV-D3 (Knowledge distillation on
online early prediction student).
Videos demonstrating the OKDAD network in action will be publicly available on
the project page.
### IV-D Ablation studies
We provide more experiments to better understand the different contributions
of our knowledge distillation framework.
#### IV-D1 Cosine similarity penalties on teacher learning
During teacher training, intraclass similarity and interclass dissimilarity
are encouraged via cosine penalty loss terms (3). In Table III, we compare the
impact of the similarity loss terms on the test set. A ”no penalty teacher”
($\alpha=\beta=0$) teacher is compared to its ”cos penalty teacher”
($\alpha=1,\beta=0.5$) counterpart. Average intraclass and interclass cosine
similarities are reported for different observation ratios. A value closer to
1 means similar orientation, closer to -1 means diametrically opposed.
The loss terms from (3) have the desired effects. When using the cosine terms,
the intraclass similarity is considerably increased by 0.19 on average. The
interclass similarity is adequately decreased by 0.05 on average. But it is
interesting to note that the network pushes interclass vectors toward
orthogonality without an explicit penalty.
The cosine similarity loss terms also improve the accuracy scores, notably on
observation ratios greater than 40%. On average, accuracy is improved by 0.7%
and by about 1.5% for ratios greater than 40%.
#### IV-D2 Teacher layer reuse on online early prediction student
Our distillation scheme consists of layer reuse (student with a teacher) and
similarity learning ($\eta>0$). Table IV shows the impact of reusing teacher
layers only (students with $\eta=0$) compared to our baseline.
Our baseline is our early prediction student with a teacher trained on
Kinetics-400 [5] and without similarity learning ($\eta=0$). In other words,
we use a generic feature extractor for our baseline, as done in previous works
[33] [14], [55], [58]. Note that for the baseline, all 3D CNN parameters are
frozen, as the network would not converge otherwise. This property is in favor
of our method. Retraining the feature extractor for its new task before using
it for the student allows it to be fine-tuned when training the student.
For our students, we use two different teachers. The first one is trained with
cosine similarity penalties, ”cos penalty teacher” ($\alpha=1,\beta=0.5$), the
second one without, ”no penalty teacher”. Note that the two teachers are
identical to the ones used in Table III.
Compared to the baseline, average accuracy nearly doubles (30.84% to about 60%
for both students). This clearly demonstrates the importance of an adequate
feature extractor.
We also compare the cosine similarity between teacher and student feature
vectors, as well as the MSE, at different observation ratios. It can be
observed that a penalized teacher naturally increases similarity and decreases
MSE between feature vectors. However, this does not seem to affect average
performance as both student networks perform similarly. Interestingly, the
student reusing the layers of the unpenalized teacher performs marginally
better, especially in the later stages, even though the latter is less
accurate as shown in Table III.
#### IV-D3 Knowledge distillation on online early prediction student
Table IV also shows the contribution of enabling teacher guidance (students
with $\eta>1$).
As hypothesized, increasing cosine similarity between teacher and student
feature vectors also reduces MSE without an explicit loss term. For the
student with a cos penalty teacher, enabling teacher guidance increases
similarity by 0.13 on average (0.80 to 0.93) and MSE reduces by half (0.49 to
0.23). Similar proportions can be observed using the unpenalized teacher.
However, the increase in performance is marginal for the student with the cos
penalty teacher (60.54% to 60.57% on average). More convincingly, the
distillation scheme increases performances by 1.4% on average (60.75% to
62.13%) using the no penalty teacher, more so for the later stages.
Additionally, the student beats all teachers with that configuration for
entire sequences, i.e. an observation ratio of 100%.
Indeed, the cos penalty teacher feature vectors are more convincingly
approximated, but the tighter class clusters may also leave less room for
error, explaining the lesser performances of the student in that case.
## V Conclusion
Action recognition, early action recognition, and action detection are often
studied separately. Moreover, it is important to distinguish between offline
and online approaches. For the latter, temporal information, such as the
duration of an action, are not used. In essence, online attempts mimic real-
time scenarios. Previous early action prediction attempts are mostly offline.
We argue that more efforts should be focused on online early prediction.
Networks that have a strong ability to correctly classify an action as it
happens should naturally perform well at detecting actions. In other words,
online action detection can be decomposed into online early action
recognition, to classify an ongoing action, and binary action detection, to
detect if a current frame is part of an action or not.
We propose a framework that builds upon this paradigm. Our online action
detection network (OKDAD) is divided into a feature extractor in the form of a
3D CNN that analyses the current frames, the outputted feature vector is used
to detect a potential ongoing action and classify it via parallel LSTM
networks. The 3D CNN is first trained in the form of an offline early action
prediction step and used as a teacher. It is then reused and fine-tuned as
part of the end-to-end training of the OKDAD network. The OKDAD student
benefits from knowledge distillation from teacher and student feature vectors
cosine similarity.
Our method achieves state-of-the-art results on the NTU RBG+D and PKU-MMD
datasets. This work focuses mainly on strong online early prediction
performances. Our network achieves results on par with the current offline
state-of-the-art attempts, despite performing online. To the best of our
knowledge, we are among the first attempts to use knowledge distillation
between networks performing different tasks, i.e. offline early prediction to
online early prediction and online action detection. Also, our experiments are
conducted on infrared data from RGB-D cameras which has never been attempted
for early prediction and action detection.
## Acknowledgment
This work was supported by research funding from the Natural Sciences and
Engineering Research Council of Canada, Prompt Québec, and industrial funding
from Aerosystems International Inc. The authors would also like to thank their
collaborators from Aerosystems International Inc.
## References
* [1] Shyamal Buch, Victor Escorcia, Bernard Ghanem, Li Fei-Fei, and Juan Carlos Niebles. End-to-end, single-stream temporal action detection in untrimmed videos. In Proceedings of the British Machine Vision Conference (BMVC), 2017.
* [2] Shyamal Buch, Victor Escorcia, Chuanqi Shen, Bernard Ghanem, and Juan Carlos Niebles. Sst: Single-stream temporal action proposals. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 2911–2920, 2017.
* [3] Cristian Buciluǎ, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 535–541, 2006.
* [4] Fabian Caba Heilbron, Juan Carlos Niebles, and Bernard Ghanem. Fast temporal activity proposals for efficient detection of human actions in untrimmed videos. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1914–1923, 2016.
* [5] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 6299–6308, 2017.
* [6] Yu-Wei Chao, Sudheendra Vijayanarasimhan, Bryan Seybold, David A Ross, Jia Deng, and Rahul Sukthankar. Rethinking the faster r-cnn architecture for temporal action localization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1130–1139, 2018.
* [7] Xiyang Dai, Bharat Singh, Guyue Zhang, Larry S Davis, and Yan Qiu Chen. Temporal context network for activity localization in videos. In Proceedings of the IEEE International Conference on Computer Vision, pages 5793–5802, 2017.
* [8] A. M. De Boissiere and R. Noumeir. Infrared and 3d skeleton feature fusion for rgb-d action recognition. IEEE Access, 8:168297–168308, 2020.
* [9] Roeland De Geest, Efstratios Gavves, Amir Ghodrati, Zhenyang Li, Cees Snoek, and Tinne Tuytelaars. Online action detection. In European Conference on Computer Vision, pages 269–284. Springer, 2016.
* [10] Roeland De Geest and Tinne Tuytelaars. Modeling temporal structure with lstm for online action detection. In 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pages 1549–1557. IEEE, 2018.
* [11] Jeffrey Donahue, Lisa Anne Hendricks, Sergio Guadarrama, Marcus Rohrbach, Subhashini Venugopalan, Kate Saenko, and Trevor Darrell. Long-term recurrent convolutional networks for visual recognition and description. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2625–2634, 2015.
* [12] Jiyang Gao, Zhenheng Yang, Kan Chen, Chen Sun, and Ram Nevatia. Turn tap: Temporal unit regression network for temporal action proposals. In Proceedings of the IEEE International Conference on Computer Vision, pages 3628–3636, 2017.
* [13] Jiyang Gao, Zhenheng Yang, and Ram Nevatia. Cascaded boundary regression for temporal action detection. arXiv preprint arXiv:1705.01180, 2017.
* [14] Jiyang Gao, Zhenheng Yang, and Ram Nevatia. Red: Reinforced encoder-decoder networks for action anticipation. arXiv preprint arXiv:1707.04818, 2017.
* [15] Ross Girshick. Fast r-cnn. In Proceedings of the IEEE international conference on computer vision, pages 1440–1448, 2015.
* [16] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra Malik. Rich feature hierarchies for accurate object detection and semantic segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 580–587, 2014.
* [17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
* [18] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
* [19] Jian-Fang Hu, Wei-Shi Zheng, Lianyang Ma, Gang Wang, Jianhuang Lai, and Jianguo Zhang. Early action prediction by soft regression. IEEE transactions on pattern analysis and machine intelligence, 41(11):2568–2583, 2018.
* [20] Haroon Idrees, Amir R Zamir, Yu-Gang Jiang, Alex Gorban, Ivan Laptev, Rahul Sukthankar, and Mubarak Shah. The thumos challenge on action recognition for videos “in the wild”. Computer Vision and Image Understanding, 155:1–23, 2017.
* [21] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
* [22] Gunnar Johansson. Visual perception of biological motion and a model for its analysis. Perception & psychophysics, 14(2):201–211, 1973.
* [23] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video classification with convolutional neural networks. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 1725–1732, 2014.
* [24] Qiuhong Ke, Mohammed Bennamoun, Senjian An, Ferdous Sohel, and Farid Boussaid. A new representation of skeleton sequences for 3d action recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3288–3297, 2017.
* [25] Qiuhong Ke, Mohammed Bennamoun, Hossein Rahmani, Senjian An, Ferdous Sohel, and Farid Boussaid. Learning latent global network for skeleton-based action prediction. IEEE Transactions on Image Processing, 29:959–970, 2019.
* [26] Leonid Keselman, John Iselin Woodfill, Anders Grunnet-Jepsen, and Achintya Bhowmik. Intel realsense stereoscopic depth cameras. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 1–10, 2017.
* [27] Tae Soo Kim and Austin Reiter. Interpretable 3d human action analysis with temporal convolutional networks. In 2017 IEEE conference on computer vision and pattern recognition workshops (CVPRW), pages 1623–1631. IEEE, 2017.
* [28] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
* [29] Yu Kong, Shangqian Gao, Bin Sun, and Yun Fu. Action prediction from videos via memorizing hard-to-predict samples. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018\.
* [30] Yu Kong, Dmitry Kit, and Yun Fu. A discriminative model with multiple temporal scales for action prediction. In European conference on computer vision, pages 596–611. Springer, 2014.
* [31] Yu Kong, Zhiqiang Tao, and Yun Fu. Deep sequential context networks for action prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1473–1481, 2017.
* [32] Tian Lan, Tsung-Chuan Chen, and Silvio Savarese. A hierarchical representation for future action prediction. In European Conference on Computer Vision, pages 689–704. Springer, 2014.
* [33] Tianwei Lin, Xu Zhao, and Zheng Shou. Single shot temporal action detection. In Proceedings of the 25th ACM international conference on Multimedia, pages 988–996, 2017.
* [34] Chunhui Liu, Yueyu Hu, Yanghao Li, Sijie Song, and Jiaying Liu. Pku-mmd: A large scale benchmark for continuous multi-modal human action understanding. arXiv preprint arXiv:1703.07475, 2017.
* [35] Jun Liu, Amir Shahroudy, Dong Xu, and Gang Wang. Spatio-temporal lstm with trust gates for 3d human action recognition. In European Conference on Computer Vision, pages 816–833. Springer, 2016.
* [36] Zelun Luo, Jun-Ting Hsieh, Lu Jiang, Juan Carlos Niebles, and Li Fei-Fei. Graph distillation for action detection with privileged modalities. In Proceedings of the European Conference on Computer Vision (ECCV), pages 166–183, 2018.
* [37] Shugao Ma, Leonid Sigal, and Stan Sclaroff. Learning activity progression in lstms for activity detection and early detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1942–1950, 2016.
* [38] Bingbing Ni, Xiaokang Yang, and Shenghua Gao. Progressively parsing interactional objects for fine grained action detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1020–1028, 2016.
* [39] Guoliang Pang, Xionghui Wang, Jian-Fang Hu, Qing Zhang, and Wei-Shi Zheng. Dbdnet: learning bi-directional dynamics for early action prediction. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, pages 897–903. AAAI Press, 2019.
* [40] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pages 91–99, 2015.
* [41] Adriana Romero, Nicolas Ballas, Samira Ebrahimi Kahou, Antoine Chassang, Carlo Gatta, and Yoshua Bengio. Fitnets: Hints for thin deep nets. arXiv preprint arXiv:1412.6550, 2014.
* [42] Michael S Ryoo. Human activity prediction: Early recognition of ongoing activities from streaming videos. In 2011 International Conference on Computer Vision, pages 1036–1043. IEEE, 2011.
* [43] Mohammad Sadegh Aliakbarian, Fatemeh Sadat Saleh, Mathieu Salzmann, Basura Fernando, Lars Petersson, and Lars Andersson. Encouraging lstms to anticipate actions very early. In Proceedings of the IEEE International Conference on Computer Vision, pages 280–289, 2017.
* [44] Amir Shahroudy, Jun Liu, Tian-Tsong Ng, and Gang Wang. Ntu rgb+ d: A large scale dataset for 3d human activity analysis. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1010–1019, 2016.
* [45] Lei Shi, Yifan Zhang, Jian Cheng, and Hanqing Lu. Skeleton-based action recognition with directed graph neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7912–7921, 2019.
* [46] Jamie Shotton, Andrew Fitzgibbon, Mat Cook, Toby Sharp, Mark Finocchio, Richard Moore, Alex Kipman, and Andrew Blake. Real-time human pose recognition in parts from single depth images. In CVPR 2011, pages 1297–1304. Ieee, 2011.
* [47] Zheng Shou, Junting Pan, Jonathan Chan, Kazuyuki Miyazawa, Hassan Mansour, Anthony Vetro, Xavier Giroi Nieto, and Shih-Fu Chang. Online action detection in untrimmed, streaming videos-modeling and evaluation. In ECCV, volume 1, page 5, 2018.
* [48] Karen Simonyan and Andrew Zisserman. Two-stream convolutional networks for action recognition in videos. In Advances in neural information processing systems, pages 568–576, 2014.
* [49] Sijie Song, Cuiling Lan, Junliang Xing, Wenjun Zeng, and Jiaying Liu. Spatio-temporal attention-based lstm networks for 3d action recognition and detection. IEEE Transactions on Image Processing, 27(7):3459–3471, 2018.
* [50] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In Proceedings of the IEEE international conference on computer vision, pages 4489–4497, 2015.
* [51] Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 6450–6459, 2018.
* [52] Heng Wang and Cordelia Schmid. Action recognition with improved trajectories. In Proceedings of the IEEE international conference on computer vision, pages 3551–3558, 2013.
* [53] Limin Wang, Yu Qiao, and Xiaoou Tang. Action recognition and detection by combining motion and appearance features. THUMOS14 Action Recognition Challenge, 1(2):2, 2014.
* [54] Pichao Wang, Wanqing Li, Philip Ogunbona, Jun Wan, and Sergio Escalera. Rgb-d-based human motion recognition with deep learning: A survey. Computer Vision and Image Understanding, 171:118–139, 2018.
* [55] Xionghui Wang, Jian-Fang Hu, Jian-Huang Lai, Jianguo Zhang, and Wei-Shi Zheng. Progressive teacher-student learning for early action prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3556–3565, 2019.
* [56] Saining Xie, Chen Sun, Jonathan Huang, Zhuowen Tu, and Kevin Murphy. Rethinking spatiotemporal feature learning: Speed-accuracy trade-offs in video classification. In Proceedings of the European Conference on Computer Vision (ECCV), pages 305–321, 2018.
* [57] Huijuan Xu, Abir Das, and Kate Saenko. R-c3d: Region convolutional 3d network for temporal activity detection. In Proceedings of the IEEE international conference on computer vision, pages 5783–5792, 2017.
* [58] Mingze Xu, Mingfei Gao, Yi-Ting Chen, Larry S Davis, and David J Crandall. Temporal recurrent networks for online action detection. In Proceedings of the IEEE International Conference on Computer Vision, pages 5532–5541, 2019.
* [59] Sijie Yan, Yuanjun Xiong, and Dahua Lin. Spatial temporal graph convolutional networks for skeleton-based action recognition. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018\.
* [60] Junho Yim, Donggyu Joo, Jihoon Bae, and Junmo Kim. A gift from knowledge distillation: Fast optimization, network minimization and transfer learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4133–4141, 2017.
* [61] Jun Yuan, Bingbing Ni, Xiaokang Yang, and Ashraf A Kassim. Temporal action localization with pyramid of score distribution features. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3093–3102, 2016.
* [62] Joe Yue-Hei Ng, Matthew Hausknecht, Sudheendra Vijayanarasimhan, Oriol Vinyals, Rajat Monga, and George Toderici. Beyond short snippets: Deep networks for video classification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4694–4702, 2015.
* [63] Pengfei Zhang, Cuiling Lan, Junliang Xing, Wenjun Zeng, Jianru Xue, and Nanning Zheng. View adaptive recurrent neural networks for high performance human action recognition from skeleton data. In Proceedings of the IEEE International Conference on Computer Vision, pages 2117–2126, 2017.
* [64] Pengfei Zhang, Cuiling Lan, Junliang Xing, Wenjun Zeng, Jianru Xue, and Nanning Zheng. View adaptive neural networks for high performance skeleton-based human action recognition. IEEE transactions on pattern analysis and machine intelligence, 2019\.
* [65] Zhengyou Zhang. Microsoft kinect sensor and its effect. IEEE multimedia, 19(2):4–10, 2012.
| Alban Main de Boissiere is following jointly a M.Eng. at Institut National
des Sciences Appliquées (INSA) Lyon and a M.A.Sc. at École de Technologie
Supérieure (ETS) Montreal. His major interests include computer vision
problems related to action recognition, early prediction, and online action
detection.
---|---
| Rita Noumeir is a full professor at the Department of Electrical
Engineering, at the École de Technologie Superieure (ETS) Montreal. Her main
research interest is in applying artificial intelligence methods to create
decision support systems. She has extensively worked in healthcare information
technology and image processing. She has also provided consulting services in
large-scale software architecture, healthcare interoperability, workflow
analysis, technology assessment, and image processing, for several
international software and medical companies including Canada Health Infoway.
Prof. Noumeir holds a Ph.D. and Master’s degrees in Biomedical Engineering
from École Polytechnique of Montreal.
---|---
|
# Analyzing and comparing door-to-door travel times for air transportation
using aggregated Uber data
Philippe Monmousseau, Aude Marzuoli, Eric Feron and Daniel Delahaye
###### Abstract
Improving the passenger air travel experience is one of the explicit goals set
by the Next Generation Air Transportation System in the United States and by
the Advisory Council for Aeronautics Research in Europe FlightPath 2050. Both
suggest door-to-door travel times as a potential metric for these objectives.
In this paper, we propose a data-driven model to estimate door-to-door travel
times and compare the reach and performance of different access modes to a
city, as well as conduct segment analysis of full door-to-door trips. This
model can also be used to compare cities with respect to the integration of
their airport within their road structure. We showcase multiple applications
of this full door-to-door travel time model to demonstrate how the model can
be used to locate where progress can be made.
## I Introduction
Both in Europe and in the United States, national or supra-national agencies
promote the need for seamless door-to-door travel and data sharing. They were
deemed as needed by the European Commission’s 2011 White Paper [1] and were
reconfirmed by the Federal Aviation Administration (FAA) in 2017 [2]. Data
sharing was already a main focus in the early 2000s; in response, Europe
created and adopted SWIM - System Wide Information Management [3] \- and the
FAA followed suit. The Next Generation Air Transportation System (NextGen) [4]
in the United States and the Advisory Council for Aeronautics Research in
Europe (ACARE) Flightpath 2050 [1] both aim to have a more passenger-centric
approach. To this end, ACARE Flightpath 2050 sets some ambitious goals, which
are not all measurable yet due to lack of available data. For example, it aims
at having 90% of travelers within Europe being able to complete their door-to-
door journey within 4 hours. In the US, the Joint Planning and Development
Office has proposed and tested metrics regarding NextGen’s goals [5], but the
passenger-centric metrics, especially regarding door-to-door travel times, are
still missing.
Cook et al. [6] first explored the shift from flight-centric to passenger-
centric metrics in the project POEM - Passenger Oriented Enhanced Metrics -
where they propose propagation-centric and passenger-oriented performance
metrics and compare them with existing flight-centric metrics. Later, Laplace
et al. [7] introduce the concept of Multimodal, Efficient Transportation in
Airports and Collaborative Decision Making (META-CDM); they propose to link
both airside CDM and landside CDM, thus taking passenger perspective into
account. In this perspective, Kim et al. [8] propose an airport gate
scheduling model for improved efficiency with a balance between aircraft,
operator and passenger objectives. Dray et al. [9] illustrate the importance
of multimodality by considering ground transportation as well during major
disturbances of the air transportation system in order to offer better
solutions to passengers.
The estimation of door-to-door travel time for multi-modal trips has been
previously studied, but for trips contained within the same metropolitan area.
Peer et al. [10] focus on commutes within a Dutch city by studying door-to-
door travel times and schedule delays for daily commuters, and show that, for
the estimation of the overall travel time, it is important to consider the
correlation of travel times across different road links. Salonen and Toivonen
[11] investigate the need for comparable models and measures for trips by car
or public transport with focus on the city of Helsinki. Their multi-modal
approach takes into account the walking and waiting times necessary to reach a
station or a parking place. Duran-Hormazabal and Tirachini [12] analyze travel
time variability for multi-modal trips within the city of Santiago, Chile,
using human surveyors and GPS data to estimate the time spent in the different
transportation modes, namely walking, driving a car, riding a bus and taking
the subway. Pels et al. [13] analyze the relative importance of access time to
airports in the passengers’ choice within the San Francisco Bay Area based on
a passenger survey, offering perspective from air transportation. These works
emphasize the importance of considering all relevant modes when estimating
door-to-door travel times, but are limited in scope with respect to the size
of the area considered and the amount of data available.
Thanks to the increasing use of mobile phones as data sources, larger scale
studies with a focus on air transportation have been possible. In the United
States, Marzuoli et al. [14] implement and validate a method to detect
domestic air passengers using mobile phone data available on a nationwide
scale. Though the main focus of this work is the passenger behavior at
airports, the granularity of the data facilitates analysis of each phase
within the full door-to-door trip. Marzuoli et al. [15] then combine mobile
phone data with social media data to analyze passenger experiences in airports
during a major disruptions. In Europe, within the BigData4ATM
project111www.bigdata4atm.eu, Garcia-Albertos et al. [16] also present a
methodology for measuring door-to-door travel times using mobile phone data,
illustrated through trips between Madrid and Barcelona. Mobile phone data are,
however, proprietary data, which are difficult to access for research.
Grimme and Martens [17] propose a model analyzing the feasibility of the
4-hour goal proposed by FlightPath 2050 based on airport-to-airport flight
times and a simplified model of access to and from airports. Sun et al. [18]
use open source maps and datasets to calculate door-to-door minimum travel
time estimations in order to study the possible competitiveness of air taxis.
In the upcoming sections, the model and analysis presented are also based on
already available online data but with a post operation approach. The aim of
this model is to create a method to measure the average door-to-door travel
times once trips are completed to analyze and compare available modes of
transportation. We have applied the first version of this method to two intra-
European multi-modal trips, thus comparing air transportation and rail
transportation [19]. We then used an improved version of the same method,
leveraging four different data sources (ride-sharing, flight, phone, and
census data) and adapted to the conditions in the United States, to compare
trips using direct flights between five US cities, three of them on the West
Coast and the other two on the East Coast [20].
In this paper, we offer a data-driven model for the computation of door-to-
door travel times that harnesses recently available data along with public
data. The data-driven methods developed can be applied for most multi-modal
trips between two cities where relevant data are available and are not limited
to the air transportation system. The range of new analyses available using
this model is illustrated with multiple modal analysis of an intra-European
trip, a per-leg analysis of multiple intra-USA trips and an analysis of the
impact of severe weather disruptions. These analyses have direct applications
for passengers, urban planners and decision makers and highlight the
difference between taking a flight-centric approach to the air transportation
system and taking a passenger-centric approach.
Section II of this paper presents the data-driven, full door-to-door travel
time model; Section III showcases a first set of analyses and applications
facilitated by this model for trips between Amsterdam and Paris; and, Section
IV focuses on a set of analyses for trips within the United States where more
data are available; Finally, Section V concludes this paper and proposes
future research directions.
## II The full door-to-door data-driven model
Similarly to [16] and [18], we can deconstruct the travel time $T$ for trips
with direct flights or direct train links into five different trip phases,
represented in Figure 1 and summarized in equation (1),
$T=t_{\text{to}}+t_{\text{dep}}+t_{\text{in}}+t_{\text{arr}}+t_{\text{from}}\leavevmode\nobreak\
,$ (1)
where
* •
$t_{\text{to}}$ is the time spent traveling from the start of the journey to
the departure station (e.g. train station or airport),
* •
$t_{\text{dep}}$ is the time spent waiting and going through security
processes (if any) at the departure station,
* •
$t_{\text{in}}$ is the time actually spent in flight or on rails,
* •
$t_{\text{arr}}$ is the time spent at the arrival station (e.g. going through
security processes),
* •
$t_{\text{from}}$ is the time spent traveling from the arrival station to the
final destination.
DeparturestationArrivalstation$t_{\text{to}}$$t_{\text{dep}}$$t_{\text{in}}$$t_{\text{arr}}$$t_{\text{from}}$
Figure 1: Model of the full door-to-door travel time.
The full model for door-to-door travel time proposed in this paper is
established by data-driven methods used to calculate the values of the
different times contained in equation (1). These data-driven methods are
described in Sections II-A through II-D.
This study focuses on air and rail transportation as main transportation
modes, which give the value of $t_{\text{in}}$, though the process can also be
applied to inter-city bus trips. Furthermore, it is assumed that passengers
travel by road when arriving or leaving the main station (airport or train
station) for the calculation of $t_{\text{to}}$ and $t_{\text{from}}$. In
response to data availability, the case studies only consider six major US
cities (Atlanta, Boston, Los Angeles, Seattle, San Francisco and Washington
D.C.) and two European capitals (Amsterdam and Paris).
### II-A Travel time from the origin location to the departure station and
from the arrival station to the final destination
We can estimate the road transit times from origin location to departure
station ($t_{\text{to}}$) and from arrival station to final destination
($t_{\text{from}}$) by using aggregated and publicly available data from taxi
or ride-sharing services. Uber [21] is a ride-sharing service launched in 2010
and located in major urban areas on six continents; it has recently released
anonymized and aggregated travel time data for certain of the urban areas
where it operates. The available data consist of the average, minimum and
maximum travel times between different zones (e.g. census tracts in the case
of US cities) within serviced area from all Uber rides aggregated over five
different periods for each considered day. The five considered periods, used
throughout this study, are defined as follows:
* •
Early Morning: from midnight to 7am
* •
AM: from 7am to 10am
* •
Midday: from 10am to 4pm
* •
PM: from 4pm to 7pm
* •
Late Evening: from 7pm to midnight
There are days when the travel times between some zones are only aggregated at
a daily level. Travel times are associated with their mean starting door time,
i.e. the mean of all the time stamps from the trip contained in the zone of
departure.
Since Uber was initially introduced in the US, the impact of Uber in US urban
transit has already been the focus of several studies prior to this data
release. Li et al. [22] concludes that, at an aggregated level, Uber tends to
decrease congestion in the US urban areas where it was introduced. Later,
Erhardt et al. [23] build a model showing that ride sharing companies do
increase congestion using the example of San Francisco. Hall et al. [24] focus
on whether Uber complemented or substituted public transit by studying the use
of public transit system before and after Uber’s entry date in different US
cities. Wang and Mu [25] study Uber’s accessibility in Atlanta, GA (US) by
using the average wait time for a ride as a proxy and conclude that the Uber
use is not associated to a specific social category. Following the release of
Uber data, Pearson et al. [26] propose a traffic flow model based on this
aggregated Uber data and use it to analyze traffic patterns for seven cities
world-wide. Assuming Uber rides as part of the road traffic flow, this study
considers that Uber’s travel times are an acceptable proxy of the actual
travel times by road. In cities where busses don’t have specific road lanes,
these travel times are a valid proxy for both car and bus trips. This paper
limits its scope to road access to and egress from considered stations. The
analysis of subway alternatives is not considered in this paper.
Each US city is divided into their census tracts; Paris into the IRIS zones
used by INSEE [27] for census, and Amsterdam into its official districts
called _wijk_.
### II-B Dwell time at stations
The dwell time at a station, either $t_{\text{dep}}$ or $t_{\text{arr}}$, is
defined as the time spent at the station, whether going through security
processes, walking through the station, or waiting. The time spent at each
station depends on the mode considered, the specific trip, and whether the
passenger is departing or arriving. The dwell time at departure can be split
into two components,
$t_{\text{dep}}=t_{\text{sec}}+t_{\text{wait}}.\leavevmode\nobreak\ ,$ (2)
a processing time, $t_{\text{sec}}$, necessary to get through security (if
any) and through the station to the desired gate or track, and an extra wait
time, $t_{\text{wait}}$, due to unanticipated delays.
Processing times at US airports are based on the average wait times at
airports extracted from the study of Marzuoli et al. [14]. The six US airports
under study in this paper are: Hartsfield-Jackson Atlanta International
Airport (ATL), Boston’s Logan International Airport (BOS) and Ronald Reagan
Washington National Airport (DCA) for the East Coast, Los Angeles
International Airport (LAX), Seattle-Tacoma International Airport (SEA) and
San Francisco International Airport (SFO) for the West Coast. Processing times
at European airports are assumed invariant between airports and determined
using most airline recommendations. The three European airports under study
are: Paris Charles de Gaulle Airport (CDG), Paris Orly Airport (ORY) and
Amsterdam Airport Schiphol (AMS).
The average dwell times at these airports are summarized in Table II for US
airports and in Table II for European airports.
Table I: Average dwell time spent at US airports in minutes.
| ATL | BOS | DCA | LAX | SEA | SFO
---|---|---|---|---|---|---
Time at departure | 110 | 105 | 100 | 125 | 105 | 105
Time at arrival | 60 | 40 | 35 | 65 | 50 | 45
Table II: Average dwell time spent at European airports in minutes.
| AMS | CDG | ORY
---|---|---|---
Time at departure | 90 | 90 | 90
Time at arrival | 45 | 45 | 45
With regard to processing times at train stations, based on the recommendation
of the train station websites, the departure dwell time is set at 15 minutes
and the arrival dwell time is set at 10 minutes for all train stations. We can
improve these estimates by gathering data from GPS or mobile phone sources as
well as WiFi beacons within airports and train stations, and by using a method
similar to Nikoue et al. [28].
We can calculate the extra wait times when the scheduled and actual departure
or arrival times are available. For US airports, these wait times are
calculated only for departure using the publicly available data from the
Bureau of Transportation Statistics (BTS) [29]. They were obtained by
subtracting the scheduled departure time from the actual flight departure
time.
### II-C Time in flight or on rail
#### II-C1 US flights
The actual flight time was calculated based on the data from BTS using the
actual departure/arrival times of all direct flights between each city pairs
from January 1${}^{\text{st}}$ 2018 to March 31${}^{\text{st}}$ 2018\.
Cancelled flights are not considered in this study and were discarded.
#### II-C2 European trips
In Europe, we assume that flights and trains are on time and follow a weekly
schedule, due to a lack of publicly centralized flight schedule data. The
weekly schedules are extracted from actual train and flight schedules gathered
over a period of several months and are assumed applicable over the full
period under study.
### II-D Full door-to-door travel time
Our model assumes that travelers plan their departure time to arrive at the
departure station exactly $t_{\text{sec}}$ minutes (eq. (2)) before the
scheduled departure time of their flight or train. We use this assumption to
determine the value of $t_{\text{to}}$ since it defines the period of the day
to consider when extracting the Uber average time from the origin location to
the departure station. We extract the value of $t_{\text{from}}$ by using the
actual arrival time of the flight or train. When only daily aggregated times
are available in the Uber data, these times are used for each period of the
day in proxy.
## III Flights versus trains: a comparison of different access modes to Paris
Let us consider a traveler leaving from Amsterdam city center to reach the
Paris area. We have chosen the city center of Amsterdam as it covers both
tourists and business travelers, but the proposed door-to-door travel time
model and subsequent analysis can be applied from any zone. Three possible
means of transportation are under study for this trip: plane from Amsterdam
Airport Schiphol to Paris Charles De Gaulle airport (CDG) or via Paris Orly
(ORY), or train from Amsterdam Centraal via Paris Gare du Nord (GDN).
### III-A Flight and train schedules
As in Section II-C2, the flight and train schedules used for this study are
weekly schedules. The weekly flight schedules between Amsterdam Airport
Schiphol (AMS) and CDG or ORY were extracted from the actual flight schedules
from December 2019 to January 2020, and it was assumed that the obtained
weekly schedules would run from January 1${}^{\text{st}}$ 2018 to September
30${}^{\text{th}}$ 2019\. These weekly schedules are summarized in Table III-A
for flights between AMS and ORY, and in Table III-A for flights between AMS
and CDG.
Table III: Extracted weekly schedule from Amsterdam to Paris via ORY.
[table head=Mo Tu We Th Fr Sa Su Ams. Paris
]tables/ams2ory.csv Table IV: Extracted weekly schedule from Amsterdam to
Paris via CDG. [table head=Mo Tu We Th Fr Sa Su Ams. Paris
]tables/ams2cdg.csv
The weekly train schedule between Amsterdam Centraal station and Paris Gare du
Nord (GDN) is similarly extracted from the actual train schedule of the year
2019 and applied to the year 2018. It is summarized in Table III-A. Night
trains are not considered for this study.
Table V: Extracted weekly schedule from Amsterdam to Paris via GDN.
[table head=Mo Tu We Th Fr Sa Su Ams. Paris
]tables/amc2gdn.csv
These schedules already highlight the major differences between the three
considered modes. The flight schedule through ORY contains the fewest
possibilities, limited to two flights daily, whereas the other two models
offer hourly scheduled transport. Another notable difference can be seen with
respect to the station-to-station travel times: flights between Amsterdam and
Paris (both CDG and ORY) take 1h20 ($\pm$ 5 minutes) while train rides between
Amsterdam Centraal and Paris GDN take 3h20 ($\pm$ 3 minutes).
### III-B Average total travel time mode comparison
The proposed data-driven door-to-door model can be used to evaluate and
compare the range of each considered mode, which helps to understand better
the urban structure and behavior from a transportation point of view.
For each trip $\tau$ (flight or rail) over the period $\mathcal{D}$ from
January 1${}^{\text{st}}$ 2018 to September 30${}^{\text{th}}$ 2019, the
associated average full door-to-door travel time $\bar{T}(\tau)$,
$\bar{T}(\tau)=\bar{t}_{\text{to}}(\tau)+t_{\text{dep}}(\tau)+t_{\text{in}}(\tau)+t_{\text{arr}}(\tau)+\bar{t}_{\text{from}}(\tau)\leavevmode\nobreak\
,$ (3)
where $\bar{t}_{\text{to}}(\tau)$ is the average ride time between Amsterdam
city center and the departure station (AMS or Amsterdam Centraal) for the trip
$\tau$, $\bar{t}_{\text{from}}(\tau)$ is the average ride time from the
arriving station (CDG, ORY or GDN) to the final arrival zone for the trip
$\tau$.
The same daily periods as those used in the Uber data (see Section II-A) are
considered here to categorize the trips into five groups depending on the time
of arrival at the final destination. For each day $d$ and each period $p$, the
mean per arrival zone $z$ of the average door-to-door travel times is
calculated for each mode $m$,
$E^{d,p}_{z}(m)=\frac{1}{|\mathcal{T}^{d,p}_{z}|}\sum_{\tau\in\mathcal{T}^{d,p}_{z}}\bar{T}(\tau)\leavevmode\nobreak\
,$ (4)
where $\mathcal{T}^{d,p}_{z}$ is the set of flight and rail trips that end at
zone $z$ on day $d$ and period $p$.
For each day $d$, each period $p$ and each arrival zone $z$, the mode with the
shortest mean travel time $E^{d,p}_{z}(m)$ is kept. The number of times
$N^{p}_{z}(m)$ a mode $m$ has had the shortest mean travel time is counted for
each zone $z$ and for each period $p$ over the twenty-one month period
$\mathcal{D}$,
$N^{p}_{z}(m)=|\\{d\in\mathcal{D}\leavevmode\nobreak\ |\leavevmode\nobreak\
m=\text{arg}\min_{m_{1}}E^{d,p}_{z}(m_{1})\\}|\leavevmode\nobreak\ .$ (5)
This distribution of modes over the different zones can help travelers choose
the mode of transport that is best suited depending on the desired arrival
zone and on the desired time of arrival. It can also help urban planners to
better understand the road network linking the different stations to the city.
Figure 2 shows the fastest mode to reach the different zones in the Paris
dataset for the five different periods of the days used by the Uber dataset.
For each zone $z$ and each period $p$, the fastest mode associated is the mode
$m$ having the highest $N^{p}_{z}(m)$, i.e. the highest number of days with
the lowest average total travel time over the considered date range. The zones
best reached through CDG are indicated in blue, ORY in red and GDN in green.
(a) Morning (AM)
(b) Midday
Figure 2: Comparison of the average total travel times to the Paris area
between the three considered arrival stations (CDG: blue, ORY: red, GDN:
green) for a trip starting from Amsterdam city center for different trip
termination periods.
We can draw several conclusions from these maps. The absence of zones best
reached by plane via ORY (in red) is particularly noticeable in the morning
period: An important area of South-West Paris is not reached by Uber rides
neither from GDN nor from CDG. These maps would advocate for an increase in
frequency for the AMS-ORY flights from a traveler’s perspective.
From a structural perspective, the interstate linking Paris to CDG is visible
on all three maps since it enables travelers through GDN to reach zones close
to CDG faster than if they flew to CDG directly. The perimeter highway
circling Paris is also a major aid to GDN and is visible on the maps where
there is an important competition between GDN and CDG. The section of the
perimeter highway farthest from GDN (i.e. in the south-west of Paris) is,
however, overtaken by either airport depending on the period of the day. The
rest of GDN influence zone is fairly invariant from a period to another.
Using a similar map representation for trips ending in the afternoon but not
shown here for space considerations, ORY’s range is limited during the
afternoon, with CDG taking over some zones close to ORY. This is essentially
due to the limited number of flights landing in the afternoon (one per week,
every Friday) compared to the daily arrival of CDG flights.
### III-C Average total travel time distribution analysis
Once the fastest mode to reach each zone is determined, it is possible to
analyze the fastest full door-to-door travel time for each zone. This approach
gives an overview of the level of integration of airports, train stations, and
road structure and can indicate zones that are less reachable than others and
would thus require more attention from urban planners.
The fastest full door-to-door travel time associated with a zone $z$ at a
period $p$ is calculated as the average fastest travel time to reach zone $z$
at period $p$ across all modes and over the period $\mathcal{D}$,
$\bar{E}_{z}^{p}=\frac{1}{|\mathcal{D}|}\sum\min_{m}E_{z}^{d,p}(m)\leavevmode\nobreak\
.$ (6)
Figure 3 displays the fastest full door-to-door travel times
$\\{\bar{E}_{z}^{p}\\}_{p,z}$ to reach the different zones in the Paris
dataset for trips finishing in the morning or at midday. The color scale that
indicates the fastest full door-to-door travel times is identical in all
subfigures. The contour of each zone indicates the fastest mode to reach it
according to the results from Section III-B using the same color code as
Figure 2, i.e. the zones reached faster through CDG are surrounded in blue,
ORY in red and GDN in green.
(a) Morning (AM)
(b) Midday
Figure 3: Comparison of the fastest full door-to-door travel times to the
Paris area between the three considered arrival stations (CDG: blue, ORY: red,
GDN: green) for a trip starting from Amsterdam city center for different trip
termination periods. The contour color of each zone indicates the fastest mode
to reach it.
For a better comparison, the distribution of the number of zones per period
reached within four time intervals (less than 4 hours, between 4 hours and 4
hours and 30 minutes, between 4 hours and 30 minutes and 5 hours, and more
than 5 hours) is presented in Table III-C.
Table VI: Number of zones per mode and period of the day grouped by full door-
to-door travel time intervals. The original dataset is the same as that used
to generate Figure 3.
Mode | Time interval | Early | AM | Midday | PM | Late
---|---|---|---|---|---|---
CDG | $t\leq$ 4h | 0 | 4 | 6 | 5 | 11
4h $<t\leq$ 4h30 | 22 | 1306 | 866 | 1189 | 845
4h30 $<t\leq$ 5h | 0 | 653 | 433 | 498 | 384
$t>$ 5h | 187 | 0 | 15 | 13 | 11
GDN | $t\leq$ 4h | 398 | 247 | 247 | 290 | 314
4h $<t\leq$ 4h30 | 775 | 818 | 731 | 719 | 641
4h30 $<t\leq$ 5h | 0 | 14 | 8 | 6 | 8
$t>$ 5h | 0 | 0 | 0 | 0 | 0
ORY | $t\leq$ 4h | 0 | 0 | 0 | 0 | 0
4h $<t\leq$ 4h30 | 0 | 0 | 906 | 563 | 997
4h30 $<t\leq$ 5h | 0 | 0 | 397 | 259 | 425
$t>$ 5h | 0 | 0 | 0 | 0 | 1
We see a dissymmetry between the north and the south of Paris by looking only
at the time color scale in Figure 3. The number of green zones, i.e. zones
reachable in less than 4 hours and 20 minutes, and the surface covered by
these green zones are more important in the northern part of Paris than in the
southern part of Paris, including in areas close to Paris Orly airport.
Combining this observation with the contour color of each zone suggests that
Paris Orly airport is not as well integrated in the Parisian road structure as
Paris Charles de Gaulle airport, which can make it less attractive for
travelers desiring to travel by air and thus less competitive.
We can complete a more quantitative analysis from Table III-C, with some of
the main findings listed here:
* •
Only 10% of the arrival zones are reached in less than 4 hours from Amsterdam
city center.
* •
Zones that can be reached in less than 4 hours from Amsterdam city center are
overwhelmingly reached by train through Paris Gare du Nord (98%).
* •
A trip from Amsterdam city center to Paris going through ORY always takes more
than 4 hours.
* •
78% of the arrival zones are reached in less than 4 hours and 30 minutes from
Amsterdam city center when combining all three possible modes.
Therefore, we can use these results to assess how well the 4-hour goal from
ACARE FlightPath 2050 is engaged.
### III-D Reliability issues
The proposed full door-to-door travel time model assumes that passengers
choose their departure time in order to arrive exactly $t_{\text{sec}}$ before
the scheduled departure of their plane or train and that they also know how
long it takes to reach the departure station. However, in reality, there is an
uncertainty in the time the traveler will spend reaching the airport and in
airport processing times. This uncertainty often leads to an additional buffer
time implying an earlier departure time for the traveler. Using the presented
model with the available data, we can find the most reliable mode to use per
arrival zone. The most reliable mode for a given arrival zone is defined as
the mode with the lowest variability in travel time, i.e. the mode where the
difference between the maximum travel time and the minimum travel time to
reach that zone is the lowest. This comparison is useful for passengers or
trips that require an accurate arrival time rather than a minimum travel time.
Figure 4 shows the most reliable mode on average to reach the different zones
in the Paris dataset for trips finishing in the morning or at midday. As for
the previous analysis, the period was determined using the departure time of
the full door-to-door trip and uses the same color code, i.e. the zones
reached most reliably through CDG are indicated in blue, ORY in red and GDN in
green. For each zone and each period of the day, the most reliable mode
associated is the mode having the highest number of days with the lowest
average variability travel time over the considered date range.
(a) Morning (AM)
(b) Midday
Figure 4: Comparison of the average variability of travel times to the Paris
area between the three considered arrival stations (CDG: blue, ORY: red, GDN:
green) for a trip starting from Amsterdam city center for different trip
termination periods.
Though Figure 4 and Figure 2 are similar, there are some major differences
between average efficiency and average reliability. For example, though it is
on average faster to reach the zones close to the highway leading to CDG by
train, after 10:00 it is safer from a time variability perspective to reach
them via CDG. From a reliability perspective, CDG has claimed the quasi
totality of the zones surrounding it, except in the early morning where trips
through GDN are still better. When we compare all three modes using this
metric, it appears that GDN has the greatest decrease in competitiveness, with
its range smaller than when considering average travel times.
### III-E Impact of faster processing times
The major difference between air and rail travel is the necessary processing
time both at departure and at arrival. In this particular study, with a flight
time of about 80 minutes, the current assumption of a departure processing
time of 90 minutes implies that travelers spend more time at their departure
airport than in flight, which greatly impacts the rapidity of air travel. With
the presented model, one can modify these assumed processing times in order to
study the impact of improving these times both from an airport perspective and
a passenger perspective. Let’s assume that the processing time at airports is
improved from 90 to 60 minutes at departure and from 45 to 30 minutes at
arrival. These modifications could be achieved in reality considering that
this is an intra-Schengen trip and that there isn’t any border control.
(a) Early morning with faster processing times
(b) Early morning with normal processing times
Figure 5: Comparison of the average total travel times to the Paris area
assuming faster airport processing times between the three considered arrival
stations (CDG: blue, ORY: red, GDN: green) for a trip starting from Amsterdam
city center for trips arriving at destination early in the morning. The
corresponding map with normal processing times is also reproduced.
Figures 5& 6 show which is the fastest mode on average to reach the different
zones in the Paris dataset for trips arriving at destination early in the
morning or at midday.
The first major difference with this processing time improvement can be seen
for trips arriving in the early morning (Figure 5): all zones previously
reached through CDG are no longer accessed at this period since they were
associated to the 21:45 flight of the previous day. This indicates that all
trips through CDG start and end on the same day, with no trips finishing after
midnight.
(a) Midday with faster processing times
(b) Midday with normal processing times
Figure 6: Comparison of the average total travel times to the Paris area
assuming faster airport processing times between the three considered arrival
stations (CDG: blue, ORY: red, GDN: green) for a trip starting from Amsterdam
city center for trips arriving at destination at midday. The corresponding
maps with normal processing times are also reproduced.
Looking at trips arriving at midday (Figure 6), trips through CDG are greatly
advantaged by this time improvement, with CDG taking over more than half of
GDN’s previous influence zone. This range increase from CDG can be explained
both by faster door-to-door travel times and by the increase of trips through
CDG arriving at midday (rather than in the afternoon). As a matter of fact,
besides in the early morning, GDN loses its competitiveness against both
airports, with its range greatly shrinking in size. The competition between
CDG and ORY remains unchanged, which is understandable since they both
received the same processing time improvement.
A quantitative analysis similar to the analysis presented in Table III-C
concludes that all trips are now conducted in less than five hours and that
99.8% of the zones reachable are reached in less than 4h30. ORY sees some
major improvements with 97.5% of the zones best reached through it reached in
less than four hours (compared to no trips in less than 4h in the initial
model), while increasing the number of zones it reaches the fastest.
Using a map representation similar to Section III-C, but not presented here
due to space considerations, it is possible to notice a 20-30 minutes shift in
the time distribution for every period except for early morning trips since
train processing times were unchanged. The upper bound travel time is also
unchanged for trips arriving in the morning, which would indicate that for
some zones, the processing time improvement resulted in no improvement or even
a worsening of the full trip travel time. Besides that exception, in this case
a 45 minutes improvement in airport processing time leads only to a maximum of
30 minutes of average total travel time improvement due to the influence of
train trips through GDN.
## IV A multi-modal analysis of the US air transportation system
Additional insights are gained from this full door-to-door travel time model
thanks to the availability of complementary data. The United States is a
federal state the size of a continent, therefore various aggregated and
centralized datasets are more easily available to all. Several of these
datasets are used in this section to add applications to the presented full
door-to-door model. This US study is limited to the period from January 1st
2018 to March 31st 2018.
### IV-A Flight schedule
As presented in the model definition in Section II-C1, both the scheduled
flight times and the actual flight schedules of most domestic flights can be
obtained via the Department of Transportation Bureau of Transportation
Statistics (BTS) [29]. This study considers only the six US airports presented
in Section II-B, three East-coast airports - Hartsfield-Jackson Atlanta
International Airport (ATL), Boston’s Logan International Airport (BOS) and
Ronald Reagan Washington National Airport (DCA) - and three West-coast
airports - Los Angeles International Airport (LAX), Seattle-Tacoma
International Airport (SEA) and San Francisco International Airport (SFO).
During this three-month period, 38,826 flights were considered, which
corresponds to 3,523 early flights, 8,170 morning flights, 13,451 midday
flights, 6.695 afternoon flights and 6,987 evening flights. The full door-to-
door travel times were then calculated for each scheduled flight from January
1${}^{\text{st}}$, 2018 to March 31${}^{\text{st}}$, 2018, using the model
presented in Section II.
### IV-B Leg analysis
We can use the full door-to-door model to better understand the time spent in
each leg proportionally to the time spent on the overall trip. For each trip,
we calculate the percentage of time spent at each phase based on the full
door-to-door travel time. We then calculate the average percentage time spent
for each phase and for each city pair trip. Figure 7 shows the bar plot of
these average percentage times for the thirty considered city pairs. The city
pairs are sorted according to the percentage of time spent in the actual
flight phase.
Figure 7: Bar plot of the average proportion of the time spent within each
phase of the full door-to-door journey for all thirty considered trips.
With the proposed full door-to-door model, for all considered trips,
passengers spend on average more time at the departure airport than riding to
and from the airports. This figure also shows that, with this model, for some
short-haul flights, such as between SFO and LAX or between BOS and DCA,
passengers spend on average more time at the departure airport than in the
plane. Refining the full door-to-door model by considering tailored airport
processing times $t_{\text{sec}}$ at departure depending on the city pair and
not only on the departure airport could lead to a different conclusion.
However, this modification of the model would require more access to passenger
data.
### IV-C Airport integration comparison
The proposed full door-to-door model allows us to compare each airport’s
integration within its metropolitan area. Each census tract is associated with
an internal point within its boundaries, and this internal point can be used
to automatically calculate the distance between airports and each census tract
of their metropolitan area. The internal points were defined using an online
database222www.usboundary.com based on the US government 2010 census. Figure 8
shows the scatter plot of the average daily ride time to each airport versus
the geodesic distance to the airport for the six considered airport. The
geodesic distance is the shortest distance between two points along the
surface of the Earth. Additionally, the plot also figures a linear regression
of these average time with respect to the distance to the airport. A steeper
slope for the linear regression indicates that it takes longer to reach the
airport from a given distance.
Figure 8: Scatter plot of the average ride time to the airport $t_{\text{to}}$
versus the distance to the airport from January 1${}^{\text{st}}$ 2018 to
March 31${}^{\text{st}}$ 2018\. Straight lines indicate the linear regression
fit for each city.
Figure 8 highlights the disparity between the range of each airport within
available data: DCA has a range limited to 20 km while SFO attracts Uber
riders from more than 120 km away. The other four airports have a similar
range. The difference in slope of their associated linear regression is
nonetheless useful to rank their integration within their region of
attraction. From this perspective, Seattle has the best integrated airport,
i.e. the smallest slope, followed by Atlanta, Boston and then Los Angeles.
### IV-D Impact of severe weather analysis
Using the same door-to-door travel time visualization process and applying it
to different days can be a tool to better analyze the effects of severe
weather perturbations on the full door-to-door journey. As an example, the
winter storm previously studied in [15] is analyzed for trips between
Washington D.C. and Boston using Figure 9. This winter storm hit the East
Coast of the United States on January 4${}^{\text{th}}$ 2018, and led to the
closure of two airports in New York City, along with the cancellation of the
majority of flights flying to or from the North-Eastern US coast.
(a) Before landfall, on January 2${}^{\text{nd}}$, 2018
(b) After landfall, on January 5${}^{\text{th}}$, 2018
Figure 9: Average door-to-door travel times from Washington D.C. city hall to
Boston over a single day, before and after the Bomb Cyclone of January 2018.
Figure 9 shows the map of the average full door-to-door travel times to reach
the Boston area starting from Washington D.C. city hall on January
2${}^{\text{nd}}$ 2018, before the landfall of this winter storm, and on
January 5${}^{\text{th}}$ 2018, after the landfall of the winter storm.
The color scales representing the full door-to-door travel time are identical
from one map to another. A shift towards the red is visible from January
2${}^{\text{nd}}$ 2018 (Figure 8(a)) to January 5${}^{\text{th}}$ 2018 (Figure
8(b)), along with some census tracts disappearing from the considered range on
January 5${}^{\text{th}}$ 2018, due to lack of sufficient Uber ride data.
These two observations indicate that the full door-to-door travel times are
closer to the maximum average travel time than from the minimum travel time on
January 5${}^{\text{th}}$ 2018 compared to January 2${}^{\text{nd}}$ 2018, and
that some zones might have been sufficiently adversely impacted by the weather
to prevent rides from the airport to reach them.
### IV-E On the importance of a passenger-centric approach to delays
A final application to the full door-to-door model presented in this paper
emphasizes the difference between flight delay and passenger delay. Since Uber
splits the day into five different periods, each with their traffic
idiosyncrasies with respect to peak times, we can calculate how much extra
travel time is required for a passenger when a flight does not arrive in the
scheduled period. For example, a flight expected to arrive in the early
morning that lands after 10:00 AM could result in the passenger getting
stranded in traffic when trying to leave the airport. Though airlines are not
responsible for road traffic, passengers can choose flights based on their
arrival time to avoid peak time traffic.
To calculate this extra travel at aggregated level, we have calculated the
difference of average travel time between the two periods concerned by flights
not arriving according to schedule for each arrival zone. These travel time
differences are then aggregated into one travel time difference per city pair
by weighing the travel time associated to each census tract with the
proportion of passengers initiating their trips from there, or finishing their
trip there. The number of passengers originating from or finishing within a
census tract is assumed to be proportional to the population density of the
considered census tract.
Another measure of sensitivity is to consider the maximum difference between
the maximum travel times of each zone between the two considered periods. This
second measure indicates the worst variation of the travel time upper bound,
i.e. the maximum difference a traveler can experience going from the airport
to their final destination zone.
Let us consider the flight UA460 from LAX to SFO scheduled to arrive on
Thursday February 15, 2018 at 18:02 local time and that landed with a minor
delay of 16 minutes. Due to the 45-minute processing time required to leave
the airport, this 16-minute delay shifts the departure period from the airport
from afternoon (PM) to late evening. The aggregated average extra travel time
is of 15 minutes and 40 seconds, i.e. a 16-minute flight delay triggered an
average 31-minute total delay for the passengers. Looking at the second
considered measure, the maximum travel time difference for this flight delay
is of 72 minutes, meaning that one passenger could experience a delay of 88
minutes resulting from this 16-minute flight delay. This first example
illustrates that passenger delays and aircraft delays are distinct.
Paradoxically, arriving earlier than scheduled for a flight does not
necessarily mean that the full door-to-door trip ends earlier. For example,
flight VX1929 from LAX to SFO scheduled to arrive on Thursday February 8, 2018
at 15:22 local time actually landed 25 minutes earlier. This implied that the
passengers were no longer leaving the airport in the afternoon (PM) period but
at midday. The aggregated average extra travel time is here of 15 minutes and
2 seconds, so on average travelers did arrive earlier than scheduled, but only
by about ten minutes and not the twenty-five minutes announced by the airline.
However, looking at the second measurement method again, the maximum ride time
difference is 66 minutes and 44 seconds, which means that a passenger could
end up arriving forty minutes later than if the flight had landed on time.
## V Discussion & Conclusion
By leveraging Uber’s recently released data and combining it with several
other available data sources, this paper proposed a data-driven model for the
estimation of full door-to-door travel times for multi-modal trips both in
Europe and in the United States. Though the model is used for one city pair in
Europe and six different cities in the United States, it can be implemented
between any world city pair with sufficient available ride-sharing or taxi
data. The proposed model can be adapted depending on how much data about the
considered travel modes are available, since a weekly schedule containing no
information relative to delays can already lead to some meaningful insights
from both a passenger perspective and a city planner perspective.
Once aggregated at a city level, the presented door-to-door travel time model
can be used for a paired comparison of the different phases of the full door-
to-door journey between two cities. It also enables us to analyze the actual
time spent while travelling between two specific cities. The evaluation on a
national level of some of the passenger-centric objectives proposed within
NextGen in the United States and within ACARE FlightPath 2050 in Europe is
possible thanks to the proposed model, especially the objectives regarding how
well integrated airports are within their cities. The model can also provide
insight to how multi-modal trips are affected by severe weather disruptions,
indicating where improvements can be made. It also brings a valuable
measurement of the difference between flight delays and passenger delays,
emphasizing the need for passenger-centric metrics to evaluate the performance
of the air transportation system, which is not solely constituted of planes.
Future studies should consider integrating additional data, such as
alternative transportation modes, e.g. the subway, in the model once they are
available and when calculating the time needed to reach the departure station
or leave from the arrival station. Additionally, the knowledge of the actual
daily proportion of passengers travelling via the different approach modes
(road or rail) would lead to a better precision of the proposed full door-to-
door travel time model. Aggregated information from GPS or mobile phone
sources could possibly be used to determine this proportion without infringing
passenger privacy.
## Acknowledgments
The authors would like to thank Nikunj Oza from NASA-Ames, the BDAI team from
Verizon Media in Sunnyvale as well as the _Ecole Nationale de l’Aviation
Civile_ and King Abdullah University of Science and Technology for their
financial support. The authors are also grateful for the help of Marine
Lebeater for her feedback. The authors would also like to thank the SESAR
Joint Undertaking for its support of the project ER4-10 ”TRANSIT: Travel
Information Management for Seamless Intermodal Transport”. Data retrieved from
Uber Movement, (c) 2020 Uber Technologies, Inc., https://movement.uber.com.
## References
* [1] M. Darecki, C. Edelstenne, E. Fernandez, P. Hartman, J.-P. Herteman, M. Kerkloch, I. King, P. Ky, M. Mathieu, G. Orsi, G. Schotman, C. Smith, and J.-D. Wörner, _Flightpath 2050: Europe’s Vision for Aviation ; Maintaining Global Leadership and Serving Society’s Needs ; Report of the High-Level Group on Aviation Research_ , E. Commission, Ed. Luxembourg: European Commission, 2011.
* [2] _NextGen Priorities - Joint Implementation Plan Update Including the Northeast Corridor_. Federal Aviation Administration, Oct. 2017.
* [3] J. Meserole and J. Moore, “What is System Wide Information Management (SWIM)?” in _2006 IEEE/AIAA 25TH Digital Avionics Systems Conference_. Portland, OR, USA: IEEE, Oct. 2006, pp. 1–8.
* [4] NextGen Integration and Implementation Office, “NextGen Implementation Plan,” in _Federal Aviation Administration_ , 2009.
* [5] Y. O. Gawdiak and T. Diana, “NextGen Metrics for the Joint Planning and Development Office,” vol. 11th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference, including the AIAA Balloon Systems Conference and 19th AIAA Lighter-Than, 2011.
* [6] A. Cook, G. Tanner, S. Cristóbal, and M. Zanin, “Passenger-Oriented Enhanced Metrics,” 2012.
* [7] I. Laplace, A. Marzuoli, and E. Feron, “META-CDM: Multimodal, Efficient Transportation in Airports and Collaborative Decision Making,” in _Airports in Urban Networks 2014 (AUN2014)_ , Paris, 2014.
* [8] S. H. Kim, A. Marzuoli, J.-P. Clarke, D. Delahaye, and E. Feron, “Airport Gate Scheduling for Passengers, Aircraft, and Operation,” in _Tenth USA/Europe Air Traffic Management Research and Development Seminar (ATM2013)_ , Chicago, Illinois, Jun. 2013.
* [9] L. Dray, A. Marzuoli, and A. Evans, “Air Transportation and Multimodal, Collaborative Decision Making during Adverse Events,” in _Eleventh USA/Europe Air Traffic Management Research and Development Seminar (ATM2015)_ , Lisboa, Portugal, Jun. 2015.
* [10] S. Peer, J. Knockaert, P. Koster, Y.-Y. Tseng, and E. T. Verhoef, “Door-to-door travel times in Revealed Preference departure time choice models: An approximation method using GPS data,” _Transportation Research Part B: Methodological_ , vol. 58, pp. 134–150, 2013.
* [11] M. Salonen and T. Toivonen, “Modelling travel time in urban networks: Comparable measures for private car and public transport,” _Journal of Transport Geography_ , vol. 31, pp. 143–153, Jul. 2013.
* [12] E. Durán-Hormazábal and A. Tirachini, “Estimation of travel time variability for cars, buses, metro and door-to-door public transport trips in Santiago, Chile,” _Research in Transportation Economics_ , vol. 59, pp. 26–39, Nov. 2016.
* [13] E. Pels, P. Nijkamp, and P. Rietveld, “Access to and competition between airports: A case study for the San Francisco Bay area,” _Transportation Research Part A: Policy and Practice_ , vol. 37, no. 1, pp. 71–83, Jan. 2003.
* [14] A. Marzuoli, E. Boidot, E. Feron, and A. Srivastava, “Implementing and Validating Air Passenger–Centric Metrics Using Mobile Phone Data,” _Journal of Aerospace Information Systems_ , vol. 16, no. 4, pp. 132–147, Apr. 2019.
* [15] A. Marzuoli, P. Monmousseau, and E. Feron, “Passenger-centric metrics for Air Transportation leveraging mobile phone and Twitter data,” in _Data-Driven Intelligent Transportation Workshop - IEEE International Conference on Data Mining 2018_ , Singapore, Nov. 2018.
* [16] P. García-Albertos, O. G. Cantú Ros, R. Herranz, and C. Ciruelos, “Understanding Door-to-Door Travel Times from Opportunistically Collected Mobile Phone Records,” in _SESAR Innovation Days 2017_ , 2017.
* [17] W. Grimme and S. Maertens, “Flightpath 2050 revisited – An analysis of the 4-hour-goal using flight schedules and origin-destination passenger demand data,” _Transportation Research Procedia_ , vol. 43, pp. 147–155, 2019.
* [18] X. Sun, S. Wandelt, and E. Stumpf, “Competitiveness of on-demand air taxis regarding door-to-door travel time: A race through Europe,” _Transportation Research Part E: Logistics and Transportation Review_ , vol. 119, pp. 1–18, Nov. 2018.
* [19] P. Monmousseau, D. Delahaye, A. Marzuoli, and E. Feron, “Door-to-door travel time analysis from Paris to London and Amsterdam using Uber data,” in _Ninth SESAR Innovation Days_ , Athens, Greece, 2019.
* [20] ——, “Door-to-door Air Travel Time Analysis in the United States using Uber Data,” in _1st International Conference on Artificial Intelligence and Data Analytics for Air Transportation (AIDA-AT 2020)_ , Singapore, Feb. 2020.
* [21] Uber Technologies Inc., “Uber, About us,” 2020. [Online]. Available: http://www.uber.com/about
* [22] Z. Li, Y. Hong, and Z. Zhang, “Do On-demand Ride-sharing Services Affect Traffic Congestion? Evidence from Uber Entry,” 2016.
* [23] G. D. Erhardt, S. Roy, D. Cooper, B. Sana, M. Chen, and J. Castiglione, “Do transportation network companies decrease or increase congestion?” _Science Advances_ , vol. 5, no. 5, May 2019.
* [24] J. D. Hall, C. Palsson, and J. Price, “Is Uber a substitute or complement for public transit?” _Journal of Urban Economics_ , vol. 108, pp. 36–50, Nov. 2018.
* [25] M. Wang and L. Mu, “Spatial disparities of Uber accessibility: An exploratory analysis in Atlanta, USA,” _Computers, Environment and Urban Systems_ , vol. 67, pp. 169–175, Jan. 2018.
* [26] M. Pearson, J. Sagastuy, and S. Samaniego, “Traffic Flow Analysis Using Uber Movement Data,” 2018.
* [27] INSEE, “Institut national de la statistique et des études économiques,” 2020\. [Online]. Available: https://www.insee.fr
* [28] H. Nikoue, A. Marzuoli, J.-P. Clarke, E. Feron, and J. Peters, “Passenger Flow Predictions at Sydney International Airport: A Data-Driven Queuing Approach,” _arXiv:1508.04839 [cs]_ , Aug. 2015.
* [29] Bureau of Transportation Statistics, “Bureau of Transportation Statistics, About BTS,” 2018. [Online]. Available: http://www.rita.dot.gov/bts/about
|
aainstitutetext: Department of Physics and Astronomy, University of British
Columbia 6224 Agricultural Road, Vancouver, B.C., V6T 1W9, Canada
# Holographic quantum tasks with input and output regions
Alex May<EMAIL_ADDRESS>
###### Abstract
Quantum tasks are quantum computations with inputs and outputs occurring at
specified spacetime locations. Considering such tasks in the context of
AdS/CFT has led to novel constraints relating bulk geometry and boundary
entanglement. In this article we consider tasks where inputs and outputs are
encoded into extended spacetime regions, rather than the points previously
considered. We show that this leads to stronger constraints than have been
derived in the point based setting. In particular we improve the connected
wedge theorem, appearing earlier in 1912.05649, by finding a larger bulk
region whose existence implies large boundary correlation. As well, we show
how considering extended input and output regions leads to non-trivial
statements in Poincaré-AdS2+1, a setting where the point-based connected wedge
theorem is always trivial.
## 1 Introduction
$c_{1}$$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{V}}_{2}$$r_{2}$$r_{1}$${\mathcal{R}_{1}}$${\mathcal{R}_{2}}$$c_{2}$
(a)
$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{C}}_{1}$$\hat{\mathcal{C}}_{2}$$\hat{\mathcal{R}}_{1}$$\hat{\mathcal{R}}_{2}$
(b)
Figure 1: (a) Bulk perspective on a particular quantum task. The task has
inputs given at bulk points $c_{1}$ and $c_{2}$, and outputs are required at
$r_{1}$ and $r_{2}$. (b) Boundary view of the same task. Now regions
$\hat{\mathcal{C}}_{i}$ whose entanglement wedge $\mathcal{C}_{i}$ contains
point $c_{i}$ become the input regions, while the $\hat{\mathcal{R}}_{i}$
whose entanglement wedge $\hat{\mathcal{R}}_{i}$ contains the $r_{i}$ become
the output regions. Define the scattering region $J^{E}_{12\rightarrow
12}\equiv J^{+}(\mathcal{C}_{1})\cap J^{+}(\mathcal{C}_{2})\cap
J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2})$ in the bulk geometry, and
the decision regions
$\hat{\mathcal{V}}_{i}\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{i})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2})$.
The connected wedge theorem states that when $J^{E}_{12\rightarrow 12}$ is
non-empty, the entanglement wedge of
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$ is connected.
Relativistic quantum tasks are quantum computations with inputs and outputs
occurring at specified spacetime locations kent2012quantum . In the context of
the AdS/CFT correspondence, the framework of holographic quantum tasks
may2019quantum considers such computations from a bulk as well as boundary
perspective. By comparing the two perspectives, it has been possible to find a
constraint on boundary entanglement placed by bulk causal features
may2019quantum ; may2020holographic . This constraint is called the connected
wedge theorem; it states that a specific entanglement wedge must be connected
when a related set of light cones overlap. In this paper we extend the
holographic quantum tasks framework and derive a stronger connected wedge
theorem.
To derive constraints for AdS/CFT from quantum tasks, we begin by defining a
task in the bulk. This is specified by sets of input and output locations,
along with a channel relating inputs to outputs. We can then discuss protocols
for completing this bulk task, and determine the probability with which the
best such protocol succeeds. Then, we identify a corresponding task in the
boundary, and note that because the boundary describes the bulk the boundary
task is completed with the same or higher probability. In some cases we can,
beginning with the success probability, show that there must be large amounts
of entanglement in the boundary. Reasoning in this way leads to the connected
wedge theorem.
In may2019quantum ; may2020holographic the tasks in the bulk considered
always had inputs and outputs taking a special form. First, the inputs and
outputs were given at locations idealized as points, and second, those points
were located at asymptotic infinity. This was convenient in that it allowed
the corresponding boundary task to be identified with no additional
assumptions: we can naturally identify a point in AdS at infinity with a point
in the CFT, and in this way define the input and output locations for the
boundary task.
In this paper we extend this procedure to include bulk tasks with input
locations that are still points, but which are not at asymptotic infinity. In
the boundary, entanglement wedge reconstruction implies the corresponding task
is one with extended regions as input locations, namely regions which have the
bulk input point in their entanglement wedge. A similar relationship relates
bulk output points and boundary output regions. An example set-up is shown in
figure 1.
By considering this broader class of holographic quantum tasks we are led to a
stronger version of the connected wedge theorem than was previously given. A
simple adaptation of the relativistic proof given for the earlier connected
wedge theorem in may2020holographic can be used to prove our theorem. As
well, the new connected wedge theorem applies non-trivially to Poincaré-
AdS2+1, where the earlier theorem was always trivial. The application to
Poincaré requires considering output regions which consist of two intervals,
and so in particular are disconnected. This gives a particularly clear example
of how the restriction of input and output locations to be points failed to
capture all useful constructions.
Note that this article emphasizes the quantum tasks perspective, however, the
reader interested in the connected wedge theorem as a geometric statement can
move directly to the theorem statement in section 3 and its relativistic proof
in section 5.
Throughout the article we refer to boundary spacetime regions with hatted
script letters
$\hat{\mathcal{X}},\hat{\mathcal{A}},\hat{\mathcal{B}},\hat{\mathcal{V}}...$
and their bulk entanglement wedges with un-hatted letters
$\mathcal{E}_{W}(\hat{\mathcal{X}})=\mathcal{X}$, etc. We will use
$J^{\pm}(\cdot)$ for the causal future or past taken in the bulk geometry, and
$\hat{J}^{\pm}(\cdot)$ for the causal future or past taken in the boundary
geometry. We will denote the Ryu-Takayanagi surface of a boundary region
$\hat{\mathcal{X}}$ by $\gamma_{\mathcal{X}}$.
### Outline of the article
The outline of this article is as follows.
In section 2, we update the quantum tasks framework to consider input and
output spacetime regions. We discuss in detail how to identify bulk and
boundary tasks in this context. We emphasize that doing so requires
identifying, for a given boundary region, a bulk region which stores the same
quantum information. In other work this has been understood to be the
entanglement wedge CKNR ; HHLR ; maximin ; JLMS ; DHW ; noisyDHW .
In section 3 we state the improved connected wedge theorem, making use of
input and output regions. We explain why this is a stronger theorem than given
previously, and comment on how to choose the inputs and output regions to
arrive at the strongest possible statement. We also point out that the
converse to the theorem does not hold. Finally we explain how to apply the
improved connected wedge theorem to Poincaré-AdS2+1, which involves taking one
of the output regions to be disconnected.
In section 4 we give the quantum tasks argument for the improved connected
wedge theorem, which exploits the expanded quantum tasks framework developed
in section 2. Aside from the generalization to input and output regions, the
treatment here also improves on may2020holographic in the way errors in the
bulk protocol are handled, in particular we derive a linear lower bound on the
mutual information even in this noisy case.
In section 5 we prove the stronger connected wedge theorem using the focusing
theorem in general relativity. The relativistic proof is a simple modification
of the proof of the earlier theorem appearing in may2020holographic .
We conclude with a brief summary and some comments in section 6.
## 2 The holographic quantum tasks framework
### 2.1 Quantum tasks
We will discuss quantum tasks where Alice is given inputs that are initially
recorded into extended spacetime regions, and must be output at extended
output regions. To make this more precise, we define a notion of quantum
information being localized to a spacetime region. Our definitions are adapted
from hayden2019localizing .
###### Definition 1
Suppose one party, Alice, holds system $X$ of a quantum state
$|\Psi\rangle_{XX^{\prime}}$. Then we say the subsystem $X$ is localized to a
spacetime region $\mathcal{R}$ if a second party, Bob, for whom the state is
initially unknown can prepare the $X$ system by acting on $\mathcal{R}$ with
some channel $\mathcal{M}_{\mathcal{R}\rightarrow X}$.
If system $X$ is localized to region $\mathcal{R}$ such that a channel
$\mathcal{M}_{\mathcal{R}\rightarrow X}$ recovers $X$, we will say that $X$ is
localized to $\mathcal{R}$ relative to $\mathcal{M}_{\mathcal{R}\rightarrow
X}$. Note that throughout this article by quantum system we mean a tensor
factor of a Hilbert space, e.g. the $X$ system of
$\mathcal{H}_{XX^{\prime}}=\mathcal{H}_{X}\otimes\mathcal{H}_{X^{\prime}}$.111It
may also be interesting to repeat our discussion in the more general setting
where $X$ is a subalgebra.
It will also be convenient to say a quantum system $X$ is excluded from a
spacetime region $\mathcal{R}$ if Bob cannot learn anything about $X$ by
accessing $\mathcal{R}$. One way to specify this precisely is to consider
$|\Psi\rangle_{XX^{\prime}}$ to be in the maximally entangled state. Then $X$
is excluded from $\mathcal{R}$ when $I(\mathcal{R}:\mathcal{X^{\prime}})=0$.
We should point out several features of these definitions. First, note that a
system $X$ is localized to $\mathcal{R}$ if and only if it is localized to the
domain of dependence of $\mathcal{R}$, and similarly it is excluded from
$\mathcal{R}$ if and only if it is excluded from the domain of dependence of
$\mathcal{R}$. This follows because all the classical and quantum data in the
domain of dependence of $\mathcal{R}$ is fixed by the data in $\mathcal{R}$ by
time evolution. Consequently we will identify regions with their domains of
dependence throughout this paper.
Second, note that a quantum system $X$ can be neither localized to nor
excluded from a region $\mathcal{R}$ if some but not all information about $X$
is available in $\mathcal{R}$. As well, notice that given a Cauchy surface
$\Sigma$ a quantum system can be excluded from both $\Sigma\cap\mathcal{R}$
and $\Sigma\setminus\mathcal{R}$. To do this, encode system $X$ using the one-
time pad ambainis2000private using a classical key $k$. This hides the state
on $X$, which can only be revealed if $k$ is known. The $X$ register can then
be passed through $\Sigma\cap\mathcal{R}$ and $k$ through
$\Sigma\setminus\mathcal{R}$, and system $X$ will be excluded from both
regions. Of course, $X$ will still be localized to the full Cauchy surface
$\Sigma$.
There are many possible ways in which a quantum system can be localized to a
given spacetime region, and a single quantum system can be localized to many
different spacetime regions. Which sets of spacetime regions the same quantum
information can be localized to is restricted however. In particular, if $X$
is localized to a region $\mathcal{R}$ then it follows that $X$ is excluded
from the spacelike complement $\mathcal{R}^{c}$. This is because otherwise we
could act independently on $\mathcal{R}$ and $\mathcal{R}^{c}$ to produce
copies of $X$, in violation of the no-cloning theorem.
As a simple example of how a single quantum system can be localized to many
regions, suppose we have three subregions
$\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{3}$ which are all spacelike
separated. Then define $\mathcal{R}_{1}=\mathcal{X}_{1}\cup\mathcal{X}_{2}$,
$\mathcal{R}_{2}=\mathcal{X}_{2}\cup\mathcal{X}_{3}$, and
$\mathcal{R}_{3}=\mathcal{X}_{3}\cup\mathcal{X}_{1}$. To localize a quantum
system $X$ to all three regions
$\\{\mathcal{R}_{1},\mathcal{R}_{2},\mathcal{R}_{3}\\}$, encode $X$ into an
error-correcting code on three subsystems $S_{1}S_{2}S_{3}$ that corrects one
erasure error. Send system $S_{i}$ to $\mathcal{X}_{i}$. Then each of the
$\mathcal{R}_{i}$ contain two of the $S_{i}$ subsystems, and $X$ can be
recovered from each of them.
Given a quantum system $X$, we can specify how it is localized in spacetime by
specifying two sets of regions, call them $\\{\mathcal{A}_{X}^{i}\\}_{i}$ and
$\\{\mathcal{U}_{X}^{i}\\}_{i}$. We specify that $X$ is localized to each of
the regions $\mathcal{A}_{X}^{i}$, and excluded from each of the regions
$\mathcal{U}_{X}^{i}$. Implicitly, each region $\mathcal{A}_{X}^{i}$ comes
with a channel $\mathcal{N}_{\mathcal{A}_{X}^{i}\rightarrow X}$ that specifies
how $X$ can be recovered from $\mathcal{A}^{i}_{X}$.222In general there could
be many such channels, though this will not be important in this article. We
summarize this in the next definition.
###### Definition 2
A quantum system $X$ is encoded into an access structure
$\mathcal{S}_{X}=(\\{\mathcal{A}_{X}^{i}\\}_{i},\\{\mathcal{U}_{X}^{i}\\}_{i})$
if $X$ is localized to each of the regions $\\{\mathcal{A}_{X}^{i}\\}_{i}$ and
excluded from each of the regions $\\{\mathcal{U}_{X}^{i}\\}_{i}$.
The term “access structure” is borrowed from the subject of quantum secret
sharing, which features a closely related object.333In quantum secret sharing
a system $X$ is recorded into shares $X_{1}...X_{n}$ such that $X$ can be
recovered from some subsets of shares, called authorized sets, and no
information about $X$ is available in another set of subsets of shares, called
the unauthorized sets. In hayden2019localizing the authors characterized
which access structures it is possible to localize a quantum system to.
Next, we give a definition of a relativistic quantum task. This builds on the
definition presented in may2019quantum by allowing for inputs and outputs to
be recorded into arbitrary access structures.
###### Definition 3
A relativistic quantum task is defined by a tuple
$\mathbf{T}=(\mathcal{M},\mathscr{A},\mathcal{S}_{\mathscr{A}},\mathscr{B},\mathcal{S}_{\mathscr{B}},\mathcal{N}_{\mathscr{A}\rightarrow\mathscr{B}})$,
where:
* •
$\mathcal{M}$ is the spacetime in which the task occurs, it is described by a
manifold equipped with a (Lorentzian) metric.
* •
$\mathscr{A}=A_{1}...A_{n_{a}}$ is the collection of all the input quantum
systems, and
$\mathcal{S}_{\mathscr{A}}=\\{\mathcal{S}_{A_{1}},...,\mathcal{S}_{A_{n_{a}}}\\}$
is the set of all access structures for the input systems.
* •
$\mathscr{B}=B_{1}...B_{n_{b}}$ is the collection of all the output quantum
systems, and
$\mathcal{S}_{\mathscr{B}}=\\{\mathcal{S}_{B_{1}},...,\mathcal{S}_{B_{n_{b}}}\\}$
is the set of all access structures for the output systems.
* •
$\mathcal{N}_{\mathscr{A}\rightarrow\mathscr{B}}$ is a quantum channel that
maps the input systems $\mathscr{A}$ to the output systems $\mathscr{B}$.
Bob encodes the input systems $A_{i}$ in such a way that the access structures
$\mathcal{S}_{A_{i}}$ are satisfied. To complete the task, Alice should apply
the channel $\mathcal{N}_{\mathscr{A}\rightarrow\mathscr{B}}$ and localize
each of the systems $B_{i}$ according to the access structure
$\mathcal{S}_{{B}_{i}}$.
In order to encode the $A_{i}$ into the appropriate regions, Bob couples the
regions $\mathcal{A}_{A_{i}}^{j}$ to some external system which initially hold
the $A_{i}$. To verify Alice has completed the task successfully, Bob will
access one or more of the regions $\mathcal{A}_{B_{j}}^{i}$,
$\mathcal{U}_{B_{j}}^{i}$ and attempt to recover system $B_{j}$. If Bob is
able to produce $B_{j}$ from the authorized region $\mathcal{A}_{B_{j}}^{i}$
he declares the task successful. Similarly if he is _unable_ to produce
$B_{j}$ from the unauthorized region $\mathcal{U}_{B_{j}}^{i}$ he declares the
task successful. The probability that Alice’s outputs pass Bob’s test is her
success probability. Alice’s success probability maximized over all possible
protocols for completing the task is the tasks success probability,
$p_{suc}(\mathbf{T})$.
Note that if Bob acts on one of the output regions $\mathcal{A}_{B_{j}}^{i}$,
$\mathcal{U}_{B_{j}}^{i}$ in performing his test, we no longer require Alice
have the correct outputs (or exclusions from) regions in the causal future of
the accessed region. Similarly, Bob will localize the inputs $A_{j}$ to
regions $\mathcal{A}_{A_{j}}^{i}$ so long as Alice never interferes. She may
choose to access some region $\mathcal{A}_{A_{j}}^{i}$ however and obtain
$A_{i}$, in which case Bob is no longer expected to localize $A_{i}$ to
regions in the future of $\mathcal{A}_{A_{j}}^{i}$.
In the application considered below, we begin with a spacetime and use tasks
as a way to probe features of that fixed geometry. Consequently, we have
defined quantum tasks to feature a fixed spacetime background. Doing so
assumes Alice’s choice of protocol does not change the geometry. It is also
possible to consider more general tasks, where we allow the spacetime geometry
to react to Alice’s protocol, which might for instance involve distributing
large numbers of qubits which then change the geometry. We leave considering
this to future work.
### 2.2 Quantum tasks in holography
In our definition of a quantum task in the last section, we have used an
operational framing. This is only for convenience however, and it is possible
to remove this language. In particular, the protocol Alice carries out is in
fact just a feature of some initial state $|\Psi\rangle$. All the instructions
for her protocol are by necessity recorded there, all that happens during the
execution of the protocol is time evolution according to the underlying
theory’s Hamiltonian. While Alice’s protocol is the internal dynamics of the
theory in question, Bob preparing the inputs and collecting outputs correspond
to couplings to some external system. Viewing quantum tasks in this way
motivates understanding them as probes of the underlying theory they are
defined in.
Because tasks probe the underlying theory, if we are given an equivalence
between two theories it is natural to try and interpret this equivalence in
the language of tasks. In particular we will consider the bulk and boundary
theories in AdS/CFT. Within each theory, there is a set of tasks that can be
defined and associated success probabilities,
$\\{(\mathbf{T}_{i},p_{suc}(\mathbf{T}_{i}))_{i}\\}$. The equivalence of bulk
and boundary theories suggests that for each task $\mathbf{T}$ defined in the
bulk there is some corresponding task $\mathbf{\hat{T}}$ in the boundary, and
further that $p_{suc}(\mathbf{\hat{T}})=p_{suc}(\mathbf{{T}})$. We will make
this more precise below.
Our first step will be to restrict attention to a bulk described by classical
geometry along with quantum fields living on a curved background (which may be
coupled to the geometry). This means that while the boundary theory completely
describes the bulk, the converse is not true. Consequently we will expect an
inequality, $p_{suc}(\mathbf{{T}})\leq p_{suc}(\mathbf{\hat{T}})$. Before
understanding this in more detail however, we need to specify how a bulk task
should be associated with a boundary task.
Given a task in the bulk
$\mathbf{T}=(\mathcal{M},\mathscr{A},\mathcal{S}_{\mathscr{A}},\mathscr{B},\mathcal{S}_{\mathscr{B}},\mathcal{N}_{\mathscr{A}\rightarrow\mathscr{B}})$,
we should identify the boundary dual of each element of the tuple. Beginning
with $\mathcal{M}$, the bulk geometry, we define $\mathbf{\hat{T}}$ to be in
the geometry $\partial\mathcal{M}$, the boundary of $\mathcal{M}$. The inputs
$\mathscr{A}$, outputs $\mathscr{B}$, and channel
$\mathcal{N}_{\mathscr{A}\rightarrow\mathscr{B}}$ we may identify trivially
across bulk and boundary. This is because while the bulk and boundary degrees
of freedom look very different, we can record the same quantum states into
these different degrees of freedom.
Next we need to discuss how to identify an access structure in the bulk with a
corresponding access structure in the boundary. Note that in principle,
because the AdS/CFT correspondence fixes the boundary description given the
bulk, the boundary access structure
$(\\{\hat{\mathcal{A}}_{A_{i}}^{j}\\},\\{\hat{\mathcal{U}}_{A_{i}}^{j}\\}\\}\\})$
is fixed by the bulk one
$(\\{{\mathcal{A}}_{A_{i}}^{j}\\},\\{{\mathcal{U}}_{A_{i}}^{j}\\}\\}\\})$. We
have not understood how to do this in the most general case, but can make some
statements which will be sufficient for the application discussed here.
First, notice that given bulk authorized regions
$\\{\mathcal{A}_{A_{i}}^{j}\\}_{j}$, we have
$\displaystyle\\{\hat{\mathcal{A}}_{A_{i}}^{k}\\}_{k}\supseteq\bigcup_{j}\\{\hat{X}:{\mathcal{A}}_{A_{i}}^{j}\subseteq\mathcal{E}_{W}(\hat{X})\\}.$
(1)
This is because the entanglement wedge $\mathcal{E}_{W}(\hat{X})$ is the
portion of the bulk which $\hat{X}$ can be used to recover CKNR ; HHLR ;
maximin ; JLMS ; DHW ; noisyDHW , so when
${\mathcal{A}}_{A_{i}}^{j}\subseteq\mathcal{E}_{W}(\hat{X})$ the boundary
region $\hat{X}$ can be used to recover $A_{i}$, which implies $\hat{X}$ is an
authorized region. The other inclusion does not follow in general since there
may be some boundary regions $\hat{\mathcal{A}}_{A_{i}}^{j}$ whose
entanglement wedge includes a portion but not all of $\mathcal{A}_{A_{i}}^{j}$
and which still construct $A_{i}$.
Given a bulk unauthorized region, we can say that
$\displaystyle\\{\hat{\mathcal{U}}_{A_{i}}^{k}\\}_{k}\supseteq\bigcup_{j}\\{\hat{X}:{\mathcal{U}}_{A_{i}}^{j}\supseteq\mathcal{E}_{W}(\hat{X})\\}.$
(2)
This follows because $\mathcal{E}_{W}(\hat{X})$ is the largest bulk region
whose quantum information can be reconstructed given $\hat{X}$, so
${\mathcal{U}}_{A_{i}}^{j}\supseteq\mathcal{E}_{W}(\hat{X})$ means $\hat{X}$
does not reconstruct $A_{i}$. Note that unless one or more of the
${\mathcal{U}}_{A_{i}}^{j}$ are anchored to the boundary, the set
$\\{\hat{X}:{\mathcal{U}}_{A_{i}}^{j}\supseteq\mathcal{E}_{W}(\hat{X})\\}$
will be empty.
We will be interested only in a special case, where the bulk tasks access
structures all have only authorized regions, and where those authorized
regions are points. In this case the inclusion 1 becomes an equality, fully
specifying the boundary authorized regions from the bulk ones. Further, there
will be no boundary unauthorized regions.444Of course the spacelike
complements $[\mathcal{A}_{A_{i}}^{k}]^{c}$ do not contain any information
about $A_{i}$, but it is not necessary to designate these as unauthorized,
since this is immediate from the $\mathcal{A}_{A_{i}}^{k}$ being authorized.
Given a bulk task $\mathbf{T}$ and associated boundary task
$\mathbf{\hat{T}}$, we’ve claimed $p_{suc}(\mathbf{{T}})\leq
p_{suc}(\mathbf{\hat{T}})$. This follows because any protocol that completes
the task in the bulk with some probability $p$ will be mapped under the
AdS/CFT duality to a protocol in the boundary. The bulk task’s success
probability is determined by the information localized to the regions
$(\\{{\mathcal{A}}_{A_{i}}^{j}\\},\\{{\mathcal{U}}_{A_{i}}^{j}\\}\\}\\})$. In
the boundary description the same information will be available in the regions
$(\\{\hat{\mathcal{A}}_{A_{i}}^{j}\\},\\{\hat{\mathcal{U}}_{A_{i}}^{j}\\}\\}\\})$,
so the boundary protocol will complete the task with probability $p$ as well.
Note that we claim only an inequality, rather than an equality, because many
protocols in the boundary theory will correspond to bulk protocols that change
the geometry $\mathcal{M}$, which we assumed should be fixed and unaffected by
the protocol. Worse, some boundary protocols might correspond to leaving a
semi-classical description of the bulk altogether.
## 3 An improved connected wedge theorem
### 3.1 Statement of the theorem
$\hat{\mathcal{C}}_{1}$$\hat{\mathcal{C}}_{2}$$\hat{\mathcal{C}}_{2}$$\hat{\mathcal{R}}_{2}$$\hat{\mathcal{R}}_{1}$
(a)
$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\mathcal{R}_{1}$ (b)
Figure 2: (a) A view of the boundary of AdS2+1. Left and right edges of the
diagram are identified. Shown is an example choice of input regions
$\hat{\mathcal{C}}_{i}$ and output regions $\hat{\mathcal{R}}_{i}$. This
particular choice is maximal for the regions $\hat{\mathcal{V}}_{i}$ defined
by $\hat{\mathcal{C}}_{i}=\hat{\mathcal{V}}_{i}$. (b) Bulk perspective on the
same choice of regions, showing the entanglement wedges $\mathcal{C}_{i}$ and
$\mathcal{R}_{i}$. Only one of the out regions is shown, to avoid cluttering
the diagram. In the bulk there is a non-empty entanglement scattering region
$J^{\mathcal{E}}_{12\rightarrow 12}$.
We state the improved connected wedge theorem below.
###### Theorem 4
(Connected wedge theorem) Pick four regions
$\hat{\mathcal{C}}_{1},\hat{\mathcal{C}}_{2},\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$
on the boundary of an asymptotically AdS spacetime. From these, define the
decision regions
$\displaystyle\hat{\mathcal{V}}_{1}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}),$
$\displaystyle\hat{\mathcal{V}}_{2}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{2})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}).$
(3)
Assume that $\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$. Define the
entanglement scattering region
$\displaystyle J_{12\rightarrow 12}^{\mathcal{E}}\equiv
J^{+}(\mathcal{C}_{1})\cap J^{+}(\mathcal{C}_{2})\cap
J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2}),$ (4)
where $\mathcal{C}_{i}=\mathcal{E}_{W}(\hat{\mathcal{C}}_{i})$ and
$\mathcal{R}_{i}=\mathcal{E}_{W}(\hat{\mathcal{R}}_{i})$. Then,
$J_{12\rightarrow 12}^{\mathcal{E}}\neq\varnothing$ implies that
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ is
connected.
Notice that this theorem generalizes the earlier one appearing in
may2020holographic . In particular choosing the $\hat{\mathcal{C}}_{i}$ and
$\hat{\mathcal{R}}_{i}$ to be points we recover the earlier theorem. Also note
that the theorem is true in arbitrary dimensions, but is trivial whenever
$\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$ overlap. We summarize what
is known about where non-trivial configurations occur in section 6.1.
It is interesting to consider starting with a choice of regions
$\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$, then pick regions
$\hat{\mathcal{C}}_{1},\hat{\mathcal{C}}_{2},\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$
to understand if $\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$ share a
connected entanglement wedge. In AdS2+1, and when the decision regions each
consist of a single diamond, there is a unique ‘best’ way to do this, in the
sense that one particular choice of regions will conclude there is a connected
wedge whenever any choice of regions does.
To find the optimal choice of $\hat{\mathcal{C}}_{i},\hat{\mathcal{R}}_{i}$,
we note first that there is a maximal choice of regions
$\hat{\mathcal{C}}_{i}$ imposed by the constraint
$\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$: choose
$\hat{\mathcal{C}}_{i}=\hat{\mathcal{V}}_{i}$. Further, there is a maximal
choice of $\hat{\mathcal{R}}_{i}$ consistent with a given
$\hat{\mathcal{V}}_{1},\hat{\mathcal{V}}_{2}$, which is illustrated in figure
2a. Since any other choice
$\hat{\mathcal{C}}_{i}^{\prime},\hat{\mathcal{R}}_{i}^{\prime}$ has
$\hat{\mathcal{C}}_{i}^{\prime}\subseteq\hat{\mathcal{C}}_{i}$ and
$\hat{\mathcal{R}}_{i}^{\prime}\subseteq\hat{\mathcal{R}}_{i}$ these maximal
choices have ${J^{\prime}}^{\mathcal{E}}_{12\rightarrow 12}\subseteq
J^{\mathcal{\mathcal{E}}}_{12\rightarrow 12}$, so whenever a non-maximal
choice has a non-empty entanglement scattering region the maximal choice will.
Thus whenever $\hat{\mathcal{C}}_{i}^{\prime},\hat{\mathcal{R}}_{i}^{\prime}$
can be used to conclude $\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$ has a
connected entanglement wedge, the maximal choice will conclude the same.
$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{V}}_{1}$$\pi/2$ (a)
$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{V}}_{1}$ (b)
Figure 3: A counterexample to the converse of Theorem 4. (a) Vacuum AdS2+1
with regions $\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$ chosen
antipodally and to each occupy $\pi/2$ of the boundary. Choosing the maximal
consistent input and output regions, the entanglement scattering region is
exactly one point, and the Ryu-Takayanagi surface is on the transition from
disconnected (blue) to connected (red). (b) Spherically symmetric matter is
added to the bulk. Now the entanglement wedges of $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ reach less deeply into the bulk hubeny2012extremal ,
and the light rays sent inward normally from their extremal surfaces are
delayed. This closes the entanglement scattering region. By spherical symmetry
however the Ryu-Takayanagi surface remains on the transition. Deforming
$\hat{\mathcal{V}}_{1}$ to be larger ensures we are in the connected phase,
and for small enough deformation ensures the scattering region remains empty.
Figure reproduced from may2020holographic .
In general, the converse to Theorem 4 does not hold. This is immediate from
the fact, pointed out in may2020holographic , that the converse to the point-
based case does not hold, which is a special case of the theorem presented
here. As commented on in the last paragraph however, the point based choice is
not the strongest choice of regions to understand if the entanglement wedge
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ is
connected. We can only expect a converse in the case where we take the optimal
choice of regions outlined above. Taking the optimal choice of regions however
the theorem still does not have a converse, as we argue in figure 3.
### 3.2 Connected wedge theorem in Poincaré-AdS2+1
$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{R}}_{1}$$\hat{\mathcal{R}}_{2}$$\hat{\mathcal{R}}_{2}$
Figure 4: A typical choice of regions
$\hat{\mathcal{C}}_{1}=\hat{\mathcal{V}}_{1}$,
$\hat{\mathcal{C}}_{2}=\hat{\mathcal{V}}_{2}$,
$\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$ in the boundary of Poincaré-AdS
which leads to a non-trivial conclusion in the connected wedge Theorem 4.
Region $\hat{\mathcal{R}}_{2}$ consists of two wedges which each extend to
infinity.
The connected wedge theorem applies to any asymptotically AdS spacetime,
including global AdS and Poincaré-AdS spacetimes in arbitrary dimensions. To
apply the theorem meaningfully however, we need to find configurations of
regions
$\hat{\mathcal{C}}_{1},\hat{\mathcal{C}}_{2},\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$
such that the bulk entanglement scattering region is non-empty, while the
boundary scattering region is empty.
It is not immediately clear how to find non-trivial configurations of input
and output regions in Poincaré AdS2+1. Indeed, at least for pure Poincaré-
AdS2+1 no such configurations exist when the input and output regions are
chosen to be points. One way to see this is to start with non-trivial
arrangements of points $c_{1},c_{2},r_{1},r_{2}$ in global AdS2+1, and chose a
Poincaré patch which includes regions $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$. Doing so one always finds that one of the four points
sit outside the patch, and consequently we cannot state the non-trivial
instances of the theorem using points directly in Poincaré AdS2+1.
For extended regions it is straightforward to find non-trivial configurations
in Poincaré AdS2+1. An example configuration is shown in figure 4.
Importantly, region $\hat{\mathcal{R}}_{2}$ consists of two disconnected
parts, where each connected component consists of a half line. In appendix B
we find configurations which are non-trivial in the case where the bulk is
pure AdS. Since many of these configurations have extended entanglement
scattering regions, and the scattering regions should be deformed only a small
amount for small perturbations to the bulk geometry, there will also be many
non-trivial configurations when matter is added.
## 4 Quantum tasks perspective on the connected wedge theorem
### 4.1 The $\mathbf{{B}}_{84}^{\times n}$ task
Following may2019quantum ; may2020holographic , we discuss the
$\mathbf{B}_{84}$ task.555The name of this task comes from its similarity to
the BB84 key distribution protocol, itself named for Bennet and Brassard
bennett2020quantum . This task has $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ as
authorized regions for inputs $A_{1}$ and $A_{2}$, and $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ as authorized regions for outputs $B_{1}$ and $B_{2}$. Alice
will be given a guarantee that $A_{1}$ is in one of the states
$H^{q}|b\rangle$, and $A_{2}$ stores the classical data $q$. Both $q$ and $b$
are bits, $q,b\in\\{0,1\\}$. Alice’s task is to localize $b$ to both
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$.
We will be interested in two strategies for completing the task: a local
strategy and a non-local strategy. The local strategy is one which makes use
of the scattering region
$\displaystyle J^{+}(\mathcal{C}_{1})\cap J^{+}(\mathcal{C}_{2})\cap
J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2})$ (5)
and so is only available when this region is non-empty. The non-local strategy
does not use this region. The two strategies are shown in figures 5a and 5b
respectively. We treat each below. Note that in arguing for Theorem 4, we will
be interested in the case where in the bulk the scattering region defined
above is available and so the local strategy can be used, while in the
boundary the corresponding scattering region is empty, so it is necessary to
use a non-local strategy.
$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\mathcal{R}_{2}$$\mathcal{R}_{1}$$J^{\mathcal{E}}_{12\rightarrow
12}$ (a)
$|\Psi^{+}\rangle$$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{C}}_{1}$$\hat{\mathcal{C}}_{2}$$\hat{\mathcal{R}}_{2}$$\hat{\mathcal{R}}_{1}$
(b)
Figure 5: Bulk and boundary perspectives on the $\mathbf{\hat{B}}_{84}$ task.
(a) Causal features present in the bulk geometry. Signals from
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ may meet in the scattering region
$J^{\mathcal{E}}_{12\rightarrow 12}$, then travel to either $\mathcal{R}_{1}$
or $\mathcal{R}_{2}$. The scattering region is a resource, useful for
completing quantum tasks. (b) Causal features present in the boundary
geometry, which lacks an entanglement scattering region.
$\hat{\mathcal{C}}_{1}$ may send signals to $\hat{\mathcal{R}}_{1}$ and
$\hat{\mathcal{R}}_{2}$, and $\hat{\mathcal{C}}_{2}$ may send signals to
$\hat{\mathcal{R}}_{1}$ and $\hat{\mathcal{R}}_{2}$. The boundary replaces the
resource of a non-empty scattering region with entanglement between
$\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$.
#### Local strategy for $\mathbf{{B}}_{84}^{\times n}$ task
The causal features of the task in the local strategy are captured by figure
5a. In this case, there is a protocol which completes the task with high
probability. In particular, Alice should bring $H^{q}|b\rangle$ from
$\mathcal{C}_{1}$ and $q$ from $\mathcal{C}_{2}$ together inside the above
causally defined region. Then, she applies $H^{q}$ to obtain
$(H^{q})^{2}|b\rangle=|b\rangle$, measures in the computational basis to learn
$b$, and then sends $b$ to both $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$.
Assuming this can be carried out as described this completes the task with
probability $p_{suc}=1$. More physically we will allow for the presence of
noise in carrying out this protocol, and say the success probability satisfies
$p_{suc}(\mathbf{\hat{B}}_{84})\geq 1-\epsilon$.
Consider repeating the $\textbf{B}_{84}$ task $n$ times in parallel. Call this
repeated task $\textbf{B}_{84}^{\times n}$. This repeated task has inputs
$A_{1}=\bigotimes_{i=1}^{n}H^{q_{i}}|b_{i}\rangle$ , $A_{2}=\\{q_{i}\\}_{i}$,
and required outputs $\\{b_{i}\\}_{i}$ at both $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$. The $q_{i}$ and $b_{i}$ are random and independent. We will
declare the task to be completed successfully if a fraction $1-2\epsilon$ of
the $n$ tasks are completed successfully.666This should be contrasted to the
condition considered in may2019quantum ; may2020holographic , where
$\mathbf{B}^{\times n}_{84}$ was declared successful only if all $n$ of the
individual $\textbf{B}_{84}$ tasks were completed successfully. Taking this
more relaxed condition is what eventually leads us to the bound 10, which
improves on the earlier bound. If each task is completed with probability
$p_{suc}=1-\epsilon$, then this occurs with probability
$\displaystyle p_{suc}(\textbf{B}^{\times n}_{84})=1-2\epsilon^{2+n}.$ (6)
This is the success probability for the $\textbf{B}^{\times n}_{84}$ when a
local strategy is available.
#### Non-local strategy for $\mathbf{\hat{B}}_{84}^{\times n}$ task
In the non-local case, where no scattering region is available, we will be
interested in strategies of the form shown in figure 5b. These do not use the
scattering region but instead use entanglement shared between the _decision_
regions
$\displaystyle\hat{\mathcal{V}}_{1}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}),$
$\displaystyle\hat{\mathcal{V}}_{2}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{2})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}),$
(7)
to complete the task. The regions $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ are relevant because they are the largest regions
which contain the inputs $A_{1}$ and $A_{2}$ but are also in the past of both
output regions. Rather than discussing specific strategies for completing the
task non-locally, we are interested in lower bounding the amount of
entanglement necessary to complete the task with high probability when using
any non-local strategy.
We begin by assuming $I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=0$, that
$\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$, and considering a
single instance of the task, $\mathbf{\hat{B}}^{\times 1}_{84}$. We will see
that this leads to a success probability bounded strictly below 1.
###### Lemma 5
Consider the $\mathbf{\hat{B}}^{\times 1}_{84}$ task with
$I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=0$. Then any strategy for
completing the task has $p_{suc}(\mathbf{\hat{B}}^{\times
1}_{84})\leq\cos^{2}(\pi/8)$.
This lemma is proven in tomamichel2013monogamy . To understand it
heuristically we can reason as follows. In region $\hat{\mathcal{C}}_{1}$
Alice holds one of the states $H^{q}|b\rangle$ for $q,b\in\\{0,1\\}$. If
$\hat{\mathcal{C}}_{1}$ also held the basis information, Alice could measure
in the computational basis $\\{|0\rangle,|1\rangle\\}$ if $q=0$ or the
Hadamard basis $\\{|+\rangle,|-\rangle\\}$ if $q=1$ to determine $b$. Since
$I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=0$ however, and $q$ is held in
$\hat{\mathcal{V}}_{2}$, Alice must act in a way independent of $q$. Doing so,
she cannot determine $b$ perfectly. Instead her optimal strategy is to measure
in an intermediate basis $\\{|\psi_{0}\rangle,|\psi_{1}\rangle\\}$ where
$\displaystyle|\psi_{0}\rangle$
$\displaystyle=\cos\left(\frac{\pi}{8}\right)|0\rangle+\sin\left(\frac{\pi}{8}\right)|1\rangle,$
$\displaystyle|\psi_{1}\rangle$
$\displaystyle=\cos\left(\frac{5\pi}{8}\right)|0\rangle+\sin\left(\frac{5\pi}{8}\right)|1\rangle.$
(8)
This leads to the $\cos^{2}(\pi/8)$ success probability, so that the bound in
Lemma 5 is actually tight. Note that the assumption
$\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$ is important for this
reasoning to hold. In particular, we used that
$q\in\hat{\mathcal{C}}_{2}\subseteq\hat{\mathcal{V}}_{2}$ in saying
$I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=0$ implies the measurement on
$A_{1}\in\hat{\mathcal{C}}_{1}\subseteq\hat{\mathcal{V}}_{1}$ is independent
of $q$.
Next we consider the parallel repetition task, $\mathbf{\hat{B}}^{\times
n}_{84}$. In tomamichel2013monogamy the following statement has been proven.
###### Lemma 6
Consider the $\mathbf{\hat{B}}^{\times n}_{84}$ task with
$I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=0$, assume that
$\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$ and that the scattering
region is empty. Require that a fraction $1-\delta$ of the individual
$\mathbf{\hat{B}}_{84}$ tasks are successful. Then any strategy for completing
the task has
$\displaystyle p_{suc}(\textbf{B}_{84}^{\times
n})\leq\left(2^{h(\delta)}\cos^{2}\left(\frac{\pi}{8}\right)\right)^{n}\equiv\left(2^{h(\delta)}\beta\right)^{n},$
(9)
where $h(\delta)$ is the binary entropy function
$h(\delta)\equiv-\delta\log_{2}\delta-(1-\delta)\log_{2}(1-\delta)$ and the
second equality defines $\beta$.
For small enough $\delta$ we have that $2^{h(\delta)}\beta<1$, so this gives a
good bound on the success probability.
In the boundary, where the scattering region is empty, the success probability
is exponentially small when the mutual information is zero. Comparing to the
bulk where the success probability is exponentially close to one suggests the
true boundary state contains large mutual information. This is indeed the
case, as we show in the next lemma.
###### Lemma 7
Suppose the $\mathbf{\hat{B}}_{84}$ task is completed with probability
$p_{suc}\geq 1-2\epsilon^{2+n}$, and that the scattering region is empty. Then
$\displaystyle\frac{1}{2}I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})\geq
n(-\log 2^{h(2\epsilon)}\beta)-1+O((\epsilon/\beta)^{n})$ (10)
This is proven in appendix A. In the next section we employ this bound to
argue for the connected wedge theorem. Note that this improves on the bound
presented in may2020holographic .
### 4.2 Connected wedge theorem from quantum tasks
In this section we give the quantum tasks argument for the connected wedge
theorem. We first recall the theorem for convenience.
Theorem 4 _(Connected wedge theorem) Pick four regions
$\hat{\mathcal{C}}_{1},\hat{\mathcal{C}}_{2},\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$
on the boundary of an asymptotically AdS spacetime. From these, define the
decision regions_
$\displaystyle\hat{\mathcal{V}}_{1}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}),$
$\displaystyle\hat{\mathcal{V}}_{2}$
$\displaystyle\equiv\hat{J}^{+}(\hat{\mathcal{C}}_{2})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2}).$
(11)
_Assume that $\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$. Define the
entanglement scattering region_
$\displaystyle J_{12\rightarrow 12}^{\mathcal{E}}\equiv
J^{+}(\mathcal{C}_{1})\cap J^{+}(\mathcal{C}_{2})\cap
J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2}),$ (12)
_where $\mathcal{C}_{i}=\mathcal{E}_{W}(\hat{\mathcal{C}}_{i})$ and
$\mathcal{R}_{i}=\mathcal{E}_{W}(\hat{\mathcal{R}}_{i})$. Then,
$J_{12\rightarrow 12}^{\mathcal{E}}\neq\varnothing$ implies that
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ is
connected. _
Argument. Consider two cases. First, supposed that
$\hat{\mathcal{V}}_{1}\cap\hat{\mathcal{V}}_{2}\neq\varnothing$. Then we
immediately have that the entanglement wedge of
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$ is connected, and we are
done.
Next, assume that
$\hat{\mathcal{V}}_{1}\cap\hat{\mathcal{V}}_{2}=\varnothing$. This is just the
statement that the boundary scattering region
$\displaystyle\hat{J}_{12\rightarrow
12}=\hat{J}^{+}(\hat{\mathcal{C}}_{1})\cap\hat{J}^{+}(\hat{\mathcal{C}}_{2})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{1})\cap\hat{J}^{-}(\hat{\mathcal{R}}_{2})$
(13)
is empty. By assumption however, the bulk entanglement scattering region
$J_{12\rightarrow 12}^{\mathcal{E}}$ is non-empty. This implies the existence
of four points $c_{1},c_{2},r_{1},r_{2}$ such that
$\displaystyle J^{+}(c_{1})\cap J^{+}(c_{2})\cap J^{-}(r_{1})\cap
J^{-}(r_{2})\neq\varnothing.$ (14)
Choose a $\mathbf{{B}}_{84}^{\times n}$ task in the bulk with $c_{1}$, $c_{2}$
as input points and $r_{1}$, $r_{2}$ as output points. Then because the above
region is non-empty, we can use the local strategy in the bulk, and we obtain
a high success probability,
$\displaystyle p_{suc}(\mathbf{\hat{B}}_{84}^{\times n})\geq
1-2\epsilon^{2+n}.$ (15)
Next, we should discuss how large we can take $n$. The obstruction to taking
$n$ arbitrarily large is that if we make use of too many qubits, the bulk
protocol may change the geometry, deforming the geometry we are attempting to
study. To avoid this we choose $n$ to be any order in $1/G_{N}$ less than
linear. Then if each qubit carries some energy $\Delta E$, Einstein’s
equations dictate that the coupling to geometry is
$\displaystyle G_{\mu\nu}=O(G_{N}\Delta En).$ (16)
Choosing $n<O(1/G_{N})$ ensures that in the $G_{N}\rightarrow 0$ limit we have
no backreaction, as needed to ensure we are studying the intended geometry.
Starting with $\mathbf{{B}}_{84}^{\times n}$, we label the corresponding
boundary task by $\mathbf{\hat{B}}_{84}^{\times n}$. Following the discussion
in section 2, we know $\mathbf{\hat{B}}_{84}^{\times n}$ has the same inputs
and outputs as the corresponding bulk task. Further, we have by assumption
that
$\displaystyle c_{i}\in\mathcal{E}_{W}(\hat{\mathcal{C}}_{i}),$ $\displaystyle
r_{i}\in\mathcal{E}_{W}(\hat{\mathcal{R}}_{i}).$ (17)
Thus $\hat{\mathcal{C}}_{i}$ is an authorized region for $A_{i}$ in the
boundary task, and $\hat{\mathcal{R}}_{i}$ is an authorized region for
$B_{i}$. Since $\hat{\mathcal{C}}_{i}\subseteq\hat{\mathcal{V}}_{i}$, and by
assumption the boundary scattering region
$\hat{\mathcal{V}}_{1}\cap\hat{\mathcal{V}}_{2}$ is empty, the boundary uses a
non-local strategy. Lemma 7 then applies and we can conclude
$\displaystyle\frac{1}{2}I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})\geq
n(-\log_{2}\beta)-1+O((\epsilon/\beta)^{n}).$ (18)
Since $n$ is any order less than $O(1/G_{N})$, we can conclude that
$I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})=O(1/G_{N})$, which occurs only
when $\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ is
connected.
Notice that the Ryu-Takayanagi formula only appears in this argument in the
last step. Without ever using the Ryu-Takayanagi formula, we can still
conclude that the mutual information is order $1/G_{N}$. We also could keep
$\epsilon$ fixed in this argument. This was possible because of the improved
bound 10. Using the earlier bound as it appeared in may2020holographic , it
was necessary to appeal to the Ryu-Takayanagi formula and to argue one can
take $\epsilon\rightarrow 0$ in the $G_{N}\rightarrow 0$ limit. It is
interesting that one can reach conclusions about boundary entanglement from
bulk geometry without using the Ryu-Takayanagi formula.
We should note that there is a gap in the argument above, which was noted also
in may2020holographic . In particular the causal diagram 5b does not include
the regions that sit between $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$. In general, these can be made use of to complete the
$\mathbf{\hat{B}}_{84}^{\times n}$ task without entanglement.777See appendix B
of may2020holographic . However, such strategies require GHZ correlations in
the CFT, which are not expected Nezami:2016zni . As well, it seems possible to
rule out such strategies by keeping Alice ignorant of the location of the
regions $\hat{\mathcal{C}}_{i}$ before the beginning of the task, in which
case she cannot coordinate actions with the intermediate regions. We leave
better understanding this to future work, and for now rely on gravitational
reasoning to provide a complete proof.
## 5 Relativistic perspective on the connected wedge theorem
### 5.1 Relativistic proof
The proof of Theorem 4 is nearly identical to the proof of the earlier
connected wedge theorem, which appears already in may2020holographic . For
readers familiar with the earlier proof, it suffices to note that the only
change is to replace the causal horizon $\partial[J^{-}(r_{1})\cap
J^{-}(r_{2})]$ with the null sheet $\partial[J^{-}(\mathcal{R}_{1})\cap
J^{-}(\mathcal{R}_{2})]$. The key point is that $\partial[J^{-}(r_{1})\cap
J^{-}(r_{2})]$ and $\partial[J^{-}(\mathcal{R}_{1})\cap
J^{-}(\mathcal{R}_{2})]$ meet the boundary along the same curves, and both
surfaces have area theorems for past directed null geodesics. This allows the
two surfaces to play similar roles in the proof.
To be self contained, we also present the proof briefly here. Assume that the
null energy condition holds. Work by contradiction by assuming that the
entanglement scattering region is non-empty and the minimal area extremal
surface homologous to $\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$
consists of a connected component $\gamma_{{\mathcal{V}}_{1}}$ homologous to
$\hat{\mathcal{V}}_{1}$ and a connected component $\gamma_{{\mathcal{V}}_{2}}$
homologous to $\hat{\mathcal{V}}_{2}$. Then, the maximin prescription maximin
for finding Ryu-Takayanagi surfaces dictates that there will exists a Cauchy
slice $\Sigma$ of the bulk in which
$\gamma_{{\mathcal{V}}_{1}}\cup\gamma_{{\mathcal{V}}_{2}}$ is minimal, where
by $\gamma_{{X}}$ we always mean the Ryu-Takayanagi surface for a boundary
region $\hat{X}$. We then construct a codimension $1$ surface we call the
_null membrane_ $\mathcal{N}_{\Sigma}$, along with the codimension $2$
_contradiction surface_ $\mathcal{C}_{\Sigma}$. The null membrane facilitates
a comparison of the area of
$\gamma_{{\mathcal{V}}_{1}}\cup\gamma_{{\mathcal{V}}_{2}}$ and the
contradiction surface. The contradiction surface will turn out to have less
area than the candidate surface
$\gamma_{{\mathcal{V}}_{1}}\cup\gamma_{{\mathcal{V}}_{2}}$, which provides the
contradiction.
$\mathcal{R}$$C_{\Sigma}$$\mathcal{\gamma}_{\mathcal{V}_{1}}$$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{V}}_{2}$
Figure 6: The null membrane. The blue surface is the lift $\mathcal{L}$, which
is generated by the null geodesics defined by the inward, future pointing null
normals to $\gamma_{\mathcal{V}_{1}}\cup\gamma_{\mathcal{V}_{2}}$, where
$\gamma_{{\mathcal{V}}_{i}}$ is the Ryu-Takayanagi surface for region
$\hat{\mathcal{V}}_{i}$. The red surfaces make up the slope
$\mathcal{S}_{\Sigma}$, which is generated by the null geodesics defined by
the inward, past directed null normals to
$\gamma_{{\mathcal{R}}_{1}}\cup\gamma_{{\mathcal{R}}_{2}}$. The ridge
$\mathcal{R}$ is where null rays from $\gamma_{{\mathcal{V}}_{1}}$ and
$\gamma_{{\mathcal{V}}_{2}}$ collide. The contradiction surface $C_{\Sigma}$
is where the slope meets a specified Cauchy surface $\Sigma$.
The null membrane is illustrated in figure 6. It is defined as the union of
two surfaces, called the _lift_ and the _slope_. The lift is defined by
$\displaystyle\mathcal{L}=\partial
J^{+}(\mathcal{V}_{1}\cup\mathcal{V}_{2})\cap J^{-}(\mathcal{R}_{1})\cap
J^{-}(\mathcal{R}_{2}).$ (19)
The slope is defined by
$\displaystyle\mathcal{S}_{\Sigma}=\partial[J^{-}(\mathcal{R}_{1})\cup
J^{-}(\mathcal{R}_{2})]\cap J^{-}[\partial
J^{+}(\mathcal{V}_{1}\cup\mathcal{V}_{2})]\cap J^{+}(\Sigma).$ (20)
The contradiction surface is defined by
$\mathcal{C}_{\Sigma}=S_{\Sigma}\cap\Sigma$. Note that $\partial
J^{+}(\mathcal{V}_{1}\cup\mathcal{V}_{2})$ is generated by geodesics starting
on the inward, future pointing, null normals to the extremal surface
$\gamma_{{\mathcal{V}}_{2}}\cup\gamma_{{\mathcal{V}}_{2}}$. Similarly,
$\partial[J^{-}(\mathcal{R}_{1})\cup J^{-}(\mathcal{R}_{2})]$ is generated by
geodesics starting on the inward, past directed, null normals to
$\gamma_{{\mathcal{R}}_{1}}\cup\gamma_{{\mathcal{R}}_{2}}$.
To repeat the proof of may2020holographic we need to establish various
features of the null membrane. The first is that the area of the past directed
null geodesics that generate the slope have decreasing area. This holds
because this congruence is defined by beginning with the inward, past directed
null normals to $\gamma_{{\mathcal{R}}_{1}}\cup\gamma_{{\mathcal{R}}_{2}}$.
Since this surface is extremal, the focusing theorem (which assumes the null
energy condition) implies that this congruence has decreasing area. Similarly,
the null normals that generate the lift also have decreasing area.
The second needed feature of the null membrane is that the contradiction
surface is homologous to $\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$.
This follows because the restriction of the contradiction surface to the
boundary is the spacelike boundary of
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$.
The null membrane can be used to establish that the contradiction surface has
less area than the candidate surface. To see this, consider pushing the
candidate surface forward along the congruence defined by the lift, removing
any generators which collide. Continue pushing the surface forward until
reaching the slope. When doing so, any generators which reach the _ridge_ will
be removed, where the ridge is defined by
$\displaystyle\mathcal{R}=\partial J^{+}(\mathcal{V}_{1})\cap\partial
J^{+}(\mathcal{V}_{2})\cap J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2}).$
(21)
See also figure 6. Assume momentarily that the ridge is non-empty. Then after
pushing forward the surface will consist of two disconnected components which
sit on the slope. Finally, push the surface backwards along the slope until it
reaches $\Sigma$, and becomes the contradiction surface. Because the null
congruences defining the lift and the slope begin as normal vectors to
extremal surfaces, moving into the future along the lift and into the past
along the slope both decrease area. Removing colliding generators also
decreases the area. Thus we can conclude the contradiction surface has less
area than the candidate surface, as needed.
To justify our assumption that the ridge is non-empty, note that this occurs
whenever the entanglement scattering region $J^{\mathcal{E}}_{12\rightarrow
12}$ is non-empty, since by assumption $J^{\mathcal{E}}_{12\rightarrow 12}$ is
non-empty and
$\displaystyle J^{\mathcal{E}}_{12\rightarrow 12}\subseteq
J^{+}(\mathcal{V}_{1})\cap J^{+}(\mathcal{V}_{2})\cap
J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2}),$ (22)
so that the region on the right is non-empty. But this region being non-empty
means $J^{+}(\mathcal{V}_{1})$ and $J^{+}(\mathcal{V}_{2})$ must meet in the
past of $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$, which means the ridge is non-
empty.
### 5.2 The entanglement scattering region is inside the entanglement wedge
In the context of the connected wedge theorem with input and output regions
taken to be points, the authors of may2020holographic noted that the
scattering region sits inside of the entanglement wedge of
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$, at least in the context of
$2+1$ bulk dimensions. We can straightforwardly adapt their argument to our
context to see that the larger entanglement scattering region is also inside
of the entanglement wedge, again in $2+1$ bulk dimensions.
To see this, define the region
$\displaystyle
X=\overline{J^{+}[\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}]^{c}\cap[\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}]^{c}}.$
(23)
This is the closure of the spacelike complement of
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}$. In $2+1$ dimensions, this
consists of the domains of dependence of two intervals which we call
$\hat{X}_{1}$ and $\hat{X}_{2}$. Note that
$\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2}\cup\hat{X}_{1}\cup\hat{X}_{2}$
is a complete Cauchy slice of the boundary, which we extend into the bulk to
some Cauchy slice $\Sigma$. We will choose this extension such that
$\gamma_{X_{1}}$ and $\gamma_{X_{2}}$ are contained in $\Sigma$.
Next we note that $\hat{\mathcal{R}}_{1}$ is inside the domain of dependence
of $\hat{\mathcal{V}}_{1}\cup\hat{X}_{1}\cup\hat{\mathcal{V}_{2}}$, which we
label $\hat{D}_{1}$, while $\hat{\mathcal{R}}_{2}$ sits inside the domain of
dependence $\hat{\mathcal{V}}_{1}\cup\hat{X}_{2}\cup\hat{\mathcal{V}_{2}}$,
which we label $\hat{D}_{2}$. Because of this, $J^{-}(\mathcal{R}_{1})$ will
be inside $J^{-}(D_{1})$, while $J^{-}(\mathcal{R}_{2})$ will be inside
$J^{-}(D_{2})$. Consequently we learn
$\displaystyle J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2})\subseteq
J^{-}(D_{1})\cap J^{-}(D_{2}).$ (24)
Notice that, assuming the entanglement wedge
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ is
connected, the future boundary of $J^{-}(D_{1})\cap J^{-}(D_{2})$ is also the
future boundary of
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$. This is
because the entangling surface for
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$ consists of
two components, one of which is homologous to $\hat{X}_{1}$ and the other
homologous to $\hat{X}_{2}$, and so these two components are the entangling
surfaces for $\hat{D}_{1}$ and $\hat{D}_{2}$ respectively. Thus equation 24
gives that $J^{-}(\mathcal{R}_{1})\cap J^{-}(\mathcal{R}_{2})$ is in the past
of the future boundary of
$\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$. Since the
scattering region is a subregion of $J^{-}(\mathcal{R}_{1})\cap
J^{-}(\mathcal{R}_{2})$, it follows that this also holds for the scattering
region.
It remains to show that the scattering region is to the future of the past
boundary of $\mathcal{E}_{W}(\hat{\mathcal{V}}_{1}\cup\hat{\mathcal{V}}_{2})$.
This is immediate, because the past of $\gamma_{X_{1}}$ and $\gamma_{X_{2}}$
meets the boundary along the past boundaries of $\mathcal{V}_{1}$ and
$\mathcal{V}_{2}$. Thus any points in the future of $\mathcal{V}_{1}$ and
$\mathcal{V}_{2}$ must be in the future of this past boundary.
## 6 Discussion
In this article we have expanded the holographic quantum tasks framework to
include inputs and outputs encoded into arbitrary access structures. We’ve
illustrated the usefulness of this framework by using this construction to
motivate the improved connected wedge theorem, which we could then verify
using a geometric proof.
### 6.1 Non-triviality of the theorem in various spacetimes
We have noted that if the boundary regions $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ overlap, then the conclusion of the connected wedge
theorem is trivial, since in that case $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ immediately have a connected entanglement wedge. Non-
trivial configurations in global AdS2+1 exist and are discussed explicitly in
may2019quantum ; may2020holographic , while may2020holographic also noted
non-trivial configurations exist in the AdS soliton, and here we have given
non-trivial configurations for Poincaré-AdS2+1 (which only exist when
considering the region based statement).
We have not explored in detail however if the region based connected wedge
theorem applies non-trivially in higher dimensions. In Poincaré-AdS3+1 it
seems straightforward to construct non-trivial configurations by defining the
input and output regions to be as defined in section 3.2, but extended
infinitely in the extra transverse direction. In global-AdS3+1 it is less
clear how to construct such non-trivial configurations, though one plausible
avenue is to begin with the Poincaré configurations and consider their
embedding into the global spacetime. We leave understanding this in detail to
future work.
### 6.2 Improved bounds on the mutual information
One technical improvement over may2020holographic made in this work is a more
robust handling of possible noise in the bulk protocol. In particular we
proved
$\displaystyle\frac{1}{2}I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})\geq
n(-\log_{2}\beta)-1+O((\epsilon/\beta)^{n}),$ (25)
where $\epsilon$ was the error in completing a single instance of the
$\textbf{B}_{84}$ task. Since we argued $n$ can be taken to be any order less
than $O(1/G_{N})$, this bound allows us to directly conclude that the mutual
information between two regions $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ which have a scattering region is $O(1/G_{N})$. In
may2020holographic , using a weaker bound, it was only possible to prove the
mutual information was $O(1/G_{N})$ by first using the HRT formula to see that
the mutual information is either $O(1/G_{N})$ or $O(1)$.
### 6.3 Towards a causal structure-entanglement theorem with a converse
From a tasks perspective the failure of Theorem 4 to have a converse is tied
to the fact that we are interested in a fixed bulk geometry. While protocols
that take place in that fixed geometric background have a boundary
description, many boundary protocols will deform the geometry. Because of
this, the bulk and boundary success probabilities are related by an
inequality, $p_{suc}(\mathbf{T})\leq p_{suc}(\mathbf{\hat{T}})$, so that
sufficient entanglement to do the task in the boundary does not imply the task
can be done in the bulk fixed geometry, and so does not signal the appearance
of a bulk scattering region.
One interesting possibility is that large boundary correlation when measured
in some way other than the mutual information will imply the existence of a
bulk entanglement scattering region. In particular, this hypothetical measure
of correlation should count only entanglement that can be made use of by
operations that preserve the bulk geometry. Then, its appearance would signal
that there should be a bulk protocol in that geometry which completes the
required task, which in turn would imply the existence of the scattering
region.888We thank Jon Sorce for discussion on these points.
Acknowledgements
I thank Kfir Dolev, Jon Sorce, and Jason Pollack for valuable discussions and
feedback on drafts of this article. I am supported by a C-GSM award given by
the National Science and Engineering Research Council of Canada.
## Appendix A Lower bound on mutual information from success probability
In this section we prove the lower bound on mutual information 10. Our
starting point is Lemma 6 which bounds the success probability for states with
zero mutual information. Our argument will show that states with high
probability must be far from these zero probability states in terms of trace
distance, which we can then translate to a bound on mutual information. Note
that the discussion here is a repetition of an argument in may2020holographic
, but, because various parameters are changed in our context, we’ve included
the proof with updated parameters here.
We begin by recalling a continuity bound on success probability for any
quantum task, stated earlier in may2020holographic .
###### Lemma 8
Consider a quantum task which takes as input a quantum system $A$. Then the
probability of completing the task, call it $p_{suc}$, satisfies the
continuity bound
$\displaystyle|p_{suc}(\rho_{A})-p_{suc}(\sigma_{A})|\leq\frac{1}{2}||\rho_{A}-\sigma_{A}||_{1}.$
(26)
The proof follows by viewing the task as a procedure for distinguishing $\rho$
from $\sigma$. The maximal success probability of distinguishing states can be
written in terms of the trace distance, which leads to the above inequality.
Intuitively, we should understand the lemma as saying that nearby states
produce nearby success probabilities.
For the task $\mathbf{\hat{B}}^{\times n}_{84}$, we have
$\displaystyle p_{suc}(\rho_{\hat{\mathcal{V}}_{1}\hat{\mathcal{V}}_{2}})$
$\displaystyle\geq 1-2\epsilon^{2+n},$ $\displaystyle
p_{suc}(\rho_{\hat{\mathcal{V}}_{1}}\otimes\rho_{\hat{\mathcal{V}}_{2}})$
$\displaystyle\leq(2^{h(\delta)}\beta)^{n}.$ (27)
The state $\rho_{\hat{\mathcal{V}}_{1}\hat{\mathcal{V}}_{2}}$ is any boundary
state where the bulk scattering region is non-empty, while
$\rho_{\hat{\mathcal{V}}_{1}}\otimes\rho_{\hat{\mathcal{V}}_{2}}$ is the
tensor product of its marginals. The second bound follows from Lemma 6.
The remainder of the argument consists of relating the trace distance to the
relative entropy. In particular we have that the trace distance and fidelity
are related by fuchs1999cryptographic ; wilde2013quantum
$\displaystyle\frac{1}{2}||\rho-\sigma||_{1}\leq\sqrt{1-F(\rho,\sigma)}$ (28)
where $F(\rho,\sigma)$ is the fidelity. Additionally,
$\displaystyle-2\log F(\rho,\sigma)\leq D(\rho||\sigma)$ (29)
where $D(\rho|\sigma)$ is the relative entropy. The final observation is that
$\displaystyle D(\rho_{AB}||\rho_{A}\otimes\rho_{B})=I(A:B)_{\rho}$ (30)
Combining inequalities 28 and 29 and Lemma 8 to lower bound the relative
entropy, and hence the mutual information, in terms of success probabilities
we find
$\displaystyle
I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})_{\rho}\geq-2\log[1-|p_{suc}(\rho_{A})-p_{suc}(\rho_{AB}\otimes\rho_{B})|^{2}].$
(31)
Finally using our bounds on success probability A we obtain
$\displaystyle\frac{1}{2}I(\hat{\mathcal{V}}_{1}:\hat{\mathcal{V}}_{2})\geq
n(-\log 2^{h(2\epsilon)}\beta)-1+O((\epsilon/\beta)^{n})$ (32)
as claimed.
## Appendix B Non-trivial configurations in Poincaré-AdS2+1
$c_{1}$$c_{2}$$\hat{\mathcal{V}}_{2}$$\hat{\mathcal{V}}_{1}$$\hat{\mathcal{R}}_{1}$$\hat{\mathcal{R}}_{2}$$\hat{\mathcal{R}}_{2}$$r_{1}$$e_{1}$$q_{1L}$$q_{1R}$$q_{2R}$$q_{2L}$
Figure 7: A typical choice of regions
$\hat{\mathcal{C}}_{1}=\hat{\mathcal{V}}_{1}$,
$\hat{\mathcal{C}}_{2}=\hat{\mathcal{V}}_{2}$,
$\hat{\mathcal{R}}_{1},\hat{\mathcal{R}}_{2}$ in the boundary of Poincaré-AdS
which leads to a non-trivial conclusion in the connected wedge Theorem 4. The
regions are conveniently specified by choosing the four points
$c_{1},c_{2},r_{1},e_{1}$.
Here we explicitly check there are non-trivial configurations for the
connected wedge theorem in (pure) Poincaré-AdS.
Begin by considering points at999Here and throughout the section coordinates
are labelled by $(t,x)$.
$\displaystyle c_{1}$ $\displaystyle=(0,-\ell),$ $\displaystyle c_{2}$
$\displaystyle=(0,\ell),$ $\displaystyle r_{1}$ $\displaystyle=(T_{r},0),$
$\displaystyle e_{1}$ $\displaystyle=(T_{e},0).$ (33)
The set-up is shown in figure 7. This defines regions,
$\displaystyle\hat{\mathcal{R}}_{1}$
$\displaystyle=\hat{J}^{+}(e_{1})\cap\hat{J}^{-}(r_{1}),$
$\displaystyle\hat{\mathcal{V}}_{1}$
$\displaystyle=[\hat{J}^{+}(c_{1})\cap\hat{J}^{-}(r_{1})]\setminus\hat{J}^{+}(e_{1}),$
$\displaystyle\hat{\mathcal{V}}_{2}$
$\displaystyle=[\hat{J}^{+}(c_{2})\cap\hat{J}^{-}(r_{1})]\setminus\hat{J}^{+}(e_{1}).$
(34)
We then define $\hat{\mathcal{R}}_{2}$ as the set of points spacelike
separated from $\hat{\mathcal{R}}_{1}$.
We are interested in finding choices of points such that the entanglement
scattering region is non-empty. Because we are in pure AdS, we find
$\displaystyle J^{+}(\mathcal{V}_{1})$ $\displaystyle=J^{+}(c_{1}),$
$\displaystyle J^{+}(\mathcal{V}_{2})$ $\displaystyle=J^{+}(c_{2}),$
$\displaystyle J^{-}(\mathcal{R}_{1})$ $\displaystyle=J^{-}(r_{1}),$
$\displaystyle J^{-}(\mathcal{R}_{2})$ $\displaystyle=[J^{+}(e_{1})]^{c}.$
(35)
Thus a non-empty entanglement scattering region amounts to configurations such
that
$\displaystyle J^{\mathcal{E}}_{12\rightarrow 12}=J^{+}(c_{1})\cap
J^{+}(c_{2})\cap J^{-}(r_{1})\cap[J^{+}(e_{1})]^{c}$ (36)
is non-empty. These light cones are given in Poincaré-AdS by
$\displaystyle J^{+}(c_{1})$
$\displaystyle=\\{(t,x,z):(x+\ell)^{2}+z^{2}-t^{2}\leq 0\\},$ (37)
$\displaystyle J^{+}(c_{2})$
$\displaystyle=\\{(t,x,z):(x-\ell)^{2}+z^{2}-t^{2}\leq 0\\},$ (38)
$\displaystyle J^{+}(r_{1})$
$\displaystyle=\\{(t,x,z):x^{2}+z^{2}-(T_{r}-t)^{2}\leq 0\\},$ (39)
$\displaystyle[J^{+}(e_{1})]^{c}$
$\displaystyle=\\{(t,x,z):x^{2}+z^{2}-(T_{e}-t)^{2}\geq 0\\}.$ (40)
By symmetry, if the scattering region is non-empty it will include at least
one point with $x=0$. Thus it suffices to find a solution to
$\displaystyle\ell^{2}+z^{2}-t^{2}$ $\displaystyle\leq 0,$ (41) $\displaystyle
z^{2}-(T_{r}-t)^{2}$ $\displaystyle\leq 0,$ (42) $\displaystyle
z^{2}-(T_{e}-t)^{2}$ $\displaystyle\geq 0.$ (43)
Eliminating $z$ and $t$ from these inequalities we find one constraint,
$\displaystyle\boxed{T_{e}T_{r}-\ell^{2}>0}.$ (44)
We need to check this can be satisfied while also having
$\hat{\mathcal{V}}_{1}\cap\hat{\mathcal{V}}_{2}=\varnothing$. In our setup
this occurs when the rightmost point in $\hat{\mathcal{V}}_{1}$ is to the left
of $x=0$ and the leftmost point in $\hat{\mathcal{V}}_{2}$ to the right of
$x=0$. Via some Minkowski space geometry we can work out the end-points of the
regions $\hat{\mathcal{V}}_{1},\hat{\mathcal{V}}_{2}$,
$\displaystyle q_{1L}$
$\displaystyle=\left(\frac{T_{r}-\ell}{2},-\frac{T_{r}+\ell}{2}\right),$
$\displaystyle q_{1R}$
$\displaystyle=\left(\frac{T_{e}+\ell}{2},\frac{T_{e}-\ell}{2}\right),$
$\displaystyle q_{2R}$
$\displaystyle=\left(\frac{T_{r}-\ell}{2},\frac{T_{r}+\ell}{2}\right),$
$\displaystyle q_{2L}$ $\displaystyle=\left(\frac{T_{e}+\ell}{2},\frac{\ell-
T_{e}}{2}\right),$ (45)
where $q_{iL}$ is the left spacelike boundary of $\hat{\mathcal{V}}_{i}$, and
$q_{iR}$ is the right spacelike boundary of $\hat{\mathcal{V}}_{i}$ (see
figure 7). We see that keeping $\hat{\mathcal{V}}_{1}$ and
$\hat{\mathcal{V}}_{2}$ disjoint just requires $T_{e}<\ell$. Given a choice of
$T_{e},\ell$ such that $T_{e}<\ell$, we can always choose $T_{r}$ sufficiently
large to satisfy 44, guaranteeing the existence of the scattering region. This
establishes that there are non-trivial configurations for Theorem 4 in
Poincaré-AdS2+1.
Given that the scattering region exists, Theorem 4 concludes that
$\hat{\mathcal{V}}_{1}$ and $\hat{\mathcal{V}}_{2}$ should have a connected
entanglement wedge. We can also check this explicitly in the simple setting
considered here. In particular we would like to understand when
$\displaystyle\mathcal{A}_{min}[(q_{1L},q_{1R})]+\mathcal{A}_{min}[(q_{2L},q_{2R})]\geq\mathcal{A}_{min}[(q_{1L},q_{2R})]+\mathcal{A}_{min}[(q_{1R},q_{2L})].$
(46)
To calculate the minimal area surface for one of the $\hat{\mathcal{V}}_{i}$,
we need the invariant length of its cross section. This is straightforward to
work out from B,
$\displaystyle\Delta x^{2}-\Delta t^{2}=\ell(T_{r}-T_{e}).$ (47)
The intervals $(q_{1L},q_{2R})$ and $(q_{1R},q_{2L})$ are on a constant $t$
surface, so we can just use their widths. The areas of the minimal surfaces
are then
$\displaystyle\mathcal{A}_{min}[(q_{iL},q_{iR})]$
$\displaystyle=2L_{AdS}\ln\left(\frac{\sqrt{\ell(T_{r}-T_{e})}}{\epsilon}\right),$
$\displaystyle\mathcal{A}_{min}[(q_{1L},q_{2R})]$
$\displaystyle=2L_{AdS}\ln\left(\frac{{\ell-T_{e}}}{\epsilon}\right),$
$\displaystyle\mathcal{A}_{min}[(q_{1R},q_{2L})]$
$\displaystyle=2L_{AdS}\ln\left(\frac{T_{r}+\ell}{\epsilon}\right).$ (48)
The condition 46 reduces then to just
$\displaystyle\boxed{T_{e}T_{r}-\ell^{2}\geq 0}$ (49)
which is the same condition we found the existence of the scattering region,
verifying the theorem explicitly in this simple case.
## References
* (1) A. Kent, Quantum tasks in minkowski space, Classical and Quantum Gravity 29 (2012), no. 22 224013.
* (2) A. May, Quantum tasks in holography, Journal of High Energy Physics 2019 (2019), no. 233 [arXiv:1902.06845].
* (3) A. May, G. Penington, and J. Sorce, Holographic scattering requires a connected entanglement wedge, Journal of High Energy Physics 2020 (2020), no. 8 1–34.
* (4) B. Czech, J. L. Karczmarek, F. Nogueira, and M. V. Raamsdonk, The gravity dual of a density matrix, Classical and Quantum Gravity 29 (2012), no. 15 [arXiv:1204.1330].
* (5) M. Headrick, V. E. Hubeny, A. Lawrence, and M. Rangamani, Causality & holographic entanglement entropy, Journal of High Energy Physics 2014 (2014), no. 12 [arXiv:1408.6300].
* (6) A. C. Wall, Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy, Classical and Quantum Gravity 31 (2014) 225007, [arXiv:1211.3494].
* (7) D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh, Relative entropy equals bulk relative entropy, Journal of High Energy Physics 2016 (2016), no. 6 4, [arXiv:1512.06431].
* (8) X. Dong, D. Harlow, and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Physical Review Letters 117 (2016) 021601, [arXiv:1601.05416].
* (9) J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle, and M. Walter, Entanglement wedge reconstruction via universal recovery channels, Physical Review X 9 (2019) 031011, [arXiv:1704.05839].
* (10) P. Hayden and A. May, Localizing and excluding quantum information; or, how to share a quantum secret in spacetime, Quantum 3 (2019) 196\.
* (11) A. Ambainis, M. Mosca, A. Tapp, and R. De Wolf, Private quantum channels, in Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 547–553, IEEE, 2000.
* (12) V. E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, Journal of High Energy Physics 2012 (2012), no. 7 93.
* (13) C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, arXiv preprint arXiv:2003.06557 (2020).
* (14) M. Tomamichel, S. Fehr, J. Kaniewski, and S. Wehner, A monogamy-of-entanglement game with applications to device-independent quantum cryptography, New Journal of Physics 15 (2013), no. 10 103002, [arXiv:1210.4359].
* (15) S. Nezami and M. Walter, Multipartite Entanglement in Stabilizer Tensor Networks, arXiv:1608.02595.
* (16) C. A. Fuchs and J. Van De Graaf, Cryptographic distinguishability measures for quantum-mechanical states, IEEE Transactions on Information Theory 45 (1999), no. 4 1216–1227, [quant-ph/9712042].
* (17) M. M. Wilde, Quantum information theory, second edition. Cambridge University Press, 2017.
|
# On a new class of series identities
Arjun K. Rathie Department of Mathematics, Vedant College of Engineering &
Technology (Rajasthan Technical University), Village: Tulsi, Post: Jakhamund,
Dist. Bundi, Rajasthan State, India<EMAIL_ADDRESS>
###### Abstract.
The aim of this paper is to provide a new class of series identities in the
form of four general results. The results are established with the help of
generalizatons of the classical Kummer’s summation theorem obtained earlier by
Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et
al. follow special cases of our main findings.
2010 Mathematics Subject Classifications : Primary : 33B20, 33C20, Secondary :
33B15, 33C05 Keywords: Generalized hypergeometric function, Kummer’s summation
theorem, product formulas, generalization, Double series
## 1\. Introduction and Results Required
We start with the following two very interesting results involving product of
generalized hypergeometric series due to Bailey[1] viz.
$\displaystyle{}_{0}F_{1}$ $\displaystyle\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho\end{array};-x\right]$ (1.1)
$\displaystyle={}_{0}F_{3}\left[\begin{array}[]{c}-\\\
\rho,\frac{1}{2}\rho,\frac{1}{2}\rho+\frac{1}{2}\end{array};-\frac{x^{2}}{4}\right]$
and
$\displaystyle{}_{0}F_{1}$ $\displaystyle\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
2-\rho\end{array};-x\right]$ (1.2)
$\displaystyle={}_{0}F_{3}\left[\begin{array}[]{c}-\\\
\frac{1}{2},\frac{1}{2}\rho+\frac{1}{2},\frac{3}{2}-\frac{1}{2}\rho\end{array};-\frac{x^{2}}{4}\right]$
$\displaystyle+\frac{2(1-\rho)x}{\rho(2-\rho)}~{}{}_{0}F_{3}\left[\begin{array}[]{c}-\\\
\frac{3}{2},\frac{1}{2}\rho+1,2-\frac{1}{2}\rho\end{array};-\frac{x^{2}}{4}\right]$
Bailey[1] established these results with the help of the following classical
Kummer’s summation theorem[2] viz.
${}_{2}F_{1}\left[\begin{array}[]{c}a,~{}b\\\
1+a-b\end{array};-1\right]=\frac{\Gamma\left(1+\frac{1}{2}a\right)~{}\Gamma\left(1+a-b\right)}{\Gamma\left(1+a\right)~{}\Gamma\left(1+\frac{1}{2}a-b\right)}$
(1.3)
Very recently, Rathie et al.[6] have obtained explicit expressions of
(i)$\displaystyle~{}{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho+i\end{array};-x\right]$
(ii) $\displaystyle{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho-i\end{array};-x\right]$
(iii) $\displaystyle{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
2-\rho+i\end{array};-x\right]$
(iv) $\displaystyle{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
\rho\end{array};x\right]\times{}_{0}F_{1}\left[\begin{array}[]{c}-\\\
2-\rho-i\end{array};-x\right]$
in the most general form for any $i\in\mathbb{Z}_{0}$ and provided the natural
generalizations of the results (1.1) and (1.2).
The aim of this paper is to obtain explicit expressions of
(a)
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho+i)_{n}~{}m!~{}n!}$
(b)
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho-i)_{n}~{}m!~{}n!}$
(c)
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(2-\rho+i)_{n}~{}m!~{}n!}$
(d)
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(2-\rho-i)_{n}~{}m!~{}n!}$
in the most general form for any $i\in\mathbb{Z}_{0}$. Here $\\{\Delta_{m}\\}$
is a sequence of arbitrary complex numbers.
The results are derived with the help of the following generalizations of
Kummer’s summation theorem obtained earlier by Rakha and Rathie[4] for
$i\in\mathbb{Z}_{0}$ viz.
$\displaystyle{}_{2}F_{1}\left[\begin{matrix}a&b\\\
1+a-b+i\end{matrix};-1\right]$
$\displaystyle=\frac{2^{-a}\Gamma\left(\frac{1}{2}\right)\Gamma(b-i)\Gamma(1+a-b+i)}{\Gamma(b)\Gamma\left(\frac{1}{2}a-b+\frac{1}{2}i+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}a-b+\frac{1}{2}i+1\right)}$
(1.4) $\displaystyle\times\sum_{r=0}^{i}{i\choose
r}(-1)^{r}\frac{\Gamma\left(\frac{1}{2}a-b+\frac{1}{2}i+\frac{1}{2}r+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}a-\frac{1}{2}i+\frac{1}{2}r+\frac{1}{2}\right)}$
and
$\displaystyle{}_{2}F_{1}\left[\begin{matrix}a&b\\\
1+a-b-i\end{matrix};-1\right]$
$\displaystyle=\frac{2^{-a}\Gamma\left(\frac{1}{2})\Gamma(1+a-b+i\right)}{\Gamma\left(\frac{1}{2}a-b+\frac{1}{2}i+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}a-b+\frac{1}{2}i+1\right)}$
(1.5) $\displaystyle\times\sum_{r=0}^{i}{i\choose
r}\frac{\Gamma\left(\frac{1}{2}a-b-\frac{1}{2}i+\frac{1}{2}r+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}a-\frac{1}{2}i+\frac{1}{2}r+\frac{1}{2}\right)}$
Results obtained earlier by Srivastava[8], Bailey[1] and Rathie et al.[6]
follow special cases of our main findings.
## 2\. Main Results
The results to be established in this paper are given in the following
theorem.
###### Theorem 2.1.
Let $\\{\Delta_{n}\\}$ be a bounded sequence of complex numbers. Then for
$i\in\mathbb{Z}_{0}$, the following general results hold true:
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho+i)_{n}~{}m!~{}n!}$
(2.1)
$\displaystyle=\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m}~{}m!}~{}\frac{2^{m}~{}\Gamma\left(\frac{1}{2}\right)\Gamma(\rho+i)\Gamma(1-\rho-
m-i)}{\Gamma(1-\rho-m)~{}\Gamma\left(\rho+\frac{1}{2}i+\frac{1}{2}m-\frac{1}{2}\right)\Gamma\left(\rho+\frac{1}{2}i+\frac{1}{2}m\right)}$
$\displaystyle\times\left(\sum_{r=0}^{i}(-1)^{r}\binom{i}{r}\frac{\Gamma\left(\rho+\frac{1}{2}m+\frac{1}{2}i+\frac{1}{2}r-\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}r-\frac{1}{2}i-\frac{1}{2}m+\frac{1}{2}\right)}\right)$
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho-i)_{n}~{}m!~{}n!}$
(2.2)
$\displaystyle=\sum_{m=0}^{\infty}(-1)^{n}\frac{\Delta_{m}~{}x^{m}}{(\rho_{m}~{}m!}~{}\frac{2^{m}~{}\Gamma\left(\frac{1}{2}\right)\Gamma(\rho-i)}{\Gamma\left(\rho-\frac{1}{2}i+\frac{1}{2}m-\frac{1}{2}\right)\Gamma\left(\rho-\frac{1}{2}i+\frac{1}{2}m\right)}$
$\displaystyle\times\left(\sum_{r=0}^{i}\binom{i}{r}\frac{\Gamma\left(\rho+\frac{1}{2}m-\frac{1}{2}i+\frac{1}{2}r-\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}r-\frac{1}{2}i-\frac{1}{2}m+\frac{1}{2}\right)}\right)$
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(2-\rho+i)_{n}~{}m!~{}n!}$
(2.3)
$\displaystyle=\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m}~{}m!}~{}\frac{2^{\rho-1+m}~{}\Gamma\left(\frac{1}{2}\right)\Gamma(2-\rho+i)\Gamma(-m-i)}{\Gamma(-m)~{}\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho+\frac{1}{2}i+1\right)\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho+\frac{1}{2}i\right)}$
$\displaystyle\times\left(\sum_{r=0}^{i}(-1)^{r}\binom{i}{r}\frac{\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho+\frac{1}{2}i+\frac{1}{2}r+1\right)}{\Gamma\left(\frac{1}{2}r-\frac{1}{2}\rho-\frac{1}{2}m-\frac{1}{2}i+1\right)}\right)$
and
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(2-\rho-i)_{n}~{}m!~{}n!}$
(2.4)
$\displaystyle=\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m}~{}m!}~{}\frac{2^{\rho-1+m}~{}\Gamma\left(\frac{1}{2}\right)\Gamma(2-\rho-i)}{\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho-\frac{1}{2}i+1\right)\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho-\frac{1}{2}i+\frac{3}{2}\right)}$
$\displaystyle\times\left(\sum_{r=0}^{i}\binom{i}{r}\frac{\Gamma\left(\frac{1}{2}m-\frac{1}{2}\rho-\frac{1}{2}i+\frac{1}{2}r+1\right)}{\Gamma\left(\frac{1}{2}r-\frac{1}{2}\rho-\frac{1}{2}m-\frac{1}{2}i+1\right)}\right)$
### Derivations :
In order to establish the first result (2.1) asserted in the theorem, we
proceed as follows. Denoting the left hand side of (2.1) by $S$, we have
$S=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho+i)_{n}~{}m!~{}n!}$
Replacing $m$ by $m-n$ and using the result[3, Equ.1, p.56] viz.
$\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}A(k,n)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}A(k,n-k)$
we have
$S=\sum_{m=0}^{\infty}\sum_{n=0}^{m}(-1)^{n}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m-n}~{}(\rho+i)_{n}~{}(m-n)!~{}n!}$
Using elementary identities[3, p.58]
$(\alpha)_{m-n}=\frac{(-1)^{n}~{}(\alpha)_{m}}{(1-\alpha-m)_{n}}$
and
$(m-n)!=\frac{(-1)^{n}~{}m!}{(-m)_{n}}$
we have, after some algebra
$S=\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m}~{}m!}~{}\sum_{n=0}^{m}(-1)^{n}~{}\frac{(-m)_{n}(1-\rho-m)_{n}}{(\rho+i)_{n}n!}$
Summing up the inner series, we have
$S=\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}x^{m}}{(\rho)_{m}~{}m!}~{}{}_{2}F_{1}\left[\begin{array}[]{c}-m,1-\rho-m\\\
\rho+i\end{array};-1\right]$
We now observe that the series ${}_{2}F_{1}$ can be evaluated with the help of
the known result (1.4) and we easily arrive at the right hand side of (2.1).
This completes the proof of the first result (2.1) asserted in the theorem.
In exactly the same manner, the results (2.2) to (2.4) can be established. So
we prefer to omit the details.
## 3\. Corollaries
In this section, we shall mention some of the interesting known as well as new
results of our main findings.
(a) In the result (2.1) or (2.2), if we take $i=0$, we have
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(\rho)_{n}~{}m!~{}n!}$
(3.1)
$\displaystyle=\sum_{m=0}^{\infty}\frac{\Delta_{2m}~{}(-x^{2})^{m}}{(\rho)_{m}~{}(\frac{1}{2}\rho)_{m}(\frac{1}{2}\rho+\frac{1}{2})_{m}~{}2^{2m}~{}m!}$
This is a known result due to Srivastava[8]. Further setting
$\Delta_{m}=1~{}(m\in\mathbb{N}_{0})$, we at once get the result (1.1) due to
Bailey.
(b) In the result (2.3) or (2.4), if we take $i=0$, we have
$\displaystyle\sum_{m=0}^{\infty}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta_{m+n}~{}x^{m+n}}{(\rho)_{m}~{}(2-\rho)_{n}~{}m!~{}n!}$
(3.2)
$\displaystyle=\sum_{m=0}^{\infty}\frac{\Delta_{2m}~{}(-x^{2})^{m}}{\left(\frac{1}{2}\right)_{m}\left(\frac{1}{2}\rho+\frac{1}{2}\right)_{m}~{}\left(\frac{3}{2}-\frac{1}{2}\rho\right)_{m}~{}2^{2m}~{}m!}$
$\displaystyle+\frac{2(1-\rho)x}{\rho(2-\rho)}~{}\sum_{m=0}^{\infty}\frac{\Delta_{m}~{}(-x^{2})^{m}}{\left(\frac{3}{2}\right)_{m}\left(\frac{1}{2}\rho+1\right)_{m}~{}\left(2-\frac{1}{2}\rho\right)_{m}~{}2^{2m}~{}m!}$
which appears to be a new result.
Further setting $\Delta_{m}=1~{}(m\in\mathbb{N}_{0})$, we at once get another
result (1.2) due to Bailey.
(c) In (2.1), if we take $i=0,1,\cdots,9$; we get known results recorded in
[7].
(d) In (2.2), if we take $i=0,1,\cdots,9$; we get known results recorded in
[7].
(e)In (2.1) to (2.4), if we get $\Delta_{m}=1~{}(m\in\mathbb{N}_{0})$, we get
known results obtained very recently by Rathie et al.[6].
We conclude the paper by remarking that the details about the result presented
in this paper together with a large number of special cases(known and new),
are given in [5].
## References
* [1] W. N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc., 2 242–254, (1928).
* [2] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Stechert-Hafner, Inc., New York, (1964).
* [3] E. D. Rainville, Special functions, Reprint of 1960 first edition, Chelsea Publishing Co., Bronx, N.Y., (1971).
* [4] M.A. Rakha, A.K. Rathie, Generalizations of classical summation theorems for the series ${}_{2}F_{1}$ and ${}_{3}F_{2}$ with applications, Integral Transforms Spec. Funct. 22 (11), 823-840, (2011).
* [5] A. K. Rathie, On a new class of series identities, Submitted for publication, (2020).
* [6] A. K. Rathie, Y. S. Kim and R. B. Paris, On some new results involving product of generalized hypergeometric series with applications, Submitted for publication, (2020).
* [7] N. Shekhawat, J. Choi, A. K. Rathie and Om Prakash, On a new class of series identities, Honam Mathematical J., 37(3), 339-352, (2015)
* [8] H. M. Srivastava, On the reducibility of Appell’s function F4, Canad. Math. Bull., 16 (2), 295-298, (1973).
|
Comments on single trace $T\bar{T}$ and other current-current deformations
Gaston Giribeta, Julio Olivab, Ricardo Stuardob
aDepartamento de Física, Universidad de Buenos Aires & IFIBA - CONICET
Ciudad Universitaria, pabellón 1 (1428) Buenos Aires, Argentina.
bDepartamento de Física, Universidad de Concepción
Casilla 160-C, Concepción, Chile.
String theory on AdS3 with NS-NS fluxes admits a solvable irrelevant
deformation which is close to the $T\bar{T}$ deformation of the dual CFT2.
This consists of deforming the worldsheet action, namely the action of the
$SL(2,\mathbb{R})$ WZW model, by adding to it the operator $J^{-}\bar{J}^{-}$,
constructed with two Kac-Moody currents. The geometrical interpretation of the
resulting theory is that of strings on a conformally flat background that
interpolates between AdS3 in the IR and a flat linear dilaton spacetime with
Hagedorn spectrum in the UV, having passed through a transition region of
positive curvature. Here, we study the properties of this string background
both from the point of view of the low-energy effective theory and of the
worldsheet CFT. We first study the geometrical properties of the semiclassical
geometry, then we revise the computation of correlation functions and of the
spectrum of the $J^{-}\bar{J}^{-}$-deformed worldsheet theory, and finally we
discuss how to extend this type of current-current deformation to other
conformal models.
## 1 Introduction
In the context of AdS3/CFT2 correspondence, it was shown in [1] that certain
type of $T\bar{T}$-deformation of the boundary CFT2, which can be regarded as
a single trace version of the one originally introduced in [2, 3, 4], gives
rise in the bulk to a string theory background that interpolates between AdS3
in the infrared limit and a flat linear dilaton background in the ultraviolet.
This construction was argued in [1] to provide a family of holographic pairs,
including a large class of string theory vacua with asymptotically linear
dilaton. The solvable irrelevant deformation of AdS3/CFT2 correspondence
studied in [1] was further studied in [5], where in particular its spectrum
was studied. It was observed that this type of deformation leads in the
ultraviolet to a theory with Hagedorn spectrum. This has been studied in [6,
7, 8, 9, 10, 11] and references therein and thereof; see also [12, 13, 14].
In [15, 16], the correlation functions in the deformed theory were studied,
and it provided an alternative way of studying the spectrum: The insertion of
the operator that realizes the deformation produces a logarithmic divergence
in the correlation functions, leading to the renormalization of the primary
operators. This yields an anomalous dimension that can be computed explicitly.
From this, one may determine the spectrum of the theory from the worldsheet
computation. The form of the correlation functions, on the other hand, permits
to investigate the properties of the dual theory [15].
The model studied in [1] was later investigated in many different context. The
entanglement entropy was first studied in [17]; in [18, 19] the theory was
studied in presence of boundaries; and the $J\bar{T}$ analog of it has also
been studied [20, 21, 22, 23]. Here, we study the properties of this string
background both from the point of view of the low-energy effective theory and
of the worldsheet CFT. In section 2, we study the geometrical properties of
the semiclassical geometry. We study the geometry as solution to the low-
energy effective field theory, its T-dual background, the main properties of
this specific deformation of AdS3, and the field probes in such an spacetime.
In section 3, we revise the computation of correlation functions of [15, 16]
and how it provides a direct way of studying the spectrum. Finally, in section
4 we discuss how to extend this type of deformation to other conformal models.
## 2 Low energy theory
### Interpolating background
Let us start by considering the effective theory describing the low-energy
limit of bosonic string theory. This is given by the field equations
$\displaystyle
R_{\alpha\beta}+2\nabla_{\alpha}\nabla_{\beta}\Phi-\frac{1}{4}H_{\alpha\mu\nu}H_{\beta}^{\mu\nu}=0,$
(1) $\displaystyle\nabla_{\alpha}\left(e^{-2\Phi}H^{\alpha\mu\nu}\right)=0,$
(2)
$\displaystyle\nabla^{\alpha}\nabla_{\alpha}\Phi-2\nabla_{\alpha}\Phi\nabla^{\alpha}\Phi+{2\alpha^{\prime}}+\frac{1}{12}H_{\alpha\mu\nu}H^{\alpha\mu\nu}=0.$
(3)
where $H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}$ is the field strengh
associated to the Kalb-Ramond $B$-field, and $\Phi$ is the dilaton. These
equations admit locally AdS3 solutions [24],
$ds^{2}=-\frac{r^{2}}{\ell^{2}}\,dt^{2}+\frac{\ell^{2}}{r^{2}}\,dr^{2}+r^{2}d\theta^{2},$
(4)
provided the other backgrounds fields take the form
$\Phi=\Phi_{0},\ \ \ \ H_{\mu\nu\rho}=\frac{2r}{\ell}\,\epsilon_{\mu\nu\rho}.$
(5)
The dilaton receives quantum (i.e. finite-$\alpha^{\prime}$) corrections.
Here, we will consider the convention $\alpha^{\prime}=1$, so that the
semiclassical limit corresponds to large
$k=\ell^{2}/\alpha^{\prime}=\ell^{2}$.
As (4) describes the universal covering of AdS3, we have $t\in\mathbb{R}$. The
radial coordinate is $r\in\mathbb{R}_{\geq 0}$, with the boundary of the space
being located at $r\rightarrow\infty$. If we take $\theta$ to be periodic with
a period $2\pi$, the metric above corresponds to that of the massless BTZ
geometry [25, 26]. It will be convenient to consider coordinates $r=\ell
e^{\phi}$ and $x=\ell\theta$. In these variables, the metric and the field
strength take the form
$\displaystyle ds^{2}=e^{2\phi}\left(-dt^{2}+dx^{2}\right)+\ell^{2}d\phi^{2},\
\ \ \
H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}=2e^{2\phi}\epsilon_{\mu\nu\rho},$
(6)
where we now consider the covering $x\in\mathbb{R}$. That is, the non-
vanishing component of the Kalb-Ramond field is $B_{xt}=e^{2\phi}$ and grows
when approaching the boundary at $\phi\rightarrow\infty$.
Now, let us consider a deformation of (6), given by
$ds^{2}=\frac{e^{2\phi}}{\lambda
e^{2\phi}+1}(-dt^{2}+dx^{2})+\ell^{2}d\phi^{2},$ (7)
with $\lambda$ being a real parameter. This metric solves the field equations
(1) for arbitrary $\lambda$ provided the Kalb-Ramond field and the dilaton are
given by
$\displaystyle B_{xt}=\frac{2e^{2\phi}}{\lambda e^{2\phi}+1}\,,\ \ \ \
\Phi=\Phi_{0}-\phi-\frac{1}{2}\log(\lambda+e^{-2\phi}),$ (8)
respectively. Near the boundary, the dilaton becomes linear in $\phi$. We will
be mostly interested in the case $\lambda\geq 0$, as for $\lambda<0$ the
geometry exhibits a singularity at $\phi=-\frac{1}{2}\log|\lambda|$. In terms
of the double null coordinates ${{u}}=(x+t)/\ell$ and $\bar{{u}}=(x-t)/\ell$,
the fields take the following form
$ds^{2}=\ell^{2}d\phi^{2}+\frac{\ell^{2}d{{u}}\,d\bar{{{u}}}}{\lambda+e^{-2\phi}}\
,\ \ B=\frac{\ell^{2}d{{u}}\wedge d\bar{{{u}}}}{\lambda+e^{-2\phi}}.$ (9)
This solution has recently attracted much attention [1, 5, 8, 9, 10, 11, 17,
15, 16, 18, 19, 20, 21, 22, 23] as it appears as an exact string background
that corresponds to a marginal deformation of the worldsheet theory on
AdS${}_{3}\times\mathcal{N}$ which is closely related to the
$T\bar{T}$-deformation of the dual CFT.
Let us go back for a moment to the more familiar coordinates $r=\ell
e^{\phi}$, namely
$ds^{2}=-\frac{r^{2}}{r^{2}\lambda+\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}}dr^{2}+\frac{r^{2}}{r^{2}\lambda+\ell^{2}}dx^{2},$
(10)
in which it becomes evident that the geometry interpolates between AdS3 and
Minkowski space: While in the limit $r\ll\ell/\sqrt{\lambda}$ one recovers the
metric (4), in the limit $r\gg\ell/\sqrt{\lambda}$ one gets
$ds^{2}=-d\hat{t}^{2}+d\hat{x}^{2}+d\hat{y}^{2}$ where
$\hat{t}=t/\sqrt{\lambda}\ell$, $\hat{x}=x/\sqrt{\lambda}\ell$,
$\hat{y}=\ell\phi$. The local isometry group of spacetime (10) for arbitrary
value of $\lambda$ is $ISO(1,1)$ and is generated by the Killing vectors
$\partial_{t}$, $\partial_{x}$ and $x\partial_{t}-t\partial_{x}$. It gets
enhanced to the full $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ for $\lambda=0$
and to $ISO(2,1)$ in the limit $\lambda\to\infty$
The interpolating geometry (9) has very interesting properties. Apart from
being fascinating in that it describes the transition between AdS3 and the
type of linear dilaton background that appears in little string theory,
geometry (9) exhibits peculiar features: It admits a supersymmetric embedding
in type IIB SUGRA, since it appears in the S-dual frame of the D1/D5 system
[27]. Besides, it is solvable in different limits: On the one hand, despite
being a geometry of non-constant curvature, it turns out that the probe fields
are integrable on it, and thus enables to gain intuition from the
semiclassical analysis. On the other hand, the string worldsheet
$\sigma$-model is an exact string solution; being a marginal deformation of a
WZW model, can be solved explicitly in the sense that analytic expressions for
the correlation functions can be obtained and the spectrum can be written
down.
### T-duality
Let us study some of the properties of (9), starting by noticing that it is
dual to $pp$-waves on AdS3. Spacetime (6) happens to be invariant under $t$\-
and $x$-translations, and so we can apply T-duality transformations along the
direction generated by $\partial_{t}$ and $\partial_{x}$ in order to obtain
new solutions $\tilde{g}_{\mu\nu},\tilde{B}_{\mu\nu},\tilde{\Phi}$ to the
field equations (1)-(3). At the level of the low-energy effective action, this
amounts to applying the Buscher rules [28, 29]. For the $t$-direction, these
transformation rules read
$\tilde{g}_{tt}=\frac{1}{g_{tt}},\ \ \ \
\tilde{g}_{ti}=\frac{B_{ti}}{g_{tt}},\ \ \ \
\tilde{g}_{ij}=g_{ij}-\frac{g_{ti}g_{tj}}{g_{tt}}-\frac{B_{ti}B_{tj}}{g_{tt}},$
(11)
together with
$\tilde{B}_{ti}=\frac{g_{ti}}{g_{tt}},\ \ \ \
\tilde{B}_{ij}=B_{ij}-2\frac{g_{t[i}B_{j]t}}{g_{tt}},\ \ \ \
\tilde{\Phi}=\Phi-\frac{1}{2}\log\left(g_{tt}\right),$ (12)
where $i,j$ correspond to the coordinates other than $t$. After performing
these transformations and renaming variables as $u=x$, $v=t$, and
$y=\ell\phi$, one obtains
$d\tilde{s}^{2}=-F(y)\,dv^{2}+2\,dudv+dy^{2},\ \ \text{with}\ \
F(y)=\lambda+e^{-2{y}/{\ell}}$ (13)
together with $\tilde{B}_{\mu\nu}=0\,,\ \ \tilde{\Phi}=\tilde{\Phi}_{0}-\phi$.
Analogously, applying similar transformations in the $x$-direction, one gets
$d\tilde{s}^{2}=F(y)\,du^{2}+2\,dudv+dy^{2}\,.$ (14)
This geometry describes a 3-dimensional version of a $pp$-wave solution in
Brinkmann type coordinates with a wave profile $F$. The non-vanishing
component of the Riemann tensor for this geometry is
$R^{vy}_{\phantom{vy}uy}=-\frac{2}{l^{2}}e^{-2y/l}.$ (15)
This solution represents an exact string background.
### Geometric properties of the interpolating spacetime
As said, spacetime (9) interpolates between AdS3 in the limit
$\phi\rightarrow-\infty$ and a flat linear dilaton background in the opposite
limit. The geometry thus has non-constant curvature. In fact, it can be shown
to have an infinite region of positive curvature. To see this, we can compute
the scalar curvature
$R=\frac{2(4\lambda r^{2}-3\ell^{2})}{(\lambda r^{2}+\ell^{2})^{2}},$ (16)
which, indeed, happens to be positive for $r>\ell\sqrt{3/(4\lambda)}$, all the
way to infinity. $R$ has a global maximum at
$r_{\text{max}}=\ell\sqrt{5/(2\lambda)}$ with a maximum value
$R_{\text{max}}=8/(7\ell^{2})$ that does not depend on the deformation
parameter $\lambda$. Figure 1 depicts the function $R$ as a function of the
radial coordinate $r$ for different values of $\lambda$.
Figure 1: Ricci scalar for different values of $\lambda$ (here, $\ell=1$).
Other curvature invariants of (9) are
$\displaystyle R^{\mu\nu}R_{\mu\nu}=\frac{4(6\lambda^{2}r^{4}-8l^{2}\lambda
r^{2}+3l^{4})}{(\lambda r^{2}+l^{2})^{4}},$ (17) $\displaystyle
R^{\mu}_{\phantom{\mu}\nu}$ $\displaystyle
R^{\nu}_{\phantom{\nu}\rho}R^{\rho}_{\phantom{\rho}\mu}=\frac{-8(-10\lambda^{3}r^{6}+18l^{2}\lambda^{2}r^{4}-12l^{4}\lambda
r^{2}+3l^{6})}{(\lambda r^{2}+l^{2})^{6}},$ (18)
and we see from these, and from (16), that the geometry is actually singular
at $r=\ell/\sqrt{-\lambda}$. It is important noticing that it is sufficient to
give the three curvature invariant $R$, $\text{Tr}(R_{\mu\nu}^{2})$ and
$\text{Tr}(R_{\mu\nu}^{3})$ to characterize them all, since any higher
curvature scalar can be obtained as a combination of powers of the latter
three quantities by virtue of the three-dimensional identities
$\delta^{\mu_{1}\cdots\mu_{n}}_{\nu_{1}\cdots\nu_{n}}\tilde{R}^{\nu_{1}}_{\
\mu_{1}}\cdots\tilde{R}^{\nu_{n}}_{\ \mu_{n}}\equiv 0\ ,n>3\ ,$ (19)
where $\tilde{R}^{\nu_{1}}_{\ \mu_{1}}$ is the traceless part of the Ricci
tensor. Despite being of non-constant curvature, geometry (9) yields vanishing
Cotton tensor
$C_{\mu\nu}=\epsilon_{\mu}^{\
\alpha\beta}\nabla_{\alpha}(R_{\nu\beta}-\frac{1}{4}Rg_{\nu\beta})=0$ (20)
which implies that it is locally conformally flat. This property makes Weyl
invariant probes being integrable on this background, as we will show below.
This permits to gain a semiclassical intuition. The conformal factor that
allows to write the interpolating background (10) in a manifestly conformally
flat form, defines an improper Weyl transformations and, therefore, imposing
boundary conditions and asymptotic behaviors on probe fields is non-trivially
related to their flat counterpart.
### Probes on the deformed geometry
Consider a conformally coupled scalar field on the geometry (9). The
corresponding equation is
$\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\Phi-\frac{1}{8}{\sqrt{-g}}R\Phi=0.$
(21)
We consider the separable ansatz
$\Phi(t,r,x)=e^{-i\omega t}e^{i\kappa x}\varphi(r).$ (22)
This problem is exactly solvable. However, we can first gain intuition from
the well-known small and large $r$ regimes, where it reduces to the AdS3 and
to the flat space computation, respectively. This does not mean that the
solution to the complete problem will be a simple junction of the two constant
curvature problems. The transmission coefficients may actually change due to
the different boundary conditions that have to be satisfied in the
$\lambda$-deformed background.
Let us consider first the case $\kappa^{2}-\omega^{2}<0$. In this case, the
solution for $\varphi(r)$ takes the form
$\begin{split}\varphi(r)&=\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}\left(A_{1}\,e^{i\sqrt{\omega^{2}-\kappa^{2}}\,\chi(r)}+B_{1}\,e^{-i\sqrt{\omega^{2}-\kappa^{2}}\,\chi(r)}\right),\end{split}$
(23)
where
$\chi(r)=\sqrt{\lambda}\log\left(\lambda
r+\sqrt{(r^{2}\lambda+1)\lambda}\right)-\frac{\sqrt{r^{2}\lambda+1}}{r}$ (24)
and where $A_{1}$ and $B_{1}$ are two constant to be determined by requiring
appropriate boundary conditions. In order to impose conditions at infinity, it
is convenient to solve (21) on the nearly flat metric
$ds^{2}\simeq-\frac{1}{\lambda}dt^{2}+\frac{dr^{2}}{r^{2}}+\frac{1}{\lambda}dx^{2}$
(25)
which is the large $r$ limit of (10) (here, we set $\ell=1$ for short). The
radial dependence of the conformal scalar on this metric reads
$\varphi(r)\propto
A_{1}\,e^{i\sqrt{\lambda}\sqrt{\omega^{2}-\kappa^{2}}\log(r)}+B_{2}\,e^{-i\sqrt{\lambda}\sqrt{\omega^{2}-\kappa^{2}}\log(r)}$.
We want to impose outgoing boundary conditions at infinity. Since for
$\lambda\neq 0$ we have
$\displaystyle\Phi(t,r,x)$ $\displaystyle\sim
A_{1}\,e^{-i\omega(t-\sqrt{\lambda}\log(r))}+B_{1}\,e^{-i\omega(t+\sqrt{\lambda}\log(r))},$
(26)
by expanding at infinity we find that imposing outgoing boundary conditions
corresponds to $B_{1}=0$. This is confirmed by considering the flux of
particles defined by the $U(1)$-current
$j^{\mu}=-i\left(\varphi^{*}\partial^{\mu}\varphi-\varphi\partial^{\mu}\varphi^{*}\right)$.
Thus,
$\varphi(r)=A_{1}\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}e^{i\sqrt{\omega^{2}-\kappa^{2}}\left(\sqrt{\lambda}\log\left(\lambda
r+\sqrt{(r^{2}\lambda+1)\lambda}\right)-\frac{\sqrt{r^{2}\lambda+1}}{r}\right)}.$
(27)
Now, let us study the behavior near $r=0$. To do so, we expand the expression
around the origin, where we find that the dominant part goes like
$\varphi(r)\sim{r^{-1/2}}{e^{-i{\sqrt{\omega^{2}-\kappa^{2}}}/{r}}}$. At this
point, we are interested in making connection between this result and the
well-known result for AdS3 (i.e. $\lambda=0$) when $\omega^{2}>\kappa^{2}$;
namely
$\varphi_{\lambda=0}(z)=A_{2}\,z\,J_{\frac{1}{2}}\left(\sqrt{\omega^{2}-\kappa^{2}}z\right)+B_{2}\,z\,J_{-\frac{1}{2}}\left(\sqrt{\omega^{2}-\kappa^{2}}z\right)$
(28)
where $z=1/r$ and where $J_{\pm 1/2}$ are Bessel functions111These Bessel
functions actually reduce to elementary functions, which is due to the fact
that on AdS our problem reduces to that of a scalar with the conformal mass
$m^{2}\ell^{2}=m_{\text{conf}}^{2}\ell^{2}=-3/4$. For arbitrary values of the
mass the Bessel functions are replaced by $J_{\pm\nu}$ with
$\nu=\sqrt{1+m^{2}}$.. In order to relate (28) with the complex exponentials
in (23) one can use
$J_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\sin(x)}{\sqrt{x}}\ ,\ \ \
J_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\cos(x)}{\sqrt{x}}$ (29)
and, therefore, in terms of $r$ this reads
$\varphi_{\lambda=0}(r)=(A_{2}+B_{2})\,\frac{e^{i\frac{\sqrt{\omega^{2}-\kappa^{2}}}{r}}}{r^{1/2}}-(A_{2}-B_{2})\,\frac{e^{-i\frac{\sqrt{\omega^{2}-\kappa^{2}}}{r}}}{r^{1/2}}\,.$
(30)
We see that asymptotic behavior of (27) is a particular linear combination of
the solutions in (30), namely with $B_{2}=-A_{2}$. This is equivalent to
setting very special mixing boundary conditions in the AdS3 region $r\ll
1/\sqrt{\lambda}$. This makes it a difference with respect to the AdS3 case
$\lambda=0$.
Now, let us see what happens in the case $\kappa^{2}-\omega^{2}>0$, where the
solution reads
$\varphi(r)=\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}\left(C_{1}e^{-\sqrt{\kappa^{2}-\omega^{2}}\chi(r)}+D_{1}e^{\sqrt{\kappa^{2}-\omega^{2}}\chi(r)}\right)$
(31)
As the wave function is now confined, we want the solution to vanish at
infinity; therefore, we set $D_{1}=0$. As before, one can first take a look at
the solution in the case $\lambda=0$; namely
$\varphi_{\lambda=0}(r)=\frac{C_{2}}{r}K_{\frac{1}{2}}\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right)+\frac{D_{2}}{r}I_{\frac{1}{2}}\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right);$
(32)
and then rewrite the modified Bessel functions $K_{1/2}$, $I_{1/2}$ as
$K_{\frac{1}{2}}(x)=\sqrt{\frac{\pi}{2}}\frac{e^{-x}}{\sqrt{x}}\ \ \ \ \
I_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\sinh(x)}{\sqrt{x}}$ (33)
to make contact with the solution for $\lambda=0$,
$\displaystyle\varphi_{\lambda=0}(r)$
$\displaystyle=\frac{C_{2}}{r^{1/2}}e^{-\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}}+\frac{D_{2}}{r^{1/2}}\sinh\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right).$
(34)
Expanding the $\lambda\neq 0$ solution near $r=0$, one obtains
$\varphi(r)=A_{1}e^{\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}}(r^{-1/2}\,{e^{-\frac{1}{2}\sqrt{k^{2}-\omega^{2}}\sqrt{\lambda}\log(\lambda)}}+\mathcal{O}(r^{1/2})\,)\,;$
(35)
so, we see that (35) corresponds to the linear combination $C_{2}=D_{2}$
above. Notice that, while (27) diverges as $\sim r^{-1/2}$ when $r$ tends to
zero, (35) is exponentially divergent in that limit. Condition $C_{2}=D_{2}$
also looks from the AdS3 viewpoint ($r\ll 1/\sqrt{\lambda}$) as mixed boundary
conditions which are induced by the presence of the unusual asymptotic of the
deformed theory.
As the conformally coupled scalar, the Dirac action is also Weyl invariant;
therefore, it is natural to study a spin-$1/2$ probe on the deformed,
conformally flat background (10). Explicitly, the Dirac equation reads
$\left(\gamma^{a}e_{a}^{\mu}\partial_{\mu}+\frac{1}{2}\omega_{\mu}^{ab}\gamma^{c}e_{c}^{\mu}J_{ab}\right)\Psi=0\,.$
(36)
The spinor $\Psi$ will split in two components
$\Psi^{T}=\left(\Psi_{1},\Psi_{2}\right)$. The defomed metric (10) is of the
form
$ds^{2}=-f^{2}(r)dt^{2}+g^{2}(r)dr^{2}+f^{2}(r)dx^{2}\,,$ (37)
for which we can choose dreibein and compute the spin connections, leading
respectively to
$e^{0}=f\,dt\text{, }e^{1}=g\,dr\text{ and }e^{2}=f\,dx\,,$ (38)
$\omega^{01}=\frac{f^{\prime}}{g}dt\text{ and
}\omega^{12}=-\frac{f^{\prime}}{g}dx\ .$ (39)
It is useful to use the explicit, real representation of the Dirac matrices
$\gamma^{0}=i\sigma^{2},\gamma^{1}=\sigma^{1},\gamma^{2}=\sigma^{3}$. With
these expressions at hand, Dirac equation leads to the coupled system
$\displaystyle\frac{1}{f}\partial_{t}\Psi_{2}+\frac{1}{g}\partial_{r}\Psi_{2}+\frac{1}{f}\partial_{x}\Psi_{1}+\frac{f^{\prime}}{fg}\Psi_{2}$
$\displaystyle=0,$ (40)
$\displaystyle\frac{1}{f}\partial_{t}\Psi_{1}-\frac{1}{g}\partial_{r}\Psi_{1}+\frac{1}{f}\partial_{x}\Psi_{2}-\frac{f^{\prime}}{fg}\Psi_{1}$
$\displaystyle=0.$ (41)
Defining $\Psi_{i}\left(t,r,x\right)=e^{-i\omega t+i\kappa
x}\psi_{i}\left(r\right)$ with $i=1,2$, we can integrate the radial profiles
for the spinor as
$\displaystyle\psi_{1}\left(r\right)=\left(\frac{r^{2}\lambda+l^{2}}{r^{2}}\right)^{1/4}\varphi(r)\text{
and
}\psi_{2}(r)=\frac{\omega}{\kappa}\psi_{1}(r)-\frac{i}{\kappa}\frac{1}{g(r)}\frac{d}{dr}\left(f(r)\psi_{1}(r)\right)\
,$ (42)
where $\varphi(r)$ is given by (23) or (31) depending on the sign of
$\kappa^{2}-\omega^{2}$, as before. Consequently, the asymptotic behavior for
the fermion is inherited by that of the scalar. When the momentum along the
direction $x$ vanishes, namely when $\kappa=0$ the integration for the Dirac
field is simpler and leads to
$\displaystyle\psi_{1}(r)$
$\displaystyle=C_{1}\left(\frac{1+r^{2}\lambda}{r^{2}}\right)^{1/2}\left(\sqrt{\lambda}r+\sqrt{1+r^{2}\lambda}\right)^{-i\omega\sqrt{\lambda}}e^{\frac{i\omega\sqrt{1+r^{2}\lambda}}{r}}$
(43) $\displaystyle\psi_{2}(r)$
$\displaystyle=C_{2}\left(\frac{1+r^{2}\lambda}{r^{2}}\right)^{1/2}\left(\sqrt{\lambda}r+\sqrt{1+r^{2}\lambda}\right)^{i\omega\sqrt{\lambda}}e^{-\frac{i\omega\sqrt{1+r^{2}\lambda}}{r}}$
(44)
The two independent solution $C_{1}=0$ or $C_{2}=0$, respectively describe an
ingoing or outgoing flux of particles. Again, from the perspective of AdS3,
these would correspond to mixed boundary conditions.
## 3 String theory
Now, let us study the string worldsheet theory. The worldsheet action on the
background (9) takes the form
$S=\frac{1}{2\pi}\int
d^{2}z\left(\frac{1}{2}\partial\phi\bar{\partial}\phi+\frac{1}{2}\partial\bar{u}\bar{\partial}u(\lambda+e^{-\sqrt{2/(k-2)}\phi})^{-1}\right),$
(45)
together with an extra linear dilaton term $-({1}/{2\pi})\int
d^{2}z\sqrt{{2}/({k-2})}{R}\phi$ with ${R}$ being here the worldsheet
curvature. We see here that, provided $\lambda\neq 0$, in the limit
$\phi\to\infty$ one recovers the free theory with a background charge term. In
the case $\lambda=0$, in contrast, the theory at $\phi\to\infty$ exhibits a
non-trivial coupling between $\phi$ and the $u,\bar{u}$ dependence. This can
be regarded as an effective potential in the $\phi$-direction. This potential
vanishes for holomorphic configurations such that
$\partial\bar{u}\bar{\partial}u=0$. This configurations are closely related to
the so-called long strings, which form a continuum in the spectrum in AdS3.
### Strings on AdS${}_{3}\times\mathcal{N}$
Let us begin by reviewing the undeformed theory ($\lambda=0$), namely bosonic
string theory on AdS${}_{3}\times\mathcal{N}$. This theory corresponds to the
level-$k$ WZW model on $SL(2,\mathbb{R})$, and so it has $\hat{sl}(2)_{k}$
affine Kac-Moody symmetry, which is generated by local currents whose modes
are usually denoted $J_{n}^{\pm}$, $J_{n}^{3}$, along with their anti-
holomorphic counterparts. Virasoro symmetry follows from the Sugawara
construction. We consider primary operators of the form
$V_{h}(p|z)=Z_{0}|p|^{2-2h}\,e^{ip{{u}}(z)+i\bar{p}\bar{{{u}}}(\bar{z})}e^{\sqrt{2/(k-2)}(h-1)\phi(z,\bar{z})}\,\times\,...$
(46)
where the ellipsis stand for the contributions of internal part $\mathcal{N}$.
These are the vertex operators of the theory. $p$ and $\bar{p}$ are the
momenta conjugate to directions $u$ and $\bar{u}$, while $h$ is related to the
radial momentum. The factor $Z_{0}|p|^{2-2h}$ stands for a normalization. The
worldsheet conformal dimension of these operators are
$\displaystyle\Delta_{\lambda=0}=\frac{h(1-h)}{k-2}+\Delta_{\mathcal{N}}+N\,$
(47)
An analogous expression holds for $\bar{\Delta}_{\lambda=0}$ with
$\bar{\Delta}_{\mathcal{N}}$ and $\bar{N}$. $\Delta_{\mathcal{N}}$ stand for
the conformal dimension of the operators of the CFT on the internal space
$\mathcal{N}$, and $N$ is the string excitation number. As just said, $p$ and
$\bar{p}$ represent the momentum in the boundary, and they relate to the
momentum in (22) as follows
$\kappa=\frac{p+\bar{p}}{\ell}\ ,\ \ \ \omega=\frac{\bar{p}-p}{\ell}\,.$ (48)
In the Euclidean theory, $t\to it$ and $\bar{p}$ is the complex conjugate of
$p$. The index $h$ labels the representations of $SL(2,\mathbb{R})$. We focus
on the long string states, which belong to the continuous series
representations, having
$h=\frac{1}{2}+is\,,\ \ \text{with}\ \ \ \ s\in\mathbb{R}.$ (49)
These long strings can reach the boundary due to the coupling to the
$B$-field. They have a continuous energy spectrum, which depends on the
spectral flow variable $w\in\mathbb{Z}_{\geq 0}$ that accounts for the winding
number of the string around the boundary. To analyze the spectrum of the
theory on AdS${}_{3}\times\mathcal{N}$ in the momentum space, it is convenient
to consider the operator basis
$V_{h,m,\bar{m}}(z)=\frac{\Gamma(h+m)}{\Gamma(1-h-\bar{m})}\int\frac{d^{2}p}{|p|^{2}}\,p^{-m}\bar{p}^{-\bar{m}}\,V_{h}(p|z)\,.$
(50)
Performing spectral flow transformation on the states created by these
operators, one obtains the states of the sector $w$, whose conformal
dimensions are
$\displaystyle\Delta_{\lambda=0}=\frac{h(1-h)}{k-2}-mw-\frac{k}{2}w^{2}+\Delta_{\mathcal{N}}+N\,.$
(51)
where the energy is given by $m+\bar{m}+kw$ and the angular momentum by
$m-\bar{m}$.
The 2-point function in the theory on AdS${}_{3}\times\mathcal{N}$ is well-
known. For long strings in the basis $V_{h}(p|z)$, this takes the form
$\langle
V_{\frac{1}{2}+is_{1}}(p_{1}|z_{1}=0)V_{\frac{1}{2}+is_{2}}(p_{2}|z_{2}=1)\rangle_{\lambda=0}=\frac{2s_{1}}{\pi
k}\,Z_{0}^{2}\nu(k)^{2is_{1}}|p_{1}|^{4is_{1}}\,\delta^{(2)}(p_{1}+p_{2})\,\delta(s_{1}-s_{2})\,e^{2i\varphi}$
(52)
where
$e^{2i\varphi}={\Gamma}/{\Gamma^{*}}\,,\ \ \text{with}\ \ \ \
\Gamma=\Gamma(-2is)\Gamma({-2is}/({k-2}))$ (53)
and
$\nu(k)=\frac{\Gamma(\frac{1}{k-2})}{\Gamma(1-\frac{1}{k-2})}.$ (54)
The subscript ${\lambda=0}$ in (52) refers to the fact that the quantity
corresponds to the undeformed AdS${}_{3}\times\mathcal{N}$ background. In the
deformed theory, the 2-point function has been computed in [15, 16], yielding
$\langle
V_{h_{1}}(p_{1}|z_{1}=0)V_{h_{2}}(p_{2}|z_{2}=1)\rangle_{\lambda}=\delta^{(2)}(p_{1}+p_{2})\delta_{h_{1}-h_{2}}\,|p_{1}|^{4h_{1}-2}\,B(h_{1})$
(55)
with
$B(h_{1})=\frac{\nu(k)^{2h_{1}-1}}{\pi}\,\frac{\Gamma(1-2h_{1})\Gamma(1-\frac{2h_{1}-1}{k-2})}{\Gamma(2h_{1}-1)\Gamma(\frac{2h_{1}-1}{k-2})},$
(56)
and where the spectrum of the theory is given by
$h=\frac{1}{2}\pm\frac{1}{2}\sqrt{8(k-2)\lambda|p|^{2}-s^{2}}\,.$ (57)
This reduces to the 2-point function of the $SL(2,\mathbb{R})_{k}$ WZW model
in the limit $\lambda=0$. Eq. (57) follows from imposing the Virasoro
constraint $\Delta_{\lambda=0}=1$ on (47).
### Turning on the deformation
To see in detail how to obtain the correlator (55)-(56) and (57), we may first
rewrite action (45) by adding auxiliary fields $v,\,\bar{v}$, yielding the
equivalent action
$S_{\lambda}=\frac{1}{2\pi}\int
d^{2}z\left(\frac{1}{2}\partial\phi\bar{\partial}\phi-{{v}}\bar{\partial}{{u}}-\bar{{{v}}}\partial\bar{{{u}}}-2{{v}}\bar{{{v}}}e^{-\sqrt{2/(k-2)}\phi}-2\lambda{{v}}\bar{{{v}}}\right),$
(58)
where now the pair $(v,u)$ forms a commuting, dimension-(1,0) $(\beta,\gamma)$
ghost system. As we work in the conformal gauge, we are omitting here a
background charge that represents the dilaton term. For $\lambda=0$, equation
(58) is, indeed, the WZW model written in Wakimoto variables [30].
Nevertheless, we prefer to keep the notation $v,\,u$ to make contact with the
spacetime interpretation (9). The action of the $\lambda$-deformed theory is
thus given by $S_{SL(2,\mathbb{R})\text{ WZW}}-2\lambda\int
dz^{2}\,{{v}}\bar{{{v}}}$. This corresponds to a current-current deformation
of the WZW model, with the deformation being realized by the operator
$\lambda{{v}}\bar{{{v}}}$. This is consistent with the fact that the specific
Kac-Moody current in these variables reads $J^{-}(z)=v(z)$.
When trying to compute a correlation function such as $\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}$ in the path
integral approach, namely
$\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}=\int\mathcal{D}\phi\mathcal{D}^{2}u\mathcal{D}^{2}v\,e^{-S_{\lambda}}\,V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\,,$
(59)
the presence of the operator $\lambda\int d^{2}z\,v\bar{v}$ induces an
ultraviolet divergent term in the effective action after integrating the
fields $u,\bar{u}$ (see [16] for more details). This makes the contribution of
the correlators coming from the undeformed theory to factorize, and the
deformation ends up contributing with an exponential that contains the
conformal integral
$I_{0}=\int d^{2}z\,|z-z_{1}|^{-2}|z-z_{2}|^{-2}.$ (60)
This divergent integral appears frequently in quantum field theory
calculations. For instance, it appears in the one-loop computation of the
anomalous dimension of the composite operator $\bar{\psi}\psi$ in the Thirring
model. The result of it is logarithmically divergent and it can be regularized
using different methods. By introducing a regulator $\epsilon$, this can be
resolved as
$I_{\epsilon}=\left(1+2\epsilon\log|z_{1}-z_{2}|+\mathcal{O}(\epsilon^{2})\right)\left(\frac{2\pi}{\epsilon}+\mathcal{O}(\epsilon^{0})\right),$
(61)
and, after renormalizing the vertex operators by choosing
$Z_{\epsilon}=e^{-2\lambda|p|^{2}/\epsilon}$, one obtains
$\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}\sim\,|z_{1}-z_{2}|^{-4\Delta_{0}-4\lambda|p_{1}|^{2}}$
(62)
From this, it is possible to read the anomalous dimension induced by the
deformation; namely
$\Delta_{\lambda=0}\rightarrow\Delta_{{\lambda}}=\Delta_{\lambda=0}+\lambda|p|^{2}.$
(63)
Finally, imposing the Virasoro constraint $\Delta_{{\lambda=1}}$ and writing
it in terms of the quanties of the undeformed theory that satisfied
$\Delta_{\lambda=0}=1$, one gets (57), which reduces to (49) in the case
$\lambda=0$. We observe that $h\in\mathbb{R}$ provided
$-4|p|\sqrt{k\lambda}\geq s\geq+4|p|\sqrt{k\lambda}$; and
$h\in\frac{1}{2}+i\mathbb{R}$ provided $|s|>4|p|\sqrt{k\lambda_{0}}$. The
overall factor $|p_{1}|^{4h_{1}-2}$ in the 2-point function and the dependence
of $h_{1}$ on $\lambda$ has been studied in detail in [15] to investigate the
properties of the dual theory, especially its non-locality encoded in a branch
cut that the 2-point function of the $\lambda$-deformed theory exhibits.
## 4 Generalizations
The advantage of the computation of the anomalous dimension described above is
that it admits a straightforward generalization to other models, such as the
$SL(N,\mathbb{R})$ WZW models or their supersymmetric extensions. Despite not
in all such cases one has a string $\sigma$-model interpretation of the CFT,
this is still interesting from the CFT point of view as it provides a set of
solvable non-rational models. The simplest extension of this sort is the
$SL(N,\mathbb{R})$ WZW model. In that case, the action can in principle be
written as a sum of a Gaussian piece and an interaction piece $S_{I}$; namely
$S_{SL(N,\mathbb{R})\text{ WZW}}=\frac{1}{2\pi}\int
d^{2}z\left(\left(\partial\phi,\bar{\partial}\phi\right)-\sum_{a=1}^{N(N-1)/2}({{{v}}}_{a}\bar{\partial}{{{u}}}_{a}+{\bar{{{v}}}}_{a}\partial{\bar{{{u}}}}_{a})\right)+S_{I}.$
(64)
This involves a set of $N-1$ scalars and $N(N-1)$ copies of $\beta,\gamma$
systems, which here we keep denoting by $v_{a},u_{a}$ with
$a=1,2,...N(N-1)/2$. The scalars, $\phi_{i}$, with $i=1,2,...N-1$ form a
vector in the space of roots of $sl(N)$. We denote $(\,.\,,\,.\,)$ the product
in this space of roots, which is defined in terms of the Cartan matrix
$K_{ij}=(e_{i},e_{j})$ with $e_{1},\,e_{2},\,...,\,e_{N-1}$ being the simple
roots, with the $N-1$ fundamental weights $w_{i}$ satisfying
$(w_{i},e_{j})=\delta_{ij}$. $\rho$ is the Weyl vector, i.e. the half-sum of
all positive roots. The Lagrangian also includes a background charge term
$\int d^{2}z\,(\rho,\phi){R}/\sqrt{k-N}$.
As before, in the appropriate basis a solvable family of current-current
deformation of the theory is given by the addition of the marginal operator
$\sum^{N-1}_{i=1}\frac{\lambda_{i}}{\pi}\int
d^{2}z\,{{v}}_{i}\bar{{{v}}}_{i},$ (65)
where the field $J^{-}_{i}(z)=v_{i}(z)$ with $i=1,2,...N-1$ correspond to the
Abelian subalgebra formed by $N-1$ lowering operators. If we denote
$H=(J^{3}_{1},J^{3}_{2},...J^{3}_{N-1})$ the generators of the Cartan
subalgebra, and $E=(J^{+}_{1},J^{+}_{2},...J^{+}_{N(N-1)/2})$ and
$F=(J^{-}_{1},J^{-}_{2},...J^{-}_{N(N-1)/2})$ the raising and lowering
operators, respectively, then there exists an ordering such that the first
$N-1$ elements $F\supset(J^{-}_{1},J^{-}_{2},...J^{-}_{N-1})$ form an Abelian
subalgebra. More importantly, there exists a free field representation such
that these $N-1$ fields are given by $J^{-}_{i}=v_{i}$ with $i=1,2,...N-1$.
For the case $A_{N-1}$ with $N=2,3,4,5$ these representations have been
explicitly constructed in the literature [31, 32, 33], and the generic case
has been extensively discussed [34, 35, 36]. Let us first show how the
argument goes for generic $N$ and then consider an illustrating particular
case.
Consider the operators
$V_{h}(p,z)=Z_{0}\,e^{\sqrt{2/(k-N)}(h,\phi(z))}e^{i\sum_{a=1}^{{N(N-1)}/{2}}(p^{a}u_{a}(z)+\bar{p}^{a}\bar{u}_{a}(z))}$
(66)
where $h=(h_{1},h_{2},...h_{N-1})$ is the vector of the space of roots, and
$p=(p^{1},p^{2},...p^{N(N-1)/2})$ are the momentum associated to the
directions $u_{a}$; and consider the correlation functions
$\left\langle
V_{h}(p_{1}|z_{1})V_{h}(p_{2}|z_{2})\right\rangle_{\lambda_{1},...\lambda_{N-1}}=\int\prod_{i=1}^{N-1}\mathcal{D}\phi_{i}\prod_{a=1}^{N(N-1)/2}\mathcal{D}^{2}u_{a}\mathcal{D}^{2}v_{a}\,e^{-S_{\lambda_{1},...,\lambda_{N-1}}}\,V_{h}(p_{1}|z_{1})V_{h}(p_{2}|z_{2})\,.$
(67)
After integrating in $u_{i}$ for some $i$ (those that correspond to the fields
$u_{i}$ that do not appear other than in the kinetic term) the action
(64)-(65) being linear in these fields, one obtains
$\bar{\partial}{{v}}_{i}=2\pi
i(p^{i}_{1}\delta^{2}(z-z_{1})+p^{i}_{2}\delta^{2}(z-z_{2})),$ (68)
The solution is
${{v}}_{i}(z)=\frac{ip^{i}_{1}}{z-z_{1}}-\frac{ip^{i}_{1}}{z-z_{2}},$ (69)
where we have used that, on the sphere, $p_{1}+p_{2}=0$ in virtue of the
Riemann-Roch theorem. This can now be inserted back in (65). When doing so,
one observes that a logarithmically divergent integral similar to (60)
appears, yielding an anomalous correction to the conformal dimension of
operators (66). To see this in detail, let us consider the case $N=3$, in
which the undeformed theory is given by the WZW model on $SL(3,\mathbb{R})$,
whose action reads
$\displaystyle S_{SL(3,\mathbb{R})\text{ WZW}}=\frac{1}{2\pi}\int d^{2}z$
$\displaystyle\left(\left(\partial\phi,\bar{\partial}\phi\right)-\sum_{a=1}^{3}({{v}}_{a}\bar{\partial}{{u}}_{a}+{\bar{v}}_{a}{\partial}{\bar{u}}_{a})+\right.$
(70)
$\displaystyle\left|{{v}}_{2}+{{v}}_{1}{{u}}_{3}\right|^{2}e^{\sqrt{2/(k-3)}\left(e_{2},\phi\right)}-{{v}}_{3}\bar{{{v}}}_{3}e^{\sqrt{2/(k-3)}\left(e_{3},\phi\right)}-$
$\displaystyle\left.{{v}}_{1}\bar{{{v}}}_{1}e^{\sqrt{2/(k-3)}\left(\rho,\phi\right)}\right).$
together with a background charge term $\int d^{2}z\ (\rho,\phi)R/\sqrt{k-3}$.
As before, this action can be written in terms of the Wakimoto variables in
such a way that two commuting currents take a simple form
$J^{-}_{1}(z)={{v}}_{1}(z)$ and $J^{-}_{2}(z)={{v}}_{2}(z)$. Therefore, in the
spirit of the deformation for $SL(2,\mathbb{R})$, we deform the
$SL(3,\mathbb{R})$ WZW model by adding to it two quadratic operators, for
${{v}}_{1}$ and ${{v}}_{2}$; namely
$S_{\lambda_{1},\lambda_{2}}=S_{SL(3,\mathbb{R})\text{
WZW}}-\frac{\lambda_{1}}{\pi}\int
d^{2}z\,{{v}}_{1}\bar{{{v}}}_{1}-\frac{\lambda_{2}}{\pi}\int
d^{2}z\,{{v}}_{2}\bar{{{v}}}_{2}.$ (71)
We consider operators (66) with $N=3$, $h=(h_{1},h_{2})$ and
$p=(p_{1},p_{2},p_{3})$, and the correlation function $\left\langle
V_{h}(p,z_{1})V_{h}(-p,z_{2})\right\rangle_{\lambda_{1},\lambda_{2}}$. After
integrating on $u_{i}$, one finds the solutions for ${{v}}_{1}$ and
${{v}}_{2}$ to be
${{v}}_{1}(z)=\frac{ip^{1}(z_{1}-z_{2})}{(z-z_{1})(z-z_{2})}\ ,\ \ \
{{v}}_{2}(z)=\frac{ip^{2}(z_{1}-z_{2})}{(z-z_{1})(z-z_{2})}.$ (72)
which, when replaced in the action, yields
$\displaystyle S_{\lambda_{1},\lambda_{2}}=\frac{1}{2\pi}\int d^{2}z$
$\displaystyle\left(\left(\partial\phi,\bar{\partial}\phi\right)-{{v}}_{3}\bar{\partial}{{u}}_{3}-\bar{{{v}}_{3}}\partial\bar{{{u}}}_{3}-{{v}}_{3}\bar{{{v}}}_{3}e^{(\phi_{2}-\sqrt{3}\phi_{3})/\sqrt{k-3}}\right.$
(73)
$\displaystyle\left.+\left|p^{2}+p^{1}{{u}}_{3}\right|^{2}e^{(\phi_{2}+\sqrt{3}\phi_{3})/\sqrt{k-3}}-|p^{1}|^{2}e^{2\phi_{2}/\sqrt{k-3}}\right)$
$\displaystyle+\frac{1}{\pi}(\lambda_{1}|p^{1}|^{2}+\lambda_{2}|p^{2}|^{2})\,|z_{1}-z_{2}|^{2}\,I_{0},$
with $I_{0}$ given by (60). Regularizing as in (61) and renormalizing the
vertices accordingly, one obtains the corrected conformal dimension
$\Delta_{\lambda_{1},\lambda_{2}}=\Delta_{\lambda_{1}=\lambda_{2}=0}+\lambda_{1}\left|p^{1}\right|^{2}+\lambda_{2}\left|p^{2}\right|^{2}.$
(74)
This follows from the last line in (73), which contains the logarithmic
dependence in (61). This manifestly shows that the method of [16] can be
straightforwardly adapted to higher-rank.
Acknowledgments
G.G. thanks Edmundo Lavia and Matías Leoni for discussions and collaborations.
This work has been partially supported by CONICET through the grant PIP
1109-2017 and by Proyecto de Cooperación Internacional 2019/13231-7
FAPESP/ANID. The work of R.S. is funded by ANID Fellowships 22191591. The work
of J.O. partially funded by FONDECYT grant 1181047.
## References
* [1] A. Giveon, N. Itzhaki and D. Kutasov, “$\mathrm{T}\overline{\mathrm{T}}$ and LST,” JHEP 1707, 122 (2017) [arXiv:1701.05576 [hep-th]].
* [2] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B 915, 363 (2017) [arXiv:1608.05499 [hep-th]].
* [3] A. Cavaglià, S. Negro, I. M. Szécsényi and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 1610, 112 (2016) [arXiv:1608.05534 [hep-th]].
* [4] L. McGough, M. Mezei and H. Verlinde, “Moving the CFT into the bulk with $T\overline{T}$,” JHEP 04, 010 (2018) [arXiv:1611.03470 [hep-th]].
* [5] A. Giveon, N. Itzhaki and D. Kutasov, “A solvable irrelevant deformation of AdS3/CFT2,” JHEP 1712, 155 (2017) [arXiv:1707.05800 [hep-th]].
* [6] R. Ben-Israel, A. Giveon, N. Itzhaki and L. Liram, “On the black hole interior in string theory,” JHEP 1705, 094 (2017) [arXiv:1702.03583 [hep-th]].
* [7] O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, “Modular invariance and uniqueness of $T\bar{T}$ deformed CFT,” JHEP 1901, 086 (2019) [arXiv:1808.02492 [hep-th]].
* [8] L. Apolo, S. Detournay and W. Song, “TsT, $T\bar{T}$ and black strings,” JHEP 06, 109 (2020) [arXiv:1911.12359 [hep-th]].
* [9] S. Chakraborty, A. Giveon and D. Kutasov, “$T\bar{T}$, Black Holes and Negative Strings,” [arXiv:2006.13249 [hep-th]].
* [10] S. Chakraborty, A. Giveon, D. Kutasov “Comments on D3-Brane Holography,” [arXiv:2006.14129 [hrp-th]].
* [11] S. Chakraborty, A. Giveon and D. Kutasov, “Strings in Irrelevant Deformations of $AdS_{3}/CFT_{2}$,” [arXiv:2009.03929 [hep-th]].
* [12] S. Chakraborty, “$\frac{SL(2,\mathbb{R})\times U(1)}{U(1)}$ CFT, NS5$+$F1 system and single trace $T\bar{T}$,” [arXiv:2012.03995 [hep-th]].
* [13] T. Araujo, E. Ó. Colgáin, Y. Sakatani, M. M. Sheikh-Jabbari and H. Yavartanoo, “Holographic integration of $T\bar{T}$ \& $J\bar{T}$ via $O(d,d)$,” JHEP 03, 168 (2019) [arXiv:1811.03050 [hep-th]].
* [14] J. L. F. Barbon and E. Rabinovici, “Remarks on the thermodynamic stability of $T\bar{T}$ deformations,” J. Phys. A 53, no.42, 424001 (2020) [arXiv:2004.10138 [hep-th]].
* [15] M. Asrat, A. Giveon, N. Itzhaki and D. Kutasov, “Holography Beyond AdS,” Nucl. Phys. B 932, 241 (2018) [arXiv:1711.02690 [hep-th]].
* [16] G. Giribet, “$T\bar{T}$-deformations, AdS/CFT and correlation functions,” JHEP 1802, 114 (2018) [arXiv:1711.02716 [hep-th]].
* [17] S. Chakraborty, A. Giveon, N. Itzhaki and D. Kutasov, “Entanglement beyond AdS,” Nucl. Phys. B 935, 290 (2018) [arXiv:1805.06286 [hep-th]].
* [18] J. P. Babaro, V. F. Foit, G. Giribet and M. Leoni, “$T\overline{T}$ type deformation in the presence of a boundary,” JHEP 1808, 096 (2018) [arXiv:1806.10713 [hep-th]].
* [19] G. Giribet and M. Leoni, “Current-current deformations, conformal integrals and correlation functions,” JHEP 04, 194 (2020) [arXiv:2003.02864 [hep-th]].
* [20] S. Chakraborty, A. Giveon and D. Kutasov, “$T\bar{T}$, $J\bar{T}$, $T\bar{J}$ and String Theory,” J. Phys. A 52, no. 38, 384003 (2019) [arXiv:1905.00051 [hep-th]].
* [21] L. Apolo and W. Song, “Strings on warped AdS3 via $\mathrm{T}\bar{\mathrm{J}}$ deformations,” JHEP 10, 165 (2018) [arXiv:1806.10127 [hep-th]].
* [22] A. Giveon, “Comments on $T\bar{T}$, $J\bar{T}$ and String Theory,” [arXiv:1903.06883 [hep-th]].
* [23] L. Apolo and W. Song, “Heating up holography for single-trace $J\bar{T}$ deformations,” JHEP 01, 141 (2020) [arXiv:1907.03745 [hep-th]].
* [24] G. T. Horowitz and D. L. Welch, “Exact three-dimensional black holes in string theory,” Phys. Rev. Lett. 71, 328-331 (1993) [arXiv:hep-th/9302126 [hep-th]].
* [25] M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69, 1849-1851 (1992) [arXiv:hep-th/9204099 [hep-th]].
* [26] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48, 1506-1525 (1993) [erratum: Phys. Rev. D 88, 069902 (2013)] [arXiv:gr-qc/9302012 [gr-qc]].
* [27] D. Israel, C. Kounnas and M. P. Petropoulos, JHEP 10, 028 (2003) doi:10.1088/1126-6708/2003/10/028 [arXiv:hep-th/0306053 [hep-th]].
* [28] T. Buscher, “Path Integral Derivation of Quantum Duality in Nonlinear Sigma Models,” Phys. Lett. B 201, 466-472 (1988)
* [29] T. Buscher, “A Symmetry of the String Background Field Equations,” Phys. Lett. B 194, 59-62 (1987)
* [30] M. Wakimoto, “Fock representations of the affine lie algebra A1(1),” Commun. Math. Phys. 104, 605 (1986).
* [31] H. Awata, A. Tsuchiya and Y. Yamada, “Integral formulas for the WZNW correlation functions,” Nucl. Phys. B 365, 680 (1991).
* [32] H. Awata, “Screening currents ward identity and integral formulas for the WZNW correlation functions,” Prog. Theor. Phys. Suppl. 110, 303 (1992) [hep-th/9202032].
* [33] A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S. L. Shatashvili, “Wess-Zumino-Witten model as a theory of free fields,” Int. J. Mod. Phys. A 5, 2495 (1990).
* [34] M. Bershadsky and H. Ooguri, “Hidden SL(n) Symmetry in Conformal Field Theories,” Commun. Math. Phys. 126, 49 (1989)
* [35] P. Bouwknegt, J. G. McCarthy and K. Pilch, “Free field approach to two-dimensional conformal field theories,” Prog. Theor. Phys. Suppl. 102, 67-135 (1990)
* [36] P. Bouwknegt, J. G. McCarthy and K. Pilch, “Some aspects of free field resolutions in 2-D CFT with application to the quantum Drinfeld-Sokolov reduction,” [arXiv:hep-th/9110007 [hep-th]].
|
# Exciton Transfer in Organic Photovoltaic Cells: A Role of Local and Nonlocal
Electron-Phonon Interactions in a Donor Domain
Mauro Cainelli<EMAIL_ADDRESS>Department of Chemistry, Graduate
School of Science, Kyoto University, Kyoto 606-8502, Japan Yoshitaka Tanimura
<EMAIL_ADDRESS>Department of Chemistry, Graduate School of
Science, Kyoto University, Kyoto 606-8502, Japan
###### Abstract
We theoretically investigate an exciton transfer process in a donor domain of
organic photovoltaic cells focusing on the roles of local and nonlocal
electron-phonon interactions. Our model consists of a three-level system
described by the Holstein-Peierls Hamiltonian coupled to multiple heat baths
for local and nonlocal molecular modes characterized by Brownian spectral
distribution functions. We chose tetracene as a reference donor molecule,
where the spectral distribution functions of the local and nonlocal modes
exist. We then employ the reduced hierarchy equations of motion (HEOM)
approach to simulating the dynamics of the system under the influence of the
environment as a function of the electron-phonon coupling strength and
temperature. We rigorously calculate the reduced density matrix elements to
explain the timescale of dynamics under the influence of the dissipative local
and nonlocal modes. The results indicate that the strong nonlocal electron-
phonon interaction under high temperature conditions favors the exciton
transfer process and enhances the efficiency of organic photovoltaic
materials, while the lifetime of the exciton becomes shorter due to a low
frequency local mode.
††preprint: AIP/123-QED
## I Introduction
The development of better photovoltaic devices is a matter of great interest
in the investigation of renewable energy sources for a wide variety of systems
including biology- and material-oriented venues. Organic solar cells are
promising materials because of its low fabrication cost, lightweight and the
flexibility of design in comparison to silicon-based materials exist in the
market.[1, 2] Typical composition consists of polymers of small molecules as
donors and fullerene derivatives as acceptors, although non-fullerene
derivatives have also exhibited promising results.[3] However, the power
conversion efficiency of organic solar cells remains with maximum values of
11-17% that are still much lower than the inorganic counterparts.[4, 5]
Moreover, the mechanism on the organic devices is more complicated, mainly
because of the strong electron-phonon interaction due to the low dielectric
constant, where the quantum coherence between the electron and phonon plays an
essential role. Thus, for the investigation to improve the efficiency of such
systems, we have to employ fully quantum mechanical descriptions of broad
validity.[6]
During the process, sunlight is absorbed predominantly by the molecular donor
with the creation of strongly bounded electron-hole pairs (excitons). The
excitons then diffuse to the heterojunction, in which they dissociate into
electrons and holes, and are then collected at the corresponding
electrodes.[7] While most of the electronic dynamics occur in the active
layer, recent studies indicate that timescales and length scales of these
processes depend on the local geometry and electrostatic field. [8, 9]
Therefore, for the design of photovoltaic devices, understanding and
controlling the morphology is significant.[10, 11, 12] Although charge
transport and charge separation processes in organic solar cells have been
studied intensively, investigations concerning the role of incoherent
(hopping) [13] and coherent (delocalization) [14, 15, 16] transfer mechanisms
in different structure and physical conditions have not been well
explored.[17, 18, 19]
As recent theoretical analysis indicated, the effect of the nonlocal
(intermolecular) electron-phonon coupling on charge transport processes, in
addition to that of the local (intramolecular) one, is essential to account
for the energy transfer mechanism on such devices.[20] In heterojunction
blends, the entire process is ultrafast and takes place within 100fs [21, 22]
due to the presence of short lived delocalized excited species,[23, 24, 25]
which associate with the formation of hot charge transfer (CT) states.[26] The
hot CT states are assumed to be the main contribution to the generated
photocurrent, because they allow us to create a charge separation before the
thermalization process occur, while incoherent charge separation mechanisms
are characterized by slower separation of charges via multiple hopping steps
between localized states. In organic solar cells, both mechanisms are assumed
to be present.
It should be noted that the predominant dynamical processes in solar cells are
a system dependent, in which the molecules used as donor and acceptor,
geometry disposition, and electron-phonon coupling strengths determine the
power conversion efficiency. In order to account for the problem of this kind,
processes such as light absorption, energy- and charge-transports in organic
semiconductors must be taken into account by simulating dynamics in both
electronic and molecular degrees of freedom simultaneously.[27] A widely used
approach in order to properly treat vibrational modes in the framework of
quantum mechanics is based on the Holstein-Peierls model,[28] where the
Holstein (local) part of the Hamiltonian describes the variation of the site
energies[29] and the Peierls (nonlocal) part of it represents the modulation
of the transfer integrals.[30] Although the importance of nonlocal electron-
phonon coupling on the charge transport properties has been realized
recently,[31] the Hamiltonian of this model is not diagonalizable and the
analysis of the system is not easy. Thus, the effects of the local and
nonlocal couplings are usually treated collectively under the mean field
polaron (MFP) approximation,[32] ignoring the interaction between the local
and nonlocal modes. Moreover, because the electronic processes occur in
condensed phases, where the surrounding molecules provide the thermal
fluctuation and dissipation, we have to adapt open quantum dynamics treatment
for the investigation of time-irreversible reaction dynamics.[33, 34]
In order to study electron transfer problems in connection to open quantum
dynamics theory, a commonly used model considers the electronic states coupled
to an intermediate harmonic oscillator, which is further coupled to a heat
bath.[35] Then, the harmonic mode can be included in the bath by carrying out
a canonical transformation, which leads to a multilevel system coupled to the
heat bath with the Brownian spectral distribution (BSD) function.[36] Such a
system can be treated using the HEOM formalism, in a numerically rigorous
manner,[37] even in the low temperature case.[38, 39, 40] The HEOM formalism
can handle not only the strong system-bath coupling but also quantum coherence
between the system and bath that is important for the investigation of the
organic materials, as demonstrated in photosynthesis antenna systems[41, 42,
43, 44] and DNA.[45] Furthermore, this method does not require any
approximation, most notably the rotating wave approximation (RWA) or the MFP
approximation.
The HEOM approach has been applied to the case of solid state materials
described by a deformation potential [46] and the Holstein Hamiltonian.[47] In
the Holstein case, because we can study only a small system with the finite
number of phonon modes associated with the system site, special treatment is
necessary to maintain the stability of the equations.[48] The situation is the
same in the present case, and thus we further introduce the heat-bath into the
regular Holstein-Peierls model.
In this paper, we present a study of the nonlocal electron-phonon coupling
effect and the relation with a local coupling effect on the charge
transference process in organic photovoltaic devices, particularly in the
donor layer. We employ the HEOM approach for a BSD function to simulate the
non-Markovian dynamics of the Holstein-Peierls model that is further coupled
to the heat-bath.
The organization of this paper is as follows: In Sec. II, we introduce the
Holstein-Peierls + bath (HPB) model. Then, the HEOM for the HPB model is
explained. In Sec. III we present the details of the calculation and the
results of the simulation for various conditions. Sec. IV is devoted to
concluding remarks.
## II The HEOM for a Holstein-Peierls + bath system
The Holstein-Peierls model employs a tight-binding picture of an electronic
system that is linearly coupled to an optical phonon mode at each site. In the
present study, we further introduce a harmonic oscillator heat-bath.
The total Hamiltonian is expressed in terms of the electron Hamiltonian
($\hat{H}_{el}$), phonon Hamiltonian ($\hat{H}_{ph}$) and electron-phonon
interaction with the heat-bath ($\hat{H}_{ph-B}$) as
$\displaystyle\hat{H}_{tot}=\hat{H}_{el}+\hat{H}_{el-ph}+\hat{H}_{ph-B},$ (1)
where
$\displaystyle\hat{H}_{el}=\sum_{i}\varepsilon_{i}^{0}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\sum_{i\neq
j}t_{ij}^{0}\hat{a}_{i}^{\dagger}\hat{a}_{j},$ (2) $\displaystyle\hat{H}_{el-
ph}$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{{\alpha}}{\hat{V}}_{\alpha}^{i}\left(\hat{b}_{\alpha}^{\dagger}+\hat{b}_{\alpha}\right)+\sum_{ij}\sum_{{\alpha}}{\hat{V}}_{\alpha}^{ij}\left(\hat{b}_{\alpha}^{\dagger}+\hat{b}_{\alpha}\right)$
(3)
with
$\displaystyle{\hat{V}}_{\alpha}^{i}=\frac{1}{\sqrt{{\cal
N}_{C}}}\hbar\Omega_{\alpha}g_{i}^{\alpha}\hat{a}_{i}^{\dagger}\hat{a}_{i},$
$\displaystyle{\hat{V}}_{\alpha}^{ij}=\frac{1}{\sqrt{{\cal
N}_{C}}}\hbar\Omega_{\alpha}g_{ij}^{\alpha}\hat{a}_{i}^{\dagger}\hat{a}_{j},$
(4)
and
$\displaystyle\hat{H}_{ph-B}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\hbar\Omega_{\alpha}\left(\hat{b}_{\alpha}^{\dagger}\hat{b}_{\alpha}+\frac{1}{2}\right)+\sum_{\alpha}\sum_{k=1}^{N_{\alpha}}\hbar\omega_{k}\left(\hat{b}_{k}^{\dagger}\hat{b}_{k}+\frac{1}{2}\right)$
(5)
$\displaystyle+\sum_{\alpha}\left(\hat{b}_{\alpha}^{\dagger}+\hat{b}_{\alpha}\right)\sum_{k=1}^{N_{\alpha}}c_{k}^{\alpha}\left(\hat{b}_{k}^{\dagger}+\hat{b}_{k}\right).$
Here, $\hat{a}_{i}^{\dagger}$ ($\hat{a}_{i}$) is the creation (annihilation)
operator of the electronic quasi-particles (electrons, holes, or Frenkel
excitons) and $\varepsilon_{i}^{0}$ and $t_{ij}^{0}$ are the on-site
electronic energy and the ampliture of the transfer integral for the $i-j$th
electronic states that correspond to the diagonal and off-diagonal elements of
the on-site Hamiltonian, respectively. The creation (annihilation) operator of
the phonon (or vibron) mode ${\alpha}$ with the frequency $\Omega_{\alpha}$ is
expressed as $\hat{b}_{\alpha}^{\dagger}$ ($\hat{b}_{\alpha}$) . The local and
nonlocal system-bath interactions are expressed as ${\hat{V}}_{\alpha}^{i}$
and ${\hat{V}}_{\alpha}^{ij}$ with the dimensionless local and nonlocal
electron-phonon couplings strength $g_{i}^{\alpha}$ and $g_{ij}^{\alpha}$,
respectively. The constant ${\cal N}_{C}$ is the number of unitary cells
considered. The operator $\hat{b}_{k}^{\dagger}$ ($\hat{b}_{k}$) is the
creation (annihilation) operator of the heat-bath that can be characterized by
the oscillator-bath coupling strength and the inverse temperature $\beta\equiv
1/k_{\mathrm{B}}T$, where $k_{\mathrm{B}}$ is the Boltzmann constant. The bath
system is typically modeled by the spectral distribution function (SDF),
defined by
$\displaystyle
J_{\alpha}(\omega)\equiv\sum_{\alpha}\sum_{k=1}^{N_{\alpha}}\left(c_{k}^{\alpha}\right)^{2}\delta(\omega-\omega_{k}).$
(6)
For the heat bath to be an unlimited heat source possessing an infinite heat
capacity, the number of heat bath oscillators $N_{\alpha}$ is effectively made
infinitely large by replacing $J_{\alpha}(\omega)$ with a continuous
distribution. This Hamiltonian can be regarded as either empirical or first-
principles model, whether the system parameters are obtained from experimental
data or quantum chemistry calculations. The electron-phonon interactions
produce a time-dependent variation of the transport parameters and thus
introduce a dynamic disorder: Local and nonlocal electron-phonon couplings
correspond to diagonal and off-diagonal dynamic disorder mechanisms.
The challenges in using the Holstein-Peierls model are the following.[49]
First, the Hamiltonian is not exactly solvable. Second, the commonly used
approximations to solve this Hamiltonian are not fully satisfied for the
investigation of organic semiconductors. Third, it is difficult to find
efficient and reliable schemes to evaluate the system parameters
($\varepsilon_{i}^{0}$, $t_{ij}^{0}$, $\omega_{\alpha}$, $g_{i}^{\alpha}$,and
$g_{ij}^{\alpha}$).
In general, the second point in the above arises as a consequence of the first
and, in order to study the model, several approximations have been introduced:
First, the electron-electron interaction is treated in a framework of mean
field theory. Second, the amplitude $t_{ij}^{0}$ must be small in comparison
with the excitation energy of the intramolecular mode. Third, the electron-
phonon coupling is considered to be a linear form. Four, the nonlocal coupling
is treated under the band renormalization theory, assuming the collective
effect of low-frequency modes (MFP approximation). Five, phonon modes are
treated under rigid-body approximation that ignores the internal molecular
vibrations with environmental degrees of freedom that consist of the
intermolecular translational and librational molecular modes, in addition to
the acoustic modes. Moreover, in order to understand the structural effects on
the charge dynamics and the role of mixing morphology on the efficient pathway
to free charge photogeneration, it is necessary to account for the individual
effects of the couplings at different phonon frequencies and the interaction
between intra- and intermolecular vibrational modes.
In order to overcome some of the limitation mentioned above, here we employ
the HEOM formalism. The MFP approximation can be avoided because the
Hamiltonian does not require to be diagonalized. Thus, the phonon modes are
treated dynamically employing the system-bath model and large values of
$t_{ij}^{0}$ can be treated by choosing an appropriate hierarchy depth.
Although in this work we limited our analysis to the linear electron-phonon
couplings case, the extension to study nonlinear coupling is also possible.
By using the canonical transformation, the Hamiltonian in Eq.(1) can be
expressed in terms of the creation and annihilation operators of the heat-bath
+ primary oscillation ($\hat{b}_{k}^{\prime\dagger}$ and
$\hat{b}_{k}^{\prime}$, respectively) in the following general form
$\displaystyle\hat{H}_{tot}$ $\displaystyle=$
$\displaystyle\hat{H}_{el}+\sum_{\alpha}\sum_{k=1}^{N_{\alpha}}\sum_{i}d_{k}^{ii\alpha}\hat{a}_{i}^{\dagger}\hat{a}_{i}(\hat{b}_{k}^{\prime\dagger}+\hat{b}_{k}^{\prime})$
(7)
$\displaystyle+\sum_{\alpha}\sum_{k=1}^{N_{\alpha}}\sum_{ij}f_{k}^{ij\alpha}\hat{a}_{i}^{\dagger}\hat{a}_{j}(\hat{b}_{k}^{\prime\dagger}+\hat{b}_{k}^{\prime})+\sum_{\alpha}\sum_{k=1}^{N_{\alpha}}\hbar\omega_{k}\left(\hat{b}_{k}^{\prime\dagger}\hat{b}_{k}^{\prime}+\frac{1}{2}\right),$
where $d_{k}^{ii\alpha}$ and $f_{k}^{ij\alpha}$ are the dimensionless
constants that represent the local and nonlocal interactions of the system
with the oscillator-bath. Assuming that the SDF of the later to be
$J_{\alpha}(\omega)=\gamma_{\alpha}\omega$, we have the BSD defined as
$\displaystyle
J_{\alpha}^{\prime}(\omega)=\frac{\hbar\lambda_{\alpha}}{2\pi}\frac{\gamma_{\alpha}\Omega_{\alpha}^{2}\omega}{(\Omega_{\alpha}^{2}-\omega^{2})^{2}+\gamma^{2}\omega^{2}},$
(8)
where $\lambda_{\alpha}$ relates to the reorganization energy and nonlocal
relaxation energy which accounts for the displacement of the excited state in
a relation to the ground state in coordinate space as a consequence of the
local and nonlocal electron-phonon interactions, respectively. The parameter
$\gamma_{\alpha}$ is the coupling strength between the oscillator and the
bath, which is related to the peak width of $J_{\alpha}^{\prime}(\omega)$.
The reduced hierarchy equations of motion for the Brownian distribution can be
expressed as (see Appendix A)
$\displaystyle\frac{\partial}{\partial t}$
$\displaystyle\hat{\rho}_{\\{n^{\alpha},m^{\alpha};j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)$
(9) $\displaystyle=$
$\displaystyle-\left[\frac{i}{\hbar}\hat{H}_{el}^{\times}+\sum_{\alpha}\left\\{\frac{(n^{\alpha}+m^{\alpha})}{2}\gamma_{\alpha}-i(n^{\alpha}-m^{\alpha})\zeta_{\alpha}+\sum_{k=1}^{K^{\alpha}}j_{k}^{\alpha}\nu_{k}^{\alpha}-\hat{\Xi}_{\alpha}\right\\}\right]\hat{\rho}_{\\{n^{\alpha},m^{\alpha};j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)$
$\displaystyle+\sum_{\alpha}\hat{V}^{\times}_{\alpha}\bigg{[}\hat{\rho}_{\\{n^{\alpha}+1,m^{\alpha};j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)+\hat{\rho}_{\\{n^{\alpha},m^{\alpha}+1;j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)\bigg{]}$
$\displaystyle+\sum_{\alpha}n^{\alpha}\hat{\Theta}_{-}^{\alpha}\hat{\rho}_{\\{n^{\alpha}-1,m^{\alpha};j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)+\sum_{\alpha}m^{\alpha}\hat{\Theta}_{+}^{\alpha}\hat{\rho}_{\\{n^{\alpha},m^{\alpha}-1;j_{1}^{\alpha}...j_{K^{\alpha}}^{\alpha}\\}}(t)$
$\displaystyle+\sum_{\alpha}\sum_{k=1}^{K^{\alpha}}\hat{V}_{\alpha}^{\times}\hat{\rho}_{\\{n^{\alpha},m^{\alpha};j_{1}^{\alpha}...j_{k+1}^{\alpha},...j_{K^{\alpha}}^{\alpha}\\}}(t)+\sum_{\alpha}\sum_{k=1}^{K^{\alpha}}j_{k}^{\alpha}\nu_{k}^{\alpha}\hat{\Psi}_{k}^{\alpha}\hat{\rho}_{\\{n^{\alpha},m^{\alpha};j_{1}^{\alpha}...j_{k-1}^{\alpha},...j_{K^{\alpha}}^{\alpha}\\}}(t),$
where the first term in brackets is the Liouvillian
($\hat{\mathcal{L}}^{(n,m)}$) that involves the temperature correction terms.
For a high number of hierarchy elements, the term
$(n^{\alpha}+m^{\alpha})\gamma_{\alpha}/2+\sum_{k=1}^{K^{\alpha}}j_{k}^{\alpha}\nu_{k}^{\alpha}$
becomes much larger than the characteristic time of the system. In this case,
the hierarchy can be truncated by the terminator.[50]
## III Results and Discussion
Figure 1: Schematic view of the model that is described by the Hamiltonian
given in Eqs.(1)-(5). The ground state and the two excited states are coupled
to a local mode that is further coupled to the local bath ($H_{Bl}$),
respectively. The excited states are coupled to the nonlocal mode that is
further coupled to the nonlocal bath ($H_{Bnl}$). Figure 2: Two configurations
of the excited state pathways. Each pair represents the dimer molecule
illustrated in Fig. 1.
We chose tetracene as a reference donor molecule under study where the SDFs of
the electron-phonon coupling have been known.[51] In order to simulate the
exciton transfer process between the molecules, we consider a multi-state
system in which each state represents a dimer in the donor layer. While we
might simulate a molecular chain in the framework of the present model by
increasing the number of excited states, here we consider a simple three-state
system in order to investigate the role of local and nonlocal modes. Thus, the
system is described by a single ground state and two excited states expressed
as $\big{|}1\big{\rangle}$, $\big{|}2\big{\rangle}$, and
$\big{|}3\big{\rangle}$, respectively. As illustrated in Fig. 1, the excited
states $\big{|}2\big{\rangle}$ and $\big{|}3\big{\rangle}$ interact with the
nonlocal and local modes, while the ground state $\big{|}1\big{\rangle}$ only
interacts with the local mode. The local and nonlocal modes are further
coupled to their own heat-bath. We then consider the transfer between the two
excited states in two directions, represented by two different pair of
geometry dispositions of the dimer (Fig. 2), whose difference is described by
the values of the transfer integral and the nonlocal coupling strength.
The exciton Hamiltonian is then expressed in the matrix form as
$\displaystyle\hat{H}_{el}=\hbar\begin{bmatrix}\omega_{1}&\Delta_{12}&0\\\
\Delta_{21}&\omega_{2}&\Delta_{23}\\\ 0&\Delta_{32}&\omega_{3}\end{bmatrix},$
(10)
where $\hbar\omega_{i}$ is the Frenkel exciton energy of the $i$th state
($\varepsilon_{i}$) and $\hbar\Delta_{ij}$ is the coupling strength between
the $i$th and $j$th states (transfer integral $t_{ij}$). For all of our
computations, we fixed the eigenenergy of the system as $\omega_{1}=0$cm-1 and
$\omega_{2}=\omega_{3}=500$cm-1. We then use the excitation energy as the
characteristic frequency of the system $\omega=500$cm-1 with the time scale
$1/{\omega}=66.71$fs. Although the band gap between the ground and singlet
state in oligoacenes is of the order of 2-2.5eV ($16000-20000$cm-1), here we
chose a much smaller value for $\omega$ to qualitatively investigate the
effect of phonon interactions upon ultrafast dynamics with suppressing
numerical costs. The off-diagonal elements representing the interaction
between the ground and excited state are $\Delta_{12}=\Delta_{21}=0.32\omega$.
It should be noted that, while similar single crystals, which include
rubrene[52, 53] and pentacene[54] have been modeled and investigated, the
parameter values of the off-diagonal element of tetracene has not been well
explored. Here we employ the value of pentacene as a representative case,
while the interactions between the excited states are chosen to be
$\Delta_{23}=\Delta_{32}=1.23\omega$ for the pair 1 and
$\Delta_{23}=\Delta_{32}=0.12\omega$ for the pair 2, respectively. We consider
the room temperature case ($T=300$K) with the value of
$\beta\hbar\omega=0.382$ and the low temperature case (T=10K) with the value
of $\beta\hbar\omega=11.45$.
The system-bath interactions for the local and nonlocal modes are defined in
the matrix form as
$\displaystyle\hat{V}^{i}=\begin{bmatrix}-0.5&0&0\\\ 0&0.5&0\\\
0&0&0.5\end{bmatrix},$ (11)
and
$\displaystyle\hat{V}^{ij}=\begin{bmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{bmatrix}.$ (12)
The diagonal and off-diagonal elements of the system-bath coupling relate to
the local and nonlocal dynamic disorder mechanisms. Although the SDFs of local
and nonlocal baths consist of two or three oscillator modes, here we consider
one primary mode to simplify the model and analysis. Then, the coupling
strength between the local/nonlocal oscillator and the bath, which acts as the
inverse noise correlation time of the BSD, is fixed to $\gamma=0.05\omega$.
The initial condition of the system at $t=0$ is set by $\rho_{22}(0)=1$ and
$\rho_{11}(0)=\rho_{33}(0)=0$. We simulate the time evolution of the reduced
density matrix by numerically integrating the HEOM with respect to the time
with a time step of ${0.001}/{\omega}$, using the fourth-order Runge-Kutta
method. The depth and the truncation number of hierarchy are chosen to $N=65$
with $K=0$ for the high temperature case and $N=12$ with $K=3$ for the low
temperature case, respectively.
Figure 3: Time-evolution of the reduced density matrix elements
$\rho_{11}(t)$ (blue), $\rho_{22}(t)$ (orange), and $\rho_{33}(t)$ (green) at
$T=300K$ for different parameter values of local and nonlocal oscillator bath.
Here, (a) and (b) represent the pair 1 case and (c) represents the pair 2
case. The very early stage of the time evolution in each figure is presented
in Figs. 5(a)-(c) in Appendix B.
The time-evolution of the reduced density matrix for different sets of bath
modes at room temperature is shown in Fig. 3. For the local/nonlocal baths, we
consider (a)
$(\Omega_{l}^{1},~{}\lambda_{l}^{1})=(160,~{}0.032\omega)$/$(\Omega_{nl}^{1},~{}\lambda_{nl}^{1})=(50,~{}0.19\omega)$
and (b)
$(\Omega_{l}^{1},~{}\lambda_{l}^{1})=(475,~{}0.005\omega)$/$(\Omega_{nl}^{1},~{}\lambda_{nl}^{1})=(50,~{}0.19\omega)$
for the pair 1, while (c)
$(\Omega_{l}^{2},~{}\lambda_{l}^{2})=(475,~{}0.005\omega)$/$(\Omega_{nl}^{2},~{}\lambda_{nl}^{2})=(50,~{}0.137\omega)$
for the pair 2, with the unit of cm-1.
As illustrated in Fig. 3(a), the local mode of lower frequency slightly
enhances the exciton relaxation at a longer time period in comparison to that
of higher frequency presented in Fig. 3(b), because the coupling strength of
the local mode $\lambda_{l}$ is larger in the former case. The population
relaxation in Fig. 3(b) is also slower than that in Fig. 3(c), because, in the
case of pair 1, the exciton transfer between $\big{|}2\big{\rangle}$ and
$\big{|}3\big{\rangle}$ is favored due to the large nonlocal electron-phonon
coupling $\lambda_{nl}$ and transfer integral (excited state interaction)
$\Delta_{23}$. The very early stage of the time-evolution in each figure is
presented in Figs. 5 (a)-(c) in Appendix B, respectively, and it is shown that
this transfer is characterized by an ultrafast coherent oscillation in the
time period of $t$=1 [1/$\omega$] for pair 1 (Figs. 5(a) and 5(b)) and $t$=1.5
[1/$\omega$] for pair 2 (Fig. 5(c)). These facts indicate that the local and
nonlocal modes play an opposite role: The nonlocal mode promotes ultrafast
exciton transfer as a function of the coupling strength of the nonlocal mode,
$\lambda_{nl}$, that suppresses the exciton relaxation to the ground sate,
whereas the local mode enhances the relaxation through the localization of the
exciton as a function of the coupling strength of the local mode,
$\lambda_{l}$.
It should be mentioned that the small oscillatory motion of the excited state
population observed at approximately $t$=20 [1/$\omega$] in Figs. 3(a) and
3(b) are numerical errors that arise from an insufficient depth of hierarchy
elements $N$ due to the computational limitation of the HEOM calculation. This
can be suppressed by increasing $N$, although the calculations become
computationally more expensive.
Figure 4: Time evolution of the reduced density matrix elements
$\rho_{11}(t)$ (blue), $\rho_{22}(t)$ (orange), and $\rho_{33}(t)$ (green) at
$T=10K$ for (a) pair 1 and (b) pair 2 with the same set of the bath parameters
presented in Figs. 3(b) and 3(c), respectively. The very early stage of the
time-evolution in each figure is presented in Figs. 6(a) and 6(b) in Appendix
B.
Next we discuss the low temperature cases ($T$=10K) of Figs. 3(b) and 3(c) in
Figs. 4(a) and 4(b), respectively. The initial stage of Figs. 4(a) and 4(b)
are separately plotted in Figs. 6(a) and 6(b) in Appendix B, respectively. We
observe ultrafast coherent oscillations between the excited states in the time
period of $t$=1.2 [1/$\omega$] for the pair 1 and $t$=7.5 [1/$\omega$] for the
pair 2, respectively. The frequency of the ultrafast oscillation in the low
temperature case is slower than that in the room temperature case, and thus
the population exchange between $\big{|}2\big{\rangle}$ and
$\big{|}3\big{\rangle}$ toward the equilibrium state takes longer time. In
fact, we observe that, in the case of the pair 2, the exciton relaxation
starts at time $t$=2.5 [1/$\omega$], which is shorter than the time period of
the nonlocal oscillation. This indicates that, under the low temperature
condition, the effect of the bath through the local and nonlocal couplings is
suppressed, while the system dynamics is determined from the interplay of the
ground and the excited states coupling $\Delta_{12}$ and the excitation
coupling $\Delta_{23}$. For the pair 1 with larger $\Delta_{23}$, the exciton
transfer between $\big{|}2\big{\rangle}$ and $\big{|}3\big{\rangle}$ is
favored and the relaxation to the ground state $\big{|}1\big{\rangle}$ becomes
slower than that for the pair 2.
In this regards, although the ground state population of the thermal
equilibrium state becomes larger for longer time period at lower temperature,
the equilibration process that arises from the local and nonlocal baths is
faster in the high temperature case.In any case, when the coupling strength
between the excited states and the nonlocal coupling strength are larger, the
relaxation to the ground state become slower. Thus, large nonlocal electron-
phonon coupling seems to favors the mechanism of the exciton transfer among
the excited states even in the weak excitation coupling strength condition,
while the lifetime of the exciton is suppressed by low frequency local modes.
Because the configuration of pair 1 satisfies both the strong excitation
coupling and strong nonlocal coupling conditions, it is more favorable than
that of pair 2.
Although it was found from investigations of polaron transport process that
the nonlocal electron-phonon coupling promotes not only delocalization, but
also localization due to a bandwidth narrowing effect, in particular for the
strong coupling case at high temperatures,[55, 56, 57] the model system
employed in this work is too small and further study in a larger system needs
to be done for the verification.
## IV Conclusion
In this work, we introduced Holstein-Peierls + bath Hamiltonian in order to
investigate the effects of local and nonlocal electron-phonon interactions for
the exciton dynamics of organic molecules in photovoltaic cells. We employed
the HEOM formalism to simulate the dynamics of excitons described by this
model under various physical conditions for local and nonlocal environmental
modes without employing approximations, most notably the RWA and MFP
approximations. It should be noted that the situations that we considered here
are classified as non-perturbative and non-Markovian system-bath interaction
case in open quantum dynamics theory.
The calculated results demonstrate the effects of dynamic disorder on the
order of the reported time scale of the exciton processes that are
characterized by the presence of ultrafast coherent oscillations, in
particular at low temperatures. We found that a stronger nonlocal electron-
phonon coupling favors the exciton transfer between excited states and may be
a key to enhancing the efficiency of exciton transfer processes even in
materials with low excitation couplings. Moreover, the effect seems to be
favored at high temperature, while the number of low frequency local bath
modes should be suppressed.
In the present work, we limited our analysis to a small system with a specific
parameter set focusing on a role of local and nonlocal system-phonon
interactions. Because the exciton transfer is a long range effect, an
extension of the present investigation to a larger system should be realized
to provide better insight in this topic, specially regarding further
discussions between the delocalization/localization mechanisms. We plan to
investigate this direction in a future study.
###### Acknowledgements.
The financial support from The Kyoto University Foundation is acknowledged.
M.C. is supported by the Japanese Government (MEXT) Postgraduate Scholarships.
## Data Availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## Appendix A Reduced Hierarchy Equations of Motion for Holstein-Peierls
Model
We consider the density matrix of the total system with the factorized initial
condition. The reduced density matrix element is then expressed in the path
integral form as
$\displaystyle\rho(z,z^{\prime},t)$ $\displaystyle=$
$\displaystyle\iint_{z(t)=z}^{z^{\prime}(t)=z^{\prime}}\mathcal{D}[z,z^{\prime}]e^{iS_{0}[z,z^{\prime}]/\hbar}$
(13)
$\displaystyle\times\rho(z_{0},z^{\prime}_{0},t_{0})e^{-iS_{0}[z,z^{\prime}]/\hbar}\mathcal{F}[z,z^{\prime},t],$
where $\iint_{z(t)=z}^{z^{\prime}(t)=z^{\prime}}\mathcal{D}[z,z^{\prime}]$
represents the functional integral of a set of Grassmann variables, which
describe the system states, $\rho(z_{0},z^{\prime}_{0},t_{0})$ is the
factorized system part density, $S_{0}$ is the action for the system
Hamiltonian (Lagrangian), and $\mathcal{F}[z,z^{\prime},t]$ is the called
Feynman-Vernon influence functional that includes all the information
regarding the bath and the system bath-interaction.[58, 59, 60, 61]
Considering the time dependent kernels $iL_{1}(t)=i\int_{0}^{\infty}d\omega
J(\omega)\sin(\omega t)$ and $L_{2}(t)=\int_{0}^{\infty}d\omega
J(\omega)\coth(\beta\hbar\omega/2)\cos(\omega t)$ that correspond to the
fluctuation and dissipation effects, respectively, introduced by the bath,[58,
62, 63] the influence functional can be written, evaluating the contour
integrals for the Brownian spectral distribution as[39, 40, 64, 65]
$\displaystyle\mathcal{F}[z,z^{\prime},t]$ $\displaystyle=$ $\displaystyle
exp\Bigg{\\{}-\int_{t_{0}}^{t}ds\int_{t_{0}}^{s}duV^{\times}(s)$ (14)
$\displaystyle\times\bigg{[}\Theta_{-}(u)e^{-\big{(}\frac{\gamma}{2}-i\zeta\big{)}(s-u)}+\Theta_{+}(u)e^{-\big{(}\frac{\gamma}{2}+i\zeta\big{)}(s-u)}\bigg{]}\Bigg{\\}}$
$\displaystyle\times\exp\Bigg{[}\int_{t_{0}}^{t}ds\int_{t_{0}}^{s}duV^{\times}(s)\sum_{k=1}^{\infty}\Psi_{k}(u)\nu_{k}e^{-\nu_{k}(s-u)}\Bigg{]}$
$\displaystyle\times\exp\bigg{[}\int_{t_{0}}^{t}ds\Xi(s)\bigg{]},$
where
$\displaystyle\Theta_{\pm}=\frac{\lambda\omega_{0}^{2}}{2\zeta}\Bigg{\\{}\pm\coth\bigg{[}\frac{\beta\hbar}{2}\Big{(}i\frac{\gamma}{2}\mp\zeta\Big{)}\bigg{]}V^{\times}(u)\mp
V^{\circ}(u)\Bigg{\\}},$ (15)
$\displaystyle\Psi_{k}(u)=\frac{4\lambda\gamma\omega_{0}^{2}}{\beta\hbar}\frac{\nu_{k}}{(\omega_{0}^{2}+\nu_{k}^{2})^{2}-\gamma^{2}\nu_{k}^{2}}V^{\times}(u),$
(16) $\displaystyle\Xi(s)=V^{\times}(s)\sum_{k=K+1}^{\infty}\Psi_{k}(s).$ (17)
Here, $\zeta=\sqrt{\omega_{0}^{2}-\gamma^{2}/4}$ for $\gamma<2\omega_{0}$ and
$\nu_{k}=(2\pi/\beta h)k$ are the Matsubara frequencies, $\beta={1}/{k_{B}T}$
is the inverse temperature, and $k_{B}$ is the Boltzmann constant.
The commutator and anticommutator of the interaction operator $V$ are
expressed as $V^{\times}(t)=V(z(t))-V(z^{\prime}(t))$ and
$V^{\circ}(t)=V(z(t))+V(z^{\prime}(t))$, and the electron-phonon couplings are
introduced in the spectral distribution. The local and nonlocal transition
rates are determined from the diagonal and off-diagonal elements of $V$,
respectively.
Thanks to the exponential function form of the kernels, any higher-order time
derivative of the correlation functions is also expressed in terms of
exponential functions. These terms are called the auxiliary density matrices
(ADMs) in the HEOM formalism and are necessary to describe the non-Markovian
effects of the environment. The HEOM for the Brownian distribution can be
derived by calculating the time derivative on the left and right side of the
factorized system density and the influence functional.[64, 65] Summing over
all bath modes results in Eq. (9).
For a high number of hierarchy elements, the hierarchy can be truncated by the
terminator
$\displaystyle\frac{\partial}{\partial
t}\hat{\rho}_{j_{1}…j_{k}}^{(n,m)}(t)=-\bigg{[}\frac{i}{\hbar}\hat{H}_{el}^{\times}-i(n-m)\zeta-\hat{\Xi}\bigg{]}\hat{\rho}_{j_{1}…j_{k}}^{(n,m)}(t).$
(18)
The total number of hierarchy elements can be evaluated as
$L=(N+M+K+1)/(K+1)!(N+M)!$ with the total number of terminator elements
$L_{term}=(N+M+K)/K!(N+M)!$, where $N$ is the hierarchy depth for $n,m_{+}$
and $m_{-}$, and $M$ is the increase of the hierarchy for each spectral mode
($M=2$ for $m_{+}$ and $m_{-}$ in the case of the Brownian oscillator). In the
HEOM formalism, each member of the hierarchy is coupled to the lower and
higher terms, and the value for the initial density element is the exact
solution of the total Hamiltonian defined by $\rho_{0…0}^{(0,0)}(t)$, which
includes all the system-bath interactions.
## Appendix B The ultrafast oscillations of the density matrix elements in
the very early stage
Here, we replot the very early stage of the time-evolution ($t\leq 10$
[1/$\omega$] of the reduced density matrix elements presented in Figs. 3 and
4, respectively. In each figure, we observe ultrafast coherent oscillations
between $\big{|}2\big{\rangle}$ and $\big{|}3\big{\rangle}$. As it is
illustrated in Figs. 5 (b) and 5 (c) and Figs. 6 (a) and 6 (b), the ultrafast
oscillation is enhanced for larger $\lambda_{nl}$, which indicates that
nonlocal electron-phonon interaction that relates to the dynamics disorder
enhances the transition between the excited states.
Figure 5: Very early stage of the time-evolution of the reduced density matrix
elements $\rho_{11}(t)$ (blue), $\rho_{22}(t)$ (orange), and $\rho_{33}(t)$
(green) at $T=300$K presented in Figs. 3 (a)-(c) as Figs. (a)-(c),
respectively, are illustrated. Here, (a) and (b) correspond to the case of
pair 1 and (c) corresponds to the case of pair 2. Figure 6: Very early stage
of the time-evolution of the reduced density matrix elements $\rho_{11}(t)$
(blue), $\rho_{22}(t)$ (orange), and $\rho_{33}(t)$ (green) at $T=10$K
presented in Figs. 4 (a) and (b) as Figs. (a) and (b), respectively, are
illustrated.
## References
* Horowitz [1998] G. Horowitz, “Organic field-effect transistors,” Adv. Mater. 10, 365–377 (1998).
* Dimitrakopoulos and Malenfant [2002] C. D. Dimitrakopoulos and P. R. L. Malenfant, “Organic thin film transistors for large area electronics,” Adv. Mater. 14, 99–117 (2002).
* Zhang _et al._ [2018] J. Zhang, H. S. Tan, X. Guo, A. Facchetti, and H. Yan., “Material insights and challenges for non-fullerene organic solar cells based on small molecular acceptors,” Nature Energy 3, 720–731 (2018).
* Scharber _et al._ [2006] M. C. Scharber, D. Mühlbacher, M. Koppe, P. Denk, C. Waldauf, A. J. Heeger, and C. J. Brabec, “Design rules for donors in bulk-heterojunction solar cells—towards 10% energy-conversion efficiency,” Adv. Mater. 18, 789–794 (2006).
* Meng _et al._ [2018] L. Meng, Y. Zhang, X. Wan, C. Li, X. Zhang, Y. Wanga, X. Ke, Z. Xiao, L. Ding, R. Xia, H.-L. Yip, Y. Cao, and Y. Chen, “Organic and solution-processed tandem solar cells with 17.3% efficiency,” Science 361, 1094–1098 (2018).
* Brédas _et al._ [2009] J.-L. Brédas, J. E. Norton, J. Cornil, and V. Coropceanu, “Molecular understanding of organic solar cells: The challenges,” Acc. Chem. Res. 42, 1691–1699 (2009).
* Heeger [2014] A. J. Heeger, “25th anniversary article: Bulk heterojunction solar cells: Understanding the mechanism of operation,” Adv. Mater. 26, 10–28 (2014).
* Devizi _et al._ [2010] A. Devizi, K. Meerholz, D. Hertel, and V. Gulbinas, “Hierarchical charge carrier motion in conjugated polymers,” Chem. Phys. Lett. 498, 302–306 (2010).
* Oviedo-Casado, Urbina, and Prior [2017] S. Oviedo-Casado, A. Urbina, and J. Prior, “Magnetic field enhancement of organic photovoltaic cells performance,” Sci. Rep. 7, 4297 (2017).
* Botiz and Stigelin [2014] I. Botiz and N. Stigelin, “Influence of molecular conformations and microstructure on the optoelectronic properties of conjugated polymers,” Materials 7, 2273–2300 (2014).
* Vandewal, Himmelberger, and Salleo [2013] K. Vandewal, S. Himmelberger, and A. Salleo, “Structural factors that affect the performance of organic bulk heterojunction solar cells,” Macromolecules 46, 6379–6387 (2013).
* Beaujuge and Frechet [2011] P. M. Beaujuge and J. M. J. Frechet, “Molecular design and ordering effects in $\pi$-functional materials for transistor and solar cell applications,” J. Am. Chem. Soc. 133, 20009–20029 (2011).
* Athanasopoulos _et al._ [2017] S. Athanasopoulos, S. Tscheuschner, H. Bässler, and A. Köhler, “Efficient charge separation of cold charge-transfer states in organic solar cells through incoherent hopping,” J. Phys. Chem. Lett. 8, 2093–2098 (2017).
* Picon, Bussac, and Zuppiroli [2007] J.-D. Picon, M. N. Bussac, and L. Zuppiroli, “Quantum coherence and carriers mobility in organic semiconductors,” Phys. Rev. B 75, 235106 (2007).
* Birech _et al._ [2014] Z. Birech, M. Schwoerer, T. Schmeiler, J. Pflaum, and H. Schwoerer, “Ultrafast dynamics of excitons in tetracene single crystals,” J. Chem. Phys. 140, 114501 (2014).
* Marciniak _et al._ [2007] H. Marciniak, M. Fiebig, M. Huth, S. Schiefer, B. Nickel, F. Selmaier, and S. Lochbrunner, “Ultrafast exciton relaxation in microcrystalline pentacene films,” Phys. Rev. Lett. 99, 176402 (2007).
* Elton and Wang [2016] J. G. S. Elton and W. L. Wang, “Ultrafast charge-transfer in organic photovoltaic interfaces: geometrical and functionalization effects,” Nanoscale 8, 15902–15910 (2016).
* Zheng _et al._ [2017] Z. Zheng, N. R. Tummala, Y.-T. Fu, V. Coropceanu, and J.-L. Brédas, “Charge-transfer states in organic solar cells: Understanding the impact of polarization, delocalization, and disorder,” ACS Appl. Mater. Interfaces 9, 18095–18102 (2017).
* Chen, Coropceanu, and Brédas [2018] X.-K. Chen, V. Coropceanu, and J.-L. Brédas, “Assessing the nature of the charge-transfer electronic states in organic solar cells,” Nat. Commun. 9, 5295 (2018).
* Li, Coropceanua, and Brédas [2013] Y. Li, V. Coropceanua, and J.-L. Brédas, “Nonlocal electron-phonon coupling in organic semiconductor crystals: The role of acoustic lattice vibrations,” J. Chem. Phys. 138, 204713 (2013).
* Bakulin _et al._ [2012] A. A. Bakulin, A. Rao, V. G. Pavelyev, P. H. M. van Loosdrecht, M. S. Pshenichnikov, D. Niedzialek, J. Cornil, D. Beljonne, and R. H. Friend, “The role of driving energy and delocalized states for charge separation in organic semiconductors,” Science 335, 1340–1344 (2012).
* Gélinas _et al._ [2014] S. Gélinas, A. Rao, A. Kumar, S. L. Smith, A. W. Chin, J. Clark, T. S. van der Poll, G. C. Bazan, and R. H. Friend, “Ultrafast long-range charge separation in organic semiconductor photovoltaic diodes,” Science 343, 512–516 (2014).
* Collini and Scholes [2009a] E. Collini and G. D. Scholes, “Coherent intrachain energy migration in a conjugated polymer at room temperature,” Science 323, 369–373 (2009a).
* Collini and Scholes [2009b] E. Collini and G. D. Scholes, “Electronic and vibrational coherences in resonance energy transfer along meh-ppv chains at room temperature,” J. Phys. Chem. A 113, 4223–4241 (2009b).
* Sio and Lienau [2017] A. D. Sio and C. Lienau, “Vibronic coupling in organic semiconductors for photovoltaics,” Phys. Chem. Chem. Phys. 19, 18813–18830 (2017).
* Grancini _et al._ [2013] G. Grancini, M. Maiuri, D. Frazzil, A. Petrozzal, H. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, “Hot exciton dissociation in polymer solar cells,” Nat. Mater. 12, 29–33 (2013).
* Tully [2012] J. C. Tully, “Perspective: Nonadiabatic dynamics theory,” J. Chem. Phys. 137, 22A301 (2012).
* Hannewald _et al._ [2004] K. Hannewald, V. M. Stojanović, J. M. T. Schellekens, P. A. Bobbert, G. Kresse, and J. Hafner, “Theory of polaron bandwidth narrowing in organic molecular crystals,” Phys. Rev. B 69, 075211 (2004).
* Holstein [1959] T. Holstein, “Studies of polaron motion: Part i. the molecular-crystal model,” Ann. Phys. 8, 325–342 (1959).
* Peierls [1930] R. E. Peierls, “Zur theorie der elektrischen und thermischen leitfähigkeit von metallen.” Ann. Phys. 396, 121–148 (1930).
* Duan _et al._ [2019] H.-G. Duan, P. Nalbach, R. J. D. Miller, and M. Thorwart, “Ultrafast energy transfer in excitonically coupled molecules induced by a nonlocal peierls phonon,” J. Phys. Chem. Lett. 10, 1206–1211 (2019).
* Florencio Jr. and Goodman [1986] J. Florencio Jr. and B. Goodman, “Mean-field approximation for the molecular polaron with q2 coupling,” Phys. Rev. B 34, 3645–3648 (1986).
* Zusman [1980] L. D. Zusman, “Outer-sphere electron transfer in polar solvents,” Chem. Phys. 49, 295–304 (1980).
* Tachiya [1989] M. Tachiya, “Relation between the electron-transfer rate and the free energy change of reaction,” J. Phys. Chem. 93, 7050–7052 (1989).
* Wolynes [1987] P. Wolynes, “Dissipation, tunneling, and adiabaticity criteria for curve crossing problems in the condensed phase,” J. Chem. Phys. 86, 1957–1966 (1987).
* Garg, Onuchic, and Ambegaokar [1985] A. Garg, J. N. Onuchic, and V. Ambegaokar, “Effect of friction on electron transfer in biomolecules,” J. Chem. Phys. 83, 4491–4503 (1985).
* Tanimura [2020] Y. Tanimura, “Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (heom),” J. Chem. Phys. 153, 020901 (2020).
* Ishizaki and Tanimura [2005] A. Ishizaki and Y. Tanimura, “Quantum dynamics of system strongly coupled to low-temperature colored noise bath: Reduced hierarchy equations approach,” J. Phys. Soc. Jpn. 74, 3131–3134 (2005).
* Tanimura and Mukamel [1994] Y. Tanimura and S. Mukamel, “Optical stark spectroscopy of a brownian oscillator in intense fields,” J. Phys. Soc. Jpn. 63, 66–77 (1994).
* Tanaka and Tanimura [2009] M. Tanaka and Y. Tanimura, “Quantum dissipative dynamics of electron transfer reaction system: Nonperturbative hierarchy equations approach,” J. Phys. Soc. Jpn. 78, 073802 (2009).
* Ishizaki and Fleming [2009] A. Ishizaki and G. R. Fleming, “Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature,” Proc. Natl. Acad. Sci. U.S.A. 106, 17255 (2009).
* Kreisbeck and Kramer [2012] C. Kreisbeck and T. Kramer, “Long-lived electronic coherence in dissipative exciton dynamics of light-harvesting complexes,” J. Phys. Chem. Lett. 3, 2828 (2012).
* Fujihashi, Fleming, and Ishizaki [2015] Y. Fujihashi, G. R. Fleming, and A. Ishizaki, “Impact of environmentally induced fluctuations on quantum mechanically mixed electronic and vibrational pigment states in photosynthetic energy transfer and 2d electronic spectra,” J. Chem. Phys. 142, 212403 (2015).
* Sakamoto and Tanimura [2017] S. Sakamoto and Y. Tanimura, “Long-lived electronic coherence in dissipative exciton dynamics of light-harvesting complexes,” J. Phys. Chem. Lett. 8, 5390 (2017).
* Dijkstra and Tanimura [2010] A. G. Dijkstra and Y. Tanimura, “Correlated fluctuations in the exciton dynamics and spectroscopy of dna,” New J. Phys. 12, 055005 (2010).
* Tanimura [1990] Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990).
* Chen, Zhao, and Tanimura [2015] L. Chen, Y. Zhao, and Y. Tanimura, “Dynamics of a one-dimensional holstein polaron with the hierarchical equations of motion approach,” J. Phys. Chem. Lett. 6, 3110–3115 (2015).
* Dunn, Tempelaar, and Reichman [2019] I. S. Dunn, R. Tempelaar, and D. R. Reichman, “Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches,” J. Chem. Phys. 150, 184109 (2019).
* Zhugayevych and Tretiak [2015] A. Zhugayevych and S. Tretiak, “Theoretical description of structural and electronic properties of organic photovoltaic materials,” Rev. Phys. Chem. 66, 305–330 (2015).
* Tanimura [2006] Y. Tanimura, “Stochastic liouville, langevin, fokker–planck, and master equation approaches to quantum dissipative systems,” J. Phys. Soc. Jpn. 75, 082001 (2006).
* Xie, Santana-Bonilla, and Troisi [2018] X. Xie, A. Santana-Bonilla, and A. Troisi, “Nonlocal electron–phonon coupling in prototypical molecular semiconductors from first principles,” J. Chem. Theory Comput. 14, 3752–3762 (2018).
* Hu _et al._ [2020] W. Hu, K. Sun, Q. Xu, L. Chen, and Y. Zhao, “Ultrafast dynamics in rubrene and its spectroscopic manifestation,” J. Chem. Phys. 153, 174105 (2020).
* Sun _et al._ [2020] K. Sun, Q. Xu, L. Chen, M. F. Gelin, and Y. Zhao, “Temperature effects on singlet fission dynamics mediated by a conical intersection,” J. Chem. Phys. 153, 194106 (2020).
* Fuemmeler _et al._ [2016] E. G. Fuemmeler, S. N. Sanders, A. B. Pun, E. Kumarasamy, T. Zeng, K. Miyata, M. L. Steigerwald, X.-Y. Zhu, M. Y. Sfeir, L. M. Campos, and N. Ananth, “A direct mechanism of ultrafast intramolecular singlet fission in pentacene dimers,” ACS Cent. Sci. 2, 316–324 (2016).
* Chen _et al._ [2011] D. Chen, J. Ye, H. Zhang, and Y. Zhao, “On the munn-silbey approach to polaron transport with off-diagonal coupling and temperature-dependent canonical transformations,” J. Phys. Chem. B 115, 5312–5321 (2011).
* Zhao, Brown, and Lindenberg [1994] Y. Zhao, D. W. Brown, and K. Lindenberg, “On the munn–silbey approach to nonlocal exciton–phonon coupling,” J. Chem. Phys. 100, 2335–2345 (1994).
* Zhou _et al._ [2016] N. Zhou, L. Chen, Z. Huang, K. Sun, Y. Tanimura, and Y. Zhao, “Fast, accurate simulation of polaron dynamics and multidimensional spectroscopy by multiple davydov trial states,” J. Phys. Chem. A 120, 1562–1576 (2016).
* Feynman and Vernon [1963] R. P. Feynman and F. L. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Ann. Phys. 271, 547–607 (1963).
* Leggett _et al._ [1987] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Rev. Mod. Phys. 59, 1–85 (1987).
* Tanimura [2014] Y. Tanimura, “Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities,” J. Chem. Phys. 141, 044114 (2014).
* Tanimura [2015] Y. Tanimura, “Real-time and imaginary-time quantum hierarchal fokker-planck equations,” J. Chem. Phys. 142, 144110 (2015).
* Tanimura and Kubo [1989] Y. Tanimura and R. Kubo, “Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath,” J. Phys. Soc. Jpn. 58, 101–114 (1989).
* Tanimura and Wolynes [1991] Y. Tanimura and P. G. Wolynes, “Quantum and classical fokker-planck equations for a gaussian-markovian noise bath,” Phys. Rev. A 43, 4131–4142 (1991).
* Tanaka and Tanimura [2010] M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
* Tanimura [2012] Y. Tanimura, “Reduced hierarchy equations of motion approach with drude plus brownian spectral distribution: Probing electron transfer processes by means of two-dimensional correlation spectroscopy,” J. Chem. Phys. 137, 22A550 (2012).
|
††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: These authors contributed equally
# Information Scrambling in Computationally Complex Quantum Circuits
Xiao Mi Google Research Pedram Roushan Google Research Chris Quintana
Google Research Salvatore Mandrà QuAIL, NASA Ames Research Center, Moffett
Field, California 94035, USA KBR, Inc., 601 Jefferson St., Houston, TX 77002,
USA Jeffrey Marshall QuAIL, NASA Ames Research Center, Moffett Field,
California 94035, USA USRA Research Institute for Advanced Computer Science,
Mountain View, California 94043, USA Charles Neill Google Research Frank
Arute Google Research Kunal Arya Google Research Juan Atalaya Google
Research Ryan Babbush Google Research Joseph C. Bardin Google Research
Department of Electrical and Computer Engineering, University of
Massachusetts, Amherst, MA Rami Barends Google Research Andreas Bengtsson
Google Research Sergio Boixo Google Research Alexandre Bourassa Google
Research Pritzker School of Molecular Engineering, University of Chicago,
Chicago, IL Michael Broughton Google Research Bob B. Buckley Google
Research David A. Buell Google Research Brian Burkett Google Research
Nicholas Bushnell Google Research Zijun Chen Google Research Benjamin
Chiaro Google Research Roberto Collins Google Research William Courtney
Google Research Sean Demura Google Research Alan R. Derk Google Research
Andrew Dunsworth Google Research Daniel Eppens Google Research Catherine
Erickson Google Research Edward Farhi Google Research Austin G. Fowler
Google Research Brooks Foxen Google Research Craig Gidney Google Research
Marissa Giustina Google Research Jonathan A. Gross Google Research Matthew
P. Harrigan Google Research Sean D. Harrington Google Research Jeremy
Hilton Google Research Alan Ho Google Research Sabrina Hong Google
Research Trent Huang Google Research William J. Huggins Google Research
L. B. Ioffe Google Research Sergei V. Isakov Google Research Evan Jeffrey
Google Research Zhang Jiang Google Research Cody Jones Google Research
Dvir Kafri Google Research Julian Kelly Google Research Seon Kim Google
Research Alexei Kitaev Google Research California Institute of Technology,
Pasadena, CA 91125 Paul V. Klimov Google Research Alexander N. Korotkov
Google Research Department of Electrical and Computer Engineering, University
of California, Riverside, CA Fedor Kostritsa Google Research David Landhuis
Google Research Pavel Laptev Google Research Erik Lucero Google Research
Orion Martin Google Research Jarrod R. McClean Google Research Trevor
McCourt Google Research Matt McEwen Google Research Department of Physics,
University of California, Santa Barbara, CA Anthony Megrant Google Research
Kevin C. Miao Google Research Masoud Mohseni Google Research Wojciech
Mruczkiewicz Google Research Josh Mutus Google Research Ofer Naaman
Google Research Matthew Neeley Google Research Michael Newman Google
Research Murphy Yuezhen Niu Google Research Thomas E. O’Brien Google
Research Alex Opremcak Google Research Eric Ostby Google Research Balint
Pato Google Research Andre Petukhov Google Research Nicholas Redd Google
Research Nicholas C. Rubin Google Research Daniel Sank Google Research
Kevin J. Satzinger Google Research Vladimir Shvarts Google Research Doug
Strain Google Research Marco Szalay Google Research Matthew D. Trevithick
Google Research Benjamin Villalonga Google Research Theodore White Google
Research Z. Jamie Yao Google Research Ping Yeh Google Research Adam
Zalcman Google Research Hartmut Neven Google Research Igor Aleiner Google
Research Kostyantyn Kechedzhi<EMAIL_ADDRESS>Google Research Vadim
Smelyanskiy<EMAIL_ADDRESS>Google Research Yu Chen<EMAIL_ADDRESS>Google Research
Interaction in quantum systems can spread initially localized quantum
information into the many degrees of freedom of the entire system.
Understanding this process, known as quantum scrambling, is the key to
resolving various conundrums in physics. Here, by measuring the time-dependent
evolution and fluctuation of out-of-time-order correlators, we experimentally
investigate the dynamics of quantum scrambling on a 53-qubit quantum
processor. We engineer quantum circuits that distinguish the two mechanisms
associated with quantum scrambling, operator spreading and operator
entanglement, and experimentally observe their respective signatures. We show
that while operator spreading is captured by an efficient classical model,
operator entanglement requires exponentially scaled computational resources to
simulate. These results open the path to studying complex and practically
relevant physical observables with near-term quantum processors.
The inception of quantum computers was motivated by their ability to simulate
dynamical processes that are challenging for classical computation Feynman
(1982). However, while the size of the Hilbert space scales exponentially with
the number of qubits, quantum dynamics can be simulated in polynomial times
when entanglement is insufficient Valiant (2002); Terhal and DiVincenzo
(2002); Vidal (2004) or when they belong to special classes such as the
Clifford group Gottesman (1996); Aaronson and Gottesman (2004); Bravyi and
Gosset (2016). A physical process that fully leverages the computational power
of quantum processors is quantum scrambling, which describes how interaction
in a quantum system disperses local information into its many degrees of
freedom Hayden and Preskill (2007); Sekino and Susskind (2008); Lashkari _et
al._ (2013); Aleiner _et al._ (2016); Zhuang _et al._ (2019). Quantum
scrambling is the underlying mechanism for the thermalization of isolated
quantum systems Deutsch (1991); Rigol _et al._ (2008) and as such, accurately
modeling its dynamics is the key to resolving a number of physical phenomena,
such as the fast-scrambling conjecture for black holes Sekino and Susskind
(2008); Lashkari _et al._ (2013), non-Fermi liquid behaviors Blake _et al._
(2017); Ben-Zion and McGreevy (2018) and many-body localization Basko _et
al._ (2006). Understanding scrambling also provides a basis for designing
algorithms in quantum benchmarking or machine learning that would benefit from
efficient exploration of the Hilbert space McClean _et al._ (2018); Knill
_et al._ (2008); Haferkamp _et al._ (2020).
A precise formulation of quantum scrambling is found in the Heisenberg
picture, where quantum operators evolve and quantum states are stationary.
Analogous to classical chaos, scrambling manifests itself as a “butterfly
effect”, wherein a local perturbation is rapidly amplified over time Roberts
_et al._ (2015); Aleiner _et al._ (2016). More specifically, the perturbation
is realized as an initially local operator (the “butterfly operator”)
$\hat{O}$, typically a Pauli operator acting on one of the qubits (the
“butterfly qubit”). When the quantum system undergoes a dynamical process
$\hat{U}$, the butterfly operator $\hat{O}$ acquires a time-dependence and
becomes $\hat{O}\left(t\right)=\hat{U}^{\dagger}\hat{O}\hat{U}$, with
$\hat{U}^{\dagger}$ being the inverse of $\hat{U}$. The resulting
$\hat{O}\left(t\right)$ can be expanded as
$\hat{O}\left(t\right)=\sum_{i=1}^{n_{\text{p}}}w_{i}{\hat{B}_{i}}$, where
$\hat{B}_{i}=\hat{\rho}_{1}^{(i)}\otimes\hat{\rho}_{2}^{(i)}\otimes...$ are
basis operators consisting of single-qubit operators $\hat{\rho}_{j}^{(i)}$
acting on different qubits and $w_{i}$ are their coefficients.
Quantum scrambling is enabled by two different mechanisms: operator spreading
and operator entanglement Aleiner _et al._ (2016); Roberts and Yoshida
(2017); Nahum _et al._ (2018); von Keyserlingk _et al._ (2018); Rakovszky
_et al._ (2018); Khemani _et al._ (2018). Operator spreading refers to the
transformation of basis operators such that on average, each $\hat{B}_{i}$
involves a higher number of non-identity single-qubit operators. Operator
entanglement, on the other hand, refers to the increase in $n_{\text{p}}$,
i.e. the minimum number of terms required to expand $\hat{O}\left(t\right)$.
Independent characterizations of these two mechanisms are essential for a
complete understanding of the nature of quantum scrambling. Quantifying the
degree of operator entanglement also holds the key to assessing the classical
simulation complexity of quantum observables Hartmann _et al._ (2009).
However, operator spreading and operator entanglement are generally
intertwined and indistinguishable in past experimental studies of quantum
scrambling Li _et al._ (2017); Gärttner _et al._ (2017); Landsman _et al._
(2019); Blok _et al._ (2020); Joshi _et al._ (2020).
Figure 1: OTOC measurement protocol. (A) Measurement scheme for the OTOC of a
quantum circuit $\hat{U}$. A quantum system (qubits $Q_{1}$ through $Q_{N}$)
is first initialized in a superposition state
$\prod_{j=1}^{j=N}|+X\rangle_{j}$, where
$|+X\rangle_{j}=\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)_{j}$ and
$|0\rangle$ ($|1\rangle$) is the ground (excited) state of individual qubits.
$\hat{U}$ and its inverse $\hat{U}^{\dagger}$ are then applied, with a
butterfly operator (realized as an $X$ gate on qubit $Q_{\text{b}}$) inserted
in-between. An ancilla qubit $Q_{\text{a}}$ is initialized along the y-axis of
the Bloch sphere and entangled with $Q_{1}$ by a pair of CZ gates. The y-axis
projection of $Q_{\text{a}}$, $\braket{\hat{\sigma}_{\text{y}}}$, is measured
at the end. (B) The structure of $\hat{U}$ consists of $K$ cycles: each cycle
includes one layer of single-qubit gates (randomly chosen from $\sqrt{X^{\pm
1}}$, $\sqrt{Y^{\pm 1}}$, $\sqrt{W^{\pm 1}}$ and $\sqrt{V^{\pm 1}}$) and one
layer of two-qubit entangling gates (EG). Here $W=\frac{X+Y}{\sqrt{2}}$ and
$V=\frac{X-Y}{\sqrt{2}}$. (C) Left panel: The filled circles represent
$\braket{\hat{\sigma}_{\text{y}}}$ measured with $Q_{\text{b}}$ successively
chosen from $Q_{2}$ through $Q_{N}$. The open grey circles are a set of
normalization values, corresponding to $\braket{\hat{\sigma}_{\text{y}}}$
without applying the butterfly operator. The data are plotted over different
numbers of cycles in $\hat{U}$ and averaged over 60 random circuit instances.
Right panel: Experimental average OTOCs $\overline{C}$ for different
$Q_{\text{b}}$, obtained from dividing the corresponding
$\braket{\hat{\sigma}_{\text{y}}}$ by the normalization values.
In this Article, we perform a comprehensive characterization of quantum
scrambling in a two-dimensional (2D) quantum system of 53 superconducting
qubits. Signatures of operator spreading and operator entanglement are clearly
distinguished in our experiment. These results are enabled by quantum circuit
designs that independently tune the degree of each scrambling mechanism, as
well as extensive error-mitigation techniques that allow us to faithfully
recover coherent experimental signals in the presence of substantial noise.
Lastly, we find that while operator spreading can be efficiently predicted by
a classical stochastic process, simulating the experimental signature of
operator entanglement is significantly more costly with a computational
resource that scales exponentially with the size of the quantum circuit.
Figure 2: OTOC propagation and speed of operator spreading. (A) Spatial
profiles of average OTOCs, $\overline{C}$, measured on the full 53-qubit
processor. The ancilla qubit $Q_{\text{a}}$ and the the measurement qubit
$Q_{1}$ are indicated by the red arrows. The colors of other filled circles
represent $\overline{C}$ with different choices of the butterfly qubit
$Q_{\text{b}}$. The two-qubit gates are iSWAP and applied between all nearest-
neighbor qubits, with the same order as done in Ref. Arute _et al._ (2019).
The dashed lines delineate the light-cone of $Q_{1}$. The data are averaged
over 38 circuit instances. (B) Similar to (A) but with $\sqrt{\text{iSWAP}}$
as the two-qubit gates. Here $\overline{C}$ is averaged over 24 circuit
instances. (C) Cycle-dependent $\overline{C}$ for 4 different choices of
$Q_{\text{b}}$. The top (bottom) panel shows data with iSWAP
($\sqrt{\text{iSWAP}}$) being the two-qubit gates. The colors of the data
points indicate the locations of $Q_{\text{b}}$ (inset to bottom panel).
Dashed lines show theoretical predictions based on a classical population
dynamics model.
Our experiment approach is based on evaluating the correlator between
$\hat{O}\left(t\right)$ and a “measurement operator”, $\hat{M}$, which is
another Pauli operator on a different qubit (the “measurement qubit”):
$C(t)=\langle\hat{O}^{\dagger}(t)\,\hat{M}^{\dagger}\,\hat{O}(t)\,\hat{M}\rangle.$
(1)
Here $\langle...\rangle$ denotes the expectation value over a particular
quantum state. $C(t)$ is commonly known as the out-of-time-order correlator
(OTOC) and related to the commutator $[\hat{O}(t),\hat{M}]$ by
$C(t)=1-\frac{1}{2}\langle|[\hat{O}(t),\hat{M}]|^{2}\rangle$ Roberts _et al._
(2015); Aleiner _et al._ (2016); Hosur _et al._ (2016); Swingle _et al._
(2016); Yoshida and Yao (2019); Vermersch _et al._ (2019). Quantum scrambling
is characterized by measuring $C$ over a collection of quantum circuits with
microscopic differences, e.g. phases of individual gates. Operator spreading
is then reflected in the average OTOC value, $\overline{C}$, which decays from
1 when $\hat{O}(t)$ and $\hat{M}$ overlap and no longer commute Landsman _et
al._ (2019); Blok _et al._ (2020). In the fully scrambled limit where the
commutation between $\hat{O}(t)$ and $\hat{M}$ is completely randomized,
$\overline{C}$ becomes $\sim$0\. If operator entanglement is also present
(i.e. $n_{\text{p}}\gg 1$), $C$ approaches 0 for all circuits and their
fluctuation $\delta_{\text{C}}$ vanishes as well. This is because each $C$
includes contributions from many basis operators $\hat{B}_{i}$ with different
phases. It is therefore sufficient to identify operator spreading through the
decay of $\overline{C}$, whereas any insight into operator entanglement
necessitates an estimate of $\delta_{\text{C}}$.
The measurement protocol for OTOC is described in Fig. 1A and consists of a
quantum circuit $\hat{U}$ and its inverse $\hat{U}^{\dagger}$, with a
butterfly operator $\hat{O}$ (Pauli operator
$\hat{\sigma}_{\text{x}}^{(\text{b})}$ on $Q_{\text{b}}$) inserted in-between.
An ancilla qubit $Q_{\text{a}}$, connected to the measurement qubit $Q_{1}$
via a controlled-phase (CZ) gate, measures $C$ between $\hat{O}$ and $\hat{M}$
(Pauli operator $\hat{\sigma}_{\text{z}}^{(1)}$ on $Q_{1}$) through its
$\braket{\hat{\sigma}_{\text{y}}}$ Swingle _et al._ (2016); foo . For this
work, we employ quantum circuits composed of random single-qubit gates and
fixed two-qubit gates (Fig. 1B) due to the wide range of quantum scrambling
that may be achieved with limited circuit depths Boixo _et al._ (2018); Arute
_et al._ (2019). The OTOC measurement protocol is first implemented on a one-
dimensional (1D) chain of 21 qubits (Fig. 1C). We use the qubit at one end of
the chain as $Q_{\text{a}}$ and successively choose qubits $Q_{2}$ through
$Q_{20}$ as $Q_{\text{b}}$. A two-qubit gate (iSWAP) is applied to each pair
$(Q_{j},Q_{j+1})$, where $j=0,2,4...$ in odd circuit cycles and $j=1,3,5...$
in even circuit cycles.
Figure 3: OTOC fluctuation and signature of operator entanglement. (A)
Transformation of a butterfly operator into products of Pauli operators (Pauli
strings) by quantum gates. In this example, an initial butterfly operator
$\hat{I}^{(1)}\hat{\sigma}_{\text{x}}^{(2)}\hat{I}^{(3)}\hat{I}^{(4)}...$ is
mapped into
$\hat{I}^{(1)}\hat{\sigma}_{\text{z}}^{(2)}\hat{\sigma}_{\text{y}}^{(3)}\hat{I}^{(4)}...$
by an iSWAP gate, then into
$\hat{I}^{(1)}\hat{\sigma}_{\text{y}}^{(2)}\hat{\sigma}_{\text{y}}^{(3)}\hat{I}^{(4)}...$
by an $\sqrt{X}$ gate and so on. In the last two steps, the butterfly operator
evolves into a superposition of multiple Pauli strings (coefficients not
shown). (B) OTOCs of individual random circuit instances, $C$, measured with
the number of non-Clifford gates in $\hat{U}$, $N_{\text{D}}$, fixed at
different values. Dashed lines are numerical simulation results. The inset
shows locations of $Q_{\text{a}}$ (open black circle), $Q_{\text{1}}$ (filled
black circle), and $Q_{\text{b}}$ (filled blue circle) as well as the number
of circuit cycles with which the data are taken. (C) The mean $\overline{C}$
(upper panel) and RMS values $\delta_{\text{C}}$ (lower panel) of $C$ for
different $N_{\text{D}}$. Dashed lines are computed from the numerically
simulated values in (B). Inset: Average numbers of Pauli strings in the time-
evolved butterfly operator $\hat{O}\left(t\right)$, $n_{\text{p}}$, for
different $N_{\text{D}}$. The scaling behavior is $n_{\text{p}}\approx
2^{0.79N_{\text{D}}}$ sup .
In the left panel of Fig. 1C, experimental values of
$\braket{\hat{\sigma}_{\text{y}}}$ are shown for different numbers of cycles
in $\hat{U}$. Here we average $\braket{\hat{\sigma}_{\text{y}}}$ over 60
circuit instances to first focus on operator spreading. It is seen that
$\braket{\hat{\sigma}_{\text{y}}}$ decays as early as cycle 1, irrespective of
the location of $Q_{\text{b}}$. This observation contradicts the 1D geometry
which requires $h+1$ circuit cycles before the time-evolved butterfly operator
$\hat{O}\left(t\right)$ overlaps with $\hat{M}$, with $h$ being the number of
qubits between $Q_{\text{b}}$ and $Q_{1}$. The signals are therefore
complicated by errors in the quantum circuits, such as mismatch between
$\hat{U}$ and $\hat{U}^{\dagger}$ or qubit decoherence Gärttner _et al._
(2017); Landsman _et al._ (2019). These deleterious effects are mitigated by
additionally measuring $\braket{\hat{\sigma}_{\text{y}}}$ without applying the
butterfly operator Swingle and Yunger Halpern (2018). These data, referred to
as normalization values, are also shown in the left panel of Fig. 1C and
approximately equal to the total fidelities of $\hat{U}$ and
$\hat{U}^{\dagger}$ Gärttner _et al._ (2017). We then divide
$\braket{\hat{\sigma}_{\text{y}}}$ for each $Q_{\text{b}}$ by the
normalization values to recover the effects of scrambling sup .
The normalized data, equal to the average OTOCs $\overline{C}$ after error-
mitigation, are shown in the right panel of Fig. 1C and exhibit features
consistent with operator spreading: For each location of $Q_{\text{b}}$,
$\overline{C}$ retains values near 1 before sufficient circuit cycles have
occurred to allow an overlap between $\hat{O}\left(t\right)$ and $\hat{M}$.
Beyond these cycles, $\overline{C}$ converges to 0, indicating that
$\hat{O}\left(t\right)$ and $\hat{M}$ have overlapped and no longer commute.
In addition, we observe that the time-evolution of $\overline{C}$ for each
$Q_{\text{b}}$ resembles a ballistically propagating wave. The front of each
wave coincides with the edge of the “light-cone” associated with $Q_{1}$, i.e.
the set of qubits that have been entangled with $Q_{1}$. This profile is
attributed to the iSWAP gates used in these circuits, which spread single-
qubit operators at the same rate as their light-cones expand Claeys and
Lamacraft (2020). For generic quantum circuits, the spreading velocity (a.k.a.
the butterfly velocity) is typically slower. Using the full 2D system, we next
demonstrate how the evolution of $\overline{C}$ may be used to diagnose the
butterfly velocity of operator spreading.
In Fig. 2A, the spatial distribution of $\overline{C}$ is shown for five
different numbers of cycles in $\hat{U}$, with iSWAP still being the two-qubit
gate. It is seen that the number of qubits with $\overline{C}<1$ rapidly
increases with the number of cycles, consistent with the spatial spread of the
time-evolved butterfly operator. Moreover, for each circuit cycle, the values
of $\overline{C}$ abruptly change across the edge of the light-cone associated
with $Q_{1}$ (dashed lines in Fig. 2A). In contrast, the spatiotemporal
evolution of $\overline{C}$ shown in Fig. 2B is significantly different. Here
the iSWAP gates are replaced with $\sqrt{\text{iSWAP}}$ gates and the decay of
$\overline{C}$ is slower. Qubits far from $Q_{1}$ still retain average OTOC
values closer to 1 even after 22 cycles. The sharp, step-like spatial
transition seen with iSWAP is also absent for $\sqrt{\text{iSWAP}}$. Instead,
$\overline{C}$ changes in a gradual fashion as $Q_{\text{b}}$ moves further
away from $Q_{1}$.
The different OTOC behaviors can alternatively be seen in the full temporal
evolution of four specific qubits (Fig. 2C). For iSWAP, the shape of the OTOC
wavefront remains sharp and relatively insensitive to the location of
$Q_{\text{b}}$, similar to the 1D example in Fig. 1C. On the other hand, the
wavefront propagates more slowly for $\sqrt{\text{iSWAP}}$ and also broadens
as the distance between $Q_{\text{b}}$ and $Q_{1}$ increases. As a result,
more circuit cycles are required before $\overline{C}$ reaches 0 for
$\sqrt{\text{iSWAP}}$. The wavefront behavior seen with $\sqrt{\text{iSWAP}}$
is similar to generic quantum circuits analyzed in past works Nahum _et al._
(2018); von Keyserlingk _et al._ (2018).
The observed features of average OTOCs are quantitatively understood by
mapping operator spreading to a classical Markov process involving population
dynamics sup . In this model, the 2D qubit lattice is populated by fictitious
particles representing two copies of a single-qubit operator. The initial
state of the entire system is a single particle at the site of $Q_{\text{a}}$.
Whenever a two-qubit gate is applied to two neighboring lattice sites, their
particle occupation changes between four possible states: $\lozenge\lozenge$
(both empty), $\lozenge\blacklozenge$ (left empty, right filled),
$\blacklozenge\lozenge$ (right empty, left filled) and
$\blacklozenge\blacklozenge$ (both filled). The transition probabilities are
described by the stochastic matrix:
$\displaystyle\Omega=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1-a-b&a&b\\\
0&a&1-a-b&b\\\ 0&\frac{b}{3}&\frac{b}{3}&1-\frac{2}{3}b\\\
\end{array}\right),$ (6)
where $a=\frac{1}{3}\sin^{4}\theta$,
$b=\frac{1}{3}\left(\frac{1}{2}\sin^{2}2\theta+2\sin^{2}\theta\right)$ with
$\theta$ being the swap angle of the two-qubit gate. The average probability
of finding a particle at the site of $Q_{1}$ is then used to estimate
$\overline{C}$. In this classical picture, the OTOC wavefront corresponds to
the boundary separating the empty region from the region populated by
particles.
The difference in OTOC propagation between iSWAP and $\sqrt{\text{iSWAP}}$ is
then captured by the dependence of $\Omega$ on $\theta$: After each
application of an iSWAP gate ($\theta=\pi/2$), the particle occupation always
changes (with the exception of $\lozenge\lozenge$). In particular, any
previously empty site will be filled and the region populated by particles
always grows. This leads to the observed maximal butterfly velocity. In
contrast, the application of any $\sqrt{\text{iSWAP}}$ gate ($\theta=\pi/4$)
can leave the particle occupation unchanged with a probability $\frac{5}{12}$.
$\overline{C}$ therefore decays more slowly in this case and its broadening is
explained by the fact that the wavefront spreads at a different velocity for
each trial of the Markov process. The predicted values of $\overline{C}$,
plotted as dashed lines in Fig. 2C, agree well with the experimental data and
indicate that the dynamics of operator spreading can be reliably predicted by
classical models. We note that the effect of noise is included when
$\overline{C}$ is calculated for $\sqrt{\text{iSWAP}}$ as it is found to
introduce deformation to the observed signals sup .
Figure 4: Classical simulation complexity of quantum scrambling. (A) Top
panels show three different circuit configurations, each having the same
$Q_{\text{a}}$ (black unfilled circle) and $Q_{1}$ (black filled circle) but
different $Q_{\text{b}}$ (colored circle) and number of cycles in $\hat{U}$.
The number of iSWAPs in $\hat{U}$ and $\hat{U}^{\dagger}$ that affect
classical simulation costs, $N_{\text{S}}$, is indicated for each
configuration. Bottom panels show the instance-dependent OTOCs measured for
each configuration. The dashed lines are simulation results using tensor-
contraction methods. $N_{\text{D}}$ is also indicated for each configuration.
(B) Top: OTOC signal size and experimental error as functions of
$N_{\text{S}}$. Bottom: Signal-to-noise ratio (SNR) as a function of
$N_{\text{S}}$. SNR equals the ratio of OTOC signal size to experimental
error. $N_{\text{D}}=64$ for the first three values of $N_{\text{S}}$ and
$N_{\text{D}}=40$ for the last two values. The reason for decreasing
$N_{\text{D}}$ is to increase the signal size such that it can be resolved
with less statistical averaging sup . (C) Estimated time $t_{\text{sim}}$
needed to simulate the OTOC of a single 53-qubit circuit with a variable
number of iSWAPs, $N_{\text{S}}$, on a single CPU core (6 Gflop/s).
Unlike operator spreading, an efficient classical description of operator
spreading is not known to exist. In particular, population dynamics cannot be
used to model the circuit-to-circuit fluctuation of OTOCs sup . Resolving the
growth of operator spreading is also difficult with a quantum processor since
it is often accompanied by increased operator spreading Aleiner _et al._
(2016); Roberts and Yoshida (2017); Nahum _et al._ (2018); von Keyserlingk
_et al._ (2018). We overcome this challenge by gradually adjusting the
composition of $\hat{U}$ and $\hat{U}^{\dagger}$, realizing a group of
circuits with predominantly Clifford gates (iSWAP, $\sqrt{X^{\pm 1}}$ or
$\sqrt{Y^{\pm 1}}$) and a small number of non-Clifford gates ($\sqrt{W^{\pm
1}}$ and $\sqrt{V^{\pm 1}}$). As illustrated by the evolution of a butterfly
operator in Fig. 3A, Clifford gates generate operator spreading by extending
the butterfly operator to other qubits while preserving the total number of
basis operators (which are products of Pauli operators, or Pauli strings, in
these cases). In contrast, non-Clifford gates generate operator entanglement
by transforming one single Pauli string into a superposition of multiple Pauli
strings, maintaining the spatial extent of operator spreading in the process.
These distinctive properties of Clifford and non-Clifford gates therefore
provide us a way to independently tune one scrambling mechanism without
affecting the other. We now focus on operator entanglement and measure the
circuit-to-circuit fluctuation of OTOCs, as shown in Fig. 3B. Here the number
of circuit cycles is fixed at 12 and the number of non-Clifford gates in
$\hat{U}$, $N_{\text{D}}$, is successively changed from 0 to 32. For each
$N_{\text{D}}$, the individual OTOCs $C$ of 130 random circuit instances are
measured using a modified normalized procedure sup . At $N_{\text{D}}=0$ where
the circuits consist of only Clifford gates, we see that $C$ takes discrete
values of 1 or $-1$. This is expected as the time-evolved butterfly operator
$\hat{O}(t)$ is a single Pauli string and therefore either commutes or anti-
commutes with the measurement operator $\hat{M}$. As more non-Clifford gates
are introduced into the circuits, $C$ starts to assume intermediate values
between $\pm 1$ and converges toward 0.
The mean $\overline{C}$ and fluctuation (i.e. root-mean-square value)
$\delta_{\text{C}}$ of $C$ are then computed from experimental data and
plotted against $N_{\text{D}}$ in Fig. 3C. We observe different behaviors for
$\overline{C}$ and $\delta_{\text{C}}$: $\overline{C}$ remains largely
constant and close to 0, confirming that operator spreading remains unaffected
by the increasing number of non-Clifford gates. On the other hand,
$\delta_{\text{C}}$ decays from an initial value of 1 and is almost suppressed
by two orders of magnitude as $N_{\text{D}}$ increases from 0 to 32. Over the
same range of $N_{\text{D}}$, we have numerically calculated the average
numbers of Pauli strings in $\hat{O}(t)$, $n_{\text{p}}$, which are seen to
increase exponentially (inset to Fig. 3C). These results demonstrate that the
decay of OTOC fluctuation allows the growth of operator entanglement to be
experimentally diagnosed.
To determine the accuracy of our measurements, the OTOCs of experimental
circuits are simulated using a Clifford-expansion method and overlaid on the
data in Fig. 3B and Fig. 3C sup . We find good agreement between experiment
and simulation even when $\delta_{\text{C}}$ is as small as 0.03, indicating
the quantum processor’s capability to resolve high degrees of operator
entanglement. This may appear surprising given that other signatures of
quantum entanglement, such as entropy, are highly susceptible to unwanted
interaction with the environment Horodecki _et al._ (2009). Instead, we find
that environmental effects are nearly absent from these data. This robustness
is a result of the effective normalization protocol and a range of other
error-mitigation techniques used in our experiment sup .
Having identified means of characterizing both operator spreading and operator
entanglement, it remains to be asked how the computational complexity of
quantum scrambling, as well as our experimental error, scale with circuit
size. This is addressed by systematically increasing the number of iSWAPs in
the quantum circuits, $N_{\text{S}}$ (here $N_{\text{S}}$ counts only iSWAP
gates that lie within the light-cones of $Q_{\text{a}}$ and $Q_{1}$ sup ). At
the same time, $N_{\text{D}}$ is kept at a large value such that the Clifford-
expansion simulation method used in Fig. 3 is challenging to perform and
tensor-contraction is the most efficient classical simulation method Arute
_et al._ (2019); Huang _et al._ (2020); sup . Figure 4A shows representative
data for three circuit configurations with different values of $N_{\text{S}}$,
along with the corresponding numerical simulation results. As $N_{\text{S}}$
and the computational complexity for tensor-contraction increase, we observe
that the OTOC fluctuation decreases and the agreement between experiment and
simulation also degrades.
To quantify these observations, we define an ideal OTOC signal as the
fluctuation $\delta_{\text{C}}$ computed from the simulated values of $C$. We
also define an experimental error as the RMS deviation between the simulated
and the measured values of $C$. Both quantities are shown as functions of
$N_{\text{S}}$ in the upper panel of Fig. 4B. It is seen that the OTOC signal
generally decreases as $N_{\text{S}}$ increases (see Extended Data in SM for
data at $N_{\text{D}}=24$ sup ). This change in signal size is difficult to
predict theoretically sup and may be related to the fact that the Pauli
strings in $\hat{O}(t)$ become less correlated with each other as the light-
cone associated with the butterfly qubit grows Claeys and Lamacraft (2020). On
the other hand, the experimental error first decreases to a minimum of
$\sim$0.01 for $N_{\text{S}}=232$ before starting to increase. The ratio of
these quantities is the experimental signal-to-noise ratio for OTOC and
plotted in the bottom panel of Fig. 4B. The SNR is seen to monotonically decay
from a value of 3 at $N_{\text{S}}=159$ to 0.55 at $N_{\text{S}}=271$.
At last, we estimate the time needed to simulate the OTOC of one circuit on a
single CPU core using tensor-contraction, $t_{\text{sim}}$ sup . The result is
plotted against $N_{\text{S}}$ in Fig. 4C. We observe an exponential increase
in $t_{\text{sim}}$ as $N_{\text{S}}$ increases, confirming that simulating
the scrambling of complex quantum circuits indeed demands exponentially scaled
classical computational resources. In particular, although simulating the
results in Fig. 4A currently requires $t_{\text{sim}}\approx 100$ hours, this
cost becomes $t_{\text{sim}}>10^{\text{10}}$ hours when $N_{\text{S}}$ reaches
400. Chartering a path to this experimental regime is the focus of our ongoing
research. In the SM, we have provided numerical simulation results showing
that a percentage decrease in the coherent or incoherent errors of iSWAP gates
will lead to at least commensurate levels of reduction for the OTOC errors sup
. Of less certainty is the size of $\delta_{\text{C}}$ in this regime, which
cannot be predicted by any known classical model sup . Nevertheless, this
difficulty also implies that resolving $\delta_{\text{C}}$ alone might reveal
scrambling dynamics beyond the simulation capacity of classical computers.
In conclusion, we characterize quantum scrambling in a 53-qubit system and
demonstrate that entanglement in the space of quantum operators is the key to
computational complexity of quantum observables. This result highlights the
importance of careful classical analysis in the ongoing quest to attain
quantum computational advantage on various problems of interest. On the other
hand, the challenge in predicting OTOC fluctuations even moderately beyond the
experimental regime also indicates that quantum processors of today can
already shed light on certain physical phenomena as well as classical
computers. Another encouraging finding of our work is that the accuracy of
quantum processors can be significantly improved through effective error-
mitigation. For example, an SNR of $\sim$1 for OTOCs is achieved for
$N_{\text{S}}=251$ where the circuit fidelity is merely $3\%$ sup . In the
immediate future, classical simulation of average OTOCs may be used to
efficiently benchmark performance of quantum processors. The experimental
framework established here can also be used to study other quantum dynamics of
interest, such as the integrability of the XY model (see SM for preliminary
data) and many-body localization Wang (2001); Basko _et al._ (2006). As the
fidelity of quantum processors continues to increase, modeling scrambling in
quantum gravity and unconventional quantum phases may become a reality as well
Sekino and Susskind (2008); Lashkari _et al._ (2013); Sekino and Susskind
(2008); Lashkari _et al._ (2013); Blake _et al._ (2017); Ben-Zion and
McGreevy (2018).
Acknowledgements P. R. and X. M. acknowledge fruitful discussions with P.
Zoller, B. Vermersch, A. Elben, and M. Knapp. S. M. and J. M. acknowledge the
support from the NASA Ames Research Center and the support from the NASA
Advanced Division for providing access to the NASA HPC systems, Pleiades and
Merope. S. M. and J. M. also acknowledge the support from the AFRL Information
Directorate under grant number F4HBKC4162G001. J. M. is partially supported by
NAMS Contract No. NNA16BD14C.
## References
* Feynman (1982) Richard P. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
* Valiant (2002) Leslie G. Valiant, “Quantum circuits that can be simulated classically in polynomial time,” SIAM Comput. 31, 1229–1254 (2002).
* Terhal and DiVincenzo (2002) Barbara M. Terhal and David P. DiVincenzo, “Classical simulation of noninteracting-fermion quantum circuits,” Phys. Rev. A 65, 032325 (2002).
* Vidal (2004) Guifré Vidal, “Efficient simulation of one-dimensional quantum many-body systems,” Phys. Rev. Lett. 93, 040502 (2004).
* Gottesman (1996) Daniel Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys. Rev. A 54, 1862–1868 (1996).
* Aaronson and Gottesman (2004) Scott Aaronson and Daniel Gottesman, “Improved simulation of stabilizer circuits,” Phys. Rev. A 70, 052328 (2004).
* Bravyi and Gosset (2016) Sergey Bravyi and David Gosset, “Improved classical simulation of quantum circuits dominated by clifford gates,” Phys. Rev. Lett. 116, 250501 (2016).
* Hayden and Preskill (2007) Patrick Hayden and John Preskill, “Black holes as mirrors: quantum information in random subsystems,” J. High Energy Phys. 2007, 120 (2007).
* Sekino and Susskind (2008) Yasuhiro Sekino and L. Susskind, “Fast scramblers,” J. High Energy Phys. 2008, 065 (2008).
* Lashkari _et al._ (2013) Nima Lashkari, Douglas Stanford, Matthew Hastings, Tobias Osborne, and Patrick Hayden, “Towards the fast scrambling conjecture,” J. High Energy Phys. 2013, 22 (2013).
* Aleiner _et al._ (2016) Igor L. Aleiner, Lara Faoro, and Lev B. Ioffe, “Microscopic model of quantum butterfly effect: Out-of-time-order correlators and traveling combustion waves,” Ann. Phys. 375, 378 – 406 (2016).
* Zhuang _et al._ (2019) Quntao Zhuang, Thomas Schuster, Beni Yoshida, and Norman Y. Yao, “Scrambling and complexity in phase space,” Phys. Rev. A 99, 062334 (2019).
* Deutsch (1991) J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A 43, 2046–2049 (1991).
* Rigol _et al._ (2008) Marcos Rigol, Vanja Dunjko, and Maxim Olshanii, “Thermalization and its mechanism for generic isolated quantum systems,” Nature 452, 854–858 (2008).
* Blake _et al._ (2017) Mike Blake, Richard A. Davison, and Subir Sachdev, “Thermal diffusivity and chaos in metals without quasiparticles,” Phys. Rev. D 96, 106008 (2017).
* Ben-Zion and McGreevy (2018) Daniel Ben-Zion and John McGreevy, “Strange metal from local quantum chaos,” Phys. Rev. B 97, 155117 (2018).
* Basko _et al._ (2006) D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states,” Ann. Phys. 321, 1126 – 1205 (2006).
* McClean _et al._ (2018) Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, “Barren plateaus in quantum neural network training landscapes,” Nat. Commun. 9, 4812 (2018).
* Knill _et al._ (2008) E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Randomized benchmarking of quantum gates,” Phys. Rev. A 77, 012307 (2008).
* Haferkamp _et al._ (2020) Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross, and Ingo Roth, “Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates,” https://arxiv.org/abs/2002.09524 (2020).
* Roberts _et al._ (2015) Daniel A. Roberts, Douglas Stanford, and Leonard Susskind, “Localized shocks,” J. High Energy Phys. 2015, 51 (2015).
* Roberts and Yoshida (2017) Daniel A. Roberts and Beni Yoshida, “Chaos and complexity by design,” J. High Energ. Phys. 2017, 121 (2017).
* Nahum _et al._ (2018) Adam Nahum, Sagar Vijay, and Jeongwan Haah, “Operator spreading in random unitary circuits,” Phys. Rev. X 8, 021014 (2018).
* von Keyserlingk _et al._ (2018) C. W. von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi, “Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws,” Phys. Rev. X 8, 021013 (2018).
* Rakovszky _et al._ (2018) Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk, “Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation,” Phys. Rev. X 8, 031058 (2018).
* Khemani _et al._ (2018) Vedika Khemani, Ashvin Vishwanath, and David A. Huse, “Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws,” Phys. Rev. X 8, 031057 (2018).
* Hartmann _et al._ (2009) Michael J. Hartmann, Javier Prior, Stephen R. Clark, and Martin B. Plenio, “Density matrix renormalization group in the heisenberg picture,” Phys. Rev. Lett. 102, 057202 (2009).
* Li _et al._ (2017) Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du, “Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator,” Phys. Rev. X 7, 031011 (2017).
* Gärttner _et al._ (2017) Martin Gärttner, Justin G. Bohnet, Arghavan Safavi-Naini, Michael L. Wall, John J. Bollinger, and Ana Maria Rey, “Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet,” Nat. Phys. 13, 781–786 (2017).
* Landsman _et al._ (2019) K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, “Verified quantum information scrambling,” Nature 567, 61–65 (2019).
* Blok _et al._ (2020) M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi, “Quantum information scrambling in a superconducting qutrit processor,” https://arxiv.org/abs/2003.03307 (2020).
* Joshi _et al._ (2020) Manoj K. Joshi, Andreas Elben, Benoît Vermersch, Tiff Brydges, Christine Maier, Peter Zoller, Rainer Blatt, and Christian F. Roos, “Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions,” Phys. Rev. Lett. 124, 240505 (2020).
* Arute _et al._ (2019) Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505–510 (2019).
* Hosur _et al._ (2016) Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida, “Chaos in quantum channels,” J. High Energy Phys. 2016, 4 (2016).
* Swingle _et al._ (2016) Brian Swingle, Gregory Bentsen, Monika Schleier-Smith, and Patrick Hayden, “Measuring the scrambling of quantum information,” Phys. Rev. A 94, 040302 (2016).
* Yoshida and Yao (2019) Beni Yoshida and Norman Y. Yao, “Disentangling scrambling and decoherence via quantum teleportation,” Phys. Rev. X 9, 011006 (2019).
* Vermersch _et al._ (2019) B. Vermersch, A. Elben, L. M. Sieberer, N. Y. Yao, and P. Zoller, “Probing scrambling using statistical correlations between randomized measurements,” Phys. Rev. X 9, 021061 (2019).
* (38) $C$ in some cases can have a non-negligible imaginary part which is measured through the $\langle\hat{\sigma}_{\text{x}}\rangle$ of $Q_{\text{a}}$. For this work, we only measure the real part of $C$ which is equal to $\langle\hat{\sigma}_{\text{y}}\rangle$ of $Q_{\text{a}}$ .
* Boixo _et al._ (2018) Sergio Boixo, Sergei Isakov, Vadim Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, John Martinis, and Hartmut Neven, “Characterizing quantum supremacy in near-term devices,” Nat. Phys. 14, 595–600 (2018).
* (40) Supplementary materials are available online. .
* Swingle and Yunger Halpern (2018) Brian Swingle and Nicole Yunger Halpern, “Resilience of scrambling measurements,” Phys. Rev. A 97, 062113 (2018).
* Claeys and Lamacraft (2020) Pieter W. Claeys and Austen Lamacraft, “Maximum velocity quantum circuits,” Phys. Rev. Research 2, 033032 (2020).
* Horodecki _et al._ (2009) Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
* Huang _et al._ (2020) Cupjin Huang, Fang Zhang, Michael Newman, Junjie Cai, Xun Gao, Zhengxiong Tian, Junyin Wu, Haihong Xu, Huanjun Yu, Bo Yuan, Mario Szegedy, Yaoyun Shi, and Jianxin Chen, “Classical simulation of quantum supremacy circuits,” https://arxiv.org/abs/2005.06787 (2020).
* Wang (2001) Xiaoguang Wang, “Entanglement in the quantum heisenberg $\mathrm{XY}$ model,” Phys. Rev. A 64, 012313 (2001).
* Calabrese _et al._ (2016) Pasquale Calabrese, Fabian H L Essler, and Giuseppe Mussardo, “Introduction to ‘quantum integrability in out of equilibrium systems’,” J. Stat. Mech. 2016, 064001 (2016).
* Nandkishore and Huse (2015) Rahul Nandkishore and David A. Huse, “Many-body localization and thermalization in quantum statistical mechanics,” Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).
* Foxen _et al._ (2020) B. Foxen, C. Neill, A. Dunsworth, P. Roushan, B. Chiaro, A. Megrant, J. Kelly, Zijun Chen, K. Satzinger, R. Barends, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, S. Boixo, D. Buell, B. Burkett, Yu Chen, R. Collins, E. Farhi, A. Fowler, C. Gidney, M. Giustina, R. Graff, M. Harrigan, T. Huang, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, P. Klimov, A. Korotkov, F. Kostritsa, D. Landhuis, E. Lucero, J. McClean, M. McEwen, X. Mi, M. Mohseni, J. Y. Mutus, O. Naaman, M. Neeley, M. Niu, A. Petukhov, C. Quintana, N. Rubin, D. Sank, V. Smelyanskiy, A. Vainsencher, T. C. White, Z. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis (Google AI Quantum), “Demonstrating a continuous set of two-qubit gates for near-term quantum algorithms,” Phys. Rev. Lett. 125, 120504 (2020).
* Nielsen and Chuang (2010) Michael A. Nielsen and Isaac L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
* McKay _et al._ (2017) David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, and Jay M. Gambetta, “Efficient $z$ gates for quantum computing,” Phys. Rev. A 96, 022330 (2017).
* Ithier _et al._ (2005) G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, “Decoherence in a superconducting quantum bit circuit,” Phys. Rev. B 72, 134519 (2005).
* Jeffrey _et al._ (2014) Evan Jeffrey, Daniel Sank, J. Y. Mutus, T. C. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. J. J. O’Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and John M. Martinis, “Fast accurate state measurement with superconducting qubits,” Phys. Rev. Lett. 112, 190504 (2014).
* Hastings (2008) M. B. Hastings, “Observations outside the light cone: Algorithms for nonequilibrium and thermal states,” Phys. Rev. B 77, 144302 (2008).
* Markov and Shi (2008) Igor L Markov and Yaoyun Shi, “Simulating quantum computation by contracting tensor networks,” SIAM Journal on Computing 38, 963–981 (2008).
* Chen _et al._ (2018) Jianxin Chen, Fang Zhang, Cupjin Huang, Michael Newman, and Yaoyun Shi, “Classical simulation of intermediate-size quantum circuits,” https://arxiv.org/abs/1805.01450 (2018).
* Villalonga _et al._ (2020) Benjamin Villalonga, Dmitry Lyakh, Sergio Boixo, Hartmut Neven, Travis S Humble, Rupak Biswas, Eleanor G Rieffel, Alan Ho, and Salvatore Mandrà, “Establishing the quantum supremacy frontier with a 281 pflop/s simulation,” Quantum Sci. Technol. 5, 034003 (2020).
* Villalonga _et al._ (2019) Benjamin Villalonga, Sergio Boixo, Bron Nelson, Christopher Henze, Eleanor Rieffel, Rupak Biswas, and Salvatore Mandrà, “A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware,” npj Quantum Inf. 5, 1–16 (2019).
* Gray and Kourtis (2020) Johnnie Gray and Stefanos Kourtis, “Hyper-optimized tensor network contraction,” https://arxiv.org/abs/2002.01935 (2020).
* Gottesman (1998) Daniel Gottesman, “The heisenberg representation of quantum computers,” https://arxiv.org/abs/quant-ph/9807006 (1998).
* Bravyi _et al._ (2019) Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard, “Simulation of quantum circuits by low-rank stabilizer decompositions,” Quantum 3, 181 (2019).
* Boixo _et al._ (2017) Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, and Hartmut Neven, “Simulation of low-depth quantum circuits as complex undirected graphical models,” https://arxiv.org/abs/1712.05384 (2017).
* Markov _et al._ (2018) Igor L Markov, Aneeqa Fatima, Sergei V Isakov, and Sergio Boixo, “Quantum supremacy is both closer and farther than it appears,” https://arxiv.org/abs/1807.10749 (2018).
* Markov _et al._ (2019) Igor Markov, Aneeqa Fatima, Sergei Isakov, and Sergio Boixo, “Faster classical sampling from distributions defined by quantum circuits,” APS 2019, K42–010 (2019).
* (64) NASA Ames Research Center, “Merope supercomputer,” https://nas.nasa.gov/hecc/resources/merope.html.
* (65) Martin Meuer, Horst Simon, Jack Dongarra, Erich Strohmaier, and Hans Werner Meuer, “Top500 List,” https://www.top500.org/lists/top500/2020/11/.
* Oak Ridge National Laboratory () (ORNL) Oak Ridge National Laboratory (ORNL), “Summit Resources,” https://www.olcf.ornl.gov/olcf-resources/compute-systems/summit/.
* Gray (2018) Johnnie Gray, “quimb: A python package for quantum information and many-body calculations,” Journal of Open Source Software 3, 819 (2018).
* Yoshida and Kitaev (2017) Beni Yoshida and Alexei Kitaev, “Efficient decoding for the hayden-preskill protocol,” https://arxiv.org/abs/1710.03363 (2017).
Supplemental Materials
###### Contents
1. I Extended Data
1. I.1 Full 53-Qubit Average OTOC Data
2. I.2 OTOCs for Non-Integrable and Integrable Quantum Circuits
3. I.3 Characterization of Experimental Errors
2. II Experimental Techniques
1. II.1 Calibration of iSWAP and $\sqrt{\text{iSWAP}}$ Gates
1. II.1.1 Reducing Conditional-Phase Errors
2. II.1.2 Arbitrary $Z$-Rotations
2. II.2 Gate Error Benchmarking and Cross-Talk Mitigation
3. II.3 Dynamical Decoupling
4. II.4 Elimination of Bias in $\braket{\hat{\sigma}_{\text{y}}}$
5. II.5 Light-cone Filter
6. II.6 Normalization via Reference Clifford Circuits
3. III Large-Scale Simulation of OTOCs of Individual Circuits
1. III.1 Numerical Calculation of the OTOC Value
2. III.2 Branching Method
3. III.3 Tensor Contraction
4. IV Markov population dynamics
1. IV.1 Symmetric single qubit gate set
2. IV.2 Efficient Population Dynamics for the Averaged OTOC
3. IV.3 Sign Problem in the Population Dynamics for OTOC Fluctuations
4. IV.4 Population Dynamics for iSWAP Gate Sets Implemented in the Main Text
1. IV.4.1 Clifford Gate Set
2. IV.4.2 Universal Gate Set
5. V Efficient Population Dynamics for Noisy Circuits
1. V.1 Inversion Error
2. V.2 Generic Error Model
3. V.3 Error Mitigation for Average OTOC
4. V.4 Theory of Error Mitigation for Individual Circuits
6. VI Numerical Simulation of Error-Limiting Mechanisms for OTOC
1. VI.1 Coherent and Incoherent Contributions
2. VI.2 Perturbative Expansion of OTOC Error
## I Extended Data
In this section, we provide data not covered in the main text. These include
the spatial distribution of average OTOCs on the 53-qubit system, average OTOC
behaviors for additional types of quantum circuits and more detailed
investigation of errors in experimental OTOC measurements.
### I.1 Full 53-Qubit Average OTOC Data
Figure S1: Full evolution of average OTOCs for iSWAP random circuits. The
average OTOCs of the 53-qubit system shown for every cycle up to a total of
15. The black unfilled (filled) circle represents the location of the ancilla
(measurement) qubit. The colors of the other filled circles represent the
values of $\overline{C}$ for different locations of the butterfly qubit. The
two-qubit gate used here is iSWAP and the data are averaged over 38 random
circuit instances. Figure S2: Full evolution of average OTOCs for
$\sqrt{\text{iSWAP}}$ random circuits. The average OTOCs of the 53-qubit
system shown for every cycle up to a total of 24. The black unfilled (filled)
circle represents the location of the ancilla (measurement) qubit. The colors
of the other filled circles represent the values of $\overline{C}$ for
different locations of the butterfly qubit. The two-qubit gate used here is
$\sqrt{\text{iSWAP}}$ and the data are averaged over 24 random circuit
instances.
Figure S1 shows experimentally measured average OTOCs, $\overline{C}$, for 51
different possible butterfly qubit locations. The data are also shown for
every circuit cycle from 1 through 15. Here iSWAP is used as the two-qubit
gate. The sharp nature of the wavefront propagation is readily visible, where
a large reduction from $\overline{C}=1$ is observed as soon as the lightcone
of the measurement qubit (black filled circle) reaches a given qubit. In
comparison, similar data are shown up to a total of 24 circuit cycles for
random circuits containing $\sqrt{\text{iSWAP}}$ gates (Fig. S2). The dynamics
is seen to be slower than iSWAP, as mentioned in the main text. In particular,
the wavefront is also broader, as seen by the more gradual spatial transition
from $\overline{C}=1$ to $\overline{C}\approx 0$.
### I.2 OTOCs for Non-Integrable and Integrable Quantum Circuits
Figure S3: Average OTOCs for non-integrable and integrable quantum circuits.
(A) Two different types of quantum circuits. Left panel shows a non-integrable
quantum circuit composed of $\sqrt{\text{iSWAP}}$ and single-qubit gates
randomly drawn from $\sqrt{X^{\pm}}$, $\sqrt{Y^{\pm}}$, $\sqrt{W^{\pm}}$,
$\sqrt{V^{\pm}}$. Right panel shows an integrable quantum circuit composed of
$\sqrt{\text{iSWAP}}$ and single-qubit gates that are rotations around the
z-axis of the Bloch sphere with random angles. (B) Average OTOCs
$\overline{C}$ for the two types of random circuits, implemented on a 1D chain
of 16 qubits. The qubit configuration is shown on top, where the unfilled
(filled) black circle represents the ancilla (measurement) qubit
$Q_{\text{a}}$ ($Q_{1}$). Left panel shows $\overline{C}$ with the butterfly
qubit corresponding to qubits $Q_{2}$ through $Q_{16}$, where the random
circuits are of the type described in the left panel of (A). Right panel shows
similar data for the random circuits of the type in the right panel of (A).
The locations of the butterfly qubit are indicated by the colors of the data
symbols and the legend on top. All data are averaged over 40 circuit
instances. Figure S4: Transition into non-integrability in the XY Model. (A) A
2D arrangement of qubits in the form of two parallel 1D chains with 5
connections between each other. Black unfilled (filled) circle denotes the
ancilla (measurement) qubit $Q_{\text{a}}$ ($Q_{1}$). (B) Order for applying
the two-qubit gates $\sqrt{\text{iSWAP}}$. The qubit connections denote the
$\sqrt{\text{iSWAP}}$ gates that are applied for a particular cycle.
Additional cycles repeat the first three cycles, e.g. cycle 4 applies the same
$\sqrt{\text{iSWAP}}$ gates as cycle 1 and so on. (C) Average OTOCs
$\overline{C}$ with the butterfly qubit corresponding to qubits $Q_{2}$
through $Q_{16}$. The locations of the butterfly qubit are indicated by the
colors of the data symbols and the colors of the qubits in (A) and (B). All
data are averaged over 36 circuit instances.
In the main text of this work, we have primarily focused on quantum circuits
that are non-integrable Calabrese _et al._ (2016), i.e. capable of evolving a
quantum system into states with maximal degrees of scrambling. In general,
many quantum circuits or dynamical processes are integrable and lead to small
degree of quantum scrambling even at long time scales. Examples include
Clifford circuits Gottesman (1996), dynamics of free fermions Terhal and
DiVincenzo (2002) and many-body localization Nandkishore and Huse (2015). For
certain integrable, pseudo-random circuits such as Clifford circuits, OTOC
fluctuation is needed to distinguish them from non-integrable circuits, as
demonstrated in the main text. In many other cases, average OTOCs behave
differently for integrable quantum circuits compared to non-integrable ones
and are sufficient to differentiate between them, which we illustrate next.
Figure. S3 shows two different types of quantum circuits. The circuit in the
left panel consists of $\sqrt{\text{iSWAP}}$ gates and single-qubit gates
randomly chosen from $\sqrt{X^{\pm}}$, $\sqrt{Y^{\pm}}$, $\sqrt{W^{\pm}}$,
$\sqrt{V^{\pm}}$. This is the example of a non-integrable circuit, which is
expected to lead to maximal scrambling at long times. The circuit in the right
panel has the same two-qubit gates, but the single-qubit gates are replaced
with $Z$ gates that have angles randomly chosen from the interval
$[-\pi,\pi]$. This is the example of an integrable circuit, where the dynamics
does not lead to maximal scrambling if implemented in 1D.
The experimentally measured average OTOCs are shown in Fig. S3B for both types
of quantum circuits. Here, the two-qubit gates are applied in a brick-work
pattern similar to what is used in Fig. 1C of the main text. For the non-
integrable circuits, we observe a clear propagation of the OTOCs along with a
diffusive broadening of the wavefronts, similar to what was observed in Fig.
2C of the main text. In particular, $\overline{C}$ monotonically decays toward
0 at large circuit cycles. On the other hand, $\overline{C}$ does not show the
wave-like propagation for the integrable circuits. For qubits closer to the
measurement qubit $Q_{1}$, $\overline{C}$ first decays but gradually increases
for longer circuit cycles. Qubits further from $Q_{1}$ barely show any
appreciate degree of OTOC decay up to 50 circuit cycles. Overall,
$\overline{C}$ for different qubits converges toward values $>0.5$ for the
largest circuit cycles probed in this experiment. These results show how one
might in some cases uses average OTOC behavior to distinguish non-integrable
quantum dynamics from integrable ones.
The integrable circuits studied in Fig. S3 in fact are the digitized
realizations of the so-called XY model Wang (2001) which is of wide interest
in condensed-matter physics due to its ability to capture a variety of
interesting physical phenomena such as quantum phase transitions. To
demonstrate an immediate application of our work, we use OTOCs to study a
particular feature of the XY-model, namely the transition from integrability
to non-integrability due to geometry.
It is well-known that XY-model in 1D exhibits integrable dynamics, as
demonstrated in Fig. S3. Dynamics for XY-model in 2D remains a highly active
area of research and is generally non-integrable. In Fig. S4, we show that the
transition into non-integrability for XY-model occurs as soon as the geometry
changes from 1D to a ladder-like geometry (Fig. S4A). Here we have arranged 16
qubits into two parallel chains of 8 qubits and connected them with 5 “cross-
links”. Next, we measure OTOCs of random circuits implemented with this new
geometry where the single-qubit gates are again randomly chosen from $Z$ gates
with angles in the interval $[-\pi,\pi]$ and the two-qubit gates are
$\sqrt{\text{iSWAP}}$. The order for applying the $\sqrt{\text{iSWAP}}$ gates
is shown in Fig. S4B.
The average OTOCs for this ladder-geometry XY model are shown in Fig. S4C. It
is seen that the wavelike propagation and long-time limits of $\overline{C}=0$
characteristic of non-integrable quantum circuits are both recovered in this
modified geometry, indicating a transition from integrability to non-
integrability for the XY-model. Detailed experimental studies of this
transition with larger numbers of qubits is a subject of future work.
### I.3 Characterization of Experimental Errors
Figure S5: Contribution of finite sampling to experimental OTOC error. (A)
Dependence of experimental OTOC error on the number of single-shots used to
estimate $\braket{\hat{\sigma}_{\text{y}}}$, $n_{\text{stats}}$. The dashed
line shows expected statistical errors for different values of
$n_{\text{stats}}$. Here the number of iSWAPs is $N_{\text{S}}=251$ and the
number of non-Cliffords is $N_{\text{D}}=40$. (B) Comparison of experimental
OTOC errors and expected statistical errors for different values of
$N_{\text{S}}$. The experimental errors are reproduced from Fig. 4C of the
main text. The statistical errors are calculated based on $n_{\text{stats}}$
and the average normalization value for each $N_{\text{S}}$. Figure S6: OTOC
errors for $N_{\text{D}}=24$. (A) OTOCs of 100 random circuit instances, $C$,
for different values of $N_{\text{S}}$. $N_{\text{D}}=24$ for all circuits.
The dashed lines are exact numerical simulation results using the branching
method. (B) Upper panel: OTOC signal size, experimental error and estimated
statistical error as functions of $N_{\text{S}}$. $N_{\text{D}}=24$ for all
data included here. Lower panel: SNR (i.e. ratio of OTOC signal size to
experimental error) as a function of $N_{\text{S}}$.
In this section, we present additional characterization data to further
corroborate and understand the experimental results in Fig. 4 of the main
text. In particular, we focus on answering two questions: 1. What fraction of
the observed experimental errors can be attributed to statistical errors due
to limited sampling? 2. How sensitive is the signal-to-noise ratio to the
number of non-Cliffords in the circuits?
We first address question 1. The role of finite sampling in experimental OTOC
measurement may be understood as follows: For $n_{\text{stats}}$ single-shot
measurements, an average error of $\frac{1}{\sqrt{n_{\text{stats}}}}$ is
expected to be present in the estimate for $\braket{\hat{\sigma}_{\text{y}}}$
of the ancilla qubit due to statistical uncertainty. In the presence of
circuit errors and a normalization value
$\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}<1$, this expected statistical
error is amplified to a value of
$\frac{1}{\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}\sqrt{n_{\text{stats}}}}$,
assuming the ideal OTOC value to be significantly smaller than 1 (i.e.
$\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}\gg|\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}|$
where $\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}$ is
$\braket{\hat{\sigma}_{\text{y}}}$ measured with the butterfly operator
applied).
In Fig. S6A, this expected statistical error is plotted as a function of
$n_{\text{stats}}$ for $N_{\text{S}}=251$ and $N_{\text{D}}=40$ where
$\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}\approx 0.04$. On the same plot,
we have included experimental OTOC errors (i.e. the RMS deviation between
simulated and experimental OTOC values of 100 random circuits) obtained from
reduced amounts of single-shot data. We observe that the experimental error
initial decreases with increasing values of $n_{\text{stats}}$, suggesting
that statistical uncertainty being a significant source of error for small
numbers of single-shot measurements. At $n_{\text{stats}}>10^{7}$, the
experimental error has a significantly weaker dependence on $n_{\text{stats}}$
and is markedly higher than the expected statistical error, indicating other
error sources are dominant in this regime. In Fig. S6A, we have re-plotted the
experimental errors in Fig. 4C of the main text along with the expected
statistical errors calculated from $n_{\text{stats}}$ and
$\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}$ of each $N_{\text{S}}$. The
increase in statistical error at higher $N_{\text{S}}$ is a result of
decreasing normalization values. It is evident that the observed experimental
errors are consistently larger than the expected statistical errors,
indicating other error mechanisms are dominant. In Section VI, we provide
numerical simulation results to further analyze the sources of experimental
error.
Next, we focus on the scaling of experimental error vs number of iSWAPs for a
different number of non-Cliffords in the quantum circuits. Fig. S6A shows $C$
of 100 circuit instances for $N_{\text{S}}=113$, 251 and 300, all of which
have $N_{\text{D}}=24$. On the same plots, exactly simulated OTOC values using
the branching method (Section IV) are also plotted. Similar to Fig. 4B of the
main text, we observe that the agreement between experimental and simulated
OTOC values degrades as $N_{\text{S}}$ increases. In addition, the OTOC signal
size (i.e. the fluctuation of the simulated OTOC values) also decreases as
$N_{\text{S}}$ increases.
In Fig. S6B, the OTOC signal size, experimental error and expected statistical
error are plotted as functions of $N_{\text{S}}$ (all with $N_{\text{D}}=24$).
Here we see that the OTOC signal size indeed monotonically decreases as
$N_{\text{S}}$ increases. The experimental error, on the other hand, shows
less dramatic changes as a function of $N_{\text{S}}$. After taking the ratio
of the OTOC signal size and the experiment error, we plot the resulting SNR as
a function of $N_{\text{S}}$ in the lower panel of Fig. S6B. The scaling of
SNR vs $N_{\text{S}}$ is roughly consistent with Fig. 4 of the main text where
higher values of $N_{\text{D}}$ (64 and 40) are used. In particularly, SNR
falls below 1 after $N_{\text{S}}$ has increased beyond $\sim$250\. The
relative insensitivity of SNR to $N_{\text{D}}$ allows us to conveniently
benchmark our system at lower values of $N_{\text{D}}$ where classical
simulation is easy. This is especially important in the future when we conduct
experiments with $N_{\text{S}}>400$ where tensor-contraction may no longer be
feasible (Section III).
## II Experimental Techniques
In this section, we describe the calibration process and metrology of quantum
gates used in the OTOC experiment. In addition, we also demonstrate a series
of error-mitigation strategies used in the compilation of experimental
circuits which significantly reduced errors from various sources such as those
related to state preparation and readout (SPAM), cross-talk, or coherent
control errors on two-qubit gates.
### II.1 Calibration of iSWAP and $\sqrt{\text{iSWAP}}$ Gates
Figure S7: Calibrating “textbook” gates. (A) Schematic illustration of the
waveforms used to realize pure iSWAP and $\sqrt{\text{iSWAP}}$ gates: The
$|0\rangle\rightarrow|1\rangle$ transition frequencies of two transmon qubits,
$f_{1}$ and $f_{2}$, are brought close to each other by adjusting the control
fluxes to their superconducting quantum interference device (SQUID) loop.
Concurrently, a pulse to the coupler’s SQUID flux changes the qubit-qubit
coupling $g$ from 0 to a maximum value of $g_{\text{max}}$. The total length
of the pulses is $t_{\text{p}}$. (B) The generic FSIM gate realized by the
waveforms in (A), composed of a partial-iSWAP gate with swap angle $\theta$, a
CPHASE gate with a conditional-phase $\phi$ and four local Z-rotations with
angles $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, $\beta_{2}$ each. (C, D)
Simulated $\theta$ (C) and $\phi$ (D) as functions of $g_{\text{max}}$ and
$t_{\text{p}}$. The simulation assumes an average qubit frequency
$\frac{1}{2}\left(f_{1}+f_{2}\right)=6.0$ GHz and an anharmonicity of 200 MHz
for each transmon qubit. The operating points for three different gates,
Sycamore, iSWAP and $\sqrt{\text{iSWAP}}$, are indicated by the the star
symbols. (E, F) Integrated histogram (empirical cumulative distribution
function, ECDF) of $\theta$ (E) and $\phi$ (F) for both the iSWAP and
$\sqrt{\text{iSWAP}}$ gates. The data include all 86 qubit pairs on the
processor. The x-axis location of each dashed line indicates the median value
of the corresponding angle.
The measurement of OTOCs requires faithful inversion of a given quantum
circuit, $\hat{U}$. The “Sycamore” gate used in our previous work Arute _et
al._ (2019), equivalent to an iSWAP gate followed by a CPHASE gate with a
conditional-phase of $\pi/6$ radians (rad), is an ill-suited building block
for $\hat{U}$ since its inversion cannot be easily created by combining
Sycamore with single-qubit gates. On the other hand, iSWAP and
$\sqrt{\text{iSWAP}}$ are commonly used two-qubit gates that can be readily
inverted by adding local Z rotations:
$G^{-1}=Z_{1}\left(\frac{\pi}{2}\right)Z_{2}\left(-\frac{\pi}{2}\right)GZ_{1}\left(-\frac{\pi}{2}\right)Z_{2}\left(\frac{\pi}{2}\right)$
(S1)
Here $G$ is the two-qubit unitary corresponding to iSWAP or
$\sqrt{\text{iSWAP}}$ and
$Z_{i}(\varphi)=e^{-i\frac{\varphi}{2}\hat{\sigma}_{z}^{(i)}}$, where
$\hat{\sigma}_{z}^{(i)}$ is the Pauli-$Z$ matrix acting on qubit $i$ ($i=1$ or
2). Compared to Sycamore, realizing pure iSWAP and $\sqrt{\text{iSWAP}}$ gates
requires the development of two additional capabilities on our quantum
processor: 1. A significant reduction of the coherent error associated with
the conditional-phase in Sycamore. 2. The abilility to implement arbitrary
single-qubit rotations around the $Z$ axis.
#### II.1.1 Reducing Conditional-Phase Errors
We first describe the calibration technique for reducing conditional phase
errors. As discussed in previous works Arute _et al._ (2019), such a phase
arises from the dispersive interaction between the $|11\rangle$ and
$|02\rangle$ (or $|20\rangle$) states of two transmon qubits while their
coupling strength $g$ is raised to a finite value to enable a resonant
interaction between the $|10\rangle$ and $|01\rangle$ states (Fig. S7A).
Although eliminating this phase is possible by concatenating a Sycamore gate
with a pure CPHASE gate having an opposite conditional-phase Foxen _et al._
(2020), this process would involve complicated waveforms and large qubit
frequency excursions which are demanding to implement on a 53-qubit processor.
Instead, we adopt an alternative approach based on the relatively simple
control waveform used in Ref. Arute _et al._ (2019) (Fig. S7A). This waveform
sequence produces a Fermionic Simulation (FSIM) gate comprising a partial-
iSWAP gate with swap angle $\theta$, a CPHASE gate with conditional-phase
$\phi$ and four local $Z$-rotations (Fig. S7B).
Next, we consider how the angles $\theta$ and $\phi$ depend on readily tunable
waveform parameters such as the pulse duration $t_{\text{p}}$ and maximum
qubit-qubit coupling $g_{\text{max}}$. This is done by numerically solving the
time-evolution of two coupled transmons with typical device parameters. The
dependences of $\theta$ and $\phi$ on $t_{\text{p}}$ and $g_{\text{max}}$ are
shown in Fig. S7C and Fig. S7D, respectively. We observe that although both
$\theta$ and $\phi$ increase linearly with $t_{\text{p}}$, their scaling with
respect to $g_{\text{max}}$ is different: $\theta\propto g_{\text{max}}$
whereas $\phi\propto g_{\text{max}}^{2}$. For a given value of $\theta$, it is
therefore possible to reduce $\phi$ by increasing $t_{\text{p}}$ and
decreasing $g_{\text{max}}$ while keeping $t_{\text{p}}g_{\text{max}}$
constant. However, since longer gate operation is more susceptible to
decoherence effects such as relaxation and dephasing, it is also desirable to
minimize values of $t_{\text{p}}$. As a result, we use $t_{\text{p}}=12$ ns,
$g_{\text{max}}/2\pi\approx 10$ MHz for calibrating the $\sqrt{\text{iSWAP}}$
gate and $t_{\text{p}}=36$ ns, $g_{\text{max}}/2\pi\approx 7$ MHz for
calibrating the iSWAP gate. Based on the simulation results, these choices
would yield values of $\theta=\pi/4$ rad, $\phi=0.14$ rad for the
$\sqrt{\text{iSWAP}}$ gate and $\theta=\pi/2$ rad, $\phi=0.19$ rad for the
iSWAP gate.
The calibrated values of $\theta$ and $\phi$ associated with every qubit pair
on the 53-qubit processor are shown in Fig. S7E and Fig. S7F, respectively.
Each angle is measured via cross entropy benchmarking (XEB), similar to
previous works Arute _et al._ (2019); Foxen _et al._ (2020). The median
value of $\theta$ is 0.783 rad (standard deviation = 0.010 rad) for
$\sqrt{\text{iSWAP}}$ and 1.563 rad (standard deviation = 0.027 rad) for
iSWAP. These median values are very close to the target values of
$\pi/4=0.785$ rad for $\sqrt{\text{iSWAP}}$ and $\pi/2=1.571$ rad for iSWAP.
In the case of $\phi$, the median value is 0.112 rad for $\sqrt{\text{iSWAP}}$
(standard deviation = 0.026 rad) and 0.136 rad for iSWAP (standard deviation =
0.055 rad). The median values of $\phi$ are close to predictions from
numerical simulation and 4 to 5 times lower compared to the Sycamore gate.
The coherent error introduced by remnant values of $\phi$ is further reduced
by adjusting other phases of a FSIM unitary
$U_{\text{FSIM}}(\theta,\phi,\Delta_{+},\Delta_{-},\Delta_{-,\text{off}})$,
defined as:
$\begin{pmatrix}1&0&0&0\\\
0&e^{i(\Delta_{+}+\Delta_{-})}\cos{\theta}&-ie^{i(\Delta_{+}-\Delta_{-,\text{off}})}\sin{\theta}&0\\\
0&-ie^{i(\Delta_{+}+\Delta_{-,\text{off}})}\sin{\theta}&e^{i(\Delta_{+}-\Delta_{-})}\cos{\theta}&0\\\
0&0&0&e^{i(2\Delta_{+}-\phi)}\end{pmatrix}.$ (S2)
Here $\Delta_{+}$, $\Delta_{-}$ and $\Delta_{-,\text{off}}$ are phases that
can be freely adjusted by local $Z$-rotations. Imperfect gate calibration
often results in an actual two-qubit unitary $U_{\text{a}}$ that differs
slightly from the target unitary $U_{\text{t}}$, leading to a Pauli error
Arute _et al._ (2019); Nielsen and Chuang (2010):
$r_{\text{p}}=1-\frac{1}{D^{2}}\left|\text{tr}\left(U_{\text{a}}^{\dagger}U_{\text{t}}\right)\right|^{2}$
(S3)
Here $D=4$ is the dimension of a two-qubit Hilbert space.
Given that our target unitaries are
$U_{\text{t}}=U_{\text{FSIM}}(\pi/4,0,0,0,0)$ for $\sqrt{\text{iSWAP}}$ and
$U_{\text{t}}=U_{\text{FSIM}}(\pi/2,0,0,0,0)$ for iSWAP, one may naively
expect that $\Delta_{+}$, $\Delta_{-}$ and $\Delta_{-,\text{off}}$ should all
be set to 0 in $U_{\text{a}}$ in order to minimize $r_{\text{p}}$. While this
is indeed the case if $\phi=0$ in $U_{\text{a}}$, it is not true when $\phi$
assumes a finite value $\phi_{\text{a}}$ in $U_{\text{a}}$. In fact, simple
algebraic calculation shows that in such a case, the minimum value of $r_{p}$
occurs at
$(\Delta_{+},\Delta_{-},\Delta_{-,\text{off}})=(\phi_{\text{a}}/2,0,0)$, where
it is a factor of 3 smaller compared to
$(\Delta_{+},\Delta_{-},\Delta_{-,\text{off}})=(0,0,0)$. By calibrating our
system such that $\Delta_{+}=\phi_{\text{a}}/2$ for every qubit pair, we
estimate that the median Pauli error introduced by the conditional-phase to be
only 0.07 % for $\sqrt{\text{iSWAP}}$ and 0.12 % for iSWAP.
#### II.1.2 Arbitrary $Z$-Rotations
Figure S8: Implementation of $Z$-rotations. (A) Circuit compilation procedure
for reducing number of required $Z$-rotations. $Z$-rotations occurring after
each FSIM gate are combined with the $Z$-rotations before the next FSIM gate.
Phases on the microwave-driven single-qubit gates are shifted to maintain the
same overall quantum evolution. $\varphi_{ij}$ denotes the angles of various
$Z$-rotations and $\chi_{i}$ denotes the phases of microwave single-qubit
gates. (B) Circuit compilation procedure for pushing $Z$-rotations through an
iSWAP gate. Combined with (A), $Z$-rotations in circuits containing only iSWAP
and single-qubit gates become entirely virtual. (C) Schematic illustration of
the waveforms used to implement $Z$-rotations before each
$\sqrt{\text{iSWAP}}$ gate. Compared to the waveforms in Fig. S7A, an
additional control flux pulse is used to detune the qubits from their idle
frequencies and thereby achieve a “physical” $Z$-gate.
We now describe the implementation of arbitrary single-qubit $Z$-rotations on
the quantum processor. In addition to constructing the inverse gates
$\sqrt{\text{iSWAP}^{-1}}$ and iSWAP-1, $Z$-rotations are also used in
removing native $Z$-rotations $Z^{\prime}$ in the FSIM gate (Fig. S7B) as well
as adjusting values of $\Delta_{+}$ to minimize conditional-phase errors. The
procedure for incorporating $Z$-rotations into quantum circuits is two-fold:
First, we recompile the circuits and combine all $Z$-rotations occurring after
each two-qubit FSIM gate with the $Z$-rotations before the next FSIM gate, as
shown in Fig. S8A. If any microwave-driven single-qubit gate such as
$R_{\pi/2}(\chi)=e^{-i\frac{\pi}{4}\left(\cos\chi\hat{\sigma}_{x}+\sin\chi\hat{\sigma}_{y}\right)}$
occurs between the two FSIM gates, where $\hat{\sigma}_{x}$ and
$\hat{\sigma}_{y}$ are Pauli-$X$ and Pauli-$Y$ matrices and $\chi$ is the
phase of the microwave drive, we apply the equivalence
$R_{\pi/2}(\chi)Z(\varphi)=Z(\varphi)R_{\pi/2}(\chi-\varphi)$ to shift the
rotation axis of the single-qubit gate and “push” the $Z$-rotation through.
This process effectively reduces the number of $Z$-rotations in the circuits
by a factor of 2 and has been demonstrated to incur negligible degradation in
the overall fidelity, since the only physical changes are modifications to the
phases of the microwave drives of single-qubit gates McKay _et al._ (2017).
The second step for implementing $Z$-rotations is different for iSWAP and
$\sqrt{\text{iSWAP}}$ gates. In the case of iSWAP, the equivalence
$U_{\text{FSIM}}(\pi/2,0,0,0,0)Z_{1}(\varphi_{1})Z_{2}(\varphi_{2})=Z_{1}(\varphi_{2})Z_{2}(\varphi_{1})U_{\text{FSIM}}(\pi/2,0,0,0,0)$
allows us to push $Z$-rotations through each two-qubit gate by simply
rearranging their phases (Fig. S8B). As a result, the $Z$-rotations occurring
in circuits with iSWAP gates are entirely virtual. In the case of
$\sqrt{\text{iSWAP}}$, such equivalence does not exist and we implement
$Z$-rotations before the two-qubit gates using additional control flux pulses,
as illustrated in Fig. S8C. Here, we detune the qubit frequencies from their
idle positions by an variable amount $\Delta f$ and for a fixed duration
$t_{\text{z}}$ = 20 ns before each $\sqrt{\text{iSWAP}}$ gate, leading to a
$Z$-rotation $\approx Z\left(2\pi\Delta ft_{\text{z}}\right)$.
### II.2 Gate Error Benchmarking and Cross-Talk Mitigation
Figure S9: Fixed-Unitary XEB and cross-talk correction. (A, B) Integrated
histograms (ECDF) of Pauli errors for both the iSWAP (A) and the
$\sqrt{\text{iSWAP}}$ (B) gates. Three sets of values are shown for each case:
the isolated (blue) curves represent the errors obtained by operating each
qubit pair individually with all other qubits at ground state. The
simultaneous, uncorrected (red) curves represent the errors obtained by
operating all qubits at the same time, without any additional correction. The
simultaneous, corrected (green) curves represent the simultaneous error rates
after additional $Z$-rotations are included in the circuits to offset unitary
shifts induced by cross-talk effects. Simultaneous error rates are obtained
from four different configurations of parallel two-qubit operations, similar
to Ref. Arute _et al._ (2019). Dashed lines indicate the median locations of
various errors. All error rates are obtained from XEB with pure iSWAP or
$\sqrt{\text{iSWAP}}$ gate used in simulation, and include contributions from
two single-qubit gates and one two-qubit gate. (C) An OTOC experimental
configuration for evaluating the effects of cross-talk correction. The empty
circle represents the ancilla qubit and the black filled circle represents the
measurement qubit. All other qubits are represented by purple spheres. (D) The
OTOC normalization value $\braket{\hat{\sigma}_{\text{y}}}$ as a function of
number of cycles in a quantum circuit $\hat{U}$ for the configuration shown in
(C), measured both with and without applying the cross-talk corrections.
$\hat{U}$ is composed of iSWAP and random single-qubit $\pi/2$ rotations
around axes on the $XY$ plane.
The Pauli errors of the calibrated iSWAP and $\sqrt{\text{iSWAP}}$ gates are
measured through XEB Arute _et al._ (2019); Boixo _et al._ (2018), which
uses a collection of random circuits comprising interleaved two-qubit and
random single-qubit gates. For each random circuit, the probability
distribution of all possible output bit-strings is both measured
experimentally and computed numerically. The statistical correlation between
the two distributions (“cross-entropy”) is then used to infer the total error
of each quantum circuit. After measuring a sufficient number of circuit
instances at different circuit depths, it has been shown that XEB reliably
yields gate errors that are very consistent with values obtained from
conventional characterization methods such as randomized benchmarking Arute
_et al._ (2019); Knill _et al._ (2008); Foxen _et al._ (2020).
A key difference between the XEB process used in our current work compared to
prior experiments Arute _et al._ (2019) is the unitaries used in the
numerical computation of the benchmarking circuits. Previously, such unitaries
are freely adjusted for each individual qubit pair, whereby values of various
FSIM angles $\theta$, $\phi$, $\Delta_{+}$, $\Delta_{-}$ and
$\Delta_{-,\text{off}}$ are optimized to obtain the lowest Pauli errors. In
this work, we do not perform such an optimization step during gate error
characterization and instead use a fixed unitary
($U_{\text{FSIM}}(\pi/2,0,0,0,0)$ for iSWAP and
$U_{\text{FSIM}}(\pi/4,0,0,0,0)$ for $\sqrt{\text{iSWAP}}$) for all qubit
pairs. The gate errors characterized by the “fixed-unitary” XEB process
include contributions from both incoherent sources such as relaxation and
dephasing and coherent sources such as remnant conditional-phases. The
adoption of a more stringent benchmarking criterion for gate errors is
motivated by the fact that both coherent and incoherent errors can lead to
imperfect reversal of quantum circuits and adversely impact the accuracy of
OTOC measurements, in contrast to previous experiment in which coherent errors
are compensated by modifying circuits in simulation Arute _et al._ (2019).
The Pauli error rate $r_{\text{p}}$ per cycle (aggregate error of two single-
qubit gates and one two-qubit gate) associated with each qubit pair is first
measured in isolation. The results are plotted as integrated histograms in
Fig. S9A and Fig. S9B, where we observe a mean (median) error of 0.0109
(0.0106) for iSWAP and 0.0106 (0.0089) for $\sqrt{\text{iSWAP}}$. These higher
errors rates compared to our previous work Arute _et al._ (2019) are a result
of enhanced incoherent errors due to longer pulses used in iSWAP and
additional detuning pulses in $\sqrt{\text{iSWAP}}$, as well as coherent
errors arising from remnant conditional-phases.
We then repeat the same process but measure the error rates of different pairs
simultaneously. We first observe, similar to previous work Arute _et al._
(2019), a sizable increase in $r_{\text{p}}$, with the mean (median) being
0.0193 (0.0163) for iSWAP and 0.0190 (0.0161) for $\sqrt{\text{iSWAP}}$. To
reduce these cross-talk effects, we first fit the XEB results to obtain the
shifts in the two-qubit unitary associated with each individual qubit pair,
which are often related to the single-qubit phases $\Delta_{+}$, $\Delta_{-}$
and $\Delta_{-,\text{off}}$. In a second step, instead of simply incorporating
these shifts into classical simulation as was done previously Arute _et al._
(2019), we add local $Z$-rotations into the quantum circuits to offset the
unitary shifts. The parallel error rates are then re-measured with these
$Z$-rotations. The mean (median) value of $r_{\text{p}}$ for simultaneous
operation is reduced to 0.0140 (0.0131) for iSWAP and 0.0142 (0.0123) for
$\sqrt{\text{iSWAP}}$.
The effects of the cross-talk correction can be readily seen in the
“normalization” values used in the OTOC experiment. Figure S9C shows the
configuration for a 53-qubit OTOC experiment. The corresponding OTOC
normalization $\braket{\hat{\sigma}_{\text{y}}}$ as a function of the number
of cycles in a quantum circuit is shown in Fig. S9D. Without applying the
additional $Z$-rotations for cross-talk correction,
$\braket{\hat{\sigma}_{\text{y}}}$ decays rapidly and falls below 0.1 after 8
cycles. After applying the additional $Z$-rotations,
$\braket{\hat{\sigma}_{\text{y}}}$ decays at a visibly slower rate and falls
below 0.1 after 10 cycles. The slower decay of
$\braket{\hat{\sigma}_{\text{y}}}$ is indicative of more accurate inversion of
the quantum circuit after cross-talk correction.
### II.3 Dynamical Decoupling
Figure S10: Dynamical coupling in OTOC experiments. (A) Test experimental
circuit for benchmarking the intrinsic coherence of the ancilla qubit.
Compared to an actual OTOC circuit, the CZ gates between the ancilla qubit and
measurement qubit is removed. (B) Same circuit as (A) with the additional of
consecutive spin echo pulses to the ancilla qubit during the quantum circuits
$\hat{U}$ and $\hat{U}^{\dagger}$, in the form of random $X$ or $Y$ gates. (C)
The projection of the ancilla qubit to the y-axis,
$\braket{\hat{\sigma}_{\text{y}}}$, at the end of the circuit, measured with
(red) and without (blue) spin echo. The bottom x-axis of the plot shows the
number of cycles in $\hat{U}$ ($\hat{U}^{\dagger}$ has the same number of
cycles), and the top x-axis shows the corresponding total circuit duration
$t_{\text{c}}$.
We now describe a series of additional error-mitigation techniques used in the
compilation of quantum circuits that further improve the accuracy of OTOC
experiments. The first such technique is dynamical decoupling, motivated by
long “idling” times at non-ground states for some of the qubits during the
experiment. The most prominent of such qubits is the ancilla qubit, which
remains idle throughout the time needed to implement the quantum circuit
$\hat{U}$ and its inverse $\hat{U}^{\dagger}$. Intrinsic decoherence of the
ancilla qubit can in principle limit the circuit depth at which OTOC can be
resolved, especially if the characteristic time $T_{2}$ is comparable to the
total duration of $\hat{U}$ and $\hat{U}^{\dagger}$.
To benchmark the intrinsic $T_{2}$ of the ancilla qubit during an OTOC
experiment, we design the test circuits shown in Fig. S10A and Fig. S10B. In
either case, we removed the two CZ gates otherwise present in an actual OTOC
experiment such that the ancilla qubit does not interact with any other qubit
apart from cross-talk effects. The difference between the two cases is that
the ancilla qubit remains completely idle during $\hat{U}$ and
$\hat{U}^{\dagger}$ in Fig. S10A, whereas a train of spin echo pulses $X-X-Y-
Y-X-X...$ is applied to the ancilla qubit during $\hat{U}$ and
$\hat{U}^{\dagger}$ in Fig. S10B.
The y-axis projection of the ancilla qubit,
$\braket{\hat{\sigma}_{\text{y}}}$, at the end of $\hat{U}$ and
$\hat{U}^{\dagger}$ is measured both with and without the added spin echoes.
The results are shown in Fig. S10C. We observe that without spin echo,
$\braket{\hat{\sigma}_{\text{y}}}$ decays rather quickly despite no
entanglement between the ancilla and other qubits in the system, falling to
$\sim$0 after 25 circuit cycles (corresponding to an evolution time
$t_{\text{c}}\approx 3.0$ $\mu$s). The Gaussian shape of the decay at earlier
times suggests that low-frequency noise is likely the dominant source of
decoherence Ithier _et al._ (2005). On the hand, with the addition of spin
echo, $\braket{\hat{\sigma}_{\text{y}}}$ decays at a much slower rate
maintaining a value of 0.78 even after 40 circuit cycles ($t_{\text{c}}\approx
4.6$ $\mu$s). By fitting $\braket{\hat{\sigma}_{\text{y}}}$ to a functional
form $Ae^{-\frac{t_{\text{c}}}{T_{2}}}$ where $A$ and $T_{2}$ are fitting
parameters, we obtain a coherence time $T_{2}=28.6$ $\mu$s for the ancilla
qubit, which is close to the $2T_{1}$ limit of our quantum processor Arute
_et al._ (2019). Given this value of $T_{2}$ is significantly longer than all
OTOC experimental circuits used in this work (the longest circuit lasts
$\sim$5 $\mu$s), we conclude that changes in the ancilla projection
$\braket{\hat{\sigma}_{\text{y}}}$ in the OTOC experiments are indeed
dominated by the many-body effects in $\hat{U}$ and $\hat{U}^{\dagger}$ rather
than the intrinsic decoherence of the ancilla qubit itself. For all
experimental results presented in the main text, spin echo is applied to the
ancilla qubit.
### II.4 Elimination of Bias in $\braket{\hat{\sigma}_{\text{y}}}$
Figure S11: Unbiased measurements of $\braket{\hat{\sigma}_{\text{y}}}$. (A)
Left panel shows the test circuits for characterizing
$\braket{\hat{\sigma}_{\text{y}}}$ arising from asymmetry in readout
fidelities. Two different schemes are implemented. The conversion between the
excited-state population(s) $P_{\uparrow}$ and
$\braket{\hat{\sigma}_{\text{y}}}$ is described in the main text. Right panel
shows the experimental values of $\braket{\hat{\sigma}_{\text{y}}}$ as a
function of the variable phase $\varphi_{\text{v}}$, obtained with both
readout schemes. Dashed lines show fits to
$\braket{\hat{\sigma}_{\text{y}}}=(1-2F_{\text{r}})\cos\left(\varphi_{\text{v}}\right)+d_{\text{r}}$,
where $F_{\text{r}}$ and $d_{\text{r}}$ are fitting parameters. (B) Schematic
of the full OTOC circuit showing two different state preparation schemes: in
the unbalanced scheme, only one phase is used for the first $R_{\pi/2}$ gate
on the ancilla qubit. In the balanced scheme, two different phases are used.
Balanced readout is used in both schemes. (C) Results of a 12-qubit OTOC
experiment. Upper panels show $\braket{\hat{\sigma}_{\text{y}}}$ at different
number of circuit cycles. The black squares represent the normalization case
and the colored spheres represent $\braket{\hat{\sigma}_{\text{y}}}$ obtained
with the butterfly operator ($Z$) successively applied to qubits 2 (Q2)
through 12 (Q12). The location of the butterfly operator for each curve is
indicated by the color legend on top. The normalized OTOCs, $\overline{C}$,
are shown in the lower panels. The data are the average values of 40 different
random circuit instances.
The accuracy of OTOC measurements is particularly susceptible to any bias in
$\braket{\hat{\sigma}_{\text{y}}}$ of the ancilla qubit, i.e. a fixed offset
$d$ to the ideal value. Such a bias can, for example, be introduced by the
different readout fidelities for the $|0\rangle$ and $|1\rangle$ states of the
ancilla qubit. To see the impact of the bias on OTOC, we consider an ideal
OTOC value of $C_{0}$ and an ideal normalization value of
$\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}$. The ideal y-projection of the
ancilla with the butterfly operator applied,
$\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}$, in such a case is
$\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}\approx
C_{0}\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}$. However, in the presence of
a bias, the measured projections become
$\braket{\hat{\sigma}_{\text{y}}}=\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}+d$
for the normalization value and
$\braket{\hat{\sigma}_{\text{y}}}=\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}+d$
with the butterfly operator applied. The experimental value for OTOC then
becomes
$C_{1}=\frac{\braket{\hat{\sigma}_{\text{y}}}_{\text{B}}+d}{\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}+d}\approx\frac{C_{0}\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}+d}{\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}+d}$.
Assuming typical values of $\braket{\hat{\sigma}_{\text{y}}}_{\text{I}}=0.05$
and $C_{0}=0.1$, even a small asymmetry $d=0.01$ would lead to a highly
erroneous value of $C_{1}\approx 0.25$. It is therefore crucial to identify
and mitigate any bias in $\braket{\hat{\sigma}_{\text{y}}}$ of the ancilla
qubit.
We begin by measuring $\braket{\hat{\sigma}_{\text{y}}}$-bias due to asymmetry
in readout errors. The test circuit is shown in the left panel of Fig. S11A
under the label “unbalanced readout”. We first project the qubit onto the
equator of the Bloch sphere with a $\pi/2$ rotation,
$R_{\pi/2}\left(\varphi_{\text{v}}\right)$, where $\varphi_{\text{v}}$ is a
variable phase. A second $\pi/2$ rotation around a fixed axis, $\sqrt{X}$, is
then applied and excited state population $P_{\uparrow}$ is finally measured.
The population is then converted to $\braket{\hat{\sigma}_{\text{y}}}$ via
$\braket{\hat{\sigma}_{\text{y}}}=2P_{\uparrow}-1$. The right panel of Fig.
S11A shows $\braket{\hat{\sigma}_{\text{y}}}$ as a function of
$\varphi_{\text{v}}$ and a fit to a functional form
$\braket{\hat{\sigma}_{\text{y}}}=(1-2F_{\text{r}})\cos\left(\varphi_{\text{v}}\right)+d_{\text{r}}$.
Here $F_{\text{r}}$ is the average of the readout fidelities of the
$|0\rangle$ and $|1\rangle$ states, and $d_{\text{r}}$ is their difference.
For this unbalanced readout scheme, we obtain $F_{\text{r}}=0.0962$ and
$d_{\text{r}}=0.055$. The observed difference in readout fidelities is
consistent with past experiments where it was shown that energy relaxation of
the qubit during dispersive readout generally leads to lower readout fidelity
for the $|1\rangle$ state compared to the $|0\rangle$ state Jeffrey _et al._
(2014).
The bias $d_{\text{r}}=0.055$ is significantly reduced by adopting a ”balanced
readout” scheme, also shown in the left panel of Fig. S11A. Here, we measure
the $P_{\uparrow}$ of the ancilla qubit with the second gate in the test
circuit being either $\sqrt{X}$ or $\sqrt{-X}$. The results are then combined
to obtain
$\braket{\hat{\sigma}_{\text{y}}}=\left(P_{\uparrow,+}-P_{\uparrow,-}\right)$,
where $P_{\uparrow,\pm}$ is $P_{\uparrow}$ measured with the second gate being
$\sqrt{\pm X}$. The averaged $\braket{\hat{\sigma}_{\text{y}}}$ is shown in
the right panel of Fig. S11A as a function of $\varphi_{\text{v}}$. A similar
fit as before yields the same average readout fidelity $F_{\text{r}}=0.0962$
and a much reduced bias $d_{\text{r}}\approx 0.0002$.
Balanced readout alone, as we will demonstrate below, is insufficient for
completely removing bias from $\braket{\hat{\sigma}_{\text{y}}}$. For all
experiments reported in the main text, we apply a second symmetrization step
shown in Fig. S11B, which is the measurement scheme for OTOC with the gates
related to SPAM explicitly shown. Here, we parametrize both the phase
$\varphi_{\text{p}}$ of the first $\pi/2$ gate and the phase
$\varphi_{\text{m}}$ of the last $\pi/2$ gate on the ancilla qubit. In an
“unbalanced state preparation” scheme, we measure the average projections
$\braket{\hat{\sigma}_{\text{y}}}=\left(P_{\uparrow,++}-P_{\uparrow,+-}\right)$,
where $P_{\uparrow,++}$ and $P_{\uparrow,+-}$ are $P_{\uparrow}$ obtained with
$(\varphi_{\text{p}},\varphi_{\text{m}})=(0,0)$ and
$(\varphi_{\text{p}},\varphi_{\text{m}})=(0,\pi)$, respectively. In a
”balanced state preparation” scheme, we measure the average projections
$\braket{\hat{\sigma}_{\text{y}}}=\frac{1}{2}\left(P_{\uparrow,++}-P_{\uparrow,+-}-P_{\uparrow,-+}+P_{\uparrow,--}\right)$,
where $P_{\uparrow,-+}$ and $P_{\uparrow,--}$ are additional populations
obtained with $(\varphi_{\text{p}},\varphi_{\text{m}})=(\pi,0)$ and
$(\varphi_{\text{p}},\varphi_{\text{m}})=(\pi,\pi)$, respectively.
The difference between the two state preparation schemes is illustrated in
Fig. S11C, where we present the results of a 12-qubit OTOC experiment. The
quantum circuit $\hat{U}$ is non-integrable and composed of
$\sqrt{\text{iSWAP}}$ and random single-qubit $\pi/2$ rotations around 8
different axes on the $XY$ plane. A total of 40 circuit instances are used and
the data shown are average values over all instances. The top panels show the
measured values of $\braket{\hat{\sigma}_{\text{y}}}$ for the normalization
case and cases where a butterfly operator $Z$ is successively applied to
qubits 2 to 12 in-between $\hat{U}$ and $\hat{U}^{\dagger}$. The y-axis scale
is intentionally limited to $<0.05$. We observe in the case of unbalanced
state preparation, the $\braket{\hat{\sigma}_{\text{y}}}$ values exhibit
sudden rise from 0 to $>$0.006 at cycles 29 to 38. This behavior is
inconsistent with the effects of scrambling and decoherence, either of which
is expected to lead to monotonic decay of $\braket{\hat{\sigma}_{\text{y}}}$
toward 0 for a non-integrable process. In contrast, with balanced state
preparation, $\braket{\hat{\sigma}_{\text{y}}}$ indeed monotonically decays
toward 0 at large cycles for all curves.
The bottom panels of Fig. S11C show the normalized average OTOCs,
$\overline{C}$, for each qubit. Here we observe that $\overline{C}$ obtained
with the unbalanced state preparation scheme again manifests unphysical jumps
from 0 at cycles 29 to 38, resulting from the finite bias
$\braket{\hat{\sigma}_{\text{y}}}$ and the mechanism outlined at the beginning
of this section. In contrast, $\overline{C}$ obtained with the balanced state
preparation scheme decays monotonically toward 0, in agreement with the
scrambling behavior of a non-integrable process. These data suggest that a
symmetrization step duration the state preparation phase of the ancilla qubit
is needed to completely remove the bias in $\braket{\hat{\sigma}_{\text{y}}}$,
in addition to the symmetrization step before readout. The physical origin of
the remnant bias seen in the left panels of Fig. S11C is not completely
understood at the time of writing, and could be related to control errors in
the single-qubit gates on the ancilla qubit, incomplete depolarization of
$T_{1}$ errors by the spin echoes, or other unknown mechanisms.
### II.5 Light-cone Filter
Figure S12: Re-compiling OTOC circuits with light-cone filter. (A) Schematic
of an OTOC measurement circuit, including the component gates of the quantum
circuits $\hat{U}$ and $\hat{U}^{\dagger}$. The ancilla qubit and its related
gates, as well as the $\sqrt{Y}$ gates used in the state preparation of all
qubits, are omitted for simplicity. Gates shown with semi-transparent colors
can be removed from the OTOC measurement circuit without altering its output.
(B) Left panel shows a configuration for evaluating experimental effects of
the light-cone filter. The unfilled circle represents the ancilla qubit and
the black filled circle represents the measurement qubit. The purple filled
circle indicates the butterfly qubit. The total number of circuit cycles is
11. Right panel shows the measured values of
$\braket{\hat{\sigma}_{\text{y}}}$ for 100 random circuit instances. Data
obtained in the normalization case are shown on top and data obtained with the
butterfly operator applied are shown at the bottom. The same number of
repetitions (1 million) is used in all cases to estimate
$\braket{\hat{\sigma}_{\text{y}}}$. (C) Normalized experimental OTOC values
$C$ for different circuit instances, plotted alongside exact numerical
simulation results. (D) Experimental errors $\epsilon$ for different circuit
instances, corresponding to the differences between experimental and simulated
values.
Analogous to classical systems, quantum perturbations often travel at a
limited speed (the “butterfly velocity”). This typically results in a “light-
cone” structure for many quantum circuits, which can be capitalized to reduce
their classical simulation costs Hastings (2008). Similarly, the light-cone
structure of these quantum circuits may also be utilized to modify their
implementations on a quantum processor and improve the fidelity of
experimental results. In this section, we describe a light-cone-based circuit
re-compilation technique that led to considerable improvements in the accuracy
of experimental OTOC measurements.
Figure S12A displays the generic structure of an OTOC measurement circuit,
where the component gates of the quantum circuit $\hat{U}$ and its inverse
$\hat{U}^{\dagger}$ are explicitly shown. The butterfly operator possesses a
pair of triangular light-cones extending from the middle of the circuit into
both $\hat{U}$ and $\hat{U}^{\dagger}$. Quantum gates outside these light-
cones may be completely removed (“filtered”) without altering the output of
the circuit. Furthermore, since the measurement at the end of the circuit is
also localized at a single qubit (Q1), one may additionally discard quantum
gates outside the light-cone of Q1 originating from the right-end of
$\hat{U}^{\dagger}$, without altering circuit output. The gates removed by the
light-cone filter are shown with semi-transparent colors in Fig. S12A. In
practice, some qubits have much longer idling times as a result of gate
removal and become more susceptible to decoherence effects such as relaxation
and dephasing. To mitigate such effects, we also apply spin echo to qubits
with long idling times, similar to the approach to the ancilla qubit in Fig.
S10.
The effects of the light-cone filter on OTOC measurements are shown in Fig.
S12B and Fig. S12C. The left panel of Fig. S12B shows the configuration for a
53-qubit OTOC experiment. Here we choose a quantum circuit $\hat{U}$ with
iSWAP and random single-qubit gates which are $\pi/2$ rotations around axes on
the $XY$ plane. The axes of rotation are chosen such that exactly 48 non-
Clifford rotations (randomly selected from $\sqrt{\pm W}$ and $\sqrt{\pm V}$)
occur in $\hat{U}$ and $\hat{U}^{\dagger}$. All other single-qubit gates are
Clifford rotations randomly selected from $\sqrt{\pm X}$ and $\sqrt{\pm Y}$).
The number of circuit cycles is fixed at 11. The right panels of Fig. S12B
show experimental results for 100 individual instances of $\hat{U}$, whereby
$\braket{\hat{\sigma}_{\text{y}}}$ for the normalization case is plotted at
the top and $\braket{\hat{\sigma}_{\text{y}}}$ with the butterfly operator
($X$) applied is plotted at the bottom. We observe significant enhancements in
the amplitudes of $\braket{\hat{\sigma}_{\text{y}}}$ after the application of
the light-cone filter, with the normalization
$\braket{\hat{\sigma}_{\text{y}}}$ values averaging to 0.024 without the
filter and 0.194 with filter.
The experimental improvement facilitated by the light-cone filter is more
clearly seen by comparing the normalized OTOC values $C$ with exact numerical
simulation of the same circuits, as shown in Fig. S12C. Without the light-cone
filter, the experimental $C$ values are substantially different from
simulation results. On the other hand, with the light-cone filter, the
agreement between experimental and simulated values is much closer. We further
quantify the effect of light-cone filter by plotting the differences between
numerical and experimental values of $C$, $\epsilon$, in Fig. S12D. Here we
observe that the root-mean-square (RMS) value for $\epsilon$ is 0.156 without
the light-cone filter, whereas it is reduced to 0.041 after filter is applied.
This four-fold improvement in accuracy of OTOC measurements is a natural
consequence of the reduction of number of gates in the overall quantum circuit
(the number of iSWAP gates is reduced from 464 to 161 for the example in Fig.
S12). Although the additional qubit idling introduced by the light-cone filter
carries errors as well, they are expected to be much less than the errors of
the removed two-qubit gates, particularly when spin echo is also applied
during the idling.
### II.6 Normalization via Reference Clifford Circuits
Figure S13: Normalization via reference Clifford circuits. (A) Schematic of
two quantum circuits: $\hat{U}$ is the actual quantum circuit of interest,
composed mostly of Clifford gates and a few non-Clifford gates ($\sqrt{\pm V}$
and $\sqrt{\pm W}$). $\hat{U}_{\text{ref}}$ is a reference circuit with the
same Clifford gates as $\hat{U}$. The non-Clifford gates in $\hat{U}$ are
replaced with random Clifford gates $\sqrt{\pm X}$ and $\sqrt{\pm Y}$ in
$\hat{U}_{\text{ref}}$. (B) Left panel shows an experimental configuration for
comparing two normalization procedures. The black unfilled (filled) circle
represents the ancilla (measurement) qubit. The purple filled circle
represents the butterfly qubit. The total number of circuit cycles is 14.
Right panel shows the measured values of $\braket{\hat{\sigma}_{\text{y}}}$
for 100 random circuit instances. $\braket{\hat{\sigma}_{\text{y}}}_{0}$
denotes values obtained without applying butterfly operator to $\hat{U}$,
$\braket{\hat{\sigma}_{\text{y}}}_{1}$ denotes values obtained with butterfly
operator applied to $\hat{U}_{\text{ref}}$, and
$\braket{\hat{\sigma}_{\text{y}}}_{2}$ denotes values obtained with butterfly
operator applied to $\hat{U}$. The same number of repetitions (4 millions) is
used in all cases to estimate $\braket{\hat{\sigma}_{\text{y}}}$. (C)
Normalized experimental OTOC values $C$ for different circuit instances,
plotted alongside exact numerical simulation results. (D) Experimental errors
$\epsilon$ for different circuit instances, corresponding to the differences
between experimental and simulated values.
The last error-mitigation technique we use for the OTOC experiment is specific
to quantum circuits $\hat{U}$ composed of predominantly Clifford gates with a
small number of non-Clifford gates. For such circuits, it is found through
numerical studies that a modified normalization procedure yields more accurate
values of OTOC (Fig. S13A): Consider a quantum circuit $\hat{U}$ composed of
mostly Clifford gates (iSWAP, $\sqrt{\pm X}$ and $\sqrt{\pm Y}$) and a small
number of non-Clifford gates ($\sqrt{\pm V}$ and $\sqrt{\pm W}$). We first
measure the $\braket{\hat{\sigma}_{\text{y}}}$ of the ancilla with a butterfly
operator applied between $\hat{U}$ and $\hat{U}^{\dagger}$ (we denote this
value as $\braket{\hat{\sigma}_{\text{y}}}_{2}$), same as before. In a second
step, instead of measuring $\braket{\hat{\sigma}_{\text{y}}}$ without applying
the butterfly operator (denoted by $\braket{\hat{\sigma}_{\text{y}}}_{0}$), we
measure $\braket{\hat{\sigma}_{\text{y}}}$ with the same butterfly operator
but a different quantum circuit $\hat{U}_{\text{ref}}$ and its inverse
$\hat{U}_{\text{ref}}^{\dagger}$ (denoted by
$\braket{\hat{\sigma}_{\text{y}}}_{1}$). The reference circuit
$\hat{U}_{\text{ref}}$ has the same Clifford gates as $\hat{U}$, whereas the
non-Clifford gates in $\hat{U}$ are replaced with Clifford gates chosen
randomly from $\sqrt{\pm X}$ and $\sqrt{\pm Y}$.
Example data showing $\braket{\hat{\sigma}_{\text{y}}}_{1}$,
$\braket{\hat{\sigma}_{\text{y}}}_{2}$ and
$\braket{\hat{\sigma}_{\text{y}}}_{3}$ are shown in Fig. S13B, where $\hat{U}$
contains a total of 8 non-Clifford rotations and 14 cycles. Similar to the
previous section, we present results from 100 circuit instances. Next, we
process the data to obtain experimental values of OTOC, $C$, in two different
ways: First, we apply
$C=\braket{\hat{\sigma}_{\text{y}}}_{2}/\braket{\hat{\sigma}_{\text{y}}}_{0}$,
which corresponds to the normalization procedure used in the previous
sections. Second, we apply
$C=\braket{\hat{\sigma}_{\text{y}}}_{2}/\left|\braket{\hat{\sigma}_{\text{y}}}_{1}\right|$,
corresponding to normalization using $\braket{\hat{\sigma}_{\text{y}}}$ of a
reference circuit. Here the absolute sign accounts for the fact that the
theoretical OTOC values of Clifford circuits are $\pm 1$. The resulting $C$
values are both plotted alongside exact simulation results in the left panel
of Fig. S13C. It is easily seen that the second normalization procedure with
reference Clifford circuits yields experimental values that are in much better
agreement with simulation results. Indeed, the experimental errors $\epsilon$
(Fig. S13D) have an RMS value of 0.250 when
$C=\braket{\hat{\sigma}_{\text{y}}}_{2}/\braket{\hat{\sigma}_{\text{y}}}_{0}$
is used and 0.157 when
$C=\braket{\hat{\sigma}_{\text{y}}}_{2}/\left|\braket{\hat{\sigma}_{\text{y}}}_{1}\right|$
is used. Given these observations, we adopt normalization via reference
Clifford circuits when measuring quantum circuits dominated by Clifford gates.
Lastly, we note that for the data in Fig. 3 and Fig. 4 of the main text, we
apply a ensemble of reference circuits $\hat{U}_{\text{ref}}$ and use the
average value of $\left|\braket{\hat{\sigma}_{\text{y}}}\right|$ obtained from
all $\hat{U}_{\text{ref}}$ to normalize $\braket{\hat{\sigma}_{\text{y}}}$ of
the actual quantum circuit $\hat{U}$. The typical number of
$\hat{U}_{\text{ref}}$ for each $\hat{U}$ is 10 in Fig. 3 and varies between
15 and 70 for Fig. 4.
## III Large-Scale Simulation of OTOCs of Individual Circuits
In the last few years, there has been a constant development of new numerical
techniques to simulate large scale quantum circuits. Among the many promising
methods, two major numerical techniques are widely used on HPC clusters for
large scale simulations: tensor contraction Markov and Shi (2008); Chen _et
al._ (2018); Villalonga _et al._ (2020, 2019); Huang _et al._ (2020); Gray
and Kourtis (2020) and Clifford gate expansion Gottesman (1998); Aaronson and
Gottesman (2004); Bravyi and Gosset (2016); Bravyi _et al._ (2019). All the
aforementioned methods have advantages and disadvantages, which mainly depend
on the underlying layout of the quantum circuits and the type of used gates.
On the one hand, tensor contraction works best for shallow circuits with a
small treewidth Boixo _et al._ (2017); Gray and Kourtis (2020). On the other
hand, Clifford gate expansion is mainly used to simulate arbitrary circuit
layouts with few non-Clifford gates. Indeed, it is well known that circuits
composed of Clifford gates only can be simulated in polynomial time Gottesman
(1998), with a numerical cost which grows exponentially with the number of
non-Clifford gates Aaronson and Gottesman (2004). Both methods can be used to
sample exact and approximate amplitudes, with a computational cost which
decreases with an increasing level of noise. For instance, approximate
amplitudes can be sampled by slicing large tensor network and contracting only
a fraction of the resulting sliced tensors Markov _et al._ (2018, 2019). The
final fidelity of the sample amplitudes is therefore proportional to the
fraction of contracted slices Villalonga _et al._ (2020, 2019), which can be
tuned to match experimental fidelity. Similarly, it is possible to sample
approximate amplitudes by only selecting the dominant stabilizer states in the
Clifford expansion Bravyi and Gosset (2016); Bravyi _et al._ (2019).
In our numerical simulations, we used tensor contraction to compute
approximate OTOC values, which are then validated using results from the
Clifford expansion for circuits with a small number of non-Clifford gates.
Both methods are described in the following sections.
### III.1 Numerical Calculation of the OTOC Value
As described in the main text and shown in Fig. 1(A), the experimental OTOC
circuits have density-matrix-like structure of the form
$\hat{C}=\hat{U}^{\dagger}\,\hat{\sigma}^{(Q_{b})}\,\hat{U}$, with
$\hat{\sigma}^{(Q_{b})}$ being the butterfly operator. In all numerical
simulations, we used iSWAPs as entangling two-qubit gates. Before and after
$\hat{C}$, a controlled-$Z$ gate is applied between the qubit $Q_{1}$ (in
$\hat{C}$) and an ancilla qubit $Q_{a}$ (external to $\hat{C}$): the OTOC
value is therefore obtained by computing the expectation value of
$\langle\hat{\sigma}_{y}\rangle$ relative to the ancilla qubit $Q_{a}$. To
reduce the computational cost, it is always possible to project the ancilla to
either $0$ or $1$ (in the computational basis). Let us call
$\hat{C}_{0}=\hat{C}$
($\hat{C}_{1}=\hat{\sigma}^{(Q_{1})}_{z}\,\hat{C}\,\hat{\sigma}^{(Q_{1})}_{z}$)
the circuit with the ancilla qubit projected on $0$ ($1$). Therefore, the OTOC
value $\langle\hat{\sigma}_{y}\rangle$ can be obtained as:
$\langle\hat{\sigma}_{y}\rangle=\mathbb{R}\left[\langle\psi_{1}|\psi_{0}\rangle\right],$
(S4)
with $|\psi_{0}\rangle=\hat{C}_{0}|+\rangle$ and
$|\psi_{1}\rangle=\hat{C}_{1}|+\rangle$ respectively.
### III.2 Branching Method
To get exact OTOC value for circuit with a small number of non-Clifford
rotations, we used a branching method based on the Clifford expansion. More
precisely, recalling that OTOC circuits have a density-matrix-like structure,
that is
$\hat{C}=\hat{U}^{\dagger}\,\hat{\sigma}^{(Q_{b})}\,\hat{U}=\big{(}\hat{g}_{t}^{\dagger}\cdots\hat{g}_{1}^{\dagger}\big{)}\,\hat{\sigma}^{(Q_{b})}\,\big{(}\hat{g}_{1}\cdots\hat{g}_{t}\big{)}$,
it is possible to apply each pair of gates
$\big{\\{}\hat{g}_{t},\,\hat{g}^{\dagger}_{t}\big{\\}}$ to
$\hat{\sigma}^{(Q_{b})}$ iteratively and “branch” only for non-Clifford gates.
At the beginning of the simulation, a Pauli “string” is initialized to all
identities except a $\hat{\sigma}_{x}^{(Q_{b})}$ operator (the butterfly
operator) on the butterfly qubit $Q_{b}$ (in this examples, the Pauli $X$ is
chosen as butterfly operator), that is
$\mathcal{P}=\hat{I}^{(1)}\hat{I}^{(2)}\cdots\hat{\sigma}_{x}^{(Q_{b})}\cdots$.
Whenever a pair of Clifford operators
$\big{\\{}\hat{g}_{t},\,\hat{g}^{\dagger}_{t}\big{\\}}$ is applied to
$\mathcal{P}$, the Pauli string is “evolved” to another Pauli string. For
instance, an iSWAP operator evolves the Pauli string
$\mathcal{P}=\hat{I}\,\hat{\sigma}_{x}$ to
$\mathcal{P}^{\prime}=\text{iSWAP}^{\dagger}\,\big{(}\hat{I}\,\hat{\sigma}_{x}\big{)}\,\text{iSWAP}=-\,\hat{\sigma}_{y}\hat{\sigma}_{z}$.
On the contrary, if a non-Clifford operator is applied to $\mathcal{P}$, the
Pauli string $\mathcal{P}$ will evolve into a superposition of multiple Pauli
strings, that can be eventually explored as independent branches. As an
example, the non-Clifford rotation $\hat{g}_{t}=\sqrt{W}=\sqrt{X+Y}$ will
branch $\mathcal{P}=\hat{\sigma}_{x}$ three times into
$\mathcal{P}^{\prime}_{1}=\frac{\hat{\sigma}_{x}}{\sqrt{2}}$,
$\mathcal{P}^{\prime}_{2}=\frac{\hat{\sigma}_{y}}{\sqrt{2}}$ and
$\mathcal{P}^{\prime}_{3}=\frac{\hat{\sigma}_{z}}{2}$ respectively. Because
$\big{\\{}\hat{g}_{t},\,\hat{g}^{\dagger}_{t}\big{\\}}$ are always applied in
pairs (one gate from $U$ and one gate from $U^{\dagger}$), the computational
complexity depends on the number $N_{D}$ of non-Clifford gates in $U$ only
($U$ and $U^{\dagger}$ may have a different number of non-Cliffords because of
the different lightcones acting on them. See Fig. S12A). More precisely, our
branching algorithms scales as the number of branches $n_{b}$ induced by the
$N_{D}$ non-Clifford rotations in $U$, that is
$\mathcal{O}\left(n_{b}\right)$.
After the applications of all $\left\\{\hat{g}_{i}\right\\}_{i=1,\,\ldots,t}$
gates, the OTOC circuit $\hat{C}$ will be then represented as a superposition
of distinct Pauli strings, each with a different amplitude. The full states
$|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ can be then obtained by applying the
initial state $|+\rangle$ to the Clifford expansion of $\hat{C}$ and,
therefore, the OTOC value from Eq. (S4). To reduce the memory footprint of our
branching method, the initial state $|+\rangle$ is always applied to all Pauli
strings once the last gate in $U$ is applied. Therefore, our branching
algorithm will output both $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ as a
superposition of binary strings. Because some of binary string composing
$|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ may have a zero amplitude, (due to
destructive interference), the number of binary strings $n_{p}$ composing
$|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ is typically smaller than the number
of explored branches $n_{b}$, that is $n_{p}\leq n_{b}$.
Figure S14: (Top) Runtime (in seconds) to explore a given number of branches
(results are for nodes with 2 six-core Intel Xeon X5670@2.93GHz). (Bottom)
Number $n_{b}$ of explored branches by varying the number $N_{D}$ of non-
Clifford rotations (boxes extend from the lower to upper quartile values of
the data, with a line at the median, while whiskers correspond to the
$5\%-95\%$ confidence interval). The inset shows the projected runtime on
Summit.
Fig S14 (top) shows the number of explored branches $n_{b}$ by varying the
number $N_{D}$ of non-Clifford rotations in $\hat{U}$. Because the only non-
Clifford gates used in the OTOC experimental circuits are $\sqrt{W^{\pm}}$ and
$\sqrt{V^{\pm}}$, with $W=X+Y$ and $V=X-Y$ respectively, one can compute the
expected scaling by assuming that, at each branching point, there is an
homogeneous probability to find any of the four Pauli operators
$\left\\{I,\,X,\,Y,\,Z\right\\}$. Because non-Clifford rotations branch only
twice on $Z$, thrice on $\left\\{X,\,Y\right\\}$ and never on $I$, the
expected scaling is $n_{b}\propto 3^{\frac{n}{2}}2^{\frac{n}{4}}$, which has
been confirmed numerically in Fig. S14 (top). Fig S14 (bottom) shows the
runtime (in seconds) to explore a given number of branches on single nodes of
the NASA cluster Merope NASA Ames Research Center . While the interface of our
branch simulator is completely written in Python, the core part is just-in-
time (JIT) compiled using numba to achieve $C-$like performance. Our branch
simulator also uses multiple threads (24 threads on the two 2 six-core Intel
Xeon<EMAIL_ADDRESS>nodes) to explore multiple branches at the same time and it
can explore a single branch in $\sim\\!\\!18\,\mu s$. Because multithreading
starts for $n_{b}>10^{3}$ only, it is possible to see the small jump caused by
the multithreading overhead. The inset of Fig. S14 shows the projected runtime
on Summit by rescaling to Summit’s $\text{{R}}_{\text{max}}$ Meuer _et al._
($\text{{R}}_{\text{max}}^{\text{Summit}}=200,\\!794.9\ \text{TFlops}$,
$\text{{R}}_{\text{max}}^{\text{Merope (single node)}}=140.64\
\text{GFlops}$), assuming that $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ can
be fully stored in Summit (seed Fig. S15). In Fig. S15, we report the number
of elements different from zero in $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$.
As one can see, the number of elements different from zeros scales as
$2^{0.79\,N_{D}}$ and can be accommodated on single nodes (shaded area
corresponds to the amount of virtual memory [RAM] used by the simulator).
Because branches are explored using a depth-first search strategy, most of the
virtual memory used by our branching algorithm is reserved to store
$|\psi_{0}\rangle$ and $|\psi_{1}\rangle$, it would be in principle possible
to simulate between $N_{D}\approx 50$ and $N_{D}\approx 60$ non-Clifford
rotations on Summit before running out of virtual memory (Summit has $10\
\text{PB}$ of available DDR4 RAM among its $4,\\!608$ nodes Oak Ridge National
Laboratory () (ORNL)).
Figure S15: Number $n_{p}$ of bitstrings in $|\psi_{0}\rangle$ and
$|\psi_{1}\rangle$ which are different from zero after applying the butterfly
operator $\hat{C}=\hat{U}^{\dagger}\hat{\sigma}^{(Q_{b})}\hat{U}$ to the
initial state $|+\rangle$, by varying the number $N_{D}$ of non-Clifford
rotations. The shaded area correspond to the peak of the amount of virtual
memory (RAM) used to simulate the OTOC circuits ($95\%$ among different
simulations).
### III.3 Tensor Contraction
Figure S16: (Top) Comparison between exact and approximate OTOC values
(circuits are ordered accordingly to the exact OTOC value over different
circuits with different depths, layouts and numbers of non-Clifford
rotations). The Pearson coefficient between exact and approximate OTOC values
is $R=0.99987$. (Bottom) Absolute error by varying the number $N_{D}$ of non-
Clifford rotations (boxes extend from the lower to upper quartile values of
the data, with a line at the median, while whiskers correspond to the
$5\%-95\%$ confidence interval).
Tensor contraction is a powerful tool to simulate large quantum circuits
Markov and Shi (2008); Chen _et al._ (2018); Villalonga _et al._ (2020,
2019); Huang _et al._ (2020); Gray and Kourtis (2020). In our numerical
simulations, we use tensor contraction to compute approximate OTOC values. It
is well known that approximate amplitudes can be sampled by properly slicing
the tensor network and only contracting a fraction of the sliced tensor
networks Markov _et al._ (2018, 2019); Villalonga _et al._ (2020, 2019).
However, in our numerical simulations, we used a different approach to compute
approximate OTOC values. More precisely, rather than computing one
(approximate) amplitude at a time using tensor contraction, we compute (exact)
“batches” of amplitudes by leaving some of the terminal qubits in the tensor
network “open”. Let us call $\kappa$ a given projection of the non-open qubits
and us define $|\psi_{0}^{(\kappa)}\rangle$ and $|\psi_{1}^{(\kappa)}\rangle$
the projection of $|\psi_{0}\rangle=\sum_{\kappa}|\psi_{0}^{(\kappa)}\rangle$
and $|\psi_{1}\rangle=\sum_{\kappa}|\psi_{1}^{(\kappa)}\rangle$ respectively.
Therefore, we can re-define the OTOC value as a “weighted” average of partial
OTOC values, that is:
$\langle\hat{\sigma}_{y}\rangle:=\frac{\sum_{\kappa}\omega_{\kappa}\langle\hat{\sigma}_{y}\rangle_{\kappa}}{\sum_{\kappa}\omega_{\kappa}},$
(S5)
with
$\langle\hat{\sigma}_{y}\rangle_{\kappa}=\frac{\mathbb{R}\big{[}\langle\psi_{1}^{(\kappa)}|\psi_{0}^{(\kappa)}\rangle\big{]}}{\big{(}\lVert\psi_{0}^{(\kappa)}\rVert^{2}+\lVert\psi_{1}^{(\kappa)}\rVert^{2}\big{)}/2}$
(S6)
and
$\omega_{\kappa}=\big{(}\lVert\psi_{0}^{(\kappa)}\rVert^{2}+\lVert\psi_{1}^{(\kappa)}\rVert^{2}\big{)}/2$.
It is interesting to observe that Eq. (S6) corresponds to the correlation
coefficient between two non-normalized states and that the exact OTOC value is
equivalent to the weighted average of single projection OTOC values. Indeed,
when all $\kappa$ are included, Eq. (S5) reduces to Eq. (S4). However, because
$\langle\sigma_{y}\rangle$ needs to be $\mathcal{O}(1)$ to be experimentally
measurable, few projection $\kappa$ may actually be sufficient to get a good
estimate of Eq. (S5).
In our numerical simulations, we left $24$ qubits open and we used $20$ random
$\kappa$ projections to compute an approximate OTOC value using Eq. (S5). If
not indicated otherwise, medians and confidence intervals are computed by
bootstrapping $1,\\!000$ times Eq. (S5) using $10$ randomly chosen projections
among the $20$ available. Fig. S16 shows the comparison of approximate and
exact OTOC values, by varying the circuit index (top) and by varying the
number of non-Clifford rotations (bottom). As one can see, there is a great
agreement between approximate and exact results across different circuits with
different depth, layout and number of non-Clifford rotations (top). The bottom
part of Fig. S16 shows the absolute error by varying the number of non-
Clifford rotations. In particular, we are comparing results by averaging over
single projections $\kappa$ using Eq. (S6) (orange/light gray boxes) or by
bootstrapping $1,\\!000$ times Eq. (S5) using $10$ randomly chosen projections
among the $20$ available (blue/dark gray boxes). As expected, results become
less noisy by increasing the number of used projections.
Figure S17: Number of slices required to fit the largest tensor in memory
during tensor contraction with a threshold of $2^{28}$ elements by varying the
number $N_{S}$ of iSWAPs (boxes extend from the lower to upper quartile values
of the data, with a line at the median, while whiskers correspond to the
$5\%-95\%$ confidence interval). Green stars correspond to expected number of
slices to compute single amplitudes of random quantum circuits (RQC) with a
similar depth Arute _et al._ (2019); Gray and Kourtis (2020). (Inset) Number
of elements to store in memory for the largest tensor during tensor
contraction. For sufficiently deep circuits, the number of elements for the
largest tensor saturates to the size of the Hilbert space $2^{n}$, with $n=53$
the number of qubits in the Sycamore chip.
Due to the limited amount of memory in HPC nodes, slicing techniques are
required to fit the tensor contraction on a single node and avoid node-to-node
memory communication overhead Villalonga _et al._ (2019); Gray and Kourtis
(2020). In our numerical simulations, we used cotengra Gray and Kourtis (2020)
and quimb Gray (2018) to identify the best contraction (including slicing) and
perform the actual tensor contraction respectively. Because each different
projection $\kappa$ may lead to a slightly different simplification of the
tensor network (in our numerical simulations, we used the rank and column
reduction included in the quimb library), we recompute the best contraction
for each single projection. For each projection (regardless of the
depth/number of iSWAPs) we fixed max_repeats = 128 in cotengra and restarted
the heuristics $10$ times to identify the optimal contraction (with a hard
limit of $2$ minutes for each run), using $24$ threads on a 2 six-core Intel
Xeon<EMAIL_ADDRESS>node. We found that the runtime to the best contraction
scales as $0.03N_{S}-3.68$ minutes, with $N_{S}$ the number of iSWAPs in the
circuit, that is less than $20$ minutes for $\sim\\!600$ iSWAPs. Fig. S17
shows the number of slices required to have the largest tensor in the tensor
contraction no larger than $2^{28}$ elements. Green stars in the figure
correspond to the number of slices to compute single amplitudes of random
quantum circuits with a similar number of iSWAPs Arute _et al._ (2019); Gray
and Kourtis (2020). Because OTOC circuits and the random quantum circuits
presented in Arute _et al._ (2019) share a similar randomized structure, the
computational complexity mainly depends on the number of iSWAPs (OTOC slicing
is slightly worse because of the open qubits). We may expect an improvement in
the slicing by using novel techniques as the “subtree reconfiguration”
proposed by Huang _et al._ Huang _et al._ (2020). It is interesting to
observe that, for deep enough circuits, the number of elements of the largest
tensor in the tensor contraction saturates to the number of qubits (inset).
Figure S18: (Top) Runtime (in seconds) to fully contract a single projection
by varying the required number of FLOPs (results are for single nodes with 2
six-core Intel Xeon X5670@2.93GHz). Because long double are used throughout
the simulation, we used the conversion factor $8$ to convert FLOPs to actual
time. (Bottom) Flops to contract a single projection by varying the number
$N_{S}$ of iSWAPs, with and without the slicing overhead. Summit’s days are
obtained by using Summit’s $\text{{R}}_{\text{peak}}$ Meuer _et al._ and the
conversion factor $8$ from FLOPs to actual time (because long double’s have
been used throughout the simulation). Green stars correspond to expected
number of FLOPs to compute single amplitudes of random quantum circuits (RQC)
with a similar number of iSWAPs Arute _et al._ (2019); Gray and Kourtis
(2020).
Fig. S18 summarizes the computational cost to compute an exact single
projection $\kappa$. Top panel of Fig. S18 reports the actual runtime to
contract a tensor network, by varying its expected total cost (including the
slicing overhead) in FLOPs (expected FLOPs are using by cotengra to identify
an optimal contraction). Dashed and dot-dashed lines correspond to the peak
performance and expected performance of a 2 six-core Intel Xeon<EMAIL_ADDRESS>node. The sustained performance of $64\%$ is consistent with similar analysis
on the NASA cluster Villalonga _et al._ (2019). Bottom panel reports the
number of FLOPs (with and without the slicing overhead) by varying the number
of iSWAPs. On the right y-axis, it is reported the extrapolated number of days
to simulate OTOC circuits of a given number of iSWAPs, by assuming a sustained
performance of $148,\\!600$ TFLOPs. Green stars corresponds to the total FLOPs
(including the slicing overhead) to compute single amplitudes of random
quantum circuits with a similar number of iSWAPs Gray and Kourtis (2020).
## IV Markov population dynamics
For a broad class of circuit ensembles the average OTOC can be computed
efficiently (in polynomial time) on a classical computer. This appears
surprising as computing the output of the random circuit is expected to
require exponential resource. This contradiction is resolved by demonstrating
an exact mapping of the average evolution of OTOC onto a Markov population
dynamics process. Such connection was identified for Hamiltonian dynamics
Aleiner _et al._ (2016) and subsequently for simplified models of random
circuits Nahum _et al._ (2018); von Keyserlingk _et al._ (2018) where
uniformly random two-qubit gate was assumed. In practice, we implement a
specific gate set consisting of “cycles” of the form $\prod_{\langle
ij\rangle}\hat{G}_{i}\hat{G}_{j}\hat{U}_{ij}(\theta,\phi)$ applied to non-
overlapping pairs of nearest neighbour qubits $\langle i,j\rangle$. For each
pair a cycle consists of two single qubit gates $\hat{G}_{i},\hat{G}_{j}$ and
an entangling two-qubit gate,
$\displaystyle\hat{U}_{i,j}(\theta,\phi)=e^{-\frac{i}{2}\theta\left(X_{i}X_{j}+Y_{i}Y_{j}\right)-\frac{i\phi}{2}Z_{i}Z_{j}},$
(S7)
parameterized by fixed angles $\theta,\phi$. The random instances are
generated by drawing single qubit gates from a specific finite set
$\left\\{\hat{G}_{i}\right\\}$. We consider two sets corresponding to
generators of single qubit Clifford group $\\{\sqrt{X^{\pm 1}},\sqrt{Y^{\pm
1}}\\}$ and the set which in conjunction with any entangling two-qubit gate
generates a universal set $\\{\sqrt{X^{\pm 1}},\sqrt{Y^{\pm 1}},\sqrt{W^{\pm
1}},\sqrt{V^{\pm 1}}\\}$ introduced in Sec. I.
Average OTOC can be calculated by first considering the dynamics of the pair
of butterfly operators $\hat{O}(t)=U^{\dagger}\hat{O}U$ evolved under circuit
$U$. We introduce the average of the pair of the operators acting in the
direct product of two Hilbert spaces of the two replicas of the same circuit:
$\displaystyle\hat{\cal
O}^{(2)}(t)=\overline{\hat{O}(t)\otimes\hat{O}(t)}\equiv\overline{\hat{O}(t)^{\otimes
2}}$ (S8)
where averaging over ensemble of the circuits is denoted by
$\overline{(...)}$, and the rightmost equation defines the short-hand
notation.
Analogously, one can introduce the higher order averages
$\displaystyle\hat{\cal
O}^{(4)}(t)=\overline{\hat{O}(t)\otimes\hat{O}(t)\otimes\hat{O}(t)\otimes\hat{O}(t)}\equiv\overline{\hat{O}(t)^{\otimes
4}},$ (S9)
as a starting point to analyze the circuit to circuit fluctuations of $C(t)$.
Then the value of OTOC is obtained by taking the matrix element of
$C(t)=\bra{\psi}\overline{\hat{M}\hat{O}(t)\hat{M}\hat{O}(t)}|\psi\rangle$
with the initial state that in our experimental setup is chosen to be
$|\psi\rangle=\bigotimes_{i=1}^{n}|+\rangle_{i}$ where $|+\rangle_{i}$ is the
symmetric superposition of the computational basis states. The average and the
second moment of $C(t)$ are obtained as a straightforward convolution of the
indices in Eqs. (S8) and (S9) respectively. Operator $\hat{\cal O}^{(4)}$ can
be also used to study the effect of the initial state to the OTOC (clearly its
average is not sensitive to the initial conditions).
### IV.1 Symmetric single qubit gate set
We first consider average over uniformly random single qubits gate such that
for any Pauli matrix,
$\displaystyle\overline{\hat{\alpha}_{i}}\equiv\overline{\hat{G}^{\dagger}\hat{\alpha}_{i}\hat{G}}=0,$
(S10)
We will analyze the specific discrete gate sets used in the experiment in
Section IV.4. In this section, we use the Latin indices to label a qubit, and
the Greek ones to denote the corresponding Pauli matrices
$\hat{\alpha}_{i}=\\{\hat{X}_{i},\hat{Y}_{i},\hat{Z}_{i}\\}$
For a pair of operators as in Eq. (S8) there exist an analog of scalar product
– a spherically symmetric combination that does not vanish after averaging,
$\displaystyle\overline{\hat{\alpha}_{i}\otimes\hat{\beta}_{j}}=\hat{G}^{\dagger}\hat{\alpha}_{i}\hat{G}\otimes\hat{G}^{\dagger}\hat{\beta}_{j}\hat{G}=\delta_{\alpha\beta}\delta_{ij}\mathcal{B}_{i},$
(S11)
where we introduced “bond” notation,
$\displaystyle\mathcal{B}_{i}\equiv\frac{1}{3}\hat{\alpha}_{i}\otimes\hat{\alpha}_{i}\equiv\frac{1}{3}\left(\hat{X}_{i}^{\otimes
2}+Y_{i}^{\otimes 2}+Z_{i}^{\otimes 2}\right),$ (S12)
and the summation over the repeated Greek indices is always implied.
Analogously one can find non-vanishing averages of the Pauli matrices acting
in four replicas space of the same qubit (we will omit the qubit label). There
are in total six bilinear combinations
$\displaystyle\overline{\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\openone}\otimes\hat{\openone}}=\delta_{\alpha\beta}\mathcal{B}^{(12)},$
$\displaystyle\mathcal{B}^{(12)}=\frac{1}{3}\hat{\alpha}\otimes\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\openone},$
(S13a)
$\displaystyle\overline{\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\beta}\otimes\hat{\openone}}=\delta_{\alpha\beta}\mathcal{B}^{(13)},$
$\displaystyle\mathcal{B}^{(12)}=\frac{1}{3}\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\openone},$
$\displaystyle\vdots$
$\displaystyle\overline{\hat{\openone}\otimes\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\beta}}=\delta_{\alpha\beta}\mathcal{B}^{(34)},$
$\displaystyle\mathcal{B}^{(34)}=\frac{1}{3}\hat{\openone}\otimes\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\alpha}.$
Four cubic invariants are possible because there is no inversion operation for
the spin:
$\displaystyle\overline{\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma}}=\epsilon^{\alpha\beta\gamma}\mathcal{C}^{(1)},$
$\displaystyle\quad\mathcal{C}^{(1)}\equiv\frac{\epsilon^{\alpha\beta\gamma}}{6}\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma},$
(S13b)
$\displaystyle\overline{\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\beta}\otimes\hat{\gamma}}=\epsilon^{\alpha\beta\gamma}\mathcal{C}^{(2)},$
$\displaystyle\quad\mathcal{C}^{(2)}\equiv\frac{\epsilon^{\alpha\beta\gamma}}{6}\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\beta}\otimes\hat{\gamma},$
$\displaystyle\vdots$
$\displaystyle\overline{\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma}\otimes\hat{\openone}}=\epsilon^{\alpha\beta\gamma}\mathcal{C}^{(4)},$
$\displaystyle\quad\mathcal{C}^{(4)}\equiv\frac{\epsilon^{\alpha\beta\gamma}}{6}\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma}\otimes\hat{\openone}.$
where $\epsilon_{\alpha\beta\gamma}$ is the three-dimensional Levi-Civita
symbol. Because they would change sign under the inversion operation, the
cubic invariants can appear only in pairs on neighboring qubits.
Finally, the three quartic invariants are
$\displaystyle\overline{\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma}\otimes\hat{\delta}}=\mathcal{D}^{(2)}\Upsilon_{\alpha\beta\gamma\delta}+\mathcal{D}^{(3)}\Upsilon_{\alpha\gamma\beta\delta}+\mathcal{D}^{(4)}\Upsilon_{\alpha\gamma\delta\beta},$
(S13c)
$\displaystyle\Upsilon_{\alpha\beta\gamma\delta}=\delta_{\alpha\beta}\delta_{\gamma\delta}-\frac{1}{4}\delta_{\alpha\gamma}\delta_{\beta\delta}-\frac{1}{4}\delta_{\alpha\delta}\delta_{\beta\gamma},$
$\displaystyle\mathcal{D}^{(2)}=\frac{2}{15}\hat{\alpha}\otimes\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\beta},$
$\displaystyle\mathcal{D}^{(3)}=\frac{2}{15}\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\alpha}\otimes\hat{\beta},$
$\displaystyle\mathcal{D}^{(4)}=\frac{2}{15}\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\beta}\otimes\hat{\alpha}.$
This simple form implies the spherical symmetry of the single qubit averaging.
For a lower symmetry (e.g. all the rotations of a cube) other quartic and
cubic invariants are possible (like $X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes
4}$ or
$|\epsilon_{\alpha\beta\gamma}|\hat{\openone}\otimes\alpha\otimes\beta\otimes\gamma$)
but we will ignore them for the sake of simplicity.
### IV.2 Efficient Population Dynamics for the Averaged OTOC
We expand average evolution of the pair of butterfly operators (S8) as,
$\displaystyle\hat{\cal
O}^{(2)}(t)=\sum_{\\{v_{i}\\}}P_{\\{v_{i}\\}}\bigotimes_{i=1}^{n}\left(v_{i}\hat{\mathcal{B}}_{i}+u_{i}\hat{\openone}_{i}^{\otimes
2}\right).$ (S14)
where normalization condition reads,
$u_{i}+v_{i}=1,$ (S15)
the variable $v_{i}=\\{0,1\\},i=1,2,...,n$ indicates whether or not Pauli
matrices occupy qubit $i$, and $P_{\\{v_{i}\\}}$ are formfactors. In other
words, each $i$th is characterized either by
$\rho^{(0)}_{i}\equiv\openone_{i}^{\otimes 2}$, ”vacuum”, or the “bond”
$\mathcal{B}_{i}$. All other terms in the right hand side of Eq. (S14) vanish
upon averaging over single qubit gates see Sec. IV.1.
Application of a two-qubit Sycamore gate to a pair of qubits $\\{i,j\\}$ is
then described by $2^{2}\times 2^{2}$ matrix in the space $\\{v_{i},v_{j}\\}$,
$\displaystyle\Omega=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1-a-b&a&b\\\
0&a&1-a-b&b\\\ 0&\frac{b}{3}&\frac{b}{3}&\left(1-\frac{2}{3}b\right)\\\
\end{array}\right),$ (S20) $\displaystyle
a=\frac{1}{3}\left(2\sin^{2}\theta+\sin^{4}\theta\right),$ $\displaystyle
b=\frac{1}{3}\left(\frac{1}{2}\sin^{2}2\theta+2\left(\sin^{2}\theta+\cos^{2}\theta\right)\right).$
where the matrix $\Omega$ acts from the right on four dimensional row vector
with the basis $(00),(01),(10),(11)$ The standard $\sqrt{\text{iSWAP}}$ gate
corresponds to $\theta=\frac{\pi}{4}$, $a=\frac{1}{12},b=\frac{1}{2}$, and
iSWAP $\theta=\frac{\pi}{2}$ with $a=\frac{1}{3},b=\frac{2}{3}$.
Each time when the two-qubit gate is applied the formfactor $P$ is updated
according to the rules
$\displaystyle
P_{v_{1}\text{...}v_{i}v_{j}\text{...}v_{n}}(t+1)=\sum_{v_{i}^{\prime}v_{j}^{\prime}}P_{v_{1}\text{...}v_{i}^{\prime}v_{j}^{\prime}\text{...}v_{n}}(t)\Omega_{v_{i}^{\prime}v_{j}^{\prime},v_{i}v_{j}}.$
(S21)
Equations (S20)–(S21) are obtained by an application of a two-qubit gate (S7)
with $\phi=0$ to a pair $(i,j)$ of factors in Eq. (S14) and averaging the
result using Eqs. (S10) – (S11).
Some additional constraints can be extracted from the exact condition
$\left[\hat{O}(t)\right]^{2}=\bigotimes_{i=1}^{n}\hat{\openone}_{n}.$ (S22)
Convoluting operators in Eq. (S14), using $\hat{\openone}_{i}^{\otimes
2}\to\hat{\openone}_{i}$, $\hat{\mathcal{B}}_{i}\to\hat{\openone}_{i}$ and the
normalization condition (S15), we obtain the requirement
$\sum_{\\{v_{i}\\}}P_{\\{v_{i}\\}}(t)=1.$ (S23)
Preserving this condition in the update rule (S21) requires the elements in
each row of the matrix $\Omega$ from Eq. (S20) to add up to one. Moreover, all
the elements of $\Omega$ are non-negative and therefore not only is
$P_{\\{v_{i}\\}}(t)$ normalized but it is also non-negative.
Therefore, rules (S21) and (S23) defines the Markov process and variables
$v_{i}=0,1$ correspond to the classical population of each qubit. As the non-
populated states $v_{i}=v_{j}=0$ do not evolve and the reproducing
$(01)\to(11)$ and destruction $(11)\to(01)$ processes are allowed the problem
(S21) is nothing but the classical population dynamics. The formfactors
$P\left(v_{1}\text{...}v_{i}^{\prime}v_{j}^{\prime}\text{...}v_{n},t\right)$
are interpreted as a distribution function over an $n$ bit register
$\left\\{v_{1}\text{...}v_{n}\right\\}$, subject to the Markov process defined
by the update, Eq. (S20).
Direct solution of Eq. (S21) would require $2^{n}$ real numbers. However,
unlike the original unitary evolution, the classical population dynamics
involves only positive numbers constrained by normalization (S23). Such
dynamics is very efficiently stimulated using a classical Monte Carlo type
algorithm.
The butterfly operator is the starting point of the Markov process Eq. (S21).
At long times the distribution $P_{\\{v_{i}\\}}$ converges to the stationary
state $\bigotimes_{i=1}^{n}\rho_{i}^{(erg)}$, where
$\rho_{i}^{(erg)}=\frac{1}{4}\left(\openone_{i}^{\otimes
2}+3\mathcal{B}_{i}\right)$, which corresponds to ”vacuum” occurring on each
site $i$ with probability $p\left(\openone_{i}^{\otimes 2}\right)=\frac{1}{4}$
and ”bond” occurring with probability $p\left(\mathcal{B}_{i}\right)=3/4$.
OTOC in this limit takes the value corresponding to the random matrix
statistics Yoshida and Kitaev (2017). Intermediate dynamics of OTOC between
these two limits is fully described by the Markov process Eq. (S21). It has a
form of shock wave spreading from the initial butterfly operator to cover the
whole system. Note that the choice of the two qubit gate parameter $\theta$
has a dramatic effect on the intermediate OTOC dynamics. As discussed in the
main text $\theta=\pi/2$ corresponds to the probability of butterfly operator
to spread equal one, and therefore spreading with maximum velocity equal to
the light cone velocity and saturating the Lieb-Robinson bound. At any other
value of $\theta$ probability to spread is less then one which results in
diffusive broadening of the front, and the center of the front propagates with
a butterfly velocity that is smaller than the light cone velocity.
### IV.3 Sign Problem in the Population Dynamics for OTOC Fluctuations
As we already mentioned, the analytic calculation of the OTOC for an
individual circuit is impossible. One can expect, however, that it is possible
to express the variance of the OTOC in terms of some products of classical
propagators similarly to the analysis of the mesoscopic fluctuations in the
disordered metals. The purpose of this subsection is to show that even such a
modest task can not be efficiently undertaken.
The starting point is the expansion in terms of the single qubit rotations
invariants (S13). Similarly to Eq. (S14), we write for $\hat{\cal O}^{(4)}$ of
Eq. (S9)
$\begin{split}\hat{\cal
O}^{(4)}(t)=\sum_{\\{{\mathbf{V}}_{i}\\}}P_{\\{{\mathbf{V}}_{i}\\}}\bigotimes_{i=1}^{n}{\mathbf{V}}_{i}\cdot\hat{\bf\cal{Q}}_{i},\end{split}$
(S24)
where $\hat{\bf\cal{Q}}_{i}$ is the vector with $14=1+6+4+3$ operator
components given by invariants of Eq. (S9):
$\begin{split}&\hat{\bf\cal{Q}}_{i}=\left[\hat{\openone}^{\otimes
4}_{i},\mathbf{b},\mathbf{c},\mathbf{d}\right].\\\
&\mathbf{b}=\left(\hat{\mathcal{B}}^{(12)}_{i},...,\hat{\mathcal{B}}^{(34)}_{i}\right),\\\
&\mathbf{c}=\left(\hat{\mathcal{C}}^{(1)}_{i},...,\hat{\mathcal{C}}^{(4)}_{i}\right),\\\
&\mathbf{d}=\left(\hat{\mathcal{D}}^{(2)}_{i},\hat{\mathcal{D}}^{(3)}_{i},\hat{\mathcal{D}}^{(4)}_{i}\right),\end{split}$
(S25)
and ${\mathbf{V}}_{i}$ are the $14$ basis unit vectors so that $13$ components
equal to zero and the remaining component is $1$. In other words, each site
can be in $14$ possible states.
A straightforward generalization of the evolution equation (S21) reads
$\displaystyle
P_{{\mathbf{V}}_{1}\cdot{\mathbf{V}}_{i}{\mathbf{V}}_{j}\cdot{\mathbf{V}}_{n}}(t+1)=\sum_{{\mathbf{V}}_{i}^{\prime}{\mathbf{V}}_{j}^{\prime}}P_{{\mathbf{V}}_{1}\cdot{\mathbf{V}}_{i}^{\prime}{\mathbf{V}}_{j}^{\prime}\cdot{\mathbf{V}}_{n}}(t){\mathbf{\Omega}}_{{\mathbf{V}}_{i}^{\prime}{\mathbf{V}}_{j}^{\prime},{\mathbf{V}}_{i}{\mathbf{V}}_{j}},$
(S26)
where ${\mathbf{\Omega}}$ is now $14^{2}\times 14^{2}$ whose explicit form is
known but not quite important for the further consideration.
Equation (S26) involves $14^{n}$ real numbers. The only feasible path to the
solution would be an efficient Monte Carlo sampling. Naively, one can hope to
map the problem to the multicolored population dynamics. However, it is not
possible as we explain below.
Reliable Monte Carlo sampling requires (1) normalizable and non-negative
weights, and (2) absence of the correlations in contribution of different
configurations. Let us demonstrate that both conditions do not hold for the
evolution of formfactors in Eq. (S24). Once again, we use exact Eq. (S22).
Convoluting third and fourth replica in $\hat{\cal O}^{(4)}(t)$ we obtain
$\hat{\cal O}^{(4)}(t)\to\hat{\cal
O}^{(2)}(t)\otimes\hat{\mathbf{\openone}},\quad\hat{\mathbf{\openone}}\equiv\left[\bigotimes_{i=1}^{n}\hat{\openone}_{i}\right].$
(S27)
Convolutions of the four operators (S13) yield
$\displaystyle\mathcal{B}^{(12)}$
$\displaystyle\to\mathcal{B}^{(3)}\equiv\frac{1}{3}\hat{\alpha}\otimes\hat{\alpha}\otimes\hat{\openone},$
(S28) $\displaystyle\hat{\openone}^{\otimes 4},\mathcal{B}^{(34)}$
$\displaystyle\to\hat{\openone}^{\otimes 3},$
$\displaystyle\mathcal{B}^{(13)},\ \mathcal{B}^{(14)}$
$\displaystyle\to\mathcal{B}^{(2)}\equiv\frac{1}{3}\hat{\alpha}\otimes\hat{\openone}\otimes\hat{\alpha},$
$\displaystyle\mathcal{B}^{(23)},\ \mathcal{B}^{(24)}$
$\displaystyle\to\mathcal{B}^{(1)}\equiv\frac{1}{3}\hat{\openone}\otimes\hat{\alpha}\otimes\hat{\alpha},$
$\displaystyle\mathcal{C}^{(3)},\mathcal{C}^{(4)}$
$\displaystyle\to\mathcal{C}\equiv\frac{\epsilon^{\alpha\beta\gamma}}{6}\hat{\alpha}\otimes\hat{\beta}\otimes\hat{\gamma},$
$\displaystyle\mathcal{C}^{(1)}$ $\displaystyle\to
i{B}^{(1)},\qquad\mathcal{C}^{(2)}\to i{B}^{(2)},$
$\displaystyle\mathcal{D}^{(2)}$
$\displaystyle\to\frac{6}{5}\mathcal{B}^{(3)},$
$\displaystyle\mathcal{D}^{(3)}$
$\displaystyle\to\frac{2}{5}\mathcal{B}^{(3)}+\frac{4i}{5}\mathcal{C},$
$\displaystyle\mathcal{D}^{(4)}$
$\displaystyle\to\frac{2}{5}\mathcal{B}^{(3)}-\frac{4i}{5}\mathcal{C}.$
Let us convolute both sides of Eq. (S24) using the rules (S27) – (S28). We
find
$\begin{split}\hat{\cal
O}^{(2)}&(t)\otimes\hat{\mathbf{\openone}}=\sum_{\\{{\mathbf{V}}_{i}\\}}P_{\\{{\mathbf{V}}_{i}\\}}\bigotimes_{i=1}^{n}\left(q_{i}^{(1)}+q_{i}^{(2)}+q_{i}^{(3)}+q_{i}^{(4)}\right),\\\
q_{i}^{(1)}&\equiv V_{i}^{(1)}\hat{\openone}^{\otimes
3}_{i}+V_{i}^{(2)}\mathcal{B}^{(3)}_{i},\\\ q_{i}^{(2)}&\equiv
V_{i}^{(7)}\hat{\openone}^{\otimes
3}_{i}+\frac{2}{5}\left(3V_{i}^{(12)}+V_{i}^{(13)}+V_{i}^{(14)}\right)\mathcal{B}^{(3)}_{i},\\\
q_{i}^{(3)}&\equiv\left(V_{i}^{(3)}+V_{i}^{(4)}+iV_{i}^{(8)}\right)\mathcal{B}^{(1)}_{i}\\\
+&\left(V_{i}^{(5)}+V_{i}^{(6)}+iV_{i}^{(9)}\right)\mathcal{B}^{(2)}_{i},\\\
q_{i}^{(4)}&\equiv\left(V_{i}^{(10)}+V_{i}^{(11)}+\frac{4i}{5}V_{i}^{(13)}-\frac{4i}{5}V_{i}^{(14)}\right)\mathcal{C}_{i}.\end{split}$
Here, the superscript in vector $V_{i}^{(\cdot)}$ enumerates components
according to Eq. (S25). The total result is real as the imaginary terms are
always generated in pairs.
Configurations with $V_{i}^{(1)}=u_{i}$, $V_{i}^{(2)}=v_{i}$,
$V_{i}^{(3,..,14)}=0$, exactly reproduce averaged result (S14). It means that
all the other terms exactly cancel each other. It is possible only if the
corresponding formfactors can be of both signs. Therefore, any finite
unconstrained sampling leads to an arbitrary result (sign problem). Moreover,
the constraints are non-local. Consider, e.g. cancellation of contribution
proportional to $\mathcal{B}^{(1)}_{i}\mathcal{B}^{(1)}_{j}$. Cancellation
occurs only if the formfactors different by replacement
$V_{i}^{(8)}V_{j}^{(8)}\to V_{i}^{(a)}V_{j}^{(b)}$, $a,b=\\{3,4\\}$ must be
kept precisely the same. The requirement quickly becomes intractable with the
increasing of the number of non-trivial matrices involved into cancellation.
To summarize, the general requirements on the evolution of the $\hat{\cal
O}^{(4)}(t)$ lead to the sign and locality problems. Those two facts render a
brute force classical Monte Carlo algorithm impossible. At present, we are not
aware of any algorithm enabling to circumvent those obstacles.
### IV.4 Population Dynamics for iSWAP Gate Sets Implemented in the Main Text
Circuits with $\theta=\pi/2$ were used to realize Clifford as well as a
universal ensemble. It is therefore instructive to study dynamics of the
specific gate sets used in the experiment in more detail. Consider conjugation
of a pair of Pauli operators $\hat{\alpha}_{i}\hat{\beta}_{j}$ by the iSWAP
gate $\hat{S}$. It maps the pair onto another pair of Pauli operators
$\hat{\gamma}_{i}\hat{\delta}_{j}$ according to the following rules,
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr\hat{\alpha}_{i}\hat{\beta}_{j}&\hat{X}_{i}\hat{\openone}_{j}&\hat{Y}_{i}\hat{\openone}_{j}&\hat{Z}_{i}\hat{\openone}_{j}&\hat{Z}_{i}\hat{X}_{j}&\hat{Z}_{i}\hat{Y}_{j}&\hat{\alpha}_{i}\hat{\alpha}_{j}&\hat{X}_{i}\hat{Y}_{j}\\\
\hline\cr
S^{\dagger}\hat{\alpha}_{i}\hat{\beta}_{j}S&-\hat{Z}_{i}\hat{Y}_{j}&\hat{Z}_{i}\hat{X}_{j}&\hat{\openone}_{i}\hat{Z}_{j}&-\hat{Y}_{i}\hat{\openone}_{j}&\hat{X}_{i}\hat{\openone}_{j}&\hat{\alpha}_{i}\hat{\alpha}_{j}&\hat{Y}_{i}\hat{X}_{j}\\\
\hline\cr\end{array}$
#### IV.4.1 Clifford Gate Set
Clifford gate set used to obtain the data in the main text is drawn form the
single qubit gate set
$\\{\hat{G}\\}=\\{\sqrt{X},\sqrt{X^{-1}},\sqrt{Y},\sqrt{Y^{-1}}\\}$. Averaging
over this gates set of a symmetric pair of Pauli operators
$\alpha_{i}\otimes\alpha_{i}$ reads,
$\displaystyle\overline{\hat{G}^{\dagger}\hat{\alpha}\hat{G}\otimes\hat{G}^{\dagger}\hat{\alpha}\hat{G}}=\Xi_{\alpha},$
(S29)
where we introduce the basis
$\Xi_{\alpha}=\\{\openone,\mathbb{X},\mathbb{Y},\mathbb{Z}\\}$,
$\displaystyle\mathbb{X}=\frac{1}{2}(\hat{Z}^{\otimes 2}+\hat{Y}^{\otimes
2}),$ $\displaystyle\mathbb{Y}=\frac{1}{2}(\hat{X}^{\otimes
2}+\hat{Z}^{\otimes 2}),$
$\displaystyle\mathbb{Z}=\frac{1}{2}(\hat{X}^{\otimes 2}+\hat{Y}^{\otimes
2}).$
Single qubit gate average can be described in terms of the invariants
$\Xi_{\alpha}\to M^{(S)}_{\alpha\beta}\Xi_{\beta}$,
$\displaystyle M^{(S)}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&\tfrac{1}{2}&\tfrac{1}{2}\\\ 0&\tfrac{1}{2}&0&\tfrac{1}{2}\\\
0&\tfrac{1}{2}&\tfrac{1}{2}&0\\\ \end{array}\right).$ (S34)
In this new basis $\\{\Xi_{\gamma}\Xi_{\delta}\\}$ instead of a $4\times 4$
matrix $\Omega$ in Eq. (S20) effect of iSWAP gate is described by $16\times
16$ matrix constructed straightforwardly from the rules Eqs. (S34) and the
rules for iSWAP.
#### IV.4.2 Universal Gate Set
Figure S19: Extended data for Fig. 2 in the main text: Average OTOC for four
different locations of butterfly operator, $X$. Solid lines correspond to
experimental data and dashed lines to the population dynamics simulation with
Eq. (S39).
The universal gates set used in the main text consists of eight choices of
single qubit gates $\\{\hat{G}\\}=\\{\sqrt{X^{\pm 1}},\sqrt{Y^{\pm
1}},\sqrt{W^{\pm 1}},\sqrt{V^{\pm 1}}\\}$. For the butterfly operator we
choose $\hat{O}_{i}=\hat{X}_{i}$ the dynamics is described in the reduced
subspace spanned by the basis $\Lambda=\\{\openone^{\otimes
2},\hat{X}^{\otimes 2},\hat{Y}^{\otimes 2},\hat{Z}^{\otimes 2}\\}$. Average of
a pair of Pauli operators over this ensemble reads $\Lambda_{\alpha}\to
M^{(U)}_{\alpha\beta}\Lambda_{\beta}$,
$\displaystyle M^{(U)}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\tfrac{3}{8}&\tfrac{1}{8}&\tfrac{1}{2}\\\
0&\tfrac{1}{8}&\tfrac{3}{8}&\tfrac{1}{2}\\\ 0&\tfrac{1}{2}&\tfrac{1}{2}&0\\\
\end{array}\right),$ (S39)
Together with the rules for iSWAP gate it is straightforward to generate
$16\times 16$ matrix defining the Markov population dynamics process.
Comparison of this noise-free population dynamics prediction for several
different circuits with different locations of the butterfly operator $X$
throughout the 53 qubit chip implemented experimentally is shown in Fig. S19.
## V Efficient Population Dynamics for Noisy Circuits
### V.1 Inversion Error
The circuit to measure OTOC require inversion of gates
$\hat{U}_{i,j}(\theta,\phi)$ which is not perfect. An important source of
error is non-invertible phase $\phi$ such that instead of
$\hat{U}_{i,j}^{\dagger}(\theta,\phi)$ the gate $\hat{U}_{i,j}(-\theta,\phi)$
is implemented. This inversion error can be included in the Markov population
dynamics process in the following way,
$\displaystyle\Omega=\left(\begin{array}[]{cccc}\cos^{2}\phi&0&0&\sin^{2}\phi\\\
0&1-\tilde{a}-b&\tilde{a}&b\\\ 0&\tilde{a}&1-\tilde{a}-b&b\\\
\tfrac{\sin^{2}\phi}{9}&\frac{b}{3}&\frac{b}{3}&\tfrac{8+\cos^{2}\phi}{9}-\frac{2}{3}b\\\
\end{array}\right),$
$\displaystyle\tilde{a}=\frac{1}{3}\left(\cos^{2}\phi\sin^{4}\theta+\sin^{2}\phi\cos^{4}\theta\right).$
### V.2 Generic Error Model
Figure S20: Average fidelity of OTOC, i.e. the circuit with no butterfly
applied shown as red line. OTOC fidelity from Monte Carlo simulation of noisy
population dynamics Sec. V. The Pauli error rate $r_{p}=0.013$ is determined
by fitting the numerics to the experimental data. This value is within $10\%$
of the gate fidelity determined in a separate experiment benchmarking two-
qubit gates, see Sec. II. This is the single parameter of the model which is
used to reproduce the rest of the OTOC data for 51 different locations of
butterfly operator on the chip. Black dashed line shows a naive estimate of
circuit fidelity via product of fidelities of gates within the light cone.
Note that naive fidelity decays much faster with time than OTOC fidelity.
For an ensemble that satisfies Eq. (S10) the effect of relaxation and
dephasing in the quantum processor can be captured by a one and two qubit
depolarizing channel noise model,
$\displaystyle\rho\to(1-p_{1})\rho+\frac{p_{1}}{3}\sum_{\hat{\alpha}=\hat{X},\hat{Y},\hat{Z}}\hat{\alpha}\rho\hat{\alpha},$
(S40)
$\displaystyle\rho\to(1-p_{2})\rho+\frac{p_{2}}{15}\sum_{\hat{\alpha},\hat{\beta}}\hat{\alpha}\hat{\beta}\rho\hat{\alpha}\hat{\beta}.$
(S41)
In this case the Markov process Eq. (S20) can be modified in a straight
forward way. For a single qubit depolarizing channel the Markov process Eq.
(S20) is supplemented by the exponential decay rate as follows,
$\displaystyle\openone_{i}^{\otimes 2}\rightarrow\openone_{i}^{\otimes 2},~{}$
(S42) $\displaystyle\mathcal{B}_{i}\rightarrow
e^{-4p_{1}/3}\mathcal{B}_{i},~{}$ (S43)
for each bond per gate cycle. Two-qubit depolarizing channel noise is
accounted by supplementing the Markov process by the following,
$\displaystyle\openone_{i}^{\otimes 2}\openone_{j}^{\otimes
2}\rightarrow\openone_{i}^{\otimes 2}\openone_{j}^{\otimes 2},~{}$ (S44)
$\displaystyle\mathcal{B}_{i}\openone_{j}^{\otimes 2}\rightarrow
e^{-\frac{16}{15}p_{2}}\mathcal{B}_{i}\openone_{j}^{\otimes 2},~{}$ (S45)
$\displaystyle\mathcal{B}_{i}\mathcal{B}_{j}\rightarrow
e^{-\frac{16}{15}p_{2}}\mathcal{B}_{i}\mathcal{B}_{j}.~{}$ (S46)
Note that the effect of noise on the Markov population dynamics cannot be
described by the global depolarizing channel model that is often conjectured
for ergodic circuits. Instead the time dependence of OTOC will demonstrate
characteristic time dependence that can be used to verify the experimental
data.
Figure S21: OTOC circuits (with no butterfly applied) for universal iSWAP gate
set. Solid red line shows experimental data and dashed blue line is the
theoretical fit with a single parameter, Pauli error rate. The extracted Pauli
error rate is $r_{p}=0.012$ within $20\%$ of the value obtained via two-qubit
gate benchmarking.
The described procedure allows us to verify the experimental results for
average fidelity of OTOC circuits, the circuit with no butterfly applied, by
direct comparison to the noisy population dynamics, see Fig. S20 and Fig. S21.
We use the two-qubit Pauli error as a fitting parameter. Best fit corresponds
to errors within $10\%$ of the average error of two qubit cycle measured
independently. We use the extracted error to predict values of OTOC for every
position of the butterfly with no additional fitting. Comparison of the values
of normalized OTOC predicted in this way with experimental data is shown in
Fig. S22. This procedure introduces substantial noise-dependent bias into the
observed OTOC values, which is illustrated in Fig. S23.
Figure S22: Extended data for Fig. 2 of the main text: Normalized OTOC for
four different locations of butterfly operator $Z$. Solid lines show
experimental data, and dashed lines correspond to the noisy population
dynamics.
### V.3 Error Mitigation for Average OTOC
We estimate circuit fidelity from the circuit shown in, Fig. 1 A, B, without
the butterfly operator. This fidelity estimate is used for the error
mitigation procedure aplied to the data in Fig. 2. In the error-free circuit
absence of the butterfly operator means that $U$ and $U^{\dagger}$ cancel each
other exactly resulting in $C_{z0}(t)=1$. In practice, inversion is imperfect
as detailed in Sec. V.1. Both errors in the unitary parameters and the effect
of noise are reflected in the time dependence of $C_{z0}(t)<1$ which serves as
circuit fidelity, an analog of Loschmidt echo for local operator $\hat{M}$.
Average over circuits reads,
$\displaystyle\overline{C}_{0z}=\mathcal{F}_{\openone^{\otimes
2}_{1}}+\mathcal{F}_{\mathcal{B}_{1}},$ (S47)
$\displaystyle\overline{C}_{zz}=\mathcal{F}_{\openone^{\otimes
2}_{1}}-\frac{1}{3}\mathcal{F}_{\mathcal{B}_{1}}.$ (S48)
where the probabilities of vacuum $\mathcal{F}_{\openone^{\otimes 2}_{1}}$ and
bond $\mathcal{F}_{\mathcal{B}_{1}}$ at the measurement site are described by
the population dynamics Eqs. (S20) with respective decay rates Eqs. (S42-S46).
Note that the decay rate grows with the extent of spreading of the butterfly
operator, resulting in the decay of fidelity that is not a simple exponent.
Moreover, in general $\mathcal{F}_{\openone^{\otimes
2}_{1}},\mathcal{F}_{\mathcal{B}_{1}}$ is not a simple probability no error
occurred, as one could naively expect, see Fig. S20 for comparison.
The ratio of $\overline{C}_{zz}/\overline{C}_{0z}$ is compared to normalized
data in Fig. S22. Note that this ratio does not correspond to the noise free
OTOC, Fig. S23. This is because the probability of vacuum at measure site
$\mathcal{F}_{\openone^{\otimes 2}_{1}}$ decays slower with respect to its
noise free value $p(\openone^{\otimes 2}_{1})$ than the probability of the
bond $\mathcal{F}_{\mathcal{B}_{1}}$. The latter corresponds to the weight of
the operators which span the distance between measure and butterfly qubit
which are more susceptible to noise. As a result normalized OTOC from noisy
circuits overestimates the value of noise-free OTOC. For individual circuits
such a simple description is no longer valid, as demonstrated by the
dependence of fidelity estimate $C_{z0}(t)$ on the circuit instance, see Sec.
VI. In many cases circuit dependent corrections are relatively small and the
normalization procedure still works for individual circuits as well.
Nonetheless, for our OTOC fluctuations data we develop a more efficient error
mitigation procedure described in the following section.
Figure S23: Comparison of noisy OTOC obtained by normalization procedure,
dashed lines, to noise-free population dynamics for the same circuit, solid
lines.
### V.4 Theory of Error Mitigation for Individual Circuits
We use a more precise error mitigation procedure for individual OTOC
measurements presented in Figs. 3, 4 of the main text. These circuits contain
iSWAP entangling gate and OTOC value can be expanded in terms of contributions
from Clifford circuits. In presence of non-Clifford gates the butterfly
operator can be conveniently expanded into Pauli strings $B_{i}$,
$\displaystyle O(t)=\sum
w_{\alpha_{1}...\alpha_{n}}B_{\alpha_{1}...\alpha_{n}},$ $\displaystyle
B_{\alpha_{1}...\alpha_{n}}=\hat{\alpha}_{1}\otimes...\otimes\hat{\alpha}_{n}$
For the initial state used in the experimental protocol
$|\psi\rangle=\bigotimes|+\rangle_{i}$, the value of OTOC is expanded as,
$\displaystyle C=\sum
w_{\alpha\alpha_{2}...\alpha_{n}}w_{\beta\alpha_{2}...\alpha_{n}}\bra{+_{1}}\sigma^{z}_{1}\sigma^{\alpha}_{1}\sigma^{z}_{1}\sigma^{\beta}_{1}|+\rangle_{1}.$
(S49)
The real part of the OTOC corresponds to $\alpha=\beta$,
$\displaystyle\mathrm{Re}C=\sum
w_{\alpha_{1}...\alpha_{n}}^{2}\kappa_{\alpha_{1}},$ (S50)
where $\kappa_{\alpha}=\\{1,-1,-1,1\\}$ for $\alpha=\\{0,x,y,z\\}$.
For Clifford circuits ideal value of OTOC can be calculated efficiently. It
can then be used to calculate circuit fidelity by comparing data with the
expected value. The fidelity of the OTOC in presence of non-Clifford gates is
then calculated by sampling a subset of OTOCs for Cliffords that appear in its
expansion, see Eq. (S49). The fidelity is calculated by averaging over
fidelities of individual Clifford contributions. Averaging reduces the circuit
specific effect of noise and gives a more accurate estimate of fidelity.
## VI Numerical Simulation of Error-Limiting Mechanisms for OTOC
In this section, we provide numerical simulation results aimed at identifying
the potential error sources that limit our experimental accuracy in resolving
OTOCs. As demonstrated in Section I, shot noise from finite statistical
sampling is unlikely the dominant mechanism. The remaining known error
channels are: 1. Incoherent, depolarizing noise in the quantum circuits, which
can arise from qubit dephasing or relaxation. 2. Coherent errors in the
quantum gates, e.g. the remnant conditional-phase $\phi$ demonstrated in
Section II. We study each of these errors below.
### VI.1 Coherent and Incoherent Contributions
Figure S24: Simulation of incoherent and coherent OTOC errors. (A) OTOCs, $C$,
of 47 quantum circuit instances simulated in three different ways: with $\phi$
error, where a conditional-phase of $0.136$ rad is added to every two-qubit
gate; with $r_{\text{p}}$ error, where a Pauli error of 0.015 is added to each
two-qubit gate; ideal, where no error is introduced. The qubit configuration
used here is a 1D chain of 10 qubits wherein $Q_{1}$ and $Q_{\text{b}}$ reside
at opposite ends of the chain. The two-qubit gate is iSWAP and
$N_{\text{D}}=12$ non-Clifford gates are used for each instance. The number of
circuit cycles is 34 ($N_{\text{S}}=250$). (B) The scaling of OTOC error (i.e.
RMS deviation from the ideal OTOC values) against $\phi$ (red) and
$r_{\text{p}}$ (blue). Dashed line shows the size of the OTOC signal (i.e. RMS
value of the ideal OTOCs).
We first describe how OTOC error from depolarizing noise may be simulated. We
consider a depolarizing channel model parameterized by an error probability
$p$,
$\mathcal{E}(\rho)=(1-p)\rho+p\frac{\mathbf{I}}{d},$ (S51)
where $d=2^{n}$ is the dimension, and $\mathbf{I}$ the identity operator. The
Kraus operators of this map are all Pauli strings of length $n$, where each
non-trivial string has weight $\frac{p}{d^{2}}$. We consider a model where
after each two qubit gate (these typically dominate the loss in fidelity
compared to single-qubit gates), a two-qubit depolarizing channel is applied.
In Fig. S24A, we show a number of instance-dependent OTOC values $C$ simulated
using full density matrix calculations and an experimentally measured Pauli
error rate $r_{\text{p}}$ of 0.015 ($r_{\text{p}}=15p/16$ in Eq. (S51), due to
the trivial Pauli string in the Kraus map from the identity matrix). Here we
have used a smaller number of qubits due to the high cost of density-matrix
simulation. To match with experiment, we have adjusted the number of circuit
cycles to yield a total number of iSWAP gates close to those shown in Fig. 4
of the main text. The same normalization protocol as the experiment is also
adopted, such that $\braket{\hat{\sigma}_{\text{y}}}$ is simulated with and
without the butterfly operator and their ratio is recorded as $C$. The
results, compared to the ideal OTOC values also plotted in Fig. S24B, show
little deviation.
Next, we simulate OTOCs of the same circuits with $r_{\text{p}}=0$ but an
experimentally measured conditional-phase of $\phi=0.136$ rad on each iSWAP
gate and its inverse. The results are also plotted in Fig. S24A and seen to
deviate much more from the ideal values. To quantify these observations, we
have plotted the OTOC error as a function of both $r_{\text{p}}$ and $\phi$ in
Fig. S24B. Here we see that at the experimental limit ($r_{\text{p}}=0.015$
and $\phi=0.136$ rad), the OTOC error is dominated by the contribution from
$\phi$ (where it is $\sim$0.03) rather than $r_{\text{p}}$ (where it is
$\sim$0.006). Interestingly, the SNR is about 0.9 for $\phi=0.136$ rad, which
is close to the value measured in Fig. 4 of the main text for
$N_{\text{S}}\approx 250$.
Figure S24B also provides preliminary indication of how the OTOC accuracy for
our experiments may be improved as we decrease both $\phi$ and $r_{\text{p}}$.
We see that the OTOC error is linearly proportional to $r_{\text{p}}$ whereas
its scaling is steeper against $\phi$. In particular, reducing $\phi$ by a
factor of 2 leads to an OTOC error that is 3 times lower. Although these
results may depend on the number of qubits and the structure of circuits,
their similarity to experimental data nevertheless provides hints that
reducing the conditional-phases in our iSWAP gates can potentially lead to
large improvements in the OTOC accuracy. This can, for example, be achieved
through concatenated pulses demonstrated in Ref. Foxen _et al._ (2020).
### VI.2 Perturbative Expansion of OTOC Error
Figure S25: Relative error $\epsilon$ of the 0’th and 1’st order error
approximation (from Eq. (LABEL:eq:single-error-approx)) calculated by a pure
state simulation, comparing to the exact value from a density matrix
calculation, as a function of depolarizing probability $p$ (Eq.(S51)). Here we
compute $C_{\text{zx}}$, varying the number of iSWAP ($N_{s}$) and non-
Clifford gates ($N_{D}$), with 11 qubits on a chain. The dots are the median
over 48 circuits with lines of best fit given by the legend. The shaded region
is the middle 50% of the data. The scaling of the error is close to
$\epsilon\propto p^{2}$ as expected for the one-error approximation in all
three cases.
The exact density-matrix calculation used in the preceding section is costly
to implement and becomes quickly intractable as the number of qubits increase.
One can more systematically estimate the instance specific noise contribution
from a perturbation theory expansion of the quantum map Eq. (S51), with an
entirely pure state calculation. Let us adopt the notation
$\sigma_{m_{1},m_{2}}^{(i,j)}=\sigma_{m_{1}}^{(i)}\sigma_{m_{2}}^{(j)}$, where
$\sigma_{m}^{(i)}$ is the $m$’th Pauli operator ($m\in\\{0,x,y,z\\}$) applied
on qubit $i$. We will also call $C[\sigma_{m_{1},m_{2}}^{(i,j)}(d)]$ the OTOC
value with the additional ‘error gate’ $\sigma_{m_{1},m_{2}}^{(i,j)}$ inserted
at layer $d$ in the circuit. Then, to accuracy $O(p^{2})$ one has
$\begin{split}&C_{p}=(1-p)^{n_{2}}C_{\mathrm{ideal}}\\\
&+\frac{p}{15}(1-p)^{n_{2}-1}\sum_{\begin{subarray}{c}(m_{1},m_{2})\neq(0,0),\\\
\langle
i,j\rangle,d\end{subarray}}C[\sigma_{m_{1},m_{2}}^{(i,j)}(d)],\end{split}$
(S52)
where $C_{\mathrm{ideal}}$ is the OTOC value in the limit of no noise, $C_{p}$
the first order approximation of $C_{\mathrm{ideal}}$ in parameter $p$, and
$n_{2}$ the total number of two-qubit gates in the circuit, occurring over
pairs defined by $\langle i,j\rangle$ (i.e. Eq. (LABEL:eq:single-error-approx)
is a sum over all error terms in the circuit). For convenience, we have also
redefined $p\rightarrow 15p/16$ from Eq. (S51) due to the trivial contribution
from the all zero Pauli string.
Eq. (LABEL:eq:single-error-approx) can be used to separate out the ideal OTOC
value of an individual circuit, from the noise contribution. This noise
contribution can be computed using $15\times n_{2}$ circuit simulations
(inserting each Pauli pair at all two-qubit gate locations in the circuit),
though in some cases symmetries can be used to reduce this. For example, for
the normalization curve $C_{\text{z0}}$, only around half of this is required,
since for each error in the reverse circuit $U^{\dagger}$, there is an
equivalent one in $U$. Of course one can continue this expansion to arbitrary
order, however the number of terms quickly becomes infeasible to compute (the
$k$’th order contains $15\times{n_{2}\choose k}$ terms).
In Fig. S25 we show the error scaling (comparing to an exact density matrix
computation) for the zero’th (i.e. only the first term in Eq.
(LABEL:eq:single-error-approx)) and first order approximation for
$C_{\text{zx}}$, giving scaling in the error as expected; the error $\epsilon$
is computed as
$|C_{p}(\text{exact})-C_{p}(\text{approx})|/|C_{p}(\text{exact})|$, where
$C_{p}(\text{approx})$ is from Eq. (LABEL:eq:single-error-approx), and
$C_{p}(\text{exact})$ the value from the density matrix calculation with noise
rate $p$. Moreover, the accuracy of the approximation remains fairly
consistent, for a fixed noise level ($p$) for the three different circuit
sizes (using a similar number of iSWAP gates as in the main text).
Here we have outlined a protocol of simulating the instance specific error
contribution to the OTOC. This can be used to more accurately separate the
instance specific OTOC value, systematic errors, and contributions from gate
noise. Of course, the overhead in the simulation is a factor of $15n_{2}$,
which can itself become challenging in deep enough circuits, although
statistical sampling may be feasible in some cases.
|
spacing=nonfrench
# Harder-Narasimhan stratification for the moduli stack of parabolic vector
bundles
Andres Fernandez Herrero
###### Abstract
For every set of parabolic weights, we construct a Harder-Narasimhan
stratification for the moduli stack of parabolic vector bundles on a curve. It
is based on the notion of parabolic slope, introduced by Mehta and Seshadri.
We also prove that the stratification is schematic, that each stratum is
complete, and establish an analogue of Behrend’s conjecture for parabolic
vector bundles. A comparison with recent $\Theta$-stratification approaches is
discussed.
###### Contents
1. 1 Introduction
1. 1.1 Notation
2. 2 Parabolic Vector Bundles
1. 2.1 Parabolic vector bundles on a curve
2. 2.2 Parabolic weights and degree
3. 2.3 Harder-Narasimhan filtrations
4. 2.4 Parabolic vector bundles in families
3. 3 Stacks of parabolic vector bundles
1. 3.1 The group scheme $\text{GL}\left(\mathcal{V}\right)$
2. 3.2 Moduli stacks of parabolic vector bundles
3. 3.3 Harder-Narasimhan strata
4. 4 Parabolic Quot Schemes
5. 5 Openness of strata
1. 5.1 Constructibility of strata
2. 5.2 Openness of strata
6. 6 Quasicompactness of strata
1. 6.1 Harder-Narasimhan stratification for the moduli of vector bundles
2. 6.2 Quasicompactness of parabolic Harder-Narasimhan strata
7. 7 Completeness of strata
8. 8 Harder-Narasimhan filtrations in families
9. A Descent for finite covers
10. B Comparison with previous work
## 1 Introduction
Let $C$ be a curve over a field $k$. Fix a finite set of $k$-points $x_{i}$ in
$C$. In this paper we investigate the geometry of moduli stacks
$\text{Bun}_{\mathcal{V}}$ of vector bundles over $C$ with the additional data
of a flag of vector spaces in the fiber at each $x_{i}$. The objects
parametrized by $\text{Bun}_{\mathcal{V}}$ are parabolic vector bundles, as in
[MS80].
The purpose of this article is to describe a stratification of
$\text{Bun}_{\mathcal{V}}$ analogous to the classical Harder-Narasimhan
stratification for the moduli of vector bundles [HN75]. It is the parabolic
analogue for $G=\text{GL}_{n}$ of the results of Gurjar and Nitsure [GN14]
when $\text{char}(k)=0$, and recently generalized in [GN18, GN16] to higher
dimensional varieties and general $k$. Our version of the Harder-Narasimhan
stratification is based on the notion of slope for a parabolic vector bundle.
This notion of slope depends on the choice of some stability parameters
$\overline{\lambda}$ known as parabolic weights. We use the slope to define a
parabolic version of the Harder-Narasimhan filtration. This filtration
provides us with an invariant of parabolic vector bundles, which we call the
HN datum. For each HN datum $P$, we define a stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ consisting of parabolic vector bundles
with HN datum bounded by $P$. We prove the following properties of the strata
in Theorems 5.11, 6.10, 7.6 and 8.6.
###### Main Theorem.
For every HN datum $P$, the following properties hold.
1. (A)
The stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$ is a quasicompact open
substack of $\text{Bun}_{\mathcal{V}}$.
2. (B)
The stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$ is complete.
3. (C)
The locus $\text{Bun}_{\mathcal{V}}^{=P}$ of parabolic vector bundles with HN
datum $P$ can be naturally equipped with the structure of a locally closed
substack of $\text{Bun}_{\mathcal{V}}$. In fact,
$\text{Bun}_{\mathcal{V}}^{=P}\hookrightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion.
The article by Mehta and Seshadri [MS80] contains a definition of parabolic
degree similar to the one we give. They use the notion of parabolic degree to
build a moduli space of semistable parabolic vector bundles on a curve of
genus $g\geq 2$ over an algebraically closed field. They also prove a version
of our Theorem 5.11 in the special case of a semistable stratum by embedding
the semistable locus into a GIT problem. In contrast we deal with the whole
moduli stack, including the unstable locus. Our proofs work intrinsically with
the stack, without appealing to GIT.
Parabolic vector bundles can be alternatively seen as torsors for a parahoric
Bruhat-Tits group scheme, as in [PR10]. See Proposition 3.7 below for a proof
of this. It follows from [Hei10][Prop. 1] that $\text{Bun}_{\mathcal{V}}$ is a
smooth algebraic stack. There has been recent interest in developing
stratifications for moduli stacks of torsors for such group schemes. Gaitsgory
and Lurie [GL19] give an ad hoc construction of stratifications in the
semisimple case in order to derive a Grothendieck-Lefschetz trace formula for
the moduli stack of torsors. Balaji and Seshadri [BS15] described Harder-
Narasimhan stratifications for parahoric Bruhat-Tits group schemes under the
assumption that the generic fiber is split. They focus on the construction of
a moduli space for semistable torsors. They obtain results valid when $C$ is a
curve of genus $g\geq 2$ over a ground field of characteristic $0$.
Heinloth [Hei17] and Alper, Halpern-Leistner, Heinloth [AHLH18][§8] develop a
theory of $\Theta$-stratifications for the moduli of torsors of a parahoric
Bruhat-Tits group scheme in arbitrary characteristic, under some mild tameness
assumptions on the generic fiber. They use a modern variation of ideas from
GIT for stacks as in [HL14] in order to obtain a stratification associated to
a line bundle on the stack. In Appendix B we explain how to build certain
$\mathbb{R}$-line bundles on $\text{Bun}_{\mathcal{V}}$ associated to a
parabolic weight $\overline{\lambda}$. We explicitly describe the subset of
parabolic weights $\overline{\lambda}$ such that the corresponding line bundle
is admissible (cf. [Hei17]). Whenever this is the case, the line bundle
induces a $\Theta$-stratification that we show coincides with our parabolic
slope stratification. It should be remarked that this admissibility condition
on parabolic weights is restrictive and we do not impose it in this text.
Our techniques to establish the Main Theorem make contact with the classical
theory for vector bundles already present in [MS80]. In order to deal with
subbundles of parabolic vector bundles in families, we define an analogue of
the Quot scheme for parabolic vector bundles (Definition 4.9). We also
construct an iterated Quot scheme $Fil^{\alpha}_{\mathcal{W}}$ parametrizing
filtrations of a parabolic vector bundle with some given fixed set of
invariants (Definition 4.16).
We also prove certain properties of the stratification that do not hold in the
generality of [Hei17] when the ground field has positive characteristic. The
obstruction in the more general case is the failure of Behrend’s conjecture
for certain reductive groups in bad characteristics (established by Heinloth
[Hei08]). Recall that Behrend’s conjecture [Beh95] states that the canonical
parabolic reduction of a principal $G$-bundle over a curve with $G$-reductive
does not admit nontrivial deformations. We prove that an analogue of Behrend’s
conjecture holds in our context of parabolic vector bundles:
###### Proposition (= Proposition 2.31).
Let $k[\epsilon]\vcentcolon=k[T]\,/\,(T^{2})$ be the ring of dual numbers. We
denote by $\sigma:\text{Spec}(k)\rightarrow\text{Spec}\;k[\epsilon]$ the
unique section of the structure map
$p:\text{Spec}\;k[\epsilon]\rightarrow\text{Spec}(k)$. Let $\mathcal{V}$ be a
parabolic vector bundle over $C$. Suppose that we are given a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=p^{*}\mathcal{V}$
of $p^{*}\mathcal{V}$ by parabolic subbundles. Assume that the pulled-back
filtration
$0=\sigma^{*}\mathcal{W}_{0}\subset\sigma^{*}\mathcal{W}_{1}\subset\cdots\,\subset\sigma^{*}\mathcal{W}_{l-1}\subset\sigma^{*}\mathcal{W}_{l}=\mathcal{V}$
over the closed point is the Harder-Narasimhan filtration of $\mathcal{V}$.
Then $\mathcal{W}_{j}=p^{*}\sigma^{*}\mathcal{W}_{j}$ for all $1\leq j\leq
l-1$.
We use this result along with the construction of the filtration scheme
$Fil^{\alpha}_{\mathcal{W}}$ in order to develop a theory of schematic Harder-
Narasimhan filtrations. This is assertion (C) of the Main Theorem (see also
Theorem 8.6). It is the parabolic analogue of the work done by Gurjar and
Nitsure in [GN14], which deals with principal $G$-bundles over a curve when
the characteristic of the ground field is $0$.
Finally, we also prove in Theorem 7.6 that for every $P$ the stratum
$\text{Bun}^{\leq P}_{\mathcal{V}}$ satisfies a strong lifting criterion for
families over a discrete valuation ring. This is assertion (B) of the Main
Theorem. Heinloth [Hei17, Remark 3.20] noted that the weaker valuative
criterion [Sta, Tag 0CLK] holds for the semistable locus for more general
parahoric Bruhat-Tits group schemes. See Section 7 for more details on this.
We conclude this introduction with an outline of the paper.
In Section 2 we recall the notions of parabolic vector bundles (Definition
2.1), parabolic weights (Definition 2.5), degree and slope (Definition 2.7)
and the existence and uniqueness of the Harder-Narasimhan filtration
(Proposition 2.18). The section concludes with a discussion of parabolic
vector bundles in families (Subsection 2.4). This includes a proof of the
compatibility of the Harder-Narasimhan filtration with extensions of the
ground field (Lemma 2.30) and a proof of the analogue of Behrend’s conjecture
for parabolic vector bundles (Proposition 2.31). In Section 3 we define the
stack $\text{Bun}_{\mathcal{V}}$ of parabolic vector bundles of a given type
(Definition 3.6) and show that it is isomorphic to the stack of torsors for a
parahoric Bruhat-Tits group scheme over $C$ (Proposition 3.7). We define the
HN datum of a vector bundle (Definition 3.12) and the strata
$\text{Bun}_{\mathcal{V}}^{\leq P}$ (Definition 3.13).
In Section 4 we construct a parabolic version
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ of the Quot scheme (Definition
4.9). More generally, we consider nested versions
$\text{Fil}^{\alpha}_{\mathcal{V}}$ of parabolic Quot schemes (Definition
4.16). The main purpose of this section is proving that
$\text{Fil}^{\alpha}_{\mathcal{V}}$ is represented by a separated scheme of
finite type over the base (Proposition 4.17). This result is used in Sections
5.1 and 8. We recommend the reader to skip the proofs in Section 4 upon their
first read.
The purpose of Section 5 is to prove that each stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is represented by an open substack of
$\text{Bun}_{\mathcal{V}}$ (Theorem 5.11). This is achieved by first showing
that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is constructible (Proposition 5.1)
and then proving that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is closed under
generalization (Proposition 5.6). In Section 6 we prove that each stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is quasicompact (Theorem 6.10). For this
purpose, we use the Harder-Narasimhan stratification of the moduli stack of
classical vector bundles, which we recall in Subsection 6.1. Our strategy is
to show that the forgetful morphism $\text{Bun}_{\mathcal{V}}^{\leq
P}\to\text{Bun}_{\text{GL}_{n}}(C)$ is quasicompact (Proposition 6.5), and
that the image lies in finitely many classical Harder-Narasimhan strata
(Corollary 6.9). In Section 7 we prove that each stratum is complete (see
Definition 7.1 and Theorem 7.6). We adapt the parabolic analogue of Langton’s
algorithm in Mehta-Seshadri [MS80] so that it applies to a general unstable
stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$.
In Section 8 we define a notion of relative Harder-Narasimhan filtration in
families (Definition 8.4). We use this to equip each locally closed stratum
$\text{Bun}_{\mathcal{V}}^{=P}$ with the structure of a locally closed
substack of $\text{Bun}_{\mathcal{V}}$ (Theorem 8.6). The strategy of proof is
to show that the natural morphism
$\text{Bun}_{\mathcal{V}}^{=P}\to\text{Bun}_{\mathcal{V}}^{\leq P}$ is proper
(using the representability of $\text{Fil}^{\alpha}_{\mathcal{V}}$), radicial
(by the uniqueness of Harder-Narasimhan filtrations) and unramified (by the
parabolic analogue of Behrend’s conjecture in Proposition 2.31).
In Appendix A we provide an exposition of descent for finite morphisms, which
we adapt from [Gro95][B.3]. In Appendix B we discuss the relation of our
stratification with the $\Theta$-stratification for the moduli stack of
parahoric torsors constructed in [Hei17, AHLH18].
Acknowledgements The notion of slope employed here is a reinterpretation of
the definition suggested by J. Lurie in the outline of his Arizona Winter
School project [Lur19], and which we futher relate to [MS80]. Some of the
arguments greatly benefited from discussion with other members of the project
during the evening sessions. I would particularly like to thank Aaron
Landesman, David Yang and Bogdam Zavyalov for helpful discussions. I am happy
to thank my advisor Nicolas Templier for encouraging me to write down these
results and providing very valuable input for the redaction of the manuscript.
I thank Nitin Nitsure for helpful comments and David Mehrle for helping out
with LaTeX. I acknowledge support by NSF grants DMS-1454893 and DMS-2001071.
### 1.1 Notation
We work over a fixed ground field $k$. All schemes are understood to be
schemes over $k$. An undecorated product of $k$-schemes (e.g. $A\times B$)
should always be interpreted as a fiber product over $k$. If $R$ is a $k$
algebra and $S$ is a $k$-scheme, we may sometimes use the notation $S_{R}$ to
denote the fiber product $S\times\text{Spec}(R)$.
We write $s\in S$ to mean that $s$ is a (set theoretic) point of $S$.
Equivalently, $s$ is the Spec of the residue field of a topological point of
$S$. For such $s\in S$, we will write $\kappa(s)$ for the residue field.
We will often deal with pullbacks of quasicoherent sheaves on schemes. Let $X$
and $Y$ be schemes and let $\mathcal{Q}$ be quasicoherent sheaf on $Y$. It
will usually be the case that there is a clear choice of a morphism from
$f:X\rightarrow Y$. In such a situation we will not mention the choice of $f$
and write $\mathcal{Q}|_{X}$ to denote the pullback $f^{*}\mathcal{Q}$ of the
quasicoherent sheaf $\mathcal{Q}$ by $f$.
We fix once and for all a curve $\pi:C\rightarrow\text{Spec}(k)$ that is
smooth, projective and geometrically irreducible over $k$. We choose a finite
set $\\{x_{i}\\}_{i\in I}$ of $k$-points $x_{i}$ of $C$. We will often refer
to these $x_{i}$ as points of degeneration. For all $i\in I$, let us denote by
$q_{i}:x_{i}\rightarrow C$ the closed immersion of $x_{i}$ into $C$. We also
fix the choice of an ample line bundle $\mathcal{O}(1)$ of degree one on $C$.
## 2 Parabolic Vector Bundles
### 2.1 Parabolic vector bundles on a curve
###### Definition 2.1.
A parabolic vector bundle $\mathcal{V}$ over $C$ with parabolic structure at
$\\{x_{i}\\}_{i\in I}$ consists of the data of:
1. (a)
A vector bundle $\mathcal{E}^{(0)}$ on $C$.
2. (b)
For each $i\in I$, a chain of vector bundles
$\mathcal{E}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})$.
We define the rank of $\mathcal{V}$ to be
$\text{rank}\,\mathcal{V}\vcentcolon=\,\text{rank}\,\mathcal{E}^{(0)}$. For
each $i\in I$, we call $N_{i}$ the chain length at $i$.
For a parabolic vector bundle $\mathcal{V}$ as above, we will use the
convention $\mathcal{E}_{i}^{(0)}\vcentcolon=\mathcal{E}^{(0)}$ for all $i\in
I$. Note that for each $i\in I$ and $1\leq m\leq N_{i}$, the quotient
$\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}$ is the pushforward
$(q_{i})_{*}(V)$ of a finite dimensional $k$-vector space $V$. We will want to
keep track of the dimension of these vector spaces. For a parabolic vector
bundle $\mathcal{V}$, we write
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$
to indicate that the $\mathcal{E}_{i}^{(m)}$s are the chains of vector bundles
appearing in Definition 2.1 and
$\text{dim}_{k}\left(\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}\right)\,=\,a_{i}^{(m)}$.
###### Definition 2.2.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(M_{i})}}{\subset}\,\mathcal{F}_{i}^{(M_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles. We say that $\mathcal{W}$ has type
$\mathcal{V}$ (or equivalently $\mathcal{V}$ has type $\mathcal{W}$) if
1. (a)
$\text{rank}\,\mathcal{W}=\text{rank}\,\mathcal{V}$.
2. (b)
For all $i\in I$, $M_{i}=N_{i}$.
3. (c)
For all $i\in I$ and $1\leq m\leq N_{i}$, we have $a_{i}^{(m)}=b_{i}^{(m)}$.
The class of parabolic vector bundles with a given fixed set of chain lengths
forms a category. For the next two definitions, we let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles with the same chain length $N_{i}$ at each
$i\in I$.
###### Definition 2.3.
A morphism of parabolic vector bundles $f:\mathcal{W}\rightarrow\mathcal{V}$
consists of the data of a morphism of sheaves
$f_{i}^{(m)}:\mathcal{F}_{i}^{(m)}\rightarrow\mathcal{E}_{i}^{(m)}$ for each
$i\in I$ and $0\leq m\leq N_{i}$ satisfying the following two conditions
1. (a)
$f_{i}^{(0)}=f_{j}^{(0)}$ for all $i,j\in I$.
2. (b)
For all $i\in I$ and $1\leq m\leq N_{i}$, the following diagram commutes
${\mathcal{E}_{i}^{(m-1)}}$${\mathcal{E}_{i}^{(m)}}$${\mathcal{F}_{i}^{(m-1)}}$${\mathcal{F}_{i}^{(m)}}$$\subset$$\subset$$\scriptstyle{f_{i}^{(m-1)}}$$\scriptstyle{f_{i}^{(m)}}$
Recall that for a given vector bundle $\mathcal{E}$ on $C$, we say that a
subsheaf $\mathcal{F}\subset\mathcal{E}$ is a subbudle if $\mathcal{F}$ and
$\mathcal{E}\,/\,\mathcal{F}$ are themselves vector bundles. In this case
$\mathcal{E}\,/\,\mathcal{F}$ is called a quotient bundle.
###### Definition 2.4.
Let $f:\mathcal{W}\rightarrow\mathcal{V}$ be a morphism.
1. (1)
We say that $\mathcal{V}$ is a parabolic quotient bundle of $\mathcal{W}$
(equivalently $f$ is an admissible epimorphism) if for each $i\in I$ and
$0\leq m\leq N_{i}$, the morphism $f_{i}^{(m)}$ witnesses
$\mathcal{E}_{i}^{(m)}$ as a quotient vector bundle of
$\mathcal{F}_{i}^{(m)}$.
2. (2)
We say that $\mathcal{W}$ is a subbundle of $\mathcal{V}$ (equivalently $f$ is
an admissible monomorphism) if it satisfies the following two conditions
1. (a)
For each $i\in I$ and $0\leq m\leq N_{i}$, the morphism $f_{i}^{(m)}$
witnesses $\mathcal{F}_{i}^{(m)}$ as a vector subbundle of
$\mathcal{E}_{i}^{(m)}$.
2. (b)
For each $i\in I$ and $0\leq m\leq N_{i}$, we have
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$.
We can use these definitions to define the structure of an exact category on
the category of parabolic vector bundles over $C$ with fixed chain lengths
$N_{i}$. All morphisms in this category admit a kernel and an image. This
follows from the fact that subsheaves of vector bundles on $C$ are always
vector bundles (because $C$ is a regular curve).
### 2.2 Parabolic weights and degree
In order to stratify moduli stacks of parabolic bundles, we will define a
notion of slope. First we need the following definition.
###### Definition 2.5.
A set of parabolic weights $\overline{\lambda}$ consists of the data of a
tuple of real numbers
$\left(\lambda_{i}^{(1)},\lambda_{i}^{(2)},\,\cdots,\lambda_{i}^{(N_{i})}\right)$
for each $i\in I$ satisfying:
1. (a)
$0<\lambda_{i}^{(m)}<1$ for all $i\in I$ and $1\leq m\leq N_{i}$.
2. (b)
$\lambda_{i}^{(m-1)}<\lambda_{i}^{(m)}$ for all $i\in I$ and $2\leq m\leq
N_{i}$.
$N_{i}$ is called the chain length of $\overline{\lambda}$ at $i$.
###### Definition 2.6.
Let $\overline{\lambda}$ be a set of parabolic weights with chain length
$N_{i}$ at $i\in I$. We denote by $\text{Vect}_{\overline{\lambda}}$ the exact
category of parabolic vector bundles with chain length $N_{i}$ at each $i\in
I$.
For the rest of this paper, we fix a choice of parabolic weights
$\overline{\lambda}$ with chain length $N_{i}$ at $i\in I$. Unless otherwise
stated, all parabolic vector bundles will be in
$\text{Vect}_{\overline{\lambda}}$.
###### Definition 2.7.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic bundle. The degree of $\mathcal{V}$ is defined to be
$\text{deg}\;\mathcal{V}=\text{deg}\;\mathcal{E}^{(0)}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,a_{i}^{(j)}\right)$
The slope of $\mathcal{V}$ is defined by
$\mu\left(\mathcal{V}\right)\vcentcolon=\frac{\text{deg}\,\mathcal{V}}{\text{rank}\,\mathcal{V}}$.
###### Remark 2.8.
In [MS80], Mehta and Seshadri consider a slightly different notion of
parabolic vector bundle. We consider flags of vector spaces inside
$\mathcal{E}(x_{i})/\mathcal{E}$. [MS80] deals with flags in the fiber
$\mathcal{E}/\mathcal{E}(-x_{i})$. The inclusion
$\mathcal{O}_{C}(-x_{i})\hookrightarrow\mathcal{O}_{C}$ of the ideal sheaf of
$x_{i}$ induces an identification
$\mathcal{E}(x_{i})/\mathcal{E}\cong\mathcal{E}/\mathcal{E}(-x_{i})$.
Therefore our notion of parabolic vector bundle is in correspondence with the
one in Mehta-Seshadri. Our altered definition of degree reflects this change
of convention.
Let us state a couple lemmas that will come in handy in later sections. They
are immediate consequences of the definitions given above.
###### Lemma 2.9.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle of rank $n$. Then, we have
$deg\left(\mathcal{E}^{(0)}\right)\leq deg\left(\mathcal{V}\right)\leq
deg\left(\mathcal{E}^{(0)}\right)+n|I|$
∎
###### Lemma 2.10.
Let
$0\rightarrow\mathcal{W}\rightarrow\mathcal{V}\rightarrow\mathcal{Q}\rightarrow
0$ be a short exact sequence in $\text{Vect}_{\overline{\lambda}}$. Then, we
have
$\text{deg}\,\mathcal{V}\,=\,\text{deg}\,\mathcal{W}\,+\,\text{deg}\,\mathcal{Q}$.
∎
We end this section with a useful technical lemma.
###### Lemma 2.11.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles. Suppose that $\mathcal{W}$ is a suboject
of $\mathcal{V}$ (not necessarily a parabolic subbundle). Then,
1. (a)
There exists a parabolic subbundle $\mathcal{W}^{sat}$ of $\mathcal{V}$
containing $\mathcal{W}$ as a subobject and satisfying
$\mu(\mathcal{W}^{sat})\geq\mu(\mathcal{W})$.
2. (b)
If we have $\mu(\mathcal{W}^{sat})=\mu(\mathcal{W})$ in the construction for
part (a), then in fact $\mathcal{W}^{sat}=\mathcal{W}$.
3. (c)
Given two parabolic subobjects
$\mathcal{W}_{1}\hookrightarrow\mathcal{W}_{2}\hookrightarrow\mathcal{V}$, we
have $\mathcal{W}_{1}^{sat}\subset\mathcal{W}_{2}^{sat}$.
###### Proof.
Set $n=\text{rank}\,\mathcal{W}$. For all $i\in I$ and $0\leq m\leq N_{i}$,
define
$\mathcal{P}_{i}^{(m)}\vcentcolon=\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{E}^{(m)}_{i}$.
Observe that $P_{i}^{(m)}\subset\mathcal{F}^{(0)}(x_{i})$ is a vector bundle,
because subsheaves of locally free sheaves in $C$ are locally free. Hence the
diagram of sheaves
$\mathcal{U}=\left[\;\mathcal{P}^{(0)}\,\overset{c_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{c_{i}^{(2)}}{\subset}\cdots\;\overset{c_{i}^{(N_{i})}}{\subset}\,\mathcal{P}_{i}^{(N_{i})}=\,\mathcal{P}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle. We have that
$\mathcal{F}^{(m)}_{i}\subset\mathcal{P}^{(m)}_{i}$ and
$\mathcal{P}^{(0)}=\mathcal{F}^{(0)}$. Therefore $\mathcal{U}$ contains
$\mathcal{W}$ as a suboject. We now claim that
$\mu(\mathcal{U})\geq\mu(\mathcal{W})$. Since
$\text{rank}\,\mathcal{U}=\text{rank}\,\mathcal{W}=n$, the claim is equivalent
to $\text{deg}\,\mathcal{U}\geq\text{deg}\,\mathcal{W}$. Since
$\mathcal{F}^{(0)}=\mathcal{P}^{(0)}$, the inequality of degrees amounts to
showing that
$\sum_{i\in I}\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,c_{i}^{(j)}\,\leq\sum_{i\in
I}\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}$
We will actually show that for all $i\in I$ we have
$\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,c_{i}^{(j)}\,\leq\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}$.
We can use summation by parts to rewrite both sides of the inequality. It
suffices to show
$n\lambda^{(N_{i})}\,+\,\sum_{j=1}^{N_{i}-1}(\lambda_{i}^{(j)}-\lambda^{(j+1)}_{i})\,\sum_{m=1}^{j}c_{i}^{(j)}\,\leq\,n\lambda^{(N_{i})}\,+\,\sum_{j=1}^{N_{i}-1}(\lambda_{i}^{(j)}-\lambda^{(j+1)}_{i})\,\sum_{m=1}^{j}b_{i}^{(j)}$
Since we have $\lambda_{i}^{(j)}<\lambda_{i}^{(j+1)}$ for all $1\leq j\leq
N_{i}-1$, it is sufficient to show that
$\sum_{m=1}^{j}c_{i}^{(m)}\,\geq\sum_{m=1}^{j}b_{i}^{(m)}$. By definition, we
have that
$\sum_{m=1}^{j}c_{i}^{(m)}=\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$.
Similarly, we have that
$\sum_{m=1}^{j}b_{i}^{(m)}=\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$.
Hence the inequality above is equivalent to
$\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}\geq\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$
This is now obvious, because
$\mathcal{P}_{i}^{(m)}\supset\mathcal{F}_{i}^{(m)}$. Suppose that
$\mu(\mathcal{U})=\mu(\mathcal{W})$. Then all of the inequalities above must
be equalities. In particular we have
$\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}=\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$
for all $i,j$. This implies that $\mathcal{U}=\mathcal{W}$. It is also clear
that inclusions of subobjects as in part (c) are preserved by this
construction.
Replacing $\mathcal{W}$ by $\mathcal{U}$ in the proposition, we can assume
from the beginning that we have
$\mathcal{F}_{i}^{(m)}=\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{E}^{(m)}_{i}$.
Let us define
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$.
This means that we have the following commutative diagram with exact columns:
${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
Figure 1: Diagram 1
The problem is that the $\mathcal{Q}_{i}^{(m)}$s do not necessarily form a
parabolic bundle, since they could have nontrivial torsion. In order to solve
this, we kill the $C$-torsion. Let $\xi$ denote the generic point of $C$. Let
$\iota:\xi\longrightarrow C$ be the corresponding inclusion. Define
$K_{i}^{(m)}\vcentcolon=\text{Ker}\left(\mathcal{Q}_{i}^{(m)}\,\xrightarrow{unit}\,i_{*}i^{*}\,\mathcal{Q}_{i}^{(m)}\right)$
$\mathcal{Q}_{i}^{(m),\,sat}\vcentcolon=\text{Im}\left(\mathcal{Q}_{i}^{(m)}\,\xrightarrow{unit}\,i_{*}i^{*}\,\mathcal{Q}_{i}^{(m)}\right)$
$\mathcal{F}_{i}^{(m),\,sat}\vcentcolon=\text{Ker}\left(\mathcal{E}_{i}^{(m)}\,\twoheadrightarrow\,\mathcal{Q}_{i}^{(m)}\rightarrow\,\mathcal{Q}_{i}^{(m),\,sat}\right)$
There is a similar commutative diagram with exact columns:
${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0),\,sat}}$${\mathcal{Q}_{i}^{(1),\,sat}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2),\,sat}}$${\mathcal{Q}_{i}^{(N_{i}-1),\,sat}}$${\mathcal{Q}^{(0),\,sat}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0),\,sat}}$${\mathcal{F}_{i}^{(1),\,sat}}$${\cdots}$${{}^{s}\mathcal{F}_{i}^{(Ni-2),\,sat}}$${\mathcal{F}_{i}^{(N_{i}-1),\,sat}}$${\mathcal{F}^{(0),\,sat}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
Figure 2: Diagram 2
We set
$\mathcal{W}^{sat}\vcentcolon=\left[\;\mathcal{F}^{(0),\,sat}\;\overset{c_{i}^{(1)}}{\subset}\;\mathcal{F}_{i}^{(1),\,sat}\;\overset{c_{i}^{(2)}}{\subset}\cdots\;\overset{c_{i}^{(N_{i})}}{\subset}\;\mathcal{F}_{i}^{(N_{i}),\,sat}=\,\mathcal{F}^{(0),\,sat}(x_{i})\;\right]_{i\in
I}$. By construction, $\mathcal{W}^{sat}$ is a parabolic subbundle of
$\mathcal{V}$ containing $\mathcal{W}$. It is clear that this construction
preserves inclusions of subobject as described in part (c). We are left to
show the claims about the slopes. By definition, we have the following
commutative diagram with exact rows:
${0}$${K_{i}^{(m-1)}}$${\mathcal{Q}_{i}^{(m-1)}}$${\mathcal{Q}_{i}^{(m-1),\,sat}}$${0}$${0}$${K_{i}^{(m)}}$${\mathcal{Q}_{i}^{(m)}}$${\mathcal{Q}_{i}^{(m),\,sat}}$${0}$
An application of the Snake Lemma yields a surjection
$\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\,\twoheadrightarrow\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}$.
Therefore, we have
$\text{dim}_{k}\,\left(\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}\right)\;\leq\;\text{dim}_{k}\,\left(\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\right)$
By applying the Snake Lemma to consecutive columns of Diagram 1 and Diagram 2
respectively, we obtain the two short exact sequences
$0\longrightarrow\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}\longrightarrow\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\longrightarrow
0$
$0\longrightarrow\mathcal{F}_{i}^{(m),\,sat}\,/\,\mathcal{F}_{i}^{(m-1),\,sat}\longrightarrow\mathcal{E}_{i}^{(m),\,sat}\,/\,\mathcal{E}_{i}^{(m-1),\,sat}\longrightarrow\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}\longrightarrow
0$
We conclude that $c_{i}^{(m)}\geq b_{i}^{(m)}$. But note that for all $i\in I$
we have
$\sum_{m=1}^{N_{i}}c_{i}^{(m)}=\text{rank}\left(\mathcal{W}^{sat}\right)=\text{rank}\left(\mathcal{W}\right)=\sum_{m=1}^{N_{i}}b_{i}^{(m)}$
Therefore, we must have $c_{i}^{(m)}=b_{i}^{(m)}$ for all $i$ and $m$. By
construction we have $\mathcal{F}^{(0)}\subset\,\mathcal{F}^{(0),\,sat}$. In
particular, $deg\,\mathcal{F}^{(0)}\leq deg\,\mathcal{F}^{(0),\,sat}$ as
vector bundles. We therefore get a chain of (in)equalities:
$\displaystyle deg\,\mathcal{W}^{sat}$
$\displaystyle=\;deg\;\mathcal{F}^{(0),\,sat}+\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}^{sat}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,c_{i}^{(m)}\right)$
$\displaystyle=\;deg\;\mathcal{F}^{(0),\,sat}+\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}^{sat}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,b_{i}^{(m)}\right)$
$\displaystyle\geq\;deg\;\mathcal{F}^{(0)}\,+\;\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,b_{i}^{(m)}\right)$
$\displaystyle=\;deg\,\mathcal{W}$
Part (a) of the lemma follows.
Suppose that $deg\,\mathcal{W}=deg\,\mathcal{W}^{sat}$. From the chain of
(in)equalities above we can conclude that
$deg\,\mathcal{F}^{(0)}=deg\,\mathcal{F}^{(0),\,sat}$. Hence we must have
$\mathcal{F}^{(0)}=\,\mathcal{F}^{(0),\,sat}$. We can use the fact that
$\mathcal{F}_{i}^{(m)}\subset\,\mathcal{F}_{i}^{(m),\,sat}$ and that
$\text{rank}\,\left(\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}^{(0)}\right)\,=\,\sum_{j=1}^{m}b_{i}^{(j)}\,=\,\text{rank}\,\left(\mathcal{F}_{i}^{(m),\,sat}\,/\,\mathcal{F}^{(0)}\right)$
in order to conclude that
$\mathcal{F}_{i}^{(m)}=\,\mathcal{F}_{i}^{(m),\,sat}$ for all $i$ and $m$.
Therefore, we have that $\mathcal{W}=\,\mathcal{W}^{sat}$, as stipulated in
part (b). ∎
### 2.3 Harder-Narasimhan filtrations
###### Definition 2.12.
A parabolic vector bundle $\mathcal{V}$ in $\text{Vect}_{\overline{\lambda}}$
is called semistable (with respect to $\overline{\lambda}$) if for all
parabolic subbundles $\mathcal{W}\subset\mathcal{V}$, we have
$\mu\left(\mathcal{W}\right)\,\leq\,\mu\left(\mathcal{V}\right)$.
###### Definition 2.13.
Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. A Harder-Narasimhan filtration of
$\mathcal{V}$ is a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by parabolic subbundles $\mathcal{W}_{j}$ satisfying the following two
conditions:
1. (a)
For all $1\leq j\leq l$, the parabolic bundle
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ is semistable.
2. (b)
For all $2\leq j\leq l$, we have
$\mu\left(\mathcal{W}_{j-1}\,/\,\mathcal{W}_{j-2}\right)\,>\,\mu\left(\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\right)$.
In order to prove the existence of such filtrations, we will need a few lemmas
along the way.
###### Lemma 2.14.
Let $\mathcal{V}$ be a parabolic bundle. There exists a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by subbundles $\mathcal{W}_{j}$ such that
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ has rank 1 for all $1\leq j\leq l$.
###### Proof.
By induction on the rank of $\mathcal{V}$, it suffices to show that
$\mathcal{V}$ has a parabolic subbundle $\mathcal{W}$ of rank 1. Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is well known that $\mathcal{E}^{(0)}$ admits a vector subbundle
$\mathcal{F}^{(0)}$ of rank 1. Indeed, by Riemann-Roch there exists some $N>0$
such that $\mathcal{E}^{(0)}(N)$ has a nonzero global section
$\mathcal{O}_{C}\rightarrow\mathcal{E}^{(0)}(N)$. Tensoring by
$\mathcal{O}(-N)$ gives us a subsheaf
$\mathcal{O}(-N)\rightarrow\mathcal{E}^{(0)}$ that is a line bundle. Let
$j:\xi\rightarrow C$ denote the inclusion of the generic point $\xi$ into $C$.
Set
$\mathcal{F}^{(0)}\vcentcolon=\text{Ker}\left(\mathcal{E}^{(0)}\rightarrow\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)\xrightarrow{unit}j_{*}j^{*}\;\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)\right)$.
Then the quotient $\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}$ is torsion free,
because it is a subsheaf of
$j_{*}j^{*}\;\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)$. So $\mathcal{F}^{(0)}$ is
a subbundle of $\mathcal{E}^{(0)}$. On the other hand, we have that
$\mathcal{F}^{(0)}\,/\,\mathcal{O}(-N)$ is torsion. Therefore
$\text{rank}\,\mathcal{F}^{(0)}=\text{rank}\,\mathcal{O}(-N)=1$.
We can now set
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,\cap\,\mathcal{F}^{(0)}(x_{i})$
and define $\mathcal{W}$ to be the diagram of sheaves
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is clear that $\mathcal{W}$ is a parabolic subbundle of rank 1,
because all $\mathcal{F}_{i}^{(m)}\subset\mathcal{F}^{(0)}(x_{i})$ are line
bundles. ∎
###### Corollary 2.15.
Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$. Then,
1. (a)
There is a constant $M$ such that any subbundle
$\mathcal{U}\subset\mathcal{V}$ satisfies $\text{deg}\,\mathcal{U}\leq M$.
2. (b)
The set of all slopes of parabolic subbundles of $\mathcal{V}$ has a maximal
element.
###### Proof.
1. (a)
Let
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
be a filtration as in Lemma 2.14. We set
$M=\underset{A\subset\\{1,2,\cdots,l\\}}{\text{max}}\,\left\\{\sum_{j\in
A}\text{deg}\,\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\right\\}$. Let
$\mathcal{U}$ be a parabolic subbundle of $\mathcal{V}$. We can take
intersections in the category of parabolic vector bundles, since
$\text{Vect}_{\overline{\lambda}}$ admits kernels. Consider the induced
filtration of $\mathcal{U}$ by parabolic subbundles
$0=\mathcal{W}_{0}\,\cap\mathcal{U}\,\subset\mathcal{W}_{1}\,\cap\mathcal{U}\,\subset\cdots\,\subset\mathcal{W}_{l-1}\,\cap\mathcal{U}\,\subset\mathcal{W}_{l}\,\cap\mathcal{U}\,=\mathcal{U}$
For all $1\leq j\leq l$ we have that
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)$
is isomorphic to a parabolic subbundle of
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$. Since
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ has rank 1, we must have either
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\,=\,0$
or
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\;\cong\;\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$.
Let
$A_{\mathcal{U}}\vcentcolon=\left\\{\;j\in\\{1,2,\cdots,l\\}\;\mid\;\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\;\cong\;\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\;\right\\}$
A repeated application of Lemma 2.10 yields
$\text{deg}\,\mathcal{U}=\sum_{j\in
A_{\mathcal{U}}}\text{deg}\,\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\leq M$.
2. (b)
Fix a rank $k$ with $1\leq k\leq n$. We claim that the set
$\left\\{\;\mu(\mathcal{U})\;\mid\;\mathcal{U}\;\text{is a subbundle of
$\mathcal{V}$ of rank $k$}\;\right\\}$ has a maximal element. Once this claim
is proven, we can take the maximum among all possible ranks $1\leq k\leq n$ to
produce the maximal element as in part (b) of the corollary. We are left to
prove the claim.
Let $\mathcal{U}$ be any subbundle of rank $k$. By definition, we have
$\mu(\mathcal{U})=\frac{1}{k}\text{deg}\,\mathcal{U}$. So it suffices to
produce a subbundle $\mathcal{U}$ of maximal degree among those of rank $k$.
Consider the finite set
$S\vcentcolon=\left\\{A_{\mathcal{U}}\;\mid\;\mathcal{U}\,\text{is a subbundle
of $\mathcal{V}$ of rank $k$}\;\right\\}$, where $A_{\mathcal{U}}$ is defined
as in part (a). Let $A_{\mathcal{U}}$ be an element of $S$ maximizing the
quantity $\sum_{j\in A_{\mathcal{U}}}\text{deg}\,\mathcal{W}_{j}$. The same
argument as in part (a) shows that $\mathcal{U}$ is a subbundle of rank $k$
with maximal degree among subbundles of the same rank.
∎
###### Lemma 2.16.
Let $\mathcal{V}$, $\mathcal{W}$ be two semistable parabolic vector bundles.
Then,
1. (a)
For any subobject $\mathcal{P}\hookrightarrow\mathcal{W}$ (not necessarily a
parabolic subbundle), we have $\mu(\mathcal{P})\leq\mu(\mathcal{W})$.
2. (b)
If $\mathcal{Q}$ is a parabolic quotient bundle of $\mathcal{W}$, then
$\mu\left(\mathcal{Q}\right)\,\geq\,\mu\left(\mathcal{W}\right)$.
3. (c)
If $\mu\left(\mathcal{W}\right)\,>\,\mu\left(\mathcal{V}\right)$, then
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{W},\,\mathcal{V}\right)=0$.
4. (d)
Let $\mathcal{P}$ fit into an exact sequence
$0\rightarrow\mathcal{V}\rightarrow\mathcal{P}\rightarrow\mathcal{W}\rightarrow
0$. If $\mathcal{U}$ is a parabolic subbundle of $\mathcal{P}$, then
$\mu(U)\leq\text{max}\left\\{\mu(\mathcal{V}),\,\mu(\mathcal{W})\right\\}$
###### Proof.
1. (a)
By Lemma 2.11, we have a subbundle $\mathcal{P}^{sat}$ of $\mathcal{W}$ with
$\mu(\mathcal{P}^{sat})\geq\mu(\mathcal{P})$. By semistability,
$\mu(\mathcal{W})\geq\mu(\mathcal{P}^{sat})$.
2. (b)
Let $m=\text{rank}\,\mathcal{Q}$ and $n=\text{rank}\,\mathcal{W}$. Define
$\mathcal{K}$ to be the kernel of the admissible epimorphism
$\mathcal{W}\twoheadrightarrow\mathcal{Q}$. By Lemma 2.10, we have
$\text{deg}\,\mathcal{W}=\text{deg}\,\mathcal{K}\,+\,\text{deg}\,\mathcal{Q}$.
We can rewrite this as
$\mu(\mathcal{W})=\frac{n-m}{n}\mu(\mathcal{K})+\frac{m}{n}\mu(\mathcal{Q})$.
Since $\mathcal{W}$ is semistable, we have
$\mu(\mathcal{K})\leq\mu(\mathcal{W})$. Therefore
$\mu(\mathcal{Q})=\frac{n}{m}\left(\mu(\mathcal{W})-\frac{n-m}{n}\mu(\mathcal{K})\right)\geq\frac{n}{m}\left(\mu(\mathcal{W})-\frac{n-m}{n}\mu(\mathcal{W})\right)=\mu(\mathcal{W})$
3. (c)
Suppose that there exists a nontrivial morphism
$f:\mathcal{W}\rightarrow\mathcal{V}$. Consider the nonzero image
$\mathcal{I}=\text{Im}\left(\,\mathcal{W}\xrightarrow{f}\mathcal{V}\,\right)$
in $\text{Vect}_{\overline{\lambda}}$. Then $\mathcal{I}$ is a parabolic
quotient bundle of $\mathcal{W}$. By part (b), we know that
$\mu(\mathcal{I})\geq\mu(\mathcal{W})$. Therefore
$\mu(\mathcal{I})>\mu(\mathcal{V})$. But $\mathcal{I}$ is a suboject of
$\mathcal{V}$. This contradicts semistability of $\mathcal{V}$ by part (a).
4. (d)
Let
$M\vcentcolon=\text{max}\left\\{\mu(\mathcal{V}),\,\mu(\mathcal{W})\right\\}$.
Define $\mathcal{Q}$ to be the image of the composition
$\mathcal{U}\,\hookrightarrow\,\mathcal{P}\,\twoheadrightarrow\,\mathcal{W}$
in $\text{Vect}_{\overline{\lambda}}$. Then $\mathcal{Q}$ is a quotient bundle
of $\mathcal{U}$. We get a short exact sequence
$0\longrightarrow\mathcal{K}\longrightarrow\mathcal{U}\longrightarrow\mathcal{Q}\longrightarrow
0$, where $\mathcal{K}\hookrightarrow\mathcal{V}$ and
$\mathcal{Q}\hookrightarrow\mathcal{W}$. By part (a), we see that
$\mu(\mathcal{K})\leq\mu(\mathcal{V})\leq M$. Similarly,
$\mu(\mathcal{Q})\leq\mu(\mathcal{W})\leq M$. Therefore, we have
$\mu(\mathcal{U})=\frac{\text{rank}\,\mathcal{K}}{\text{rank}\,\mathcal{U}}\,\mu(\mathcal{K})\,+\,\frac{\text{rank}\,\mathcal{Q}}{\text{rank}\,\mathcal{U}}\,\mu(\mathcal{Q})\,\leq\,\frac{\text{rank}\,\mathcal{K}}{\text{rank}\,\mathcal{U}}\,M\,+\,\frac{\text{rank}\,\mathcal{Q}}{\text{rank}\,\mathcal{U}}\,M\,=\,M$
∎
###### Lemma 2.17.
Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Let $\mu$ denote the maximal element of
the set of all slopes of subbundles of $\mathcal{V}$, as in part (b) of
Corollary 2.15. Then,
1. (a)
There exists a unique maximal semistable subbundle $\mathcal{U}$ of
$\mathcal{V}$ such that $\mu(\mathcal{U})=\mu$ and $\mathcal{U}$ contains all
other subbundles of slope $\mu$. We will call $\mathcal{U}$ the maximal
destabilizing subbundle of $\mathcal{V}$.
2. (b)
If $\mathcal{U}$ is as in part (a), then all subobjects
$\overline{\mathcal{W}}\,\hookrightarrow\,\mathcal{V}\,/\,\mathcal{U}$ satisfy
$\mu(\overline{\mathcal{W}})<\mu$.
3. (c)
For $\mathcal{U}$ as in part (a), we have
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{U},\;\mathcal{V}\,/\,\mathcal{U}\,\right)=0$.
###### Proof.
Uniqueness of such subbundle $\mathcal{U}$ is clear. It suffices to prove
existence. Let $\mathcal{U}$ be a parabolic subbundle of $\mathcal{V}$ with
$\mu(\mathcal{U})=\mu$ that has maximal rank among the subbundles with slope
$\mu$. By the definition of $\mu$, it is clear that $\mathcal{U}$ is
semistable.
We claim that $\mathcal{U}$ satisfies the statement in part (b) of the
proposition. Suppose that $\overline{\mathcal{W}}$ is a subobject of
$\mathcal{V}\,/\,\mathcal{U}$ with slope $\mu(\overline{\mathcal{W}})\geq\mu$.
By the axioms of exact categories, there is a subobject $\mathcal{W}$ of
$\mathcal{V}$ with $\mathcal{U}\subset\mathcal{W}$ and a short exact sequence
$0\longrightarrow\mathcal{U}\longrightarrow\mathcal{W}\longrightarrow\overline{\mathcal{W}}\rightarrow
0$.
By additivity of degree in short exact sequences (Lemma 2.10), we have
$\mu(\mathcal{W})=\frac{\text{rank}\,\mathcal{U}}{\text{rank}\,\mathcal{W}}\,\mu(\mathcal{U})\,+\,\frac{\text{rank}\,\overline{\mathcal{W}}}{\text{rank}\,\mathcal{W}}\,\mu(\overline{\mathcal{W}})\,\geq\,\mu$
contradicting maximality of the rank of $\mathcal{U}$.
Now let $\mathcal{P}$ be any parabolic subbundle of $\mathcal{V}$ with slope
$\mu(\mathcal{P})=\mu$. By the definition of $\mu$, $\mathcal{P}$ is
semistable. We claim that $\mathcal{P}\subset\mathcal{U}$. In order to see
this, it suffices to show that the composition
$\mathcal{P}\hookrightarrow\mathcal{V}\twoheadrightarrow\mathcal{V}\,/\,\mathcal{U}$
is 0. Let us show that
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{P},\;\mathcal{V}\,/\,\mathcal{U}\,\right)=0$.
This also implies part (c) as a special case.
For the sake of contradiction, suppose that
$\psi\in\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{P},\;\mathcal{V}\,/\,\mathcal{U}\,\right)$
is nonzero. Let $\mathcal{I}$ be the nontrivial image $\text{Im}(\psi)$ of
$\mathcal{P}$ in $\mathcal{V}\,/\,\mathcal{U}$. $\mathcal{I}$ is a parabolic
quotient bundle of $\mathcal{P}$. By part (b) of Lemma 2.16 we have
$\mu(\mathcal{I})\geq\mu(\mathcal{P})$. Therefore $\mathcal{I}$ is a suboject
of $\mathcal{V}\,/\,\mathcal{U}$ with slope $\mu(\mathcal{I})\geq\mu$. We get
a contradiction, since we have proven in the paragraph above that this is not
possible. ∎
Now we are ready to prove the main proposition of this section.
###### Proposition 2.18.
Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vec}_{\overline{\lambda}}$. Then $\mathcal{V}$ admits a unique Harder-
Narasimhan filtration.
###### Proof.
Let $\mathcal{U}_{1}$ be the maximal destabilizing subbundle of $\mathcal{V}$.
Set $\mu_{1}=\mu(\mathcal{U}_{1})$. We can iterate this process by recursion,
setting $\overline{\mathcal{U}_{j}}$ to be the maximal destabilizing subbundle
of $\mathcal{V}\,/\,\mathcal{U}_{j-1}$ and letting $\mathcal{U}_{j}$ be the
unique subbundle of $\mathcal{V}$ containing $\mathcal{U}_{j-1}$ such that
$\overline{\mathcal{U}_{j}}=\mathcal{U}_{j}\,/\,\mathcal{U}_{j-1}$. Define
$\mu_{j}\vcentcolon=\mu(\overline{\mathcal{U}_{j}})$. By part (b) of Lemma
2.11, we see that $\mu_{j}>\mu_{j-1}$. Since the rank of $\mathcal{V}$ is
finite, this process must terminate. Hence $\mathcal{U}_{l}=\mathcal{V}$ for
some $l$. We get a filtration
$0\subset\mathcal{U}_{1}\subset\cdots\,\subset\mathcal{U}_{l-1}\subset\mathcal{U}_{l}=\mathcal{V}$
which by construction is a Harder-Narasimhan filtration.
For uniqueness, we can induct on the length $l$ of the filtration above. We
are reduced to showing that for any Harder-Narasimhan filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{m-1}\subset\mathcal{W}_{m}=\mathcal{V}$
we have that $\mathcal{W}_{1}$ is the maximal destabilizing subsheaf
$\mathcal{U}_{1}$ of $\mathcal{V}$. By applying part (d) of Lemma 2.16 several
times, we conclude that we must have
$\mu(\mathcal{W}_{1})=\mu(\mathcal{U}_{1})=\mu_{1}$. Therefore, by maximality
of $\mathcal{U}_{1}$, we must have $\mathcal{U}_{1}\supset\mathcal{W}_{1}$.
Consider $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$. This is a subbundle of
$\mathcal{V}\,/\,\mathcal{W}_{1}$. Another sequence of applications of part
(d) of Lemma 2.16 shows that if $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$ is
nontrivial, then we must have
$\mu\left(\mathcal{U}_{1}\,/\,\mathcal{W}_{1}\right)<\mu_{1}$. This
contradicts part (b) of Lemma 2.16 applied to the semistable bundle
$\mathcal{U}_{1}$. Hence $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$ is trivial, and
therefore $\mathcal{U}_{1}=\mathcal{W}_{1}$. ∎
### 2.4 Parabolic vector bundles in families
In this subsection we extend the theory of parabolic vector bundles to the
relative setting.
###### Definition 2.19.
Let $S$ be a $k$-scheme. A parabolic vector bundle $\mathcal{V}$ of rank $n$
over $C\times S$ (with parabolic structure at $\\{x_{i}\\}$) consists of the
data of:
1. (a)
A vector bundle $\mathcal{E}^{(0)}$ of rank $n$ on $C\times S$.
2. (b)
For each $i\in I$, a chain of vector bundles
$\mathcal{E}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})$.
such that for all $i\in I$ and all $1\leq m\leq N_{i}$, the quotient
$\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}$ is $S$-flat and the
restriction $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times
S}$ is a vector bundle of constant rank on $x_{i}\times S$.
For each $i\in I$, the number $N_{i}$ will be called the chain length at $i$.
###### Remark 2.20.
Since $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times S}$ is
finitely presented and $S$-flat, it must be a vector bundle. The reason why we
require $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times S}$
to have constant rank is to simplify notation. Note that this always true
locally on $S$.
For a parabolic vector bundle $\mathcal{V}$ over $C\times S$, we will write
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$
to convey the information that
$a_{i}^{(m)}=\text{rank}\,\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times
S}$.
###### Definition 2.21.
Let $S$ be a $k$-scheme and let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$. For any $S$-scheme
$f:T\rightarrow S$, the pullback of $\mathcal{V}$ is defined to be the
parabolic vector bundle $f^{*}\mathcal{V}$ on $C\times T$ given by
$\displaystyle f^{*}\mathcal{V}=\left[\;(id_{C}\times
f)^{*}\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,(id_{C}\times
f)^{*}\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,(id_{C}\times
f)^{*}\mathcal{E}_{i}^{(N_{i})}=(id_{C}\times
f)^{*}\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in I}$
It will often be the case that the choice of a map $f$ is understood and
unambiguous, and then we will write $\mathcal{V}|_{C\times T}$ instead of
$f^{*}\mathcal{V}$.
###### Remark 2.22.
It is necessary to know that
$\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}$ is $S$-flat in order to be
able to conclude that the inclusion
$\mathcal{E}_{i}^{(m-1)}\subset\mathcal{E}_{i}^{(m)}$ remains a monomorphism
after pulling back by $f$. Note that the ranks $a_{i}^{(m)}$ are preserved by
pullbacks.
###### Definition 2.23.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic bundle over $C$. Let $S$ be a $k$-scheme and let
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(M_{i})}}{\subset}\,\mathcal{F}_{i}^{(M_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$. We say that $\mathcal{W}$
is of type $\mathcal{V}$ if all of the following conditions are satisfied
1. (a)
$\text{rank}\,\mathcal{W}=\text{rank}\,\mathcal{V}$.
2. (b)
For all $i\in I$, $N_{i}=M_{i}$ (they have the same chain lengths).
3. (c)
For all $i\in I$ and $1\leq m\leq N_{i}$, we have $a_{i}^{(m)}=b_{i}^{(m)}$.
###### Remark 2.24.
The property of being equivalent to a given $\mathcal{V}$ is preserved by
pullbacks.
For the next two definitions, we fix a $k$-scheme $S$ and we let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$
be two parabolic vector bundles over $C\times S$ with the same chain length
$N_{i}$ at each $i\in I$.
###### Definition 2.25.
A morphism of parabolic Vector bundles $f:\mathcal{W}\rightarrow\mathcal{V}$
consists of of the data of a morphism of sheaves
$f_{i}^{(m)}:\mathcal{F}_{i}^{(m)}\rightarrow\mathcal{E}_{i}^{(m)}$ for each
$i\in I$ and $0\leq m\leq N_{i}$ satisfying the following two conditions:
1. (a)
$f_{i}^{(0)}=f_{j}^{(0)}$ for all $i,j\in I$.
2. (b)
For all $i\in I$ and $1\leq m\leq N_{i}$, the following diagram commutes
${\mathcal{E}_{i}^{(m-1)}}$${\mathcal{E}_{i}^{(m)}}$${\mathcal{F}_{i}^{(m-1)}}$${\mathcal{F}_{i}^{(m)}}$$\subset$$\subset$$\scriptstyle{f_{i}^{(m-1)}}$$\scriptstyle{f_{i}^{(m)}}$
###### Definition 2.26.
Let $f:\mathcal{W}\rightarrow\mathcal{V}$ be a morphism of parabolic bundles.
1. 1.
We say that $\mathcal{V}$ is a parabolic quotient bundle of $\mathcal{W}$
(equivalently $f$ is an admissible epimorphism) if the morphism $f_{i}^{(m)}$
witnesses $\mathcal{E}_{i}^{(m)}$ as a quotient vector bundle of
$\mathcal{F}_{i}^{(m)}$ for all $i\in I$ and $0\leq m\leq N_{i}$.
2. 2.
We say that $\mathcal{W}$ is a parabolic subbundle of $\mathcal{V}$
(equivalently $f$ is an admissible monomorphism) if it satisfies the following
two conditions:
1. (a)
For each $i\in I$ and $0\leq m\leq N_{i}$, we have
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$.
2. (b)
The naive quotient (as a diagrams of quasicoherent sheaves)
$\mathcal{Q}\vcentcolon=\left[\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\,/\,\mathcal{F}_{i}^{(1)}\,\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}\,/\,\mathcal{F}_{i}^{(N_{i})}\,=\,\left(\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}\right)(x_{i})\right]_{i\in
I}$
is a parabolic vector bundle over $C\times S$.
We can use this definitions to impose the structure of an exact category on
the class of parabolic vector bundles over $C\times S$ with a fixed set of
chain lengths the structure. In this case we don’t have kernels and images in
general, because it is no longer true that a subsheaf of a locally free sheaf
is locally free.
Let us fix a set of parabolic weights $\overline{\lambda}$ with chain length
$N_{i}$ at $i\in I$.
###### Definition 2.27.
Let $S$ be a $k$-scheme. We write $\text{Vect}_{\overline{\lambda}}(S)$ to
denote the exact category of parabolic vector bundles over $C\times S$ with
chain length $N_{i}$ for each $i\in I$.
###### Remark 2.28.
Since the pullbacks of parabolic vector bundles preserve exact sequences,
$\text{Vect}_{\overline{\lambda}}$ is a pseudofunctor from $k$-schemes into
the 2-category of exact categories (with exact functors as 1-morphisms).
###### Lemma 2.29.
Let $S$ be a scheme over $k$ and let $\mathcal{W}$ be a parabolic vector
bundle in $\text{Vect}_{\overline{\lambda}}(S)$. Then, the function
$f:S\longrightarrow\mathbb{R}$ given by
$f(s)=\text{deg}\;\mathcal{W}\,|_{C_{s}}$ is locally constant in the Zariski
topology.
###### Proof.
Suppose that
$\mathcal{W}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. By the formula for the degree in Definition 2.7, we have
$f(s)=\text{deg}\;\mathcal{E}^{(0)}\,|_{C_{s}}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,a_{i}^{(j)}\right)$
The corollary follows, because the degree of the fibers of $\mathcal{E}^{(0)}$
is know to be a locally constant function. ∎
The following lemma tells us that the Harder-Narasimhan filtration behaves
well under extension of scalars. The analogous result for classical vector
bundles was proven in [Lan75]. See also [HL97][Thm 1.3.7]. Our proof is a
modification of the argument found there.
###### Lemma 2.30.
Let $\mathcal{V}$ be a parabolic vector bundle over $C$ that belongs to
$\text{Vect}_{\overline{\lambda}}$. Let $K$ be a field extension of $k$. If
$\mathcal{V}$ is semistable, then $\mathcal{V}|_{C_{K}}$ is semistable.
In particular, the pullback of the Harder-Narasimhan filtration of
$\mathcal{V}$ to $C_{K}$ is the Harder-Narasimhan filtration of
$\mathcal{V}|_{C_{K}}$.
###### Proof.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a semistable parabolic vector bundle over $C$ as in the statement.
Suppose that $\mathcal{V}|_{C_{K}}$ is not semistable. Let $\mathcal{W}_{K}$
be the maximal destabilizing parabolic subbundle as in Lemma 2.17. It is clear
by the formula in Definition 2.7 that the slope of a parabolic vector bundle
is preserved under extension of ground fields. We will try to descend
$\mathcal{W}_{K}$ to a parabolic subbundle of $\mathcal{W}$ of $\mathcal{V}$
in order to contradict semistability of $\mathcal{V}$.
Suppose that $\mathcal{W}_{K}$ is given by
$\mathcal{W}_{K}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. The definition of parabolic subbundle implies
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}|_{C_{K}}\,\cap\,\mathcal{F}^{(0)}$.
Hence $\mathcal{W}_{K}$ is completely determined by the vector subbundle
$\mathcal{F}^{(0)}\subset\,\mathcal{E}^{(0)}|_{C_{K}}$. This vector subbundle
gives us a $K$-point in $\text{Quot}_{\mathcal{E}^{(0)}/C/k}$ (see [Nit05] or
Section 4 below). Since $\text{Quot}_{\mathcal{E}^{(0)}/C/k}$ is locally of
finite type, this actually comes from a $K^{\prime}$-point
$\mathcal{F}^{(0)}_{K^{\prime}}\subset\,\mathcal{E}^{(0)}|_{C_{K^{\prime}}}$,
where $K^{\prime}$ is a subextension of $K\supset k$ that is finitely
generated over $k$. Define
$\mathcal{F}_{i,\,K^{\prime}}^{(m)}\vcentcolon=\mathcal{F}^{(0)}_{K^{\prime}}\,\cap\,\mathcal{E}_{i}^{(m)}|_{C_{K^{\prime}}}$.
Set
$\mathcal{W}_{K^{\prime}}=\left[\;\mathcal{F}_{K^{\prime}}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i,\,K^{\prime}}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i,\,K^{\prime}}^{(N_{i})}=\,\mathcal{F}_{K^{\prime}}^{(0)}(x_{i})\;\right]_{i\in
I}$. This is a parabolic subbundle of $\mathcal{V}|_{C_{K^{\prime}}}$, because
$C_{K^{\prime}}$-flatness of the subsheaves and the quotients can be checked
fpqc-locally. $\mathcal{W}_{K^{\prime}}\subset\mathcal{V}|_{C_{K^{\prime}}}$
is the maximal parabolic subbundle with maximal slope. This is because there
are more subbundles to check over $C_{K}$ than over $C_{K^{\prime}}$. Hence we
can assume without loss of generality that $K\supset k$ is a finitely
generated field extension.
By enlarging the field if necessary, we can express $K\supset k$ as the
composition of a finitely generated purely transcendental extension, followed
by a finite Galois extension and a finite purely inseparable extension. We are
thus reduced to analyze each of these three cases.
1. C 1
$K\supset k$ is a finite Galois extension Let $G=\text{Gal}(K\,/\,k)$. By the
discussion in Appendix A, it suffices to show that for all $\sigma\in G$ we
have $\sigma^{*}(\mathcal{W}_{K})=\mathcal{W}_{K}$ as parabolic subbundles of
$\mathcal{V}|_{C_{K}}$. From the formula for degree in Definition 2.7, it is
easy to see that parabolic degree is preserved by elements of the Galois
group. Since the rank is also preserved, $\sigma^{*}(\mathcal{W}_{K})$ is
still the unique maximal subbundle with maximal slope. Therefore
$\mathcal{W}_{K}$ descends.
2. C 2
$K=k\left(a^{\frac{1}{p}}\right)$ for some $a\in k$ Let Der be the set of
$k$-derivations of $K$. For each $D\in\text{Der}$, we can extend $D$ to a
$\mathcal{O}_{C}$-linear derivation $D_{C}$ of $\mathcal{O}_{C_{K}}$. This
uniquely defines an $\mathcal{O}_{C}$-linear endomorphism of each
$\mathcal{E}_{i}^{(m)}|_{C_{K}}=\mathcal{E}_{i}^{(m)}\otimes\mathcal{O}_{C_{K}}$
by acting on the second component of the tensor product. This endomorphism
preserves the inclusion relations among the $\mathcal{E}_{i}^{(m)}$s, so it
gives a $\mathcal{O}_{C}$-linear endomorphism $D_{\mathcal{V}}$ of
$\mathcal{V}$. This describes the canonical descent data of
$\mathcal{V}|_{C_{K}}$ viewed as a diagram of quasicoherent sheaves, as
explained in Appendix A. By the discussion in Appendix A, it follows that
$\mathcal{W}_{K}\subset\mathcal{V}|_{C_{K}}$ descends to a parabolic subbundle
of $\mathcal{V}$ if and only if
$D_{\mathcal{V}}(\mathcal{W}_{K})\subset\mathcal{W}_{K}$ for all
$D\in\text{Der}$. This is equivalent to the composition
$\phi:\mathcal{W}_{K}\xrightarrow{D_{\mathcal{V}}}\mathcal{V}|_{C_{K}}\twoheadrightarrow\mathcal{V}|_{C_{K}}\,/\,\mathcal{W}_{K}$
being $0$. But, by the Leibniz rule, this composition is actually
$\mathcal{O}_{C_{K}}$-linear (because we are killing elements of
$\mathcal{W}_{K}$). Hence $\phi$ is a genuine morphism of parabolic vector
bundles. Since $\mathcal{W}_{K}$ is the maximal destabilizing sheaf, part (c)
of Lemma 2.17 implies that
$\text{Hom}\left(\mathcal{W}_{K},\,\mathcal{V}|_{C_{K}}\,/\;\mathcal{W}_{K}\right)=0$.
Therefore we must have $\phi=0$, as desired.
3. C 3
$K=k(t)$ The same trick with the Quot scheme allows us to spread
$\mathcal{W}_{K}$ to a parabolic subbundle $\mathcal{W}_{U}$ of
$\mathcal{V}|_{C\times U}$, where $U$ is an open subset of
$\mathbb{A}^{1}_{k}$. We can first extend as a set of chains of vector
bundles. Then using opennes of the flat locus [Sta, Tag 0399] we can decrease
$U$ to ensure that all quotients in the chain are $U$-flat. Let $t\in U$ be a
closed point of $U$. Consider the parabolic subbundle
$\mathcal{W}_{U}|_{C_{t}}$ of $\mathcal{V}|_{C_{t}}$. By local constancy of
degree (Lemma 2.29), $\mathcal{W}_{U}|_{C_{t}}\subset\mathcal{V}|_{C_{t}}$
implies that $\mathcal{V}|_{C_{t}}$ is not semistable. Since the residue field
$\kappa(t)$ is finite over $k$, we are reduced to the case of finite field
extensions as in Cases 1,2 above.
For the second part of the proposition, suppose that
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
is the Harder-Narasimhan filtration of $\mathcal{V}$. The first part of the
proposition implies that
$\mathcal{W}_{j}|_{C_{K}}\,/\,\mathcal{W}_{j-1}|_{C_{K}}$ is semistable for
all $1\leq j\leq l$. Since slopes are preserved by field extensions, we
conclude that
$0=\mathcal{W}_{0}|_{C_{K}}\subset\mathcal{W}_{1}|_{C_{K}}\subset\cdots\,\subset\mathcal{W}_{l-1}|_{C_{K}}\subset\mathcal{W}_{l}|_{C_{K}}=\mathcal{V}|_{C_{K}}$
is a Harder-Narasimhan filtration of $\mathcal{V}|_{C_{K}}$. ∎
Next, we show that Harder-Narasimhan filtrations have no nontrivial first
order deformations. We denote by $k[\epsilon]\vcentcolon=k[T]\,/\,(T^{2})$ the
ring of dual numbers. We denote by
$\sigma:\text{Spec}(k)\rightarrow\text{Spec}\;k[\epsilon]$ the unique section
of the structure map $p:\text{Spec}\;k[\epsilon]\rightarrow\text{Spec}(k)$.
###### Proposition 2.31.
Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Suppose that we are
given a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=p^{*}\mathcal{V}$
of $p^{*}\mathcal{V}$ by parabolic subbundles. Assume that the pulled-back
filtration
$0=\sigma^{*}\mathcal{W}_{0}\subset\sigma^{*}\mathcal{W}_{1}\subset\cdots\,\subset\sigma^{*}\mathcal{W}_{l-1}\subset\sigma^{*}\mathcal{W}_{l}=\mathcal{V}$
over the closed point is the Harder-Narasimhan filtration of $\mathcal{V}$.
Then $\mathcal{W}_{j}=p^{*}\sigma^{*}\mathcal{W}_{j}$ for all $1\leq j\leq
l-1$.
###### Proof.
By induction on the length $l$ of the filtration, we are reduced to showing
that $\mathcal{W}_{1}=p^{*}\sigma^{*}\,\mathcal{W}_{1}$. Let us show that
$\mathcal{W}_{1}\subset p^{*}\sigma^{*}\,\mathcal{W}_{1}$. Observe that
$p^{*}\sigma^{*}\,\mathcal{W}_{1}$ is a parabolic subbundle of
$p^{*}\mathcal{V}$. It suffices to prove that the composition
$f:\,\mathcal{W}_{1}\,\hookrightarrow\,p^{*}\mathcal{V}\,\twoheadrightarrow\,p^{*}\mathcal{V}\,/\,p^{*}\sigma^{*}\,\mathcal{W}_{1}$
is $0$. Restricting to the closed point gives us the map
$\sigma^{*}f:\,\sigma^{*}\mathcal{W}_{1}\,\hookrightarrow\,\mathcal{V}\,\twoheadrightarrow\,\mathcal{V}\,/\,\sigma^{*}\,\mathcal{W}_{1}$,
which is clearly $0$. Therefore, basic deformation theory tells us that
$f=\epsilon\,\psi$ for some homomorphism of parabolic bundles
$\psi\in\text{Hom}\left(\,\sigma^{*}\mathcal{W}_{1},\;\mathcal{V}\,/\,\sigma^{*}\mathcal{W}_{1}\,\right)$.
But note that, by assumption, $\sigma^{*}\mathcal{W}_{1}$ is the maximal
destabilizing parabolic subbundle of $\mathcal{V}$. Hence part (c) of Lemma
2.17 implies that
$\text{Hom}\left(\,\sigma^{*}\mathcal{W}_{1},\;\mathcal{V}\,/\,\sigma^{*}\mathcal{W}_{1}\,\right)=0$.
We conclude that $f=0$.
The same reasoning shows that
$p^{*}\sigma^{*}\,\mathcal{W}_{1}\subset\mathcal{W}_{1}$, thus concluding the
proof. ∎
## 3 Stacks of parabolic vector bundles
### 3.1 The group scheme $\text{GL}\left(\mathcal{V}\right)$
Let $J$ be a small category and let $X$ be a scheme. A diagram of sheaves on
$X$ of shape $J$ is a functor $F$ from $J$ into the abelian category of
quasicoherent sheaves $\text{QCoh}(X)$. Diagrams of sheaves on $X$ of shape
$J$ form an abelian category, where the morphisms are natural transformations.
Given a morphism of schemes $f:Y\rightarrow X$ and a diagram of sheaves
$F:J\rightarrow\text{QCoh}(X)$, the pullback $f^{*}F$ is defined to be the
diagram of sheaves on $Y$ obtained by postcomposing $F$ with the functor
$f^{*}:\text{QCoh}(X)\rightarrow\text{QCoh}(Y)$. As usual, we will write
$F|_{Y}$ instead of $f^{*}F$ when the morphism $f$ is implicitly understood.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C$. We can interpret $\mathcal{V}$ as a
diagram of sheaves on $C$. The source category $J_{\mathcal{V}}$ can be
described as follows. $J_{\mathcal{V}}$ has $1+\sum_{i\in I}N_{i}$ objects.
There is a initial object $X^{(0)}$, and the rest of the objects can be
labeled $X_{i}^{(m)}$ for $i\in I$ and $1\leq m\leq N_{i}$. For each $i\in I$
and $1\leq m_{1}\leq m_{2}\leq N_{i}$, there is a unique morphism
$X_{i}^{(m_{1})}\rightarrow X_{i}^{(m_{2})}$. These, along with the initial
morphims with source $X^{(0)}$, account for all morphisms in the category.
Here is a picture of $J_{\mathcal{V}}$ (we fix an enumeration
$I=\\{i_{1},i_{2},\cdots,i_{k}\\}$ for convenience):
${X_{i_{1}}^{(1)}}$${X_{i_{1}}^{(2)}}$${\cdots}$${X_{i_{1}}^{(N_{i_{1}}-1)}}$${X_{i_{1}}^{(N_{i_{1}})}}$${X^{(0)}}$${X_{i_{2}}^{(1)}}$${X_{i_{2}}^{(2)}}$${\cdots}$${X_{i_{1}}^{(N_{i_{2}}-1)}}$${X_{i_{2}}^{(N_{i_{2}})}}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${X_{i_{k}}^{(1)}}$${X_{i_{k}}^{(2)}}$${\cdots}$${X_{i_{k}}^{(N_{i_{k}}-1)}}$${X_{i_{k}}^{(N_{i_{k}})}}$
This description is compatible with some of the definitions given in Section
2. For example, morphisms of parabolic vector bundles are the same as
morphisms of diagrams of sheaves. Fixing the chain lengths amounts to fixing
the shape $J$ of the diagram.
###### Definition 3.1.
Let $\mathcal{V}$ a parabolic vector bundle as above. Denote by
$\text{GL}(\mathcal{V})$ the contravariant functor from the category of flat
$C$-schemes into Set given as follows. For $f:T\rightarrow C$ a flat
$C$-scheme, we set
$\text{GL}(\mathcal{V})\,(T)\vcentcolon=\text{Aut}(f^{*}\,\mathcal{V})$. Here
the pullbacks and automorphisms are to be interpreted in the category of
diagrams of sheaves of shape $J_{\mathcal{V}}$, as described in the paragraphs
above.
###### Proposition 3.2.
$\text{GL}(\mathcal{V})$ is represented by a smooth group scheme over $C$.
###### Proof.
By descent for morphisms of quasicoherent sheaves, we know that
$\text{GL}(\mathcal{V})$ is a sheaf in the Zariski topology. In particular, it
suffices to check the claim after passing to a Zariski cover of $C$. Let
$\bigsqcup_{j\in J}U_{j}\rightarrow C$ be an open cover of $C$ such that for
all $j\in J$ we have that
$\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus n}$. It suffices
to show that the automorphism functor $\text{GL}(\mathcal{V}|_{U_{j}})$ of the
diagram of sheaves $\mathcal{V}|_{U_{j}}$ is a smooth group scheme over
$U_{j}$ for any $j\in J$. If $U_{j}$ does not contain any of the points of
degeneration $\\{x_{i}\\}_{i\in I}$, then we have
$\text{GL}(\mathcal{V}|_{U_{j}})\,\cong\,\text{GL}(\mathcal{E}^{(0)}|_{U_{j}})\,\cong\,\text{GL}(\mathcal{O}_{U_{j}}^{\oplus
n})\,\cong\,\text{GL}_{n}\times U_{j},$
where $n=\text{rank}\,\mathcal{V}$. (Here we are abusing notation and writing
$\mathcal{E}^{(0)}$ to denote the constant diagram with identities as
morphisms). Hence we are done in this case.
Suppose that $U_{j}$ contains some of the points of degeneration
$\\{x_{i}\\}_{i\in I}$. By passing to a finer Zariski cover, we can assume
that $U_{j}$ contains only one such point of degeneration. Fix $i\in I$ such
that $x_{i}\in U_{j}$. For all indexes $s\neq i$, the inclusions in the $s$th
chain of vector bundles
$\mathcal{E}^{(0)}|_{U_{j}}\,\overset{a_{s}^{(1)}}{\subset}\,\mathcal{E}_{s}^{(1)}|_{U_{j}}\overset{a_{s}^{(2)}}{\subset}\cdots\;\overset{a_{s}^{(N_{s})}}{\subset}\,\mathcal{E}_{s}^{(N_{s})}|_{U_{j}}=\mathcal{E}^{(0)}(x_{s})|_{U_{j}}$
become isomorphisms. We conclude that the functor of automorphisms of the
diagram $\mathcal{V}|_{U_{j}}$ is the same of the functor of automorphisms
preserving the single chain of vector bundles
$\mathcal{E}^{(0)}|_{U_{j}}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{U_{j}}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{s})}|_{U_{j}}=\mathcal{E}^{(0)}(x_{i})|_{U_{j}}.$
By making $U_{j}$ smaller, we can assume that $U_{j}$ is the spectrum of a
Dedekind domain $R$ over $k$ and that the Cartier divisor $x_{i}$ is given by
the vanishing of a nonzerodivisor $t\in R$. We can also assume that
$\mathcal{E}_{i}^{(m)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus n}$ for all
$1\leq m\leq N_{i}$. Now $\mathcal{E}^{(0)}(x_{i})|_{U_{j}}$ is a free
$R$-module of rank $n$, let’s denote it by $F$. Similarly, for $1\leq m\leq
N_{i}-1$, let’s write $F_{m}$ for the free $R$-module
$\mathcal{E}^{(m)}_{i}|_{U_{j}}$. Our chain of vector bundles is given by a
set of inclusions
$t\,F\,\subset\,F_{1}\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\,\subset\,F$.
Observe that all maps in this chain become isomorphisms after inverting $t$.
Hence we get a collection of canonical isomorphisms $F_{l}|_{U_{j}\setminus
x_{i}}\xrightarrow{\sim}F|_{U_{j}\setminus x_{i}}$ for each $1\leq l\leq
N_{i}-1$. Consider the smooth group scheme $G$ over $R$ given by
$G\vcentcolon=\text{GL}(t\,F)\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F_{l})\,\times\text{GL}(F)$.
We have a diagonal morphism $\overline{\Delta}:\text{GL}(F)|_{U_{j}\setminus
x_{i}}\rightarrow G|_{U_{j}\setminus x_{i}}$ obtained by using the
aforementioned canonical isomorphisms $F_{l}|_{U_{j}\setminus
x_{i}}\xrightarrow{\sim}F|_{U_{j}\setminus x_{i}}$. To be precise,
$\overline{\Delta}$ is given by the composition
$\displaystyle\text{GL}(F)|_{U_{j}\setminus x_{i}}\,\xrightarrow{\Delta}$
$\displaystyle\;\text{GL}(F)|_{U_{j}\setminus
x_{i}}\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F)|_{U_{j}\setminus
x_{i}}\,\times\text{GL}(F)|_{U_{j}\setminus x_{i}}$
$\displaystyle\xrightarrow{\sim}$
$\displaystyle\;\text{GL}(t\,F)|_{U_{j}\setminus
x_{i}}\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F_{l})|_{U_{j}\setminus
x_{i}}\,\times\text{GL}(F)|_{U_{j}\setminus x_{i}}$
Let $S$ be a flat $R$-algebra. Automorphisms of the diagram of sheaves
$\mathcal{V}|_{\text{Spec}(S)}$ are the same as automorphisms the chain of
free modules $t\,F\otimes S\,\subset\,F_{1}\otimes
S\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\otimes S\,\subset\,F\otimes S$. By
$R$-flatness of $S$, we know that $t$ is still a nonzerodivisor in $S$. So the
map $S\rightarrow S[\frac{1}{t}]$ is injective. By unraveling the definition,
an automorphism of the chain of modules above is an element of $G(S)$ such
that its localization in $G(S[\frac{1}{t}])=G|_{U_{j}\setminus
x_{i}}(S[\frac{1}{t}])$ belongs to the image of the diagonal map
$\overline{\Delta}(S[\frac{1}{t}])$. These are exactly the $S$-points of the
scheme theoretic closure of the composition $D:\,\text{GL}(F)|_{U_{j}\setminus
x_{i}}\,\xrightarrow{\Delta}\,G|_{U_{j}\setminus x_{i}}\,\hookrightarrow\,G$.
This scheme theoretic closure $\text{Im}(D)$ must be $R$-flat, because it is
the Spec of a subalgebra of the torsion-free $R$-module
$\mathcal{O}_{\text{GL}(F)|_{U_{j}\setminus x_{i}}}$. We conclude that the
functor $\text{GL}(\mathcal{V})|_{U_{j}}$ is represented by an affine flat
$U_{j}$-scheme. We are left to show smoothness.
$\text{Im}(D)$ is of finite type, because it is a closed subscheme of $G$.
Also, we have seen that $\text{GL}(\mathcal{V})|_{U_{j}\setminus
x_{i}}\,\cong\,\text{GL}_{n}\times(U_{j}\setminus x_{i})$. We are left to show
that the fiber $\text{Im}(D)|_{x_{i}}$ is smooth. Since taking scheme
theoretic image of quasicompact maps commmutes with flat base-change, we can
pass to the completion of the local ring at $x_{i}$ without altering the fiber
$\text{Im}(D)|_{x_{i}}$. Therefore we can assume that $R$ is a complete
discrete valuation with residue field $k$. By the Cohen structure theorem, we
have $R=k[[{t}]]$.
Now we have put ourselves in a familiar situation, since the scheme theoretic
closure is by definition the parahoric group scheme of $\text{GL}_{n}$ defined
over $k[[{t}]]$ that preserves the chain of lattices
$t\,F\,\subset\,F_{1}\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\,\,\subset\,F$.
This group scheme is known to be smooth over $R$, see [BT84b, §3]. ∎
###### Remark 3.3.
We have seen in the proof above that $\text{GL}(\mathcal{V})$ is generically
isomorphic to the constant group scheme $\text{GL}_{n}$ (via choosing a
trivialization of $\mathcal{E}^{(0)}$ at the generic point). At the finitely
many points of degeneration $\\{x_{i}\\}_{i\in I}$, the base-change to the
completion of the local ring
$\text{GL}(\mathcal{V})|_{\widehat{\mathcal{O}}_{C,\,x_{i}}}$ is a parahoric
group scheme of $\text{GL}_{n}$ in the sense of Bruhat-Tits [BT84a, Def.
5.2.6]. $\text{GL}(\mathcal{V})$ is a special case of what is sometimes called
a parahoric Bruhat-Tits group scheme in the literature, see [PR10], [BS15],
[Hei10].
We briefly recall the definition of the moduli stack of torsors for a group
scheme such as $\text{GL}(\mathcal{V})$ above. Let $X$ be a scheme. Let
$\mathcal{G}\rightarrow X$ be a relatively affine group scheme that is smooth
over $X$. Let $\mathcal{P}$ be a $X$-scheme. A right $\mathcal{G}$-action on
$\mathcal{P}$ consists of a map of $X$\- schemes
$a:\,\mathcal{P}\times_{X}\mathcal{G}\rightarrow\mathcal{P}$ such that for all
$X$-schemes $T$ the map induced on $T$-points
$a(T):\,\mathcal{P}(T)\times\mathcal{G}(T)\rightarrow\mathcal{P}(T)$ is a
right action of $\mathcal{G}(T)$ on the set $\mathcal{P}(T)$. A (right)
$\mathcal{G}$-torsor over $X$ is a scheme $\pi:\mathcal{P}\rightarrow X$
together with a right $\mathcal{G}$-action that makes $\mathcal{P}$ a
$\mathcal{G}$-torsor in the ètale topology [Sta, Tag 0497]. This means that
there is an ètale cover $T\rightarrow X$ such that the base-change $\pi\times
id_{T}:\mathcal{P}\times_{X}T\rightarrow T$ is $\mathcal{G}$-equivariantly
isomorphic to the trivial $\mathcal{G}$-torsor $\mathcal{G}\times_{X}T$. An
isomorphism between $\mathcal{G}$-torsors is an isomorphism of $X$-schemes
that intertwines the $\mathcal{G}$-actions.
Let $\mathcal{G}$ be a relatively affine smooth group scheme over $C$.
###### Definition 3.4.
The moduli stack of $\mathcal{G}$-torsors $\text{Bun}_{\mathcal{G}}(C)$ is
defined to be the pseudofunctor from $k$-schemes into groupoids given as
follows. For every $k$-scheme $T$, we let
$\text{Bun}_{\mathcal{G}}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of $\mathcal{G}\times T$-torsors over $C\times T$}\end{matrix}\right\\}$
See Proposition 1 in [Hei10] for a proof of the following using the Artin
criteria.
###### Proposition 3.5.
$\text{Bun}_{\mathcal{G}}$ is a smooth algebraic stack over $k$. ∎
### 3.2 Moduli stacks of parabolic vector bundles
Fix a parabolic vector bundle $\mathcal{V}$ over $C$. Let us define the stack
of parabolic vector bundles of type $\mathcal{V}$.
###### Definition 3.6.
We define the pseudofunctor $\text{Bun}_{\mathcal{V}}$ from $k$-schemes into
groupoids as follows. For any $k$-scheme $S$, we set
$\text{Bun}_{\mathcal{V}}(S)\;\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of parabolic vector bundles $\mathcal{W}$}\\\ \text{of type $\mathcal{V}$ over
$C\times S$}\end{matrix}\right\\}$
For any morphism $f:S\rightarrow T$ of $k$-schemes, we set
$\text{Bun}_{\mathcal{V}}(f)$ to be the pullback $f^{*}$ of parabolic vector
bundles.
For the next proposition, we write $\text{Bun}_{\text{GL}(\mathcal{V})}(C)$ to
denote the moduli stack of torsors of $\text{GL}(\mathcal{V})$ over $C$. See
Definition 3.4 and [Hei10] Example (2) pg. 500.
###### Proposition 3.7.
There is an equivalence of pseudofunctors
$\text{Bun}_{\text{GL}(\mathcal{V})}(C)\xrightarrow{\sim}\text{Bun}_{\mathcal{V}}$.
###### Proof.
Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Recall that the group scheme $\text{GL}(\mathcal{V})$ acts on
$\mathcal{V}$ by automorphisms of diagrams of sheaves. Let $T$ be a
$k$-scheme. Let $\mathcal{P}$ be a $\text{GL}(\mathcal{V})\times T$-torsor
over $C\times T$. We can use the natural action of $\text{GL}(\mathcal{V})$ on
$\mathcal{V}$ in order to define an associated parabolic vector bundle
$\mathcal{P}\times_{\text{GL}(\mathcal{V})\times T}\,\mathcal{V}|_{C\times T}$
on $C\times T$. Let us describe how this is defined. Let $f:S\rightarrow
C\times T$ be an ètale cover of $C$ such that $f^{*}\mathcal{P}$ is isomorphic
to the trivial torsor. Choosing such an isomorphims, the descent datum for
$\mathcal{P}$ is given by an element
$h\in\text{GL}(\mathcal{V})(S\times_{C\times T}S)$. Here $h$ is viewed as an
automorphism of the trivial torsor $\text{GL}(\mathcal{V})|_{S}$ via left
multiplication. Now $h$ gives us an automorphims of
$\mathcal{V}|_{S\times_{C\times T}S}$, which we can interpret as a descent
datum for the diagram of sheaves $\mathcal{V}|_{S}$ relative to the cover
$f:S\rightarrow C\times T$. We can use effectivity of ètale descent for
quasicoherent sheaves in order to obtain a diagram of sheaves
$\mathcal{P}\times_{\text{GL}(\mathcal{V})\times T}\,\mathcal{V}|_{C\times T}$
such that its canonical descent datum is given by $h$. Recall that being a
vector bundle of a specified degree is a property of quasicoherent sheaves
that can be checked after passing to a flat cover. We can use this fact to
conclude that $\mathcal{P}\times_{\text{GL}(\mathcal{V})\times
T}\,\mathcal{V}|_{C\times T}$ is a parabolic vector bundle of type
$\mathcal{V}$ over $C\times T$.
Let us now define a functor
$F:\text{Bun}_{\text{GL}(\mathcal{V})}(C)\rightarrow\text{Bun}_{\mathcal{V}}$
as follows. For every $k$ scheme $T$ and for every
$\text{GL}(\mathcal{V})\times T$-torsor $\mathcal{P}$ on $C\times T$, we set
$F(T)\,(\mathcal{P})\vcentcolon=\mathcal{P}\times_{\text{GL}(\mathcal{V})\times
T}\mathcal{V}|_{C\times T}$.
We want to define a quasi-inverse functor $G$. Let $T$ be a $k$-scheme.
Suppose that we are given a parabolic vector bundle $\mathcal{W}$ of type
$\mathcal{V}$ over $C\times T$. We can define a presheaf of sets
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ on the category of flat $C\times
T$-schemes as follows. For every scheme $f:S\rightarrow C\times T$ that is
flat over $C\times T$, we define
$\text{Iso}(\mathcal{V},\,\mathcal{W})\,(S)\vcentcolon=\text{Iso}(f^{*}(\mathcal{V}|_{C\times
T}),\,f^{*}\mathcal{W})$. Here the isomorphisms are taken in the category of
diagrams of sheaves on $S$. It is clear by definition that
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ admits a right action of
$\text{GL}(\mathcal{V})$ given by precomposition.
We claim that $\text{Iso}(\mathcal{V},\,\mathcal{W})$ is in fact a
$\text{GL}(\mathcal{V})\times T$-torsor over $C\times T$. By ètale descent for
morphisms of quasicoherent sheaves, we know that
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ is a sheaf in the ètale topology.
Therefore, in order to prove the claim it suffices to show that there is an
ètale cover $f:S\rightarrow C\times T$ such that $f^{*}\mathcal{W}\cong
f^{*}(\mathcal{V}|_{C\times T})$ as diagrams of sheaves.
Say
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $\bigsqcup_{j\in J}U_{j}\rightarrow C\times T$ be an open cover of
$C\times T$ such that for all $j\in J$ we have
$\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U}^{\oplus
n}\cong\mathcal{F}^{(0)}|_{U_{j}}$. By refining the cover, we can furthermore
assume that each $U_{j}$ intersects at most one of the divisors $x_{i}\times
T$ of $C\times T$.
Let’s focus our attention on a single open subset $U_{j}$. If $U_{j}$ does not
intersect any of the divisors $x_{i}\times T$, then we have a chain of
isomorphisms
$\mathcal{V}|_{U_{j}}\cong\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus
n}\cong\mathcal{F}_{i}^{(0)}|_{U_{j}}\cong\mathcal{W}|_{U_{j}}$
(Here we are abusing notation for the intermediate terms; they are constant
diagrams of vector bundles with identities as morphisms). Hence we are done in
this case.
Suppose now that $U_{j}$ intersects one of the divisors $x_{i}\times T$. By
construction, it only intersects one such divisor. Fix $i\in I$ such that
$U_{j}\,\cap\,(x_{i}\times T)\neq\emptyset$. Let us denote by
$q_{i}:x_{i}\rightarrow C$ the closed immersion of the point $x_{i}$ into $C$.
Let $\pi:U_{j}\rightarrow T$ be the structure morphism. Let $W$ denote the
open subset of $T$ given by the inverse image of $U_{j}$ under the section
$q_{i}\times Id_{T}:\,T\rightarrow C\times T$. Define
$V\vcentcolon=\pi^{-1}(W)$. We can cover $U_{j}=V\,\bigcup\,(U_{j}\setminus
x_{i})$. Since $U_{j}\setminus x_{i}$ does not intersect any of the divisors
of degeneration, we have seen above that
$\mathcal{V}|_{U_{j}\setminus x_{i}}\cong\mathcal{E}^{(0)}|_{U_{j}\setminus
x_{i}}\cong\mathcal{O}_{U_{j}\setminus x_{i}}^{\oplus
n}\cong\mathcal{F}_{i}^{(0)}|_{U_{j}\setminus
x_{i}}\cong\mathcal{W}|_{U_{j}\setminus x_{i}}$
It remains to check the claim on $V$. Let $D\hookrightarrow V$ be the divisor
given by the intersection $V\,\cap\,(x_{i}\times T)$. By construction, we have
a structure morphims $\pi:V\rightarrow W$ and $D$ is given by a section
$s:W\rightarrow V$ of $\pi$. Since $V$ does not intersect any of the other
divisors of degeneration, for all $u\neq i$ the chain of inclusions with index
$u$ gives us canonical identifications
$\mathcal{E}_{u}^{(m)}|_{V}\cong\mathcal{E}^{(0)}|_{V}$ for all $m$.
Similarly, there are canonical identifications
$\mathcal{F}^{(m)}_{u}|_{V}\cong\mathcal{F}^{(0)}|_{V}$ for $u\neq i$. By
choosing isomorphisms
$\mathcal{E}^{(0)}|_{V}\cong\mathcal{F}^{(0)}|_{V}\cong\mathcal{O}_{V}^{\oplus
n}$ we get compatible isomorphisms
$\mathcal{E}_{u}^{(m)}|_{V}\cong\mathcal{F}_{u}^{(m)}|_{V}\cong\mathcal{O}_{V}^{\oplus
n}$ for all indexes $u\neq i$ and $m$ via the aforementioned canonical
identifications. Under these identifications, we see that $\mathcal{V}|_{V}$
and $\mathcal{W}|_{V}$ are determined by the following two chains of vector
bundles on $V$ respectively:
$\mathcal{O}_{V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{V}(D)^{\oplus
n}$ $\mathcal{O}_{V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}|_{V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{V}(D)^{\oplus
n}$
The claim is reduced to the problem of finding (maybe after passing to an
ètale cover of $V$) an automorphism $\text{GL}_{n}(V)$ of
$\mathcal{O}_{V}^{\oplus n}$ that sends one chain to the other. Consider the
corresponding flags of vector bundles on $W$
$0\,\overset{a_{i}^{(1)}}{\subset}\;s^{*}\,\left(\mathcal{E}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
$0\,\overset{a_{i}^{(1)}}{\subset}\;s^{*}\,\left(\mathcal{F}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{M_{l}}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
These two flags represent two $W$-points of the partial flag variety
$\text{GL}_{n}\,/\,P$, where $P$ is the standard parabolic corresponding to
the partition $(a_{i}^{(m)})_{m}$ of $n$. Let us denote these points by
$p_{1},p_{2}\in(\text{GL}_{n}\,/\,P)(W)$. We know that the quotient
$\text{GL}_{n}\,/\,P$ represents the ètale sheafification of the naive coset
functor. So there is an ètale cover $w:Y\rightarrow W$ and an element
$g\in\text{GL}_{n}(Y)$ such that $g\cdot w^{*}p_{1}=w^{*}p_{2}$. In other
words, the element $g$ relates the following two flags:
$0\,\overset{a_{i}^{(1)}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{E}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
$0\,\overset{a_{i}^{(1)}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{F}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
The base-change $pr_{2}:Y\times_{W}V\rightarrow V$ is an ètale cover. We have
a structure morphism $pr_{1}:Y\times_{W}V\rightarrow Y$. By construction, the
automorphism $pr_{1}^{*}\,g$ of $\mathcal{O}_{Y\times_{W}V}^{\oplus n}$
relates the two chains of vector bundles over $Y\times_{W}V$
$\mathcal{O}_{Y\times_{W}V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{Y\times_{W}V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{Y\times_{W}V}(D)^{\oplus
n}$ $\mathcal{O}_{Y\times_{W}V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}|_{Y\times_{W}V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{Y\times_{W}V}(D)^{\oplus
n}$
We conclude that
$\mathcal{W}|_{Y\times_{W}V}\cong\mathcal{V}|_{Y\times_{W}V}$, and the claim
is proven.
We can now define a quasi-inverse to $F$. We define
$G:\text{Bun}_{\mathcal{V}}\rightarrow\text{Bun}_{\text{GL}(\mathcal{V})}(C)$
as follows. For every $k$-scheme $T$ and every parabolic vector bundle
$\mathcal{W}$ of type $\mathcal{V}$ over $C\times T$, we let
$G(T)\,(\mathcal{W})\vcentcolon=\text{Iso}(\mathcal{V},\,\mathcal{W})$.
Let us prove that $F$ and $G$ are quasi-inverse functors. Fix a $k$-scheme
$T$. By construction, we have a canonical identification
$FG(T)\,(\mathcal{V}|_{C\times T})\cong\mathcal{V}|_{C\times T}$. It is also
clear by construction that under this canonical identification we have
$FG(T)\,(g)=g$ for any automorphism $g\in\text{Aut}(\mathcal{V}|_{C\times
T})$. This remains true for $\mathcal{V}|_{S}$, where $S$ is any ètale cover
of $C\times T$. Technically our functor $G$ is not defined for such $S$, but
the construction using the scheme
$\text{Iso}(\mathcal{V}|_{S},\,\mathcal{V}|_{S})$ still makes sense. So we
abuse notation and write $G$ for such construction.
Let $\mathcal{W}$ be any parabolic vector bundle of type $\mathcal{V}$ over
$C\times T$. We know that there exists an ètale cover $f:S\rightarrow C\times
T$ such that $f^{*}\mathcal{W}\cong f^{*}(\mathcal{V}|_{C\times T})$.
Therefore, $\mathcal{W}$ can be described by some ètale descent datum. In this
case this is just an element of
$\text{Aut}\left(\mathcal{V}|_{S\times_{C\times T}S}\right)$. We have seen
that we have a canonical isomorphism $FG\,(\mathcal{V}|_{S\times_{C\times
T}S})\cong\mathcal{V}|_{S\times_{C\times T}S}$. Under this canonical
identification, $FG(T)\,(\mathcal{W})$ is described by the same ètale descent
datum with respect to the cover $S\rightarrow C\times T$ (since we said that
$FG$ induces the identity on morphisms). So we can canonically identify
$\mathcal{W}$ and $FG(T)\,(\mathcal{W})$. The fact that $FG$ induces the
identity on automorphisms of $\mathcal{V}|_{C\times T}$ implies the same for
automorphims of $\mathcal{W}$ under the identification
$FG(T)\,(\mathcal{W})\,\cong\,\mathcal{W}$ we just constructed. The argument
for $GF(T)\cong id_{\text{Bun}_{\text{GL}(\mathcal{V})}(C)\,(T)}$ follows in a
similar manner. ∎
###### Corollary 3.8.
$\text{Bun}_{\mathcal{V}}$ is a smooth algebraic stack over $k$.
###### Proof.
This is an immediate consequence of Proposition 3.7 and Proposition 3.5. ∎
###### Remark 3.9.
It should be remarked that Corollary 3.8 can be done directly by hand. One can
describe a concrete atlas in terms of Quot schemes, just as is usually done
for the moduli stack of vector bundles. We wanted to prove the isomorphism in
Proposition 3.7 in order to show explicitly how the moduli of parabolic vector
bundles fits into the framework of moduli of torsors for Bruhat-Tits group
schemes over $C$.
### 3.3 Harder-Narasimhan strata
Fix a parabolic vector bundle $\mathcal{V}$ of rank $n$ over $C$. We will use
the existence of the Harder-Narasimhan filtration in order to attach
invariants to parabolic bundles of type $\mathcal{V}$. Let us first define the
kind of invariants we will be working with.
###### Definition 3.10.
A parabolic HN datum of rank $n$ is an $n$-tuple $(\mu_{l})_{l=1}^{n}$ of real
numbers $\mu_{l}$ such that $\mu_{l+1}\geq\mu_{l}$ for all $l$.
We can define a partial order on the set of Harder-Narasimhan data of a given
rank.
###### Definition 3.11.
Let $P_{1}=(\mu_{l})_{l=1}^{n}$ and $P_{2}=(\nu_{l})_{l=1}^{n}$ be two
parabolic HN data of rank $n$. We say $P_{1}\leq P_{2}$ if
1. (1)
$\sum_{l=1}^{n}\mu_{l}=\sum_{l=1}^{n}\nu_{l}$.
2. (2)
For all $1\leq k<n$, we have $\sum_{l=1}^{k}\mu_{l}\leq\sum_{l=1}^{k}\nu_{l}$.
For any parabolic bundle of type $\mathcal{V}$, we can associate a Harder-
Narasimhan datum.
###### Definition 3.12.
Let $\mathcal{W}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Suppose that it has Harder-Narasimhan
filtration given by
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l}=\mathcal{W}$.
Then, we define the Harder-Narasimhan datum $HN(\mathcal{W})$ associated to
$\mathcal{W}$ to be the parabolic HN datum given by
$\displaystyle
HN(\mathcal{W})=(\;\underbrace{\mu(\mathcal{W}_{1}),\,\cdots,\,\mu(\mathcal{W}_{1})}_{\text{rank}\;\mathcal{W}_{1}/0\,\text{times}},\;\underbrace{\mu(\mathcal{W}_{2}/\,\mathcal{W}_{1}),\,\cdots,\,\mu(\mathcal{W}_{2}/\,\mathcal{W}_{1})}_{\text{rank}\;\mathcal{W}_{2}/\,\mathcal{W}_{1}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{W}_{l}/\,\mathcal{W}_{l-1}),\;\cdots,\;\mu(\mathcal{W}_{l}/\,\mathcal{W}_{l-1})}_{\text{rank}\;\mathcal{W}_{l}/\mathcal{W}_{l-1}\,\text{times}}\;)$
We are now ready to define some relevant subfunctors of
$\text{Bun}_{\mathcal{V}}$.
###### Definition 3.13.
Let $P$ be a HN-datum. We define $\text{Bun}_{\mathcal{V}}^{\leq P}$ to be the
pseudofunctor from $k$-schemes into groupoids given as follows. For every
$k$-scheme $S$,
$\text{Bun}_{\mathcal{V}}^{\leq
P}(S)\;\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of
}\mathcal{W}\in\text{Bun}_{\mathcal{V}}(S)\text{ such that}\\\
HN\left(\mathcal{W}\,|_{C_{s}}\right)\leq P\;\;\text{for all }s\in
S\end{matrix}\right\\}$
###### Remark 3.14.
The locus of semistable bundles of a given degree $d$ is
$\text{Bun}_{\mathcal{V}}^{\leq P}$, where $P$ is the the rank $n$ parabolic
HN datum given by $P=\left(\frac{d}{n},\frac{d}{n},\cdots,\frac{d}{n}\right)$.
Our main goal is to analyze the geometry of these subfunctors. It will turn
out that all $\text{Bun}_{\mathcal{V}}^{\leq P}$s are quasicompact open
substacks of $\text{Bun}_{\mathcal{V}}$. As we vary $P$, they form a
stratification of $\text{Bun}_{\mathcal{V}}$. We will refer to the subfunctors
$\text{Bun}_{\mathcal{V}}^{\leq P}$ collectively as Harder-Narasimhan strata.
## 4 Parabolic Quot Schemes
The main goal of this section is to prove the representability of the two
functors described in Definition 4.9 and Definition 4.16. These can be seen as
parabolic analogues of the Quot scheme and the iterated Quot scheme
respectively. The reader who is only interested in the main theorems described
in the introduction can safely skip this section and refer back to it when
needed.
###### Definition 4.1.
Let $\mathcal{Q}$ be a coherent sheaf on $C$. The Hilbert polynomial
$HP(\mathcal{Q})$ of $\mathcal{Q}$ with respect to $\mathcal{O}(1)$ is the
polynomial with rational coefficients whose value at an integer
$n\in\mathbb{Z}$ is given by
$HP(\mathcal{Q})(n)=\text{dim}_{k}\,H^{0}\left(\mathcal{Q}(n)\right)\,-\,\text{dim}_{k}\,H^{1}\left(\mathcal{Q}(n)\right)$.
###### Remark 4.2.
If $\mathcal{Q}$ is a vector bundle of rank $r$ and degree $d$, Riemann-Roch
tells us that $HP(\mathcal{Q})(n)=(r\cdot d)\,n+r\cdot\chi(\mathcal{O_{C}})$.
So the Hilbert polynomial contains precisely the information of the degree and
the rank of the vector bundle.
Let $S$ be a scheme of finite type over $k$. Let $\mathcal{E}$ be a coherent
sheaf on $C\times S$ that is flat over $S$. Choose a polynomial
$P\in\mathbb{Q}[T]$ with rational coefficients.
###### Definition 4.3.
We define $\text{Quot}_{\mathcal{E}/C\times S/S}^{P}$ to be the contravariant
functor from $S$-schemes to sets given as follows. For any $S$-scheme $T$,
$\text{Quot}_{\mathcal{E}/C\times S/S}^{P}(T)$ is the set of isomorphism
classes of quotients $\mathcal{Q}$ of $\mathcal{E}|_{C\times T}$ such that
1. (a)
$\mathcal{Q}$ is flat over $T$
2. (b)
Let $\mathcal{F}$ denote the kernel of $\mathcal{E}|_{C\times
T}\twoheadrightarrow\mathcal{Q}$. For all points $t\in T$, the Hilbert
polynomial of the fiber $\mathcal{F}|_{C_{t}}$ is $P$.
In this context, we have the following classical theorem of Grothendieck. See
[Nit05] for a proof.
###### Theorem 4.4.
$\text{Quot}_{\mathcal{E}/C\times S/S}^{P}$ is represented by a projective
$S$-scheme. ∎
We want to define a variation of the Quot scheme that will serve our needs in
latter sections. The set-up is as follows. We fix a finite set
$\\{x_{i}\\}_{i\in I}$ of $k$-points in $C$. Let $S$ be a scheme of finite
type over $k$. Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$ with parabolic structure at
$\\{x_{i}\\}_{i\in I}$. We want to describe a functor that classifies
subbundles of $\mathcal{V}$ with some given invariants. Let us first describe
the invariants we will be looking at.
###### Definition 4.5.
A parabolic Quot datum of rank $k$ is a tuple $\left(P,\,b_{i}^{(m)}\right)$
where
1. (a)
$P\in\mathbb{Q}[T]$ is a polynomial with rational coefficients.
2. (b)
For each $i\in I$ and $1\leq m\leq N_{i}$, $b_{i}^{(m)}$ is a nonnegative
integer. These are required to satisfy $\sum_{m=1}^{N_{i}}b_{i}^{(m)}=k$ for
all $i\in I$.
We build our parabolic Quot scheme in stages. We start with the following
moduli problem:
###### Definition 4.6.
Let $S$ and $\mathcal{V}$ as above. Fix $P\in\mathbb{Q}[T]$. We denote by
$\text{Quot}_{\mathcal{V}}^{P}$ the subfunctor of
$\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$ given as follows. For a
$S$-scheme $T$, define $\text{Quot}_{\mathcal{V}}^{P}(T)$ to be the set of
isomorphism classes of quasicoherent quotients $\mathcal{Q}^{(0)}$ in
$\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}(T)$ such that for all $i\in
I$ and $0\leq m\leq N_{i}$, the sheaf $\mathcal{L}_{i}^{(m)}$ given as the
pushout of the following diagram is $T$-flat.
$\textstyle{\mathcal{Q}^{(0)}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{i}^{(k)}}$$\textstyle{\mathcal{E}^{(0)}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
###### Remark 4.7.
Since $\mathcal{L}_{i}^{(m)}$ is given as a pushout, its formation commutes
with base-change. This is therefore a well defined functor.
###### Proposition 4.8.
$\text{Quot}_{\mathcal{V}}^{P}$ is a finite disjoint union of locally closed
subschemes of $\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$.
###### Proof.
Let
$\mathcal{Q}^{(0)}_{univ}\in\text{Coh}\left(C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\right)$ be the universal quotient of $\mathcal{E}^{(0)}$ on
$C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. Define
$\mathcal{L}_{i,\,univ}^{(k)}$ to be the pushout diagram:
$\textstyle{\mathcal{Q}^{(0)}_{univ}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{i,\,univ}^{(m)}}$$\textstyle{\mathcal{E}^{(0)}(x_{i})|_{C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\right)|_{C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
By definition,
$\text{Quot}_{\mathcal{V}}^{P}\longrightarrow\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}$ is the functor from $S$-schemes into set given as follows. For a
$S$-scheme $T$, we have
$\text{Quot}_{\mathcal{V}}^{P}(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\text{ such that}\\\ (id_{C}\times
f)^{*}\mathcal{L}_{i,\,univ}^{(m)}\;\text{is $T$-flat for all $i$ and
$m$}\end{matrix}\right\\}$
Now we can apply the existence of flattening stratifications (see [Nit05]) to
the projective map $pr_{2}:\,C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\;\longrightarrow\;\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$
and the coherent sheaves $\mathcal{L}_{i,\,univ}^{(m)}$ on
$C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. We conclude that
this functor is represented by a finite disjoint union of locally closed
subschemes of $\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. ∎
Suppose that we have $S$, $\mathcal{V}$, $T$ and $\mathcal{Q}^{(0)}$ as in
Definition 4.6. We denote by $\mathcal{F}^{(0)}$ the kernel of the quotient
$\mathcal{E}^{(0)}\longrightarrow\mathcal{Q}^{(0)}$. Furthermore, define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$
and
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$.
This means that we have the following commutative diagram with exact columns:
${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
By construction, we have a short exact sequence
$0\longrightarrow\,\mathcal{Q}_{i}^{(m)}\,\longrightarrow\,\mathcal{Q}^{(0)}(x_{i})\,\longrightarrow\,\mathcal{L}_{i}^{(m)}\,\longrightarrow
0$. By the long exact sequence of Tors at each stalk, we conclude that
$\mathcal{Q}_{i}^{(m)}$ is $T$-flat. The same reasoning applied to the short
exact sequence
$0\longrightarrow\,\mathcal{F}_{i}^{(m)}\,\longrightarrow\,\mathcal{E}_{i}^{(m)}\,\longrightarrow\,\mathcal{Q}_{i}^{(m)}\,\longrightarrow
0$ shows that $\mathcal{F}_{i}^{(m)}$ is $T$-flat. In addition, the snake
lemma gives us a short exact sequence
$0\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{L}_{i}^{(m)}\longrightarrow
0$
By $T$-flatness of $\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}$ we
conclude that $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}$ is
$T$-flat. Since all of these sheaves are $T$-flat, the formation of all of
$\mathcal{F}_{i}^{(m)}$, $\mathcal{Q}_{i}^{(m)}$,
$\mathcal{F}_{i}^{(m+1)}\,/\,\mathcal{F}_{i}^{(m)}$ and
$\mathcal{Q}_{i}^{(m+1)}\,/\,\mathcal{Q}_{i}^{(m)}$ commutes with base-change
on $T$. Hence the diagram above pulls back under base-change. This in turn
shows that the following is a well-defined subfunctor.
###### Definition 4.9.
Fix $S$ and $\mathcal{V}$ as above. Let $\left(P,\,b_{i}^{(m)}\right)$ a
parabolic Quot datum. We let $\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ be
the subfunctor of $\text{Quot}_{\mathcal{V}}^{P}$ defined as follows. For any
$S$-scheme $T$, $\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)$ is the set of
iso. classes of quotients $\mathcal{Q}^{(0)}\in\text{Quot}_{\mathcal{V}}^{P}$
satisfying
1. (a)
For all $i\in I$ and $1\leq m\leq N_{i}$,
$\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}|_{{x_{i}}\times T}$ is a
vector bundle of rank $b_{i}^{(m)}$.
2. (b)
$\mathcal{Q}_{i}^{(m)}$ and $\mathcal{F}_{i}^{(m)}$ are vector bundles on
$C\times T$.
Here we define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times
T}\cap\mathcal{F}^{(0)}(x_{i})$ and
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$,
as explained in the paragraphs above.
###### Proposition 4.10.
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ is an open subscheme of
$\text{Quot}_{\mathcal{V}}^{P}$. We call this the parabolic Quot scheme with
parabolic Quot datum $\left(P,b_{i}^{(m)}\right)$.
Before proving the proposition, let us state a lemma that will be used
repeatedly.
###### Lemma 4.11 (Fiberwise flatness criterion).
Let $B$ be a scheme and let $X$ be a scheme locally of finite presentation
over $B$. Let $\mathcal{F}\in\text{QCoh}(X)$ be a sheaf that is locally of
finite presentation as an $\mathcal{O}_{X}$-module. Suppose that $\mathcal{F}$
is flat over $B$. Let $x\in X$. Set $s=f(x)$. If the fiber
$\mathcal{F}|_{X_{s}}$ is $\mathcal{O}_{X_{s}}$-flat at $x$, then
$\mathcal{F}$ is $\mathcal{O}_{X}$-flat at $x$.
###### Proof.
This follows from [Sta, Tag 039C] by letting $X=Y$, $f=id_{X}$ and $B=S$. ∎
###### Proof of Proposition 4.10.
Let
$\mathcal{Q}^{(0)}_{univ}\in\text{Coh}(C\times\text{Quot}_{\mathcal{V}}^{P})$
be the universal quotient of $\mathcal{E}^{(0)}$ on
$C\times\text{Quot}_{\mathcal{V}}^{P}$. We denote by
$\mathcal{F}^{(0)}_{univ}$ the kernel of the quotient
$\mathcal{E}^{(0)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\longrightarrow\mathcal{Q}^{(0)}_{univ}$.
Furthermore, define
$\mathcal{F}_{i,\,univ}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\,\cap\,\mathcal{F}^{(0)}_{univ}(x_{i})$
and
$\mathcal{Q}_{i,\,univ}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\,/\,\mathcal{F}_{i,\,univ}^{(m)}$.
By definition, the functor
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow\text{Quot}_{\mathcal{V}}^{P}$
is given as follows. For every $S$-scheme $T$,
$\displaystyle\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such
that}\\\ \\\ (1)\quad\text{For all $i$ and $m$ we have that }\;(q_{i}\times
f)^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\\\
\text{is a vector bundle of rank $b_{i}^{(m)}$}\\\ \\\ (2)\quad\text{For all
$i$ and $m$, we have that both}\;(id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}\\\ \text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}\;\text{are $C\times
T$-flat}\end{matrix}\right\\}$
Let us define an intermediate functor $L$ given by imposing only the first
condition. In other words, we let $L$ be the functor from $S$-schemes $T$ into
sets given by
$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $i$, $m$ we have that}\\\ f^{*}\,(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\;\text{is
a vector bundle of rank $b_{i}^{(m)}$}\end{matrix}\right\\}$
By the discussion before Definition 4.9, we already know that $(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is a vector bundle. Hence for any morphism
$f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}$, we have that $f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is a vector bundle. Note that the condition for the rank can be checked at
each point $t\in T$. In other words, $L$ can be rewritten as
$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $i$, $m$ and $t\in T$ we have}\\\
\text{rank}_{\kappa(t)}\,\left(f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{t}\;=\;b_{i}^{(m)}\end{matrix}\right\\}$
But we know that
$\displaystyle\text{rank}_{\kappa(t)}\,\left(f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{t}\;=\;\text{rank}_{\kappa(f(t))}\left((q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{f(t)}$
Denote by $U$ the set of points $s\in\text{Quot}_{\mathcal{V}}^{P}$ such that
$\text{rank}_{\kappa(s)}\left((q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{s}\,=\,b_{i}^{(m)}$
for all $i$ and $m$. Then $L$ is given by the set theoretic condition
$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $t\in T$ we have $f(t)\in U$}\\\
\end{matrix}\right\\}$
Since $(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is locally free, the rank of the fibers is a locally constant function on
$\text{Quot}_{\mathcal{V}}^{P}$. In particular, we see that $U$ is an open
subset of $\text{Quot}_{\mathcal{V}}^{P}$. It follows that the functor $L$ is
represented by $U$ viewed as an open subscheme of
$\text{Quot}_{\mathcal{V}}^{P}$.
We can now go back to our functor
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$. We see that it is given by
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $i$ and $m$, we have that both}\\\ (id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}\;\text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}\;\text{are $C\times
T$-flat}\end{matrix}\right\\}$
By the discussion before Definition 4.9, the sheaves
$\mathcal{F}_{i,\,univ}^{(m)}$ and $\mathcal{Q}_{i,\,univ}^{(m)}$ are
$\text{Quot}_{\mathcal{V}}^{P}$-flat. So for any morphism $f:T\rightarrow U$
the sheaves $(id_{C}\times f)^{*}\mathcal{F}_{i,\,univ}^{(m)}$ and
$(id_{C}\times f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}$ are $T$ flat. Hence we can
apply Lemma 4.11 with $X=C\times T$ and $B=T$ to conclude that it suffices to
check flatness for the fibers $C_{t}$ at each $t\in T$. So our fuctor can be
described as:
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $i$ and $m$, for every $t\in T$ we have that both}\;\\\ (id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}|_{C_{t}}\;\text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{t}}\;\text{are $C_{t}$
flat}\end{matrix}\right\\}$
Since flatness can be checked after faithfully flat base-change, we see that
$(id_{C}\times f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{t}}$ is flat if and
only if $\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{f(t)}}$ is flat. The same reasoning
applies to $(id_{C}\times f)^{*}\mathcal{F}_{i,\,univ}^{(m)}|_{C_{t}}$. Let
$V$ be the set of points $s\in U$ such that
$\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{s}}$ and
$\mathcal{F}_{i,\,univ}^{(m)}|_{C_{s}}$ are $C_{s}$-flat for all $i$ and $m$.
Then we have just seen that the functor can described as
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $t\in T$, we have that $f(t)\in V$}\end{matrix}\right\\}$
We are reduced to check that $V$ is an open subset of $U$. For a fixed choice
of indexes $i$ and $m$, let $V_{i,m}^{\mathcal{Q}}$ be the set of points $s\in
U$ such that $\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{s}}$ is flat. We shall show
that $V_{i,m}^{\mathcal{Q}}$ is open.
Suppose that $s\in V_{i,m}^{\mathcal{Q}}$. The sheaf
$\mathcal{Q}_{i,\,univ}^{(m)}$ is $U$-flat (see the discussion before
Definition 4.9). Hence we can apply Lemma 4.11 to the morphism
$pr_{2}:\,C\times U\,\rightarrow U$ (here $X=C\times U$ and $B=U$) in order to
conclude that $\mathcal{Q}_{i,\,univ}^{(m)}$ is flat at all points $x\in
C\times U$ lying over $s$. Since the flatness locus of
$\mathcal{Q}_{i,\,univ}^{(m)}$ is open in $C\times U$ [Sta, Tag 0399], there
is an open $N\subset C\times U$ containing the fiber $C_{s}$ such that
$\mathcal{Q}_{i,\,univ}^{(m)}|_{N}$ is flat. Since the morphism
$pr_{2}:\,C\times U\,\rightarrow U$ is projective, $pr_{2}$ is in particular
closed. So we can find an open neighborhood $W$ of $s$ in $U$ such that
$(pr_{2})^{-1}(W)\subset N$. We conclude that $W\subset
V_{i,\,m}^{\mathcal{Q}}$, which shows that $V_{i,\,m}^{\mathcal{Q}}$ is open.
The same argument applies verbatim to the locus $V_{i\,m}^{\mathcal{F}}$ of
points $s\in U$ such that $\mathcal{F}_{i,\,univ}^{(m)}|_{C_{s}}$ is flat.
Since
$V=\bigcap_{i,\,m}\left(V_{i\,m}^{\mathcal{F}}\,\cap\,V_{i\,m}^{\mathcal{Q}}\right)$,
we conclude that $V$ is open. ∎
###### Corollary 4.12.
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ is a separated scheme of finite
type over $S$.
###### Proof.
This is an immediate consequence of Theorem 4.4, Proposition 4.8 and
Proposition 4.10. ∎
###### Definition 4.13.
Let $\mathcal{V}$ is a parabolic vector bundle over $C$. Let $\mathcal{W}$ be
a parabolic subbundle of $\mathcal{V}$. There is a unique Quot datum
$\theta(\mathcal{W})$ such that $\mathcal{W}$ is a $k$-point of
$\text{Quot}_{\mathcal{V}}^{\theta(\mathcal{W})}$. We call
$\theta(\mathcal{W})$ the parabolic Quot datum associated to $\mathcal{W}$.
More explicitly, if we have
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then
$\theta(\mathcal{W})=\left(HP(\mathcal{F}^{(0)}),\,b_{i}^{(m)}\right)$.
###### Remark 4.14.
1. (i)
Let $\mathcal{W}$ be a parabolic vector bundle over $C$. Recall that the rank
and degree of a vector bundle can be recovered from its Hilbert polynomial via
Riemann-Roch. By the degree formula for parabolic vector bundles, it follows
that $\mu(\mathcal{W})$ is determined by $\theta(\mathcal{W})$.
2. (ii)
Let $T$ be a connected $k$-scheme. Let $\mathcal{W}$ be a parabolic vector
bundle over $C\times T$. Since the degree and rank of a vector bundle are
locally constant on families, it follows that $\theta(\mathcal{W}|_{C_{t}})$
is the same for all $t\in T$.
We now want to iterate our construction in order to parametrize filtrations of
a parabolic vector bundle by parabolic subbundles with some given parabolic
Quot data.
###### Definition 4.15.
Let $S$ and $\mathcal{V}$ be as before. A filtration datum of length $l$ is a
pair of tuples
$\alpha=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}1\leq j\leq
l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{1\leq j\leq l-1}\right)$, where
1. (a)
Each ${}_{j}b_{i}^{(m)}$ is a nonnegative integer.
2. (b)
Each ${}_{j}P\in\mathbb{Q}[T]$ is a polynomial with rational coefficients.
###### Definition 4.16.
Let $S$ and $\mathcal{V}$ as usual. Fix a filtration datum $\alpha$ of length
$l$. Then, we define the functor $\text{Fil}_{\mathcal{V}}^{\alpha}$ over $S$
as follows. For any $S$-scheme $T$, $\text{Fil}^{\alpha}_{\mathcal{V}}(T)$
will denote the set of iso. classes of filtrations of $\mathcal{V}|_{T\times
C}$ by parabolic subbundles
$0=_{0}\mathcal{V}\,\subset\,_{1}\mathcal{V}\,\subset\,_{2}\mathcal{V}\,\subset\,\cdots\,\subset\,_{l}\mathcal{V}=\mathcal{V}$
such that if
${}_{j}\mathcal{V}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\subset\,_{j}\mathcal{F}_{i}^{(1)}\subset\cdots\;\subset\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then we have:
1. (a)
For points $t\in T$ and all $1\leq j\leq l-1$, the Hilbert Polynomial of
${}_{j}\mathcal{F}^{(0)}\,|_{C_{t}}$ is ${}_{j}P$.
2. (b)
For all $1\leq j\leq l-1$, $i\in I$ and $1\leq m\leq N_{i}$, we have that
${}_{j}\mathcal{F}_{i}^{(m)}\,/\,_{j}\mathcal{F}_{i}^{(m-1)}\,|_{T\times{x_{i}}}$
is a vector bundle of rank ${}_{j}b_{i}^{(m)}$.
###### Proposition 4.17.
$\text{Fil}^{\alpha}_{\mathcal{V}}$ is represented by a scheme that is
separated and of finite type over $S$.
###### Proof.
We induct on the length $l$ of the filtration datum. The case $l=1$ is just a
parabolic Quot scheme, so it follows from Corollary 4.12. Suppose that the
theorem has been proved for filtration data of size $l-1$. Let $\alpha$ be as
in the statement of the proposition. We can define
$\beta\vcentcolon=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}2\leq
j\leq l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{2\leq j\leq l-1}\right)$, which
is a filtration datum of length $l-1$. By the induction hypothesis, we know
that $Fil_{\mathcal{V}}^{\beta}$ is a scheme that is separated and of finite
type over $S$.
There is a universal filtration
$0\subset\,\mathcal{W}_{2,\,univ}\,\subset\,\mathcal{W}_{3,\,univ}\,\subset\cdots\,\subset\,\mathcal{W}_{l,\,univ}=\mathcal{V}|_{C\times
Fil^{\beta}_{\mathcal{V}}}$ by parabolic subbundles. Here each
$\mathcal{W}_{j,\,univ}$ has the required associated parabolic data
$\theta(\mathcal{W}_{j,\,univ})=\left({}_{j}P,\,_{j}b_{i}^{(m)}\right)$ for
$2\leq j\leq l-1$. We can define a map $Fil^{\alpha}_{\mathcal{V}}\rightarrow
Fil^{\beta}_{\mathcal{V}}$ by forgetting the first element in the parabolic
filtration. By tracing back the definitions, the functor
$Fil^{\alpha}_{\mathcal{V}}$ is represented by the scheme
$\text{Quot}_{\mathcal{W}_{2,\,univ}}^{{}_{1}P,\,_{1}b_{i}^{(m)}}$. Since we
know that
$\text{Quot}_{\mathcal{W}_{2,\,univ}}^{{}_{1}P,\,_{1}b_{i}^{(m)}}\longrightarrow
Fil^{\beta}_{\mathcal{V}}$ is a separated morphism of finite type, we conclude
that the composition $Fil^{\alpha}_{\mathcal{V}}\longrightarrow
Fil^{\beta}_{\mathcal{V}}\longrightarrow S$is separated and of finite type. ∎
###### Definition 4.18.
Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Suppose that we have
a filtration $\\{_{j}\mathcal{W}\\}$ of length $l$
$0=\,_{0}\mathcal{W}\subset\,_{1}\mathcal{W}\subset\cdots\,\subset\,_{l-1}\mathcal{W}\subset\,_{l}\mathcal{W}=\mathcal{V}$.
We know that such filtration is a $k$-point of
$\text{Fil}^{\,\psi(\mathcal{W}_{j})}_{\mathcal{V}}$ for some filtration datum
$\psi(\mathcal{W}_{j})$ of length $l$. We call this the filtration datum
associated to the filtration $\\{\mathcal{W}_{j}\\}$.
More explicitly, if we have
${}_{j}\mathcal{W}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\overset{{}_{j}b_{i}^{(1)}}{\subset}\,_{j}\mathcal{F}_{i}^{(1)}\overset{{}_{j}b_{i}^{(2)}}{\subset}\cdots\;\overset{{}_{j}b_{i}^{(N_{i})}}{\subset}\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then
$\psi(\mathcal{W}_{j})=\left({}_{j}b_{i}^{(m)},\,HP(_{j}\mathcal{F}^{(0)})\right)$.
## 5 Openness of strata
The goal of this section is to prove that each stratum
$\text{Bun}_{\mathcal{V}}(C)^{\leq P}$ is open (Theorem 5.11). This is
achieved by first proving that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is
constructible (Proposition 5.1), and then proving that
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is closed under generalization
(Proposition 5.9). A treatment of the analogous result in the case of
principal $G$-bundles over a curve of characteristic $0$ can be found in
[GN14][§2]. Our general approach is different from [GN14] in that we prove
constructibility of all strata directly without treating the openness of the
semistable locus first.
### 5.1 Constructibility of strata
The goal of this subsection is to prove the following proposition.
###### Proposition 5.1.
Let $T$ be a scheme of finite type over $k$. Let $\mathcal{V}$ be a parabolic
vector bundle of rank $n$ over $C\times T$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. Then, the subset
of $T$ given by $T^{\leq P}=\left\\{t\in T\,\mid\,HN(\mathcal{V}|_{C_{t}})\leq
P\right\\}$ is constructible in the Zariski topology.
Before proving Proposition 5.1, we need a couple of lemmas.
###### Lemma 5.2.
Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$ over $C$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum with
$\sum_{j=1}^{n}\mu_{j}=\text{deg}\,\mathcal{V}$. Suppose that
$HN(\mathcal{V})\nleq P$. Then, there exists a parabolic subbundle
$\mathcal{W}$ of $\mathcal{V}$ such that
$\mu(\mathcal{W})>\frac{1}{r}\sum_{l=1}^{r}\mu_{l}$, where $r$ is the rank of
$\mathcal{W}$.
###### Proof.
Suppose that $\mathcal{V}$ has Harder-Narasimhan filtration given by
$0\subset\mathcal{V}_{0}\subset\mathcal{V}_{1}\subset\cdots\,\subset\mathcal{V}_{l}=\mathcal{V}$.
Recall that
$HN(\mathcal{V})=(\;\underbrace{\mu(\mathcal{V}_{0}),\,\cdots,\,\mu(\mathcal{V}_{0})}_{\text{rank}\;\mathcal{V}_{0}/0\,\text{times}},\;\underbrace{\mu(\mathcal{V}_{1}/\,\mathcal{V}_{0}),\,\cdots,\,\mu(\mathcal{V}_{1}/\,\mathcal{V}_{0})}_{\text{rank}\;\mathcal{V}_{1}/\,\mathcal{V}_{0}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{V}_{l}/\,\mathcal{V}_{l-1}),\;\cdots,\;\mu(\mathcal{V}_{l}/\,\mathcal{V}_{l-1})}_{\text{rank}\;\mathcal{V}_{l}/\mathcal{V}_{l-1}\,\text{times}}\;)$
Let us write $HN(\mathcal{V})=(\nu_{1},\nu_{2},\cdots,\nu_{n})$. Then we know
$\sum_{j=1}^{n}\nu_{j}=\text{deg}\,\mathcal{V}=\sum_{j=1}^{n}\mu_{j}$. By
definition, $HN(\mathcal{V})\nleq P$ implies that there exists $1\leq m<n$
such that $\sum_{j=1}^{m}\nu_{j}>\sum_{j=1}^{m}\mu_{j}$. Let $m_{0}$ the
maximal such $m$. Let $1\leq q<l$ be such that
$\text{rank}\,\mathcal{V}_{q}\leq m_{0}<\text{rank}\,\mathcal{V}_{q+1}$.
Define $r\vcentcolon=\text{rank}\,\mathcal{V}_{q}$. We claim that we have
$\sum_{j=1}^{r}\nu_{j}>\sum_{j=1}^{r}\mu_{j}$. Once we prove this claim we
will be done, because we can then set $\mathcal{W}=\mathcal{V}_{q}$.
In order to prove the claim, note that we have
$0<\sum_{j=1}^{m_{0}}\nu_{j}-\sum_{j=1}^{m_{0}}\mu_{j}=\left(\sum_{j=1}^{r}\nu_{j}-\sum_{j=1}^{r}\mu_{j}\right)+\sum_{j=r+1}^{m_{0}}(\nu_{j}-\mu_{j})$
So it suffices to show that $\nu_{j}\leq\mu_{j}$ for all $r+1\leq j\leq
m_{0}$. For all $r+1\leq j\leq m_{0}$ we have
$\nu_{j}=\mu(\mathcal{V}_{q+1})=\nu_{m_{0}+1}$. Using
$\mu_{j}\geq\mu_{m_{0}+1}$, we are reduced to showing that
$\nu_{m_{0}+1}\leq\mu_{m_{0}+1}$.
By maximality of $m_{0}$, we have
$\sum_{j=1}^{m_{0}}\nu_{j}>\sum_{j=1}^{m_{0}}\mu_{j}$ and
$\sum_{j=1}^{m_{0}+1}\nu_{j}\leq\sum_{j=1}^{m_{0}+1}\mu_{j}$. This means that
we must have $\nu_{m_{0}+1}<\mu_{m_{0}+1}$, as desired. ∎
###### Definition 5.3.
If $\mathcal{V}$, $P$ and $\mathcal{W}$ are as in Lemma 5.2, we say that
$\mathcal{W}$ is a $P$-destabilizing subbundle of $\mathcal{V}$.
For the next lemma, we need to talk about $P$-destabilizing subbundles for the
fibers of a family.
###### Definition 5.4.
Let $T$ be a $k$-scheme of finite type and let $\mathcal{V}$ be a parabolic
vector bundle over $T\times_{k}C$. Let $P$ be a HN datum. For any topological
point $t\in T$, we denote by $Dest_{t}^{P}(\mathcal{V})$ the set of
$P$-destabilizing subbundles of $\mathcal{V}|_{C_{t}}$.
###### Lemma 5.5.
Let $T$ , $\mathcal{V}$ and $P$ as in the definition above. The set of all
Hilbert polynomials
$\displaystyle\left\\{HP\left(\mathcal{F}^{(0)}\right)\;\mid\;\exists t\in
T,\;\exists\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\in Dest_{t}^{P}(\mathcal{V})\right\\}$
is finite.
###### Proof.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $t\in T$. Recall (Remark 4.2) that the Hilbert polynomial of a vector
bundle on $C_{t}$ is completely determined by the degree and the rank of the
bundle. For any
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\in Dest_{t}^{P}(\mathcal{V})$
we know that
$1\leq\text{rank}\,\mathcal{F}^{(0)}\leq\text{rank}\,\mathcal{V}$. So there
are only finitely many possible ranks for such $\mathcal{F}^{(0)}$. It
therefore suffices to fix some rank $r$ and show that there are finitely many
possible values of $\text{deg}\,\mathcal{F}^{(0)}$ for
$\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})$ of rank $r$ as above.
By definition of $P$-destabilizing bundle, we know that for such a
$\mathcal{W}$ we have $\text{deg}\,\mathcal{W}>\sum_{l=1}^{r}\mu_{l}$. By
Lemma 2.9 we know that $deg\,\mathcal{W}\,\leq
deg\left(\mathcal{F}^{(0)}\right)+r|I|$. Hence we must have
$\text{deg}\,\mathcal{F}^{(0)}>\sum_{l=1}^{r}\mu_{l}-r|I|$. It therefore
suffices to find a uniform upper bound for the possible values of
$\text{deg}\,\mathcal{F}^{(0)}$. We argue by contradiction. Suppose that the
degrees of all possible subbundles of $\mathcal{E}^{(0)}|_{C_{t}}$ for $t\in
T$ are not uniformly bounded above. Since
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}|_{C_{t}}\right)\supset
H^{0}\left(C_{t},\,\mathcal{F}^{(0)}\right)$, Riemann-Roch for
$\mathcal{F}^{(0)}$ implies that the dimension of
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}|_{C_{t}}\right)$ is not bounded above.
This in turn implies that for each $m\geq 0$ the dimension of
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}(m)|_{C_{t}}\right)$ is not bounded.
Serre vanishing and cohomology and base-change imply that there exists some
$m>>0$ such that $H^{0}\left(C_{t},\,\mathcal{E}^{(0)}(m)|_{C_{t}}\right)$ is
the fiber of the pushforward $(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ for all
$t\in T$. Since $\pi_{T}$ is a projective morphism, the sheaf
$(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ is coherent on the Noetherian scheme
$T$. This means that the dimension of the fibers of
$(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ are uniformly bounded, a contradition. ∎
We are now ready to prove the main proposition of this section.
###### Proof of Proposition 5.1.
The question is local, so we can restrict to each (open) connected component
of $T$. We can therefore assume that the degree of the fibers
$\mathcal{V}|_{C_{t}}$ is the same for all $t\in T$. Let us denote this
commond degree by $d$. We can assume without loss of generality that
$\sum_{j=1}^{n}\mu_{i}=d$, because otherwise we would have that $T^{\leq P}$
is empty. In order to prove constructibility, we shall show that the
complement $T^{\nleq P}\vcentcolon=|T|\setminus T^{\leq P}$ is constructible.
For each $t\in T$ and $\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})$, we can
associate a parabolic Quot datum $\theta(\mathcal{W})$ as in Definition 4.5.
Define
$\mathfrak{S}\vcentcolon=\left\\{\,\theta(\mathcal{W})\;\mid\;\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})\;\text{for
some $t\in T$}\,\right\\}$. Lemma 5.5 tells us that the set of possible first
coordinates of tuples $\left(HP(\mathcal{F}^{(0)}),(b_{i}^{(m)})\right)$ in
$\mathfrak{S}$ is finite. But there are only finitely many possible choices of
nonnegative integers $b_{i}^{(m)}$, because
$\sum_{m=1}^{N_{i}}b_{i}^{(m)}=\text{rank}\,\mathcal{W}$ for all $i\in I$ and
the rank of $\mathcal{W}$ is always smaller than the rank of $\mathcal{V}$.
Hence $\mathfrak{S}$ is finite.
The $T$-scheme
$\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$
is of finite type over $T$, because it is a finite disjoint union of schemes
of finite type over $T$. By Chevalley’s Theorem, the set theoretic image of
the structure morphism
$\pi:\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow
T$ is constructible. We claim that this image is precisely $T^{\nleq P}$.
Let $t\in T^{\nleq P}$. By Lemma 5.2, $\mathcal{V}|_{C_{t}}$ admits a
$P$-destabilizing subbundle $\mathcal{W}$ over $C_{t}$. By the definition of
the parabolic Quot scheme, $\mathcal{W}$ represents a $t$-point of
$\text{Bun}_{\mathcal{V}}^{\theta(\mathcal{W})}$. Therefore $t$ is in the
image of the structure morphism
$\pi:\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow
T$.
Conversely, let $s$ be a point of $\text{Bun}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$
for some $(P,\,b_{i}^{(m)})\in\mathfrak{S}$. By definition, this means that
$\mathcal{V}|_{C_{s}}$ has a subbundle $\mathcal{W}$ over $C_{s}$ of the form
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$
such that $HP(\mathcal{F}^{(0)})=P$. But we know that
$(P,\,b_{i}^{(m)})\in\mathfrak{S}$. We can compute the slope of $\mathcal{W}$
from the parabolic Quot data $(P,\,b_{i}^{(m)})$, and we see that
$\mathcal{W}$ must be a $P$-destabilizing subbundle of $\mathcal{V}|_{C_{s}}$.
Therefore $HN(\mathcal{V}|_{C_{s}})\nleq P$. Lemma 2.30 implies that
$HN(\mathcal{V}|_{C_{s}})=HN(\mathcal{V}|_{C_{\pi(s)}})$, so we must also have
$HN(\mathcal{V}|_{C_{\pi(s)}})\nleq P$. We conclude that $\pi(s)\in T^{\nleq
P}$. ∎
### 5.2 Openness of strata
The following proposition will be used to prove the main theorem of this
section.
###### Proposition 5.6.
Let $R$ be a discrete valuation ring over $k$. Let $\eta$ (resp. $s$) denote
the generic (resp. special) point of $\text{Spec}(R)$. Let $\mathcal{V}$ be a
parabolic vector bundle of rank $n$ over $C_{R}$. Let $P$ be a parabolic HN
datum. If $HN\left(\mathcal{V}|_{C_{s}})\right)\leq P$, then
$HN\left(\mathcal{V}|_{C_{\eta}}\right)\leq P$.
Before proving Proposition 5.6, let’s state a lemma that we will use in the
proof.
###### Lemma 5.7.
Let $R$ be a $k$-algebra. Let $\mathcal{E}$ be a vector bundle on $C_{R}$.
Suppose that $\mathcal{F}$ is a coherent subsheaf of $\mathcal{E}$ such that
the quotient $\mathcal{E}\,/\,\mathcal{F}$ is $R$-flat. Then, $\mathcal{F}$ is
a vector bundle on $C_{R}$.
###### Proof.
We want to show that $\mathcal{F}$ is $C_{R}$-flat. There is a short exact
sequence
$0\longrightarrow\mathcal{F}\longrightarrow\mathcal{E}\longrightarrow\mathcal{E}\,/\,\mathcal{F}\longrightarrow
0$. We can use the Tor long exact sequence at each stalk to conclude that
$\mathcal{F}$ is $R$-flat. By Lemma 4.11 with $X=C_{R}$ and
$B=\text{Spec}\,R$, it suffices to check flatness at each fiber. Choose a
point $t\in\text{Spec}\,R$. Proving flatness of $\mathcal{F}|_{C_{t}}$ amounts
to showing that $\mathcal{F}|_{C_{t}}$ is torsion-free, because $C_{t}$ is a
regular curve. Since $\mathcal{E}\,/\,\mathcal{F}$ is $R$-flat, the fiber
$\mathcal{F}|_{C_{t}}$ remains a subsheaf of $\mathcal{E}|_{C_{t}}$. Since
$\mathcal{E}|_{C_{t}}$ is torsion-free, we conclude that
$\mathcal{F}|_{C_{t}}$ is torsion-free. ∎
###### Proof of Proposition 5.6.
For the purpose of this proof, it will be convenient to fix the following
notation. For every sheaf $\mathcal{F}$ over $C_{R}$, we will write
${}^{\eta}\mathcal{F}\vcentcolon=\mathcal{F}|_{C_{\eta}}$ and
${}^{s}\mathcal{F}\vcentcolon=\mathcal{F}|_{C_{s}}$. We will use the same
notation for parabolic vector bundles.
Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C_{R}$. Suppose that
$HN\left({}^{\eta}\mathcal{V}\right)\nleq P$, where
${}^{\eta}\mathcal{V}=\left[\;{}^{\eta}\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,^{\eta}\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,^{\eta}\mathcal{E}_{i}^{(N_{i})}=\,^{\eta}\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$
By Lemma 5.2, there exists a $P$-destabilizing parabolic subbundle
${}^{\eta}\mathcal{W}\subset\,^{\eta}\mathcal{V}$. Set
${}^{\eta}\mathcal{W}=\left[\;{}^{\eta}\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,^{\eta}\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,^{\eta}\mathcal{F}_{i}^{(N_{i})}=\,^{\eta}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$
Let
${}^{\eta}\mathcal{Q}^{(m)}_{i}\vcentcolon=\,^{\eta}\mathcal{E}_{i}^{(m)}\,/\,^{\eta}\mathcal{F}_{i}^{(m)}$,
and set
${}^{\eta}\mathcal{L}_{i}^{(m)}\vcentcolon=\,^{\eta}\mathcal{Q}^{(0)}(x_{i})\,/\,^{\eta}\mathcal{Q}^{(m)}_{i}$.
Let $j$ denote the open immersion $j:C_{\eta}\hookrightarrow C_{R}$. Define
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\text{Im}\left(\;\mathcal{E}_{i}^{(m)}\xrightarrow{unit}j_{*}\,^{\eta}\mathcal{E}_{i}^{(m)}\longrightarrow
j_{*}\,^{\eta}\mathcal{Q}_{i}^{(m)}\;\right)$. Also, define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\text{Ker}\left(\;\mathcal{E}_{i}^{(m)}\twoheadrightarrow\mathcal{Q}_{i}^{(m)}\;\right)$.
We have the following commutative diagram with exact columns:
${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
Figure 3: Diagram 3
Define
$\mathcal{L}_{i}^{(m)}\vcentcolon=\mathcal{Q}^{(0)}(x_{i})\,/\,\mathcal{Q}_{i}^{(m)}$.
Since $j$ is affine, the pushforward functor $j_{*}$ is exact on quasicoherent
sheaves. This implies that $\mathcal{L}_{i}^{(m)}\subset
j_{*}\,^{\eta}\mathcal{L}_{i}^{(m)}$, and hence $\mathcal{L}_{i}^{(m)}$ is
$R$-torsion free. Therefore $\mathcal{L}_{i}^{(m)}$ is $R$-flat, because $R$
is a discrete valuation ring.
We can use the short exact sequence,
$0\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{L}_{i}^{(m)}\longrightarrow
0$
and the flatness of $\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}$ to
see that $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}$ is $R$-flat for
all $i$, $m$. Now we can use the short exact sequence
$0\longrightarrow\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow
0$
to conclude that $\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}$ is $R$\-
flat for all $i$ and $m$. A similar argument shows that
$\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}$ is $R$ flat. Therefore,
Diagram 3 remains exact and with horizontal arrows given by inclusions after
any base-change with respect to $R$.
Since $\mathcal{Q}_{i}^{(m)}\subset j_{*}\,^{\eta}\mathcal{Q}_{i}^{(m)}$, we
have that $\mathcal{Q}_{i}^{(m)}$ is $R$-torsion free. This implies that
$\mathcal{Q}_{i}^{(m)}$ is $R$-flat, because $R$ is a discrete valuation ring.
By construction we have the short exact sequence
$0\longrightarrow\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{Q}_{i}^{(m)}\longrightarrow
0$
By Lemma 5.7 we conclude that $\mathcal{F}_{i}^{(m)}$ is a vector bundle on
$C_{R}$. The discussion above implies that
$\mathcal{W}\vcentcolon=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle on $C_{R}$. We don’t necessarily know that
$\mathcal{W}$ is a parabolic subbundle of $\mathcal{V}$, because we have no
control on the quotients $\mathcal{Q}_{i}^{(m)}$. We do however know that
$\mathcal{Q}_{i}^{(m)}$ $R$-flat. Therefore, the base-change to the special
fiber ${}^{s}\mathcal{W}\hookrightarrow\,^{s}\mathcal{V}$ remains a
monomorphism.
We can now appeal to Lemma 2.11 to obtain a subbundle
${}^{s}\mathcal{W}^{sat}$ of ${}^{s}\mathcal{V}$ containing the subobject
${}^{s}\mathcal{W}$ and satisfying
$\mu(^{s}\mathcal{W}^{sat})\geq\mu(^{s}\mathcal{W})$. By local constancy of
degree (Lemma 2.29) we have $\mu(^{s}\mathcal{W})=\mu(^{\eta}\mathcal{W})$.
Hence $\mu(^{s}\mathcal{W}^{sat})\geq\,\mu(^{\eta}\mathcal{W})$. We conclude
that ${}^{s}\mathcal{W}^{sat}$ is $P$-destabilizing, because
$\text{rank}\,^{s}\mathcal{W}^{sat}=\text{rank}\,^{\eta}\mathcal{W}$ and
${}^{\eta}\mathcal{W}$ is $P$-destabilizing. This implies that
$HN\left({}^{s}\mathcal{V}\right)\nleq P$, as desired. ∎
###### Remark 5.8.
Suppose that $\mu(^{s}\mathcal{W}^{sat})=\mu(^{s}\mathcal{W})$ in the proof
above. Then part (b) of Lemma 2.11 implies that
${}^{s}\mathcal{\mathcal{W}}=\,^{s}\mathcal{W}^{sat}$. In this case
${}^{s}\mathcal{Q}_{i}^{(m)}$ is a vector bundle. Applying Lemma 4.11 to the
$R$-flat sheaf $\mathcal{Q}_{i}^{(m)}$, we see that $\mathcal{Q}_{i}^{(m)}$ is
a vector bundle over $C_{R}$. Therefore $\mathcal{W}\subset\mathcal{V}$ is a
parabolic subbundle over $C_{R}$ in this case.
###### Proposition 5.9.
Let $T$ be a scheme of finite type over $k$. Let $\mathcal{V}$ a parabolic
vector bundle over $C\times S$. Let $P$ be a HN datum. The set $T^{\leq
P}\vcentcolon=\left\\{t\in T\,\mid\,HN(\mathcal{V}|_{C_{t}})\leq P\right\\}$
is Zariski open in $T$.
###### Proof.
By Proposition 5.1, we know that $T^{\leq P}$ is constructible. In order to
prove that it is open we are left to show that it is closed under
generalization. Let $p,t\in T$ be two topological points with
$t\in\overline{\\{p\\}}$. Assume that $t\in T^{\leq P}$. By [Sta, Tag 054F],
there exists a discrete valuation $k$-algebra $R$ and a morphism
$f:\text{Spec}(R)\rightarrow T$ such that $f(\eta)=p$ and $f(s)=t$. (Here we
are using the same notation as in Proposition 5.6, so $s$ is the special point
and $\eta$ is the generic point). By Lemma 2.30, we know that
$HN(\mathcal{V}|_{C_{s}})=HN(\mathcal{V}|_{C_{t}})$. Since $t\in T^{\leq P}$,
we have $HN\left(\mathcal{V}|_{C_{t}}\right)\leq P$. Hence
$HN\left(\mathcal{V}|_{C_{s}}\right)\leq P$. We can apply Proposition 5.6 to
deduce that $HN(\mathcal{V}|_{C_{\eta}})\leq P$. By Lemma 2.30 again, we have
$HN(\mathcal{V}|_{C_{p}})=(\mathcal{V}|_{C_{\eta}})\leq P$. So $p\in T^{\leq
P}$, as desired. ∎
###### Remark 5.10.
In particular, Proposition 5.9 implies that parabolic HN data are upper
semicontinuous.
###### Theorem 5.11.
Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Let $P$ be a
parabolic HN datum. Then $\text{Bun}_{\mathcal{V}}^{\leq P}$ is an open
substack of $\text{Bun}_{\mathcal{V}}$.
###### Proof.
The theorem amounts to showing that for any $k$-scheme $T$ and morphism
$T\rightarrow\text{Bun}_{\mathcal{V}}$, the base-change
$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\;\rightarrow T$ is represented by an open
subscheme of $T$. Corollary 3.8 tells us that $\text{Bun}_{\mathcal{V}}$ is
locally of finite type over $k$. So we can assume that $T$ is of finite type
over $k$.
A map $T\rightarrow\text{Bun}_{\mathcal{V}}$ is by definition a parabolic
vector bundle $\mathcal{W}$ of type $\mathcal{V}$ on $C\times T$. We can
describe the base-change $\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T$ as follows. For any scheme $U$ we have
$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\,(U)\;=\;\left\\{\begin{matrix}\text{morphisms}\;f:U\rightarrow
T\text{ such that}\\\ \text{ for all $u\in
U$},\;HN\left(\mathcal{W}\,|_{C_{u}}\right)\,\leq P\end{matrix}\right\\}$
For any $u\in U$, Lemma 2.30 implies that
$HN\left(\mathcal{W}\,|_{C_{u}}\right)=HN\left(\mathcal{W}\,|_{C_{f(u)}}\right)$.
We conclude that the functor $\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T$ can be alternatively described by the
set theoretic condition
$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\,(U)\;=\;\left\\{\begin{matrix}\text{morphisms}\;f:U\rightarrow
T\text{ such that}\\\ \text{ for all $u\in U$, $f(u)\in T^{\leq
P}$}\end{matrix}\right\\}$
By Proposition 5.9, the set $T^{\leq P}$ is open. Hence the functor is
represented by $T^{\leq P}$ viewed as an open subscheme of $T$. ∎
The analogue of Theorem 5.11 in the context of vector bundles without
parabolic structure when $P=0$ is classical. This amounts to proving that the
semistable locus of a family of vector bundles is open. See e.g. [HL97][Prop.
2.3.1].
## 6 Quasicompactness of strata
### 6.1 Harder-Narasimhan stratification for the moduli of vector bundles
We recall the Harder-Narasimhan stratification for the classical moduli stack
of vector bundles on a curve.
###### Definition 6.1.
The moduli stack of vector bundles of rank $n$ over $C$ is the pseudofunctor
$\text{Bun}_{\text{GL}_{n}}(C)$ from $k$-schemes into groupoids given as
follows. For each $k$-scheme $T$, we define
$\text{Bun}_{\text{GL}_{n}}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of rank $n$ vector bundles over $C\times T$}\end{matrix}\right\\}$.
It is well known that $\text{Bun}_{\text{GL}_{n}}(C)$ is a smooth algebraic
stack over $k$. This is just a special case of Proposition 3.5.
Recall that the slope of a rank $n$ vector bundle $\mathcal{E}$ over $C$ is
defined to be $\mu(\mathcal{E})=\frac{1}{n}\,\text{deg}\,\mathcal{E}$. A
vector bundle is called semistable if all nontrivial subbundles
$\mathcal{F}\subset\mathcal{E}$ satisfy
$\mu(\mathcal{F})\leq\mu(\mathcal{E})$. For any given vector bundle
$\mathcal{E}$ over $C$, there exists a filtration
$0=\mathcal{E}_{0}\,\subset\,\mathcal{E}_{1}\,\subset\,\cdots\,\subset\,\mathcal{E}_{l}=\mathcal{E}$
by subbundles such that $\mathcal{E}_{j}\,/\,\mathcal{E}_{j-1}$ is semistable
for all $j$, and such that for all $1\leq j\leq l-1$ we have
$\mu\left(\mathcal{E}_{j}\,/\,\mathcal{E}_{j-1}\right)>\mu\left(\mathcal{E}_{j+1}\,/\,\mathcal{E}_{j}\right)$.
There is a unique filtration satisfying these conditions; it is called the
Harder-Narasimhan filtration.
By a classical HN datum of length $n$ we mean a tuple of $n$ rational numbers
$(\mu_{1},\mu_{2},\cdots,\mu_{n})\in\mathbb{Q}^{n}$ satisfying
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}$. To any vector bundle $\mathcal{E}$
on $C$ we can associate a HN datum $HN(\mathcal{E})$ given as follows. Let
$0=\mathcal{E}_{0}\,\subset\,\mathcal{E}_{1}\,\subset\,\cdots\,\subset\,\mathcal{E}_{l}=\mathcal{E}$
be the Harder-Narasimhan filtration of $\mathcal{E}$. Then, we set
$\displaystyle
HN(\mathcal{E})=(\;\underbrace{\mu(\mathcal{E}_{1}),\,\cdots,\,\mu(\mathcal{E}_{1})}_{\text{rank}\;\mathcal{E}_{1}/0\,\text{times}},\;\underbrace{\mu(\mathcal{E}_{2}/\,\mathcal{E}_{1}),\,\cdots,\,\mu(\mathcal{E}_{2}/\,\mathcal{E}_{1})}_{\text{rank}\;\mathcal{E}_{2}/\,\mathcal{E}_{1}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{E}_{l}/\,\mathcal{E}_{l-1}),\;\cdots,\;\mu(\mathcal{E}_{l}/\,\mathcal{E}_{l-1})}_{\text{rank}\;\mathcal{E}_{l}/\mathcal{E}_{l-1}\,\text{times}}\;)$
If $\mathcal{E}$ has rank $n$, then
$HN(\mathcal{E})\in\left(\frac{1}{n!}\mathbb{Z}\right)^{n}$.
Let $P_{1}=(\mu_{l})$ and $P_{2}=(\nu_{l})$ be two classical HN-data of rank
$n$. We say that $P_{1}\leq P_{2}$ if the following two conditions are
satisfied
1. (a)
$\sum_{l=1}^{n}\mu_{l}=\sum_{l=1}^{n}\nu_{l}$.
2. (b)
For all $1\leq m<n$, we have $\sum_{l=1}^{m}\mu_{l}\leq\sum_{l=1}^{m}\nu_{l}$.
###### Definition 6.2.
Let $Q$ be a classical HN datum of rank $n$. We define
$\text{Bun}_{\text{GL}_{n}}^{\leq Q}(C)$ to be the pseudofunctor from
$k$-schemes into groupoids given as follows. For any $k$-scheme $T$,
$\text{Bun}_{\text{GL}_{n}}^{\leq
Q}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of rank $n$
vector bundles $\mathcal{E}$ over $C\times T$ such that}\\\ \text{for all
$t\in T$, we have $HN(\mathcal{E}|_{C_{t}})\leq Q$}\end{matrix}\right\\}$
The following is a well known result. See e.g. [Beh91] or [Sch15] for proofs
in the generality of reductive algebraic groups.
###### Theorem 6.3.
Let $Q$ be a classical HN datum. The subfunctor
$\text{Bun}_{\text{GL}_{n}}^{\leq Q}(C)$ is a quasicompact open substack of
$\text{Bun}_{\text{GL}_{n}}(C)$. ∎
### 6.2 Quasicompactness of parabolic Harder-Narasimhan strata
Let us now return to our parabolic setting. We have the following natural map
of stacks.
###### Definition 6.4.
We define
$Forget:\text{Bun}_{\mathcal{V}}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$
to be the map of stacks given as follows. Suppose $T$ is a $k$-scheme and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle in $\text{Bun}_{\mathcal{V}}(T)$. We set
$\phi(T)\,(\mathcal{W})=\mathcal{F}^{(0)}$.
###### Proposition 6.5.
$Forget:\text{Bun}_{\mathcal{V}}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$
is schematic and proper.
In order to prove this proposition, we first recall the definition and
properties of generalized flag varieties.
###### Definition 6.6.
Let $S$ be a scheme and let $\mathcal{G}$ be a locally free sheaf of constant
rank on $S$. Let $(a^{(l)})_{l}$ be a tuple of nonnegative integers
$a^{(1)},a^{(2)},\cdots,a^{(N)}$. The generalized flag variety
$Flag^{(a^{(l)})_{l}}(\mathcal{G})$ is defined to be the functor from
$S$-schemes to sets given as follows. Let $X$ be an $S$-scheme. Then,
$\displaystyle
Flag^{(a^{(l)})_{l}}(\mathcal{G})(X)=\;\left\\{\begin{matrix}\text{filtrations
of $\mathcal{G}|_{X}$ by vector
subbundles}\;\left[\;0\,=\mathcal{P}^{(0)}\,\subset\,\mathcal{P}^{(1)}\subset\cdots\;\subset\mathcal{P}^{(N)}=\mathcal{G}|_{X}\;\right]\\\
\\\ \text{such that}\;\mathcal{P}^{(l)}\,/\,\mathcal{P}^{(l-1)}\;\text{is
locally free of rank $a^{(l)}$ for every $1\leq l\leq
N$}\end{matrix}\right\\}$
The map on morphisms is given by pulling back the filtrations.
###### Proposition 6.7.
Let $S$ be a $k$-scheme and $\mathcal{G}$ a vector bundle of constant rank $n$
on $S$. Let $(a^{(l)})_{l}$ be a tuple of nonnegative integers.
$Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is represented by a proper scheme over
$S$.
###### Proof.
Zariski descent for quasicoherent sheaves implies that
$Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is a sheaf in the Zariski topology. In
particular, it suffices to check representability by a proper scheme after
passing to a Zariski cover of $S$. We can assume that $\mathcal{G}$ is trivial
by passing to an open cover. Then $Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is the
base-change of the classical partial flag variety of flags of type
$(a^{(l))})_{l}$ in the vector space $k^{n}$. This is known to be projective
over $k$. ∎
###### Proof of Proposition 6.5.
Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $T$ be a $k$-scheme and $\mathcal{E}$ a vector bundle of rank $n$ on
$C\times T$. In order to ease notation, let us write
$L\vcentcolon=T\times_{\text{Bun}_{\text{GL}_{n}}(C)}\text{Bun}_{\mathcal{V}}$.
We want to show that $L$ is represented by a proper $T$-scheme. By definition,
$L$ is the functor from $T$-schemes into groupoids given as follows. If $S$ is
a $T$-scheme, then
$\displaystyle L(S)=\;\left\\{\begin{matrix}\text{parabolic vector
bundles}\;\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\\\ \text{ over $C\times S$ of type $\mathcal{V}$},\text{ with an
isomorphism}\;\phi:\mathcal{E}|_{C\times
S}\xrightarrow{\sim}\mathcal{F}^{(0)}\end{matrix}\right\\}$
The isomorphism $\phi$ is part of the data. Automorphisms in the groupoid are
required to be compatible with the isomorphism $\phi$. So they must be the
identity on $\mathcal{F}^{(0)}$. This in turn implies that they must be the
identity as a morphism of parabolic vector bundles. Therefore, $L$ is
naturally equivalent to the sheaf of sets:
$\displaystyle L(S)=\;\left\\{\begin{matrix}\text{parabolic vector
bundles}\;\mathcal{W}=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}\\\ \text{ over $C\times S$ of type
$\mathcal{V}$}\end{matrix}\right\\}$
We claim that this sheaf is represented by the $T$-scheme $\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)$. By Proposition 6.7, this claim concludes the proof.
In order to prove the claim, we present two natural transformations that are
inverse to each other. We start with $f:L\rightarrow\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)$ given as follows. For a $T$-scheme $S$, and a parabolic bundle
$\mathcal{W}=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}$ in $L(S)$, we define
$\displaystyle
f\left(\mathcal{W}\right)\;\vcentcolon=\;\left(\;0\,\overset{a_{i}^{(1)}}{\subset}\,\left(\mathcal{F}_{i}^{(1)}\,/\,\mathcal{E}|_{C\times
S}\right)|_{x_{i}\times
S}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\left(\mathcal{F}_{i}^{(N_{i})}\,/\,\mathcal{E}|_{C\times
S}\right)|_{x_{i}\times
S}=\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\right)|_{x_{i}\times
S}\;\right)_{i\in I}$
Let $q_{i}:x_{i}\rightarrow C$ denote the inclusion of the closed point
$x_{i}$ into $C$. The inverse $g:\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)\rightarrow L$ is defined as follows. Let $S$ be a $T$-scheme. Let
$(fl_{i})_{i\in I}$ be at tuple of flags in $\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)(S)$. Suppose that $fl_{i}$ is given by a chain of vector bundles
over $S$
$fl_{i}=\;\left[\;0\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{P}_{i}^{(N_{i})}=\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\right)|_{x_{i}\times
S}\;\right]$
We define $g(\,(fl_{i})_{i\in I}\,)=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}$, where
$\displaystyle\mathcal{F}^{(l)}_{i}\vcentcolon=\text{Ker}\left(\;\mathcal{E}(x_{i})|_{C\times
S}\xrightarrow{unit}\,(q_{i}\times id_{S})_{*}(q_{i}\times
id_{S})^{*}\mathcal{E}(x_{i})|_{C\times S}\,\twoheadrightarrow\,(q_{i}\times
id_{S})_{*}\left(\mathcal{E}(x_{i})|_{x_{i}\times
S}\,/\,\mathcal{P}^{(l)}_{i}\right)\;\right)$
These functors are inverses of each other by construction. The only nontrivial
thing to check is that $g$ well defined (i.e. $g(\,(fil_{i})_{i\in I}\,)$ is a
parabolic bundle). In order to show this, we need to check that the
$\mathcal{F}_{i}^{(l)}$s are vector bundles on $C\times S$. Each sheaf
$\mathcal{F}^{(l)}_{i}$ is locally finitely presented over $C\times S$ and
flat over $S$, because it fits into the short exact sequence
$0\rightarrow\mathcal{E}|_{C\times
S}\rightarrow\mathcal{F}^{(l)}_{i}\rightarrow(q_{i}\times
id_{S})_{*}\left(\mathcal{P}^{(l)}_{i}\right)\,\rightarrow 0$ and the other
two sheaves in the sequence are finitely presented over $C\times S$ and flat
over $S$. By Lemma 4.11, it suffices to show that for every point $s\in S$ we
have that $\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is flat over $C_{s}$. Since $C_{s}$
is a smooth curve, this is equivalent to proving that
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is torsion free. Since
$\mathcal{E}(x_{i})|_{C\times
S}\,/\,\mathcal{F}^{(l)}_{i}\,\cong\,(q_{i}\times
id_{S})_{*}\left(\mathcal{E}|_{x_{i}\times
S}\,/\,\mathcal{P}_{i}^{(l)}\right)$ is $S$-flat, the inclusion
$\mathcal{F}_{i}^{(l)}\subset\mathcal{E}(x_{i})|_{C\times S}$ remains a
monomorphism after passing to the fiber $C_{s}$. Hence
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is a subsheaf of the locally free sheaf
$\mathcal{E}(x_{i})|_{C_{s}}$. We conclude that
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is torsion free, as desired. ∎
We want to leverage the quasicompactness of the classical Harder-Narasimhan
strata for the moduli of vector bundles in order to prove quasicompactness of
HN strata for parabolic vector bundles. As a first step, we need to understand
how the map $Forget$ interacts with the strata in the source and target. This
is achieved by the following lemma.
###### Lemma 6.8.
Let $P$ be a parabolic HN datum. There exists a finite set
$F(P)\subset\frac{1}{n!}\mathbb{Z}$ of classical HN data of rank $n$ such that
for all fields $K\supset k$ and all parabolic vector bundles
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}^{\leq P}(K)$, the classical HN datum
of the vector bundle $Forget(\mathcal{W})$ belongs to $F(P)$.
###### Proof.
Suppose that $P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$. Let $K\supset k$ be a field
extension. Let
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle of type $\mathcal{V}$ over $C_{K}$. Let
$HN(\mathcal{W})=(\nu_{1},\nu_{2},\cdots,\nu_{n})$. Suppose that
$HN(\mathcal{W})\leq P$. This means in particular that $\mu_{1}\geq\nu_{1}$.
Let
$0=\,_{0}\mathcal{F}^{(0)}\subset\,_{1}\mathcal{F}^{(0)}\,\subset\,_{2}\mathcal{F}^{(0)}\,\subset\cdots\,\subset\,_{l}\mathcal{F}^{(0)}=\mathcal{F}^{(0)}$
be the (classical) Harder-Narasimhan filtration for $\mathcal{F}^{(0)}$.
Suppose that the classical Harder-Narasimhan datum for $\mathcal{F}^{(0)}$ is
given by $HN(\mathcal{F}^{(0)})=(\xi_{1},\xi_{2},\cdots,\xi_{n})$.
Define
${}_{1}\mathcal{F}_{i}^{(m)}\vcentcolon=\,_{1}\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{F}_{i}^{(m)}$.
The diagram of sheaves ${}_{1}\mathcal{W}$ given by
${}_{1}\mathcal{W}=\left[\;{}_{1}\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,_{1}\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;_{1}\overset{a_{i}^{(N_{i})}}{\subset}\,_{1}\mathcal{F}_{i}^{(N_{i})}=_{1}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$
is a parabolic subbundle of $\mathcal{W}$, because any subsheaf of a locally
free sheaf on $C_{K}$ is locally free. By the proof of Proposition 2.18, we
know that $\nu_{1}$ is the maximal slope among subbundles of $\mathcal{W}$. In
particular, $\mu(_{1}\mathcal{W})\leq\nu_{1}\leq\mu_{1}$. By Lemma 2.9, we
know that $\mu(_{1}\mathcal{F}^{(0)})\leq\mu(_{1}\mathcal{W})$. So we have
$\mu_{1}\geq\xi_{1}$.
Next we will bound $\xi_{n}$. By Lemma 2.9, we have
$\text{deg}\,\mathcal{F}^{(0)}\geq\text{deg}\,\mathcal{W}-n|I|$. By additivity
of degree for vector bundles in short exact sequences, we have
$\text{deg}\,\mathcal{F}^{(0)}=\sum_{j=1}^{n}\xi_{j}$. Similarly, additivity
of degree for parabolic bundles gives
$\text{deg}\,\mathcal{W}=\sum_{j=1}^{n}\nu_{j}$. Replacing these in the
inequality above yields $\sum_{j=1}^{n}\xi_{j}\geq\sum_{j=1}^{n}\nu_{j}-n|I|$.
But we know that $\xi_{1}\geq\xi_{2}\geq\cdots\geq\xi_{n}$. Therefore,
$\sum_{j=1}^{n}\xi_{j}\leq(n-1)\xi_{1}+\xi_{n}$. Also, we know that
$HN(\mathcal{W})\leq P$. This means in particular that
$\sum_{j=1}^{n}\nu_{j}=\sum_{j=1}^{n}\mu_{j}$. Hence the inequality above
becomes $(n-1)\xi_{1}+\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|$.
We have seen that $\xi_{1}\leq\mu_{1}$. So we can rearrange the last
inequality to obtain $\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|-(n-1)\mu_{1}$.
Hence, we get the uniform bounds
$\mu_{1}\geq\xi_{1}\geq\xi_{2}\geq\cdots\geq\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|-(n-1)\mu_{1}.$
Since the $\xi_{j}$s must lie in the lattice $\frac{1}{n!}\mathbb{Z}$, there
are finitely many possibilities for
$HN(\mathcal{F}^{(0)})=(\xi_{1},\xi_{2},\cdots,\xi_{n})$. ∎
###### Corollary 6.9.
Let $P$ a parabolic HN datum. There exists finitely many classical HN data
${Q_{j}}$ such that the restriction $Forget:\text{Bun}_{\mathcal{V}}^{\leq
P}\longrightarrow\text{Bun}_{\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
GL}\crcr}}}}_{n}}(C)}$ factors through the open inclusion
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq
Q_{j}}(C)\hookrightarrow\text{Bun}_{\text{GL}_{n}}(C)$.
###### Proof.
We can take the finite set $\\{Q_{j}\\}$ to be $F(P)$ as in Lemma 6.8 above.
It suffices to show that for any $k$-scheme $T$ and a vector bundle
$\mathcal{E}$ of rank $n$ on $C\times T$, the fiber product
$Forget:\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\text{GL}_{n}}(C)}\,T\,\rightarrow T$ factors through
the open subset $\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq
Q_{j}}(C)\times_{\text{Bun}_{\text{GL}_{n}}(C)}\,T$ of $T$.
In order to show that it factors through an open subset, it suffices to show
that it does so set theoretically. So we can check it at the level of points.
We have to show that if $t\in T$ is such that $\mathcal{E}|_{C_{t}}$ does not
satisfy $HN(\mathcal{E}|_{C_{t}})\leq Q_{j}$ for some $Q_{j}\in F(P)$, then
there is no field extesion $K\supset\kappa(t)$ and parabolic vector bundle
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}^{\leq P}(K)$ satisfying
$Forget(\mathcal{W})\cong\mathcal{E}|_{C_{K}}$. This is just a restatement of
Lemma 6.8, because the classical HN type of a vector bundle is preserved under
scalar extension (see [Lan75] for a proof of this last fact). ∎
###### Theorem 6.10.
For any parabolic HN datum $P$, the open substack
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is quasicompact.
###### Proof.
By Corollary 6.9, there are finitely many classical HN data $Q_{j}$ such that
the map $Forget$ factors as $\text{Bun}_{\mathcal{V}}^{\leq
P}\longrightarrow\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$. By
Proposition 6.5, this map is the composition of an open immersion and a proper
schematic map. Since everything is locally of finite type over $k$, this map
is quasicompact. Now Theorem 6.3 tells us that
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$ is a finite union of
quasicompact open substacks of $\text{Bun}_{\text{GL}_{n}}(C)$. Therefore
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$ is itself a quasicompact
algebraic stack. We conclude that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is
quasicompact. ∎
## 7 Completeness of strata
In this section we deal with discrete valuation rings $R$ over $k$. We always
write $\eta$ (resp. $s$) for the generic (resp. special) point of
$\text{Spec}(R)$. We will also work with the base-change $C_{R}$. We denote by
$j:C_{\eta}\hookrightarrow C_{R}$ the open immersion of the generic fiber, and
$\iota:C_{s}\hookrightarrow C_{R}$ the closed immersion of the special fiber.
###### Definition 7.1.
Let $\mathfrak{X}$ be a stack of finite type over $k$. We say that
$\mathcal{X}$ is complete over $k$ if for all complete discrete valuation
$k$-algebras $R$, it satisfies the following lifting criterion. For every map
$f:\eta\rightarrow\mathfrak{X}$ there exists a morphism
$g:\text{Spec}\,R\rightarrow\mathfrak{X}$ making the following diagram commute
$\textstyle{\text{Spec}\,R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\eta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathfrak{X}}$
The goal of this section is to show Theorem 7.6, which states that all the HN-
strata $\text{Bun}_{\mathcal{V}}^{\leq P}$ are complete over $k$. Let us
briefly describe some of the work done in this direction. Mehta and Seshadri
[MS80] prove completeness of the moduli of parabolic vector bundles that are
semistable of parabolic degree $0$. On the other hand, Heinloth [Hei17, Remark
3.20] notes that the valuative criterion for universal closedness [Sta, Tag
0CLK] can be proven for the semistable locus of the moduli stack of torsors
for a parahoric Bruhat-Tits group scheme (under some mild tameness conditions
on the generic fiber). Recall that the semistable locus consits of the union
of the minimal open strata. In our case these are given by
$\text{Bun}_{\mathcal{V}}^{\leq P}$, where $P=(r,r,r,\cdots,r)$ for some real
number $r$. We note that this definition of completeness is slightly stronger
than the valuative criterion for universal closedness, because we do not
require passing to an extension of the fraction field.
We extend the proof in [MS80] so that it applies to unstable strata of
arbitrary degree. First we need a series of lemmas.
###### Lemma 7.2.
Let $Q$ be a parabolic HN datum of rank $n$. Consider the set $B_{Q}$ of
parabolic HN data $P$ satisfying
1. (a)
$P\leq Q$
2. (b)
$P=HN(\mathcal{W})$ for some $\mathcal{W}$ of rank $n$ in
$\text{Vect}_{\overline{\lambda}}$
The set $B_{Q}$ is finite.
###### Proof.
Set $Q=(\xi_{1},\xi_{2},\cdots,\xi_{n})$. Choose
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})\in B_{Q}$. Define the finite set of real
numbers
$X\vcentcolon=\left\\{-\sum_{i\in
I}\sum_{m=1}^{N_{i}}b_{i}^{(m)}\lambda_{i}^{(m)}\;\mid\;0\leq b_{i}^{(m)}\leq
n\text{\; for $i\in I$ and $1\leq m\leq N_{i}$}\right\\}$
Let $\mathcal{W}$ in $\text{Vect}_{\overline{\lambda}}$ such that
$HN(\mathcal{W})=P$. Fix $1\leq k\leq n$. By definition, we have that
$\mu_{k}=\mu(\mathcal{P})$ for some parabolic subbundle
$\mathcal{P}\subset\mathcal{W}$. If we have
$\mathcal{P}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$
then by definition
$\mu_{k}=\frac{1}{\text{rank}\,\mathcal{P}}\left(\text{deg}\;\mathcal{F}^{(0)}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}\right)\,\right)$.
So $\mu_{k}$ belongs to the discrete subset of real numbers
$\frac{1}{n!}\left(\mathbb{Z}+X\right)$. Since $P\leq Q$, we have
$\mu_{1}\leq\xi_{1}$. By definition, we must also have
$\sum_{j=1}^{n}\xi_{j}=\sum_{j=1}^{n}\mu_{j}$. Since $\mu_{j}\leq\mu_{1}$ for
all $j$, we know that
$\sum_{j=1}^{n-1}\mu_{j}\leq(n-1)\mu_{1}\leq(n-1)\xi_{1}$. So we get the
inequality
$\sum_{j=1}^{n}\xi_{j}=\sum_{j=1}^{n}\mu_{j}\,\leq\,(n-1)\xi_{1}+\mu_{n}$
We conclude that we have the uniform bounds
$\sum_{j=1}^{n}\xi_{j}-(n-1)\xi_{1}\,\leq\,\mu_{k}\,\leq\,\xi_{1}$ for all
$k$. Since $\mu_{k}$ belongs to a discrete set, this shows that there are
finitely many possible values for each $\mu_{k}$. ∎
In order to prove completeness, we need to be able to extend a given parabolic
vector bundle on $C_{\eta}$. This is what the next lemma achieves.
###### Lemma 7.3.
Let $R$ be a discrete valuation $k$-algebra. Let $\mathcal{P}$ be a parabolic
vector bundle over $C_{\eta}$. There exists a parabolic vector bundle
$\mathcal{W}$ over $C_{R}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$.
###### Proof.
Suppose that
$\mathcal{P}=\left[\;\mathcal{P}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{P}_{i}^{(N_{i})}=\mathcal{P}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is well known that exists a vector bundle $\mathcal{F}^{(0)}$ over
$C_{R}$ such that $\mathcal{F}^{(0)}|_{C_{\eta}}\cong\mathcal{P}^{(0)}$ (for
example see Proposition 6 in [Lan75]). Define $\mathcal{F}^{(m)}_{i}$ to be
the kernel
$\mathcal{F}^{(m)}_{i}\vcentcolon=\text{Ker}\,\left(\mathcal{F}^{(0)}(x_{i})\xrightarrow{unit}j_{*}\mathcal{P}^{(0)}(x_{i})\twoheadrightarrow
j_{*}\left(\mathcal{P}^{(0)}(x_{i})\,/\,\mathcal{P}^{(m)}_{i}\right)\,\right)$
We have that
$\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\,\subset\,j_{*}\left(\mathcal{P}^{(0)}(x_{i})\,/\,\mathcal{P}^{(m)}_{i}\right)$
is $R$-torsion free, and hence $R$-flat. By Lemma 5.7, $\mathcal{F}_{i}^{(m)}$
is a vector bundle on $C_{R}$. Also, we have a short exact sequence
$0\longrightarrow\,\mathcal{F}^{(m)}_{i}\,/\,\mathcal{F}^{(m-1)}_{i}\,\longrightarrow\,\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m-1)}_{i}\,\longrightarrow\,\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m)}_{i}\,\longrightarrow
0$
By the $R$-flatness of $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m-1)}_{i}$
and $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m)}_{i}$ plus the Tor long
exact sequence at each stalk we see that
$\mathcal{F}^{(m)}_{i}\,/\,\mathcal{F}^{(m-1)}_{i}$ is $R$-flat. We conclude
that
$\mathcal{W}\vcentcolon=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle over $C_{R}$. By construction
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. ∎
###### Lemma 7.4.
Let $\mathcal{W}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Let $\mathcal{U}$ be the maximal
destabilizing parabolic subbundle of $\mathcal{W}$. Set
$\mathcal{Q}\vcentcolon=\mathcal{W}\,/\,\mathcal{U}$. Suppose that there is a
parabolic vector bundle $\mathcal{W}_{1}$ fitting in a short exact sequence
$0\longrightarrow\mathcal{Q}\longrightarrow\mathcal{W}_{1}\longrightarrow\mathcal{U}\longrightarrow
0$. Then, $HN(\mathcal{W}_{1})\leq HN(\mathcal{W})$.
###### Proof.
Set $P=(\mu_{1},\cdots,\mu_{n})\vcentcolon=HN(\mathcal{W})$. Suppose for the
sake of contradiction that $HN(\mathcal{W}_{1})\nleq P$. The short exact
sequence above show that
$\text{deg}\,\mathcal{W}_{1}=\text{deg}\,\mathcal{W}=\sum_{j=1}^{n}\mu_{j}$.
So we can use Lemma 5.2 to conclude that there is a $P$-destabilizing
parabolic subbundle $\mathcal{M}$ of $\mathcal{W}_{1}$. Let us denote by
$\mathcal{L}$ the image of the composition
$\mathcal{M}\rightarrow\mathcal{W}_{1}\twoheadrightarrow\mathcal{U}$. We have
a short exact sequence
$0\longrightarrow\mathcal{Q}\,\cap\,\mathcal{M}\longrightarrow\mathcal{M}\longrightarrow\mathcal{L}\longrightarrow
0$ of parabolic vector bundles. Since $\mathcal{U}$ is semistable and
$\mathcal{L}\subset\mathcal{U}$ is a parabolic subbundle, we have
$\mu(\mathcal{L})\leq\mu(\mathcal{U})$. We conclude that
$\deg(\mathcal{L})\leq\text{rank}\,\mathcal{L}\cdot\mu_{1}=\sum_{j=1}^{\text{rank}\,\mathcal{L}}\mu_{j}$.
We can also look at the decompostion of $\mathcal{Q}\,\cap\,\mathcal{M}$
obtained by intersecting with the Harder-Narasimahn filtration of
$\mathcal{Q}$. By a similar argument we conclude that
$\text{deg}\,(\mathcal{Q}\,\cap\,\mathcal{M})\leq\sum_{j=\text{rank}\,\mathcal{U}+1}^{\text{rank}\,(\mathcal{Q}\,\cap\,\mathcal{M})+\text{rank}\,\mathcal{U}}\mu_{j}$.
We can put these together to obtain
$\text{deg}\,\mathcal{M}\leq\sum_{j=1}^{\text{rank}\,\mathcal{L}}\mu_{j}\,+\,\sum_{j=\text{rank}\,\mathcal{U}+1}^{\text{rank}\,(\mathcal{Q}\,\cap\,\mathcal{M})+\text{rank}\,\mathcal{U}}\mu_{j}$
Since the $\mu_{j}$ are decreasing, the right hand side is less than or equal
to $\sum_{j=1}^{\text{rank}\,\mathcal{M}}\mu_{j}$. By definition, this
contradicts the fact that $\mathcal{M}$ is $P$-destabilizing. ∎
###### Proposition 7.5.
Let $R$ be a complete discrete valuation ring over $k$. Let $P$ be a HN datum.
Let $\mathcal{P}$ be a parabolic vector bundle of rank $n$ over $C_{\eta}$.
Suppose that $HN(\mathcal{P})=P$. Then there exists a parabolic bundle
$\mathcal{W}$ over $C_{R}$ such that $\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$
and $HN\left(\mathcal{W}|_{C_{s}}\right)=P$.
###### Proof.
Suppose $\mathcal{P}$ has Harder-Narasimhan filtration
$0\subset\,\mathcal{P}_{1}\,\subset\,\cdots\,\subset\,\mathcal{P}_{l}=\mathcal{P}$.
By Lemma 7.3, there exists a parabolic vector bundle $\mathcal{W}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. Let $\overline{\mathcal{U}}$ be the
maximal destabilizing parabolic subbundle of $\mathcal{W}|_{C_{s}}$. Write
$\overline{\mathcal{Q}}\vcentcolon=\mathcal{W}|_{C_{s}}\,/\,\overline{\mathcal{U}}$.
We have a map $g$ of diagrams of sheaves given by the composition
$g:\,\mathcal{W}\mathrel{\ooalign{$\xrightarrow[\mkern 4.0mu]{unit\mkern
4.0mu}$\cr$\rightarrow\mkern
4.0mu$}}\iota_{*}\mathcal{W}|_{C_{s}}\twoheadrightarrow\iota_{*}\overline{\mathcal{Q}}$.
Define $\mathcal{W}_{1}\vcentcolon=\text{Ker}(g)$. We have a short exact
sequence of diagrams of sheaves
$0\longrightarrow\mathcal{W}_{1}\longrightarrow\mathcal{W}\longrightarrow\iota_{*}\overline{\mathcal{Q}}\longrightarrow
0$. The pullback
$\mathcal{W}_{1}|_{C_{\eta}}\hookrightarrow\mathcal{W}|_{C_{\eta}}$ becomes an
isomorphism. In order to pass to the special fiber we use the Tor long exact
sequence for each short exact sequence of sheaves in the diagrams. We get an
exact sequence of diagrams of sheaves over the special fiber
$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\mathcal{W}|_{C_{s}}\longrightarrow\overline{\mathcal{Q}}\longrightarrow
0$. To see this recall that
$\text{Tor}^{1}_{\mathcal{O}_{s}}(\mathcal{O}_{s},\,\mathcal{M})$ is given by
the $R$-torsion submodule of $\mathcal{M}$ for any $R$-module $M$. This can be
applied to the stalks of the short exact sequence of sheaves above.
We therefore get short exact sequences
$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\overline{\mathcal{U}}\longrightarrow
0$
$0\longrightarrow\overline{\mathcal{U}}\longrightarrow\mathcal{W}|_{C_{s}}\longrightarrow\overline{\mathcal{Q}}\longrightarrow
0$
Using this we see that $\mathcal{W}_{1}|_{C_{s}}$ is a vector bundle over
$C_{s}$. All the sheaves in the diagram
$\mathcal{W}_{1}\hookrightarrow\mathcal{W}$ are $R$-torsion free. Therefore
they are $R$-flat. We can then use Lemma 4.11 and an argument similar to the
one in Lemma 7.3 to conclude that $\mathcal{W}_{1}$ is a parabolic vector
bundle. Lemma 7.4 implies that $HN(\mathcal{W}_{1}|_{C_{s}})\leq
HN(\mathcal{W}|_{C_{s}})$.
Now we can replace $\mathcal{W}$ with $\mathcal{W}_{1}$ and iterate the same
construction. By Lemma 7.2, the Harder-Narasimhan datum
$HN(\mathcal{W}|_{C_{S}})$ can’t decresase indefinitely. So it must eventually
stabilize. Therefore, we can assume without loss of generality that
$HN(\mathcal{W}_{1}|_{C_{s}})=HN(\mathcal{W}|_{C_{s}})$ in the construction
above. Let $\overline{\mathcal{U}}_{1}$ be the maximal destabilizing parabolic
subbundle of $\mathcal{W}_{1}|_{C_{s}}$. Since
$HN(\mathcal{W}_{1}|_{C_{s}})=HN(\mathcal{W}|_{C_{s}})$, we must have that
$\overline{\mathcal{U}}$ and $\overline{\mathcal{U}}_{1}$ have the same rank
and slope. By Lemma 2.17, we see that $\overline{\mathcal{U}}_{1}$ maps
isomorphically to $\overline{\mathcal{U}}$ under the surjection
$\mathcal{W}_{1}\twoheadrightarrow\overline{\mathcal{U}}$. So the short exact
sequence
$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\overline{\mathcal{U}}\longrightarrow
0$ splits. Hence we see that
$\mathcal{W}_{1}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.
Repeating this construction for $\mathcal{W}_{1}$, we can obtain a parabolic
suboject $\mathcal{W}_{2}\hookrightarrow\mathcal{W}_{1}$ such that
$\mathcal{W}_{2}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.
We can iterate this construction to get an infinite chain of parabolic
subojects
$\cdots\,\mathcal{W}_{n}\,\xhookrightarrow{f_{n}}\,\mathcal{W}_{n-1}\,\xhookrightarrow{f_{n-1}}\,\cdots\,\mathcal{W}_{1}\,\xhookrightarrow{f_{1}}\mathcal{W}$
such that
$\mathcal{W}_{n}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.
The pullback
$\iota^{*}f_{n}:\mathcal{W}_{n}|_{C_{s}}\rightarrow\mathcal{W}_{n-1}|_{C_{s}}$
induces an isomorphims between the corresponding maximal destabilizing
parabolic subbundles $\overline{\mathcal{U}}$. The proof of Lemma 3.5 in
[MS80] now implies that there exists a parabolic subbundle $\mathcal{U}$ of
$\mathcal{W}$ over $C_{R}$ such that $\mathcal{U}|_{C_{s}}$ is the maximal
destabilizing parabolic subbundle $\overline{\mathcal{U}}$ of
$\mathcal{W}|_{C_{s}}$. By local constancy of slope we know that
$\mu(\mathcal{U}|_{C_{\eta}})=\mu(\overline{\mathcal{U}})$. But
$\mathcal{U}|_{C_{\eta}}$ is a parabolic subbundle of $\mathcal{P}$. Hence the
definition of the maximal destabilizing parabolic subbundle $\mathcal{P}_{1}$
of $\mathcal{P}$ implies that
$\mu(\mathcal{U}|_{C_{\eta}})\leq\mu(\mathcal{P}_{1})$. On the other hand, we
have that $HN(\mathcal{P})\leq HN(\mathcal{W}|_{C_{s}})$ by Proposition 5.6.
So we must actually have $\mu(\overline{U})=\mu(\mathcal{P}_{1})$. The
definition of the maximal destabilizing parabolic subbundle $\mathcal{P}_{1}$
of $\mathcal{P}$ now implies that
$\text{rank}\,\mathcal{U}|_{C_{\eta}}\leq\text{rank}\,\mathcal{P}_{1}$. We can
use again the fact that $HN(\mathcal{P})\leq HN(\mathcal{W}|_{C_{s}})$ to
conclude that we must actually have
$\text{rank}\,\mathcal{U}|_{C_{\eta}}=\text{rank}\,\overline{\mathcal{U}}=\text{rank}\,\mathcal{P}_{1}$.
Hence $\mathcal{U}|_{C_{\eta}}=\mathcal{P}_{1}$, and so the subbundle
$\mathcal{U}$ restricts to the maximal destabilizing parabolic subbundle at
both the special and the generic fiber.
Consider now the parabolic quotient
$\mathcal{M}\vcentcolon=\mathcal{W}\,/\,\mathcal{U}$. We can repeat the same
construction as above to obtain a parabolic subobject
$\mathcal{K}\hookrightarrow\mathcal{M}$ such that
1. 1.
$\mathcal{K}|_{C_{\eta}}\cong\mathcal{P}\,/\,\mathcal{P}_{1}$.
2. 2.
There exists a parabolic subbundle $\mathcal{L}$ of $\mathcal{K}$ over $C_{R}$
that restricts to the maximal destabilizing parabolic subbundle at both the
special and the generic fiber.
We can consider the corresponding parabolic suboject
$\mathcal{W}^{1}\hookrightarrow\mathcal{W}$ containing $\mathcal{U}$ such that
$\mathcal{W}^{1}\,/\,\mathcal{U}=\mathcal{M}$. Similarly we can lift
$\mathcal{L}$ to a subbundle $\mathcal{U}^{1}$ of $\mathcal{W}^{1}$ containing
$\mathcal{U}$. By construction the two step filtration
$\mathcal{U}\subset\mathcal{U}^{1}\subset\mathcal{W}^{1}$ restricts to the
first two elements of the Harder-Narasimhan filtration at both the special and
the generic fiber. Now we can repeat the construction with
$\left(\mathcal{W}^{1},\,\mathcal{U}^{1}\right)$ in place of
$\left(\mathcal{W},\,\mathcal{U}\right)$.
Since the length of the Harder-Narasimhan filtration of $\mathcal{P}$ is
finite, this process must eventually stop. We end up with a parabolic vector
bundle $\mathcal{W}$ over $C_{R}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$, and $\mathcal{W}$ admits a
filtration by parabolic subbundles
$0\subset\mathcal{U}\subset\mathcal{U}_{1}\subset\cdots\subset\mathcal{W}^{l}$
that restricts to the Harder-Narasimhan filtration at both the special and the
generic fiber. By local constancy of slope, this implies that
$HN(\mathcal{W}^{l}|_{C_{s}})=HN(\mathcal{P})=P$. ∎
###### Theorem 7.6.
Let $P$ be a parabolic HN datum. The moduli stack
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is complete over $k$.
###### Proof.
Let $R$ be a complete discrete valuation ring over $k$. Let
$\mathcal{P}:\eta\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a parabolic
vector bundle over $C_{\eta}$ such that $HN(\mathcal{P})\leq P$. By
Proposition 7.5, there exists a parabolic vector bundle $\mathcal{W}$ of type
$\mathcal{V}$ over $C_{R}$ such that
$HN(\mathcal{W}|_{C_{s}})=HN(\mathcal{P})\leq P$ and
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. This means that $\mathcal{W}$
represents a morphism
$\mathcal{W}:\text{Spec}\,R\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$
lifting $\mathcal{P}:\eta\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$. ∎
## 8 Harder-Narasimhan filtrations in families
In this section we use the filtration schemes $Fil^{\alpha}_{\mathcal{V}}$
defined in Section 4 in order to give the structure of an algebraic stack to
the locus $\text{Bun}^{=P}_{\mathcal{V}}$ for each HN datum $P$. Since we have
proved that conjecture holds for parabolic vector bundles in arbitrary
characteristic (Proposition 2.31), we have established the result in its
strongest form that
$\text{Bun}_{\mathcal{V}}^{=P}\hookrightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion for every field $k$. See Theorem 8.6 below. This
idea of schematic Harder-Narasimhan stratifications first appeared in [Nit11]
for the moduli of sheaves. Gurjar and Nitsure proved an analogous result for
the moduli stack of principal $G$-bundles on a curve over a field of
characteristic $0$ [GN14], and later for higher dimensional varieties [GN18,
GN16].
We start with a technical definition.
###### Definition 8.1.
Let $T$ be a $k$-scheme. Let $\mathcal{W}$ be a parabolic vector bundle over
$C\times T$. Let $\alpha$ be a filtration datum, and let $P$ be a HN datum. We
write $HN_{\mathcal{W}}(\alpha)=P$ if for all points $x\in
Fil^{\alpha}_{\mathcal{W}}$ we have
1. (a)
$HN(\mathcal{W}|_{C_{x}})=P$
2. (b)
$x$ represents the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{x}}$.
###### Lemma 8.2.
Let $T$ be a connected $k$-scheme. Let $\mathcal{W}$ be a parabolic vector
bundle over $C\times T$. Let $P$ be a HN datum. Fix a point $t\in T$. Suppose
that $HN(\mathcal{W}|_{C_{t}})=P$. Let
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{W}|_{C_{t}}$
denote the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{t}}$. Let
$\alpha=\psi(\mathcal{W}_{j})$ denote the associated filtration datum, as in
Definition 4.18. Then, we have $HN_{\mathcal{W}}(\alpha)=P$.
###### Proof.
Suppose that
$\alpha=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}1\leq j\leq
l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{1\leq j\leq l-1}\right)$. Choose
$x\in Fil^{\alpha}_{\mathcal{W}}$. Let
$0=\mathcal{P}_{0}\subset\mathcal{P}_{1}\subset\cdots\,\subset\mathcal{P}_{l-1}\subset\mathcal{P}_{l}=\mathcal{W}|_{C_{x}}$
be the filtration of $\mathcal{W}|_{C_{x}}$ corresponding to $x$. We know that
$\theta(\mathcal{P}_{j})=\theta(\mathcal{W}_{j})=(_{j}b_{i}^{(m)},\,_{j}P)$
for all $1\leq j\leq l-1$. By connectedness of $T$, we know that
$\theta(\mathcal{P}_{l})=\theta(\mathcal{W}|_{C_{x}})=\theta(\mathcal{W}|_{C_{t}})=\theta(\mathcal{W}_{l})$
(see Remark 4.14). Since the rank and slope of a parabolic vector bundle can
be determined from its associated parabolic Quot datum, we conclude that
$\mu(\mathcal{P}_{j})=\mu(\mathcal{W}_{j})$ and
$\text{rank}\,\mathcal{P}_{j}=\text{rank}\,\mathcal{W}_{j}$ for all $1\leq
j\leq l$. Since $\\{\mathcal{W}_{j}\\}$ is a Harder-Narasimhan filtration, we
must have that $\\{\mathcal{P}_{j}\\}$ is a Harder-Narasimhan filtration. We
have $HN(\mathcal{W}|_{C_{x}})=HN(\mathcal{W}|_{C_{t}})=P$, because both
filtrations have the same ranks and slopes. ∎
###### Proposition 8.3.
Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$ over $C$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. Let $\alpha$ be
filtration datum. Fix a scheme $S$ of finite type over $k$. Let
$\mathcal{W}:\,S\longrightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a
parabolic vector bundle. Suppose that $HN_{\mathcal{W}}(\alpha)=P$. Then, the
structure map $\pi:\,Fil^{\alpha}_{\mathcal{W}}\longrightarrow S$ is a closed
immersion.
###### Proof.
We will use [Sta, Tag 04XV], which says that a morphism that is universally
injective, unramified and proper is a closed immersion. By Proposition 4.17,
we know that $\pi:\,Fil^{\alpha}_{\mathcal{W}}\longrightarrow S$ is of finite
type. Let $t\in S$ such that the fiber
$\pi^{*}(t)=\text{Fil}^{\alpha}_{\mathcal{W}}\times_{S}t$ is nonempty. By
definition, the $\kappa(t)$-scheme
$\pi^{*}(t)=\text{Fil}^{\alpha}_{\mathcal{W}|_{C_{t}}}$ classifies filtrations
of $\mathcal{W}|_{C_{t}}$ with filtration datum $\alpha$. A point
$x\in\pi^{*}(t)$ would give us a filtration of $\mathcal{W}|_{C_{\kappa(x)}}$
with associated HN datum $P$. By Lemma 2.30, we know that
$HN(\mathcal{W}|_{C_{\kappa(x)}})=HN(\mathcal{W}_{C_{t}})\leq P$. Hence, we
must have $HN(\mathcal{W}|_{C_{\kappa(x)}})=P$. So the filtration must be the
Harder-Narasimhan filtration of $\mathcal{W}|_{C_{\kappa(x)}}$. By the
uniqueness of the Harder-Narasimhan filtration for any field extension (Lemma
2.30 again), $\pi^{*}(t)$ has a single point and this point is defined over
$\kappa(t)$. So $\pi^{*}(t)$ must be of the form $\text{Spec}(A)$ for some
local artinian $\kappa(t)$-algebra $A$ with residue field $\kappa(t)$. This
shows that $\pi$ radicial. Hence $\pi$ is universally injective.
We claim that $A=\kappa(t)$. In order to see this, it suffices to show that
$\Omega_{A\,/\,\kappa(t)}^{1}=0$. We have that $\Omega_{A\,/\,\kappa(t)}^{1}$
is dual to the tangent space
$\text{Hom}_{\kappa(t)}\left(\text{Spec}(\kappa(t)[\epsilon]),\,\pi^{*}(t)\right)$.
By Proposition 2.31, the Harder-Narasimhan filtration has no notrivial first
order deformations. This means that
$\text{Hom}_{\kappa(t)}\left(\text{Spec}(\kappa(t)[\epsilon]),\,\pi^{*}(t)\right)=0$.
Therefore $A=\kappa(t)$. We conclude that $\pi$ is unramified.
We are left to show that $\pi$ is proper. We will use the valuative criterion
for properness. The argument is similar to the proof of 5.6. We use the same
notation as in that proof (look there for any piece of unexplained notation).
Let $R$ be a discrete valuation $k$-algebra with generic point $\eta$ and
special point $s$. Let $f:\text{Spec}\;R\rightarrow S$. The fiber product
$\text{Fil}^{\alpha}_{\mathcal{W}}\times_{S}\text{Spec}\;R$ is by definition
the $R$-scheme
$\pi_{R}:\text{Fil}^{\alpha}_{\mathcal{W}|_{C_{R}}}\rightarrow\text{Spec}\;R$.
To simplify notation, let us set
$\mathcal{U}\vcentcolon=\mathcal{W}|_{C_{R}}$. Suppose that we have a section
$p:\eta\rightarrow\text{Fil}^{\alpha}_{\mathcal{U}}$ of $\pi_{R}$ defined over
$\eta$. We want to show that $p$ extends to a unique section over
$\text{Spec}\;R$. Uniqueness follows from the fact that
$\text{Fil}^{\alpha}_{\mathcal{U}}$ is a separated $R$-scheme (Proposition
4.17). We are left to show existence.
By hypothesis, we know that $HN(^{\eta}\mathcal{U}),HN(^{s}\mathcal{U})\leq
P$. The point $p$ is a filtration of ${}^{\eta}\mathcal{U}$ by parabolic
subbundles
$0=\,^{\eta}\mathcal{W}_{0}\,\subset\,^{\eta}\mathcal{W}_{1}\,\subset\,\cdots\,\subset\,^{\eta}\mathcal{W}_{l}=\,^{\eta}\mathcal{U}$
with filtration datum $\alpha$. Set
$k_{j}\vcentcolon=\text{rank}^{\eta}\mathcal{W}_{j}$. Arguing as in the proof
of Proposition 5.6, we get set of inclusions
$0=\,\mathcal{W}_{0}\,\hookrightarrow\,\mathcal{W}_{1}\,\hookrightarrow\,\cdots\,\hookrightarrow\,\mathcal{W}_{l}=\,\mathcal{U}$
of parabolic subobjects over $C_{R}$. These are not parabolic subbundles a
priori. As in the proof of Proposition 5.6, passing to the special fiber
preserves the subobject inclusions
$0=\,^{s}\mathcal{W}_{0}\,\hookrightarrow\,^{s}\mathcal{W}_{1}\,\hookrightarrow\,\cdots\,\hookrightarrow\,^{s}\mathcal{W}_{l}=\,^{s}\mathcal{U}$.
Using Lemma 2.11, we get a filtration of ${}^{s}\mathcal{U}$ by parabolic
subbundles
$0=\,^{s}\mathcal{W}^{sat}_{0}\,\subset\,^{s}\mathcal{W}_{1}^{sat}\,\subset\,\cdots\,\subset\,\mathcal{W}_{l}^{sat}=\,^{s}\mathcal{U}$
with $\mu(^{s}\mathcal{W}_{j})\leq\mu(^{s}\mathcal{W}^{sat}_{j})$ for all
$1\leq j\leq l$. By local constancy of degree we have
$\mu(^{s}\mathcal{W}_{j})=\mu(^{\eta}\mathcal{W}_{j})$. Since $HN(\alpha)=P$,
we have $\mu(^{s}\mathcal{W}_{j})=\mu_{k_{j}}$. So
$\mu_{k_{j}}\leq\mu(^{s}\mathcal{W}^{sat}_{j})$. We know that
$HN(^{s}\mathcal{U})\leq P$, hence we must have
$\mu_{k_{j}}=\mu(^{s}\mathcal{W}^{sat}_{j})$. By Remark 5.8, this implies that
$\mathcal{W}_{j}$ is actually a parabolic subbundle of $\mathcal{U}$ over
$C_{R}$. The filtration
$0=\,\mathcal{W}_{0}\,\subset\,\mathcal{W}_{1}\,\subset\,\cdots\,\subset\,\mathcal{W}_{l}=\,\mathcal{U}$
gives us a section of $\pi_{R}$ over $\text{Spec}\;R$, as desired. ∎
###### Definition 8.4.
Let $S$ be a $k$-scheme and $\mathcal{V}$ be a parabolic vector bundle on
$C\times S$. A relative Harder-Narasimhan filtration (over $S$) is a
filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by parabolic subbundles $\mathcal{W}_{j}$ over $C\times S$ such that for all
$s\in S$, the restriction
$0=\mathcal{W}_{0}|_{C_{s}}\subset\mathcal{W}_{1}|_{C_{s}}\subset\cdots\,\subset\mathcal{W}_{l-1}|_{C_{s}}\subset\mathcal{W}_{l}|_{C_{s}}=\mathcal{V}|_{C_{s}}$
to the fiber $C_{s}$ is the Harder-Narasimhan filtration of
$\mathcal{V}|_{C_{s}}$.
###### Definition 8.5.
Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Let $P$ be a HN
datum. $\text{Bun}_{\mathcal{V}}^{=P}$ denotes the functor from $k$-schemes
into groupoids given as follows. For a $k$-scheme $S$,
$\text{Bun}_{\mathcal{V}}^{=P}(S)$ is the groupoid of parabolic bundles over
$C\times S$ of type $\mathcal{V}$ along with a relative Harder-Narasimhan
filtration (over $C\times S$) such that for all $s\in S$, we have
$HN(\mathcal{V}|_{C_{s}})=P$.
We will denote by
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$ the
morphism given by forgetting the relative Harder-Narasimhan filtration.
###### Theorem 8.6.
Let $P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. The morphism
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$
exhibits $\text{Bun}_{\mathcal{V}}^{=P}$ as a locally closed substack of
$\text{Bun}_{\mathcal{V}}$.
###### Proof.
By definition
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$
factors through the open substack $\text{Bun}_{\mathcal{V}}^{\leq P}$. We
shall show that
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion.
Let $T$ be a connected scheme of finite type over $k$. Let
$\mathcal{W}:T\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a parabolic
vector bundle over $C\times T$. Define the set
$T^{=P}\vcentcolon=\left\\{\;t\in
T\;\mid\;HN(\mathcal{W}|_{C_{t}})=P\right\\}$. Recall that for any $t\in T$
and any filtration $\\{\mathcal{W}_{j}^{t}\\}$ of $\mathcal{W}|_{C_{t}}$, we
have an associated filtration datum $\psi(\mathcal{W}^{t}_{j})$ as in
Definition 4.18. In particular, we can do this with the Harder-Narasimhan
filtration $HN\text{Fil}\,\mathcal{W}|_{C_{t}}$ of $\mathcal{W}|_{C_{t}}$.
Let’s define a set
$\mathfrak{S}=\left\\{\;\psi\left(\,HN\text{Fil}\,\mathcal{W}|_{C_{t}}\,\right)\;\mid\;t\in
T^{=P}\right\\}$. We claim that $\mathfrak{S}$ is a finite set. Let $t\in
T^{=P}$ and let
$0=\mathcal{W}_{0}^{t}\subset\mathcal{W}_{1}^{t}\subset\cdots\,\subset\mathcal{W}_{l-1}^{t}\subset\mathcal{W}_{l}^{t}=\mathcal{W}|_{C_{t}}$
be the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{t}}$. Suppose that
${}_{j}\mathcal{W}^{t}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\overset{{}_{j}b_{i}^{(1)}}{\subset}\,_{j}\mathcal{F}_{i}^{(1)}\overset{{}_{j}b_{i}^{(2)}}{\subset}\cdots\;\overset{{}_{j}b_{i}^{(N_{i})}}{\subset}\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. We know that $HN(\mathcal{W}|_{C_{t}})=P$. Therefore the length $l$ of
the filtration and the rank $k_{j}$ of each $\mathcal{W}_{j}^{t}$ do not
depend on $t$. Since there is a finite number of possibilities for the numbers
${}_{j}b_{i}^{(m)}$, it suffices to show that for each $j$ there is a finite
number of possibilities for $\text{deg}\,_{j}\mathcal{F}^{(0)}$. We know that
$\text{deg}\,_{j}\mathcal{W}=k_{j}\,\mu_{k_{j}}$ is fixed. By Lemma 2.9, we
have
$k_{j}\,\mu_{k_{j}}-\,k_{j}\,|I|\leq\text{deg}\,_{j}\mathcal{F}^{(0)}\leq\,k_{j}\,\mu_{k_{j}}$
Hence there are finitely many possible choices for the integer
$\text{deg}\,_{j}\mathcal{F}^{(0)}$. We conclude that the set $\mathfrak{S}$
is finite.
Consider the $T$-scheme
$X\vcentcolon=\sqcup_{\alpha\in\mathfrak{S}}\,\text{Fil}_{\mathcal{V}}^{\alpha}$.
By construction, $X$ represents the fiber product
$\text{Bun}_{\mathcal{V}}^{=P}\times_{\text{Bun}_{\mathcal{V}}^{\leq P}}T$. It
therefore suffices to show that $X\rightarrow T$ is a closed immersion. For
every $\alpha\in\mathfrak{S}$, we have $HN_{\mathcal{W}}(\alpha)=P$ by Lemma
8.2. By Proposition 8.3, $X$ is a finite disjoint union of closed subschemes
of $T$. It therefore suffices to show that if $\alpha\neq\beta$ are two
filtration data in $\mathfrak{S}$, then the closed subschemes
$\text{Fil}_{\mathcal{V}}^{\alpha}$ and $\text{Fil}_{\mathcal{V}}^{\beta}$ are
set theoretically disjoint inside $T$. But we can’t have a point $t\in T$ in
the intersection, since that would violate the uniqueness of the Harder-
Narasimhan filtration for $\mathcal{W}|_{C_{t}}$. ∎
The result that the relative Harder-Narasimhan stacks are locally closed
substacks does not hold for $\text{Bun}_{\mathcal{G}}$ for exceptional
parahoric Bruhat-Tits group schemes $\mathcal{G}$ over $C$ when the
characteristic of $k$ is small. This is because of the failure of Behrend’s
conjecture, even in the case when the group scheme is reductive (see [Hei08]).
We end this section by stating an explicit corollary of Theorem 8.6. A weaker
version of this corollary in the context of coherent sheaves without parabolic
structure can be found in [HL97][Thm. 2.3.2].
###### Corollary 8.7.
Fix a parabolic vector bundle $\mathcal{V}$ over $C$. Let $S$ be a $k$-scheme
and let $\mathcal{W}$ be a parabolic vector bundle of type $\mathcal{V}$ over
$C\times S$. There is a surjective morphism $\bigsqcup_{P}S^{P}\rightarrow S$
where
1. (i)
The $P$ run over the set of all HN data of rank $n$.
2. (ii)
Each $S^{P}$ is a locally closed subscheme of $S$.
3. (iii)
$\mathcal{W}|_{S^{P}}$ admits a relative Harder-Narasimhan filtration.
4. (iv)
For all $s\in S^{P}$, we have $HN(\mathcal{W}|_{C_{s}})=P$.
###### Proof.
The parabolic vector bundle $\mathcal{W}$ is the same a as morphism
$S\rightarrow\text{Bun}_{\mathcal{V}}$. Theorem 8.6 implies that there is a
surjective map of stacks
$\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\,\rightarrow\text{Bun}_{\mathcal{V}}$,
where the $P$ run over the set of HN data of rank $\text{rank}\,\mathcal{V}$.
Moreover, each $\text{Bun}_{\mathcal{V}}^{=P}$ is a locally closed substack of
$\text{Bun}_{\mathcal{V}}$.
Define $S^{P}$ to be the locally closed subscheme of $S$ given by
$S^{P}\vcentcolon=S\times_{\text{Bun}_{\mathcal{V}}}\text{Bun}_{\mathcal{V}}^{=P}$.
We have a fiber product diagram
$\textstyle{\bigsqcup_{P}S^{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Bun}_{\mathcal{V}}}$
Since the map
$\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\,\rightarrow\text{Bun}_{\mathcal{V}}$
is surjective, we conclude that the base-change $\bigsqcup_{P}S^{P}\rightarrow
S$ is surjective as well. Properties $(iii)$ and $(iv)$ follow directly from
the definition of $\text{Bun}_{\mathcal{V}}^{=P}$ above. ∎
## Appendix A Descent for finite covers
For the convenience of the reader, we give an explicit description of descent
data for certain finite flat covers as done in [Gro95][B.3]. All sheaves
considered are quasicoherent.
Let $\pi:Y\rightarrow X$ be a faithfully-flat finite morphims of schemes such
that $\pi_{*}(\mathcal{O}_{Y})$ is a locally free $\mathcal{O}_{X}$-module. We
want to explicitly describe descent data for quasicoherent sheaves on $Y$
relative to $X$. Since $\pi$ is affine, the functor $\pi_{*}$ sets up an
equivalence between $\text{QCoh}(Y)$ and the category of quasicoherent sheaves
on $X$ with a $\pi_{*}(\mathcal{O}_{Y})$-module structure. Consider the fiber
product:
$\textstyle{Y\times_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;\;\;\;pr_{1}}$$\scriptstyle{pr_{2}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X}$
By the same reasoning, a quasicoherent sheaf on $Y\times_{X}Y$ is the same as
a quasicoherent
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-module.
Given a $\pi_{*}(\mathcal{O}_{Y})$-module $\mathcal{F}$, we can obtain two
different
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-modules via pullback:
$pr_{1}^{*}(\mathcal{F})=\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
$pr^{*}_{2}(\mathcal{F})=\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}$
A descent datum for $\mathcal{F}$ would correspond to an homorphism of
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-modules $\phi:pr_{2}^{*}(\mathcal{F})\rightarrow pr_{1}^{*}(\mathcal{F})$
satisfying a cocycle condition.
By assumption $\pi_{*}(\mathcal{O}_{Y})$ is locally free as a
$\mathcal{O}_{X}$-module, so it is $\mathcal{O}_{X}$-dualizable. Using this
fact plus the tensor-Hom adjunction, we get a chain of isomorphisms:
$\begin{split}\text{Hom}_{\mathcal{O}_{X}}\left(pr_{2}^{*}(\mathcal{F}),\,pr_{1}^{*}(\mathcal{F})\right)&=\,\text{Hom}_{\mathcal{O}_{X}}\left(\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F},\;\;\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\right)\\\
&\cong\,\text{Hom}_{\mathcal{O}_{X}}\left(\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})^{\vee}\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F},\;\;\;\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\right)\\\
&\cong\,\text{Hom}_{\mathcal{O}_{X}}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)\end{split}$
What is the image of the set of
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$\-
homomorphisms? Tracing back the chain isomorphisms above, one can check that
this is
$\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)$.
Here
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
acts on each End group by $(s\otimes t)\cdot f(x)\vcentcolon=tf(sx)$ for any
sections $s,t$ and $x$ of the corresponding sheaves over a given open set.
We are left to describe the cocycle condition for a given
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\,$-homomorphims
$\phi:pr_{2}^{*}(\mathcal{F})\rightarrow pr_{1}^{*}(\mathcal{F})$. Recall that
the cocycle condition amounts to the commutativity of the following diagram:
$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\otimes\phi}$$\scriptstyle{id\otimes\text{swap}}$$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes
id}$$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes
id}$$\textstyle{\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}$
Tracing back the construction of the isomorphim
$\psi:\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(pr_{2}^{*}(\mathcal{F}),\,pr_{1}^{*}(\mathcal{F})\right)\,\xrightarrow{\sim}\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)$
we see that the cocycle condition corresponds to $\psi(\phi)$ being an
homomorphism of unital algebras.
Let us give two examples when there is a nice explicit presentation of the
unital algebra $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$.
###### Example A.1 (Galois descent).
Suppose that $\pi:Y\rightarrow X$ is a Galois cover with Galois group $G$. By
definition, $G$ is a group of $\mathcal{O}_{X}$-linear automorphisms of
$\pi_{*}(\mathcal{O}_{Y})$. Then it can be checked that we have
$\text{End}\left(\,\pi_{*}(\mathcal{O}_{Y})\,\right)=\bigoplus_{g\in
G}\left(1\otimes\pi_{*}(\mathcal{O}_{Y})\right)\,g$
Since everything is $\mathcal{O}_{X}$ coherent, it suffices to check this
equality on fibers, This reduces the claim to Galois theory for separable
field extensions.
By definition of the
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$-module
structure on $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$, we have
$(s\otimes 1)\cdot g=(1\otimes g(s))\cdot g$ for any $g\in G$ and any section
$s$ of $\pi_{*}(\mathcal{O}_{Y})$ on an open. So in this case descent data
correspond to homomorphisms of groups
$\psi:G\rightarrow\text{Aut}_{\mathcal{O}_{X}}\,\mathcal{F}$ such that
$\psi(g)$ is $g$-semilinear for all $g\in G$.
###### Example A.2 (Inseparable descent/ Frobenius descent).
Let us suppose that $X$ is a scheme over a finite field $\kappa$ of
characteristic $p$ (in other words, $p\,\mathcal{O}_{X}=0$). Let
$\pi:Y\rightarrow X$ as above. Suppose that
$\pi_{*}(\mathcal{O}_{Y})^{p}\subset\mathcal{O}_{X}$, so the cover is of
”exponent 1”. Furthermore, suppose that $\pi_{*}(\mathcal{O}_{Y})$ locally
admits a $p$-basis over $\mathcal{O}_{X}$ (for Noetherian schemes this is
known to be equivalent to $\Omega^{1}_{Y\,/\,X}$ being locally free, see
[Tyc88]).
Let $\mathcal{D}$ be the $\pi_{*}(\mathcal{O}_{Y})$-module of
$\mathcal{O}_{X}$-linear derivations of $\pi_{*}(\mathcal{O}_{Y})$. Here the
$\pi_{*}(\mathcal{O}_{Y})$-module structure comes from the action of the
second coordinate in
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
when we view $\mathcal{D}$ as a subset of
$\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$. By assumption, we
know that $\mathcal{D}=(\Omega^{1}_{Y\,/\,X})^{\vee}$ is locally free. Observe
that $\mathcal{D}$ has the structure of a restricted p-Lie algebra over
$\mathcal{O}_{X}$. We will denote by $[-,-]$ and $(-)^{(p)}$ the p-Lie algebra
operations in $\mathcal{D}$.
We claim that $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$ is
generated as a
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$-algebra
by $\mathcal{D}$ subject to the relations
$(1\otimes s)\cdot\delta\,=\,(1\otimes\delta(s))\cdot\,id\,+\,(s\otimes
1)\cdot\delta$ $\delta\gamma-\gamma\delta\,=\,[\delta,\,\gamma]$
$\delta^{p}\,=\,\delta^{(p)}$
Here $\delta$ and $\gamma$ are sections of $\mathcal{D}$ and $s$ is a section
of $\pi_{*}(\mathcal{O}_{Y})$. The claim is a local statement. Since
everything is $\mathcal{O}_{X}$ coherent, we can check it on fibers. We are
reduced to the case of a finite algebra of exponent 1 over a field. This can
be done by hand by using the result for field extensions ([Jac89] Theorem
8.45) and a computation for the ring of dual numbers.
Let $\mathcal{F}$ be a quasicoherent sheaf on $Y$. The discussion above shows
that a descent datum
$\phi:\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})\,\rightarrow\,\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}$
is equivalent to a map
$\nabla:\,\mathcal{D}\,\rightarrow\,\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}$
of restricted p-Lie algebras over $\mathcal{O}_{X}$ satisfying the Leibniz
condition
$(s\otimes 1)\cdot\nabla(\delta)=(1\otimes\delta(s))\cdot id\,+\,(1\otimes
s)\cdot\nabla(\delta)$
for all sections $\delta$ of $\mathcal{D}$ and sections $s$ of
$\pi_{*}(\mathcal{O}_{Y})$. This is by definition a $X$-relative connection on
$\mathcal{F}$ with vanishing $p$-curvature.
We will use this version of descent in the special case of inseparable field
extensions of exponent 1. This goes back to the work of Jacobson (see [Jac89]
Theorem 8.45). Another example is the case when $\pi$ is a Frobenius morphism.
This is worked out in [Kat70] Theorem 5.1.
Finally, let us describe descent for morphims of quasicoherent sheaves in this
language. Let quasicoherent $\pi_{*}(\mathcal{O}_{Y})$-modules
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, along with descent data
$\phi_{i}\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{i}\,\right)$
for $i=1,2$. A homomorphims of $\pi_{*}(\mathcal{O}_{Y})$-modules
$f:\mathcal{F}_{2}\rightarrow\mathcal{F}_{2}$ descends if and only if we have
$f^{*}(\phi_{2})=f_{*}(\phi_{1})$, where
$\displaystyle
f^{*}:\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{2}\,\right)\,\longrightarrow\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{Hom}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\,\mathcal{F}_{2})\,\right)$
$\displaystyle
f_{*}:\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{1}\,\right)\,\longrightarrow\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{Hom}_{\mathcal{O}_{X}}(\mathcal{F_{1}},\,\mathcal{F}_{2})\,\right)$
are given by precomposing (resp. postcomposing) with $f$. Let us give a few
examples/consequences of this.
###### Example A.3 (Subsheaves).
Let $\mathcal{E}$ be a $\pi_{*}(\mathcal{O}_{Y})$-module along with a descent
datum
$\phi\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{E}\,\right)$
A subsheaf $\mathcal{F}\subset\mathcal{E}$ descends if and only if for all
sections $s\in\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})(U)$ over
an open $U\subset X$, we have that $\mathcal{F}|_{U}$ is stable under the
corresponding homomorphism $\phi(s)$.
If $\text{End}_{\mathcal{O}_{X}}$ is generated by global sections, it suffices
to check global sections. This is the case for descent along field extensions
(or base-changes of field extensions).
###### Example A.4 (Diagrams of sheaves).
Let $I$ be a small category. A diagram of sheaves on $Y$ of shape $I$ is a
covariant functor $F:I\rightarrow\text{QCoh}(Y)$. A morphism of diagrams of
sheaves of shape $I$ is a natural transformation. Descent data for a diagram
of sheaves $F$ correspond to homomorphisms of unital algebras
$\phi\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}F\,\right)$
Here $\text{End}_{\mathcal{O}_{X}}F$ is the algebra of endomorphisms of $F$
viewed as a diagram in $\text{QCoh}(X)$ via $\pi_{*}$.
###### Example A.5 (Parabolic vector bundles).
Parabolic vector bundles are a particular case of diagrams of sheaves. Given a
parabolic vector bundle $\mathcal{W}$ over $C\times Y$, descent data
correspond to homomorphisms of unital algebras from
$\text{End}_{\mathcal{O}_{C\times X}}\,(id\times\pi)_{*}(\mathcal{O}_{C\times
Y})$ into the algebra of $\mathcal{O}_{C\times X}$-linear endomorphisms of
$\mathcal{W}$.
## Appendix B Comparison with previous work
In this appendix we compare the stratification described in this article with
the $\Theta$-stratifications defined in [Hei17] and [AHLH18][§8]. For this
appendix we will freely use the notation from these papers.
For simplicity of notation, we will stick with the case when there is a single
point of degeneration $x\in C(k)$. Our arguments can be modified easily to
deal with the case when there are several points of degeneration. Let’s denote
by $q:x\hookrightarrow C$ the closed immerion of $x$ into $C$. Let
$\mathcal{V}$ be a parabolic vector bundle on $C$ with parabolic structure at
$x$. Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a^{(1)}}{\subset}\,\mathcal{E}^{(1)}\overset{a^{(2)}}{\subset}\cdots\;\overset{a^{(N)}}{\subset}\,\mathcal{E}^{(N)}=\mathcal{E}^{(0)}(x)\;\right]$.
Fix a collection of parabolic weights
$0<\lambda^{(1)}<\lambda^{(2)}<\cdots<\lambda^{(N)}<1$. This collection
defines a Harder-Narasimhan stratification on $\text{Bun}_{\mathcal{V}}$, as
in Definition 3.13. Our goal in this appendix is to explain how, in the case
when $\overline{\lambda}$ is “admissible”, one can view this stratification as
the $\Theta$-stratification coming from a $\mathbb{R}$-line bundle on the
stack $\text{Bun}_{\mathcal{V}}$, as in [AHLH18][§8].
Set $n=\text{rank}\,\mathcal{V}$. By Definition 3.6,
$\text{Bun}_{\mathcal{V}}$ only depends on the type of the parabolic bundle
$\mathcal{V}$. Hence we can, without loss of generality, assume that
$\mathcal{E}^{(0)}=\mathcal{O}_{C}^{\oplus n}$ and
$\mathcal{E}^{(i)}=\mathcal{O}_{C}(x)^{\oplus\sum_{j=1}^{i}a^{(j)}}\,\oplus\,\mathcal{O}_{C}^{\oplus\left(n-\sum_{j=1}^{i}a^{(j)}\right)}$
By Proposition 3.7, we have
$\text{Bun}_{\mathcal{V}}\simeq\text{Bun}_{\text{GL}(\mathcal{V})}$. Set
$\mathcal{G}\vcentcolon=\text{GL}(\mathcal{V})$. Let $\text{Lie}(\mathcal{G})$
denote the Lie algebra of $\mathcal{G}$. Since $\mathcal{G}$ is smooth over
$C$, this Lie algebra is a vector bundle on $C$. Following [Hei17], we define
$\mathcal{L}_{det}$ to be the line bundle on the stack
$\text{Bun}_{\mathcal{G}}$ defined as follows. For any $k$-scheme $T$ and
every map $f:T\rightarrow\text{Bun}_{\mathcal{G}}$ represented by a
$\mathcal{G}_{T}$-torsor $\mathcal{P}$ on $C\times T$, we let
$f^{*}(\mathcal{L}_{det})=\text{det}\left(R(\pi_{T})_{*}\,\mathcal{P}\times^{\mathcal{G}}\text{Lie}(\mathcal{G})\right)^{\vee}$
Here det denotes the determinant in the $K$-theoretic sense. This makes sense
because the derived pushforward is a perfect complex [Sta, Tag 0A1H].
As in [HL14], we let
$\Theta=\left[\,\mathbb{A}^{1}_{k}/\,\mathbb{G}_{m}\,\right]$. We write $0$
for the origin in $\mathbb{A}^{1}_{k}$. We can view $0$ as a point of
$\Theta$. Let $\varphi:\Theta\rightarrow\text{Bun}_{\mathcal{G}}$ be a
filtration of a parabolic vector bundle
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}$. We would like to compute the weight
of the $0$-fiber of $\mathcal{L}_{det}$. Say
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a^{(1)}}{\subset}\,\mathcal{F}^{(1)}\overset{a^{(2)}}{\subset}\cdots\;\overset{a^{(N)}}{\subset}\,\mathcal{F}^{(N)}=\,\mathcal{F}^{(0)}(x)\;\right]$
Via the Rees construction, a filtration of $\mathcal{W}$ corresponds to a
decreasing $\mathbb{Z}$-filtration of $\mathcal{F}^{(0)}$ by subbundles
$\left(\mathcal{F}^{(0)}_{m}\right)_{m\in\mathbb{Z}}$. More precisely, for all
$m\in\mathbb{Z}$ we have that $\mathcal{F}^{(0)}_{m}$ is a subbundle of
$\mathcal{F}^{(0)}$ and $\mathcal{F}_{m}^{(0)}\supset\mathcal{F}^{(0)}_{m+1}$.
Moreover we require that $\mathcal{F}^{(0)}_{m}=0$ for $m>>0$ and
$\mathcal{F}^{(0)}_{m}=\mathcal{F}^{(0)}$ for all $m<<0$. Alternatively, we
can view this as a filtration of $\mathcal{W}$ by parabolic subbundles
$(\mathcal{W}_{m})_{m\in\mathbb{Z}}$ by setting
$\mathcal{F}^{(i)}_{m}\vcentcolon=\mathcal{F}_{m}^{(0)}\cap\mathcal{F}^{(i)}$
and defining
$\mathcal{W}_{m}\vcentcolon=\left[\;\mathcal{F}_{m}^{(0)}\,\overset{a_{m}^{(1)}}{\subset}\,\mathcal{F}_{m}^{(1)}\overset{a_{m}^{(2)}}{\subset}\cdots\;\overset{a_{m}^{(N)}}{\subset}\,\mathcal{F}_{m}^{(N)}=\,\mathcal{F}_{m}^{(0)}(x)\;\right]$
In order to compute $wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)$, we
will set up a short exact sequence for $\text{Lie}(\mathcal{G})$. There is a
monomorphism of group schemes
$\mathcal{G}\hookrightarrow\text{Aut}(\mathcal{E}^{(0)}(x))$, given by
forgetting the automorphisms of $\mathcal{E}^{(i)}$ for $i<N$. By
[MRR20][Lemma 3.7], we have a short exact sequence of pointed ètale sheaves
$1\,\rightarrow\,\mathcal{G}\,\rightarrow\,\text{Aut}(\mathcal{E}^{(0)}(x)\,\rightarrow\text{Aut}(\mathcal{E}^{(0)}(x)|_{x})\,/\,P\,\rightarrow\,1$
where $P$ is the parabolic subgroup of $\text{Aut}(\mathcal{E}^{(0)}(x)|_{x})$
that preserves the flag
$0\subset\mathcal{E}^{(1)}/\,\mathcal{E}^{(0)}\,\subset\,\mathcal{E}^{(2)}/\,\mathcal{E}^{(0)}\,\subset\cdots\,\subset\mathcal{E}^{(N)}/\,\mathcal{E}^{(0)}\cong\mathcal{E}^{(0)}(x)|_{x}$
We remark that in this case the short exact sequence can be proven by hand.
Under a choice of isomorphism
$\mathcal{E}^{(0)}(x)|_{x}\cong\mathcal{O}^{\oplus n}_{C}(x)|_{x}\cong k^{n}$,
the group $P$ is the standard parabolic subgroup of $\text{GL}_{n}$
corresponding to the partition $(a^{(1)},a^{(2)},\cdots,a^{(N)})$. In other
words, $P$ consists of block upper-triangular matrices in $\text{GL}_{n}$ with
blocks of sizes given by the tuple $(a^{(1)},a^{(2)},\cdots,a^{(N)})$.
It is known that $\text{GL}_{n}/P$ admits sections Zariski locally. Let
$\mathfrak{gl}_{n}$ (resp. $\mathfrak{p}$) denote the Lie algebra of
$\text{GL}_{n}$ (resp. $P$). By looking at points over the dual numbers, one
can see that the short exact sequence of étale sheaves above induces a short
exact sequence of coherent sheaves on $C$
$0\,\rightarrow\,\text{Lie}(\mathcal{G})\,\rightarrow\,\text{Lie}(\text{Aut}(\mathcal{E}^{(0)}(x))\,\rightarrow\,q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\,\rightarrow\,0$
Note that
$\text{Lie}(\text{Aut}(\mathcal{E}^{(0)}(x))=\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))$.
The group scheme $\mathcal{G}$ acts on both
$\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))$ and
$q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})$ via the adjoint representation. The
action on $q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})$ factors through the
composition $\mathcal{G}\rightarrow q_{*}(\mathcal{G}|_{x})\rightarrow
q_{*}(P)$.
Let’s compute the weight of $\varphi^{*}(\mathcal{L}_{det})|_{0}$. The
composition $0\rightarrow\Theta\xrightarrow{\varphi}\text{Bun}_{\mathcal{G}}$
is represented by the $\mathbb{Z}$-graded parabolic vector bundle
$\bigoplus_{m\in\mathbb{Z}}\mathcal{W}_{m}/\mathcal{W}_{m+1}$. Here
$\mathbb{G}_{m}$ acts with weight $m$ on $\mathcal{W}_{m}/\mathcal{W}_{m+1}$.
Let $\mathcal{P}_{univ}$ denote the universal $\mathcal{G}$-torsor on
$C\times\text{Bun}_{\mathcal{G}}$. We can use flat base-change to write
$wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)=-wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\text{Lie}(\mathcal{G})\right)$
After using the short exact sequence of vector bundles above and the
additivity of the (K-theoretic) determinant, we get
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)=-wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))\right)+wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\right)$
We are left to compute each term separately. For the first term, one can
unravel the definitions to obtain
$\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))=\mathcal{E}\mathit{nd}\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{(0)}_{m}(x)/\,\mathcal{F}^{(0)}_{m+1}(x)\right)$
The computation in [Hei17][1.E.c] shows that
$\displaystyle-
wt\left(\text{det}\,R\pi_{*}\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{(0)}_{m}(x)/\,\mathcal{F}^{(0)}_{m+1}(x)\right)\right)=2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$
This takes care of the first term above. On the other hand, we can write
$\displaystyle\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\text{det}\,R\pi_{*}q_{*}\left(\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\mathfrak{gl}_{n}/\mathfrak{p})\right)$
where we use the natural map $\mathcal{G}|_{x}\rightarrow P$ to form the
associated $P$-torsor $\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P$. We can further rewrite this as
$\text{det}\,R(\pi q)_{*}\left(\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\mathfrak{gl}_{n}/\mathfrak{p})\right)\,\cong\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$
The character $\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$ of $P$ factors
through the Levi quotient $M=P/\,U=\prod_{i=1}^{N}\text{GL}_{a^{(i)}}$.
Therefore we can rewrite the expression above as
$\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}M\times^{M}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$
As a $M$-representation, we have
$\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\text{det}\left(\bigoplus_{i<j}\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\right)$
Here $St_{\text{GL}_{a^{(i)}}}$ denotes the standard $a^{(i)}$-dimensional
representation of the $\text{GL}_{a^{(i)}}$-component of $M$. Using this, we
get
$\displaystyle\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}M\times^{M}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\bigotimes_{i<j}\,\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)$
Here $\mathbb{G}_{m}$ acts on
$\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)$
with weight $l-m$. We are reduced to computing for each $i,j$ the weight
$wt\left(\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\right)$
This is the weight of the $0$-fiber of a line bundle
$\mathcal{L}_{\chi_{i,j}}$ in $\text{Bun}_{\mathcal{G}}$ corresponding to a
character $\chi_{i,j}$ of $\mathcal{G}|_{x}$, as in [Hei17][3.B]. Indeed we
can take
$\chi_{i,j}=\text{det}\left(\,\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\,\right)$,
which we can view as a representation of $\mathcal{G}|_{x}$ via the natural
map $\mathcal{G}|_{x}\rightarrow M$. Let’s compute the corresponding weight.
We have
$\displaystyle
wt\left(\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\right)\,=\,\sum_{m,l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)$
Let’s rewrite this as
$\displaystyle\sum_{m<l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\,-\,\sum_{m<l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{l}/\mathcal{F}^{(i)}_{l+1}}{\mathcal{F}^{(i-1)}_{l}/\mathcal{F}^{(i-1)}_{l+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{m}/\mathcal{F}^{(j)}_{m+1}}{\mathcal{F}^{(j-1)}_{m}/\mathcal{F}^{(j-1)}_{m+1}}\right)$
We can use the same trick as in [Hei17][1.E.c] to express this as a sum
$\displaystyle\sum_{h\in\mathbb{Z}}\left(\sum_{m\leq
h}\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\right)\cdot\left(\sum_{l>h}\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\,-\,\sum_{h\in\mathbb{Z}}\left(\sum_{l>h}\text{rank}\left(\frac{\mathcal{F}^{(i)}_{l}/\mathcal{F}^{(i)}_{l+1}}{\mathcal{F}^{(i-1)}_{l}/\mathcal{F}^{(i-1)}_{l+1}}\right)\right)\cdot\left(\sum_{m\leq
h}\text{rank}\left(\frac{\mathcal{F}^{(j)}_{m}/\mathcal{F}^{(j)}_{m+1}}{\mathcal{F}^{(j-1)}_{m}/\mathcal{F}^{(j-1)}_{m+1}}\right)\right)$
This can be rewritten as
$\displaystyle\sum_{h\in\mathbb{Z}}a^{(i)}_{h}\cdot\left(a^{(j)}-a^{(j)}_{h}\right)\,-\,\sum_{h\in\mathbb{Z}}a^{(j)}_{h}\cdot\left(a^{(i)}-a^{(i)}_{h}\right)$
After some cancellation, we end up with
$wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)=\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot a^{(i)}\right)$. So after all of our computations, we
conclude that
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,+\,\sum_{i<j}wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$
Set $\chi$ to be the element in
$\text{Hom}\left(\mathcal{G}|_{x},\mathbb{G}_{m}\right)\otimes_{\mathbb{Z}}\mathbb{R}$
given by
$\chi=\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})^{\vee}\,\otimes\,\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}$.
By linearizing over $\mathbb{R}$ the definition in [Hei17][3.B], we can define
an element $\mathcal{L}_{\chi}$ in the $\mathbb{R}$-Picard group of the stack
$\text{Bun}_{\mathcal{G}}$. Using our computations from above, we can get
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,-\,2\sum_{i,j}\lambda^{(i)}\cdot
wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$
We can further plug in our formula for
$wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$ to get
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,-\,2\sum_{i,j}\lambda^{(i)}\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot a^{(i)}\right)$
The second term can be rewritten as
$\displaystyle
2\sum_{i,j}\lambda^{(i)}\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot
a^{(i)}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\left(\sum_{i}a^{(i)}_{h}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}^{(0)})-\left(\sum_{i}a^{(i)}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$
Plugging this back in, we end up with
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))-\sum_{i}a^{(i)}_{h}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}^{(0)})-\left(\text{deg}(\mathcal{F}^{(0)}(x))-\sum_{i}a^{(i)}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$
By our definition of parabolic degree, this is the same as
$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\,\text{deg}(\mathcal{W}_{h})\cdot\text{rank}(\mathcal{W})-\text{deg}(\mathcal{W})\cdot\text{rank}(\mathcal{W}_{h})\,\right)$
Using this formula, one can see that a parabolic vector bundle is
$\mathcal{L}_{det}\otimes\mathcal{L}_{\chi}$-(semi)stable in the sense of
[Hei17] if and only if it is slope (semi)stable as in our definition. In order
to see that the HN filtrations match up, one can use the deletion lemma in
[HL14][Lemma 5.32], which can be verified to work the same way in our context
of parabolic vector bundles.
It should be noted that the results in [Hei17] [AHLH18][§8] are valid whenever
the character $\chi$ used to define $\mathcal{L}_{\chi}$ is “admissible” (see
[Hei17][3.F] for a definition). Let’s study whether the character we defined
above is admissible. The diagonal torus $T$ of $\text{GL}_{n}$ over $k(C)$ can
be viewed as a maximal split torus of the generic fiber of $\mathcal{G}$ under
the identification provided by the proof of Proposition 3.2. This torus
extends to a split torus $\mathcal{T}$ of $\mathcal{G}$ defined over all of
$C$ (take the scheme theoretic closure). The fiber
$\mathcal{T}|_{x}\subset\mathcal{G}|_{x}$ maps isomorphically onto the
diagonal torus $T_{x}$ of the Levi quotient
$M=\prod_{i=1}^{N}\text{GL}_{a^{(i)}}$. We will use the standard basis of the
character lattice of $T_{x}$, namely
$X^{*}(T_{x})=\bigoplus_{j=1}^{n}\mathbb{Z}e_{j}$, where $e_{j}$ is the
character that projects to the $j$th component of $T_{x}$ (under the canonical
ordered basis of $k^{n}$). We can view each $\chi_{i,j}$ as a
$T_{x}$-representation via restriction. Then, we can write
$\chi_{i,j}=\text{det}\left(\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\right)=-\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}a^{(j)}e_{l}\;\;+\sum_{l=(\sum_{m=1}^{j-1}a^{(m)})+1}^{\sum_{m=1}^{j}a^{(m)}}a^{(i)}e_{l}$
We conclude that
$\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}=-2\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(\left(\sum_{m=1}^{N}a^{(m)}\lambda^{(m)}\right)-n\lambda^{(i)}\right)$
and
$\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,=\,\bigotimes_{i<j}\chi_{i,j}\,=\,\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(2\sum_{m=1}^{i-1}a^{(m)}+a^{(i)}-n\right)$
Therefore
$\displaystyle\chi=\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})^{\vee}\,\otimes\,\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}=\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(n-a^{(i)}-2\sum_{m=1}^{i-1}a^{(m)}-2(\sum_{m=1}^{N}a^{(n)}\lambda^{(m)})+2n\lambda^{(i)}\right)$
To determine admissibility, now we need to compute the inner product with each
coroot of $\text{GL}_{n}$. We know that the coroots are of the form
$\alpha_{ij}=e_{i}-e_{j}$ for some $i\neq j$ with $1\leq i,j\leq n$. In this
case the admissibility condition means that
$\langle\chi,\alpha_{i,j}\rangle\leq 1$ for all $i\neq j$. Note that if the
indices $i,j$ lie in the same “block” determined by the partition
$(a^{(1)},a^{(2)},\cdots,a^{(N)})$, then $\langle\chi,\alpha_{i,j}\rangle=0$.
Otherwise, if $i$ is in the $l$-th block and $j$ is in the $k$-th block we
have
$\langle\chi,\alpha_{i,j}\rangle=2n\lambda^{(l)}-2\sum_{m=1}^{l-1}a^{(m)}-a^{(l)}-2n\lambda^{(m)}+2\sum_{m=1}^{k-1}a^{(m)}+a^{(k)}$
Hence the admissibility condition reads
$2n\lambda^{(l)}-2\sum_{m=1}^{l-1}a^{(m)}-a^{(l)}-2n\lambda^{(m)}+2\sum_{m=1}^{k-1}a^{(m)}+a^{(k)}\leq
1$
This condition can be expressed as follows. For all $1\leq k\leq l\leq N$, we
must have the pair of inequalities
$-\frac{1}{2n}+\frac{1}{n}\sum_{m=k}^{l-1}a^{(m)}+\frac{a^{(l)}-a^{(k)}}{2n}\,\leq\,\lambda^{(l)}-\lambda^{(m)}\,\leq\,\frac{1}{2n}+\frac{1}{n}\sum_{m=k}^{l-1}a^{(m)}+\frac{a^{(l)}-a^{(k)}}{2n}$
In particular this implies that for all $l$ we must have
$\frac{a^{(l)}+a^{(l-1)}}{2n}-\frac{1}{2n}\,\leq\,\lambda^{(l)}-\lambda^{(l-1)}\,\leq\,\frac{a^{(l)}+a^{(l-1)}}{2n}+\frac{1}{2n}$
From these inequalities we conclude that $\lambda^{(l)}>\lambda^{(l-1)}$ and
that the maximum difference $\lambda^{(N)}-\lambda^{(1)}$ is strictly smaller
than $1$. The notion of
$\mathcal{L}_{det}\otimes\mathcal{L}_{\chi}$-(semi)stability and the
admissibility of $\chi$ do not change if we translate all stability parameters
$\lambda^{(l)}$ by the same constant
$\lambda^{(l)}\,\mapsto\,\lambda^{(l)}+c$. In order to normalize, we can
translate so that $0<\lambda^{(1)}<\lambda^{(2)}<\cdots<\lambda^{(N)}<1$. We
therefore see that, up to translation, all of the admissible tuples of
parameters $\lambda^{(l)}$ in the sense of [Hei17] are parabolic weights in
our sense. However the admissibility conditions described above are not
satisfied by all parabolic weights. In other words, the admissibility of the
stability parameters from [Hei17] is more restrictive that the conditions in
the definition of parabolic weights (Definition 2.5).
## References
* [AHLH18] Jarod Alper, Daniel Halpern-Leistner, and Jochen Heinloth. Existence of moduli spaces for algebraic stacks. https://arxiv.org/abs/1812.01128, 2018. arXiv preprint.
* [Beh91] Kai Achim Behrend. The Lefschetz trace formula for the moduli stack of principal bundles. 1991\. Thesis (Ph.D.)–University of California, Berkeley.
* [Beh95] Kai A. Behrend. Semi-stability of reductive group schemes over curves. Math. Ann., 301(2):281–305, 1995.
* [BS15] V. Balaji and C. S. Seshadri. Moduli of parahoric $\mathcal{G}$-torsors on a compact Riemann surface. J. Algebraic Geom., 24(1):1–49, 2015.
* [BT84a] F. Bruhat and J. Tits. Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math., (60):197–376, 1984.
* [BT84b] F. Bruhat and J. Tits. Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. France, 112(2):259–301, 1984.
* [GL19] Dennis Gaitsgory and Jacob Lurie. Weil’s conjecture for function fields. Vol. 1, volume 199 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2019.
* [GN14] Sudarshan Gurjar and Nitin Nitsure. Schematic Harder-Narasimhan stratification for families of principal bundles and $\Lambda$-modules. Proc. Indian Acad. Sci. Math. Sci., 124(3):315–332, 2014.
* [GN16] Sudarshan Gurjar and Nitin Nitsure. Harder-Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics. https://arxiv.org/abs/1605.08997, 2016. arXiv preprint.
* [GN18] Sudarshan Gurjar and Nitin Nitsure. Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions. Math. Z., 289(3-4):1121–1142, 2018.
* [Gro95] Alexander Grothendieck. Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. In Séminaire Bourbaki, Vol. 5, pages Exp. No. 190, 299–327. Soc. Math. France, Paris, 1995.
* [Hei08] Jochen Heinloth. Bounds for Behrend’s conjecture on the canonical reduction. Int. Math. Res. Not. IMRN, (14):Art. ID rnn045, 17, 2008.
* [Hei10] Jochen Heinloth. Uniformization of $\mathcal{G}$-bundles. Math. Ann., 347(3):499–528, 2010.
* [Hei17] Jochen Heinloth. Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves. Épijournal Geom. Algébrique, 1:Art. 11, 37, 2017.
* [HL97] Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997\.
* [HL14] Daniel Halpern-Leistner. On the structure of instability in moduli theory. https://arxiv.org/abs/1411.0627, 2014. arXiv preprint.
* [HN75] G. Harder and M. S. Narasimhan. On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann., 212:215–248, 1974/75.
* [Jac89] Nathan Jacobson. Basic algebra. II. W. H. Freeman and Company, New York, second edition, 1989.
* [Kat70] Nicholas M. Katz. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math., (39):175–232, 1970.
* [Lan75] Stacy G. Langton. Valuative criteria for families of vector bundles on algebraic varieties. Ann. of Math. (2), 101:88–110, 1975.
* [Lur19] Jacob Lurie. Weil’s Conjecture for Function Fields. http://swc.math.arizona.edu/aws/2019/2019LurieNotes.pdf, 2019.
* [MRR20] Arnaud Mayeux, Timo Richarz, and Matthieu Romagny. Néron blowups and low-degree cohomological applications. https://arxiv.org/abs/2001.03597, 2020.
* [MS80] V. B. Mehta and C. S. Seshadri. Moduli of vector bundles on curves with parabolic structures. Math. Ann., 248(3):205–239, 1980.
* [Nit05] Nitin Nitsure. Construction of Hilbert and Quot schemes. In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr., pages 105–137. Amer. Math. Soc., Providence, RI, 2005.
* [Nit11] Nitin Nitsure. Schematic Harder-Narasimhan stratification. Internat. J. Math., 22(10):1365–1373, 2011.
* [PR10] Georgios Pappas and Michael Rapoport. Some questions about $\mathcal{G}$-bundles on curves. In Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), volume 58 of Adv. Stud. Pure Math., pages 159–171. Math. Soc. Japan, Tokyo, 2010.
* [Sch15] Simon Schieder. The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld’s compactifications. Selecta Math. (N.S.), 21(3):763–831, 2015.
* [Sta] The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu.
* [Tyc88] Andrzej Tyc. Differential basis, $p$-basis, and smoothness in characteristic $p>0$. Proc. Amer. Math. Soc., 103(2):389–394, 1988.
Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New
York 14853, USA
E-mail address<EMAIL_ADDRESS>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.